We analyze the global dynamics of Bianchi type I solutions of the Einstein equations with anisotropic matter. The matter model is not specified explicitly but only through a set of mild and physically motivated assumptions; thereby our analysis covers matter models as different from each other as, e.g., collisionless matter, elastic matter and magnetic fields. The main result we prove is the existence of an `anisotropy classification' for the asymptotic behaviour of Bianchi type I cosmologies. The type of asymptotic behaviour of generic solutions is determined by one single parameter that describes certain properties of the anisotropic matter model under extreme conditions. The anisotropy classification comprises the following types. The convergent type \Aplus: Each solution converges to a Kasner solution as the singularity is approached and each Kasner solution is a possible past asymptotic state. The convergent types \Bplus and \Cplus: Each solution converges to a Kasner solution as the singularity is approached; however, the set of Kasner solutions that are possible past asymptotic states is restricted. The oscillatory type \Dplus: Each solution oscillates between different Kasner solutions as the singularity is approached. Furthermore, we investigate non-generic asymptotic behaviour and the future asymptotic behaviour of solutions.
We present diagonal Einstein metrics for Bianchi III and V, both for minkowskian and euclidean signatures and we show that the Einstein Bianchi III metrics have an integrable geodesic flow.
We study Bianchi type $I$ cosmological model in the presence of magnetized anisotropic dark energy. The energy-momentum tensor consists of anisotropic fluid with anisotropic EoS $p=ωρ$ and a uniform magnetic field of energy density $ρ_B$. We obtain exact solutions to the field equations using the condition that expansion is proportional to the shear scalar. The physical behavior of the model is discussed with and without magnetic field. We conclude that universe model as well as anisotropic fluid do not approach isotropy through the evolution of the universe.
This paper is devoted to study the Bianchi type III model in the presence of anisotropic fluid in f(R) gravity. Exponential and power-law volumetric expansions are used to obtain exact solutions of the field equations. We discuss the physical behavior of the solutions and anisotropy behavior of the fluid, the expansion parameter and the model in future evolution of the universe.
Spatially homogeneous and anisotropic Bianchi type $VI_0$ cosmological models with cosmological constant are investigated in the presence of anisotropic dark energy. We examine the effects of electromagnetic field on the dynamics of the universe and anisotropic behavior of dark energy. The law of variation of the mean Hubble parameter is used to find exact solutions of the Einstein field equations. We find that electromagnetic field promotes anisotropic behavior of dark energy which becomes isotropic for future evolution. It is concluded that the isotropic behavior of the universe model is see…
We prove a global in time existence theorem, for the initial value problem for the Einstein-Boltzmann system, with arbitrarily large initial data, in the homogeneous case, in a Bianchi type I space-time
The Bianchi type III dark energy models with constant deceleration parameter are investigated. The equation of state parameter $ω$ is found to be time dependent and its existing range for this model is consistent with the recent observations of SN Ia data, SN Ia data (with CMBR anisotropy) and galaxy clustering statistics. The physical aspect of the dark energy models are discussed.
Intrinsic time-dependent invariants are constructed for classical, flat, homogeneous, anisotropic cosmology with a massless scalar material source. Invariance under the time reparameterization-induced canonical symmetry group is displayed explicitly.
The main purpose of this paper is to explore the solutions of Bianchi type $III$ cosmological model in Brans Dicke theory of gravity in the background of anisotropic dark energy. We use the assumption of constant deceleration parameter and power law relation between scalar field $ϕ$ and scale factor $a$ to find the solutions. The physical behavior of the solutions has been discussed using some physical quantities.
A spatially homogeneous and anisotropic Bianchi Type I universe has been studied with ω <-1 without Big Smash. It is demonstrated that if cosmic dark energy behaves like a fluid with equation of state p = ωρ (p and ρ being pressure and energy density respectively) as well as generalized chaplygin gas simultaneously, Big Rip or Big Smash problem does not arise even for equation of state parameter ω <-1 unlike other phantom models, here, the scale factor for Bianchi Type I universe is found regular for all time. The present model is derived by using law of variation of Hubble's parameter from Bi…
In this article, dedicated to one of the best specialist of the FOSH systems, we couple the Bianchi equations with the equations satisfied by the dynamical acceleration of a charged fluid and the derivatives of the associated Maxwell field.
