Bayesian model selection of Primordial Black Holes and Dressed Primordial Black Holes with lensed Gravitational Waves

Abstract

If particle dark matter (DM) and primordial black holes (PBHs) coexist, PBHs will be surrounded by particle DM, forming celestial objects known as dressed PBHs (dPBHs). These structures suggest a scenario in which PBHs and DM can exist simultaneously. However, in the high-frequency regime, the gravitational lensing effect of bare PBHs is similar to that of dPBHs. Ground-based gravitational wave (GW) detectors are particularly sensitive to high-frequency GW signals. In this regime, the lensing effect of a point-mass lens with a mass in the range of $10^{-1} \sim 10^2 M_{\odot}$ becomes significant. In this work, we incorporate dPBH models with GW observations and employ Bayesian inference techniques to distinguish PBHs from dPBHs. Using the third-generation ground-based GW detectors, Einstein Telescope (ET) and Cosmic Explorer (CE), as examples, we demonstrate that these detectors can effectively differentiate the lensing effects of dPBHs from those of PBHs across a broad frequency range. Furthermore, we find that with a larger black hole (BH) mass inside the surrounding particle DM, ET and CE can distinguish these two lensed models with even greater precision.

Figures (4)

• Figure 1: The amplification factors $F$ as a function of frequency $f$ for both dPBHs and PBHs across different regimes. The upper panel illustrates the absolute value $|F(f)|$, while the lower panel displays the corresponding phase $\text{arg}~F$. The dashed blue curves show $F(f)$ for dPBHs with $M_{\mathrm{PBH}} = 30~M_{\odot}$ in the wave-optics regime ($w_0 \leq 6$), while the solid orange curves reflect the geometric-optics regime ($w_0 > 6$). The dotted red curves depict PBHs with $M_{\mathrm{PBH}} = 30~M_{\odot}$ in the wave-optics regime. The dashdot purple curves display $F(f)$ for PBHs with $M_{\mathrm{PBH}} = 180~M_{\odot}$ in wave-optics ($w \leq 10$), in contrast to the solid green curves showing the geometric-optics behavior ($w > 10$).
• Figure 2: The posterior distribution of $\eta$, $M$, $z_S$, and $M_{\text{PBH}}$. The blue curves and the ligth orange curves correspond to the dPBH detection hypothesis and the PBH detection hypothesis, respectively. The black lines indicate the injected values.
• Figure 3: The dependence of Bayes factors on $M$. The dotted blue, dashed orange, and solid green lines correspond to the Bayes factors for datasets with $m_{\text{PBH}} = 10~M_{\odot}$, $m_{\text{PBH}} = 30~M_{\odot}$, and $m_{\text{PBH}} = 50~M_{\odot}$, respectively.
• Figure 4: The dependence of Bayes factors on $\eta$. The dotted blue, dashed orange, and solid green lines correspond to the Bayes factors for datasets with $m_{\text{PBH}} = 10~M_{\odot}$, $m_{\text{PBH}} = 30~M_{\odot}$, and $m_{\text{PBH}} = 50~M_{\odot}$, respectively.