Inhomogeneous Cosmological Models Research Articles

Review on exact and perturbative deformations of the Einstein–Straus model: uniqueness and rigidity results

The Einstein–Straus model consists of a Schwarzschild spherical vacuole in a Friedman–Lemaître–Robertson–Walker (FLRW) dust spacetime (with or without $$\Lambda $$ ). It constitutes the most widely accepted model to answer the question of the influence of large scale (cosmological) dynamics on local systems. The conclusion drawn by the model is that there is no influence from the cosmic background, since the spherical vacuole is static. Spherical generalizations to other interior matter models are commonly used in the construction of lumpy inhomogeneous cosmological models. On the other hand, the model has proven to be reluctant to admit non-spherical generalizations. In this review, we summarize the known uniqueness results for this model. These seem to indicate that the only reasonable and realistic non-spherical deformations of the Einstein–Straus model require perturbing the FLRW background. We review results about linear perturbations of the Einstein–Straus model, where the perturbations in the vacuole are assumed to be stationary and axially symmetric so as to describe regions (voids in particular) in which the matter has reached an equilibrium regime.

Monodromy transform and the integral equation method for solving the string gravity and supergravity equations in four and higher dimensions

The monodromy transform and corresponding integral equation method described here give rise to a general systematic approach for solving integrable reductions of field equations for gravity coupled bosonic dynamics in string gravity and supergravity as well as for pure vacuum gravity in four and higher dimensions. For physically different types of fields in space-times of $D\ensuremath{\ge}4$ dimensions with $d=D\ensuremath{-}2$ commuting isometries---stationary fields with spatial symmetries, interacting waves or evolution of partially inhomogeneous cosmological models---the string gravity equations govern the dynamics of interacting gravitational, dilaton, antisymmetric tensor, and any number $n\ensuremath{\ge}0$ of Abelian vector gauge fields (all depending only on two coordinates). The equivalent spectral problem constructed earlier allows one to parametrize the entire infinite-dimensional space of (normalized) local solutions of these equations by two pairs of arbitrary coordinate-independent holomorphic $d\ifmmode\times\else\texttimes\fi{}d$- and $d\ifmmode\times\else\texttimes\fi{}n$- matrix functions ${{\mathbf{u}}_{\ifmmode\pm\else\textpm\fi{}}(w),{\mathbf{v}}_{\ifmmode\pm\else\textpm\fi{}}(w)}$ of a spectral parameter $w$ which constitute a complete set of monodromy data for a normalized fundamental solution of this spectral problem. The ``direct'' and ``inverse'' problems of such monodromy transform---calculating the monodromy data for any local solution and constructing the field configurations for any chosen monodromy data---always admit unique solutions. We construct the linear singular integral equations which solve this inverse problem. For any rational and analytically matched [i.e. ${\mathbf{u}}_{+}(w)\ensuremath{\equiv}{\mathbf{u}}_{\ensuremath{-}}(w)$ and ${\mathbf{v}}_{+}(w)\ensuremath{\equiv}{\mathbf{v}}_{\ensuremath{-}}(w)$] monodromy data the solution of these integral equations and corresponding solution for string gravity equations can be found explicitly. Simple reductions of the space of monodromy data leads to the similar constructions for solving of other integrable symmetry reduced gravity models, e.g. 5D minimal supergravity or vacuum gravity in $D\ensuremath{\ge}4$ dimensions.

Lagrangian theory of structure formation in relativistic cosmology. II. Average properties of a generic evolution model

Kinematical and dynamical properties of a generic inhomogeneous cosmological model, spatially averaged with respect to free-falling (generalized fundamental) observers, are investigated for the matter model irrotational dust. Paraphrasing a previous Newtonian investigation, we present a relativistic generalization of a backreaction model based on volume-averaging the Relativistic Zeldovich Approximation. In this model we investigate the effect of kinematical backreaction on the evolution of cosmological parameters as they are defined in an averaged inhomogeneous cosmology, and we show that the backreaction model interpolates between orthogonal symmetry properties by covering subcases of the plane-symmetric solution, the Lemaitre-Tolman-Bondi solution and the Szekeres solution. We so obtain a powerful model that lays the foundations for quantitatively addressing curvature inhomogeneities as they would be interpreted as Dark Energy or Dark Matter in a quasi-Newtonian cosmology. The present model, having a limited architecture due to an assumed FLRW background, is nevertheless capable of replacing 1/4 of the needed amount for Dark Energy on domains of 200 Mpc in diameter for typical (one-sigma) fluctuations in a CDM initial power spectrum. However, the model is far from explaining Dark Energy on larger scales (spatially), where a 6% effect on 400 Mpc domains is identified that can be traced back to an on average negative intrinsic curvature today. One drawback of the quantitative results presented is the fact that the epoch when backreaction is effective on large scales and leads to volume acceleration lies in the future. We discuss this issue in relation to the initial spectrum, the Dark Matter problem, the coincidence problem, and the fact that large-scale Dark Energy is an effect on the past light cone (not spatial), and we pinpoint key elements of future research.

