GRAVITATIONAL WAVES FROM INTERMEDIATE-MASS BLACK HOLES IN YOUNG CLUSTERS

So far, there are no definitive observational proofs of the existence of intermediate-mass black holes (IMBHs), i.e., of black holes (BHs) with masses ranging from ~10² to ~10⁵ M_☉ (see van der Marel 2004 for a review). However, the formation of IMBHs is predicted by various theoretical scenarios. In the early universe, IMBHs might form via the direct collapse of metal-free stars (Heger et al. 2003) or via the collapse of gaseous disks in the center of pre-galactic halos (Begelman et al. 2006). IMBHs might form even in more recent epochs, via the direct collapse of metal-poor stars (Mapelli et al. 2009; Zampieri & Roberts 2009; Mapelli et al. 2010), the repeated mergers of stellar mass BHs in globular clusters (GCs; Miller & Hamilton 2002), and the runaway growth of IMBHs (Portegies Zwart & McMillan 2002, hereafter PZM02) in massive (≥10⁴ M_☉) young (3 × 10⁷ yr) clusters (YCs). In particular, the runaway growth scenario predicts that star clusters with initial half-mass relaxation times t_h 25 Myr are dominated, during the core collapse, by star collisions, which build up one or even two (Gürkan et al. 2006) very massive objects, which likely evolve into IMBHs (PZM02). Thus, YCs are among the best environments where we can search for IMBHs.

However, the observational features of IMBHs are difficult to pinpoint. There are hints (Strohmayer & Mushotzky 2003; Kaaret et al. 2004; Strohmayer et al. 2007) that IMBHs might power some of the brightest ultra-luminous X-ray sources (ULXs; i.e., those sources with an X-ray luminosity higher than the one predicted for a ~10 M_☉ BH). Other studies point out that the presence of IMBHs may be inferred from anomalies in the acceleration of millisecond pulsars (Colpi et al. 2002; Colpi et al. 2003) or from the observation of high-velocity stars in GCs (Mapelli et al. 2005). Furthermore, the combination of kinematic and photometric data with dynamical models suggests the presence of IMBHs in the GC G1 (Gebhardt et al. 2002, 2005) in the Andromeda galaxy and in the GC M15 (Gerssen et al. 2002; van den Bosch et al. 2006) in the Milky Way (MW).

Finally, gravitational waves (GWs) will provide a powerful tool for detecting IMBHs. In fact, IMBHs lying in the center of dense clusters are expected to form binaries with stars or stellar mass BHs (e.g., Sigurdsson & Hernquist 1993). Such binaries harden (i.e., progressively reduce their orbital separation) due to gravitational encounters (Colpi et al. 2003 and references therein). When the orbital separation is sufficiently small and if the companion of the IMBH is a compact object, such as a stellar mass BH or a neutron star (NS), the binary enters a regime in which the orbital evolution is driven by radiation reaction induced by GW emission. This further reduces the orbital separation until the system merges. GWs emitted by binaries hosting IMBHs are mostly in the frequency range accessible to current or forthcoming detectors, such as the Laser Interferometer Gravitational-Wave Observatory (LIGO; Barish & Weiss 1999; Abbott et al. 2009) and Virgo (Acernese et al. 2004) in initial and advanced configuration (also known as first and second-generation laser interferometers). GWs from IMBH mergers also fall in the frequency range of future GW observatories, such as the Laser Interferometer Space Antenna (LISA; Bender et al. 1998) and third-generation ground-based instruments, for which we adopt, as an example, the concept of the Einstein gravitational-wave Telescope (ET; Freise et al. 2009; Hild et al. 2008). In particular, Miller (2002), considering IMBHs in GCs, predicts that tens of sources connected with Galactic IMBHs will be detectable with LISA in a five-year integration. More recent studies investigate the emission of GWs from IMBH–IMBH binaries (Amaro-Seoane 2006; Amaro-Seoane & Freitag 2006; Fregeau et al. 2006; Amaro-Seoane et al. 2009; Amaro-Seoane & Santamaria 2009), whose existence is predicted by simulations (Gürkan et al. 2006). Massive stellar mass BHs (15–20 M_☉) are also found to be important sources of GWs, detectable by LIGO and by Advanced LIGO (O'Leary et al. 2007). Inspirals of an NS or of a stellar mass BH into an IMBH in GCs are detectable by Advanced LIGO, with rates up to tens per year (Brown et al. 2007; Mandel et al. 2008). Finally, ET might be able to observe GWs from IMBHs produced by seed BHs at high redshift (Sesana et al. 2009; Gair et al. 2009a, 2009b).

In this paper, we extend the previous studies by Miller (2002) and by Mandel et al. (2008) to IMBHs hosted in YCs, making predictions for the detection rate of GWs from IMBHs in YCs. YCs are an ideal environment in which GWs from IMBHs can be studied, as YCs are a key place for the formation of IMBHs and because the density of YCs is expected to rapidly increase with redshift (Hopkins & Beacom 2006). For this reason, we pay particular attention to the integration with redshift of the detection rate. We consider, as representatives of the science capability of first-, second- and third-generation ground-based instruments, Initial LIGO, Advanced LIGO, and ET, respectively, and for space-borne instruments, LISA. We note that space-based instruments with peak sensitivity in the frequency region ~0.1 Hz that fills the gap between LISA and ground-based laser interferometers would be ideal for studying IMBHs, due to the emission frequency of systems in the mass range ~10²–10³ M_☉. Such instruments, such as DECIGO (Kawamura et al. 2006), ALIA (Bender et al. 2005), and the Big-Bang-Observer (Phinney et al. 2003) are being investigated, but their timescale is currently very uncertain and for this reason we will not consider them in this paper.

