We discuss the capability of a third-generation ground-based detector such as the Einstein Telescope to detect mergers of intermediate-mass black holes that may have formed through runaway stellar collisions in globular clusters. We find that detection rates of 500 events per year are plausible.
The Einstein Telescope (ET), a proposed third-generation ground-based gravitational-wave (GW) detector, will be able to probe GWs in a frequency range reaching down to ∼ 1 Hz. 2 This bandwidth will allow the ET to probe sources with masses of hundreds or a few thousand M ⊙ which are out of reach of LISA or the current ground-based detectors LIGO, Virgo, and GEO-600.
Globular clusters may host intermediate-mass black holes (IMBHs) with masses in the ∼ 100 -1000 M ⊙ range (see Ref. 3 and references therein). If the stellar binary fraction in a globular cluster is sufficiently high, two or more IMBHs can form. 4 These IMBHs then sink to the center in a few million years, where they form a binary and merge via three-body interactions with cluster stars followed by gravitational radiation reaction (see 4,5 for more details). Therefore, the rate of IMBH binary mergers is just the rate at which pairs of IMBHs form in clusters. The rate of detectable coalescences is (1) Here M tot is the total mass of the coalescing IMBH-IMBH binary and q ≤ 1 is the mass ratio between the IMBHs; z max (M tot , q) is the maximum redshift to which the ET could detect a merger between two IMBHs of total mass M tot and mass ratio q; dt e /dt o = (1 + z) -1 is the relation between local time and our observed time, and dV c /dz is the change of comoving volume with redshift, given by
We assume a flat universe (Ω k = 0), and use Ω M = 0.27, Ω Λ = 0.73, H 0 = 72 km s -1 Mpc -1 , and D H = c/H 0 ≈ 4170 Mpc, so that the luminosity distance can be written as a function of redshift as:
We make the following assumptions. 1. IMBH pairs form in a fraction g of all globular clusters. 2. We neglect the delay between cluster formation and IMBH coalescence. 3. When an IMBH pair forms in a cluster, its total mass is a fixed fraction of the cluster mass, M tot = 2 × 10 -3 M cl , consistent with simulations. 7 The mass ratio is uniform in [0, 1]. We restrict our attention to systems with a total mass between M tot,min = 100M ⊙ and M tot,max = 20000M ⊙ . Thus,
- The distribution of cluster masses scales as (dN cl /dM cl ) ∝ M -2 cl independently of redshift. We confine our attention to clusters with masses ranging from M cl,min = 5 × 10 4 M ⊙ to M cl,max = 10 7 M ⊙ . The total mass formed in all clusters in this mass range at a given redshift is a redshift-independent fraction g cl of the total star formation rate per comoving volume:
(5)
The star formation rate as a function of redshift z rises rapidly with increasing z to z ∼ 2, after which it remains roughly constant:
.17 e 3.4z e 3.4z + 22
Rather than computing z max (M tot , q) [Eq. 1] for all values of M tot and q, we rely on the following fitting formula for the luminosity-distance range D L,max as a function of the redshifted total mass M z = M tot (1 + z), obtained by using the effectiveone-body, numerical relativity (EOBNR) gravitational waveforms 9 to model the inspiral, merger, and ringdown phases of coalescence:
where A = 500, M 0 = 600M ⊙ for q = 1 and A = 281, M 0 = 450M ⊙ for q = 0.25. We use ρ = 8 as the SNR threshold for a “single ET” configuration. We determine the sky-location and orientation averaged range by dividing the horizon distance by 2.26, 10 ignoring redshift corrections to this factor.
We can compute z(D L ) by inverting Eq. ( 3). For a given choice of M tot and q, the maximum detectable redshift z max (M tot , q) is then obtained by finding a self-consistent solution of z D L,max M tot (1 + z max ) = z max .
In order to compute the rate of detectable coalescences, we carry out the integrals over M tot and z in Eq. ( 1) for two specific values of q. For q = 1, we find the total rate to be R = 7.5 × 10 4 g g cl yr -1 ; for q = 0.25, it is R = 2.7 × 10 4 g g cl yr -1 . The range varies smoothly with q; therefore, we estimate that full rate, including the integral over q is
zmax(Mtot,q) 0 dz 0.17 e 3.4z e 3.4z + 22 4π(D H /Mpc) 3 (1 + z)
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