Szekeres Models Research Articles

Studying the precision of ray tracing techniques with Szekeres models

The simplest standard ray tracing scheme employing the Born and Limber approximations and neglecting lens-lens coupling is used for computing the convergence along individual rays in mock N-body data based on Szekeres swiss cheese and onion models. The results are compared with the exact convergence computed using the exact Szekeres metric combined with the Sachs formalism. A comparison is also made with an extension of the simple ray tracing scheme which includes the Doppler convergence. The exact convergence is reproduced very precisely as the sum of the gravitational and Doppler convergences along rays in Lemaitre-Tolman-Bondi models. This is not the case when the models are based on non-symmetric Szekeres models. For such models, there is a significant deviation between the exact and ray traced paths and hence also the corresponding convergences.

Methods for studying the accuracy of light propagation inN-body simulations

It is proposed to use exact, cosmologically relevant solutions to Einstein's equations to accurately quantify the precision of ray tracing techniques through Newtonian N-body simulations. As an initial example of such a study, the recipe in (Green & Wald, 2012) for going between N-body results and a perturbed FLRW metric in the Newtonian gauge is used to study light propagation through quasi-spherical Szekeres models. The study is conducted by deriving a set of ODEs giving an expression for the angular diameter distance in the Newtonian gauge metric. The accuracy of the results obtained from the ODEs is estimated by using the ODEs to determine the distance-redshift relation in mock N-body data based on quasi-spherical Szekeres models. The results are then compared to the exact relations. From this comparison it is seen that the obtained ODEs can accurately reproduce the distance-redshift relation along both radial and non-radial geodesics in spherically symmetric models. The reproduction of geodesics in non-symmetric Szekeres models is slightly less accurate, but still good. These results indicate that the employment of perturbed FLRW metrics for standard ray tracing techniques yields fairly accurate results, at least regarding distance-redshift relations. It is possible though, that this conclusion will be rendered invalid if other typical ray tracing approximations are included and if light is allowed to travel through several structures instead of just one.

Stringent Restriction from the Growth of Large-Scale Structure on Apparent Acceleration in Inhomogeneous Cosmological Models

Probes of cosmic expansion constitute the main basis for arguments to support or refute a possible apparent acceleration due to different expansion rates in the Universe as described by inhomogeneous cosmological models. We present in this Letter a separate argument based on results from an analysis of the growth rate of large-scale structure in the Universe as modeled by the inhomogeneous cosmological models of Szekeres. We use the models with no assumptions of spherical or axial symmetries. We find that while the Szekeres models can fit very well the observed expansion history without a Λ, they fail to produce the observed late-time suppression in the growth unless Λ is added to the dynamics. A simultaneous fit to the supernova and growth factor data shows that the cold dark matter model with a cosmological constant (ΛCDM) provides consistency with the data at a confidence level of 99.65%, while the Szekeres model without Λ achieves only a 60.46% level. When the data sets are considered separately, the Szekeres with no Λ fits the supernova data as well as the ΛCDM does, but provides a very poor fit to the growth data with only 31.31% consistency level compared to 99.99% for the ΛCDM. This absence of late-time growth suppression in inhomogeneous models without a Λ is consolidated by a physical explanation.

EXACT INHOMOGENEOUS MODELS AND THE DRIFT OF LIGHT RAYS INDUCED BY NONSYMMETRIC FLOW OF THE COSMIC MEDIUM

After introducing the Szekeres and Lemaître–Tolman cosmological models, the real-time cosmology program is briefly mentioned. Then, a few widespread misconceptions about the cosmological models are pointed out and corrected. Investigation of null geodesic equations in the Szekeres models shows that observers in favorable positions would see galaxies drift across the sky at a rate of up to 10-6 arc s per year. Such a drift would be possible to measure using devices that are under construction; the required time of monitoring would be ≈10 years. This effect is zero in the FLRW models, so it provides a measure of inhomogeneity of the Universe. In the Szekeres models, the condition for zero drift is zero shear. But in the shearfree normal models, the condition for zero drift is that, in the comoving coordinates, the time dependence of the metric completely factors out.

