User:Andrzej Krasiński/Proposed/Exact inhomogeneous cosmology

Inhomogeneous cosmological models are generalisations of the standard Friedmann–Lemaître–Robertson–Walker (FLRW) models, which are homogeneous and isotropic solutions of the Einstein field equations. Inhomogeneous cosmological models are also exact solutions of the Einstein equations but, unlike the FLRW models, they allow for physical quantities (such as matter density, expansion field, or curvature) to vary from point to point in space. The assumption of spatial homogeneity that leads to the FLRW models is just the first approximation introduced to simplify equations. So far this assumption is believed to have worked well, but future, more precise observations will not be properly analysed unless inhomogeneities are taken into account.

Introduction

Definitions

We define inhomogeneous cosmological models as follows: they are those exact solutions of Einstein's equations that are inhomogeneous and contain at least a subclass of nonvacuum and nonstatic FLRW solutions as a limit. The necessary and sufficient conditions for a spacetime to be FLRW are

1. The metric obeys the Einstein equations with a perfect fluid source.
2. The velocity field of the perfect fluid source has zero rotation, shear and acceleration.

Sometimes it is convenient to consider other necessary conditions, which follow from the two above:

• The Weyl curvature vanishes,
• The hypersurfaces orthogonal to the velocity field have constant curvature.

All models considered here violate some or all of the above conditions. For example, the LemaîtreTolman and Szekeres models have non-zero shear and Weyl curvature, and their spatial curvature is not constant. On the other hand, the Lemaître model violates all of the above conditions.

Purpose/scope

We aim to describe only these inhomogeneous cosmological solutions of the Einstein equations that are the basis for the greatest number of papers aimed at physical and astrophysical interpretation. These are the LemaîtreTolman (LT), Lemaître and Szekeres models. In addition to these models, there are also other inhomogeneous solutions, which have limited applications in cosmology because of some of their properties. They are briefly mentioned further on; readers who are interested in expanding their knowledge about other cosmological models are referred to the monograph by Krasiński (1997).

Importance

The LT and Szekeres models could be used to model the evolution of the Universe in the post-recombination era, in which only gravitational interactions play a role. The matter source in them is dust, i.e. a perfect fluid with zero pressure. They are meant to be a replacement for the linearised perturbations of the FLRW models and for backreactions considered in averaging schemes. Because of their symmetries (LT) and quasi-symmetries (Szekeres) they apply to less general situations than the perturbative calculations, but their advantage is that they fulfil the Einstein equations exactly.

While the evolution of the early universe can be studied using the perturbative framework, in the post-recombination era, the small scale inhomogeneities quickly enter the non-linear regime and cannot be traced using the linear approximation. In such a case, cosmologists most often turn to numerical N-body simulations. This has the advantage of tracing the evolution of matter in the non-linear regime, but on the other hand N-body simulations also face a number of limitations. Namely, the interactions are strictly Newtonian, the large scale evolution is assumed to obey the FLRW equations, and the spatial curvature is uniform throughout the entire N-body box. In addition to this, using the N-body simulations one cannot follow the evolution of geometrical quantities such as the Ricci and Weyl curvature, which, as we will see later on, are essential for the light propagation equations.

The inhomogeneous cosmological models provide us with tools to study the evolution of matter and light propagation in the non-linear regime, which, as will be shown, occurs quite early in the Universe's evolution. Although these models are restricted by their symmetries or quasi symmetries, they still can successfully be employed to study some parts of our Universe; also the methods developed within the framework on inhomogeneous cosmology can be successfully implemented in parts of the N-body simulations.