In this section, we will focus on the properties of IMBHs in YCs and on their role as sources of GWs. In particular, we will consider two galaxies: the MW, for which we have the best observational data about YCs, and the Antennae, which host a large population of YCs. These two galaxies can also be considered as prototypical of, respectively, a bright disk galaxy with a typical star formation rate (SFR) for the current cosmic epoch, and a merging system involving two massive disk galaxies undergoing enhanced star formation, a configuration that is expected to be increasingly common at higher redshift. We will subsequently generalize our results and make predictions for the detection rate of GWs by LIGO, Advanced LIGO, LISA, and ET.

2.1. IMBH Mergers in the MW

First, we derive an estimate of the number of IMBHs which may exist in the YCs of the MW. As the YCs reside in the high-density regions of our Galaxy, their detection and observation are made difficult by Galactic absorption. Thus, only data for ~10 massive (≥10⁴ M_☉) Galactic YCs are available (Davies et al. 2007 and references therein). However, Gvaramadze et al. (2008) estimate that the total number of massive Galactic YCs is ~70–100, much higher than the number of the detected ones. The runaway growth of IMBHs is possible only in those YCs with half-mass relaxation time t_h 25 Myr (PZM02). Extrapolating from the properties of the YCs listed in Table 1 of Portegies Zwart et al. (2002), we find that a fraction f_tot~0.5–1.0 of the Galactic massive YCs have t_h 25 Myr and thus may host IMBHs. Therefore, ~30–100 IMBHs may exist at present in the YCs of the MW. In the following sections, we adopt the average value f_tot = 0.75 as a fiducial value.

Table 1. Detection Rates from Equation (5) Assuming a Lada & Lada (2003) MF and f_tot = 0.75 for the IMBHs (see Sections 2.1 and 2.3 for Details)

Companion	LIGO	Advanced LIGO	LISA	ET
	R (yr−1)	R (yr−1)	R (yr−1)	R (yr−1)
NS	2 × 10−5	0.2	2 × 10−5	200
Stellar mass BHs	3 × 10−4	4	0.01	400

Download table as:  ASCIITypeset image

IMBHs born via runaway collapse in the center of dense YCs likely form binaries with other stars or stellar mass BHs (see Colpi et al. 2003 and references therein). Such binaries harden due to three-body (or even four-body) interactions at a rate (Sigurdsson & Phinney 1993; Colpi et al. 2003)

$$\begin{eqnarray} \nu _{3b}&\sim & 2{\pi} {G}\,m_{\rm BH}\,n_{c}\,a\,\sigma _{c}^{-1} \sim 10^{-6}\;{\rm yr}^{-1}\nonumber \\ &&\times \left(\frac{m_{\rm BH}}{500\,{M}_\odot }\right)\,\left(\frac{n_{c}}{5\times 10^5\;{\rm pc}^{-3}}\right)\,\left(\frac{a}{0.4\;{\rm AU}}\right)\nonumber \\ &&\times \left(\frac{20\;{\rm km}\;{\rm s}^{-1}}{\sigma _{c}}\right), \end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant, m_BH is the mass of the IMBH, n_c is the core density of the YC, a is the orbital separation of the binary, and σ_c is the velocity dispersion in the core of the YC. In Equation (1), we take n_c = 5 × 10⁵ stars pc⁻³ and σ_c = 20 km s⁻¹ as fiducial values for the MW, which are the average values for the YCs listed in Table 1 of Portegies Zwart et al. (2002). The masses of IMBHs formed by runaway collapse (m_BH) likely range from 10² to 10³ M_☉ (PZM02; Gürkan et al. 2006). Finally, a typical initial orbital separation of ~0.4 AU is assumed, according to Devecchi et al. (2007).⁴ Typically, a merger between the IMBH and its companion occurs after a few hundred three-body interactions (Miller 2002). Since it is hard to give a more accurate expression for the merger rate, in the following we will assume that the merger rate for a single IMBH binary is ν_mrg = 10⁻² ν_3b (Miller 2002), neglecting its possible dependence on various quantities, such as m_BH. Thus, the merger rate for a single IMBH binary in the MW is ν_mrg ~ 10⁻⁸(m_BH/500 M_☉)(n_c/5 × 10⁵ pc⁻³)(a/0.4 AU)(20 km s⁻¹/σ_c) yr⁻¹. We expect a Galactic total merger rate of ν_mrg,tot ≈ 10⁻⁶ yr⁻¹, considering all the ~100 IMBHs hosted in the YCs of the MW.

2.2. IMBH Mergers in the Antennae

2.3. GWs from IMBHs in the Local Universe

From the calculations reported in the previous sections, we can predict a detection rate of GWs emitted by merging IMBHs in the local universe expected for LISA, LIGO, Advanced LIGO, and ET. In particular, we consider all the mergers involving IMBH–BH and IMBH–NS binaries on the past light cone, within a sphere of fixed radius, derived from the detection range of a given instrument, through the expression of the maximum observable redshift z_max; see the description below. Under the assumption that all the IMBHs have the same mass, such detection rate can be expressed as (Miller 2002)

$$\begin{eqnarray} R &=& 4\,\pi \,\left(\frac{c}{H_0}\right)^3\,f_{\rm tot}\,\nu _{\rm mrg}(m_{\rm BH})\nonumber \\ &&\times\int ^{z_{\rm max}(m_{\rm BH},\,m_{\rm co})}_{0}\left(\int ^{z}_{0}\frac{{d}\tilde{z}}{E(\tilde{z})}\right)^2 \frac{n_{\rm YC}(z, m_{\rm BH})}{E(z)}\,\frac{{d}t_{e}}{{d}{t_{o}}}\,{d}z, \end{eqnarray} \tag{ 2 }$$

where c is the light speed, H₀ (=72 km s⁻¹ Mpc⁻¹; Spergel et al. 2007) the Hubble constant, f_tot ~ 0.75 is the fraction of massive YCs hosting an IMBH (see Section 2.1), ν_mrg(m_BH) is the merger rate per IMBH (see Section 2.1), and E(z) = [(1 + z)³Ω_M + Ω_Λ]^1/2 (where Ω_M = 0.27 and Ω_Λ = 0.73; Spergel et al. 2007). The factor $\frac{{d}t_{e}}{{d}{t_{o}}}=(1+z)^{-1}$ accounts for the difference between the time in the rest frame of the source (t_e) and of the observer (t_o).

