Primordial Black Hole Formation in Dust-Radiation Bouncing Cosmologies

TL;DR

This work develops a unified framework to assess primordial black hole formation during the contracting phase of dust–radiation bouncing cosmologies. By extending the two-fluid perturbation theory and generalizing the collapse model to include radiation, the authors compute the curvature perturbation spectrum via an adiabatic semi-analytical approach, derive a relativistic Jeans length, and apply two sound-wave–based collapse criteria to two-fluid patches. They find an extremely small and nearly mass-independent collapse threshold $oldsymbol{ ilde{oldsymbol{ ext{ζ}}}}_c$ (about $1.5 imes10^{-21}$) and a correspondingly tiny curvature power at formation, yielding negligible PBH mass fractions $eta$ for representative masses, even before the transition to the Jeans- or Hubble-dominated regime. The results imply that, without additional perturbation amplification, dust–radiation bouncing cosmologies are unlikely to produce significant PBH abundances, though the developed framework can be applied to explore other multi-fluid or amplification scenarios.

Abstract

Primordial black holes (PBHs) provide a unique probe of the early Universe and may have an enhanced abundance in bouncing cosmologies, where a long contracting phase can amplify perturbations. We develop a unified framework to study PBH formation in dust-radiation bouncing cosmologies, focusing on the classical contracting phase so that the results are insensitive to bounce details. We compute the curvature power spectrum for an extremely small dust equation of state using a stable semi-analytical (adiabatic) method, derive the Jeans length of the two-fluid system using dynamical-system analysis and the WKB approximation, and extend the three-zone model from the single- to the two-fluid case to model local collapse. We implement two collapse criteria to obtain the curvature perturbation threshold for PBH formation and estimate PBH mass fractions for benchmark masses spanning low-mass ($10^{-17} M_{\odot}$) to supermassive ($10^{13} M_{\odot}$) scales. The critical curvature threshold is extremely small and nearly mass-independent over a broad range $(ζ_c \sim 10^{-21}$ for $10^{-14}$ to $10^{13} M_{\odot})$, with deviations only near dust-radiation equality. Nevertheless, the square root of the curvature power spectrum at the relevant formation times is many orders of magnitude smaller, yielding vanishingly small PBH mass fractions across the benchmark masses. Compared with the pure-dust case, radiation pressure and the two-fluid collapse conditions significantly suppress PBH production, implying that substantial PBH formation in dust-radiation bouncing cosmologies would require additional mechanisms to amplify curvature perturbations.

Figures (9)

• Figure 1: Blue: rescaled scale factor $\bar{a}/\bar{a}_c$ as a function of the dimensionless time $(-x)$. Red: dust-radiation equality. The evolution of the scale factor should be understood from the right hand side to the left in the contracting phase where $x<0$. The same applies to Figs.\ref{['WKBvalid']}, \ref{['powerspectrumofiin']}, and \ref{['compare']}.
• Figure 2: $\Omega_k^{(0)}/\Omega_k^{(2)}$ as a function of the dimensionless time. Green: PBH with mass $10^{13} M_{\odot}$. Blue: PBH with mass $10^{2} M_{\odot}$. Red: PBH with mass $10^{-14} M_{\odot}$. Black: PBH with mass $10^{-17} M_{\odot}$. The value of $\Omega_k^{(0)}/\Omega_k^{(2)}$ at the formation time is indicated by the vertical dashed lines.
• Figure 3: Blue: Evolution of spectrum ${\cal P}_{\zeta}(k_H,x)$ at a given scale. Red: dust-radiation equality. Black: PBH formation time. (a) $M=10^{-14} M_{\odot}$. (b) $M=10^{2} M_{\odot}$ (c) $M=10^{13} M_{\odot}$. (d) $M=10^{-17} M_{\odot}$.
• Figure 4: Comparison between the Jeans wavenumber and the Hubble wavenumber.
• Figure 5: Scale factor of the overdense region for $\eta_\text{eq}/ L=1$.