We perform N-body simulations on some of the most massive galaxies extracted from a cosmological simulation of hierarchical structure formation with total masses in the range $10^{12} M_{\odot} < M_{tot} < 3\times 10^{13} M_{\odot}$ from $4\geq z \geq 0$. After galactic mergers, we track the dynamical evolution of the infalling black holes (BHs) around their host's central BHs. From 11 different simulations, we find that, of the 86 infalling BHs with masses > $10^4 M_{\odot}$, 36 merge with their host's central BH, 13 are ejected from their host galaxy, and 37 are still orbiting at $z=0$. Across all galaxies, 33 BHs are kicked to a higher orbit after close interactions with the central BH binary or multiple, after which only one of them merged with their hosts. These orbiting BHs should be detectable by their anomalous (not Low Mass X-ray Binary) spectra. The X-ray luminosities of the orbiting massive BHs at z=0 are in the range $10^{28}-10^{43}$ $\mathrm{erg}~\mathrm{s}^{-1}$, with a currently undetectable median value of $10^{33}$ $\mathrm{erg}~\mathrm{s}^{-1}$. However, the most luminous $\sim$5\% should be detectable by existing X-ray facilities.
Our Galaxy and the nearby Andromeda Galaxy (M31) form a bound system, even though the relative velocity vector of M31 is currently not well constrained. Their orbital motion is highly dependent on the initial conditions, but all the reliable scenarios imply a first close approach in the next 3-5 Gyrs. In our study, we simulate this interaction via direct N-body integration, using the HiGPUs code. Our aim is to investigate the dependence of the time of the merger on the physical and dynamical properties of the system. Finally, we study the dynamical evolution of the two Supermassive Black Holes…
The formation of supermassive stars (SMSs) is a possible pathway to seed supermassive black holes in the early universe. This chapter summarizes recent theoretical efforts to understand their evolution, highlighting effects of very rapid accretion at the rates of $\dot{M}_* \gtrsim 0.1~$M$_\odot$ yr$^{-1}$. Stellar evolution calculations predict that such an accreting SMS has a characteristic feature, i.e., a very large radius that monotonically increases as the stellar mass increases. The radius exceeds $7000$ R$_\odot$ (or $30$ AU) after the star accretes the gas of more than $10^3$ M$_\odot…
One of the main possible way of creating the supermassive black hole (SMBH) is a so call hierarchical merging scenario. Central SMBHs at the final phase of interacting and coalescing host-galaxies are observed as SMBH binary (SMBHB) candidates at different separations from hundreds of pc to mpc. Today one of the strongest SMBHB candidate is a ULIRG galaxy NGC 6240 which was X-ray spatially and spectroscopically resolved by Chandra. Dynamical calculation of central SMBHBs merging in a dense stellar environment allows us to retrace their evolution from kpc to mpc scales. The main goal of our dyn…
The evolution of a dark matter halo in a dark matter only simulation is governed purely byNewtonian gravity, making a clean testbed to determine what halo properties drive its fate.Using machine learning, we predict the survival, mass loss, final position, and merging time of subhalos within a cosmological N-body simulation, focusing on what instantaneous initial features of the halo, interaction, and environment matter most. Survival is well predicted, with our model achieving 96.5% accuracy using only 3 model inputs from the initial interaction.However, the mass loss, final location, and mer…
The evolution of main sequence binaries that reside in the Galactic Centre can be heavily influenced by the central supermassive black hole (SMBH). Due to these perturbative effects, the stellar binaries in dense environments are likely to experience mergers, collisions or ejections through secular and/or non-secular interactions. More direct interactions with the central SMBH are thought to produce hypervelocity stars (HVSs) and tidal disruption events (TDEs). In this paper, we use N-body simulations to study the dynamics of stellar binaries orbiting a central SMBH primary with an outer SMBH …
Supermassive black holes are a key ingredient of galaxy evolution. However, their origin is still highly debated. In one of the leading formation scenarios, a black hole of $\sim100$ M$_{\odot}$ results from the collapse of the inner core of a supermassive star ($\gtrsim 10^{4-5}$ M$_{\odot}$), created by the rapid accumulation ($\gtrsim 0.1 $ M$_{\odot}$ yr$^{-1}$) of pristine gas at the centre of newly formed galaxies at $z\sim 15$. The subsequent evolution is still speculative: the remaining gas in the supermassive star can either directly plunge into the nascent black hole, or part of it c…
