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

arXiv commentAccepted in JCAP

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