Revisiting cosmologies in teleparallelism

We discuss the most general field equations for cosmological spacetimes for theories of gravity based on non-linear extensions of the non-metricity scalar and the torsion scalar. Our approach is based on a systematic symmetry-reduction of the metric-affine geometry which underlies these theories. While for the simplest conceivable case the connection disappears from the field equations and one obtains the Friedmann equations of General Relativity, we show that in $f(\mathbb{Q})$ cosmology the connection generically modifies the metric field equations and that some of the connection components become dynamical. We show that $f(\mathbb{Q})$ cosmology contains the exact General Relativity solutions and also exact solutions which go beyond. In $f(\mathbb{T})$~cosmology, however, the connection is completely fixed and not dynamical.

Applied Mathematics of Space-time & Space+time: Problems in General Relativity and Cosmology

<p>Cosmography is the part of cosmology that proceeds by making minimal dynamic assumptions. That is, one does not assume the Friedmann equations (Einstein equations) unless and until absolutely necessary. On the other hand, cosmodynamics is the part of cosmology that relates the geometry to the density and pressure using the Friedmann equations. In both frameworks, we consider the amount of information and the nature of the constraints we can obtain from the Hubble flow in a FLRW universe. Indeed, the cosmological parameters contained in the Hubble relation between distance and redshift provide information on the behaviour of the universe (expansion, acceleration etc...). In the first framework, it is possible to concentrate more directly on the observational situation in a model-independent manner. We perform a number of inter-related cosmographic fits to supernova datasets, and pay particular attention to the extent to which the choice of distance scale and manner of representing the redshift scale affect the cosmological parameters. In the second framework, we use the class of w-parameter models which has become increasingly popular in the last decade. We explore the extent to which a constraint on the w-parameter leads to useful and non-trivial constraints on the Hubble flow in terms of cosmological parameters H(z), density p(z), density parameter O(z), distance scales d(z), and lookback time T(z). On another front, Numerical Relativity has experienced many breakthroughs since 2005, with full inspiral-merger-ringdown simulations now possible. One of the main goals is to provide very accurate templates of gravitational waves for ground-based and space-based interferometers. We explore the potential of a very recent and accurate numerical method, the Spectral Element Method (SEM), for Numerical Relativity, by treating a singular Schwarszchild black hole evolution as a test case. Spectral elements combine the theory of spectral and pseudo-spectral methods for high order polynomials and the variational formulation of finite elements and the associated geometric flexibility. We use the BSSN formulation of the Einstein equations with the method of the moving punctures. After applying the variational formulation to the BSSN system, we present several possible weak forms of this system and its spectral element discretization in space. We use a Runge-Kutta fourth order time discretization. The accuracy of high order methods can deteriorate in the presence of discontinuities or sharp gradients. We show that we can treat the element that contains the puncture with a filtering method to avoid artificial and spurious oscillations. These might form and propagate into the domain coming from discontinuous initial data from the BSSN system.</p>

Applied Mathematics of Space-time & Space+time: Problems in General Relativity and Cosmology

<p>Cosmography is the part of cosmology that proceeds by making minimal dynamic assumptions. That is, one does not assume the Friedmann equations (Einstein equations) unless and until absolutely necessary. On the other hand, cosmodynamics is the part of cosmology that relates the geometry to the density and pressure using the Friedmann equations. In both frameworks, we consider the amount of information and the nature of the constraints we can obtain from the Hubble flow in a FLRW universe. Indeed, the cosmological parameters contained in the Hubble relation between distance and redshift provide information on the behaviour of the universe (expansion, acceleration etc...). In the first framework, it is possible to concentrate more directly on the observational situation in a model-independent manner. We perform a number of inter-related cosmographic fits to supernova datasets, and pay particular attention to the extent to which the choice of distance scale and manner of representing the redshift scale affect the cosmological parameters. In the second framework, we use the class of w-parameter models which has become increasingly popular in the last decade. We explore the extent to which a constraint on the w-parameter leads to useful and non-trivial constraints on the Hubble flow in terms of cosmological parameters H(z), density p(z), density parameter O(z), distance scales d(z), and lookback time T(z). On another front, Numerical Relativity has experienced many breakthroughs since 2005, with full inspiral-merger-ringdown simulations now possible. One of the main goals is to provide very accurate templates of gravitational waves for ground-based and space-based interferometers. We explore the potential of a very recent and accurate numerical method, the Spectral Element Method (SEM), for Numerical Relativity, by treating a singular Schwarszchild black hole evolution as a test case. Spectral elements combine the theory of spectral and pseudo-spectral methods for high order polynomials and the variational formulation of finite elements and the associated geometric flexibility. We use the BSSN formulation of the Einstein equations with the method of the moving punctures. After applying the variational formulation to the BSSN system, we present several possible weak forms of this system and its spectral element discretization in space. We use a Runge-Kutta fourth order time discretization. The accuracy of high order methods can deteriorate in the presence of discontinuities or sharp gradients. We show that we can treat the element that contains the puncture with a filtering method to avoid artificial and spurious oscillations. These might form and propagate into the domain coming from discontinuous initial data from the BSSN system.</p>

Cosmological dynamics

The chapter deals with the large-scale dynamics of the universe. First the Friedmann equations are obtained from the Einstein field equation, and they are interpreted with the aid of a Newtonian comparison. Then the application to the universe modelled as a collection of ideal fluids is described. Density parameters and the equation of the state are defined, and the main features of the evolution of matter, radiation and the vacuum are obtained. Analytic solutuions in various simple cases are found. Dark matter and dark energy are defined through their observational evidence. The particle horizon is defined and discussed. The density and temperature at last scattering are calculated by a model involving Thomson scattering, expansion, and the Saha equation.

