Dancing to ChaNGa: A Self-Consistent Prediction For Close SMBH Pair Formation Timescales Following Galaxy Mergers
Michael Tremmel, Fabio Governato, Marta Volonteri, Thomas R. Quinn, Andrew Pontzen
MMon. Not. R. Astron. Soc. , 1–11 (2015) Printed 18 September 2018 (MN L A TEX style file v2.2)
Dancing to ChaNGa: A Self-Consistent Prediction For Close SMBHPair Formation Timescales Following Galaxy Mergers
M. Tremmel , (cid:63) , F. Governato , M. Volonteri , T. R. Quinn , A. Pontzen Astronomy Department, University of Washington, Box 351580, Seattle, WA, 98195-1580 Yale Center for Astronomy & Astrophysics, Physics Department, P.O. Box 208120, New Haven, CT 06520, USA Sorbonne Universitès, UPMC Univ Paris 6 et CNRS, UMR 7095, Institut d‘Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France Department of Physics and Astronomy, University College London, 132 Hampstead Road, London, NW1 2PS, United Kingdom
18 September 2018
ABSTRACT
We present the first self-consistent prediction for the distribution of formation timescalesfor close Supermassive Black Hole (SMBH) pairs following galaxy mergers. Using R omu - lus
25, the first large-scale cosmological simulation to accurately track the orbital evolution ofSMBHs within their host galaxies down to sub-kpc scales, we predict an average formationrate density of close SMBH pairs of 0.013 cMpc − Gyr − . We find that it is relatively rare forgalaxy mergers to result in the formation of close SMBH pairs with sub-kpc separation andthose that do form are often the result of Gyrs of orbital evolution following the galaxy merger.The likelihood and timescale to form a close SMBH pair depends strongly on the mass ratioof the merging galaxies, as well as the presence of dense stellar cores. Low stellar mass ratiomergers with galaxies that lack a dense stellar core are more likely to become tidally disruptedand deposit their SMBH at large radii without any stellar core to aid in their orbital decay, re-sulting in a population of long-lived ‘wandering’ SMBHs. Conversely, SMBHs in galaxiesthat remain embedded within a stellar core form close pairs in much shorter timescales onaverage. This timescale is a crucial, though often ignored or very simplified, ingredient tomodels predicting SMBH mergers rates and the connection between SMBH and star forma-tion activity. Key words:
Supermassive black holes: cosmological simulations: Gravitational Waves
Despite their importance to galaxy evolution theory, the mech-anisms driving the co-evolution of supermassive black holes(SMBHs) and their host galaxies, and indeed the processes thatform SMBHs in the first place, are highly uncertain. SMBHs areubiquitous in galaxies ranging from massive ellipticals and bulge-dominated galaxies (e.g. Gehren et al. 1984; Kormendy & Rich-stone 1995; Kormendy & Ho 2013) to smaller, bulge-less diskgalaxies and dwarfs (Shields et al. 2008; Filippenko & Ho 2003;Reines et al. 2011; Reines & Deller 2012; Reines et al. 2013; Moranet al. 2014). Empirical scaling relationships between the mass ofSMBHs and that of their host galaxies are indicative of coevalgrowth (Häring & Rix 2004; Gültekin & et al. 2009; Schramm& Silverman 2013; Kormendy & Ho 2013; Volonteri & Bellovary2012).Future observations of gravitational waves emitted from bi-nary and merging SMBHs via pulsar timing arrays (Sesana 2013)and the planned LISA mission (Klein et al. 2016) will provideunique information on the SMBH population and its dynamical (cid:63) email: [email protected] evolution. Pulsar timing arrays probe relatively low-redshift (z < > M (cid:12) ), while LISA can detectmergers of SMBHs with mass ∼ − M (cid:12) out to the high-est redshift. LISA has therefore the capability to provide uniqueconstraints to the SMBH mass function across cosmic time as wellas critical insight into their possible formation mechanisms (Sesanaet al. 2007; Volonteri & Natarajan 2009; Klein et al. 2016) and theirgrowth and spin evolution (Berti & Volonteri 2008; Barausse 2012).Further, on-going observations, as well as large-scale cosmologicalsimulations, of active SMBHs that are o ff set from the centre of theirhost galaxies, possibly in galaxies with multiple luminous SMBHs,can potentially help constrain the extent to which galaxy mergersdrive SMBH growth (Comerford et al. 2015; Steinborn et al. 2016;Barrows et al. 2017).The formation of a SMBH binary and subsequent merger oftwo SMBHs can be described in a number of stages. First, a darkmatter halo falls into a halo of larger mass. It then sinks to thecentre via dynamical friction and the two central galaxies then be-gin to strongly interact and merge. Following the merger of twogalaxies hosting SMBHs, dynamical friction acting on the SMBHscauses them to sink to galactic centre and form a close pair withsub-kiloparsec (kpc) separation. The close pair, through dynami- © a r X i v : . [ a s t r o - ph . GA ] J a n M. Tremmel et al. cal interactions with gas and stars, then forms a bound SMBH bi-nary ( D <
10 pc), which then itself hardens to the point wheregravitational wave emission causes rapid orbital decay and the twoSMBHs merge ( D < . ff ects of di ff erent morphology and merger dynamicsare naturally accounted for without a priori assumptions, as eachgalaxy in the simulation has a cosmologically realistic accretionand merger history. However, past simulations generally had poorresolution, which required simplified assumptions such as ‘advec-tion’, where SMBHs quickly sink into the deepest nearby potentialwell, resulting in unrealistic, nearly instantaneous SMBH orbitaldecay. This approximation contrasts with the above numerical re-sults as it assumes that the orbital sinking timescale on kpc scalesis e ff ectively zero. In previous works we have shown that this tech-nique often results in inaccurate SMBH dynamics within galaxiesand a drastic underestimate of sinking timescales (Tremmel et al.2015). While current simulations are beginning to employ more de-tailed approaches to SMBH dynamics (e.g. Hirschmann et al. 2014;Dubois et al. 2016; Steinborn et al. 2016), accurate orbital evolutiondown to sub-kpc scales remains a challenge.In this Paper, using the R omulus
25 cosmological simulation(Tremmel et al. 2017) which is uniquely able to predict the or-bital evolution of SMBHs down to sub-kpc scales (Tremmel et al.2015), we present the first robust estimate of SMBH sinking andsubsequent close SMBH pair formation timescales over a range ofcosmic epochs and galaxy properties.
