Kasner Regime Research Articles

Semiclassical study of the mixmaster model: The quantum Kasner map

According to the Belinski-Khalatnikov-Lifshitz conjecture, close to a spacelike singularity different spatial points decouple, and the dynamics can be described in terms of the Mixmaster (vacuum Bianchi IX) model. In order to understand the role played by quantum-gravity effects in this context, in the present work we consider the semiclassical behavior of this model. Classically, this system undergoes a series of transitions between Kasner epochs, which are described by a specific transition law. This law is derived based on the conservation of certain physical quantities and the rotational symmetry of the system. In a quantum scenario, however, fluctuations and higher-order moments modify these quantities, and consequently also the transition rule. In particular, we perform a canonical quantization of the model and then analytically obtain the modifications of this transition law for semiclassical states peaked around classical trajectories. The transition rules for the quantum moments are also obtained, and a number of interesting properties are derived concerning the coupling between the different degrees of freedom. More importantly, we show that, due to the presence of quantum-gravity effects and contrary to the classical model, there appear certain finite ranges of the Kasner parameters for which the system will not undergo further transitions and will follow a Kasner regime until the singularity. Even if a more detailed analysis is still needed, this feature points toward a possible resolution of the classical chaotic behavior of the model. Finally, a numerical integration of the equations of motion is also performed in order to verify the obtained analytical results.

Kasner geometries inside holographic superconductors

The recent study of holographic superconductors has shown the emergence of a Kasner universe behind the event horizon. This paper serves to add to the discussion by introducing two modifications to the holographic superconductor model: an axion field term and an Einstein-Maxwell-scalar (EMS) coupling term. We first discuss the effect the modification parameters have on the condensate then explore the black hole interior dynamics. Features previously identified in the interior are found in the model presented, including the collapse of the Einstein-Rosen bridge, Josephson oscillations and Kasner inversions/transitions. However, we find that by increasing the EMS coupling parameter, the collapse does not occur near the axion-Reissner-Nordström horizon and the oscillations are no longer present; the geometry entering into a Kasner regime after a large-r collapse instead.

The final Kasner regime inside black holes with scalar or vector hair

The final (close to the singularity) dynamical behavior of the metric inside black holes with massive charged scalar or vector hair is analyzed for general anisotropic and inhomogeneous initial conditions. These solutions are relevant to a holographic realization of superconductivity. It is shown that the dynamics falls within the scope of the “cosmological billiard” description and that in both cases, the corresponding hyperbolic billiard region has infinite volume so that the system ultimately settles down to a final Kasner regime. For massive vector hair, the conclusion holds because the longitudinal mode plays the same role as a scalar field. There exists, however, a measure-zero subset of solutions characterized by vanishing longitudinal modes that exhibit a chaotic behavior with an infinite number of BKL oscillations as one goes to the singularity.

Anisotropic cosmological dynamics in f(T) gravity in the presence of a perfect fluid

We consider the cosmological evolution of a flat anisotropic Universe in f(T) gravity in the presence of a perfect fluid. It is shown that the matter content of the Universe has a significant impact of the nature of a cosmological singularity in the model studied. Depending on the parameters of the f(T) function and the equation of state of the perfect fluid in question the well-known Kasner regime of general relativity can be replaced by a new anisotropic solution, or by an isotropic regime, or the cosmological singularity changes its nature to a non-standard one with a finite values of Hubble parameters. Six possible scenarios of the cosmological evolution for the model studied have been found numerically.

Realistic compactification in spatially flat vacuum cosmological models in cubic Lovelock gravity: high-dimensional case

In this paper we perform systematic investigation of all possible regimes in spatially flat vacuum cosmological models in cubic Lovelock gravity. The spatial section is considered as a product of three- and extra-dimensional isotropic subspaces, and the former represents our Universe. As the equations of motion are different for D=3, 4, 5 and general D ge 6 cases, we considered them all separately. This is the second paper of the series, and we consider D=5 and general D ge 6 cases here. For each D case we found critical values for alpha (Gauss–Bonnet coupling) and beta (cubic Lovelock coupling) which separate different dynamical cases, isotropic and anisotropic exponential solutions, and study the dynamics in each region to find all regimes for all initial conditions and for arbitrary values of alpha and beta . The results suggest that in all D ge 3 there are regimes with realistic compactification originating from so-called “generalized Taub” solution. The endpoint of the compactification regimes is either anisotropic exponential solution (for alpha > 0, mu equiv beta /alpha ^2 < mu _1 (including entire beta < 0)) or standard Kasner regime (for alpha > 0, mu > mu _1). For D ge 8 there is additional regime which originates from high-energy (cubic Lovelock) Kasner regime and ends as anisotropic exponential solution. It exists in two domains: alpha > 0, beta < 0, mu le mu _4 and entire alpha > 0, beta > 0. Let us note that for D ge 8 and alpha > 0, beta < 0, mu < mu _4 there are two realistic compactification regimes which exist at the same time and have two different anisotropic exponential solutions as a future asymptotes. For D ge 8 and alpha > 0, beta > 0, mu < mu _2 there are two realistic compactification regimes but they lead to the same anisotropic exponential solution. This behavior is quite different from the Einstein–Gauss–Bonnet case. There are two more unexpected observations among the results – all realistic compactification regimes exist only for alpha > 0 and there is no smooth transition from high-energy Kasner regime to the low-energy regime with realistic compactification.

