Primordial black holes as dark matter in supergravity inflation models

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 framework. 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.

Figure 1

The power spectra of the curvature perturbations for parameters (17) (solid line) and (26) (dashed line). The red line represents the constraints on the power spectrum from the nondetection of the CMB $\mu$ distortion estimated by Eq. (18). The modes $k\lesssim 10^6{\mathrm{Mpc}}^{-1}$ actually reenter the horizon between the two inflations and the treatments for them are described in the Appendix.

Figure 2

The PBH fraction to DM in each logarithmic mass bin with the parameters (17) (solid line) and (26) (dashed line). Several constraints for PBH abundance are also shown as red regions: from left to right, extragalactic $\gamma$ rays from evapolation [5, 11], femtolensing of $\gamma$-ray bursts [8], existence of white dwarfs in our local galaxy [10], Kepler microlensing and millilensing [9], EROS and MACHO collaborations [6], and accretion effects on CMB [7]. 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 $\sim 30M_{\odot }$, which might cause GW150914 [29, 30].

Figure 3

The schematic image of the relation between the horizon scale $aH$ and the multiple horizon crossing mode $k$. The time before $a_{\text{pre},\mathrm{f}}$ corresponds with that during the preinflation, while $a_{\text{new},\mathrm{i}}$ 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 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 $k\sim 10^6{\mathrm{Mpc}}^{-1}$ shown in Fig. 1, which can lead the second peak to the PBH fraction on $\sim 30M_{\odot }$. 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 $\frac{H_{\text{new}}}{2\pi }$ in spite of their complicated horizon crossing process.