In Equation (2), n_YC(z, m_BH) is the comoving number density of YCs which are sufficiently massive to host an IMBH. Assuming that a constant fraction of stars (f_SFC) forms in clusters, n_YC(z, m_BH) can be approximately estimated as

$$\begin{eqnarray} n_{\rm YC}(z,m_{\rm BH})&=&4.5\times 10^{-2}\;{\rm Mpc}^{-3}\nonumber \\ &&\times \left(\frac{\dot{\rho }_{\ast }(z)}{1.5\times 10^{-2}\,{M}_\odot \;{\rm yr}^{-1}\;{\rm Mpc}^{-3}}\right) \left(\frac{t_{\rm max}}{10^9\;{\rm yr}}\right)\nonumber \\ &&\times \left(\frac{f_{\rm surv}}{10^{-2}}\right) \left(\frac{\langle m_{\rm YC}\rangle }{260\,{M}_\odot }\right)^{-1}\,\left(\frac{f_{\rm SFC}}{0.8}\right)\nonumber \\ &&\times \left(\frac{f_{\rm YC}(m_{\rm BH})}{10^{-4}}\right), \end{eqnarray} \tag{ 3 }$$

where t_max is the maximum lifetime of a YC: we adopt t_max = 10⁹ yr, as we account for the fact that the IMBH–BH and IMBH–NS mergers occur ~2 × 10⁸ yr after the formation of the central binary. We must also include a correction f_surv which represents the fraction of YCs that avoid disruption and survive up to t_max = 10⁹ yr. f_surv is quite uncertain; we adopt a conservative value of f_surv = 10⁻² from the literature (Lada & Lada 2003; Fall et al. 2005). For a discussion about the possible mechanisms of cluster infant mortality, see, e.g., Gieles (2010) and references therein.

$\dot{\rho }_{\ast }(z)=10^{\left[\beta \,\log _{10}{(1+z)}+\alpha \right]}$ is the comoving density of SFR (where α = −1.82 and β = 3.28 for z ≤ 1.04, and α = −0.724 and β = −0.26 for 1.04 ≤ z ≤ 4.48; Hopkins & Beacom 2006). These values of $\dot{\rho }_{\ast }(z)$ are based on the most updated available data.⁶f_SFC is the fraction of star formation which occurs in clusters (f_SFC = 0.7–0.9, Lada & Lada 2003). We assume that  f_SFC is constant with redshift, although this cannot be proved on the basis of currently available data.

Zoom In Zoom Out Reset image size

〈m_YC〉 ~ 260 M_☉ is the average mass of a YC, and has been derived assuming that the masses of the YCs range from m_YC, min ~ 20 M_☉ to m_YC, max ~ 10⁷ M_☉ and are distributed according to an MF $\frac{{d}N}{{d}m}\propto m^{-2}$ (Lada & Lada 2003). Finally, f_YC(m_BH) is the fraction of YCs that are massive enough to host IMBHs:

$$\begin{equation} f_{\rm YC}(m_{\rm BH})=\left(\int _{m_{\rm YC,\,min}}^{m_{\rm YC,\,max}} {\frac{{d}N}{{d}m}\,{d}m}\right)^{-1}\,\int _{10^3\,m_{\rm BH}}^{m_{\rm YC,\,max}}{\frac{{d}N}{{d}m}\,{d}m}\,; \end{equation} \tag{ 4 }$$

f_YC depends on m_BH, as the simulations show that the IMBH mass scales with the cluster mass (m_BH ~ 10⁻³ m_YC; PZM02). In the following calculations, we assume that m_YC, min = 20 M_☉, m_YC, max = 10⁷ M_☉, and $\frac{{d}N}{{d}m}\propto m^{-2}$ (Lada & Lada 2003). Under these assumptions, f_YC(m_BH) goes from ~2 × 10⁻⁴ (for an IMBH mass m_BH = 100 M_☉) to ~1.8 × 10⁻⁵ (for an IMBH mass m_BH = 1000 M_☉).

