Abstract

Ultraluminous X-ray sources (ULXs) with X-ray luminosities larger than the Eddington luminosity of stellar-mass objects may be powered by intermediate-mass black holes (IBHs) of masses M∼ 103 M. If IBHs form in young dense stellar clusters, they can be fed by Roche lobe overflow from a tidally captured massive (M > 10 M) stellar companion. After the donor leaves the main sequence it forms a compact remnant, which spirals in as a result of gravitational wave (GW) emission. We show that space-based detectors such as the Laser Interferometer Space Antenna are likely to detect several of these sources. GW sources stemming from this scenario have small eccentricities which give distinct GW signals. Detection of such a GW signal will unambiguously prove the existence of IBHs, and support the hypothesis that some ULXs are powered by IBHs with captured companions.

Introduction

The counterparts of ultraluminous X-ray sources (ULXs), which have X-ray luminosities LX larger than the Eddington luminosity of a stellar-mass object of mass M, LX > LE= 1.3 × 1039 erg s-1M/10 M, are not known. Most likely there is no universal engine for ULXs. Some may be powered by anisotropic radiation (e.g. King et al. 2001; Rappaport, Podsiadlowski & Pfahl 2005) or super-Eddington luminosity (Begelman 2002) from stellar-mass black holes.

A third possibility is offered by the hypothesized existence of intermediate-mass black holes (IBHs; see Miller & Colbert 2004 for a review) of masses 103M≲ 105 M, which could radiate isotropically at a sub-Eddington rate to account for the observed luminosities. There has been some observational evidence that at least some ULXs are powered by IBHs. In some cases the mass–temperature relation (Miller et al. 2003; Miller, Fabian & Miller 2004) or quasi-periodic oscillations (Fiorito & Titarchuk 2004; Liu et al. 2005a) indicate a high accreting mass.

Numerical N-body and Monte Carlo simulations indicate that the stellar collision rate in young dense stellar clusters during core collapse can become very large, giving rise to a hierarchical merger. In that case an object of thousands of solar masses may form (Portegies Zwart et al. 1999, 2004a; Gürkan, Freitag & Rasio 2004; Freitag, Rasio & Baumgardt 2005a; Freitag, Gürkan & Rasio 2005b). The fate of such a massive star is unclear, but it might lead to the formation of an IBH. This scenario is supported by the fact that ULXs correlate positively with star formation (Swartz et al. 2004; Liu, Bregman & Irwin 2005b) and that some ULXs are associated with stellar clusters (e.g. Zezas et al. 2002).

The young stars in the host cluster of the IBH have strong winds which blow out the gas from the cluster, and there is insufficient free gas available to power a ULX. However, the IBH can acquire a companion star by dynamical capture or tidal capture. Here we discuss the latter possibility. Hopman, Portegies Zwart & Alexander (2004) showed that tidal capture of a main-sequence star of mass M and radius R can lead to circularization close to the tidal radius  

(1)
formula
which is the distance from the IBH where the tidal forces equal the forces that keep the star bound. As the star evolves it starts to fill its Roche lobe, causing it to lose mass to the IBH. This leads to high X-ray luminosities, provided that the donor is sufficiently massive (M≳ 10 M: Hopman et al. 2004; Portegies Zwart, Dewi & Maccarone 2004b; Li 2004). Interestingly, for some ULXs an optical counterpart has been identified, indicating that these ULXs are binary systems (Liu, Bregman & Seitzer 2004; Kuntz et al. 2005).

In this scenario a ULX may turn on for at most the life-time t∼ 10 Myr of the captured star. In addition to strong X-ray emission, the star emits gravitational waves (GWs). Portegies Zwart (2004) discussed the possibility of observing X-rays and GWs simultaneously. Only if the captured star is sufficiently light (M≲ 2 M) is its tidal radius small enough for it to emit GWs in the Laser Interferometer Space Antenna (LISA) band during the mass transfer phase (Portegies Zwart 2004). However, in order to account for the high ULX luminosities the companion star should be considerably more massive than 2 M (Hopman et al. 2004; Portegies Zwart et al. 2004b; Li 2004). In young and dense star clusters, mass segregation causes most massive stars to accumulate in the cluster centre, and the combination of a high mass and large size makes these stars excellent candidates for tidal capture. When after time t the massive donor explodes, it turns into a compact remnant (CR): a neutron star (NS) or a stellar-mass black hole (SBH). From that moment the ULX, deprived of its source of gas, turns off.

Here we focus on the subsequent — post-supernova — evolution of the (IBH, CR) binary. In-spiral of the CR into an IBH due to GW emission results in a strong signal in frequencies measurable by space detectors such as LISA, providing a wealth of information about the system (see Miller 2002 for a discussion). We show that LISA may be able to observe such (IBH, CR) binaries.

Stellar capture and binary evolution

When stars orbit IBHs on wide, but highly eccentric, orbits, with periapse close to the tidal radius of the IBH, rprt, orbital energy is invested in tidal distortions of the star, causing it to spiral in. Hopman et al. (2004) showed that if the IBH is less massive than Mmax≈ 105 M, the star may be able to cool down by radiating the excess energy efficiently, and survive the strong tidal forces. Eventually it circularizes near the IBH. We first estimate the fraction fmerge of isolated binaries that merge as a result of GW emission within the age of the Universe, while accounting for angular momentum conservation during the mass transfer phase and an isotropic velocity kick. We then discuss the consequences of the gravitational interaction of cluster stars with the (IBH, CR) binary.

Evolution of an isolated IBH—star binary

After tidal circularization near the IBH, the star may fill its Roche lobe, possibly after a period of stellar evolution and expansion. Roche lobe overflow from a M≳ 10 M donor can then give rise to high luminosities for a time limited by the lifetime t of the star (Hopman et al. 2004; Portegies Zwart et al. 2004b; Li 2004).

The main-sequence star fills its Roche lobe at a distance ∼2rt before the terminal-age main sequence. We assume that the entire hydrogen envelope is transferred to the IBH, while the binary conserves angular momentum. The mass of the stellar core of the donor star is given by Mc= 0.08 M (M/M)1.4 (Iben, Tutukov & Yungelson 1995), while the mass of the hydrogen envelope is MH=MMc. Angular momentum conservation implies that the binary separation increases during the mass transfer phase.

The remaining helium star subsequently explodes in a supernova after a time t, and forms a CR. In this event a star of 10 < M < 20 M forms a NS of Chandrasekhar mass MCh= 1.4 M. Stars more massive than 20 M collapse to SBHs; we assume that the mass distribution is that found by Fyer & Kalogera (2001).

The semi-major axes and eccentricities of the CR are determined by the post-mass-transfer orbital elements while accounting for the mass lost in the supernova and for the velocity kick, imparted to the CR at the time of the explosion. The kick velocity is taken in a random direction with magnitude from the distribution of pulsar velocities, which is well fitted by a double Gaussian distribution  

(2)
formula
where w= 0.4, σ1= 90 km s-1 and σ2= 500 km s-1 (Arzoumanian, Chernoff & Cordes 2002). For SBHs we adopt the same distribution, but with velocities smaller by a factor of MCh/MSBH (White & Van Paradijs 1996; Gualandris et al. 2005). The velocity kick leads to an increase of the eccentricity of the binary and a change in its energy; the kicks are generally insufficient to ionize the binary systems (van den Heuvel et al. 2000).

The surviving binary loses orbital energy as a result of the emission of GWs. A CR of mass MCR on an orbit with semi-major axis a and eccentricity e around an IBH of mass MMCR loses energy owing to the emission of GWs at rate of  

(3)
formula
with  
(4)
formula
If the orbit is circular (e= 0), spiral-in occurs on a time-scale of (Peters 1964)  
(5)
formula
For non-zero eccentricities the in-spiral time is shorter by a factor of ∼(1 −e)7/2.

We perform binary population synthesis with the above-described scenario to compute the fraction of high-mass binaries that become potential LISA sources. The mass of the stellar companion is selected from the initial mass function d N/d MM−α, where we assume α= 2, consistent with the mass function in the cores of clusters in which a runaway merger occurred (Portegies Zwart et al. 2004a). We assume a minimal mass of 10 M, since lighter donors cannot account for ULX luminosities (Hopman et al. 2004; Portegies Zwart et al. 2004a), and a maximal donor mass of 100 M.

LISA is likely to observe (IBH, CR) binaries in the phase before the merger, rather than the merger event itself. However, the CRs spent only a very short time in the LISA frequency band as compared with the Hubble time (see Section 3). The question of whether these sources are detectable therefore depends on how many merge within a the Hubble time.

In Figs 1 and 2 we show the results for an IBH of M= 3 × 103 and 104 M, respectively. Only a fraction of the star spirals in within a Hubble time. For M= 3 × 103 M nearly all objects that spiral in are NSs, with only very few SBHs. For more massive IBHs, a significant fraction of SBHs also spiral in quickly enough, and the total fraction of merging objects exceeds 10 per cent for M≳ 104 M. In Fig. 3 we show fmerge, the fraction of stars that spiral in within the age of the Universe, as a function of M. After the velocity kick, the CRs have eccentricities up to e≲ 0.9. While this decreases tmerge, these eccentricities are much smaller than the e∼ 0.995 eccentricities found by Hopman & Alexander (2005) for direct GW capture. By the time that the stars spiral in to orbital frequencies ν > 10−4 s, which is in the LISA band, the orbits are close to circular.

Figure 1

Scatter diagram of the merger time tmerge as a function of the zero-age main-sequence mass of the donor MZAMS. Events below the horizontal solid line merge within the age of the Universe. For this image we generated 5000 binaries with an α= 2 power-law initial mass function for the donor mass and we selected an IBH mass of M= 3000 M. A small fraction (144 or 2.9 per cent) of the objects generated experienced merger within a Hubble time. The sudden jump at 20 M indicates the transition between NS and SBH formation.

Figure 1

Scatter diagram of the merger time tmerge as a function of the zero-age main-sequence mass of the donor MZAMS. Events below the horizontal solid line merge within the age of the Universe. For this image we generated 5000 binaries with an α= 2 power-law initial mass function for the donor mass and we selected an IBH mass of M= 3000 M. A small fraction (144 or 2.9 per cent) of the objects generated experienced merger within a Hubble time. The sudden jump at 20 M indicates the transition between NS and SBH formation.

Figure 2

As Fig. 1, but with M= 104 M. In this case many more (∼15 per cent) stars spiral in within the age of the Universe, and in particular there is a significant contribution from SBHs.

Figure 2

As Fig. 1, but with M= 104 M. In this case many more (∼15 per cent) stars spiral in within the age of the Universe, and in particular there is a significant contribution from SBHs.

Figure 3

The fraction fmerge of CRs that spiral in because of GW emission within the age of the Universe as a function of the mass of the IBH. The error bars are estimated by the 1σ Poissonian uncertainties of the simulations.

Figure 3

The fraction fmerge of CRs that spiral in because of GW emission within the age of the Universe as a function of the mass of the IBH. The error bars are estimated by the 1σ Poissonian uncertainties of the simulations.

Interactions of the binary with cluster stars

The two-body relaxation time tr of star clusters that host a runaway merger is short (tr∼ 30 Myr: Portegies Zwart & McMillan 2002), and these evaporate on a time-scale of 108 yr if tidally limited. As long as the cluster has not yet evaporated, the (IBH, CR) binary has interactions with cluster stars. Since the (IBH, CR) binary is ‘hard’ (Heggie 1975), these interactions tend to decrease the orbital separation between the IBH and the CR (Miller 2002). In addition, scattering changes the eccentricity of the binary. The angular momentum vector J of the binary performs a random walk, and its magnitude J samples angular momenta 0 ≲JJm on the relaxation time-scale of the cluster (Alexander & Hopman 2003; Hopman & Alexander 2005). Here Jm is the maximum angular momentum of a binary of given energy. When J decreases the orbit becomes more eccentric, and the in-spiral time decreases (see equation 5).

It is not straightforward to quantify the effect of gravitational two-body scattering, since the cluster is not in a steady state, and in particular the relaxation time in the core can vary wildly. For simplicity we therefore neglect the effect of scattering in the following discussion on the in-spiral rate and the number of observable LISA sources. We note, however, that scattering may significantly increase the number of GW sources, so that the following LISA detection rates should be regarded as lower limits.

Observable gravitational waves from IBHs

The dimensionless strain of the GWs emitted at a frequency ν= 10−3ν−3 s-1 from a source at a distance d=dMpc Mpc is  

(6)
formula
(e.g. Sigurdsson & Rees 1997). LISA is sensitive to frequencies in the range 10−4≲ν≲ 1 Hz. At ν= 10−3 Hz, LISA can detect sources with strains larger than graphic, where graphic. This estimate is based on a 1-yr observation with signal-to-noise ratio S/N = 1 (see e.g. ). In the following we assume for concreteness a GW source with frequency ν= 10−3 Hz; application to other frequencies is straightforward. Sources can be observed to distances up to  
(7)
formula

We assume that only ULXs with luminosities LX > 1040 erg s-1 contain IBHs (Portegies Zwart et al. 2004b). Presently the average number of >1040 erg s-1 ULXs per galaxy is ∼0.1 (Swartz et al. 2004). The star formation rate dropped by an order of magnitude since z∼ 2 (Madau, Pozzetti & Dickinson 1998). ULXs correlate with star formation; here we assume that the number of ULXs is proportional to the star formation rate, in which case the number of luminous ULXs per galaxy at an earlier time was NULX∼ 1. Since the ULX lives for a time t= 107 yr t7, this yields an ULX formation rate per galaxy of 10−7 yr-1NULXt−17. As was discussed in the previous section, only a small fraction fmerge= 0.1f−1 of the ULXs with M < 104 M leave behind a remnant binary that spirals in within a Hubble time. The rate at which observable GW sources are produced per relic ULX is then  

(8)
formula

At the point where the period P= 2πa3/2/(GM)1/2 equals 103 s, the in-spiral time is  

(9)
formula
where we assumed that the orbit has circularized when the frequency is this high. We thus find that the mean number of sources emitting GWs with frequencies of ν∼ 10−3 s-1 per galaxy is graphic, or  
(10)
formula

The local galaxy density is estimated to be ngal≈ 3 × 10−2 Mpc-3 (Marinoni et al. 1999). If the maximal distance at which the GW source can be observed is dmax, the number of LISA sources in the sky at any moment is given by graphic, or, using equation (7),  

(11)
formula

Discussion

For IBHs of M≲ 3 × 103 M, f−1∼ 0.3 and almost no SBHs spiral in (Fig. 1). The main contribution to LISA sources comes from NSs, i.e. MCR/10 M∼ 0.14. We estimate that the merger rate for NSs with IBHs is too low to be likely to be seen by LISA. However, in this estimate we have ignored the effect of three-body scattering on the binary orbit. Although hard to quantize, its result may be a marginal detection rate for NSs. We also note that we estimated that SBHs have a velocity kick distribution similar to that of NSs, but with velocities smaller by the mass ratio MCh/MSBH of the objects. This estimate is rather uncertain, and may be too conservative. Larger kick velocities cause more SBHs to merge within a Hubble time, in which case the contribution of M≲ 3 × 103 M IBHs to LISA would increase.

For IBHs of M≳ 3 × 103 M we find that f−1∼ 1 and for SBHs tmerge decreases to well within a Hubble time. In this case SBHs (MCR/10 M∼ 1) give the most promising GW sources, with a detection rate of about 10 per year, in the case that three-body scattering is ignored. Including three-body scattering in this estimate may boost the detection rate by a sizable fraction.

Our estimate in equation (11) for the number of sources that LISA can detect is conservative. First, in Section 3 the (IBH, CR) binary is considered to be isolated. Indeed, the host cluster eventually evaporates, but this is preceded by a phase during which the binary interacts with other cluster stars (Section 2.2). These interactions tend to harden the binary, and change its eccentricity. As a result tmerge decreases, and thus the number of potential LISA sources increases. It is not implausible that nearly all (IBH, CR) binaries merge within a Hubble time, in which case f−1= 10. Secondly, the lifetime of ULXs is probably significantly shorter than the main-sequence lifetime t of the star. A more realistic assumption would be that ULX luminosities are only achieved when the donor is near the terminal-age main-sequence, in which case t7≈ 0.1, boosting the predicted detection rate for LISA by an order of magnitude. In conclusion, the number of observable GW sources could easily be orders of magnitude larger than the expression (11) indicates.

Previous studies of the number of potential LISA sources from binaries with an extremely small mass ratio focused mainly on cases in which the orbital energy is dissipated by the GWs themselves (e.g. Hils & Bender 1995; Sigurdsson & Rees 1997; Freitag 2001, 2003; Ivanov 2002; Miller 2002; Alexander & Hopman 2003; Hopman & Alexander 2005). In that case the event rate is of the order of a few per Gyr, and the orbits are highly eccentric, with typical eccentricities as large as e∼ 0.995 for ν= 10−4 s-1 near an IBH of M= 103 M (Hopman & Alexander 2005). In that case stars spiral in very quickly, and emit GWs in the LISA band only for a short time, in contrast to the tidal capture scenario discussed here.

GW sources originating from tidal capture sources have nearly circular orbits when they enter the LISA frequency band, leading to a distinctly different signal from the highly eccentric sources originating from direct capture (e.g. Barack & Cutler 2004; Wen & Gair 2005). LISA can determine both the mass of the IBH and the eccentricity of the orbit. Detection of GWs from an IBH will give proof of the existence of these objects. If the signal stems from an orbit with low eccentricity, this supports the scenario that ULXs are accreting IBHs in binary systems.

Acknowledgments

We thank T. Alexander for discussions. We are grateful to the Dutch Royal Academy of Arts and Sciences (KNAW), the Dutch Organization for Scientific Research (NWO) and the Dutch Advanced School for Astronomy (NOVA).

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