# Cosmic microwave background constraints on primordial black hole dark matter

###### Abstract

We revisit cosmic microwave background (CMB) constraints on primordial black hole dark matter. Spectral distortion limits from COBE/FIRAS do not impose a relevant constraint. Planck CMB anisotropy power spectra imply that primordial black holes with are disfavored. However, this is susceptible to sizeable uncertainties due to the treatment of the black hole accretion process. These constraints are weaker than those quoted in earlier literature for the same observables.

## I Introduction and result

Primordial black holes (BHs) accrete matter in the early Universe, releasing accretion luminosity that heats and ionises hydrogen leading to potentially observable effects in the spectrum and anisotropy pattern of cosmic microwave background (CMB) radiation.
Ref. Ricotti et al. (2008) analyzed these effects, concluding that BHs with mass are excluded as dark matter (DM) candidates (see e.g. Carr et al. (2016) for a recent review).
We revisit cosmological aspects of the analysis, finding weaker limits. Considering spectral distortions we find no constraint^{1}^{1}1This is consistent with footnote in Ref. Clesse and García-Bellido (2016)..
Considering CMB anisotropies, using Planck data we find, for BHs making up all of the DM, likelihood ratios of and for and , respectively. For reference, for a Gaussian likelihood a likelihood ratios of and correspond to and constraints, respectively.

The limit on is susceptible to theoretical uncertainties that are difficult to quantify, at the level of a factor of few at least, due to the modelling of BH accretion rate and luminosity.

A recent re-evaluation of the CMB anisotropy constraint was presented in Ref. Chen et al. (2016) which, however, repeated the accretion analysis of Ricotti et al. (2008) including inaccuracies that we explain below, especially regarding the relative bulk velocity between the BH and the baryonic plasma.

While this paper was being typed for publication, Ref. Ali-Haïmoud and Kamionkowski (2016) appeared, dealing with the same topic. Compared with our simple analysis that adopts the accretion modelling from Ref. Ricotti et al. (2008) and refines the cosmology, Ref. Ali-Haïmoud and Kamionkowski (2016) took on the more challenging task of also redoing the accretion astrophysics. The bound on found in that work is weaker than ours by a factor of between 2 to 20, depending on the details assumed in the description of the accretion process.

## Ii Analysis

We modify RECFAST Seager et al. (2000); Wong et al. (2008) to include the BH accretion luminosity in the cosmological ionization history, and use CAMB Lewis et al. (2000); Howlett et al. (2012); Lewis () to calculate the effect on the CMB anisotropies. We use the formulae in Ref. Ricotti et al. (2008) to relate the BH mass accretion rate to the sound speed and to the relative velocity between the BH and the plasma, and to parametrise the accretion luminosity per BH, , giving power emitted per unit volume

(1) |

where is the BH mass density.

The relative bulk velocity between a BH and the plasma affects the BH accretion rate. Ref. Ricotti et al. (2008) used the “cosmic Mach number” defined in Ostriker and Suto (1990) to derive . However, in the linear regime, we obtain a different expression for the RMS relative velocity that is given by Tseliakhovich and Hirata (2010)

(2) |

Here is the comoving wavenumber, is the input curvature perturbation variance per , and is the velocity divergence of the baryon and dark matter fluids Ma and Bertschinger (1995). Eq. (2) was evaluated in Ref. Dvorkin et al. (2014) using CAMB. A simple analytic approximation of the result, that we use in numerical calculations below, is shown by the orange line in Fig. 1. Comparing to Fig. 2 of Ricotti et al. (2008), our result for is larger by about a factor of five at , leading to suppressed accretion.

Following Ricotti et al. (2008), we define the Bondi-Hoyle effective velocity such as to incorporate the statistical nature of in cosmological perturbation theory and its interplay with the parametric dependence of the BH accretion luminosity^{2}^{2}2See Eq. (13) in Ricotti et al. (2008).,

(3) |

where denotes averaging with a Maxwellian velocity distribution with RMS speed given by . The resulting is shown by the blue line in Fig. 1.

We are now in position to calculate the effect of BH accretion luminosity on the state of the plasma. Before reporting the results, we note that Ref. Ricotti et al. (2008) used numerical simulations to estimate a number of effects involved in the calculation, that we treat more simplistically. The effects and our differences in treating them are as follows.

(i) BH accretion luminosity produces a spectrum of radiation. Photons of different energy induce different effects in the plasma, including ionisation, atomic excitations, heating, and the delayed subsequent absorption of redshifted X-ray photons. In place of the simulations of Ricotti et al. (2008), we simply assume that a fraction of the total luminosity goes into instantaneous ionisation of the plasma, a fraction goes to atomic excitations, and a fraction goes to heat. Assuming instantaneous deposition of the total BH luminosity in the plasma should lead to an over-estimate of the significance of the CMB constraint we derive, because (a) a large fraction of the photons emitted at escape to redshift without incurring any ionisations, and (b) the differential contribution to the Thomson optical depth scales as , so a given increase in the ionised fraction contributes more to the optical depth at high .

(ii) Ref. Ricotti et al. (2008) included in their simulations back-reaction processes where heating of the plasma near the BH temporarily halts the accretion, leading to suppressed duty cycle of the emission. We neglect this effect, which could relax our bounds further.

