INTRODUCTION {#section:introduction}
============

Irrefutable dynamical evidence indicates that supermassive black holes (SMBHs) with masses ranging from $10^{6}$ to above $10^{9}{{\rm M_\odot}}$ reside at the centers of most galaxies hosting spheroids [e.g., @kormendy_richstone95]. The available data show the existence of a remarkably tight correlation between the mass of the SMBHs, $M_{\rm BH}$, and the stellar velocity dispersion of the host galaxy spheroidal component, $\sigma$ [@ferrarese_merritt00; @gebhardt_etal00], suggesting a fundamental mechanism that connects SMBH assembly and galaxy formation [e.g., @silk_rees98; @burkert_silk01].

According to the currently favored cold dark matter cosmological models, structures in the universe are the end result of a complex hierarchy of mergers and accretion of smaller subunits. Thus, the hierarchical buildup of SMBHs by massive seed black holes present at the center of protogalaxies and the formation of SMBH binaries appear as a natural consequence in any “bottom-up” cosmogony. Recently, the discovery of a binary active galactic nucleus in the interacting system NGC 6240 by @komossa_etal03 has lent support to this picture. Although substantial gas accretion is also required for the SMBHs to reach their present-day mass density and satisfy the $M_{\rm BH}$–$\sigma$ relation [@volonteri_etal03], the link between these two main modes of SMBH growth is still missing.

The formation and subsequent orbital evolution of a SMBH pair depends on how efficiently the host galactic cores lose angular momentum by dynamical friction during the merging process. Once the two cores have merged, the SMBHs may decay further and eventually form a binary owing to the drag exerted by the background mass distribution. These two evolutionary phases have been investigated numerically by a number of authors [@governato_etal94; @makino_ebisuzaki96; @milosavljevic_merritt01; @makino_funato04]. However, the galaxy models used in these studies were idealized, spherical stellar systems that could at most faithfully represent the central spheroidal components of real galaxies. Hence, both the larger scale dynamical evolution of the merging systems and the cosmological framework were missing. Moreover, these studies did not explore the role of a dissipative component in driving the evolution of a SMBH pair. Notable exceptions are the studies by @escala_etal04a [@escala_etal04b], who showed that gas causes continuing loss of angular momentum and rapidly reduces the relative SMBH separation to distances where gravitational radiation and coalescence is efficient. However, if and how their initial conditions are related to the larger scale dynamics involved in galaxy merging is still unclear.

In this Letter we report on the effects of gaseous dissipation on the fate of SMBHs using high-resolution binary disk galaxy merger simulations. @governato_etal94 have already highlighted the difficulty of forming a close SMBH pair when tidal disruption of one of the host systems intervenes. As we illustrate below, gaseous dissipation is vital for the survival of the host galaxies by deepening their potential wells and provides the necessary fuel for the SMBH growth. Finally, we analyze the stellar kinematics and distribution of gas in the central regions of the merger remnants to investigate for the first time how merging galaxies move with respect to their initial location on the $M_{\rm BH}$–$\sigma$ plane.

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NUMERICAL SIMULATIONS {#section:num_sim}
=====================

We perform four types of high-resolution simulations of binary disk galaxy mergers with mass ratios of $1$:$1$ and $4$:$1$ including different physical processes. We simulate purely collisionless mergers and mergers in which we follow the gas dynamics “adiabatically,” i.e., without radiative cooling. A third set of simulations includes radiative cooling, while in the fourth set we also allow the cold gas to form stars. The simulations were performed with GASOLINE, a multi-stepping, parallel TreeSPH $N$-body code [@wadsley_etal04]. We include radiative and Compton cooling for a primordial mixture of hydrogen and helium. The star formation algorithm is based on that by @katz92, where gas particles in dense, cold Jeans unstable regions and in convergent flows spawn star particles at a rate proportional to the local dynamical time [see also @governato_etal04], and reproduces the Schmidt law. The star formation efficiency was set to $0.1$ which yields a star formation rate of $1-2\, {{\rm M_{\odot}yr^{-1}}}$ for models in isolation that have a disk gas mass and surface density comparable to those of the Milky Way.