The R omulus
Simulations are a set of large-scale, high resolutioncosmological simulations with emphasis on implementing a novelapproach to SMBH formation, dynamics, and accretion. For thiswork, we focus on R omulus
25, our flagship 25 Mpc per side vol-ume simulation, as it provides a uniform sample of galaxies withina wide range of halo masses (3 × to 2 × M (cid:12) ). The sim-ulation is run assuming a Λ CDM cosmology following the most recent results from Planck ( Ω = . Λ = . = . σ = .
77; Planck Collaboration et al. 2016), a Plummer equiva-lent force softening of 250 pc (a 350 pc spline force softening isused), and mass resolution for dark matter and gas of 3 . × and 2 . × M (cid:12) respectively. The simulation was run using thenew Tree + SPH code, C ha NG a (Menon et al. 2015), which in-cludes an updated SPH implementation that accurately simulatesshearing flows with Kelvin-Helmholtz instabilities. The Simula-tions also include the standard physics modules previously usedin GASOLINE, such as a cosmic UV background, star formation,‘blastwave’ SN feedback, low temperature metal cooling (Wadsleyet al. 2004, 2008; Stinson et al. 2006; Shen et al. 2010), as well as anovel implementation of SMBH formation, growth, and dynamics(Tremmel et al. 2015, 2017).As described in more detail in Tremmel et al. (2017), thefree parameters within our sub-grid models for star formation andSMBH physics (see §2.1) are optimized and held constant. Thiswas achieved using a large set of ‘zoomed-in’ simulations of galax-ies within dark matter halos with masses 10 . , 10 . , and 10 M (cid:12) . Each set of galaxies was 1) run using a di ff erent set of param-eters and 2) graded against di ff erent z = ffi ciency, gas fraction, angular momentum, andblack hole growth. This resulted in fully specified sub-grid modelsgoverning star formation, stellar feedback, and SMBH accretionand feedback that are optimized to provide realistic z = h > M (cid:12) ). R omulus
25 has been shown to reproduce thez = predicts cosmicstar formation and SMBH accretion histories at high redshift thatare consistent with observations (Tremmel et al. 2017). Accretion of gas onto SMBHs is governed by a modified Bondi-Hoyle prescription. Using the same energy balance argument as inthe derivation of Bondi-Hoyle, we re-derive the SMBH accretionradius to include the e ff ect of angular momentum support based onthe resolved dynamics of gas in the simulation. We also apply adensity dependent boost factor to account for the unresolved mul-tiphase nature of the ISM near a SMBH (Booth & Schaye 2009),giving us the final equation˙ M = (cid:32) nn th , ∗ (cid:33) β π ( GM ) ρ ( v + c s ) / if v bulk > v θπ ( GM ) ρ c s ( v θ + c s ) if v bulk < v θ . (1)The tangential velocity, v θ , is estimated at the smallest resolvedscales in the simulation and compared to v bulk , the overall bulkmotion of the gas that already enters into the Bondi-Hoyle model.When the bulk motion dominates over the nearby rotational mo-tion, or the energetics are dominated by the internal energy of thegas, the accretion reverts to the normal Bondi-Hoyle prescription.The threshold for star formation, n th , ∗ , also determines the thresholdbeyond which we assume gas becomes multiphase and poorly re-solved, requiring a boost to the approximated accretion rate. Forlower densities, we assume that the gas is not su ffi ciently mul-tiphase to require such a boost, as in Booth & Schaye (2009). © , 1–11 ancing to ChaNGa How much this boost increases with density is governed by β , con-strained by our parameter search to be 2.An accreting SMBH converts a fraction of that mass, (cid:15) r , intoenergy. A fraction of this energy, (cid:15) f , is thermally coupled to the 32nearest gas particles according to the smoothing kernel. We assumethe common value of 10% for (cid:15) r and take (cid:15) f as a free parameteragain set by our parameter search technique to be 0.02. For moredetails on SMBH accretion and feedback in R omulus , we refer tothe reader to Tremmel et al. (2017). We note that while there stillexists issues with the Bondi-Hoyle formalism even in the regime ofnon-rotating gas (e.g. Hobbs et al. 2012), for the spatial and timeresolution of these simulations, it still represents the best way ofapproximating long term accretion onto SMBHs based on large-scale gas properties without requiring additional assumptions. SMBHs are seeded in the simulation based on gas properties, form-ing in rapidly collapsing, low metallicity regions in the early Uni-verse. We isolate pristine gas particles ( Z < × − ) that havereached densities 15 times higher than what is required by ourstar formation prescription without forming a star or cooling be-yond 9 . × K (just below the temperature threshold used forstar formation, 10 K). These regions are collapsing on timescalesmuch shorter than the cooling and star formation timescales and aremeant to approximate the regions that would form large SMBHs,regardless of the details of their formation mechanism. Tremmelet al. (2017) show how this method forms most SMBHs withinthe first Gyr of the simulation, compared with the later seedingtimes inherent to common approaches that seed based on halo massthresholds (e.g. Di Matteo et al. 2008; Genel et al. 2014; Schayeet al. 2015).The seed SMBH mass is set to 10 M (cid:12) and is justified by ourchoice of formation criteria, which would produce SMBHs that areable to attain higher masses quickly, as there is a lot of dense, col-lapsing gas nearby that is unlikely to form stars. Critically for ouranalysis presented here, this initial mass guarantees that SMBHsalways have a mass significantly larger than DM and gas particles,allowing us to correctly resolve their dynamics without resortingto ad hoc simplifications (Tremmel et al. 2015). This approach re-sults in an evolving occupation fraction. At early times, small halos(M vir ∼ − M (cid:12) ) host massive, newly formed SMBHs. The occu-pation fraction evolves due to hierarchical merging and the fact thathalos in less dense regions are less likely to host such dense collaps-ing regions at early times. Less than 10% of halos with 3 × < M vir < M (cid:12) host a SMBH of mass at least 10 M (cid:12) at z = Following the merger of two galaxies hosting SMBHs, the accretedSMBHs sink toward the centre of the descendant galaxy throughdynamical friction, the force exerted by the gravitational wakecaused by a massive body moving through an extended medium(Chandrasekhar 1943; Kazantzidis et al. 2005; Binney & Tremaine2008). However, the limited mass and gravitational force resolution
