Running Spectral Index and Formation of Primordial Black Hole in Single Field Inflation Models

TL;DR

This paper evaluates various single-field inflation models to determine their potential for primordial black hole formation, emphasizing the importance of the spectral index's derivatives and identifying the running mass model as a viable candidate.

Contribution

It provides a comprehensive analysis of inflation models considering recent data, highlighting the conditions under which primordial black holes can form based on spectral index behavior.

Findings

Only the running mass model can produce PBHs within certain parameters.

PBH formation requires a positive second derivative of the spectral index.

Current data do not support large negative spectral index derivatives.

Abstract

A broad range of single field models of inflation are analyzed in light of all relevant recent cosmological data, checking whether they can lead to the formation of long-lived Primordial Black Holes (PBHs). To that end we calculate the spectral index of the power spectrum of primordial perturbations as well as its first and second derivatives. PBH formation is possible only if the spectral index increases significantly at small scales, i.e. large wave number $k$. Since current data indicate that the first derivative $\alpha_S$ of the spectral index $n_S(k_0)$ is negative at the pivot scale $k_0$, PBH formation is only possible in the presence of a sizable and positive second derivative ("running of the running") $\beta_S$. Among the three small-field and five large-field models we analyze, only one small-field model, the "running mass" model, allows PBH formation, for a narrow range of parameters. We also note that none of the models we analyze can accord for a large and negative value of $\alpha_S$, which is weakly preferred by current data.