In Equation (2), z_max(m_BH,  m_co) is the maximum redshift at which an event can be detected with a sky location and orientation averaged signal-to-noise ratio 〈S/N〉 ≥10 by a single interferometer. In observations with a network of instruments, the S/N scales as the square root of the number of instruments, and in this respect the results presented here should be considered as conservative. The maximum redshift depends on the mass of the IMBH, m_BH, and on the mass of the companion that merges with the IMBH (m_co), as well as on the sensitivity of the instrument. The expressions adopted to calculate z_max(m_BH,  m_co) (for LIGO, Advanced LIGO, LISA, and ET) are summarized in Appendix A. As a model for the gravitational waveform, we consider the analytical phenomenological inspiral-merger-ringdown waveform approximant for non-spinning BHs in circular orbits derived by Ajith et al. (2008). This waveform model together with the Effective-One-Body-Numerical-Relativity (EOBNR) waveform family (Buonanno et al. 2007) provides a prescription for the signal from the whole coalescence of binary systems. These waveform approximants have both been calibrated on full numerical relativity simulations for binaries for approximately equal mass systems, and yield S/Ns that are consistent to within ≈25%. However, in this paper we will apply the waveforms on a much larger mass-ratio regime that extends up to ~10⁻². For this mass ratio we still do not have reliable inspiral-merger-ringdown waveforms, and the phenomenological and EOBNR approximants yield significantly different S/Ns that differ by a factor of ≈2–10 depending on the mass ratio, with the phenomenological waveforms producing the larger S/N; see, e.g., Figure 2 of Gair et al. (2009a). As the detection rate scales as the cube of the maximum distance at which a source can be detected, the amplitude differences lead to rate uncertainties of a factor of 10 or larger. Furthermore, if the BHs are (rapidly) spinning, the S/N at which a source is observed can be significantly (by a factor of ≈2, see, e.g., Ajith et al. 2009) affected. However, the lack of astrophysical predictions for the likely spins of IMBH binaries and the lack of full coalescence waveform approximants for generic spin magnitudes and geometries (see however Ajith et al. 2009; Pan et al. 2009) prevent us from considering this possibly important physical effect. The resulting values of z_max(m_BH,  m_co) are shown in the upper panels of Figure 1. In this figure, we consider mergers of the IMBHs with two different compact objects, i.e., NSs (left-hand panel) and stellar mass BHs (right-hand panel).⁷ It is evident that there is a large difference between ET and the other interferometers. In fact, while for ET z_max(m_BH,  m_co) is always higher than 0.5 and 1 in the case of NSs and stellar mass BHs, respectively, for Advanced LIGO z_max(m_BH,  m_co) < 0.2 and < 0.5 in the case of NSs and stellar mass BHs, respectively. For the other interferometers z_max(m_BH,  m_co) is even smaller.

The detection rates R derived from Equation (2) for events detectable with Advanced LIGO, LISA, and ET are shown in the bottom panels of Figure 1. In particular, the detection rates R for mergers involving IMBH–NS and IMBH–BH binaries are shown in the left-hand panel and in the right-hand panel, respectively. The different lines show the values of R for different detectors, as a function of the IMBH mass, ranging from 10² to 10³ M_☉. The results obtained for LIGO are not shown in Figure 1, as they are orders of magnitude lower than those obtained for the other interferometers. In particular, the rate of IMBH–NS (IMBH–BH) mergers detectable with LIGO is lower than 5 × 10⁻⁵ (1 × 10⁻³) per year, even in the most favorable case (corresponding to m_BH = 100 M_☉).

Figure 1 shows that ET (dotted line, red on the web) is expected to detect a large number of events per year: R>10 yr⁻¹ and R>60 yr⁻¹ in the case of IMBH–NS and IMBH–BH mergers, respectively. Thus, ET will be essential to assess the existence of IMBHs in the local universe, and to distinguish among different theories of IMBH formation. Advanced LIGO (solid black line) is expected to detect a few mergers involving IMBH–BH binaries per year and less than one merger involving IMBH–NS binaries per year. Advanced LIGO is particularly suitable for observations of low-mass IMBHs (300 M_☉) merging with stellar mass BHs. In the case of m_BH = 100 M_☉, Advanced LIGO is expected to detect up to ~5 IMBH–BH mergers per year.

On the other hand, LISA (dashed line, green on the web) is more efficient in detecting massive (500 M_☉) IMBHs merging with stellar mass BHs due to its sensitivity to frequencies lower than 1 Hz. However, LISA is not particularly suited for detecting GWs from IMBHs, as less than one event per year is expected to be detected, even in the most optimistic case (i.e., for m_BH = 10³ M_☉). In the case of IMBH–NS mergers, LISA is very unlikely to observe any such system during the mission lifetime (nominally set to five years), even for m_BH = 10³ M_☉; the detection rate is in fact 10⁻⁴ yr⁻¹.

In Equation (2), we assume that all the IMBHs have the same mass. However, this is an oversimplification, as IMBH masses will be distributed according to a certain MF. Accounting for such MF, Equation (2) becomes

$$\begin{eqnarray} R &=& 4\,\pi \,\left(\frac{c}{H_0}\right)^3\,f_{\rm tot}\,\left(\,\int ^{m_2}_{m_1}\,\frac{{d}N}{{d}m_{\rm BH}}\,{d}m_{\rm BH}\right)^{-1} \nonumber \\ &&\times \int ^{m_2}_{m_1}\,\nu _{\rm mrg}(m_{\rm BH})\,\left[\int ^{z_{\rm max}(m_{\rm BH},\,m_{\rm co})}_{0}\left(\int ^{z}_{0}\frac{{d}\tilde{z}}{E(\tilde{z})}\right)^2\right. \nonumber \\ &&\times \left. \frac{n_{\rm YC}(z,m_{\rm BH})}{(1+z)\,E(z)}\,{d}z\vphantom{\left(\int ^{z}_{0}\frac{{d}\tilde{z}}{E(\tilde{z})}\right)^2}\right]\,\frac{{d}N}{{d}m_{\rm BH}}\,{d}m_{\rm BH}, \end{eqnarray} \tag{ 5 }$$

where $\frac{{d}N}{{d}m_{\rm BH}}$ is the MF of the IMBHs, whereas m₁ and m₂ are the minimum and the maximum IMBH mass, respectively.

The MF of IMBHs is unknown. However, as we already mentioned for the estimate of f_YC(m_BH), PZM02 shows that there is a correlation between the mass of the IMBH and the mass of the host YC (m_YC), and in particular that the mass of the IMBH is m_BH ~ 10⁻³m_YC. Since the YCs in the MW have an MF $\frac{{d}N}{{d}m}\propto m^{-2}$ (Lada & Lada 2003), we can assume that the IMBHs have the same MF. Adopting this MF and assuming m₁ = 100 M_☉ and m₂ = 1000 M_☉, we find 〈m_BH〉 = 256 M_☉.