Global existence to the coupled Einstein-Maxwell system which rules the dynamics of a kind of charged matter with a pseudo-tensor of pressure is proved, in Bianchi I-VIII spacetimes. We study the geodesics completeness, the asymptotic behavior, the positivity conditions, and we prove that the problem is well-posed in the sense of Hadamard.
We will present a complete set of equations, in the form of an Einstein-Bianchi system, that describe the evolution of generic smooth lattices in spacetime. All 20 independent Riemann curvatures will be evolved in parallel with the leg-lengths of the lattice. We will show that the evolution equations for the curvatures forms a hyperbolic system and that the associated constraints are preserved. This work is a generalisation of our previous paper arXiv:1101.3171 on the Einstein-Bianchi system for the Schwarzschild spacetime to general 3+1 vacuum spacetimes.
Using the methods developed for different Bianchi class A cosmological models we treat the simplest Bianchi class B model, namely Bianchi type V. The future non-linear stability for solutions of the Einstein-Vlasov system is demonstrated and it is shown that these solutions are asymptotically stable to the Milne solution. Within the isotropic solutions of the Einstein-Vlasov system the spatially flat Friedmann solution is unstable within this class, and expanding models tend also to the Milne solution.
The late-time behaviour of the Einstein-dust system is well understood for homogeneous spacetimes. For the case of Bianchi I we have been able to show that the late-time behaviour of the Einstein-Vlasov system is well approximated by the Einstein-dust system assuming that one is close to the unique stationary solution which is the attractor of the Einstein-dust system.
Here, we have investigated the interaction of Bianchi type I anisotropic cloud string cosmological model universe with electromagnetic field in the context of general relativity. In this paper, the energy momentum tensor is assumed to be the sum of the rest energy density and string tension density with an electromagnetic field. To obtain exact solutions of Einstein field equations, we take average scale factor as an integrating function of time. Also, the dynamics and significance of various physical parameters of model are discussed.
Using the methods developed for the Bianchi I case we have shown that a boostrap argument is also suitable to treat the future non-linear stability for reflection symmetric solutions of the Einstein-Vlasov system of Bianchi types II and VI$_0$. These solutions are asymptotic to the Collins-Stewart solution with dust and the Ellis-MacCallum solution respectively. We have thus generalized the results obtained by Rendall and Uggla in the case of locally rotationally symmetric Bianchi II spacetimes to the reflection symmetric case. However we needed to assume small data. For Bianchi VI$_0$ there i…
This paper is a study of the effects of anisotropic matter sources on the qualitative evolution of spatially homogenous cosmologies of Bianchi type VIII. The analysis is based on a dynamical system approach and makes use of an anisotropic matter family developed by Calogero and Heinzle which generalises perfect fluids and provides a measure of deviation from isotropy. Thereby the role of perfect fluid solutions is put into a broader context. The results of this paper concern the past and future asymptotic dynamics of locally rotationally symmetric solutions of type VIII with anisotropic matter…
Within the scope of an anisotropic Bianchi type-VI cosmological model we have studied the evolution of the universe filled with perfect fluid and dark energy. To get the deterministic model of Universe, we assume that the shear scalar $(σ)$ in the model is proportional to expansion scalar $(\vartheta)$. This assumption allows only isotropic distribution of fluid. Exact solution to the corresponding equations are obtained. The EoS parameter for dark energy as well as deceleration parameter is found to be the time varying functions. Using the observational data qualitative picture of the evoluti…
The dynamics of cosmological models with isotropic matter sources (perfect fluids) is extensively studied in the literature; in comparison, the dynamics of cosmological models with anisotropic matter sources is not. In this paper we consider spatially homogeneous locally rotationally symmetric solutions of the Einstein equations with a large class of anisotropic matter models including collisionless matter (Vlasov), elastic matter, and magnetic fields. The dynamics of models of Bianchi types I, II, and IX are completely described; the two most striking results are the following: (i) There exis…
Using the one-parameter internal symmetry group in the Bianchi type-I spacetime for cosmological models with a perfect fluid, we show that a system of coordinates exists in the associated internal space where two scale factors become equal. We find the general solution for an anisotropic model containing a perfect fluid with constant baryotropic index and investigate the asymptotic regimes. We obtain exact solutions for a set of anisotropic fluids which includes an anisotropic stiff fluid.
