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The gravitational potential of a gas of initially randomly distributedprimordial black holes (PBH) can induce a stochastic gravitational-wave (GW)background through second-order gravitational effects. This GW background canbe abundantly generated in a cosmic era dominated by ultralight primordialblack holes, with masses $m_\mathrm{PBH}<10^{9}\mathrm{g}$. In this work, weconsider $f(R)$ gravity as the underlying gravitational theory and we study itseffect at the level of the gravitational potential of Poisson distributedprimordial black holes. After a general analysis, we focus on the $R^2$ gravitymodel. In particular, by requiring that the scalar induced GWs (SIGWs) are notoverproduced, 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$\Omega_\mathrm{PBH,f}<5.5\times10^{-5}\left(\frac{10^9\mathrm{g}}{m_\mathrm{PBH}}\right)^{1/4}$, 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 asphysical given the fact that the Cosmic Microwave Background (CMB) constraintson $R^2$ gravity are quite tight. Finally, we conclude that the portal of SIGWsassociated to PBH Poisson fluctuations can act as a novel complementary probeto constrain alternative gravity theories.

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