Class Of Cosmological Solutions Research Articles

On stable exponential cosmological solutions with two factor spaces in (1+ m + 2)-dimensional Einstein-Gauss-Bonnet model with Λ-term.

A [Formula: see text]-dimensional Einstein-Gauss-Bonnet gravitational model including the Gauss-Bonnet term and the cosmological term [Formula: see text] is considered. Exact solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters [Formula: see text] and [Formula: see text], corresponding to factor spaces of dimensions [Formula: see text] and [Formula: see text], respectively, are found. Under certain restrictions on [Formula: see text], the stability of the solutions in a class of cosmological solutions with diagonal metrics is proved. A subclass of solutions with small enough variation of the effective gravitational constant [Formula: see text] is considered and the stability of all solutions from this subclass is shown. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.

On Stability of Exponential Cosmological Type Solutions with Two Factor Spaces in the Einstein–Gauss–Bonnet Model with a $$\boldsymbol{\Lambda}$$-Term

We study a $$D$$-dimensional Einstein–Gauss–Bonnet gravitational model including the Gauss-Bonnet term and the cosmological constant $$\Lambda$$. We find a class of cosmological type solutions with exponential dependence of two scale factors on the variable $$u$$ (either cosmological time or a spatial coordinate), governed by two Hubble-like parameters $$H\neq 0$$ and $$h$$, corresponding to factor spaces of dimensions $$m>2$$ and $$l>2$$, respectively, and depending on the sign parameter $$\varepsilon=\pm 1$$ ($$\varepsilon=1$$ corresponds to cosmological solutions and $$\varepsilon=-1$$ to static ones). These solutions are governed by a certain master equation $$\Lambda\alpha=\lambda(x)$$ and the restriction $$\alpha\varepsilon(x-x_{+})(x-x_{-})<0$$ ($$x_{-}<x_{+}<0$$) for the ratio $$h/H=x$$, where $$\alpha=\alpha_{2}/\alpha_{1}$$ is the ratio of two constants of the model . The master equation is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals. Imposing certain restrictions on $$x$$, and we prove the stability of the solutions for $$u\to\pm\infty$$ in a certain class of cosmological solutions with diagonal metrics.

Exponential cosmological solutions with two factor spaces in EGB model with Lambda = 0 revisited

We study exact cosmological solutions in D-dimensional Einstein–Gauss–Bonnet model (with zero cosmological term) governed by two non-zero constants: alpha _1 and alpha _2 . We deal with exponential dependence (in time) of two scale factors governed by Hubble-like parameters H >0 and h, which correspond to factor spaces of dimensions m >2 and l > 2, respectively, and D = 1 + m + l. We put h ne H and mH + l h ne 0. We show that for alpha = alpha _2/alpha _1 > 0 there are two (real) solutions with two sets of Hubble-like parameters: (H_1, h_1) and (H_2, h_2), which obey: h_1/ H_1< - m/l< h_2/ H_2 < 0, while for alpha < 0 the (real) solutions are absent. We prove that the cosmological solution corresponding to (H_2, h_2) is stable in a class of cosmological solutions with diagonal metrics, while the solution corresponding to (H_1, h_1) is unstable. We present several examples of analytical solutions, e.g. stable ones with small enough variation of the effective gravitational constant G, for (m, l) = (9, l >2), (12, 11), (11,16), (15, 6).

Exact exponential cosmological solutions with two factor spaces of dimension m in EGB model with a Λ-term

A [Formula: see text]-dimensional Einstein–Gauss–Bonnet (EGB) model with the cosmological term [Formula: see text] is considered. We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters [Formula: see text] and [Formula: see text], corresponding to two factor spaces of dimension [Formula: see text] and obeying [Formula: see text]. We prove that the solutions, obeying [Formula: see text], are stable (in a class of cosmological solutions with diagonal metrics) for [Formula: see text] and they are unstable for [Formula: see text]. A subclass of solutions with small enough variation of the effective gravitational constant [Formula: see text] is considered. It is shown that all solutions from this subclass are stable.