The Szekeres models

The metric of the Szekeres solutions is 1

$$\tag{1} {\rm d} s^2 = {\rm d} t^2 - \frac {\left(R' - R {\cal E}'/{\cal E}\right)^2} {\varepsilon + 2E} {\rm d} r^2 - \frac {R^2} {{\cal E}^2} \left({\rm d} x^2 + {\rm d} y^2\right),$$ where prime denotes the partial derivative with respect to $r$, i.e. $R' = \partial R / \partial r$. The function ${\cal E}$ is defined as $$\tag{2} {\cal E} = \frac S 2 \left[\left(\frac {x - P} S\right)^2 + \left(\frac {y - Q} S\right)^2 + \varepsilon\right],$$ where $E(r)$, $P(r)$, $Q(r)$ and $S(r)$ are arbitrary functions, and $\varepsilon = \pm 1, 0$. The velocity field of matter is $u^{\mu} = {\delta^{\mu}}_0$. The function $R(t, r)$ is determined by $$\tag{3} \dot{R}^2 = 2E(r) + \frac {2 M(r)} {R} + \frac 1 3 \Lambda R^2,$$ where dot denotes the partial derivative with respect to $t$, i.e. $\dot{R} = \partial R / \partial t$, $\Lambda$ is the cosmological constant, while $M(r)$ is one more arbitrary function. The mass density in energy units is $$\tag{4} \kappa \rho = \frac {2 \left(M' - 3 M {\cal E}' / {\cal E}\right)} {R^2 \left(R' - R {\cal E}' / {\cal E}\right)},$$ where $\kappa = 8 \pi G / c^4$, $G$ is the gravitational constant, and $c$ is the speed of light. The solution of (3) may be written as $$\tag{5} t - t_B(r) = \pm \int\limits_0^{R}\frac{{\rm d} \widetilde{R}}{\sqrt{2E + 2M / \widetilde{R} + \frac 1 3 \Lambda \widetilde{R}^2}},$$ where $t_B(r)$ is one more arbitrary function called the bang time (the $+$ in (5) is for expansion, the $-$ is for collapse). The instant $t = t_B(r)$ is the initial or final singularity, at which $R(t, r) = 0$; in general it thus occurs at different times in different places2. The Szekeres metric has in general no symmetry, but acquires a 3-dimensional symmetry group with 2-dimensional orbits when $P$, $Q$ and $S$ are constant.

Equation (3) is the same as the evolution equation of the Friedmann models, except that the arbitrary constants of Friedmann are replaced by arbitrary functions of $r$ here, and also coincides with the Newtonian equation of radial motion in a spherically symmetric gravitational field. The function $M$ is an analogue of the Newtonian active gravitational mass, and $E$ is the analogue of Newtonian total energy per unit mass of the moving object.

The value of $\varepsilon$ determines the geometry of the 2-surfaces of constant $t$ and $r$. The geometry is spherical, planar or hyperbolic (pseudo-spherical) when $\varepsilon > 0$, $\varepsilon = 0$ or $\varepsilon < 0$, respectively. Accordingly, there are three classes of the Szekeres models, called quasi-spherical, quasi-plane and quasi-hyperbolic. The surfaces $r =$ const within a single space $t =$ const may be spheres in one part of the space and surfaces of constant negative curvature elsewhere, the curvature being zero at the boundary, but the parametrisation used in (1) – (2) does not cover this possibility, see (Plebański and Krasiński 2006).

The sign of $E(r)$ determines the type of evolution when $\Lambda = 0$; with $E<0$ the model expands away from an initial singularity and then recollapses to a final singularity; with $E > 0$ the model is either ever-expanding or ever-collapsing; $E = 0$ is the intermediate spatially flat case. Also $E$ can have different signs in different regions of the same space (Plebański and Krasiński 2006).

Since $\varepsilon + 2E$ in (1) must be non-negative, we have the following: with $\varepsilon > 0$, all three types of evolution are allowed; with $\varepsilon = 0$, $E$ must be positive (only parabolic or hyperbolic evolutions are allowed); and with $\varepsilon < 0$, $E$ must be positive, so only the hyperbolic evolution is allowed. The geometry of the latter two classes is poorly understood.

