# Did LIGO detect dark matter?

###### Abstract

We consider the possibility that the black-hole (BH) binary detected by LIGO may be a signature of dark matter. Interestingly enough, there remains a window for masses where primordial black holes (PBHs) may constitute the dark matter. If two BHs in a galactic halo pass sufficiently close, they radiate enough energy in gravitational waves to become gravitationally bound. The bound BHs will rapidly spiral inward due to emission of gravitational radiation and ultimately merge. Uncertainties in the rate for such events arise from our imprecise knowledge of the phase-space structure of galactic halos on the smallest scales. Still, reasonable estimates span a range that overlaps the Gpc yr rate estimated from GW150914, thus raising the possibility that LIGO has detected PBH dark matter. PBH mergers are likely to be distributed spatially more like dark matter than luminous matter and have no optical nor neutrino counterparts. They may be distinguished from mergers of BHs from more traditional astrophysical sources through the observed mass spectrum, their high ellipticities, or their stochastic gravitational wave background. Next generation experiments will be invaluable in performing these tests.

The nature of the dark matter (DM) is one of the most longstanding and puzzling questions in physics. Cosmological measurements have now determined with exquisite precision the abundance of DM Hinshaw:2012aka ; Ade:2015xua , and from both observations and numerical simulations we know quite a bit about its distribution in Galactic halos. Still, the nature of the DM remains a mystery. Given the efficacy with which weakly-interacting massive particles—for many years the favored particle-theory explanation—have eluded detection, it may be warranted to consider other possibilities for DM. Primordial black holes (PBHs) are one such possibility Carr:1975qj ; Carr:1974nx ; Meszaros:1974tb ; Clesse:2015wea .

Here we consider whether the two black holes detected by LIGO Abbott:2016blz could plausibly be PBHs. There is a window for PBHs to be DM if the BH mass is in the range Carr:2009jm ; Monroy-Rodriguez:2014ula . Lower masses are excluded by microlensing surveys Allsman:2000kg ; Tisserand:2006zx ; Wyrzykowski:2011tr . Higher masses would disrupt wide binaries Yoo:2003fr ; Quinn:2009zg ; Monroy-Rodriguez:2014ula . It has been argued that PBHs in this mass range are excluded by CMB constraints Ricotti:2007au ; Ricotti:2007jk . However, these constraints require modeling of several complex physical processes, including the accretion of gas onto a moving BH, the conversion of the accreted mass to a luminosity, the self-consistent feedback of the BH radiation on the accretion process, and the deposition of the radiated energy as heat in the photon-baryon plasma. A significant (and difficult to quantify) uncertainty should therefore be associated with this upper limit AliHaimoud:2016 , and it seems worthwhile to examine whether PBHs in this mass range could have other observational consequences.

In this Letter, we show that if DM consists
of BHs, then the rate for mergers of such
PBHs falls within the merger rate inferred from GW150914.
In any galactic halo, there is a chance two BHs
will undergo a hard scatter, lose energy to a
soft gravitational wave (GW) burst and become gravitationally
bound. This BH binary will merge via emission of
GWs in less than a Hubble time.^{1}^{1}1In our analysis, PBH binaries
are formed inside halos at . Ref. Nakamura:1997sm
considered instead binaries which form at early times and merge over a Hubble time.
Below we first estimate roughly the rate of such mergers and then present
the results of more detailed calculations. We
discuss uncertainties in the calculation and some possible
ways to distinguish PBHs from BH binaries from more traditional
astrophysical sources.

Consider two PBHs approaching each other on a hyperbolic orbit with some impact parameter and relative velocity . As the PBHs near each other, they produce a time-varying quadrupole moment and thus GW emission. The PBH pair becomes gravitationally bound if the GW emission exceeds the initial kinetic energy. The cross section for this process is Quinlan:1989 ; Mouri:2002mc ,

(1) | |||||

where is the PBH mass, and the PBH mass in units of , is its Schwarzschild radius, is the relative velocity of two PBHs, and is this velocity in units of km sec.

We begin with a rough but simple and illustrative estimate of the rate per unit volume of such mergers. Suppose that all DM in the Universe resided in Milky-Way like halos of mass and uniform mass density pc with . Assuming a uniform-density halo of volume , the rate of mergers per halo would be

(2) | |||||

The relative velocity is specified by a characteristic halo velocity. The mean cosmic DM mass density is Mpc, and so the spatial density of halos is Mpc. The rate per unit comoving volume in the Universe is thus

(3) |

The normalized halo mass drops out, as it should. The merger rate per unit volume also does not depend on the PBH mass, as the capture cross section scales like .

