Primordial black holes from inflaton and spectator field perturbations in a matter-dominated era

We study production of primordial black holes (PBHs) during an early matter-dominated phase. As a source of perturbations, we consider either an inflaton field with a running spectral index or a spectator field that has a blue spectrum and thus provides a significant contribution to PBH production at small scales. First, we identify the region of the parameter space where a significant fraction of the observed dark matter can be produced, taking into account all current PBH constraints. Then, we present constraints on the amplitude and spectral index of the spectator field as a function of the reheating temperature. We also derive constraints on the running of the inflaton spectral index, $\mathrm{d}n/\mathrm{d}\ln k\lesssim 0.001$, which are comparable to those from the Planck satellite for a scenario where the spectator field is absent.

Figure 1

Contributions of the inflaton (black dashed line) and the spectator field (gray dashed line) to the total curvature power spectrum (black solid line). The thick and thin lines correspond to $(n_s,{\alpha }_s,A_s/A)=(1.6,0,0.01)$ and $(n_s,{\alpha }_s,A_s/A)=(2.1,-0.034,0.01)$, respectively.

Figure 2

The upper panel shows the power spectrum for $n_s=3.22$, ${\alpha }_s=-0.131$, $A/A_s=0.1$, $T_{\mathrm{reh}}=6\mathrm{MeV}$, and $a_{\mathrm{reh}}/a_{\mathrm{md}}=40$. The lower panel shows the corresponding PBH mass function. The smallest and largest modes which produce PBHs during the matter-dominated era are indicated in the upper panel by the vertical lines.

Figure 3

The lines show the different constraints for monochromatic PBH mass function. The purple region on the left is excluded by evaporation [63], the red region by femtolensing of gamma-ray bursts [64], the brown region by neutron-star capture [65], the green region by white dwarf explosions [66], the blue, yellow, and purple regions by microlensing results from Subaru [67], EROS [68], and MACHO [69], respectively, and the dark blue region by Planck [70]. The regions to the right of the dashed lines are excluded by survival of a stars in Segue I [71] and Eridanus II [72], and distribution of wide binaries [73]. PBHs to the left of the gray vertical line have evaporated before today.

Figure 4

Top panel: Minimum and maximum PBH masses. Black and gray lines correspond to $a_{\mathrm{reh}}/a_{\mathrm{md}}=40$ and $a_{\mathrm{reh}}/a_{\mathrm{md}}=100$, respectively. The red region is excluded by the BBN constraint on the reheating temperature. Bottom panel: The red contours correspond to 100% (upper) and 1% (lower) of PBH DM. The dark and light gray regions are excluded without and with dynamical constraints, respectively. The dashed contours from left to right correspond to $M_{\max }=30M_{\odot }$, $100M_{\odot }$, and $300M_{\odot }$. In both plots the values used in Fig. 2 are depicted by the solid gray lines and other parameters are fixed to the same values.

Figure 5

Constraints on the amplitude of power spectrum. The gray dashed line corresponds to PBH production in a radiation-dominated era and the red lines in a matter-dominated era for reheating temperatures $10^5\mathrm{GeV}$, 30 GeV, and 0.01 GeV from darkest to lightest. The red dashed lines show the smallest $k$ which become nonlinear during the matter-dominated era, and the vertical black dashed lines the largest $k$ which cross the horizon after reheating. The black solid lines show the power spectrum with (thin line) and without (thick line) the spectator field. For the thin line the running of both the inflaton and spectator spectral indices are fixed to zero and the amplitude of the spectator field power spectrum to $A_s=0.1A$. Here the constraints in the matter-dominated era are calculated without the factor ${\sigma }^{3/2}$.

Figure 6

Constraints on the running of the inflaton spectral index. The thin black line shows the constraint as a function of the reheating temperature from PBH production during an early matter-dominated phase. The thick line shows the same constraint calculated with the extra factor ${\sigma }^{3/2}$. The horizontal gray line in the upper right corner shows the constraint from PBH production during radiation dominance. The red lines show the Planck result $\mathrm{d}n/\mathrm{d}\ln k=-0.0033\pm 0.0074$.

Figure 7

Constraints on the spectral index of a generic spectator field. The thin black lines show the constraint as a function of the reheating temperature arising from PBH production during an early matter-dominated phase for $A_s=0.1A$ (solid line) and $A_s=0.005A$ (dashed line). The thick lines show the same constraints calculated with the factor ${\sigma }^{3/2}$. The horizontal gray lines in the upper right corner show the constraint arising from PBH production during radiation dominance.