Relativistic dynamics and extreme mass ratio inspirals
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Abstract
It is now wellestablished that a dark, compact object, very likely a massive black hole (MBH) of around four million solar masses is lurking at the centre of the Milky Way. While a consensus is emerging about the origin and growth of supermassive black holes (with masses larger than a billion solar masses), MBHs with smaller masses, such as the one in our galactic centre, remain understudied and enigmatic. The key to understanding these holes—how some of them grow by orders of magnitude in mass—lies in understanding the dynamics of the stars in the galactic neighbourhood. Stars interact with the central MBH primarily through their gradual inspiral due to the emission of gravitational radiation. Also stars produce gases which will subsequently be accreted by the MBH through collisions and disruptions brought about by the strong central tidal field. Such processes can contribute significantly to the mass of the MBH and progress in understanding them requires theoretical work in preparation for future gravitational radiation millihertz missions and Xray observatories. In particular, a unique probe of these regions is the gravitational radiation that is emitted by some compact stars very close to the black holes and which could be surveyed by a millihertz gravitationalwave interferometer scrutinizing the range of masses fundamental to understanding the origin and growth of supermassive black holes. By extracting the information carried by the gravitational radiation, we can determine the mass and spin of the central MBH with unprecedented precision and we can determine how the holes “eat” stars that happen to be near them.
Keywords
Black holes Gravitational waves Stellar dynamicsGlossary
 \(1\,M_\odot \)
1 solar mass = \(1.99\times 10^{30}\,\mathrm{kg}\)
 \({\mathscr {M}}_{\bullet }\)
Mass of super or massive black hole
 \(1\,\mathrm{pc}\)
1 parsec \( \approx 3.09\times 10^{16}\,\mathrm{m}\)
 1 Myr/Gyr
One million/billion years
 AGN
Active galactic nucleus
 BH
Black hole
 CO
Compact object (a white dwarf or a neutron star), or a stellarmass black hole. In general, a collapsed star with a mass \(\in [1.4,\,10]\,M_{\odot }\) in this work
 DCO
Dark compact object
 DF
Dynamical friction
 EMRI
Extreme mass ratio inspiral
 GC
Galactic centre
 GPU
Graphics processing unit
 GW/GWs
Gravitational wave/s
 HB
Giant stars in the horizontal branch
 HST
Hubble space telescope
 IMBH
Intermediatemass black hole (\(M \in \,[10^2,\,10^5]\,M_\odot \))
 IMF
Initial mass function
 IMRI
Intermediate mass ratio inspiral
 LISA
Laser interferometer space antenna
 LSO
Last stable orbit
 MBH
Massive black hole (\(M \approx 10^6 M_\odot \))
 MC
Monte carlo
 MW
Milky Way
 NB6
Directsummation Nbody6
 NS
Neutron star
 PN
PostNewtonian
 RG
Red giant
 RMS
Root mean square
 SMBH
Super massive black hole (\(M > 10^6 M_\odot \))
 SNR
Signaltonoise ratio
 SPH
Smoothed particle hydronamics
 TDE
Tidal disruption event
 UCD
Ultracompact dwarf galaxy
 z
Redshift
1 Foreword
The volume where capture orbits are produced is so small in comparison to other typical length scales of interest in astrodynamics that it has usually been seen as unimportant and irrelevant to the global dynamical evolution of the system. The only exception has been the tidal disruption of stars by massive black holes. Only when it transpired that the slow, adiabatic inspiral of compact objects onto massive black holes provides us with valuable information, did astrophysicists start to address the question in more detail. Since the problem of EMRIs (extreme mass ratio inspiral) started to draw our attention, there has been a notable progress in answering fundamental questions of stellar dynamics. The discoveries have been numerous and some of them remain puzzling. The field is developing very quickly and we are making important breakthroughs even before a millihertz mission flies.
When I was approached and asked to write this review, I was glad to accept it without realising the dimensions of the task. I was told that it should be similar to a plenary talk for a wide audience. I have a personal problem with instructions like this. I remember that when I was nine years old, our Spanish teacher asked us to summarise a story we had read together in class. I asked her to define “summarise”, because I could easily produce a summary of one, two or fifty pages, depending on what she was actually expecting from us. She was confused and I never got a clear answer. She replied that “A summary is a summary and that’s it”. On this occasion, I am afraid that I have run into the same snag and I have gone for the manypages approach, to be sure that any newcomer will have a good overview of the subject, with relevant references, in a single document. If the document is too long, please address your complains to her, because she is solely responsible.
However, I would like to note that I have not focused on gathering as much information as possible from different sources. I think it is more interesting for the reader, though harder for the writer, to have a consistent document. This can be done by introducing the subject step by step, rather than working out a compendium of citations of the related literature. For instance, I present results that I have not previously published that will, I hope, enlighten the reader. Figures that I prepared myself and are not published elsewhere do not have a reference.
From the point of view of millihertz gravitationalwave (GW) missions, as the reader probably knows, the laser interferometer space antenna (LISA), see AmaroSeoane et al. (2017), is now the official ESA L3 mission, already entering the phase A.
2 Massive dark objects in galactic nuclei
That dark star would hence not be directly observable, but if it is in a binary system, one could use the kinematics of a companion star. He even derived the corresponding radius, which corresponds to exactly the Schwarzschild radius.If the semidiameter of a sphere of the same density as the sun is in the proportion of five hundred to one, and by supposing light to be attracted by the same force in proportion to its mass with other bodies, all light emitted from such a body would be made to return towards it, by its own proper gravity.
A “black hole”^{1} means the observation of phenomena which are associated with matter accretion on to it, for we are not able to directly observe it electromagnetically. Emission of electromagnetic radiation, accretion discs and emerging jets are some, among many, kinds of evidence we have for the existence of such massive dark objects, lurking at the centre of galaxies.
On the other hand, spectroscopic and photometric studies of the stellar and gas dynamics in the inner regions of local spheroidal galaxies and prominent bulges suggest that nearly all galaxies harbour a central massive dark object, with a tight relationship between its mass and the mass or the velocity dispersion of the host galaxy spheroidal component (as we will see below). Nonetheless, even though we do not have any direct evidence that such massive dark objects are black holes, alternative explanations are sorely constrained (see, e.g., Kormendy 2004 and also AmaroSeoane et al. 2010 for an exercise on constraining the properties of scalar fields with the observations in the galactic centre, although the authors conclude that one needs a mixed configuration with a black hole at the centre).
Supermassive black holes are ensconced at the centre of active galaxies. What we understand by active is a galaxy in which we can find an important amount of emitted energy which cannot be attributed to its “normal” components. These active galactic nuclei (AGNs) are powered by a compact region in their centres.
We will embark in the next sections of this review on a study of the dynamics of stellar systems harbouring a central massive object in order to extract the dominant physical processes and their parameter dependences, for instance, dynamical friction and mass segregation, as a precursor to the astrophysics of extreme mass ratio inspirals.
2.1 Active galactic nuclei
In this section, and to motivate the introduction of the concept of massive black holes, I give a succinct introduction to active galactic nuclei, but I refer the reader to the book by Krolik (1999) on this topic.
The expression “active galactic nucleus” of a galaxy (AGN henceforth) is referring to the energetic phenomena occurring at the central regions of galaxies which cannot be explained in terms of stars, dust or interstellar gas. The released energy is emitted across most of the electromagnetic spectrum, UV, Xrays, as infrared, radio waves and gamma rays. Such objects have large luminosities (\(10^4\) times that of a typical galaxy) coming from tiny volumes (\(\ll 1\, \mathrm{pc}^3\)); in the case of a typical Seyfert galaxy the luminosity is about \(\sim 10^{11} ~L_{\odot }\) (where \(L_{\odot } :=3.83 \cdot 10^{33}\) erg/s is the luminosity of the sun), whilst for a typical quasar it is brighter by a factor 100 or even more; actually they can emit as much as some thousand galaxies like our MilkyWay. They are, therefore, the most powerful objects in the universe. There is a connection between young galaxies and the creation of active nuclei, because the luminosity can strongly vary with the redshift.
In anticipation of something that I will elaborate on later, nowadays one explains the generation of energy as a product of matter accreting on to a supermassive black hole in the range of mass \({\mathscr {M}}_{\bullet }\sim 10^{\,610}\,M_{\odot }\) (where \({\mathscr {M}}_{\bullet }\) is the black hole mass). In this process, angular momentum flattens the structure of the infalling material to a socalled accretion disc.
For some alternative and interesting schemes to that of MBHs, see Ginzburg and Ozernoy (1964) for spinars, Arons et al. (1975) for clusters of stellar mass BHs or neutron stars, and Terlevich (1989) for warmers: massive stars with strong massloss spend a significant amount of their Heburning phase to the left of the ZAMS on the HR diagram. The ionisation spectrum of a young cluster of massive stars will be strongly influenced by extremely hot and luminous stars.
It is frequent to observe jets, which may arise from the accretion disc, although we do not dispose of direct observations that corroborate this. Accretion is a very efficient channel for turning matter into energy. Whilst nuclear fusion reaches only a few percent, accretion can transfer almost 50% of the massenergy of a star into energy.
Being a bit more punctilious, we should say that hallmark for AGNs is the frequency range of their electromagnetic emission, observed from \(\lesssim 100\) MHz (as low frequency radio sources) to \(\gtrsim 100\) MeV (which corresponds to \(\sim 2 \cdot 10^{22}\) Hz gamma ray sources). Giant jets give the upper size of manifest activity \(\lesssim 6~\mathrm{Mpc} \sim 2 \cdot 10^{25}\) cm,^{2} and the lower limit is given by the distance covered by light in the shortest Xray variability times, which is \(\sim 2\cdot 10^{12}\) cm.
With regard to the size, we can envisage this as a radial distance from the very centre of the AGN where, ostensibly, a supermassive black hole (SMBH) is harboured along with the different observed features of the nucleus. From the centre outwards, we have first a UV ionising source amidst the optical continuum region. This, in turn, is enclosed by the emission line clouds and the compact radio sources and these between another emitting region.
The radiated power at a certain frequency per dex^{3} frequency ranges from \(\sim 10^{39}\) erg/s (radio power of the MW) to \(\sim 10^{48}\) erg/s, the emitted UV power of the most powerful, highredshifted quasars. Such broad frequency and radius ranges for emission causes us to duly note that they are far out of thermal equilibrium. This manifests in two ways: first, smaller regions are hotter; second, components of utterly different temperature can exist together, even though components differ by one or two orders of magnitude in size.
2.2 Massive black holes and their possible progenitors
The quest for the source of the luminosities of \(L \approx 10^{12}\, \mathrm{L}_{\odot }\) produced on such small scales, jets and other properties of quasars and other types of active galactic nuclei led in the 1960s and 1970s to thorough research that pointed to the inkling of “supermassive central objects” or “dark compact objects” (DCO) harboured at their centres.
These objects were suggested to be the main source of such characteristics LyndenBell (1967), LyndenBell and Rees (1971), Hills (1975). LyndenBell (1969) showed that the release of gravitational binding energy by stellar accretion on to a MBH could be the primary powerhouse of an AGN LyndenBell (1969). Following the same argument, 13 years later Sołtan related the quasars luminosity to the accretion rate of mass on to MBHs, so that if we use the number of observed quasars at different redshifts, we can obtain an integrated energy density Sołtan (1982). This argument strengthened the thought that MBHs are found at the centre of galaxies and acted in the past as the engines that powered ultraluminous quasars.
In the last decade, observational evidence has been accumulating that strongly suggests that MBHs are indeed present at the centre of most galaxies with a significant spheroidal component. Mostly thanks to the Hubble Space Telescope (HST), the kinematics of gas or stars in the presentday universe has been measured in the central parts of tens of nearby galaxies. In almost all cases,^{4} proper modelling of the measured motions requires the presence of a central compact dark object with a mass of a few \(10^{6}\) to \(10^{9}\,M_{\odot }\), see Ferrarese et al. (2001), Gebhardt et al. (2002), Pinkney et al. (2003), Kormendy (2004), Genzel et al. (2010) and references therein. Note, however, that the conclusion that such an object is indeed a MBH rather than a cluster of smaller dark objects (like neutron stars, brown dwarfs etc) has only been reached for a two galaxies. The first one is the Milky Way itself at the centre of which the case for a 3–\(4\times 10^{6}\,M_{\odot }\) MBH has been clinched, mostly through groundbased IR observations of the fast orbital motions of a few stars (Ghez et al. 2005; Schödel et al. 2003 and see Genzel et al. 2010 for a review). The second case is NGC4258, which possesses a central Keplerian gaseous disc with \(\mathrm {H_2O}\) MASER strong sources allowing high resolution VLBI observations down to 0.16 pc of the centre Miyoshi et al. (1995), Herrnstein et al. (1999), Moran et al. (1999).
It is, hence, largely accepted that the central dark object required to explain kinematics data in local active and nonactive galaxies should be a MBH. The large number of galaxies surveyed has allowed us to study the demographics of the MBHs and, in particular, to look for correlations with properties of the host galaxy. Indeed, a deep link exists between the central MBH and its host galaxy Kormendy and Ho (2013), illuminated by the discovery of correlations between the mass of the MBH, \(M_{\bullet }\), and global properties of the surrounding stellar system, e.g., the velocity dispersion \(\sigma \) of the spheroid of the galaxy, known as the \(M\sigma \) relation. In spite of some progress in recent decades, many fundamental questions remain open. There is still no clear evidence of MBH feedback in galaxies, and the low mass end of the \(M\sigma \) relation is very uncertain. These facts certainly strike a close link between the formation of the galaxy and the massive object harboured at its centre.
It is also important to note that claims of detection of “intermediatemass” black holes (IMBHs) at the centre of globular clusters raise the possibility that these correlations could extend to much smaller systems, see e.g., Gebhardt et al. (2002), Gerssen et al. (2002). The origin of these (I)MBH is still shrouded in mystery, and many aspects of their interplay with the surrounding stellar cluster remain to be elucidated.
2.3 Tidal disruptions
The centremost part of a galaxy, its nucleus consists of a cluster of a few \(10^7\) to a few \(10^8\) stars surrounding the DCO, assumed from now onward to be a MBH, with a size of a few pc. The nucleus is naturally expected to play a major role in the interaction between the DCO and the host galaxy, as we mentioned before. In the nucleus, stellar densities in excess of \(10^6\,\mathrm{pc}^{3}\) and relative velocities of order a few 100 to a few \(1000\,\mathrm{km\,s}^{1}\) are reached. In these exceptional conditions, unlike anywhere else in the bulk of the galaxy, collisional effects come into play. These include 2body relaxation, i.e., mutual gravitational deflections, and genuine contact collisions between stars.
These processes may contribute significantly to the mass of the MBH, see e.g., Murphy et al. (1991), Freitag and Benz (2002). Tidal disruptions trigger phases of bright accretion that may reveal the presence of a MBH in an otherwise quiescent, possibly very distant, galaxy (Hills 1975; Gezari et al. 2003).
2.4 Extreme mass ratio inspirals
On the other hand, stars can be swallowed whole if they are kicked directly through the horizon of the MBH (the socalled direct plunges) or gradually inspiral due to the emission of GWs The latter process, known as an “extreme mass ratio inspiral” (EMRI) is one of the main objects of interest for LISA, see AmaroSeoane et al. (2017), eLISA Consortium et al. (2013), AmaroSeoane et al. (2012a, 2013a). A compact object, such as a star so dense that it will not be disrupted by the tidal forces of the MBH, (say, a neutron star, a white dwarf or a small stellarmass black hole), is able to approach very close to the central MBH. When the compact object comes very close to the MBH, a large amount of orbital energy is radiated away, causing the semimajor axis shrink. This phenomenon will be repeated thousand of times as the object inspirals until is swallowed by the central MBH.
The “doomed” object spends many orbits around the MBH before it is swallowed. When doing so, it radiates energy which can be conceptualised as a snapshot containing detailed information about spacetime and all the physical parameters that characterise the binary, the MBH and the stellarmass black hole: their masses, spins, inclination and their sky position. The emitted GWs encode a map of the spacetime. If we can record and decode it, then we will be able to test the theory that massive dark objects are indeed Kerr black holes as the theory of general relativity predicts, and not exotic objects such as boson stars. This would be the ultimate test of general relativity.
The theoretical study of the structure and evolution of a stellar cluster (galactic nucleus or globular cluster) harbouring a central MBH started a few decades years ago. However, due to the complex nature of the problem which includes many physical processes and span a huge range of time and length scales, our understanding of such systems is still incomplete and, probably, subjected to revision. As in many fields of astrophysics, analytical computations can only been applied to highly idealised situations and only a very limited variety of numerical methods have been developed so far that can tackle this problem. In the next sections I will address the most relevant astrophysical phenomena for EMRIs and in the last section I give a description of a few different approaches to study these scenarios with numerical schemes.
3 GWs as a probe to stellar dynamics and the cosmic growth of SMBHs
3.1 GWs and stellar dynamics
The challenge of detection and characterisation of gravitational waves is strongly coupled with the dynamics of dense stellar systems. This is especially true in the case of the capture of a compact object by a MBH.
In order to estimate how many events one can expect and what we can assess about the distribution of parameters of the system, we need to have a very detailed comprehension of the physics. In this regard, the potential detection of GWs is an incentive to dive into a singular realm otherwise irrelevant for the global dynamics of the system.
