Minisuperspace Model Research Articles

Quantum dynamics of the black hole interior in loop quantum cosmology

It has been suggested that the homogeneous black hole interior spacetime, when quantized following the techniques of loop quantum cosmology, has a resolved singularity replaced by a black-to-white hole transition. This result has however been derived so far only using effective classical evolution equations, and depends on details of the so-called polymerization scheme for the Hamiltonian constraint. Here we propose to use the unimodular formulation of general relativity to study the full quantum dynamics of this mini-superspace model. When applied to such cosmological models, unimodular gravity has the advantage of trivializing the problem of time by providing a true Hamiltonian which follows a Schr\"odinger evolution equation. By choosing variables adapted to this setup, we show how to write semi-classical states agreeing with that of the Wheeler-DeWitt theory at late times, and how in loop quantum cosmology they evolve through the would-be singularity while remaining sharply peaked. This provides a very simple setup for the study of the full quantum dynamics of these models, which can hopefully serve to tame regularization ambiguities.

Quantum fluctuations of the compact phase space cosmology

In the recent article (2019 Phys. Rev. D 100 043533) a compact phase space generalization of the flat de Sitter cosmology has been proposed. The main advantages of the compactification is that physical quantities are bounded, and the quantum theory is characterized by finite dimensional Hilbert space. Furthermore, by considering the phase space, quantum description is constructed with the use SU(2) representation theory. The purpose of this article is to apply effective methods to analyze quantum dynamics of the model at the semi-classical regime. The analysis is performed both without prior solving of the quantum constraint and by extracting physical Hamiltonian of the model. At the effective level, the results of the two procedures are shown to be equivalent. We find a nontrivial behavior of the fluctuations around the recollapse of the Universe, which is distinct from what is found after quantization with the standard flat phase space. The behavior is reflected at the level of the modified Friedmann equation with quantum back-reaction effects, which is derived. Finally, a relation between quantum fluctuations in the minisuperspace model under consideration and the Bousso bound has been found. It has been shown that the Bousso bound can be violated for certain choices of semiclassical states.

Supersymmetric minisuperspace models in self-dual loop quantum cosmology

In this paper, we study a class of symmetry reduced models of mathcal{N} = 1 super- gravity using self-dual variables. It is based on a particular Ansatz for the gravitino field as proposed by D’Eath et al. We show that the essential part of the constraint algebra in the classical theory closes. In particular, the (graded) Poisson bracket between the left and right supersymmetry constraint reproduces the Hamiltonian constraint.For the quantum theory, we apply techniques from the manifestly supersymmetric approach to loop quantum supergravity, which yields a graded analog of the holonomy-flux algebra and a natural state space.We implement the remaining constraints in the quantum theory. For a certain subclass of these models, we show explicitly that the (graded) commutator of the supersymmetry constraints exactly reproduces the classical Poisson relations. In particular, the trace of the commutator of left and right supersymmetry constraints reproduces the Hamilton constraint operator. Finally, we consider the dynamics of the theory and compare it to a quantization using standard variables and standard minisuperspace techniques.

How round is the quantum de Sitter universe?

We investigate the quantum Ricci curvature, which was introduced in earlier work, in full, four-dimensional quantum gravity, formulated nonperturbatively in terms of Causal Dynamical Triangulations (CDT). A key finding of the CDT approach is the emergence of a universe of de Sitter-type, as evidenced by the successful matching of Monte Carlo measurements of the quantum dynamics of the global scale factor with a semiclassical minisuperspace model. An important question is whether the quantum universe exhibits semiclassicality also with regard to its more local geometric properties. Using the new quantum curvature observable, we examine whether the (quasi-)local properties of the quantum geometry resemble those of a constantly curved space. We find evidence that on sufficiently large scales the curvature behaviour is compatible with that of a four-sphere, thus strengthening the interpretation of the dynamically generated quantum universe in terms of a de Sitter space.

