Limits on Runaway Growth of Intermediate Mass Black Holes from Advanced LIGO

Black holes have so far been detected in two separate mass ranges. One is the supermassive black hole (SMBH) range, roughly $10^{5}-10^{9}\,M_{\odot}$. SMBHs have been indirectly observed throughout the cosmos, from the center of the Milky Way (by studying the kinematics of stars near the Galactic center [1]) and up to redshifts of more than 7 (detecting emission at various wavelengths originating from the centers of their host galaxies [2, 3]). The other is the stellar-mass range, from a few to a few tens of solar masses, where BHs were first detected through the observation of X-rays emitted as a result of their accretion from a binary partner [4, 5, 6, 7, 8, 9, 10, 11], and more recently have been detected by measuring gravitational waves originating from BH binary coalescences [12, 13, 14]. An interesting regime left to explore is the intermediate range, which we take here to be $\sim 500-10^{5}\,M_{\odot}$. The lower mass limit is chosen in order to avoid confusion with “second generation” BHs [16, 15]—the product of single mergers of stellar BHs—which can be as massive as $\simeq 100M_{\odot}$.

There are various indications, albeit mostly circumstantial, for the existence of such intermediate-mass black holes (IMBHs) [17]. Some empirical evidence comes from the observation of Ultra-Luminous X-ray sources, with luminosities as high as $10^{40}\,{\rm ergs/sec}$, larger than the Eddington luminosity of stellar-mass BHs, hinting towards more massive accreting objects [18, 19, 20, 21, 22]. Evidence of ULX variability, especially for sources in M82 [24], suggests some may be rapidly accreting highly super-Eddington neutron stars, based on periodicity, while others are quasi-periodic oscillations that are most likely accreting IMBHs of a few hundred solar masses [25]. While one could possibly explain this by considering a cluster of small sources instead of an IMBH, the $O(10)$ variability of the radiation over periods of months makes this unlikely [23]. In addition, the sources are often detected away from the host galaxy centers [17], and are therefore unlikely to be due to SMBHs. Moreover, the inferred masses of these sources are consistent with an extrapolation of the $M_{\rm BH}-\sigma$ relation, which holds across decades of masses in the SMBH range [26]. Recently, indirect evidence was reported for the presence of IMBHs in NGC 104—a globular cluster (GC) we shall investigate more closely below—based on dynamical analysis of its pulsars [27] (see however [28]), as well as for NGC 6624 [29].

Theoretical arguments also support the existence of IMBHs, with an occupation fraction as high as unity in dwarf galaxies (as early feedback they induce can provide a solution to a number of pressing dwarf-galaxy anomalies [30]), as well as in GCs (where they can potentially account for the “missing link” in generating SMBHs at high redshift [31]).

Advanced LIGO (aLIGO) has already detected a series of stellar mass black hole mergers in its first two observing runs, with pre-merger masses as massive as $\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}30\,M_{\odot}$, and as its sensitivity sharply increases for higher masses, one might hope that IMBHs would pose an easy target for detection. However, IMBHs lie in a problematic regime for aLIGO. To detect a merger involving an IMBH, we could either hope to see it merging with another IMBH, or with abundant compact objects in the host GC (such as neutron stars or stellar-mass BHs). For the former case, the signal from its inspiral and merger phases lies outside the frequency range of aLIGO, if $M_{\rm IMBH}\gtrsim 500\,M_{\odot}$. The latter case is termed an intermediate mass-ratio inspiral (IMRI). While it may be accessible at slightly higher frequencies, the exact waveforms are hard to model [32, 33], as the post-Newtonian approximation—an expansion in $v/c$—fails when the velocity approaches the speed of light, a regime win which IMRIs spend many cycles [34]. Thus accurate templates are hard to calculate¹¹ 1 Note that the case for extreme mass-ratio inspirals (EMRIs) is somewhat simpler, as linearization based on a perturbative expansion in $m/M$ is valid for mass ratios $>O(10^{6})$ [35, 36].. It is worth noting that a quantum of solace, so to speak, lies in the fact that the ringdown process following massive mergers includes non-negligible GW emission at higher frequencies (the “quasi-normal modes”) [37, 38], some within the LIGO range [39, 40]. This prevents its detectable volume from shrinking quickly to zero as the total mass approaches the IMBH range [41, 42, 43].

