Brajesh Gupt

Contrasting features of anisotropic loop quantum cosmologies: The Role of spatial curvature

A characteristic feature of loop quantization of the isotropic and Bianchi-I spacetimes is the existence of universal bounds on the energy density and the expansion and shear scalars, independent of the matter content. We investigate the properties of these physical quantities in Bianchi-II and Bianchi-IX spacetimes, which have been recently loop quantized using the connection operator approach. Using the effective Hamiltonian approach, we show that for Bianchi-II spacetime, energy density and the expansion and shear scalars turn out to be bounded, albeit not by universal values. In Bianchi-IX spacetime, when the approach to the classical singularity is isotropic, above physical quantities are bounded. In addition, for all other cases, where the approach to singularities is not isotropic and effective dynamics can be trusted, these quantities turn out to be finite. These results stand in sharp distinction to general relativity, where above physical quantities are generically unbounded, leading to the break down of geodesic equations. In contrast to the isotropic and Bianchi-I models, we find the role of energy conditions for Bianchi-II model and the inverse triad modifications for Bianchi-IX to be significant to obtain above bounds. These results bring out subtle physical distinctions between the quantization using holonomies over closed loops performed for isotropic and Bianchi-I models, and the connection operator approach. We find that qualitative differences in physics exist for these quantization methods even for the isotropic models in the presence of spatial curvature. We highlight these important differences in the behavior of the expansion scalar in the holonomy based quantization and connection operator approach for isotropic spatially closed and open models.

Numerical simulations of a loop quantum cosmos: robustness of the quantum bounce and the validity of effective dynamics

A key result of isotropic loop quantum cosmology is the existence of a quantum bounce which occurs when the energy density of the matter field approaches a universal maximum close to the Planck density. Though the bounce has been exhibited in various matter models, due to severe computational challenges, some important questions have so far remained unaddressed. These include the demonstration of the bounce for widely spread states, its detailed properties for the states when matter field probes regions close to the Planck volume and the reliability of the continuum effective spacetime description in general. In this manuscript we rigorously answer these questions using the Chimera numerical scheme for the isotropic spatially flat model sourced with a massless scalar field. We show that, as expected from an exactly solvable model, the quantum bounce is a generic feature of states even with a very wide spread, and for those which bounce much closer to the Planck volume. We perform a detailed analysis of the departures from the effective description and find some expected, and some surprising results. At a coarse level of description, the effective dynamics can be regarded as a good approximation to the underlying quantum dynamics unless the states correspond to small scalar field momenta, in which case they bounce closer to the Planck volume or are very widely spread. Quantifying the amount of discrepancy between the quantum and the effective dynamics, we find that the departure between them depends in a subtle and non-monotonic way on the field momentum and different fluctuations. Interestingly, the departures are generically found to be such that the effective dynamics overestimates the spacetime curvature, and underestimates the volume at the bounce.

Quantum gravitational Kasner transitions in Bianchi-I spacetime

Due to non-perturbative quantum gravitational effects, the classical big bang singularity is replaced by a quantum big bounce of the mean scale factor in loop quantization of Bianchi-I spacetime. An important issue is to understand various differences in the physical properties of the spacetime across the bounce. We investigate this issue in the context of various geometrical structures, identified by Kasner exponents of the metric, which arise on approach to the singularity in the classical theory. Using effective spacetime description of Bianchi-I model in loop quantum cosmology with dust, radiation and stiff matter, we find that as in the classical theory, geometrical structures such as a cigar or a pancake form, but they are finite and non-singular. Depending on the initial conditions of matter and anisotropies, different geometric structures are possible in the pre- and post-bounce phases in physical evolution. Thus, quantum gravitational effects can cause a Kasner transition in Bianchi-I spacetime, which is not possible at the classical level. Interestingly, we find that not all transitions are allowed at the level of effective dynamics in loop quantum cosmology. We find the selection rules and underlying conditions for all allowed and forbidden transitions. Selection rules suggest that for a given set of initial conditions on anisotropies, occurrence of Kasner transitions follows a distinct pattern, and certain transitions are more favored than others.

Numerical evolution of squeezed and non-Gaussian states in loop quantum cosmology

In recent years, numerical simulations with Gaussian initial states have demonstrated the existence of a quantum bounce in loop quantum cosmology in various models. A key issue pertaining to the robustness of the bounce and the associated physics is to understand the quantum evolution for more general initial states, which may depart significantly from Gaussianity and may have no well defined peakedness properties. The analysis of such states, including squeezed and highly non-Gaussian states, has been computationally challenging until now. In this paper, we overcome these challenges by using the Chimera scheme for the spatially flat, homogeneous and isotropic model sourced with a massless scalar field. We demonstrate that the quantum bounce in this model occurs even for states that are highly squeezed or are non-Gaussian with multiple peaks and with little resemblance to semi-classical states. The existence of the bounce is found to be robust, being independent of the properties of the states. The evolution of squeezed and non-Gaussian states turns out to be qualitatively similar to that of Gaussian states, and satisfies strong constraints on the growth of the relative fluctuations across the bounce. We also compare the results from the effective dynamics and find that, although it captures the qualitative aspects of the evolution for squeezed and highly non-Gaussian states, it always underestimates the bounce volume. We show that various properties of the evolution, such as the energy density at the bounce, are in excellent agreement with the predictions from an exactly solvable loop quantum cosmological model for arbitrary states.