Log Primary Galaxy Stellar Mass [M (cid:12) ] S t e ll a r M a ss R a t i o . . . . . . . . . F r a c t i o n o f G a l a xy M e r g e r s R e s u l t i n g i n C l o s e S M B H P a i r s Figure 1. L ikelihood of C lose SMBH P air F ormation . The fraction of allgalaxy mergers that result in a close SMBH pair as a function of the stellarmass of the primary galaxy and the stellar mass ratio at the time of firstsatellite in-fall. In addition to the colours, the fraction and, in parenthesis,associated uncertainty ( n . pairs , i / n i , where n i is the total number of galaxymergers in each bin and n pairs , i the number of close SMBH pairs resultingfrom mergers in each bin) are labeled. Considered are galaxy mergers re-sulting from initial satellite in-fall at z <
5. The formation of a close SMBHpair is not a common result of galaxy mergers. The likelihood of a closeSMBH pair forming is sensitive to both stellar mass and mass ratio, andmost likely to occur in massive, major mergers. of cosmological simulations leaves this process largely unresolved.The Romulus simulations uniquely include the sub-grid correc-tion accounting for this unresolved dynamical friction described inTremmel et al. (2015) that has been shown to produce realisticallysinking SMBHs (see appendix A for tests of this prescription at thespecific resolution of R omulus (cid:15) g , and assuming a locally isotropic velocitydistribution and constant density out to (cid:15) g . a DF = − π G M ρ ( < v BH )ln Λ v BH v BH . (2)The velocity of the SMBH, v BH , is taken relative to the local cen-tre of mass (COM) velocity of stars and DM. We have also as-sumed that the contribution from objects moving faster than theSMBH is negligible, where ρ ( < v BH ) is the density of particlesmoving slower than the SMBH relative to the local COM. Thisis a good assumption to make for dynamical evolution on scalesmuch larger than 1 pc (Antonini & Merritt 2012). The coulomblogarithm, ln Λ , is taken to be ln( b max b min ), where b max = (cid:15) g to avoiddouble counting the resolved dynamical friction that is already oc-curring on larger scales and b min is the 90 ◦ deflection radius, with alower limit set to the Schwarzschild Radius, R Sch . The calculationis done based the 64 nearest star and DM particles to each SMBH.R omulus
25 achieves mass resolution such that the ambient darkmatter, gas, and star particles are significantly less massive than thesmallest SMBHs, allowing it to avoid the numerical e ff ects that per-sist at low resolution even with this dynamical friction prescription(Tremmel et al. 2015).This is a critical improvement over standard approaches tocorrecting SMBH dynamics that involve repositioning or pushingSMBHs toward their local potential minima (e.g. Di Matteo et al.2005, 2008; Genel et al. 2014; Schaye et al. 2015). Such methods ©000
25 achieves mass resolution such that the ambient darkmatter, gas, and star particles are significantly less massive than thesmallest SMBHs, allowing it to avoid the numerical e ff ects that per-sist at low resolution even with this dynamical friction prescription(Tremmel et al. 2015).This is a critical improvement over standard approaches tocorrecting SMBH dynamics that involve repositioning or pushingSMBHs toward their local potential minima (e.g. Di Matteo et al.2005, 2008; Genel et al. 2014; Schaye et al. 2015). Such methods ©000 , 1–11 M. Tremmel et al. force an un-physically fast sinking timescale for accreted SMBHs,leading to a nearly immediate formation of a close SMBH pairthat does not sample the properties or kinematics of the merginggalaxies (Tremmel et al. 2015). With this technique, the dynamicsand morphology of the merging galaxies are self-consistently ac-counted for in the SMBH sinking timescales and the subsequentformation (or not) of a close SMBH pair.
SMBHs are assumed to form a close pair when they become closerthan two softening lengths ( ≈
700 pc in our simulations) withrelative velocities small enough such that they can be consideredbound, i.e. ∆ v < ∆ a · ∆ r , where ∆ v , ∆ a , and ∆ r are the relativevelocity, acceleration, and distance vectors between two SMBHparticles. Below this distance limit, the simulation fails to resolvethe relevant stellar and gas dynamical processes involved in SMBHpair evolution and such calculations are not attempted.In the simulation, once two SMBHs form a close pair, theyare taken to act as a single SMBH with the sum of the masses.While there are still many theoretical uncertainties in the timescalesto form and merge a binary SMBH system, binary hardeningtimescales can be relatively quick, on the order of 10 − yrs,if even a small amount of gas is present (Armitage & Natarajan2002; Haiman et al. 2009; Colpi 2014), and even in some cases forgas poor systems (Holley-Bockelmann & Khan 2015). If the bi-nary hardening timescales are significant compared to the relevanttimescales of the simulation, because the smallest resolved scalesare much larger than the typical binary separation, taking the pair toact as a single object with respect to accretion and feedback is still areasonable approximation for those processes. The timescales thatwe predict in the following sections are therefore a lower limit tothe timescales to form a SMBH binary and subsequent merger.We predict that the formation of close SMBH pairs is a rel-atively rare occurrence, with an average formation rate per co-moving volume of 0 .
