Distant foreground and the Planck-derived Hubble constant

It is possible to reduce the discrepancy between the local measurement of the cosmological parameter $H_0$ and the value derived from the $Planck$ measurements of the Cosmic Microwave Background (CMB) by considering contamination of the CMB by emission from some medium around distant extragalactic sources, such as extremely cold coarse-grain dust. Though being distant, such a medium would still be in the foreground with respect to the CMB, and, as any other foreground, it would alter the CMB power spectrum. This could contribute to the dispersion of CMB temperature fluctuations. By generating a few random samples of CMB with different dispersions, we have checked that the increased dispersion leads to a smaller estimated value of $H_0$, the rest of the cosmological model parameters remaining fixed. This might explain the reduced value of the $Planck$-derived parameter $H_0$ with respect to the local measurements. The signature of the distant foreground in the CMB traced by SNe was previously reported by the authors of this paper -- we found a correlation between the SN redshifts, $z_{\rm SN}$, and CMB temperature fluctuations at the SNe locations, $T_{\rm SN}$. Here we have used the slopes of the regression lines $T_{\rm SN}\,/\,z_{\rm SN}$ corresponding to different {\it Planck} wave bands in order to estimate the possible temperature of the distant extragalactic medium, which turns out to be very low, about 5\,K. The most likely ingredient of this medium is coarse-grain ($grey$) dust, which is known to be almost undetectable, except for the effect of dimming remote extragalactic sources.