Bianchi type I magnetized cosmological models in the presence of a bulk viscous fluid are investigated. The source of the magnetic field is due to an electric current produced along x-axis. The distribution consists of an electrically neutral viscous fluid with an infinite electrical conductivity. The coefficient of bulk viscosity is assumed to be a power function of mass density. The cosmological constant $Λ$ is found to be positive and is a decreasing function of time which is supported by results from recent supernovae observations. The behaviour of the models in presence and absence of mag…
We present a general study of the dynamical properties of Anisotropic Bianchi Universes in the context of Einstein General Relativity. Integrability results using Kovalevskaya exponents are reported and connected to general knowledge about Bianchi dynamics. Finally, dynamics toward singularity in Bianchi type VIII and IX universes are showed to be equivalent in some precise sence.
Locally rotationally symmetric (LRS) Bianchi Type I cosmological models are examined in the presence of dynamically anisotropic dark energy and perfect fluid. We assume that the dark energy (DE) is minimally interacting, has dynamical energy density, anisotropic equation of state parameter (EoS). The conservation of the energy-momentum tensor of the DE is assumed to consist of two separately additive conserved parts. A special law is assumed for the deviation from isotropic EoS, which is consistent with the assumption on the conservation of the energy-momentum tensor of the DE. Exact solutions…
In this work we investigate the asymptotic behaviour of solutions to the Einstein equations with a minimally coupled scalar field. The primary focus of the present paper here establishing under what conditions a solution becomes "asymptotically Kasner" sufficiently close to the initial singularity. To address this question we restrict our attention to Bianchi I space-times. By restricting our attention to a strictly monotonic scalar field we are able to provide necessary conditions on a potential so that the resulting solution is asymptotically Kasner. Moreover, we provide both explicit and nu…
The 1+3 covariant equations, embedded in an extended tetrad formalism and describing a space-time with an arbitrary energy-momentum distribution, are reconsidered. It is shown that, provided the 1+3 splitting is performed with respect to a generic timelike congruence with tangent vector u, the Einstein field equations can be regarded as the integrability conditions for the Jacobi and Bianchi equations together with the Ricci equations for u. The same conclusion holds for a generic null congruence in the Newman-Penrose framework.
The general form of the anisotropy parameter of the expansion for Bianchi type-III metric is obtained in the presence of a single diagonal imperfect fluid with a dynamically anisotropic equation of state parameter and a dynamical energy density in general relativity. A special law is assumed for the anisotropy of the fluid which reduces the anisotropy parameter of the expansion to a simple form. The exact solutions of the Einstein field equations, under the assumption on the anisotropy of the fluid, are obtained for exponential and power-law volumetric expansions. The isotropy of the fluid, sp…
By employing the Bianchi identities for the Riemann tensor in conjunction with the Einstein equations, we construct a first order symmetric hyperbolic system for the evolution part of the Cauchy problem of general relativity. In this system, the metric evolves at zero speed with respect to observers at rest in a foliation of spacetime by spacelike hypersurfaces while the curvature and connection propagate at the speed of light. The system has no unphysical characteristics, and matter sources can be included.
