Scale dependence of cosmological backreaction

Because of the noncommutation of spatial averaging and temporal evolution, inhomogeneities and anisotropies (cosmic structures) influence the evolution of the averaged Universe via the cosmological backreaction mechanism. We study the backreaction effect as a function of averaging scale in a perturbative approach up to higher orders. We calculate the hierarchy of the critical scales, at which 10% effects show up from averaging at different orders. The dominant contribution comes from the averaged spatial curvature, observable up to scales of $\sim 200\mathrm{Mpc}$. The cosmic variance of the local Hubble rate is 10% (5%) for spherical regions of radius 40 (60) Mpc. We compare our result to the one from Newtonian cosmology and Hubble Space Telescope Key Project data.

Figure 1

Scale dependence of the cosmic variance of the Hubble rate. Data are from the Newtonian CDM model in Ref. 23, with $h=0.5$, ${\Omega }_{\mathrm{m}}=1$ and a COBE-normalized power spectrum. Thick and thin lines correspond to the relativistic result Eq. (18) for a scale-invariant power spectrum with cutoffs at $k_c=1/\mathrm{kpc}$ (simulation) and $1/\mathrm{pc}$ (physical), respectively.

Figure 2

Relative fluctuation of the Hubble rate from cosmological backreaction and its cosmic variance band (thick lines) compared to the empirical mean and variance of ${\delta }_H$ obtained from the HST Key Project data [5] as a function of averaging radius. The thin line shows the ensemble mean of ${\delta }_H$. The band enclosed by the thick lines indicates the effect of the inhomogeneities ($\propto 1/r^2$), and the dashed lines are the effect from sampling with given measurement errors in an otherwise perfectly homogeneous Universe.