On the absence of the usual weak-field limit, and the impossibility of embedding some known solutions for isolated masses in cosmologies withf(R)dark energy

The problem of matching different regions of spacetime in order to construct inhomogeneous cosmological models is investigated in the context of Lagrangian theories of gravity constructed from general analytic functions f(R), and from non-analytic theories with f(R)=R^n. In all of the cases studied, we find that it is impossible to satisfy the required junction conditions without the large-scale behaviour reducing to that expected from Einstein's equations with a cosmological constant. For theories with analytic f(R) this suggests that the usual treatment of weak-field systems as perturbations about Minkowski space may not be compatible with late-time acceleration driven by anything other than a constant term of the form f(0), which acts like a cosmological constant. In the absence of Minkowski space as a suitable background for weak-field systems, one must then choose and justify some other solution to perform perturbative analyses around. For theories with f(R)=R^n we find that no known spherically symmetric vacuum solutions can be matched to an expanding FLRW background. This includes the absence of any Einstein-Straus-like embeddings of the Schwarzschild exterior solution in FLRW spacetimes.

Large-scale growth evolution in the Szekeres inhomogeneous cosmological models with comparison to growth data

We use the Szekeres inhomogeneous cosmological models to study the growth of large-scale structure in the universe including nonzero spatial curvature and a cosmological constant. In particular, we use the Goode and Wainwright formulation of the solution, as in this form the models can be considered to represent exact nonlinear perturbations of an averaged background. We identify a density contrast in both classes I and II of the models, for which we derive growth evolution equations. By including $\ensuremath{\Lambda}$, the time evolution of the density contrast as well as kinematic quantities of interest can be tracked through the matter- and $\ensuremath{\Lambda}$-dominated cosmic eras up to the present and into the future. In class I, we consider a localized cosmic structure representing an overdensity neighboring a central void, surrounded by an almost Friedmann-Lema\^{\i}tre-Robertson-Walker background, while for class II, the exact perturbations exist globally. In various models of class I and class II, the growth rate is found to be stronger in the matter-dominated era than that of the standard lambda-cold dark matter ($\ensuremath{\Lambda}\mathrm{CDM}$) cosmology, and it is suppressed at later times due to the presence of the cosmological constant. We find that there are Szekeres models able to provide a growth history similar to that of $\ensuremath{\Lambda}\mathrm{CDM}$ while requiring less matter content and nonzero spatial curvature, which speaks to the importance of including the effects of large-scale inhomogeneities in analyzing the growth of large-scale structure. Using data for the growth factor $f$ from redshift space distortions and the Lyman-$\ensuremath{\alpha}$ forest, we obtain best fit parameters for class II models and compare their ability to match observations with $\ensuremath{\Lambda}\mathrm{CDM}$. We find that there is negligible difference between best fit Szekeres models with no priors and those for $\ensuremath{\Lambda}\mathrm{CDM}$, both including and excluding Lyman-$\ensuremath{\alpha}$ data. We also find that the standard growth index $\ensuremath{\gamma}$ parametrization cannot be applied in a simple way to the growth in Szekeres models, so a direct comparison of the function $f$ to the data is performed. We conclude that the Szekeres models can provide an exact framework for the analysis of large-scale growth data that includes inhomogeneities and allows for different interpretations of observations.

Inhomogeneous Cosmological Models in Bimetric Theory of Gravitation

Plane symmetric inhomogeneous cosmological models for perfect fluid distribution are investigated in the frame work of Rosen's bimetric theory of gravitation. A negative pressure corresponding to false vacuum state is considered. Some of the physical and kinematical properties of the models are discussed.