Rather than try to observe IMBHs directly, for which we likely have to wait until the 2030s (or later), when next-generation experiments can more easily access IMBH-IMBH mergers or EIMRIs [44], in this paper we take an alternative approach. We focus on the prospects of using existing and upcoming results from aLIGO to probe the formation of these IMBHs.

The formation process of IMBHs is highly uncertain. Several scenarios have hitherto been suggested, for example describing how massive BH seeds can be formed from direct collapse of Population III stars [17] or following consecutive mergers of stars in young dense stellar clusters [31, 45, 46, 47]. An obvious route to IMBH formation is repeated mergers of stellar remnants such as BHs in (old) dense clusters, which can be efficient if a non-linear runaway process can take place [48]. Over the past decade, numerical simulations of such mergers in GCs, showing that most tight binaries are ejected from the cluster before or after merging, have cast heavy doubts on this scenario [49, 50]. However, recent works taking into account higher-order post-Newtonian dynamics have shown both analytically [51] and numerically [52] that significantly(!) more (by up to two orders of magnitude) binaries are retained and as much as half of the total binaries can merge within the cluster, reviving the scenario and motivating attempts to test it observationally.

Fortunately, if indeed an IMBH is formed by a runaway merger starting from small seed BHs, aLIGO may be sensitive enough to detect the early stages of this process. We show that provided the fraction of GCs hosting such IMBHs is non-negligible, aLIGO will detect at a minimum several mergers per year involving BHs with masses larger than $100\,M_{\odot}$ resulting from their formation. Alternatively, as our analysis demonstrates, a null detection of mergers involving $\sim 50-300\,M_{\odot}$ black holes by aLIGO running at design sensitivity can place useful limits on $f_{\rm occ}$, the occupation fraction of IMBHs formed by a runaway merger of stellar black holes in GCs. Our conclusion is that with six years of aLIGO running at design sensitivity, using the most conservative assumptions about the signal and background, aLIGO will be able to robustly probe down to $f_{\rm occ}\lesssim 3\%$ at 99.9% confidence.

This paper is constructed as follows: In Section II we describe the setup for our analysis and introduce our target observable, the total observed mass distribution of merging BHs by aLIGO. This includes contributions from standard mergers of stellar BHs, which form a background, and those stemming from runaway mergers leading up to IMBH formation in GCs, which is the signal we are after. In Section III we model the signal, considering opposite limiting cases for the possible merger trees leading up to the IMBH and using a simple ansatz for the rate of such events and its redshift dependence. In Section IV we derive an expected value for $f_{\rm occ}$, based on analytic estimates of the rates of binary capture and merger in the centers of GCs, which we calculate for a representative sample of Milky-Way GCs. In Section V, we account for the BH merger events not associated with the runaway process that act as a background to its observation, following the prescription in Ref. [53]. In Section VI, we present our calculations for the detectable spacetime volume that aLIGO has access to for mergers of BHs of given masses. Our results are presented and explained in Section VII. We conclude in Section VIII.

In a nutshell, we wish to place a limit on the abundance of IMBHs in GCs that were formed by a runaway merger of smaller black holes by studying the mass distribution of merging BHs detected by aLIGO. We leave a discussion of the spin distribution, which may provide an additional means of probing hierarchical mergers [16], to future work. The number of observed mergers with a heavier component of mass $M$ is given by

where $dn/dM$ is the mass function of merging black holes, $\bar{R}$ is the overall event rate, which in general can be redshift dependent, and $\langle VT\rangle$ is the space-time volume detectable by aLIGO. To calculate this observable, we distinguish between the signal coming from the formation of IMBHs and the background, which is due to mergers occurring otherwise throughout the Universe.

The signal contribution depends (linearly) on $f_{\rm occ}$, the occupation fraction of runaway-formed IMBHs in GCs. After we present our prescription for calculating the signal in the next Section, we will discuss what is the expected value for this quantity, to be later compared with our final results. Next we will review the formalism presented in Ref. [53] for calculating the background contribution. We will then describe our use of the LIGO Scientific Collaboration Algorithm Library Suite (LALSuite) [54] to calculate the sensitive volume of aLIGO to mergers of given masses with noise levels corresponding to its completed O1 and O2 runs, its upcoming O3 run in late 2018 and the final run at design sensitivity planned for 2020-2025.

