Extended mass distribution of PBHs during QCD phase transition: SGWB and mini-EMRIs

TL;DR

This work links an extended PBH mass distribution, generated by a broadly peaked inflationary scalar power spectrum during the QCD phase transition, to two SGWB channels and a population of mini-EMRIs. It develops a comprehensive framework connecting the primordial spectrum to PBH formation thresholds, mass functions, and merger rates, and then confronts these predictions with NANOGrav and LVK data via Bayesian analysis. The study finds that a broad PBH mass distribution naturally yields detectable mini-EMRIs and a merger SGWB in future detectors, while the low-frequency induced SGWB can accommodate NANOGrav's signal; in many parameter regions, the SGWB is masked by astrophysical backgrounds, making mini-EMRIs a more robust probe. The results highlight a promising, multi-band observational pathway to test primordial scenarios and distinguish them from astrophysical black hole populations.

Abstract

Primordial black holes (PBHs) are one of the most important tracers of cosmic history. In this work, we investigate the formation of PBHs around the time of the QCD phase transition from a broadly peaked inflationary scalar power spectrum, which naturally produces an extended PBH mass function. This scenario yields two distinct stochastic gravitational wave backgrounds (SGWB): (i) scalar-induced, second-order tensor perturbations generated at PBH formation, and (ii) a merger-driven SGWB from the subsequent PBH binary population. Using Bayesian analysis, we examine both SGWB channels with the data from the NANOGrav 15-year dataset and the first three observing runs of LVK. We also forecast continuous-wave signals from mini extreme mass ratio inspirals (mini-EMRIs) for direct comparison with NANOGrav and LVK constraints. Our parameter scans identify regions of the parameter space where the combined SGWB is detectable in future ground-based and space-based detectors. A broad PBH mass distribution naturally gives rise to mini-EMRIs, which future ground-based observatories, such as LVK A+, ET, and CE, can detect. For a large part of the PBH parameter space, the SGWB of astrophysical origin masks the primordial SGWB in the frequency band of ground-based detectors. Thus, for extended PBH mass distributions, we find that the detection of mini-EMRIs is a more robust channel for probing the PBH parameter space than the corresponding SGWB.

Figures (10)

• Figure 1: Left panel: Here, we plot the equation of state as a function of temperature, $T$Borsanyi:2016ksw. Right panel: We plot the resultant deviation in the critical density contrast from the QCD phase transition, as obtained in Musco:2023dak.
• Figure 2: Here we plot the PBH mass fraction $f_\text{PBH}$ in Peaks Theory (solid lines) formalism for three different sets of parameters. These parameter choices illustrate the variation in mass fraction resulting from changes in the height, width, and location of the inflationary scalar power spectrum peak. Evidently, a broader inflationary spectrum leads to a more extended PBH mass distribution, while the change in the height of the inflationary power spectrum peak significantly changes the PBH abundance. Please note that we are limiting ourselves to exactly flat scalar power spectrum amplification $n_0=1$.
• Figure 3: Here, we plot the PBH Merger Rate Density per Logarithmic Mass Interval for Different Power Spectrum Models. The plot shows the calculated merger rate density in logarithmic mass interval ${dR_{E2}}/({d(\ln m_1) d(\ln m_2)})$ [$\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$], as a function of the two merging black hole masses, $m_1$ and $m_2$, in solar mass units ($M_{\odot}$). Each set of colored contours corresponds to a different set of parameters for the primordial power spectrum, as detailed in the "Datasets" legend. We plot the contours for 4 different levels of merger rate with respect to the maximum merger rate, as described in the "contour levels" legend.
• Figure 4: Comparison of waveform models for a binary system with a primary mass of $M_1 = 10\, M_\odot$ at a distance of 1 Mpc. Each panel shows a different mass ratio, $Q = M_1/M_2$. Panels (A)-(D) compare the IMRPhenomXASPratten:2020fqn model with Ajith:2009bn. In panels (E) and (F), we compare the mini-EMRI waveform obtained from FastEMRIWaveformsChua:2020stfKatz:2021yftSperi:2023jteChapman-Bird:2025xtd with Eq. \ref{['eq:CharStrain']} for m=2, which follows from Finn:2000sy which also takes into account the relativistic correction factors based on the Teukolsky-Sasaki-Nakamura formalism Teukolsky:1973haSasaki:1981sx.
• Figure 5: We plot the SGWB spectra for three different sets of parameters. The amplification in SGWB in the left part of the plot is for second-order tensor perturbation during the PBH formation, while the right side of the plot corresponds to the SGWB from PBH mergers. The solid SGWB lines make use of IMRPhenomXASPratten:2020fqn waveform of pycbcalex_nitz_2024_10473621 for $M_1/M_2 < 10^3$ and EMRI waveform Finn:2000sy for $M_1/M_2 > 10^3$ while the dashed lines correspond to using Ajith:2009bn for all mass ratios. Here we also plot the Kernel Density Estimations (KDE) for NANOGrav, $2\sigma$ power law spectrum sensitivity lines for LVK O3 and O4a LIGOScientific:2025bgj and projected power law integrated sensitivity curves Schmitz:2020syl for LVK A+, LIGOScientific:2016fpeLIGOScientific:2019vicKAGRA:2021kbbLIGO-G2001287, ET Punturo:2010zzHild:2010id, LISA Bartolo:2016amiCaprini:2019pxzLISACosmologyWorkingGroup:2022jokSchmitz:2020rag, AION Badurina:2019hst CE Evans:2021gyd and ET Punturo:2010zzHild:2010id and Taiji Luo:2021qji with a cutoff SNR $\ge$ 10 and assuming operation time $\mathcal{T}=4$ yr for LISA, Taiji, CE and ET; 1 yr for LVK detectors.