Figure 1: An example of matter distribution in the Szekeres model, which has the structure of a mass-dipole superposed on a monopole. Figure (a) shows various 2D cross-sections, which are presented in the subsequent insets. Figure (b) shows a colour-coded density distribution on a vertical cross-section as depicted in Figure (a). Underdense regions are represented by a dark region in the centre, while the adjacent overdensity is represented by bright colours. The structure's diameter is approximately 80 Mpc, beyond which the density becomes homogeneous ($\rho/\rho_b \rightarrow 1$). Figures (c) and (d) present the matter density distribution on a 2D horizontal cross section that passes through the origin (green plane in Figure (a)). In Figure (c) the density is depicted by the amplitude (the higher amplitude the larger the density), in Figure (d) density is depicted by colours (darker represents underdense regions with $\rho/\rho_b < 1$, and brighter overdense regions with $\rho/\rho_b > 1$). Figures (e) and (f) present the matter density distribution on a 2D horizontal cross section (lower, red horizontal plane in Figure (a)). In Figure (e) the density is depicted by the amplitude, in Figure (f) density is depicted by colours.

The quasi-spherical Szekeres models are such deformations of the spherically symmetric models after which the spheres (still identifiable in the Szekeres geometry) are no longer concentric. The mass-density distribution may be interpreted as a superposition of a monopole and a dipole – see (de Souza 1985) and (Plebański and Krasiński 2006). An example of mass distribution in the Szekeres model is presented in Figure 1. In this case the mass monopole describes an underdensity (such as a cosmic void) and the mass dipole describes an overdense region (such as galaxy supercluster). Converse examples are also known, where the overdensity is described in terms of the mass monopole, while the mass dipole is used to model the underdensity.

The mater velocity field of in the quasi-spherical Szekeres model has vanishing vorticity and zero acceleration $$\tag{6} \omega_{ab} = 0, \quad u^a{}_{;b}u^b = 0.$$ However, the shear of the mater velocity filed is not zero $$\tag{7} \Sigma^a{}_{b} = \frac{1}{3} \frac{\dot{R}' - R' \dot{R}/R}{R'-R {\cal E}' / {\cal E}} {\rm~diag}(0,2,-1,-1).$$ Also the Weyl curvature is does not vanish – while the magnetic part of the Weyl tensor is zero, the electric part is $$\tag{8} C^a{}_{bcd}u^b u^d = \frac{1}{3} \frac{M}{R^3} \frac{3R' - R M'/M}{R'-R {\cal E}' / {\cal E}} {\rm~diag}(0,2,-1,-1).$$ Another property that distinguishes the quasi-spherical Szekeres model from the FLRW models is non-constant curvature of hypersurfaces orthogonal to the velocity field $$\tag{9} ^3{\cal R} = - 4 \frac{E}{R^2} \left( \frac{ R E'/E - R {\cal E}' / {\cal E}}{R'-R {\cal E}' / {\cal E}} + 1\right).$$ Similarly as with the mass-dipole (Figure 1) the function ${\cal E}' / {\cal E}$ induces dipolar variation of the curvature. In the special case of vanishing dipole (${\cal E}' = 0$) the quasi-spherical Szekeres model reduces to the spherically symmetric Lemaître–Tolman model.

The Lemaître–Tolman (LT) models

The LemaîtreTolman model (LT) is a spherically symmetric dust solution of the Einstein equations. It was derived by Lemaître in 1933 and further investigated by Tolman in 19343. The LT metric is $$\tag{10} {\rm d}s^2 = {\rm d}t^2 - \frac{{R'}^2(t,r)}{1 + 2E(r)} {\rm d}r^2 - R^2(t,r) \left({\rm d}\vartheta^2 + \sin^2 \vartheta {\rm d}\varphi^2 \right).$$ The Einstein equations reduce here to: \begin{eqnarray}\tag{11} \kappa \rho &=& \frac{2M'}{R^2 R'}, \\ \dot{R}^2 &=& 2E + \frac{2M}{R} + \frac 1 3\ \Lambda R^2, \tag{12} \end{eqnarray} where $M(r)$ and $E(r)$ are arbitrary functions. This model follows from (1)–(3) as the limit of constant $(P, Q, S)$ and $\varepsilon = +1$; also the coordinates on the spheres of constant $t$ and $r$ have to be transformed from stereographic in (1) to spherical in (10).

