We consider the dynamics towards the initial singularity of Bianchi type IX vacuum and orthogonal perfect fluid models with a linear equation of state. The `Bianchi type IX attractor theorem' states that the past asymptotic behavior of generic type IX solutions is governed by Bianchi type I and II vacuum states (Mixmaster attractor). We give a comparatively short and self-contained new proof of this theorem. The proof we give is interesting in itself, but more importantly it illustrates and emphasizes that type IX is special, and to some extent misleading when one considers the broader context of generic models without symmetries.
Bianchi models are posited by the BKL picture to be essential building blocks towards an understanding of generic cosmological singularities. We study the behaviour of spatially homogeneous anisotropic vacuum spacetimes of Bianchi type VIII and IX, as they approach the big bang singularity. It is known since 2001 that generic Bianchi IX spacetimes converge towards the so-called Mixmaster attractor as time goes towards the singularity. We extend this result to the case of Bianchi VIII vacuum. The BKL picture suggests that particle horizons should form, i.e. spatially separate regions should cau…
We consider the asymptotic behaviour of spatially homogeneous spacetimes of Bianchi type IX close to the singularity (we also consider some of the other Bianchi types, e. g. Bianchi VIII in the stiff fluid case). The matter content is assumed to be an orthogonal perfect fluid with linear equation of state and zero cosmological constant. In terms of the variables of Wainwright and Hsu, we have the following results. In the stiff fluid case, the solution converges to a point for all the Bianchi class A types. For the other matter models we consider, the Bianchi IX solutions generically converge …
It is shown that the generalized Collins--Stewart radiation and Milne solutions are attractors of the massless Einstein--Vlasov system for Bianchi types II and V spacetimes, respectively. The proof is based on an energy method and bootstrap argument which are used to determine the decay rates of the perturbations away from the attractors.
We examine the Hamiltonian dynamics of bouncing Bianchi IX cosmologies in Hořava-Lifshitz gravity. The $6$-dim phase space presents two critical points, one asymptotic de Sitter attractor at infinity and a $2$-dim invariant plane. We identified four distinct parameter domains $A$, $B$, $C$ and $D$ for which the pair of critical points engenders distinct features in the dynamics. In the domain $A$ the dynamics consists basically of periodic bouncing orbits, or oscillatory orbits with a finite number of bounces. The center with multiplicity two engenders in its neighborhood the topology of stabl…
Recently cosmological dynamics of anisotropic Universe in $f(T)$ gravity became an area of intense investigations. Some earlier papers devoted to this issue contain contradictory claims about the nature and propertied of vacuum solutions in this theory. The goal of the present paper is to clarify this situation. We compare properties of $f(T)$ and $f(R)$ vacuum solutions and outline differences between them. The Kasner solution appears to be an exact solution for the $T=0$ branch, and an asymptotic solution for $T \ne 0$ branch. It is shown that Kasner solution is the past attractor if $T<0$, …
By applying a standard solution-generating transformation to an arbitrary vacuum Bianchi type II solution, one generates a new solution with spikes commonly observed in numerical simulations. It is conjectured that the spike solution is part of the generalized Mixmaster attractor.
SU(2) gauge fields coupled to an axion field can acquire an isotropic background solution during inflation. We study homogeneous but anisotropic inflationary solutions in the presence of such (massless) gauge fields. A gauge field in the cosmological background may pose a threat to spatial isotropy. We show, however, that such models $\textit{generally}$ isotropize in Bianchi type-I geometry, and the isotropic solution is the attractor. Restricting the setup by adding an axial symmetry, we revisited the numerical analysis presented in Wolfson et.al (2020). We find that the reported numerical b…
We study the asymptotic dynamics of $f(T, B)$-theory in an anisotropic Bianchi III background geometry. We show that an attractor always exists for the field equations, which depends on a free parameter provided by the specific $f(T, B)$ functional form. The attractor is an accelerated spatially flat FLRW or non-accelerated LRS Bianchi III geometry. Consequently, the $f(T, B)$-theory provides a spatially flat and isotropic accelerated Universe.