As mentioned, a harbinger in this respect has been the tidal disruption of stars as a way to feed the central MBH. About 50% of the star is bound to the MBH and accreted on to it, producing an electromagnetic flare which tops out in the UV/Xrays, emitting a luminosity close to Eddington. Nonetheless, the complications of accretion are particularly intricate, tight on many different timescales to the microphysics of gaseous processes. Even on local, galactic accreting objects the complications of accretion are convoluted. It is thus extremely difficult to understand how to extract very detailed information about extragalactic MBHs from the flare. The question of feeding a MBH is a statistical one. We do not care about individual events to understand the growth in mass of the hole, but about the statistics of the rates on cosmological timescales. Obviously, if we tried to understand the individual processes, we would fail.
As for the fate a compact object which approaches the central MBH, this was never addressed before we had the incentive of direct detection of gravitational radiation. Astrophysical objects such as a black hole binary, generate perturbations in space and time that spread like ripples on a pond. Such ripples, known as “gravitational waves” or “gravitational radiation”, travel at the speed of light, outward from their source. These gravitational waves are predicted by general relativity, first proposed by Einstein. Measurement of these gravitational waves give astrophysicists a totally new and different way of studying the Universe: instead of analysing the propagation and transformation of particles such as photons, we have direct information from the fabric of spacetime itself. The information carried by the gravitational radiation will tell us in exquisite detail about the history, behaviour and structure of the universe: from the Big Bang to black holes.
When we started to look into this problem, we realised that there were many questions of stellar dynamics that either did not have an answer or that had not even been addressed at all. In this review I will discuss the relaxation processes that we know to play a major role in the dynamics of this particular regime. This involves twobody as well as manybodycoherent or noncoherent relaxation, and relativity. The list of processes is most likely incomplete, for there can still be additional, even more complicated processes unknown to us. We now have more questions than answers.
3.2 The mystery of the growth of MBHs
The close examination of the Keplerian orbits of the socalled “Sstars” (also called S0stars, where the letter S stands simply for source) has revealed the nature of the central dark object located at the Galactic Centre. By following one of them, S2 (S02), the mass enclosed by the orbit, a volume with radius no larger than 6.25 lighthours, was estimated to be about \(3.7\times 10^6\,M_{\odot }\) Schödel et al. (2003), Ghez et al. (2003). More recent data based on many years of observations set the mass of the central MBHs to \(\sim 4 \times 10^{6} \, M_{\odot }\).
Observations of other galaxies indicate that the masses of SMBH can reach a few billion solar masses (\(M_{\odot }\)), they correlate tightly with the stellar properties of the host galaxies (e.g., the velocity dispersion \(\sigma \) of galaxy bulge). The existence of such a SMBH population in the presentday universe is strongly supported by Sołtan’s argument that the average mass density of these SMBHs agrees with expectations from integrated luminosity of quasars Sołtan (1982), Yu and Tremaine (2002). Claims of detection of “intermediatemass” black holes (IMBHs, with masses ranging between \(10010^4\,M_{\odot }\)) at the centre of globular clusters Gebhardt et al. (2002), Gerssen et al. (2002) raise the possibility that these correlations extend to much smaller systems, but so far the strongest, although not conclusive, observational support for the existence of IMBHs are ultraluminous Xray sources Miller and Colbert (2004), Kong et al. (2010).
Although there is an emerging consensus regarding the growth of largemass MBHs thanks to Sołtan’s argument, MBHs with masses up to \(10^7\,M_{\odot }\), such as our own MBH in the Galactic Centre (with a mass of \(\sim 4\times 10^6\,M_{\odot }\)), are enigmatic. There are many different explains of their masses: accretion of multiple stars from arbitrary directions, see Phinney (1989), Magorrian and Tremaine (1999), Syer and Ulmer (1999), Hills (1975), Rees (1988), mergers of compact objects such as stellarmass black holes and neutron stars, see Quinlan and Shapiro (1990), or IMBHs falling on to the MBH, Portegies Zwart et al. (2006). Other more peculiar means are accretion of dark matter Ostriker (2000) or collapse of supermassive stars Hara (1978), Shapiro and Teukolsky (1979), Rees (1984), Begelman (2010). The origin of these lowmass MBHs and, therefore, the early growth of all MBHs, remains a conundrum.
The centremost part of a galaxy, its nucleus, consists of a nuclear star cluster of a few millions of stars surrounding the MBH, see Schödel et al. (2014). The nucleus is naturally expected to play a major role in the interaction between the MBH and the host galaxy. In the nucleus, stellar densities in excess of a million stars per cubic parsec and relative velocities of the order \(\sim \) 100–1000 \(\mathrm{km\,s}^{1}\)can be reached. In these conditions, as mentioned before, collisional effects are important come into play. This is true except in globular clusters, but one important difference is that the SMBH gives the central part of the cluster almost a Keplerian potential, and thus very tricky resonance characteristics. This is one reason it has been difficult to analyse the stars here.
3.3 A magnifying glass
The laser interferometer space antenna (LISA), see in particular the document prepared in response to the call for missions for the L3 slot in the Cosmic Vision Programme, AmaroSeoane et al. (2017), but also Danzmann (2000), AmaroSeoane et al. (2012a, 2013a), will be our reference point throughout my review. LISA consists of three spacecraft arranged in an equilateral triangle with sides of length 2.5 million kilometre. LISA will scan the entire sky and covers a band from below \(10^{4}\,\)Hz to above \(10^{1}\,\)Hz. In this frequency band, the Universe is populated by strong sources of GWs such as binaries of supermassive black holes merging in the centre of galaxies, massive black holes “swallowing” entirely small compact objects like stellarmass black holes, neutron stars and white dwarfs. The information is encoded in the gravitational waves: the history of galaxies and black holes, the physics of dense matter and stellar remnants like stellarmass black holes, as well as general relativity and the behaviour of space and time itself. Chinese mission study options, such as Taiji, Bender et al. (2005a), Gong et al. (2011, 2015), Huang et al. (2017) will also be able to catch these systems with good signaltonoise ratios.
For the full success of a mission such as LISA, it is important that we understand the systems that we expect to observe. A deep theoretical comprehension of the sources which will populate LISA’s field of view is important to achieve its main goals.
Whilst mainsequence stars are tidally disrupted when approaching the central MBH, compact objects (stellarmass black holes, neutron stars, and white dwarfs) slowly spiral into the MBH and are swallowed whole after some \(\sim 10^5\) orbits in the LISA band. At the closest approach to the MBH, the system emits a burst of GWs which contains information about spacetime and the masses and spins of the system. We can envisage each such burst as a snapshot of the system. This is what makes EMRIs so appealing: a set of \(\sim 10^5\) bursts of GWs radiated by one system will tell us with the utmost accuracy about the system itself, it will test general relativity, it will tell us about the distribution of dark objects in galactic nuclei and globular clusters and, thus, we will have a new understanding of the physics of the process. New phenomena, unknown and unanticipated, are likely to be discovered.
If the central MBH has a mass larger than \(10^7\,M_{\odot }\), then the signal of an inspiraling stellarmass black hole, even in its last stable orbit (LSO) will have a frequency too low for detection. On the other hand, if it is less massive than \(10^4\,M_{\odot }\), the signal will also be quite weak unless the source is very close. This is why one usually assumes that the mass range of MBHs of interest in the search of EMRIs for LISA is between \([10^7,\,10^4]\,M_{\odot }\). Nonetheless, if the MBH is rotating rapidly, then even if it has a mass larger than \(10^7\,M_{\odot }\), the LSO will be closer to the MBH and thus, even at a higher frequency, the system should be detectable. This would push the total mass to a few \(\sim 10^7\,M_{\odot }\).
For a binary of a MBH and a stellarmass black hole to be in the LISA band, it has to have a frequency of between roughly \(10^{5}\) and 1 Hz. The emission of GWs is more efficient as they approach the LSO, so that LISA will detect the sources when they are close to the LSO. The total mass required to observe systems with frequencies between 0.1 Hz and \(10^{4}\) is of \(10^4\)–\(10^7\,M_{\odot }\). For masses larger than \(10^7\,M_{\odot }\), the frequencies close to the LSO will be too low, so that their detection will be very difficult. On the other hand, for a total mass of less than \(10^3\,M_{\odot }\) we could in principal detect them at an early stage, but then the amplitude of the GW would be rather low.
On top of this, the measurement of the emitted GWs will give us very detailed information about the spin of the central MBH. With current techniques, we can only hope to measure MBH spin through Xray observations of Fe K\(\alpha \) profiles, but the numerous uncertainties of this technique may disguise the real value. Moreover, such observations can only rarely be made.
This means that LISA will scrutinise exactly the range of masses fundamental to the understanding of the origin and growth of supermassive black holes. By extracting the information encoded in the GWs of this scenario, we can determine the redshifted mass and spin of the central MBH with an astonishing relative precision. Additionally, the mass of the compact object which falls into the MBH and the eccentricity of the orbit will be recovered from the gravitational radiation with a tiny fractional accuracy. All this means that LISA will not be “just” the ultimate test of general relativity, but an exquisite probe of the spins and range of masses of interest for theoretical and observational astrophysics and cosmology.
3.3.1 A problem of \(\sim \) 10 orders of magnitude
For the particular problem of how does a compact object end up being an extreme mass ratio inspiral, we have to study very different astrophysical regimes, spanning over many orders of magnitude.
Galactic or cosmological dynamics
Figure 6 depicts the three different realms of stellar dynamics of relevance for the problem of EMRIs. At the largest scale exists the galaxy, with a size of a few kiloparsecs. Just as a point of reference, the gravitational radius of a MBH of \(10^6\,M_{\odot } \sim 5\cdot 10^{8}\) pc. The relaxation time, \(t_\mathrm{rlx}\) which I will introduce with more detail ahead, is a timescale which can be envisaged as the required time for the stars to exchange energy and angular momentum between them: it is the time that the stars need to “see” each other individually and not only the average, background stellar potential of the whole stellar system. For the galaxy, \(t_\mathrm{rlx}\) is larger than the Hubble time, which means that, on average, it has no influence on the galaxy at all. A test star will only feel the mean potential of the rest of the stars and it will never exchange either energy or angular momentum with any other star. The system is “collisionless”, meaning that twobody interactions can be neglected. This defines the realm of stellar galactic dynamics, the one investigated in Cosmological simulations using, e.g., Nbody integrators. Since we do not have to take into account the strong interactions between stars, one can easily simulate ten billion particles with these integrators.
Cluster dynamics
If we zoom in by typically a factor of \(10^3\), we enter the (mostly Newtonian) stellar dynamics of galactic nuclei. There, \(t_\mathrm{rlx} \sim 10^8{}10^{10}\) yrs. In this realm stars do feel the graininess of the stellar potential. The closer we get to the central MBH, the higher \(\sigma \) will be, if the system is in centrifugal equilibrium; the stars have to orbit around the MBH faster. In particular, S2 (or S02), one of the Sstars (S0stars) for which we have enough data to reconstruct the orbit to a very high level of confidence—as we saw in the previous section—has been observed to move with a velocity of \(15\cdot 10^{3}\,\mathrm{km\,s}^{1}\). Typically, \(t_\mathrm{rlx}\) is (on occasion much) shorter than the age of the system, of a few \(\sim 10^{8}  10^{10}\) yrs. For these kind of systems one has to take into account relaxation, exchange of energy and angular momentum between stars. The system is “collisional”. When we have to take into account this in the numerical simulations, the result is that we cannot simulate with Nbody integrators more than some thousands of stars on a regular computer. To get to more realistic particle numbers one has to resort to many computers operating in parallel, specialpurpose hardware or the graphic processor units. I will discuss this later.
Relativistic stellar dynamics
Last, in the right panel of Fig. 6, we have the relativistic regime of stellar dynamics when we enlarge the previous by a factor of ten million. There the role of relativistic effects is of paramount importance for the evolution of the system. In this zone, generally, there are no stars. Even at the densities which characterise a galactic nucleus, the probability of having a star in such a tiny volume is extremely small. Moreover, even if we had a significantly larger volume, or a much higher density for the galactic nucleus, so that we had a few stars close to the MBH, these would quickly merge with the MBH due to the emission of GWs, which is what defines an EMRI. But they do it too fast. These systems can be collisional or collisionless, depending on how many stars we have at a given time. If they are there, they will exchange energy and angular momentum between them. Nevertheless, relaxation is not welldefined in this regime.
3.4 How stars distribute around MBHs in galactic nuclei
4 A taxonomy of orbits in galactic nuclei
Before we address the physics and event rate estimates of EMRIs, it is crucial that we have a good understanding of the kind of orbits that we might expect in the environments natural to COs in dense systems around a MBH. An important factor in understanding how a star can become an EMRI is the shape and evolution of its orbit. In this section, I will address these two aspects. First, we will not take into account the role of relaxation. The stellar potential in which our test star \(m_{\bullet }\) is moving is completely smooth. For any purpose, the test star will not feel any individual star, but a background potential.
4.1 Spherical potentials
Consider now Fig. 8; there we have two orbits which differ in their eccentricity. The rosettes are characterised by their energy and angular momentum. Since the test stars do not suffer any individual gravitational tug from the stellar system (at least not on a noticeable timescale), the orbital elements are kept constant. The periapsis^{6} is fixed because the angular momentum is conserved, so that the test star will never come arbitrarily close to the central MBH. In order to achieve anything interesting, one needs to perturbate the system.
4.2 Nonspherical potentials
As for the implications of the detection rates of EMRIs, this could have a huge impact, but the problem should probably be revisited due to the enormous difficulties that force us to make broad simplifications. For instance, we should explore the behaviour of the potential very close to the MBH because, by definition, at some point the potential is completely dominated by the MBH and, thus, spherically symmetric. The only realistic hope here are those stars that typically are on orbits with semimajor axis much larger than the radii of interest to us, so that even if they spend most of the time very far away from the MBH, they will be set on a centrophilic orbit due to the triaxiality of the system, but it is unclear whether these can contribute significantly to the local density around the MBH. As an example of the kind of orbits one can get in a triaxial galactic nucleus, in Fig. 12 I show some representative examples of centrophobic orbits from Poon and Merritt (2001) (cases b, c, d, e). This means that the stars never reach the centre. The lack of conservation of the angular momentum can set stars on either centrophilic orbits or, alternatively, on centrophobic orbits. These can be envisaged as a generalisation of rosette orbits. Nevertheless, since we are interested in EMRIs, we will focus on centrophilic orbits and leave the further description of centrophobic orbits aside. I refer the interested reader to the work by HolleyBockelmann et al. (2001, 2002), and also to the more recent one by Merritt and Vasiliev (2011).
We have two different kinds of centrophilic orbits: (i) pyramid or box orbits. These are still regular but a star on such an orbit can reach arbitrarily small distances in its periapsis; (ii) stochastic orbits, which also come arbitrarily close to the centre. The probability for an orbit to get within a distance d from the central MBH, the very centre of the potential, is proportional to d.
This is nonintuitive. If you have a target with a mass and you shoot a projectile from random directions, the probability of coming within a distance d of the target \(R_\mathrm{p}<d\) is proportional to d itself and not \(d^2\) (which would have been the case for a totally random experiment, without focusing). In the case of a star on an orbit towards the MBH, the number of times you have to “throw” it to get to a periapsis distance closer than d is, \(N_\mathrm{pass}\,(R_\mathrm{p}<d) \propto d\). The reason for this is that our target is a particular one and influences the projectile through a process called gravitational focusing. The projectile, the star, is attracted by the target, the MBH.
Something to also bear in mind is that all of these simulations are limited by a particular resolution, which is still far from being close to reality, so that we are not in the position of extrapolating these results to the distances where the star will be captured by the MBH and become and EMRI.
5 Twobody relaxation in galactic nuclei
5.1 Introduction
We are now back to a spherical system world, in which orbits such as those in the previous section do not exist. Therefore, one needs an additional factor to bring stars close to the MBH. As I have already discussed before, a possibility, is to have a source of exchange of energy and angular momentum. We use and abuse the term collisional to refer to any effect not present in a smooth, static potential, including secular effects. Among these, standard twobody relaxation excells not due to its relevance of contributing to EMRI sources, but due to the fact that this is the beststudied effect; namely the exchange of energy and angular momentum between stars due to gravitational interactions.
Another possibility is physical collisions.^{7} The stars come so close to each other that they collide, they have a hydrodynamic interaction; the outcome depends on a number of factors, but the stars involved in the collision could either merge with each other or destroy each other completely or partially. Contrary to what one could expect, the impact of these processes for the global evolution of the dynamics of galactic nuclei is negligible Freitag and Benz (2002). In most of the cases, when these extended stars, such as mainsequence stars (MS) collide, they do not merge due to the very high velocity dispersion, and they will also not be totally destroyed, because for that they would need a nearly headon collision, so that they have a partial massloss and are for our purposes uninteresting. For the kind of objects of interest to us in this review, stellarmass black holes, the probability that they physically collide is negligible.
A third way of altering the angular momentum of stars are secular effects. They do nevertheless not modify the energy. If we assume that the orbits around the MBH are nearly Keplerian, the shape, an ellipse, does not change, and the orientation will not change much. If we have another orbit with a different orientation, both orbits will exert a torque \(\mathcal T\) on each other. This will change angular momentum but not energy. A Keplerian orbit can be described in terms of its semimajor axis and eccentricity. The semimajor axis is only connected to energy and, for a given semimajor axis the eccentricity is connected to the angular momentum. If one changes the angular momentum but not the energy, the eccentricity will vary but not the semimajor axis. By decreasing the angular momentum, one increases the eccentricity.
In this section, however, I introduce the fundamentals of relaxation theory, focusing on the aspects that will be more relevant for the main interest of this review. Further ahead, in Sect. 7, I will address resonant relaxation and other “exotic” (in the sense that they are not part of the traditional twobody relaxation theory) processes. For a comprehensive discussion on twobody relaxation, I recommend the textbooks by Spitzer Jr (1969) and Binney and Tremaine (2008) or, for a shorter but very nice introduction, the article by Freitag and Benz (2001).
