problem of standard big-bang cosmology. We then ... relies on the argument that winding string states generically in- tersect .... factor oscillates about the minimum of its potential while the winding ... If the initial equilibrium winding number in the ...... sustain inflation, so in order to construct inflationary models it is necessary.
Aspects of String Gas Cosmology: Cyclic Models and Decompactification
Stefanos Marnerides
Professor Brian Greene
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2010
c 2010
Stefanos Marnerides All Rights Reserved
Abstract
Aspects of String Gas Cosmology: Cyclic Models and Decompactification Stefanos Marnerides
String gas cosmology is a natural approach to cosmology within string theory. Two of its novelties are that it provides means to replace the singular evolution of the early universe with a smooth cyclic phase and that it suggests a dynamical mechanism for selecting at most three spatial dimensions that can grow large cosmologically. This thesis presents progress in these two directions. By implementing corrections to gravitational dynamics, we show how string gas cosmology gives rise to cyclic and bouncing models in the early universe. In these models entropy production naturally leads to an exit from the cyclic phase and drives an eventual transition to a radiation dominated, expanding universe. Further, between the cyclic and radiation phases, the scale factors can have long loitering phases that can address the horizon
problem of standard big-bang cosmology. We then address the decompactification mechanism, as originally suggested by Brandenberger and Vafa. This mechanism relies on the argument that winding string states generically intersect, and therefore interact appreciably, in at most three spatial dimensions. When these strings are not energetically favored they can decay to unwound strings and if not, they oppose the expansion of the dimensions they wrap. Since they interact efficiently in at most three dimensions this is the maximum number of dimensions that is allowed to expand. We show that a semiclassical treatment of fundamental string interactions makes this dimensional dependence manifest. This allows us to demonstrate in a cosmological setting that decompactification of d = 3 dimensions is largely favored over d > 3. We conclude with a study of the mechanism on anisotropic backgrounds.
Contents List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
I
String Theory
1
1 String Theory Preliminaries 1.1
1.2
1.3
3
The String Action . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.1
The Conformal Gauge . . . . . . . . . . . . . . . . . .
4
1.1.2
Quantized Closed Strings . . . . . . . . . . . . . . . . .
8
One-Loop Vacuum Amplitudes . . . . . . . . . . . . . . . . . 12 1.2.1
Toroidal Worldsheet and the Partition Function . . . . 14
1.2.2
The Bosonic One-Loop Amplitude . . . . . . . . . . . . 18
Toroidal Compactification in Spacetime . . . . . . . . . . . . . 22 1.3.1
T-duality . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.2
Generalized Toroidal Compactification and T-duality . 27 i
1.4
Closed Superstrings . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4.1
One Loop Amplitudes . . . . . . . . . . . . . . . . . . 33
2 String Thermodynamics 2.1
39
The Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1.1
Singularities of the Free Energy . . . . . . . . . . . . . 43
2.1.2
The Hagedorn Temperature . . . . . . . . . . . . . . . 46
2.2
The Microcanonical Ensemble and the Hagedorn Phase . . . . 48
2.3
The Large Volume - Low Temperature Limit and the Radiation Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
II
Cosmology
59
3 String Gas Cosmology
60
3.1
Motivation and The Basic Picture . . . . . . . . . . . . . . . . 61
3.2
The Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3
The Problem of Exiting the Hagedorn Phase . . . . . . . . . . 75 3.3.1
Surveying the Space of Initial Conditions . . . . . . . . 77
4 Bouncing and Cyclic Cosmologies
83
4.1
Finite or Eternal Universe? . . . . . . . . . . . . . . . . . . . . 84
4.2
A Modified Action . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3
A First Pass at Thermodynamics . . . . . . . . . . . . . . . . 95
4.4
Bouncing and Cyclic Cosmologies . . . . . . . . . . . . . . . . 99 ii
4.5
Interactions and Entropy Production . . . . . . . . . . . . . . 103
4.6
Shrinking Cycles and Exit . . . . . . . . . . . . . . . . . . . . 109 4.6.1
Shrinking Cycles . . . . . . . . . . . . . . . . . . . . . 110
4.6.2
After the Hagedorn Era . . . . . . . . . . . . . . . . . 112
4.7
On the BV Mechanism . . . . . . . . . . . . . . . . . . . . . . 115
4.8
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5 Dynamical Decompactification and Three Large Dimensions124 5.1
Introduction and Previous Results . . . . . . . . . . . . . . . . 125
5.2
Interaction Amplitudes and Impact Parameter Picture . . . . 131
5.3
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 135 5.3.1
Matter Content and Boltzmann Equations . . . . . . . 136
5.3.2
Thermodynamic Phases and Interaction Rates . . . . . 137
5.3.3
Initial Conditions . . . . . . . . . . . . . . . . . . . . . 140
5.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.5
Summary and Discussion . . . . . . . . . . . . . . . . . . . . . 145
6 Anisotropy and the Cases d = 1 and d = 2 6.1
150
The Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.1.1
Equilibrium Phases . . . . . . . . . . . . . . . . . . . . 154
6.2
Fluctuations, Initial Conditions and Procedure . . . . . . . . . 156
6.3
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 160
References
171 iii
Appendices
180
A Covariant Action for Isotropic Cyclic Models
181
B Energy Conditions in Dilaton Gravity
185
C Root Mean Square Velocity of Winding Modes
187
iv
List of Figures
1.1
The fundamental domain F0 of the torus (shaded region). The τ2 direction extends to infinity. Boundaries mirrored about τ1 = 0 are identified . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1
Contour deformation that yields the leading behaviour of Ω(E) from the Hagedorn singularity. . . . . . . . . . . . . . . . . . . 50
2.2
A singularity due to a light mode approaching the hagedorn singularity as a compactification parameter changes, giving subleading corrections to Ω(E). This could correspond to a Kaluza-Klein mode as the universe grows. . . . . . . . . . . . 52
2.3
Contour deformation yielding the leading terms in Ω(E), due to light Kaluza-Klein modes and the zero-point energy (Hagedorn singularity) . . . . . . . . . . . . . . . . . . . . . . . . . 54 v
3.1
The temperature - radius relation for a gas of strings at constant entropy. The top curve corresponds to higher entropy. Near the self-dual radius the temperature plateus close to the Hagedorn temperature. At large radii the Kaluza Klein modes become light and the graph resembles that of pure radiation. At small radii the relevant modes are the winding modes. Because of T-duality the graph is symmetric about the self-dual radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
√
3.2
Surveying the initial condition space (x ≡
dλ˙ 0 , ϕ0 , ϕ˙ 0 ) |ϕ˙ 0 |
for
fixed volumes V0 and d = 3. For smaller initial volumes, corresponding to λ0 = 1 (left) the system can exit the Hagedorn phase only with small initial energies. For larger volumes corresponding to λ0 = 3 (right) larger energies are allowed but exit still occurs for fine-tuned initial condition. In both cases, the system has to begin near the critical density ρH . . . . . . . 81
4.1
Phase space flows for w = 0 (top), w = 1/d (middle), w = −1/d √ ˙ = (±1, ±1/ d) and (ϕ, ˙ = (bottom). The five fixed points (ϕ, ˙ λ) ˙ λ) (0, 0) are connected smoothly. Some typical trajectories are also shown. For w = 1/d and w = −1/d they represent bounces of the scale factor due to KK and winding modes respectively. . . . . . . 94
vi
4.2
Numerical solution with Hagedorn and frozen phases and no potential for the dilaton. The oscillations have constant amplitude as there is no entropy production. The oscillations stop when the universe reaches weak coupling. We use d = 3, as in all graphs that follow. . . . . . . . . . . . . . . . . . . . 104
4.3
A plot of the energy in the string gas for Fig. 4.2. The energy is constant during the Hagedorn phases. During a frozen phase it increases until the scale factor bounces. It then decreases and the system re-enters the Hagedorn phase. . . . . . . . . . 104
4.4
Same as Fig. 4.2, but with a dilaton potential of the form Aeϕ that yields a single bounce for the dilaton. . . . . . . . . . . . 105
4.5
A potential of the form Aeϕ + Be−ϕ can confine the dilaton at weak coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.6
An integration of the equations of motion (5.9), (4.8), (4.34), (4.35) for d = 3. As the entropy increases the energy during the Hagedorn phases increases towards Emax and the size of the oscillations in the scale factor gets smaller (the dashed lines are drawn at constant λ). vii
. . . . . . . . . . . . . . . . . 113
4.7
The log of the scale factor and the matter energy in a typical numerical solution. For t < 400 the universe cycles between Hagedorn and radiation phases. For 400 < t < 640 the scale factor oscillates about the minimum of its potential while the winding strings gradually annihilate (in practice we use a cutoff value W = 1/2 to specify winding mode annihilation). For t > 640 the universe is radiation-dominated. . . . . . . . . . . 116
5.1
Number of cases unwinding for d = 9, 8, 7, 6, 5, 4 as a function of ϕ0 and λ0 . The z-axis is clipped at 200 to make the few decompactifying cases (other than those in the Hagedorn phase where hW i = 0) visible. . . . . . . . . . . . . . . . . . . . . . . 146
5.2
Number of cases unwinding for d = 3 as a function of ϕ0 and λ0 . While in the Hagedorn phase, as is the case for all d, the system stays trapped, in the radiation phase, in contrast to the cases d > 3, any remining winding modes decay. . . . . . . 147 viii
5.3
A plot of behavior in the (λ0 , ϕ0 ) plane contrasting the cases d = 3, 4, 6, 9. If the initial equilibrium winding number in the Hagedorn phase is non-zero the system typically stays trapped in the Hagedorn phase, unless the initial Hubble rate is large and the initial winding number is small (thin dark region labeled ‘Equilibrate out of Hagedorn for λ˙ 0 = 10σ’). If the initial equilibrium winding number is zero in the Hagedorn phase then the universe typically decompactifies in any number of dimensions (regions in the upper left corner, labeled by dimension). But if the universe begins in a radiation phase with a dilute gas of winding strings then only d = 3 will decompactify (orange region to the right of the grey line). . . . . 148
6.1
Number of cases decompactifying (out of 100) as a function of ϕ0 and r, for ν0 = 3, 5 and tf = 1, 10, 100 when d1 = 1. . . . . 163
6.2
Number of cases decompactifying (out of 100) as a function of ϕ0 and r, for ν0 = 3, 5 and tf = 1, 10, 100 when d1 = 2. . . . . 164
6.3
Contours of constant number of decompactification cases. The thick black line is for constant energy density at the critical density between radiation and Hagedorn phases. The thin red line is for a constant energy density below which more than 90% cases decompactify. Here, d1 = 1 . . . . . . . . . . . . . . 166
6.4
Same as in figure 6.3 but for d1 = 2. . . . . . . . . . . . . . . . 167
ix
List of Tables 6.1
The range of densities in units of α0−5 resulting to decompactification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.2
Number density of decompactification cases normalized to 100, when the universe exits to a radiation era. . . . . . . . . . . . 169
x
Acknowledgments I would like to thank my advisor, Brian Greene, for his support and guidance throughout my studies, and in particular for his openmindedness, clarity of thought, pedagogy and for teaching me how to look at the big picture of physics. Dan Kabat for his immeasurable help and patience throughout our collaboration. Lam Hui and Marilena LoVerde, and also Bowen Xiao, Simon Judes, Pontus Ahlqvist, Amanda Weltman, Kurt Hinterbichler, Adam Brown and especially C ¸ a˘glar Girit, with whom studying physics was a pleasure. Allan Blaer for being an inspiring figure, just two doors down the hall. My thesis committee members for their participation in my defense. My brothers Andreas, Angelos and Demetris for their outlook on life. Anton and especially Yiannis for being an irreplaceable roommate for five years. My work was supported by Cyprus State scholarship fund, A.G Leventis foundation and Columbia University. And last but not least I would like to thank Susan, who explained to me how you eat an elephant: one bite at the time.
xi
To my parents, Kyriacos and Elsie
xii
Part I String Theory
1
2 In the first two chapters, we review some aspects of string theory that are relevant to this thesis. We are ultimately interested in the thermodynamics of a gas of type II closed strings on toroidal backgrounds. To that end we guide the reader through derivations of relevant quantities from first principles. We emphasize on the string spectrum, T-duality, one-loop amplitudes and the free energy, and the microcanonical density of states derived as a Laplace transform of the thermal partition function. The results will be applied to early universe string cosmology in part II.
3
Chapter 1 String Theory Preliminaries 1.1
The String Action
The trajectory of a string in spacetime is two-dimensional and its motion is described by functions X µ (τ, σ) that map a two-dimensional worldsheet, parametrized by τ and σ, onto D dimensional spacetime. Here, µ ranges from 0 to D − 1. In analogy to the action for the relativistic point particle which is proportional to its proper time, we can write down an action for the string proportional to the area of the worldsheet: Z 1 SN G [X] = − dτ dσ(−dethab )1/2 2πα0 M
(1.1)
where, hab = ∂a X µ ∂b Xµ ,
a, b running over values (τ, σ)
(1.2)
is the induced metric on the worldsheet. The constant α0 which has units of length squared is called the Regge slope1 and is related to the string tension 1
This name is a remnant of the early days of string theory, developed as a dual model in to address strong interactions.
4 T by T =
1 . 2πα0
(1.3)
The action in (1.1) is called the Nambu-Goto action. The peculiar square root in the Nambu-Goto action makes it difficult to use in a path integral formulation, so instead we introduce an independent worldsheet metric γ(τ, σ) with Lorentzian signature (−, +)2 and write down the Polyakov action 3 Z 1 dτ dσ(−detγab )1/2 γ ab ∂a X µ ∂b Xµ (1.4) SP [X, γ] = − 0 4πα M The equations of motion for γab can be used to eliminate it from SP and one then arrives back to the Nambu-Goto action. Hence the two are equivalent. The Polyakov action has the following symmetries: 1. D-dimensional (spacetime) Poincare invariance, X µ → Λµν X ν + aµ with Λ the usual Lorentz matrices and aµ translations. 2. Diffeomorphism or reparametrization invariance on the worldsheet (τ, σ) → (τ 0 (τ, σ), σ 0 (τ, σ)). 3. Worldsheet Weyl invariance, γ → e2ω(τ,σ) γ for arbitrary ω(τ, σ).
1.1.1
The Conformal Gauge
The worldsheet metric γab has three independent components. We may use the reparametrization invariance above to fix two of those so that only one 2
Note that for Lorentzian metrics in general, we will be using a ’mostly positive’ signature. 3
Or named after its founders, the Brink-Di Vecchia-Howe-Deser-Zumino action.
5 independent component remains. But the latter can be gauged away using the invariance of the action under a Weyl rescaling. As a result, γab can be chosen to be the two-dimensional Euclidean metric δab 4 . The action now takes the simple form of D free scalar fields in two dimensions, (with Euclidean signature and an extra minus sign as a convention for the Euclidean path integral): 1 SE [X] = 4πα0
Z
dτ dσ(∂τ X µ ∂τ Xµ + ∂σ X µ ∂σ Xµ )
(1.5)
It is useful to adopt complex coordinates z = τ + iσ,
z¯ = τ − iσ
(1.6)
and the action takes the form (note the factor of 2 from the Jacobian) Z 1 ¯ µ SE [X] = (1.7) d2 z∂X µ ∂X 2πα0 ¯ denoting partial derivative with respect only to z (¯ with ∂ (∂) z ). The classical equations of motion are µ ¯ µ ) = ∂(∂X ¯ ∂(∂X )=0
(1.8)
¯ µ = ∂X ¯ µ (¯ and it follows that ∂X µ = ∂X µ (z) is holomorphic and ∂X z ) is antiholomorphic, or holomorphic in z¯. Consequently they admit the Laurent expansions µ
∂X (z) = −i 4
α0 2
1/2 X ∞ −∞
µ αm ¯ µ (¯ , ∂X z ) = −i z m+1
α0 2
1/2 X ∞ −∞
α ˜µ m z¯m+1
(1.9)
This flat metric is a local gauge fixing and it works as long as there is no global topological obstruction, as is the case for cylindrical or toroidal worldsheets that we will encounter.
6 µ µ The coefficients αm and α ˜m are given by inverting the expansions giving 1/2 I 1/2 I 2 2 dz m d¯ z m¯ µ µ µ µ αm = z ∂X , α ˜m = − z¯ ∂X (1.10) 0 0 α 2π α 2π
The divergence theorem and single valuedness of X µ imply that α0µ = α ˜ 0µ . Integrating the expansions in (1.9) above yields 0 1/2 0 1/2 X µ µ 1 αm α α α ˜m µ 2 µ µ X (z, z¯) = x − i α0 ln |z| + i + m (1.11) 2 2 m zm z¯ m6=0 where xµ is an integration constant. So far we merely solved locally the classical equations of motion of our two-dimensional free-field theory on the plane. We have not specified the global properties of the worldsheet, or its topology. To see how we can describe strings, let us map the plane to the cylinder. A cylindrical worldsheet with the “spatial” identification σ∼ = σ + 2π
(1.12)
−∞ < τE < ∞
(1.13)
and Euclidean time running
would describe a free closed string. We can map the z-plane to the cylinder z = e−iw ,
w ≡ σ + iτE
(1.14)
Substituting this to the solution (1.11) and moving to Minkowski time τE → iτ gives µ
µ
0 µ
X (σ, τ ) = x + α p τ + i
α0 2
1/2 X
1 µ −im(τ +σ) µ −im(τ −σ) e αm e +α ˜m m m6=0 (1.15)
7 where we identified pµ =
2 1/2 α0
α0µ . This solution now gives a more intuitive
picture. The free closed string has a center of mass position xµ and center of mass momentum pµ (zero-modes), and left-moving and right-moving waves propagating along it. Reality of X µ requires that xµ and pµ are real, while positive and negative modes are conjugate to each other: µ µ ∗ α−m = (αm ),
µ µ ∗ α ˜ −m = (˜ αm )
(1.16)
But since we haven’t fully fixed the gauge of the original action, we are not properly counting independent solutions. We can do this by choosing light-cone coordinates in spacetime, X ± =
√1 (X 0 2
± X 1 ). Once we fully fix
the gauge of the Polyakov action, the equations of motion of the original γab should be imposed as constraints to the resulting equations of motion. The advantage of the light-cone gauge is that the constraints become linear and easy to solve. This results in the vanishing of the oscillatory modes of X + and further allows us to express the oscillatory modes of X − in terms of the oscillatory modes of the transverse directions X i , i = 2, ..., D − 1. The resulting physical modes are then the zero modes of X ± (or X 0,1 ) and the zero modes and D − 2 oscillatory modes of the X i . The string oscillates only transverly to its worldsheet while oscillations tangent to the worldsheet (timelike and longitudinal) are just oscillations of the coordinate system (that can be gauged away). The residual symmetries of the action SE [X] consist of an infinite-dimensional group of transformations z → f (z) , z¯ → f (¯ z ). This is the conformal group
8 in two dimensions generated by an infinite set of operators Ln , n = 0, 1, 2..., called the Virasoro operators. They generate infinitesimal transformations z → z − n z n+1 and z¯ → z¯ − n z¯n+1
(1.17)
¯ n = −¯ Ln = −z n+1 ∂ and L z n+1 ∂¯
(1.18)
via
and classically they satisfy the algebra [Lm , Ln ] = (m − n)Lm+n
(1.19)
˜ m satisfy the same algebra. In terms of the oscillators αµ the Virasoro The L m operators have the following form: +∞
+∞
X µ 1X µ ˜m = 1 Lm = αm−n αnµ L α ˜ α ˜ nµ 2 −∞ 2 −∞ m−n
(1.20)
˜ m rightNote that the Lm involve exclusively left-moving modes and the L moving modes. We are interested in two particular generators in the closed string theory. These are the worldsheet Hamiltonian that corresponds to translations in the worldsheet “time” τ and the worldsheet momentum that corresponds to translations along the worldsheet “spatial” direction σ: ˜ 0, H = L0 + L 1.1.2
˜0 P = L0 − L
(1.21)
Quantized Closed Strings
In the quantum theory the Virasoro algebra is modified to [Lm , Ln ] = (m − n)Lm+n +
c m(m2 − 1)δm+n,0 12
(1.22)
9 and ˜ m, L ˜ n ] = (m − n)L ˜ m+n + [L
c˜ m(m2 − 1)δm+n,0 12
(1.23)
The term proportional to c is called the central extension and c is called the central charge. It appears solely due to quantization. From canonical quantization it can be seen to arise due to normal ordering ambiquities in the oscillator modes. The commutation relations are µ µ [αm , αnν ] = [˜ αm ,α ˜ nν ] = mη µν δm+n,0 ,
µ [αm ,α ˜ nν ] = 0
(1.24)
By calculating explicitly, we find that every oscillating field (in light-cone gauge we know that these would be the D − 2 transverse fields) contributes 1 unit to c. ˜ 0 will be affected by From the relations (1.20) we see that only L0 and L normal ordering, up to an additive constant (proportional to c and c˜). This will add an additive constant to the hamiltonian H while it will cancel out in P . If we keep the quantum definitions of the Lm the same as in (1.20) then the normal ordering in the hamiltonian gives ˜ 0 − c + c˜. H = L0 + L 24
(1.25)
As mentioned above, each oscillating transverse field contributes 1 unit to c and also, c˜ = c. The constant a ≡
c+˜ c 24
is equal to
D−2 . 12
We still haven’t spec-
ified the spacetime dimensionality D. When requiring a consistent quantum theory the normal ordering constant turns out to be a = 2 and D = 26. In light cone quantization this comes about by requiring manifest spacetime
10 Lorentz invariance, since that was lost in fixing the noncovariant light-cone gauge. In covariant quantization this follows when we require the absense of unphysical negative norm states. A path-integral BRST quantization yields the same results. D = 26 is called the critical dimension of the bosonic string. In anticipating later results, it is worth commenting on our emphasis on the central charge and the constant a. It is what essentially gives rise to the Hagedorn temperature in our formal derivation that is to follow. In fact, for any string theory with left moving normal ordering constant aL and right moving normal ordering constant aL (here they are the same and equal to 1) √ √ the Hagedorn temperature is proportional to α0−1/2 ( aL + aR )−1 . In the quantum theory, the fact that H and P are generators of symmetries translates to their annihilating physical states. In fact the whole set of ˜ m annihilate physical states. It amounts to enforcVirasoro operators Lm , L ing the equations of motion of γab from the Polyakov action.5 To specify our states, first define 1 i aim = √ αm m
and
1 i† aim = √ αm m
for
m>0
(1.26)
with i ranging over the physical transverse polarizations as above. Then, using (1.24), the algebra satisfied by these modes is essentially the algebra of raising and lowering operators for a simple harmonic oscillator (for each i 5
To be more presice, the Virasoro generators are the fourier modes of the worldsheet stress-energy tensor which has to vanish. This is what the equations of motion of γab require.
11 and for each level n): ij ˜j† aim , a [aim , aj† n ] = δ δm,n n ] = [˜
for
˜ 0 , using our expression α0µ = α Then L0 and L ˜ 0µ = L0 = The operators N ≡
m, n > 0 q
α0 µ p , 2
take the form
∞ ∞ X α0 α0 2 X i† i ˜ ˜in p + nan an L0 = p2 + n˜ ai† na 4 4 n=1 n=1
P∞
n=1
(1.27)
(1.28)
i i† i ˜ P∞ n˜ nai† ˜n are called the number n an and N ≡ n=1 an a
operators in analogy to the quantum harmonic oscillator. They count the overall number of excitations, for each level n and transverse dimension i. The Hamiltonian then takes the form α0 2 ˜ −a H = p +N +N 2
(1.29)
The fact that it should vanish on physical states gives the mass shell contraint for the closed bosonic string: M 2 = −p2 =
2 ˜ − a) (N + N α0
(1.30)
States then can be labeled by their left-right oscillation numbers and spatial momentum. As in the harmonic oscillator, we assume a vacuum state E ~ 0, 0; k that is annihilated by all lowering operators. A general state can be E build by acting on 0, 0; ~k with the raising operators: "D−1 ∞ # ˜in E E i N Y Y (ai−n )Nin (˜ a ) −n ˜ ; ~k = p N, N 0, 0; ~k ˜in ! Nin !N i=1 n=1
(1.31)
Note that states are built by acting with the aµm ’s and a ˜µm ’s on the ground state. As in the quantum harmonic oscillator, the ground state is assumed to
12 be unique and it forms a scalar representation of the spacetime Lorentz group. Since then, the aµm ’s and a ˜µm ’s are spacetime vectors the states of this theory are spacetime bosons. This is why this is called a bosonic string theory. We will see how fermions can be incorporated within the superstring formulation ˜ 0 should vanish on physical states later on. The constraint that P = L0 − L gives the level matching condition on left and right excitations: ˜ N =N
1.2
(1.32)
One-Loop Vacuum Amplitudes
As mentioned in the introduction, we are ultimately interested in the one loop free energy of strings. Before introducing the finite temperature case, let us see the generic case for closed string one loop vacuum amplitudes. In general, a closed string amplitude is specified by the topology of the worldsheet. A cylinder represents a free propagating closed string, as we saw earlier. A three point amplitude at tree level, that is an interaction with one string in the initial state decaying to two in the final state would be a cylinder with one leg on one side splitting in two on the other end. With a conformal transformation this can be turned into a sphere with three punctures, representing the three asymptotic states. A two → two tree level amplitude would then be a sphere with four punctures and any generic tree level amplitude involving n closed strings in the initial state and m in the final would be a sphere with n + m puncture. From the 2-dimensional field theory
13 point of view this would be evaluated with a path integral of the Polyakov action with insertions of vertex operators (operators that create states on the worldsheet and map to states in space time) in the path integral. At one loop, it is easy to visualize that the closed string worldsheet is a torus with punctures representing asymptotic states. More generally, closed (oriented) string amplitudes are given by path integrals on two-dimensional compact surfaces with genus corresponding to the number of loops. Ofcourse the correct path integral of the Polyakov action should be divided with the appropriate “gauge volume” of Weyl and diffeomorphism invariance. The great simplicity in string theory is that such an integral takes into account all the different feynman diagrams that correspond to the same order in perturbation theory, unlike point particle field theory where one can encounter distinct integrals for different diagrams of the same order. A note on the string coupling is in order here. There is one more term that we can write in the action for the two-dimensional theory, that preserves all the symmetries. That would be Z p λ Sλ [γ] = dτ dσ −detγab R(γab ) 4π
(1.33)
with R(γab ) the Ricci scalar calculated from γab . This is an Einstein-Hilbert term on the worldsheet. According to the Gauss-Bonet theorem, in two dimensions this term is a constant: Z p 1 dτ dσ −detγab R(γab ) = 2(1 − g) 4π
(1.34)
with g the genus of the two-dimensional compact surface. Being a constant
14 it does not affect the local solutions we discussed earlier, but in the perturbative theory it results to an extra factor of e2λ at every order. Since this corresponds to “emmiting” and “absorbing” a string, eλ would be the string coupling. When the theory is generalized to include dynamical spacetime backgrounds λ is promoted to a dynamical field as well, and the string coupling becomes the exponential of the expectation value of this dynamical field, known as the dilaton. The fact that the string coupling is dynamical is of major importance in early universe cosmology and it will be central to later chapters. We know turn to the one loop vacuum amplitude, i.e a worldsheet torus with no insertions. As we will see, a lot of interesting physics comes with this quantity. In particular, it knows of the whole spectrum of the theory.
