Cosmic Variance and the Measurement of the Local Hubble Parameter

There is an approximately 9% discrepancy, corresponding to $2.4\sigma$, between two independent constraints on the expansion rate of the Universe: one indirectly arising from the cosmic microwave background and baryon acoustic oscillations and one more directly obtained from local measurements of the relation between redshifts and distances to sources. We argue that by taking into account the local gravitational potential at the position of the observer this tension—strengthened by the recent Planck results—is partially relieved and the concordance of the Standard Model of cosmology increased. We estimate that measurements of the local Hubble constant are subject to a cosmic variance of about 2.4% (limiting the local sample to redshifts $z>0.010$) or 1.3% (limiting it to $z>0.023$), a more significant correction than that taken into account already. Nonetheless, we show that one would need a very rare fluctuation to fully explain the offset in the Hubble rates. If this tension is further strengthened, a cosmology beyond the Standard Model may prove necessary.

Figure 1

Function $\Theta$ which corrects the relation of Eq. (1) when the density contrast is not linear. The plot assumes the $\mathrm{\text{Planck}}+\mathrm{BAO}$ best-fit value of ${\Omega }_m=0.3086$, but the dependence of $\Theta$ on cosmological parameters is very weak.

Figure 2

The 68%, 95%, and 99.7% confidence-level probabilities of Gaussian matter fluctuations (right vertical axis) and consequently of the local Hubble parameter (left vertical axis), as a function of co-moving size of the matter fluctuation (top ticks) or, equivalently, redshift (bottom ticks). The relation between $\delta H/H$ and $\delta \rho /\rho$ is given by Eq. (1). The range $z_{\min }\le z\le z_{\max }$ corresponds to the range of observation of Ref. 2. Also shown is the $1\mathrm{\text{−}}\sigma$ emerald band relative to the value $H_0^{\mathrm{local}}/H_0^{\mathrm{CMB}}-1$, which shows the $2.4\sigma$ tension between CMB and local measurements of the Hubble constant.

Figure 3

The 68%, 95%, and 99.7% confidence-level probabilities of log-normally distributed matter fluctuations (right vertical axis) and consequently of the local Hubble parameter (left vertical axis), as a function of co-moving size of the matter fluctuation (top ticks) or, equivalently, redshift (bottom ticks). As in Fig. 2, we show the $1\mathrm{\text{−}}\sigma$ band relative to the value $H_0^{\mathrm{local}}/H_0^{\mathrm{CMB}}-1$.

Figure 4

Probability of having an inhomogeneity that induces a $\delta H/H$ (left vertical axis) or $\delta H$ (right vertical axis) greater than a given value for the cases listed in the legend and in Table . Also shown is the $1\mathrm{\text{−}}\sigma$ band relative to the value $H_{0,\mathrm{unc}}^{\mathrm{local}}/H_0^{\mathrm{CMB}}-1$.