013 cMpc − Gyr − . Figure 1 shows the likeli-hood that the merger of two galaxies will result in the formation of aclose SMBH pair within a Hubble time. With our formation scheme(§2.2), lower mass galaxies are naturally less likely to host SMBHsand so often their mergers do not result in any close pairs, as onemore of the galaxies do not host any SMBHs to begin with. In addi-tion, as we explore in the next section, galaxies in lower mass ratiomergers are more likely to become tidally disrupted and deposittheir SMBHs on very wide orbits with larger sinking timescales.While we will focus in the following sections on close SMBH pairsthat do form in the simulation, it is important to note that only afraction of galaxy mergers result in a close SMBH pair formingwithin a Hubble time. While several di ff erent timescales are important for understand-ing the formation and evolution of SMBH pairs, the evolution ofSMBH orbits on kpc scales is often simplified, relying on analyticapproximations that do not self consistently account for the kine-matics and internal properties of the merging galaxies (e.g. Dvorkin& Barausse 2017), which previous studies have shown can have animportant role in determining how the SMBHs will evolve follow-ing a galaxy merger (.e.g Governato et al. 1994; Callegari et al. . . . . . C u m u l a t i v e F r a c t i o n o f C l o s e S M B H P a i r s D BH < kpcAll Close PairsFormed at z < Galaxy Mergers at z < Time Before Close SMBH Pair Formation [Gyr] . . . . All Close Pairs
D < kpc D < kpc D < kpc Figure 2. T he T imescale to F orm C lose SMBH P airs . Top:
The cumulativedistribution of time that SMBH pairs spend separated by less than 10 kpcprior to close pair formation for all close SMBH pairs formed in R omulus / black solid) While about half of the close pairs form relatively quickly( < . / blue, solid) aremostly very far removed from their progenitor galaxy merger event. Alsoshown is the subset of close SMBH pairs resulting satellites in-falling after z = Bottom:
The cumulative distribution of timescales that SMBH pairs spentat 5, (red), 10 (green), and 20 (orange) kpc separations before forming aclose pair with sub-kpc separation. As expected, closer proximity impliesfaster sinking timescales, as the dynamical time of the galaxy at smallerradii decreases. Overall, the distributions are quite similar, implying that ourresults are insensitive to the specific choice of separation scales explored.Vertical dashed lines show the 75th percentiles. omulus
25, the simulation is uniquely capable of estimating thistimescale for a realistic population of galaxy mergers taking placewithin a fully cosmological environment.For our analysis we measure the time that each eventual closepair of SMBHs spends at ‘galaxy-scale’ ( ∼ −
10 kpc) separations.Position information for each SMBH is recorded every 1.6 Myrand simulation snapshots are recorded every 10 −
100 Myr, withhigher time resolution at earlier epochs. In our analysis we onlyinclude close SMBH pairs formed within resolved DM halos, withat least 10,000 DM particles, resulting in a lower mass limit of ∼ × M (cid:12) . We also only include close pairs that form at least 100Myr after each SMBH has been seeded, in order to avoid countingpairings that occur as a result of multiple SMBHs forming fromthe same cloud of gas, a rare but possible result of our formationscheme and should be considered degenerate to a single SMBHgrowing quickly from a particularly large, dense cloud of gas. Weconfirm that our results are insensitive to the specific choice of thistime threshold.The top panel of Figure 2 shows the cumulative distribution oftime that SMBH pairs spend within 10 kpc of one another beforeforming a close pair. The distance is small enough that the two tar-get SMBHs must be within the same galaxy or interacting pair ofgalaxies. For the overall population (black line) most of the close © , 1–11 ancing to ChaNGa pairs form with less than 1 Gyr spent at these intermediate sep-arations, consistent with many studies of isolated galaxy mergers(e.g. Mayer et al. 2007). However, there is a significant populationof pairs that remain at galactic-scale separations for several Gyr.Taking only the population of close pairs that form at low redshift( z <
2; blue line) we see that the majority of these close pairs formseveral Gyr after their original galaxy merger event. We thereforepredict that a significant fraction of low redshift SMBH pairs (andtherefore subsequent SMBH binaries and SMBH merger events)are formed from a population of long-lived, ‘wandering’ SMBHs(Schneider et al. 2002; Volonteri & Perna 2005; Bellovary et al.2010) born out of early galaxy mergers.This result can have critical implications for gravitationalwave analysis in the future, a ff ecting how such signal is interpretedin terms of connecting SMBH mergers to galaxy evolution. It canalso be important for interpreting dual and o ff set AGN observa-tions, as it becomes unclear how connected they may be to actualgalaxy mergers. Though beyond the scope of this paper, we willexplore in more detail the implications of these results to gravita-tional wave predictions as well as the population of o ff set and dualAGN in future work.The bottom panel of Figure 2 shows the cumulative distribu-tion of timescales that SMBH pairs spend at 5, 10, and 20 kpcseparations. As expected, the evolution of SMBH pairs occurson slightly shorter timescales for smaller separations. The sinkingtimescale due to dynamical friction depends on the local dynam-ical time, which decreases toward galactic centre. Still, we findSMBH pairs that spend several Gyrs separated by 5 kpc or less.This shows that our results are insensitive to our specific choice ofseparation threshold. In the following sections, we choose 10 kpc asour galaxy-scale separation threshold, as it corresponds to the sizeof the Galactic disk and is a good representation of the inner regionof a dark matter halo that is dominated by baryonic processes. Ad-ditionally, we have confirmed that our other conclusions are alsoinsensitive to this chosen scale.The distribution of timescales presented in Figure 2 is likelydue to several variables, including the kinematics of the merginggalaxies, the morphology of the galaxy merger remnant, the massof the SMBHs, and where within that galaxy the SMBHs are de-posited. Callegari et al. (2011) find that the behaviour of in-fallingsatellite galaxies and their host SMBHs depend strongly on the an-gle of the interaction. How SMBHs are deposited within a galacticdisk can also a ff ect the e ffi ciency of dynamical friction. If the hostgalaxy has a cored density profile, delay timescales can also bemade longer (Read et al. 2006; Di Cintio et al. 2017). Similarly, alarge stellar core with high velocity dispersion could also make dy-namical friction less e ff ective, as there would be more stars movingtoo fast to contribute. All of these merger and galaxy properties area natural consequence of the simulation volume and are folded intothe timescale distributions we predict.Because these timescales are the result of many di ff erent vari-ables interacting with one anther, we find little overall dependenceon single parameters like SMBH mass or halo mass. However, wedo find a strong dependence on the morphology of the accretedgalaxy and its stellar mass relative to the primary galaxy, which weexplore in the following section. In this section, we examine how the close SMBH pair formationtimescale depends on the properties of the interacting galaxies. Wetake a sub-set of our close SMBH pair population that result fromgalaxy mergers initiated by in-falling satellites at z <
