Constraints on amplitudes of curvature perturbations from primordial black holes

TL;DR

This paper addresses how enhanced curvature perturbations at small scales can produce primordial black holes (PBHs) and how such PBHs imprint extragalactic photon and neutrino backgrounds. It develops a PBH mass-spectrum calculation within a Press-Schechter framework that accounts for the explicit time dependence of the gravitational potential, and explores two inflationary scenarios: a peaked curvature spectrum and a running-mass model. By computing Hawking evaporation spectra and integrating over the PBH population, the authors derive constraints on the peak parameters and on the running of the spectral index from gamma-ray and neutrino data, as well as a bound on the PBH energy density fraction. They also discuss the possibility of PBHs constituting dark matter and potentially explaining the Galactic center 511 keV line via PBH clustering, highlighting the broader implications for inflationary model building and early-universe constraints.

Abstract

We calculate the primordial black hole (PBH) mass spectrum produced from a collapse of the primordial density fluctuations in the early Universe using, as an input, several theoretical models giving the curvature perturbation power spectra with large (~ 0.01 - 0.1) values at some scale of comoving wave numbers k. In the calculation we take into account the explicit dependence of gravitational (Bardeen) potential on time. Using the PBH mass spectra, we further calculate the neutrino and photon energy spectra in extragalactic space from evaporation of light PBHs, and the energy density fraction contained in PBHs today (for heavier PBHs). We obtain the constraints on the model parameters using available experimental data (including data on neutrino and photon cosmic backgrounds). We briefly discuss the possibility that the observed 511 keV line from the Galactic center is produced by annihilation of positrons evaporated by PBHs.

Figures (11)

• Figure 1: The dependencies of the gravitational potential $\Psi_k$ (upper panel) and the density contrast $\delta_k$ (lower panel) on time. H.C. denotes the moment of horizon entry for the mode. The wave number taken is $k=0.2 \tau_i^{-1}$, and normalization is ${\cal R}_k^{\rm (inf)}(\tau_i) = 0.1$.
• Figure 2: PBH mass spectra calculated for the models of the Carr-Hawking collapse (solid line) and the critical collapse (dashed line). The following set of the parameters was used: $\Sigma=3, {\cal P}_{\cal R}^0=0.02, M_h^0=10^{14} {\rm g}.$
• Figure 3: PBH mass spectra for the standard collapse case with ${\cal P}_{\cal R}^0=0.0172, M_h^0=10^{17} {\rm g}$ (for all curves); from bottom to top, $\Sigma = 1, 3, 5$.
• Figure 4: Absorption cross section $\sigma_s$ for photons, in units of $\pi r_g^2$. Dashed line is the result of the numerical calculation MacGibbon:1990zk, solid line is the approximation (\ref{['gaga']}).
• Figure 5: Red shift distribution of the integrand $F(E,z)$. Thick lines are for neutrinos, thin lines are for photons (absorption of $\gamma$-rays at $z\gtrsim 700$ is not shown in this figure). Dashed curves represent the case of the critical collapse, solid ones correspond to the standard collapse. PBH mass spectra shown in Fig. \ref{['nBHfig']} were used in the calculation, $E=1$GeV for all cases.