Loop Quantum Cosmology and Phenomenology

Loop Quantum Cosmology and Phenomenology Martin Bojowald The Pennsylvania State University Institute for Gravitation and the Cosmos University Park, PA

Loop Quantum Cosmology and Phenomenology – p.1

Microscopic degrees of freedom What are the quantum degrees of freedom of space-time? In the absence of direct observations, further input needed. Example from particle physics: renormalizability leads to electroweak theory from 4-fermion interaction of β -decay. n

n

ν

ν

W p

e

p

e

Quantized field: W-boson and its interactions. Loop Quantum Cosmology and Phenomenology – p.2

Principles for quantum gravity Renormalizability −→ string theory (roughly speaking). Quantization of gravitational (and other) excitations on space-time. Strong gravitational fields of big bang and black holes: interaction of matter with space-time structure, must consider quantum nature of full space-time. Alternative principle: background independence. Quantize full metric gµν , not just perturbations hµν on (e.g. Minkowski) background where gµν = ηµν + hµν . Realized in loop quantum gravity. Loop Quantum Cosmology and Phenomenology – p.3

Background independence Quantum field theory on background space-time: Operators ak and a†k describe annihilation and creation of particles of momentum k. n

ν

a†k

Using introduces a new particle, and increases the total energy. W

Products of operators for interactions.

e

p

Problem: Particles can only be created on a given space-time, whose metric is used in the definition of ak and a†k . Solution: Define operators for space-time itself. Increase distances, areas and volumes, not energy.


Holonomy-flux representation [Ashtekar, Rovelli, Smolin]

Loop quantum gravity: Holonomies hI = P exp(∫eI dte˙ aI Aia τi ) for Ashtekar connection Aia = Γia + γKai , spatial curves eI . (Spatial spin connection Γia , extrinsic curvature Kai , γ > 0.) Define basic state |0i by hAia |0i = 1: independent of connection. Excited states, simplified U(1)-example where R hI = exp(i eI dte˙ aI Aa ):

ˆ n1 · · · h ˆ nk |0i |e1 , n1 ; . . . ; ek , nk i = h 1 k

General state labeled by graph g and integers ne as quantum numbers on edges ψg,n (Aa ) =

Y e∈g

ne

he (Aa )

=

Y e∈g

exp(ine ∫ dte˙ a Aa ) e


Excitations of geometry Holonomies hI , create spatial geometry. Excitations in two ways: (i) use operators for the same loop

or (ii) use different loops.

Strong excitation necessary for macroscopic geometry: “many particles.” Loop Quantum Cosmology and Phenomenology – p.6

Discrete Geometry Canonical framework: Ashtekar connection has momentum Eia such that Eia Eib = (det q)q ab gives the spatial metric qab . Flux S d2 yEia Na quantized as derivative operator, measures excitation level (again U(1)-simplification): Z Z X δψ γG~ g,n 2 2 2 ˆ a ψg,n = = γℓP ne Int(S, e)ψg,n d yNa d yNa E i δAa (y) S S e∈g R

with intersection number Int(S, e). (Na co-normal to surface S .) Discrete geometry: for gravity, flux represents spatial metric (area, volume). Graphs represent atomic nature of space; geometry from intersections. Loop Quantum Cosmology and Phenomenology – p.7

Dynamics [Thiemann]

Cgrav [N ] =

1 16πγG

Z

Σ

a b 3 i √Ej Ek d xN ǫijk Fab

| det E| [a b] Ei Ej

− 2(1 + γ −2 )(Aia − Γia )(Ajb − Γjb ) √

| det E|


Dynamics [Thiemann]

Cgrav [N ] =

1 16πγG

Z

Σ



− 2(1 + γ −2 )(Aia − Γia )(Ajb − Γjb ) √

| det E|

Requires inverse determinant, from relation Z p b c E j Ek i 3 ijk Aa , | det E|d x = 2πγGǫ ǫabc p | det E|


Dynamics [Thiemann]

Cgrav [N ] =

1 16πγG

Z

Σ



− 2(1 + γ −2 )(Aia − Γia )(Ajb − Γjb ) √

| det E|


For curvature:

i τ sa1 sb2 Fab i

=

∆−1 (h

λ

− 1) + O(∆)

~s2 ∆ ~s1

λ


Dynamics [Thiemann]

Cgrav [N ] =

1 16πγG

Z

Σ



− 2(1 + γ −2 )(Aia − Γia )(Ajb − Γjb ) √

| det E|


For curvature:

i τ sa1 sb2 Fab i

Extrinsic curvature: (

=

∆−1 (h

(Z

Kai = γ −1 (Aia −Γia ) ∝ Aia ,

λ

− 1) + O(∆)

ijk E a E b ǫ j k i p d3 xFab

| det E|

~s2 ∆ ~s1

λ

)) Z p , | det E|d3 x Loop Quantum Cosmology and Phenomenology – p.8

Quantum corrections −→ Inverse volume corrections from quantizing

Aia ,

Z p

| det E|d3 x

bEc E j k = 2πγGǫijk ǫabc p | det E|

2.5

(with quantization ambiguities, e.g. r)