5, where bothhalos are resolved ( M vir > × M (cid:12) ) at the time of satellite in-fall. This time of satellite in-fall is taken as the time the secondarygalaxy’s host dark matter halo crosses the virial radius of the mainhalo. For halos that cross the virial radius multiple times, the finalcrossing time is used. The initial properties of each galaxy prior tothe merger are taken at this final in-fall time. We do not includemergers at higher redshift, as often the details of these interactionsare not fully captured by our snapshots, with halos attaining a massthat passes our strict definition of what is resolved and falling intothe main halo in between snapshots. This sub-sample consists of330 close SMBH pairs resulting from 196 unique galaxy mergers.Note that, because individual galaxies can host multiple SMBHs,it is common for single galaxy mergers to result in multiple closeSMBH pairs. The dashed black line in Figure 2 shows the distri-bution of delay timescales for this subset of close SMBH pairs,showing that this population is indeed representative of the whole.In Figure 3 we plot the cumulative distribution of time thateventual close SMBH pairs spend within 10 kpc of one another.We group these pairs based on the central stellar density of the in-falling galaxy and the stellar mass ratio of the two merging galax-ies. The stellar density is calculated within the central kpc of eachin-falling satellite galaxy. Figure 3 shows the results in units of bothGyr (right) and number of dynamical times (left), where the dynam-ical time is calculated at a radius of 10 kpc of the main galaxy atthe approximate time the two SMBHs come within 10 kpc of oneanother. The median values for the central stellar density and stel-lar mass ratio are 3 . × M (cid:12) kpc − and 0.43 respectively. Onlysystems where the accreted SMBH is within the central 1 kpc of itshost galaxy at in-fall time are considered. Initially o ff set SMBHsare considered in the next section.It is clear from this figure that accreted galaxies with high cen-tral densities and higher stellar mass ratios result in significantlyshorter delay times. Galaxies with either low central densities orlow stellar mass ratios experience longer times spent at galaxy-scale separations, implying that tidal disruption of the host galaxyis important for determining the timescale for close SMBH pairformation. During a galaxy interaction, ram pressure stripping candisrupt gas within galactic disks at larger radii and tidal heating candisrupt the inner core of the galaxies. Dense stellar cores withinhigh mass ratio mergers are more likely to avoid disruption throughboth ram pressure stripping and tidal heating (Gnedin & Ostriker1999; Callegari et al. 2009; Van Wassenhove et al. 2014), so thecentral SMBHs remain embedded in a dense stellar core that aidsin their orbital decay. In galaxies lacking a dense stellar core, orthose involved in more minor mergers, tidal heating is more e ffi -cient at disrupting the inner parts of the galaxy, resulting in SMBHsdeposited at large radii without any stellar core to assist in theirorbital decay. This is consistent with analytical experiments show-ing how the orbital evolution of SMBHs is highly dependent onwhether they are embedded in a stellar core or ‘naked’ within theirnew host galaxy (Yu 2002; Dosopoulou & Antonini 2017).Figure 4 shows a series of snapshots from two example galaxymergers taking place with both primary and secondary galaxies ini-tially within a factor of 2 of one another in stellar mass. However, ©000
5, where bothhalos are resolved ( M vir > × M (cid:12) ) at the time of satellite in-fall. This time of satellite in-fall is taken as the time the secondarygalaxy’s host dark matter halo crosses the virial radius of the mainhalo. For halos that cross the virial radius multiple times, the finalcrossing time is used. The initial properties of each galaxy prior tothe merger are taken at this final in-fall time. We do not includemergers at higher redshift, as often the details of these interactionsare not fully captured by our snapshots, with halos attaining a massthat passes our strict definition of what is resolved and falling intothe main halo in between snapshots. This sub-sample consists of330 close SMBH pairs resulting from 196 unique galaxy mergers.Note that, because individual galaxies can host multiple SMBHs,it is common for single galaxy mergers to result in multiple closeSMBH pairs. The dashed black line in Figure 2 shows the distri-bution of delay timescales for this subset of close SMBH pairs,showing that this population is indeed representative of the whole.In Figure 3 we plot the cumulative distribution of time thateventual close SMBH pairs spend within 10 kpc of one another.We group these pairs based on the central stellar density of the in-falling galaxy and the stellar mass ratio of the two merging galax-ies. The stellar density is calculated within the central kpc of eachin-falling satellite galaxy. Figure 3 shows the results in units of bothGyr (right) and number of dynamical times (left), where the dynam-ical time is calculated at a radius of 10 kpc of the main galaxy atthe approximate time the two SMBHs come within 10 kpc of oneanother. The median values for the central stellar density and stel-lar mass ratio are 3 . × M (cid:12) kpc − and 0.43 respectively. Onlysystems where the accreted SMBH is within the central 1 kpc of itshost galaxy at in-fall time are considered. Initially o ff set SMBHsare considered in the next section.It is clear from this figure that accreted galaxies with high cen-tral densities and higher stellar mass ratios result in significantlyshorter delay times. Galaxies with either low central densities orlow stellar mass ratios experience longer times spent at galaxy-scale separations, implying that tidal disruption of the host galaxyis important for determining the timescale for close SMBH pairformation. During a galaxy interaction, ram pressure stripping candisrupt gas within galactic disks at larger radii and tidal heating candisrupt the inner core of the galaxies. Dense stellar cores withinhigh mass ratio mergers are more likely to avoid disruption throughboth ram pressure stripping and tidal heating (Gnedin & Ostriker1999; Callegari et al. 2009; Van Wassenhove et al. 2014), so thecentral SMBHs remain embedded in a dense stellar core that aidsin their orbital decay. In galaxies lacking a dense stellar core, orthose involved in more minor mergers, tidal heating is more e ffi -cient at disrupting the inner parts of the galaxy, resulting in SMBHsdeposited at large radii without any stellar core to assist in theirorbital decay. This is consistent with analytical experiments show-ing how the orbital evolution of SMBHs is highly dependent onwhether they are embedded in a stellar core or ‘naked’ within theirnew host galaxy (Yu 2002; Dosopoulou & Antonini 2017).Figure 4 shows a series of snapshots from two example galaxymergers taking place with both primary and secondary galaxies ini-tially within a factor of 2 of one another in stellar mass. However, ©000 , 1–11 M. Tremmel et al.