At this point, armed with a prediction for $N^{\rm Signal}_{\rm obs}(M)+N^{\rm BG}_{\rm obs}(M)$ where $N^{\rm Signal}_{\rm obs}(M)\propto f_{\rm occ}$, we will be in a position to derive an upper bound on $f_{\rm occ}$ based on the null hypothesis that there are no detectable events originating from runaway mergers. Since the expected number of background events at high masses is low, we will have to go beyond the Gaussian approximation for estimating the significance of a peak in an individual mass bin and use Poisson statistics to calculate a $99.9\%$-confidence upper bound on $f_{\rm occ}$. To achieve robust conclusions, we will repeat our calculations for wide a range of models (and their free parameters) for the background distribution.

Referring to Eq. (1), in order to calculate the signal we are after, we need to estimate $dn/dM$, the mass distribution of merging black holes in the runaway process, and $\bar{R}$, the overall rate of IMBH formation in GCs.

The goal of this Section is to derive a reasonable expectation for the range of values the quantity we focus on in this work—the occupation fraction $f_{\rm occ}$ of runaway-formed IMBHs in GC centers—can take. We do this based on analytic estimates of the BH merger rates due to various binary capture mechanisms, calculated for a representative sample of observed Milky-Way (MW) GCs.

Inevitably, the mergers we are after will have to be distinguished from the background events that have nothing to do with IMBH formation. Referring again to Eq. (1), we require a prescription for the mass function and rate of background merger events in order to calculate $N^{\rm BG}_{\rm obs}(M)$. To quantify the mass distribution, we employ a simple ansatz for the mass function of a merging BH binary with component masses $M_{1}>M_{2}$ (see [53, 79])

where

Here $\alpha$ is a power law with a fiducial value of $2.35$ (to match the Kroupa mass function [60]), $M_{\rm gap}$ is the lowest stellar-BH mass possible, which we set to $5\,M_{\odot}$ (this has no effect on our results), $M_{\rm cap}$ is a double-exponential upper cutoff on the stellar-BH mass, $\beta$ is the power-law index of the mass ratio, whose choice generally depends on the binary-BH progenitor model, $\mathcal{H}$ is the Heaviside function and $A_{M_{1}},A_{M_{2}}$ are normalization constants. As default values we take throughout $M_{\rm cap}=40\,M_{\odot}$ and $\beta=0$, both consistent with current aLIGO observations.

For the rate of mergers, we take the central value from Ref. [80], $\bar{R}_{\rm BG}=103\,{\rm Gpc^{-3}~yr^{-1}}$. As we did when calculating the signal, in the simplest case we will take the rate to be redshift independent. To account for time evolution, we will correct the background rate at $z=0$ according to [81, 82, 83]

where $z_{f}$ is the redshift at formation, $R_{f}(z_{f})$ is the best-fit function found in Ref. [84] for the star-formation-rate history and the convolution is with a delay-time distribution of the form $P(t_{d})=1/t_{d}$, with $t_{\rm min}=50\,{\rm Myrs}$ and $t_{\rm max}$ equal to the Hubble time.

Our fiducial model includes a double-exponential cutoff at high mass—supported both by current data from aLIGO and by theoretical models of pair-instabillity (and pulsational pair-instability) SNe which predict a dearth of stellar BHs above $M\sim 50\,M_{\odot}$ [85, 86])—and $\beta=0$, so that $M_{2}$ takes values uniformly from $M_{\rm gap}$ to $M_{1}$. However, to account for the uncertainty regarding these choices, we will calculate our results for a variety of different models, including a shallower exponential cutoff as well as a sharper cutoff, different values for $\beta$, and a value of $R_{\rm BG}$ corresponding to the upper bound at $90\%$ confidence in the recent aLIGO analysis [80].

Our observable, introduced in Eq. (1) in Section II, depends on the quantity $\langle VT\rangle$, the detectable space-time volume of aLIGO, which in turn is given by [41]

where $T_{0}$ is the coincident online time of the LIGO interferometers, $V_{c}(z)$ is the comoving volume in a sphere that extends to redshift $z$, $s(\theta)$ is an injected distribution of binary parameters which include masses, spins, tilt angles and the orientation of the orbital plane, and $f(z,\theta)$ is the fraction of injections with redshift $z$ and parameters $\theta$ that are detectable by aLIGO with a given sensitivity (the standard criterion for detection is a signal-to-noise threshold of $8$ per detector). To evaluate the integral in Eq. (17) we follow Ref. [87] and use the IMRPhenomD waveform approximant given in the LALSimulation package [88, 89] to perform a Monte Carlo simulation. To get the noise power spectra for the different versions of aLIGO we consider, we digitized the curves in Ref. [90]. As a consistency check, we compared our results with the online GW Distance Calculator [91] and verified that our calculations are consistent to within less than $\sim 10\%$.