The multicomponent galaxy models are constructed using the technique described by @hernquist93 and their structural parameters are consistent with the paradigm for structure formation [@mo_etal98]. To this end, each galaxy consists of a spherical and isotropic @navarro_etal96 dark matter (DM) halo [@kazantzidis_etal04a], an exponential disk, and a spherical, @hernquist90 non-rotating bulge. For the basic galaxy model we adopted parameters from the Milky Way model A1 of [@klypin_etal02]. Specifically, the DM halo had a virial mass of $M_{\rm vir}=10^{12}{{\rm M_\odot}}$, a concentration parameter of $c=12$, and a dimensionless spin parameter of $\lambda=0.031$. The mass, thickness and resulting scale length of the disk were $M_{\rm d}=0.04 M_{\rm vir}$, $z_{0}=0.1 R_{\rm d}$, and $R_{\rm d}=3.5\kpc$, respectively. The bulge mass and scale radius were $M_{\rm b}=0.008 M_{\rm vir}$ and $a=0.2 R_{\rm d}$, respectively. The DM halo was adiabatically contracted to respond to the growth of the disk and bulge [@blumenthal_etal86] resulting into galaxy models with a central total density slope close to isothermal (Figure \[fig1\]). The companion galaxy is either a replica of the same model in equal-mass mergers or a system containing one-fourth of the mass in each component in which lengths and velocities are renormalized according to the cosmological scaling for virialized systems [@mo_etal98]. The disk scale length of the satellite galaxy is equal to $R_{\rm d}=2.2\kpc$. Given the uncertainties in the adopted M/L, our galaxies are consistent with the stellar mass Tully-Fisher and size-mass relations. To each of the galaxy models we added a particle representing a SMBH at the center of the bulge component that was initially at rest. For the larger galaxy model we used a SMBH mass equal to $M_{\rm BH}=3 \times 10^{6} M_{\odot}$.

We considered two values for the gas fraction, $f_{\rm g}$, namely 10% and 50% of the total disk mass. We shut off radiative cooling at temperatures below $2 \times 10^{4}$ K that is about a factor of $2$ higher than the temperature at which atomic radiative cooling would drop sharply due to the adopted cooling function. With this choice we take into account non-thermal pressures and approximate the simulated gas with the warm ISM of a real galaxy [@barnes02]. Moreover, this choice enables us to simulate very gas-rich disks that would otherwise become strongly gravitationally unstable and undergo widespread fragmentation. Two main encounter geometries were adopted: prograde coplanar mergers and mergers in which one of the disks was inclined with respect to the orbital plane. We also simulated a single retrograde equal-mass merger in which both disk spin vectors were antiparallel to the orbital angular momentum vector. The galaxies approached each other on parabolic orbits with pericentric distances that were 20% of the more massive galaxy’s virial radius, typical of cosmological mergers [@khochfar_burkert03]. The initial separation of the halo centers was twice their virial radii and their initial relative velocity was determined from the corresponding Keplerian orbit of two point masses. In collisionless mergers, we used $N=1.2 \times 10^6$ particles in total, with each galaxy consisting of $10^5$ disk particles, $10^5$ bulge particles, and $10^6$ DM particles. The gas component in gasdynamical simulations was represented by $10^5$ particles. We adopted a gravitational softening of $\epsilon = 0.1\kpc$ for both the DM and baryonic particles of the larger galaxy, and for its SMBH $\epsilon=0.03\kpc$, which is small enough to follow its orbital evolution. For the satellite galaxy, all softening lengths were scaled according to $\epsilon \propto m^{1/3}$. In all simulations the merger remnants were allowed to settle into equilibrium $\sim 15$ dynamical times at the disk half-mass radius after the merger was completed.

RESULTS {#section:results}
=======

SMBH Pairing in Binary Disk Galaxy Mergers
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The galaxies merge in three to five orbits (between $5.5$ to $7$ Gyr) depending on the mass of the companion, a timescale that is set by dynamical friction and tidal stripping [e.g., @taffoni_etal03]. These timescales are longer than those reported in earlier studies [e.g., @barnes_hernquist96] due to the larger and more realistic pericentric distances adopted here.