Asymptotically de Sitter universe inside a Schwarzschild black hole

Extending our previous analysis, we study the interior of a Schwarzschild black hole derived from a partial gauge fixing of the full Loop Quantum Gravity Hilbert space, this time including the inverse volume and coherent state subleading corrections. Our derived effective Hamiltonian differs crucially from the ones introduced in the minisuperspace models. This distinction is reflected in the class of homogeneous bouncing geometries that replace the classical singularity and are labeled by a set of quantum parameters associated with the structure of coherent states used to derive the effective Hamiltonian. By fixing these quantum parameters through geometrical considerations, the post-bounce interior geometry reveals a high sensitivity to the value of the Barbero-Immirzi parameter $\gamma$. Surprisingly, we find that $\gamma\approx 0.274$ results in an asymptotically de Sitter geometry in the interior region, where now a cosmological constant is generated purely from quantum gravitational effects. The striking fact is the exact coincidence of this value for $\gamma$ with the one derived from the SU(2) black hole entropy calculations in Loop Quantum Gravity. The emergence of this value in two entirely unrelated theoretical frameworks and computational setups is strongly suggestive of deep ties between the area gap in Loop Quantum Gravity, black hole physics, and the observable Universe. In connection to the latter, we point out an intriguing relation between the measured value of the cosmological constant and the observed mass in the Universe from a proposal for a spin quantum number renormalization effect associated to the microscopic dynamics.

“Time”-Covariant Schrödinger Equation and the Canonical Quantization of the Reissner–Nordström Black Hole

A “time”-covariant Schrödinger equation is defined for the minisuperspace model of the Reissner–Nordström (RN) black hole, as a “hybrid” between the “intrinsic time” Schrödinger and Wheeler–DeWitt (WDW) equations. To do so, a reduced, regular, and “time(r)”-dependent Hamiltonian density was constructed, without “breaking” the re-parametrization covariance r→f(r˜). As a result, the evolution of states with respect to the parameter r and the probabilistic interpretation of the resulting quantum description is possible, while quantum schemes for different gauge choices are equivalent by construction. The solutions are found for Dirac’s delta and Gaussian initial states. A geometrical interpretation of the wavefunctions is presented via Bohm analysis. Alongside this, a criterion is presented to adjudicate which, between two singular spacetimes, is “more” or “less” singular. Two ways to adjudicate the existence of singularities are compared (vanishing of the probability density at the classical singularity and semi-classical spacetime singularity). Finally, an equivalence of the reduced equations with those of a 3D electromagnetic pp-wave spacetime is revealed.

Evidence of Time Evolution in Quantum Gravity

In this paper, we argue that the problem of time is not a crucial issue inherent in the quantum picture of the universe evolution. On the minisuperspace model example with the massless scalar field, we demonstrate four approaches to the description of quantum evolution, which give similar results explicitly. The relevance of these approaches to building a quantum theory of gravity is discussed.

Euclidean wormholes in Hořava-Lifshitz gravity

Euclidean wormholes first appeared in the Euclidean path integral approach to quantum gravity. In a more general way, Hawking and Page interpreted such configurations as solutions to the Wheeler-DeWitt equation with appropriate boundary conditions. We study Euclidean wormholes in the framework of the Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz theory of gravity. We use its projectable version to obtain the Wheeler-DeWitt equation of a minisuperspace model considering a closed Friedmann Universe with a massless scalar field. For large values of the scale factor we find that the solution of the Wheeler-DeWitt equation coincides with the one obtained by Hawking. Whereas in the limit corresponding to the early Universe we find a new set of solutions, which agree with the Hawking and Page boundary conditions for wormholes.

Perspectives on the dynamics in a loop quantum gravity effective description of black hole interiors

In the loop quantum gravity context, there have been numerous proposals to quantize the reduced phase space of a black hole, and develop a classical effective description for its interior which eventually resolves the singularity. However, little progress has been made towards understanding the relation between such quantum/effective minisuperspace models and what would be the spherically symmetric sector of loop quantum gravity. In particular, it is not clear whether one can extract the phenomenological predictions obtained in minisuperspace models, such as the singularity resolution and the spacetime continuation beyond the singularity, based on results in full loop quantum gravity. In this paper, we present an attempt in this direction in the context of Kantowski-Sachs spacetime, through the proposal of two new effective Hamiltonians for the reduced classical model. The first is derived using Thiemann classical identities for the regularized expressions, while the second is obtained as a first approximation of the expectation value of a Hamiltonian operator in loop quantum gravity in a semi-classical state peaked on the Kantowski-Sachs initial data. We then proceed with a detailed analysis of the dynamics they generate and compare them with the Hamiltonian derived in General Relativity and the common effective Hamiltonian proposed in earlier literature.

