Shadow of a black hole at cosmological distances

Cosmic expansion is expected to influence the size of a black hole shadow observed by a comoving observer. Except for the simplest case of the Schwarzschild black hole in the de Sitter universe, an analytical approach for the calculation of shadow size in an expanding universe is still not developed. In this paper, we present the approximate method based on using the angular size redshift relation. This approach is appropriate for the general case of any multicomponent universe (with matter, radiation, and dark energy). In particular, we show that supermassive black holes at large cosmological distances in the Universe with matter may give a shadow size approaching the shadow size of the black hole in the center of our Galaxy, and present sensitivity limits.

Figure 1

The dependence of the angular radius ${\alpha }_{\mathrm{sh}}$ on $z$ for different spacetimes: the de Sitter model without matter (${\mathrm{\Omega }}_{m0}=0$, ${\mathrm{\Omega }}_{\mathrm{\Lambda }0}=1$), the Einstein–de Sitter spacetime filled by matter only (${\mathrm{\Omega }}_{m0}=1$, ${\mathrm{\Omega }}_{\mathrm{\Lambda }0}=0$), and the mixed case with matter and dark energy presence (${\mathrm{\Omega }}_{m0}=0.3$, ${\mathrm{\Omega }}_{\mathrm{\Lambda }0}=0.7$). Angular radius is presented in units of $m{H}_0/c$. These curves are based on use of the angular size redshift relation with effective “physical size” of the BH shadow and provide the approximate solution for the angular size of the shadow in expanding Universe; see details in the text.

Figure 2

Zoom of previous figure. Here we have also shown the additional dashed curve plotted according to formula (10). Formula (10) is only valid for small $z$, but in the whole range of $z$ it can be interpreted as the Schwarzschild case without cosmological expansion. It can be understood from transformation of (10) to (11). Circles on curves show the results of the numerical calculation of the shadow size by the integration of the null geodesics in McVittie spacetime and represent the exact numerical solution for the angular size of the shadow in expanding Universe; see details in the text.