We study the future orbital evolution and merging of the MW-M31-M33 system, using a combination of collisionless N-body simulations and semi-analytic orbit integrations. Monte-Carlo simulations are used to explore the consequences of varying the initial phase-space and mass parameters within their observational uncertainties. The observed M31 transverse velocity implies that the MW and M31 will merge t = 5.86 (+1.61-0.72) Gyr from now, after a first pericenter at t = 3.87 (+0.42-0.32) Gyr. M31 may (probability p=41%) make a direct hit with the MW (defined here as a first-pericenter distance le…
Our Galaxy and the nearby Andromeda galaxy (M31) are the most massive members of the Local Group, and they seem to be a bound pair, despite the uncertainties on the relative motion of the two galaxies. A number of studies have shown that the two galaxies will likely undergo a close approach in the next 4$-$5 Gyr. We used direct $N$-body simulations to model this interaction to shed light on the future of the Milky Way - Andromeda system and for the first time explore the fate of the two supermassive black holes (SMBHs) that are located at their centers. We investigated how the uncertainties on…
The coalescence of supermassive black holes (SMBHs) should generate the strongest sources of gravitational waves (GWs) in the Universe. However, the dynamics of their coalescence is the subject of much debate. In this study, we use a suite of $N$-body simulations to follow the merger of two nuclear star clusters (NSCs), each hosting a SMBH in their centre. We find that the presence of distinct star clusters around each SMBH has important consequences for the dynamical evolution of the SMBH binary: (i) The separation between the SMBHs decreases by a few orders of magnitude in the first few Myrs…
In hierarchical structure formation scenarios, merging galaxies are expected to be seen in different phases of their coalescence. Simulations suggest that simultaneous activity of the supermassive black holes (SMBHs) in the centres of the merging galaxies may be expected at kpc-scale separations. Currently, there are no direct observational methods which allow the selection of a large number of such dual active galactic nuclei (AGN) candidates. SDSS J142507.32+323137.4 was reported as a promising candidate source based on its optical spectrum. Here we report on our sensitive e-MERLIN observati…
Given its velocity dispersion, the early-type galaxy NGC 1600 has an unusually massive ($M_\bullet = 1.7 \times 10^{10} M_\odot$) central supermassive black hole (SMBH), surrounded by a large core ($r_\mathrm{b} = 0.7$ kpc) with a tangentially biased stellar distribution. We present high-resolution equal-mass merger simulations including SMBHs to study the formation of such systems. The structural parameters of the progenitor ellipticals were chosen to produce merger remnants resembling NGC 1600. We test initial stellar density slopes of $ρ\propto r^{-1}$ and $ρ\propto r^{-3/2}$ and vary the i…
We propose a new mechanism for the growth of supermassive black hole (BH) seeds in the star-forming progenitors of local early-type galaxies (ETGs) at $z\gtrsim 1$. This envisages the migration and merging of stellar compact remnants (neutron stars and stellar-mass BHs) via gaseous dynamical friction toward the central high-density regions of such galaxies. We show that, under reasonable assumptions and initial conditions, the process can build up central BH masses of order $10^4-10^6\, M_\odot$ within some $10^7$ yr, so effectively providing heavy seeds before standard disk (Eddington-like) a…
We present the highest resolution --- 15 pc (0.03'') --- ALMA $^{12}$CO(2-1) line emission and 1.3mm continuum maps, tracers of the molecular gas and dust, respectively, in the nearby merging galaxy system NGC 6240, that hosts two supermassive black holes growing simultaneously. These observations provide an excellent spatial match to existing Hubble optical and near-infrared observations of this system. A significant molecular gas mass, $\sim$9$\times$10$^9$M$_\odot$, is located in between the two nuclei, forming a clumpy stream kinematically dominated by turbulence, rather than a smooth rota…
