Cosmic microwave background limits on accreting primordial black holes

Interest in the idea that primordial black holes (PBHs) might comprise some or all of the dark matter has recently been rekindled following LIGO’s first direct detection of a binary-black-hole merger. Here we revisit the effect of accreting PBHs on the cosmic microwave background (CMB) frequency spectrum and the angular temperature and polarization power spectra. We compute the accretion rate and luminosity of PBHs, accounting for their suppression by Compton drag and Compton cooling by CMB photons. We estimate the gas temperature near the Schwarzschild radius and, hence, the free-free luminosity, accounting for the cooling resulting from collisional ionization when the background gas is mostly neutral. We account approximately for the velocities of PBHs with respect to the background gas. We provide a simple analytic estimate of the efficiency of energy deposition in the plasma. We find that the spectral distortions generated by accreting PBHs are too small to be detected by FIRAS, as well as by future experiments now being considered. We analyze Planck CMB temperature and polarization data and find, under our most conservative hypotheses, and at the order-of-magnitude level, that they rule out PBHs with masses $\gtrsim 10^2{M}_{\odot }$ as the dominant component of dark matter.

Figure 1

Schematic temperature profile for the gas accreting onto a BH. If Compton cooling is efficient ($\gamma \gg 1$), the gas temperature remains close to the CMB temperature down to $r\sim {\gamma }^{-2/3}{r}_{\mathrm{B}}$, where $r_{\mathrm{B}}$ is the Bondi radius. The temperature then increases adiabatically as $T\propto {\rho }^{2/3}\propto 1/r$. If photoionizations can be neglected, and if the background gas is partially neutral, the gas gets collisionally ionized at nearly constant temperature once it reaches $T_{\mathrm{ion}}\approx 1.5\times 10^4\mathrm{K}$. Once the gas is fully ionized, the temperature resumes increasing adiabatically as $T\propto 1/r$ until electrons become relativistic, at which point the change in the adiabatic index implies $T\propto {\rho }^{4/9}\propto r^{-2/3}$. If the luminosity of the accreting PBH is large enough, the gas is photoionized instead of collisionally ionized. In that case the gas temperature reaches larger values near the black hole horizon.

Figure 2

Characteristic dimensionless Compton drag rate $\beta$ [Eq. (7), upper panel] and Compton cooling rate $\gamma$ [Eq. (8), lower panel], as a function of redshift, and for PBH masses $M=1,10^2$ and $10^4{M}_{\odot }$, from bottom to top. Both are evaluated for a standard recombination and thermal history, with the substitution $v_{\mathrm{B}}\to v_{\mathrm{eff}}$ as described in Sec. 2f.

Figure 3

Dimensionless accretion rate $\lambda$ as a function of the dimensionless Compton cooling rate $\gamma$. Black circles are our numerical results and the purple line is our analytic fit, Eq. (27).

Figure 4

Characteristic dimensionless accretion rate $\lambda$ (upper panel) and accretion rate normalized to the Eddington value $\dot{m}\equiv \dot{M}{c}^2/L_{\text{Edd}}$ (lower panel) as a function of redshift, for PBH masses 1, $10^2$ and $10^4{M}_{\odot }$. These quantities are evaluated with substitution $v_{\mathrm{B}}\to v_{\mathrm{eff}}$ as described in Sec. 2f.

Figure 5

Characteristic temperature of the accreting gas near the Schwarzschild radius, evaluated with the substitution $v_{\mathrm{B}}\to v_{\mathrm{eff}}$ as described in Sec. 2f.

Figure 6

Radiative efficiency $\epsilon \equiv L/\dot{M}{c}^2$ divided by the dimensionless accretion rate $\dot{m}\equiv \dot{M}{c}^2/L_{\text{Edd}}$, evaluated with the substitution $v_{\mathrm{B}}\to v_{\mathrm{eff}}$ as described in Sec. 2f.

Figure 7

Characteristic velocities in the problem at hand: the isothermal sound speed $v_{\mathrm{B}}$ (dotted), rms BH-baryon relative velocity $⟨v_{\mathrm{L}}^2{⟩}^{1/2}$ (dashed) and effective velocity $v_{\mathrm{eff}}$ defined in Eq. (61) (solid), used in Figs. 2, 4, 5 and 6 to illustrate characteristic values of intermediate quantities.

Figure 8

Luminosity of accreting PBHs as a function of redshift, averaged over the Gaussian distribution of large-scale relative velocities.

Figure 9

Estimated maximum fractional importance of local thermal feedback from Compton heating by the PBH radiation.

Figure 10

Ratio of the cross-section-averaged energy loss per Compton scattering event to ${\sigma }_{\mathrm{T}}E$, as a function of photon energy.

Figure 11

Ratio of the energy deposition rate to the instantaneous energy injection rate (equivalent of the dimensionless efficiency $f(z)$ usually computed in the context of dark-matter annihilation), as a function of redshift. We only show the case $M=10^2{M}_{\odot }$ as other cases are very similar.

Figure 12

Upper panel: global free electron fraction $x_e(z)$ in the standard scenario (lower black curve), and accounting for PBHs with parameters $(M_{\mathrm{pbh}}/M_{\odot },f_{\mathrm{pbh}})=(10^2,1),(10^3,{10}^{-2}),(10^4,{10}^{-4})$, in that order from bottom to top at low redshift. Lower panel: change in the ionization history due to accreting PBHs for the same parameters. We only show the collisional ionization casehere.

Figure 13

Fractional change in the CMB temperature (upper panel) and $E$-mode polarization (lower panel) power spectra resulting from accreting PBHs. The parameters are $(M_{\mathrm{pbh}}/M_{\odot },f_{\mathrm{pbh}})=(10^2,1),(10^3,{10}^{-2}),(10^4,{10}^{-4})$, in that order with increasing overall amplitude. We only show the collisional ionization case here.

Figure 14

Approximate CMB-anisotropy constraints on the fraction of dark matter made of PBHs derived in this work (thick black curves). The “collisional ionization” case assumes that the radiation from the PBH does not ionize the local gas, which eventually gets collisionally ionized. The “photoionization” case assumes that the local gas is ionized due to the PBH radiation, up to a radius larger than the collisional ionization region, yet smaller than the Bondi radius. The former case is the most conservative, as collisional ionization leads to a smaller temperature near the black hole horizon, hence a smaller luminosity, and weaker bounds. The correct result lies somewhere between these two limiting cases. For comparison, we also show the CMB bound previously derived by ROM (thin dashed curve), as well as various dynamical constraints: microlensing constraints from the EROS [15] (purple curve) and MACHO [14] (blue curve) collaborations (but see Ref. [53] for caveats), limits from Galactic wide binaries [17], and ultrafaint dwarf galaxies [54] (in all cases we show the most conservative limits provided in the referenced papers).