The present study deals with a spatially homogeneous and anisotropic Bianchi-II cosmological model representing massive strings. The energy-momentum tensor, as formulated by Letelier (1983), has been used to construct a massive string cosmological model for which the expansion scalar is proportional to one of the components of shear tensor. The Einstein's field equations have been solved by applying a variation law for generalized Hubble's parameter that yields a constant value of deceleration parameter in Bianchi-II space-time. A comparative study of accelerating and decelerating modes of the…
The anisotropic Bianchi type I in multi-scalar field cosmology is studied with a particular potential of the form $\rm V= V_0 e^{-\left[λ_1 ϕ_1 + \cdots + λ_n ϕ_n \right]}\,,$ which emerges as a condition between the time derivatives of their corresponding momenta. Using the Hamiltonian formalism for the inflation epoch with a quintessence framework we find the exact solutions for the Einstein-Klein-Gordon (EKG) system with different scenarios specified by the parameter $\rm λ^2= \sum_{i=1}^n λ_i^2$. For the quantum scheme of this model, the corresponding Wheeler-DeWitt (WDW) equation is solve…
In this work, we study the effect of a magnetic field on the growth of cosmological perturbations. We develop a mathematical consistent treatment in which a perfect fluid and a uniform magnetic field evolve together in a Bianchi I universe. We then study the energy density perturbations on this background with particular emphasis on the effect of the background magnetic field. We develop a full relativistic solution which refines previous analysis in the relativistic limit, recovers the known ones in the Newtonian treatment with adiabatic sound speed, and it adds anisotropic effects to the rel…
I study the dynamical effects due to the Brans-Dicke scalar $ϕ$-field at the early stages of a supposedly anisotropic Universe expansion in the scalar-tensor cosmology of Jordan-Brans-Dicke. This is done considering the behaviour of the general solutions for the homogeneous model of Bianchi type VII in the vacuum case. I conclude that the Bianchi-VII$_0$ model shows an isotropic expansion and that its only physical solution is equivalent to a Friedman-Robertson-Walker spacetime whose evolution can, depending on the value of the JBD coupling constant, begin in a singularity and, after expanding…
We investigate the dynamics of spatially homogeneous solutions of the Einstein-Vlasov equations with Bianchi type I symmetry by using dynamical systems methods. All models are forever expanding and isotropize toward the future; toward the past there exists a singularity. We identify and describe all possible past asymptotic states; in particular, on the past attractor set we establish the existence of a heteroclinic network, which is a new type of feature in general relativity. This illustrates among other things that Vlasov matter can lead to quite different dynamics of cosmological models as…
In order to investigate if the anisotropy of the spacetime may induce future singularities at a finite value of the cosmic time on the evolution of cosmological models, we study vacuum and non-vacuum Bianchi type I spacetimes exhibiting such future singularities. We show that in the case of Kasner vacuum cosmologies the spacetime may rip itself apart in a finite time, and only in the direction corresponding to the unique scale factor which diverges at this finite value of the cosmic time. The other two directional scale factors and the average scale factor do not diverge and tend to zero at th…
We present the solution space of the field equations in the Einstein-aether theory for the case of a vacuum Bianchi Type V space-time. We also find that there are portions of the initial parameters space for which no solution is admitted by the reduced equations. Whenever solutions do exist, their physical interpretation is examined through the behavior of Ricci and/or Kretsmann scalar, as well as with the identification of the effective energy momentum tensor in terms of a perfect fluid. There are cases in which no singularities appear and others where the effective fluid is isotropic.
The dynamics of solutions of the Einstein-Vlasov system with Bianchi I symmetry is discussed in the case of massive or massless particles. It is shown that in the case of massive particles the solutions are asymptotic to isotropic dust solutions at late times. The initial singularity is more difficult to analyse. It is shown that the asymptotic behaviour there must be one of a small set of possibilities but it is not clear whether all of these possibilities are realized. One solution is exhibited in the case of massless particles which behaves quite differently near the singularity from any Bi…
The second Bianchi identity can be recast as an evolution equation for the Riemann curvatures. Here we will report on such a system for a vacuum static spherically symmetric spacetime. This is the first of two papers. In the following paper we will extend the ideas developed here to general vacuum spacetimes. In this paper we will demonstrate our ideas on a Schwarzschild spacetime and give detailed numerical results. For suitable choices of lapse function we find that the system gives excellent results with long term stability.
