Geometry of the quasi-hyperbolic Szekeres models

Geometric properties of the quasi-hyperbolic Szekeres models are discussed and related to the quasi-spherical Szekeres models. Typical examples of shapes of various classes of 2-dimensional coordinate surfaces are shown in graphs; for the hyperbolically symmetric subcase and for the general quasi-hyperbolic case. An analysis of the mass function $M(z)$ is carried out in parallel to an analogous analysis for the quasi-spherical models. This leads to the conclusion that $M(z)$ determines the density of rest mass averaged over the whole space of constant time.Comment: 19 pages, 13 figures. This version matches the published tex

Apparent horizons in the quasi-spherical Szekeres models

The notion of an apparent horizon (AH) in a collapsing object can be carried over from the Lema\^{\i}tre -- Tolman (L--T) 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 \textit{geodesics} are turned toward decreasing $R$. The name "light collapse region" (LCR) is proposed for this region (which is 3-dimensional in every space of constant $t$); its boundary coincides with the AAH. The AH and AAH coincide in the L--T models. In the quasispherical Szekeres models, the AH is different from (but not disjoint with) the AAH. Properties of the AAH and LCR 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.Comment: 14 pages, 9 figures, includes little extensions and style corrections made after referee's comments, the text matches the published versio

Volume averaging in the quasispherical Szekeres model

This paper considers the volume averaging in the quasispherical Szekeres model. The volume averaging became of considerable interest after it was shown that the volume acceleration calculated within the averaging framework can be positive even though the local expansion rate is always decelerating. This issue was intensively studied within spherically symmetric models. However, since our Universe is not spherically symmetric similar analysis is needed in non symmetrical models. This papers presents the averaging analysis within the quasispherical Szekeres model which is a non-symmetrical generalisation of the spherically symmetric Lema\^itre--Tolman family of models. Density distribution in the quasispherical Szekeres has a structure of a time-dependent mass dipole superposed on a monopole. This paper shows that when calculating the volume acceleration, $\ddot{a}$, within the Szekeres model, the dipole does not contribute to the final result, hence $\ddot{a}$ only depends on a monopole configuration. Thus, the volume averaging within the Szekeres model leads to literally the same solutions as obtained within the Lema\^itre--Tolman model.Comment: 8 pages; calculation of the spatial Ricci scalar added; accepted for publication in Gen. Rel. Gra

Generalized Swiss-Cheese Cosmologies II: Spherical Dust

The generalized Swiss - cheese model, consisting of a Lema\^itre - Tolman (inhomogeneous dust) region matched, by way of a comoving boundary surface, onto a Robertson-Walker background of homogeneous dust, has become a standard construction in modern cosmology. Here we ask if this construction can be made more realistic by introducing some evolution of the boundary surface. The answer we find is no. To maintain a boundary surface using the Darmois - Israel junction conditions, as opposed to the introduction of a surface layer, the boundary must remain exactly comoving. The options are to drop the assumption of dust or allow the development of surface layers. Either option fundamentally changes the original construction.Comment: 5 pages revtex 4.1 Final form to appear in Phys. Rev.

A (giant) void is not mandatory to explain away dark energy with a Lemaitre - Tolman model

Lema\^itre - Tolman (L--T) toy models with a central observer have been used to study the effect of large scale inhomogeneities on the SN Ia dimming. Claims that a giant void is mandatory to explain away dark energy in this framework are currently dominating. Our aim is to show that L-T models exist that reproduce a few features of the $\Lambda$CDM model, but do not contain the giant cosmic void. We propose to use two sets of data - the angular diameter distance together with the redshift-space mass-density and the angular diameter distance together with the expansion rate - both defined on the past null cone as functions of the redshift. We assume that these functions are of the same form as in the $\Lambda$CDM model. Using the Mustapha-Hellaby-Ellis algorithm, we numerically transform these initial data into the usual two L-T arbitrary functions and solve the evolution equation to calculate the mass distribution in spacetime. For both models, we find that the current density profile does not exhibit a giant void, but rather a giant hump. However, this hump is not directly observable, since it is in a spacelike relation to a present observer. The alleged existence of the giant void was a consequence of the L-T models used earlier because their generality was limited a priori by needless simplifying assumptions, like, for example, the bang-time function being constant. Instead, one can feed any mass distribution or expansion rate history on the past light cone as initial data to the L-T evolution equation. When a fully general L-T metric is used, the giant void is not implied.Comment: 25 pages, 20 figures, substantially revised to match the published version, figs. 9 and 19 changed to match the caption

