Disentangling the potential dark matter origin of LIGO's black holes

TL;DR

This paper explores the hypothesis that primordial black holes could constitute all dark matter and examines how LIGO's gravitational wave observations could detect such black holes, especially those below one solar mass, providing insights into dark matter's nature.

Contribution

It proposes a primordial black hole mass distribution consistent with dark matter and LIGO observations, predicting detectable sub-solar mass black holes.

Findings

LIGO may detect black holes smaller than the Sun within a year.

A significant fraction (~1%) of detected black holes could be sub-solar mass.

Detection of sub-solar mass black holes would confirm primordial black holes as dark matter.

Abstract

The nature of dark matter remains one of the biggest open questions in physics. Intriguingly, it has been suggested that dark matter may be explained by another recently observed phenomenon: the detection of gravitational waves by LIGO. LIGO's detection of gravitational waves from merging stellar mass black holes renewed attention toward the possibility that dark matter consists solely of black holes created in the very early universe and that these primordial black holes are what LIGO is presently observing. Subsequent work on this topic has ruled out the possibility that dark matter could consist solely of black holes similar to those that LIGO has detected with masses above 10 solar masses. However, LIGO's connection to dark matter remains an open question and in this work we consider a distribution of primordial black holes that accounts for all of the dark matter, is consistent with LIGO's observations arising from primordial black hole binaries, and resolves tension in previous surveys of microlensing events in the Milky Way halo. The primordial black hole mass distribution that we consider offers an important prediction--LIGO may detect black holes smaller than have ever been observed with ~1% of the black holes it detects having a mass less than the mass of our Sun. Approximately one year of operating advanced LIGO at design sensitivity should be adequate to begin to see a hint of a primordial black hole mass distribution. Detecting primordial black hole binaries below a solar mass will be readily distinguishable from other known compact binary systems, thereby providing an unambiguous observational window for advanced LIGO to pin down the nature of dark matter.

Tables

Table 1. Table 1: The average mass, dark matter fraction, and relative rates of detection for LIGO in three regions of interest: I [ 0 , .9 ​ M ⊙ ) 0 .9 subscript 𝑀 direct-product [0,.9M_{\odot}) , II [ .9 ​ M ⊙ , 5 ​ M ⊙ ) .9 subscript 𝑀 direct-product 5 subscript 𝑀 direct-product [.9M_{\odot},5M_{\odot}) , and III [ 5 ​ M ⊙ , 100 ​ M ⊙ ) 5 subscript 𝑀 direct-product 100 subscript 𝑀 direct-product [5M_{\odot},100M_{\odot}) . These are arranged from left to right for the models color coded in Fig.  2 and Fig.  3 as the orange ( M ∗ = .1 , α = 2.65 formulae-sequence superscript 𝑀 .1 𝛼 2.65 M^{*}=.1,\alpha=2.65 ), purple ( M ∗ = 2.25 , α = 3.25 formulae-sequence superscript 𝑀 2.25 𝛼 3.25 M^{*}=2.25,\alpha=3.25 ), and turquoise ( M ∗ = .08 , α = 2.35 formulae-sequence superscript 𝑀 .08 𝛼 2.35 M^{*}=.08,\alpha=2.35 ) diamonds and lines, respectively. For each model, the relative rate of detection by LIGO is set to 1 for region III and calculated for the other regions by ( 5 ).

Region	Avg. Mass ($M_{\odot }$)	$f$	Relative rate
I	.1, .6, .1	.75, .12, .53	.01, .01, .003
II	1.7, 1.9, 1.8	.17, .62, .21	.13, .22, .06
III	11, 9.3, 12.8	.07, .25, .17	1, 1, 1