Since the metric (10) is covariant with the coordinate transformations $r = f(\tilde{r})$, one of the functions $(M, E, t_B)$ may be chosen to define the $r$-coordinate. Usually, this is $M(r) = {\rm constant} \times r^3$, but sometimes it is more convenient to use $M$ as the radial coordinate (Krasiński and Hellaby 2002).

For reviews of applications of the LT models see (Krasiński 1997), (Plebański and Krasiński 2006), (Bolejko et al. 2009), and (Bolejko, Célérier and Krasiński, 2011).

The Lemaître models

The most general form of a spherically symmetric metric is

$$\tag{13} {\rm d} s^2 = {\rm e}^{A}{\rm d} t^2-\frac{R'^{2}}{1+2E} {\rm d} r^2 - R^2 {\rm d} \vartheta^2 - R^2 \sin^2 \vartheta {\rm d} \varphi^2,$$ where the functions $A,R,$ and $E$ depend on $t$ and $r$. We presume that the source the gravitational field is an imperfect fluid. From the Einstein equations we obtain the following evolution equations

\begin{eqnarray} && \dot{R}^{2}=2E + \frac{2M}{R}+ \frac{1}{3}\Lambda R^2, \tag{14} \\ && \dot{M} = -\frac{1}{2} \kappa p \dot{R}R^{2},\tag{15} \\ && \dot{E} = \frac{A'}{2} (1+2E) \frac{\dot{R}}{R'}, \tag{16} \end{eqnarray} where $p$ is pressure, and a dot denotes the directional derivative along the velocity field, $\dot{R} \equiv {\rm e}^{-A/2} \partial R / \partial t$ (when $A=0$, as in the case of dust, this reduces to the ordinary time derivative, cf. the LT and Szekeres models). The gradient of the function $A$ follows from $T^{ab}{};_b =0$ and is $$\frac{A'}{2} = \frac{-p' + \frac{2}{\sqrt{3}} ( \lambda \Sigma)' +2 \sqrt{3} \lambda \Sigma R'/R }{\rho+p}, \tag{17}$$ where $\lambda$ is the viscosity coefficient, $\Sigma$ is the shear scalar $\Sigma^2 = \Sigma_{a b}\Sigma^{a b}/2$ (Eckart's method of modelling the viscous stress has been implemented above (Eckart 1940)). Finally the energy density is given by $$\kappa \rho = \frac{2M'}{R^2R'} \tag{18}.$$

As seen from (14)-(16) in contrast to the dust solution the mass ($M$) and curvature/energy ($E$) functions depend on time. The rate of change of mass depends on pressure, and the rate of change of the function $E$ depends on the gradient of pressure and viscosity. In the special case of zero pressure and viscosity the model reduces to the LT model.

Other inhomogeneous models

In addition to the LT and Szekeres models, a few other classes of generalisations of the FLRW metrics are known. The drawback of several of them is that they contain arbitrary functions of the comoving time coordinate, and so do not determine any evolution law for the Universe. To obtain an evolution law, one would have to impose an equation of state on them. However, the only equations of state that cosmologists know how to handle are $p = 0$ or $\rho = \rho(p)$, and they both reduce the models mentioned in this section to the FLRW limit. To keep the models nontrivial, a more general equation of state is needed, for example $\rho = \rho(p, T)$, with a position-dependent temperature $T$. So far, no-one has had an idea what function to use for $T$.

However, these models have proven useful as a basis for various theoretical considerations (see, for example Krasiński (1997)), so we list a selection of them here. Detailed descriptions of these classes are given in Krasiński (1997).