Integrating Equation (5) for these values of the MF and of m₁ and m₂, we obtain the detection rates R shown in Table 1. In Figure 1, we also plot the results of Equation (5), using a filled square (red on the web), a filled black circle, and a filled triangle (green on the web) for ET, Advanced LIGO, and LISA, respectively. These points are drawn, for simplicity, in correspondence to m_BH = 256 M_☉, but they have been obtained integrating Equation (5) over a Lada & Lada (2003) MF, with an average IMBH mass 〈m_BH〉 = 256 M_☉. We note that the results obtained from Equation (5), under these assumptions, are very similar to those obtained from Equation (2), assuming that all the IMBHs have mass m_BH = 256 M_☉.

Figure 1 and Table 1 show that, if the IMBHs are distributed according to a Lada & Lada (2003) MF, ET is expected to detect a large number of events per year, involving both IMBH–NS (R ~ 200 yr⁻¹) and IMBH–BH (R ~ 400 yr⁻¹) binaries. Advanced LIGO is expected to detect ~4 mergers involving IMBH–BH binaries and ~0.2 mergers involving IMBH–NS binaries per year. LISA is not suited for detecting IMBHs distributed according to the Lada & Lada (2003) MF, as such MF favors relatively "light" IMBHs.

In Appendix B, we derive an alternative calculation of R, based on Equation (14) of Gair et al. (2009a). In Figure 1, the results of the alternative calculation reported in Appendix B are shown as open points: an open square in the case of ET, an open circle for Advanced LIGO, and an open triangle for LISA. The results of the alternative calculation are similar (within a factor of 5) to those of Equation (5). Given the large uncertainties in both calculations, the results are consistent between them. Furthermore, a more simplified, order-of-magnitude derivation of the detection rate R is provided in Appendix C. We stress that the calculation in Appendix C is much more approximated than the one presented in this section and the one reported in Appendix B, but it is a useful tool in order to estimate the order of magnitude of R with "back of the envelope" calculations.

Our results are consistent with those indicated by previous studies. In particular, in the case of LISA, our estimates of the detection rate R are approximately 1 order of magnitude higher than those obtained in Will (2004), which adopts the noise curves by Larson et al. (2000), and a factor of 5 lower than those derived by Miller (2002), based on the noise curves in Flanagan & Hughes (1998a, 1998b). The results obtained for Advanced LIGO are consistent with the recent study by Mandel et al. (2008). In the case of ET, as we discuss in Appendix C, the results of our approximate calculation in Equation (C1) are very similar (within a factor of 2) to those reported in Table 1 of Gair et al. (2009a). The results derived from the more refined Equation (5) and from the alternative model in Equation (B4) are a factor of 4–10 larger than those reported in Gair et al. (2009a), mainly due to the fact that we integrate over the IMBH mass function and that we account for the dependence of n_YC on the redshift.

In conclusion, GWs from IMBHs are hardly detectable with LIGO and LISA. Instead, Advanced LIGO has chances of detecting GWs from IMBHs in clusters. ET is expected to observe hundreds of IMBHs formed via the runaway collapse mechanism.

In this paper, we study the occurrence of mergers between IMBHs and compact objects (NSs and stellar mass BHs) in YCs. These are found to be important sources of GWs. Our study shows that GWs from IMBHs are unlikely to be detected with first-generation instruments, such as the Initial LIGO, and even with LISA. Advanced LIGO instead offers the ability to observe these sources: a few merger events are expected to be detected by Advanced LIGO in one-year integration. ET is by far the best interferometer projected to study GWs from IMBHs: ~10 to ~300 (~60 to ~600) merger events of IMBH–NS (IMBH–BH) binaries might be detected in a one-year integration, according to the scenario of runaway collapse for IMBH formation. Thus, ET will be a powerful tool to check the runaway collapse and the other scenarios for IMBH formation. However, our estimates are affected by large uncertainties (more than a factor of 10), because of our insufficient knowledge of the masses and density of IMBHs in the local universe. In particular, the fraction of IMBHs per cluster f_tot is highly uncertain, as it is based only on the properties of the ~5 Galactic YCs for which enough data are available. Furthermore, f_tot = 0.75 adopted in this paper must be considered as an upper limit, as we are assuming that all the YCs which can undergo runaway collapse host an IMBH, and that all the mass involved in the collapse ends up into the IMBH. There is no evidence that the runaway collapse necessarily leads to the formation of an IMBH (PZM02) and there are hints that a large fraction of the mass is lost due to winds and/or to recoil (Gaburov et al. 2010). Thus, there is no evidence that the MF of IMBHs is the same as that of the host YCs. Furthermore, even the density (n_YC(z, m_BH)) and the MF of YCs are uncertain: in order to derive such quantities, we extrapolated to massive (≥10⁴ M_☉) YCs various relationships (e.g., the slope of the MF) derived by Lada & Lada (2003) for a sample of smaller clusters (20–1000 M_☉).

Finally, the models adopted to derive the instrumental range, and the corresponding z_max, of the considered interferometers assume that the binary has zero eccentricity, the BHs are non-spinning, and consider waveforms in a mass-ratio regime in which we still do not have reliable signal approximants. These are all factors that may significantly alter the results presented in this paper; in particular, YCs should produce rapidly spinning and/or highly eccentric binaries. Full inspiral-merger-ringdown gravitational-waveform families for (generic) binary systems with a mass ratio of the order of 1–100 are not available at present, and we will be able to address this problem more rigorously only in the future.