I will first introduce handy timescales in Sect. 4.2 that will allow us to pinpoint the relevant physical phenomena that reign the process of bringing stars (extended or compact) close to the central MBH. I will then address a particular case of relaxation, in Sect. 4.3, dynamical friction. Later, in Sect. 4.4, I will define more concisely the region of spacephase in which we expect stars to interact with the central MBH. Once we have all these concepts, we can cope with the problem of how mass segregates in galactic nuclei, in Sect. 5. We will first see in detail the “classical” although academic solution, and later a more recent and physical result, the socalled strong mass segregation, in Sect. 7.3
5.2 Twobody relaxation
I introduce now some useful timescales to which I will refer often throughout this review; namely the relaxation time, the crossing time and the dynamical time. These three timescales allow us to delimit our physical system.
The relaxation time
In Chandrasekhar (1942) a timescale was defined which stems from the 2body smallangle encounters and gives us a typical time for the evolution of a stellar system.
It must, nevertheless, be noted that the way in which I have introduced the concept of the relaxation time is a particular one. In Eq. (11), I have introduced the “encounter relaxation time” to stress that it depends on the characteristics of a peculiar class of encounter: a star of mass \(m_1\) with “field stars” of mass \(m_2\) with a local density \(n_{\star }\) and a relative velocity \(v_\mathrm{rel}\). It can be envisaged as the required time to deflect gradually the motion of star \(m_1\) due to encounters with field stars by a root mean square (RMS) angle \(\pi /2\). This definition is useful to understand the fundamentals of relaxation, but it must be noted that it is subject to this very peculiar type of encounter.
However, in a general case, we define relaxation by simplifying the problem: (i) We restrict to the radius of influence for a system in which the distribution of stars is spherically symmetric, (ii) stars are treated as single objects, with a twobody relaxation as the only mechanism that can change the angular momentum, and (iii) we neglect mass segregation.
For typical density profiles, \(t_\mathrm{rlx}\) decreases slowly with decreasing r inside \(r_\mathrm{infl}\). It should be noted that the exchange of energy between stars of different masses—sometimes referred to as dynamical friction, as we will see ahead, in Sect. 4.3 in the case of one or a few massive bodies in a field of much lighter objects—occurs on a timescale shorter than \(t_\mathrm{rlx}\) by a factor of roughly \(M/\langle m\rangle \), where M is the mass of a heavy body.
As we will see later, relaxation redistributes orbital energy amongst stellarmass objects until the most massive of them (presumably stellarmass black holes) form a powerlaw density cusp around the MBH, \(n(r)\propto r^{\gamma }\) with \(\gamma \) ranging between \(\simeq \) 1.75–2.1, which depends on the solution to mass segregation considered, while less massive species arrange themselves into a shallower profile, with \(\alpha \simeq 1.4{}1.5\) (Bahcall and Wolf 1976; Lightman and Shapiro 1977; Duncan and Shapiro 1983; Freitag and Benz 2002; AmaroSeoane et al. 2004; Baumgardt et al. 2004a; Preto et al. 2004; Freitag et al. 2006b; Hopman and Alexander 2006a; Alexander and Hopman 2009; Merritt 2010; Preto and AmaroSeoane 2010; AmaroSeoane and Preto 2011; see also Sect. 8.7). Nuclei likely to host MBHs in the LISA mass range (\({\mathscr {M}}_{\bullet }\lesssim \text{ few }\times 10^6\,M_{\odot }\)) probably have relaxation times comparable to or less than a Hubble time, so that it is expected that their heavier stars form a steep cusp.
Collision time
The crossing time
Contrary to gas dynamics, the thermodynamical equilibrium timescale \(t_\mathrm{rlx}\) in a stellar system is large compared with the crossing time \(t_\mathrm{cross}\). In a homogeneous, infinite stellar system, we expect some kind of stationary state to be established in the limit \(t\rightarrow \infty \). The decisive feature for such a virial equilibrium is how quickly a perturbation of the system will be smoothed down.
5.3 Dynamical friction
5.4 The difussion and losscone angles
6 “Standard” mass segregation
6.1 Introduction
In order to address the question of how many objects a year get close enough to the central MBH to be tidally destroyed, in the case of an extended star, or captured, if a compact object, the zeroth order problem we must solve is how stars distribute around MBHs.
In a system with a spectrum of masses initially distributed uniformely, the more massive ones have a higher kinetic energy than the lighter ones, simply due to the fact that they have the same velocity dispersion but a higher mass. The heavy stars exchange energy with each other and with the light stars through relaxation. The exchange of energy goes in the direction of equipartition, because the system searches the equilibrium. The heavy stars will lose energy to the light ones. When they do so, since they feel their own potential or the potential well of the MBH, their semimajor axis shrinks and they segregate to the centre of the system. When doing so, their kinetic energy will become higher. The system tries to reequilibrate itself; the velocity dispersion is larger as it was when the massive stars were at larger distances from the centre. As they approach the MBH, their kinetic energy will be higher as compared to the light stars, which are pushed to the outskirts of the system.
In Fig. 21, we have a density profile that shows us the evolution of a singlemass galactic nucleus with a MBH while letting relaxation play a role (i.e., the simulations were run for at least a \(T_\mathrm{rlx}\)). The initial density profile is depicted in red and shows already a cusp because the authors were using a King model (Freitag et al. 2006a, b), so that it diverges at the centre. When we let it evolve, the profile obtains a much steeper cusp, the blue curve, reaching later a powerlaw cusp of \(\rho \propto R^{1.75}\). This cusp is kept as the system continues to evolve and the cluster expands. The time units are expressed in Fokker–Plank units.^{8}
In this section, I will illustrate the different phenomena with numerical simulations published for the first time in this review.Our other excuse for leaving out high order correlations is that only a fool tries the harder problem when he does not understand the simplest special case.
6.2 Singlemass clusters
Peebles (1972) was the first to realise^{9} that the statistical thermal equilibrium in a stellar cluster, i.e., the fact that the distribution of energy in the cluster is \(f(E) \propto e^{E/\sigma ^2}\), with \(\sigma \) the velocity dispersion, must be violated when we are close to the MBH, because we have three characteristic radii within which stars are lost for the system. These are the tidal radius, \(R_\mathrm{t}\), the “Schwarzschild radius” \(R_\mathrm{Schw}\) (i.e., the capture radius via gravitational loss), and the collisional radius \(R_\mathrm{coll}\). Peebles found that there should be a steady state with a net inward flux of stars and energy in the stellar system. Nevertheless, well within the influence radius \(R_{h}\) of the MBH but far from \(R_\mathrm{t}\), the stars should have nearlyisotropic velocities. Peebles derived a solution in the form of a powerlaw for a system in which all stars have the same mass. The quasisteady solution takes the form (for an isotropic distribution function) \(f(E)\sim E^p\), \(\rho (r)\sim r^{\gamma }\), with \(\gamma = 3/2 + p\). Nevertheless, Peebles derived the wrong exponent. A few years later, Bahcall and Wolf (1976) did an exhaustive kinematic treatment for singlemass systems and found that the exponent should be \(\gamma =7/4\) and \(p=\gamma 3/2=1/4\). This solution has been corroborated in a number of semi/analytical approaches, and approximative numerical schemes, see e.g., Shapiro and Marchant (1978), Marchant and Shapiro (1979), Marchant and Shapiro (1980), Shapiro and Teukolsky (1985), Freitag and Benz (2001), AmaroSeoane et al. (2004), as well as directsummation Nbody simulations, of which the work of Preto et al. (2004) was the first one.
This is one of the most important phenomena in the production of EMRIs, since the galactic nuclei of interest for us, the ones which are thought to be harbouring EMRIs in their cores and are in the range of frequencies of interest, are relaxed. These nuclei are relatively small and are likely to have at least gone through at least one full relaxation time. In general, nuclei in the range of interest for LISA are relaxed (see the rule of thumb introduced in Preto 2010).
6.3 Mass segregation in two masscomponent clusters
As we have just seen, the processes that onecomponent clusters bring about are nowadays relatively well understood and has been plentifully studied by different authors to check for the quality of their approaches. Nonetheless, the properties of multimass systems are only very poorly represented by idealised models in which all stars have a single mass. New features of these systems’ behaviour arise when we consider a stellar system in which masses are divided into two groups. Hence, since the idealised situation in which all stars in a stellar cluster have the same mass has been arduously examined in literature, we have the right to extend the analysis a further step. Here I address more realistic configurations in which the stellar system is split into various components. The second integer immediately after one is two, so we will first extend, cautious and wary as we are, our models to twocomponent star clusters.
Initial mass functions (IMFs), introduced with more detail in Sect. 5.4, ranging between \([0.1,\, \sim 120] M_\odot \) can be approximated to first order by two wellseparated mass scales : one with a mass of the order of \({\mathscr {O}}(1 M_\odot )\) (which could represent mainsequence stars, MS, white dwarfs, WD, or neutron stars NS) and \({\mathscr {O}}(10 M_\odot )\) (stellarmass black holes). Depending on how the system taken into consideration is configured we will exclude dynamical equilibrium (meaning that the system is not stable on dynamical timescales) or equipartition of different components kinetic energies is not allowed (thermal equilibrium).
The work of Spitzer Jr (1969) was in this respect pioneering. For some clusters, it seemed impossible to find a configuration in which they enjoy dynamical and thermal equilibrium together. The heavy components sink into the centre because they cede kinetic energy to the light ones when reaching equipartition. The process will carry on until equipartition is fully gained. In most of the cases, equipartition happens to be impossible, because the subsystem of massive stars will undergo core collapse before equipartiton is reached. Anon, a gravothermal collapse will happen in this component and, as a result, a small dense core of heavy stars is formed Spitzer Jr (1969), Lightman and Fall (1978). This gravothermal contraction is a product of negative heat capacity, a typical property of gravitationally bound systems Elson et al. (1987).
Different authors have addressed the problem of thermal and dynamical equilibrium in such systems, using techniques such as direct Nbody Portegies Zwart and McMillan (2000), Khalisi et al. (2007) and Monte Carlo simulations Watters et al. (2000) to direct integration of the Fokker–Planck equation Inagaki and Wiyanto (1984), Kim et al. (1998) or moments of it AmaroSeoane et al. (2004), including Monte Carlo approaches to the numerical integration of this equation Spitzer Jr and Hart (1971). For a general and complete overview of the historical evolution of twostars stellar components, see Watters et al. (2000), AmaroSeoane et al. (2004) and references therein.
If we do not have any energy source in the cluster and stars do not collide (physically), the contraction carries on selfsimilarly indefinitely; in such a case, one says that the system undergoes core collapse. This phenomenon has been observed in a large number of works using different methods Hénon (1973, 1975), Spitzer Jr and Shull (1975), Cohn (1980), Marchant and Shapiro (1980), Stodołkiewicz (1982), Takahashi (1993), Giersz and Heggie (1994), Takahashi (1995), Spurzem and Aarseth (1996), Makino (1996), Quinlan (1996), Drukier et al. (1999), Joshi et al. (2000). Core collapse is not just a characteristic of multimass systems, but has been also observed in single mass analysis.
At this point, the question looms up of whether for very young clusters mass segregation is due to relaxation, like in our models, or rather reflects the fact that massive stars are formed preferentially around the centre of the cluster, as some models predict.
Raboud and Mermilliod (1998) addressed the radial structure of Praesepe and of the very young open cluster NGC 6231. There they find evidence for mass segregation among the cluster members and between binaries and single stars. They put it down to the greater average mass of the multiple systems. Figure 24 reproduces a plot of Raboud and Mermilliod (1998), where again we have clear evidence for mass segregation in NGC 6231. In the two first panels, the mass intervals are set in a different way to those in the bottom.
The two lefthand panels of Fig. 24 include the 9 bright stars of the cluster Corona, while on the right do not. The manifestation of mass segregation for massive stars (triangles) is clearly displayed, while stars with masses between \([5,\,20]\,M_{\odot }\) are spatially well mixed (open squares and crosses); i.e., mass segregation is not yet established over a rather large mass interval. This population is more concentrated than the lowermass population (here shown with filled squares). They derive from Fig. 24 that only a dozen, bright, massive, mainly binary stars are well concentrated toward the cluster centre.
Now, I introduce the quantity \(\zeta \equiv 1q\), and we let \(\zeta \) vary from \(10^{4}\) to \(9.99 \cdot 10^{1}\). For each \(\zeta \) value, we let \(\mu \) vary between 1.03 and \(10^3\). The values for q are regularly distributed in \(\log {(\zeta )}\). For \(\zeta \approx 1\) we have added a series of values in \(\log {(\zeta 1)}\). The mean particle mass is \(1,M_{\odot }\) and the total mass \(10^6 \,M_{\odot }\), but this is not important for our study, because the physics of the system is driven by relaxation and therefore the only relevant concept is the relaxation time. We can always extend the physics to any other system containing more particles, with the proviso that only relaxation is at play. The mean mass is therefore just a normalisation. What really determines the dynamics of the system are the mass ratios, q and \(\mu \), which is the reason why I use them to explore the system.
In Fig. 25, I show the whole \((q,\,\mu )\)parameter space in a plot where the time at which the corecollapse begins is included. The green zone corresponds to the quasi singlemass case. In the red zone we have the largest difference between masses and blue is an intermediate case.
We have followed in the curves the evolution of the system until a deep collapse of the system. These figures show the evolution until the most massive component dominates the centre.
In order to compare our plots with those of Khalisi et al. (2007), one should look at their diagrams in the region during core contraction. At this point, we can observe in Fig. 28 a selfsimilarity after corecollapse (Giersz and Heggie 1996). Binaries are responsible for interrupting corecollapse and driving core reexpansion in the Nbody simulations. The flattening in the Nbody plots at the moment of corecollapse is due to the binary energy generation. This means that we can only compare the steep rise, but not the saturation.
For instance, in the second plot of the Nbody set (second column on the top of Fig. 28), we have to look at the point at which the average mass of the Nbody system is about 1.20 in the 0–1% shell. This establishes the limit until which we can really compare the behaviour as given by both methods. Our simulations yield a very similar evolution until that point. The gaseous model behaves (it clearly shows the tendency) like the Nbody result.
By converting the Fokker–Planck units, we find that the conversion factor is the same; namely, for \(\gamma =0.11\), \(\ln (\gamma \cdot {\mathscr {N}}_{\star })/ {\mathscr {N}}_{\star }= 0.0022\). On the other hand, the value of \(\gamma \) is not so well defined and depends on the mass spectrum (Hénon 1975). This means that potentially it is not the same for the different models. For a broader mass spectrum, \(\gamma \) is about 0.01 and, unfortunately, in the case of having a small particle number, it will definitively make an important difference despite the “smoothing” effect of the logarithm, viz \(\ln (\gamma \cdot {\mathscr {N}}_{\star })/ {\mathscr {N}}_{\star } = 0.0013\). Thus, in order to be able to compare the different models, one should consider \(\gamma \) as a free parameter ranging between 0.01 and 0.2 and look for the best fit for the majority of cases. On the other hand, we must bear in mind that the Nbody simulations of Khalisi et al. (2007) do not go into deep core collapse and so, the moment at which the core radius reaches a minimum is not the same as for our model. To sum up, although we cannot say exactly to what point we can compare the two methods (the Gas Model and directsummation simulations), because the core collapse time will be different, the physics of the system is the same in the two cases. This should provide the reader with a good understanding of the phenomena in play, as well as a proof that they are independent of the details of the algorithm used.
6.4 Clusters with a broader mass spectrum with no MBH
In order to understand the phenomena that I will describe later, which is crucial for EMRI formation, it is of relative relevance to understand first the physics behind cluster dynamics without a central MBH. This section is also interested in interpreting observations of young stellar clusters extending to a larger number of mass components. In clusters with realistic IMFs, equipartition cannot be reached, because the most massive stars build a subsystem in the cluster’s centre as the process of segregation goes on thanks to the kinetic energy transfer to the light mass components until the cluster undergoes core collapse (Spitzer Jr 1969; Inagaki and Wiyanto 1984; Inagaki and Saslaw 1985). Although the case in which the MBH is lurking at the centre of the host cluster is more attractive for EMRI production and from a dynamical point of view, one should study, in a first step, more simple models.
In this section we want, thus, to go a step further and evaluate stellar clusters with a broad mass function (MF hereafter). For this, I will again be using the Gas Model, because it is a good compromise between accuracy and integration time for this review.
We study those clusters for which the relaxation time is relatively short, because the most massive stars will sink to the centre of the system due to mass segregation before they have time to leave the main sequence (MS). In this scenario we can consider, as an approximation, that stellar evolution plays no role; stars did not have time to start evolving. The configuration is similar to that of Gürkan et al. (2004), but they employ a rather different approach based on a Monte Carlo code (MC), using the ideas of Hénon (1973) that allow one to study various aspects of the stellar dynamics of a dense stellar cluster with or without a central MBH. Our scheme, although being more approximate than MC codes (and directsummation Nbody ones) and unable, in its present version, to account for collision has the advantage, as we will see in the Sect. 8, of being much faster to run, and of providing data that has no numerical noise. It captures the essential features of the physical systems considered in our analysis and is an interesting, powerful tool for illustrating the different scenarios in this review.
One of the first questions we should address is the maximum number of components one should take into consideration when performing our calculations. Since the computational time becomes larger and larger when adding more and more components to the system even for an approximative scheme such as the Gas Model, we should first find out what is a realistic number of components in our case. For this end I have performed different computations with different number of stellar components.
