Primordial Black Holes and their Mass Spectra: The Effects of Mergers and Accretion within Stasis Cosmologies

TL;DR

The paper investigates how mergers and accretion affect primordial black hole (PBH) induced cosmological stasis in scenarios with an extended PBH mass spectrum. By formulating a Boltzmann-like evolution for the PBH differential number density $f_{BH}(M,t)$ that includes Hawking evaporation, accretion, and mergers, and by adopting a non-standard four-epoch expansion history, the authors derive a three-body merger rate $\mathcal{R}_3$ and analyze the associated two-body merger rates $\Gamma_+(M)$ and $\Gamma_-(M)$. They find that mergers stay well below the Hubble rate across the stasis-relevant parameter space, implying negligible distortion of the PBH mass spectrum, while accretion is typically negligible except in regions with very broad spectra or large PBH abundances where stasis can be abridged or avoided. Accretion can, however, introduce distortions in the mass spectrum and modestly shorten stasis in limited corners of parameter space, and the authors provide criteria and numeric contours (e.g., $(\Delta M/M)_{max}$ and $\Gamma_{ac}/\Gamma_e$) to delineate these regions. Overall, PBH-induced stasis remains viable with characteristic gravitational-wave signatures tied to the spectrum and expansion history, offering potential observational probes for early-universe dynamics.

Abstract

A variety of processes in the very early universe can give rise to a population of primordial black holes (PBHs) with an extended mass spectrum. For certain mass spectra of this sort, it has been shown that the evaporation of these PBHs into radiation can drive the universe toward an epoch of cosmological stasis which can persist for a significant number of $e$-folds of cosmological expansion. However, in general, the initial mass spectrum which characterizes a population of PBHs at the time of production can subsequently be distorted by processes such as mergers and accretion. In this paper, we examine the effects that these processes have on the spectra that lead to a PBH-induced stasis. Within such stasis models, we find that mergers have only a negligible effect on these spectra within the regime of interest for stasis. We likewise find that the effect of accretion is negligible in many cases of interest. However, we find that the effect of accretion on the PBH mass spectrum is non-negligible in situations in which this spectrum is particularly broad. In such situations, the stasis epoch is abridged or, in extreme cases, does not occur at all. Thus accretion plays a non-trivial role in constraining the emergence of stasis within scenarios which lead to extended PBH mass spectra.

Figures (6)

• Figure 1: Contours of the maximum value $\mathcal{N}_{\rm PBH}^{(\rm max)}$ of $\mathcal{N}_{\rm PBH}$ consistent with our stasis cosmology, shown in the $(M_{\rm min},M_{\rm max})$-plane for $\alpha = -1$. The black region of the plot is inconsistent with our cosmology for all $\mathcal{N}_{\rm PBH} \geq 0$ and therefore excluded. The corresponding upper bound on $\mathcal{N}_{\rm PBH}$ is weaker for all $\alpha < -1$ throughout the entirety of the upper half-plane, though the value $\mathcal{N}_{\rm PBH}^{(\rm max)}$ is largely insensitive to the choice of $\alpha$.
• Figure 2: The scale factor $a_B$ at the time at which a pair of PBHs decouple from the Hubble flow and form a binary in a cosmology involving an epoch of PBH-induced stasis, shown for different values of $\alpha$ (solid colored curves). All of the results shown correspond to the parameter choices $M_{\rm min} = 10$ g, $M_{\rm max} = 10^9$ g, and $\mathcal{N}_{\rm PBH} = 5$. The black dashed curve indicates the corresponding result for $a_B$ in the standard cosmology. The circular dot at the end of each curve indicates the minimum value of $x/\widetilde{x}$ for a binary with $M_B = 2M_{\rm min}$, given that a binary cannot form before its constituent PBHs are produced via gravitational collapse. The square that appears further to the right along each curve indicates the corresponding minimum value of $x/\widetilde{x}$ for a binary with $M_B = 2M_{\rm max}$.
• Figure 3: The ratio $\Gamma_3(M_1,M_2,M_3)/H_{\rm PBH}$ of the differential merger rate defined in Eq. (\ref{['eq:Gamma3']}) to the value $H_{\rm PBH} = H (t_{\rm PBH})$ of the Hubble expansion rate at the beginning of the stasis epoch for the parameter choices $M_{\rm min} = 1$ g, $M_{\rm max} = 10^5{\rm g}$, $\mathcal{N}_{\rm PBH} = 2$, $\alpha=-2$, and $M = 10^5$ g. Each panel is a density plot of this ratio within the $(M_2,M_3)$-plane evaluated at a different time $t$ within the range $t_i < t < t_s$. The gray region of each panel indicates the region of the plane wherein either $M_2$ or $M_3$ lies below the corresponding value of $M_{\rm cut}(t)$.
• Figure 4: The merger rates $\Gamma_+(M)$ (solid colored curves) and $\Gamma_-(M)$ (corresponding dashed curves), as well as the expansion rate $H$, shown as functions of time within the range $t_i < t < t_s$ for a variety of different values of $M$. The different panels of the figure correspond to different choices of $\alpha$, and the results shown in all panels correspond to the parameter choices $M_{\rm min} = 1$ g, $M_{\rm max} = 10^5$ g, and $\mathcal{N}_{\rm PBH} = 2$. The four red dots indicated along the $M = 10^5$ g curve in the upper left panel correspond to the four $t$ values for which $\Gamma_3(M_1,M_2,M_3)/H_{\rm PBH}$ is plotted for this $M_1$ value in Fig. \ref{['fig:ScanDiffMrgRateMinus']}. Since it is always the case that $\Gamma_+(M_{\min}) = 0$, only a dashed curve appears in any of the four panels for $M = 1$ g.
• Figure 5: Contours of $(\Delta M/M)_{\rm max}$ within the $(\alpha,M_{\rm min})$-plane at the time $t = \tau_e(M_{\rm max})$ at which a PBH with $M_i = M_{\rm max}$ would have evaporated completely in our stasis cosmology in the absence of accretion --- i.e., at the time at which the stasis epoch would have ended. The left panel shows the results for $M_{\rm max} = 10^7$ g and $\mathcal{N}_{\rm PBH} = 2$, the middle panel shows the results for $M_{\rm max} = 10^7$ g and $\mathcal{N}_{\rm PBH} = 8$, and the right panel shows the results for $M_{\rm max} = 10^9$ g and $\mathcal{N}_{\rm PBH} = 2$. Within the gray region at the bottom of each panel, $(\Delta M/M)_{\rm max} > 0.01$, indicating a non-negligible change in $M$ for at least the heaviest PBHs. The red hatched region in each panel indicates the region within which PBHs with $M_i = M_{\rm max}$ form sufficiently close to the end of the PBH-formation epoch that the quasi-stationary approximation underlying Eq. (\ref{['eq:AccretRateGen']}) may not be reliable. The yellow hatched region at the bottom of the middle panel indicates the region of the $(\alpha,M_{\rm min})$-plane within which the consistency condition $\mathcal{N}_{\rm PBH} < \mathcal{N}_{\rm PBH}^{({\rm max})}$ is violated. Throughout the region above the orange dashed contour in each panel, the condition $J < 0.01$ is satisfied for $\theta_{\rm QS} = 100$.