Isometric embeddings of 2-spheres by embedding flow for applications in numerical relativity**AEI preprint number: AEI-2011-096, University of Vienna preprint number: UWThPh-2011-42.

We present a numerical method for solving Weyl’s embedding problem which consists in finding a global isometric embedding of a positively curved and positive-definite spherical 2-metric into the Euclidean 3-space. The method is based on a construction introduced by Weingarten and was used in Nirenberg’s proof of Weyl’s conjecture. The target embedding results as the endpoint of an embedding flow in R3 beginning at the unit sphere’s embedding. We employ spectral methods to handle functions on the surface and to solve various (non)linear elliptic PDEs. The code requires no additional input or steering from the operator and its convergence is guaranteed by the Nirenberg arguments. Possible applications in 3 + 1 numerical relativity range from quasi-local mass and momentum measures to coarse-graining in inhomogeneous cosmological models.

Some clarifications about Lemaître-Tolman models of the Universe used to deal with the dark energy problem

Context. During the past fifteen years, inhomogeneous cosmological models have been put forward to explain the observed dimming of the SNIa luminosity without resorting to dark energy. The simplest models are the spherically symmetric Lemaître-Tolman (LT) solutions with a central observer. Their use must be considered as a mere first step towards more sophisticated models. Spherical symmetry is but a mathematical simplification and one must consider spherical symmetric models as exhibiting an energy density smoothed out over angles around us. However, they have been taken at face value by some authors who tried to use them for either irrelevant purposes or to put them to the test as if they were robust models of our Universe.

Local and nonlocal measures of acceleration in cosmology

Current cosmological observations, when interpreted within the framework of a homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) model, strongly suggest that the Universe is entering a period of accelerating expansion. This is often taken to mean that the expansion of space itself is accelerating. In a general spacetime, however, this is not necessarily true. We attempt to clarify this point by considering a handful of local and non-local measures of acceleration in a variety of inhomogeneous cosmological models. Each of the chosen measures corresponds to a theoretical or observational procedure that has previously been used to study acceleration in cosmology, and all measures reduce to the same quantity in the limit of exact spatial homogeneity and isotropy. In statistically homogeneous and isotropic spacetimes, we find that the acceleration inferred from observations of the distance-redshift relation is closely related to the acceleration of the spatially averaged universe, but does not necessarily bear any resemblance to the average of the local acceleration of spacetime itself. For inhomogeneous spacetimes that do not display statistical homogeneity and isotropy, however, we find little correlation between acceleration inferred from observations and the acceleration of the averaged spacetime. This shows that observations made in an inhomogeneous universe can imply acceleration without the existence of dark energy.

Growth of structure in the Szekeres class-II inhomogeneous cosmological models and the matter-dominated era

This study belongs to a series devoted to using the Szekeres inhomogeneous models in order to develop a theoretical framework where cosmological observations can be investigated with a wider range of possible interpretations. While our previous work addressed the question of cosmological distances versus redshift in these models, the current study is a start at looking into the growth rate of large-scale structure. The Szekeres models are exact solutions to Einstein's equations that were originally derived with no symmetries. We use here a formulation of the Szekeres models that is due to Goode and Wainwright, who considered the models as exact perturbations of a Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) background. Using the Raychaudhuri equation we write, for the two classes of the models, exact growth equations in terms of the under/overdensity and measurable cosmological parameters. The new equations in the overdensity split into two informative parts. The first part, while exact, is identical to the growth equation in the usual linearly perturbed FLRW models, while the second part constitutes exact nonlinear perturbations. We integrate numerically the full exact growth rate equations for the flat and curved cases. We find that for the matter-dominated cosmic era, the Szekeres growth rate is up to a factor of three to five stronger than the usual linearly perturbed FLRW cases, reflecting the effect of exact Szekeres nonlinear perturbations. We also find that the Szekeres growth rate with an Einstein-de Sitter background is stronger than that of the well-known nonlinear spherical collapse model, and the difference between the two increases with time. This highlights the distinction when we use general inhomogeneous models where shear and a tidal gravitational field are present and contribute to the gravitational clustering. Additionally, it is worth observing that the enhancement of the growth found in the Szekeres models during the matter-dominated era could suggest a substitute to the argument that dark matter is needed when using FLRW models to explain the enhanced growth and resulting large-scale structures that we observe today.