(iii) Ref. Ricotti et al. (2008) considered cases in which BHs make up only a small fraction of the DM, with . In this case, a dark halo of DM accreting into the BH increases the effective BH mass, leading to enhanced accretion luminosity. We neglect this effect. The cost is that one cannot trust our limits when . However, at the point there is no room for a dark halo and our calculation determines the value of below which BHs could make up all of the DM consistent with CMB data.

## Iii Spectral distortions and anisotropy correlation functions

Ref. Ricotti et al. (2008) calculated the contribution of BH accretion luminosity to the Compton parameter, that was estimated by the contribution obtained from energy injected in the redshift interval . We can write this estimate as

(4) |

The result is shown in Fig. 2. The constraint from COBE/FIRAS Fixsen (2009); Fixsen and Mather (2002) is at 95%CL; the BH contribution cannot be constrained in the mass range of interest.

Moving on to CMB anisotropies, in Fig. 3 we show results for the recombination history (left panel) and TT power spectrum (right panel) for sample values of . Performing a likelihood analysis Dunkley et al. (2005); Lewis and Bridle () for the 6 usual CDM parameters augmented by another parameter for , using the latest TT, TE, EE anisotropy data from Planck Aghanim et al. (2015); Ade et al. (2015), leads to the constraint quoted in the introduction. In Fig. 4 we show part of the likelihood triangle.

## Iv Summary

We have re-analyzed the CMB constraints on primordial black holes (BHs) playing the role of cosmological dark matter. We find that primordial black holes with masses are disfavored. This limit is subject to large, and difficult to quantify, theory uncertainty arising from the treatment of accretion and accretion luminosity of the BHs. Assuming, for concreteness, the same accretion prescription as in the earlier analysis of Ref. Ricotti et al. (2008), our limit is weaker despite the fact that we use Planck CMB data of far superior quality compared to the WMAP3 data considered in Ricotti et al. (2008).

###### Acknowledgements.

We thank Jens Chluba and, especially, Yacine Ali-Hamoud for useful discussions. KB is incumbent of the Dewey David Stone and Harry Levine career development chair, and is supported by grant 1507/16 from the Israel Science Foundation and by a grant from the Israeli Centres Of Excellence (ICORE) program. RF is supported in part by the Alfred P. Sloan Foundation.## References

- Ricotti et al. (2008) M. Ricotti, J. P. Ostriker, and K. J. Mack, Astrophys. J. 680, 829 (2008), eprint 0709.0524.
- Carr et al. (2016) B. Carr, F. Kuhnel, and M. Sandstad, Phys. Rev. D94, 083504 (2016), eprint 1607.06077.
- Clesse and García-Bellido (2016) S. Clesse and J. García-Bellido (2016), eprint 1610.08479.
- Chen et al. (2016) L. Chen, Q.-G. Huang, and K. Wang (2016), eprint 1608.02174.
- Ali-Haïmoud and Kamionkowski (2016) Y. Ali-Haïmoud and M. Kamionkowski (2016), eprint 1612.05644.
- Seager et al. (2000) S. Seager, D. D. Sasselov, and D. Scott, Astrophys. J. Suppl. 128, 407 (2000), eprint astro-ph/9912182.
- Wong et al. (2008) W. Y. Wong, A. Moss, and D. Scott, MNRAS 386, 1023 (2008), eprint arXiv:0711.1357 [astro-ph].
- Lewis et al. (2000) A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538, 473 (2000), eprint astro-ph/9911177.
- Howlett et al. (2012) C. Howlett, A. Lewis, A. Hall, and A. Challinor, JCAP 1204, 027 (2012), eprint 1201.3654.
- (10) A. Lewis, http://cosmologist.info/notes/CAMB.pdf.
- Ostriker and Suto (1990) J. P. Ostriker and Y. Suto, Astrophys. J. 348, 378 (1990).
- Tseliakhovich and Hirata (2010) D. Tseliakhovich and C. Hirata, Phys. Rev. D82, 083520 (2010), eprint 1005.2416.
- Ma and Bertschinger (1995) C.-P. Ma and E. Bertschinger, Astrophys. J. 455, 7 (1995), eprint astro-ph/9506072.
- Dvorkin et al. (2014) C. Dvorkin, K. Blum, and M. Kamionkowski, Phys. Rev. D89, 023519 (2014), eprint 1311.2937.
- Fixsen (2009) D. Fixsen, Astrophys.J. 707, 916 (2009), eprint 0911.1955.
- Fixsen and Mather (2002) D. J. Fixsen and J. C. Mather, Astrophys. J. 581, 817 (2002).
- Dunkley et al. (2005) J. Dunkley, M. Bucher, P. G. Ferreira, K. Moodley, and C. Skordis, Mon. Not. Roy. Astron. Soc. 356, 925 (2005), eprint astro-ph/0405462.
- (18) A. Lewis and S. Bridle, http://cosmologist.info/notes/CosmoMC.pdf.
- Aghanim et al. (2015) N. Aghanim et al. (Planck) (2015), eprint 1507.02704.
- Ade et al. (2015) P. Ade et al. (Planck) (2015), eprint 1502.01589.