Phenomenological investigation of a quantum gravity extension of inflation with the Starobinsky potential

We investigate the pre-inflationary dynamics of inflation with the Starobinsky potential, favored by recent data from the Planck mission, using techniques developed to study cosmological perturbations on quantum spacetimes in the framework of loop quantum cosmology. We find that for a large part of the initial data, inflation compatible with observations occurs. There exists a subset of this initial data that leads to quantum gravity signatures that are potentially observable. Interestingly, despite the different inflationary dynamics, these quantum gravity corrections to the powerspectra are similar to those obtained for inflation with a quadratic potential, including suppression of power at large scales. Furthermore, for super horizon modes the tensor modes show deviations from the standard inflationary paradigm that are unique to the Starobinsky potential and could be important for non-Gaussian modulation and tensor fossils.

Chimera: A hybrid approach to numerical loop quantum cosmology

The existence of a quantum bounce in isotropic spacetimes is a key result in loop quantum cosmology (LQC), which has been demonstrated to arise in all the models studied so far. In most of the models, the bounce has been studied using numerical simulations involving states which are sharply peaked and which bounce at volumes much larger than the Planck volume. An important issue is to confirm the existence of the bounce for states which have a wide spread, or which bounce closer to the Planck volume. Numerical simulations with such states demand large computational domains, making them very expensive and practically infeasible with the techniques which have been implemented so far. To overcome these difficulties, we present an efficient hybrid numerical scheme using the property that at the small spacetime curvature, the quantum Hamiltonian constraint in LQC, which is a difference equation with uniform discretization in volume, can be approximated by a Wheeler–DeWitt differential equation. By carefully choosing a hybrid spatial grid allowing the use of partial differential equations at large volumes, and with a simple change of geometrical coordinate, we obtain a surprising reduction in the computational cost. This scheme enables us to explore regimes which were so far unachievable for the isotropic model in LQC. Our approach also promises to significantly reduce the computational cost for numerical simulations in anisotropic LQC using high performance computing.

A quantum gravitational inflationary scenario in Bianchi-I spacetime

We investigate the 2 inflationary model in the Bianchi-I spacetime using the effective spacetime description of loop quantum cosmology to understand the issues of the resolution of initial singularity, isotropization, effect of anisotropies on the amount of inflation, and the phase-space attractors in the presence of non-perturbative quantum gravitational modifications. A comparative analysis with the classical theory by including more general initial conditions than the ones previously considered in the latter is also performed. We show that, in general, the classical singularity is replaced by a bounce of the mean scale factor in loop quantum cosmology. Due to the underlying quantum geometric effects, the energy density of the inflaton and the anisotropic shear remain bounded throughout the non-singular evolution. Starting from arbitrary anisotropic initial conditions, a loop quantum universe isotropizes either before or soon after the onset of slow-roll inflation. We find a double attractor behavior in the phase-space dynamics of loop quantum cosmology, similar to the one in classical theory, but with some additional subtle features. Quantum modifications to the dynamical equations are such that, unlike the classical theory, the amount of inflation does not monotonically depend on the initial anisotropy in loop quantum cosmology. Our results suggest that a viable non-singular inflationary model can be constructed from highly anisotropic initial conditions in the Planck regime.

Tensor perturbations during inflation in a spatially closed Universe

In a recent paper [16], we studied the evolution of the background geometry and scalar perturbations in an inflationary, spatially closed Friedmann-Lema\^itre-Robertson-Walker (FLRW) model having constant positive spatial curvature and spatial topology

Nonsingular AdS-dS transitions in a landscape scenario

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Phenomenology of inflationary scenario in loop quantum cosmology

. Due to the spatial curvature, the early phase of slow-roll inflation is modified, leading to suppression of power in the scalar power spectrum at large angular scales. In this paper, we extend the analysis to include tensor perturbations. We find that --- similarly to the scalar perturbations --- the tensor power spectrum also shows power suppression for long wavelength modes. The correction to the tensor spectrum is limited to the very long wavelength modes, therefore the resulting observable CMB B-mode polarization spectrum remains practically the same as in the standard scenario with flat spatial sections. However, since both the tensor and scalar power spectra are modified, there are scale dependent corrections to the tensor-to-scalar ratio that lead to violation of the standard slow-roll consistency relation.