6.5 Corecollapse evolution
One notes that, during core collapse, the central regions of the cluster become completely dominated by the most massive stars. But, contrary to the case of singlemass clusters, the central velocity dispersion decreases (see Fig. 35).
6.6 Clusters with a broader mass spectrum with a MBH
Afer having addressed the systems studied in previous sections we now look into the dynamical problem of a multimass component cluster harbouring a central seed MBH that grows due to stellar accretion.
 (i)
Main sequence stars of 0.1–\(1\,M_{\odot }\) (\(\sim \) 7 components)
 (ii)
White dwarfs of \(\sim \, 0.6\,M_{\odot }\) (1 component)
 (iii)
Neutron stars of \(\sim \,1.4\,M_{\odot }\) (1 component)
 (iv)
Stellar black holes of \(\sim \,10M_{\odot }\) (1 component)
 (a)
White dwarfs in the range of \(1 \le m_\mathrm{MS}/M_{\odot }< 8\)
 (b)
Neutron stars for masses \(8 \le m_\mathrm{MS}/M_{\odot }< 30\)
 (c)
Stellar black holes for bigger masses, \(\ge 30 M_{\odot }\)
The presence of a small fraction of stellar remnants may greatly affect the evolution of the cluster and growth of the MBH because they segregate to the centre, and in doing so, they expel MS stars from it but, being compact, they cannot be tidally disrupted. This kind of evolution is shown in Figs. 36 and 37.
Figure 36 shows us the time evolution of different Lagrange radii with 0.1, 10, 50, 80% of the mass of each component. Here, the core collapse happens at about \(T=0.18\,T_\mathrm{rh}(0)\). The later reopening out is due MBH accretion.
We can study how the system evolves from the point of view of the distribution of kinetic energies between the different components of the clusters during the process of mass segregation.
7 Twobody extreme mass ratio inspirals
After the first sections we have a good understanding of the fundamentals of twobody relaxation in dense stellar systems, including mass segregation and dynamical friction, which could be roughly described as “relaxation when we have a large mass ratio”. In this section, I address the subject of capture of compact objects by a massive black hole considering that the driving mechanism in the production is twobody relaxation.
7.1 A hidden stellar population in galactic nuclei
The question about the distribution and capture of stellarmass black holes at the Galactic Centre has been addressed a number of times by different authors, from both a semi or analytical and numerical standpoint, see e.g., Sigurdsson and Rees (1997), MiraldaEscudé and Gould (2000), Freitag (2001, 2003b), Freitag et al. (2006a, b), Hopman and Alexander (2006b), AmaroSeoane et al. (2007), Preto and AmaroSeoane (2010), AmaroSeoane and Preto (2011). Addressing this problem has implications for a variety of astrophysical questions, including of course inspirals of compact objects onto the central MBH, but also on the distribution of Xray binaries at the Galactic Centre, tidal disruptions of main sequence stars, and the behaviours of the socalled “source” stars, which were introduced in Sect. 2.2. Even if we only consider single stellarmass black holes, the impact they can have on the Sstars is not negligible; a distribution of nonluminous matter around the Galactic Centre would have a clear fingerprint on their orbits. Current data are insufficient to detect such an extended nonluminous cusp which typically would induce a slight Newtonian retrograde precession (Mouawad et al. 2005), so that we will have to wait for future telescopes before we can hope to see such trajectory deflections. The study of Weinberg et al. (2005) estimated that proposed 30–100 m aperture telescopes will allow us to observe about three trajectory deflections per year between any of the monitored “source” stars and a stellarmass black hole.
The centermost part of the stellar spheroid, the galactic nucleus, constitutes an extreme environment for stellar dynamics. With stellar densities higher than \(10^6\,M_{\odot }\,\mathrm {pc}^{3}\), relative velocities in excess of 100 \(\mathrm{km\,s}^{1}\) the nucleus (unlike most of the rest of the galaxy) is the site of a variety of “collisional processes”—both close encounters and actual collisions between stars, as we have seen in the previous sections. The central MBH and the surrounding stellar environment interact through various mechanisms: some are global, like the accretion of gases liberated by stellar evolution or the adiabatic adaptation of stellar orbits as the mass of the MBH increases; others, which involve the close interaction between a star and the MBH—EMRIs and stellar disruptions—are local in nature. As we have seen in Sect. 4.4, to interact closely with the central MBH, stars have to find themselves on “losscone” orbits, which are orbits elongated enough to have a very closein periapsis (Frank and Rees 1976; Lightman and Shapiro 1977; AmaroSeoane and Spurzem 2001).
The rate of tidal disruptions can be established (semi)analytically if the phase space distribution of stars around the MBH is known, see Magorrian and Tremaine (1999), Syer and Ulmer (1999), Wang and Merritt (2004) for estimates in models of observed nearby nuclei. However, in order to account for the complex influence of mass segregation, collisions and the evolution of the nucleus over billions of years, detailed numerical simulations are required, as in the work of David et al. (1987a, b), Murphy et al. (1991), Freitag and Benz (2002), Baumgardt et al. (2004b), Freitag et al. (2006b), Khalisi et al. (2007), Preto and AmaroSeoane (2010), AmaroSeoane and Preto (2011).
In the case of a gradual inspiral following the “capture” of a compact object (i.e., an EMRI), the situation becomes even more complex, even in the idealised case of a spherical nucleus with stars all of the same mass. As the star spirals down towards the MBH, it has many opportunities to be deflected back by twobody encounters on to a “safer orbit”, i.e., an orbit which does not lead to gravitational capture (Alexander and Hopman 2003) hence even the definition of a losscone is not straightforward. Once again, the problem is a compound of the effects of mass segregation, general relativity and resonant relaxation, to mention three main complications. As as result, considerable uncertainties are attached to the (semi)analytical predictions of capture rates and orbital parameters of EMRIs.
Only selfconsistent stellar dynamical modeling of galactic nuclei will provide us with a better understanding of these questions. Some steps in that direction have been made by Freitag (2001, 2003a, b) using Monte Carlo simulations. Later, Freitag et al. (2006a, b) improved upon these results. Yet these studies neglected a direct estimation of EMRIs or “direct plunges”, due in part to the fact that, to follow stars on very eccentric orbits, one needs the combined effects of GW emission and relaxation on timescales much shorter than the capabilities of the numerical Monte Carlo code. Much work remains to be done to confirm these results and improve on them with a more accurate treatment of the physics, to extend them to a larger domain of the parameter space and to more general situations, including nonspherical nuclei.
Classical studies based on approximate stellar dynamics methods that neglect, in particular, the motion of the central MBH and strong 2body interactions, indicate that, in dense enough clusters, a “seed” MBH (in the IMBH mass range) could swallow a significant fraction of the cluster mass, and thus become a MBH over the span of a few Gyrs (Murphy et al. 1991; Freitag and Benz 2002; AmaroSeoane et al. 2004). More detailed, higher fidelity Nbody simulations of relatively small clusters (Baumgardt et al. 2004a, b) have not confirmed this classical result, calling for a critical reexamination and improvement of approximation techniques, the only ones that can cope with the high particle numbers found in massive clusters such as galactic nuclei. It has also been suggested that some processes, such as the effects of chaotic orbits in a slightly nonspherical potential, may effectively keep the losscone orbits populated. In this case disruptions and captures can efficiently feed the central MBH and produce the \(M\sigma \) relation (Zhao et al. 2002; Merritt and Poon 2004).
Understanding the astrophysical processes within galactocentric clusters that give rise to EMRI events has significant bearing on LISA’s applicability to this science. Accurate predictions of the event rate are important for preparing LISA data analysis and design —many events lead to sourceconfusion, which must be dealt with, while a few events necessitate identifying weak sources in the presence of instrumental noise (AmaroSeoane et al. 2007). More importantly, LISA observations alone cannot decouple the mass distribution of the galactic black hole population from the massdependence of the EMRI rate within a single system. If we can improve our understanding of the latter, we improve LISA’s utility as a probe of the former. In this section I elaborate in detail on the “standard” physics leading to sources of gravitational radiation in the millihertz regime—i.e., in the bandwidth of a LISAlike detector—originating in twobody relaxation processes.
7.2 Fundamentals of EMRIs
In the simplest idealisation, an EMRI consists of a binary of two compact objects, a massive black hole (MBH) and a—typically—stellar black hole (SBH) describing a large number of cycles around the MBH as it approaches the LSO, emitting important, coherent amounts of GWs at every periapsis passage.^{11} After every \(2\pi \) around the orbit, the semimajor axis decays a fraction proportional to the energy loss. After typically some \(10^{4{}5}\) cycles, the small body, the CO, plunges through the horizon of the MBH and is lost. The emission of GW finishes. This is what makes this system so attractive. We can regard it as a camera flying around a MBH taking extremely detailed pictures of the space and time around it. With one EMRI we are provided with a set of \(\sim 10^{4{}5}\) pictures from a binary, and the information contained in them will allow us also to know with an unprecedented accuracy in the history of astronomy about the mass of the system, the inclination, the semimajor axis, the spin, to mention some, and it will also be an accurate test of the general theory of relativity.
At first glance the task seems simple and, of course, worth doing; we just have to analyse a binary which decays slowly in time proportionally to \(a^4\), where a is the semimajor axis. The work seems to be easy for such a big gain. The only problem is that it is not as easy as it seems, because we need to understand how a star can become an EMRI in such a dynamically complex system as a galactic nucleus. Also, the EMRI might suffer perturbations either from gas or from the stellar system (Kocsis et al. 2011; AmaroSeoane et al. 2012b; Barausse et al. 2014).
In Fig. 40, I show what systems would missions such as LISA be more sensitive to. Obviously, this is only an illustration and the data analysis of the signal will be much more complicated in reality, but it is just an indication already that if the central MBH has a mass larger than \(10^7\,M_{\odot }\), then the signal, even at the LSO, will have a frequency too low for detecting the system. On the other hand, if it is less massive than \(10^4\,M_{\odot }\), the signal will also be quite weak unless the source is very close. This is why one usually assumes that the mass range of MBHs of interest in the search of EMRIs for LISA is between \([10^4,\,10^7]\,M_{\odot }\). We note that this picture is shifted towards lighter masses in the eLISA configuration, as explained in AmaroSeoane et al. (2012a, 2013a). Nonetheless, if the MBH is rotating fast, then even if it has a mass larger than \(10^7\,M_{\odot }\), the LSO will be closer to the MBH and thus, even at a higher frequency the system should be detectable. This would push to the left the total mass to a few \(\sim 10^7\,M_{\odot }\). Indeed, in Fig. 1 of Gair (2009) we can see how the sensitivity varies as we vary the spin of the MBH. The sensitivity limit for nonspinning black holes is about \(5\times 10^6\,M_{\odot }\), but this goes up to a few times \(10^7\,M_{\odot }\) for prograde inspirals into rapidly spinning black holes. More recently, in Fig. 5 of Babak et al. (2017), we have skyaverage horizons for prograde inspirals into maximally spinning black holes. The authors show that we can see inspirals out to \(z \sim 1\) even if the MBH has a mass of \(10^7\,M_{\odot }\). From the point of view of astrophysics, this range of masses corresponds to lowmass SMBHs. They are not easily detectable and we do not know much about them.
In a spherical potential, at any given time, the stars and compact objects in the nucleus simply orbit the MBH with their semimajor axes and eccentricities changing slowly, owing to 2body relaxation. For an EMRI to occur, in this standard picture, 2body relaxation has to bring a compact remnant on to an orbit with such a small periapsis distance that dissipation of energy by emission of GWs becomes significant.
Not all objects with an inspiral time by GW emission shorter than a Hubble time will end up as EMRIs. This is because, although relaxation can increase the eccentricity of an object to very high values, it can also perturb the orbit back to a more circular one for which GW emission is completely negligible. Typically, neglecting GW emission, it takes a time of the order \(t_\mathrm{rlx}\ln (1e)\) for an orbit to reach a (large) eccentricity e through the effects of 2body relaxation. However, the periapsis distance \(R_\mathrm{p}=a(1e)\) can be significantly altered by relaxation on a timescale \(t_\mathrm{rel,p} \simeq (1e)\,t_\mathrm{rlx}\), so the condition for a star to become an EMRI is that it moves onto an orbit for which the timescale for orbital decay by GW emission, \(\tau _\mathrm {GW}\) [see Eq. (73)] is sufficiently shorter than \((1e)\,t_\mathrm{rlx}\). If the semimajor axis of the orbit is too large, this condition cannot be obeyed unless the star actually finds itself on an unstable, plunging orbit, with \(e\ge e_\mathrm{pl}(a) \equiv 1{}4R_\mathrm{Schw}/a\) where \(R_\mathrm{Schw}\) is the Schwarzschild radius of the MBH. The very short burst of gravitational radiation emitted during a plunge through the horizon can only be detected if originating from the Galactic centre (Hopman et al. 2007). Coherent integration of the GW signal for \(>10^4\) cycles with a frequency in LISA band is required for detection of extragalactic EMRIs. Therefore a central concern in the determination of EMRI rates is to distinguish between plunges and progressive inspirals (Hils and Bender 1995; Hopman and Alexander 2005).
The situation for EMRI production in the standard picture is more complicated than that of tidal disruptions by the MBH (e.g., Rees 1988; Magorrian and Tremaine 1999; Syer and Ulmer 1999; Wang and Merritt 2004) or GW bursts from stars on very eccentric orbits (Rubbo et al. 2006; Hopman et al. 2007) because these processes require a single passage within a welldefined distance \(R_\mathrm{enc}\) from the MBH to be “successful”. In such cases, at any distance from the centre and for any given modulus of the velocity, as mentioned in Sect. 4.4 and later, there exists a “loss cone” inside which the velocity vector of a star has to point for it to pass within \(R_\mathrm{enc}\) of the MBH (Frank and Rees 1976; Bahcall and Wolf 1977; Lightman and Shapiro 1977; AmaroSeoane and Spurzem 2001). In contrast, an EMRI is a progressive process which will only be successful (as a potential source for LISA) if the stellar object experiences a very large number of successive dissipative close encounters with the MBHs (Alexander and Hopman 2003). There is no welldefined loss cone for such a situation.
Implementing this basic scenario in various ways (see Sect. 8.7), several authors have estimated the rate at which stellar remnants are captured by the central MBH, with results between \(\sim 10^{6}\,{\hbox {and}}\,10^{8}~\mathrm{yr}^{1}\) for a \(10^6\,M_\odot \) central black hole (Hils and Bender 1995; Sigurdsson and Rees 1997; Ivanov 2002; Hopman and Alexander 2005). When combined with the uncertainty in the number density of massive black holes with \({\mathscr {M}}_{\bullet }<\mathrm{few}\times 10^6\,M_\odot \), the net predicted number of detections that LISA can make spans over three orders of magnitude, from a few to a few thousand events per year.
We note, incidentally, that even in the LISA band (in the final year of inspiral), the eccentricity of the typical EMRI in the standard picture is high enough that a large number of harmonics are likely to contribute to the gravitational waves (Freitag 2003b; Barack and Cutler 2004; Hopman and Alexander 2005). In addition, the orbital plane of the EMRIs is unlikely to be significantly correlated with the spin plane of the MBH. These characteristics are distinct from those in nonstandard scenarios (discussed below), leading to optimism that some aspects of the nuclear dynamics could be inferred from just a few events.
7.3 Orbital evolution due to emission of gravitational waves
A classic EMRI, with \(M=10^4{}10^7\,M_\odot \) and \(\mu =1{}10\,M_\odot \), could have a significant eccentricity if (as expected in galactic nuclei) the orbits come in from large distances, \(a>10^{2}\,\) pc with \(e\gtrsim 0.9999\). Hopman and Alexander (2005) made an estimate of the distribution of eccentricities for one body inspiral and their results showed that it is skewed to highe values, with a peak of the distribution at \(e \sim 0.7\), at an orbital period of \(10^4\) s. On the other hand, following a binary separation event (and possibly the tidal capture of giant’s core), the compact star is deposited on an orbit with semimajor axis of order a few tens to a few hundreds of AU. In this case, the GWdominated regime is reached with an eccentricity smaller than 0.99 and the orbit should be very close to circular when it has shrunk into the LISA band. Such typical orbital evolutions for EMRIs are shown in Fig. 43.
7.4 Decoupling from dynamics into the relativistic regime
In Fig. 41, I follow the signal emitted by a binary consisting of a Milky Waylike MBH and a stellar BH during their GWdriven inspiral without taking into account any possible dynamical interaction; i.e. we only allow the system to evolve via gravitational radiation emission. I plot the five lowest harmonics of the quadrupolar emission in a rough approximation (Peters and Mathews 1963), only useful for illustrative purposes. In this figure, I assume a distance of 1 Gpc.
Figure 43 displays the last stable orbit in the effective Keplerian approximation (\(R_\mathrm{p}\simeq 4\,R_\mathrm{Schw}\) for \(e\ll 0.1\), see Cutler et al. (1994) with a solid, thick diagonal line. The thin dotted blue lines are contours of constant time left until the final coalescence, \(T_\mathrm{GW}\) in the Peters (1964) approximation. The years are show on the right. The thin diagonal green lines are the inspiral, capture orbits due only to the emission of GWs. The upper dashdotted red line shows \(\tilde{e}(a)\), defined by \(t_e=T_\mathrm{GW}\) [Eq. (65) with \(C_\mathrm{EMRI}=1\)] assuming a constant value \(t_\mathrm{rlx}=1\) Gyr. The lower dashdotted red lines depict the same threshold times a factor 10, 100, 1000, 10,000 and 100,000. On the righthand side of these lines the evolution of the binary is driven mainly by relaxation, GW emission is totally negligible and viceversa; i.e., on the lefthand side the evolution is led by the loss of energy in GWs. An interesting point is the intersection of the first of these red lines (the uppermost one) with the last stable orbit line. This is the transition between the socalled direct plunges and the EMRIs.
