Shadows of CPR black holes and tests of the Kerr metric

We study the shadow of the Cardoso-Pani-Rico (CPR) black hole for different values of the black hole spin $a_*$, the deformation parameters $\epsilon_3^t$ and $\epsilon_3^r$, and the viewing angle $i$. We find that the main impact of the deformation parameter $\epsilon_3^t$ is the change of the size of the shadow, while the deformation parameter $\epsilon_3^r$ affects the shape of its boundary. In general, it is impossible to test the Kerr metric, because the shadow of a Kerr black hole can be reproduced quite well by a black hole with non-vanishing $\epsilon_3^t$ or $\epsilon_3^r$. Deviations from the Kerr geometry could be constrained in the presence of high quality data and in the favorable case of a black hole with high values of $a_*$ and $i$. However, the shadows of some black holes with non-vanishing $\epsilon_3^r$ present peculiar features and the possible detection of these shadows could unambiguously distinguish these objects from the standard Kerr black holes of general relativity.