Aurélien Barrau- Loop Quantum Cosmology and the CMB

8/3/2019 Aurlien Barrau- Loop Quantum Cosmology and the CMB

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1 Aurlien Barrau LPSC-Grenoble (CNRS / UJF)

Loop Quantum Cosmologyand the CMB

Aurlien Barrau

Laboratoire de Physique Subatomique et de CosmologyCNRS/IN2P3 University Joseph Fourier Grenoble, France


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Why going beyond GR ?

Dark energy (and matter) / quantum gravity

Observations : the acceleration of the Universe Theory : singularity theoremsSuccessful techniques of QED do not apply to gravity. A new paradigm must be

invented.

Which gedenkenexperiment ? (as is QM, SR and GR) Which paradoxes ?

Quantum black holes and the early universe are privileged places to

investigate such effects !

* Entropy of black holes

* End of the evaporation process, IR/UV connection

* the Big-Bang

Many possible approaches : strings, covariant approaches (effective theories,the renormalization group, path integrals), canonical approaches (quantum

geometrodynamics, loop quantum gravity), etc. See reviews par C. Kiefer


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The observed acceleration

SNLS, Astier et al.

WMAP, 5 ans

SDSS, Eisenstein et al. 2005


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Alam et al., MNRAS

344 (2003) 1057


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Lovelock (Gauss-Bonnet) : IR or UV ?

A. B. et al., Class. Quantum Grav. 19 (2002) 4444


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Toward the Planck era: LQG

Strings vs loops or. SU(3) X SU(2) X U(1) vs g!

Can we construct a quantum theory of spacetime

based only on the experimentally well confirmed

principles of general relativity and quantum

mechanics ? L. Smolin, hep-th/0408048

DIFFEOMORPHISM INVARIANCE

Loops (solutions to the WDW) = space

-Mathematically well defined-Singularities-Black holes


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How to build Loop Quantum Gravity ?

- Foliation space metric and conjugate momentum- Constraints (diffomorphism, hamiltonian + SO(3))- Quantification la Dirac WDWAshtekar variables- smearing holonomies and fluxes

See e.g. the book Quantum Gravity by C. Rovelli

LQC :

-IR limit-UV limit (bounce)

-inflation


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Experimental tests

- High energy gamma-ray (Amlino-Camelia et al.)

Not very conclusive however


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Experimental tests

- Discrete values for areas and volumes (Rovelli et al.)- Observationnal cosmology (, et al.)


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Toward LQC

Within the Wheeler, Misner and DeWitt QGD, the BB singularity is not resolved

could it be different in the specific quantum theory of Riemannian geometrycalled LQG?

KEY questions:

- How close to the BB does smooth space-time make sense ? Is inflation safe ?- Is the BB singularity solved as the hydrogen atom in electrodynamics (Heinsenberg)?- Is a new principle/boundary condition at the BB essential ?- Do quantum dynamical evolution remain deterministic through classical singularities ?- Is there an other side ?The Hamiltonian formulation generally serves as the royal road to quantum theory. But

absence of background metric constraints, no external time.- Can we extract, from the arguments of the wave function, one variable which can serve

as emergent time ?

- Can we cure small scales and remain compatible with large scale ? 14 Myr is a lot oftime ! How to produce a huge repulsive force @ 10^94 g/cm^3 and turn it off quickly.

Following Ashtekar


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FLRW and the WDW theory

k=0 and k=1 models: every classical solution has a singularity.

No prefered physical time variable relational time scalar field as a clockHomogeneity finite number of degrees of freedom. But elementary cell q0abWDW approach : hamiltonian constraint

The IR test is pased with flying colors.

But the singularity is not resolved.

Plots from Ashtekar


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LQC: preliminariesPessimistic view von Neumann theorem.

LQG : well established kinematical framework. LQC is INEQUIVALENT to WDW already

at the kinematical level.

1) Bojowald showed that the quantum Hamiltonian constraint of LQC does not breakdown at a=0. But K assumed to be periodic !

2) 0 scheme. Requirement of diffeomorphism covariance leads to a uniquerepresentation of the algebra of fundamental operators. If one mimics the procedure

in LQCK takes values on the real line.BUT. No clear Hilbert space nor well-defined Dirac observables (the

Hamiltonian constraint failed to be self ajoint : no group averaging)

3) _bar scheme. Pbs : a) the density at which the bounce occurs depends on the quantum

state (the more classical the smaller the density !). b) Large deviations from GR when

0. c) dependance on the fiducial cell. Solution : dont use q0ab but qab. fullsingularity resolution AND 0 scheme problems resolution.