Equations

$$\rho (M_1,M_2)={\int }_{M_1}^{M_2}Mn(M)dM.$$

$$f(M_1,M_2)=\frac{\rho (M_1,M_2)}{{\rho }_{CDM}}$$

$$n(M)$$

$$\frac{df}{dM}$$

$$r$$

Full text

dark matter — early universe — gravitational waves — stars: black holes

1 Introduction

Advanced LIGO’s first observing run detected gravitational waves from the mergers of two separate binary black hole systems (BBHs), GW150914  (Abbott et al., 2016b) and GW151226 (Abbott et al., 2016c). A third candidate event, LVT151012 (Abbott et al., 2016d), was also observed. The second observing run is currently underway and at least one more merger has been confirmed, GW170104 (Abbott et al., 2017). These detections prove the existence of compact binary systems with component masses between $~{}7-35M_{\odot}$ and demonstrate not only that they merge over a time scale less than the age of the universe, but also that BBHs are relatively common and LIGO should expect to continue to detect their coalescences (Abbott et al., 2016e). LIGO’s detections are all consistent with relatively low spin black holes, and together they are in agreement with a power-law distribution for the number density per unit mass of black holes, $dN/dM\propto M^{-\alpha}$, with a 90% credible interval $\alpha=2.3\begin{subarray}{c}+1.3\\ -1.4\end{subarray}$ (Abbott et al., 2016f, 2017). LIGO is or will be sensitive to binary black holes between $\sim.01-100M_{\odot}$ at extra-galactic distances, and while the mass distribution was calculated assuming a minimum black hole mass of $5M_{\odot}$, it remains unknown over what mass range the presently observed power law will hold.

The notion that LIGO could detect gravitational waves from the merger of two primordial black holes has existed for nearly two decades (Nakamura et al., 1997), though interest in PBHs has been around for much longer (Zel’dovich & Novikov, 1967). Until 2004, the LIGO Scientific Collaboration actively searched for BBHs with component masses below $1M_{\odot}$ (Abbott et al., 2005, 2008). Then, the average range to a binary with component masses $(0.35,0.35)M_{\odot}$ was approximately 3 Mpc  (Abbott et al., 2008). Advanced LIGO should already be able to improve on that range by a factor of ten (Abbott et al., 2016a). The PBH theory of dark matter fell out of favor with stronger constraints from microlensing searches placing strict limits on primordial black holes below a solar mass  (Lasserre, 2000; Alcock et al., 2000; Tisserand et al., 2007), though the idea recently resurfaced with the first model suggesting that all dark matter consisted of PBH with a nearly monochromatic distribution around $\sim 30M_{\odot}$ (Bird et al., 2016; Clesse & García-Bellido, 2017). It has since been pointed out that the monochromatic scenario has two problems. First, constraints from dwarf galaxy dynamics and radio emissions imply that not all of dark matter can be explained by $\sim 30M_{\odot}$ black holes (Brandt, 2016; Koushiappas & Loeb, 2017; Gaggero et al., 2017). Second, the expected merger rate predicted by this model would be above the inferred merger rate provided by LIGO (Sasaki et al., 2016; Eroshenko, 2016; Abbott et al., 2017). These concerns suggest that if LIGO is probing a population of PBHs with its current detections, PBHs might only make up a small fraction of the dark matter in the universe. We address both concerns in this letter.

2 An overview of constraints on primordial black hole dark matter

For a given PBH distribution, $n(M)$, we can define the cumulative mass density as,

[TABLE]

We can then write the fraction of DM made up of PBHs in some mass range as

[TABLE]

where $\rho_{CDM}$ is the presently observed cold dark matter density and $f\leq 1$. Constraints on PBH dark matter are typically presented assuming a monochromatic distribution, but the consistency of LIGO’s detections with a power law distribution raises the question: is it possible for an extended PBH distribution compatible with LIGO’s observations to account for 100% of the dark matter? Constraints on monochromatic distributions can still provide meaningful limits on extended spectra such as the one we later consider. The constraint curves (shown in Fig. 1 for the LIGO region) bound $df/dM$, and they also limit the cumulative fraction contained across any given interval; for example, the EROS2 curve in Fig. 1 implies that the total fraction of dark matter in PBHs of mass $.3M_{\odot}<M<3M_{\odot}$ is $\lesssim 30\%$ and in masses $M>\sim 3M_{\odot}$ is $\lesssim 30\%$.  Carr et al. (2016) points out that these two tests are not enough to guarantee the compliance of a given extended mass function with the constraints, however. In general, this is a difficult task and the most rigorous test is to explicitly recalculate every constraint for a given distribution, which several groups have already examined for various functional forms. The primary constraints that we consider in this work are already in a cumulative form that limits the fraction of dark matter that can be composed of primordial black holes above a given mass.

Quasi log-normal forms have been extensively analyzed by Green (2016) and Kühnel & Freese (2017), with the former ruling out this family of mass functions as a source of all DM for $10^{-7}M_{\odot}<M<10^{5}M_{\odot}$ while the latter showed that for a narrow window where $10^{-10}M_{\odot}<M<10^{-8}M_{\odot}$ this remains a possibility. These results all assume some of the strictest interpretations of the microlensing constraints.