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…
We study tilted perfect fluid cosmological models with a constant equation of state parameter in spatially homogeneous models of Bianchi type VI_h using dynamical systems methods and numerical experimentation, with an emphasis on their future asymptotic evolution. We determine all of the equilibrium points of the type VI_h state space (which correspond to exact self-similar solutions of the Einstein equations, some of which are new), and their stability is investigated. We find that there are vacuum plane-wave solutions that act as future attractors. In the parameter space, a `loophole' is sho…
It is given a simple proof of the collapse conjecture for a diagonal Bianchi type-IX vacuum space-time. It is shown that the codimension of the infinity stable attractor, restricted to the anisotropy plane, is not zero, thus proving that "escape along a channel" is impossible.
We carry out a thorough analysis on a class of cosmological spacetimes which admit three space-like Killing vectors of Bianchi class B and contain electromagnetic fields. Using dynamical system analysis, we show that a family of vacuum plane-wave solutions of the Einstein-Maxwell equations is the stable attractor for expanding universes. Phase dynamics are investigated in detail for particular symmetric models. We integrate the system exactly for some special cases to confirm the qualitative features. Some of the obtained solutions have not been presented previously to the best of our knowledg…
In this paper we give, for the first time, a qualitative description of the asymptotic dynamics of a class of non-tilted spatially homogeneous (SH) cosmologies, the so-called exceptional Bianchi cosmologies, which are of Bianchi type VI$_{-1/9}$. This class is of interest for two reasons. Firstly, it is generic within the class of non-tilted SH cosmologies, being of the same generality as the models of Bianchi types VIII and IX. Secondly, it is the SH limit of a generic class of spatially inhomogeneous $G_{2}$ cosmologies. Using the orthonormal frame formalism and Hubble-normalized variables, …
We show that the tilted perfect fluid Bianchi VI$_0$ family of self-similar models found by Rosquist and Jantzen [K. Rosquist and R. T. Jantzen, \emph{% Exact power law solutions of the Einstein equations}, 1985 Phys. Lett. \textbf{107}A 29-32] is the most general class of tilted self-similar models but the state parameter $γ$ lies in the interval $(\frac 65,\frac 32) $. The model has a four dimensional stable manifold indicating the possibility that it may be future attractor, at least for the subclass of tilted Bianchi VI$_0$ models satisfying $n_α^α=0$ in which it belongs. In addition the a…
An asymptotic stability analysis of spatially homogeneous models of Bianchi type containing tilted perfect fluids is performed. Using the known attractors for the non-tilted Bianchi type universes, we check whether they are stable against perturbations with respect to tilted perfect fluids. We perform the analysis for all Bianchi class B models and the Bianchi type VI_0 model. In particular, we find that none of the non-tilted equilibrium points are stable against tilted perfect fluids stiffer than radiation. We also indicate parts of the phase space where new tilted exact solutions might be f…
Using dynamical systems theory and a detailed numerical analysis, the late-time behaviour of tilting perfect fluid Bianchi models of types IV and VII$_h$ are investigated. In particular, vacuum plane-wave spacetimes are studied and the important result that the only future attracting equilibrium points for non-inflationary fluids are the plane-wave solutions in Bianchi type VII$_h$ models is discussed. A tiny region of parameter space (the loophole) in the Bianchi type IV model is shown to contain a closed orbit which is found to act as an attractor (the Mussel attractor). From an extensive nu…
We use a dynamical systems approach to study Bianchi type VI$_0$ cosmological models containing two tilted $γ$-law perfect fluids. The full state space is 11-dimensional, but the existence of a monotonic function simplifies the analysis considerably. We restrict attention to a particular, physically interesting, invariant subspace and find all equilibrium points that are future stable in the full 11-dimensional state space; these are consequently local attractors and serve as late-time asymptotes for an open set of tilted type VI$_0$ models containing two tilted fluids. We find that if one of …
We study the asymptotic behaviour of the Bianchi type VI$_0$ universes with a tilted $γ$-law perfect fluid. The late-time attractors are found for the full 7-dimensional state space and for several interesting invariant subspaces. In particular, it is found that for the particular value of the equation of state parameter, $γ=6/5$, there exists a bifurcation line which signals a transition of stability between a non-tilted equilibrium point to an extremely tilted equilibrium point. The initial singular regime is also discussed and we argue that the initial behaviour is chaotic for $γ<2$.