Spinors in supersymmetric dS/CFT

We study fermionic bulk fields in the dS/CFT dualities relating mathcal{N} = 2 su- persymmetric Euclidean vector models with reversed spin-statistics in three dimensions to supersymmetric Vasiliev theories in four-dimensional de Sitter space. These dualities specify the Hartle-Hawking wave function in terms of the partition function of deforma- tions of the vector models. We evaluate this wave function in homogeneous minisuperspace models consisting of supersymmetry-breaking combinations of a half-integer spin field with either a scalar, a pseudoscalar or a metric squashing. The wave function appears to be well-behaved and globally peaked at or near the supersymmetric de Sitter vacuum, with a low amplitude for large deformations. Its behavior in the semiclassical limit qualitatively agrees with earlier bulk computations both for massless and massive fermionic fields.

Some aspects of the canonical analysis of Reuter-Weyer RG improved Einstein-Hilbert action

A canonical analysis of RG improved action of the Einstein-Hilbert functional is performed. The gravitational and cosmological constants as function of the space-time coordinates are treated as external non-geometrical fields. Dirac’s constraint analysis is performed, in the general case, up to secondary constraints. The constraints are second class and, in general, the problem appears to be technically complicated. This fact suggests studying the Dirac’s constraint analysis of the related Brans-Dicke theory, which shows the Poisson Brackets between Hamiltonian-Hamiltonian constraints do not close.A simplified FLRW minisuperspace model based on the RG improved Einstein Hilbert action contains Bouncing and Emergent Universes for values of K = –1, 0,1

Elimination of cosmological singularities in quantum cosmology by suitable operator orderings

Canonical quantization of general relativity does not yield a unique quantum theory for gravity. This is in part due to operator ordering ambiguities. In this paper, we investigate the role of different operator orderings on the question of whether a big bang or big crunch singularity occurs. We do this in the context of the minisuperspace model of a Friedmann-Lemaitre-Robertson-Walker universe with Brown-Kuchar dust. We find that for a certain class of operator orderings such a singularity is eliminated without having to impose boundary conditions.

Constraints on non-minimal coupling from quantum cosmology

Quantum cosmology is investigated in a de Sitter minisuperspace model with a quantized scalar field non-minimally coupled to curvature. Quantum states of the scalar field must satisfy the regularity condition, which requires that the probability of field fluctuations should not increase with their amplitude. We show that this condition imposes constraints on the allowed values of the curvature coupling parameter ξ. This is a surprising result, since the field dynamics depends only on the combination m2+ξ R, where m is the field mass and R = const is the curvature, and does not depend on ξ separately.

No-boundary proposal in biaxial Bianchi IX minisuperspace

We implement the no-boundary proposal for the wave function of the universe in an exactly solvable Bianchi IX minisuperspace model with two scale factors. We extend our earlier work (Phys. Rev. Lett. 121, 081302, 2018 / arXiv:1804.01102) to include the contribution from the $\mathbb{C}\text{P}^2 \setminus B^4$ topology. The resulting wave function yields normalizable probabilities and thus fits into a predictive framework for semiclassical quantum cosmology. We find that the amplitude is low for large anisotropies. In the isotropic limit the usual Hartle-Hawking wave function for the de Sitter minisuperspace model is recovered. Inhomogeneous perturbations in an extended minisuperspace are shown to be initially in their ground state. We also demonstrate that the precise mathematical implementation of the no-boundary proposal as a functional integral in minisuperspace depends on detailed aspects of the model, including the choice of gauge-fixing. This shows in particular that the choice of contour cannot be fundamental, adding weight to the recent proposal that the semiclassical no-boundary wave function should be defined solely in terms of a collection of saddle points. We adopt this approach in most of this paper. Finally we show that the semiclassical tunneling wave function of the universe is essentially equal to the no-boundary state in this particular minisuperspace model, at least in the subset of the classical domain where the former is known.