Inhomogeneities as a possible factor responsible for the acceleration of the Universe: a (2+1)-gravity study

Here we present an inhomogeneous cosmological model, in 2+1 gravity, which satisfy all the energy conditions, although it generates non deaccelerated universes. We work in a self-similar 2+1 gravity scenario, in order to simplify the equations system and to allow us to find analytical and simple solutions. Our propose is basically to improve our understanding on the role of inhomogeneities on the acceleration of the Universe.

Observational constraints on inhomogeneous cosmological models without dark energy

It has been proposed that the observed dark energy can be explained by the effect of large-scale nonlinear inhomogeneities. In this paper, we discuss how observations constrain cosmological models featuring large voids. We start by considering Copernican models, in which the observer is not occupying a special position and homogeneity is preserved on a very large scale. We show how these models, at least in their current realizations, are constrained to give small, but perhaps not negligible in certain contexts, corrections to the cosmological observables. We then examine non-Copernican models, in which the observer is close to the center of a very large void. These models can give large corrections to the observables which mimic an accelerated Friedmann–Lemaître–Robertson–Walker model. We carefully discuss the main observables and tests able to exclude them.

Inhomogeneous cosmological models: exact solutions and their applications

Recently, inhomogeneous generalizations of the Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological models have gained interest in the astrophysical community and are more often employed to study cosmological phenomena. However, in many papers the inhomogeneous cosmological models are treated as an alternative to the FLRW models. In fact, they are not an alternative, but an exact perturbation of the latter, and are gradually becoming a necessity in modern cosmology. The assumption of homogeneity is just a first approximation introduced to simplify equations. So far this assumption is commonly believed to have worked well, but future and more precise observations will not be properly analysed unless inhomogeneities are taken into account. This paper reviews recent developments in the field and shows the importance of an inhomogeneous framework in the analysis of cosmological observations.

Exact nonlinear inhomogeneities inΛCDMcosmology

At a time when galaxy surveys and other observations are reaching unprecedented sky coverage and precision it seems timely to investigate the effects of general relativistic nonlinear dynamics on the growth of structures and on observations. Analytic inhomogeneous cosmological models are an indispensable way of investigating and understanding these effects in a simplified context. In this paper, we develop exact inhomogeneous solutions of general relativity with pressureless matter (dust, describing cold dark matter) and cosmological constant $\Lambda$, which can be used to model an arbitrary initial matter distribution along one line of sight. In particular, we consider the second class Szekeres models with $\Lambda$ and split their dynamics into a flat $\Lambda$CDM background and exact nonlinear inhomogeneities, obtaining several new results. One single metric function $Z$ describes the deviation from the background. We show that $F$, the time dependent part of $Z$, satisfies the familiar linear differential equation for $\delta$, the first-order density perturbation of dust, with the usual growing and decaying modes. In the limit of small perturbations, $\delta \approx F$ as expected, and the growth of inhomogeneities links up exactly with standard perturbation theory. We provide analytic expressions for the exact nonlinear $\delta$ and the growth factor in our models. For the case of over-densities, we find that, depending on the initial conditions, the growing mode may or may not lead to a pancake singularity, analogous to a Zel'dovich pancake. This is in contrast with the $\Lambda=0$ pure Einstein-de-Sitter background where, at any given point in comoving (Lagrangian) coordinates pancakes will always occur.

Luminosity distance and redshift in the Szekeres inhomogeneous cosmological models

The Szekeres inhomogeneous models can be used to model the true lumpy universe that we observe. This family of exact solutions to Einstein's equations was originally derived with a general metric that has no symmetries. In this work, we develop and use a framework to integrate the angular diameter and luminosity distances in the general Szekeres models. We use the affine null geodesic equations in order to derive a set of first-order ordinary differential equations that can be integrated numerically to calculate the partial derivatives of the null vector components. These equations allow the integration in all generality of the distances in the Szekeres models and some examples are given. The redshift is determined from simultaneous integration of the null geodesic equations. This work does not assume spherical or axial symmetry, and the results will be useful for comparisons of the general Szekeres inhomogeneous models to current and future cosmological data.

