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 < 109g. 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 ΩPBH,f as a function of their mass, namely that
ΩPBH,f < 5.5 × 10-5 (109g/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.