Primordial black holes with mass ratio-modulated initial clustering: merger suppression and projected constraints

TL;DR

The paper tackles how initial PBH clustering and broad mass distributions affect merger rates and gravitational-wave constraints. It develops a mass-ratio dependent framework by modifying the local neighbor count $N(y)$ with a clustering term $\xi(r)$ and a mass-ratio factor $G(q;\delta_c,\,n_p)$, enabling refined calculations of merger-rate suppression. Applying this to a two-body PBH merger rate with a broad mass function, the work finds that, for $q$-ratio binaries, merger suppression is reduced and the low-mass end of ET/LISA constraints, as well as SGWB constraints, are extended to smaller $\langle M\rangle$. These results suggest that PBH clustering could be testable at lower abundances and mass ranges than previously anticipated, though a full treatment would require incorporating three-body merger dynamics and alternative clustering models.

Abstract

We present a modification of the expected local primordial black hole (PBH) count $N(y)$, typically seen in the context of the early PBH binary merger rate as a term in the merger rate suppression. We utilize recent results in small-scale PBH clustering to formulate $N(y)$ in such a way that accounts for variations in the binary mass ratio $q$. We then examine how this change affects the projected constraints on PBH abundance from simulated Einstein Telescope (ET) and LISA mergers. Our results indicate that for broadly extended mass distributions, the merger suppression is greatly reduced for binaries with $q \gg 1$. This leads to an enhanced merger rate for binary distributions favoring a lighter average mass. This change is best reflected by an extension of the low mass end of the constraint windows derived from the resolvable merger channel, although this result is present as well in the stochastic gravitational wave background (SGWB) constraints. Our results imply that the assumption of PBH clustering is testable at abundances and mass ranges much lower than anticipated. Note however that we have only considered the two-body merger channel in this work; a more thorough analysis of our scenario will require a study on the dynamics involved in the three-body merger channel for broad mass distributions, which we leave to future work.

Figures (5)

• Figure 1: Local count mass ratio dependence $\mathcal{G}(q)$ with varying values of $\delta_c$ (top panel, fixed $n_p = 0$) and varying values of spectral tilt $n_p$ (bottom panel, fixed $\delta_c = 0.56$). In both panels, the black solid line corresponds to the case where $\delta_c = 0.56$ and $\gamma = -1/2$, while the black dashed line is approximately $\mathcal{G}(q; \delta_c=0.56, n_p = 0) \approx q^{-6/5}$. Note that for the top panel, $\delta_c = 0.10, 0.90$ are both represented by dash-dotted lines, while $\delta_c = 0.33, 0.66$ are represented by dotted lines. For the bottom panel, all dotted curves ($n_p = -0.5, \,-0.01,\,0.01,\,0.5$) almost completely overlap with the $n_p = 0$ case.
• Figure 2: Expected local count $N(y)$ for varying clustering models, plotted against PBH abundance $f_\mathrm{PBH}$ for power law exponent $\gamma = -1/2$. Black lines represent Poisson-distributed PBHs modeled by Eq. \ref{['eq:raidal_neighbor']}, with the black dotted lines showing the monochromatic case and the black solid lines showing the power law case. Colored dash-dotted lines show the clustered case with Eq. \ref{['eq:general_neighbor_count']} as the model, with blue, red, and green representing mass ratios $q = 1, 32, 1000$ respectively. All power law plots assume $M_\mathrm{max} = M_\mathrm{tot} = 30\,M_\odot$ and maximum mass ratio $q_\mathrm{max} = 1000$.
• Figure 3: Initial merger suppression $S_1$ for varying clustering models, plotted against PBH abundance $f_\mathrm{PBH}$ for power law exponent $\gamma = -1/2$. Black lines represent Poisson-distributed PBHs modeled by Eq. \ref{['eq:raidal_neighbor']}, with the dotted and dash-dotted lines showing the monochromatic and power law cases respectively. Colored dash-dotted lines show the clustered case with Eq. \ref{['eq:general_neighbor_count']} as the model, with blue, red, and green representing mass ratios $q = 1, 32, 1000$ respectively. All non-monochromatic plots assume power law distributions where $M_\mathrm{max} = M_\mathrm{tot} = 30\,M_\odot$ and maximum mass ratio $q_\mathrm{max} = 1000$.
• Figure 4: Projected constraints on PBH abundance $f_\mathrm{PBH}$ as a function of the average PBH mass ${\left\langle M \right\rangle}$ from simulated resolvable mergers at ET (green lines) and LISA (red lines) for monochromatic and $\gamma = -1/2$ power law distributions. Top panel shows constraints for merger events occurring in the redshift range $0.01 \leq z < 30$, while bottom panel shows constraints for mergers occurring in the redshift range $30 < z \leq 300$. The solid lines represent the constraints assuming monochromatic mass distributions, while the dashed and dash-dotted lines represent constraints for Poisson power law and correlated power law cases, respectively. The dotted lines represent the case where $\mathcal{G}(q)$ is set to $q^{-6/5}$. The gray regions are monochromatic constraints from other observations (see main text for the corresponding legend).
• Figure 5: Projected constraints on PBH abundance $f_\mathrm{PBH}$ as a function of the average PBH mass ${\left\langle M \right\rangle}$ from simulated SGWB mergers at ET (green lines) and LISA (red lines) for monochromatic and $\gamma = -1/2$ power law distributions. The solid lines represent the constraints assuming monochromatic mass distributions, while the dashed and dash-dotted lines represent constraints for Poisson power law and correlated power law cases, respectively. The dotted lines represent the case where $\mathcal{G}(q)$ is set to $q^{-6/5}$. The gray regions are monochromatic constraints from other observations (see main text for the corresponding legend).