Communication The Merger Rate of Black Holes in a Primordial Black Hole Cluster Viktor D. Stasenko * and Alexander A. Kirillov Moscow Engineering Physics Institute, National Research Nuclear University MEPhI, 115409 Moscow, Russia; [email protected] * Correspondence: [email protected] Abstract: In this paper, the merger rate of black holes in a cluster of primordial black holes (PBHs) is investigated. The clusters have characteristics close to those of typical globular star clusters. A cluster −2 that has a wide mass spectrum ranging from 10 to 10 M (Solar mass) and contains a massive 3 central black hole of the mass M• = 10 M is considered. It is shown that in the process of the evolution of cluster, the merger rate changed significantly, and by now, the PBH clusters have passed the stage of active merging of the black holes inside them. Keywords: black hole cluster; black hole merging; merger rate; primordial black holes; Fokker– Planck equation 1. Introduction Since the registration of gravitational waves from black hole mergers, interest in primor- dial black holes (PBHs) and their clusters has increased. Models of the formation of PBH clusters are considered with a narrow mass spectrum [1–3] and a wide one [4–6]. The first type of clusters appears in the early Universe due to the collapse of density perturbations. The second type describes cluster formation via gravitational collapse of domain walls appearing Citation: Stasenko, V.D.; Kirillov, in the early Universe as a result of the evolution of scalar fields in the inflationary epoch; see A.A. The Merger Rate of Black Holes review [7] and references therein. A cluster is formed after detachment from the Hubble flow in a Primordial Black Hole Cluster. and consequent collapse of domain walls into PBHs. Finally, it forms at the redshift z ∼ 104 Physics 2021, 3, 372–378. https:// and obtains characteristics close to those of globular star clusters. Despite the similarities, the doi.org/10.3390/physics3020026 considered PBHs clusters might have black holes with a wide range of masses and a massive black hole at the center. We discussed the evolution of these systems in Refs. [8,9] in the Received: 19 March 2021 Accepted: 10 May 2021 framework of two-body relaxation; however, the question of the black hole merger rate was Published: 17 May 2021 not studied. This paper continues the previous investigation and explores the merger rate of black holes during the evolution of such clusters. Publisher’s Note: MDPI stays neutral There are two possibilities: mergers of black holes with each other and with a massive with regard to jurisdictional claims in central black hole (CBH). The first possibility was discussed in Refs. [7,10–13] in the context 4 published maps and institutional affil- of PBH clusters with narrow mass spectra (with the number of PBH NPBH . 10 ) and in iations. Refs. [14–22] in the context of a dense globular star cluster. The second possibility was considered for globular star clusters [23–28] and galactic nuclei [29–33] in the loss-cone treatment [34,35]. In the calculations described in this paper, it is shown that during cluster evolution, mergers mainly occur with a central black hole; nevertheless, the merger rate is low. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. 2. Estimations This article is an open access article distributed under the terms and Let us estimate the merger rate of black holes for a single-mass PBH cluster. For the 5 conditions of the Creative Commons cluster with the total mass M = 10 M (Solar mass)p and the characteristic radius R ∼ 1 pc, −1 Attribution (CC BY) license (https:// the characteristic velocity of the PBHs is v = GM/R ≈ 20 km s [7], where G is the creativecommons.org/licenses/by/ 4.0/). Physics 2021, 3, 372–378. https://doi.org/10.3390/physics3020026 https://www.mdpi.com/journal/physics Physics 2021, 3 373 Newtonian gravitational constant. The merger cross-section of two black holes with masses m and m0 is given by [14,36]: 85p 2/7 G2(m + m0)10/7m2/7m02/7 s = 2p p . (1) 6 2 c10/7jv − v0j18/7 This cross-section describes the formation of PBHs of binaries due to gravitational radiation. In the present calculations, it is assumed that the lifetime of a binary is less than the dynamical time of the PBH cluster, the approximation similar to one used in Refs. [14,17,22]. The merger rate can be estimated as G ≈ Nvsn, where N is the