2

1.5

α(r) 1

−→ Higher order corrections from use of holonomies

r=1/2 r=3/4 r=1 r=3/2 r=2

0.5

0 0

−→ Quantum back-reaction

0.5

1

1.5

2

µ flux eigenvalues Loop Quantum Cosmology and Phenomenology – p.9

Quantum Hamiltonian Cˆgrav [N ] =

X

−1 ˆ −1 N (v)ǫIJK tr(hv,I hv+I,J h−1 h [h h v+J,I v,J v,K v,K , V ])

v,IJK

as (simplified) many-body Hamiltonian: excitations of geometry take place dynamically. Depends on geometry through volume operator Vˆ . Modified gravity: Corrections to general relativity. Main questions: → Observable? (Or already ruled out?) → Consistent with general covariance/Lorentz symmetry? → Singularity problem of general relativity resolved? Loop Quantum Cosmology and Phenomenology – p.10

Spectroscopy of geometry Several results known in model systems (symmetry reduction and perturbative inhomogeneities).

Symmetry: Level splitting, e.g. in volume spectrum.

4

V

2

Most highly symmetric systems easiest to analyze, also concerning dynamics: quantum cosmology. 0

1 h Loop Quantum Cosmology and Phenomenology – p.11

Symmetry reduction Hamiltonian isotropic cosmology in Ashtekar variables (spatially flat): Aia = cδai with c = γ a˙ , Eia = pδia with |p| = a2 , canonically conjugate: {c, p} = 8πγG/3. Hamiltonian constraint c2

p |p| 8πG C := − + Hmatter = 0 2 γ 3

equivalent to Friedmann equation. Generates Raychaudhuri equation from p˙ = {p, C}, c˙ = {c, C}. ˆ Loop quantum cosmology: C|ψi = 0, but with exp(iδc) for c: 2

p sin (δc) |p| 8πG − + Hmatter = 0 2 2 γ δ 3 Loop Quantum Cosmology and Phenomenology – p.12

Loop quantum cosmology Difference equation for wave function of the universe: ˆ φ (µ)ψµ (φ) C+ (µ)ψµ+δ (φ) − C0 (µ)ψµ (φ) + C− (µ)ψµ−δ (φ) = H ˆ φ ; “internal time” µ. depending on matter energy H

Non-singular: wave function evolves uniquely across classical singularity (µ = 0).

Physical explanation: limited storage for energy in discrete space-time −→ repulsive force. Geometrical picture: bounce, but in general not available in strong quantum regime. Use effective equations.


Harmonic oscillator Simple system in quantum mechanics: harmonic oscillator, ˆ = i~ pˆ , [ˆ ˆ = −i~mω 2 qˆ [ˆ q , pˆ] = i~ , [ˆ q , H] p, H] m spreading wave packets do not disturb mean position. Closed sets of equations for expectation values of qˆ and pˆ: ˆ ˆ H]i/i~ ˆ dhOi/dt = h[O, , fluctuations/correlations, . . . unsqueezed state:

squeezed state:


Harmonic cosmology Similarly solvable system exists in loop quantum cosmology. (Conditions: isotropic, flat space; free, massless scalar φ) [Earlier numerical studies by Ashtekar, Pawlowski, Singh]

−→ Deparameterization: Use φ as internal time.

Hamiltonian H = pφ (V, P ) obtained from Friedmann equation (Hamiltonian constraint p2φ − V Cgrav = 0). −→ Momentum P of V : Hubble parameter, J = V exp(iδP ). Linear Hamiltonian H ∝ ImJ for loop quantum cosmology.

ˆ −→ sl(2, R) algebra of operators Vˆ (volume), Jˆ (“holonomy”), H (Hamiltonian for evolution with respect to φ): ˆ = iδ~H ˆ [Vˆ , J]

,

ˆ = −iδ~Jˆ , [Vˆ , H]

ˆ H] ˆ = iδ~Vˆ [J,


Harmonic cosmology Similarly solvable system exists in loop quantum cosmology. (Conditions: isotropic, flat space; free, massless scalar φ) [Earlier numerical studies by Ashtekar, Pawlowski, Singh]

−→ Deparameterization: Use φ as internal time.