Space-time quasicrystal structures and inflationary and late time evolution dynamics in accelerating cosmology

We construct new classes of cosmological solutions in modified and Einstein gravity theories encoding space-time quasicrystal, STQC, configurations modeled by nonlinear self-organized and pattern forming quasi-periodic structures. Such solutions are defined by generic off-diagonal locally anisotropic and inhomogeneous metrics depending via generating and integration functions on all spacetime coordinates. There are defined nonholonomic variables and conditions for the generating/integration functions and sources for effective descriptions, or approximations, as ‘quasi’ Friedmann–Lamaître–Robertson–Walker (FLRW) metrics. Such (off-) diagonal STQC-FLRW configurations contain memory on nonlinear classical and/or quantum interactions and may describe new acceleration cosmology scenarios. For special time-periodic conditions on nonlinear gravitational and matter field interactions, we can model at cosmological scales certain analogous of time crystal like structures originally postulated by Frank Wilczek in condensed matter physics. We speculate how STQC quasi-FLRW configurations could explain modern cosmology data and provide viable descriptions for the inflation and structure formation in our Universe. Finally, it is discussed systematically and critically how a unified description of inflation with dark energy era can be explained by (modified) cosmological STQC-scenarios.

Stable exponential cosmological solutions with two factor spaces in the Einstein–Gauss–Bonnet model with a $$\Lambda $$ Λ -term

We study $D$-dimensional Einstein-Gauss-Bonnet gravitational model including the Gauss-Bonnet term and the cosmological term $\Lambda$. We find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters $H >0$ and $h$, corresponding to factor spaces of dimensions $m >2$ and $l > 2$, respectively. These solutions contain a fine-tuned $\Lambda = \Lambda (x, m, l, \alpha)$, which depends upon the ratio $h/H = x$, dimensions of factor spaces $m$ and $l$, and the ratio $\alpha = \alpha_2/\alpha_1$ of two constants ($\alpha_2$ and $\alpha_1$) of the model. The master equation $\Lambda(x, m, l,\alpha) = \Lambda$ is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals. The explicit solution for $m = l$ is presented in Appendix. Imposing certain restrictions on $x$, we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. We also consider a subclass of solutions with small enough variation of the effective gravitational constant $G$ and show the stability of all solutions from this subclass.

Stable exponential cosmological solutions with 3- and l-dimensional factor spaces in the Einstein\u2013Gauss\u2013Bonnet model with a Lambda -term

A D-dimensional gravitational model with a Gauss–Bonnet term and the cosmological term Lambda is studied. We assume the metrics to be diagonal cosmological ones. For certain fine-tuned Lambda , we find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters H >0 and h, corresponding to factor spaces of dimensions 3 and l > 2, respectively and D = 1 + 3 + l. The fine-tuned Lambda = Lambda (x, l, alpha ) depends upon the ratio h/H = x, l and the ratio alpha = alpha _2/alpha _1 of two constants (alpha _2 and alpha _1) of the model. For fixed Lambda , alpha and l > 2 the equation Lambda (x,l,alpha ) = Lambda is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals (the example l =3 is presented). For certain restrictions on x we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. A subclass of solutions with small enough variation of the effective gravitational constant G is considered. It is shown that all solutions from this subclass are stable.

On non-exponential cosmological solutions with two factor spaces of dimensions m and 1 in the Einstein–Gauss–Bonnet model with a Λ-term

We consider a [Formula: see text]-dimensional Einstein–Gauss–Bonnet (EGB) model with the cosmological [Formula: see text]-term. We restrict the metrics to be diagonal ones and find for certain [Formula: see text] class of cosmological solutions with non-exponential time dependence of two scale factors of dimensions [Formula: see text] and 1. Any solution from this class describes an accelerated expansion of [Formula: see text]-dimensional subspace and tends asymptotically to isotropic solution with exponential dependence of scale factors.