4) - more general situations

- observationnal consequences


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LQC: a few resultsvon Neumann theorem ? OK in non-relativistic QM. Here, the holonomy operators fail to

be weakly continuous no operators corresponding to the connections! new QM

Dynamics studied:

- Numerically- With effective equations- With exact analytical results

- Trajectory defined by expectation values of the observable V is in good agreement withthe classical Friedmann dynamics for


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LQC: a few results

- The volume of the Universe take its minimum value at the bounce and scales as p()- The recollapse happens at Vmax which scales as p()^(3/2). GR is OK.- The states remain sharply peaked for a very large number of cycles. Determinisme is

kept even for an infinite number of cycles.

- The dynamics can be derived from effective Friedmann equations

- The LQC correction naturally comes with the correct sign. This is non-trivial.- Furthermore, one can show that the upper bound of the spectrum of the density

operator coincides with crit

The matter momentum and instantaneous volumes form a complete set of Dirac observables. The

density and 4D Ricci scalar are bounded. precise BB et BC singularity resolution. No finetuning of initial conditions, nor a boundary condition at the singularity, postulated from outside.No violation of energy conditions (What about Penrose-Hawking th ? LHS modified !).Quantum corrections to the matter hamiltonian plays no role. Once the singularity is resolved, a

new world opens.

Role of the high symmetry assumed ? (string entropy ?)


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-Inflation-success (paradoxes solved, perturbations, etc.)-difficulties (no fundamental theory, initial conditions, etc.)

-LQG-success (background-independant quantization of GR)-difficulties (LQC LQG)Main results of LQC in this framework

(2 kinds of quantum corrections)

-Inverse volume-excitation of the field up its self interaction potential-superinflationary phase

-Holonomies-negative 2 term (bounce)-superinflationary phase

Bojowald, Hossain, Copeland, Mulryne, Numes, Shaeri, Tsujikawa,

Singh, Maartens, Vandersloot, Lidsey, Tavakol, Mielczarek .

LQC & inflation


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Our hypothesis

standard inflation

-decouples the effects-happens after superinflation

1) Holonomy correctionsBojowald & Hossain, Phys. Rev. D 77, 023508 (2008)


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Holonomy corrections, basic picturedS background

Which translates, in a cosmological framework, in:

Redifining the field:

Which should be compared (pure general relativity) to:

A.B., Grain, Phys. Rev. Lett. , 102, 081321, 2009


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With

nt ~ (H/Mpl)^2 in de Sitter inflation blue spectrum easilly tounderstand (amplification occurs later but the difference incritical time decreases with increasing wavenumber)

Power law inflation (and slow-roll : beta+1

-1-epsilon)

one can consider two specific cases.

1 n=-1/2 (Ashtekar, Pawlowski, Singh scheme)

Analytical solution

P = cte * k^3 in the IR region


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Grain & Barrau, Phys. Rev. Lett. 102,081301 (2009)


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2 for n>-1/2

Confusing situation :

- the modes are not amplified anymore in the limit k infinity- the modes start to re-oscillate at the end inflation


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J. Grain, A.B., A. Gorecki, Phys. Rev. D , 79, 084015, 2009

Inverse volume


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IV

Holonomy


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Case (1+)=2 with S(q)=1+*q^(-/2)


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Excellent numerical / analytical agreement


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Holonomies + inverse volume

J. Grain, T. Cailleteau, A.B., A. Gorecki, Phys. Rev. D. , 2009


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Holonomies + inverse volume

Holonomies dominate the background and inverse-volume dominate the modes


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Background corrections

J. Mielczarek, arXiv0908.4329v1

Pure holonomy corrections

- conditions for inflation are naturally met- Bounce- Analytical study (Bogoliubov transformations)- Numerical study (full system of coupled differential equations)

AB, Cailleteau, Grain, J. Mielczarek, in preparation


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Avec perturbations du fond

Contraction, bounce, superinflation, inflation, RD


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To do

-Take fully into account corrections for both the backgroundand the modes

-Clarify the status of inverse-volume corrections-Compute holonomy corrections for SCALAR modes-Compare with alternative theories

Aurélien Barrau- Loop Quantum Cosmology and the CMB

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