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

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 $m_\mathrm{PBH}<10^{9}\mathrm{g}$. 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 $\Omega_\mathrm{PBH,f}$ as a function of their mass, namely that $\Omega_\mathrm{PBH,f}<5.5\times 10^{-5}\left(\frac{10^9\mathrm{g}}{m_\mathrm{PBH}}\right)^{1/4}$, 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.