This rate is small compared with the Gpc yr estimated by LIGO for a population of mergers Abbott:2016nhf , but it is a very conservative estimate. As Eq. (3) indicates, the merger rate is higher in higher-density regions and in regions of lower DM velocity dispersion. The DM in Milky-Way like halos is known from simulations Moore:1999nt and analytic models Kamionkowski:2008vw to have substructure, regions of higher density and lower velocity dispersion. DM halos also have a broad mass spectrum, extending to very low masses where the densities can become far higher, and velocity dispersion far lower, than in the Milky Way. To get a very rough estimate of the conceivable increase in the PBH merger rate due to these smaller-scale structures, we can replace and in Eq. (3) by the values they would have had in the earliest generation of collapsed objects, where the DM densities were largest and velocity dispersions smallest. If the primordial power spectrum is nearly scale invariant, then gravitationally bound halos of mass , for example, will form at redshift . These objects will have virial velocities km sec and densities pc Kamionkowski:2010mi . Using these values in Eq. (3) increases the merger rate per unit volume to

(4) |

This would be the merger rate if *all* the DM resided in the smallest haloes.
Clearly, this is not true by the present day; substructures are at
least partially stripped as they merge to form larger objects, and so
Eq. (4) should be viewed as a conservative upper limit.

Having demonstrated that rough estimates contain the merger-rate range Gpc yr suggested by LIGO, we now turn to more careful estimates of the PBH merger rate. As Eq. (3) suggests, the merger rate will depend on a density-weighted average, over the entire cosmic DM distribution, of . To perform this average, we will (a) assume that DM is distributed within galactic halos with a Navarro-Frenk-White (NFW) profile Navarro:1995iw with concentration parameters inferred from simulations; and (b) try several halo mass functions taken from the literature for the distribution of halos.

The PBH merger rate within each halo can be computed using

(5) |

where is the NFW density profile with characteristic radius and characteristic density . is the virial radius at which the NFW profile reaches a value times the comoving mean cosmic density and is cutoff. The angle brackets denote an average over the PBH relative velocity distribution in the halo. The merger cross section is given by Eq. (1). We define the concentration parameter . To determine the profile of each halo, we require as a function of halo mass . We will use the concentration-mass relations fit to DM N-body simulations by both Ref. Prada:2011jf and Ref. Ludlow:2016ifl .

We now turn to the average of the cross section times relative velocity. The one-dimensional velocity dispersion of a halo is defined in terms of the escape velocity at radius , the radius of the maximum circular velocity of the halo. i.e.,

(6) |

where , and . We approximate the relative velocity distribution of PBHs within a halo as a Maxwell-Boltzmann (MB) distribution with a cutoff at the virial velocity. i.e.,

(7) |

where is chosen so that . This model provides a reasonable match to N-body simulations, at least for the velocities substantially less than than the virial velocity which dominate the merger rate (e.g., Ref. Mao:2012hf ). Since the cross-section is independent of radius, we can integrate the NFW profile to find the merger rate in any halo:

(8) |

where

(9) |

comes from Eq. (7).

Eq. (1) gives the cross section for two PBHs to form a binary. However, if the binary is to produce an observable GW signal, these two PBHs must orbit and inspiral; a direct collision, lacking an inspiral phase, is unlikely to be detectable by LIGO. This requirement imposes a minimum impact parameter of roughly the Schwarzschild radius. The fraction of BHs direct mergers is and reaches a maximum of for km s. Thus, direct mergers are negligible. We also require that once the binary is formed, the time until it merges (which can be obtained from Ref. O'Leary:2008xt ) is less than a Hubble time. The characteristic time it takes for a binary BH to merge varies as a function of halo velocity dispersion. It can be hours for or kyrs for , and is thus instantaneous on cosmological timescales. Given the small size of the binary, and rapid time to merger, we can neglect disruption of the binary by a third PBH once formed. BH binaries can also form through non-dissipative three-body encounters. The rate of these binary captures is non-negligible in small halos Quinlan:1989 ; Lee:1993 , but they generically lead to the formation of wide binaries that will not be able to harden and merge within a Hubble time. This formation mechanism should not affect our LIGO rates. The merger rate is therefore equal to the rate of binary BH formation, Eq. (8).

Fig. 1 shows the contribution to the merger rate, Eq. (8), for two concentration-mass relations. As can be seen, both concentration-mass relations give similar results. An increase in halo mass produces an increased PBH merger rate. However, less massive halos have a higher concentration (since they are more likely to have virialized earlier), so that the merger rate per unit mass increases significantly as the halo mass is decreased.

