  Over the last few years, primordial black holes (PBHs) have emerged as a
strong candidate for cold dark matter. A significant number of PBHs are
produced when the strength of the primordial scalar power spectrum is enhanced
on small scales (compared to the COBE normalized values on large scales). Such
primordial spectra also inevitably lead to strong amplification of the
scalar-induced, secondary gravitational waves (GWs) at higher frequencies. The
recent detection of the stochastic gravitational wave background (SGWB) by the
pulsar timing arrays (PTAs) has opened up the possibility of directly probing
the very early universe. Different studies have shown that, when PBHs are
assumed to have been formed during the epoch of radiation domination, the
mechanism for the amplification of the scalar-induced GWs that is required to
explain the PTA data can overproduce the PBHs over some ranges of masses. In
this work, we assume a specific functional form for the primordial scalar power
spectrum and examine the production of PBHs and the scalar-induced secondary
GWs during the phase of reheating, which precedes the standard epoch of
radiation domination. Specifically, we account for the uncertainties in the
conditions for the formation of PBHs and ensure that the extent of PBHs
produced remains within the observational bounds. We find that the
scalar-induced SGWB generated during a phase of reheating with a steeper
equation of state (than that of radiation) fit the NANOGrav 15-year data with a
stronger Bayesian evidence than the astrophysical scenario involving GWs
produced by merging supermassive binary black holes.