1.2.1
Toroidal Worldsheet and the Partition Function
To avoid confusion with our previous labeling of coordinates, we will use the coordinate names (σ1 , σ2 ) = (σ, τE ) from now on. Also, recall the complex coordinate w = σ1 + iσ2 that we used for the cylinder in (1.14). A twodimensional torus can be described with one complex parameter τ = τ1 + iτ2 called the modulus. In terms of the local diffeormorphism and Weyl symmetries, a theorem states that a (flat) metric on a torus can be brought to the form ds2 = |dσ1 + τ dσ2 |2
(1.35)
15 For τ = i this would be the metric δab we encountered before. A torus is then defined as the complex plane with the above metric and identifications w∼ = w + 2π ∼ = w + 2πτ
(1.36)
or in terms of the real coordinates (σ1 , σ2 ) ∼ = (σ1 + 2π, σ2 ) ∼ = (σ1 + 2πτ1 , σ2 + 2πτ2 )
(1.37)
We can think of a torus as a cylinder of circumference 2π and length 2πτ2 with the ends rotated by an angle of 2πτ1 and then sewn together. On the plane, this is simply a flat parallelogram, with edges at 0, 2π, 2πτ and 2π(τ + 1) and opposite sides identified. Consider now the action in conformal gauge as in (1.7) and for now with just one scalar field (X µ = X) . We can write the action with a flat metric, with coordinates on the parallelogram described above, and the periodicity of the field implicit. Z SE [X] =
d2 σ∂a X∂ a X
(1.38)
We are interested in the one loop amplitude given by the Euclidean path integral Z
dX|p e−SE [X]
(1.39)
The subscript p reminds us that the field X is periodic in both directions on the torus. We can think of this amplitude as given by taking a field theory on a circle, evolving for Euclidean time (along σ2 ) 2πτ2 , translating in σ1 by
16 2πτ1 and then identifying the ends. In operator languange this gives a trace6 Z dX|p e−SE [X] ≡ ZX (τ ) = Tr[e(2πiτ1 P −2πτ2 H) ] (1.40) ˜ 0 generates translations of Recall that the worldsheet momentum P = L0 − L ˜ 0 − 1 (c + c˜) generates transσ1 and the world sheet hamiltonian H = L0 + L 24 lations of σ2 . Given the resemblance to statistical mechanics, this quantity is termed the partition function. It contains all the information of the spectrum of the theory. With only one scalar field, c = c˜ = 1. Defining q = e2πiτ we have ˜
ZX (τ ) = (q q¯)−1/24 Tr[q L0 q¯L0 ]
(1.41)
Recalling the expressions in terms of momentum and left/right number operators L0 =
α0 2 p + N, 4
0 ˜ 0 = α p2 + N ˜ L 4
(1.42)
˜n and the trace breaks up into a sum over occupation numbers Nn and N over the momenta p and a product over the levels n. In other words, each level yields it’s own independent tower of oscillators. We can consider a P continuum normalization for the momentum, turning the sum p to an R integral V /(2π) dp and introducing a spacetime volume V (even if in one dimension for now) we get −1/24
ZX (τ ) = V (q q¯)
Z
∞ dp −πτ2 α0 p2 Y X nNn nN˜n e q q¯ 2π n ˜
(1.43)
Nn ,Nn =0
R R ˆ ˆ This is the usual relation in one dimension Tre−HT = dq hq| e−HT |qi = q|P e−SE with q periodic paths on the interval [0, T ] generalized to one more minkowskian direction σ1 (hence the i infront of P ) 6
17 The sums are geometric7 ∞ X
q nN =
N =0
1 1 − qn
(1.44)
and using the conventional definition of the Dedekind eta function η(τ ) = q
1/24
∞ Y
(1 − q n )
(1.45)
n=1
we finally get ZX (τ ) = V (4π 2 α0 τ2 )−1/2 |η(τ )|−2
(1.46)
This is the partition function8 , or the one loop amplitude for a single boson on a torus with modulus τ . The partition function is invariant under the transformations τ 0 = τ + 1,
τ 0 = −1/τ
(1.47)
This symmetry is called modular invariance. It is easy to check given the modular transformations of the eta function: η(τ + 1) = eiπ/12 η(τ ), η(−1/τ ) = (−iτ )1/2 η(τ )
(1.48)
Repeated application of the two transformations above generates the mappings τ0 =
aτ + b cτ + d
(1.49)
7
We have implicitly made a choice of fixed τ with positive real and imaginary parts. The sums then converge as |q| < 1 8
Note that ZX is given here with the factor of V . Later on it will be more convenient to drop this.
18 for any integers a, b, c, d such that ad − bc = 1. Writing it in terms of the real coordinates as
0 σ1 d b σ1 = σ2 c a σ20
(1.50)
takes the metric in (1.35) for σ1,2 with modulus τ into a metric with the same 0 form for σ1,2 with modulus τ 0 . This is called the group of large diffeomor-
phisms of the torus as it contains transformations that are not connected to the identity. Using the modular transformations one can identify the region in the complex τ -plane containing points that represent inequivalent tori. It is called the fundamental region, F0 , for the moduli space of the torus (figure 1.1): 1 1 F0 : − ≤ Reτ ≤ , Imτ > 0, |τ | ≥ 1 2 2
(1.51)
Infact the boundaries mirrored about Reτ = 0 are identified so this region is open only at Imτ → ∞.
1.2.2
The Bosonic One-Loop Amplitude
We now want to calculate the full vacuum amplitude at one-loop for the bosonic string described by the Polyakov action in (1.4) namely Z dXdγ W1l oop = e−SP [X,γ] Vdif f ×W eyl
(1.52)
with γ Euclidean such that, as usual, the integral is well defined. Vdif f ×W eyl accounts for the overcounting due to the local gauge freedom, here in diffeomorphisms and Weyl transformations on the worldsheet. Formally this
19
Figure 1.1: The fundamental domain F0 of the torus (shaded region). The τ2 direction extends to infinity. Boundaries mirrored about τ1 = 0 are identified can be done using the Faddeev-Popov ghost gauge fixing techniques, but our results and discussion in the section above can lead us to the right answer. One gets precisely the same results with a ghost gauge fixing technique ([1]). The answer should first involve a trace just like for the partition function of the single scalar above. Since the partition function is multiplicative in different fields it will involve ZX (τ ) to some power. We need to trace over the correct degrees of freedom (after accounting for gauge redundancy) and we saw earlier that those are the zero modes (momentum) of all the X µ fields and the oscillator modes only of the transverse 24 fields9 . So we have a factor ZX (τ )24 and an extra factor V0,1 (4π 2 α0 )−1 from the momenta of the 9
With Faddeev-Popov gauge fixing, the ghosts cancel the contribution of two of the X ’s oscillators. µ
20 longitudinal, or X 0 and X 1 fields. There is a subtlety in the integration of the temporal momentum p0 though, giving a wrong-sign gaussian due to Lorentzian spacetime signature that diverges. To define that integral correctly we can rotate p0 → ip0 so this gives an overall factor of i to the trace. So far then we have iV26 (4π 2 α0 )−13 |η(τ )|−48
(1.53)
As this is the quantity for a single torus, labeled by τ , to consider all inequivalent tori we need to integrate over the fundamental domain F0 . There is though a further redundancy. Requiring the metric to take the form (1.35) with τ in F0 does not fully fix the dif f × W eyl invariance. The metric and periodicity are left invariant under rigid translations σa → σa + va and reflections σa → −σa . This amounts to deviding the measure dτ1 dτ2 by twice the volume of the torus 2(2π)2 τ 2 (the factor of 2 for the reflections). Also, as we are integrating over moduli space and not metrics as in the path integral, we need to include a factor of the jacobian between dτ and dγab which is (2π)2 . Including these factors and noting that d2 τ ≡ dτ d¯ τ = 2dτ1 dτ2 we end up with the final expression for the one loop vacuum amplitude Z d2 τ (4π 2 α0 τ2 )−13 |η(τ )|−48 W1-loop = iV26 4τ 2 F0
(1.54)
The amplitude ofcourse has the important property of modular invariance. Every τ2 |η(τ )|4 factor is invariant using (1.48) and it is easy to check for the remaining dτ d¯ τ /τ22 .
21 The bosonic 1-loop amplitude is a great means to reveal the essential difference between the ultraviolet behaviours of string theory and (nonsupersymmetric) field theory. Let’s compare (1.54) with the corresponding quantity in field theory, the sum over particle paths with the topology of a circle10 . For particles of mass m this is Z Z ∞ dd k dl −(k2 +m2 )l/2 Zcircle (m) = Vd e d (2π) 0 2l Z ∞ dl 2 = iVd (2πl)−d/2 e−m l/2 2l 0
(1.55)
This is completely analogous to (1.54): k 2 + m2 is the world-line hamiltonian, l is the circle’s metric’s modulus (the circumference), and the 2l in the denominator removes the overcounting from translation and reversal of the proper-time coordinate (the world-line parameter). In the UV / short distance l → 0 limit, this is divergent, and reflects the usual UV divergence of point particle field theory. The analogus τ2 → 0 (small torus limit) in string theory is simply absent as it is not included in the domain F0 . The modular invariance built in string theory’s gauge symmetries renders it UV finite! This is even more strikingly exemplified if one assignes to the point particle field theory the mass spectrum of string theory, m2 = and trace over it. With an integral Z δN,N˜ =
π
−π
dτ1 i(N −N˜ )τ1 e 2π
2 (N α0
˜ − 2) +N
(1.56)
˜ and defining τ = τ1 + il/α0 to enforce the level matching constraint N = N 10
The description of a field theory as a sum over particle paths is equivalent to the description in terms of field histories.
22 one gets for the field theory case an expression identical to (1.54) with the sole difference being the τ integration region, which is now the full strip τ2 > 0, |τ1 | < 21 . The τ2 → 0 UV divergence is now clear. Note though that in the IR limit τ2 → ∞ the amplitude (1.54) diverges, ˜ = 0 and due to the presence of a tachyon in the spectrum, namely for N = N m2 = − α40 and reflects the usual tachyon instabilities of field theories. The leading term in the integrant as τ2 → ∞ goes as e4πτ2 giving the divergence. This is only an artifact of the bosonic string theory and it can be cured in the superstring theory that includes spacetime fermions as we will see below. It is worth noting that this divergence is identical to the divergence of the free energy at the Hagedorn temperature, namely the appearance of a temporal tachyon, as we will see later. Finally, the important physical interpretation of (1.54) is that it gives the leading order cosmological constant or vacuum energy density via ρvac = iW1−loop /V26 .
1.3
Toroidal Compactification in Spacetime
As string theory involves extra spatial dimensions beyond the 3 that we observe at large scales, it is common practice to treat the extra dimensions as being small, near the yet-unexplored planck scale, forming a compactified manifold. The simplest (and solvable) manifold that we can have is the torus. In this thesis such compactification is relevant in two ways. First, a
23 background where all spatial dimensions are toroidal is the starting point of string gas cosmology, and the stringy winding modes that we discover here are a crucial ingredient for the early universe scenaria that we present later. Second, we are interested in finite temperature physics, which is formally done by compactifying the temporal dimension on a 1-torus, i.e a circle. Consider then one (spatial) X scalar field, compactified on a circle of radius R. The field is periodic, namely X∼ = X + 2πR
(1.57)
This periodicity has two effects. The first is that the center of mass momentum is quantized as in Kaluza-Klein theories. p=
n R
(1.58)
with n any integer. In other words, the operator e2πiRp translating once around the periodic dimension leaves states invariant. The second effect is special to string theory. A closed sting can wind around the compact direction: X(σ1 + 2π) = X(σ1 ) + 2πRw
(1.59)
The integer w, with allowed values in (−∞, ∞) is the winding number. In a compact space it must be conserved, for example a string with winding number 1 can interact with an (anti)wound string with w = −1 to give an unwound string with w = 0.
24 Recall that the condition on the left/right zero modes of the closed string in the first section was α0 = α˜0 which resulted in equal left/right center of mass momenta. This was a consequence of the single valuedness of X and the laurent expansions of the modes. In the case at hand X is not single-valued, but the change in X in going around the string (σ1 coordinate) is I
I
∆Xσ1 =
dσ1 ∂σ1 X = σ1
¯ (dz∂X + d¯ z ∂X) = 2π(α0 /2)1/2 (α0 − α ˜0)
σ1
(1.60) = 2πRw
hence (α0 − α ˜ 0 ) = (2/α0 )1/2 Rw. The center of mass momentum is as before p = (2α0 )1/2 (α0 + α˜0 ), hence for a periodic dimension we now have the left and right center of mass momenta n wR + 0 R α wR n pR = (2/α0 )1/2 α0 = − 0 R α pL = (2/α0 )1/2 α0 =
(1.61)
The Virasoro operators become ∞
α0 p2L X L0 = + α−n αn , 4 n=1
∞
0 2 X ˜ 0 = α pR + L α ˜ −n α ˜n 4 n=1
(1.62)
The partition function for a single periodic X is now ˜
ZX|p (τ ) = (q q¯)−1/24 Tr[q L0 q¯L0 ] −2
∞ X
−2
n,w=−∞ ∞ X
= |η(−τ )| = |η(−τ )|
0 2
0 2
q α pL /4 q¯α pR /4
0 2 w 2 R2 αn exp −πτ2 ( + + 2πiτ1 nw R2 α0 n,w=−∞ (1.63)
25 Modular invariance of this partition function is not manifest but one can check that it is present using the poisson resummation formula. For later purposes, it is useful to note the change to the total partition (1.46) function when one dimension is compactified. We merely multiply with a factor 0 2 ∞ X αn w 2 R2 2 0 1/2 exp −πτ2 ( F2 (τ, R) = (4π α ) + 2πiτ1 nw + R2 α0 n,w=−∞ (1.64) We undo the continuous momentum integral with the factor (4π 2 α0 )1/2 and replace it with the compact trace given by the sum. Also, since there is no momentum integral, the volume factor is reduced by the volume of the compact dimension, Vd → Vd−1 . The oscillator sum is the same as before giving the factor |η(τ )|−2 from a compact dimension as for a noncompact one. To see how the spectrum is modified, consider the critical D = 26 case with only one dimension compactified, say X 25 . The total Hamiltonian (with the oscillators only in the transverse directions as before) is now H=
α0 α0 ˜ −2 pµ pµ + (p2L + p2R ) + N + N 2 4
(1.65)
with µ = 0, ..., 24 running over only the non-compact dimensions. We have then an effective mass m2 = −pµ pµ =
n2 w 2 R2 2 ˜ − 2) + + 0 (N + N 2 02 R α α
(1.66)
˜ 0 is now The level matching condition L0 = L ˜ = nw N −N
(1.67)
26 1.3.1
T-duality
In the mass formula above, one identifies 4 contributions: along with the oscillators and zero point energy (central charge) we have Kaluza Klein modes with mKK ∼
1 R
and winding modes with mw ∼ R. Note that the mass of
the latter is what one would expect classicaly for the potential energy stored in a string with tension T = 1/(2πα0 ) wrapping a circle with circumference 2πR, w times. Now as R → ∞ the winding modes become infinitely massive, while the compact momentum go over to a continuous spectrum, as would be expected in a non-compact dimension. The crucial stringy feature here is what one finds in the opposite R → 0 limit: The compact momenta become infinitely massive while the winding momenta very light as it does not cost energy to wrap a string around a very small circle. This is a symmetry of the theory, and infact the two limits are physically identical. The spectrum is invariant under R↔
α and n ↔ w R
This symmetry is called T − duality and the special radius
(1.68) √
α0 is called the
self-dual radius. It is an essential ingredient for the string gas cosmology scenario that we encounter later on. This symmetry has an interesting field theory interpretation. In particle Kaluza-Klein theories the isometries of the compactified space translate to gauge symmetries in the effective lower dimensional description. With one compact dimension, µ = 5 for example, the isometries of the circle give
27 a U (1) symmetry. The corresponding gauge boson is the g µ5 component of the metric and the charges are the Kaluza-Klein momenta n as above. String theory, in addition to the metric, has an antisymmetric tensor field Bµν . These, along with the scalar trace are the irreducible representations ˜ = 1. Upon of the massless tensor that appears in the spectrum for N = N compactification of say X 25 , string theory gives two copies of the symmetry, that is U (1) × U (1). The new gauge boson is B µ25 and the new charges are the winding numbers w. This is the case at a generic radius R. At the √ self-dual radius R = α0 though, there are extra massless states, 4 of which are vector bosons in the effective theory. These are the states with ˜ = 1, n = w = ±1, N = 0, N
˜ =0 n = −w = ±1, N = 1, N
(1.69)
In fact these carry the KK and antisymetric tensor charges w and the only consistent theory of charged massless vectors is non-Abelian gauge theory. These new massless gauge bosons combine with the old into a theory of SU (2) × SU (2) symmetry. For this reason, the self-dual radius is also called a point of enhanced gauge symmetry. Away from this special point, the extra gauge bosons acquire mass and the symmetry is broken spontaneously as in the higgs mechanism.
1.3.2
Generalized Toroidal Compactification and T-duality
There is an elegant description for the case of the general compactification of d dimensions, due to Narain ([2]). Consider the dimensionless momenta
28 lL,R =
p
α0 /2pL,R . For any compactification the spectrum of these forms a
lattice Γ of points l = (lL , lR ) in a 2d-dimensional space R2d . The partition function for the compact dimensions in this case is ZΓ (τ ) = |η(τ )|−2d
X
2
2
eπiτ lL −πiτ lR
(1.70)
l∈Γ
In our one dimensional example the lattice was 2-dimensional, consisting of points labeled by the Kaluza-Klein and winding numbers. In the space R2d , consider an inner product of signature (d, d). That is, 0 the product of two points l = (lL , lR ) and l0 = (lL0 , lR ) is given by 0 l ◦ l0 = lL · lL0 − lR · lR
(1.71)
A lattice is called even if all points on this lattice have even integer norm under this product, i.e for l ∈ Γ, l ◦ l = 2n, n = 0, ±1, ... Also, define the self dual lattice to Γ to be the lattice consisting of all points that have integer such product with points in Γ. The statement is that consistency of a string theory compactified on such lattice requires that the lattice is even and self dual. All these lattices can be classified. Then T-duality is simply the group of discrete rotations, that preserve the signature and send one compactification lattice to itself (by permuting the points). This group is labeled O(d, d, Z). Note that a generic rotation O(d, d, R) on some lattice generates a new inequivalent lattice and hence an inequivalent compactified theory. This is true modulo a subgroup O(d, R)×O(d, R) that preserves seperately the products lL ·lL and lR ·lR that appear in the partition function and would yield the same theory. Taking the
29 generalized T-duality into account then, the space of inequivalent lattices or compactification backgrounds is given by O(d, d, R) O(d, R) × O(d, R) × O(d, d, Z)
1.4
(1.72)
Closed Superstrings
In this section we will extend out results above to include fermions in spacetime. In string theory the inclusion of fermions turns out to require supersymmetry and the resulting theories are called superstring theories. This can be done in two ways: By adding anticommuting grassman spacetime coordinates Θa making manifest spacetime supersymmetry, or by adding fermions and supersymmetry on the worldsheet, called the RNS (Ramond Neveu Schwarz) formalism. Here we will follow the latter as it is more illustrative for our purposes. The desired action is obtained by adding to the X µ action above (in conformal gauge) the following action: 1 S[ψ] = 4π
Z
¯ µ + ψ˜µ ∂ ψ˜µ ) d2 z(ψ µ ∂ψ
(1.73)
This describes left moving ψ µ (z) and right moving ψ˜µ (¯ z ) Majorana-Weyl spinors on the world sheet. Like before, let’s consider closed strings and a cylindrical worldsheet in the complex coordinate w with w ∼ = w +2π. A lot of the richness in superstring theories arises from the two possible periodicities
30 for ψ: Ramond (R): ψ µ (w + 2π) = +ψ µ (w) (1.74) µ
µ
Neveu-Schwarz (NS): ψ (w + 2π) = −ψ (w) Similarly for the right movers ψ˜µ (w). ¯ If we define constants ν and ν˜ taking the values 0 for the R sector and 1/2 for the NS sector we can summarize the boundary conditions as ψ µ (w + 2π) = e2πiν ψ µ (w) (1.75) ψ˜µ (w¯ + 2π) = e−2πi˜ν ψ˜µ (w) ¯ This simply reflects the fact that when a spinor is parallel transported around a closed loop it returns to itself up to a sign ambiguity. The total of four combinations that we can combine these boundary conditions from left and right moving spinors gives rise to four different sectors in the Hilbert space, usually labeled R-R, R-NS, NS-NS, NS-R. The expansions in fourier modes of these fields are ψ µ (w) = i−1/2
X
ψrµ eirw ,
ψ˜µ (w) ¯ = i1/2
r∈Z+ν
X
ψ˜rµ e−irw¯
(1.76)
r∈Z+˜ ν
with the phase factors kept as conventions. On each R sector (left or right moving) the sum runs over integers and on each NS sector over half integers. Canonical quantization gives the fermionic anti-commutation rules: {ψrµ , ψsν } = {ψ˜rµ , ψ˜sν } = η µν δr,−s
(1.77)
The critical dimension for superstrings is D = 10. As with the bosonic string, only the eight transverse directions oscillate.
31 When constructing the spectrum, we define the ground state to be annihilated by all the r > 0 modes. In the NS sector there are no r = 0 modes and modes with r < 0 act as raising operators. Since these are anticommuting, each mode can only be excited once. The interesting case, which yields spacetime fermions, is the R sector. This is due to the presence of the ψ0µ ’s who’s commutation relations translate to the spacetime Dirac gamma matrix algebra. Since they anticommute with all the ψrµ ’s they take ground states to ground states, thus the ground states form a representation of the gamma matrix algebra which turns out to have half-integral spin values (in D=10 spacetime dimensions). As the raising operators are spacetime vectors all the excited states remain fermionic. In the NS sector the ground state is a singlet (as in the bosonic theory) and the whole spectrum is bosonic. Since spins are additive, in the closed string theory we get fermions in the R-NS and NS-R sectors, while we get bosons in the NS-NS and R-R sectors. This infact could motivate the whole construction of superstrings: in the bosonic theory the mass shell condition was pµ pµ + m2 = 0 and the pµ ’s were the zero modes α0µ . To get spacetime fermions and have a Dirac equation ipµ Γµ + m = 0 we have fields whose zero modes ψ0µ become the Γµ matrices. As in the bosonic case where the mass shell condition resulted from the Virasoro constraints on physical states, the Dirac equation arises by enforcing the constraints of the extended now worldsheet symmetry, called super-conformal invariance.