We perform magnetohydrodynamic simulations in full general relativity of disk accretion onto nonspinning black hole binaries with mass ratio 36:29. We survey different disk models which differ in their scale height, total size and magnetic field to quantify the robustness of previous simulations on the initial disk model. Scaling our simulations to LIGO GW150914 we find that such systems could explain possible gravitational wave and electromagnetic counterparts such as the Fermi GBM hard X-ray signal reported 0.4s after GW150915 ended. Scaling our simulations to supermassive binary black holes…
There are increasing interests in binary supermassive black holes (SMBHs), but merging binaries with separations smaller than ~1 light days (~10^2 gravitational radii for 10^8 Msun), which are rapidly evolving under control of gravitational waves, are elusive in observations. In this paper, we discuss fates of mini-disks around component SMBHs for three regimes: 1) low rates (advection-dominated accretion flows: ADAFs); 2) intermediate rates; 3) super-Eddington accretion rates. Mini-disks with intermediate rates are undergoing evaporation through thermal conduction of hot corona forming a hybr…
Our Galaxy hosts a very massive object at its centre, often referred to as the supermassive black hole Sgr A*. Its gravitational tidal field is so intense that can strip apart a binary star passing its vicinity and accelerate one of the components of the binary as hypervelocity star (HVS) and grab the other star as S-star. Taken into consideration that many binary star systems are known to host planets, in this paper we aim to broaden the study of the close interaction of binary stars and their planetary systems with Sgr A* massive object. Results are obtained via a high precision $N-$body cod…
Recently we discovered an unprecedented supermassive black hole binary (SMBHB) candidate in the nearby Seyfert galaxy SDSS J1430+2303, which is predicted to merge within three years. X-ray spectroscopy may bring unique kinematic evidence for the last inspiraling stage, when the binary is too close to allow each of them to hold an individual broad line region. We try to confirm the unique SMBHB merger event and understand the associated high-energy processes from a comprehensive X-ray view. We observed SDSS J1430+2303 with XMM-Newton, NuSTAR, Chandra, and Swift spanning the first ~200 days sinc…
With projects such as Laser Interferometer Space Antenna (LISA) and Pulsar Timing Arrays expected to detect gravitational waves from supermassive black hole mergers in the near future, it is key that we understand what we expect those detections to be, and maximize what we can learn from them. To address this, we study the mergers of supermassive black holes in the Illustris simulation, the overall rate of mergers, and the correlation between merging black holes and their host galaxies. We find that these mergers occur in typical galaxies along the $M_{\rm{BH}}-M_*$ relation, and that between …
The main idea of our research is to estimate the physical coalescence time of the double supermassive black hole (SMBH) system in the centre of NGC 6240 based on the X-ray observations from the Chandra space observatory. The spectra of the Northern and Southern nuclei were fitted by spectral models from Sherpa and both presented the narrow component of the Fe K$α$ emission line. It enabled us to apply the spectral model to these lines and to find relative offset $\approx0.02$ keV. The enclosed dynamical mass of the central region of NGC 6240 with radius 1 kpc was estimated $\approx 2.04\times …
We simulate shells created by supernovae expanding into the interstellar medium (ISM) of the nuclear region of a galaxy, and analyze how the shell evolution is influenced by the supernova (SN) position relative to the galactic center, by the interstellar matter (ISM) density, and by the combined gravitational pull of the nuclear star cluster (NSC) and supermassive black hole (SMBH).We adopted simplified hydrodynamical simulations using the infinitesimally thin layer approximation in 3D (code RING) and determined whether and where the shell expansion may bring new gas into the inner parsec arou…
RABBITS IV: Stellar feedback and SMBH merging time-scales in the sub-Milky Way mass regime
University of Western Australia（西澳大学）；University of Helsinki（赫尔辛基大学）；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）；Max-Planck-Institut für Astrophysik（马克斯·普朗克天体物理学研究所）；Excellence Cluster ORIGINS（ORIGINS卓越集群）；OzGrav: The ARC Centre of Excellence for Gravitational-wave Discovery（引力波发现ARC卓越中心）