To compute the expected LIGO event rate, we convolve the merger rate per halo with the mass function . Since the redshifts () detectable by LIGO are relatively low we will neglect redshift evolution in the halo mass function. The total merger rate per unit volume is then,

(10) |

Given the exponential falloff of at high masses, despite the increased merger rate per halo suggested in Fig. 1, the precise value of the upper limit of the integrand does not affect the final result.

At the lower limit, discreteness in the DM particles becomes important, and the NFW profile is no longer a good description of the halo profile. Furthermore, the smallest halos will evaporate due to periodic ejection of objects by dynamical relaxation processes. The evaporation timescale is Binney:1987

(11) |

where is the number of individual BHs in the halo, and we assumed that the PBH mass is . For a halo of mass , the velocity dispersion is km sec, and the evaporation timescale is Gyr. In practice, during matter domination, halos which have already formed will grow continuously through mergers or accretion. Evaporation will thus be compensated by the addition of new material, and as halos grow new halos will form from mergers of smaller objects. However, during dark-energy domination at , Gyr ago, this process slows down. Thus, we will neglect the signal from halos with an evaporation timescale less than Gyr, corresponding to . This is in any case PBHs, and close to the point where the NFW profile is no longer valid.

The halo mass function is computed using both semi-analytic fits to N-body simulations and with analytic approximations. Computing the merger rate in the small halos discussed above requires us to extrapolate both the halo mass function and the concentration-mass relation around six orders of magnitude in mass beyond the smallest halos present in the calibration simulations. High-resolution simulations of cold dark matter micro-halos Ishiyama:2010es ; Ishiyama:2014uoa suggest that our assumed concentration-mass relations underestimate the internal density of these halos, making our rates conservative.

The mass functions depend on the halo mass through the perturbation amplitude at the virial radius of a given halo. Due to the scale invariance of the window functions on small scales, varies only by a factor of two between and . Thus the extrapolation in the mass function is less severe than it looks. We also note that the scale-invariant nature of the initial conditions suggests that the shape of the halo mass function should not evolve unduly until it reaches the scale of the PBH mass, or evaporation cutoff.

To quantify the uncertainty induced by the
extrapolation, we obtained results with two different
mass functions: the classic analytic Press-Schechter
calculation Press:1973iz and one calibrated to numerical
simulations from Tinker et al. Tinker:2008ff . The agreement
between the two small-scale behaviors suggests that extrapolating
the mass functions is not as blind as it might otherwise seem.
We also include a third mass function,
due to Jenkins et. al. Jenkins:2000bv , that includes an artificial
small-scale mass cutoff at a halo mass .
This cutoff is inserted to roughly model the mass
function arising if there is no power on scales smaller than
those currently probed observationally. We include it
to provide a *very* conservative lower limit to the
merger rate if, for some reason, small-scale power were
suppressed. We do not, however, consider it likely that this
mass function accurately represents the distribution of halo masses
in our Universe.

Fig. 2 shows the merger rate per logarithmic interval in halo mass. In all cases, halos with dominate the signal, due to the increase in concentration and decrease in velocity dispersion with smaller halo masses. The Tinker mass function, which asymptotes to a constant number density for small masses, produces the most mergers. Press-Schechter has fewer events in small halos, while the Jenkins mass function results in merger rates nearly four orders of magnitude smaller (and in rough agreement with Eq. (3)).

We integrate the curves in Fig. 2 to compute the total merger rate . All mass functions give a similar result, Gpc yr, from halos of masses , representing for the Tinker and Press-Schechter mass function a small fraction of the events. When we include all halos with , the number of events increases dramatically, and depends strongly on the lower cutoff mass for the halo mass. Both the Press-Schechter and Tinker mass functions are for small halos linear in the integrated perturbation amplitude at the virial radius of the collapsing halo. In small halos, is roughly constant. Thus for a mass function , we have

(12) |

The concentration is also a function of and it too becomes roughly constant for small masses. Assuming a constant concentration, the merger rate per halo scales as . Thus, Eq. (10) suggests that . This compares well to numerical differentiation of Fig. 2, which yields .

The integrated merger rate is thus

(13) |

with for the Tinker mass function, and for the Press-Schechter mass function (the Jenkins mass function results in an event rate Gpc yr, independent of ).

A variety of astrophysical processes may alter the mass function in some halos, especially within the dwarf galaxy range, . However, halos with are too small to form stars against the thermal pressure of the ionized intergalactic medium Efstathiou:1992zz and are thus unlikely to be affected by these astrophysical processes. Inclusion of galactic substructure, which our calculation neglects, should boost the results. However, since the event rate is dominated by the smallest halos, which should have little substructure, we expect this to make negligible difference to our final result.

