Probing primordial black holes from a first order phase transition through pulsar timing and gravitational wave signals

In this work, we assess the sensitivity reach of pulsar timing array (PTA) measurements to probe pointlike primordial black holes (PBHs), with an extended mass distribution, which originate from collapsed Fermi balls that are formed through the aggregation of asymmetric U(1) dark fermions trapped within false vacuum bubbles during a dark first order phase transition (FOPT). The PBH formation scenario is mainly characterized by the dark asymmetry, strength of the FOPT, rate of FOPT, and the percolation temperature. Meanwhile, for PBH masses of interest lying within $10^{-10} M_\odot - 10^{2}M_\odot$, the relevant signal for PTA measurements is the Doppler phase shift in the timing signal, due to the velocity change induced by transiting PBHs on pulsars. Taking the dark asymmetry parameter to be $10^{-4}$ and $10^{-5}$, we find that percolation temperatures within the 0.1-10 keV range, FOPT rates above $10^3$ times the Hubble parameter at percolation, and FOPT strengths within $10^{-6}-0.1$ can give rise to PBHs that can be probed by an SKA-like PTA observation. On the other hand, the accompanying gravitational wave (GW) signal from the FOPT can be used as a complementary probe, assuming that the peak frequency lies within the $\mathcal{O}(10^{-10})-\mathcal{O}(10^{-7})$ Hz range, and the peak GW abundance is above the peak-integrated sensitivity curves associated with pulsar timing observations that search for stochastic GWs. At the fundamental level, a quartic effective potential for a dark scalar field can trigger the FOPT. By performing a parameter scan, we obtained the class of effective potentials that lead to FOPT scenarios that can be probed by SKA through pulsar timing and GW observations.