In the LIGO region, microlensing  (Alcock et al., 1996, 2000; Allsman et al., 2001; Lasserre, 2000; Tisserand et al., 2007; Wyrzykowski et al., 2011) and star cluster dynamics in the Eridanus II (Brandt, 2016) and Segue I dwarf galaxies (Koushiappas & Loeb, 2017) place the tightest constraints on PBH dark matter (Fig. 1). If the Eri II and Segue I constraints are relaxed, power-law, log-normal, and critical collapse mass functions could allow for 100% PBH dark matter for two small windows surrounding $5\times 10^{-16}M_{\odot}$ and $2\times 10^{-14}M_{\odot}$ as well as for $~{}25-100M_{\odot}$, though Carr finds that no more than 10% can be accounted for by these mass functions when the dwarf galaxy constraints are considered (Carr et al., 2017). The first two windows (as well as the one Kühnel & Freese (2017) describe) are unreachable by LIGO, and the $\sim 25-100M_{\odot}$ interval could yield LIGO detection rates inconsistent with the works of  Sasaki et al. (2016) and Eroshenko (2016). Additionally, new constraints by Gaggero et al. (2017) place similar limits to Eri II and Segue I on PBH DM in the window $25-100M_{\odot}$ using observations with independent systematics, making this window less likely to contain all of the dark matter.

We explore an extended PBH mass function with the following properties. First, it will be single-peaked. Second, it will have a functional form and predicted rate consistent with LIGO observations to date. Third, it will not violate the tightest constraints on primordial black hole binaries in the mass range already observed by LIGO (Koushiappas & Loeb, 2017; Brandt, 2016) or the microlensing constraints set by (Alcock et al., 2000), but it will violate those set by the EROS2 collaboration  (Tisserand et al., 2007). Finally, our proposed mass function will account for 100% of the dark matter. While we do not explicitly recalculate the constraints for our test function, we use our results to generate predictions that will be directly testable by LIGO within the next few years. The monochromatic EROS constraints shown in Fig. 2 for the mass range $.1M_{\odot}<M<7M_{\odot}$ imply PBH can account for at most $\sim 25\%$ of the dark matter, while our models place $\sim 60-80\%$ in that range. We note, however, that microlensing constraints have recently been called into question since there are several possibilities for systematic errors that would make them far less constraining in the region of interest for this work (Hawkins, 2015).

3 Tension in the microlensing regime

Paczynski (1986) first proposed that massive compact halo objects (MACHOs) could be detected by searching for evidence of stellar microlensing in the Large Magellanic Cloud (LMC). Since then, the MACHO, EROS, and OGLE collaborations have all sought to identify microlensing candidates within the galaxy through multi-year observations of millions of stars. These collaborations were able to place tight constraints on the amount of compact object dark matter for $10^{-7}M_{\odot}<M<10M_{\odot}$, though the limits depend strongly on galactic halo models and small variations in the model parameters can have a large effect on the derived constraints. Many of these parameters remain poorly constrained and have changed since the initial microlensing results; for example, the rotational speed of the Milky Way at $50\text{kpc}$ was formerly assumed to be $\sim 200\text{km/s}$, whereas current observations suggest a speed of $136\pm 56\text{km/s}$ (Sofue, 2013) and $178\pm 17\text{km/2}$ (Bhattacharjee et al., 2014) at $53\pm 15\text{kpc}$ and $\sim 50\text{kpc}$ respectively. Similarly, stellar velocity dispersion observations now appear to show a rotation curve that falls off with distance (Deason et al., 2012) rather than remaining flat as was previously assumed, and Bratek et al. (2014) showed that determining the mass of the Milky Way is strongly dependent on the assumed galaxy model. These studies show that the accepted values of important astrophysical parameters have changed and that these variations can affect predictions of other halo parameters.

Any deviation from the standard halo model has profound impacts on the dark matter constraints, which is abundantly clear from the MACHO first year  (Alcock et al., 1996) and 5.7 year results  (Alcock et al., 2000), which were unable to rule out a 100% compact object halo at 90% confidence for certain models.  Hawkins (2015) acknowledges that the models that did allow for 100% MACHO dark matter may not be an accurate depiction of the Milky Way, but suggests that ruling out MACHO dark matter is premature.  Green (2017) also explicitly explored this model dependency for the tighter EROS2 constraints and tracked changes resulting from the variation of several parameters, confirming Hawkins’ claim that current constraints are extremely model dependent.