We also note that, although we have consistently taken into account the non-linear dynamics of the binary during the final merger by adopting full inspiral-merger-ringdown waveforms (Ajith et al. 2008), calibrated on actual numerical relativity simulations of the coalescence of BH binaries, we have however neglected in the merger rate estimates the possibility that the GW recoil (see, e.g., Baker et al. 2008; Lousto et al. 2010) may impart to the binary a sufficient recoil velocity to eject it from the cluster. This would prevent future mergers, and in this respect, the results presented in this paper should be regarded as upper limits, as GW recoil can only reduce the merger rate. On the other hand, assuming a typical escape velocity of 20 km s⁻¹ from the core of the host cluster (see, e.g., Colpi et al. 2003), the GW recoil (calculated according to Baker et al. 2008) should not be able to eject IMBHs more massive than ~250 M_☉ (~35 M_☉) merging with a 10 M_☉ stellar mass BH (with a 1.4 M_☉ NS). We notice, however, that the available expressions of the recoil velocity were obtained from simulations with mass ratio m_co/m_BH = 1–1/3 (e.g, Baker et al. 2008), whereas we consider systems for which m_co/m_BH = 0.1–0.0014. No simulations have been carried out for such extreme mass ratio. Furthermore, even in a pessimistic scenario, accounting for GW recoils should reduce the detection rate R by at most a factor of a few, as in our model most of IMBHs undergo 5 merger events within the lifetime of the cluster.

All these caveats must be taken into account when comparing the rates R derived in this paper with the forthcoming observational data. However, our results will provide the basis for future studies exploring these issues further with improved data on the nature and properties of YCs.

We thank the referee for a critical reading of the manuscript. M.M. acknowledges support from the Swiss National Science Foundation, project 200020-109581/1.

We define z_max(m_BH,  m_co) as the maximum redshift at which an event can be detected with a sky location and orientation averaged 〈S/N〉 ≥10 by a single interferometer. The coherent S/N at which a GW h(t) can be detected by an instrument characterized by a (zero mean) Gaussian, stationary noise with a one-sided noise power spectral density S_n(f), is given by

$$\begin{equation} {\rm {(S/N)}}^2= (h|h). \end{equation} \tag{ A1 }$$

Here, (.|.) is the noise-weighted inner product, defined as (Cutler & Flanagan 1994)

$$\begin{equation} (a|b)=2\int _0^\infty \frac{\tilde{a}^\ast (f)\,\tilde{b}(f)+\tilde{a}(f)\,\tilde{b}^\ast (f)}{S_n(f)}\,df\,, \end{equation} \tag{ A2 }$$

where $\tilde{a}(f)$ and $\tilde{b}(f)$ are two generic functions defined in the Fourier domain.

For the waveform h(t), we use the phenomenological inspiral-merger-ringdown waveform model by Ajith et al. (2008) for non-spinning BHs in circular orbit. For an optimally orientated binary the waveform emitted during the whole coalescence is described by

$$\begin{equation} u(f)\equiv A_{\rm eff}(f)\,e^{\rm i \Psi _{\rm eff}(f)}, \end{equation} \tag{ A3 }$$

where

$$\begin{equation} A_{\rm eff}(f)\equiv C\left\lbrace \begin{array}{@{}l} (f/f_{\rm merg})^{-7/6}\; \hbox{ if }\; f<f_{\rm merg}\\[4pt] (f/f_{\rm merg})^{-2/3}\; \hbox{ if }\; f_{\rm merg}\le f < f_{\rm ring}\\[4pt] w\mathcal {L}(f,f_{\rm ring},\sigma)\; \hbox{ if }\; f_{\rm ring}\le f<f_{\rm cut}. \end{array} \right. \end{equation} \tag{ A4 }$$

Expressions for f_merg, f_ring, f_cut, Ψ_eff(f), C, w, and $\mathcal {L}(f,f_{\rm ring},\sigma)$ are given in Equations (4.14)–(4.19) and Tables I and II of Ajith et al. (2008). The fitting coefficients reported in Ajith et al. (2008) have been revised in Ajith (2008), but have no effect on the actual S/N, and we have therefore adopted the original values. The angle-averaged S/N is obtained by dividing the S/N (Equation (A1)), from an optimally oriented source, described by Equation (A4), by a factor of 2.26, which accounts for the varying response of GW instruments to sources in different locations of the sky and with different orientations of the orbital angular momentum:

$$\begin{equation} \langle \mathrm{{S/N}}\rangle = \frac{(u|u)}{2.26}. \end{equation} \tag{ A5 }$$

In observations carried out with a network the total coherent network S/N is 〈S/N〉² = ∑_k〈S/N〉²_k, where 〈S/N〉²_k is the S/N at each instrument. We caution the reader that if sources are not uniformly distributed in distance, then considering angle-averaged S/Ns (Equation (A5)) introduces some errors in the estimate of the detection rates; however, considering the other (large) uncertainties (from the astrophysics and waveform modeling) that enter the computation of this quantity, the simplification introduced by Equation (A5) has a negligible effect on the final results.