The (2+1) dimensional metric f (R) gravity non-minimally coupled with fermion field

In this article, we examine a gravitational theory including a fermion field that is non-minimally coupled to metric f (R) gravity in (2+1) dimensions. We give the field equations for fermion fields and Friedmann equations. In this context, we study cosmological solutions of the field equations using these forms obtained by the existent of Noether symmetry.

Cosmological Milestones and Gravastars - Topics in General Relativity

<p>In this thesis, we consider two different problems relevant to general relativity. Overthe last few years, opinions on physically relevant singularities occurring in FRWcosmologies have considerably changed. We present an extensive catalogue of suchcosmological milestones using generalized power series both at the kinematical anddynamical level. We define the notion of “scale factor singularity” and explore its relationto polynomial and differential curvature singularities. We also extract dynamicalinformation using the Friedmann equations and derive necessary and sufficient conditionsfor the existence of cosmological milestones such as big bangs, big crunches, bigrips, sudden singularities and extremality events. Specifically, we provide a completecharacterization of cosmological milestones for which the dominant energy conditionis satisfied. The second problem looks at one of the very small number of seriousalternatives to the usual concept of an astrophysical black hole, that is, the gravastarmodel developed by Mazur and Mottola. By considering a generalized class of similarmodels with continuous pressure (no infinitesimally thin shells) and negative centralpressure, we demonstrate that gravastars cannot be perfect fluid spheres: anisotropcpressures are unavoidable. We provide bounds on the necessary anisotropic pressureand show that these transverse stresses that support a gravastar permit a higher compactnessthan is given by the Buchdahl–Bondi bound for perfect fluid stars. We alsocomment on the qualitative features of the equation of state that such gravastar-likeobjects without any horizon must have.</p>

Cosmological Milestones and Gravastars - Topics in General Relativity

<p>In this thesis, we consider two different problems relevant to general relativity. Overthe last few years, opinions on physically relevant singularities occurring in FRWcosmologies have considerably changed. We present an extensive catalogue of suchcosmological milestones using generalized power series both at the kinematical anddynamical level. We define the notion of “scale factor singularity” and explore its relationto polynomial and differential curvature singularities. We also extract dynamicalinformation using the Friedmann equations and derive necessary and sufficient conditionsfor the existence of cosmological milestones such as big bangs, big crunches, bigrips, sudden singularities and extremality events. Specifically, we provide a completecharacterization of cosmological milestones for which the dominant energy conditionis satisfied. The second problem looks at one of the very small number of seriousalternatives to the usual concept of an astrophysical black hole, that is, the gravastarmodel developed by Mazur and Mottola. By considering a generalized class of similarmodels with continuous pressure (no infinitesimally thin shells) and negative centralpressure, we demonstrate that gravastars cannot be perfect fluid spheres: anisotropcpressures are unavoidable. We provide bounds on the necessary anisotropic pressureand show that these transverse stresses that support a gravastar permit a higher compactnessthan is given by the Buchdahl–Bondi bound for perfect fluid stars. We alsocomment on the qualitative features of the equation of state that such gravastar-likeobjects without any horizon must have.</p>

D-dimensional self-gravitating lattice gas in general relativity

Using a lattice equation of state combined with the D-dimensional Tolman–Oppenheimer–Volkoff equation and the Friedmann equations, we investigate the possibility of the formation of compact objects as well as the time evolution of the scale factor and the density profile of a self-gravitating material cluster. The numerical results show that in a ([Formula: see text])-dimensional space–time, the mass is independent of the central pressure. Hence, the formation of only compact objects with a finite constant mass similar to the white dwarf is possible. However, in a ([Formula: see text])-dimensional space–time, self-gravity leads to the formation of compact objects with a large gap of mass and the corresponding phase diagram has the same structure as the one for Neutron Star. The results also show that beyond certain critical central pressure, the star is unstable against gravitational collapse, and it may end in a black hole. Analysis of space–times of higher dimensions shows that gravity has the stronger effect in [Formula: see text] dimensions. Numerical solutions of the Friedmann equations show that the effect of the curvature of space–time increases with the increasing temperature, but decreases with the increasing dimensionality beyond [Formula: see text].

Continuous cosmic evolution with diffusive barotropic fluid: First-order thermodynamic phase transition

In the background of homogeneous and isotropic flat FLRW model, a complete cosmic scenario from nonsingular emergent scenario to the present late time acceleration through inflationary era and matter-dominated epoch has been presented in this work with cosmic matter in the form of diffusive barotropic fluid. By proper choices of the diffusion parameter and using Friedmann equations, it is possible to show the transitions: Emergent scenario[Formula: see text]Inflationary era[Formula: see text]matter-dominated phase[Formula: see text]Late time acceleration epoch. In analogy with analytic continuation, it is found that the above evolution will be continuous for suitable values of the parameters involved. Finally, possible first-order thermodynamic phase transition has been analyzed for such cosmic evolution.

The cosmology of quadratic torsionful gravity

We study the cosmology of a quadratic metric-compatible torsionful gravity theory in the presence of a perfect hyperfluid. The gravitational action is an extension of the Einstein–Cartan theory given by the usual Einstein–Hilbert contribution plus all the admitted quadratic parity even torsion scalars and the matter action also exhibits a dependence on the connection. The equations of motion are obtained by regarding the metric and the metric-compatible torsionful connection as independent variables. We then consider a Friedmann–Lemaître–Robertson–Walker background, analyze the conservation laws, and derive the torsion modified Friedmann equations for our theory. Remarkably, we are able to provide exact analytic solutions for the torsionful cosmology.