There is another scenario that would loosen constraints in the microlensing region independently from the halo model. As pointed out by Clesse & García-Bellido (2017), microlensing surveys would be far less constraining if PBHs clustered into sub-halos, causing spikes in the local dark matter density. If PBHs are clustered, then the likelihood of one passing directly in front of a star under observation is much lower than for free populations. Clusters could be detected by microlensing surveys through caustic crossings, and there may already be evidence for this; the MACHO collaboration identified several candidate binary microlensing events with mostly sub-solar lens masses (Alcock et al., 1999; MACHO & Alcock, 2000).

The MACHO collaboration claimed positive detections of compact halo objects with most likely masses of $.15-.9M_{\odot}$ (Alcock et al., 2000), and recent microlensing surveys have hinted at similar results (Lee et al., 2015). If we interpret even one detection as a possible dark matter candidate, then we need a distribution that spans the MACHO and LIGO mass ranges. LIGO will be sensitive to binaries in the MACHO range, and will provide a way of making a definitive detection or establishing constraints with completely independent systematics from those used to derive current limits on PBH DM.

4 An extended primordial black hole mass function

Motivated by the LIGO and MACHO observations, we consider a modified Schechter function for the PBH mass distribution given by,

[TABLE]

where the constant $C$ has units of $\text{vol}^{-1}\text{mass}^{-1}$.

This implies that the differential mass fraction can be written as,

[TABLE]

which is also a modified Schechter function. We stress that we are not suggesting any connection to the Press-Schechter formalism with our choice of this functional form.

We choose this distribution because it has a power-law tail that fits LIGO observations and exhibits a natural exponential cutoff at mass $M^{*}$. This allows us to place the mass function’s peak within the bounds set by the MACHO collaboration. The only additional free parameter is the overall normalization, which we constrain by considering normalizations which integrate to the overall mass density of dark matter so that we might account for all of dark matter with this population of black holes.

In Fig. 2 we show the $M^{*}$ and $\alpha$ values that are consistent with LIGO observations and the dwarf galaxy constraints111Strictly speaking, we only mandate consistency up to $100M_{\odot}$. Though our distributions violate these constraints at higher masses, the excess is so small ($\sim 1\%$) that uncertainties in the dark matter density itself would need to be taken into account., as well as an interpretation of the MACHO results which allows for a peak in the mass function with objects of masses between 0.06–1$M_{\odot}$222We use the fact that the function defined in (4) peaks at $M^{*}/(\alpha-1)$. Among the regions permitted by these constraints we favor those with the lowest $\alpha$ to be consistent with the present most likely LIGO estimate of $\alpha=2.3$, though we note that all $\alpha$ in Fig. 2 are within LIGO’s 90% confidence interval estimate. From the highest $\alpha$ contour, we examine the two end points $M^{*}=.10,2.25M_{\odot}$ and $\alpha=2.65,3.25$ as well as the point $M^{*}=.08,\alpha=2.35$, which is the lowest value consistent with one version of the Eridanus II constraints. With these parameters, we consider three observational windows defined in Table 1. Region I includes masses up to $.9M_{\odot}$, which is the lower limit on measured neutron star masses (Lattimer, 2012). Region II contains binaries that could plausibly be neutron stars or objects in the mass gap, and region III defines black holes consistent with current LIGO observations. The average masses, fraction of dark matter, and relative rates of detection by LIGO for these windows and values of $M^{*}$ and $\alpha$ are shown in Table  1. The first two models allow for $\sim 99\%$ of the dark matter to be concentrated below $100M_{\odot}$, while the last model accounts for $\sim 90\%$. In both cases, the remaining dark matter is accounted for by contributions above $100M_{\odot}$.

5 LIGO primordial black hole merger rates

The space-time volume that LIGO is sensitive to scales approximately as $M^{15/6}$, assuming equal mass binary systems with an average mass below $\sim 10M_{\odot}$. Therefore, the relative LIGO detection fraction $r$ can be approximated as

[TABLE]

Using the average mass for each region of interest in Fig. 3, this suggests relative rates of $\mathcal{O}({1\%})$ and $\mathcal{O}({10\%})$ for the sub-solar and BNS plus mass gap regions, respectively. Our calculations implicitly assume that the binary black hole population is identical to the isolated population.

