Loop Quantum Cosmology and Phenomenology

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perturbations hµν on (e.g. Minkowski) background where gµν = ηµν + hµν. Realized in loop quantum gravity. 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.

Loop Quantum Cosmology and Phenomenology – p.4

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

Loop Quantum Cosmology and Phenomenology – p.5

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|



Loop Quantum Cosmology and Phenomenology – p.8

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|



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|

Loop Quantum Cosmology and Phenomenology – p.8

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|

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|

For curvature:

i τ sa1 sb2 Fab i

=

∆−1 (h

λ

− 1) + O(∆)

~s2  ∆ ~s1



λ

Loop Quantum Cosmology and Phenomenology – p.8

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|

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|

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.

Loop Quantum Cosmology and Phenomenology – p.13

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:

Loop Quantum Cosmology and Phenomenology – p.14

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,

Loop Quantum Cosmology and Phenomenology – p.15

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,

Loop Quantum Cosmology and Phenomenology – p.15

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.

Loop Quantum Cosmology and Phenomenology – p.19

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]

Loop Quantum Cosmology and Phenomenology – p.21

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).

Loop Quantum Cosmology and Phenomenology – p.23

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

Loop Quantum Cosmology and Phenomenology – p.25

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

Loop Quantum Cosmology and Phenomenology – p.26

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?

Loop Quantum Cosmology and Phenomenology – p.27

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.

Loop Quantum Cosmology and Phenomenology – p.29