Weighed scalar averaging in LTB dust models: part I. Statistical fluctuations and gravitational entropy

We introduce a weighed scalar average formalism (‘q-average’) for the study of the theoretical properties and the dynamics of spherically symmetric Lemaître–Tolman–Bondi (LTB) dust models. The ‘q-scalars’ that emerge by applying the q-averages to the density, Hubble expansion and spatial curvature (which are common to FLRW models) are directly expressible in terms of curvature and kinematic invariants and identically satisfy FLRW evolution laws without the back-reaction terms that characterize Buchert's average. The local and non-local fluctuations and perturbations with respect to the q-average convey the effects of inhomogeneity through the ratio of curvature and kinematic invariants and the magnitude of radial gradients. All curvature and kinematic proper tensors that characterize the models are expressible as irreducible algebraic expansions on the metric and 4-velocity, whose coefficients are the q-scalars and their linear and quadratic local fluctuation. All invariant contractions of these tensors are quadratic fluctuations, whose q-averages are directly and exactly related to statistical correlation moments of the density and Hubble expansion scalar. We explore the application of this formalism to a definition of a gravitational entropy functional proposed by Hosoya et al (2004 Phys. Rev. Lett. 92 141302–14). We show that a positive entropy production follows from a negative correlation between fluctuations of the density and Hubble scalar, providing a brief outline on its fulfilment in various LTB models and regions. While the q-average formalism is specially suited for LTB (and Szekeres) models, it may provide a valuable theoretical insight into the properties of scalar averaging in inhomogeneous spacetimes in general.

CMB dipoles and other low-order multipoles in the quasispherical Szekeres model

Several authors have previously shown that Gpc-scale void based on the spherically symmetric LTB model can provide a good fit to certain cosmological data, including the SNIa data, but it is only consistent with the observed CMB dipole if we are located very close to the center, in violation of the Copernican principle. In this work we investigate the more general quasispherical Szekeres model, which does not include spherical symmetry, in order to determine whether this option may be less constricting. We find that the observer is still constrained to a small region, but it is not as geometrically ``special'' as the center of an LTB void. Furthermore, whereas the quadrupole and octupole near the center of an LTB void are necessarily small, certain Szekeres models can include a significant quadrupole while still being consistent with the observed dipole, hinting that Szekeres models may be able give an explanation for the observed quadrupole/octupole anomalies.

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.

Constructing realistic Szekeres models from initial and final data

The Szekeres family of inhomogeneous solutions, which are defined by six arbitrary metric functions, offers a wide range of possibilities for modelling cosmic structure. Here we present a model construction procedure for the quasispherical case using given data at initial and final times. Of the six arbitrary metric functions, the three which are common to both Szekeres and Lemaître-Tolman models are determined by the model construction procedure of Krasinski & Hellaby. For the remaining three functions, which are unique to Szekeres models, we derive exact analytic expressions in terms of more physically intuitive quantities — density profiles and dipole orientation angles. Using MATLAB, we implement the model construction procedure and simulate the time evolution.

Geometry of the quasihyperbolic Szekeres models

Geometric properties of the quasihyperbolic Szekeres models are discussed and related to the quasispherical Szekeres models. Typical examples of shapes of various classes of two-dimensional coordinate surfaces are shown in graphs, for the hyperbolically symmetric subcase and for the general quasihyperbolic case. An analysis of the mass function $M(z)$ is carried out in parallel to an analogous analysis for the quasispherical models. This leads to the conclusion that $M(z)$ determines the density of rest mass averaged over the whole space of constant time.