32 Now let us tale a look at the spectrum. In each left/right sector we have 8 oscillatory bosons and 8 fermions. Each boson contributes −1/24 to the normal ordering constant as before, while each periodic fermion contributes 1/24 (the vacuum energies of fermions comes with the opposite sign) and each anti-periodic fermion contributes −1/48. So for the zero point constants we have (for either left or right movers) 1 1 1 1 1 NS sector: 8 − − = − , R sector: 8 − + = 0 (1.78) 24 48 2 24 24 Given ν as defined above, we can summarize this by identifying the zero point constant with −ν (or −˜ ν for right moves). In ten flat dimensions the Hamiltonian is then H = L0 − ν =
α0 2 p +N −ν 4
(1.79)
We wrote the expression for left movers, but the same is true for right movers ˜ 0, N ˜ , ν˜). The oscillator numbers N include contributions with (L0 , N, ν) → (L from both bosonic and fermionic modes N=
∞ X n=1
i α−n αni
+
∞ X
i rψ−r ψri
(1.80)
r=1−ν
with i taken only in the transverse directions. Note that the excitations in the NS sector (ν = 1/2) come in half-integral units. The mass formula for the closed string is m2 =
2 ˜ − ν − ν˜) (N + N α0
(1.81)
˜ − ν˜. We note two while the level matching constraint is now N − ν = N important facts about the spectrum: First, the NS-NS sector has a tachyon
33 ˜ = 0 with mass m2 = −2/α0 (the R-NS or NS-R sectors do when N = N not have a tachyon because the level matching constraint does not allow it). This is just like the bosonic theory and it is an inconvinient feature. Second, the spectrum is not spacetime supersymmetric. For example there is no fermion in the spectum with the same mass as the (bosonic) tachyon. Absence of supersymmetry gives an inconsistency when one notices that the spectrum contains a massless gravitino (spin 3/2) in the R-NS (fermionic) ˜ = 1/2. A gravitino has consistent interactions only in a sector for N = 1, N supersymmetric theory. Both of these unwanted features can be removed with the GSO projection. One defines an operator (−1)F where F counts the number of worldsheet fermions modulo 2. By projecting the spectrum appropriately on either the 1 or -1 eigenvalues of (−1)F in each sector, we can get rid of the tachyon and render the theory supersymmetric. Although such projections might seem ad-hoc, they can be justified through the main object of this chapter: the one loop amplitude that should be modular invariant.
1.4.1
One Loop Amplitudes
The path integral over a toroidal worlsheet for the fermionic action in (1.73) has the same operator form of a trace as in the bosonic case. For our worldsheet fermions though, we need to further specify the boundary conditions along both cycles of the toroidal worldsheet. We define A±± to indicate whether we are considering periodic (+) or anti-periodic (-) boundary condi-
34 tions along σ1 and σ2 respectively. For example, A−+ would be the amplitude over fermionic paths that are anti-periodic along σ1 and periodic along σ2 . How do we express this in operator language? This will determine how we take the traces to correctly evaluate the path integral. The periodic condition along σ1 is considered by specifying what sector we take the trace in, that is anti-periodic (-) boundary conditions in the NS sector and periodic (+) in the R sector. This is how we defined the NS and R sectors in the first place. To see how we specify the boundary conditions about σ2 it is essential to remember that in a path integral formulation of statistical mechanics, the partition function of fermions is computed with path integrals over antiperiodic boundary conditions. The factor
Tre−2πτ2 H
(1.82)
in the loop amplitutes (for propagation through imaginary time 2πτ2 ) naturally represents the path integral with anti-periodic boundary conditions for the fermions. If we now consider the quantity
Tre−2πτ2 H (−1)F
(1.83)
with F counting the worldsheet fermions modulo 2 as above, then the minus sign for each fermion twists the boundary conditions (at either σ or σ + 2πτ2 ) so the path integral now goes over periodic paths. So considering only left-moving fermions (the right-movers contribute the
35 complex conjugate of whatever we find for the left-movers) we have A−− (τ ) ∝ TrN S [q H ] A+− (τ ) ∝ TrR [q H ] (1.84) H
F
A−+ (τ ) ∝ TrN S [q (−1) ] A++ (τ ) ∝ TrR [q H (−1)F ] The total amplitute is the sum of the above. Note that we have determined these amplitudes only up to relative phases/ normalization constants. This wasn’t the case for the bosonic string where we could fix the one constant to be 1. What determines these phases infact is requiring modular invariance. Under modular transformations the amplitutes above are permuted between them. For example, the modular transformation (σ1 , σ2 ) → (σ2 , −σ1 ) exchanges the role of σ1 and σ2 and turns A+− to A−+ . It turns out that modular invariance fixes the phases ([3]) and gives the total amplitude11 Zψ± (τ )
F F H1 H1 = TrN S q (1 − (−1) ) − TrR q (1 ± (−1) ) 2 2
(1.85)
The last factor can have either a + or a - sign. This is what gives rise to two different closed superstring theories, type IIA and type IIB. While for the left movers both theories’ spectrum is defined through Z + , type IIA is ∗
defined to have a right-moving spectrum given by Z − and type IIB a right ∗
moving spectrum given by Z + . 11
To be precise, it is modular invariance of the two-loop amplitude that determines the phase of A++ so it is at second order that the GSO projection is defined in the R sector.
36 The modular invariant form in (1.85) defines the GSO projection: 12 (1 − (−1)F ) keeps only states with (−1)F = −1 in the NS sector. In the R sector, type IIB has left and right moving states with (−1)F = +1 while type IIA has left-moving states with (−1)F = +1 and right-moving states with (−1)F = −1. As a result, the spacetime fermion sectors NS-R and R-NS have modes with the same chirality for IIB and opposite for IIA. The terminology II refers to the fact that these theories each have two gravitinos. The opposite sign between the two sectors R and NS in (1.85) reflects what we should expect from spacetime statistics. The states in the R sector are spacetime fermions and their contribution to the one-loop graph has the opposite sign to that of the spacetime bosons from the NS sector. Explicitly evaluating the traces is a simple task. They brake up into products for each transverse dimension and for each oscillator level sector. Each sectors has 2 states, the vacuum and that of one (worldsheet) fermionic i . The operator (−1)F gives a minus sign infront of the latter. excitation ψ−r
The A++ in fact vanishes for a reason that we need to explain. Recall the degeneracy in the ground states of the R sector. Only two of these survive the Virasoro constraints, but under the action of (−1)F when properly defined (the detailed definitions are not necessary for our purposes and are only relevant for this point) they have opposite eigenvalues. As a result the trace for A++ has a factor (1 − 1) = 0 which makes the whole amplitude vanish. Recall here that in the anti-periodic NS sector each fermion contributes −1/48 to the zero-point energy, in the R sector each periodic fermion 1/24,
37 NS states are half-integrally moded, R states integrally moded, and there are 8 transverse directions in the 10-dimensional superstring theories. We then have " TrN S [q H ] = q −1/48
∞ Y
#8 (1 + q m−1/2 )
m=1
" H
TrR [q ] = 16 q
1/24
∞ Y
#8 m
(1 + q ) (1.86)
m=1
" TrN S [q H (−1)F ] = q −1/48
∞ Y
#8 (1 − q m−1/2 )
m=1
TrR [q H (−1)F ] = 0 The factor of 16 in the R sector is due to the degeneracy of the fermionic ground state. It is common to write these in terms of the eta function and the elliptic theta functions: θ00 (ν, τ ) = θ01 (ν, τ ) =
∞ Y m=1 ∞ Y
(1 − q m )(1 + e2πiν q m−1/2 )(1 + e−2πiν q m−1/2 ) (1 − q m )(1 − e2πiν q m−1/2 )(1 − e−2πiν q m−1/2 )
(1.87)
m=1
θ10 (ν, τ ) = 2q 1/8 cos(πν)
∞ Y
(1 − q m )(1 + e2πiν q m )(1 + e−2πiν q m )
m=1
We then have Zψ (τ ) =
1 4 4 4 θ (0, τ ) − θ (0, τ ) − θ (0, τ ) 00 01 10 η(τ )4
(1.88)
The extra step to get the full path integral that gives the one loop amplitude for type II theories is simple. We integrate over the fundamental domain of the torus with the measure d2 τ /(4τ2 ) as before, and the total partition function comes from the product of all the modes, bosonic and left and
38 right fermionic. Removing the factor of the one-dimensional volume V from the expression ZX (τ ) in (1.46) we get Z W1−loop(II) = iV10 F0
d2 τ Z 8 (τ )Zψ (τ )Zψ (τ )∗ 4τ2 (4π 2 α0 τ2 ) X
(1.89)
Recall that the factor of i comes from rotating the k0 integral and the factor of (4π 2 α0 τ2 ) from the k0 ,k1 integrals. The Z’s contain only the contribution of the transverse modes. We have left but one important comment. In 1829, Jacobi proved the “abstruse” identity 4 4 4 θ00 (0, τ ) − θ01 (0, τ ) − θ10 (0τ ) = 0
(1.90)
The 1-loop amplitude vanishes. The partition function of the spacetime bosons cancels that of spacetime fermions. In fact, at each mass level their contributions are equal and opposite. The theory is supersymetric as a result of the GSO projection or modular invariance. Also, the NS-NS tachyon was eliminated from the spectrum by the GSO projection (it has (−1)F = 1).
39
Chapter 2 String Thermodynamics 2.1
The Free Energy
Given the results of the previous chapter, it is straight forward to discuss the finite temperature case. In field theory, the thermal ensemble at temperature T can be studied by studying the propagation of fields on RD−1 ×S 1 where S 1 is the Euclidean time with circumference β = 1/T . It was shown by explicit computation ([1]) that at the one-loop level, the free energy of a thermal gas of (bosonic) strings can likely be calculated by considering string propagation on RD−1 × S 1 . This is an intuitive, but not very obvious result. We have already seen the partition function and one loop amplitude for the case of a compactified dimension. We can then take X 0 to be compactified (and Euclidean so there is no factor of i to the one loop amplitude) on a circle of radius R = β/(2π) and given our discussion in section 1.3 the one loop amplitude becomes Z W1-loop (β) = V25 F0
d2 τ β (4π 2 α0 τ2 )−13 |η(τ )|−48 F2 τ, 4τ2 2π
(2.1)
40 The function F2 accounts for the compactification of one dimension as explained below equation (1.63). The free energy of a gas of bosonic strings F (β) is given by1 βF (β) = −W1-loop (β)
(2.2)
There is an elegant way of extending this treatment to superstrings. Recall that in point particle field theories the finite temperature treatment results in Matsubara frequencies which are essentially the Kaluza Klein modes of the compact temporal dimension. But while bosonic thermal frequencies are integrally moded, fermionic frequencies are half-integrally moded. In superstring theory this effect can be achieved by twisting the temporal direction in a spin-dependent way ([4]). We require the operator T ≡ eip0 β+i2πJ12
(2.3)
to be 1 on physical states. where J12 is the rotation operator for the X 1 and X 2 coordinates (it could be any pair of coordinates). This is a change in the boundary conditions along the temporal direction. Since J12 eigenvalues are integers for spacetime bosons and fermions for spacetime fermions, requiring T = 1 yields the desired moding for bosons and fermions, 2πn/β and 2π(n + 1/2)/β respectively. The fact that this is a string theory though, yields further complications as there is a winding number associated with the compact temporal dimension. 1
It is useful to check that our expressions for W1-loop are unitless: The volume factor √ cancels the powers of α0 which has units of length squared. F2 has units of length ∼ α0
41 Since our discussion is focusing on spacetime and not worldsheet bosons and fermions, let us set up appropriate notation. We will refer to the spacetime bosonic and fermionic partition functions with PB and PF . For the type II superstrings in particular, PB is the trace involving the bosonic NS-NS and R-R sectors and PF the trace in the R-NS and NS-R sectors. Specifically, setting the proportionality constants to 1 in (1.84) and including the X µ contributions, we have NS
NS
R
R
NS
R
R
NS
8 PB (τ ) = ZX [(A−− − A−+ )(A−− − A−+ )∗ + (A+− ± A++ )(A+− ± A++ )∗ 8 PF (τ ) = ZX [(A−− − A−+ )(A+− ± A++ )∗ + (A+− ± A++ )(A−− − A−+ )∗ (2.4)
The one loop amplitude is then Z W1-loop = iV10
d2 τ (PB − PF ) 16πα0 τ22
(2.5)
With the minus sign reflecting the spacetime statistics. Again, this formally vanishes due to supersymmetry with PB = PF (Jacobi’s identity) but we need the formal expression as a starting point. As expected, the finite temperature expression will break the supersymmetry (simply due to the different Matsubara moding between fermions and bosons). Now if we compactify one dimension on a periodic torus (e2πiRp0 = 1) we get Z W1-loop = iV9
d2 τ (PB − PF )F2 (τ, R) 16πα0 τ2
(2.6)
Consider now the twisted torus with the projection T = ei2πiRp0 +i2πJ12 = 1. We think of this as twisting the boundary conditions every time we wind
42 around the circumference 2πR. Winding w times amounts to applying the twist T w . But it is easy to see that T 2w = 1 and T 2w+1 = T and this tells us that only the odd-wound sectors will be affected. Recall that F2 (τ, R) involves a sum over winding w and KK-momentum n. Now we want to split it into two sums one over odd and one over even winding. Further, as was the aim of this procedure, the KK moding will be either integer or halfinteger valued depending whether we are in the bosonic or fermionic sector respectively. So we split F2 into four sums and define
E0 to be F2 with E ven w and integer n E1/2 to be F2 with E ven w and half-integer n (2.7) O0 to be F2 with Odd w and integer n O1/2 to be F2 with Odd w and half-integer n Then the amplitude takes the form Z W1-loop = iV9
d2 τ (PB E0 − PF E1/2 + O0 PB0 − O1/2 PF0 ) 16πα0 τ2
(2.8)
We denote with primes (PB0 and PF0 ) the modified partition functions in the odd-wound sector (O0 and O1/2 , where the twist T is not the identity) with the labels for bosons or fermions depending on the thermal KK moding (0 or 1/2). The form of these modified partition functions is determined by modular invariance. It turns out that they can be derived with a simple transformation, namely taking the initial PF and PB in (2.4) and exchanging
43 the NS and R sectors (A±± → A∓± ). We then have ∗
∗
∗
∗
8 PB0 (τ ) = ZX [(A+− − A++ )(A+− − A++ ) + (A−− ± A−+ )(A−− ± A−+ )
8 PF0 (τ ) = ZX [(A+− − A++ )(A−− ± A−+ ) + (A−− ± A−+ )(A+− − A++ ) (2.9)
This then gives for the free energy (R → β/2π and iV9 → V9 ) the expression Z βF (β) = −V9
d2 τ 0 0 P E (β) − P E (β) + O (β)P − O (β)P B 0 F 0 1/2 1/2 B F 16πα0 τ2 (2.10)
This is a general recipe for obtaining the free energy from flat space partition functions, for any string theory.
2.1.1
Singularities of the Free Energy
Despite the possibly complicated expression for the free energy, like above, the underlying construction principle is simple. It is a trace integrated over the fundamental domain of the torus with the general form:
Z F (β) = − F0
d2 τ Tr{Pˆ e−H(β)2πτ2 ei∆τ1 } 4τ2
(2.11)
The trace must be taken over all sectors of the Hilbert space of the theory and Pˆ schematically represents any projection in each such sector (for example it includes the GSO projection in NS and R sectors). Note that ei∆τ1 is just another projection, that in practice enforces level matching of left and right modes of a closed string. In particular, as we saw in the previous section,
44 the trace includes states that arise due to compactification of the targetspace temporal direction on a twisted torus over circumference β (hence the β dependence), broken up in sectors that arise due to the twist. Recall here the definition of the canonical thermal partition function Z (not to be confused with the stringy partition functions that we encountered earlier), related to the free energy via Z = e−βF . If we now we consider d non-compact spatial dimensions of volume Vd , once the traces are taken we can write Z Vd d2 τ X 2 −βF = lnZ = gα e−2πMα (β)τ2 e2πiCα τ1 (d/2+1) 2 0 d/2 (4π α ) F0 4τ2 α
(2.12)
with α indexing sets of all the allowed quantum numbers to specify the mass Mα2 (oscillators, spatial and temporal windings, KK modes, e.t.c) including normal ordering constants or zero point energies depending on the sector. gα are constants reflecting the degeneracies of each (β dependent) mass level, with the appropriate sign for spacetime bosons or fermions. Cα = 0 is the level matching constraint enforced through the τ1 integral. We want to consider the analytic behaviour of F (or Z) as a function of the complex inverse temperature β. The reason for this will become evident soon. The singular behaviour of lnZ(β) is given in the τ2 → ∞ limit. Recall that this can be thought as an IR in this representation (long worldsheet loops). Each term in the sum in this limit then behaves as Z ∞ d/2 dτ2 −2πMα2 (β)τ2 e = 2πMα2 (β) Γ −d/2, 2πMα2 (β) (d/2+1) τ2 1
(2.13)
with Γ(a, b) the incomplete gamma function. The analytic behaviour is read
45 off the right hand side: lnZ has branch cut singularities at Mα2 = 0 , i.e at effective (β dependent) massless modes. For d odd, these are square root singularities and for d even they are logarithmic singularities contained in the asymptotic expansion of the gamma function. For Z, and the particular case of interest for all dimensions compact (d = 0), these become simple pole singularitites. The overall Mα2 is the sum of the usual left and right mass modes of the string and the temperature dependent part. Defining the radius β¯ = β/(2π) we have L2 W 2 β¯2 Mα2 = m2L + m2R + ¯2 + α0 2 β 2 m2L = 0 (N + aL ) + m2χL α 2 ˜ + aR ) + m2χR m2R = 0 (N α
(2.14)
We denote with L and W the thermal KK and winding modes (recall that the former can be half integers). The constants aL,R are the left and right zero-point energies, for example a = 0 in the R sector of the superstring, a = −1/2 in the N S sector and a = −1 for the bosonic string. To be general, we denote with m2χ mass modes that arise from compactification, χ denoting compactification moduli. For the case that we will be mostly interested in, with all dimensions compactified on square tori with radii Ri (in which case d = 0, Vd = 1) we have 9
m2χL ,R =
1 X ni wi Ri 2 ( ± ) 2 i=1 Ri α0
(2.15)
46 The level matching constraint reads Cα = m2L − m2R +
2 LW = 0 α0
(2.16)
By solving the conditions Mα2 = 0 and Cα = 0 we can then find all values of β in terms of mass levels that correspond to complex temperature singularities of the free energy. The solutions are √
q q 2πα0 2 ( −mR ± −m2L ) βs = ± W 2.1.2
(2.17)
The Hagedorn Temperature
Real valued singularities correspond to physical temperatures. The most important, and the one that the physical string gas encounters first as the temperature is raised is the one given by the largest real βs . This is the Hagedorn singularity, given by W = 1 and no other excitations, in which case −m2L,R equal the (negative of the) smallest possible zero-point energy from the string sectors. For type II superstrings, this comes from the NS sector on both sides and the Hagedorn temperature is r r √ 2 2 1 1 √ √ βHtype II = 2πα0 − 0 (− ) + − 0 (− ) = 8π α0 α 2 α 2
(2.18)
√ For the bosonic string we get βHbosonic = 4π α0 . Our treatment sheds light to some interesting features of the Hagedorn singularity. Firstly, it evidently arises due to the fact that a thermal winding mode W = 1 (thus purely a stringy feature) becomes massless and above the Hagedorn temperature it is tachyonic. In the literature it has been suggested
47 that this signals a phase transition analogous to the deconfinement transition in QCD at which point the theory must be handled with new degrees of freedom ([5]). This point of view has not been yet established. Secondly, in this treatment the Hagedorn singularity appears as an IR effect, in the τ2 → ∞ limit and due solely to the present of central charges (the lightest “masses”) or zero point energies, thus purely a quantum effect. But alternatively, it can be seen as a high energy effect (at a high temperature of order α0−1/2 ). When a lot of energy is pumped into a gas of strings oscillators are highly excited. As we will see in the microcanonical treatment that follows, the phase-space density of these modes increases exponentially as opposed to a power law as is usually the case in point particle theories. As a result, above a certain temperature the thermal partition function diverges. The reason why the Hagedorn divergence appears as an IR effect here can be attributed to modular invariance, a symmetry of the world-sheet. As is the case with T-duality which manifests itself as a spacetime symmetry, strings do not distinguish between large and small scales and here this shows up from the point of view of the worldsheet. A modular transformation like τ → −1/τ turns small worldsheet distances to large and vice versa. We take the point of view that the Hagedorn singularity merely signals the breakdown of the canonical ensemble. For thermodynamic studies near the string scale (as is relevant to the string gas model) one can revert to the more fundamental microcanonical ensemble. In fact this can be elegantly done with a method due to Deo, Jain and Tan ([6, 7]) that we review below,
48 according to which one utilizes the singularity structure of the free energy to derive the microcanonical ensemble.
2.2
The Microcanonical Ensemble and the Hagedorn Phase
The relevant quantity in the microcanonical ensemble is the density of states Ω(E) which counts the number of states available to a system at fixed energy E within a given volume V . It is convenient to define the entropy2 S ≡ log Ω(E). Given the entropy, a number of thermodynamical properties of the system can be determined. For example, from dE = T dS − P dV it follows that the temperature T and pressure P are given by T −1 =
∂S |V , ∂E
P ≡T
∂S |E ∂V
(2.19)
One can define the canonical ensemble through the partition function Z(β) as a function of a parameter β via Z Z(β) = dEΩ(E)e−βE
(2.20)
The reason why this provides an equivalent description of the thermodynamics is that for usual (in particular point particle) systems the integral above is dominated by a saddle point. If the saddle point exists then it is located at a value E0 which is a solution to β= 2
∂S(E0 ) . ∂E0
We will keep any volume dependence implicit.
(2.21)
49 This coincides with the definition of temperature above and also the energy E0 (β) coincides up to small corrections with the average energy calculated through the partition function. It is this correspondence that breaks down for strings, since the partition function does not any longer converge for β > βH . Relevant thermodynamical quantities can still be determined but we merely lose the ability to recover them through the canonical ensemble. What we want to utilize here is the relationship in (2.20). If Z(β) exists and is known as a function of β then one can obtain Ω(E) through an inverse laplace transform: Z dβ βE Ω(E) = e Z(β) C 2πi
(2.22)
The contour C lies in a domain where Z(β) is analytic. This is why the singularity structure of Z(β) is relevant, so we proceed to describe some of its general features. The Hagedorn singularity is an isolated singularity and is the first real singularity the string gas encounters as β → 0. This is a universal feature of all string theories (modulo the value of the Hagedorn singularity which depends on the central charge of the underlying CFT) independent of compactification. With all the m2χ set to zero, the closest singularities to βH consist of a tower to the left of it due to either left or right oscillatory modes, which for the superstring are given q p 2 by β = π(1 ± (2N − 1)i). The contour C then could lie anywhere α0 in the complex β plane to the right of βH . By appropriately deforming the contour to surround singularites we can get the correponding dependence in Ω(E). For example, with the contour wrapping βH as in figure 2.1 we get a
50 contribution Ω(E) ∼ O(eβH E ) + O(eLE )
(2.23)
Since by choice βH > L it follows that the second term is exponentially suppressed, in particular at high energies, and thus we recover the leading dependence of the density of states3 . Ω(E) ∼ eβH E
(2.24)
Figure 2.1: Contour deformation that yields the leading behaviour of Ω(E) from the Hagedorn singularity. This is precisely why the partition function in the usual expression (2.20) diverges for β < βH . This exponential increase in the density of available states is directly related to the large number of states available to a string with oscillator number N ∼ E 2 for large E. This is the number of integer 3
up to powers of E that we will clarify later
51 partitions of N that grows as ea
√
N
with a constant (the precise form is
known as the Hardy - Ramanujan formula). As mentioned earlier, in the high energy limit most of the energy flows into exciting oscillators on the strings. The leading behaviour for Ω(E) is not always sufficient. Continuing with the singularity structure of the free energy, consider light states due to compactification with masses (for type II strings) −m2L,R =
1 − m2χL,R with m2χL,R λexit . Using (3.17) and (3.20) this can be written conveniently in terms of the variable x as x > (r − 1)/(r + 1),
r ≡ ρ(λ0 )/ρH
1/√d
(3.21)
Note that r is the ratio of the initial density to the “exit” density, so in order to begin in the Hagedorn era at t = 0 we need r > 1.
78 The equations of motion have four initial conditions, which can be chosen to be the initial volume V0 = (2π)d edλ0 (instead of λ0 , ϕ0 , ϕ˙ 0 and x (instead of λ˙0 , once ϕ˙ 0 is fixed). The variable x reflects the initial “boost” of the scale factor and it is in practice the measure of thermal fluctuations.11 Fixing the first three initial conditions, we then ask whether a given x can drive the system out of the Hagedorn phase. It is worth commenting on what we really mean by a thermal fluctuation. The total energy in matter and gravity is fixed. By a thermal fluctuation we mean that energy flows from one to the other. In essence we are treating the large number of heavy oscillator modes that characterizes the Hagedorn phase as a “thermal bath”. We draw then energy from this bath and re˙ As distribute it to the other energetic modes, including the expansion rate λ. derived earlier, in the Hagedorn phase the entropy is to a good approximation proportional to the energy. With S = E/TH , the distribution of x once the other three initial conditions are fixed is a Gaussian,
d(x) ∼ e∆S ∝ e−(4π
2 e−ϕ0 ϕ ˙ 20 /TH )x2
.
(3.22)
It is convenient to write the conditions on initial data such that the system starts in the Hagedorn phase (r > 1) and exits to the large radius (or 11
This is a subtle point: We are in a sense focusing on metric and curvature fluctuations within a “mini-superspace”. More general metric fluctuations would be extremely hard to trace. This applies throughout this thesis. Within this approach, it costs no energy for a fluctuation in λ until the exponentially supressed corrections to the entropy become important, which is far outside the domain of values of λ and E that we consider.
