Black hole formation and slow-roll inflation

TL;DR

The paper investigates primordial black hole formation from inflationary perturbations by linking PBH abundance to the small-scale growth of the curvature perturbation spectrum $P_\zeta(k)$ and its running $n'$. It argues that achieving PBH production generally requires $n'\sim10^{-2}$, which can be realized in the running mass model of slow-roll inflation, and checks this against current data. The authors show that current observations still allow such running and thus PBH formation, but also explore alternative routes, including two forms of the inflaton potential and curvaton-type scenarios where PBHs could form even if running on cosmological scales is small. They provide observational bounds on the running mass model (e.g., $n'_0<0.026$ at 95% C.L.) and discuss the implications if future data constrain $n'$ to be negligible on large scales, highlighting the potential need for model-building flexibility or paradigm-switching involving curvatons. Overall, the work connects inflationary microphysics to PBH phenomenology, offering concrete predictions and constraints that tie small-scale PBH viability to the slow-roll dynamics and possible curvaton contributions.

Abstract

Black hole formation may occur if the spectrum of the curvature perturbation ζincreases strongly as the scale decreases. As no such increase is observed on cosmological scales, black hole formation requires strongly positive running n' of the spectral index n, though the running might only kick in below the `cosmological scales' probed by the CMB anisotropy and galaxy surveys. A concrete and well-motivated way of producing this running is through the running mass model of slow roll inflation. We obtain a new observational bound n' < 0.026 on the running provided by this model, improving an earlier result by a factor two. We also discuss black hole production in more general scenarios. We show that the usual conditions ε<< 1 and |η| << 1 are enough to derive the spectrum {\cal P}_ζ(k), the introduction of higher order parameters ξ^{2} etc. being optional.

Figures (6)

• Figure 1: $68\%$ and $95 \%$ c.l. likelihoods in the $n-n'$ plane from the WMAP data alone (Top Panel) and WMAP+LSS (Bottom Panel).
• Figure 2: The band corresponds to the spectrum ${{\cal P}_\zeta}$ versus $\ln k$, with constant slope corresponding to $n=0.94$. The width of the band corresponds to a fractional uncertainty $0.025$. The range of $\ln k$ corresponds to the range of cosmological scales, explored by observation of the cmb anisotropy and galaxy surveys. We see that the band is narrow enough to make the variation of ${{\cal P}_\zeta}$ significant over the cosmological range.
• Figure 3: Schematic pictures of functional form of y(N) in Case I (top) and Case II (bottom), respectively. For reference we also plot the constant case, $n = 0.3$ (see the text).
• Figure 4: Form of the potential $V$ as a function of the field $\phi$. The horizontal axis is the normalized value of $\phi$, $\tilde{\phi} = \left[\frac{ \phi - \phi_{0} }{M_p}\right]/ \left[\frac{V_{0}^{1/4}}{M_p}/10^{-3} \right]^2$ with $\phi_{0} \equiv \phi(0)$ and $V_{0} \equiv V(\phi_{0})$. The vertical axis means $\tilde{V} = \left[ \frac{V}{V_{0}}-1 \right] / \left[ \frac{V_{0}^{1/4} }{M_{p}}/ 10^{-3} \right]^{4}$.
• Figure 5: Derivatives of $d (\ln \epsilon)/dN$ with respect to $N$ for Case I (top) and Case II (bottom), respectively. Here we plot the higher derivatives (ln $\epsilon)^{(\ell)} = d^{(\ell)} (\ln \epsilon)/dN^{(\ell)}$ for $\ell$ = 2, 3, and 4.