From formation to evaporation: Induced gravitational wave probes of the primordial black hole reheating scenario

We study the primordial black hole (PBH) reheating scenario, where PBHs originate in a general cosmological background. In this scenario, ultralight PBHs with masses <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>M</a:mi><a:mo>≲</a:mo><a:msup><a:mrow><a:mn>10</a:mn></a:mrow><a:mrow><a:mn>8</a:mn></a:mrow></a:msup><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi mathvariant="normal">g</a:mi></a:mrow></a:math> 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>w</d:mi></d:math> (after inflation). We first derive the transfer functions of curvature fluctuations for general <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>w</f:mi></f:math> 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 <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>w</h:mi></h:math>, 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 <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>w</j:mi></j:math>, 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.