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Abstract

We study the Primordial Black Hole (PBH) reheating scenario, where PBHs
originate in a general cosmological background. In this scenario, ultralight
PBHs with masses $M\lesssim 10^8$g temporarily dominate the Universe and reheat
it via Hawking radiation before Big Bang Nucleosynthesis (BBN). We investigate
whether the induced Gravitational Wave (GW) spectrum associated with PBH
reheating contains information about the pre-PBH-dominated stage, namely the
initial equation of state $w$ (after inflation). We first derive the transfer
functions of curvature fluctuations for general $w$ with adiabatic and
isocurvature initial conditions. We find that, in general, a stiffer equation
of state enhances the induced GW amplitude as it allows for a longer PBH
dominated phase compared to the radiation dominated case. We also find that the
spectral slope of GWs induced by primordial curvature fluctuations is sensitive
to $w$, while the spectral slope of GWs induced by PBH number density
fluctuations is not. Lastly, we derive constraints of the initial PBH abundance
as a function of $w$, using BBN and Cosmic Microwave Background (CMB)
observations. A stiffer equation of state leads to stricter constraints on the
initial energy density fraction, as induced GWs are enhanced. Interestingly, we
find that such induced GW signals may enter the observational window of several
future GW detectors, such as LISA and the Einstein Telescope. Our formulas,
especially the curvature fluctuation transfer functions, are applicable to any
early matter-dominated universe scenario.