Realistic compactification in spatially flat vacuum cosmological models in cubic Lovelock gravity: low-dimensional case

In this paper we begin to perform systematic investigation of all possible regimes in spatially flat vacuum cosmological models in cubic Lovelock gravity. We consider the spatial section to be a product of three- and extra-dimensional isotropic subspaces, with the former considered to be our Universe. As the equations of motion are different for D=3, 4, 5 and general D geqslant 6 cases, we considered them all separately. Due to the quite large amount different subcases, in the current paper we consider only D=3, 4 cases. For each D case we found values for alpha (Gauss–Bonnet coupling) and beta (cubic Lovelock coupling) which separate different dynamical cases, all isotropic and anisotropic exponential solutions, and study the dynamics in each region to find all possible regimes for all possible initial conditions and any values of alpha and beta . The results suggest that in both D cases the regimes with realistic compactification originate from so-called “generalized Taub” solution. The endpoint of the compactification regimes is either anisotropic exponential (for alpha > 0, mu equiv beta /alpha ^2 < mu _1 (including entire beta < 0)) or standard low-energy Kasner regime (for alpha > 0, mu > mu _1); as it is compactification regime, both endpoints have expanding three and contracting extra dimensions. There are two unexpected observations among the results – all realistic compactification regimes exist only for alpha > 0 and there is no smooth transition between high-energy and low-energy Kasner regimes, the latter with realistic compactification.

Effects of spatial curvature and anisotropy on the asymptotic regimes in Einstein\u2013Gauss\u2013Bonnet gravity

In this paper we address two important issues which could affect reaching the exponential and Kasner asymptotes in Einstein–Gauss–Bonnet cosmologies—spatial curvature and anisotropy in both three- and extra-dimensional subspaces. In the first part of the paper we consider the cosmological evolution of spaces that are the product of two isotropic and spatially curved subspaces. It is demonstrated that the dynamics in D=2 (the number of extra dimensions) and D ge 3 is different. It was already known that for the Lambda -term case there is a regime with “stabilization” of extra dimensions, where the expansion rate of the three-dimensional subspace as well as the scale factor (the “size”) associated with extra dimensions reaches a constant value. This regime is achieved if the curvature of the extra dimensions is negative. We demonstrate that it takes place only if the number of extra dimensions is D ge 3. In the second part of the paper we study the influence of the initial anisotropy. Our study reveals that the transition from Gauss–Bonnet Kasner regime to anisotropic exponential expansion (with three expanding and contracting extra dimensions) is stable with respect to breaking the symmetry within both three- and extra-dimensional subspaces. However, the details of the dynamics in D=2 and D ge 3 are different. Combining the two described effects allows us to construct a scenario in D ge 3, where isotropization of outer and inner subspaces is reached dynamically from rather general anisotropic initial conditions.

Realistic Compactification Models in Einstein–Gauss–Bonnet Gravity

We report the results of a study on the dynamical compactification of spatially flat cosmological models in Einstein–Gauss–Bonnet gravity. The analysis was performed in the arbitrary dimension in order to be more general. We consider both vacuum and Λ -term cases. Our results suggest that for vacuum case, realistic compactification into the Kasner (power law) regime occurs with any number of dimensions (D), while the compactification into the exponential solution occurs only for D ⩾ 2 . For the Λ -term case only compactification into the exponential solution exists, and it only occurs for D ⩾ 2 as well. Our results, combined with the bounds on Gauss–Bonnet coupling and the Λ -term ( α , Λ , respectively) from other considerations, allow for the tightening of the existing constraints and forbid α &lt; 0 .

