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

TL;DR

The paper tackles whether single-field inflation models can produce long-lived Primordial Black Holes (PBHs) by examining the scale dependence of the curvature perturbation spectrum through the spectral index $n_S$, its running $α_S$, and the running of running $β_S$. It combines a model-independent analysis with detailed investigations of three small-field and five large-field inflationary potentials, linking slow-roll parameters to observables and enforcing current observational bounds from CMB and LSS data. The main finding is that PBH formation requires a significant positive $β_S$, and among the models considered, only the running-mass small-field model can realize PBH formation in a narrow parameter window; all large-field models generally predict negligible PBH production and none accommodates a large negative $α_S$. The results imply that PBH abundances serve as a powerful discriminator among single-field inflation scenarios, and future refinements of $α_S$ and $β_S$ constraints could decisively rule out broad classes of models.

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 $α_S$ 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") $β_S$. 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 $α_S$, which is weakly preferred by current data.

Figures (5)

• Figure 1: Fraction of the energy density of the universe collapsing into PBHs as a function of the PBH mass.
• Figure 2: Contours of $\alpha_S + 2\xi^2$ (left) and of $\beta_S - 2\sigma^3$ (right); the right frame assumes negligible tensor modes, $r = 0$.
• Figure 3: Illustrating the dependence according to eqs.(\ref{['model11']}) of $n_S$ (solid curve), $10 \alpha_S$ (dashed curve) and $100 \beta_S$ (dotted curve), on the number of $e$--folds before the end of inflation, for the fixed value of $p=4$.
• Figure 4: Spectral parameters as a function of the number of $e$--folds $N\propto \ln(k)$ for $p=1$. Note especially the rapid approach to scale--invariance at short wavelengths (small $N$).
• Figure 5: Scatter plot of allowed values of $\alpha_S$ and $\beta_S$ assuming that $n_S$ lies in its currently allowed $2\,\sigma$ range and $45$$e$--folds of inflation occurred after the pivot scale for potential (\ref{['lf21']}).