Radial asymptotics of Lemaitre-Tolman-Bondi dust models

We examine the radial asymptotic behavior of spherically symmetric Lemaitre-Tolman-Bondi dust models by looking at their covariant scalars along radial rays, which are spacelike geodesics parametrized by proper length $\ell$, orthogonal to the 4-velocity and to the orbits of SO(3). By introducing quasi-local scalars defined as integral functions along the rays, we obtain a complete and covariant representation of the models, leading to an initial value parametrization in which all scalars can be given by scaling laws depending on two metric scale factors and two basic initial value functions. Considering regular "open" LTB models whose space slices allow for a diverging $\ell$, we provide the conditions on the radial coordinate so that its asymptotic limit corresponds to the limit as $\ell\to\infty$. The "asymptotic state" is then defined as this limit, together with asymptotic series expansion around it, evaluated for all metric functions, covariant scalars (local and quasi-local) and their fluctuations. By looking at different sets of initial conditions, we examine and classify the asymptotic states of parabolic, hyperbolic and open elliptic models admitting a symmetry center. We show that in the radial direction the models can be asymptotic to any one of the following spacetimes: FLRW dust cosmologies with zero or negative spatial curvature, sections of Minkowski flat space (including Milne's space), sections of the Schwarzschild--Kruskal manifold or self--similar dust solutions.Comment: 44 pages (including a long appendix), 3 figures, IOP LaTeX style. Typos corrected and an important reference added. Accepted for publication in General Relativity and Gravitatio

Imitating accelerated expansion of the Universe by matter inhomogeneities - corrections of some misunderstandings

A number of misunderstandings about modeling the apparent accelerated expansion of the Universe, and about the `weak singularity' are clarified: 1. Of the five definitions of the deceleration parameter given by Hirata and Seljak (HS), only $q_1$ is a correct invariant measure of acceleration/deceleration of expansion. The $q_3$ and $q_4$ are unrelated to acceleration in an inhomogeneous model. 2. The averaging over directions involved in the definition of $q_4$ does not correspond to what is done in observational astronomy. 3. HS's equation (38) connecting $q_4$ to the flow invariants gives self-contradictory results when applied at the centre of symmetry of the Lema\^{\i}tre-Tolman (L-T) model. The intermediate equation (31) that determines $q_{3'}$ is correct, but approximate, so it cannot be used for determining the sign of the deceleration parameter. Even so, at the centre of symmetry of the L-T model, it puts no limitation on the sign of $q_{3'}(0)$. 4. The `weak singularity' of Vanderveld {\it et al.} is a conical profile of mass density at the centre - a perfectly acceptable configuration. 5. The so-called `critical point' in the equations of the `inverse problem' for a central observer in an L-T model is a manifestation of the apparent horizon - a common property of the past light cones in zero-lambda L-T models, perfectly manageable if the equations are correctly integrated.Comment: 15 pages. Completely rewritten to match the published version. We added discussion of 2 key papers cited by VFW and identified more clearly the assumptions, approximations and mistakes that led to certain misconceptions

Generalized Swiss-cheese cosmologies: Mass scales

We generalize the Swiss-cheese cosmologies so as to include nonzero linear momenta of the associated boundary surfaces. The evolution of mass scales in these generalized cosmologies is studied for a variety of models for the background without having to specify any details within the local inhomogeneities. We find that the final effective gravitational mass and size of the evolving inhomogeneities depends on their linear momenta but these properties are essentially unaffected by the details of the background model.Comment: 10 pages, 14 figures, 1 table, revtex4, Published form (with minor corrections

Conditions for spontaneous homogenization of the Universe

The present-day Universe appears to be homogeneous on very large scales. Yet when the casual structure of the early Universe is considered, it becomes apparent that the early Universe must have been highly inhomogeneous. The current paradigm attempts to answer this problem by postulating the inflation mechanism However, inflation in order to start requires a homogeneous patch of at least the horizon size. This paper examines if dynamical processes of the early Universe could lead to homogenization. In the past similar studies seem to imply that the set of initial conditions that leads to homogenization is of measure zero. This essay proves contrary: a set of initial conditions for spontaneous homogenization of cosmological models can form a set of non-zero measure.Comment: 7 pages. Fifth Award in the 2010 Gravity Research Foundation essay competitio

Covariant coarse-graining of inhomogeneous dust flow in General Relativity

A new definition of coarse-grained quantities describing the dust flow in General Relativity is proposed. It assigns the coarse--grained expansion, shear and vorticity to finite-size comoving domains of fluid in a covariant, coordinate-independent manner. The coarse--grained quantities are all quasi-local functionals, depending only on the geometry of the boundary of the considered domain. They can be thought of as relativistic generalizations of simple volume averages of local quantities in a flat space. The procedure is based on the isometric embedding theorem for S^2 surfaces and thus requires the boundary of the domain in question to have spherical topology and positive scalar curvature. We prove that in the limit of infinitesimally small volume the proposed quantities reproduce the local expansion, shear and vorticity. In case of irrotational flow we derive the time evolution for the coarse-grained quantities and show that its structure is very similar to the evolution equation for their local counterparts. Additional terms appearing in it may serve as a measure of the backreacton of small-scale inhomogeneities of the flow on the large-scale motion of the fluid inside the domain and therefore the result may be interesting in the context of the cosmological backreaction problem. We also consider the application of the proposed coarse-graining procedure to a number of known exact solutions of Einstein equations with dust and show that it yields reasonable results.Comment: 17 pages, 5 figures. Version accepted in Classical and Quantum Gravity