Backreaction of cosmological perturbations in covariant macroscopic gravity

The problem of corrections to Einstein’s equations arising from averaging of inhomogeneities (backreaction) in the cosmological context has gained considerable attention recently. We present results of analyzing cosmological perturbation theory in the framework of Zalaletdinov’s fully covariant macroscopic gravity. We show that this framework can be adapted to the setting of cosmological perturbations in a manner which is free from gauge related ambiguities. We derive expressions for the backreaction which can be readily applied in any situation (not necessarily restricted to the linear perturbations considered here) where the metric can be brought to the perturbed Friedmann-Lemaître-Robertson-Walker form. In particular, these expressions can be employed in toy models studying nonlinear structure formation, and possibly also in $N$-body simulations. Additionally, we present results of example calculations which show that the backreaction remains negligible well into the matter dominated era.

Figure 1

The evolution of two large scale modes. Also shown is the Kodama-Sasaki analytical solution in the large scale limit $k\eta \ll 1$, given by Eq. (79).

Figure 2

The transfer function ${\Phi }_k$ normalized by its constant value at large scales, at the epoch $a=500a_{\mathrm{eq}}$. The dotted line is the BBKS transfer function (71).

Figure 3

The dimensionless integrand of ${\mathcal{S}}^{(1)}$, namely, the function $(k/H_0{)}^2{\Phi }_k^2$, at three sample values of the scale factor. The function dies down rapidly for large $k$, with the value at some $k$ being progressively smaller with increasing scale factor. The declining behavior of the curves for $a=a_{\mathrm{eq}}$ and $a=200a_{\mathrm{eq}}$ extrapolates to large $k$.

Figure 4

The dimensionless CDM density contrast. Together with Fig. 3 this shows that nonlinear scales do not impact the backreaction integrals significantly.

Figure 5

The correlation scalars (backreaction) for the sCDM model, normalized by $H^2(a)$. ${\mathcal{S}}^{(1)}$, ${\mathcal{P}}^{(1)}$, and ${\mathcal{S}}^{(2)}$ are negative definite and their magnitudes have been plotted. The vertical line marks the epoch of matter radiation equality $a=a_{\mathrm{eq}}$.