The Stephani and Barnes models

These are perfect fluid solutions with zero shear, zero rotation, nonzero acceleration and nonzero expansion. There are two collections of solutions in this set:

1.  The conformally flat solution of Stephani (1967)
The matter density in it depends only on the comoving time, while the pressure depends on all the coordinates. In general, the solution has no symmetry.
2.  The Petrov type D solutions
These models were found by Barnes (1973), but the spherically symmetric case was known much earlier (Kustaanheimo and Qvist 1948), and rediscovered many times over (Krasiński 1997). They necessarily have one of the three symmetries: spherical, planar or pseudo-spherical. In general, they have mass density and pressure depending on time and on the spatial coordinate that is constant on the symmetry orbits. One member of the Barnes family of solutions, found by McVittie (1933), was recently discussed in several papers as a model of a black hole embedded in an expanding Universe (see also Bolejko, Célérier and Krasiński, 2011).

Generalisations of the LT and Barnes models

For the LT model and for the Barnes class generalisations were found in which the matter source is a charged dust, or, respectively, a charged perfect fluid obeying the Einstein–Maxwell equations – see books by Krasiński (1997) and (Plebański and Krasiński 2006) for overviews. The charged LT model has interesting geometrical properties (Krasiński and Bolejko 2007). In addition, several generalisations of the LT and Barnes models were found, in which the source has nonzero viscosity or heat conduction. The physical interpretation of these in a cosmological context is less clear.

Models with null radiation

These are superpositions of the FLRW models with various vacuum solutions, like those of Schwarzschild, Kerr, Kerr–Newman, etc. Their energy-momentum tensors are mixtures of perfect fluid with null radiation, sometimes also with electromagnetic field. The different contributions to the source are coupled through common constants so that, for example, the null radiation can vanish only if either the perfect fluid component or the inhomogeneity on the FLRW background go away.

The "stiff-fluid" models

These are solutions of the Einstein equations with a 2-dimensional Abelian symmetry group acting on spacelike orbits, in which the perfect fluid source obeys the "stiff equation of state", $\rho = p$ (the source can be alternatively interpreted as a massless scalar field).

Cosmological applications

Formation of cosmological black holes

Figure 2: Formation of a black hole in the Szekeres model. Figure a) shows the radial cross-section through the apparent horizon (AH) and the absolute apparent horizon (AAH). Horizontal solid lines show the instants for which the curves in the next figure (figure b) are drawn, these are: t = 4.0, 10.0, 11.5, 11.9772, 12.3 and 13.0. Figure b) shows 2D cross-sections of the horizons at various time instants. Figure c) shows 3D profiles of the AAH and AH horizons at t = 11.5.

Black holes are usually described using the Schwarzschild or Kerr metrics. However, (1) these space-times are asymptotically flat while the real Universe is not; (2) these black holes do not evolve, they exist unchanged from $t = -\infty$ to $t = + \infty$, while real black holes can accrete mass.

Inhomogeneous cosmological models can solve both these problems. They can reproduce both the FLRW and Schwarzschild solutions in appropriate limits, so they can be used to describe the formation of a black hole in an asymptotically FLRW model. Already in 1947 Bondi predicted, using the LT model, that rapidly collapsing matter forces the light rays to also converge toward the final singularity. A more detailed model of a process of black hole formation in the cosmological context was presented by Krasiński and Hellaby (2004). It is also based on the LT model and describes the formation of a black hole with mass comparable to those observed at the centres of galaxies. Three scenarios were considered: black holes that evolve from either localised mass-density perturbations, or out of localised velocity perturbations, or around pre-existing wormholes.

