Session D10: Quantum Gravity and Cosmology

The loop quantum gravity black hole

We study the quantization of vacuum spherically symmetric space-times. We use variables adapted to spherical symmetry but do not fix the gauge further. One is left with a diffeomorphism constraint and a Hamiltonian constraint. Rescaling the latter turns the constraint algebra into a true Lie algebra and allows to implement the Dirac quantization procedure. We find exactly the physical states annihilated by all constraints using loop quantum gravity techniques. The space-time metric can be recovered as an evolving constant of the motion in terms of Dirac observables. The singularity is resolved as was anticipated in previous semiclassical studies. The quantum theory has new observables with respect to the classical theory that may play a role in discussions of ``firewalls'' during black hole evaporation.   

Black Hole Entropy from complex Ashtekar variables

In loop quantum gravity, black holes can be described in terms of an SU(2) Chern-Simons theory on a punctured 2-sphere. The level $k$ of the Chern-Simons theory depends on both the Barbero-Immirzi parameter $\gamma$ and the horizon area $a_H$. In this framework, the number of microstates of the black hole is a function which is expressed in terms of the dimension of the SU(2) Chern-Simons theory Hilbert space. We propose an analytic continuation of this number of microstates to a purely imaginary value of $\gamma$, and we give an interpretation based on the analytic continuation of SU(2) Chern-Simons theory to a complex gauge group. We show that the number of microstates behaves as exp$(a_H/(4lp^2))$ for large area $a_H$ if $\gamma=\pm i$, and finally discuss the relation between this striking result and quantum gravity in terms of the original complex Ashtekar variables.   

Quantum gravitational inflationary scenario in Bianchi-I spacetime

We investigate the inflationary scenario in Bianchi-I anisotropic spacetime in the effective description of loop quantum cosmology (LQC). We discuss important differences between the inflationary evolution of LQC and classical theory. We also explore the effects of the anisotropic shear and initial conditions on the amount of inflation. We find that, unlike in the classical theory, the amount of inflation in LQC inflationary Bianchi-I spacetime does not vary monotonically with increasing shear. The attractor behavior of the dynamical trajectories of Bianchi-I LQC spacetime is also studied. In deep Planck regime, LQC trajectories show distinctly different attraction behavior than classical trajectories.   

Chimera: A hybrid numerical approach for isotropic loop quantum cosmology

Loop quantum cosmology (LQC) is one approach to the resolution of the problem of singularities in classical cosmologies. The evolution of a cosmological model in LQC is governed by a set difference equations. In the isotropic cosmology (1+1 dimensions) the discretization is uniform in the spatial dimension. The stable simulation of a widely spread semi-classical state requires a very large computational domain and would therefore be computationally very expensive. In this talk we present an efficient hybrid numerical scheme based on the fact that the difference equations can be approximated by a set of partial differential equations (PDE's) in the limit of large spatial volume. We therefore introduce a hybrid scheme where we solve the LQC difference equations in the small volume and the PDE's in the large volume regime. By a simple change of coordinates in the large volume regime, we can significantly reduce the computational cost and explore regions of parameter space previously unachievable. We will describe the numerical implementation, present selected results and discuss the extension of the scheme to other models.   

Observational consequences of Loop Quantum Cosmology

I will discuss possible tests of Loop Quantum Cosmology (LQC) through cosmological perturbations and their consequences on the Cosmic Microwave background (CMB). It has been known for quite a long time that tensor perturbations are a promising way to possibly investigate the predictions of the theory. However, we have recently understood that the algebra usually assumed was in fact not correct. The requirement of anomaly-freedom for vector and scalar perturbations leads to modifications of the tensor-mode algebra. I will explain this for the two main corrections implied by LQC, namely the holonomy and inverse-triad terms. Then, I will focus on holonomy corrections in the bouncing scenario. I will explain a possible generic way to derive the anomaly-free and gauge-invariant variables. I will show that a specific scheme (the so-called mu-bar scheme), usually assumed for other reasons, naturally appears. This approach will also be briefly discussed for inverse-triads. Finally I will show some cosmological consequences of this approach (power spectrum, possible change of signature of the metric, ...) and a possible way to take into account both corrections.   

Loop quantum gravity without the Hamiltonian constraint

We construct a version of loop quantum gravity on a fixed constant mean curvature Cauchy slice for general relativity conformally coupled to a scalar field. The key input is to gauge fix the Hamiltonian constraint classically with the generator of local conformal transformations. This generator coincides with the constant mean curvature condition in the presence of a conformally coupled scalar field. The resulting quantum theory is a reduced phase space quantization with respect to the Hamiltonian constraint. As an application, it allows to calculate the entropy of a certain class of black holes based on counting physical states. In addition, the interpretation of the geometric operators of loop quantum gravity changes in this framework, and an effective Planck scale emerges with a dependence on the scalar field.   

Unimodular quantum gravity

Unimodular gravity is classically equivalent to standard Einstein gravity, but differs when it comes to the quantum theory: The conformal factor is non-dynamical, and the gauge symmetry consists of transverse diffeomorphisms only. Furthermore, the cosmological constant is not renormalized. Thus the quantum theory is distinct from a quantization of standard Einstein gravity. Here I show that within a truncation of the full Renormalization Group flow of unimodular quantum gravity, there is a non-trivial ultraviolet-attractive fixed point, yielding a UV completion for unimodular gravity.   

Riemann observables of the noncancellation of spacetime geometry fluctuations

A model of the noncancellation of spacetime geometry fluctuations is considered in the presence of an oscillating background metric. Quantum fluctuations originate in the stress-energy tensor and drive fluctuations in the oscillating metric, leading to a change in the correlation function of observables based on the Riemann tensor, and to a secularly growing effect. One such observable, the rate of the fractional redshift, will be presented.   

Breakdown of the Equivalence between Gravitational Mass and Energy for a Quantum Body in Metric Theories of Gravity

We define passive gravitational mass operator of a hydrogen atom in the post-Newtonian approximation of metric theories of gravity, including general relativity [1,2], and show that it does not commute with energy operator, taken in the absence of gravitational field. Nevertheless, the equivalence between the expectation values of passive gravitational mass and energy is shown to survive for stationary quantum states. Inequivalence between passive gravitational mass and energy at a macroscopic level results in time dependent oscillations of the expectation values of passive gravitational mass for superpositions of stationary quantum states, where the equivalence restores after averaging over time. Inequivalence between gravitational mass and energy at a microscopic level reveals itself as unusual electromagnetic radiation, emitted by the atoms, supported and moved in the Earth gravitational field with constant velocity using spacecraft or satellite, which can be experimentally measured.\\[4pt] [1] A.G. Lebed, arXiv:1111.5365v1 [gr-qc]; arXiv:1205.3134v1 [gr-qc].\\[0pt] [2] A.G. Lebed, to be published in proceedings of MG-13 Meeting (arXiv:1208.5756v1 [gr-qc]).