Hamiltonian H = pφ (V, P ) obtained from Friedmann equation (Hamiltonian constraint p2φ − V Cgrav = 0). −→ Momentum P of V : Hubble parameter, J = V exp(iδP ). Linear Hamiltonian H ∝ ImJ for loop quantum cosmology.

ˆ −→ sl(2, R) algebra of operators Vˆ (volume), Jˆ (“holonomy”), H (Hamiltonian for evolution with respect to φ): ˆ = iδ~H ˆ [Vˆ , J]

,

ˆ = −iδ~Jˆ , [Vˆ , H]

ˆ H] ˆ = iδ~Vˆ [J,


Quantum back-reaction Generic models more complicated, e.g. with cosmological constant (for now ignoring quantum geometry corrections): p pφ ∝ V P 2 − Λ =: H(V, P ) Equations of motion for physical observables ˆ H]i ˆ ˆ h[O, dhOi = dφ i~ couple to fluctuations, etc. dhVˆ i dφ

dhPˆ i dφ

=

3 hVˆ ihPˆ i hVˆ ihPˆ i 9 q (∆P )2 + Λ 2 4 (hPˆ i2 − Λ)5/2 2 ˆ hP i − Λ

CV P 3 + ··· − Λ 3/2 2 ˆ 2 (hP i − Λ) q 2 (∆P ) 3 3 + ··· = − hPˆ i2 − Λ + Λ 2 3/2 ˆ 2 4 (hP i − Λ) Loop Quantum Cosmology and Phenomenology – p.16

Fluctuations d(∆P )2 dφ dCV P dφ d(∆V )2 dφ

hPˆ i

= −3 q (∆P )2 + · · · hPˆ i2 − Λ

3 hVˆ i (∆P )2 + · · · = − Λ 2 (hPˆ i2 − Λ)3/2 = −3Λ

hVˆ i

CV P

(hPˆ i2 − Λ)3/2 hPˆ i +3 q (∆V )2 + · · · hPˆ i2 − Λ

Canonical procedure for effective equations. Higher moments become important in some regimes. [Work in progress with Brizuela, Hernández2 , Mena-Marugán, Morales-Técotl] Loop Quantum Cosmology and Phenomenology – p.17

Quantum Friedmann equation With scalar mass/potential: 2 a˙ = a

ρQ 8πG ρ 1− 3 ρcrit r 2 ρQ (ρ − p) 2 1 η η(ρ − p) + 1− ± 2 2 ρcrit (ρ + p)

where p is pressure and η parameterizes quantum correlations, ρQ := ρ + ǫ0 ρcrit + (ρ − p)

∞ X k=0

ǫk+1 (ρ − p)k /(ρ + p)k

with fluctuation parameters ǫk ; ρcrit = 3/8πGδ 2 with scale δ . Simple bounce if ρ = p (e.g. free, massless scalar). Otherwise? Loop Quantum Cosmology and Phenomenology – p.18

Cosmology With matter interactions and inhomogeneities, back-reaction results: systematic perturbation theory around solvable model. Indirect effects of atomic space-time: small individual corrections even at high energies, must add up coherently. −→ cosmology, high energy density, long evolution −→ high energy particles from distant sources.


Inhomogeneity Contracted Bianchi identity ∇µ Gµν = 0 implies ∂0 G0µ = −∂a Gaµ − Γννκ Gκµ + Γκµν Gνκ

Right hand side at most of second order in time derivatives, so G0µ of first order; provides constraints C[N ] = D[N a ] =

Z

Z

d3 xN (x)(G00 − 8πGT00 ) = 0 d3 xN a (x)(G0a − 8πGTa0 ) = 0

Must be preserved under second order equations provided by Gab . Loop Quantum Cosmology and Phenomenology – p.20

Constraint algebra Conservation laws lead to symmetries. {D[N a ], D[M a ]} = D[LM a N a ] {C[N ], D[M a ]} = C[LM a N ]

{C[N ], C[M ]} = D[q ab (N ∂b M − M ∂b N )]

as algebra of gauge transformations. C[N ] + D[N a ] generates infinitesimal space-time diffeomorphism along ξ µ = (N, N a ). “It would be permissible to look upon the Hamiltonian form as the fundamental one, and there would then be no fundamental four-dimensional symmetry in the theory. . . . The usual requirement of four-dimensional symmetry in physical laws would then get replaced by the requirement that the functions have weakly vanishing P.b.’s . . . ” [P.A.M. Dirac, The theory of gravitation in Hamiltonian form, Proc. Roy. Soc. A 246 (1958) 333]


Anomaly-freedom “Effective” constraints including some of the corrections from quantum geometry can be made anomaly-free: {D[N a ], D[M a ]} = D[LM a N a ] {C(α) [N ], D[M a ]} = C(α) [LM a N ]

