Minisuperspace Model Research Articles

Hořava-Lifshitz quantum cosmology

In this work, a minisuperspace model for the projectable Ho\v{r}ava-Lifshitz (HL) gravity without the detailed balance condition is investigated. The Wheeler-deWitt equation is derived and its solutions are studied and discussed for some particular cases where, due to HL gravity, there is a "potential barrier" nearby $a=0$. For a vanishing cosmological constant, it is found a normalizable wave function of the universe. When the cosmological constant is non-vanishing, the WKB method is used to obtain solutions for the wave function of the universe. Using the Hamilton-Jacobi equation, one discusses how the transition from quantum to classical regime occurs and, for the case of a positive cosmological constant, the scale factor is shown to grow exponentially, hence recovering the GR behaviour for the late universe.

Decoherent histories analysis of minisuperspace quantum cosmology

Recent results on the decoherent histories quantization of simple cosmological models (minisuperspace models) are described. The most important issue is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation as the probability. This gives sensible semiclassical results but lacks an underlying operator formalism. Here, we supply the underlying formalism by deriving probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory, building on the generalized quantum mechanics formalism developed by Hartle. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in non-relativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the S-matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit and are closely related to an intersection number operator. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures.

Topologically massive gravity and Ricci–Cotton flow

We consider topologically massive gravity (TMG), which is three-dimensional general relativity with a cosmological constant and a gravitational Chern–Simons term. When the cosmological constant is negative the theory has two potential vacuum solutions: anti-de Sitter space and warped anti-de Sitter space. The theory also contains a massive graviton state which renders these solutions unstable for certain values of the parameters and boundary conditions. We study the decay of these solutions due to the condensation of the massive graviton mode using Ricci–Cotton flow, which is the appropriate generalization of Ricci flow to TMG. When the Chern–Simons coupling is small the AdS solution flows to warped AdS by the condensation of the massive graviton mode. When the coupling is large the situation is reversed, and warped AdS flows to AdS. Minisuperspace models are constructed where these flows are studied explicitly.

Quantum cosmological metroland model

Relational particle mechanics is useful for modelling whole-universe issues such as quantum cosmology or the problem of time in quantum gravity, including some aspects outside the reach of comparably complex mini-superspace models. In this paper, we consider the mechanics of pure shape and not scale of four particles on a line, so that the only physically significant quantities are ratios of relative separations between the constituents' physical objects. Many of our ideas and workings extend to the N-particle case. As such models' configurations resemble depictions of metro lines in public transport maps, we term them ‘N-stop metrolands’. This 4-stop model's configuration space is a 2-sphere, from which our metroland mechanics interpretation is via the ‘cubic’ tessellation. This model yields conserved quantities which are mathematically SO(3) objects like angular momenta but are physically relative dilational momenta (i.e. coordinates dotted with momenta). We provide and interpret various exact and approximate classical and quantum solutions for 4-stop metroland; from these results one can construct expectations and spreads of shape operators that admit interpretations as relative sizes and the ‘homogeneity of the model universe's contents’, and also objects of significance for the problem of time in quantum gravity (e.g. in the naïve Schrödinger and records theory timeless approaches).

Black holes and phase-space noncommutativity

We use the solutions of the noncommutative Wheeler-De Witt equation arising from a Kantowski-Sachs cosmological model to compute thermodynamic properties of the Schwarzschild black hole. We show that the noncommutativity in the momentum sector introduces a quadratic term in the potential function of the black hole minisuperspace model. This potential has a local minimum and thus the partition function can be computed by resorting to a saddle point evaluation in the neighbourhood of the minimum. The temperature and the entropy of the black hole are derived, and they have an explicit dependence on the inverse of the momentum noncommutative parameter, $\eta$. Moreover, we study the $t=r=0$ singularity in the noncommutative regime and show that in this case the wave function of the system vanishes in the neighbourhood of $t=r=0$.

Probabilities in quantum cosmological models: A decoherent histories analysis using a complex potential

In the quantization of simple cosmological models (minisuperspace models) described by the Wheeler-DeWitt equation, an important step is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation. This gives sensible semiclassical results but lacks an underlying operator formalism. In this paper, we address the issue of constructing probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in nonrelativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the $S$ matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit, and are closely related to an intersection number operator. The definitions of class operators given here handle the key case in which the underlying classical system has multiple crossings of the boundaries of the regions of interest. We show that oscillatory WKB solutions to the Wheeler-DeWitt equation give approximate decoherence of histories, as do superpositions of WKB solutions, as long as the regions of configuration space are sufficiently large. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures. In brief, we exhibit the well-defined operator formalism underlying the usual heuristic interpretational methods in quantum cosmology.