Upon entering the tidal sphere of a supermassive black hole, a star is ripped apart by tides and transformed into a stream of debris. The ultimate fate of that debris, and the properties of the bright flare that is produced and observed, depends on a number of parameters, including the energy of the center of mass of the original star. Here we present the results of a set of smoothed particle hydrodynamics simulations in which a $1~M_\odot $, $γ= 5/3$ polytrope is disrupted by a $10^6 ~M_\odot$ supermassive black hole. Each simulation has a pericenter distance of $r_{\rm p} = r_{\rm t}$ (i.e.,…
Fuzzy dark matter (FDM) consisting of ultra-light axions has been invoked to alleviate galactic-scale problems in the cold dark matter scenario. FDM fluctuations, created via the superposition of waves, can impact the motion of a central supermassive black hole (SMBH) immersed in an FDM halo. The SMBH will undergo a random walk, induced by FDM fluctuations, that can result in its ejection from the central region. This effect is strongest in dwarf galaxies, accounting for wandering SMBHs and the low detection rate of AGN in dwarf spheroidal galaxies. In addition, a lower bound on the allowed ax…
We follow trajectories of recoiling supermassive black holes (SMBHs) in analytical and numerical models of galaxy merger remnants with masses of $10^{11} \rm{M_{sun}}$ and $10^{12} \rm{M_{sun}}$. We construct various merger remnant galaxies in order to investigate how the central SMBH mass and the mass ratio of progenitor galaxies influence escape velocities of recoiling SMBHs. Our results show that static analytical models of major merger remnant galaxies overestimate the SMBHs escape velocities. During major mergers violent relaxation leads to the decrease of galaxy mass and lower potential …
Supermassive black hole binaries (SMBHBs) in the 10 million to 10 billion $M_\odot$ range form in galaxy mergers, and live in galactic nuclei with large and poorly constrained concentrations of gas and stars. There are currently no observations of merging SMBHBs--- it is in fact possible that they stall at their final parsec of separation and never merge. While LIGO has detected high frequency GWs, SMBHBs emit GWs in the nanohertz to millihertz band. This is inaccessible to ground-based interferometers, but possible with Pulsar Timing Arrays (PTAs). Using data from local galaxies in the 2 Micr…
JWST spectra are consistent with the edge-on star-forming galaxy scenario for the "runaway supermassive black hole"
Instituto de Astrofísica de Canarias（加那利群岛天体物理研究所）；Universidad de La Laguna（拉古纳大学）；Universidad Nacional Autónoma de México（墨西哥国立自治大学）；Institute of Space Sciences (ICE, CSIC)（空间科学研究所）
Investigating the Spectral Properties of Dual Nuclei in Galaxy Mergers from the GOTHIC survey: Supermassive Black Hole Growth, metal enrichment and Dual AGN
Indian Institute of Astrophysics（印度天体物理研究所）；Observatoire de Paris, LERMA, College ’ de France, PSL University, Sorbonne University, CNRS（巴黎天文台，LERMA，法兰西学院，PSL大学，索邦大学，法国国家科学研究中心）；Siebel School of Computing and Data Science, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校赛贝尔计算与数据科学学院）；APPCAIR, Department of CSIS, Birla Institute of Technology & Science（Birla理工学院CSIS系APPCAIR）；HappyMonk AI（HappyMonk人工智能）；Department of Physics, Indian Institute of Science Education and Research, Bhopal（博帕尔印度科学教育研究系物理系）
Supermassive black hole (SMBH) binaries residing at the core of merging galaxies are recently found to be strongly affected by the rotation of their host galaxies. The highly eccentric orbits that form when the host is counterrotating emit strong bursts of gravitational waves that propel rapid SMBH binary coalescence. Most prior work, however, focused on planar orbits and a uniform rotation profile, an unlikely interaction configuration. However, the coupling between rotation and SMBH binary evolution appears to be such a strong dynamical process that it warrants further investigation. This st…
We investigate the origin and fate of the recently discovered gas cloud G2 close to the Galactic Center. Our hydrodynamical simulations focussing on the dynamical evolution of the cloud in combination with currently available observations favor two scenarios: a Compact Cloud which started around the year 1995 and a Spherical Shell of gas, with an apocenter distance within the disk(s) of young stars and a radius of a few times the size of the Compact Cloud. The former is able to explain the detected signal of G2 in the position-velocity diagram of the Br gamma emission of the year 2008.5 and 20…
