Scalar induced gravitational waves from primordial black hole Poisson
  fluctuations in Starobinsky inflation
  The gravitational potential of a gas of initially randomly distributed
primordial black holes (PBH) can induce a stochastic gravitational-wave
background through second-order gravitational effects. This gravitational-wave
background can be abundantly generated in a cosmic era of domination of
ultralight primordial black holes, with masses
$m_\mathrm{PBH}<10^{9}\mathrm{g}$, which evaporate before Big Bang
Nucleosynthesis. Hence, the condition to avoid overproduction of gravitational
waves at PBH evaporation time, can act as a novel method to extract constraints
on cosmological models and gravitational theories. 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 the general analysis we focus on Starobinsky $R^2$ gravity model and we
extract strong constraints on the involved mass parameter, denoted as $M$, as a
function of the initial primordial black hole abundance,
$\Omega_\mathrm{PBH,f}$ and the black hole mass, $m_\mathrm{PBH}$. In
particular, one finds that in general $5\times
10^{-14}\lesssim\frac{M_\mathrm{min}}{M_\mathrm{Pl}}\lesssim 10^{-5}$, and only
in the extreme possible regime where $\Omega_\mathrm{PBH,f}>10^{-3}$ we get
that $ 10^{-5}\lesssim\frac{M_\mathrm{min}}{M_\mathrm{Pl}}\lesssim 10^{-1}$.
