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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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2 Aurlien Barrau LPSC-Grenoble (CNRS / UJF)
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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8 Aurlien Barrau LPSC-Grenoble (CNRS / UJF)
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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10 Aurlien Barrau LPSC-Grenoble (CNRS / UJF)
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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11 Aurlien Barrau LPSC-Grenoble (CNRS / UJF)
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