Cosmic spherical void via coarse-graining and averaging non-spherical structures

Inhomogeneous cosmological models are able to fit cosmological observations without dark energy under the assumption that we live close to the “center” of a very large-scale under-dense region. Most studies fitting observations by means of inhomogeneities also assume spherical symmetry, and thus being at (or very near) the center may imply being located at a very special and unlikely observation point. We argue that such spherical voids should be treated only as a gross first approximation to configurations that follow from a suitable smoothing out of the non-spherical part of the inhomogeneities on angular scales. In this Letter we present a toy construction that supports the above statement. The construction uses parts of the Szekeres model, which is inhomogeneous and anisotropic thus it also addresses the limitations of spherical inhomogeneities. By using the thin-shell approximation (which means that the Israel–Darmois continuity conditions are not fulfilled between the shells) we construct a model of evolving cosmic structures, containing several elongated supercluster-like structures with underdense regions between them, which altogether provides a reasonable coarse-grained description of cosmic structures. While this configuration is not spherically symmetric, its proper volume average yields a spherical void profile of 250 Mpc that roughly agrees with observations. Also, by considering a non-spherical inhomogeneity, the definition of a “center” location becomes more nuanced, and thus the constraints placed by fitting observations on our position with respect to this location become less restrictive.

Constraints on smoothness parameter and dark energy using observational H(z) data

With large-scale homogeneity, the universe is locally inhomogeneous, clustering into stars, galaxies and larger structures. Such property is described by the smoothness parameter α which is defined as the proportion of matter in the form of intergalactic medium. If we consider the inhomogeneities over a small scale, there should be modifications of the cosmological distances compared to a homogenous model. Dyer and Roeder developed a second-order ordinary differential equation (D-R equation) that describes the angular diameter distance-redshift relation for inhomogeneous cosmological models. Furthermore, we may obtain the D-R equation for observational H(z) data (OHD). The density-parameter ΩM, the state of dark energy ω, and the smoothness-parameter α are constrained by a set of OHD in a spatially flat ΛCDM universe as well as a spatially flat XCDM universe. By using a χ2 minimization method, we get α = 0.81+0.19−0.20 and ΩM = 0.32+0.12−0.06 at the 1σ confidence level. If we assume a Gaussian prior of ΩM = 0.26 ± 0.1, we get α = 0.93+0.07−0.19 and ΩM = 0.31+0.06−0.05. For the XCDM model, α is constrained to α ≥ 0.80 but ω is weakly constrained around −1, where ω describes the equation of state of the dark energy (pX = ωρX). We conclude that OHD constrains the smoothness parameter more effectively than the data of SNe Ia and compact radio sources.

PARAMETRIC SOLUTION OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS IN COSMOLOGY

We obtain in terms of the Weierstrass elliptic ℘-function, sigma function, and zeta function an explicit parametrized solution of a particular nonlinear, ordinary differential equation. This equation includes, in special cases, equations that occur in the study of both homogeneous and inhomogeneous cosmological models, and also in the dynamic Bose–Einstein condensates–cosmology correspondence, for example.

A New Class of Inhomogeneous Cosmological Models with Electromagnetic Field in Normal Gauge for Lyra’s Manifold

A new class of exact solutions of Einstein’s modified field equations in inhomogeneous space-time for perfect fluid distribution with electromagnetic field is obtained in the context of normal gauge for Lyra’s manifold. We have obtained solutions by considering the time dependent displacement field. The source of the magnetic field is due to an electric current produced along the z-axis. Only F12 is a non-vanishing component of the electromagnetic field tensor. It has been found that the displacement vector β(t) behaves like the cosmological constant Λ in the normal gauge treatment and the solutions are consistent with the recent observations of Type Ia supernovae. Physical and geometric aspects of the models are also discussed in the presence of magnetic field.

Cosmic age test in inhomogeneous cosmological models mimickingΛCDMon the light cone

The possibility of reconstructing a spherically symmetric inhomogeneous Lemaitre-Tolman-Bondi (LTB) model with $\Lambda$CDM observations has drawn a lot of attention. Recently, a LTB model having the same luminosity-distance and light-cone mass density as the $\Lambda$CDM model was reconstructed. We calculate the cosmic age at our position in this model, and get the age $t_{LTB}<11.7$Gyr in 2$\sigma$ confidence level from the Seven-Year WMAP observations. This result is inconsistent with some Milky Way observations.