A viable compactification scenario in Gauss-Bonnet gravity

In this talk we address two important issues which could affect reaching the exponential and Kasner asymptotes in Einstein-Gauss-Bonnet cosmologies – spatial curvature and anisotropy in both three- and extra-dimensional subspaces. In the first part we consider cosmological evolution of spaces being the product of two isotropic and spatially curved subspaces. We consider all possible number of spatial dimensions and provide description of the curvature effects in these dimensions. It is demonstrated that the dynamics in D = 2 (the number of extra dimensions) and D ≥ 3 is different. In particular, the regime with the “stabilization” of extra dimensions could be reached only if D ≥ 3. In the second part we study the influence of initial anisotropy. Our study of reveals that transition from Gauss-Bonnet Kasner regime to anisotropic exponential expansion (with expanding three and contracting extra dimensions) is stable with respect to breaking the symmetry within both three- and extra-dimensional subspaces in any number of extra dimensions. This allows us to construct a scenario where isotropisation of outer and inner subspaces is reached dynamically from rather general anisotropic initial conditions.

Cosmological dynamics of spatially flat Einstein\u2013Gauss\u2013Bonnet models in various dimensions: high-dimensional Lambda -term case

In this paper we perform a systematic study of spatially flat [(3+D)+1]-dimensional Einstein–Gauss–Bonnet cosmological models with Lambda -term. We consider models that topologically are the product of two flat isotropic subspaces with different scale factors. One of these subspaces is three-dimensional and represents our space and the other is D-dimensional and represents extra dimensions. We consider no ansatz of the scale factors, which makes our results quite general. With both Einstein–Hilbert and Gauss–Bonnet contributions in play, D=3 and the general Dge 4 cases have slightly different dynamics due to the different structure of the equations of motion. We analytically study the equations of motion in both cases and describe all possible regimes with special interest on the realistic regimes. Our analysis suggests that the only realistic regime is the transition from high-energy (Gauss–Bonnet) Kasner regime, which is the standard cosmological singularity in that case, to the anisotropic exponential regime with expanding three and contracting extra dimensions. Availability of this regime allows us to put a constraint on the value of Gauss–Bonnet coupling alpha and the Lambda -term – this regime appears in two regions on the (alpha , Lambda ) plane: alpha < 0, Lambda > 0, alpha Lambda le -3/2 and alpha > 0, alpha Lambda le (3D^2 - 7D + 6)/(4D(D-1)), including the entire Lambda < 0 region. The obtained bounds are confronted with the restrictions on alpha and Lambda from other considerations, like causality, entropy-to-viscosity ratio in AdS/CFT and others. Joint analysis constrains (alpha , Lambda ) even further: alpha > 0, D ge 2 with (3D^2 - 7D + 6)/(4D(D-1)) ge alpha Lambda ge - (D+2)(D+3)(D^2 + 5D + 12)/(8(D^2 + 3D + 6)^2).

Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in various dimensions: Vacuum case

In this paper we perform a systematic study of vacuum spatially flat ((3+D)+1)-dimensional Einstein-Gauss-Bonnet cosmological models. We consider models which topologically are the product of two flat isotropic subspaces with different scale factors. One of these subspaces is 3D and represents our space and the other is D-dimensional and represents extra dimensions. We consider no ansatz of the scale factors, which makes our results quite general. With both Einstein-Hilbert and Gauss-Bonnet contributions in play, the cases with D=1, D=2, D=3 and $D\geqslant 4$ have different dynamics due to different structure of the equations of motion. We analytically study equations of motion in all cases and describe all possible regimes. It appears that the only regimes with nonsingular future asymptotes are the Kasner regime in GR as well as exponential regimes. As of the past asymptotes, for a smooth transition only Kasner regime in Gauss-Bonnet is an option. With that at hand, we are down only to two viable regimes -- "pure" Kasner regime (transition from high- to low-energy Kasner regime) and a transition from high-energy Kasner to anisotropic exponential solution. It appears that these regimes take place for different signs of the Gauss-Bonnet coupling $\alpha$: "pure" Kasner regime occur for $\alpha > 0$ at low D and $\alpha < 0$ for high D; anisotropic exponential regime is reached only for $\alpha > 0$. So if we restrain ourselves with $\alpha > 0$ solutions, the only late-time regimes are Kasner for D=1, 2 and anisotropic exponential for $D\geqslant 2$. Also, low-energy Kasner regimes ($a(t)\propto t^p$) have expansion rates for (3+1)-dimensional subspace ("our Universe") ranging from p=0.5 (D=1) to $p=1/\sqrt{3} \approx 0.577$ ($D\to\infty$), which contradicts with dust-dominated Friedmann prediction (p=2/3).