Unimodular f(R,T) gravity

We extend the idea of unimodular gravity to the modified $f(R,T)$ theories. A new class of cosmological solutions, that the unimodular constraint on the metric imposes on the $f(R,T)$ theories, are studied. This extension is done in both Jordan and Einstein frames. We show that while the Lagrange multiplier (that imposes the unimodular constraint on the action) depends on the cosmic time in Jordan frame and therefore, can act as an evolving scalar field in the universe history, in the Einstein frame it acts as a cosmological constant. Then a general reconstruction method is used to realize an explicit form of the unimodular $f(R,T)$ corresponding to a given cosmological solution. By adopting a specific form of $f(R,T)$, the issue of cosmological inflation is studied in this setup. To see the observational viability of this model, a numerical analysis on the model parameter space is done in the background of Planck2015 observational data.

Stable exponential cosmological solutions with zero variation of G and three different Hubble-like parameters in the Einstein\u2013Gauss\u2013Bonnet model with a Lambda -term

We consider a D-dimensional gravitational model with a Gauss–Bonnet term and the cosmological term Lambda . We restrict the metrics to diagonal cosmological ones and find for certain Lambda a class of solutions with exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters H >0, h_1 and h_2, corresponding to factor spaces of dimensions m > 2, k_1 > 1 and k_2 > 1, respectively, with k_1 ne k_2 and D = 1 + m + k_1 + k_2. Any of these solutions describes an exponential expansion of 3d subspace with Hubble parameter H and zero variation of the effective gravitational constant G. We prove the stability of these solutions in a class of cosmological solutions with diagonal metrics.

Stable exponential cosmological solutions with zero variation of G in the Einstein\u2013Gauss\u2013Bonnet model with a Lambda -term

A D-dimensional gravitational model with a Gauss–Bonnet term and the cosmological term Lambda is considered. By assuming diagonal cosmological metrics, we find, for a certain fine-tuned Lambda , a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters H >0 and h < 0, corresponding to factor spaces of dimensions m > 3 and l > 1, respectively, with (m,l) ne (6,6), (7,4), (9,3) and D = 1 + m + l. Any of these solutions describes an exponential expansion of three-dimensional subspace with Hubble parameter H and zero variation of the effective gravitational constant G. We prove the stability of these solutions in a class of cosmological solutions with diagonal metrics.

On stable exponential solutions in Einstein–Gauss–Bonnet cosmology with zero variation of G

A D-dimensional gravitational model with a Gauss-Bonnet term and the cosmological term Lambda is considered. Assuming diagonal cosmological metrics, we find, for certain non-zero Lambda new examples of solutions with an exponential time dependence of two scale factors, governed by two Hubble-like parameters H > 0 and h < 0, corresponding to submanifolds of dimensions m and l, respectively, with (m,l) = (4,2), (5,2), (5,3), (6,7), (7,5), (7,6) and D = 1 + m + l. Any of these solutions describes an exponential expansion of "our" 3-dimensional factor-space with Hubble parameter H and zero variation of the effective gravitational constant G. We also prove the stability of these solutions in the class of cosmological solutions with diagonal metrics.

Tensor modes in bigravity: Primordial to present

Massive bigravity, a theoretically consistent modification of general relativity with an additional dynamical rank two tensor, successfully describes the observed accelerated expansion of the Universe without a cosmological constant. Recent analyses of perturbations around a cosmological background have revealed power law instabilities in both scalar and tensor perturbations, motivating an analysis of the initial conditions, evolution, and cosmological observables to determine the viability of these theories. In this paper we focus on the tensor sector, and study a primordial stochastic gravitational wave background in massive bigravity. The phenomenology can differ from standard General Relativity due to non-trivial mixing between the two linearized tensor fluctuations in the theory, only one of which couples to matter. We study perturbations about two classes of cosmological solutions in bigravity, computing the tensor contribution to the temperature anisotropies in the Cosmic Microwave Background radiation and the present stochastic gravitational wave background. The result is strongly dependent on the choice of cosmological background and initial conditions. One class of background solution generically displaying tremendous growth in the amplitude of large-wavelength gravitational waves, while the other remains observationally indistinguishable from standard General Relativity for a wide variety of initial conditions. We analyze the initial conditions for tensor modes expected in an inflationary cosmology, finding again that there is a strong dependence on the assumed background. For one choice of background, the semi-classical theory is beyond the perturbative regime. For the other choice, inflation generically yields initial conditions that, when evolved, give rise to a stochastic background observationally indistinguishable from standard General Relativity.