We study tilted perfect fluid cosmological models with a constant equation of state parameter in spatially homogeneous models of Bianchi type VI$_{-1/9}$ using dynamical systems methods and numerical simulations. We study models with and without vorticity, with an emphasis on their future asymptotic evolution. We show that for models with vorticity there exists, in a small region of parameter space, a closed curve acting as the attractor.
Continuing previous work, we show the existence of stable, anisotropic future attractors in Bianchi invariant sets with a $p$-form field ($p\,\in\,\{1,3\}$) and a perfect fluid. In particular, we consider the not previously investigated Bianchi invariant sets $\mathcal{B}$(II), $\mathcal{B}$(IV), $\mathcal{B}$(VII$_0$) and $\mathcal{B}$(VII$_{h})$ and examine their asymptotic behaviour. We find that the isolated equilibrium set Wonderland is a future attractor on all of its existence ($2/3<\,γ\,<2$) in all these sets except in $\mathcal{B}$(II), where the peculiar equilibrium sets Edge and Rop…
In this paper we study the future asymptotics of spatially homogeneous Bianchi type II cosmologies with a tilted perfect fluid with a linear equation of state. By means of Hamiltonian methods we first find a monotone function for a special tilted case, which subsequently allows us to construct a new set of monotone functions for the general tilted type II cosmologies. In the context of a new partially gauge invariant dynamical system, this then leads to a proof for a theorem that for the first time gives a complete description of the future asymptotic states of the general tilted Bianchi type …
We consider the critical behavior at the threshold of black hole formation for the five dimensional vacuum Einstein equations satisfying the cohomogeneity-two triaxial Bianchi IX ansatz. Exploiting a discrete symmetry present in this model we predict the existence of a codimension-two attractor. This prediction is confirmed numerically and the codimension-two attractor is identified as a discretely self-similar solution with two unstable modes.
We prove the cosmic no-hair conjecture for all orthogonal Bianchi cosmologies with matter in the $R+βR^2$ theory using the conformally equivalent Einstein field equations, with the scalar field having the full self-interacting potential, in the presence of the conformally related matter fields. We show, in particular, that the Bianchi IX universe asymptotically approaches de Sitter space provided that initially the scalar three-curvature does not exceed the potential of the scalar field associated with the conformal transformation. Our proof relies on rigorous estimates of the possible bounds …
We present the full set of evolution equations for the spatially homogeneous cosmologies of type VI$_h$ filled with a tilted perfect fluid and we provide the corresponding equilibrium points of the resulting dynamical state space. It is found that only when the group parameter satisfies $h>-1$ a self-similar solution exists. In particular we show that for $h>-{\frac 19}$ there exists a self-similar equilibrium point provided that $γ\in ({\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}},{\frac 32}) $ whereas for $h<-{\frac 19}$ the state parameter belongs to the interval $γ\in (1,{\frac{2(3+\sqrt{-h})}{5+3…
We apply the dynamical systems approach to ever-expanding Bianchi type VIII cosmologies filled with a tilted $γ$-fluid undergoing velocity diffusion on a scalar field. We determine the future attractors and investigate the late-time behaviour of the models. We find that at late times the normalized energy density $Ω$ tends to zero, while the scalar potential $Φ$ approaches 1 and dominates the evolution. Moreover, we demonstrate that in presence of diffusion fluids with $γ<3/2$, which includes physically important cases of dust $(γ=1)$ and radiation $(γ=4/3)$, are asymptotically non-tilted; the…
It is shown that all contracting, spatially homogeneous, orthogonal Bianchi cosmologies that are sourced by an ultra-stiff fluid with an arbitrary and, in general, varying equation of state asymptote to the spatially flat and isotropic universe in the neighbourhood of the big crunch singularity. This result is employed to investigate the asymptotic dynamics of a collapsing Bianchi type IX universe sourced by a scalar field rolling down a steep, negative exponential potential. A toroidally compactified version of M*-theory that leads to such a potential is discussed and it is shown that the iso…