79 radiation) era (3.21) as f1 ≡ K0 (1 − x2 ) − 1 > 0 √ 1/ d
f2 ≡ x −
√
K0
(1 − x2 )1/
K0
(1 − x2 )1/
√ 1/ d
d
√
d
−1
(3.23) >0
+1
where K0 (ϕ0 , ϕ˙ 0 , V0 ) ≡
(2π)2 e−ϕ0 ϕ˙ 20 . ρH V0
(3.24)
Now let us be more precise about the condition for Hagedorn exit and specify ρH . In the Hagedorn phase the string gas is taken to be in equilibrium at temperature TH . Here, all the modes, massive and massless, are in equilibrium. This is similar to having a black hole in thermal equilibrium with the surrounding radiation, something which can only happen within a finite volume. Past a critical volume, massive modes should decay. The constraints on the volumes for which such equilibrium can be maintained have been studied, and for a string gas in d spatial dimensions they amount to ([19, 20, 21]) Vd
0 and x < 3σ with σ the standard deviation in the distribution (3.22). Even for large hubble rates ˙ fluctuations the space of initial conditions that allow an exit from the (λ) Hagedorn phase is considerably small. This conclusion does not change if we vary d, V0 or even the critical transition density ρH by one or two orders of magnitude. Exiting the Hagedorn phase requires extreme fine tuning, in particular, the system needs to begin very near the critical density ρH with relatively large hubble rates. In the context of string gas cosmology, we are therefore facing a troublesome restriction. This is what motivates the approaches in the following chapters. The dynamics make it difficult to exit the initial phase of the string gas scenario and to do so, we have to either consider large fluctuations or modify the dynamics. In chapters 5 and 6 we take the former approach and in chapter 4 the latter. These two directions coincide with the attempts to investigate the possibilities of decompactification and the realization of
81
0
j0 -5
-10 0.0
0.0 0 -0.5 j0
-5 j0
-1.0 0.0
0.5
x
-0.5 j0
-10 1.0
-1.0 0.0
0.5
1.0
x
√
Figure 3.2: Surveying the initial condition space (x ≡
dλ˙ 0 , ϕ0 , ϕ˙ 0 ) |ϕ˙ 0 |
for fixed
volumes V0 and d = 3. For smaller initial volumes, corresponding to λ0 = 1 (left) the system can exit the Hagedorn phase only with small initial energies. For larger volumes corresponding to λ0 = 3 (right) larger energies are allowed but exit still occurs for fine-tuned initial condition. In both cases, the system has to begin near the critical density ρH .
82 a non-singular cyclic universe within string gas cosmology. In order for the decompactification mechanism to work, we consider large fluctuations to exit the Hagedorn phase and enter a large radius (radiation) regime where the heavy winding modes can decay. In order to avoid an initial singularity we need to move beyond the lowest order gravitational dynamics (if we choose not to modify the matter content in a rather unnatural way) and add corrections to dilaton-gravity.
83
Chapter 4
Bouncing and Cyclic Cosmologies
In this chapter we show that, in the presence of a string gas, simple higherderivative modifications to the effective action for gravity can lead to bouncing and cyclic cosmological models. These modifications bound the expansion rate and avoid singularities at finite times. In these models the scale factors can have long loitering phases that solve the horizon problem. Adding a potential for the dilaton gives a simple realization of the pre-big bang scenario. Entropy production in the cyclic phase drives an eventual transition to a radiation-dominated universe. As a test of the Brandenberger-Vafa scenario, we comment on the probability of decompactifying three spatial dimensions in this class of models.
84
4.1
Finite or Eternal Universe?
A fundamental question in cosmology is whether the universe has always existed, or whether it came into being a finite time in our past. It could be that the age of the universe is finite; at the classical level the singularity theorems of general relativity make such an assumption seem unavoidable [22]. The other possibility is that the universe has infinite age. A number of past-eternal models have been developed, exploiting the fact that quantum effects or other modifications to general relativity can get around the singularity theorems [23]. String theory should ultimately provide a framework for deciding between a moment of creation and an eternal universe.1 On the one hand various toy models for cosmological singularities in string theory have been developed, and considerable effort has been devoted to studying them, but a complete understanding is still lacking [24]. On the other hand several string-inspired models for eternal cosmologies have been proposed, most notably the ekpyrotic [25] and pre-big bang [26] scenarios, but it is not clear to what extent these proposals capture the generic (or even allowed) behavior of string theory. In view of this situation it is worthwhile developing additional scenarios for eternal cosmologies in string theory. Here, we consider a simple class of higher-derivative modifications to the effective action for gravity. These 1
Eternal inflation, despite its name, is not past-eternal and cannot by itself address this issue [12, 13].
85 modifications have the effect of bounding the expansion rate and limiting dilaton gradients, thereby avoiding singularities at any finite time. In the absence of matter the universe would approach a de Sitter phase at early times. But when coupled to a gas of string winding and momentum modes the scale factors can oscillate or bounce as functions of time. By introducing a dilaton potential the dilaton can be made to oscillate or bounce as well. Our work has several motivations. Bouncing and cyclic cosmologies Eternal cosmologies in which the scale factors bounce or oscillate as functions of time have been extensively studied, and within field theory a variety of mechanisms for realizing this type of behavior have been developed [23]. Our work provides a simple string-inspired mechanism for obtaining bouncing and cyclic cosmologies. For other studies in this direction see [27, 28, 29]. Pre-big bang scenario In the pre-big bang scenario the universe is assumed to begin from a cold flat weakly-coupled initial state. The dilaton rolls towards strong coupling and bounces, and the universe emerges in an expanding FRW phase [26]. However at the level of the two-derivative effective action the two branches of pre-big bang cosmology cannot be smoothly connected [30]. There are various ways around this, reviewed in section 8 of [26]. Our work leads to a particularly simple realization of the pre-big bang scenario, in a manner similar to the proposal [31]. Horizon problem
86 One of the main puzzles of conventional FRW cosmology is the horizon problem: how did causally-disconnected regions of the universe come to be in thermal equilibrium? Inflation explains this by postulating a rapid growth of the scale factor at early times. But an alternative way to address the horizon problem is to postulate a loitering phase in which the scale factor is roughly constant (that is, the Hubble length diverges). If the loitering phase lasts long enough the universe has time to come to thermal equilibrium. The models we discuss can have long loitering phases, a phenomenon observed in a similar context in [32]. Brandenberger-Vafa mechanism As a zeroth-order goal, one might hope that string cosmology could account for the three large spatial dimensions we observe. Within the string gas scenario this can potentially be realized via the BV decompactification mechanism presented in the previous chapter. Although appealing, at the level of the two-derivative effective action we have already seen that this scenario has a problem: there is only a small window for opportunity for the universe to exit the initial Hagedorn phase and enter a radiation phase where the winding modes would decay. Recall that the lowest order dilaton-gravity equations forbid ϕ˙ = 0 at any time, for positive matter energy.2 Their solutions therefore fall in two branches, with ϕ˙ < 0 and ϕ˙ > 0 respectively. Our observation that an exit from the Hagedorn phase is not generic, regarded the direction 2
This is analogous to the condition for a bounce of the scale factor in Einstein gravity. With H 2 ∝ ρ, H cannot be zero for ρ > 0.
87 of time defined by ϕ˙ < 0. In the ϕ˙ > 0 direction we encounter an even more fundamental problem: the universe has a singularity at a finite time in the past with ϕ → ∞. The eternal cosmologies we discuss would seem to provide a natural setting for realizing the Brandenberger-Vafa mechanism, since, by extending time infinetely to the past, we allow infinite time for the necessary fluctuations to take place. We will find that, due to possibility of long loitering phases, our models do not preferentially decompactify three dimensions. Further, in order to recover the semiclassical nature of strings, necessary for the BV mechanism to operate as explained later, we need some additional ingredients and this is the subject of chapter 5. Several of the results we will obtain have antecedents in the literature, in particular in the works [31, 32], although to our knowledge the phenomena we will discuss have never appeared in combination before. An outline of this chapter is as follows. In section 4.2 we introduce a modified action for Einstein-dilaton gravity which has the effect of bounding the expansion rate and dilaton gradient. In section 4.3 we introduce matter degrees of freedom and give a preliminary discussion of their thermodynamics. In section 4.4 we consider the coupled gravity – matter system and show how bouncing and cyclic cosmologies result. In section 4.5 we study string interactions and entropy production in these cosmologies. In section 4.6 we show that entropy production drives an eventual transition to a conventional radiation-dominated cosmology. Section 4.7 studies the extent to which the Brandenberger-Vafa mechanism is operative in these models. We conclude
88 in section 4.8.
4.2
A Modified Action
We consider type II string theory compactified on a torus with metric 2
2
ds = −dt +
d X
e2λi (t) dx2i
xi ≈ xi + 2π .
(4.1)
i=1
Recall that we set α0 = 1. The self dual radius is at λi = 0. Although we have in mind that all spatial dimensions are compactified, we will allow d to vary to study the dimension dependence of our results. At the two-derivative level the string-frame effective action for homogeneous fields takes the form Z S0 =
h X i dt 4π 2 e−ϕ λ˙ 2i − ϕ˙ 2 + Lmatter .
(4.2)
i
The action includes standard kinetic terms for the radii and dilaton; we’re working in terms of a shifted dilaton ϕ, related to the usual dilaton φ by ϕ = 2φ −
X
λi .
(4.3)
i
Lmatter is the effective Lagrangian for matter degrees of freedom. In thermal equilibrium we’ll identify Lmatter = −F with the negative of the matter free energy. Besides the equations of motion which follow from this action we have the Hamiltonian constraint that the total energy in the universe vanishes. ϕ˙ 2 −
X i
1 λ˙ 2i = 2 Eeϕ 4π
(4.4)
89 Here E is the energy in matter. These equations are invariant under Tduality, which acts according to λi → −λi ϕ, Lmatter , E
for some i invariant
Provided E satisfies certain energy conditions the action S0 leads to cosmologies that have initial singularities: at a finite proper time in the past the shifted dilaton diverges and the λi → ±∞. These singularities have mostly been studied in the context of pre-big bang cosmology, see for example [33] and [34]. But it is tempting to speculate that stringy effects (α0 corrections to the effective action) will lead to non-singular cosmologies.3 As a model which captures this sort of behavior, we introduce the following modified action for the metric and dilaton. Z S=
s i p X 2 −ϕ 2 2 ˙ λi + Lmatter dt 8π e 1 − ϕ˙ − 1 − h
(4.5)
i
Again in equilibrium we’ll identify Lmatter = −F with the negative of the matter free energy. There are several motivations for writing down this action. As a simple way to think about it, note that in the action (4.2) both ϕ and λi appear as non-relativistic particles of mass 8π 2 e−ϕ (although ϕ has a wrong-sign kinetic energy which is not a problem in this frame). In going from (4.2) to (4.5) we have promoted ϕ and λi to become relativistic particles of the same mass. 3
For a study of α0 corrections in string gas models see [35].
90 This clearly bounds their velocities, ϕ˙ 2 < 1
X
λ˙ 2i < 1
i
which has the desired effect of ruling out singularities at any finite proper time. In this sense the action we have written down incorporates a “limiting curvature hypothesis” in a manner similar to [31, 36]. In fact, for homogeneous and isotropic spaces this action can be obtained by a manifestly covariant action applying a technique suggested in [31, 36], that can be readily extended to generalize our model. This is shown in Appendix A. It’s also amusing to note the resemblance of S to the DBI action for open strings, which is related by T-duality to the action for a relativistic particle [37]. A note on α0 corrections to the effective action of string theory is in order here. In principle, the whole series of α0 corrections is rigidly prescribed by the condition of conformal invariance applied to the scattering amplitudes determined by the string S-matrix (at tree level in gs ). This is what, after all, resulted in the massless mode dynamics to leading order in (3.3), (3.4). At each approximation level, however, there is an intrinsic and anavoidable ambiguity due to field redefinitions which preserve the general covariance and gauge invariance of the action ([38, 39]). From a world-sheet path integral point of view the ambiguity is due to the freedom in the choice of a renormalization scheme and a Weyl gauge. This freedom is what in essence justifies our seemingly arbitrary choice of α0 corrections in (4.5). We proceed to analyze the dynamics of (4.5) and for simplicity we specialize to a square torus with all λi = λ. Then the equations of motion which
91 follow from S are
Here Pϕ =
∂F ∂ϕ
d 1 (γϕ ϕ) ˙ = γϕ − γλ−1 + 2 eϕ Pϕ dt 8π 1 d ˙ = γλ λ˙ ϕ˙ + (γλ λ) eϕ Pλ dt 8π 2
(4.6)
is the force on the dilaton and Pλ = − d1 ∂F is the pressure ∂λ
(or more accurately, the pressure times the volume of the torus).4 We have defined the relativistic factors γϕ = p
1 1 − ϕ˙ 2
γλ = p
1 1 − dλ˙ 2
.
(4.7)
The Hamiltonian constraint (Friedmann equation) is γϕ − γλ =
1 Eeϕ 8π 2
(4.8)
where E is the matter energy. Compared to the original equations of motion (3.14) and (6.3), we see that the changes are in a sense minimal (yet they serve the non-trivial task of bounding derivatives). For one, the equations of motion are still 2nd order; no further initial data is needed to solve them. Next, they preserve the effects of matter, with positive pressure accelerating and negative pressure decelerating expansion. Further, the Hamiltonian constraint (4.8) dictates √ ˙ just as in lowest order dilaton that the positive energy region is |ϕ| ˙ ≥ d|λ|, gravity (6.3). In effect, we merely absorb the large growth in derivatives that lead to singularities into the relativistic factors γϕ and γλ . 4
These are derivatives at fixed temperature. Entropy will be conserved, until we consider out-of-equilibrium processes in section 4.5, so it’s perhaps more appropriate to write Pλ = − d1 ∂E ∂λ S as a derivative at fixed entropy.
92 For later use, it is convenient to re-write the equations of motion in the more symmetric form 1 ϕe ˙ ϕ Pϕ 8π 2 1 ˙ ϕ γ˙λ = ϕ(γ ˙ λ − γλ−1 ) + 2 dλe Pλ 8π
γ˙ϕ = ϕ(γ ˙ ϕ − γλ−1 ) +
(4.9)
To get oriented, consider a simple equation of state Pλ = wE, w constant, with Pϕ = 0. Of particular interest are the cases w = 0, w = 1/d and w = −1/d which correspond to a Hagedorn era, a radiation dominated era and a winding mode dominated era respectively. One can get an idea of how ˙ and λ( ¨ ϕ, ˙ and studying the system evolves by writing equations for ϕ( ¨ ϕ, ˙ λ) ˙ λ) the phase space flow. Using the above equation of state, and substituting the Hamiltonian constraint in the equation for λ we can write ϕ¨ = (1 − ϕ˙ 2 )(1 − γλ−1 γϕ−1 )
(4.10)
¨ = (1 − dλ˙ 2 )(ϕ˙ λ˙ − w(1 − γ −1 γϕ )) λ λ It is easy to see that these equations have fixed points at the constant √ ˙ = (±1, ±1/ d). These can be curvature, linear dilaton solutions (ϕ, ˙ λ) ˙ = (0, 0) in the sense smoothly connected to the trivial fixed point (ϕ, ˙ λ) that no singularity stands between them. This is an attractive feature that α0 corrections to the low energy effective action are conjectured to have, perhaps to all orders in α0 [40]. It is particularly relevant to pre-big bang models. The phase space flows and some trajectories for d = 3 are shown in figure 4.1.5 In such a smooth and “connected” phase space, the system can move 5
For w 6= 0 the equations of motion are singular when |ϕ| ˙ = 1. This is not problematic
93 around the phase space towards the attractors without encountering singularities, independently of initial conditions. This feature is hard to obtain with generic α0 corrections to dilaton gravity and is crucial for the cyclic and bouncing solutions we will study below. For example, with the conventional two-derivative action for dilaton gravity one could at most hope for a single bounce before encountering a singularity. In essence, with the new action, we have replaced these singularities with the constant velocity fixed points. For general matter content there is a simple way to see how the modified equations of motion capture the desired behavior. Assuming that E is positive the Friedmann equation requires γϕ > γλ ≥ 1 so ϕ˙ can never vanish. Orienting time so that ϕ˙ < 0, the dilaton rolls monotonically from strong to weak coupling. Since ϕ and λ behave like relativistic particles of mass ∼ e−ϕ , at early times they are massless and move at the speed of light: ϕ˙ → −1
√ λ˙ → ±1/ d
as t → −∞.
(4.11)
Thus at early times the scale factors grow exponentially and the metric (4.1) approaches de Sitter space in planar coordinates, with the spatial coordinates periodically identified to make a torus. This early-time de Sitter phase is what replaces the big bang singularity in these models.6 This is very rembecause trajectories never quite reach points where |ϕ| ˙ = 1. Instead the γλ−1 √ γϕ term in the ¨ equation for λ eventually pushes the trajectories towards the line ϕ˙ = ± dλ˙ (depending on the sign of w) where γλ−1 γϕ → 1. 6
The dilaton diverges at t = −∞, so strictly speaking we have not eliminated the singularity, just moved it infinitely far into the past. As we will see even this can be cured by adding a potential for the dilaton.
94 Λ 0.6
0.4
0.2
-1.0
-0.5
0.5
1.0
j
-0.2
-0.4
-0.6
Λ 0.6
0.4
0.2
-1.0
-0.5
0.5
1.0
j
-0.2
-0.4
-0.6
Λ 0.6
0.4
0.2
-1.0
-0.5
0.5
1.0
j
-0.2
-0.4
-0.6
Figure 4.1: Phase space flows for w = 0 (top), w = 1/d (middle), w = −1/d √ ˙ = (±1, ±1/ d) and (ϕ, ˙ = (0, 0) are (bottom). The five fixed points (ϕ, ˙ λ) ˙ λ) connected smoothly. Some typical trajectories are also shown. For w = 1/d and w = −1/d they represent bounces of the scale factor due to KK and winding modes respectively.
95 iniscent of the behavior obtained in [36]. It is also similar to pre-big bang models where the kinetic energy of the dilaton dominates and drives inflation. One might worry about the fact that the coupling diverges at early times; as we will see we can cure this behavior by introducing a potential for the dilaton which violates positivity of E.
4.3
A First Pass at Thermodynamics
To proceed further we need to specify the matter content of the universe. This way of modeling string gas matter will be common, modulo slight variations, to the rest of this thesis. We take matter to consist of the following ingredients. 1. There may be a gas of string winding modes, characterized by winding numbers Wi that count the number of strings wound with positive orientation around the ith dimension of the torus.7 For simplicity we set all Wi = W . Then the energy in winding modes is EW = 2dW eλ . 2. Likewise there may be a gas of Kaluza-Klein momentum modes, characterized by positively-oriented momentum numbers Ki . With all Ki = K, the energy in Kaluza-Klein modes is EK = 2dKe−λ . 7
Since we work in a compact space there must be an equal number of strings wound with the opposite orientation.
96 3. We allow for a gas of string oscillator modes which we will model as pressureless dust with energy Edust . To be precise, W and K refer to the winding and momentum numbers in the first d dimensions. Thus we take Edust to represent the energy, not only in string oscillators, but also in winding and momentum modes in the remaining 9 − d dimensions. These modes can be modeled as dust since they do not contribute to the pressure in the first d dimensions. As the remaining component of the energy budget, we may introduce a potential for the dilaton V (ϕ). The total energy is then the sum E = EW + EK + Edust + V .
(4.12)
Treating the system adiabatically the “pressures” are Pϕ =
∂E ∂V = ∂ϕ ∂ϕ
Pλ = −
1 ∂E = 2Ke−λ − 2W eλ d ∂λ
(4.13) (4.14)
We will assume that the dilaton potential is independent of temperature. However the other components of the energy budget behave thermodynamically. To make the distinction, we refer to Es ≡ EW + EK + Edust as the energy in the string gas, the thermodynamical component of the total energy. The following phases will be of interest to us. Hagedorn phase In the Hagedorn phase we assume that all matter degrees of freedom are in √ thermal equilibrium at the type II Hagedorn temperature TH = 1/( 8π).
97 The pressure vanishes, Pλ = 0 and the energy Es is conserved; since Es = TH S the entropy is also conserved. In equilibrium the winding and momentum numbers are (see section 2.2) √ Es hW i = √ e−λ 12 π
√ Es hKi = √ eλ 12 π
(4.15)
As expected these values make the pressure in (4.14) vanish. Radiation phase As in section 3.3, the radiation phase describes the equilibrium situation when the energy density in matter drops below the critical density ρH = cd THd+1 , that is Es < ρH Vd
(4.16)
Here cd is a constant appropriate to a gas of 128 massless Bose and 128 massless Fermi degrees of freedom, cd = 128
2d!ζ(d + 1) (2 − 2−d ) . (4π)d/2 Γ(d/2)
(4.17)
Alternatively, the condition (4.16) can be expressed in terms of the temperature T of the gas of massless modes with energy Es in a volume Vd as T < TH
(4.18)
That is, in the radiation phase the “would be” radiation temperature drops below the Hagedorn temperature and becomes the true temperature of the universe.
98 In thermal equilibrium at this phase, the universe is dominated by a gas of massless modes with energy Es = cd Vd T d+1 .
(4.19)
For the volume we will use Vd = (2π)d ed |λ| , the T-duality invariant “volume” of the torus. This takes into account the fact that the energy could be stored in either momentum or winding modes depending on the size of the torus (see section 2.3). In other words this reflects the fact that the energy of the string gas is T-duality invariant. In the radiation phase we have the standard thermodynamic results 1 F = Es − T S = − cd Vd T d+1 d
(4.20)
Pλ = sign(λ)Es /d
(4.21)
leading as usual to a conserved entropy. We also have the equilibrium values λ>0:
hW i = 0
hKi = 21 Pλ eλ (4.22)
λ 0. Since we’re at strong coupling the fields ϕ and λ behave like massless particles.8 Moreover there is no force on these particles: with no dilaton potential Pϕ = 0, and in the Hagedorn phase 8
We will be more precise about this in (4.25) below.
100 Pλ = 0. So the particles move at nearly the speed of light, ϕ˙ ≈ −1
√ λ˙ ≈ 1/ d .
(4.23)
But this behavior cannot persist indefinitely. As the universe expands eventually it will cool below the Hagedorn temperature. To see when this happens we compute the energy E in matter using the Friedmann equation (4.8). Then we compute the equilibrium radiation temperature Trad using (4.19). If Trad < TH the universe is no longer in the Hagedorn phase. But rather than go to an equilibrium radiation phase, we assume the universe makes a transition to a frozen phase in which the momentum and winding numbers K and W are conserved, equal to whatever values they had on Hagedorn exit. In the frozen phase the pressure does not vanish. Instead there is an effective potential for the scale factor, V (λ) = EW + EK = 2dW eλ + 2dKe−λ .
(4.24)
At some point λ bounces off this potential. The universe shrinks and eventually re-enters a Hagedorn phase. It subsequently emerges from this new Hagedorn phase and undergoes a T-dual bounce, driven by momentum modes, at λ < 0. The whole cycle repeats, resulting in an oscillating scale factor. This is indeed as envisioned in [14]. The temperature bounce in figure 3.1 is realized, but also repeats in cycles due to the scalefactor bounces in the presence of winding modes in large volumes (off-equilibrium radiation phase) or the presence of Kaluza-Klein modes in the T-dual phase.
101 However the oscillations cannot persist indefinitely. When the dilaton reaches weak coupling the ϕ and λ particles become very massive and come to rest, putting an end to the oscillations. This can be seen in a numerical solution in figure 4.2. Note that at strong coupling the oscillations have constant amplitude. This is a consequence of neglecting interactions, which implies no entropy production in the frozen phase: the system always reenters the Hagedorn phase with the same values of λ and E, which in the Hagedorn phase corresponds to the system having the same entropy. We will relax this approximation in section 4.5. What we need for a cyclic scale factor is not strong coupling, necessarily, but rather a large amount of energy stored in the dilaton. This can be seen from the Friedmann equation γλ =
1 (Eϕ − Es )eϕ 8π 2
(4.25)
where Eϕ = 8π 2 γϕ e−ϕ − V (ϕ) is the (negative of) the total energy stored in the dilaton, and Es is the energy in the string gas. As long as Eϕ is large enough the scale factor is relativistic and can undergo bounces in a suitable potential. It is useful to note that when γλ >> 1 the equations of motion (5.9) imply γ˙ϕ ≈ ϕγ ˙ ϕ+
1 ϕe ˙ ϕ ∂V∂ϕ(ϕ) , 8π 2
so
d (8π 2 γϕ e−ϕ dt
− V (ϕ)) ≈ 0 and Eϕ is conserved.
In the Hagedorn phase Es is conserved as well, so (4.25) gives a clear picture of the dynamics: in the frozen phase, as λ grows the winding modes (or KK modes in the dual picture) absorb energy and increase Es until γλ drops to 1 and the universe bounces. A plot of Es is shown in figure 4.3.
102 So far we have discussed solutions in which the dilaton evolves monotonically. However the dilaton need not run to infinite coupling in the far past. A past state for the universe could be one where the expansion rate is arbitrarily small and the string coupling is arbitrarily weak. Provided ϕ˙ & 0 at early times, a simple dilaton potential of the form V (ϕ) = Aeϕ with A < 0 can generate a bounce for ϕ and turn it back toward weak coupling at late times. This is the basic idea of the pre-big bang scenario. A numerical solution is shown in figure 4.4.
As a further example, an upside down potential of the form V (ϕ) = Aeϕ + Be−ϕ , with A and B negative, can restrict the dilaton to vary within a finite range. The dilaton will undergo bounces, just like the scale factor, with Eϕ converting between large negative kinetic energy and large negative potential energy. A typical numerical solution is shown in figure 4.5.9
So far we have discussed bouncing and cyclic behavior using the string frame metric. Since we have in mind coupling to stringy matter this is the physically relevant frame to use. However one might be interested in the
9
The initial conditions in figures 4.2, 4.4 and 4.5 are chosen such that when γλ >> 1, Eϕ has the same value in all three examples. In the Hagedorn phase Es is also chosen to be the same (smaller than Eϕ ). These two energies determine the amplitude of the cycles, as we will see in more detail in section 4.6, so the maximum value of λ is the same in all three figures.