There is also the issue of the NFW density profile assumed. The results are fairly insensitive to the detailed density profile as long as the slope of the density profile varies no more rapidly than as . For example, suppose we replace the NFW profile with the Einasto profile Einasto:1965 ,

(14) |

with , which has a core as . The reduction in the merger rate as is more than compensated by an increased merger rate at larger radii leading to a total merger rate that is raised by relative to NFW, to Gpc yr.

Our assumption of an isotropic MB-like velocity distribution in the halo may also underestimate the correct answer, as any other velocity distribution would have lower entropy and thus larger averaged . Finally, the discreteness of PBH DM will provide some Poisson enhancement of power on scales. More small-scale power would probably lead to an enhancement of the event rate beyond Eq. (13).

The recent LIGO detection of two merging black holes suggests a 90% C.L. event rate Abbott:2016nhf of Gpc yr if all mergers have the masses and emitted energy of GW150914. It is interesting that—although there are theoretical uncertainties—our best estimates of the merger rate for PBHs, obtained with canonical models for the DM distribution, fall in the LIGO window.

The possibility that LIGO has seen DM thus cannot be immediately excluded. Even if the predicted merger rates turn out, with more precise treatments of the small-scale galactic phase-space distribution, to be smaller, conservative lower estimates of the merger rate for PBH DM suggests that the LIGO/VIRGO network should see a considerable number of PBH mergers over its lifetime.

We have assumed a population of PBHs with the same mass. The basic results obtained here should, however, remain unaltered if there is some small spread of PBH masses, as expected from PBH-formation scenarios, around the nominal value of .

PBH mergers may also be interesting for LIGO/VIRGO even if PBHs make up only a fraction of the DM, as implied by CMB limits from Refs. Ricotti:2007jk ; Ricotti:2007au or the limits in Ref. Carr:2009jm . In this case, the number density of PBHs will be reduced by . The cutoff mass will increase as if we continue to require PBHs in each halo to avoid halo evaporation. The overall event rate will be Gpc yr. Advanced LIGO will reach design sensitivity in 2019 Aasi:2013wya ; TheLIGOScientific:2016zmo , and will probe , an increase in volume to Gpc (comoving). Thus over the six planned years of aLIGO operation, while we should expect to detect events with , we will expect at least one event if .

Distinguishing whether any individual GW event, or even some population of events, are from PBH DM or more traditional astrophysical sources will be daunting. Still, there are some prospects. Most apparently, PBH mergers will be distributed more like small-scale DM halos and are thus less likely to be found in or near luminous galaxies than BH mergers resulting from stellar evolution. Moreover, PBH mergers are expected to have no electromagnetic/neutrino counterparts whatsoever. A DM component could conceivably show up in the BH mass spectrum as an excess of events with BH masses near over a more broadly distributed mass spectrum from astrophysical sources (e.g. Belczynski:2016obo, ).

Since the binary is formed on a very elongated orbit, the GW waveforms will initially have high ellipticity, exhibited by higher frequency harmonics in the GW signal O'Leary:2008xt . We have verified that the ellipticities become unobservably small by the time the inspiral enters the LIGO band, but they may be detectable in future experiments ellipticity . Mutiply-lensed quasars Pooley:2008vu ; Mediavilla:2009um , pulsar timing arrays Bugaev:2010bb , and FRB lensing searches Munoz:2016tmg may also allow probes of the PBH mass range.

Another potential source of information is the stochastic GW background. Models for the stochastic background due to BH mergers usually entail a mass distribution that extends to smaller BH masses and a redshift distribution that is somehow related to the star-formation history. Given microlensing limits, the PBH mass function cannot extend much below . Moreover, the PBH merger rate per unit comoving volume is likely higher for PBHs than for traditional BHs at high redshifts. Together, these suggest a stochastic background for PBHs that has more weight at low frequencies and less at higher ones than that from traditional BH sources.

The results of this work provide additional motivation for more sensitive next-generation GW experiments such as the Einstein Telescope ET , DECIGO Seto:2001qf and BBO BBO , which will continuously extend the aLIGO frequency range downwards. These may enable the tests described above for excesses in the BH mass spectrum, high ellipticity and low-frequency stochastic background that are required to determine if LIGO has detected dark matter.

###### Acknowledgements.

We thank Liang Dai for useful discussions. SB was supported by NASA through Einstein Postdoctoral Fellowship Award Number PF5-160133. This work was supported by NSF Grant No. 0244990, NASA NNX15AB18G, the John Templeton Foundation, and the Simons Foundation.## References

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