In this paper the dynamics of free gauge fields in Bianchi type I-VII$_{h}$ space-times is investigated. The general equations for a matter sector consisting of a $p$-form field strength ($p\,\in\,\{1,3\}$), a cosmological constant ($4$-form) and perfect fluid in Bianchi type I-VII$_{h}$ space-times are computed using the orthonormal frame method. The number of independent components of a $p$-form in all Bianchi types I-IX are derived and, by means of the dynamical systems approach, the behaviour of such fields in Bianchi type I and V are studied. Both a local and a global analysis are perform…
In this paper we investigate, from the dynamical systems perspective, the evolution of an scalar field with arbitrary potential trapped in a Randall-Sundrum's Braneworld of type 2. We consider an homogeneous but anisotropic Bianchi I (BI) brane filled also with a perfect fluid. We also consider the effect of the projection of the five-dimensional Weyl tensor onto the three-brane in the form of a positive Dark Radiation term. Using the center manifold theory we obtain sufficient conditions for the asymptotic stability of de Sitter solution with standard 4D behavior. We also prove that there are…
We present the derivation, for these vacuum metrics, of the Painlevé VI equation first obtained by Christodoulakis and Terzis, from the field equations for both minkowskian and euclidean signatures. This allows a complete discussion and the precise connection with some old results due to Kinnersley. The hyperkähler metrics are shown to belong to the Multi-Centre class and for the cases exhibiting an integrable geodesic flow the relevant Killing tensors are given. We conclude by the proof that for the Bianchi B family, excluding type III, there are no hyperkähler metrics.
We give a very short proof that the vacuum Einstein equations in 4+1 dimensions have no cohomogeneity-two Bianchi IX continuously self-similar solutions.
A rationality result previously proved for Robertson-Walker metrics is extended to a homogeneous anisotropic cosmological model, namely the Bianchi type-IX minisuperspace. It is shown that the Seeley-de Witt coefficients appearing in the expansion of the spectral action for the Bianchi type-IX geometry are expressed in terms of polynomials with rational coefficients in the cosmic evolution factors $w_1(t), w_2(t), w_3(t),$ and their higher derivates with respect to time. We begin with the computation of the Dirac operator of this geometry and calculate the coefficients $a_0, a_2, a_4$ of the s…
We develop a Hamiltonian formulation of the Bianchi type I space-time in conformal gravity, i.e. the theory described by a Lagrangian that is defined by the contracted quadratic product of the Weyl tensor, in a four-dimensional space-time. We derive the explicit forms of the super-Hamiltonian and of the constraint expressing the conformal invariance of the theory and we write down the system of canonical equations. To seek out exact solutions of this system we add extra constraints on the canonical variables and we go through a global involution algorithm which eventually leads to the closure …
The dissipative dynamics anticipated in the proof of 't Hooft's existence theorem -- "For any quantum system there exists at least one deterministic model that reproduces all its dynamics after prequantization" -- is constructed here explicitly. We propose a generalization of Liouville's classical phase space equation, incorporating dissipation and diffusion, and demonstrate that it describes the emergence of quantum states and their dynamics in the Schroedinger picture. Asymptotically, there is a stable ground state and two decoupled sets of degrees of freedom, which transform into each other…
We consider spatially homogeneous Hořava-Lifshitz (HL) models that perturb General Relativity (GR) by a parameter $v\in (0,1)$ such that GR occurs at $v=1/2$. We describe the dynamics for the extremal case $v=0$, which possess the usual Bianchi hierarchy: type $\mathrm{I}$ (Kasner circle of equilibria), type $\mathrm{II}$ (heteroclinics that induce the Kasner map) and type $\mathrm{VI_0},\mathrm{VII_0}$ (further heteroclinics). For type $\mathrm{VIII}$ and $\mathrm{IX}$, we use a computer-assisted approach to prove the existence of periodic orbits which are far from the Mixmaster attractor and…
Recently an inflationary model with a vector field coupled to the inflaton was proposed and the phenomenology studied for the Bianchi type I spacetime. It was found that the model demonstrates a counter-example to the cosmic no-hair theorem since there exists a stable anisotropically inflationary fix-point. One of the great triumphs of inflation, however, is that it explains the observed flatness and isotropy of the universe today without requiring special initial conditions. Any acceptable model for inflation should thus explain these observations in a satisfactory way. To check whether the m…