The thick, dashed black line shows the tidal disruption radius. Any extended star fording that radius will be torn apart by tidal forces of the MBH, which we assume to have a mass \({\mathscr {M}}_{\bullet } = 4 \times 10^6\,M_{\odot }\) (\(\mathrm{M}_{\mathrm{MBH}}\) in the plot). Then, as an illustration, I depict the trajectory of a \(10\,M_{\odot }\) stellar BH (\(m_{bh}\) in the plot) inspiralling into the MBH. We can separate two kind of sources according to their astrophysical origin; namely loweccentricity captures, stars captured by tidal binary separation, and higheccentricity captures, stemming from “simple” twobody relaxation. The latter initially have semimajor axis values of order 100–1000 AU [\(5\times (10^{4}{}10^{3})\,\)pc] and \(e=0.9{}0.99\) (Miller et al. 2005). The evolution of the eccentricity is a random walk leading to nearlycircular orbits after a timescale of about \(T_\mathrm{rlx}\ln (1\tilde{e})^{1}\). The latter correspond to stars on capture orbits due to diffusion form large radii or capture by GW emission and have initially have a much larger value of semimajor axis and hence a higher eccentricity. If a star has a semimajor axis \(\gtrsim 5\times 10^{2}\) pc, it will not reach small orbital periods, i.e., it will not enter a millihertz detector such as LISA unless the semimajor axis is reduced considerably, which in the context of “normal” relaxation theory, takes about a time \(t_\mathrm{rlx}\).
However, if we have a dissipation process acting on to the star, which could be energy loss in the form of GWs as well as drag forces originating in an accretion disc or, obviously tidal forces created by the central MBH, the picture changes significantly. The process follows the same path and, at some point, the star reaches the region in which it is on a very radial orbit, i.e. where the zigzag stops and we can approximate the curve by a horizontal line. Nonetheless, in this case, at every periapsis passage, the star will emit an intense burst of GWs and, thus, shrink its semimajor axis. If this happens “efficiently enough”, i.e., “fast enough” (we will elaborate on this later), the star is more and more bound to the central MBH and drifts away (goes up in the energy axis of the figure). The danger of being scattered away from the capture orbit by other stars decreases more and more and the compact object finds itself on a safe inspiraling EMRI orbit. The precise details of the dynamics that lead to this situation determines the distribution of eccentricities that we can expect. The semimajor axis shrinks to the point that the source enters the “Detectable GW” regime (lightblue band in the right panel of Fig. 42). As the source advances in that band, the period becomes shorter and shorter and, hence, the power (emitted energy per unit of time) grows larger and larger, so that the gravitational radiation can be measured when it enters the frequency band of the observatory.
The statistical orbital properties of the EMRI in the region where GW emission is prominent are fully determined by the transition phase between the region dominated by 2body scattering processes (the right part of the curve) of the random walk in phasespace and the deterministic dissipation part of the capture trajectory, i.e., where the energy loss occurs.
As described in Hopman and Alexander (2005), in this statistical treatment there is a critical energy, i.e., a certain distance from the central MBH, of the order \(\sim 10^{2}\) pc, that can be envisaged as the threshold between the two regions. This means that stars with energy below the yellow dashed line of the right panel of Fig. 42 will have “longer horizontal segments”, they will scatter faster in angular momentum than in energy and then they will end up as direct plunges. They approach the central MBH in such a radial orbit that they are swallowed after one or, at most, a few intense bursts of GWs. This situation is reverted if the energy of the star is above the line; the star will spiral in adiabatically and it will not be perturbed out of the EMRI trajectory, with a significant amount of GW bursts at periapsis before coalescing with the MBH.
8 Beyond the standard model of twobody relaxation
8.1 The standard picture
Although Sterling was not directly referring to our standard model, of course. This means that, illustrating and enlightening as it might be, the standard model we have been describing so far must be regarded as a (probably very well) educated guess.Do you know what the standard (American) model is? : One gallon per flush.
As the interest in a millihertz mission started to grow and develop, astrophysicists started to dedicate more and more time to a problem that, naively, was not very difficult. How do you get a small black hole into a massive black hole in a galactic nucleus? Now, some decades after the very first estimates, we have a much better and clear vision of the main phenomena at play in the process. Well before any spaceborne mission is launched, our understanding of theory related to stellar dynamics has become much broader and new, unexpected effects have emerged.
8.2 Coherent or resonant relaxation^{14}
As I have discussed previously, in a gravitational potential with a high degree of symmetry, a test star will receive gravitational tugs from the rest of the field stars which are not totally arbitrary and hence do not add up in a random walk way, but coherently. As we have seen in Sect. 3, the potential will prevent stellar orbits from evolving in an erratic way. In a twobody Keplerian system, a SBH will orbit around the MBH in a fixed ellipse. The stellar BH will not feel random gravitational tugs. It evolves coherently as the result of the action of the gravitational potential. When an EMRI approaches the periapsis of its orbit, we can envisage the situation as a pure twobody problem; initially Newtonian but later GR effects must be taken into account as the periapsis grows smaller and smaller. Nonetheless, as the stellar BH goes back to the apoapsis, it will feel the surrounding stellar system, distributed in the shape of a cusp which grows in mass the further away we are from the periapsis. The time spent in the region in which we can regard this as a twobody problem is much shorter than the time in which the stellar BH will feel the rest of the stellar system. This is particularly true for the kind of objects of our interest, since the very high eccentricity implies a large semimajor axis. The time spent on periapsis is negligible as compared with the time spent on apoapsis, so that the stellar BH can feel the graininess of the potential. The gravitational tugs from other stars will alter its orbit. The mean free path in angular momentumspace of that test stellar BH is very large and thus, it has a fast random walk. Both the magnitude and direction of angular momentum of the stellar black hole are altered. When the magnitude changes but not the direction, we talk of “scalar” resonant relaxation, and correspondingly when the direction is changed but not the size, “vector” resonant relaxation.
A very radial orbit can become a very eccentric one, so that a compact object initially set on a potential EMRI orbit can be “pushed out” of it. In a more general case, a spherical potential that is nonKeplerian, the orbits, as we have described before, are rosettes and averaged over time they are circular anuli. In that case we can change the direction of angular momentum but not the modulus. An eccentric orbit will stay eccentric, but any coherence that was there will be washed out.
In a more general case, if we have a potential that is simply spherical but not necessarily Keplerian (a point mass), the field stars, the perturbing orbits to the test star, describe rosettes as we have seen and averaged over time they can be approximated by a set of anuli that share a centre. From a secular point of view, the masses are smeared over those anuli which create torques that do not change the magnitude of angular momentum but they do change the orientation because of reasons of symmetry (Rauch and Tremaine 1996; Rauch and Ingalls 1998; Hopman and Alexander 2006b). Hence a circular test star will keep a negligible eccentricity and it will not approach the central MBH. Any coherence that was present in the system will nonetheless be destroyed. I will refer to this as vectorial coherent relaxation. From the standpoint of EMRI production, though, this process is not as relevant and we will not elaborate on it further, though it can be very relevant for phenomena related to galactic nuclei, for instance, warping of accretion discs (Bregman and Alexander 2009).
The impact of coherent relaxation on the production of EMRIs is important. While the underlying physics of the process is very robust, it is a rather difficult task to ponder the efficiency of the different parameters involved in the process. A possible way of evaluating it is given by Hopman and Alexander (2006b) and Eilon et al. (2009). Figure 45, which is Fig. 6 of Eilon et al. (2009), shows the rate of EMRIs and plunges in a system in which we take into account both orthodox or regular relaxation and coherent relaxation normalised to what one can expect when only taking into account normal relaxation as function of the \(\varXi \) parameter, which gives us the efficiency of coherent relaxation. The units of \(\varXi \) are such that the value suggested in Rauch and Tremaine (1996) is unity. We note that the work of these authors was limited to a very low number of particles, but we can consider it as a reference point to refer to. Thus, if coherent relaxation is more efficient than what they found, \(\varXi >1\) and viceversa, i.e., we approach the regime in which there is not coherent relaxation. It is very remarkable to see that by choosing the value suggested (Rauch and Tremaine 1996), we achieve the maximum of the EMRI rate curve. If the “real” value of \(\varXi \) happened to be a factor 10 larger, then we would be drastically dropping the rates and increasing the direct plunges and, of course, also the tidal disruptions event rate, since these occur at larger radii.
At first glimpse, everything seems to boil down to calculating the precession of coherent relaxation. One obvious way is to do largeparticle number simulations, since the first attempt of Rauch and Tremaine (1996) was really very limited and difficult to interpret (they were using fewer than 100 particles). However, the systems we are trying to simulate are much more complicated than something a simplified approach will be able to investigate. From a numerical point of view the complications are big and nonnegligible. Nevertheless, there has been an important and impressive advance in this front recently but, before we address it, the results and interpretations, it is probably better to have a look at a very familiar system for us, Sgr A*. Hopman and Alexander (2006b) have done this interesting and useful exercise, which is summarised in their Fig. 6 (see Fig. 46). In this figure, the authors display the relevance of different dynamical components in an attempt to constrain the strength of coherent relaxation.
On the vertical axes, we have the age of different systems found in the GC as function of the semimajor axis of the stars with the object in Sgr A*. On the top of the figure we see a line giving us the timescale for normal relaxation, \(T_\mathrm{NR}\) to use the same nomenclature as the authors and their plot, which is shorter than the Hubble time but not much shorter. The following two curves from the top give the timescale for scalar coherent relaxation for two cases, the first curve from the top corresponds to a system of \(1\,M_{\odot }\) stars and \(10\,M_{\odot }\) stars at large values the effect is quenched by the presence of an extended mass, i.e., Newtonian precession and at short distances it is periapsis shift that decreases its strength. The minima displayed in the figure fence in the potential range of values for the efficiency. The “real” value probably lies somewhere in the middle.
It is, nevertheless, important to note that the authors did not take into account the effect of a mass spectrum. In this respect, while it is easier to understand the fundamentals of the scenario, the system lacks an important ingredient in realism that could significantly change the narrative.
On the lower right corner of the figure, we have vector coherent relaxation, which is much more efficient with associated timescales shorter than a million years for a short enough semimajor axis.
As I have already explained previously, there are different populations of stars in the GC that we can observe. One of these is the disc stars, some \(\sim \) 50–100 very massive and young stars observed to be orbiting on discs and almost circularly. The upper limit on the edge is of a few \(10^6\,M_{\odot }\) and, thus, the strip in the figure is very narrow. These discs are characterised by having a relatively welldefined and sharp inner cutoff. It is remarkable to note that the cutoff happens to be exactly at the place in the figure where the timescale associated with vectorial coherent relaxation (\(T^\mathrm{V}_\mathrm{RR}\) in the plot) crosses the strip, without a fit, as Hopman and Alexander (2006b) claim. On the left side of the line, we have the Sstars, which are not on circular orbits, nor aligned with the disc, but randomly orientated. They are sometimes envisaged as the lowmass members of the disc of stars. In any case, it is intriguing that these stars lie exactly on the left of the curve, where we expect any disc structure to be destroyed by vectorial coherent relaxation. This would imply that the values derived by Rauch and Tremaine (1996) are very close to the real ones. While it is probably too early to make any strong statement from this fact, it is encouraging enough to keep us studying and trying to understand normal as well as coherent relaxation in galactic nuclei. Another interpretation of Fig. 46 is that we can expect some of the Sstars to have random eccentricities due to the fact that those which are close enough are affected by scalar coherent relaxation. Also, we can in principle explain why latetype giants do not have any particular orientation in their orbits, since they are in that part of the plot.
The numerical simulations of Eilon et al. (2009) show that coherent relaxation can enhance the EMRI rate by a factor of a few over the rates predicted assuming only slow stochastic twobody relaxation, as the authors prove.
8.3 Strong mass segregation
Inspired by their work, Preto and AmaroSeoane (2010) and AmaroSeoane and Preto (2011) used directsummation simulations as a calibration to Fokker–Planck experiments that allowed them to explore this new solution. This is a priori not obvious, since we are in a regime in which scattering is dominated by uncorrelated, 2body, encounters and dense stellar cusps are robust against ejections. The authors proved that the agreement between both methods is quite good.
8.4 The cusp at the Galactic Centre
Indeed, the work by GallegoCano et al. (2018), Schödel et al. (2018) suggests that the observational data of the Galactic Centre had to be reanalysed. They show that the red and brighter giants display a corelike surface density profile within a projected radius of \(R<0.3\) pc of the central MBH, in agreement with previous studies, but show a cusplike surface density distribution at larger radii. The authors conclude that the observed stellar density at the Galactic Centre is consistent with the existence of a stellar cusp around the Milky Way’s MBH, and that it is well developed inside its influence radius. It is remarkable that this observational study agrees very well with the numerical work of Baumgardt et al. (2018). The authors of the paper ran a series of directsummation Nbody simulations of the Galactic Centre and found that the distribution of stars is what one might expect from usual twobody relaxation, without the need of invoking exotic phenomena. The comparison between the numerical simulation and the observational data is shown in Fig. 48.
We can see in the figure that by \(t \sim 0.25 \ t_\mathrm{rlx}(r_h)\), cusps with \(\gamma _L\sim 1.5\) and \(\gamma _H \sim 2\) (\(p_L\sim 0.05\) and \(p_H\sim 0.5\), where the subscript “L” refers to the light species and “H” to the heavy stars) are fully developed (\(\sim 0.02\) pc if scaled to a Milky Waylike nucleus). For masses similar to Sgr A*, \({\mathscr {M}}_{\bullet } \lesssim 5 \times 10^6 M_\odot \), this is shorter than a Hubble time. Hence, if indeed a carving event depleted the inner agglomeration of stars around the MBH, as soon as only 6 Gyr later a very steep cusp of stellarmass black holes would have had time to regrow.
I must note that this result is different to what Merritt (2010) finds, but this is probably due to the fact that the author only takes into account the effect of dynamical friction from the light stars over the heavy stars, and he neglects the scattering of the heavy stars. In this respect, he is limited in his approach to the early evolution of the system, when the heavy stars only represent a minor perturbation on the light stars. As a matter of fact, very similar results to ours were derived later by Gualandris and Merritt (2012).
The impact on EMRI production is the following: If carved nuclei were common in the range of masses relevant to an observatory like LISA, then we would be cutting down production of old remnants significantly. However, even if our Milky Way had a hole in its stellar cusp, LISA EMRI rates peak around \({\mathscr {M}}_{\bullet } \sim 4 \times 10^5  10^6 M_\odot \) and regrowth times are \(\lesssim 1\) Gyr for \({\mathscr {M}}_{\bullet } \lesssim 1.2 \times 10^6 M_\odot \), so that we still expect that a substantial fraction of EMRI events will originate from segregated stellar cusps
On the other hand, strong mass segregation not only “comes to the rescue” in the case of carved nuclei. It helps in the production of EMRIs. AmaroSeoane and Preto (2011) estimate that thanks to strong mass segregation one might expect EMRI even rates to be \(\sim 12\) orders of magnitude larger than one would expect from using the Bahcall and Wolf solution, as they show.
Their solution for the weak branch is physically unrealistic, since it predicts a too high event rate because it uses unreasonably high number fractions of stellarmass black holes \(f_\bullet \) (\(\ge 0.05\)). In a more realistic case, when \(\varDelta \sim 0.03\), (\(f_\bullet \sim 10^{3}\)) the Bahcall and Wolf solution would lead to a strong suppression of the EMRI rate to—at best—a few tens of events per Gyr.
The new solution of strong mass segregation implies a higher \(\rho _{\bullet }\) well inside the influence radius of the MBH, so that we have a boost in the diffusion of stellarmass black holes close to the MBH. When going from number fractions that are based on unrealistic IMF, such as in Bahcall and Wolf (1977) (say \(\varDelta =3\)) to realistic values (\(\varDelta =0.03\)), the event rate is suppressed by factors of \(\sim 100\)–150, if we ignore strong mass segregation. Thanks to this new solution, based on more realistic physics, even for low values of \(\varDelta =0.03\), we boost the rates from few tens to a few hundred per Gyr, \(\sim 250\)/Gyr if we consider a mass ratio of 10 between the stellarmass black holes and the MS stars and if we take a fractional number for stellarmass black holes of \(f_{\bullet }=0.001\).
8.5 Tidal separation of binaries
His work, about the tidal separation of binaries by a MBH, did not have a big impact for some 15 years until the discovery of the socalled “hypervelocity stars”, stars with a velocity of \(> 10^3\,\mathrm{km\,s}^{1}\), which he had predicted. Indeed, several such stars have been discovered in the last years. I refer the reader to Brown et al. (2009) for a discussion of the properties of these stars, as well as for references.A close but Newtonian encounter between a tightly bound binary and a million solar mass black hole causes one binary component to become bound to the black hole and the other to be ejected at up to 4000 km/s. The discovery of even one such hypervelocity star coming from the Galactic center would be nearly definitive evidence for a massive black hole. The new companion of the black hole has a high orbital velocity which increases further as its orbit shrinks by tidal dissipation. The gravitational energy released by the orbital shrinkage of such a tidal star can be comparable to its total nuclear energy release.