New exact cosmologies on the brane

We develop a method for constructing exact cosmological solutions in brane world cosmology. New classes of cosmological solutions on Randall-Sandrum brane are obtained. The superpotential and Hubble parameter are represented in quadratures. These solutions have inflationary phases under general assumptions and also describe an exit from the inflationary phase without a fine tuning of the parameters. Another class solutions can describe the current phase of accelerated expansion with or without possible exit from it.

New class of cosmological solutions for a self-interacting scalar field

New cosmological solutions are found to the system of Einstein scalar field equations using the scalar field φ as the argument. For a homogeneous and isotropic Universe, the system of equations is reduced to two equations, one of which is an equation of Hamilton–Jacobi type. Using the hyperbolically parameterized representation of this equation together with the consistency condition, explicit dependences of the potential V of the scalar field and the Hubble parameter H on φ are obtained. The dependences of the scalar field and the scale factor a on cosmic time t have also been found. It is shown that this scenario corresponds to the evolution of the Universe with accelerated expansion out to times distant from the initial singularity.

Nonlinear stability of cosmological solutions in massive gravity

We investigate nonlinear stability of two classes of cosmological solutions in massive gravity: isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) solutions and anisotropic FLRW solutions. For this purpose we construct the linear cosmological perturbation theory around axisymmetric Bianchi type-I backgrounds. We then expand the background around the two classes of solutions, which are fixed points of the background evolution equation, and analyze linear perturbations on top of it. This provides a consistent truncation of nonlinear perturbations around these fixed point solutions and allows us to analyze nonlinear stability in a simple way. In particular, it is shown that isotropic FLRW solutions exhibit nonlinear ghost instability. On the other hand, anisotropic FLRW solutions are shown to be ghost-free for a range of parameters and initial conditions.

Oscillations toward the Singularity of Locally Rotationally Symmetric Bianchi Type IX Cosmological Models with Vlasov Matter

We analyze the dynamics of a class of cosmological solutions of the Einstein–Vlasov equations. These equations describe an ensemble of collisionless particles (which represent galaxies or clusters of galaxies) that interact gravitatively through Einstein's equations of general relativity. The cosmological models we consider are spatially homogeneous, of Bianchi type IX, and locally rotationally symmetric (LRS). We prove that generic solutions exhibit an oscillatory behavior close to the singularities (the “big bang” in the past and the “big crunch” in the future); this is in contrast to the behavior of Einstein-vacuum or Einstein–Euler solutions. To establish this result we make use of dynamical systems theory; we introduce dimensionless dynamical variables that are defined on a compact state space. In this formulation the oscillatory behavior of generic solutions is represented by the existence of heteroclinic cycles on the boundary of the state space.

A class of cosmological solutions in induced matter theory with conformally flat bulk space

Some solutions of 5D conformally flat extension of induced-matter theory are derived. The corresponding induced matter solutions on the 4D hypersurfaces and their properties are presented.

Bianchi-IX string cosmological model in Lyra geometry

A class of cosmological solutions of massive strings for the Bianchi-IX space-time are obtained within the framework of Lyra geometry. Various physical and kinematical properties of the models are discussed

Viscosity effects in (1 + 1)-dimensional cosmologies

We investigate the behaviour of the solutions to the gravitational (1 + 1)-dimensional equations with a viscous fluid source in the case of a Robertson - Walker metric. In our model the bulk viscosity coefficient is related to the energy density as and we also take into account the effects of a nonlinear viscosity . We find exact solutions of the field equations for several physically interesting cases which include Friedmann cosmologies and the time reversal of Murphy's class of cosmological solutions. We find that the global effect of bulk viscosity is to decrease the expansion rate while the nonlinear viscosity does not qualitatively change the models. On the other hand, the bulk viscosity is able to remove the initial singularities which are present in the corresponding viscosity free models as in the four-dimensional case.