Non-symmetrical models, such as the Szekeres models, have also been used to describe the formation of non-spherical black holes. In fact the very first application of the Szekeres model was to study a non-symmetrical collapse (Szekeres 1975b). Formation of various types of horizons within the quasispherical Szekeres model was studied further by Krasiński and Bolejko (2012). An example of this study is presented in Figure 2. Inset (a) shows the radial cross-section through the apparent horizon (AH) and the absolute apparent horizon (AAH). For each shell of mass M, we see that as time increases shells approach the big crunch singularity. Initially there is no singularity. As time passes and shells collapse towards the center, various horizons start to appear before the singularity at the origin forms. We show the formation of horizons and of the singularity in two different radial directions; these directions correspond to two extremes in the Szekeres matter dipole. As seen, in one direction the absolute apparent horizon (AAH) appears first, and in the other the apparent horizon (AH) appears first. The dashed-dotted line represents the singularity. Horizontal solid lines show the instants for which the curves in the next figure (inset b) are drawn, these are: t = 4.0, 10.0, 11.5, 11.9772, 12.3 and 13.0. Inset (b) shows 2D cross-sections of the horizons at various time instants. Inset (c) shows 3D profiles of the AAH and AH horizons at t = 11.5. The example presented here is extreme, with parameters of the Szekeres model chosen so as to maximise the differences and asymmetry between the horizons.

Formation of cosmic structures

Figure 3: The evolution of the density contrast inside cosmic voids. Comparison between the linear approximation and the Szekeres models, which can trace the evolution in the non-linear regime.
Figure 4: The evolution of the density contrast inside condensations. Comparison between the linear approximation and the Szekeres models, which can trace the evolution in the non-linear regime.

Astronomical observations reveal inhomogeneous structures in the Universe. It appears that all these structures evolved from small initial fluctuations that were present in the early Universe. The evolution of these structures, usually modelled in terms of the density contrast ($\delta = \rho/\bar{\rho} - 1$) is currently highly non-linear. This is where the exact solutions of the Einstein equations come very useful, as they can trace the evolution of structures into the non-linear regime.

The comparison of the evolution of a negative density contrast (such as inside cosmic voids) between the exact Szekeres model and the linear approximation is presented in Figure 3. In both cases the initial conditions are the same. As seen, the evolution of cosmic voids enters the non-linear regime less than 500 My after the Big Bang. After 2,000 My the linear approximation breaks down completely as it predicts unphysical values of the density contrast (which cannot be smaller than -1 in voids).

The comparison of the evolution of a positive density contrast (such as inside clusters of galaxies) between the exact Szekeres model and the linear approximation is presented in Figure 4. In both cases the initial conditions are the same. Here, as in the case of cosmic voids, the evolution quickly enters a non-linear regime and needs to be traced using the exact methods.

Light propagation

Figure 5: The role of the expansion rate, and the Weyl and Ricci focusing on the distance-redshift relation.

Our knowledge about the Universe comes from astronomical observations. While the light propagation is governed by the geodesic equations ($k^\alpha_{;\mu} k^\mu =0$, where $k^\alpha$ is the null vector), in cosmology we also need two other quantities: redshift and distance.

The angular diameter distance $D_A$ is proportional to the cross-section of the light bundle. Its logarithmic rate of change is proportional to the expansion rate of the null bundle, hence

$$\frac{{\rm d} \ln D_A}{{\rm d} s} = \theta, \tag{19}$$ where $\theta$ is the expansion of the null bundle, and is given by the Sachs optical equations (Sachs 1961) \begin{eqnarray} \frac{{\rm d} \theta}{{\rm d} s} + \theta^2 + |\sigma|^2 = - \frac{1}{2} R_{\alpha \beta} k^{\alpha} k^{\beta}, \tag{20} \\ \frac{{\rm d} \sigma}{{\rm d} s} + 2 \theta \sigma = C_{\alpha \beta \mu \nu} \epsilon^{*\alpha} k^{\beta} \epsilon^{*\mu} k^{\nu}. \tag{21} \end{eqnarray} where $R_{\alpha \beta}$ and $C_{\alpha \beta \mu \nu}$ are respectively the Ricci and Weyl tensors, $\epsilon^\mu$ is perpendicular to $k^\mu$ and is tangent to a wave front, and $\sigma$ is the shear of the null bundle.