Time within 10 kpc / t dyn (10 kpc) . . . . . C u m u l a t i v e F r a c t i o n o f C l o s e S M B H P a i r s Time within 10 kpc / Gyr
Top 50% ρ ∗ T Top 50% M ∗ , /M ∗ , Top 50% ρ ∗ T Bottom 50% M ∗ , / M ∗ , Bottom 50% ρ ∗ T Top 50% M ∗ , / M ∗ , Bottom 50% ρ ∗ T Bottom 50% M ∗ , / M ∗ , Figure 3. C lose P air F ormation T imescales and M erging G alaxy P roperties . The cumulative distribution of the number of dynamical times (left) and totaltime (right) that SMBH pairs spend within 10 kpc of one another before forming a close pair. The data is taken from 196 unique galaxy mergers taking placeat z <
5, resulting in 330 close SMBH pairs. Shown here are only those close pairs where the accreted SMBH is initially within the central 1 kpc of its hostsatellite galaxy (159 total pairs). The distributions are split up based on the 50th percentiles in central stellar density of the accreted galaxy and the stellarmass ratio (3 . × M (cid:12) kpc − and 0.43 respectively) calculated at the in-fall time of the satellite halo. Accreted galaxies that have both high central stellardensities and high stellar mass ratios compared to the main galaxy are significantly more likely to result in a quick formation of a close SMBH pair. the stellar mass ratio of the top and bottom examples is 0.45 and0.22 respectively. This, combined with the fact that the secondarygalaxy in the top example has an initial central stellar density nearly5 times higher than that in the bottom case, results in very di ff erentSMBH orbital evolution. In the bottom case, the secondary galaxy’score becomes tidally heated and eventually disrupted by the maingalaxy, no longer maintaining its structure. In the top case, thedenser core is able to avoid disruption and maintains its integrity upuntil the two cores merge, bringing the SMBHs along with them.The bottom example of a disrupted galaxy forms a close SMBHpair only after the SMBHs spend 1 . ff set SMBHs In the previous section, we focused on central SMBHs, those thatare at the centre of their host galaxy at the time of satellite in-fall. However, approximately half of the close SMBH pairs in oursub-sample from R omulus
25 form from accreted SMBHs initiallyo ff set from the centre of their host galaxy. As we have seen, theorbital decay of SMBHs can often take several Gyr and galaxymergers often never result in a close SMBH pair. Massive galaxiesin the R omulus
25 simulation therefore often have several SMBHsthat are o ff set from galactic centre, gathered throughout the hostgalaxy’s merger history. In some rare cases, galaxies only have o ff -set SMBHs.Figure 5 is similar to the left panel of Figure 3, with SMBHbinaries binned based on whether the accreted host galaxy is more likely to avoid complete disruption due to a dense stellar coreand high mass ratio (orange / solid), less likely to avoid disruption(blue / dashed), or whether the target SMBH is o ff set from the cen-tre of their host satellite galaxy by more than 1 kpc as it crosses themain halo’s virial radius (green / dotted).The close pair formation timescale distribution for initiallyo ff set SMBHs is similar to that for more easily disrupted satellitegalaxies. When the SMBH is not central, it is likely not embed-ded within a dense stellar core, even if its host galaxy has one. Itwill therefore become accreted onto the main galaxy without a stel-lar core to aid in dynamical friction, just like SMBHs in galaxieswhose cores become tidally disrupted. The previous sections have shown that SMBH pairs can spend sig-nificant time at kpc-scale distances before forming a close pair.Their evolution on kpc scales is a phase that is very di ffi cult to fullycapture analytically, as it straddles the separations where galaxiesare still merging and those where the sinking concerns the SMBHsthemselves, naked or surrounded by the core of their satellite (see adiscussion in McWilliams et al. (2014)). When estimating the timeof binary formation (for which the time of pair formation studiedhere is a lower limit) semi-analytical models normally use satel-lite merging timescales that should account for the full ‘amalga-mation’ of the satellite (see a discussion in Boylan-Kolchin et al.(2008)), typically estimated from large suites of dark matter-onlysimulations. Boylan-Kolchin et al. (2008) argue that the inclusionof baryons (specifically, bulges, that are denser than dark matter © , 1–11 ancing to ChaNGa High Central DensityHigh Mass Ratio
Low Central DensityLow Mass Ratio
Figure 4. A n I llustrative E xample . Two examples of galaxy mergers taking place around the same time and with galaxies of similar mass. Each set of plotsshows the spatial distribution and colour of stars at five di ff erent times leading up to and following the merger of the two galaxies. Colours are based on thecontribution of di ff erent bands within each pixel using U (blue), V (green), J (red) assuming a Kroupa IMF, so young stars look blue and older stars lookyellow. The stellar emission is calculated using tables generated from population synthesis models (http: // stev.oapd.inaf.it / cgi-bin / cmd; Marigo et al. 2008;Girardi et al. 2010). Red and black crosses mark the positions of the SMBHs and the green cross in the top final frame represents a close pair of SMBHs. Theinitial stellar masses of the accreted galaxies in the top and bottom cases are 1 . × and 1 . × M (cid:12) respectively and, for the main galaxies, stellarmasses of 2 . × and 4 . × M (cid:12) respectively. The accreted galaxy in the top case originally has a stellar core nearly five times denser than that of thebottom galaxy. This, combined with the higher stellar mass ratio, allows the core of the galaxy to avoid disruption, quickly resulting in a close SMBH pair. Inthe bottom case, the core of the original galaxy is tidally heated, becomes more di ff use, and is quickly assimilated into the main galaxy, leaving the SMBHto sink on its own. Despite the close passage shown in the last frame, the SMBHs will not form a close pair until t = .
34 Gyr, after 1 . . ©000
34 Gyr, after 1 . . ©000 , 1–11 M. Tremmel et al.
Time within 10 kpc / t dyn (10 kpc) C u m u l a t i v e F r a c t i o n o f C l o s e S M B H P a i r s Top 50% ρ ∗ T Top 50% M ∗ , /M ∗ , (N =29)Bottom 50% ρ ∗ S Bottom 50% M ∗ , /M ∗ , (N =130)Initially Offset SMBHs (D > kpc; N =171) Figure 5. C lose P air F ormation T imescales for I nitially O ffset SMBH s .The cumulative distribution of the number of dynamical times SMBH pairsspend within 10 kpc of one another before forming a close pair. The solidorange line represents SMBHs from galaxies that are less susceptible todisruption (same as in Figure 3) and the blue dashed line represents SMBHsfrom galaxies that are more likely to become tidally disrupted due to a lowerstellar mass ratio and / or low central density (the union of the other threelines shown in Figure 3). The green dotted line represents SMBHs that wereinitially o ff set from the centres of their host satellite galaxies by more than1 kpc at the time of in-fall. The green and blue distributions are very similar,which is to be expected. In both cases, the SMBHs lack the extra support ofa stellar core when making their way to the centre of their new galaxy. and thus more resistant to disruption) would shorten the timescalescompared to estimates for dark matter haloes alone. The results ofthe previous sections, however, show a more complex picture whendealing with SMBHs, rather than halos and galaxies only.In order to test the approach of semi-analytic models, we es-timate the close pair formation times that would be predicted frommore simplistic models. We approximate the halo sinking timescalevia the analytic fit derived by Boylan-Kolchin et al. (2008), giventhe halo masses of the primary and satellite halos and the virialradius of the primary halo taken from the simulation at the timeof satellite in-fall. Following a procedure similar to modern semi-analytic models (e.g. Barausse 2012), we give each halo pair a cir-cularity, (cid:15) = j / j circ , sampled from a normal distribution centredat ¯ (cid:15) = . σ = .
23 (Khochfar & Burkert 2006). Thecircular radius is calculated from the periastron radius, approxi-mated by r peri = R vir (cid:15) . (Khochfar & Burkert 2006). In order to re-main in the regime where the fit from Boylan-Kolchin et al. (2008)is accurate, we only allow (cid:15) to vary between 0.2 and 1.0. Below (cid:15) = .
