Value of $H_0$ in the Inhomogeneous Universe

Local measurements of the Hubble expansion rate are affected by structures like galaxy clusters or voids. Here we present a fully relativistic treatment of this effect, studying how clustering modifies the mean distance- (modulus-)redshift relation and its dispersion in a standard cold dark matter universe with a cosmological constant. The best estimates of the local expansion rate stem from supernova observations at small redshifts ($0.01<z<0.1$). It is interesting to compare these local measurements with global fits to data from cosmic microwave background anisotropies. In particular, we argue that cosmic variance (i.e., the effects of the local structure) is of the same order of magnitude as the current observational errors and must be taken into account in local measurements of the Hubble expansion rate.

Figure 1

The average $\overline{⟨d_L^{-2}⟩}(z)$ of Eq. (5) in units of ${\mathrm{Mpc}}^{-2}$ (thick solid curve), its dispersion (shaded region), and the homogeneous value (dashed curve) are computed within a range $z=0.01$ and $z=0.03$. We have used a best-fit cosmology from Planck [4] and a UV cutoff of $k_{\mathrm{UV}}=0.1h{\mathrm{Mpc}}^{-1}$.

Figure 2

The redshift distribution of the 155 SNe of the $\mathrm{CfA}3+\mathrm{OLD}$ sample [22, 23] with redshift within 0.01 and 0.1 considered here.