number of PBHs in the cluster, n is the number density of the PBHs. For simplicity, let us consider a cluster with a constant number density of n = 3N/4pR3. Thus, the merger rate is: M 17/14 R −31/14 G ∼ 10−10 yr−1. (2) 105 M 1 pc During the cluster evolution, black holes evaporate from the core. This process is a slow diffusion in the velocity space. Therefore, one can assume that black holes escaping the cluster carry away zero energy. Hence, the total energy of the core is conserved. According to the virial theorem, the total energy is E = −G(mN)2/2R. Therefore, in the process of BH evaporation, the core size is R ∝ N2. As a result, the relationship between the number of black holes in the cluster core and the merger rate is: G ∝ N−3.2. In this study, one considers the clusters with a massive CBH. As a result of the capture of less massive black holes, the CBH mass grows as M˙ • = rscv, where r is the density of 2 2 black holes in the cluster, and sc = 4prg(c/v) is the cross-section for particle capture by a Schwarzschild black hole [37], rg is the gravitational radius of the CBH. Estimation of M˙ • gives −5/2 1/2 2 −11 R M M• −1 M˙ • ∼ 10 M yr . (3) 1 pc 105 M 103 M If the mass of the CBH M• is much less than the core mass, Mc, then the CBH does −4.5 not affect the core collapse. Hence, the growth rate of the CBH evolves as M˙ • ∝ N . Hence, the merger rate of the BHs may play a significant role in both cases. Note that when the core contracts significantly, the CBH stops the collapse and causes an expansion phase [27,28,33,38,39]. Note that there is also another mechanism of creating PBH binaries in a cluster based on three-body interaction. The third body can carry away enough energy to form a binary from the two others. The formed binaries lose energy through interactions with single black holes and gravitational radiation. As a result, pairs of mergers are caused. This possibility was studied for globular star clusters with black holes in Refs. [17–22]. In addition, these binaries might significantly affect the cluster evolution, stop the core collapse, and cause gravothermal oscillations [40–44]. These binaries will descend towards the cluster center under the influence of dynamical friction. In the case considered in this paper, they will be 3 destroyed as a result of interactions with the massive central black hole (∼10 M )[45,46]. The characteristic time scale for this process is [34]: v3 v 3 r −1 m −1 ∼ B ≈ B B tdf 2 1.7 −1 5 −1 Myr, (4) 4pG mBr ln L 20 km s 10 M pc 20 M where vB and mB are the velocity and the mass of the PBH binary, and ln L = 10, where ln L is the Coulomb logarithm. On the other hand, the merger time tgw of two black holes of the same mass m in a circular orbit with the separation a due to an emission of gravitational waves is given by the expression [37]: − 5c5a4 a 4 m 3 tgw = ≈ 1.6 Myr. (5) 512G3m3 0.01 au 10 M Physics 2021, 3 374 The condition tgw < tdf leads to the maximum value of a for a binary system that merges before its destruction: 1/4 3/4 −2 tdf m a . 10 au. (6) 2 Myr 10 M The probability of the formation of such hard binaries requires a careful analysis. Thus, we assume that all binaries that are forming are destroyed by the CBH. The destruction mechanism also remains true if the PBH binaries are primordial in a cluster. These binaries may have a similar origin that of those formed in the early Universe [47–49]. 3. Merger Rate of Black Holes The merger rate of black holes with the mass m per unit of phase space, per unit of mass, and per unit of time is given by [14]: Z Z g = f (r, v, m) dm0 dv0 f (r, v0, m0) jv − v0j sm, m0, jv − v0j, (7) where f is the distribution function of black holes in the cluster and s is the merger cross section is defined by (1). The expression (1) shows that the cross-section is s = s(m, m0)jv − v0j−18/7. In the case of spherical symmetry, Equation (7) reads: 14p f (r, v, m) Z Z h i g = dm0 dv0 v0s(m, m0) f (r, v0, m0) (v + v0)3/7 − jv − v0j3/7 . (8) 3v Let us take the delta-functional approximation for the distribution function of the i-th black hole type: n (r) f (r, v, m) = i d v2 − v2(r) d(m − m ), (9) i 2pv i i 2 where vi (r) is the mean-squared velocity [50]: 0 Z 3/2 2 4p h i vi (r) = dE fi(E) 2 E − f(r) , (10) ni(r) f(r) and mi and ni are, respectively, the mass and the number density of the i-th PBH type and f(r) is the gravitational potential of the cluster.