103 behavior of the Einstein frame metric10 , with scale factor λE = −
1 (ϕ + λ) . d−1
If the matter energy is positive, implying that the dilaton evolves monotonically, then the Einstein frame scale factor will evolve monotonically as well: the Friedmann equation (4.8) requires ϕ˙ 2 > dλ˙ 2 . However the models with a bouncing dilaton lead to a bouncing scale factor in Einstein frame. Generically each bounce of the dilaton will correspond to a bounce of λE . One might worry that we have introduced dilaton potentials which are unbounded below. However note that our solutions only explore a limited range of ϕ, and one could easily imagine obtaining the same behavior from a stable potential, just by modifying V (ϕ) outside the range of variation of the dilaton. One might also worry that bouncing and cyclic cosmologies require violation of certain energy conditions. We address this in appendix 6.3.
4.5
Interactions and Entropy Production
In this section we study the effect of interactions on an out-of-equilibrium string gas. We will continue to assume that thermal equilibrium holds during the Hagedorn phase, but we will allow the momentum and winding modes to go out of equilibrium in the radiation phase, where the temperature is 2φ
This is the frame resulting from a conformal transormation gµν → gµν e− d−1 such that the dilaton enters as a canonically normalized scalar field, minimally coupled to gravity. It is in general useful for comparison of different cosmological scenarios. 10
104
j
Λ
200
15 10
150
5
100 50
100
150
200
t
250
50
-5
0
-10
-50
-15
50
100
150
200
250
t
Figure 4.2: Numerical solution with Hagedorn and frozen phases and no potential for the dilaton. The oscillations have constant amplitude as there is no entropy production. The oscillations stop when the universe reaches weak coupling. We use d = 3, as in all graphs that follow.
E 7.01 ´ 1010 7.005 ´ 1010 7. ´ 1010 6.995 ´ 1010 6.99 ´ 1010 6.985 ´ 1010 50
100
150
200
250
t
Figure 4.3: A plot of the energy in the string gas for Fig. 4.2. The energy is constant during the Hagedorn phases. During a frozen phase it increases until the scale factor bounces. It then decreases and the system re-enters the Hagedorn phase.
105
Λ
j
15 10
200
5
150 100
200
300
400
500
600
t
700
-5
100 50
-10 100
200
300
400
500
600
700
-15
Figure 4.4: Same as Fig. 4.2, but with a dilaton potential of the form Aeϕ that yields a single bounce for the dilaton.
j
Λ 15
50
100
150
200
250
t
10 -5
5
-5
50
100
150
200
250
t -10
-10 -15
-15
Figure 4.5: A potential of the form Aeϕ + Be−ϕ can confine the dilaton at weak coupling.
t
106 below Hagedorn. We first take a macroscopic thermodynamic perspective and discuss entropy production, then present Boltzmann equations for the winding and momentum numbers. For simplicity in this section we will neglect the possibility of having a dilaton potential. Our goal is to understand how the momentum and winding numbers K and W evolve towards their equilibrium values. One constraint comes from energy conservation. The equations of motion (5.9) along with the Hamiltonian constraint (4.8) imply that E˙ = −dPλ λ˙ .
(4.26)
Here the dot indicates a time derivative and d is the number of dimensions, not a differential. Breaking up the matter energy as in (4.12), namely E = EW + EK + Edust , and likewise breaking up the pressure, the energy conservation equation (4.26) becomes E˙ W + E˙ K + E˙ dust = −d(PW + PK )λ˙ .
(4.27)
For the individual species we have d ˙ eλ E˙ W = (2dW eλ ) = 2dW eλ λ˙ + 2dW dt
(4.28)
˙ eλ = −dPW λ˙ + 2dW and
d ˙ −λ E˙ K = (2dKe−λ ) = −2dKe−λ λ˙ + 2dKe dt
(4.29)
˙ = −dPK λ˙ + 2dKe
−λ
Combining (4.27), (4.28) and (4.29), we must have ˙ eλ + Ke ˙ −λ ) = 0 E˙ dust + 2d(W
(4.30)
107 in order for energy to be conserved. Another constraint comes from the second law of thermodynamics. To illustrate what’s required let’s temporarily model the universe as filled with two fluids at different temperatures. One fluid consists of pressureless dust and winding modes and is held at the Hagedorn temperature TH , the other consists of radiation (i.e. momentum modes) held at temperature TK . The two fluids are out of equilibrium when TK < TH which is what we expect to occur when we exit the Hagedorn phase. The resulting entropy production rate is S˙ = S˙ K + S˙ W + S˙ dust E˙ K + dPK λ˙ E˙ W + dPW λ˙ E˙ dust + + TK TH TH λ −λ ˙ ˙ ˙ Edust + 2dW e 2dKe + = TK TH
=
(4.31)
and using (4.30) ˙ S˙ = 2dKe
−λ
1 1 − TK TH
.
(4.32)
Provided K˙ is positive (radiation is produced) whenever TK < TH , we will have S˙ > 0 consistent with the second law of thermodynamics. So in the frozen phase of section 4.3, where K was conserved, there was no entropy production. But any sensible evolution equation for K and W will lead to an increase in entropy. We now present such an evolution equation for the winding number W . At weak string coupling the appropriate Boltzmann equation was derived in [41], based on the cross section for winding – anti-winding annihilation
108 obtained in [42]. This derivation is discussed in more detail in chapter 5. 2λ+ϕ
˙ = −e W
π
W 2 − hW i2
(4.33)
Here h · i denotes a thermal expectation value, given in (4.22) for temperatures below Hagedorn. This expression makes intuitive sense: the factor e2λ captures the fact that longer strings are more likely to annihilate, while the factor eϕ = gs2 /V takes into account both enhancement by the string coupling gs and suppression by the volume of the torus V . The result (4.33) is reliable at weak coupling, but we will often be interested in behavior at strong coupling. At strong coupling we adopt the following modified Boltzmann equation. 2λ−d|λ|
˙ = −e W
π
W 2 − hW i2
(4.34)
This equation can be obtained from the previous weak-coupling Boltzmann equation (4.33) by making the replacement ϕ → −d|λ|, that is, by dropping the unshifted dilaton from the cross section but keeping the dependence on the T-duality-invariant “volume” exp(−d|λ|). This can be thought of purely phenomenologically, as describing winding strings (such as cosmic strings) whose interactions do not depend on the unshifted dilaton. It can also be regarded as describing fundamental strings, but with a potential for the dilaton that fixes the unshifted dilaton to φ ≈ 0. For momentum modes at strong coupling we use the T-dual equation −2λ−d|λ|
e K˙ = −
π
K 2 − hKi2 .
(4.35)
109
4.6
Shrinking Cycles and Exit
We now study how entropy production in an out-of-equilibrium string gas affects the cyclic cosmologies of section 4.4. For simplicity we set the dilaton potential to zero.
Recall that in section 4.4 we neglected interactions during the frozen phase; the momentum and winding numbers were taken to be conserved. This led to a constant entropy and oscillations of fixed amplitude. Taking interactions into account we will see that the resulting entropy production leads to oscillations of decreasing amplitude. Oscillating models often exhibit this sort of behaviour, but the details depend on the mechanism that drives the bounce [43]. For example in a recent bouncing cosmology scenario, in which an equilibrium Hagedorn era was also used, the oscillations grew with time [44]. But in this model there was no dilaton and the bounce was driven by positive spatial curvature and negative Casimir energy.
In our models eventually so much entropy is produced that it is no longer thermodynamically possible for the universe to re-enter the Hagedorn phase. At this point the universe transitions to a loitering phase in which the scale factors are roughly constant, oscillating about a minimum in their potential. Eventually the loitering phase also ends and the universe transitions to a standard radiation-dominated cosmology.
110 4.6.1
Shrinking Cycles
The dynamics are largely governed by the energy stored in the dilaton. We are neglecting any dilaton potential, so as noted in section 4.4 the (negative of) the dilaton kinetic energy Emax ≡ 8π 2 e−ϕ γϕ
(4.36)
is essentially constant. We have denoted this Emax because it’s equal to the maximum matter energy during a cycle. To see this recall that the Friedmann equation (4.8) states that the energy in matter is E = 8π 2 e−ϕ (γϕ − γλ ) .
(4.37)
At a bounce we have γϕ γλ = 1 and therefore E ≈ Emax . During the radiation phase of the nth cycle the energy in matter starts at En , the (conserved) matter energy during the Hagedorn phase of the nth cycle. It increases to Emax as the wound strings are stretched.11 After the bounce the matter energy decreases down to the value En+1 associated with the next Hagedorn phase. These Hagedorn phases serve as reference equilibrium points in phase space where the entropy is well defined, given by Sn = En /TH . Since entropy is produced during the radiation phase, Sn+1 > Sn as we saw above, and since we return to the same (equilibrium) temperature TH when re-entering the Hagedorn phase, the matter energy must increase during each 11
For simplicity we discuss bounces at large radius. At small radius T-duality would exchange momentum and winding.
111 radiation phase as well, En+1 > En . This means the radius at which we exit the Hagedorn phase also increases with each cycle. To see this recall that the condition for exit is that the equilibrium radiation temperature drops below Hagedorn.12 From (4.19) this means that at Hagedorn exit En = cd (2π)d edλn THd+1 .
(4.38)
Since En increases with each cycle, so does the scale factor at exit eλn . We can also estimate the maximum scale factor reached during each cycle max
eλn . From the moment of Hagedorn exit to the bounce, matter energy increases by an amount Z
λmax n
Emax − En = −d
Z
λmax n
dλ Pλ ≈ 2d λn
dλ Wn eλ − Kn e−λ
(4.39)
λn
where we’ve assumed interactions are weak so the values at Hagedorn exit √ En Wn = √ e−λn 12 π
√ En Kn = √ eλn 12 π
(4.40)
p ∼ En1/d α(En ) + α(En )2 − 1
(4.41)
are roughly conserved. This leads to max
eλn where
3 α(En ) = d 12
r
π (Emax − En ) + 1 . En
Note that no real temperature is dropping here since during the Hagedorn phase the temperature is constant at TH . By equilibrium radiation temperature we mean the temperature that radiation alone would have in a universe of volume V = (2π)d edλ and energy En . It is the volume that grows and signals a transition to a radiation phase.
112 λmax is a decreasing function of En , so the maximum radius shrinks with each n cycle. The features we have discussed can be seen in figure 4.6, which shows a numerical solution to the combined gravitational equations of motion (5.9), (4.8) and the strong-coupling Boltzmann equations (4.34), (4.35). The matter energy E has plateaus which correspond to Hagedorn phases of vanishing pressure. During the radiation phases the matter energy jumps to Emax before falling to the next Hagedorn plateau.13 In figure 4.6 one can also see the slight decrease in the amplitude of the oscillations with time.14
4.6.2
After the Hagedorn Era
The dilaton kinetic energy Emax sets the maximum possible entropy that the system can have and still be in the Hagedorn phase, namely Smax = Emax /TH . As entropy is produced during the radiation phases eventually a bounce will occur during which S exceeds Smax . At this point a return to a Hagedorn phase is no longer possible. Instead the universe enters a new era which resembles the loitering phase 13
The small dips in the energy on either side of the plateaus is due to the redshift of energy in an expanding radiation-dominated universe if λ > 0, or the T-dual phenomenon if λ < 0. Eventually either the stretching of winding strings its T-dual takes over and leads to the large spikes in energy. Note that time-reversal invariance is only broken by entropy production during the radiation phases. 14
In figure 4.6 we used by hand a slightly larger value for TH (larger by a factor of 1.7). This allows us to illustrate the desired effects over a shorter integration time as the phase transitions between Hagedorn and radiation phases occur earlier (smaller λ) and the interactions are more efficient. The qualitative picture is not altered.
113
E
Λ
50 000 5
40 000
50
100
150
200
250
300
350
t
30 000
20 000 -5 50
100
150
200
250
300
350
Figure 4.6: An integration of the equations of motion (5.9), (4.8), (4.34), (4.35) for d = 3. As the entropy increases the energy during the Hagedorn phases increases towards Emax and the size of the oscillations in the scale factor gets smaller (the dashed lines are drawn at constant λ). discussed in [32]. The scale factor undergoes oscillations about the minimum of the potential (4.24), namely15 V (λ) = 2dW eλ + 2dKe−λ . Assuming λ > 0, and using the strong-coupling Boltzmann equations (4.34), (4.35), the winding strings will gradually annihilate and radiation (momentum modes) will be produced. Eventually all the winding strings will be gone. At that point the oscillations stop and the universe transitions to a radiation-dominated cosmology. With our modified gravity action we may 15
As can be seen from (4.40), at the moment of Hagedorn exit W and K are such that the scale factor sits at the minimum of the potential. For λ > 0 the subsequent evolution of W and K will tend to shift the minimum to larger radii.
t
114 not have the usual radiation-dominated expansion, as one could enter the radiation-dominated era while the scale factors and shifted dilaton are still √ relativistic (ϕ˙ ≈ −1, λ˙ ≈ 1/ d). This is similar to inflation models. It is the large energy stored in the dilaton that dominates and drives accelerated expansion eventhough matter consists of radiation. But eventually, and in particular at weak coupling the higher-derivative modifications to the action are unimportant and we go over to a standard radiation-dominated cosmology. The dilaton continues to roll to weak coupling, while the scale factor grows according to eϕ ∼
1 t2d/(d+1)
eλ ∼ t2/(d+1) .
Somewhat curiously the unshifted dilaton is constant and the scale factor grows just as it would in Einstein gravity. The whole story can be seen in figure 4.7 which is simply an extension of figure 4.6 to later times. It shows the log of the scale factor and the matter energy for a universe evolving through an era of Hagedorn oscillations and a loitering era of potential oscillations before finally entering a radiationdominated era. In the Hagedorn era the scale factor oscillates about λ = 0, while in the loitering era it oscillates about the minimum in the potential, and in the radiation-dominated era it starts out growing relativistically. The amplitude of the oscillations decreases during the Hagedorn era and increases during the loitering era. The behavior of the matter energy also changes. It has plateaus during the era of Hagedorn oscillations which disappear during the loitering era. (The spikes in the matter energy during the loitering era
115 are simply conversion between kinetic and potential energy.)
4.7
On the BV Mechanism
One might expect the models we have been discussing to provide an ideal setting for realizing the Brandenberger-Vafa mechanism. Indeed this was our original motivation for developing these models. The original BV scenario runs into two difficulties [41, 45]: as the dilaton rolls to weak coupling, the standard Boltzmann equation (4.33) predicts that string interactions turn off, and one is generically left with a gas of non-interacting strings on a torus of fixed size. Also with the two-derivative effective action (4.2) the universe has a singularity a finite time in the past, so there is only a limited amount of time for the necessary thermal fluctuations to take place. Both of these difficulties would seem to be cured in the models we have considered. With the modified Boltzmann equations (4.34), (4.35) string interactions do not turn off at late times.16 Moreover with the modified gravity action (4.5) the singularity is pushed infinitely far into the past. The oscillating scale factors we have found can be thought of as repeated attempts at decompactifying; if on each bounce there was some probability of decompactifying for d ≤ 3, but vanishing probability for d ≥ 4, then the Brandenberger-Vafa mechanism would work. This is not, however, the behavior we generically find. Instead in any 16
This could also be achieved with the standard Boltzmann equations by introducing a potential to confine the dilaton.
116
Λ 15
10
5
100
200
300
400
500
400
500
t
600
!5
E 50 000
45 000
40 000
35 000
30 000
25 000
20 000 100
200
300
600
t
Figure 4.7: The log of the scale factor and the matter energy in a typical numerical solution. For t < 400 the universe cycles between Hagedorn and radiation phases. For 400 < t < 640 the scale factor oscillates about the minimum of its potential while the winding strings gradually annihilate (in practice we use a cutoff value W = 1/2 to specify winding mode annihilation). For t > 640 the universe is radiation-dominated.
117 number of dimensions the era of Hagedorn oscillations eventually ends and the universe transitions to a loitering phase of oscillations about the minimum of the effective potential for λ. Taking λ > 0 for purposes of discussion, with the modified Boltzmann equation the winding strings will eventually annihilate and the universe will decompactify. This chain of events can happen for any d. In this sense the Brandenberger-Vafa mechanism is not operative. The reason why the BV mechanism seems to be failing is that quantum √ fluctuations give the winding strings an effective thickness of order α0 in all spatial dimensions, hence their probability to interact is non-vanishing for any d and only decreases with d through a “per volume,” ∼ e−dλ , dependence. The original BV argument rested on interactions via (classical) string intersections which did not take into account this (quantum) thickness. It is the subject of the next chapter to consider this crucial point and eventually identify how d = 3 is the special dimensionality. Despite that, one might still hope that d ≤ 3 can be still favored because the universe might not follow the expected behavior we discussed above. Imagine that due to a thermal fluctuation the universe exits a Hagedorn phase with an unusually small number of winding strings. In the subsequent radiation phase perhaps all these strings will annihilate and the universe will decompactify immediately, without additional bounces and without going through a loitering era. Since the annihilation rate (4.34) falls off rapidly with d, perhaps this fluctuation-driven mechanism will preferentially decompactify d ≤ 3?
118 To address this issue let’s study the conditions for decompactifying in a single cycle in the framework we have been using. The probability of decompactifying depends not only on the energy during the Hagedorn phase En , which determines the number of winding strings present at Hagedorn exit, but also on Emax , which determines how long the subsequent radiation phase will last. For fixed Emax , smaller values of En – that is, less winding on Hagedorn exit and a larger value of λ˙ – will give an increased probability of decompactifying. It’s convenient to express this in terms of c = (Emax − En )/Emax . Since En > 0 we have c < 1. Taking Emax = 107 as an example, we find that 3 dimensions decompactify promptly on Hagedorn exit for c & 0.984. To decompactify 4 dimensions requires c & 0.9994, and to decompactify more dimensions requires slightly larger c. To translate this into winding numbers on Hagedorn exit we use (4.38), (4.40). We find that 3 dimensions decompactify promptly if hWn i < 0.507, while 4 dimensions decompactify promptly if hWn i < 0.504. We conclude that, with this value of Emax , strings are only slightly more efficient at annihilating in d = 3 compared to d = 4. The only way to decompactify promptly is to exit Hagedorn with essentially no winding (recall that our criterion for no winding was W < 0.5). One could imagine choosing special initial conditions – say a small value of En – to make the winding number small. But this seems against the spirit of the BV mechanism, which should operate starting from generic initial conditions. To quantify just how special the initial conditions have to be, note
119 that the number of Hagedorn-era microstates which decompactify promptly (proportional to the probability of decompactifying) is eSn = eEn /TH ∼ e−cEmax /TH . With Emax = 107 the probabilities of having sufficiently small En are tiny, although they do fall off rapidly with d. Other types of fluctuations are more likely. For example, even for large En and hWn i, there might be fluctuations away from the mean that make the winding number vanish. To estimate the probability of this happening, note that for reasonable distributions of winding numbers the probability of having zero winding on Hagedorn exit scales as d Prob.(no winding) ∼ 1/hWn i . If hWn i is large then the probabilities are tiny (although again they fall off rapidly with d). There are other interesting types of fluctuations to consider, for example fluctuations in the initial value of λ. For large initial λ strings should be more likely to annihilate in d = 3 than d = 4, due to the dimension dependence of the cross-section. Although we have not estimated the probability of this happening, it seems unlikely to us that the basic picture will be modified: fluctuations which are large enough to favor d = 3 are also very unlikely to take place. As an alternative approach, one could set initial conditions such that three dimensions decompactify, but this requires careful tuning and violates the spirit of the BV scenario.
120 We conclude with a few comments on the robustness of our results. We have assumed that the string gas is in thermodynamic equilibrium during the Hagedorn phase. Let’s consider the alternative possibility that the winding modes fall out of equilibrium during the Hagedorn phase, well before exit to the radiation phase. To test this we should modify the Boltzmann equations to take into account the fact that string oscillators are highly excited. This was considered in [41] and it amounts to putting a factor of E in the string cross-sections. With this enhancement, numerical tests for a wide range of energies (103 – 107 ) showed that during the Hagedorn phase the winding number indeed closely tracked its thermodynamic average. This supports our assumption of a string gas in thermal equilibrium during the Hagedorn phase.17
4.8
Conclusions
To summarize, we studied the dynamics of a string gas coupled to a modified gravity action. The modified gravity action was set up to avoid singularities, and when coupled to a string gas we found that bouncing and cyclic cosmologies naturally result. Several aspects of our analysis deserve comment and further investigation. • We postulated a particular form for the modified gravity action (4.5). It 17
Recall however that in these models the collective degrees of freedom λ, φ remain essentially out of equilibrium with the rest of the string gas, evolving with classical equation of motion.
121 would be interesting to understand to what extent our action captures the effect of α0 corrections in string theory. But we expect that any action which avoids singularities and respects T-duality should lead to qualitatively similar results. • Modified gravity theories generically have ghosts [46]. A question for future investigation is whether our action is ghost-free. • We described the string gas using modified Boltzmann equations (4.34), (4.35) in which we simply dropped the dependence on the (unshifted) dilaton. This could be thought of quasi-phenomenologically, as describing cosmic strings whose interactions do not depend on the dilaton. It could also be thought of as a crude representation of the behavior of either fundamental strings or D-strings [47], given a potential which confines the dilaton to string couplings gs = O(1). • Although we developed our models to illustrate some of the features that result from a non-singular string gas cosmology, it would be interesting to study whether they provide a basis for a realistic cosmology. An important step would be to study the spectrum of scalar perturbations resulting from early Hagedorn and loitering eras, extending the work of [48, 49, 50] to the present context. The models we have discussed provide a remarkably simple realization of bouncing and cyclic cosmologies. With a suitable potential for the dilaton,
122 they also provide a simple realization of the pre-big bang scenario. Let us comment on the two other motivations given in the introduction. Horizon problem As we have seen the universe can evolve to a loitering phase in which the scale factor oscillates about the minimum of its potential. If the loitering phase lasts long enough the entire universe will be in causal contact and might be expected to become quite homogeneous. This would provide a solution to the horizon problem. There are two conditions that must be met. ¯
1. The time-averaged scale factor eλ depends on initial conditions while the duration of the loitering phase t also depends on the string crosssection. The condition for the universe to come in causal contact is ¯
eλ t which can easily be satisfied by going to weak coupling. 2. Even if the universe is in causal contact we still need to make sure it becomes homogeneous. The condition is that the universe be smaller √ ¯ than the Jeans length, eλ 1/ Gρ. Again this can easily be satisfied by going to weak coupling.18 Provided these conditions are satisfied any inhomogeneities generated during the Hagedorn phase transitions will be washed out and the universe will 18
Here, we are interested in the √ Jeans length in the near-radiation loitering phase and we adopt the naive estimate 1/ Gρ. Jeans instability in the Hagedorn regime is a subtle and debatable issue. Eventhough the speed of sound is small, it is unlikely that the usual equations for evolution of matter fluctuations in particle theories can apply to extended non-localized objects like strings, that are highly excited in the Hagedorn phase. See [51] for a related discussion.
123 eventually approach radiation domination in a state very near thermal equilibrium.19 In this way our models provide a simple natural resolution of the horizon problem. Brandenberger-Vafa mechanism The Brandenberger-Vafa mechanism is predicated on the idea that winding strings can only annihilate efficiently in d ≤ 3 dimensions. In the models developed here, these wound strings will inevitably annihilate, and the universe will transition to radiation domination, no matter the number of dimensions. The necessary ingredients for the Brandenberger-Vafa mechanism to operate, in the sense that the interaction rates single out three dimensions, are presented in the following chapter.
19
For a study of perturbations in bouncing models see [52].
124
Chapter 5 Dynamical Decompactification and Three Large Dimensions In this chapter we study string gas dynamics in the early universe and seek to realize the Brandenberger-Vafa mechanism that singles out three or fewer spatial dimensions as the number which grow large cosmologically. Considering wound string interactions in an impact parameter picture, we find a strong exponential suppression in the interaction rates for d > 3 spatial dimensions that reflects the classical argument that string worldsheets generically only intersect in four or fewer spacetime dimensions. An impact parameter picture is appropriate when the winding modes in the early universe are diluted enough such that their mean transverse separation, and also their length, is significantly larger than their effective quantum thickness. In that case they behave as effectively 1-dimensional objects something necessary for the Brandenberger-Vafa mechanism to operate. We consider the dynamics of a string gas coupled to dilaton-gravity and find that a) for any number of dimensions the universe generically stays trapped in the Hagedorn regime if
125 initially found there and b) if the universe fluctuates to a radiation regime any remaining winding modes are indeed diluted enough such that their interactions freeze-out in d > 3 large dimensions while they generically annihilate for d = 3. In this sense the Brandenberger-Vafa mechanism is operative.
