  It is believed that primordial black holes (PBHs), if they exist, can serve
as a powerful tool to probe the early stage of the cosmic history. Essentially,
in the radiation dominated universe, PBHs could form by the gravitational
collapse of overdense primordial perturbations produced during inflation. In
this picture, one important ingredient is the threshold of density contrast,
which defines the onset of PBH formation. In the literature, most of the
estimations of threshold, no matter numerically or analytically, are
implemented in the framework of general relativity (GR). In this paper, by
performing analytic estimations, we point out that the threshold for PBH
formation depends on the gravitational theory under consideration. In GR, given
a fixed equation of state, the analytic estimations adopted in this paper give
a constant value of the formation threshold. If the theory is characterized by
additional mass scales other than the Planck mass, the estimated threshold of
density contrast may depend on the energy scale of the universe at the time of
PBH formation. In this paper, we consider the Eddington-inspired-Born-Infeld
gravity as an example. We find that the threshold would be enhanced if the
Born-Infeld coupling constant is positive, and would be suppressed for a
negative coupling constant. Also, we show explicitly that the threshold depends
on the energy scale of the universe at the PBH formation time. This conclusion
is expected to be valid for any gravitational theory characterized by
additional mass scales, suggesting the possibility of testing gravitational
theories with PBHs.