In equal-mass mergers, the cuspy potentials of both galaxies are deep enough to allow the survival of their inner regions until orbital decay by dynamical friction is complete. This result holds for both collisionless and gasdynamical mergers. In the latter, during the first two orbits a strong spiral pattern appears in both the stellar and the gaseous component and mild non-axisymmetric torques redistribute mass and angular momentum driving approximately 20% of the gas towards the center. However, the central bulges stabilize the galaxies against bar formation. Shortly before the galaxies merge, a second much stronger gas inflow occurs caused by strong tidal torquing as the galaxies perform a close fly-by. However, only in the simulations with radiative cooling does this inflow result in a significant central concentration of cold ($T \sim 2 \times 10^{4}$ K) gas. In this case, more than 80% of the gas that was originally in the disks is collected within the central $500\pc$. This results in a considerable steepening of the central density slope of the combined distribution of DM and baryons (Figure \[fig1\]). In adiabatic simulations, only 20-30% of the initial gas mass collects within the central $\kpc$. This gas is heated to temperatures $T >10^5$ K by shocks and compressions during the merger and subsequently expands since it cannot radiate away its energy.

Large central gas inflows were already noticed in lower resolution merger simulations [e.g. @barnes_hernquist96; @barnes02]. In simulations with star formation these large inflows result in a central starburst and more than 90% of the central gas distribution is converted into stars in less than $10^8$ yr. The peak star formation rates range from $30$ to $>100\ {{\rm M_{\odot}yr^{-1}}}$, comparable to those of luminous infrared galaxies (LIRGs) and ultraluminous infrared galaxies (ULIRGs), suggesting that our modeling of star formation and the amount of cold gas present are realistic. The two SMBHs end up orbiting at the center of the remnant on eccentric orbits and form a [*close*]{} pair at a separation comparable to the adopted force resolution of $\sim 100\pc$ (Figure \[fig2\]). Significantly higher mass and force resolution would be needed to follow the SMBHs at relative separations where they would form a binary system.

In unequal-mass mergers, the outcome depends sensitively on how the internal structure of the merging galaxies is modified by dissipation. In collisionless simulations, the tidal disruption of the satellite galaxy at about $6$ kpc from the center leaves its SMBH wandering at a distance that prohibits the formation of a close pair. This “naked” SMBH would contribute to a population of wandering SMBHs in galactic halos [@volonteri_etal03] At such a distance the timescale for orbital decay predicted by dynamical friction would exceed a Hubble time. In simulations with cooling, the situation is completely reversed as the gas inflow becomes particularly strong in the companion galaxy. We measured an inflow of about $3\ {{\rm M_{\odot}yr^{-1}}}$ within the central $\sim$ kpc compared with $<0.5\ {{\rm M_{\odot}yr^{-1}}}$ for the larger galaxy. This striking difference in the gas inflows is attributed to a tidally induced strong stellar bar observed in the satellite galaxy approximately $1.5$ Gyr before the final passage. More than $50\%$ of the gas is funneled into the center owing to gas shocking in the bar potential. At this stage the companion galaxy has about 20% more gas within its central $\sim$ kpc relative to the larger galaxy. Dissipation enables the core of the infalling galaxy to survive complete tidal disruption by deepening its potential well resulting in a SMBH pair separated by $\sim 100\pc$ as in equal-mass mergers (Figure \[fig2\]).