Using these equations, we can derive the luminosity distance D_L(z_max(m_BH,  m_co)) at which an event can be detected with an 〈S/N〉 ≥ 10 by a certain interferometer. We can, thus, derive z_max(m_BH,  m_co) by inverting the expression of the luminosity distance in the Λ cold dark matter (CDM) model:

$$\begin{eqnarray} D_L(z_{\rm max}(m_{\rm BH},\,m_{\rm co}))&=&\frac{c}{H_0}(1+z_{\rm max}(m_{\rm BH},\,m_{\rm co}))\nonumber \\ &&\times \int _0^{z_{\rm max}{(m_{\rm BH},\,m_{\rm co})}}{\frac{{d}z}{E(z)}}. \end{eqnarray} \tag{ A6 }$$

The range of an instrument, Equation (A6), and therefore the IMBH binary detection rates, are entirely determined by the noise spectral density S_n(f); see Equation (A1). For ground-based interferometers we consider representative sensitivity curves for the three generations of instruments. For instruments now in operation (first generation), we adopt the Initial LIGO design sensitivity curve, which well approximates the sensitivity achieved during the last science run (Abbott et al. 2009). Consistently, we set the low frequency cutoff, the minimum of integration in Equation (A2), to f_min = 40 Hz; the upper frequency cutoff is irrelevant, as the ringdown signal for the IMBH mass range of interest is de facto zero in the high-frequency region of the instrumental sensitivity window f ~ 1 kHz.

For second-generation (or advanced) interferometers, we use the broadband target design sensitivity curve of Advanced LIGO with a low frequency cutoff f_min = 10 Hz (Adhikari et al. 2006); Advanced Virgo is expected to operate on the same timescale and has a similar noise performance.

For third-generation interferometers, which are currently undergoing conceptual design studies, we adopt the noise curve of a single right-angle ET instrument and a low frequency cutoff f_min = 1 Hz (Hild et al. 2008). We note that other configurations, such as the Xylophone (Hild et al. 2010), have been proposed and are currently under study.

For LISA, we adopt the current best estimate of the instrumental noise spectral density; see Barack & Cutler (2004). As we show below, the frequency range relevant for observations of IMBH binaries is always above several mHz; as a consequence, the confusion noise generated by stochastic foregrounds of close white-dwarf binaries (and possibly extreme mass-ratio inspirals) does not contribute to the total noise budget and we therefore ignore it in the calculation. In the case of low-frequency observations, the binary system lifetime may be longer than the mission lifetime. In fact, the time to coalescence (at the leading quadrupole Newtonian order) for a binary radiating at frequency f is

$$\begin{equation} \tau (f) = 5\,c^5\,G^{-5/3}\,(8{\pi}{f})^{-8/3}\, \mu ^{-1}\,m_{\rm tot}^{-2/3}\,, \end{equation} \tag{ A7 }$$

where G is the gravitational constant, c is the speed of light, and m_tot = m_BH + m_co and μ = m_BH m_co/m_tot are the total and reduced mass, respectively, of the coalescing binary. Moreover, LISA will not be able to observe the final merger-ringdown phase, as it takes place at a frequency higher than the observable window. Here, we assume that the highest frequency that LISA can observe is 1 Hz and the last stable orbit around a Schwarzschild BH corresponds to the GW frequency of ≈ 4(10³ M_☉/m_tot) Hz. The integration limits in Equation (A1) are obtained with the following procedure. The time of observability T_obs for a merger event is given by (Peters 1964)

$$\begin{equation} {T}_{\rm obs}= \tau (f_\mathrm{min}) - \tau (f_\mathrm{max})\,. \end{equation} \tag{ A8 }$$

From Equation (A8), we can derive f_min as a function of f_max, assuming T_obs = 5 yr. Figure 2 shows f_min as a function of f_max. We note that f_min is almost constant for f_max 3 × 10⁻² Hz. We then calculate 〈S/N〉 from Equation (A5) for different values of the couple f_min, f_max. Figure 3 shows 〈S/N〉 as a function of the couple f_min, f_max. We find that 〈S/N〉 is maximum when we take f_max = 1 Hz and we consistently derive f_min from Equation (A8). The results presented in Table 1 and in Figure 1 are obtained with such choice of f_max = 1 Hz and f_min derived from Equation (A8). As a consequence the results presented for LISA in this paper should be considered as upper limits, assuming that the binary is in the optimal stage of the merger for detection. However, from Figure 3 we also note that 〈S/N〉 is almost constant for f_max ≥ 10⁻² Hz. Thus, we expect that the upper limit for LISA is not far from the average value. The integration frequency range also justifies neglecting the confusion noise from unresolved stochastic foregrounds.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Gair et al. (2009a) report in their Equation (14) a derivation of the detectable merger rate for IMBH–IMBH binaries. In the present paper, we do not consider the mergers of IMBH–IMBH binaries, because the possibility of forming more than one IMBH in the same cluster is still debated (see, e.g., Gürkan et al. 2006). However, it is possible to perform a calculation similar to Equation (14) of Gair et al. (2009a) for IMBH–BH and IMBH–NS binaries, as follows:

$$\begin{equation} R=\int _{m_1}^{m_2}{d}m_{\rm BH}\int _0^{z_{\rm max}(m_{\rm BH}, m_{\rm co})}{d}\tilde{z}\frac{{d}^3N_{\rm merg}}{{d}m_{\rm BH}\,{d}t_{e}\,{d}V_{c}}\,\frac{{d}t_{e}}{{d}{t_{o}}}\,\frac{{d}V_{c}}{{d}\tilde{z}}, \end{equation} \tag{ B1 }$$

where m₁ and m₂ are the minimum and the maximum IMBH mass, respectively, $\frac{{d}t_{e}}{{d}{t_{o}}}=(1+z)^{-1}$ (since t_e and t_o are the time in the rest frame of the source and of the observer, respectively), V_c is the comoving volume, and N_merg is the number of mergers. Assuming that m_BH = 10⁻³m_YC (PZM02),

$$\begin{equation} \frac{{d}^3N_{\rm merg}}{{d}m_{\rm BH}\,{d}t_{e}\,{d}V_{c}}=\nu _{\rm mrg}(m_{\rm BH})\,f_{\rm tot}\,t_{\rm max}\,f_{\rm surv}\,10^3\frac{{d}^3N_{\rm YC}}{{d}m_{\rm YC}\,{d}t_{e}\,{d}V_{c}}, \end{equation} \tag{ B2 }$$