Several groups have already considered the formation of PBH binary systems and predicted detection rates for LIGO. Though most assume a monochromatic, $30M_{\odot}$ mass distribution, our distribution should yield similar results for the surrounding mass range,  $20-40M_{\odot}$. These groups have found that LIGO’s observed rates are consistent with their results only if the fraction of dark matter is small, $\sim.002<f<.02$  (Eroshenko, 2016; Sasaki et al., 2016). Both mass functions consistent with the Segue I constraints predict a comparable fraction of $\sim.01-.03$ for this window, with a total fraction of $\sim.02-.04$ above $30M_{\odot}$. Performing this same analysis using the looser Eridanus II constraints at $\alpha=2.35$, however, yields fractions in tension with these estimates with $\sim.03-.04$ for $20M_{\odot}<M<40M_{\odot}$ and $\sim.14$ above $30M_{\odot}$. Thus while LIGO’s most probable $\alpha$ is marginally consistent with the Eridanus II constraints for our distribution, it appears to violate theoretical formation and merger rate calculations. Consequently we favor the $M^{*}=.1$, $\alpha=2.65$ distribution since it has the lowest value of $\alpha$ that satisfies the Segue I constraints and contains the smallest fraction at or above $30M_{\odot}$.

Directly applying the restrictions set by  Eroshenko (2016) and  Sasaki et al. (2016) yields comparable results. Mandating that no more than 2% of the DM can exist at and above $30M_{\odot}$ predicts $\alpha\in[2.7,3.4]$; this does not substantially change our rate predictions. If we instead demand that no more than $\sim.002$ exist above $30M_{\odot}$, our model predicts $\alpha\simeq 3.75$. This falls outside of the LIGO 90% confidence region for $\alpha$ and introduces an additional source of tension, suggesting that a distribution of that form would be extremely unlikely.

At the end of its first observing run, advanced LIGO predicted a BBH merger rate of $9-240\,\text{Gpc}^{-1}\text{yr}^{-1}$(Abbott et al., 2016f). This implies that once LIGO has surveyed 40 times the space-time volume of its first observing run there will be a $\sim 50\%$ chance of detecting 100 binary black hole mergers. Advanced LIGO hopes to improve its reach by a factor of three, leading to 27 times the volume accumulated per unit time. That suggests that a year long observing run with advanced LIGO at design sensitivity could detect a sub-solar mass binary black hole and provide smoking gun evidence for the primordial black hole dark matter hypothesis.

In principle, LIGO should detect a binary within the BNS + mass gap region prior to the sub-solar regime which might already hint at the origin of dark matter. Work done by Littenberg et al. (2015), however, shows that few detection scenarios provide certainty that a detected binary has a component within the mass-gap. They found that to confidently identify component masses between $2-5M_{\odot}$, hundreds of detections could be needed. Even then, it may be difficult to conclude that the black holes detected had primordial origins. A sub-solar detection, however, would unambiguously provide evidence for black holes created by non-stellar processes. We recommend a binary search in that mass range to better study primordial black hole dark matter by systematics independent of those used by microlensing surveys.

6 Discussion

Though there are many constraints on a primordial black hole model of dark matter, there is tension in the microlensing region. We find that a modified Schechter functional form for the differential number density can satisfy constraints above $\sim 1M_{\odot}$, remain consistent with both the 90% likely region for several MACHO models and LIGO observations, and explain all of dark matter. This distribution matches calculations made for the merger rate of binary PBHs at $\sim 30M_{\odot}$ and predicts a sizeable number of LIGO detections for both the subsolar and mass gap regions, though the latter will be difficult to distinguish from neutron stars for the foreseeable future. In future work, it may be worth considering the spin distribution of primordial black hole populations. A recent study by Chiba & Yokoyama (2017) suggests that PBHs may be predominately slowly spinning. As more research is done into models of PBH spin distributions and as LIGO continues to refine its measurements, PBH spin analysis will become another strong channel for testing this model. LIGO has the unique opportunity to place its own constraints on PBH DM abundance while also testing for interesting new physics and we recommend searching for binary systems below one Solar mass with LIGO in order to provide the most definitive evidence for the PBH dark matter hypothesis.

7 Acknowledgments

We thank Sarah Shandera for useful discussions. We also thank the LIGO Scientific Collaboration and the VIRGO Collaboration, especially the Compact Binary Coalescence working groups for their many helpful comments and suggestions. We are grateful for the useful comments provided by the referees. This research was supported by the National Science Foundation through PHY-1454389. Funding for this project was provided by the Charles E. Kaufman Foundation of The Pittsburgh Foundation.