Redshift drift in axially symmetric quasispherical Szekeres models

Models of inhomogeneous universes constructed with exact solutions of Einstein's General Relativity have been proposed in the literature with the aim of reproducing the cosmological data without any need for a dark energy component. Besides large scale inhomogeneity models spherically symmetric around the observer, Swiss-cheese models have also been studied. Among them, Swiss-cheeses where the inhomogeneous patches are modeled by different particular Szekeres solutions have been used for reproducing the apparent dimming of the type Ia supernovae (SNIa). However, the problem of fitting such models to the SNIa data is completely degenerate and we need other constraints to fully characterize them. One of the tests which is known to be able to discriminate between different cosmological models is the redshift-drift. This drift has already been calculated by different authors for Lema\^itre-Tolman-Bondi (LTB) models. We compute it here for one particular axially symmetric quasi-spherical Szekeres (QSS) Swiss-cheese which has previously been shown to reproduce to a good accuracy the SNIa data, and we compare the results to the drift in the $\Lambda$CDM model and in some LTB models that can be found in the literature. We show that it is a good discriminator between them. Then, we discuss our model's remaining degrees of freedom and propose a recipe to fully constrain them.

Apparent horizons in the quasispherical Szekeres models

The notion of an apparent horizon (AH) in a collapsing object can be carried over from the Lema\^{\i}tre-Tolman to the quasispherical Szekeres models in three ways: 1. Literally by the definition---the AH is the boundary of the region, in which every bundle of null geodesics has negative expansion scalar. 2. As the locus, at which null lines that are as nearly radial as possible are turned toward decreasing areal radius $R$. These lines are in general nongeodesic. The name ``absolute apparent horizon'' (AAH) is proposed for this locus. 3. As the boundary of a region, where null geodesics are turned toward decreasing $R$. The name ``light collapse region'' is proposed for this region (which is three-dimensional in every space of constant $t$); its boundary coincides with the AAH. The AH and AAH coincide in the Lema\^{\i}tre-Tolman models. In the quasispherical Szekeres models, the AH is different from (but not disjoint with) the AAH. Properties of the AAH and light collapse region are investigated, and the relations between the AAH and the AH are illustrated with diagrams using an explicit example of a Szekeres metric. It turns out that an observer who is already within the AH is, for some time, not yet within the AAH. Nevertheless, no light signal can be sent through the AH from the inside. The analogue of the AAH for massive particles is also considered.

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.

A novel approach to the dynamics of Szekeres dust models

We obtain an elegant and useful description of the dynamics of Szekeres dust models (in their full generality) by means of ‘quasi-local’ scalar variables constructed by suitable integral distributions that can be interpreted as weighted proper volume averages of the local covariant scalars. In terms of these variables, the field equations and basic physical and geometric quantities are formally identical to their corresponding expressions in the spherically symmetric Lemaître–Tolman–Bondi (LTB) dust models. Since we can map every Szekeres model to a unique LTB model, rigorous results valid for the latter models can be readily generalized to a non-spherical Szekeres geometry. The new variables lead naturally to an initial value formulation in which all scalars are expressed as scaling laws in terms of their values at an arbitrary initial space slice. These variables also yield a significant simplification of numerical work, since the fluid-flow evolution equations become a system of autonomous ordinary differential equations subjected to algebraic constraints containing the information on the deviations from spherical symmetry. As an example of how this formalism can be applied, we show that spherical symmetry is stable against small dipole-like perturbations. This new approach to the dynamics of the Szekeres solutions has an enormous potential for dealing with a wide variety of theoretical issues and for constructing non-spherical models of cosmological inhomogeneities to fit observational data.

NON-SPHERICAL VOIDS: THE BEST ALTERNATIVE TO DARK ENERGY?

If our cosmic location lies within a large-scale under–dense region or "void", then current cosmological observations can be explained without resorting to a cosmological constant or to an exotic and elusive source like "dark energy". If we further assume this void region to be spherical (as almost all current void models do), then fitting observational data severely constrains our position to be very near the void center, which is a very special and unlikely observation point. We argue in this article that existing spherical void models must be regarded as gross approximations that arise by smoothing out more realistic non–spherical configurations that may fit observations without the limitations imposed by spherical symmetry. In particular, the class of quasi–spherical Szekeres models provides sufficient degrees of freedom to describe the evolution of non–spherical inhomogeneities, including a configuration consisting of several elongated supercluster–like overdense filaments with large underdense regions between them. We summarize a recently published example of such configuration, showing that it yields a reasonable coarse-grained description of realistic observed structures. While the density distribution is not spherically symmetric, its proper volume average yields a spherical density void profile of 250 Mpc that roughly agrees with observations. Also, once we consider our location to lie within a non-spherical void, the definition of a "center" location becomes more nuanced, and thus the constraints placed by the fitting of observations on our position with respect to this location become less restrictive.

