Primordial Black Holes Place the Universe in Stasis

TL;DR

The work shows that a broad PBH mass spectrum with an initial abundance after inflation can drive a finite cosmic stasis epoch in which PBH and radiation energy densities remain constant while the universe expands. This stasis arises as a global attractor with fixed-point abundances and an effective equation of state $ar{w}=-(\alpha+1)/(\alpha+7)$, lasting approximately $oxed{\mathcal{N}_s \approx \frac{\alpha+7}{3}\ln\left(\frac{M_{ m max}}{M_{ m min}}\right)}$ e-folds before the heaviest PBHs evaporate. The altered expansion history impacts inflationary observables $(n_s,r)$, reshapes the stochastic gravitational-wave background, and modifies production of dark radiation, dark matter, and the baryon asymmetry, with accretion/merger effects largely negligible across viable parameter space. Overall, PBH-induced stasis offers a natural cosmological epoch with distinctive, testable signatures across CMB, GW, and early-universe phenomenology. $

Abstract

A variety of scenarios for early-universe cosmology give rise to a population of primordial black holes (PBHs) with a broad spectrum of masses. The evaporation of PBHs in such scenarios has the potential to place the universe into an extended period of "stasis" during which the abundances of matter and radiation remain absolutely constant despite cosmological expansion. This surprising phenomenon can give rise to new possibilities for early-universe dynamics and lead to distinctive signatures of the evaporation of such PBHs. In this paper, we discuss how this stasis epoch arises and explore a number of its phenomenological consequences, including implications for inflationary observables, the stochastic gravitational-wave background, baryogenesis, and the production of dark matter and dark radiation.

Figures (7)

• Figure 1: Trajectories (blue curves) in the $(\Omega_{\rm BH},\langle\Omega_{\rm BH}\rangle)$-plane for the system of differential equations in Eq. (\ref{['eq:2Dsystem']}) with $\alpha = -7/4$. The red dot indicates the point at which $\Omega_{\rm BH} = \langle\Omega_{\rm BH}\rangle = \overline{\Omega}_{\rm BH}$, where in this case $\overline{\Omega}_{\rm BH} \approx 0.57$.
• Figure 2: The total abundance $\Omega_{\rm BH}$ of a population of PBHs with an initial mass spectrum of the form given in Eq. (\ref{['eq:dist']}), plotted as a function of the number of $e$-folds $\mathcal{N}$ of cosmic expansion since $t_i$ for several different choices of the power-law exponent $\alpha$ (solid curves). For all of the curves shown, we have taken $M_{\rm min}=10^{-1}$ g and $M_{\rm max}=10^{9}$ g. In each case, we see that the universe experiences a period of stasis stretching over many $e$-folds. Indeed, for each curve, the dotted line of the corresponding color represents the theoretical prediction in Eq. (\ref{['eq:omegabar']}) for the PBH abundance $\overline{\Omega}_{\rm BH}$ during stasis.
• Figure 3: The PBH abundance ${\overline{\Omega}}_{\rm BH}$ during stasis, the equation-of-state parameter $\overline w$ during stasis, and the equation-of-state parameter $w_c$ during the PBH-formation epoch, each plotted as a function of $\alpha$.
• Figure 4: Qualitative sketch illustrating how the comoving Hubble horizon evolves in cosmologies involving an epoch of PBH-induced stasis as a function of the number of $e$-folds for several different values of $\alpha$. All curves shown correspond to the parameter choices $M_{\rm max}/M_{\rm min} = 10^{7}$ and $\mathcal{N}_{\rm PBH} = 10$. The evolution of $(aH)^{-1}$ in the standard cosmology is indicated by the dotted black curve.
• Figure 5: The spectral index $n_s$ and tensor-to-scalar ratio $r$ in cosmologies involving an epoch of PBH-induced stasis, shown as points in the $(n_s,r)$-plane for the two classes of inflaton potential given in Eqs. (\ref{['eq:PolyV']}) and (\ref{['eq:TModelV']}). Each sequence of points appearing on the left side of the figure corresponds to a potential of the form given in Eq. (\ref{['eq:TModelV']}) with a different value $\alpha_{\rm inf}$. Each sequence of points appearing on the right side of the figure corresponds to a different value of the parameter $p$ in Eq. (\ref{['eq:PolyV']}). The color of each point within a particular such sequence indicates the value of the parameter $\alpha$ in Eq. (\ref{['eq:dist']}). For all points shown in the figure, we have taken $M_{\rm min} = 5\,{\rm g}$ and $M_{\rm max} = 10^9\,{\rm g}$, such that the value of $\mathcal{N}_s$ is maximized for each choice of $\alpha$. We have also taken $\mathcal{N}_{\rm PBH} = 2$. The regions within which the values of $n_s$ and $r$ are consistent with Planck data Planck:2018jri at the 68% and 95% CL are shaded dark and light blue, respectively. The dark- and light-purple regions represent the corresponding 68% and 95% CL projections for CMB-S3 experiments collectively, while the dark- and light-red regions represent the corresponding projections for CMB-S4 Abazajian:2019eic.