2, baryonic e ff ects dominate due to the satellite galaxy’s veryradial orbit, making the approximation less accurate. This simpleapproach allows us to compare the sinking times predicted fromR omulus
25 to the average halo sinking timescales that would beincluded in most semi-analytic models.We find that galaxy-scale orbital evolution is an important bot-tleneck to close SMBH pair formation for high redshift galaxymergers. In Figure 6 we plot the close pair formation times di-rectly from the R omulus
25 cosmological simulation against the in-fall redshift of the parent satellite galaxy for the secondary SMBH(orange points). We compare this time to that which would be pre-dicted solely using the analytic halo sinking timescale describedabove (blue points). In other words, these points represent thetime for close pair formation if galaxy-scale orbital evolution andother baryonic e ff ects were ignored, as they often are in both semi- analytic models and other cosmological simulations. We find thatthe orbital evolution of SMBHs from 10 kpc to sub-kpc scales is animportant bottleneck to close pair formation (and the subsequentbinary formation and merger) for high redshift galaxy interactions,where the dynamical timescale for satellite halos is comparativelysmall. At redshift less than ∼ ff erence between the two types of points, indicating that halosinking timescales are more similar to or even sometimes dominantcompared to galaxy-scale SMBH orbital evolution. Semi-analyticmodels of SMBH binary evolution find similar results, with binaryevolution timescales acting as a dominant bottleneck at high red-shift and increasingly less important when compared to satellitesinking timescales at low redshift (Volonteri et al. 2016).Examining the halo in-fall times and the predicted closeSMBH pair formation it is clear that close pairs that form at latertimes are often the consequence of high redshift mergers, an ef-fect also seen in Figure 2. These results show that SMBH orbitalevolution on galaxy scales is a very important bottleneck for theformation of close SMBH pairs and, therefore, SMBH binaries andmergers, and must be accounted for when predicting the popula-tion of binary SMBHs and gravitational wave events across cosmictime. Using the R omulus
25 cosmological simulation, which is uniquelycapable of tracking the dynamics of SMBHs within galaxies downto sub-kpc scales, we examine the timescale for SMBH pairs toevolve from galaxy-scale separations (1 −
10 kpc) to form closepairs with separations less than a kpc, the precursor phase to abound SMBH binary and (possible) future SMBH merger. The for-mation of close SMBH pairs is a relatively rare occurrence, be-coming more common in major mergers of more massive galax-ies. We find that galaxy mergers across cosmic time result in closeSMBH pairs that often form several Gyr after the original galaxymerger event. SMBHs often accrete onto a new host galaxy viagalaxy merger at high redshift, but only form a close SMBH pair atmuch lower redshift, resulting in a long lived population of ‘wan-dering’ SMBHs (Schneider et al. 2002; Volonteri & Perna 2005;Bellovary et al. 2010). This can a ff ect how we predict and interpretfuture observations of gravitational waves and dual / o ff set AGN, aswell as the observational signatures of gravitational recoil events(Blecha et al. 2016).Using a set of 330 SMBH close pairs resulting from 196unique galaxy mergers within R omulus
25, we show that thetimescales for the formation of a close SMBH pair is dependenton galaxy morphology and stellar mass ratio. Galaxy mergers withsimilar mass and dense stellar cores result in faster close pair for-mation, as the secondary galaxy is less likely to become tidallydisrupted. SMBHs that are embedded in stellar cores that are ableto avoid disruption will be aided in sinking to galactic centre (Yu2002; Dosopoulou & Antonini 2017). Satellite galaxies that aremore susceptible to tidal disruption result in longer SMBH sinkingtimescales and close SMBH pairs that form long after the galaxymerger event (if they form at all). A similar situation is true forSMBHs that are initially o ff set from the centre of satellite galaxies.These SMBHs are not likely to be within the central stellar core, ifone exists, of their host galaxy and so are deposited on their own atrelatively large radii during the galaxy interaction.The resolution limit of the R omulus
25 simulation a ff ects the © , 1–11 ancing to ChaNGa Satellite Infall Redshift C l o s e P a i r F o r m a t i o n T i m e [ G y r ] Predicted from
Romulus25
Estimate from Boylan-Kolchin+ 2008Hubble Time
Figure 6.
SMBH vs . H alo S inking T imescales . The formation time of closeSMBH pairs as a function of satellite in-fall redshift. The black line de-notes the time as a function of redshift. The orange points plot the time ofclose SMBH pair formation predicted directly from the R omulus
25 simula-tion. The blue points estimate what the close pair formation time would beonly accounting for halo sinking timescales approximated by the analytic fitfrom Boylan-Kolchin et al. (2008), as described in the text. The in-fall red-shifts are shifted slightly between the two in order to make the distinctionmore clear. For high redshift halo mergers, the timescale for SMBH orbitsto decay from galaxy-scale separations is a critical bottleneck to close pairformation, resulting in formation times that are often much later than thosepredicted solely based on halo sinking timescales. At lower redshift ( z < ff erence betweenthe two types of points. scale at which tidal heating and disruption can be captured. Tidalprocesses become important when the impact parameter is simi-lar to the e ff ective radius of the disrupting object. With a Plummerequivalent gravitational force resolution of 250 pc, the e ff ective ra-dius of galaxies are well resolved for a wide range of masses andredshifts (Graham & Worley 2008; van der Wel et al. 2014) andso disruption occurring on large scales is captured, but the internalstructure on scales very close to the SMBHs remains unresolved.Dense cusps of stars can form in galaxies, particularly during gasrich mergers. These dense regions would persist for longer, as theyrequire closer interactions to tidally heat. These unresolved stel-lar remnants can have an important e ff ect on SMBH dynamics onscales much lower than 700 pc (Van Wassenhove et al. 2014), thelimit beyond which we do not attempt to follow them in this work.SMBHs deposited on larger scales may still have a dense stel-lar core or nuclear star cluster (Wehner & Harris 2006; Ferrareseet al. 2006) around them that R omulus
25 is unable to resolve, e ff ec-tively increasing their dynamical mass. However, for the sample ofclose SMBH pairs formed from galaxy mergers where disruptionlikely takes place, we find that the sinking time does not show aclear dependence on SMBH mass. This indicates that the existenceof an unresolved, dense stellar component around these SMBHswill only have a secondary e ff ect on their orbital evolution. Rather,the sinking times depend more on the details of the galaxy merger,i.e. where and with what orbital energy the SMBHs deposited.We show that orbital evolution of SMBHs within galaxies onscales between 1-10 kpc are a major bottle neck for forming closeSMBH pairs, particularly for high redshift galaxy interactions. Inagreement with the arguments by Volonteri et al. (2016), at lowerredshifts ( z <