In Fig. 3 we plot the horizon distance—the farthest luminosity distance at which a source could ever be detected—for mergers involving equal-mass binaries, matching our bottom-heavy Isosceles Triangle merger tree, as well as for ones that involve a massive BH merging with a seed BH, corresponding to the top-heavy Right Triangle merger tree. We see that for equal-mass binaries, the sensitivity peaks when the two BHs are roughly $100\,M_{\odot}$, while in the Right Triangle case the sensitivity peaks earlier and then slowly decreases. It is important to note that in the latter case, when the massive BH approaches $300\,M_{\odot}$, the mass ratio starts to become large, and the waveforms for the signal from these mergers should be considered less reliable [33].

To incorporate a redshift dependence for the signal and background, we correct the detectable volume calculated via Eq. (17) by a factor $\bar{V_{z}}/\bar{V_{0}}$. Here, $\bar{V_{0}}$ is the average detectable volume for all sources, calculated by weighting the comoving volume at redshift $z$ by the unit detectable volume at each redshift slice, and $\bar{V_{z}}$ is a similar average volume with the weights multiplied by the appropriate redshift-dependence for either the signal (i.e. $n_{\rm GC}(z)$) or the background (i.e. Eq. (16), which accounts for a time-delayed star-formation-rate history); see Fig. 4.

Lastly, we wish to conservatively take into account the mass-measurement uncertainty in the observed mass distribution of detected events, especially at the high-mass end [92]. To do so, we follow Ref. [53] and convolve the mass functions for the signal and background with a log-normal distribution

Here, $M_{\rm th}$ is the true value of the mass (which follows the theoretical mass functions given in Eqs. (2), (4) and (14)), $M_{\rm obs}$ is the observed mass, and the relation between them is given by $M_{\rm obs}=xM_{\rm th}$, where $x$ follows a normal distribution, $x\sim\mathcal{N}(1,\sigma^{2})$ and $P_{\rm G}=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-(x-1)^{2}/2\sigma^{2}}$. We set the relative measurement error to $\sigma=0.1$ ($10\%$).

With prescriptions in hand for calculating the signal and background, as well as taking into account the experimental sensitivity and measurement uncertainty properly, we are ready to present our results. We start from the observed mass distribution, Eq. (1), and then proceed to calculate the limits on $f_{\rm occ}$ that aLIGO will be able to set in its future runs (as discussed below, we find that the O1 and O2 runs that have recently been completed do not yet place meaningful constraints on the runaway formation of IMBHs in GCs).

In this work we addressed a question involving IMBHs that as we demonstrated, can be effectively answered with upcoming measurements of advanced LIGO at its design sensitivity. The question is whether IMBHs can form in globular clusters via runaway merger of stellar seed BHs. Our results show that aLIGO will be able to probe the occupation fraction $f_{\rm occ}$ of IMBHs formed in this way in GCs down to values as low as $f_{\rm occ}\lesssim 3\%$ (at $99.9\%$ significance), if no GW events are detected from mergers involving BHs with masses in the range $\gtrsim 50-300\,M_{\odot}$. This will place a robust bound on this scenario, which comes directly from observations and is independent of the numerous assumptions about the structure and dynamics of GCs that inevitably enter any attempt to calculate this quantity from first principles. The rather strict limit we forecast is especially interesting, as analytical estimates we carry out above, using empirical measurements of a representative sample of Milky-Way GCs, show that $f_{\rm occ}$ can be as large as $10\%$ or more.

There is growing evidence that IMBHs may be ubiquitous in dwarf galaxies and GCs. If indeed most GCs harbor an IMBH at their centers, the limit we show can be derived from aLIGO measurements would mean that alternative scenarios to the runaway merger of stellar BHs are required to explain how they formed (such as POP III stars, direct collapse of massive gas clouds or mergers of stars followed by direct collapse). It is exciting that this conclusion can be achieved with an experiment that has almost no direct access to mergers that involve IMBHs themselves. Naturally, future ground-based gravitational-wave observatories such as the Einstein Telescope [94, 95] and Cosmic Explorer [96], as well as the proposed space interferometers DECIGO [97] and LISA [98, 99], will be able to achieve more direct constraints, however those will require decades of suspense. Meanwhile results from aLIGO are already pouring in.