In this article we compute the density of Dirac particles created by a cosmological anisotropic Bianchi I universe in the presence of a constant electric field. We show that the particle distribution becomes thermal when one neglects the electric interaction.
We derive the metric for a Bianchi type I space-time with energy density that is dominated by that of a perfect fluid with equation of state $p=wρ$ and whose anisotropy is seeded by a fixed norm spacelike vector field. We solve for the evolution of perturbations about this space-time. In particular, the Jeans instability in an expanding flat Friedmann-Robertson-Walker universe is modified by the presence of the vector field so that energy density perturbations develop direction-dependent growth. We also briefly consider observational limits on the vector field vacuum expectation value, $m$. We…
We construct numerical solutions to the higher-dimensional Einstein-Maxwell theory. The solutions are based on embedding the four dimensional Bianchi type IX space in the theory. We find the solutions as superposition of two functions, which one of them can be found numerically. We show that the solutions in any dimensions, are almost regular everywhere, except a singular point. We find that the solutions interpolate between the two exact analytical solutions to the higher dimensional Einstein-Maxwell theory, which are based on Eguchi-Hanson type I and II geometries. Moreover, we construct the…
Evolution of an anisotropic universe described by a Bianchi type I (BI) model in presence of nonlinear spinor field has been studied by us recently in a series of papers. On offer the Bianchi models, those are both inhomogeneous and anisotropic. Within the scope of Bianchi type VI (BVI) model the self-consistent system of nonlinear spinor and gravitational fields are considered. The role of inhomogeneity in the evolution of spinor and gravitational field is studied.
In one-loop string effective action, we study a generality of non-singular cosmological solutions found in the isotropic and homogeneous case. We discuss Bianchi I and IX type spacetimes. We find that nonsingular solutions still exist in Bianchi I model around nonsingular flat Friedmann solutions. On the other hand, we cannot find any nonsingular solutions in Bianchi IX model. Non-existence of nonsingular Bianchi IX universe may be consistent with the analysis by Kawai, Sakagami and Soda, i.e. the tensor mode perturbations against nonsingular flat Friedmann universe are unstable, because Bianc…
In a recent paper arXiv:1208.4231 we have treated the future non-linear stability for reflection symmetric solutions of the Einstein-Vlasov system of Bianchi types II and VI$_0$. We have been able now to remove the reflection symmetry assumption, thus treating the non-diagonal case. Apart from the increasing complexity the methods have been essentially the same as in the diagonal case, showing that they are thus quite powerful. Here the challenge was to put the equations in a form that permits the use of the previous results. We are able to conclude that after a possible basis change the futur…
Cosmic Microwave Background anomalies suggest that the Universe is slightly anisotropic, so a Cosmological Model that does not satisfy the Cosmological Principle is necessary. Anisotropic Bianchi models have been studied given that they would be necessary to understand the early stages of the Universe. In this paper, we analyze qualitatively the behavior of the cosmological redshift in a Bianchi type I model after a coordinate transformation in three cases: vacuum with no vanishing cosmological constant $Λ\not= 0$, universe dominated by dust and universe dominated by radiation.
In this article we compute the density of scalar and Dirac particles created by a cosmological anisotropic Bianchi type I universe in the presence of a time varying electric field. We show that the particle distribution becomes thermal when one neglects the electric interaction.