5.1
Introduction and Previous Results
As mentioned in chapter 3, one of the few mechanisms aiming to explain the hierarchy between three large and six small spatial dimensions within superstring theory is due to a suggestion, some two decades ago, by Brandenberger and Vafa [14]. In this scenario the early universe consists of a hot string gas in thermal equilibrium near the Hagedorn temperature. The topology of space has non-trivial cycles supporting winding modes in the gas. The background metric and string coupling evolve with the low energy effective dilaton-gravity equations of motion according to which the winding modes resist the expansion of the spatial directions they wrap. If due to a thermal fluctuation a number of dimensions starts growing then eventually the equilibrium number of winding modes will drop to zero. The winding modes have the capacity to relax to equilibrium through annihilations with anti-winding modes; if these interactions are efficient then at large volumes the winding numbers will vanish allowing the corresponding dimensions to grow. The Brandenberger-Vafa (BV) mechanism relied on a simple dimension-counting argument that wound strings generically intersect in at most 3 spatial di-
126 mensions, singling this out as the maximum number of dimensions in which winding numbers have the capacity to track their equilibrium values, thereby dropping to zero and allowing the dimensions to grow large. In [53] this argument was supported using numerical simulations of a network of classical strings, though gravitational dynamics was not taken into account. Over time it became clear that strings are not the only fundamental degrees of freedom of string theory and that higher dimensional objects (membranes) are also fundamental states of the theory; superstring theory was shown to result from compactification of a higher dimensional theory, Mtheory. In a paper by Alexander, Brandenberger and Easson [54] the setup of [14] was extended to include p-branes for p = 0, 1, 2, 4, 5, 6, 8 in the weakcoupling limit of M-theory with one small dimension compactified on S 1 (type IIA string theory). The other spatial dimensions were compactified on a 9-torus. The authors argued that fundamental string winding modes are still the decisive objects regarding decompactification and that the conclusions of [14] still hold. They also pointed out that a further hierarchy between dimensions could arise. Past the string scale, as the universe grows, more and more energy is needed to support wound branes of highest p, hence highest p branes would tend to decay first. As two p-branes can intersect in at most 2p + 1 spatial dimensions, there is no obstacle for the disappearance of p-branes for p > 2. But 2-branes can allow for a 5-dimensional subspace to grow first. Further, within this subspace, 1-branes will only allow for a 3-dimensional space to continue expanding, as in [14], hence one is left with
127 a 3-2-4 dimensional hierarchy. These claims relied on heuristic thermodynamic and topological arguments. Aiming to carry out a more rigorous investigation, Easther et al. [55] considered the full equations of motion for 11D supergravity on a homogeneous but anisotropic toroidal background, coupled to a gas of branes and supergravity particles. Focusing on the late time behavior of the system, they justifiably ignored excitations on the branes and included only M2-branes, since M5-branes (the other fundamental states of M-theory) would annihilate efficiently in the full 11-dimensional spacetime. Motivated by the BV mechanism and the arguments of [54], the authors of [55] chose initial states resulting from fluctuations that would leave 3 dimensions unwrapped, some number of dimensions partially wrapped and some fully wrapped. The conclusion was that indeed the dynamics leverage the topological reasoning and a hierarchy among dimensions is established. This conclusion was supported further by [56], where in addition nontrivial fluxes were included (the 3-form gauge field of 11D supergravity). In fact, the presence of fluxes seemed to enlarge the possible space of initial conditions that lead to three large dimensions at late times. Specifically, for the case of six initially unwrapped dimensions, the dynamics of fluxes introduced a new hierarchy suppressing the growth of 3 out of the 6 unwrapped dimensions. An apparent limitation of the BV argument is that it seems to depend crucially on non-contractible spatial cycles and their associated topologically stable winding modes. Phenomenologically viable compactifications of string
128 theory, however, may not have such cycles. Nonetheless the authors of [57] surmised that these more general spaces might still support “pseudo-wound” modes, long strings that extend around a dimension but are contractible. If these strings are stable over time scales larger than the cosmological Hubble scale, then as far as the dynamics are concerned they play the same role as stable wound strings. In [57], using numerical simulations for string networks on toroidal orbifolds with trivial fundamental group, the authors showed that pseudo-wound strings generically do persist for many Hubble times, suggesting that the requirement of non-contractible cycles can be relaxed. The results up to this point seemed promising, but it remained to actually test the heart of the argument: whether at early times, thermal fluctuations near the Hagedorn era and string (or brane) interactions really lead to annihilation of winding modes in a 3-dimensional subspace. An early attempt to investigate this was carried out in [58]. The authors considered a gas of 2branes and supergravity particles, along with excitations on the branes that lead to a limiting Hagedorn temperature. This setup was within the lowenergy limit of M-theory compactified on a 10-torus, with an anisotropic and homogeneous metric evolving according to 11-dimensional Einstein gravity. The winding numbers of 2-branes evolved according to Boltzmann equations. The authors assumed initial conditions in which the total volume of the torus was fixed but otherwise assumed that all states were equally likely. By numerically solving the coupled Boltzmann-gravity equations the authors concluded that the number of unwrapped dimensions at late times depended
129 crucially on the initial volume of the torus. Typically a large (and monotonically increasing) overall volume would decrease the interaction cross-section of branes too quickly, eventually leading to brane number freeze-out. If the initial volume was constrained according to holographic arguments, the initial winding numbers proved so small that all dimensions would decompactify early on. Three dimensions was not found to be singled out by the dynamics. Similar all-or-nothing behaviour was found in [41, 59] for IIA theory compactified on T 9 . In these papers the dilaton-gravity equations for the background were coupled to Boltzmann equations for winding modes and radiation. The decisive role of the volume in M-theory was played by the dimensionally reduced string coupling which would monotonically run to weak values. This meant interactions were not efficient enough for the winding modes to annihilate unless the initial conditions were fine-tuned. Note that this was a generic conclusion and independent of the number of dimensions growing large. The reason was that, after averaging over impact parameters, the rate at which wound strings annihilated only fell off like the inverse volume of the transverse dimensions. This behavior again failed to single out three large dimensions as special, suggesting that the Brandenberger-Vafa argument might not be supported by the dynamics underlying string/Mtheory. Here, we re-examine this conclusion and suggest a possible way in which string dynamics may indeed favor three large dimensions. Our basic approach is this: According to the Brandenberger-Vafa dimension-counting argument,
130 one expects that string interaction rates should be dramatically suppressed when the number of large spatial dimensions is bigger than three. Moreover, as the dimension-counting argument is purely classical, one expects it to be valid in a regime where the wound strings behave nearly classically and can be regarded as one-dimensional extended objects tracing a two-dimensional worldvolume. In such a regime the quantum thickness of the strings should be small compared to their length along the dimension they wrap and also small compared to the size of the transverse space. This suggests that we consider a semi-classical impact parameter representation of the string scattering amplitude. As we will see, this makes manifest the distinction between three and more large spatial dimensions and allows us to realize the mechanism within a suitable cosmological model. While a number of important issues remain, this appears to be the first demonstration of dynamical string theory decompactification that generically yields three large spatial dimensions.
By way of outline we begin with a discussion of the impact parameter representation, proceed to set up our model for the string gas, and finish with a numerical simulation, along the lines of [41], that will allow us to identify the regions of phase space in which three or more spatial dimensions decompactify.
131
5.2
Interaction Amplitudes and Impact Parameter Picture
In this section we derive interaction rates for wound strings in a semiclassical impact parameter picture. We will show that when long strings interact at impact parameters larger than their thickness, there is an exponential suppression in the interaction rates for d > 3. The starting point is the Virasoro-Shapiro amplitude, in terms of the usual Mandelstam variables s and t, for wound strings in d = D − 1 large dimensions given by [60, 61] A(s, t) = −κ2D−2
s2 0 0 0 (α s/4)α t/2 e−iπα t/4 . t
(5.1)
The center of mass momentum s is computed either from the right-moving or left-moving momenta of the closed string, s ≈ 4R2 /α02 with R the radius of the dimension the strings wrap. The imaginary part of the amplitude as t → 0 (exchange of massless modes dominates) is Im A(s, t = 0) =
α0 π 2 κ s2 4 D−2
(5.2)
Here κ2D−2 = κ2 /V is the gravitational coupling in D−2 dimensions, where V is the transverse compactification volume times the area of the torus wrapped by the strings [60]. By the optical theorem 1s Im(A(s, t))|t=0 ∼ α0 κ2D−2 s controls string interactions. For D = 4 this quantity is dimensionless and gives the probability for two colliding winding strings to interconnect and unwind, while for D > 4 it has units of (length)D−4 and represents a cross section in
132 the D − 4 dimensions transverse to the moving strings. This reflects the fact that long strings generically intersect in D = 4, like point particles moving on a line, while they generically miss in D > 4 and the relevant quantity becomes a cross-section. One can consider the interaction probability in an impact parameter picture. As shown above, long wound strings have an effective impact parameter in the D − 4 directions transverse to the motion of both strings. The impact √ parameter b is the conjugate variable to the transverse momentum q = −t and the amplitude in this representation is obtained by the following transform in the transverse directions. dD−4 q −iqb A(s, t) e (2π)D−4 s
Z A(s, b) =
For the Virasoro-Shapiro amplitude (5.1), using q −2 = α0
(5.3) R1 0
0 2 −1
dx xα q
, this
gives 1
dD−4 q −(Y −i π −log(x))α0 q2 −ibq 4 e (2π)D−4 0 κ2D−2 s 6−D D b2 /(4α0 ) = (D/2)−2 b γ − 3, 4π 2 Y − i π4
A(s, b) =
α0 κ2D−2 s
Z
dx x
Z
(5.4)
0
where Y = log( α4s ) and γ(a, x) is the lower incomplete gamma function. The imaginary part of the above amplitude in the limit b2 Y α0 is Im A(s, b) →
2 πα0 κ2D−2 s − 4Yb α0 e 4(4πY α0 )D/2−2
(5.5)
These results are similar to those in [62], the difference being that the authors of [62] consider graviton scattering and take the number of transverse directions to be D − 2. In fact the interpretation of b as a classical impact
133 parameter in (5.3) can be justified along the lines of [62]. In the high energy limit (s → ∞ which for wound strings is R → ∞) the authors of [62] sum up the amplitude to all loop orders to a unitary eikonal form. The large R or large energy limit localizes strings in the transverse directions and reveals classical behaviour, much as the eikonal treatment in quantum mechanics (or optics) reveals semiclassical particle (or ray) behavior. Note that A(s, b) is dimensionless for any D. It determines the annihilation probability P (b) via P (b) = v1 Im(A(s, b)), with ImA(s, b) as in equation (5.5) and v the velocity of the colliding strings in their center of mass frame. This prescription can be shown to satisfy the usual unitarity conditions in the large s limit [63, 64]. The quantity ∆x2 ≡ 4Y α0 = 4α0 log(R2 /α0 ) appearing in (5.5) is interpreted as the quantum thickness of the string. It measures the fluctuations about the classical straight string configuration. The fact that it increases logarithmically with the string’s length reflects the fact that it is energetically less costly to excite oscillators on a long string. Similar string spreading effects occur in high energy collisions and for strings falling into black holes [65]. Note that this string spreading does not include the effect of real (as opposed to virtual) oscillator excitations as would be appropriate in the Hagedorn phase of a string gas. In the Hagedorn phase wound strings are highly excited and their spread in the transverse directions is comparable to the length of the dimension they wrap [66].1 These wiggly strings are 1
Recall (section 2.2) that the Hagedorn phase strings perform a random walk in all directions. As their energy scales with their length, their mean extent in all directions
134 very likely to intersect, leading to rapid interactions which keep the strings in equilibrium. But as the universe expands and cools down the equilibrium phase becomes one of pure radiation. Then the oscillator excitations decay away and the spread of the wound strings approaches ∆x. This justifies our use of the amplitude (5.5) if b ∆x. It is useful to contrast the impact parameter picture to the more standard method of obtaining a scattering probability. Typically one derives a cross section σ and the collision probability is simply nσ where n is the number of targets per transverse volume. If one has a collision probability in impact parameter space, P (b), then the scattering cross section is obtained via [67, 68] Z σ=
d⊥ b P (b)
(5.6)
d⊥ b nP (b)
(5.7)
The quantity Z nσ =
is then an averaged probability in impact parameter space. There are two ways in which this approach can be justified. First, if the targets are dense and uniform as in collider experiments. A test particle in that case will interact with targets at all impact parameters so one can integrate as in (5.6). Second, if the time between collisions is much smaller than the total time over which collisions take place. Then the test particle is given enough time to interact with targets at all impact parameters (assuming each collision is √ scales as E. This is the dependence of the winding number on energy as we saw in section 2.2 and review below.
135 at a random impact parameter) and the averaging over impact parameters is essentially a time average. But if the winding modes in a string gas are diluted, with a mean separation much larger than their thickness, the dense target assumption above does not apply. It could still be that, since the strings move in a compact space, they collide repeatedly with each other and a time average is appropriate. It then becomes a matter of timescales. We need to compare the mean time between collisions with the recollapse time, the time required for winding modes to pull the universe back to a small-radius regime where winding modes are no longer dilute. An additional effect which must be taken into account is that the string coupling is time dependent. This could also invalidate the use of a time-averaged cross section. We thus have to develop a model for the distribution of interactions over impact parameters. We will return to this in the next section after we set up the rest of the dynamics.
5.3
Equations of Motion
In this section we write down coupled dilaton-gravity and Boltzmann equations for the matter degrees of freedom. For further details on the thermodynamic phases and energy conservation the reader can refer to chapters 3 and 4. We consider d growing dimensions all with the same radius, and hold the
136 remaining 9 − d dimensions frozen at the self-dual radius. By removing the randomness in the choice of initial radii as was the case in [41] we can see more clearly the dependence of the winding annihilations on the number of growing dimensions. We consider type IIA string theory with a flat FRW metric on a torus for the d = D − 1 growing dimensions, ds2 = −dt2 + α0 e2λ(t)
2 i dxi
P
0 ≤ xi ≤ 2π
(5.8)
and a homogeneous shifted dilaton ϕ(t). From now on we set α0 = 1. When the metric and dilaton are coupled to matter, the equations of motion are 1 ϕ¨ = (ϕ˙ 2 + dλ˙ 2 ) 2 ¨ = ϕ˙ λ˙ + 1 eϕ P λ 8π 2
(5.9)
and the Hamiltonian constraint (Friedmann equation) is E = (2π)2 e−ϕ (ϕ˙ 2 − dλ˙ 2 )
(5.10)
Here E is the total energy in the string gas and P is the pressure (times the volume) in d dimensions.
5.3.1
Matter Content and Boltzmann Equations
The background equations of motion are coupled to phenomenological Boltzmann equations that govern the evolution of matter as was the case in chapter 4. The difference lies in the interaction rates, as here we are strictly working
137 at weak coupling accounting also for the possibility of dilution. As before, we model matter with three species: • Winding modes that evolve according to ˙ = −ΓW (W 2 − hW i2 ) W
(5.11)
We specify the interaction rates Γ and the thermal equilibrium values h·i below. The total energy in winding and anti-winding modes is EW = 2dW eλ and their contribution to the pressure is PW = −2W eλ . • Radiation, or pure Kaluza-Klein modes, evolving according to K˙ = −ΓK (K 2 − hKi2 )
(5.12)
The energy in KK and anti-KK modes is EK = 2dKe−λ and their pressure is PK = 2Ke−λ . • Finally we include string oscillators, or massive string modes, as pressureless matter. The oscillator modes fill up the energy budget via Eosc = E − (EW + Ek ). We do not need a Boltzmann equation for these modes since the dilaton-gravity equations of motion automatically conserve energy, dE = −P dV . 5.3.2
Thermodynamic Phases and Interaction Rates
Near the self-dual radius the gas of strings is in a high density Hagedorn phase. The quantities of interest here are the equilibrium values of the winding and KK numbers 1 hW i = 12
r
E −λ e π
1 hKi = 12
r
E λ e π
(5.13)
138 Since E 1 most of the energy in the Hagedorn phase resides in oscillator √ modes (Eosc ' E − E). As the volume of the universe grows and the energy density drops, the equilibrium state should be one with only radiation. The condition for the transition is E ≤ cd THd+1 Vd
(5.14)
with TH the Hagedorn temperature, Vd = (2π)d edλ the volume of the large dimensions, and cd the Stefan-Boltzmann constant appropriate to the IIA gas of 128 massless Bose and Fermi degrees of freedom cd = 128
2d!ζ(d + 1) (2 − 1/2d ) (4π)d/2 Γ(d/2)
(5.15)
In this phase the equilibrium values are hW i = 0
hKi =
1 Eeλ 2d
(5.16)
That is, at equilibrium all the energy is in radiation (KK and anti-KK modes). Now we need to specify the interaction rates entering the Boltzmann equations. Recall that for winding modes, with an impact parameter b in D − 4 dimensions, the interaction probability is P (b) =
1 ImA(s, b) v
with
ImA(s, b) given in (5.5). For two wound strings moving in the x1 direction and with opposite winding along x2 , the right moving momenta are pR1,2 = (E, ±Ev, ±R/α0 ) so sR = −(pR1 +pR2 )2 = 4E 2 ' (2R/α0 )2 for slowly moving strings. Putting things together, the interaction probability per unit time
139 (per winding mode in the direction of motion) can be written as ΓW = Γ0 × Γb 0
≡
κ210
πα 4 V
2R α0
2
×
!D−4
2πR 1
(π∆x2 α0 ) 2
b2
(5.17)
e− ∆x2 α0
with V the total spatial volume of the 9-torus. In terms of our variables, and with α0 = 1, we have
κ210 V
=
1 eϕ 2(2π)
and R = eλ . Note that Γ0 is
the interaction rate used in chapter 4 where it was modified for the strong coupling approximation. As explained earlier, an impact parameter representation is only appropriate in the radiation phase, when the separation between winding modes r is larger than the string thickness ∆x. From thermodynamics, we can estimate how the mean velocity v¯ of a single winding mode depends on R and E (see appendix C).2 The mean time between collisions, or recollision time, is then tr ' vr¯ . In practice, as we numerically integrate the equations of motion, once we are in the dilute regime we randomly choose an impact parameter b on every recollision time. The impact parameter is chosen at random, from a uniform distribution in the transverse D − 4 dimensions, up to the maximum value b = r. This gives enough information to determine Γb . Another concern, raised earlier, is that in the dilute regime the winding strings might not have time to collide before the universe re-collapses to a 2
Even though we are working off-equilibrium we consider the equilibrium velocities to be a good approximation. To be precise we are using the root mean square velocity.
140 dense Hagedorn phase. In principle this could happen even in D = 4. We test for this as follows. Upon entering the dilute regime we estimate the recollision time tr and turn off interactions, i.e. set Γ = 0. If the negative pressure from the frozen winding modes recollapses the universe in a time smaller than tr it means that freezing the interactions was consistent, that is, the winding modes truly had no time to collide. On the other hand, if after time tr we are still in the dilute regime, then string interactions must be taken into account. Thus (5.17) provides our description of string interactions in the dilute regime. In the Hagedorn phase the strings have highly excited oscillator modes which enhance the interaction rates since more string is available. This was studied in [41] and it amounts to inserting an overall factor of
16 E 9
in the Boltzmann equations. We also need to specify the interaction rate for KK modes. Since the wavelength of these modes grows with R a semiclassical impact parameter picture at large R is not appropriate. Instead we should average over impact parameters. Since we already know the averaged interaction rate Γ0 for winding modes, by T-duality we can take ΓK = Γ0 |λ→−λ .
5.3.3
Initial Conditions
We need to solve the coupled equations (5.9), (6.4), (6.5) subject to the constraint (5.10). We need 6 initial conditions: ϕ0 , ϕ˙ 0 , λ0 , λ˙ 0 , W0 and K0 . In this section we describe our method for sampling from the space of initial
141 conditions. The basic idea is to fix the value of ϕ, ˙ scan over allowed values of λ0 and ϕ0 , and average over the values of λ˙ 0 , W0 and K0 using a suitable probability distribution. For effective supergravity to be valid and to ensure weak coupling we take ϕ˙ 0 = −1 and ϕ0 . −1. Recall that the dilaton evolves monotonically to weaker coupling with the absolute value of its velocity decreasing [41]. With ϕ˙ 0 and ϕ0 fixed we consider the universe at equilibrium with λ˙ = 0. determines the equilibrium number of Calculating the energy Eeq = E|λ=0 ˙ winding and KK modes per dimension the universe would have at the self dual radius, 1 hW isd = hKisd = 12
r
Eeq . π
We will use these values to set upper (lower) bounds for choosing W0 (K0 ) below. Next we choose the scale factor λ0 of the d-torus.3 We take λ0 > 0 since by T-duality we need not consider smaller volumes. Having already determined Eeq , choosing λ0 fixes the energy density and thus the equilibrium thermodynamic phase. The Hubble rate λ˙ can fluctuate away from zero. In the Hagedorn phase the entropy is given to a good approximation by S = E/TH . Thus λ˙ 0 is chosen randomly from the Gaussian distribution ˙2
2
eS ∝ e−λ0 /(2σH ) 3
(5.18)
We consider the dynamics of d dimensions, while the remaining 9 − d are kept frozen at the self dual radius, λ = 0 with λ˙ = 0 for all times.
142
2 with σH =
TH eϕ0 . 2(2π)2 d
In the radiation phase the entropy is S =
d+1 c V d d d
E cd Vd
d d+1
Using (5.10), to leading order in λ˙ 0 we have the distribution ˙2
2
eS ∝ e−λ0 /(2σr )
(5.19)
with σr2
=
ρ ρH
1 d+1
2 , σH
ρ=
E|λ=0 ˙ , Vd
ρH ≡ cd THd+1
(5.20)
It remains to choose the initial winding and KK numbers. Depending on whether we are in the radiation or Hagedorn phase, the equilibrium number of winding modes could be zero or not. Since we do not want to begin with zero winding (we would not be testing the BV mechanism in that case) the lowest value of W we may pick is 0.5, our chosen threshold between zero and non-zero winding. The furthest we can fluctuate from equilibrium (the largest winding) is Wsd . However it’s possible that the volume is so large that there isn’t enough energy to support that much winding. This occurs if Wsd > Ee−λ /(2d). Putting everything together, the initial winding number is chosen randomly in the range
E −λ Max{0.5, hW i}, Min{hW isd , e } 2d
(5.21)
In the Hagedorn phase the KK number can fluctuate between hKisd and the equilibrium value at the given λ0 , so we choose a value randomly in this range. In the radiation phase, given W0 , we compute the energy in winding EW = 2dW0 eλ . The rest of the energy should be available to radiation with the maximum KK number being Kmax = (E − EW )eλ /(2d). Therefore K0
.
143 is chosen randomly in the range (hKisd , Kmax ). If Kmax < hKisd we set K0 = Kmax .4 Once initial conditions are fixed we integrate the equations of motion until either the winding modes annihilate (W < 0.5) or the interactions freeze out, which we define as Γ0 W < 0.1H. We use the maximum rate Γ0 instead of the total ΓW to allow for the possibility that, depending on the randomly chosen value for b, strings could interact even for D > 4.
5.4
Results
With ϕ˙ 0 fixed at -1 we can scan initial conditions over a two-dimensional lattice of points (λ0 , ϕ0 ). For each lattice point we do 1000 runs to average over different choices of λ˙ 0 , W0 and K0 . The results for d = 3 and d = 4, 6 are shown in figures 5.2 and 5.1 and summarized in figure 5.3. For all values of d we see the common feature that equilibrium is maintained during the Hagedorn phase. A consequence of this is that for a large range of initial conditions, as long as the system starts out in the Hagedorn phase, it will remain forever trapped in the Hagedorn phase (see figure 5.3). This is related to the limiting value of λ(t) as t → ∞ in the solution to the equations of motion (5.9) in thermal equilibrium as shown in 4
When integrating the equations of motion we need to be careful not to produce more radiation than energy conservation allows. If at some point E = EK + EW , that is all the oscillators decay yet there is still winding (possibly frozen-in), we set hKi = (E − EW )eλ /(2d).
144 section 3.3. When volume is large enough that5 hW i → 0 yet the system is still in the Hagedorn phase, then the universe decompactifies for any d. This region gets narrower as d increases as seen in figure 5.3. But for d > 3 this is essentially the only region in parameter space where decompactification occurs. If the system gets to the radiation era, as the oscillators decay to massless modes and the long strings start diluting, hardly any choice of initial conditions leads to decompactification for d > 3. As shown for the cases d = 4 and d = 6 of figure 5.1, outside the Hagedorn phase there are only a few occurences (of order ≤ 1%) for which the universe decompactifies. Those rare cases have very small winding number (very near equilibrium to begin with) and happen to have collisions at small impact parameters. All other cases d > 3 give a similar picture. By contrast, for the d = 3 case shown in figure 5.2, we see that even in the radiation phase long strings are for the biggest part able to annihilate. It is interesting to note that the effects of large λ enter in two competing ways. First, due to the factor of s ∼ R2 = e2λ in the amplitude, long strings interact more efficiently, even at weak coupling. But at large λ the effect of dilution is more dramatic and strongly (exponentially) suppresses interactions for d > 3. Finally, a comment on the choice ϕ˙ 0 = −1. We could have considered smaller (absolute) values for ϕ˙ 0 , still valid within supergravity. This can be compensated by a small (logarithmic) shift in the initial dilaton to smaller 5
In practice this would be hW i < 0.5. We round W to zero and not hW i.
145 values such that the initial energy remains the same. The qualitative results should be unaltered.