While one of the objects is ejected into the stellar system, the other binary member can remain bound to the MBH on a rather tight orbit. If this star happens to be a compact object, then we would have an EMRI which would be rather “immune” to the problems of EMRIs caused by twobody relaxation. Since the tidal separation happens very close to the MBH, the CO will have a shorter apoapsis (usually only tens of times the periapsis distance) and thus, potential tugs that lead it out of the capture orbit are reduced. This process was described by Miller et al. (2005). The properties of these EMRIs are very interesting and I describe the process in this section, both from an astrophysical point of view and the observational signature.
Hence, the stellarmass BH is bound to the MBH and the escaping star leaves the system with a high velocity, which is of the order of the velocity in the binary, typically of about \(\sim 10\,\hbox {km}\,s\), multiplied by the same mass ratio as in Eq. (83) but to a different power, as we can see in Eq. (88).
8.6 A barrier for captures ignored by rotating MBHs
A number of authors have addressed the question of EMRI event rates in a Milky Waylike galaxy. The numbers differ but a common denominator to all estimates is that the number of “direct plunges” is much larger than slowly decaying, “adiabatic” EMRIs. This is so simply because the region of the galaxy from which potential plunges originate contains many more stars than the volume within which we expect EMRIs, as we have seen in Sect. 6.
For a long time “plunges” have been considered to be irrelevant for the purposes for which EMRIs are best. After one intense burst of radiation, the source would be lost along with, obviously, the SBH. Some studies have looked into that, such as Hopman et al. (2007), which is probably one of the most meticulous one since it incorporates a high realism of the physics in that regime. However, the conclusions of the authors are that these sources are not interesting because they could only be detected if they originated in our own Galactic Centre. Later, Berry and Gair (2013) addressed the possible constraints on parameters of our Milky Way’s MBH if one of this bursting sources was to be observed with LISA.
In contrast, a few years later, AmaroSeoane et al. (2013b) showed that since MBH are likely to be spinning, it is actually very hard for a SBH on a plunge orbit to “hit” the MBH. They show that the majority of plunging orbits for spinning MBHs are actually not plunging but EMRI orbits. They prove that since spin allows for stable orbits very near the LSO in the case in which the EMRI is prograde, the contribution of each cycle to the SNR is much bigger than each cycle of an EMRI around a nonspinning MBH. On the other hand, retrograde orbits “push the LSO outwards” and hence, it is easier for a SBH to plunge, and the EMRI is lost. However, this situation is not symmetric, resulting in an effective enhancement of the rates. These results have been also confirmed by Will and Maitra (2017) using a different method based in a postNewtonian algorithm. In this approach these EMRI spend a lower number of cycles in the band of the detector. However, as Will and Maitra (2017) state, “(...) the PN approximation is being pushed up to or beyond its limit of validity, so we do not wish to claim too much accuracy for our values of \(T_\mathrm{{plunge}}\) in Table III.”
AmaroSeoane et al. (2013b) also show that vectorial coherent relaxation is not efficient enough to turn a prograde orbit into a retrograde one, which would be fatal for this scenario, once the evolution is dominated by GW emission. This result is crucial in the formation of EMRI sources. To understand why, first we need to introduce the problem of the socalled “Schwarzschild barrier”.
This finding has been confirmed and quantified by Brem et al. (2014) using a statistical sample of 2500 directsummation Nbody simulations using both a postNewtonian but also, and for the first time, a geodesic approximation for the relativistic orbits. However, in their work, the authors do not find a sharp transition “barrier”, but an area in phase space within which particles (stars) spend more time than outside of it.
A better way of displaying this barrier is not by following a few individual orbits, which are not representative of the phenomenon, but to depict a full presence density map. Indeed, in Figs. 55 and 56, we have the normalised presence density as a histogram in the \((a, 1e)\) plane for the Newtonian case, Fig. 56 (left panel) and the relativistic case (right panel), and I give the theoretical distribution in Fig. 55. In these figures, we see that on the right of the blue line there is a region within which stars significantly spend more time than in other areas. If we consider our specific setup, there are 3 different regions in the \((a,1e)\) plane where different mechanisms are efficient. In the right region, where pericenters are large, coherent relaxation plays the dominant role. The left border of this region is roughly given by the appearance of the Schwarzschild precession which inhibits stellarmass black holes from experiencing coherent torques (Brem et al. 2014).
This interesting and pioneering scenario would obviously imply a priori a severe suppression of EMRI event rates, if we relied on resonant relaxation. While this is true for EMRIs originating at these distances, the whole picture looks much more different at larger semimajor axis and eccentricities.
We have seen in Sect. 6.2 that the small compact object will be on a socalled “plunging orbit” if \(e\ge e_\mathrm{plunge} \equiv 14\,R^{}_\mathrm{Schw}/a\). It has been claimed a number of times by different authors that this would result in a too short burst of gravitational radiation which could only be detected if it was originated in our own Galactic Centre (Rubbo et al. 2006; Hopman et al. 2007; Yunes et al. 2008; Berry and Gair 2013) because one needs a coherent integration of some few thousand repeated passages through the periapsis in the LISA bandwidth.
Therefore, such “plunging” objects would then be lost for the GW signal, since they would be plunging “directly” through the horizon of the MBH and only a final burst of GWs would be emitted, and such burst would be very difficult to recover, since the very short signal would be buried in a sea of instrumental and confusion noise and the information contained in the signal would be practically nil.
AmaroSeoane et al. (2013b) estimated that the number of cycles that certain EMRI orbital configurations, which were thought to be plunging orbits (or orbits with no sufficient cycles), in the case of nonspinning MBHs, can spend in a frequency regime of \(f\in [10^{4},1]\) Hz during their last year(s) of inspiral before plunging into the MBH. This is important to assess how many of these EMRIs will have sufficient SignaltoNoise Ratio (SNR) to be detectable. It was found that (prograde) EMRIs that are in a “plunge” orbit actually spend a significant number of cycles, more than sufficient to be detectable with good SNR. The number of cycles has been associated with \(N^{}_{\varphi }\) (the number of times that the azimuthal angle \(\varphi \) advances \(2\pi \)) which is usual for binary systems. However, as I have discussed above, the structure of the waveforms from EMRIs is quite rich since they contain harmonics of three different frequencies. Therefore, the waveforms have cycles associated with the three frequencies \((f^{}_{r},f^{}_{\theta },f^{}_{\varphi })\) which makes them quite complex and in principle this is good for detectability (assuming we have the correct waveform templates). Moreover, these cycles happen just before plunge and take place in the strong field region very near the MBH horizon. Then, these cycles should contribute more to the SNR than cycles taking place farther away from the MBH horizon.
To sum up, the existence of the barrier prevents “traditional EMRIs” from approaching the central MBH, but if the central MBH is spinning the rate will be dominated by highlyeccentric extreme mass ratio inspirals anyway, which insolently ignore the presence of the barrier, because they are driven by chaotic twobody relaxation.
8.7 Extended stars EMRIs
In this section, I review the idea described in Freitag (2003b) that MS stars can be potential sources of GWs in our Galactic Centre. I include this in this section because in the whole review our standard CO is considered to be a SBH and so, it falls into the category of “not in the standard model”.
Indeed, a MS star can reach close enough distances to the central MBH depending on its average density and stellar structure. For a mass of around \(0.07\,M_{\odot }\), the density of the MS star is maximum and corresponds to the transition to a substellar object (Chabrier and Baraffe 2000). For masses smaller than \(0.3  0.4\,M_{\odot }\), the core is totally convective and can be described with a polytrope of index \(n = 3/2\).
8.8 The butterfly effect
An interesting effect, described in AmaroSeoane et al. (2012b), is the lack of determinism in an EMRI system if a perturbing star is close enough to the binary formed by the MBH and the SBH. One immediate question that arises is how realistic it is to assume that we can have a second star so close to the EMRI so as to perturb it.
We find that the interloper introduces an observable modification in the orbit of the EMRI when using a code that uses loss of energy via gravitational radiation at periapsis. The interesting result, though, is that when taking into account also the two firstorder nondissipative postNewtonian contributions, the orbital evolution is not deterministic. We do not know what the stellar distribution around a MBH is at such short radii, but if this scenario was possible, then the detection of EMRIs would be much more challenging than it was thought, because the waveforms developed for detection would be of little use. There has also been work about the role of a massive perturber on an EMRI. I refer the reader to Chen et al. (2011), Yunes et al. (2011), Seto (2012).
8.9 Role of the gas
Another proposal is related to the presence of massive accretion discs around MBHs. At distances of \(\sim 0.1{}1\) pc from the MBH and with typical accretion rates, these discs can be unstable to star formation (Collin and Zahn 1999; Levin and Beloborodov 2003; Goodman 2003; Goodman and Tan 2004; Milosavljević and Loeb 2004; Levin 2003, 2006; Nayakshin 2006). If, as in some calculations, there is a bias towards the production of massive stars in the disc, they could evolve to become black holes, which are then dragged in along with the disc matter. Alternately, massive stars on orbits that cross the disc could be captured and then evolve into black holes (Syer et al. 1991; Rauch 1995; Šubr and Karas 1999; Karas and Šubr 2001). Rates are highly uncertain as well as the mass of the stellar remnants formed (which could even be IMBHs). However these events would likely have a different signature waveform than those of the other two classes because they should occur on corotating, circular orbits lying in the equatorial plane of the spinning MBH if it has gained a significant fraction of its mass by accreting from the disc (Bardeen 1970; King et al. 2005; Volonteri et al. 2005). Moreover, there is the exciting possibility that in such a scenario the compact object would open a gap in the disc, which could lead to an optical counterpart to the EMRI event (Levin 2006).
Barausse et al. (2007) address the imprint on the waveform of compact, massive tori close to the central MBH. The kludge waveforms generated in their study were indistinguishable from pure Kerr waveforms in the regime on which they focused. Barausse and his collaborators later extended the study to a non selfgravitating torus with constant specific angular momentum and found that typically one should not expect big differences, although for a certain region of the parameter space the hydrodynamic drag acting on the EMRI does have an impact comparable to the radiationreaction, so that it could, in principle, be measurable (Barausse and Rezzolla 2008). Later, this work was expanded in Barausse et al. (2014). Nevertheless, it is not clear what the appropriate gas distribution around the MBH is in the regime of their study. Perturbations to the SBH are likely to be negligible if accretion onto the hole happens in a low density, radiatively inefficient flow. Such flows are much more common than dense accretion discs, which in principle could yield observable phase shifts during the inspiral (Kocsis et al. 2011), at least within the redshift range in which we expect to observe EMRIs.
9 Integration of dense stellar systems and EMRIs
9.1 Introduction
In this section, we give a summary of the current numerical approaches available for studying stellar dynamics in systems for which relaxation is an important factor.^{18}
As of writing this article, only approximate methods using a number of simplifying assumptions have been used to estimate the rates and characteristics of EMRIs. I review these approaches, their accomplishments and limitations. Thanks to the rapid computational power increase and the development of new algorithms, it is most likely that direct Nbody techniques will soon be able to robustly confirm or disprove these approximate results and extend them. One of the main issues is that exceptionally long and accurate integrations are required to account correctly for secular effects such as coherent relaxation or Kozai oscillations. These requirements, and the extreme mass ratio pose new challenges to developers of Nbody codes.
We can approximately classify the different kinds of techniques employed for studying stellar dynamics according to the dynamical regime(s) they can cope with. In Fig. 62 we have a classification of these techniques. (Semi)analytical methods are generally sufficient only to study systems which are in dynamical equilibrium and which are not affected by collisional (relaxational) processes. In all other cases, including those of importance for EMRI studies, the complications that arise if we want to extend the analysis to more complex (realistic) situations, force us to resort to numerical techniques.
The Nbody codes are the most straightforward approach from a conceptual point of view. In those, one seeks to integrate the orbital motion of N particles interacting gravitationally. It is necessary to distinguish between the direct Nbody approaches which are extremely accurate but slow and the fast Nbody approaches, which less accurate and therefore generally deemed inadequate for studying relaxing systems because relaxation is the cumulative effect of small perturbations of the overall, smooth, gravitational potential. Fast Nbody codes are usually based on either TREE algorithms (Barnes and Hut 1986) or on an FFT (Fast Fourier Transform) convolution to calculate the gravitational potential and force for each particle (Fellhauer et al. 2000) or on an SCF (selfconsistentfield) (CluttonBrock 1973; Hernquist and Ostriker 1992) approach. I will not describe these numerical techniques in this section because they have never been used to study E/IMRIs and the approximations on which they are based make them unsuitable for an accurate study of such systems, since relaxation plays a role of paramount importance. Fast Nbody algorithms can only be employed in situations in which relaxation is not relevant or over relatively short dynamical times, such as in studying bulk dynamics of whole galaxies.
On the other hand, if we want to study a system including both collisional effects and dynamical equilibrium, we can employ direct Nbody codes or use faster approaches, like the Monte Carlo, Fokker Planck and Gas methods, which we will describe below. The only technique that can cope with all physical inputs is the direct Nbody approach, in which we make no strong assumptions other than that gravity is Newtonian gravity (although nowadays postNewtonian corrections have also been incorporated, see Sect. 8.8).
It is unclear whether this effect could enhance the replenishment of the loss cone, see Murphy et al. (1991), Freitag and Benz (2002), AmaroSeoane et al. (2004), Baumgardt et al. (2004a, b), Merritt and Vasiliev (2011), Vasiliev and Merritt (2013); and Vasiliev et al. (2014) in particular for the even more complex of binaries of massive black holes, in the context of the “final parsec problem”. The problem is further compounded, for example, by the presence of multiple stellar populations whose spatial distributions are segregated (“mass segregation”), with more massive stars sinking deeper into the potential well and approaching closer to the central black hole. Besides, two interacting stars may become gravitationally bound (become a binary) so that during the subsequent interactions with other stars or massive black holes they behave differently from single stars, or they may collide into each other, then the subsequent evolution will be determined by gasdynamics. As these “microphysical” effects are usually not incorporated into the global modeling of the entire nuclear star clusters, considerable uncertainties are attached to the theoretical predictions of the abundance and orbital parameters of the stars in the relativistic regime.
Whilst assuming sphericity will probably not have any impact on the estimate of capture rates, it is of huge relevance for “tidal processes”, since this is the region in which binary tidal separation and the tidal capture of giant cores will happen. For these processes the critical radius is beyond the influence radius of the central MBH and so triaxiality can probably play an important role. Due to computer power and the limitations of simulation codes, galactic nuclei have so far been modelled only as isolated spherical clusters with purely Newtonian gravity (e.g., Murphy et al. 1991; Freitag and Benz 2002). Vasiliev (2015) used the Princeton approach to derive a new Monte Carlo code, which presents a scheme to deal with asphericity (with other issues remaining open), with the limitation that it assumes isotropy of background stars population, so that it cannot model a highly flattened system with significant rotation support.
Figure 63 shows a schematic illustration of the current available codes for stellar dynamics including relaxation. The physical realism of the codes increases from the left to the right while the computational speed decreases. The twodimensional numerical direct solutions of the Fokker–Planck equation (Takahashi 1997, 1996, 1995) probably require the least computational time, but these are followed closely by the gaseous model. The idea behind it is to treat twobody relaxation as a transport process such as in a conducting plasma (Hachisu et al. 1978; LyndenBell and Eggleton 1980). Multimass models have been implemented(Louis and Spurzem 1991; Spurzem 1992; Giersz and Spurzem 1994; Spurzem and Takahashi 1995) and improved for the detailed form of the conductivities by comparing to direct Nbody models (described below). The addition of a central accreting MBH and a treatment for losscone effects was done by AmaroSeoane et al. (2004) (a comprehensive description of the code is in the appendix of the same work) for the singlemass case, and also for a stellar mass spectrum (AmaroSeoane 2004). The advantage of these two codes is the computational time required to perform a simulation (typically of the order of one minute on a regular PC for a Hubble time) and since they are not particlebased, the resolution can be envisaged as infinite, so that they are not limited by the particle number of the system and there is practically no numerical noise. Nevertheless, although they should be envisaged as powerful tools to make an initial, fast exploration of the parameter space, the results give us tendencies of the system, rather than an accurate answer (AmaroSeoane 2004). Studying the astrophysical I/EMRI problem requires a meticulous characterisation of the orbital parameters, so that approximate techniques should be regarded as exploratory only (de Freitas Pacheco et al. 2006).
9.2 The Fokker–Planck approach
 1.
Diffusion step. The change in the distribution function F for a discrete timestep \(\varDelta t\) is computed by using the FP equation assuming the potential \(\phi \) is fixed, i.e., setting \(D_t N = {\partial N}/{\partial t} = \left. {\partial N}/{\partial t}\right _\mathrm{coll}\). The FP equation is discretized on an energy grid. The flux coefficients are computed using the DF(s) of the previous step; this makes the equations linear in the values of F on the grid points. The finitedifferencing scheme is the implicit Chang and Cooper (1970) algorithm, based on a finite difference scheme for initial value problems, which is first order in time and energy.
 2.
Poisson step. Now, the change of potential resulting from the modification in the DF F is computed and F is modified to account for the term \(\left. dE/dt\right _\phi \), i.e., assuming \(D_t N = {\partial N}/{\partial t} + {\partial N}/{\partial E}\left. {dE}/{dt}\right _{\phi } = 0\). This can be done implicitly because, as long as the change in \(\phi \) over \(\varDelta t\) is very small, the actions of each orbit are adiabatic invariants. Hence, during the Poisson step, the distribution function, expressed in terms of the actions, does not change. In practice, an iterative scheme is used to compute the modified potential, determined implicitly by the modified DF, through the Poisson equation. The iteration starts with the values of \(\phi \), \(\rho \), etc. computed before the previous diffusion step.