Using (19) together with (20) one gets the following relation for the angular diameter distance $$\frac{{\rm d^2} D_A}{{\rm d} s^2} = - ( |\sigma|^2 + \frac{1}{2} R_{\alpha \beta} k^{\alpha} k^{\beta}) D_A, \tag{22}$$ where two terms on the RHS ($\sigma$ and $R_{\alpha\beta} k^{\alpha} k^{\beta}$) are usually referred to as the Weyl focusing and the Ricci focusing respectively.

The source of the Weyl focusing is the Weyl curvature. In the LT model it is given by $$\tag{23} C_{\alpha \beta \mu \nu} \epsilon^{*\alpha} k^{\beta} \epsilon^{*\mu} k^{\nu} = \frac{1}{2} \left(\frac{c_\phi}{R^2}\right)^2 \left( \rho - \bar{\rho} \right),$$ where $c_\phi$ is related to the impact parameter, $\rho$ and $\bar{\rho}$ are respectively local and average matter densities. In special cases, such as for example the radial geodesics in the LT model, or axial geodesics in the axially-symmetric Szekeres model, the Weyl focusing is zero. In all other, generic cases it is not zero (for details see Bolejko and Ferreira 2012).

The Ricci focusing in many cases can be expressed using a simple formula. For the energy-momentum tensor of dust it becomes $$\tag{24} R_{\alpha \beta} k^{\alpha} k^{\beta} = \rho (1+z)^2,$$ where the above can be derived by contracting the Einstein equations twice with the null vector (Bolejko and Ferreira 2012).

The last remaining part of evaluating the redshift-distance relation is to convert the affine parameter to redshift. This is given by the following formula $$\tag{25} \frac{1}{(1+z)^2} \frac{ {\rm d} z } { {\rm d} s } = \frac{1}{3} \Theta + \Sigma_{ab} n^a n^b + u^a{}_{;b} u^b n_a,$$ where $\Theta$ is the scalar of the matter velocity field, $\Theta = u^a{}_{;a}$, and $n^a$ is the unit spatial vector in the direction of light propagation. For most cosmological models $u^a{}_{;b} u^b = 0$. As for the role of shear of the matter velocity field, it has been argued that for a sufficiently long line of sight the shear may cancel out (Räsänen 2010). However, on small scales the contribution to redhsift from the matter shear need not be small, and can lead to observable effects, such as those reported by Bolejko et al. (2013).

An example of how these factors affect the distance-redshift relation is presented in Figure 5. Large Weyl and Ricci focusing tend to decrease the distance for a given redshift compared to the case of a homogeneous FLRW model. The expansion rate affects the redshift – when the expansion rate of the velocity field of matter is high (higher than in the FLRW model) the redshift for a given distance increases; if the expansion rate is low then the redshift to a given distance is lower than in the FLRW case. In the inhomogeneous Universe all these factors play a role: for example inside cosmic voids the expansion rate is high but the Ricci focusing is very weak, on the otehr hand when light propagates through overdense regions, such as clusters of galaxies, the expansion rate of the matter velocity field is low but the Ricci and Weyl focusing is very strong.

Observational consequences of inhomogeneities

There are a number of potentially observable effects that could occur only when inhomogeneities are present. The obvious one is the presence of structures in the Universe (galaxies, clusters, and cosmic voids). However, some other effects (related to light propagation) would never occur if the geometry of our universe were the same as in the FLRW models.

The best-known example is gravitational lensing, which does not exist in the FLRW models. The FLRW spacetime is conformally flat, so the null geodesics in the FLRW model are conformal images of the null geodesics from the Minkowski spacetime. In this spacetime, rays sent from a common origin never intersect again (unless there is a singularity in the conformal mapping). Hence, the presence of the gravitational lensing phenomenon shows that our Universe cannot be conformally flat.

