Testing supermassive primordial black holes with lensing signals of binary black hole merges

TL;DR

This work proposes using the rate and time-delay distribution of strongly lensed gravitational-wave events to constrain the abundance of supermassive primordial black holes (SMPBHs) within a $Λ$CDM framework. It models both Poisson and clustered SMPBH scenarios, computes the PBH-induced isocurvature power and its impact on the halo mass function, and then derives lensing probabilities and time-delay distributions for lensed GW signals. Through mock data and hierarchical Bayesian inference, the authors show that next-generation detectors could constrain $f_{ m PBH}$ to approximately $10^{-4}$ for $M_{ m PBH} \,\gtrsim\ 10^8 M_\odot$, with clustered SMPBHs yielding stronger, mass-independent bounds. The method provides a complementary constraint to large-scale structure and other PBH probes, though it relies on idealized lens models and will benefit from accounting for selection effects and more realistic SMPBH scenarios in future work.

Abstract

Next-generation ground-based gravitational wave (GW) detectors are expected to observe millions of binary black hole mergers, a fraction of which will be strongly lensed by intervening galaxies or clusters, producing multiple images with characteristic distribution of time delay. Importantly, the predicted rate and properties of such events are sensitive to the abundance and distribution of strong lensing objects which directly depends on cosmological models. One such scenario posits the existence of supermassive primordial black holes (SMPBHs) in the early universe, which would enhance the formation of dark matter halos. This mechanism has been proposed to explain the abundance of high-redshift galaxies observed by James Webb Space Telescope. Crucially, the same cosmological model with SMPBHs would also leave a distinct imprint on the population of strongly lensed GWs. It predicts both an increased event rate and a modified distribution of time delays between the multiple images. Therefore, we propose statistical measurements of the rate and time delay distribution of strong lensing GW events as a powerful probe to directly constrain the abundance of SMPBHs. Considering $Λ$CDM cosmology with (non-)clustered SMPBHs, we find that the abundance of SMPBHs $f_{\rm PBH}$ with masses above $10^8~M_{\odot}$ is constrained to be $\sim10^{-4}$ at $95\%$ confidence level. It will be comparable and complementary to the currently available constraint from large scale structure observations.

Figures (5)

• Figure 1: Left: The linear matter power spectrum at $z=0$ for $\Lambda$CDM cosmology (black solid); for $\Lambda$CDM+PBH cosmology with initially Poisson distributed SMPBHs ($\xi_0=0$, red dashed); and for initially clustered SMPBHs ($x_{\rm cl}=1~{\rm Mpc}$, $\xi_0=10$, green dash-dotted) respectively, where $M_{\rm PBH}=10^9~M_{\odot}$, $f_{\rm PBH}=10^{-3}$. The blue dashed and dash-dotted lines represent the isocurvature perturbations originating from the Poisson distributed and clustered SMPBHs with cutoff scale $k_{\rm cut}$, respectively. Middle: Corresponding halo mass function for $\Lambda$CDM and $\Lambda$CDM+PBH cosmology at redshift $z=7$. Right: Halo mass function as function of the velocity dispersion of lenses in singular isothermal sphere model.
• Figure 2: Left: Expected number of strong lensing GW events $\Lambda_{\rm L,GW}(f_{\rm PBH})$ for both of $\Lambda$CDM and $\Lambda$CDM+PBH cosmology ($x_{\rm cl}=1~{\rm Mpc}$, $\xi_0=[0,10]$, $M_{\rm PBH}=10^9~M_{\odot}$), assuming a merger rate $R=5\times10^5~{\rm yr^{-1}}$ and observation duration $T_{\rm obs}=10~{\rm yrs}$. Right: The strong lensing time delay distributions $p(\Delta t|f_{\rm PBH})$ at different values of $f_{\rm PBH}$ for different cosmology.
• Figure 3: We simulate the population of observable strong lensing GW events based on the $\Lambda$CDM cosmology as fiducial model, considering different BBH detection rates and observing times $T_{\rm obs}$. Upper Left: Detectable time delay distributions $p(\Delta t|f_{\rm PBH},T_{\rm obs})$ and corresponding event counts $\Lambda_{\rm L,GW}(f_{\rm PBH},T_{\rm obs})$ for a fixed observational duration $T_{\rm obs} = 10~{\rm yrs}$ under four different BBH detection rates $R \in[5 \times 10^5,\; 1 \times 10^5,\; 5 \times 10^4,\; 1 \times 10^4]~\rm{yr^{-1}}$. Here, $n$ denotes the realized number of observed lensed events $N_{\rm obs}$ in each mock sample. Upper Right: Comparison between the theoretical cumulative distribution functions (CDFs; black solid curve) and the sample-derived CDFs for the corresponding cases. The maximum vertical distance between each pair of CDFs is quantified by the Kolmogorov-Smirnov (K-S) statistic $D$ indicated in the plot. Lower: Similar to the upper panel but with the detection rate fixed at $R = 1\times10^5~\rm{yr^{-1}}$ and for varying observational durations $T_{\rm obs}\in[10,\;5,\;2,\;1]~\rm{yrs}$.
• Figure 4: Constraints on $f_{\rm{PBH}}$ at $95\%$ confidence level in $\Lambda$CDM+PBH cosmology with $M_{\rm{PBH}} = 10^9~M_{\odot}$. Upper Left: Initially Poisson distributed SMPBHs (i.e., $\xi_0 = 0$) under different BBH detection rates $R\in[5\times10^5, 1\times10^5,5\times10^4,1\times10^4]~{\rm yr^{-1}}$, for a fixed observational duration $T_{\rm obs}=10~{\rm yrs}$. Upper Right: The same Poisson-distributed scenario, but with fixed detection rate $R=1\times10^5~{\rm yr^{-1}}$ and varying observational durations $T_{\rm obs}\in[10,\;5,\;2,\;1]~\rm{yrs}$. Lower Left: Initially clustered SMPBHs ($x_{\mathrm{cl}} = 1~\rm{Mpc}$, $\xi_0 = 10$) under the same set of detection rates $R$ as in the upper left panel. Lower Right: The same clustered scenario, under the same varying $T_{\mathrm{obs}}$ as in the upper right panel.
• Figure 5: The upper limit of $f_{\rm PBH}$ at $95\%$ confidence level versus SMPBH mass $M_{\rm PBH}\in[10^6,10^{10}]~M_{\odot}$ for fixed BBH detection rates $R=5\times10^5~{\rm yr^{-1}}$ and observational durations $T_{\rm obs}=10~{\rm yrs}$. The red dashed and green dash-dotted curves denote the $\Lambda$CDM+PBH cosmology with initially Poisson distributed ($\xi_0=0$) and clustered SMPBHs ($x_{\rm cl}=10~{\rm Mpc}$, $\xi_0=10$), respectively. Other constraints are compiled from existing reviews Green:2020jorCarr:2020goxCarr:2021bzv: these include limits from Hawking radiation evaporation (Evaporation), micro-lensing surveys (Microlensing), Stochastic GW background (GWs), effect of accretion (Accretion), dynamical effects (Dynamical), and the imprint of PBHs on large-scale structure (LSS).