Example of an inhomogeneous cosmological model in the context of backreaction

In this article, we present an example of an inhomogeneous cosmological model, which is inspired by the linear perturbation theory. The metric of this model can be described as the Einstein–de Sitter background with periodically distributed dust overdensities. The model construction enables application of the Green-Wald averaging scheme and the Buchert averaging technique simultaneously. We compare the angular diameter distance function of the considered model to the angular diameter distances corresponding to the average space-times given by the Green-Wald and the Buchert frameworks respectively.

Figure 1

Isodensity surfaces of the function ${\sin }^2(x)+{\sin }^2(y)+{\sin }^2(z)$, which is a distribution of the first order density perturbation ${\rho }^{(1)}$.

Figure 2

Comparison between the effective scale factor $a_{\mathcal{D}}$ (blue) and the background scale factor $a(t)$ (black), for a model with parameters given in the text.

Figure 3

Comparison between the volume of the elementary cell $x\in [0,\pi )$, $y\in [0,\pi )$, $z\in [0,\pi )$, calculated for the metric with parameters described in the text $V_{\mathcal{D}}$ (blue), with the volume of the elementary cell in the background metric $V={\pi }^3{a}^3(t)$ (black).

Figure 4

The components of the first Buchert equation as a function of time $t$. The left-hand side of the equation is plotted in blue, while the right-hand side is plotted by the red curve.

Figure 5

The components of the second Buchert equation as a function of time $t$. The left-hand side of the equation is plotted in blue, while the right-hand side is plotted by the red curve.

Figure 6

The relation between the scale factor and the redshift. In blue we plotted the family of the one hundred $a(z)$ curves, each of them corresponding to the particular geodesic. This result is compared with the standard relation $a(z)=1/(1+z)$ for a background Einstein–de Sitter universe (black).

Figure 7

The angular diameter distance as a function of redshift calculated along each of the one hundred null geodesics of a random direction (the set of the blue curves). This result is compared with the Einstein–de Sitter (black) model, the FLRW model with parameters ${\mathrm{\Omega }}_m=1.0497$, ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.0$ (red) and the Buchert average space-time (green dashes).

Figure 8

The angular diameter distance relative to the Einstein–de Sitter model. The result of the numerical procedure described in the text (the set of the blue curves) is compared with the FLRW model with parameters ${\mathrm{\Omega }}_m=1.0497$, ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.0$ (red) and the Buchert average space-time (green dashes).

Figure 9

The angular diameter distance relative to the Einstein–de Sitter model for the case when the observer is located at the density maximum. The result of the numerical procedure described in the text (the set of the blue curves) is compared with the FLRW model with parameters ${\mathrm{\Omega }}_m=1.0497$, ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.0$ (red) and the Buchert average space-time (green dashes).