where N_YC is the number of YCs. Following Equation (12) of Gair et al. (2009a),

$$\begin{equation} \frac{{d}^3N_{\rm YC}}{{d}m_{\rm YC}\,{d}t_{e}\,{d}V_{c}}=\frac{g_{\rm YC}(m_{\rm YC})}{\ln {(m_{\rm YC,max}/m_{\rm YC,min})}}\,\dot{\rho }_{\ast }(z)\,m_{\rm YC}^{-2}, \end{equation} \tag{ B3 }$$

where $\dot{\rho }_{\ast }(z)$ is the comoving density of SFR (see Section 2.3) and g_YC(m_YC) is the fraction of the total stellar mass which is formed in YCs of mass m_YC.

Substituting Equation (B3) into Equation (B1), and assuming that ν_mrg(m_BH) = 10⁻² ν_3b(m_BH) (see Section 2.1) and m_BH = 10⁻³m_YC, we finally obtain

$$\begin{eqnarray} R&=&\frac{2{\pi}{G}\,f_{\rm tot}\,t_{\rm max}\,f_{\rm surv}}{\ln {(m_{\rm YC,max}/m_{\rm YC,min})}}\,10^{-5}\,n_{c}\,a\,\sigma _{c}^{-1}\,g_{\rm YC}\,\int _{m_{\rm YC,minBH}}^{m_{\rm YC,maxBH}}\frac{{d}m_{\rm YC}}{m_{\rm YC}}\nonumber \\ &&\times \int _0^{z_{\rm max}(10^{-3}\,m_{\rm YC}, m_{\rm co})}{d}\tilde{z}\, \frac{\dot{\rho }_{\ast }(\tilde{z})}{(1+\tilde{z})}\frac{{d}V_{c}}{{d}\tilde{z}}, \end{eqnarray} \tag{ B4 }$$

where m_YC,minBH and m_YC,maxBH are the minimum and the maximum YC mass in order to form an IMBH in the considered mass range, respectively (we adopt m_YC,minBH = 10⁵ M_☉ and m_YC,maxBH = 10⁶ M_☉, corresponding to m_BH between 10² and 10³ M_☉). As in Section 2.3, m_YC,min = 20 M_☉ and m_YC,max = 10⁷ M_☉ are the minimum and the maximum YC mass, respectively. In Equation (B4), we assume that n_c, a, σ_c, and g_YC do not depend on the cluster mass. Adopting n_c = 5 × 10⁵ pc⁻³, a = 0.4 AU, σ_c = 20 km s⁻¹, and g_YC = 0.8 (the same values as in the alternative calculation reported in the main text), we derive the values of R shown in Figure 1 as open points (open triangles for LISA, open circles for Advanced LIGO, and open squares for ET). The difference between open and filled points (derived from Equation (B4) and from Equation (5), respectively) is within the large uncertainties of the two calculations. In Table 2, we report the values of R derived from Equation (B4), for a comparison with those listed in Table 1: differences are less than a factor of 7 in the case of LISA and less than a factor of 2 for the other instruments.

An order-of-magnitude estimate of R may be obtained with a much simpler calculation than those reported in Section 2.3 and in Appendix B. In particular, assuming that all the IMBHs have the same mass m_BH, we can write

$$\begin{equation} R\approx f_{\rm tot}\,\frac{n_{\rm YC}(z=0, m_{\rm BH})}{T_{\rm mrg}(m_{\rm BH})}\,V_{c}(z_{\rm max}), \end{equation} \tag{ C1 }$$

where n_YC(z = 0, m_BH) is the comoving density of YCs which are sufficiently massive to host an IMBH, defined by Equation (3), calculated at z = 0. We note that we are considering a lower limit of the YC density, as the value of n_YC(z, m_BH) is a minimum for z = 0. T_mrg = ν⁻¹_mrg = 2 × 10⁸ yr (256 M_☉/m_BH) (5 × 10⁵ pc⁻³/n_c) (σ_c/20 km s⁻¹) is the timescale for a merger. Finally, V_c(z_max) is the comoving volume up to redshift z_max(m_BH) (with z_max(m_BH) derived as described in Appendix A). The results of Equation (C1) as a function of the fiducial IMBH mass are shown in Figure 4. Table 3 reports the value of R for a fixed IMBH mass m_BH = 256 M_☉. We note that the detection rates predicted by Equation (C1) are a factor of ~2–10 lower than those derived with the other two more refined methods (see Figure 1). This is likely due to the fact that in Equation (C1) n_YC does not increase with redshift. For this reason, Equation (C1) is a sort of lower limit for our estimates.

Zoom In Zoom Out Reset image size

Finally, Table 4 is focused on the case of ET, and shows both the quantities adopted in Equation (C1) and the derived detection rates for some choices of m_BH and m_co. Table 4 was introduced for a comparison between our procedure and the similar one reported in Section 3.1 and in Table 1 of Gair et al. (2009a). We note that the values of the comoving density of YCs, which are sufficiently massive to host an IMBH (n_YC), adopted in our calculations, are a factor of 3–40 lower than those adopted by Gair et al. (2009a), who consider the comoving space density of GCs and assume that ~10% of them host an IMBH. On the other hand, the usage of the Ajith et al. (2008) model for the gravitational waveform leads to an overestimate of z_max by a factor of 2–3 with respect to the EOBNR models adopted by Gair et al. (2009a; see the discussion in the main text for details). For these reasons, our estimates of R are very similar (within a factor of 2) to those reported in Table 1 of Gair et al. (2009a).