Influence of structure formation on the cosmic expansion

We investigate the effect that the average backreaction of structure formation has on the dynamics of the cosmological expansion, within the concordance model. Our approach in the Poisson gauge is fully consistent up to second order in a perturbative expansion about a flat Friedmann background, including a cosmological constant. We discuss the key length scales which are inherent in any averaging procedure of this kind. We identify an intrinsic homogeneity scale that arises from the averaging procedure, beyond which a residual offset remains in the expansion rate and deceleration parameter. In the case of the deceleration parameter, this can lead to a quite large increase in the value, and may therefore have important ramifications for dark energy measurements, even if the underlying nature of dark energy is a cosmological constant. We give the intrinsic variance that affects the value of the effective Hubble rate and deceleration parameter. These considerations serve to add extra intrinsic errors to our determination of the cosmological parameters, and, in particular, may render attempts to measure the Hubble constant to percent precision overly optimistic.

Figure 1

The averaged Hubble rate as a function of redshift as a fractional change to the background Hubble rate. The top two curves show the Hubble rate ${\breve{H}}_{\mathcal{D}}$ calculated from the ensemble-averaged Friedmann equation, and the bottom two show the Hubble rate calculated directly, $\overline{H_{\mathcal{D}}}$. Both concordance and Einstein–de Sitter (EdS) models are considered, and the domain size is set at the Silk scale, and the smoothing scale is set to zero.

Figure 2

Plots for ${\breve{H}}_{\mathcal{D}}$ with $R_{\mathcal{D}}=1/k_{\mathrm{equality}}$, $R_{\mathcal{S}}=0$, with the variance included, as a function of redshift.

Figure 3

The averaged Hubble rate today as a function of domain size, with the smoothing scale removed, $R_{\mathcal{S}}=0$, and with the variance included.

Figure 4

The averaged Hubble rates today as a function of domain size, and the variance (top), with the smoothing scale removed, $R_{\mathcal{S}}=0$. Note that both EdS and concordance models become independent of domain size when it becomes sufficiently large. Intriguingly this happens around 100–200 Mpc; at this scale, for both models, we also have ${\breve{H}}_{\mathcal{D}}\sim \overline{H_{\mathcal{D}}}$ and the variance reduces to the fiducial ${10}^{-5}$ level.

Figure 5

Homogeneity scale in the averaged Hubble rate, as a function of present-day matter density. As the domain size is increased the averaged Hubble rate settles down to a constant value which is different from $H_0$ (see Fig. 4). We may investigate this scale as a function of ${\Omega }_0$ by deciding when $H_{\mathcal{D}}$ is sufficiently independent of $R_{\mathcal{D}}$ [here, $|d\ln ({\breve{H}}_{\mathcal{D}}/H_0-1)/d\ln R_{\mathcal{D}}|\lesssim ϵ$—i.e., when the slope in the log-log plot of Fig. 4 levels off]. To illustrate, we have arbitrarily fixed this $ϵ$ so that for ${\Omega }_0=0.1$, the homogeneity scale coincides with the equality scale, but this curve can be moved up and down to taste. Finally, another useful scale is when the variance becomes negligible: here we show the scale for which $2\sqrt{\mathrm{Var}[H_{\mathcal{D}}]}=0.01H_0$. Note how the latter of these measures behaves in the opposite way to the scales intrinsic to the models—the equality and Silk scales. We have fixed the baryon fraction at 5% here.

Figure 6

The ensemble-averaged deceleration parameter and its variance, for the EdS and concordance models. We can see from the graphs on the left that the difference from normal behavior grows during dust domination, and decays when the dark energy accelerated the expansion rate. Interestingly, the variance also starts to decrease then, reaching a minimum for $z\approx 0.22$. Note the divergence with the smoothing scale becomes irrelevant if it is bigger than tens of Mpc (right). The dependence on domain scale depends crucially on the model, however (middle), and no longer matters if $R_{\mathcal{D}}<R_{\mathcal{S}}$ (shown for the concordance model only, where the curves flatten off).

Figure 7

The ensemble-averaged deceleration parameter today as a function of the smoothing scale (in Mpc). This gives the residual contribution to the deceleration parameter for very large domains—clearly the smoothing scale is crucial, and suggests that higher-order contributions might be necessary in order quantify the backreaction effect in $q_0$.