The large inflows observed in the cooling and star formation simulations always produce a rotationally supported nuclear disk with a size in the range $1-2\kpc$ and a temperature set by our temperature floor ($T = 2 \times 10^{4}$ K). In adiabatic simulations, this gas remains in a hot phase and forms a pressure supported cloud of similar size. The nuclear disks result from the coalescence of two central disk-like structures formed at the centers of the galaxies primarily by the strong inflow just before they merge. These disks are tilted by several to a few tens of degrees relative to the orbital plane of the galaxies. They have peak rotational velocities and gas masses in the range of $250-300{\>{\rm km}\,{\rm s}^{-1}}$ and $\sim 10^8-10^9 M_{\odot}$, respectively, and they are well-resolved ($N \gtsim10^4$) in our high mass resolution simulations. Nuclear disks of [*molecular*]{} gas of a few hundred pc to a $\sim$ kpc scale have been identified spectroscopically for several AGNs and ULIRGs [e.g. @downes_solomon98; @davies_etal04]. The observed sizes, masses, and rotational velocities are comparable to those measured in our simulations. The nuclear disks reported here provide the reservoir of gas that fuels the central SMBHs.

$M_{\rm BH}$–$\sigma$ Relation in Binary Disk Galaxy Mergers
------------------------------------------------------------

For each remnant, we calculate line-of-sight aperture stellar velocity dispersions $\sigma$ within one effective radius $R_{\rm e}$ [@gebhardt_etal00] and we use a range of viewing orientations, namely inclinations $i=0$, $60\degrees$, and $90\degrees$. For $i=60\degrees$ and $i = 90\degrees$, we rotate the system about an axis perpendicular to the inclination axis through an angle $\phi$ set to 0, $45\degrees$, $90\degrees$, and $135\degrees$. Thus, for each simulation we obtain $9$ different orientations. For each of these viewing angles, we measured the mass density profiles in circular apertures out to two radii ($8$ and $25\kpc$) under two assumptions: (1) the system is a disk galaxy and decompose the mass density profile into a Sérsic bulge and an exponential disk; (2) the system is a pure spheroid and fit it with a pure Sérsic profile. This procedure ensures that the measurements of $R_{\rm e}$, and therefore of $\sigma$, that we obtain are very robust.

In cases when the two SMBHs form a close pair, the $M_{\rm BH}$ assigned to each remnant is calculated by summing the masses of the two SMBHs and adding the gas mass contained within the resolution of our gasdynamical simulations. In doing this we implicitly assume that any close pair will actually form a SMBH binary that will subsequently coalesce. Indeed, this is supported by the smaller scale simulations of @escala_etal04a [@escala_etal04b], who showed that SMBH pairs starting at a separation of few hundred pc in a gaseous background will typically merge within $\sim 10^7$ yr. When the SMBHs do not form a close pair, we only consider the mass of the SMBH located closer to the center of the remnant. Figure \[fig3\] shows the location of all merger remnants on the $M_{\rm BH}$–$\sigma$ plane together with real galaxy data and their best-fit correlation taken from @tremaine_etal02. We note that the initial galaxy models start very close to the mean correlation by construction. The error bars show the spread about the mean value of $\sigma$ and are smaller in the cooling and star formation simulations likely due to the fact that these remnants are more spherical [@kazantzidis_etal04b]. Mergers lead to an increase of the stellar velocity dispersion that is remarkably different depending on the physics included in the simulations. The largest increase occurs in simulations with radiative cooling which exhibit the deepest potentials. When star formation is included, we measure velocity dispersions closer to that of the collisionless simulations, since the gas does not build a significant central mass concentration.

Figure \[fig3\] illustrates that collisionless mergers move remnants away from the mean $M_{\rm BH}$–$\sigma$ relation. In adiabatic simulations, the remnants have in principle enough gaseous fuel to remain close to the relation, but the gas is too hot to accrete onto the SMBHs. In mergers with radiative cooling, the total amount of nuclear gas is more than one order of magnitude larger than needed to keep the galaxies close to the relation. This should also be regarded as an upper limit on the gas mass that may accrete onto the SMBHs since feedback processes ensuing once the AGN becomes active will regulate the inflow and allow the accretion of only a fraction of this gas, thus limiting the SMBH growth [@silk_rees98]. If AGN feedback is strong enough to supress cooling a hot gas phase similar to that found in our adiabatic simulations is expected to form. Thus, the results from cooling and adiabatic simulations likely provide upper and lower bounds to the true physical behavior of the resulting remnants. Note that the point corresponding to the $4$:$1$ merger is the highest with respect to the best-fit correlation in Figure \[fig3\]. This is the product of the particularly strong gas inflow occurring in the satellite galaxy just before the merger is completed. Overall a higher fraction of the available gas is driven towards the center relative to the equal-mass galaxy mergers. When star formation is included the amount of cold gas in the nuclear disks is reduced by more than 90% and the merger remnants move nearly parallel to the relation. This suggests that star formation during mergers is a key ingredient for maintaining the tightness of the $M_{\rm BH}$–$\sigma$ relation.