The use of the FP approach to determine the distribution of stars around a MBH requires a few modifications. First the (Keplerian) contribution of the MBH to the potential has to be added. Several authors have made use of the FP or similar formalisms to study the dynamics well within the influence radius under the assumption of a fixed potential (Bahcall and Wolf 1976, 1977; Lightman and Shapiro 1977; Cohn and Kulsrud 1978; Hopman and Alexander 2006b, a; Merritt et al. 2006), which is a significant simplification. The static potential included a contribution for the stellar nucleus in the last study (Merritt et al. 2006) but was limited to a Keplerian MBH potential in the other cases. The presence of the MBH also constitutes a central sink as stars are destroyed or swallowed if they come very close to it. This has to be implemented into FP codes as a boundary condition. Lightman and Shapiro (1977) and Cohn and Kulsrud (1978) have developed detailed (and rather complex) treatments of the loss cone for the anisotropic FP formalism. It can be used in a simplified way in an isotropic FP analysis (Bahcall and Wolf 1977) to obtain a good approximation to the distribution of stars around a MBH and of the rates of consumption of stars by the MBH. However, additional analysis is required to determine what fraction of the swallowed stars are EMRIs and what their orbital properties are (Hopman and Alexander 2005, 2006a).
9.3 Moment models
Another way to approximately solve the (collisional) Boltzmann equation is to take velocity moments of it. The moment or order \(n=0\) of the DF is the density, the moments of order \(n=1\) are bulk velocities and \(n=2\) corresponds to (anisotropic) pressures (or velocity dispersions). This is analogous to the derivation of the Jeans equation from the collisionless Boltzmann equation (Binney and Tremaine 1987) but the collision term introduces moments of order \(n+1\) in the equations for moments of order n.
In statistical moment models, we employ velocity moments to characterise the local velocity distribution function. The nth moment of a velocity distribution f(v) is defined as \(\langle v^n \rangle = \int (v)^n\,f(v) \,\, \mathrm {d}v\). The accuracy of these models is then limited by the order of the highest moment included to describe the velocity distribution, as discussed in detail in Schneider et al. (2011).
 0th moment:

The zeroth moment of a velocity distribution is 1 due to normalisation.
 1st moment:

The first moment of a velocity distribution is the mean velocity \(\bar{v}_r\) and denotes the bulk mass transport velocity.
 2nd moment:

The second moment of a velocity distribution is the variance \(\sigma \) and is equal to the velocity dispersion. It determines the width of \(f(v_r)\) and thus the scattering of stellar velocities around the mean velocity \(\bar{v}_r\). If \(f(v_r)\) is fully determined by \(\bar{v}_r\) and \(\sigma \) and \(h_3=h_4=0\) it is a Gaussian (upper left panel in Fig. 64) corresponding to thermal equilibrium. Then the symmetry of the onedimensional velocity distribution \(f(v_r)\) to \(\bar{v}_r\) reflects isotropy.
 3rd moment:

The third moment, denotes the transport of random kinetic energy and depends on \(h_3\). If the third moment of the velocity distribution does not vanish, implying that \(h_3\ne 0\), then the shape of the velocity distribution is a skewed Gaussian (Fig. 64, upper right panel). The asymmetry indicates the direction of the energy flux, and the uneven distribution of velocities in different directions denotes anisotropy.
 4th moment:

The fourth moment is a measure of the excess or deficiency of particles/stars with high velocities as compared to thermodynamical equilibrium, and depends on the value of \(h_4\). An excess of particles with high velocities results in thicker wings of the velocity distribution and a more pointed maximum (Fig. 64, lower left panel). A deficiency of high velocities causes a broader shape around the mean and thinner wings of the velocity distribution (Fig. 64, lower right panel).
The socalled “gaseous model”, is a particular case of moment models.^{19} In this approach, one assumes spherical symmetry (but not necessarily dynamical equilibrium) and stops the infinite set of moment equations at \(n=2\). The system is closed with the assumption that energy exchanges between stars through 2body relaxation can be approximated by an ad hoc (local) heat conduction prescription (Hachisu et al. 1978; LyndenBell and Eggleton 1980). This reduces the study of the stellar system to that of a selfgravitating conducting gas sphere. Multimass models have been implemented (Louis and Spurzem 1991; Spurzem 1992; Giersz and Spurzem 1994; Spurzem and Takahashi 1995) and the detailed forms for the conductivities have been improved by comparing to direct Nbody models (described below). The addition of a central accreting MBH and a treatment for losscone effects was done in AmaroSeoane et al. (2004) for the singlemass case (a comprehensive description of the code is in the appendix of the same work), and in AmaroSeoane (2004) for a stellar mass spectrum.
The system is treated as a continuum, which is only adequate for a large number of stars and in well populated regions of the phase space. Here, I consider spherical symmetry and singlemass stars. We handle relaxation in the Fokker–Planck approximation, i.e., like a diffusive process determined by local conditions. We also make use of the hydrodynamical approximation; that is to say, only local moments of the velocity dispersion are considered, not the full orbital structure. In particular, the effect of the twobody relaxation can be modelled by a local heat flux equation with an appropriately tailored conductivity.
\(F = (F_\mathrm{r} + F_\mathrm{t})/2\) is a radial flux of random kinetic energy. In the notion of gas dynamics it is just an energy flux. Whilst for the \(\theta \) and \(\phi \) components in the set of Eqs. (125) are equal in spherical symmetry, for the r and t quantities this is not true. In stellar clusters the relaxation time is larger than the dynamical time and so any possible difference between \(p_\mathrm{r}\) and \(p_\mathrm{t}\) may survive many dynamical times. We shall call such differences anisotropy. In case of weak isotropy (\(p_\mathrm{r}\)=\(p_\mathrm{t}\)), \(2F_\mathrm{r}\) = \(3F_\mathrm{t}\), and thus \(v_\mathrm{r}\) = \(v_\mathrm{t}\), i.e., the (radial) transport velocities of radial and tangential random kinetic energy are equal.
It has been argued that for the classical approach \(\varLambda \propto \bar{\lambda }^2/\tau \), one has to choose the Jeans’ length \(\lambda _J^2 = \sigma ^2/(4\pi G\rho )\) and the standard Chandrasekhar local relaxation time \(t_\mathrm{rlx}\propto \sigma ^3/\rho \) (LyndenBell and Eggleton 1980), where \(\bar{\lambda }\) is the mean free path and \(\tau \) the collisional time. In this context we obtain a conductivity \(\varLambda \propto \rho / \sigma \). We shall consider this as a working hypothesis. For the anisotropic model we use a mean velocity dispersion \(\sigma ^2 = (\sigma _\mathrm{r}^2 + 2\sigma _\mathrm{t}^2)/3\) for the temperature gradient and assume \(v_\mathrm{r} = v_\mathrm{t}\) (Bettwieser and Spurzem 1986).
9.3.1 Equation of continuity
9.3.2 Momentum balance equation
9.3.3 Radial energy equation
9.3.4 Tangential energy equation
9.4 Solving conducting, selfgravitating gas spheres
In this subsection, I explain briefly how the gaseous model is solved. The algorithm used is a partially implicit Newton–Raphson–Henyey iterative scheme, see Henyey et al. (1959), Kippenhahn and Weigert (1994), their Sect. 11.2.
9.5 The local approximation
There are two alternative methods for further simplification of FP or moment models. One is the orbit average, which uses the fact that that any distribution function, being a steady state solution of the collisionless Boltzmann equation, can be expressed as a function of the constants of motion of an individual particle (Jeans’ theorem). For the sake of simplicity, it is assumed that all orbits in the system are regular, as it is the case for example in a spherically symmetric potential; thus the distribution function f now only depends maximally on three independent integrals of motion (strong Jeans’ theorem). Let us transform the Fokker–Planck equation to a new set of variables, which comprise the constants of motion instead of the velocities \(v_i\). Since in a spherically symmetric system the distribution only depends on energy and the modulus of the angular momentum vector, the number of independent coordinates in this example can be reduced from six to two, and all terms in the transformed equation containing derivatives of other variables than energy and angular momentum vanish (in particular those containing derivatives of the spatial coordinates \(x_i\)). Integrating the remaining parts of the Fokker–Planck equation over the spatial coordinates is called orbit averaging, because in our present example (a spherical system) it would be an integration over accessible coordinate space for a given energy and angular momentum (which is a spherical shell between \(r_{\min }(E,\,J)\) and \(r_{\max }(E,\,J)\), the minimum and maximum radius for stars with energy E and angular momentum J). Such volume integration is, since f does not depend anymore on \(x_i\) carried over to the diffusion coefficients D, which become orbitaveraged diffusion coefficients.
Orbitaveraged Fokker–Planck models effectively deal with the diffusion of orbits according to the changes of their constants of motion, taking into account the potential and the orbital structure of the system in a selfconsistent way. However, they are not free of any problems or approximations. They require checks and tests, for example by comparisons with other methods, like the one described in the following. We treat relaxation like the addition of a big noncorrelated number of twobody encounters. Close encounters are rare and, thus, I suppose that each encounter produces a very small deflection angle. Thence, relaxation can be regarded as a diffusion process.^{21}
A typical twobody encounter in a large stellar system takes place in a volume whose linear dimensions are small compared to other typical radii of the system (total system dimension, or scaling radii of changes in density or velocity dispersion). Consequently, it is assumed that an encounter only changes the velocity, not the position of a particle. Thenceforth, encounters do not produce any changes \({\varDelta \mathbf x}\), so all related terms in the Fokker–Planck equation vanish. However, the local approximation goes even further and assumes that the entire cumulative effect of all encounters on a test particle can approximately be calculated as if the particle were surrounded by a very big homogeneous system with the local distribution function (density, velocity dispersions) everywhere. We are left with a Fokker–Planck equation containing only derivatives with respect to the velocity variables, but still depending on the spatial coordinates (a local Fokker–Planck equation).
Before going ahead the question is raised, why such approximation can be reasonable, regarding the longrange gravitational force, and the impossibility to shield gravitational forces as in the case of Coulomb forces in a plasma by opposite charges. The key is that logarithmic intervals in impact parameter p contribute equally to the mean square velocity change of a test particle, provided \(p\gg p_0\) (see, e.g., Spitzer Jr 1987, Sect. 2.1). On one side, the lower limit of impact parameters (\(p_0\), the \(90^o\) deflection angle impact parameter) is small compared to the mean interparticle distance d but, on the other side, D is a typical radius connected with a change in density or velocity dispersions (e.g., the scale height in a disc of a galaxy), and R is the maximum total dimension of the system.
Let us assume \(D=100\,d\), and \(R=100\,D\). In that case the volume of the spherical shell with radius between D and R is \(10^6\) times larger than the volume of the shell defined by the radii d and D. Nevertheless the contribution of both shells to diffusion coefficients or the relaxation time is approximately equal. This is a heuristic illustration of why the local approximation is not so bad; the reason is that there are a lot more encounters with particles in the outer, larger shell, but the effect is exactly compensated by the larger deflection angle for encounters happening with particles from the inner shell. If we are in the core or in the plane of a galactic disc the density would fall off further out, so the actual error will be smaller than outlined in the above example. By the same reasoning one can see, however, that the local approximation for a particle in a lowdensity region, which suffers from relaxation by a nearby density concentration, is prone to failure.
These simple examples should illustrate that under certain conditions the local approximation is a priori not bad. On the other hand, it is obvious from our previous arguments that, if we are interested in relaxation effects on particles in a lowdensity environment, whose orbit occasionally passes distant, highdensity regions, the local approximation could be completely wrong. One might think here, for example, of stars on radially elongated orbits in the halo of globular clusters or of stars, globular clusters, or other objects as massive black holes, on spherical orbits in the galactic halo, passing the galactic disc. In these situations an orbitaveraged treatment seems much more appropriate.
9.6 Monte Carlo codes
The Monte Carlo (MC) numerical scheme is intermediate in realism and numerical efficiency between Fokker–Planck or moment/gas approaches, which are very fast but based on a significantly idealised description of the stellar system, and direct Nbody codes, which treat (Newtonian) gravity in an essentially assumptionfree way but are extremely demanding in terms of computing time. The MC scheme was first introduced by Hénon to follow the relaxational evolution of globular clusters (Hénon 1971a, b, 1973, 1975). To my knowledge, there exist three independent codes in active development and use that are based on Hénon’s ideas. The first is the one written by M. Giersz (see Giersz 2006), which implements many of the developments first introduced by Stodołkiewicz (Stodołkiewicz 1982, 1986). The second code is the one written by K. Joshi (Cluster Monte Carlo, MCM), see Joshi et al. (2000), Joshi et al. (2001) and greatly improved and extended by A. Gürkan and J. Fregeau (see, e.g., Fregeau et al. 2003; Gürkan et al. 2004; Fregeau et al. 2005; Gürkan et al. 2006 and Pattabiraman et al. 2013 describing the latest parallel version). Finally, M. Freitag developed an MC code specifically aimed at the study of galactic nuclei containing a central MBH (Freitag and Benz 2001, 2002; Freitag et al. 2006b).^{22} The description of the method given here is based on this particular implementation.
The MC technique assumes that the cluster is spherically symmetric^{23} and represents it as a set of particles, each of which may be considered as a homogeneous spherical shell of stars sharing the same orbital and stellar properties. The number of particles may be lower than the number of stars in the simulated cluster but the number of stars per particle has to be the same for each particle. Another important assumption is that the system is always in dynamical equilibrium so that orbital time scales need not be resolved and the natural timestep is a fraction of the relaxation (or collision) time. Instead of being determined by integration of its orbit, the position of a particle (i.e., the radius R of the shell) is picked up at random, with a probability density for R that reflects the time spent at that radius: \(\mathrm {d}P/\mathrm {d}R\propto 1/V_\mathrm {r}(R)\) where \(V_\mathrm {r}\) is the radial velocity. The Freitag scheme adopts time steps that are a small fraction f of the local relaxation (or collision) time: \(\delta t(R) \simeq f_{\delta t} \left( t_\mathrm {rlx}^{1} + t_\mathrm {coll}^{1}\right) ^{1}\). Consequently the central parts of the cluster, where evolution is faster, are updated much more frequently than the outer parts. At each step, a pair of neighbouring particles is selected randomly with probability \(P_\mathrm {selec} \propto 1/\delta t(R)\). This ensures that a particle stays for an average time \(\delta t(R)\) at R before being updated.
Using a binary tree structure which allows quick determination and updating of the potential created by the particles, the self gravity of the stellar cluster is included accurately. This potential is not completely smooth because the particles are infinitesimally thin spherical shells whose radii change discontinuously. Test computations have been used to verify that the additional, unwanted, relaxation is negligible provided the number of particles is larger than a few tens of thousands.
Although Hénon’s method is based on the assumption than all departures from the smooth potential can be treated as 2body small angle scatterings, it is flexible enough to incorporate more realism. The dynamical effect of binaries (i.e., the dominant 3 and 4body processes), which may be important in the evolution of globular clusters, have been included in various MC codes through the use of approximate analytical crosssections (Stodołkiewicz 1986; Giersz and Spurzem 2000; Rasio et al. 2001). Fregeau et al. (2005), Gürkan et al. (2006), and Hypki and Giersz (2013) introduced a much more realistic treatment of binaries by onthefly, explicit integrations of the 3 or 4body interactions, a brute force approach that is necessary to deal with the full diversity of unequalmass binary interactions. This approach was pioneered by Giersz and Spurzem (2003) in a hybrid code where binaries are followed as MC particles while single stars are treated as a gaseous component. In particular, the code MCM of Joshi et al. (2000), Joshi et al. (2001) has been further developed to integrate larger numbers of particles than earlier attempts with the integrator of Fregeau et al. (2004), named RAPID, see Rodriguez et al. (2015), Fregeau and Rasio (2007), but it is limited to CPUs, and the code does not account for a central MBH in its current status.
The few 2body encounters that lead to large angle (\(> \pi /10\), say) deflections are usually neglected. In globular clusters, these “kicks” have a negligible imprint on the overall dynamics (Hénon 1975; Goodman 1983) but it has been suggested that they lead to a high ejection rate from the density cusp around a central (I)MBH (Lin and Tremaine 1980). Kicks can be introduced in the MC code, where they are treated in a way similar to collisions, with a cross section \(\pi b_\mathrm {l.a.}^2\), where \(b_\mathrm {l.a.}=f_{\mathrm {l.a.}}G(M_1+M_2)v_\mathrm {rel}^{2}\). \(f_{\mathrm {l.a.}}\) is a numerical factor to distinguish between kicks and “normal” small angle scatterings (impact parameter \(> b_\mathrm {l.a.}\)). However, simulations seem to indicate that such kicks have little influence on the evolution of a stellar cusp around a MBH (Freitag et al. 2006b).
The MC code is much faster than a direct Nbody integration: a simulation of a MilkyWaytype galactic nucleus represented by \(10^7\) particles requires between a few days and a few weeks of computation on a single CPU. Furthermore, with the proper scaling with the number of stars, the number of stars represented is independent of the number of particles. A high particle number is obviously desirable for robust statistics, particularly when it comes to rare events such as starMBH interactions. In contrast, because they treat gravitational (Newtonian) interactions on a elementary level, without relying on any theory about their collective and/or longterm effects, the results of direct Nbody codes can generally be applied only to systems with a number of stars equal to the number of particles used.
