PBH Dark Matter in Supergravity Inflation Models

TL;DR

This work addresses how primordial black holes (PBHs) can serve as dark matter within a supergravity-based double-inflation framework. It introduces a second-phase new inflation with a potential containing linear, quadratic, and cubic terms that, under an inflection-point condition, amplifies small-scale curvature perturbations and yields PBHs with a main mass peak around $M_{\rm PBH} \sim 10^{22}$ $\mathrm{g}$; a secondary peak near $30\ M_\odot$ is also possible, potentially relating to LIGO/Virgo detections. The model ties the linear term to SUSY breaking via $c \sim \mu_{\rm SUSY}^3$ and employs a preinflation phase to stabilize the inflaton and set initial conditions, while respecting a vanishing cosmological constant after inflation. The PBH abundances are computed from the curvature spectrum and horizon-mass relations, showing that the total PBH fraction can approach unity within observational constraints, with the smaller-scale perturbations constrained by CMB $\mu$-distortion limits. Overall, the paper provides a concrete mechanism within supergravity for generating PBH dark matter and offers a potential link to observed gravitational-wave events, while highlighting the need for detailed treatment of horizon-crossing modes across the inflationary sequence in future work.

Abstract

We propose a novel scenario to produce abundant primordial black holes (PBHs) in new inflation which is a second phase of a double inflation in the supergravity frame work. In our model, some preinflation phase before the new inflation is assumed and it would be responsible for the primordial curvature perturbations on the cosmic microwave background scale, while the new inflation produces only the small scale perturbations. Our new inflation model has linear, quadratic, and cubic terms in its potential and PBH production corresponds with its flat inflection point. The linear term can be interpreted to come from a supersymmetry-breaking sector, and with this assumption, the vanishing cosmological constant condition after inflation and the flatness condition for the inflection point can be consistently satisfied.

Figures (3)

• Figure 1: The power spectra of the curvature perturbations for parameters (\ref{['single peak parameter']}) (solid line) and (\ref{['DP parameter']}) (dashed line). The red line represents the constraints on the power spectrum from the non-detection of the CMB $\mu$-distortion estimated by Eq. (\ref{['mu-dist']}). The modes $k\raisebox{-0.5ex}{$\:\stackrel{ <}{\sim}\:$}10^6\,\mathrm{Mpc^{-1}}$ actually reenter the horizon between the two inflations and the treatments for them are described in Appendix \ref{['multiple horizon crossing']}.
• Figure 2: The PBH fraction to DM in each logarithmic mass bin with the parameters (\ref{['single peak parameter']}) (solid line) and (\ref{['DP parameter']}) (dashed line). Several constraints for PBH abundance are also shown as red regions: from left to right, extragalactic $\gamma$-rays from evapolation Carr:2009jmCarr:2016drx, femtolensing of $\gamma$-ray bursts Barnacka:2012bm, existence of white dwarfs in our local galaxy Graham:2015apa, Kepler microlensing and millilensing Griest:2013esa, EROS/MACHO microlensing Tisserand:2006zx, and accretion effects on CMB Ricotti:2007au. For both the solid and dashed lines, the total integrated fraction is about unity, therefore these models predicts abundant PBHs as a main component of DM. Moreover in the dashed line case there is another peak at $\sim30M_\odot$, which might cause GW150914 Sasaki:2016jopEroshenko:2016hmn.
• Figure 3: The schematic image of the relation between the horizon scale $aH$ and the multiple horizon crossing mode $k$. The time before $a_\mathrm{pre,f}$ corresponds with that during the preinflation, while $a_\mathrm{new,i}$ represents the beginning of the new inflation. Between the two inflations, the horizon scale generally decreases as $aH\propto a^{-\frac{1+3\omega}{2}}$ with $\omega=p/\rho$. Therefore the modes which exit the horizon at near the end of the preinflation can reenter the horizon during the preinflaton oscillation phase, and then reexit the horizon at the beginning of the new inflation. Those modes contribute to the sharp peaks of the power spectra around $k\sim10^6\,\mathrm{Mpc^{-1}}$ shown in Fig. \ref{['power']}, which can lead the second peak to the PBH fraction on $\sim30M_\odot$. In the super or subhorizon limits, it can be shown that the amplitude of the perturbations at the second horizon exit $a_3$ is equal to the standard value $\frac{H_\mathrm{new}}{2\pi}$ in spite of their complicated horizon crossing process.