Chaos removal inR+qR2gravity: The mixmaster model

We study the asymptotic dynamics of the Mixmaster Universe, near the cosmological singularity, considering $f(R)$ gravity up to a quadratic corrections in the Ricci scalar $R$. The analysis is performed in the scalar-tensor framework and adopting Misner-Chitr\'e-like variables to describe the Mixmaster Universe, whose dynamics resembles asymptotically a billiard-ball in a given domain of the half-Poincar\'e space. The form of the potential well depends on the spatial curvature of the model and on the particular form of the self-interacting scalar field potential. We demonstrate that the potential walls determine an open domain in the configuration region, allowing the point-Universe to reach the absolute of the considered Lobachevsky space. In other words, we outline the existence of a stable final Kasner regime in the Mixmaster evolution, implying the chaos removal near the cosmological singularity. The relevance of the present issue relies both on the general nature of the considered dynamics, allowing its direct extension to the BKL conjecture too, as well as the possibility to regard the considered modified theory of gravity as the first correction to the Einstein-Hilbert action as a Taylor expansion of a generic function $f(R)$ (as soon as a cut-off on the space-time curvature takes place).

Demonstration of the spike phenomenon using the LTB models

We demonstrate the occurrence of permanent spikes using the Lemaître–Tolman–Bondi models, chosen because the solutions are exact and can be analyzed by qualitative dynamical systems methods. Three examples are given and illustrated numerically. The third example demonstrates that spikes can form directly in the matter density, as opposed to indirectly in previous studies of spikes in the Kasner regime. Spikes provide an alternative general relativistic mechanism for generating exceptionally large structures observed in the Universe.

Dynamics of flat anisotropic models in Lovelock gravity: The even-dimensional case

In this article we give a full description of the dynamics of the flat anisotropic (4+1)-dimensional cosmological model in the presence of both Gauss-Bonnet and Einstein contributions. This is the first complete description of this model with both terms taken into account. Our data is obtained using the numerical analysis, though, we use analytics to explain some features of the results obtained, and the same analytics could be applied to higher-dimensional models in higher-order Lovelock corrections. Firstly, we investigate the vacuum model and give a description of all regimes; then, we add a matter source in the form of perfect fluid and study the influence the matter exerts upon the dynamics. Thus, we give a description of matter regimes as well. Additionally, we demonstrate that the presence of matter not only "improves" the situation with a smooth transition between the standard singularity and the Kasner regime, but also brings additional regimes and even partially "erases" the boundaries between different regimes inside the same triplet. Finally, we discuss the numerical and analytical results obtained and their generalization to the higher-order models.

A note on the viability of Gauss–Bonnet cosmology

In this Letter, we analyze the viability of a vacuum Gauss–Bonnet cosmology by examining the dynamics of the homogeneous and anisotropic background in 4+1 dimensions. The trajectories of the system either originate from the standard singularity or from non-standard type, the later is characterized by the divergence of time derivative of the Hubble parameters for its finite value. At the onset, the system should relax to Einstein phase at late times as the effect of Gauss–Bonnet term becomes negligible in the low energy regime. However, we find that most of the trajectories emerging from the standard big-bang singularity lead to future re-collapse whereas the system beginning its evolution from the non-standard singularity enters the Kasner regime at late times. This leads to the conclusion that the measure of trajectories giving rise to a smooth evolution from a standard singularity to the Einstein phase is negligibly small for generic initial conditions.

Dynamics of multidimensional generalization of Bianchi type-IX cosmological models.

We investigate the dynamics of an 11-dimensional homogeneous cosmological model. We assume that the t=const hypersurfaces are products of a 3-dimensional Bianchi type-IX space and a 7-dimensional torus. Most results of our investigation hold when the 7-dimensional torus is replaced by an m-dimensional torus ${T}^{m}$. We show that for a large class of vacuum solutions the physical space expands while the microspace contracts providing a natural mechanism of dimensional reduction. Matter satisfying a simple barotropic equation of state always breaks the process of dynamical dimensional reduction. With special attention we study the behavior of our model close to the initial singularity. In contrast with the 4-dimensional Bianchi type-IX cosmological model the Kasner solution always describes an approach to the initial singularity. We study the transition from the Kasner regime to the oscillatory regime. We show that matter does not significantly change this property. We have found some exact solutions; they describe isotropic physical space, and axially symmetric physical space. We also investigate high-temperature quantum effects and their effects on the dynamics of our model. We show that the high-temperature quantum corrections do not depend on the geometrical properties of space, provided it is compact. The high-temperature quantum corrections can be described by a macroscopic equation of state satisfied by radiation.