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.

Redshift propagation equations in theβ′≠0Szekeres models

The set of differential equations obeyed by the redshift in the general ${\ensuremath{\beta}}^{\ensuremath{'}}\ensuremath{\ne}0$ Szekeres spacetimes is derived. Transversal components of the ray's momentum have to be taken into account, which leads to a set of 3 coupled differential equations. It is shown that in a general Szekeres model, and in a general Lema\^{\i}tre-Tolman (L-T) model, generic light rays do not have repeatable paths (RLPs): two rays sent from the same source at different times to the same observer pass through different sequences of intermediate matter particles. The only spacetimes in the Szekeres class in which all rays are RLPs are the Friedmann models. Among the proper Szekeres models, RLPs exist only in the axially symmetric subcases, and in each one the RLPs are the null geodesics that intersect each $t=\mathrm{\text{constant}}$ space on the symmetry axis. In the special models with a 3-dimensional symmetry group (L-T among them), the only RLPs are radial geodesics. This shows that RLPs are very special and in the real Universe should not exist. We present several numerical examples which suggest that the rate of change of positions of objects in the sky, for the studied configuration, is ${10}^{\ensuremath{-}6}--{10}^{\ensuremath{-}7}$ arc sec per year. With the current accuracy of direction measurement, this drift would become observable after approximately 10 years of monitoring. More precise future observations will be able, in principle, to detect this effect, but there are basic problems with determining the reference direction that does not change.

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.

The Szekeres Swiss Cheese model and the CMB observations

This paper presents the application of the Szekeres Swiss Cheese model to the analysis of observations of the cosmic microwave background (CMB) radiation. The impact of inhomogeneous matter distribution on the CMB observations is in most cases studied within the linear perturbations of the Friedmann model. However, since the density contrast and the Weyl curvature within the cosmic structures are large, this issue is worth studying using another approach. The Szekeres model is an inhomogeneous, non-symmetrical and exact solution of the Einstein equations. In this model, light propagation and matter evolution can be exactly calculated, without such approximations as small amplitude of the density contrast. This allows to examine in more realistic manner the contribution of the light propagation effect to the measured CMB temperature fluctuations. The results of such analysis show that small-scale, non-linear inhomogeneities induce, via Rees-Sciama effect, temperature fluctuations of amplitude 10−7–10−5 on angular scale ϑ < 0.24° (ℓ > 750). This is still much smaller than the measured temperature fluctuations on this angular scale. However, local and uncompensated inhomogeneities can induce temperature fluctuations of amplitude as large as 10−3, and thus can be responsible the low multipoles anomalies observed in the angular CMB power spectrum.

Dark energy or apparent acceleration due to a relativistic cosmological model more complex than the Friedmann-Lemaitre-Robertson-Walker model?

We use the Szekeres inhomogeneous relativistic models in order to fit supernova combined data sets. We show that with a choice of the spatial curvature function that is guided by current observations, the models fit the supernova data almost as well as the Lambda-CDM model without requiring a dark energy component. The Szekeres models were originally derived as an exact solution to Einstein's equations with a general metric that has no symmetries, and they are regarded as good candidates to model the true lumpy universe that we observe. The null geodesics in these models are not radial. The best fit model found is also consistent with the requirement of spatial flatness at cosmic microwave background scales. The first results presented here seem to encourage further investigations of apparent acceleration using various inhomogeneous models, and other constraints from cosmic microwave background and large structure need to be explored next.