Primordial Black Holes from inflationary models with and without Broken Scale Invariance

TL;DR

The paper identifies a systematic overestimate in the mass variance used to predict primordial black hole production and provides the corrected expression involving the transfer function and window effects, requiring numerical evaluation. It then analyzes both scale-free and scale-featured inflationary spectra, showing that accurate calculations weaken constraints on the spectral index $n$ for the scale-free case and can dramatically strengthen constraints when a step in the variance is present. The work translates PBH overclosure requirements into bounds on inflationary parameters, demonstrating the potential to use PBH observations to limit features in the primordial power spectrum. Overall, the results underscore the need for precise, model-dependent treatment of $\sigma_H(t_k)$ and offer a framework to constrain features in the inflationary spectrum from PBH data.

Abstract

We review the formalism of primordial black holes (PBHs) production and show that the mass variance at horizon crossing has been systematically overestimated in previous studies. We derive the correct expression. The difference is maximal at the earliest formation times and still very significant for PBH masses $\sim 10^{15}$g, an accurate estimate requiring numerical calculations. In particular, this would lead to weaker constraints on the spectral index $n$. We then derive constraints on inflationary models from the fact that primordial black holes must not overclose the Universe. This is done both for the scale-free case of the power spectrum studied earlier and for the case where a step in the mass variance is superimposed. In the former case we find various constraints on $n$, depending on the parameters. In the latter case these limits can be much more strengthened, so that one could find from an observational limit on $n$ a constraint on the allowed height of the step.

Figures (4)

• Figure 1: The quantity $\beta\equiv\beta(M=10^{15}\hbox{g})$ is shown as a function of the spectral index $n$, for several values of $\alpha\equiv\alpha(M= 10^{15}\hbox{g})$. The straight lines represent the gravitational constraint (\ref{['gc2']}) and, for the purpose of illustration, the same constraint weakened and strengthened by a factor of $100$, respectively. The resulting maximally allowed value for $n$ clearly depends only weakly on the precise value of the prefactor of the gravitational constraint (\ref{['gc2']}), whereas it depends rather sensitively on the value of $\alpha$. The value usually used in the literature, $\alpha\approx 4.5$, overestimates the initial PBH abundance significantly and thus leads to much stronger constraints on $n$ (i.e. $n<1.27$) than would be expected for the more realistic choice of $\alpha\approx 1.5$ (see section \ref{['secalpha']}), which results in $n\lesssim 1.32$.
• Figure 2: Dependence of the constraint on the spectral index, $n_{max}$, on $\delta_{min}$ for the case of a scale-free spectrum, with $h=0.5, M_H=10^{15}$ g. The constraint on $n$ is clearly weakened for larger values of $\delta_{min}$.
• Figure 3: This figure shows $\beta(M)$ for a step spectrum as in (\ref{['sig2']}), with (arbitrarily chosen) $n=1.27$, $M_H(t_{k_s})=10^{17}$ g and $p=0.5$. The straight line is the gravitational constraint (\ref{['gc2']}), which applies only for $M\gtrsim 10^{15}$ g. The figure illustrates that the gravitational constraint is to be evaluated at $10^{15}$ g and that the result does not depend on the choice of $k_s$ (as long as $M_H(t_{k_s})>10^{15}$ g).
• Figure 4: Dependence of $n_{max}$ on $p$ for the step spectrum (\ref{['sig2']}). For $p=1$ the result for the scale-free case is recovered. For $p<1$ the constraint on $n$ is considerably strengthened.