{C(α) [N ], C(α) [M ]} = D[α2 q ab (N ∂b M − M ∂b N )]

with correction α from inverse volume. Algebra closed, but deformed. No local symmetries broken. [with Hossain, Kagan, Shankaranarayanan]

In an effective action, these corrections cannot be purely of higher curvature type: would still produce the classical algebra. [e.g. Deruelle, Sasaki, Sendouda, Yamauchi: arXiv:0908.0679]

More general effective action. (Non-commutative space-time?) Loop Quantum Cosmology and Phenomenology – p.22

New types of quantum corrections Potential for quantum corrections larger than often expected, i.e. larger than ℓP H. Main physical mechanism: Non-conservation of power on large scales. Classically: conservation from Bianchi identity, but modified for corrected constraint algebra. ¨ + 3(1 + w + ǫ1 )HΨ ˙ − (w + ǫ2 )∇2 Ψ + ǫ3 H2 Ψ = 0 Ψ

Constant large-scale mode of Ψ disappears when ǫ3 6= 0. Small slope during long evolution times between Hubble exit and re-entrance (for scalars and tensors).


Cosmological perturbation equations [MB, Hossain, Kagan, Shankaranarayanan: PRD 2009]

¯ ˙ + H(1 + f )Φ = 4πG α ∂c Ψ ϕ∂ ¯˙ c δϕ ν¯ ˙ + HΦ(1 + f ) α2 Ψ) − 3H(1 + f ) Ψ ∆(¯ α ¯ 2 ¯ ¯˙ ϕ˙ − ϕ¯˙ (1 + f1 )Φ + ν¯a2 V,ϕ (ϕ)δϕ = 4πG (1 + f3 ) ϕδ ν¯ ¯ a dα ¨ ˙ ˙ + f) Ψ + H 2Ψ 1 − + Φ(1 2¯ α da a df a dα ¯ 2 ˙ Φ(1 + f ) + 2H + H 1 + − 2 da 2¯ α da α ¯ 2 = 4πG ϕδ ¯˙ ϕ˙ − a ν¯V,ϕ (ϕ)δϕ ¯ ν¯ as well as Φ = (1 + h)Ψ (metric perturbation) and Klein–Gordon equation for δϕ (scalar matter field). Loop Quantum Cosmology and Phenomenology – p.24

Primordial gravitational waves 1020 λ = 0.01

T

Tensorial power spectrum P (k)

[Grain, Barrau, Gorecki, arXiv:0902.3605]

α = 1 + λ(ρdisc ℓ3P )κ/3 κ=2

10

18

λ = 0.1 λ=1

1016

λ = 10

1014

GR (H = 0.1) 0

1012 1010 108 106 104 102 1 10−2 10−4

10−2

10−1

1

102 10 Modes wave number k


Primordial gravitational waves 1020 1019

κ=1

1017

κ=2

1015

κ=3

T

Tensorial power spectrum P (k)

[Grain, Barrau, Gorecki, arXiv:0902.3605]

α = 1 + λ(ρdisc ℓ3P )κ/3

10

GR (H = 0.1) 0

13

1011 109 107

λ=1

105 103 10 10−1 10−3 10−2

10−1

1

102 10 Modes wave number k


Gravitational waves Implications of consistency: Propagation of gravitational waves compared to light. Hamiltonian (for inverse volume correction) 1 HG = 16πG

Z

c d E E j k i k ǫi jk Fcd − 2(1 + γ 2 )K[cj Kd] d3 xα(Eia ) p |det E| Σ

implies linearized wave equation 2adα/da ˙ i a˙ 1 1 ¨i 1− ha − α∇2 hia = 8πGΠia ha + 2 2 α a α for tensor mode hia on cosmological background with scale factor a and source-term Πia . Dispersion relation for gravitational waves: ω 2 = α2 k 2 α > 1 from perturbative corrections: super-luminal?


Causality Compare with electrodynamics, Hamiltonian: Z √ q 2π a b 3 HEM = Fab Fcd q ac q bd d x αEM (qcd ) √ E E qab + βEM (qcd ) q 16π Σ wave equation −1 ∂t αEM ∂t Aa

dispersion relation

− βEM ∇2 Aa = 0

ω 2 = αEM βEM k 2

Also “super-luminal” compared to classical speed of light. Anomaly-freedom: α2 = αEM βEM

Physically, no super-luminal propagation. Loop Quantum Cosmology and Phenomenology – p.28

Outlook → Consistent deformations in model systems of canonical

quantum gravity: discreteness without spoiling covariance.

→ Tools for canonical effective equations.

Completion for quantum field theories to be carried out. New computational tools for numerical implementation.

→ Link to cosmological, astrophysical and particle

observations.

Space-time structure important in high-energy regimes, but not necessarily just at the Planck scale.
