Constraints on primordial black holes by distortions of the cosmic microwave background

The possible influence of primordial black hole (PBH) evaporations on cosmic microwave backgrounds (CMB) is investigated. The spectrum distortions of CMB from the blackbody spectrum are described by the chemical potential $\mu$ and the Compton parameter $y$. From COBE/FIRAS limits on $\mu$ and $y$, the power-law index $n$ of primordial density fluctuations and the mass fraction of PBHs $\beta$ are constrained by employing the peak theory for the formation process of PBHs. Constraints set here are $n<1.304$ and $n<1.333$ in the thresholds of peaks ${\zeta }_{\mathrm{th}}=0.7$ and ${\zeta }_{\mathrm{th}}=1.2$, respectively, for the PBH mass range between $2.7\times {10}^{11}\mathrm{g}$ and $1.6\times {10}^{12}\mathrm{g}$, and $n<1.312$ and $n<1.343$ in the thresholds of peaks ${\zeta }_{\mathrm{th}}=0.7$ and ${\zeta }_{\mathrm{th}}=1.2$, respectively, for the PBH mass range between $1.6\times {10}^{12}\mathrm{g}$ and $3.5\times {10}^{13}\mathrm{g}$, which correspond to the comoving scales between $3\times {10}^{-18}\mathrm{Mpc}$ and $4\times {10}^{-17}\mathrm{Mpc}$. The constraint on the PBH fraction, which is the direct probe of the amplitude of density fluctuations on these scales, stays at almost the same value as $\beta <{10}^{-21}$ in these mass ranges. It is also found that, with these constraints, UV photons injected by PBH evaporations are unlikely to ionize the majority of hydrogen atoms.

Figure 1

The chemical potential for each spectral index. We assume the reheating temperature $T_{\mathrm{rh}}\gg T_{\mathrm{DC}}$ (see text). The critical thresholds of PBH formation are taken as ${\zeta }_{\mathrm{th}}=0.7$ and 1.2 in the left and right panels, respectively. The left and right panels correspond to the cases of PBH formation at the density peaks surrounded by the average and low density regions, respectively. The dotted line is the upper limit of the COBE/FIRAS observation, $\mu =9.0\times {10}^{-5}$. The region under the dotted line is allowed. We find the upper limit on the spectral index $n$ as $n<1.304$ for ${\zeta }_{\mathrm{th}}=0.7$ and as $n<1.333$ for ${\zeta }_{\mathrm{th}}=1.2$.

Figure 2

The production rate of the chemical potential per PBH mass. We assume the reheating temperature $T_{\mathrm{rh}}\gg T_{\mathrm{DC}}$ (see text). The thick solid and thick dotted lines correspond to the cases with ${\zeta }_{\mathrm{th}}=0.7$ and $n=1.304$, and ${\zeta }_{\mathrm{th}}=1.2$ and $n=1.333$, respectively. The vertical thin dashed line represents the characteristic PBH mass scale $M_{\mathrm{DC}}$, while the thin solid line represents $M_{\mathrm{freeze}}$. It is clear that PBHs with a mass around $M_{\mathrm{DC}}$ give a dominant contribution.

Figure 3

The constraint on the spectral index as a function of the reheating temperature from the $\mu$ distortion. The thick solid line and the thick dotted line are the upper limits with the PBH critical thresholds ${\zeta }_{\mathrm{th}}=0.7$ and ${\zeta }_{\mathrm{th}}=1.2$, respectively. The vertical thin dashed and thin solid lines represent the reheating temperatures whose horizon scales correspond to PBHs with masses $M_{\mathrm{DC}}$ and $M_{\mathrm{freeze}}$, respectively.

Figure 4

The constraint on the PBH abundance from the $\mu$ distortion. The allowed region is under the thick line. The vertical thin dashed and thin solid lines represent the locations of $M_{\mathrm{DC}}$ and $M_{\mathrm{freeze}}$, respectively.

Figure 5

The $y$ distortion as a function of the spectral index. We assume $T_{\mathrm{rh}}\gg T_{\mathrm{freeze}}$. The critical thresholds of PBH formation are taken as ${\zeta }_{\mathrm{th}}=0.7$ and 1.2 in the left and right panels, respectively. The dotted line is the upper limit of the COBE/FIRAS observation. The region under the dotted line is allowed. The upper limits on the spectral index $n$ are $n<1.312$ for ${\zeta }_{\mathrm{th}}=0.7$ and $n<1.343$ for ${\zeta }_{\mathrm{th}}=1.2$.

Figure 6

The production rate of the $y$ parameter per PBH mass. We assume $T_{\mathrm{rh}}\gg T_{\mathrm{freeze}}$. The thick solid and thick dotted lines correspond to the cases with ${\zeta }_{\mathrm{th}}=0.7$ and $n=1.312$, and ${\zeta }_{\mathrm{th}}=1.2$ and $n=1.343$, respectively. The vertical thin solid and dotted lines represent $M_{\mathrm{freeze}}$ and $M_{\mathrm{RC}}$, respectively.

Figure 7

The constraint on the spectral index as a function of the reheating temperature from the $y$ distortion. The thick solid line and the thick dotted line are the upper limits with the PBH critical thresholds ${\zeta }_{\mathrm{th}}=0.7$ and ${\zeta }_{\mathrm{th}}=1.2$, respectively. The vertical thin dotted and thin solid lines represent the reheating temperatures whose horizon scales correspond to PBHs with masses $M_{\mathrm{RC}}$ and $M_{\mathrm{freeze}}$, respectively.

Figure 8

The constraint on the PBH abundance from the $y$ distortion (thick solid line). The allowed region is under the thick solid line. The vertical thin solid and thin dotted lines represent the locations of $M_{\mathrm{freeze}}$ and $M_{\mathrm{RC}}$, respectively. The thick dotted line is the constraint on the PBH abundance from the $\mu$ distortion. The thick dashed line is obtained by Eq. (36). Here we adopt the COBE/FIRAS upper limit for $y$, and describe $z_{\mathrm{e}}$ and $z_{\mathrm{BH}}$ in terms of $M_{\mathrm{BH}}$.