5.5
Summary and Discussion
The Brandenberger-Vafa mechanism relies on a classical dimension-counting argument, namely that the worldvolumes of one-dimensional objects will generically intersect in at most three spatial dimensions. To see the mechanism at work one needs to be in a regime where strings behave as semiclassical one-dimensional objects. That is, one needs their length to be much larger than their effective quantum thickness, and also their thickness to be much smaller than the size of the transverse directions (the space is compact). With oscillators excited, as in the Hagedorn phase, strings have a significant spread in all directions and the classical picture fails. But when oscillators decay, as √ in the radiation phase, the thickness of strings grows as ∆x ∼ log R. Thus √ strings begin to behave classically as R/ log R grows. To model string interactions in this regime we developed an impact parameter representation of the string scattering amplitude. This allowed us to show that in this regime the BV mechanism indeed operates and favors decompactification of three spatial dimensions. To enter this regime we had to consider departures from equilibrium, often large. Clearly in the radiation phase this is necessary since the equilibrium number of winding strings is zero. In the Hagedorn phase strings rapidly come
146
d=8
d=9
200 150 100 50 0
-2 -4 -6
1 2
Λ0
200 150 100 50 0
j0
-2 -4 2
-8
3 4 5
-6
1
Λ0
-10
4 5
d=7
-2 -4 -6
1 2
200 150 100 50 0
j0
-2 -4
3 4
-6
1 2
-8 5
Λ0
-10
j0
-8
3 4 5
-10
d=4
d=5
200 150 100 50 0
-2 -4 -6
1 2
Λ0
-10
d=6
200 150 100 50 0
Λ0
j0
-8
3
-8
3 4 5
-10
j0
200 150 100 50 0
-2 -4 1
-6 2
Λ0
j0
-8
3 4 5
-10
Figure 5.1: Number of cases unwinding for d = 9, 8, 7, 6, 5, 4 as a function of ϕ0 and λ0 . The z-axis is clipped at 200 to make the few decompactifying cases (other than those in the Hagedorn phase where hW i = 0) visible.
147
d=3
1000 500
-2 -4
0 -6
1 2
Λ0
j0
-8
3 4 5
-10
Figure 5.2: Number of cases unwinding for d = 3 as a function of ϕ0 and λ0 . While in the Hagedorn phase, as is the case for all d, the system stays trapped, in the radiation phase, in contrast to the cases d > 3, any remining winding modes decay.
148
-2
No Winding
Hagedorn -> Radiation HΛ0 =0, d=4L
d=9
d=6
d=3 d=4 Hagedorn -> Radiation HΛ0 =0, d=3L
j0
-4
-6
Equilibriate out of Hagedorn for Λ0 =10Σ
Trapped in Hagedorn
-8
-10 1
2
3
4
5
Λ0
Figure 5.3: A plot of behavior in the (λ0 , ϕ0 ) plane contrasting the cases d = 3, 4, 6, 9. If the initial equilibrium winding number in the Hagedorn phase is non-zero the system typically stays trapped in the Hagedorn phase, unless the initial Hubble rate is large and the initial winding number is small (thin dark region labeled ‘Equilibrate out of Hagedorn for λ˙ 0 = 10σ’). If the initial equilibrium winding number is zero in the Hagedorn phase then the universe typically decompactifies in any number of dimensions (regions in the upper left corner, labeled by dimension). But if the universe begins in a radiation phase with a dilute gas of winding strings then only d = 3 will decompactify (orange region to the right of the grey line).
149 to equilibrium and the pressure vanishes. This means the universe tends to remain stuck in the Hagedorn phase, and for some number of dimensions to decompactify a large fluctuation is needed, either in the Hubble rate or in the initial volume, to send the system to a regime where the equilibrium winding number is zero. As the distribution (5.18) typically allows for only small fluctuations in the Hubble rate, we had to consider large fluctuations in the volume to realize the BV mechanism. An important next step would be to understand the likelihood of such a fluctuation taking place in the early universe. We emphasize that, given the tendency of the system to remain trapped in the Hagedorn phase, these fluctuations are necessary even when one is not interested in the possibility of decompactification. One shortcoming of our framework was that, even though in the radiation regime we dealt with a dilute gas of winding strings, we modeled the resulting pressure as a homogeneous term in the gravity equations of motion. This led us to consider their back-reaction on spacetime in an all-or-nothing manner, in which any amount of winding would oppose expansion while zero winding would not. As far as testing the interactions and eventual annihilation of strings, which was our focus, this shouldn’t be a concern. But a more detailed investigation of string gas cosmology should address the issue of spatial inhomogeneity. Finally it would be interesting to extend the analysis of this paper to the more general context of M-theory, taking into account the effects of the full p-brane spectrum.
150
Chapter 6 Anisotropy and the Cases d = 1 and d = 2 In the previous chapter we did not consider anisotropic cases, or scenaria for which the number of initially large dimensions is less than three. Clearly it should be easier for one or two spatial dimensions to decompactify via the BV mechanism, as it is even more likely for winding modes to find each other in this lower dimensional subspace. If the BV mechanism is indeed the way with which the universe has 3 cosmologically large dimensions, we should address the question of why we don’t end up with an effectively 2-dimensional universe, since that seems to be equally, if not more, likely. Often answers to this question are based on anthropic arguments. However, in the original suggestion of the BV mechanism ([14]) the authors argued that continuously sampling thermal fluctuations in the Hagedorn phase may very likely lead to the maximum allowed number of large dimensions, which is three. Here, we would like to examine this possibility and investigate the likelyhood of three dimensions eventually decompactifying via fluctuations, once one or
151 two dimensions have initially grown large.
Essential to this discussion is that, while there might be a window of opportunity for three dimensions to decompactify once one or two dimensions are initially large, there will not be such window for more than three dimensions. Further, no other anisotropic expansion of any number of dimensions will favor decompactification in more than three. To see this, consider first the isotropic case of d > 3 dimensions. The interaction amplitude of winding modes depends on the size R of these dimensions in two ways. First, there is an enchancement ∼ R2 reflecting the fact that longer strings carry more momentum and are more likely to annihilate. Second, with W the wind√ ing number and ∆x ∼ log R the characteristic quantum thickness of long strings with radius R, the amplitude has an impact parameter suppression ∼ exp(−( WR∆x )2 ) for each of the d − 3 large directions, transverse to string collisions. This exponential suppression is the reason why the BV mechanism operates and the winding modes freeze-out for d > 3. Now, if we pick any subspace of these dimensions and let them grow larger to effect anisotropy then this will only further (and exponentially) suppress the annihilation of winding modes in the smaller dimensions by increasing the impact parameters along the large transverse dimensions. On the other hand, the large dimensions will only get rid of winding modes if the small ones are of order string scale and the winding modes (as was the implicitly assumed in chapter 5) continuously collide along them. Given the results of chapter 5, it is
152 not hard then to conclude that only d ≤ 3 dimensions can decompactify1 via anisotropic fluctuations. Interestingly enough, even if this large subspace is initially anisotropic, it was shown in [69] that string gas cosmology can naturally lead to isotropization. Hence studying anisotropy within a three-dimensional subspace covers the relevant cases to decompactification. Also, to be more general, we will consider non-simultaneous fluctuations. As mentioned above, we will focus on investigating the likelyhood of 3-dimensional decompactification, given that initially one or two dimensions have grown large. To keep this investigation more tractable, we only consider one degree of anisotropy, namely between one large and two small dimensions, or between one small and two large dimensions.
6.1
The Dynamics
The general setup will follow the lines of the previous chapter and the reader may refer to that for more details. The difference is that here we are considering two (logarithmic) scale factors, ν and λ, respectively for the d1 < 3 dimensions initially unwound and growing, and for the d2 = 3 − d1 dimensions subsequently expanding after a fluctuation, but wrapped with winding modes. We are working within type IIA string theory, on a flat toroidal back1
As mentioned in section 5.1, considering p−branes in the spectrum may yield a further dimensional hierarchy due to 2−branes.
153 ground and metric ansatz (in α0 = 1 units) 2
2
2ν(t)
ds = −dt + d1 e
d1 X
dx2i
2λ(t)
+ d2 e
i=1
3 X
dx2i
0 ≤ xi ≤ 2π
(6.1)
i=d1 +1
All other dimensions are frozen at the self-dual radius. In the gravitational sector we also consider the homogeneous shifted dilaton ϕ(t). When the metric and dilaton are coupled to matter, the equations of motion are 1 ϕ¨ = (ϕ˙ 2 + d1 ν˙ 2 + d2 λ˙ 2 ) 2 1 ν¨ = ϕ˙ ν˙ + 2 eϕ Pν 8π ¨ = ϕ˙ λ˙ + 1 eϕ Pλ λ 8π 2
(6.2)
These satisfy the Hamiltonian constraint, or Friedmann equation, E = (2π)2 e−ϕ (ϕ˙ 2 − d1 ν˙ 2 − d2 λ˙ 2 )
(6.3)
E is the total energy in string-gas matter and Pν ,Pλ the pressures along the d1 and d2 dimensions respectively. We will not consider a potential for the dilaton. In this setup matter consists of: • Winding modes with W denoting the winding number. Since the d1 dimensions are assumed unwound to begin with, the winding modes only wrap the d2 dimensions with logarithmic scale-factor λ. The winding modes evolve according to ˙ = −ΓW (W 2 − hW i2 ) W
(6.4)
hW i denotes the equilibrium average, and ΓW the interaction rate that we specify in the relevant thermodynamic phases later on. The contribution to
154 the energy from winding and anti-winding modes is EW = 2d2 W eλ and their contribution to the (off-equilibrium) pressure Pλ is PW = −2W eλ . • Radiation, or pure Kaluza-Klein modes Kν and Kλ along the d1 and d2 dimensions respectively. These evolve according to K˙ λ = −ΓKλ (Kλ2 − hKλ i2 )
(6.5)
K˙ ν = −ΓKν (Kν2 − hKν i2 ) The energy in Kaluza-Klein modes is EK = 2d1 Kν e−ν + 2d2 Kλ e−λ and their contributions to the pressures Pν and Pλ is 2Kν e−ν and 2Kλ e−λ respectively. • String oscillator modes that we model as pressureless matter.
6.1.1
Equilibrium Phases
In the d-dimensional isotropic case the early universe string gas could be found in two thermodynamic phases. In the Hagedorn phase massive and massless modes are in thermal equilibrium, the pressure vanishes and the energy in matter is conserved. At larger volumes or smaller densities, massive modes decay to radiation and this transition occurs when the effective density in d dimensions satisfies ρd =
E < ρH = cd THd+1 Vd
(6.6)
with TH the Hagedorn temperature and cd the Stefan-Boltzman constant in d dimensions (2.39). This can be alternatively expressed in terms of temperatures: in a volume Vd and with energy E, a radiation gas has temperature
155 1
Td = ( cdEVd ) d+1 . The condition (6.6) can be then expressed as Td < TH
(6.7)
that is, the would-be radiation temperature in d + 1 dimensions falls below the Hagedorn temperature. In the anisotropic case, however, there is an additional possibility to the above. Outside the Hagedorn phase and with 3 large dimensions, d1 of them larger than the remaining d2 = 3 − d1 , the system can be found either in a 3-dimensional or in a d1 -dimensional radiation regime. To decide what the equilibrium phase of the universe would be we generalize the condition in (6.7) above. Given the energy of the system and the 3-dimensional and d1 1
dimensional volumes, the lowest of the two temperatures Td1 = ( cd EVd ) d1 +1 1
and T3 =
1 ( c3EV3 ) 4
1
determines the equilibrium phase. If none of these temper-
atures is lower than the Hagedorn temperature TH then the system is in the Hagedorn phase. For brevity, we will refer to these phases as Rd1 , R3 and H. When solving the equations of motion, we will assume that if at any time the universe is found in Rd1 then all of the rest (D − 1) − d1 dimensions will remain frozen-in and the universe will never be effectively three-dimensional. This poses a limitation to this scenario in particular as it applies to the instant t = 0, where only the d1 dimensions are considered large. It reduces the region in initial-condition space where we can hope for three-dimensional decompactification but as we will see, it affects more the case d1 = 2
156 In Rd1 all the energy flows in the massless Kaluza-Klein modes Kν . In thermal equilibrium we then have hKν i = Rd1 :
E ν e , 2d1
(6.8)
hKλ i = hW i = 0 In R3 the energy is split between Kλ and Kν , E = 2d1 Kν e−ν + 2d2 Kλ e−λ . Recalling that d = 3 = d1 + d2 we have E ν e , 6 E hKλ i = eλ , hW i = 0 6 hKν i =
R3 :
(6.9)
In the Hagedorn phase we have
H:
r 1 E ν hKν i = e , 12 r π r 1 E λ 1 E −λ hKλ i = e , hW i e 12 π 12 π
(6.10)
Finally recall the off-equilibrium interaction rates. In the radiation phases they are given by ΓKν =
1 1 1 −2ν+ϕ e , ΓKλ = e−2λ+ϕ and ΓW = e2λ+ϕ . π π π
(6.11)
In the Hagedorn phase, these get multiplied by a factor of 16E/9 reflecting the enhancement due to highly excited oscillator modes.
6.2
Fluctuations, Initial Conditions and Procedure
We consider the following scenario: At time t = 0 the system begins with d1 < 3 dimensions expanding. Their initial size is given by ν0 and their
157 initial expansion rate is ν˙ 0 . All the other dimensions are frozen with vanishing expansion rates. At some time tf a number of dimensions d2 = 3−d1 fluctuate ˙ f ). to a radius λ(tf ) ≤ ν(tf ) and begin expansion with a non-zero rate λ(t We think of this as an energy fluctuation with energy flowing from matter to gravitation. We will be interested in the possibility of decompactification, that is, annihilation of the winding modes wrapping the d2 dimensions as a function of initial conditions, tf and the anisotropy parameter r≡
Rλ (tf ) = eλ(tf )−ν(tf ) Rν
(6.12)
Now depending on the energy density, the system at any time could be in any of the tree phases Rd1 , R3 or H. In particular, at time t = 0 the system may be in Rd1 . In that case, as mentioned earlier, we consider that the d2 dimensions are frozen in and no eventual growth of three dimensions is allowed. Therefore, at time t = 0 the system should find itself in H. Given ν0 , this puts a lower bound in the energy at t = 0, E ≥ c2 V2 TH3
(6.13)
In principle this should not be a concern2 but for practical purposes it essentially constraints the initial values of the dilaton as we will see shortly. One would worry that the condition to be found in H at t = 0 sets another limitation when we require that the d1 dimensions are initially unwound. At equilibrium in the Hagedorn phase the winding number does not necessarily 2
For the thermodynamical treatment to even make sense, we are formally working in the E → ∞ limit.
158 vanish, unless the energy is small enough and the radius eν large enough such that no winding is energetically favored. It turns out that we do need to worry about this, as, in order for decompactification of overall three dimensions to eventually take place, the initial value of ν (at t = 0) must be large enough, and the initial energy low enough (or the initial value of the dilaton large enough, once ϕ˙ is chosen) that the window for decompactification is always consistent with the condition that the d1 dimensions are initially unwound. Another way of seeing this is assumeing the contrary as follows. Suppose that we start deep in the Hagedorn era where the d1 are wound. Then when the d2 fluctuate, it is more common that they fail to push the system out of H and into R3 . We are then left trapped in the Hagedorn phase as was the case for the isotropic scenaria. This will become clearer as we present our results.
At this point let us view the initial conditions and the procedure we use to solve the coupled system of gravity and matter, equations (6.2), (6.4), (6.5) subject to the contraint (6.3). The maximum allowed value for |ϕ| ˙ consistent with the supergravity approximation is 1. We set then ϕ˙ 0 = ϕ(t ˙ = 0) = −1. The equations of motion guarantee that |ϕ(t)| ˙ < 1 and ϕ(t) ˙ < 0 for all t. A different choice in ϕ˙ 0 should not change the qualitative picture of our results, as it can be undone by slightly shifting the initial value for the dilaton. For the latter, we investigate values consistent with weak coupling and lying in the range ϕ0 < −1. Given ϕ˙ 0 and ϕ0 , the expansion rate ν˙ 0 is chosen at one
159 standard deviation from its gaussian distribution 4π 2 d1 e−ϕ0 2 exp(S) ∝ exp − ν˙0 TH
(6.14)
The quantities above are sufficient to determine the energy in matter at t = 0. Given the energy and ν0 the value of Kν (t = 0) is chosen at equilibrium (recall that we are in the Hagedorn phase). We then evolve the system to time tf . At this point, we need to effect the fluctuation for the expansion of the d2 dimensions with (logarithmic) scale factor λ. We specify the value of λ(tf ) by choosing a value of r as defined in (6.12), in the range 0 < r < 1 (in practice, we scan over (∼ 0.1, 1). Then, using the values of ϕ(t ˙ f ), ϕ(tf ) and ν(t ˙ f ) we evaluate the energy in matter with λ˙ = 0 and determine the equilibrium phase in order to choose the ˙ f ) from a distribution e−λ˙ 2 /(2σ2 ) . In H, σ 2 = TH eϕ2 0 and expansion rate λ(t H 2(2π) d2 1 d+1 2 2 ˙ f ) is chosen we re-calculate the ˙ f )2 σH . Once λ(t in R3 , σR = ρρH ϕ(t 3 energy available to matter. We are then left with determining the values for Kλ (tf ) and W (tf ). These are chosen with respect to their values at the self dual radius λ = 0 which sets a lower bound for Kλ and un upper bound for W . We randomly select Kλ (tf ) in the range (Kself dual , Kequilibrium ) and W (tf ) in the range (Max{0.5, hW i}, Wself dual ). The value 0.5 is our cutoff value for 0 and thus we do not allow for the fluctuation to result in vanishing of the winding, as our ultimate purpose is to test whether the winding vanishes as a consequence of interactions. In essence, to model a fluctuation we draw energy from the bath of heavy oscillator modes and re-distribute that energy to the other matter and metric
160 modes, i.e the expansion rates. In summary, ϕ0 , ϕ˙ 0 , ν0 , ν˙ 0 , Kν (t = 0) and ˙ f ), Kλ (tf ) and W (tf ) from λ( tf ) (or r) are fixed and we randomly choose λ(t the distributions given above. These are all the initial conditions needed to proceed with solving the equations of motions until the winding modes either freeze out or annihilate completely. In practice, the winding modes are considered annihilated when W < 0.5 and frozen-in when ΓW W < 0.1ν. ˙ The choice of ν˙ here reflects the fact that the winding modes move along and collide in the larger d1 dimensions. In a sense this lies in the heart of this particular test of the BV mechanism. The d1 dimensions with scalefactor ν are unwound and free to expand. We would like to test to what extend the winding modes wrapping the smaller dimensions can remain in equilibrium as the compete with the expansion rate of the former.
6.3
Results and Discussion
For ν0 we choose to test two values, 3 and 5. We will see that the results only mildly depend on the choice of ν0 , with everything else fixed, and slightly favor smaller values. As values of ν0 larger than 5 are in a sense “too large”, since we move way far from the string scale, we trust that these two values are good representatives of the whole picture. Once ν0 is chosen, the condition 6.13 that the system starts in H at t = 0 translates to the following condition for the values of ϕ0 : ϕ0 . −d1 ν0 − log[(2π)d1 −2 cd1 THd1 +1 ]
(6.15)
161 This is valid as the hubble rates are much smaller than 1 (the fluctuations are exponentially suppressed). The above is not a severe constraint since it is consistent with the prerequisite of weak coupling, but we will see that it constraints the case d1 = 2 more than d1 = 1 as we are pushed to weaker values for the coupling. The condition that the d1 dimensions are unwound, hW i < 0.5, translates to ϕ0 & −2ν0 − log[9/π]
(6.16)
Given ν0 then, these are the allowed values of the coupling we can test. What happens outside this range? In the small window in the strong coupling regime, which corresponds to less initial energy, the system at t = 0 is found in Rd1 . For the weaker coupling values lying outside the range in (6.16), which also corresponds to larger initial energies, the system stays trapped in the Hagedorn phase, with all dimensions still wound. For the fluctuation time tf we consider three values, tf = 1, 10, and 100 in string units, α0 = 1. This turns out to be the most decisive parameter, with the value tf = 1 yielding the largest window for decompactification of three dimensions. Let us once again list the initial conditions that are fixed for any given integration of the equations of motion: ϕ˙ 0 = −1, ν0 , ν˙ 0 (6.14) and Kν (t = 0) (chosen at equilibrium). We then scan over 0 < r < 1 and the range of values of ϕ0 as dictated in (6.15) and (6.16). For each of these sets of initial conditions, we perform 100 integrations to sample values of the Hubble rate
162 ˙ f ), K(tf ) and W (tf ) from the distributions mentioned above. λ(t The results are shown in figures 6.1 and 6.2 with the z−axis showing the count of cases for which the winding modes annihilate, for each pair of values of ϕ0 and r. One can immediately see the dependence on tf . The reason that decompactification is less favorable for larger values of tf is the rolling of the dilaton to very weak values. Recall that the condition (6.16) requires us to use smaller values for ϕ0 . This does not change the results significantly for larger ν0 as the enhancement in the amplitudes e2λ(tf ) = r2 e2ν(tf ) counteracts the suppression due to weaker coupling for fixed r. However, the number of initially large dimensions is more sensitive to (6.16), with d1 = 2 being less facilitating than d1 = 1. There is more “room” for a radiation equilibrium phase in 2 dimension than for 1. Finally, the dependence on r is as expected, with higher values more favored. It is infact useful to notice that the deciding factor is the total energy density of the universe. First, it determines whether we are found in the Hagedorn or in the radiation phase. As expected from the results of the previous chapter, the system generically stays trapped in H, but note that here what we mean by this is that winding modes freeze-out while the d1 dimensions keep growing, though very slowly as we will explain shortly. It is also possible that in a narrow region for the density (near the critical density between H and R3 ) hW i drops to zero while interactions are still efficient and the winding modes annihilate. Further, note that in the radiation phase contours of constant number of decompactification cases coincide very well
163
d1=1, Ν0=3, t f =1
d1 =1, Ν0 =5, t f =1
100
100 1.0 0.8 0.6 0.4
50 0 -2
j0
-6
r
-4 -6
1.0 0.8 0.6 0.4
50 0 -4
j0 -8
0.2
d1 =1, Ν0=3, t f =10
r
-10
0.2
d1 =1, Ν0 =5, t f =10
100
100 1.0 0.8 0.6 0.4
50 0 -2
j0
-6
r
-4 -6
1.0 0.8 0.6 0.4
50 0 -4
j0 -8
0.2
d1 =1, Ν0=3, t f =100
r
-10
0.2
d1 =1, Ν0 =5, t f =100
100
100 1.0 0.8 0.6 0.4
50 0 -2
r
-4
j0
-6
0.2
1.0 0.8 0.6 0.4
50 0 -4 -6
j0 -8
r
-10
0.2
Figure 6.1: Number of cases decompactifying (out of 100) as a function of ϕ0 and r, for ν0 = 3, 5 and tf = 1, 10, 100 when d1 = 1.
164
d1=2, Ν0=3, t f =1
100
d1 =2, Ν0 =5, t f =1
100 1.0 0.8 0.6
50 0 -4 -5
j0
0.4 -6
1.0 0.8 0.6
50 -8 0 -9
r
0.4
j0 -10
0.2 -7
r
0.2 -11
d1 =2, Ν0=3, t f =10
100
d1 =2, Ν0 =5, t f =10
100 1.0 0.8 0.6
50 -4 0 -5
j0
0.4
-6
1.0 0.8 0.6
50 -8 0 -9
r
0.4
j0 -10
0.2
r
0.2
-7
-11
d1 =2, Ν0=3, t f =100
d1 =2, Ν0 =5, t f =100
100
100 1.0 0.8 0.6
50 -4 0 -5
j0
0.4
-6
0.2 -7
r
1.0 0.8 0.6
50 -8 0 -9 0.4
j0 -10
r
0.2 -11
Figure 6.2: Number of cases decompactifying (out of 100) as a function of ϕ0 and r, for ν0 = 3, 5 and tf = 1, 10, 100 when d1 = 2.