The inhomogeneities not only distort images of distant objects, but also affect photons' trajectories. In the homogeneous FLRW models two rays sent from the same source at different times to the same observer pass through the same sequence of intermediate matter particles. In inhomogeneous models this does not occur (unless under very special circumstances), hence photons sent from point A to point B at two different time instants will intersect different collections of matter world-lines. This property is known as non-repeatable light paths or the position drift effect. As a consequence, cosmological objects change their positions in the sky, what could be in principle detectable by future observations (Krasiński and Bolejko 2011).

Inhomogeneities not only cause image distortions and position drift, they also affect the energy of the photons. In inhomogeneous models the energy of photons changes accordingly to physical conditions along the line of sight (25). In the perturbative framework this effect is known as the Integrated Sachs–Wolfe effect or the Rees–Sciama effect. In addition, inhomogeneities also contribute to the redshift drift, i.e. the change of the redshift with time (this particular effect exists also in the FLRW models, but in inhomogeneous models the amplitude and sign of this effect will depend on the inhomogeneities along the line of sight). As as result, the position of cosmological objects changes with time, also their redshift changes, and all of these changes are not uniform across the whole sky, but depend on physical conditions along a particular line of sight.

Discussion and future prospects

Inhomogeneous cosmological models are very useful as they allow us to follow the evolution of cosmic structures in the non-linear regime. They are also essential when tracing light propagation. The inhomogeneous cosmological models are advantageous compared to the linear perturbations, which can only follow the evolution and trace the light when inhomogeneities are small $|\delta| \ll 1$. The disadvantage of exact models is their symmetry or quasi-symmetry, which does not allow us to describe all forms and shapes of cosmic structures.

The prospect for future development is twofold. One can seek new solutions of the Einstein equations that would generalize the known ones, using either analytical methods or computational methods, such as for example The Cactus Framework. The other front is based on employing exact methods in the N-body simulations.

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• Krasiński, A and Hellaby, C (2002). Structure formation in the Lemaître-Tolman model. Phys. Rev. D 65: 023501.
• Krasiński, A and Hellaby, C (2004). Formation of a galaxy with a central black hole in the Lemaître–Tolman model. Phys. Rev. D. 69: 043502.
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• Plebański, J and Krasiński, A (2006). An Introduction to General Relativity and Cosmology. Cambridge University Press, Cambridge. ISBN-13: 9780511654411
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Footnotes

[1]There are two families of Szekeres solutions; the other one is the limit $R' \to 0$ of (1). So far it has found no application in astrophysical cosmology, and we shall not discuss it here – see (Plebański and Krasiński 2006) for an extended description.

[2]The coordinate $t$ used in (1) is singled out geometrically: the hypersurfaces of constant $t$ are orthogonal to the cosmic velocity field $u^{\alpha}$, which in turn is uniquely determined as the unit timelike eigenvector of the Ricci tensor. The non-simultaneity of the Big Bang is meant with respect to this time, and is thus defined unambiguously.

[3]Tolman did not independently derive this solution. Tolman's paper focuses on the evolution of inhomogeneities, using Lemaître's solution and citing his (1933) paper. That said, the equations that Tolman uses more closely resemble the now-standard notation than Lemaître's.

• Books:
• Krasiński (1997) – a systematic review of exact inhomogeneous cosmological solutions of the Einstein equations up to 1997.
• Stephani et al. (2003) – a systematic review of exact solutions of the Einstein equations (some of them include a class of inhomogeneous cosmological models).
• Plebański and Krasiński (2006) – a GR textbook that also discusses inhomogeneous cosmological models and their applications. It can serve as a pedagogical introduction to the subject of exact inhomogeneous cosmology for newcomers to the field.
• Bolejko et al. (2009) – a book, which discusses cosmological applications of the LT and Szekeres models.
• Review articles: