The MEGaN project II. Gravitational waves sources around a super-massive black hole
In this paper we investigate the formation of coalescing stellar black hole binaries (BHBs) and extreme mass ratio inspirals (EMRIs) during the assembly of a massive nucleus around an SMBH with mass M. Using direct -body modeling, we show that this phase of the galaxy life is characterised by strong interactions between the SMBH and a population of massive star clusters (GCs) either formed locally or orbitally segregated due to dynamical friction. We show that if the infalling clusters contain a significant population of BHBs, the GCs-SMBH interactions can reduce the merger time-scale to Gyr. Calibrating our results to the sensitivity of the LIGO experiment, we found a detection rate of Gpc yr for a BHB similar to the observed GW150914 source. We also investigated the rate of EMRIs caused by the GCs debris deposited around the SMBH, finding a value of yr. Finally, we show that if two or more of the infalling GCs host in their centre an intermediate mass black hole (IMBH), the three-body system composed by the two IMBHs and the SMBH causes the merging between one of the IMBH and the SMBH in more than of the cases within Gyr. This may give an explanation for the observed lack of IMBHs signatures in galactic nuclei.
keywords:galaxies:nuclei, star clusters: general; galaxies: super-massive black holes; stars: black holes
The recent detection of gravitational waves (GWs) operated by the LIGO-Virgo collaboration (Abbott et al., 2016a), opened new chapters in astronomy and astrophysics. The LIGO detections spotted for the first time the merging of two black holes (BHs) with masses and , providing for the first time the evidence of the existence of BHs with masses above as well as of stellar black hole binaries (BHBs) (Abbott et al., 2016c). The subsequent discovery of two new mergers events by two BHBs with final mass (Abbott et al., 2016b) and (The LIGO Scientific Collaboration et al., 2017), pulled the scientific community in spending a large effort in understanding the origin of the merging BHB process.
One of the most credited scenarios, called “dynamical”, suggest that BHB form and merge in dense stellar systems by continuous interactions with other stars. The repeated stellar scatterings cause either the BHB direct merger, that occurs well deep into the host system, or its ejection. In the latter case, the BHB is thrown away from the stellar system and its orbital properties remain frozen. Hence, the only process that causes its later shrinking is due to GW emission.
Within the framework of the dynamical scenario, globular clusters (GCs) are thought to be the perfect birthplace for coalescing BHBs, as they may contain a huge quantity of binary stars. These are either primordial binaries (PB) or dynamically formed (DB), since they formed through three or multiple gravitational scatterings. While DBs form within the GC core, on a time-scale comparable to the dynamical friction (df) time-scale of the most massive stars of the GC (Portegies Zwart & McMillan, 2002; Gaburov et al., 2008), PBs would have birth during the GC formation and then sink toward its centre due to df braking. Binaries may constitute a non-negligible fraction of the GC mass and are thought to be at the origin of several observational “anomalies”, such as the presence of the so called blue straggler stars in several H-R diagrams of GCs (Sandage, 1953; Bailyn, 1995).
A number of papers proposed that the bursts of GWs detected by Ligo originated by the coalescence of a BHB, either kicked out from, or placed in the centre of, massive globular clusters (Rodriguez et al., 2015, 2016; Askar et al., 2017), young massive clusters (Mapelli, 2016; Banerjee, 2016), or even dwarf spheroidal galaxies in the local volume (Schneider et al., 2017).
Other authors proposed, instead, that such events come from isolated binaries, through a complex binary stellar evolution (Belczynski et al., 2016), through the axisymmetric fragmentation of a rapidly rotating massive star into two heavy BHs (Loeb, 2016) or through a rapid merging driven by the interaction with an SMBH in a galactic nucleus (Bartos et al., 2016; Antonini & Rasio, 2016).
GCs are thought to be also the birthplace of intermediate mass black holes (IMBHs), a class of objects with masses in the range that should fill the gap between stellar BHs and SMBHs. Due to their relatively small masses, IMBHs are still controversial astrophysical objects, quite hard to observe (Noyola et al., 2010; van der Marel & Anderson, 2010; Haggard et al., 2013; Lanzoni et al., 2013; Lützgendorf et al., 2013, 2015a, 2015b; Kızıltan et al., 2017). IMBHs are thought to originate from repeated stellar collisions on time-scales Gyr (Portegies Zwart & McMillan, 2002, 2007; Giersz et al., 2015; Arca-Sedda, 2016), depending on the host GC structural properties.
Therefore, while BHB mergers are more frequent in an earlier phase of the host GC life-time, the formation of an IMBH is a slower process, and possibly shape significantly the host cluster properties (Zocchi et al., 2015).
For the sake of clarity, hereby we refer to massive clusters as GCs, independently on their age, mass or metallicity.
Most of the studies presented in literature investigate the formation and evolution of BHBs and IMBHs in GCs model usually orbiting very far from the centre of their hosting galaxy, where the galactic gravitational field has a marginal effect on the star cluster evolution.
On the other hand, if a massive cluster form in an inner portion of its galaxy, for instance within the inner kpc, its motion can be substantially altered by two competing processes: dynamical friction, which drags the infalling clusters toward the centre, and tidal forces, that strip away their stars. The competing action of dynamical friction and galactic tidal forces are thought to be the basis of the nuclear star cluster (NSC) formation (Tremaine et al., 1975a; Tremaine, 1976; Tremaine et al., 1975b; Capuzzo-Dolcetta, 1993; Antonini et al., 2012; Antonini, 2013; Perets & Mastrobuono-Battisti, 2014; Arca-Sedda & Capuzzo-Dolcetta, 2014a; Gnedin et al., 2014; Arca-Sedda et al., 2015, 2016; Arca-Sedda & Capuzzo-Dolcetta, 2017a).
NSCs are massive and compact stellar systems, with masses in between and half-light radii pc, observed in the centre of a large fraction of galaxies of different masses and Hubble types (Côté et al., 2006; Turner et al., 2012). They are characterised by complex star formation history, possibly connected with their origin (Rossa et al., 2006; Walcher et al., 2006; Carson et al., 2015). Often the NSC harbours a central SMBH, provided that the (Graham, 2012; Neumayer & Walcher, 2012; Georgiev et al., 2016).
The Milky Way does no exception, it hosts a NSC with mass and effective radius pc (Schödel et al., 2014; Gallego-Cano et al., 2017; Schödel et al., 2017), harboring an SMBH with mass (Schödel et al., 2002; Ghez et al., 2008; Gillessen et al., 2009; Schödel et al., 2009).
The MW NSC is characterised by two stellar population: younger, mostly concentrated in the NSC inner region, likely formed in-situ Myr ago; and older, with an age Gyr (Lu et al., 2009; Bartko et al., 2009; Do et al., 2013; Lu et al., 2013; Pfuhl et al., 2011; Feldmeier-Krause et al., 2015; Minniti et al., 2016).
During the NSC formation, GCs undergo a series of strong encounters with the central SMBH, if present, that can strongly affect the shape and properties of the galactic centre, (Perets et al., 2007; Aharon et al., 2016; Arca-Sedda et al., 2016; Arca-Sedda & Capuzzo-Dolcetta, 2017b).
In this context, the effects of the interaction between an SMBH and an infalling GC on its binaries has not yet been quantitatively investigated due to the heavy computational load required to deeply investigate on such a problem.
The recent finding in the Henize 2-10 starburst dwarf spheroidal of a population of GCs with ages Myr (Chandar et al., 2003) and masses in between (Nguyen et al., 2014), orbiting within pc from a SMBH (Reines & Deller, 2012) suggests that the formation of a NSC can be an extremely rapid process, with time-scales Gyr (Arca-Sedda et al., 2015), depending on the host galaxy formation history.
If a population of GCs form well deep the galaxy innermost regions, their orbital decay and subsequent interaction with the central SMBH will occur before the BHB population had time to evolve. Hence, it is possible that part of the BHB population of an infalling GC can be affected by the intense tidal forces exerted from a central SMBH before they are ejected away from their parent cluster or they merge in the parent cluster core.
If the GCs formed in an outer region of the galaxy, instead, their segregation time-scale can easily exceed several Gyr, allowing the GCs to possibly be the site for IMBH formation. If this is the case, the subsequent orbital decay of multiple GCs can bring together several IMBHs around an SMBH. The specific topic of IMBH transport toward galactic centre has been already investigated by few authors for the MW (Mastrobuono-Battisti et al., 2014; Arca-Sedda & Gualandris, 2017) and slightly heavier systems (Baumgardt et al., 2006). Mastrobuono-Battisti et al. (2014) outlined that a population of IMBHs, dragged to the MW NSC during its formation, should have left evident fingerprints in the kinematics of stars moving in the inner few pc of the MW which, however, are not observed.
As noted by Giersz et al. (2015), the formation of an IMBH is a highly stochastic phenomenon, strongly dependent on the host cluster properties. By modeling a huge set of massive star clusters, with masses in the range , the authors were able to show that only of GCs are expected to host an IMBH. Hence, the presence of many IMBHs surrounding an SMBH is unlikely while having just a few them, perhaps one or two, seems more probable.
In this paper, we use direct -body modeling of a heavy galactic nucleus hosting an SMBH with mass , orbited by 42 massive GCs, to investigate the fate of their stellar content.
We focus on two possible scenarios:
the GC population formed in an inner portion of the galaxy. In this case, the GCs - SMBH gravitational interactions occur when GCs are in an early phase of their life;
the GC population formed outside the galaxy bulge, and slowly spiralled to the inner regions over a time in the interval Gyr. In this case, all the processes related to GC internal evolution may have already taken place, possibly leading to the formation of an IMBH.
It is important noting that the two above scenarios are not mutually exclusive, and are not at odds with our current knowledge of NSC formation. If a population of GCs formed around the galactic centre several Gyr ago, they would have spiralled to the galactic centre (provided they were massive enough), populating it with stars that, as of today, are old. If, on another hand, GCs formed outside the galaxy core, again their slow decay would lead to the formation of an NSC populated by old stars.
Currently, it is hard to determine whether an NSC formed from GCs coming far out from the galactic centre, on a long time-scale, or formed on a much shorter time-scale from GCs born well inside the galactic inner bulge. Likely, in the dry merger scenario, NSC formation arises from a combination of both cases: in a first phase the clusters born closer to the galactic center rapidly segregate and merge, leading to the formation of an “NSC seed”, and, in a second stage, further clusters infall and merge into this growing up NSC, thanks also to the enhanced effect of dynamical friction. Hopefully, observations at high-resolution of the gamma and X-ray flux coming from the MW NSC and their interpretation will shed light on which is the dominant phase in the NSC build up. (Hooper & Goodenough, 2011; Perez et al., 2015; Brandt & Kocsis, 2015; Arca-Sedda et al., 2017; Fermi-LAT Collaboration, 2017).
In this framework, this paper investigates the consequences of strong scattering interactions between infalling GCs and a central SMBH in a massive elliptical galaxy.
We focus the attention on three different processes. In first place, we followed the evolution of BHBs during and promptly after the GCs flyby over the SMBH, aiming at understanding whether such interaction can facilitate its merging. The second process considered is related to the possibility that the deposit of stellar debris around the SMBH lead some of these stars to tightly bind to the SMBH and undergo a slow spiralling due to GW emission. Due to very large mass ratio, , the binary formed would likely give rise to what is referred to as an extreme mass-ratio inspiral (EMRI, Hils & Bender (1995); Merritt et al. (2011)). Smaller mass ratios are referred to as intermediate mass-ratio inspirals (IMRIs) (Amaro-Seoane et al., 2007). EMRIs and IMRIs are the most promising sources of GWs to be detected with the next generation of space-based GW observatories, Laser Interferometer Space Antenna (LISA Mapelli et al., 2012; Amaro-Seoane et al., 2013; Vitale, 2014) and the Chinese space interferometer “TianQin” (Luo et al., 2016). Thus, characterising their occurrence in galactic nuclei can represent a key to correctly interpret the future observations.
In the case in which the GCs formed outside the galaxy core and reached the inner region on a longer time-scale, we investigated the evolution of triple interactions between 2 IMBHs, dragged by the most massive and dense GCs, and the central SMBH. Our results show that the probability that the IMBH falls onto the SMBH within a few hundred Myr is not negligible, partially explaining the dearth of IMBHs observational and kinematic signatures in galactic nuclei.
The paper is organised as follows: in Section 2 we discuss the galaxy model and the GCs orbital and structural properties, although a more detailed description of the model has been already given in the companion paper Arca-Sedda & Capuzzo-Dolcetta (2017b); in Section 3 we discuss the results of the numerical simulations, with particular focus on the impact of GC-SMBH interaction in the production of coalescing BHBs (Sect. 3.1), as well as on its role in favouring or preventing EMRIs (Sect. 3.3) and IMBHs interactions (Sect. 3.4); finally, Sect. 4 is devoted to the conclusions.
2 Model and numerical strategy
2.1 The galaxy model and its globular cluster system
Our analysis is based on the numerical simulation discussed in Arca-Sedda & Capuzzo-Dolcetta (2017b), who modelled the evolution of 42 GCs orbiting around an SMBH with mass hosted in the central region of a massive elliptical galaxy.
The galaxy model is represented by a truncated Dehnen’s density profile (Dehnen, 1993):
where is the total galaxy mass, kpc its scale radius, , and is a truncation radius. The choice pc allows us to represent both the galaxy nucleus and each of the 42 GCs by a suitable sample of particles. As discussed in Arca-Sedda & Capuzzo-Dolcetta (2017b), this choice of parameters permitted us to provide a reliable galaxy environment for studying the GC orbital evolution.
All the GCs are represented according to single mass King (1962) profiles, and their initial internal and orbital properties are summarized in Tab. 1. Further details can be found in the above cited Arca-Sedda & Capuzzo-Dolcetta (2017b) paper.
Properties of the GCs sample
Column 1: GCs name. Column 2: value of the adimensional potential well. Column 3: GC tidal radius. Column 4: GC core radius. Columns 5-7: GC mass, initial position and velocity. Column 8: GC orbital eccentricity. Column 9: dynamical friction timescale according to Arca-Sedda & Capuzzo-Dolcetta (2014b) and Arca-Sedda et al. (2015). Column 10: GCs mass percentage deposited within the inner 10 pc around the SMBH. Column 11: number of particles used to model the GCs.
2.2 The numerical approach
To model the evolution of the GCs around the galactic centre and the SMBH, we used the HiGPUs code (Capuzzo-Dolcetta et al., 2013), a direct summation, 6 order Hermite integrator with block time-steps, which runs efficiently on composite GPU+CPU platforms. The galaxy nucleus and GCs have been modelled with a total number of particles , a number sufficiently high to ensure a correct representation of the GCs “internal” and orbital evolution. Each “star” in our GC models has a mass , a value which represents almost the state-of-art in the field of direct -body galaxy modelling. However, this mass value is still much larger than the average mass expected in a real GC, which is according to the standard (Kroupa, 2001) initial mass function (IMF). Moreover, HiGPUs does not allow treating the evolution of tight systems, such as binaries and triples, thus we could not model the complex subsystems that are expected to form during the GC evolution.
Due to this, we decided to use the detailed information provided by the MEGAN simulation to follow the evolution of their BHBs with ARGdf. Based on the ARCHAIN code, ARGdf is a few-body integrator which implements the algorithmic regularization scheme, a treatment adapted to model strong gravitational encounters (Mikkola & Tanikawa, 1999) including post-Newtonian terms up to the 2.5 order (Mikkola & Merritt, 2008). Our modified version takes into account also the gravitational field of the galaxy, treated as an external static potential field, and dynamical friction, which was considered in its classic local approximation (Chandrasekhar, 1943).
The strong interactions occurring during the GCs fly-by over the SMBH can drive several astrophysical phenomena. In this paper, we focus the attention on three interesting possible outcomes:
the possible hardening of BHBs residing within the GC core during and after the scattering;
the possible enhancement of EMRIs by GCs flyby;
the evolution of 2 IMBHs borrowed by the 2 most massive GCs in our sample and moving in the SMBH vicinity, comparing this possibility with the effect of a larger number of GCs carrying an IMBH in their core.
3.1 SMBH - stellar BHBs interactions during the GC fly-by: BHBs break-up and mergers
In order to model the evolution of a BHB after the flyby of its host cluster with the SMBH, we used the detailed orbital parameters of all the GCs modelled through the MEGAN simulation (Arca-Sedda & Capuzzo-Dolcetta, 2017b). We decided to use one of the most massive GCs in our sample (GC4, see Tab. 1) as “host” cluster. GC4 has an initial mass of and a high initial orbital eccentricity, . Using the GC4 initial conditions, we use ARGdf to integrate the motion of a heavy binary in the GC as this moves toward the SMBH.
We assume that the BHB is composed of two BHs with masses and , similar to those of the first GW source observed by LIGO, named GW150914.
We run 1000 simulations at varying BHB position, initial semi-major axis and eccentricity, in order to filter out statistical fluctuations in our results.
The BHB initial separation was selected by mean of sampling the semi-major axis probability distribution (Rodriguez et al., 2015), assuming as maximum value pc, and as minimum value the sum of the two BHs Schwarzschild radii. To select BHB eccentricities we assumed a thermal distribution (Jeans, 1919).
In our few-body integrator, the GC is considered as a point-like object. We selected the initial BHB position in the range times the GC core radius, which, for GC4, is pc. This partly alleviates the strong approximation to point-like objects, that remains valid as long as the particles move around the GC core.
By mean of ARGdf we then simulate the evolution of the BHB around its parent GC as a four-body problem, the fourth body being the SMBH, with the whole system embedded in the external potential of the galaxy. The reason for this crude approximation is the otherwise overwhelming computational complexity of modelling the binary in a massive star cluster which is scattering over an SMBH with a mass 100 times larger. The approximation of the GC as a point-like particle could lead to spurious effects related to its interaction with the BHB, like resonances that can make unreliable the BHB evolution. To limit this effect, we restricted our simulations to one close GC-SMBH passage, a process that lasts only a few Myr, corresponding to many orbits for the BHB but only a few orbits for the BHB centre of mass inside the GC host.
After the scattering, the BHB can either i) breaks up, ii) be ejected from the parent cluster, or iii) be ejected from the galaxy. In cases i) and ii) the BHB will move in an environment characterized by a density lower than in its parent cluster. In this conditions, the BHB orbital properties are almost frozen, and GWs emission becomes the dominant process determining the BHB late evolution. In this regime, it is possible to calculate the coalescence time-scale (Peters, 1964)
where is the speed of light, is the binary semi-major axis, is the eccentricity, and is the gravitational constant. Substituting our BHB masses, Eq. 2 can be written in the more suitable form
To highlight the effect of this four-body interaction, BHB+GC+SMBH, we show in Fig. 1 for all our models the correlation between the ratio of the final and initial value of the semi-major axis, and of the eccentricity, .
In of the cases, after the GC-SMBH interactions decreases and increases at the same time, making the merging time to decrease. Such a decrease leads to a GW coalescence time Gyr only in the of the cases. We note here that represents a lower limit to the probability that a BHB coalesces within a Hubble time, since we integrated only one single GC flyby. Figure 2 shows the aftervelocity the GC-SMBH interaction.
In of our models the BHB breaks up and at least one of the two components is ejected away from the galaxy, reaching velocities in the range km s at distances above 1 kpc. We identified between the wandering BHs the real escapers from the galaxy as those having a positive energy and a velocity larger than the local escape velocity, calculated in their current position. The velocity distribution of BHs components after BHB unbound is shown in Figure 3 Following the same procedure, we identified 11 BHBs, still bounded after the flyby, moving away from the galaxy with velocities km s at pc from the central SMBH, above the local escape velocity. 7 out of the 11 escaping BHBs undergo a significant decrease in the semi-major axis and an increase in eccentricity, leading to yr.
Using these results, we can give an estimate of the expected number of BHB merger events produced in GCs-SMBH close encounters. A possible way to define such a number is:
where represents the fraction of BHB within the cluster core, the fraction of stars with mass above a given value , represents the fraction of orbits that allow the production of coalescing BHB, is the probability for the BHB to coalesce, is the total number of stars in the cluster and, finally, represents the number of GCs that have a strong interaction with the SMBH.
The most uncertain parameter is , as it is still poorly known how the population of BHBs forms and evolves in dense star clusters.
However, in order to provide a reliable estimate of we should consider both the GC infall, which drags it toward the SMBH, and the evolution of its internal structure, which drives the formation of BHBs. The relevant time-scale for GC infall is the dynamical friction time-scale, which can be written as (Arca-Sedda & Capuzzo-Dolcetta, 2014b; Arca-Sedda et al., 2015)
Since primordial binaries seem to play a minor role in determining the total number of BHBs, as they are easily disrupted on relatively small time-scales (Downing et al., 2010), we can connect BHBs formation time-scale to the BHs mass segregation, namely their own dynamical friction time-scale, which, also, can be estimated (Arca-Sedda, 2016) with Eq. 5.
The formation of the first BHB will occur then over this typical time, although its precise value depends on specific orbit and GC characteristics (Gaburov et al., 2008; Downing et al., 2010; Askar et al., 2017).
We show in Fig. 4 the comparison between and at varying BH orbital properties and GC mass and orbital pericentre. If does not exceed severely , then the formation of BHBs within the host cluster is a process still working when it impacts the SMBH. This, in turn, implies the possibility that a consistent population of BHBs is present in the GC core at the moment of its close encounter with the central SMBH.
Another critical parameter is . In our study, we focus on an ”impulsive” perturbation induced by the central SMBH over the BHB, as it is maximized during the first GC flyby. This forces us to exclude nearly circular orbits, since for them the effect of the SMBH on the GC and its components acts on secular time-scale. As shown by Arca-Sedda et al. (2016), the GC evolution is strongly affected by the central SMBH when its orbital eccentricity exceeds , leading to the ejection of high- and hyper velocity stars (HVSs). Although HVSs cannot be directly associated with the BHB evolution, we set as threshold value needed to impact the BHB orbit. Therefore, assuming a thermal distribution for GC eccentricities leads to .
We set , as it came out from our previously described simulations. According to these parameters, we found as a number of merging events per galactic nucleus varying in the range .
Figure 5 shows how varies for different values of , calculated for stars with masses above , and . The value has been assumed. Note that is the minimum mass in the main sequence stage which allows BH formation (Hurley et al., 2000). Note also that in the interval for progenitors masses, the BH masses range between and , depending on the cluster metallicity and the stellar evolution tracks adopted (Hurley et al., 2000; Belczynski et al., 2010; Spera et al., 2015).
The first GW source detected by LIGO, named GW150914, was composed of two BHs with masses and , whose progenitors likely had masses in the case of low metallicity, , according to the most recent stellar evolution recipes (Spera et al., 2015), or much higher masses, up to for nearly solar metallicities (Spera et al., 2015; Belczynski et al., 2010; Banerjee, 2016). Limiting the analysis to stars with initial masses above , which actually may lead to BHs with masses above , we show in Fig. 6 the variation of as a function of and .
In our sample, 30 clusters have eccentricities such to have a close flyby over the SMBH; assuming , this would imply events per galaxy.
As suggested recently by Askar et al. (2017), the number of mergers involving BHBs with masses above is roughly given by
with and for low metallicities, and and for solar values of . Then, for a GC, , a value comparable to our estimate for this channel, although other works pointed out higher values for star clusters, up to a factor 2 (Rodriguez et al., 2016).
3.2 The BHB merger rate
In order to estimate the BHB merger rate for the “GC-SMBH collision channel”, we must calculate:
the rate at which GCs form in a given galaxy;
the number of GCs that can undergo a strong interaction with the central galactic SMBH;
the number of galaxies of a given mass in a given cosmological volume.
In this calculation, the most uncertain factor is related to the GCs formation history, a topic still poorly understood. In the following, we will assume that GC formation process is similar to the galaxy ordinary star formation rate (Antonini et al., 2015).
Following Sesana et al. (2014), the rate at which stars form in a galactic bulge is given by
where is the fraction of the available gas surface density that is converted in stars, is the galaxy volumetric density evaluated at distance to the galaxy centre, and is the typical time-scale over which giant molecular clouds fragment into stars. To calculate we followed Sesana et al. (2014) and Krumholz et al. (2009).
The star formation time-scale is roughly given by (Sesana et al., 2014)
with , the galaxy density profile, and pc. In our model , thus implying and (Antonini et al., 2015).
Using as a proxy for the actual star formation rate, we can estimate the GC formation rate in the galaxy bulge as (Antonini et al., 2015):
being the efficiency factor for star cluster formation (Kruijssen, 2012).
Assuming a Dehnen model for the galactic bulge, the total star cluster formation rate is thus given by
The number of BHB mergers produced by SMBH-GC scattering depends on the total number of stars in the GC, as shown in Equation 4. In order to provide general results, we calculate this number as
being the mean mass of GCs with masses above M, and the mean stellar mass. In order to calculate we assumed for the GCs mass function (Baumgardt, 1998; de Grijs et al., 2003; Arca-Sedda & Capuzzo-Dolcetta, 2014a).
Under the assumption that star cluster formation occurs continuously during the galaxy lifetime, (Antonini et al., 2015), the total number of GC with mass above formed within a given galactic radius is thus given by
where the term enclosed between brackets represents the total number of GCs born during the galaxy life, is the fraction of GCs with mass above a given value . The time over which SMBH-GCs strong interactions take place is of the order of the dynamical friction time-scale calculated at the galactic radius of interest, . In our calculations, we use as given by Eq. 5.
In order to estimate the BHB merger rate for massive BHBs similar to the GW150914 and the GW170104 events observed by LIGO, we set the BHB total mass to .
Assuming , a lower limit to is obtained by the assumption that the GC hosts at least one BHB with mass , that means it hosts at least two progenitors with an initial mass . Therefore, the minimum GC mass to host at least one massive stellar BHB is given by
To compare the occurrence of these events with the expected rate for BHB merging calculated on the basis of LIGO detection (Abbott et al., 2016c), we should calculate the number of galaxies in which at least one strong SMBH-GCs scattering can take place.
According to Abbott et al. (2016c), the aLIGO experiment can detect BHB merging out to redshift . Therefore, we calculated the galaxy number density at redshift for galaxies with masses above following Conselice et al. (2016), who provided a parametrized number density function depending on the redshift.
Combining all the equations above, we show in Fig. 7 the total rate calibrated on BHBs with total mass per unit time and volume, at varying and the value of the GC mass above which the GC-SMBH interaction can boost the BHB merger. We stress here that our calculations focus on galaxies with masses .
For typical values of the GC mass , we found a merger rate of
This value is consistent with the expected massive BHB merger rate estimated in GCs (Rodriguez et al., 2016; Askar et al., 2017) and in NSCs (Antonini & Rasio, 2016; Hoang et al., 2017), and to the expected rate for GW150914 estimated in Abbott et al. (2016d), thus placing the formation channel proposed here among the other for massive stellar BHs coalescence.
3.3 Extreme mass ratio inspirals
One consequence of GC-SMBH strong interaction is the possibility that stars fall directly onto the SMBH, the so-called direct plunge, or bind to the SMBH, forming binaries with a very high mass ratio, and, later, slowly inspirals toward it by GW energy loss, forming an extreme mass ratio inspirals (EMRI) (Hils & Bender, 1995; Merritt et al., 2011).
According to Amaro-Seoane et al. (2007), the formation of an EMRI requires that the time-scale over which the star orbit changes due to GW emission is shorter than the 2-body relaxation time-scale, which regulates the time interval needed to substantially modify the orbital parameters.
This condition can be written as
where regulates the validity of the inequality. Manipulating the two sides of the inequality allows finding the limiting eccentricity required for an EMRI to form
Given the eccentricity lower limit for directly plunging orbits, where is the SMBH Schwarzschild radius and is the EMRI semi-major axis, clearly an EMRI can develop only if .
This yields to definition of a critical semi-major axis as such below which the star can be captured by the SMBH as an EMRI, given by
so that heavier stars are most likely captured as EMRI. Another important parameter is the star Roche radius. Indeed, stars having a pericentre smaller than their Roche radius are tidally disrupted and cannot give origin to an EMRI.
As discussed in the companion paper (Arca-Sedda & Capuzzo-Dolcetta, 2017b), the stellar pericentre distribution is well described by a simple relation, , thus allowing calculating in an easy way the number of stars having a pericentre, , such to give rise to an EMRI.
being the galactic velocity dispersion, the mean stellar mass, the galactic mass density distribution and the usual Couloumb logarithm.
The number of stars with such an orbit to form an EMRI is then given by:
so that the corresponding EMRIs rate can be calculated as
We found that does not depend significantly on the choice of the pericentral distribution, or , because above pc the two functions are similar.
At varying stellar mass between and , we found
with the maximum values corresponding to .
The GC infall and disruption processes, operated by dynamical friction and tidal forces, lead to a significant increase of the overall density profile. As shown in Arca-Sedda & Capuzzo-Dolcetta (2017b), the inner density increases from to pc after the GCs evolution.
At this regard, we show in Fig. 8 how changes due to the GC debris deposited around the SMBH. The plot highlights the efficiency of GCs disruption in determining an enhancement (factor 10 increase) of GWs emission by EMRIs in massive galactic nuclei.
3.4 Multiple IMBHs-SMBH scattering: formation of IMRIs
Another intriguing possibility is that the infalling GC transports an IMBH into its core. This would have an implication on the possible formation of an IMRI driven by the interaction of the central SMBH with the IMBH carried there by infalling GCs.
The IMBH orbital infall toward the galactic centre can be schematized in two phases:
the IMBH is transported within the GC, whose orbital pericentre, , reduces substantially over a time-scale ;
the GC is disrupted by tidal forces after a few passages at pericentre, leaving the IMBH freely wandering within the galactic nucleus;
the interaction between the field stars and the IMBH drags it toward the SMBH due to dynamical friction, acting in this case on a time-scale , being the IMBH distance to the SMBH.
If GCs mergers drive NSC formation and GCs drag an IMBH in their core, one would expect that the interaction between IMBHs and SMBH must start shortly after the NSC formation. The absence of observational signatures of IMBHs around an SMBH implies either that the formation of an IMBH has a very small probability, or that they have already merged with the central SMBH, i.e. in a time shorter than the time-scale of NSC formation.
Giersz et al. (2015) developed 2000 Monte Carlo models of GCs, showing that the formation of an IMBH is an extremely stochastic process, strongly dependent on the GC initial mass and properties. In their sample, an IMBH develops in of the cases. In our sample, nearly 10 clusters out of 2000 have eccentricities above and a mass above , which translates into only 2 IMBHs expected. Therefore, we followed the evolution of 2 IMBHs dragged toward the SMBH by GC infall, in order to understand whether their evolution can lead to SMBH-IMBH merger within a Hubble time.
The IMBH masses were assigned according to the scaling law proposed by Arca-Sedda (2016), which links the GC mass and the central mass accumulated into the GC centre:
with the coefficients and depending on the IMF and the metallicity of the host GC. For a low-metal cluster characterised by a Kroupa IMF, Arca-Sedda (2016) found and . Note that this estimate is in good agreement with the general results proposed by Portegies Zwart & McMillan (2007). According to Eq. 24, the corresponding IMBH mass is and , respectively.
We simulated the evolution of the triple system composed by the SMBH and two IMBHs, using the ARGdf code, taking into account the external background and the Chandrasekhar dynamical friction coefficient.
We assumed that the heaviest IMBH segregates faster than the other and binds to the SMBH, forming a binary system moving in a circular orbit with separation pc.
This latter assumption is justified by that, in a nearly spherical configuration, the inner binary initially shrinks due to the interaction with field stars, circularize and eventually stalls due to the inefficient replenishment of the so called loss-cone. The coalescence time-scale in such configuration is yr.
Regarding the outer IMBH, we varied both its initial eccentricity between 0.5 and 1.0, assuming that it is inherited from the parent GC orbit, and the relative inclination between the inner and outer binary planes between 0 and 90 degrees.
In the end, we gathered a total number of 100 simulations, characterised by an inner binary with total mass and mass ratio , and an outer binary with semi-major axis in the range pc and eccentricity .
An interesting outcome of these models is that the arrival of a second IMBH may lead to an exchange of the binary components, with the primary IMBH shifted to a larger orbit.
The three BHs form rapidly a triple system, whose evolution can be easily followed by defining an inner and an outer binary. At the beginning of our modelling, the inner binary is comprised of the SMBH and the heavier IMBH, which arrives in the galactic centre before the other. The outer binary, instead, is represented by the inner binary centre of mass and the third infalling IMBH.
In our models, we found that in of the cases the binary exchanges its components and coalesces within a Hubble time. In almost all these cases the secondary IMBH substitutes the primary, leading to the formation of a new binary having a significantly smaller separation.
Fig. 9 shows the evolution of the semi-major axis of the inner and outer binary in some of the models investigated. In all the 3 panels a sharp transition is evident, which marks the moment in which the component exchange takes place.
Figure 10 shows the time at which the merger occurs as a function of the outer binary initial pericentre. The plot also outlines the initial inclination between the inner and outer binary.
It seems evident a weak dependence of the coalescence time by the outer binary initial pericentre, while it less trivial a possible connection with its inclination.
For the sake of comparison, we also run a few models in which the of the whole GC population hosts an IMBH in its centre, thus implying 7 IMBHs orbiting the central SMBH.
We kept the same inner binary as above, composed of an IMBH with mass moving on a circular orbit around the central SMBH, while the other IMBHs are distributed randomly in the space, due to the fact that we cannot predict their exact position and velocity shortly after the host GC disruption. In order to focus on the IMBHs-SMBH interactions, we set 50 times the inner binary separation as the maximum distance from the central SMBH allowed.
We gathered 474 models, divided into 4 different groups, which differ each other in two important features: i) dynamical friction and ii) central SMBH spin. Regarding the first point, we ran a half of the models taking into account dynamical friction, while in the remaining models we only considered the mutual gravitational interactions among the IMBHs and the galactic external background. Regarding the second point, instead, in a half of the models we assigned to the central SMBH a spin , oriented along the y-axis. Such value is compatible with spins of SMBHs with mass above , as inferred by observations of AGNs (Reynolds, 2013). In the following, we refer to the 4 groups as DFnSn, DFySn, DFnSy, DFySy.
For each model in each group, we calculated the number of merger events, the mass of the merged IMBH and the time at which the event occurs. Our models, carried out up to 13 Gyr, suggest that at most three IMBH can merge into the SMBH, while the other remain orbiting in its surrounding or are ejected away from the galaxy with velocities up to 1000 km s. Figure 11 shows the IMBHs half-mass radius as a function of the time in four of the models investigated taking into account alternatively the SMBH spin and the dynamical friction coefficient as explained above. We stress here that for each model, the same IC set has been modelled in the four different configurations (dynamical friction on/off, zero/non-zero SMBH spin). It is evident the crucial role of initial conditions and these physical features in shaping the evolution of such a complex system, although it is composed only of 8 particles. In some cases, dynamical friction tends to keep the IMBH population more concentrated, while the SMBH spin, when the dynamical friction is “turned on”, determines a stronger contraction of the IMBH system, leading to a half-mass radius smaller than 0.05 pc. In other models, however, dynamical friction causes stronger interactions between the IMBHs which, in turn, causes a rapid ejection of the lighter components, while this does not occur when it is not taken into account, due to the longer time-scales over which the 8-body system evolves.
Table 2 outlines the number of mergers occurring between the IMBHs and the SMBH, specifying the fraction of cases in which the event involves the initial inner binary. Note that in several models the SMBH swallow two or even three IMBHs, leading the total number of mergers to exceeds the total number of runs. Despite a detailed investigation of the parameter space and the role played by different physical processes is beside the scope of this work, we note here that including dynamical friction and a Kerr SMBH seems to slightly decrease the merger probability, while is less trivial the effect that they have on the evolution of the inner binary. This is likely due to the fact that when dynamical friction is turned on, a larger number of IMBHs interact together, causing complex multi-body interactions that may lead to the ejection of one or more components before it undergoes a merger.
Column 1: group ID. Column 2: number of simulated models. Column 3: total number of merger events. Column 4: fractional number of mergers normalized to the total number of simulations in each group. Column 5-6: number of inner binary coalescence and their percentage over the total number of merger events.
Figure 12 shows, for each group, the mass distribution of the merged IMBHs. The distribution peaks in correspondence of the largest bin value, where the mass of the IMBH in the inner binary lies, due to the fact that in all the groups nearly of the mergers involve the inner binary.
The time at which a merger occurs has a weak dependence on the model group considered, as shown in Figure 13. Comparing groups DFnSn and DFySnit seems quite evident that a merger occurs earlier when dynamical friction is taken into account. When a spinning SMBH is considered (models DFnSn and DFnSy) the merger occurs earlier. Indeed, while in absence of dynamical friction and with a Schwarzschild SMBH the time distribution peaks at Gyr, this slightly decreases down to Gyr when a Kerr SMBH is considered and the dynamical friction is calculated accordingly to the Chandrasekhar theory.
In this paper we used the data produced by the direct -body simulation called MEGaN (Arca-Sedda & Capuzzo-Dolcetta, 2017b), representing the evolution of 42 GCs and an SMBH in a massive elliptical nucleus, to investigate the possible implication of GC-SMBH interactions on the production of highly eccentric BHBs and on the possible formation of a tight massive binary (MBHB) composed of an IMBH and the SMBH, and its interaction with another IMBH transported by an infalling GC. This investigation has been performed using a suited numerical code, called ARGdf, that implements the algorithmic regularization, a treatment that allows avoiding the divergence in the evaluation of the gravitational force when the mutual distance approaches zero.
Our main results can be summarised as follows:
if the infalling GC hosts a population of stellar BHBs in its core during the flyby over the SMBH, we found that there is a tiny possibility that some BHBs are ejected from the cluster, and in some cases from the galaxy, after increase the orbital eccentricity and reducing separation to undergo coalescence within a Hubble time;
for a massive galactic nucleus, this process leads to the production of BHB merger events per galaxy, in dependence on the fraction of BHs that bind in binary;
we used our results to estimate the merger rate for BHBs in galaxies heavier than that coalesce through this channel. Considering the cosmological volume enclosed within , which is currently the limit achievable with the aLIGO experiment, we found a rate yr Gpc, within the range of the observed aLIGO merger rate;
the enhancement in the stellar density around the SMBH caused by the GC disruption leads to a significant increase of the EMRIs rate, which increases by an order of magnitude after GCs flybys, reaching values yr. This outcome is interesting in the view of the next generation of space based GWs detectors;
we investigated the possibility that two of the most massive clusters host in their centre an IMBH with mass . Assuming that the heavier IMBH is already bound to the SMBH when the second IMBH arrives, we found that the triple (IMBHs+SMBH) system undergoes very often a swap between the IMBHs, and lead to coalescence in of the cases.
we modelled the evolution of 7 IMBHs around the SMBH, following the theoretical findings that IMBH formation occurrence in GCs can be as high as 20. We grouped 474 simulations in four groups, which differ in the spin of the central SMBH and the presence or not of dynamical friction in the IMBHs equation of motion. We found that in most of the cases 1, 2 or 3 IMBHs are swallowed by the SMBH within 12 Gyr. In a large fraction of cases, the merge occurs with the closest IMBH, which is the most massive in the IMBH sample. As expected, dynamical friction drives the merger events at earlier times. The merger time distribution peaks at around Gyr, with a few cases occurring in between 1 and 10 Gyr.
MAS aknowledges Sapienza, University of Rome, which funded the research program “MEGaN: modelling the evolution of galactic nuclei” via the grant 52/2015, and the Sonderforschungsbereich SFB 881 ”The Milky Way System” of the German Research Foundation (DFG) for the financial support provided. MAS also thanks the Zentrum fur Astronomie - Astronomisches Rechen Institut of Heidelberg for the hospitality during the development of part of this research. The authors acknowledge Abraham Loeb, Daniel D’Orazio and Monica Colpi, who provided useful comments and suggestions that helped in improving the manuscript.
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