Constraints on the density perturbation spectrum from primordial black holes

TL;DR

This work revisits how primordial black holes (PBHs) constrain the primordial density perturbation spectrum, focusing on the spectral index $n$ under standard radiation domination and under a subsequent period of thermal inflation.The authors correct two errors in prior analyses, obtaining a stronger standard-cosmology bound $n \lesssim 1.25$, and show that thermal inflation weakens the bound to $n \lesssim 1.3$ while introducing a missing mass window for PBHs in the range $10^{18}$ g to $10^{26}$ g.Non-Gaussianity in density perturbations can alter PBH formation probabilities, but its effect on the $n$ bounds is modest, typically up to about $0.05$ in the worst cases.Overall, PBH constraints provide a powerful, history-sensitive probe of the early universe, tightly linking the small-scale perturbation spectrum to the thermal evolution and potential relics from PBH evaporation.

Abstract

We re-examine the constraints on the density perturbation spectrum, including its spectral index $n$, from the production of primordial black holes. The standard cosmology, where the Universe is radiation dominated from the end of inflation up until the recent past, was studied by Carr, Gilbert and Lidsey; we correct two errors in their derivation and find a significantly stronger constraint than they did, $n \lesssim 1.25$ rather than their 1.5. We then consider an alternative cosmology in which a second period of inflation, known as thermal inflation and designed to solve additional relic over-density problems, occurs at a lower energy scale than the main inflationary period. In that case, the constraint weakens to $n \lesssim 1.3$, and thermal inflation also leads to a `missing mass' range, $10^{18} g \lesssim M \lesssim 10^{26} g$, in which primordial black holes cannot form. Finally, we discuss the effect of allowing for the expected non-gaussianity in the density perturbations predicted by Bullock and Primack, which can weaken the constraints further by up to 0.05.

Figures (4)

• Figure 1: The tightest limits on the initial mass fraction of PBHs, $\alpha_{\rm_{i}}$. The relic constraint is shown as a dotted line, emphasizing that it is not compulsory.
• Figure 2: A schematic of the variation of the comoving Hubble radius $(H^{-1}/a)$ with time for the standard evolution of the universe (solid line) and with thermal inflation (dashed line). Points A and B correspond to the end of the original period of inflation in the standard evolution and with thermal inflation respectively. Thermal inflation begins at point C and finishes at D, after which time the comoving Hubble radii must coincide. Between D and E the scales which are entering the Hubble radius are doing so for the second time so that no PBHs are formed in this region. The values of the comoving Hubble radius and the horizon mass at these points are displayed in Table \ref{['comhub']}. We denote the current comoving Hubble radius and horizon mass as $(H^{-1}/a)_0$ and $M_{\rm{H0}}$ respectively.
• Figure 3: The tightest limits on the initial mass fraction of PBHs $\alpha_{\rm{i}}$ if thermal inflation occurs, on the same vertical scale as Fig. \ref{['fstand']}. The gap $10^{18} \, {\rm g} <M<10^{26} \, {\rm g}$ is the excluded mass range. For $M<10^{9} \rm{g}$ a large initial mass fraction $\sim 1$ of PBHs is allowed.
• Figure 4: The variation of the limits on $n$ with reheating temperature from the relic constraint (lower line) and from $\delta(M_{\rm{min}})<1$.