Replication regulates volume weighting in quantum cosmology

Probabilities for observations in cosmology are conditioned both on the Universe's quantum state and on local data specifying the observational situation. We show the quantum state defines a measure for prediction through such conditional probabilities that is well-behaved for spatially large or infinite universes when the probabilities that our data are replicated are taken into account. In histories where our data are rare volume weighting connects top-down probabilities conditioned on both the data and the quantum state to the bottom-up probabilities conditioned on the quantum state alone. We apply these principles to a calculation of the number of inflationary e-folds in a homogeneous, isotropic minisuperspace model with a single scalar field moving in a quadratic potential. We find that volume weighting is justified and the top-down probabilities favor a large number of e-folds, hereby predicting the curvature of our Universe at the present time to be approximately zero.

Effective polymer dynamics ofD-dimensional black hole interiors

We consider two different effective polymerization schemes applied to $D$-dimensional, spherically symmetric black hole interiors. It is shown that polymerization of the generalized area variable alone leads to a complete, regular, single-horizon spacetime in which the classical singularity is replaced by a bounce. The bounce radius is independent of rescalings of the homogeneous internal coordinate, but does depend on the arbitrary fiducial cell size. The model is therefore necessarily incomplete. It nonetheless has many interesting features: After the bounce, the interior region asymptotes to an infinitely expanding Kantowski-Sachs spacetime. If the solution is analytically continued across the horizon, the black hole exterior exhibits asymptotically vanishing quantum corrections due to the polymerization. In all spacetime dimensions except four, the falloff is too slow to guarantee invariance under Poincar\'e transformations in the exterior asymptotic region. Hence, the four-dimensional solution stands out as the only example which satisfies the criteria for asymptotic flatness. In this case it is possible to calculate the quantum-corrected temperature and entropy. We also show that polymerization of both phase space variables, the area and the conformal mode of the metric, generically leads to a multiple horizon solution which is reminiscent of polymerized minisuperspace models of spherically symmetric black holes in loop quantum gravity.

Noncommutative quantum cosmology

We present a phase-space noncommutative extension of Quantum Cosmology in the context of a Kantowski-Sachs (KS) minisuperspace model. We obtain the Wheeler-DeWitt (WDW) equation for the noncommutative system through the ADM formalism and a suitable Seiberg-Witten map. The resulting WDW equation explicitly depends on the phase-space noncommutative parameters, θ and η. Numerical solutions of the noncommutative WDW equation are found and, interestingly, also bounds on the values of the nonommutative parameters. Moreover, we conclude that the noncommutativity in the momenta sector lead to a damped wave function implying that this type of noncommutativity can be relevant for a selection of possible initial states for the universe.

Deformed phase space in a two-dimensional minisuperspace model

We study the effects of noncommutativity and deformed Heisenberg algebra on the evolution of a two-dimensional minisuperspace cosmological model in classical and quantum regimes. The phase-space variables turn out to correspond to the scale factor of a flat FRW model with a positive cosmological constant and a dilatonic field with which the action of the model is augmented. The exact classical and quantum solutions in commutative and noncommutative cases are presented. We also obtain some approximate analytical solutions for the corresponding classical and quantum cosmology in the presence of the deformed Heisenberg relations between the phase-space variables, in the limit where the minisuperspace variables are small. These results are compared with the standard commutative and noncommutative cases, and similarities and differences of these solutions are discussed.

Quantum gravity equation in largeNYang-Mills quantum mechanics

We propose a new interpretation of the BFSS large N matrix quantum mechanics analogous to a novel interpretation of the IKKT matrix model where infinitely large N matrices act as differential operators in a curved space. In this picture, the Schrodinger equation in the BFSS model is regarded as the Wheeler-DeWitt equation which determines the wave function of universe. An explicit solution of wave function is studied in a simple two-dimensional minisuperspace model.

THE PROBABILITY OF THE CREATION OF EXTRA DIMENSIONS IN NUCLEAR COLLISIONS

The minisuperspace model in 3+d spatial dimensions with matter described by the bag model is considered with the aim of estimating the probability of creation of compactified extra dimensions in nuclear collisions. The amplitude of transition from three- to (3+d)-dimensional space has been calculated both in the case of completely confined matter, when the contribution of radiation is ignored, and in the case of radiation domination, when the bag constant is negligible. It turns out that the number of additional dimensions is limited in the first regime, while it is infinite in the second one. It is shown that the probability of creation of extra dimensions is finite in the both regimes.