9.7 Applications of Monte Carlo and Fokker–Planck simulations to the EMRI problem
MC and FP codes are only appropriate for studying how collisional effects (principally relaxation) affect spherical systems in dynamical equilibrium. These assumptions are probably valid within the radius of influence of MBHs with masses in the LISA range. Indeed, assuming naively that the Sgr A* cluster at the centre of our Galaxy is typical (as far as the total stellar mass and density is concerned) and that one can scale to other galactic nuclei using the \(M  \sigma \) relation in the form \(\sigma = \sigma _\mathrm{MW} ({\mathscr {M}}_{\bullet }/3.6\times 10^6\,M_{\odot })^{1/\beta }\) with \(\beta \approx 45\) (Ferrarese and Merritt 2000; Tremaine et al. 2002), one can estimate the relaxation time at the radius of influence to be \(t_\mathrm {rlx}(R_\mathrm{infl}) \approx 25\times 10^9\,\mathrm{yr}\,({\mathscr {M}}_{\bullet }/3.6\times 10^6\,M_{\odot })^{(23/\beta )}\).
Although observations suggest a large spread amongst the values of the relaxation time at the influence radius of MBHs with similar masses (see, e.g., Fig. 4 of Merritt et al. 2006), most galactic nuclei hosting MBHs less massive than a few \(10^6\,M_{\odot }\) are probably relaxed and amenable to MC or FP treatment. Even if the age of the system is significantly smaller than its relaxation time, such approaches are valid as long as the nucleus is in dynamical equilibrium, with a smooth, spherical distribution of matter. In such conditions, relaxational processes are still controlling the EMRI rate, no matter how long the relaxation time is, but one cannot assume a steadystate rate of diffusion of stars onto orbits with small periapsis, as is often done in FP codes (see the discussion in Milosavljević and Merritt (2003), in the different context of the evolution of binary MBHs).
The Hénontype MC scheme of Freitag and Benz (2002) has been used to determine the structure of galactic nuclei (Freitag and Benz 2002; Freitag et al. 2006b). Predictions for the distribution of stars around a MBH have also been obtained by solving some form of the Fokker–Planck equation (Bahcall and Wolf 1977; Murphy et al. 1991; Hopman and Alexander 2006b, a; Merritt et al. 2006) or using the gaseous model (AmaroSeoane 2004; AmaroSeoane et al. 2004). These methods have proved useful to determine how relaxation, collisions, largeangle scatterings, MBH growth, etc., shape the distribution of stars around the MBH, which is an obvious prerequisite for the determination of the rate and characteristics of EMRIs. Of particular importance is the inward segregation of stellar BHs as they lose energy to lighter objects. This effect, combined with the fact that stellar BHs produce GWs with higher amplitude than lowermass stars, explains why they are expected to dominate the EMRI detection rate (Sigurdsson and Rees 1997; Hopman and Alexander 2006a). An advantage of the MC approach is that it can easily and realistically include a continuous stellar mass spectrum and extra physical ingredients. However, the first point might not be critical here as MC results suggest that, for models where all the stars were born \(\sim \) 10 Gyr ago, the pattern of mass segregation can be well approximated by a population of two components only, one representing the stellar BHs and the other representing all other (lighter) objects (Freitag et al. 2006b). Furthermore, the uncertainties are certainly dominated by our lack of knowledge about where and when stellar formation takes place in galactic nuclei, what the masses of the stars which form might be, and what type of compact remnants they become.
The most recent FP results concerning mass segregation were obtained under the assumptions of a fixed potential and an isotropic velocity dispersion, with the effects of (standard or resonant) relaxation being averaged over angular momentum at a given energy. The MC code includes the selfgravity of the cluster so the simulated region can extend past the radius of influence, allowing a more natural outer boundary condition. We note that one has to impose a steeper density dropoff at large radii than what is observed to limit the number of particles to a reasonable value while keeping a good resolution in the region of influence. The MC code naturally allows anisotropy and implicitly follows relaxation in both energy and angular momentum. Anisotropic FP codes for spherical selfgravitating systems exist (Takahashi 1996, 1997; Drukier et al. 1999) but, to our knowledge, none are currently in use that also include a central MBH. Unique amongst all stellar dynamical codes based on the Chandrasekhar theory of relaxation is Fopax, a FP code which assumes axial rather than spherical symmetry, thus permitting the study of clusters and nuclei with significant global rotation (see Fiestas et al. 2006 and references therein) and which has been adapted to include a central MBH (Fiestas 2006).
Determining the EMRI rates and characteristics is a harder challenge for statistical stellar dynamics codes because these events are intrinsically rare and critically sensitive to rather fine details of the stellar dynamics around a MBH. As I explained previously, the main difficulty, in comparison with, for example, tidal disruptions, is that EMRIs are not “onepassage” events but must be gradual. The first estimate of EMRI rates was performed by Hils and Bender (1995). Assuming a static cusp profile, they followed the evolution of the orbits of testparticles subject to GW emission, Eqs. (71) and (72), and 2body relaxation introduced by random perturbations of the energy and angular momentum according to precomputed “diffusion coefficients”. Hopman and Alexander (2005) have used a refined version of this “singleparticle Monte Carlo method”, as well as the Fokker–Planck equation, to make a more detailed analysis. It was found that no more than \(\sim 10\%\) of the compact objects swallowed by the MBH are EMRIs, while the rest are direct plunges.
Determination of EMRI rates and characteristics were also attempted with Freitag’s MC code (Freitag 2001, 2003a, b). Despite its present limitations, this approach might serve to inspire future, more accurate, computations and is therefore worth describing in some detail. The MC code does not include GW emission explicitly (or any other relativistic effects). At the end of each step in which two particles have experienced an encounter (to simulate 2body relaxation), each particle is tested for entry into the “radiationdominated” regime, defined by Eq. (65) (with \(C_\mathrm{EMRI}=1\)). A complication arises because the time step \(\delta t\) used in the MC code is a fraction \(f_{\delta t}=10^{3}10^{2}\) of the local relaxation time \(t_\mathrm {rlx}(R)\), which is generally much larger than the critical timescale defined by the equality \(\tau _\mathrm {GW}(e,a) = C_\mathrm{EMRI}\, (1e)t_\mathrm {rlx}\). In other words, the effective diffusion angle \(\theta _\mathrm{eff}\) is generally much larger than the opening angle of the “radiation cone”, \(\tilde{\theta }\equiv (1\tilde{e})^{1/2}\). So that the entry of the particle into the radiation cone (corresponding to a possible EMRI) is not missed, it is assumed that, over \(\delta t\), the energy of a given particle does not change. Hence, each time it comes back to a given distance from the centre, its velocity vector has the same modulus but relaxation makes its direction execute a random walk with an individual step per orbital period of \(\theta _\mathrm{orb} = \theta _\mathrm{eff} (P_\mathrm{orb}/\delta t)^{1/2}\). Entry into the unstable or radiation cone is tested at each of these substeps. If the particle is found on a plunge or radiationdominated orbit, it is immediately removed from the simulation and its mass is added to the MBH.
Unfortunately, in addition to this approximate way of treating relaxation on small time scales, there are a few reasons why the results of these simulations may be only indicative. One is the way \(t_\mathrm {rlx}\) is estimated, using the coefficient in front of \(\delta t\) in Eq. (177), i.e., an estimate based on the neighbouring particle. Even if it is correct on average, this estimate is affected by a very high level of statistical noise and its value can be far too long in some cases (e.g., when the relative velocity between the particles in the pair is much larger than the local velocity dispersion). This could lead one to conclude erroneously that a star has reached the radiationdominated regime and will become an EMRI. To improve on this one could base the \(t_\mathrm {rlx}\) estimate on more than one point on the orbit and on more than one “fieldparticle” (the number of stars within a distance of \(10^{2}\) pc of Sgr A* is probably larger than 1000, so \(t_\mathrm {rlx}\) is a welldefined quantity even at such small scales). Another limitation is that GW emission is not included in the orbital evolution, which forces one to assume an abrupt transition when \(\tau _\mathrm{GW} = (1e)t_\mathrm {rlx}\). Hopman and Alexander (2005) have also shown that a value of \(C_\mathrm{EMRI}\) as small as \(10^{3}\) might be required to be sure the EMRI will be successful. Furthermore, the MC simulations carried out so far suffer from relatively poor resolution, with each particle having the statistical weight of a few tens of stars. To improve this one would need to limit the simulation to a smaller volume (such as the influence region) or develop a parallel implementation of the MC code to use \(\sim 10^8\) particles.
9.8 Directsummation Nbody codes
We finally consider the direct Nbody approach (Aarseth 1999, 2003; Portegies Zwart et al. 2001). This is the most expensive method because it involves integrating all gravitational forces for all particles at every time step, without making any a priori assumptions about the system. The Nbody codes use the improved Hermite integration scheme as described in Aarseth (1999, 2003), which requires computation of not only the accelerations but also of their time derivatives. Since these approaches integrate Newton’s equations directly, all Newtonian gravitational effects are included naturally. More relevant for this subject is that the family of the direct Nbody codes of Aarseth also includes versions in which both KS regularisation and chain regularisation are employed, so that when particles are tightly bound or their separation becomes too small during a hyperbolic encounter, the system is regularised (as described first in Kustaanheimo and Stiefel 1965; Aarseth 2003) to prevent dangerous small individual time steps. This means that we can accurately follow and resolve individual orbits in the system. Other schemes which make use of a softening in the gravitational forces (i.e., \(1/(r^2+\epsilon ^2)\) instead of \(1/r^2\), where \(\epsilon \) is the softening parameter) cannot be employed because \(\epsilon \) can induce unacceptable errors in the calculations. The Nbody codes scale as \(N_{\star }^2\), or \(\varDelta t \propto t_\mathrm{dyn}\), which means that even with specialpurpose hardware, a simulation can take of the order of weeks if not months. This hardware is the GRAPE (short for GRAvity PipE), a family of hardware which acts as a Newtonian force accelerator. For instance, a GRAPE6A PCI card has a peak performance of 130 Gflop, roughly equivalent to 100 single PCs (Fukushige et al. 2005). It is possible to parallelise basic versions of the direct Nbody codes (without including regularisation schemes) on clusters of PCs, each equipped with one GRAPE6A PCI card. This leads to efficiencies greater than 50% and speeds in excess of 2 TFlops and thus the possibility of simulating up to \(N_{\star } = 2\cdot 10^6\) stars (Harfst et al. 2006). Nevertheless, when we consider the situation relevant to an EMRI, in which mass ratios are large and we need to follow thousands of orbits, the Hermite integrator is not suitable and problems show up even in the Newtonian regime. Aarseth (2006, 2003) summarise different methods developed to cope with large systems with one or more massive bodies. The problem becomes even more difficult when including relativistic corrections to the forces when the stellarmass black hole approaches the central MBH, because extremely small timescales are involved in the integration. Progress is being made in this direction with a developed timetransformed leapfrog method (Mikkola and Aarseth 2002) (for a description of the leapfrog integrator see Mikkola and Merritt 2006) and the even more promising wheelspoke regularisation, which was developed to handle situations in which a very massive object is surrounded by strongly bound particles, precisely the situation for EMRIs (Zare 1974; Aarseth 2003). Additionally, one must include postNewtonian corrections in the direct Nbody code because secular effects such as Kozai or resonant relaxation may be smoothed out significantly by relativistic precession and thus have an impact on the number of captures, see, e.g., Merritt et al. (2011).
9.8.1 Relativistic corrections: the postNewtonian approach
Direct Nbody have been modified to take into account the role of relativity. The first inclusion of relativistic corrections at 1PN, 2PN (periapsis shifts) and 2.5PN (energy loss in the form of gravitationalwave emission) in an Nbody code was presented in Kupi et al. (2006). Later, in Brem et al. (2013), we presented the first implementation of the effect of spin in mergers in a directsummation code, NBODY6. We employ nonspinning postNewtonian (PN) corrections to the Newtonian accelerations up to 3.5 PN order as well as the spinorbit coupling up to nexttolowest order and the lowest order spinspin coupling.
These KS pairs are only formed when the interaction between two bodies becomes strong enough so that the pair, as mentioned, has to be regularised. During the KS regularisation the relative motion of the companions is still far from relativistic. Hence, only a small, relativistic subset of all regularised KS pairs will need postNewtonian corrections.
Whilst the gauge choice was not a problem for the system studied in Kupi et al. (2006), since we were interested in the global dynamical evolution, for the EMRI problem the centreofmass frame (located at the origin of the coordinates) must be employed. The integration cannot be extended to velocities higher than \(\sim \) 0.3 c, because at these velocities the postNewtonian formalism can no longer be applied accurately. This means that we cannot reach the final coalescence of the stellar BH with the MBH, but this is not a big issue, because this part of the evolution does not contribute significantly to the SNR of the GW signal. We note that it will not be possible to include in Nbody codes all the \(\mathrm{PN}\) corrections that are required for accurate modelling of the phase evolution of the EMRI during the last few years before plunge. However, the Nbody codes are not required in that regime, since the system is then decoupled from the rest of the stellar cluster.
9.8.2 Relativistic corrections: a geodesic solver
Brem et al. (2014) presented, for the first time, a geodesic approximation for the relativistic orbits in an Nbody code. I show in this section, the geodesic equations of motion in a form that is suitable to be included in an Nbody code that uses a Newtoniantype formulation of the equations of motion (initially presented in the appendix of Brem et al. 2014). Also, so as to be able to compare results with postNewtonian approach, I show the geodesic equations using harmonic coordinates for Schwarzschild, which are compatible with the harmonic gauge condition of postNewtonian theory.
9.8.3 Nbody units and conversion
Footnotes
 1.
This term was first employed by John Archibald Wheeler (b. 1911).
 2.
If we do not take into account the ionising radiation on intergalactic medium.
 3.
The number of orders of magnitude between two numbers. This means that if we have two numbers within one dex, the ratio between the larger and the smaller number is less than one order of magnitude.
 4.
 5.
 6.
In the related literature there exist other terms to refer to the distance of maximum or minimum approach to a black hole; namely peribarathron and apobarathron, respectively. There seems to be a confusion and wrong use of the later. I discuss this in Sect. A.
 7.
The terminology is somehow, and as forewarned, misleading; whilst in general we refer to “collisional” to any effect leading to exchange of energy and angular momentum among stars, here I mean real collisions between two stars. For a thorough discussion of the mechanism and an extremely detailed numerical study, I refer the reader to Freitag and Benz (2002).
 8.
We can relate standard Nbody time units \(T_\mathrm{NB}\) as defined in, e.g., Heggie and Hut (2003) to Fokker–Planck time units \(T_\mathrm{FP}\) as follows: \(T_\mathrm{FP} = T_\mathrm{NB} \cdot {N}_{\star } / \ln (\gamma \cdot {N}_{\star })\), with \({N}_{\star }\) the number of stars in the system.
 9.
We note that eight years earlier, the article by Gurevich (1964) had an interesting first idea of this concept: The authors obtained a similar solution for how electrons distribute around a positively charged Coulomb centre.
 10.
The zero age main sequence (ZAMS) corresponds to the position of stars in the Hertzsprung–Russell diagram where stars begin hydrogen fusion.
 11.
The systems emits gravitational radiation all the time, but the most important bursts of energy occur at periapsis.
 12.
This is not strictly true, the spin of the MBH might “push out” the LSO and so Schwarzschild plunges are Kerr EMRIs; see AmaroSeoane et al. (2013b).
 13.
 14.
The reason for the title of this section is that probably the choice of “resonant” for this process is not a good one. Rauch and Tremaine (1996) coined this term thinking of the effect of a resonance between the radial and azimuthal periods in a Keplerian orbit.
 15.
 16.
For the derivation and some examples of values for \({\mathscr {W}}\), I refer the reader to AmaroSeoane et al. (2013b).
 17.
The authors obtained a similar solution for how electrons distribute around a positively charged Coulomb centre.
 18.
A part of this section profits from AmaroSeoane et al. (2007), though some parts have been significantly expanded and improved.
 19.
 20.
Memory usage is also reduced, scaling like \({\mathscr {O}}(N_\mathrm{r})\) rather than \({\mathscr {O}}(N_\mathrm{r}^2)\).
 21.
Anyhow, it has been argued that rare deflections with a large angle may play a important role in the vicinity of a BH (Lin and Tremaine 1980).
 22.
 23.
But see Vasiliev (2015), who has developed a MC code base on the Princeton approach. The code features a scheme to deal with asphericity, with the limitation that it assumes isotropy of the population of background stars, so that it cannot model a highly flattened system with significant rotation support.
 24.
Unfortunately, I was unable to reproduce the IndoEuropean root with the appropriate diacritical marks, due to limitations of typography, so I have approximated it.
Notes
Acknowledgements
I first of all thank my children, Antón and Natalia, for being the nicest children in the world, and for their daily present of smiles during the long preparation of this work, in spite of my spastic and even spasmodic changes of mood as the article progressed (or retrogressed).
I am particularly thankful to one of my best and closest friends, and also one of the brightest minds in this field, Tal Alexander. Tal has given to me not just the present of his friendship, humour, company and countless hours of smiles and laugh, which have made my life more beautiful, but also his wit and sharpness in virtually any subject we discussed, in science or any other intellectual topic. He is a reference to me as human being and scientist. Many of the ideas I had would have never happened if it had not been for my discussions with him.
It is a pleasure for me to show my most sincere gratitude to Francine Leeuwin and Marc Dewi Freitag. The many discussions contributed enormously to the writing up of this review. More importantly, they “also” contributed enormously to my human formation during my freshman years of PhD student in Heidelberg. I am also thankful to Bernard Schutz, Carlos F. Sopuerta, Xian Chen, Steve Drasco, Rainer Spurzem, Rainer Schödel, Simos Konstantinidis, Miguel Preto, and Cole Miller both for discussions and their friendship (and waveforms, in the case of Steve). I am indebted to Emily Davidson for her titanic work of checking my abhorrent English, but also to Emma Robinson, Jon Gair, Melissa and Taka Tanaka. Part of this work has been finished at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, Germany.
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