Tunneling wave function of the universe. II. The backreaction problem

The tunneling wave function of the universe is calculated exactly for a de Sitter minisuperspace model with a massless conformally coupled scalar field, both by solving the Wheeler-DeWitt equation and by evaluating the Lorentzian path integral. The same wave function is found in both approaches. The back-reaction of quantum field fluctuations on the scale factor amounts to a constant renormalization of the vacuum energy density. This is in contrast to the recent suggestion of Feldbrugge $et$ $al.$ that the back-reaction should diverge when the scale factor gets small, $a \to 0$. Similar results are found for a massive scalar field in the limit of a large mass. We also verified that the tunneling wave function can be expressed as a transition amplitude from a universe of vanishing size with the scalar field in the state of Euclidean vacuum, as it was suggested in our earlier work.

Bouncing unitary cosmology I. Mini-superspace general solution

We offer a new proposal for cosmic singularity resolution based upon a quantum cosmology with a unitary bounce. This proposal is illustrated via a novel quantization of a mini-superspace model in which there can be superpositions of the cosmological constant. This possibility leads to a finite, bouncing unitary cosmology. Whereas the usual Wheeler–DeWitt cosmology generically displays pathological behaviour in terms of non-finite expectation values and non-unitary dynamics, the finiteness and unitarity of our model are formally guaranteed. For classically singular models with a massless scalar field and cosmological constant, we show that well-behaved quantum observables can be constructed and generic solutions to the universal Schrödinger equation are singularity-free. Generic solutions of our model displays novel features including: (i) superpositions of values of the cosmological constant; (ii) universal effective physics due to non-trivial self-adjoint extensions of the Hamiltonian; and (iii) bound ‘Efimov universe’ states for negative cosmological constant. The last feature provides a new platform for quantum simulation of the early universe. A companion paper provides detailed interpretation and analysis of particular cosmological solutions that display a cosmic bounce due to quantum gravitational effects, a well-defined FLRW limit far from the bounce, and a semi-classical turnaround point in the dynamics of the scalar field which resembles an effective inflationary epoch.

The BKL scenario, infrared renormalization, and quantum cosmology

A discussion of inhomogeneity is indispensable to understand quantum cosmology, even if one uses the dynamics of homogeneous geometries as a first approximation. While a full quantization of inhomogeneous gravity is not available, a broad framework of effective field theory provides important ingredients for quantum cosmology. Such a setting also allows one to take into account lessons from the Belinski-Khalatnikov-Lifshitz (BKL) scenario. Based on several new ingredients, this article presents conditions on various parameters and mathematical constructions that appear in minisuperspace models. Examples from different approaches demonstrate their restrictive nature.

Integrable minisuperspace models with Liouville field: energy density self-adjointness and semiclassical wave packets

The homogeneous cosmological models with a Liouville scalar field are investigated in classical and quantum contexts of Wheeler–DeWitt geometrodynamics. In the quantum case of quintessence field with potential unbounded from below and phantom field, the energy density operators are not essentially self-adjoint, and self-adjoint extensions contain ambiguities. Therefore the same classical actions correspond to a family of distinct quantum models. For the phantom field the energy spectrum happens to be discrete. The probability conservation and appropriate classical limit can be achieved with a certain restriction of the functional class. The appropriately localized wave packets are studied numerically using the Schrödinger’s norm and a conserved Mostafazadeh’s norm introduced from techniques of pseudo-Hermitian quantum mechanics. These norms give a similar packet evolution that is confronted with analytical classical solutions.

Superpositions of the cosmological constant allow for singularity resolution and unitary evolution in quantum cosmology

A novel approach to quantization is shown to allow for superpositions of the cosmological constant in isotropic and homogeneous mini-superspace models. Generic solutions featuring such superpositions display unitary evolution and resolution of the classical singularity. Physically well-motivated cosmological solutions are constructed. These particular solutions exhibit characteristic features of a cosmic bounce including universal phenomenology that can be rendered insensitive to Planck-scale physics in a natural manner.

Wavepacket Evolution in Unimodular Quantum Cosmology

The unimodular theory of gravity admits a canonical quantization of minisuperspace models without the problem of time. We derive instead a kind of Schr\"odinger equation. We have found unitarily evolving wave packet solutions for the special case of a massless scalar field and a spatially flat Friedmann universe. We show that the longterm behaviour of the expectation values of the canonical quantities corresponds to the evolution of the classical variables. The solutions provided in an explicit example can be continued beyond the singularity at t=0, passing a finite minimal extension of the universe.