165 with contours of constant energy density. For each case labeled by d1 , ν0 and tf then, there is a density in the radiation phase below which the system three dimensions generically decompactify. These observations are summarized in figures 6.3 and 6.4. On each graph we show contour plots for the number of decompactification occurences along with contours corresponding to densities equal to the critical density between H and R3 (thick black line) and density below which more than about 90% of integrations result in three-dimensional decompactification (thin red line). It is interesting then to contrast the results in terms of the ranges in energy density that yield three-dimensional decompactification. These are summarized in table 6.1. Note that these are densities at tf . It is implicit that at t = 0, for decompactification of three dimensions to result, the system is constrained according to (6.13) or alternatively (6.15). Finally we contrast the results as a function of d1 , ν0 and tf in terms of the overall percentage of cases yielding decompactification of three dimensions in R3 in table 6.2. In what sense then are our results of consequence to the BV mechanism? First of all let us note that in the most generic sense, fluctuations in the Hagedorn regime will be anistropic and sporadic (not simultaneous). Given our results, in order for the mechanism to have substantial grounds to operate, we have to require that fluctuations in the Hagedorn phase occur at √ timescales of order ∼ α0 , precisely so that the dilaton does not roll to very weak values. It is perhaps reasonable to assume that this is the case3 as this 3
This discussion is strictly within the homogeneous minisuperspace approach we take here, where hubble-rate fluctuations are distributed normally and scalefactor fluctuations
166
d1 =1, Ν0=3, t f =1
d1 =1, Ν0 =5, t f =1
R3
-2
-5
-3
-6
j0
j0
R3
-4
-4
-7 -8
-5 -6
-9
H 0.2
-11 0.4
0.6
0.8
0.2
1.0
0.4
0.6
0.8
1.0
r
r
d1=1, Ν0=3, t f =10
d1 =1, Ν0 =5, t f =10
R3
-2
R3
-4 -5
-3
-6
j0
j0
H
-10
-4
-7 -8
-5 -6
-9
H 0.2
-10 0.4
0.6
0.8
-11 0.0
1.0
H 0.2
0.4
r
1.0
d1 =1, Ν0 =5, t f =100
R3
R3
-4 -5
-3
-6
j0
j0
0.8
r
d1=1, Ν0=3, t f =100 -2
0.6
-4
-7 -8
-5 -6
-9
H 0.2
-10 0.4
0.6
r
0.8
1.0
-11 0.0
H 0.2
0.4
0.6
0.8
1.0
r
Figure 6.3: Contours of constant number of decompactification cases. The thick black line is for constant energy density at the critical density between radiation and Hagedorn phases. The thin red line is for a constant energy density below which more than 90% cases decompactify. Here, d1 = 1
167
d1 =2, Ν0 =5, t f =1
d1=2, Ν0=3, t f =1 -4.0
-8.0
R3
-4.5
-9.0
j0
j0
-5.0 -5.5
-9.5
-6.0
-10.0
-6.5
-10.5
-7.0
R3
-8.5
H 0.2
0.4
0.6
0.8
-11.0 0.0
1.0
H
0.2
0.4
r
-9.0
j0
j0
R3
-8.5
-5.0 -5.5
-9.5
-6.0
-10.0
-6.5
-10.5
-7.0
H 0.2
0.4
0.6
0.8
H
-11.0 0.0
1.0
0.2
0.4
d1 =2, Ν0=3, t f =100 -4.0
-9.0
j0
j0
1.0
R3
-8.5
-5.0 -5.5
-9.5 -10.0
-6.0
-7.0
0.8
d1 =2, Ν0 =5, t f =100 -8.0
R3
-4.5
0.6
r
r
-6.5
1.0
d1 =2, Ν0 =5, t f =10 -8.0
R3
-4.5
0.8
r
d1 =2, Ν0=3, t f =10 -4.0
0.6
H 0.2
-10.5
0.4
0.6
r
0.8
1.0
H
-11.0 0.0
0.2
0.4
0.6
0.8
1.0
r
Figure 6.4: Same as in figure 6.3 but for d1 = 2.
168
d1 = 1
ν0 = 3
tf = 1
tf = 10
tf = 100
(1.5 ∼ 65) × 10−9
(1.5 ∼ 15) × 10−9
(1.5 ∼ 2.8) × 10−9
(2.4 ∼ 195) × 10−11
(2.4 ∼ 7.1) × 10−11
ν0 = 5 (2.4 ∼ 640) × 10−11
d1 = 2 tf = 1
tf = 10
tf = 100
ν0 = 3
(1.4 ∼ 6.5) × 10−8
(1.4 ∼ 3.1) × 10−8
0
ν0 = 5
(2.3 ∼ 10) × 10−9
(2.3 ∼ 3.9) × 10−9
0
Table 6.1: The range of densities in units of α0−5 resulting to decompactification. is the natural scale of the system. Next, as argued in the introduction, and given the results of the previous chapter, the only fluctuations with considerable probability for decompactifications are the ones involving a total of d ≤ 3 growing dimensions. Our study has focused on this relevant space of fluctuations. Imagine then that an initial fluctuation occurs at time t = 0 and the next after a short time t ∼ 1. Our assumption is that dilaton gravity is in effect for t ≥ 0. The value of the dilaton at time t = 0 determines the subsequent possibilites: First, there is a relatively narrow range, at the stronger coupling regime, where, depending on d1 , the equilibrium phase at t = 0 favors Rd1 . In cost nearly no entropy.
169
d1 = 1 tf = 1 tf = 10 tf = 100 ν0 = 3
82
53
20
ν0 = 5
65
47
17
d1 = 2 tf = 1 tf = 10 tf = 100 ν0 = 3
70
26
1
ν0 = 5
43
13
0.1
Table 6.2: Number density of decompactification cases normalized to 100, when the universe exits to a radiation era. that case, other than perhaps invoking anthropic arguments, we have no other way to argue for an effectively higher-dimensional universe. If this is not the case, and for the most part, at t = 0 we are in H and the following possibilities remain. In the worst possible scenario, and what seems to occupy the smallest volume in phase-space in particular for smaller initial fluctuations in ν0 and d1 = 1, the system lands in the part of R3 where the winding modes wrapping the d2 smaller dimensions cannot annihilate. In that case, we again expect only d1 dimensions to grow large. But for what seems to be the largest possibility, for tf ∼ 1 fluctuations, the system ends up either in the part of R3 where a total of three dimensions decompactify, or it stays in the Hagedorn phase. We would like to argue that this latter large possibility can partially work
170 in favor of the mechanism in the sense that, roughly speaking, it brings us back to step one and if the fluctuations are frequently sampled, it is inconsequential modulo the rolling of the dilaton per fluctuation. The important thing to note here is that even if the d1 dimensions are unwound and experience a net-positive pressure, when the system is at equilibrium in H their growth over many string times is by far insignificant and the system persists in H where it may keep sampling fluctuations, without exiting to Rd1 . This is because the situation resembles much more a matter dominated regime with vanishing pressure (where the equations of motion literally leave the system trapped in H with a limiting value for the scalefactor and an insignificant overall change – see section 3.3) than a radiation dominated regime. The √ √ √ energy in radiation is ∼ E and in matter ∼ E − E E and changes little with a change in ν. This nearly constant and large energy is what keeps the velocity of the dilaton large, resulting in large “friction” and negligible change for ν. More precisely, observe that the equations of motion can be written as −
d −ϕ 1 (e ϕ) ˙ = 2E dt 4π √ d −ϕ 1 √ (e ν) ˙ = E dt 48π 2 π
(6.17)
The limiting case (when winding modes are present at equilibrium in H) has E constant and the right hand side of the second equation zero. In the numerical solutions we encountered, the most ν changed over tf = 100 was one part in a hundred and the energy remained constant by one part in thousands. The d1 dimensions then can “hover” in the Hagedorn phase for a
171 long time where more fluctuations can be sampled. One may argue, however, that if this is the case, subsequent fluctuations may also make ν larger. This is possible but less likely than a fluctuation of any other dimension, since we may reasonably assume that all dimensions are on equal footing. To conclude, we find that there is a considerable window for which anisotropic fluctuations favor three dimensions, in particular when we require that they √ are sampled over time scales α0 . A significant limitation comes from the possibity that a two-dimensional volume gets large enough such that equilibrium favors its further expansion. The requirement that fluctuations should eventully push the system out of the Hagedorn phase and into a threedimensional phase persists. To proceede further and solidify any conclusions, however, one needs deeper knowledge of the dynamics of the early Hagedorn era, that is, the nature of fluctuations for more general metric configurations and most importantly, the dynamics of the dilaton at strong coupling.
172
References [1] Joseph Polchinski. Evaluation of the One Loop String Path Integral. Commun. Math. Phys., 104:37, 1986. [2] K. S. Narain. New Heterotic String Theories in Uncompactified Dimensions ¡ 10. Phys. Lett., B169:41, 1986. [3] N. Seiberg and Edward Witten. Spin Structures in String Theory. Nucl. Phys., B276:272, 1986. [4] Michael McGuigan. Finite Temperature String Theory and Twisted Tori. Phys. Rev., D38:552, 1988. [5] Joseph J. Atick and Edward Witten. The Hagedorn Transition and the Number of Degrees of Freedom of String Theory. Nucl. Phys., B310:291– 334, 1988. [6] Nivedita Deo, Sanjay Jain, and Chung-I Tan. Strings at high energy densities and complex temperature. Phys. Lett., B220:125, 1989. [7] Nivedita Deo, Sanjay Jain, and Chung-I Tan. The ideal gas of strings.
173 In Bombay quantum field theory 1990, pp. 112–148. Based on talk given at Int. Colloq. on Modern Quantum Field Theory, Bombay, India, Jan 8-14, 1990. [8] Bruce A. Bassett, Monica Borunda, Marco Serone, and Shinji Tsujikawa. Aspects of string-gas cosmology at finite temperature. Phys. Rev., D67:123506, 2003. [9] Nivedita Deo, Sanjay Jain, and Chung-I Tan. String statistical mechanics above Hagedorn energy density. Phys. Rev., D40:2626, 1989. [10] Nivedita Deo, Sanjay Jain, Onuttom Narayan, and Chung-I Tan. The effect of topology on the thermodynamic limit for a string gas. Phys. Rev., D45:3641–3650, 1992. [11] Thorsten Battefeld and Scott Watson. String gas cosmology. Rev. Mod. Phys., 78:435–454, 2006. [12] Arvind Borde and Alexander Vilenkin. Eternal inflation and the initial singularity. Phys. Rev. Lett., 72:3305–3309, 1994. [13] Arvind Borde, Alan H. Guth, and Alexander Vilenkin.
Inflation-
ary space-times are incomplete in past directions. Phys. Rev. Lett., 90:151301, 2003. [14] Robert H. Brandenberger and C. Vafa. Superstrings in the Early Universe. Nucl. Phys., B316:391, 1989.
174 [15] Andreas Karch and Lisa Randall. Relaxing to three dimensions. Phys. Rev. Lett., 95:161601, 2005. [16] William Nelson and Mairi Sakellariadou. Space-time dimensionality from brane collisions. Phys. Lett., B674:210–212, 2009. [17] J. Polchinski. String theory. Vol. 1: An introduction to the bosonic string. Cambridge, UK: Univ. Pr. (1998) 402 p. [18] E. S. Fradkin and Arkady A. Tseytlin. Effective Field Theory from Quantized Strings. Phys. Lett., B158:316, 1985. [19] Mark J. Bowick and L. C. R. Wijewardhana. Superstrings at high temperature. Phys. Rev. Lett., 54:2485, 1985. [20] Minos Axenides, Stephen D. Ellis, and Costas Kounnas.
Universal
behavior of D-dimensional superstring models. Phys. Rev., D37:2964, 1988. [21] S. W. Hawking. Black holes and thermodynamics. Phys. Rev., D13:191– 197, 1976. [22] S. W. Hawking and R. Penrose. The singularities of gravitational collapse and cosmology. Proc. Roy. Soc. Lond., A314:529–548, 1970. [23] M. Novello and S. E. Perez Bergliaffa. Bouncing cosmologies. 2008. [24] Micha Berkooz and Dori Reichmann. A short review of time dependent solutions and space-like singularities in string theory. Nucl. Phys. Proc.
175 Suppl., 171:69–87, 2007. [25] Justin Khoury, Burt A. Ovrut, Paul J. Steinhardt, and Neil Turok. The ekpyrotic universe: Colliding branes and the origin of the hot big bang. Phys. Rev., D64:123522, 2001. [26] M. Gasperini and G. Veneziano. The pre-big bang scenario in string cosmology. Phys. Rept., 373:1–212, 2003. [27] J. C. Fabris, R. G. Furtado, N. Pinto-Neto, and Patrick Peter. Regular cosmological solutions in low energy effective action from string theories. Phys. Rev., D67:124003, 2003. [28] Tirthabir Biswas, Anupam Mazumdar, and Warren Siegel. Bouncing universes in string-inspired gravity. JCAP, 0603:009, 2006. [29] Tirthabir Biswas, Robert Brandenberger, Anupam Mazumdar, and Warren Siegel. Non-perturbative gravity, Hagedorn bounce and CMB. JCAP, 0712:011, 2007. [30] Ram Brustein and G. Veneziano. The graceful exit problem in string cosmology. Phys. Lett., B329:429–434, 1994. [31] Robert H. Brandenberger, Richard Easther, and J. Maia. Nonsingular dilaton cosmology. JHEP, 08:007, 1998. [32] Robert Brandenberger, Damien A. Easson, and Dagny Kimberly. Loitering phase in brane gas cosmology. Nucl. Phys., B623:421–436, 2002.
176 [33] Ram Brustein and Richard Madden. A model of graceful exit in string cosmology. Phys. Rev., D57:712–724, 1998. [34] Ram Brustein and Richard Madden. Graceful exit and energy conditions in string cosmology. Phys. Lett., B410:110–118, 1997. [35] Monica Borunda and Lotfi Boubekeur. The effect of alpha’ corrections in string gas cosmology. JCAP, 0610:002, 2006. [36] Viatcheslav F. Mukhanov and Robert H. Brandenberger. A nonsingular universe. Phys. Rev. Lett., 68:1969–1972, 1992. [37] R. G. Leigh. Dirac-Born-Infeld action from Dirichlet sigma model. Mod. Phys. Lett., A4:2767, 1989. [38] Arkady A. Tseytlin. Ambiguity in the Effective Action in String Theories. Phys. Lett., B176:92, 1986. [39] R. R. Metsaev and Arkady A. Tseytlin. Order alpha-prime (Two Loop) Equivalence of the String Equations of Motion and the Sigma Model Weyl Invariance Conditions: Dependence on the Dilaton and the Antisymmetric Tensor. Nucl. Phys., B293:385, 1987. [40] M. Gasperini, M. Maggiore, and G. Veneziano. Towards a non-singular pre-big bang cosmology. Nucl. Phys., B494:315–330, 1997. [41] Richard Easther, Brian R. Greene, Mark G. Jackson, and Daniel N. Kabat. String windings in the early universe. JCAP, 0502:009, 2005.
177 [42] Joseph Polchinski. Collision of macroscopic fundamental strings. Phys. Lett., B209:252, 1988. [43] T. Clifton and John D. Barrow. The ups and downs of cyclic universes. Phys. Rev., D75:043515, 2007. [44] Tirthabir Biswas. The Hagedorn soup and an emergent cyclic universe. 2008. [45] Rebecca Danos, Andrew R. Frey, and Anupam Mazumdar. Interaction rates in string gas cosmology. Phys. Rev., D70:106010, 2004. [46] Takeshi Chiba. Generalized gravity and ghost. JCAP, 0503:008, 2005. [47] Mark G. Jackson, Nicholas T. Jones, and Joseph Polchinski. Collisions of cosmic F- and D-strings. JHEP, 10:013, 2005. [48] Ali Nayeri, Robert H. Brandenberger, and Cumrun Vafa. Producing a scale-invariant spectrum of perturbations in a Hagedorn phase of string cosmology. Phys. Rev. Lett., 97:021302, 2006. [49] Robert H. Brandenberger, Ali Nayeri, Subodh P. Patil, and Cumrun Vafa. String gas cosmology and structure formation. Int. J. Mod. Phys., A22:3621–3642, 2007. [50] Nemanja Kaloper, Lev Kofman, Andrei D. Linde, and Viatcheslav Mukhanov. On the new string theory inspired mechanism of generation of cosmological perturbations. JCAP, 0610:006, 2006.
178 [51] Nima Lashkari and Robert H. Brandenberger. Speed of sound in string gas cosmology. 2008. [52] Jerome Martin and Patrick Peter. On the ’causality argument’ in bouncing cosmologies. Phys. Rev. Lett., 92:061301, 2004. [53] Mairi Sakellariadou. Numerical Experiments in String Cosmology. Nucl. Phys., B468:319–335, 1996. [54] S. Alexander, Robert H. Brandenberger, and D. Easson. Brane gases in the early universe. Phys. Rev., D62:103509, 2000. [55] Richard Easther, Brian R. Greene, Mark G. Jackson, and Daniel N. Kabat. Brane gas cosmology in M-theory: Late time behavior. Phys. Rev., D67:123501, 2003. [56] Antonio Campos. Dynamical decompactification from brane gases in eleven- dimensional supergravity. JCAP, 0501:010, 2005. [57] Richard Easther, Brian R. Greene, and Mark G. Jackson. Cosmological string gas on orbifolds. Phys. Rev., D66:023502, 2002. [58] Richard Easther, Brian R. Greene, Mark G. Jackson, and Daniel N. Kabat. Brane gases in the early universe: Thermodynamics and cosmology. JCAP, 0401:006, 2004. [59] Rebecca Danos, Andrew R. Frey, and Anupam Mazumdar. Interaction rates in string gas cosmology. Phys. Rev., D70:106010, 2004.
179 [60] Mark G. Jackson, Nicholas T. Jones, and Joseph Polchinski. Collisions of cosmic F- and D-strings. JHEP, 10:013, 2005. [61] Joseph Polchinski. Collision of macroscopic fundamental strings. Phys. Lett., B209:252, 1988. [62] D. Amati, M. Ciafaloni, and G. Veneziano. Classical and Quantum Gravity Effects from Planckian Energy Superstring Collisions. Int. J. Mod. Phys., A3:1615–1661, 1988. [63] T. Adachi and T. Kotani. An impact parameter representation of the scattering problem. Prog. Theor. Phys., 39:430–452, 1968. [64] W. N. Cottingham and Ronald F. Peierls. Impact-parameter expansion of high-energy elastic-scattering amplitudes. Phys. Rev., 137(1B):B147– B154, Jan 1965. [65] Leonard Susskind. Strings, black holes and Lorentz contraction. Phys. Rev., D49:6606–6611, 1994. [66] N. Turok. String Statistical Mechanics. Physica, A158:516–535, 1989. [67] Michael Edward Peskin and Daniel V. Schroeder. An Introduction to quantum field theory. 1995. Reading, USA: Addison-Wesley (1995) 842 p. [68] J.R. Taylor and J. Robert. Scattering Theory: The Quantum Theory on Nonrelativistic Collisions. Wiley New York;, 1972.
180 [69] Scott Watson and Robert H. Brandenberger. Isotropization in brane gas cosmology. Phys. Rev. D, 67(4):043510, Feb 2003. [70] Carmen Molina-Paris and Matt Visser. Minimal conditions for the creation of a Friedman-Robertson-Walker universe from a ‘bounce’. Phys. Lett., B455:90–95, 1999. [71] Nemanja Kaloper and Scott Watson. Geometric precipices in string cosmology. Phys. Rev., D77:066002, 2008.
181
Appendix A Covariant Action for Isotropic Cyclic Models In this appendix, we construct the manifestly covariant action for the homogeneous and isotropic models of chapter 4. This is done along the lines of [31, 36] where a “limiting curvature hypothesis” (LCH) is implemented. In short, the LCH claims that at high spacetime curvatures, a fundamental length scale should bound all curvature invariants. This insight is drawn from other physical situations, such as the theory of a point particle where the speed of light bounds the particle’s velocity, or quantum mechanics, where Planck’s constant bounds the size of elements in phase space. In practice, to see how this principle can be incorporated in the dynamics, consider the case of a point particle. The action of a non-relativistic point particle (of unit mass) is Z SN R =
1 dt x˙ 2 2
(A.1)
This can be promoted to the relativistic case by coupling the particle’s velocity to an auxiliary field, or Laplace multiplier ψ, subject to a suitable
182 potential: Z SN R →
1 dt x˙ 2 − ψ x˙ 2 − V (ψ) 2
(A.2)
With the choice V (ψ) =
2ψ 2 2ψ + 1
(A.3)
if the auxiliary field is eliminated at the level of the action we end up with the familiar action for the relativistic particle, Z SR =
√ dt 1 − x˙ 2
(A.4)
Alternatively, and yielding equivalent dynamics, the equations of motion for x and ψ can be considered simultaneously, in which case the ψ equation is treated as a constraint on the trajectory of the particle. We can apply this prescription to the case of gravity. Consider a function of curvature invariants I(R, R2 ...) that we wish to bound. This could be, for example, the Ricci scalar I = R or any other covariant function of the metric. Then we may extend the Einstein-Hilbert action 1 2κ2
Z R
(A.5)
R + ψI − V (ψ)
(A.6)
to the action 1 2κ2
Z
Since the equation of motion for the auxiliary field reads I = V 0 (ψ), the curvature invariant I is guaranteed to be bounded as long as the potential is chosen to satisfy V 0 (ψ) < ∞ for all values of ψ. This prescription can
183 easily be extended to several invariants Ii coupled (if necessary) to different auxiliary fields ψi . In the case of dilaton-gravity we want to modify the action (note that we are using the original, unshifted dilaton) 1 S(gµν , φ) = 2 2κ
Z
√
−ge−2φ R + 4(∇φ)2
(A.7)
Consider the invariants 2 I1 = R − (R2 − c1 RGB )1/2
with c1 =
d(d+1) (d−1)(d−2)
and c2 =
I2 = (∇φ)2 + (c2 Gµν ∇µ φ∇ν φ)1/2
2d . d−1
(A.8)
2 2 = is the Gauss-Bonnet invariant RGB RGB
R2 − 4Rµν Rµν + Rµνσρ Rµνσρ and Gµν = Rµν − 12 Rgµν is the Einstein tensor. By introducing two auxiliary fields ψ1 and ψ2 we write down the action Z √ 1 S(gµν , φ, ψi ) = 2 −ge−2φ R + 4(∇φ)2 2κ (A.9) + c3 (ψ1 − dψ2 )I1 + 4ψ2 I2 + V (ψ1 ) − V (ψ2 ) where c3 =
1 d+1
and V (ψ) =
ψ2 ψ+1
(A.10)
In an isotropic and homogeneous ansatz, ds2 = −dt2 + eλ(t)
X
dx2i ,
φ = φ(t)
(A.11)
i
this action yields the dynamics in chapter 4 encoded in equations (4.6) and (4.8) once ψ1 , ψ2 are eliminated. This is easier to see if the metric and dilaton ansatz is directly substituted in the action, with the invariants giving I1 = d(d + 1)λ˙ 2 and I2 = λ˙ φ˙ − φ˙ 2 . Note that this holds for any d > 2.
184 The form (A.9) shows how our model can be extended to a more general class of models, namely by modifying the potentials for the auxiliary fields. As mentioned above, it is necessary that V 0 (ψ) is bounded. For example, this can be achieved as in our case (A.10), where V 0 (ψ) → 1 as ψ → ±∞ and V 0 (ψ) → 0 as ψ → 0.
185
Appendix B Energy Conditions in Dilaton Gravity Within Einstein gravity a bouncing cosmology requires ρ + p < 0, a violation of the null energy condition [23, 70]. Here we make the analogous statements for dilaton gravity. A more detailed discussion can be found in [71]. With conventional (two-derivative) dilaton gravity the Friedmann equation and the equations of motion are 1 2 1 ˙2 1 ϕ˙ = dλ + 2 eϕ E 2 2 8π 1 2 1 ˙2 1 ϕ¨ = ϕ˙ + dλ + 2 eϕ Pϕ 2 2 8π 1 ¨ = ϕ˙ λ˙ + eϕ Pλ λ 8π 2
(B.1) (B.2) (B.3)
To study a bounce in the scale factor we set λ˙ = 0. Then (B.1) requires E ≥ 0. Equation (B.2) gives no constraint, while (B.3) implies that Pλ has ¨ A string gas can exert pressure of either sign, so it is the same sign as λ. easy to obtain a bouncing or cyclic scale factor in dilaton gravity coupled to a string gas. To obtain a bounce in the dilaton is more difficult. Setting ϕ˙ = 0 note that (B.1) requires E ≤ 0. Indeed we had to introduce negative potentials in section 4.4 to make the dilaton bounce. Equation (B.3) gives no constraint,
186 while (B.2) can be rewritten as ϕ¨ = −
1 ϕ e (E − Pϕ ) . 8π 2
(B.4)
The sign of ϕ¨ is correlated with the sign of E − Pϕ . In particular ϕ¨ > 0 requires E − Pϕ < 0, the dilaton gravity analog of violating the null energy condition.4 One can likewise study the conditions for a bounce in the Einstein-frame scale factor λE = −(ϕ + λ)/(d − 1). When λE bounces we have ϕ˙ = −λ˙ and (B.1) requires E < 0. Adding (B.2) and (B.3) gives ¨ E = d − 1 eϕ (E − Pϕ − Pλ ) . λ 8π 2 ¨ E is correlated with the sign of E − Pϕ − Pλ . Thus the sign of λ The use of our higher-derivative modified action for dilaton gravity does not significantly change these string-frame results. In fact the only change is that (B.4) is replaced with γλ ϕ¨ = −
1 ϕ e (E − Pϕ γλ ) 8π 2
so the sign of ϕ¨ at a bounce is correlated with the sign of E − Pϕ γλ .
4
Due to the wrong-sign kinetic term for the dilaton we inserted a minus sign in our definition of Pϕ below (5.9).
187
Appendix C Root Mean Square Velocity of Winding Modes To calculate the root mean square thermal velocity of winding modes, we will use the generalized form of the distribution in equation (2.33) as calculated in [10]. The average number of strings with winding charge vector w, KaluzaKlein charge vector k and energy on a D-torus (we are using d for the number of large dimensions) with total energy E is given by D(, w, k, E) =
N d −uqT A−1 q/4 u e
(C.1)
where u=
E (E − )
q = (w, k) √ (2 π)−2D N= √ detA 1 4π2 Ri2 δij A= 0
0 Ri2 δ 4π 2 ij
One can consider a unit winding mode, w1 = (1, 0, ...) along one of the d large dimensions, (Ri = R, i = 1, ..., d and Rj = 1, j = d + 1, ..., D) and
188 calculate the mean momentum squared of the string. RE R D d d k k 2 D(, w = w1 , k, E) hk 2 i = 0R E R d dD k D(, w = w1 , k, E) 0 RE 2 dR2 0 d uD/2−1 e−uπR = R 2π 2 E d uD/2 e−uπR2 0
(C.2)
Given the total energy E and the radius R the two integrals above can be evaluated with saddle point methods or numerically. For a heavy winding mode (large R) we have hv 2 i ' p hv 2 i.
hk2 i R4
and the RMS velocity is given by v¯ =