Classical universes of the no-boundary quantum state

We analyze the origin of the quasiclassical realm from the no-boundary proposal for the Universe's quantum state in a class of minisuperspace models. The models assume homogeneous, isotropic, closed spacetime geometries, a single scalar field moving in a quadratic potential, and a fundamental cosmological constant. The allowed classical histories and their probabilities are calculated to leading semiclassical order. For the most realistic range of parameters analyzed, we find that a minimum amount of scalar field is required, if there is any at all, in order for the Universe to behave classically at late times. If the classical late time histories are extended back, they may be singular or bounce at a finite radius. The ensemble of classical histories is time symmetric although individual histories are generally not. The no-boundary proposal selects inflationary histories, but the measure on the classical solutions it provides is heavily biased towards small amounts of inflation. However, the probability for a large number of e-foldings is enhanced by the volume factor needed to obtain the probability for what we observe in our past light cone, given our present age. Our results emphasize that it is the quantum state of the Universe that determines whether or not it exhibits a quasiclassical realm and what histories are possible or probable within that realm.

Tomographic representation of minisuperspace quantum cosmology and noether symmetries

The probability representation, in which cosmological quantum states are described by a standard positive probability distribution, is constructed for minisuperspace models selected by Noether symmetries. In such a case, the tomographic probability distribution provides the classical evolution for the models and can be considered an approach to select “observable” universes. Some specific examples, derived from Extended Theories of Gravity, are worked out. We discuss also how to connect tomograms, symmetries and cosmological parameters.

Minisuperspace models in M-theory

We derive the full canonical formulation of the bosonic sector of eleven-dimensional supergravity and explicitly present the constraint algebra. We then compactify M-theory on a warped product of homogeneous spaces of constant curvature and construct a minisuperspace of scale factors. First classical behaviour of the minisuperspace system is analysed and then a quantum theory is constructed. It turns out that there are similarities with the ‘pre-big-bang’ scenario in string theory.

Gravitational Collapse in Loop Quantum Gravity

In this paper we study the gravitational collapse applying methods of loop quantum gravity to a minisuperspace model. We consider the space-time region inside the Schwarzschild black hole event horizon and we divide this region in two parts, the first one where the matter (dust matter) is localized and the other (outside) where the metric is Kantowski–Sachs type. We study the Hamiltonian constraint obtaining a set of three difference equations that give a regular and natural evolution beyond the classical singularity point in “r=0” localized.

Quantum collapse of a self-gravitating thin shell and statistical model of quantum black hole

Abstract The quantum collapse of a self-gravitating thin shell in the minisuperspace models is revisited on the assumption that the shell is composed of N distinguishable identical particles. The ground state of the shell is found and defined as a quantum black hole (QBH). We show that the energy of single particle in the QBH is dependent on N , and N has an up-limit for a stable QBH. The effective exciting energy of single particle is determined, which is universally 1 / 2 of the Planck energy for the full-filled QBHs. We also propose a simple statistical model of QBH and show that a QBH is full-filled at low temperatures and half-filled at high temperatures. The specific heat of QBH is found to be positive at low temperatures and the relation of the QBH mass with its temperature is obtained in the high-temperature limit of our model.

Gauge-noninvariance of quantum cosmology and vacuum dark energy

We address the question how to adapt cosmological constant Λ for description of a vacuum dark energy density jumping from the big initial value to the small today value suggested by observations. We find such a possibility in the gauge-noninvariance of quantum cosmology which leads to a connection between a choice of the gauge and quantum spectrum for a certain physical quantity which can be specified in the framework of the minisuperspace model. We introduce a particular gauge in which the cosmological constant Λ is quantized and show that making a measurement of Λ today one can find its small value with the biggest probability, while at the beginning of the evolution, the biggest probability corresponds to its biggest value. Transitions between quantum levels of Λ in the course of the Universe evolution, could be related to several scales for symmetry breaking.

Landscape of string theory and the wave function of the universe

We explore the possibility that quantum cosmology considerations could provide a selection principle in the landscape of string vacua. We propose that the universe emerged from the string era in a thermally excited state and determine, within a mini-superspace model, the probability of tunneling to different points on the landscape. We find that the potential energy of the tunneling endpoint from which the universe emerges and begins its classical evolution is determined by the primordial temperature. By taking into account some generic properties of the moduli potential we then argue that the tunneling to the tail of the moduli potentials is disfavored, that the most likely emergence point is near an extremum, and that this extremum is not likely to be in the outer region of moduli space where the compact volume is very large and the string coupling very weak. As a concrete example we discuss the application of our arguments to the Kachru, Kallosh, Linde and Trivedi model of moduli stabilization.

Minisuperspace models in histories theory

We study the Robertson–Walker minisuperspace model in histories theory, motivated by the results that emerged from the histories approach to general relativity. We examine, in particular, the issue of time reparametrization in such systems. The model is quantized using an adaptation of reduced state space quantization. We finally discuss the classical limit, the implementation of initial cosmological conditions and estimation of probabilities in the histories context.