Shadows of black holes at cosmological distances in the co-varying physical couplings framework

The co-varying physical couplings (CPC) framework states that physical parameters like the speed of light in vacuum $c$, the Newtonian constant $G$, and the cosmological constant $\mathrm{\Lambda }$ could indeed vary with the spacetime coordinates $x^{\mu }$. Here, we assume a temporal variation, that is, $c(t),G(t)$ and $\mathrm{\Lambda }(t)$. We show that the McVittie spacetime, a black hole in an expanding universe, is a solution of the CPC framework providing naturally an important parameter of the model. Then, we calculate the shadow angular radius of this black hole at cosmological distances. A black hole shadow in the CPC context could be either larger or smaller than the same shadow in the standard cosmology. It depends on how the set $\{c,G,\mathrm{\Lambda }\}$ varies with time or with the cosmic expansion.