2) the sinking timescale of satellite halos becomes a more dominant factor and the specific e ff ect of galaxy-scale orbitaldecay is less important, though still not trivial. How much of an ef-fect this timescale plays in the overall prediction for SMBH mergerrates will also depend on the hardening timescales after formationof the binary. While there is evidence that such hardening times canbe relatively short, on the order of 10 − yr (Armitage & Natara-jan 2002; Haiman et al. 2009; Colpi 2014; Holley-Bockelmann& Khan 2015), other recent work suggests that these hardeningtimescales may be very long in some cases (Vasiliev et al. 2015;Kelley et al. 2017; Tamburello et al. 2017). Further, it is importantto note that we do not include the e ff ects of gravitational recoil, northree-body SMBH encounters, both of which can further a ff ect theformation of SMBH binaries.It is clear that this stage of SMBH pair evolution plays a cru-cial role in determining when and where close SMBH pairs occur,and therefore the SMBH binaries and mergers that may result fromsuch pairs. It is also important to understanding the time connectionbetween AGN activity and galaxy interaction induced star forma-tion, as the SMBH sinking timescale may be much larger than thatof the typical observed starburst timescale, found to be on the or-der of 0.1 Gyr (Marcillac et al. 2006; Pereira-Santaella et al. 2015).As illustrated in Figure 4, close SMBH pairs often form in relaxedgalaxies that show no morphological disturbances indicative of arecent merger.In future work, we will examine in more detail how this addi-tional timescale can a ff ect SMBH merger predictions from state-of-the-art SAMs, exploring in particular how the close pair formationtimescale explored in this work compares with other a ff ects such asthree body interactions and binary hardening rates in determiningthe predicted signals for future gravitational wave observatories.We will also explore the occurrence of dual and o ff set AGN (e.g.Comerford & Greene 2014; Comerford et al. 2015; Barrows et al.2017), to examine in more detail the phase of galaxy and SMBHevolution traced by these events. ACKNOWLEDGMENTS
The Authors thank the anonymous referee for a thorough read-ing of the manuscript and their helpful comments. FG, TQ andMT were partially supported by NSF award AST-1514868. APwas supported by the Royal Society. This research is part of theBlue Waters sustained-petascale computing project, which is sup-ported by the National Science Foundation (awards OCI-0725070and ACI-1238993) and the state of Illinois. Blue Waters is a jointe ff ort of the University of Illinois at Urbana-Champaign and its Na-tional Center for Supercomputing Applications. This work is alsopart of a PRAC allocation support by the National Science Foun-dation (award number OCI-1144357). MV acknowledges fundingfrom the European Research Council under the European Com-munity’s Seventh Framework Programme (FP7 / ©000
The Authors thank the anonymous referee for a thorough read-ing of the manuscript and their helpful comments. FG, TQ andMT were partially supported by NSF award AST-1514868. APwas supported by the Royal Society. This research is part of theBlue Waters sustained-petascale computing project, which is sup-ported by the National Science Foundation (awards OCI-0725070and ACI-1238993) and the state of Illinois. Blue Waters is a jointe ff ort of the University of Illinois at Urbana-Champaign and its Na-tional Center for Supercomputing Applications. This work is alsopart of a PRAC allocation support by the National Science Foun-dation (award number OCI-1144357). MV acknowledges fundingfrom the European Research Council under the European Com-munity’s Seventh Framework Programme (FP7 / ©000 , 1–11 M. Tremmel et al.
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APPENDIX A: DYNAMICAL FRICTION TEST ATROMULUS25 RESOLUTION
In this section we explicitly confirm that the dynamical friction pre-scription presented in Tremmel et al. (2015) is able to correctlytrack the orbital decay of a SMBH at the resolution of R omulus .The simulation is dark matter only with a particle mass of 3 . × M (cid:12) and gravitational softening, (cid:15) g , of 342 pc, within 10% and 3%of the values used in R omulus
25 respectively. We set up and run theinitial overdensity collapse until its virial mass is 3 . × M (cid:12) , butlarger scales are still actively collapsing, as in a cosmological sim-ulation. At this time, the density profile is consistent with an NFWprofile (Navarro et al. 1996) of concentration 6 and a virial radiusof 143 kpc. In order to test pair formation timescales, we run onesimulation with two SMBHs. One is a central SMBH and one is ano ff -center SMBH on an eccentric orbit. Both SMBHs are 10 M (cid:12) .The o ff -center SMBH is placed at 2 kpc from the center of the halowith a tangential velocity of 4.7 km / s relative to the center of massvelocity of the inner 5 kpc of the halo. This is approximately 0 . v circ for an NFW halo at this radius with the test halo’s size, mass, andconcentration. The SMBH placed at the center is given no rela-tive velocity. In order to more accurately model the conditions forSMBH pair formation in R omulus
25, we ensure that both SMBHshave a timestep of ∼ yrs, similar to the largest timesteps forSMBHs in R omulus
25. In the simulation, two SMBHs are allowedto form a close pair when they are within 2 (cid:15) g of one another becausebelow this scale the orbital evolution is poorly resolved. Their rel-ative velocities must also be consistent with being mutually bound.This avoids having two SMBHs form a close pair when one is onan eccentric orbit that may bring it into close proximity of a centralSMBH, as in this test scenario.The result of this simulation is shown in Figure A1. TheSMBH pair forms (shown as the red star) at a time nearly equalto the analytic prediction from Ta ff oni et al. (2003) (black verticalline) with no tuning at all of our sub-grid physics . This experimentconfirms that the method used for correcting for unresolved dynam-ical friction, combined with our pair formation criteria, works at theresolution attained in R omulus https: // github.com / mtremmel / ICInG.git
Time [Gyr] D i s t a n c e f r o m H a l o C e n t e r [ p c ] Figure A1. C lose P air F ormation T imescale T est . Two SMBHs evolvedwithin an isolated, actively collapsing dark matter halo. One SMBH is ini-tially in the center and the other on an eccentric orbit with apocenter of2 kpc. To ensure accurate representation of the R omulus
25 simulation, thetime steps for SMBHs are forced to be ∼ yrs, similar to the largest timesteps for SMBHs in R omulus
25. The dashed horizontal line represents 2 (cid:15) g from halo center and the vertical line the theoretical dynamical friction sink-ing timescale, τ DF from Ta ff oni et al. (2003). The two lines correspond tothe two SMBHs and the red star the position and time when the two SMBHsform a close pair and are then tracked by a single particle with mass 2 × M (cid:12) . The merger occurs at a time very nearly equal to τ DF . ©000