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DISCUSSION AND CONCLUSIONS {#section:discussion_conclusions}
==========================

Gaseous dissipation influences considerably the outcome of binary mergers of disk galaxies containing SMBHs. Most importantly, it controls the SMBH pairing process in unequal-mass mergers by modifying the central structure of the companion galaxy, enabling it to survive complete tidal disruption. Dissipationless, unequal-mass mergers are also expected to produce close SMBH pairs when the satellite galaxy is dominated by a dense stellar central component, as in nucleated dwarf elliptical (e.g. M32) and S0 galaxies. Our results suggest that semi-analytic models of hierarchical SMBH growth that neglect the effect of dissipation on the orbital evolution of SMBHs underestimate their pairing efficiency and subsequent coalescence. This consequently leads into overestimating the number of wandering SMBHs in MW-sized galaxies [@volonteri_etal03]. The higher pairing efficiency of $\sim 10^6 M_{\odot}$ SMBHs reported here has important implications for the probability of observing coalescence events with space-based gravitational waves experiments such as [LISA]{}[@sesana_etal04].

Gaseous dissipation also determines how merging galaxies constructed to satisfy the $M_{\rm BH}$–$\sigma$ relation move relative to their initial location on the $M_{\rm BH}$–$\sigma$ plane. The collisionless simulations reveal that when galaxies become gas poor which is likely at low $z$, their merger remnants will tend to move away from the mean relation. This suggests that gas-poor mergers act as a possible source of scatter in the mean $M_{\rm BH}$–$\sigma$ relation. The natural scaling between the amount of cold nuclear gas and the increase in the central stellar velocity dispersion explains why merger remnants in simulations with radiative cooling move above the mean relation. Star formation serves to move merger remnants nearly parallel to the relation. The fact that unequal-mass mergers appear effective at building a reservoir of gas for SMBH accretion is noteworthy, since in hierarchical structure formation models they are significantly more frequent than equal-mass ones.

The nuclear disks at the center of the remnants show significant spiral patterns (Figure \[fig2\]) which transport angular momentum outwards and sustain radial inflows towards the center. We measure typical inflows in the range $10^{-2}-10^{-3}\ {{\rm M_{\odot}yr^{-1}}}$ within $1\kpc$. The spiral instabilities are sustained by the self-gravity of the disks. In star formation simulations which contain the lightest disks, the minimum Toomre Q parameter [@toomre64] is $\sim 2$ within the same distance. Since gravity is softened at scales corresponding to a significant fraction of the disk size ($200\pc$), short wavelength modes are stabilized and the disks behave effectively as if they had a higher Q parameter [@mayer_etal04]. Therefore, spiral modes should be more vigorous with increasing resolution and promote even stronger inflows. Such inflows could feed the SMBHs and bridge the gap between the large scale flows and the viscous accretion taking place once the gas has reached the AGN accretion disk at subparsec scales [@heller_shlosman94; @fukuda_etal00]. Future simulations will be used to explore the feasibility of such mechanism. Interestingly, observed nuclear disks also have Q values between $1$ and $2$ and some of them exhibit significant non-axisymmetric structure [@downes_solomon98].

S. K. would like to thank Aaron Dutton, Andrey Kravtsov and David Merritt for useful discussions. P. M. acknowledges support by NASA grants NAG5-11513 and NNG04GK85G, and by NSF grant AST-0205738. The numerical simulations were performed on the zBox supercomputer at the University of Zürich and on the Intel cluster at the Cineca Supercomputing Center in Bologna (Italy).

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