In this paper, the cosmic no hair theorem for anisotropic Bianchi models which admit inflation with a scalar field is studied in the framework of Brane world. It is found that all Bianchi models except Bianchi type IX, transit to an inflationary regime with vanishing anisotropy. In the Brane world, anisotropic universe approaches the inflationary era much faster than that in the general theory of relativity. The form of the potential does not affect the evolution in the inflationary epoch. However, the late time behaviour is controlled by a constant additive factor in the potential for the inf…
We construct new classes of cosmological solution to the five dimensional Einstein-Maxwell-dilaton theory, that are non-stationary and almost conformally regular everywhere. The base geometry for the solutions is the four-dimensional Bianchi type IX geometry. In the theory, the dilaton field is coupled to the electromagnetic field and the cosmological constant term, with two different coupling constants. We consider all possible solutions with different values of the coupling constants, where the cosmological constant takes any positive, negative or zero values. In the ansatzes for the metric,…
The paper deals with Bianchi type V Universe, which has dynamical energy density. We consider Bianchi type V space-time, introducing three different skewness parameters along spatial directions to quantify the deviation of pressure from isotropy. To study the anisotropic nature of the dynamical dark energy, we assume that the skewness parameters are time dependent. It is found that the Universe achieves flatness in quintessence model. The physical behavior of the Universe has been discussed in detail.
In the context of modified tele-parallel theory of gravity, we undertake cosmological anisotropic models and search for their solutions. Within a suitable choice of non-diagonal tetrads, the decoupled equations of motion are obtained for Bianchi-I, Bianchi-III and Kantowski-Sachs models, from which we obtain the correspondent solutions. By the way, energy density and pressures are also obtained, showing, as an important result, that our universe may live a quintessence like universe even still anisotropic models are considered.
Some features of the Bianchi type-I universes in the presence of a fluid that wields an anisotropic equation of state (EoS) parameter are discussed in the context of general relativity. The models that exhibit de Sitter volumetric expansion due to the constant effective energy density (the sum of the energy density of the fluid and the anisotropy energy density) are of particular interest. We also introduce two locally rotationally symmetric models, which exhibit de Sitter volumetric expansion in the presence of a hypothetical fluid that has been obtained by minimally altering the conventional…
We solve the Wheeler-DeWitt equation for the minisuperspace of a cosmological model of Bianchi type I with a minimally coupled massive scalar field $ϕ$ as source by generalizing the calculation of Lukash and Schmidt [1]. Contrarily to other approaches we allow strong anisotropy. Combining analytical and numerical methods, we apply an adiabatic approximation for $ϕ$, and as new feature we find a period-doubling bifurcation. This bifurcation takes place near the cosmological quantum boundary, i.e., the boundary of the quasiclassical region with oscillating $ψ$-function where the WKB-approximatio…
The dynamics of Dvali-Gabadadze-Porrati Cosmology (DGP) braneworld with an anisotropic brane is studied. The Friedmann equations and their solutions are obtained for two branches of anisotropic DGP model. The late time behavior in DGP cosmology is examined in the presence of anisotropy which shows that universe enters a self-accelerating phase much later compared to the isotropic case. The acceleration conditions and slow-roll conditions for inflation are obtained.
We analyse in a systematic way the (non-)compact four dimensional Einstein-Weyl spaces equipped with a Bianchi metric. We show that Einstein-Weyl structures with a Class A Bianchi metric have a conformal scalar curvature of constant sign on the manifold. Moreover, we prove that most of them are conformally Einstein or conformally Kähler ; in the non-exact Einstein-Weyl case with a Bianchi metric of the type $VII_0, VIII$ or $IX$, we show that the distance may be taken in a diagonal form and we obtain its explicit 4-parameters expression. This extends our previous analysis, limited to the diago…
In this work, we study a cosmological model of Bianchi type-I Universe in teleparallel gravity for a perfect fluid. To obtain the cosmological solution of the model, we assume that the deceleration parameter is a linear function of the Hubble parameter $H$ i.e. $q=-1+βH$ (where $β$ as a positive constant). Consequently, we get a model of our Universe, where it goes from the initial phase of deceleration to the current phase of acceleration. We have discussed some physical and geometric properties such as Hubble parameter, deceleration parameter, energy density, pressure, and equation of state …
