Scalar induced gravitational waves from primordial black hole Poisson fluctuations in f(R) gravity

Abstract The gravitational potential of a gas of initially randomly distributed primordial black holes (PBH) can induce a stochastic gravitational-wave (GW) background through second-order gravitational effects. This GW background can be abundantly generated in a cosmic era dominated by ultralight primordial black holes, with masses m PBH < 10 9 g. In this work, we consider f ( R ) gravity as the underlying gravitational theory and we study its effect at the level of the gravitational potential of Poisson distributed primordial black holes. After a general analysis, we focus on the R 2 gravity model. In particular, by requiring that the scalar induced GWs (SIGWs) are not overproduced, we find an upper bound on the abundance of PBHs at formation time Ω PBH,f as a function of their mass, namely that Ω PBH,f < 5.5 × 10 -5 (10 9 g/ m PBH ) 1/4 , which is 45% tighter than the respective upper bound in general relativity. Afterwards, by considering R 2 gravity as an illustrative case study of an f ( R ) gravity model, we also set upper bound constraints on its mass parameter M. These mass parameter constraints, however, should not be regarded as physical given the fact that the Cosmic Microwave Background (CMB) constraints on R 2 gravity are quite tight. Finally, we conclude that the portal of SIGWs associated to PBH Poisson fluctuations can act as a novel complementary probe to constrain alternative gravity theories.