It has been suggested for quite a long time that galaxy mergers trigger activities of supermassive black holes (SMBHs) on the grounds of imaging observations of individual galaxies. To quantitatively examine this hypothesis, we calculate quasar luminosity functions (LFs) by manipulating the observed galaxy LFs ($z\lesssim 2$) and theoretical merger rates from semi-analytical formulations. We find that the model reproduces the observed quasar LFs provided that the mass ratio ($q$) of the secondary galaxy to the newly formed one changes with cosmic time. The results show that the fraction of maj…
SDSS J150243.09+111557.3 is a merging system at z = 0.39 that hosts two confirmed AGN, one unobscured and one dust-obscured, offset by several kiloparsecs. Deane et al. recently reported evidence from the European VLBI Network (EVN) that the dust-obscured AGN exhibits two flat-spectrum radio sources, J1502SE/SW, offset by 26 mas (140 pc), with each source being energized by its own supermassive black hole (BH). This intriguing interpretation of a close binary BH was reached after ruling out a double-hotspot scenario, wherein both hotspots are energized by a single, central BH, a configuration …
Using numerical hydrodynamics code, we perform various idealized galaxy merger simulations to study the star formation (SF) of two merging disk galaxies. Our simulations include gas accretion onto supermassive black holes and active galactic nucleus (AGN) feedback. By comparing AGN simulations with those without AGNs, we attempt to understand when the AGN feedback effect is significant. With ~70 simulations, we investigated the SF with the AGN effect in mergers with variety of mass ratios, inclinations, orbits, galaxy structures and morphologies. Using these merger simulations with AGN feedbac…
We present an analytic model to describe the supermassive black hole binary (SMBHB) merger rate in the Universe with astrophysical observables: galaxy stellar mass function, pair fraction, merger timescale and black hole - host galaxy relations. We construct observational priors and compute the allowed range of the characteristic spectrum $h_c$ of the gravitational wave background (GWB) to be $10^{-16}<h_c<10^{-15}$ at a frequency of $f=1/{\rm yr}$. We exploit our parametrization to tackle the problem of astrophysical inference from Pulsar Timing Array (PTA) observations. We simulate a series …
We have carried out a systematic search for recoiling supermassive black holes (rSMBH) using the Chandra Source and SDSS Cross Matched Catalog. From the survey, we have detected a potential rSMBH, 'CXO J101527.2+625911' at z=0.3504. The CXO J101527.2+625911 has a spatially offset (1.26$\pm$0.05 kpc) active SMBH and kinematically offset broad emission lines (175$\pm$25 km s$^{\rm -1}$ relative to systemic velocity). The observed spatial and velocity offsets suggest this galaxy could be a rSMBH, but we also have considered a possibility of dual SMBH scenario. The column density towards the galax…
A large fraction of galactic nuclei is expected to host supermassive black hole binaries (BHB), likely formed during the early phase of galaxies assembly and merging. In this paper, we use a large set of state-of-art numerical models to investigate the interplay between a BHB and a massive star cluster (GCs) driven toward the galactic centre by dynamical friction. Varying the BHB mass and mass ratio and the GC orbit, we show that the reciprocal feedback exerted between GCs and the BHB shapes their global properties. We show that, at GC-to-BHB mass ratios above 0.1, the GC affects notably the B…
We estimate the expected event rate of gravitational wave signals from mergers of supermassive black holes that could be resolved by a space-based interferometer, such as the Evolved Laser Interferometer Space Antenna (eLISA), utilising the reference cosmological hydrodynamical simulation from the EAGLE suite. These simulations assume a $Λ$CDM cosmogony with state-of-the-art subgrid models for radiative cooling, star formation, stellar mass loss, and feedback from stars and accreting black holes. They have been shown to reproduce the observed galaxy population with unprecedented fidelity. We c…
We present calculations on the formation of massive black holes with 10^5 Msun at z > 6 that can be the seeds of supermassive black holes at z > 6. Under the assumption of compact star cluster formation in merging galaxies, star clusters in haloes of 10^8 ~ 10^9 Msun can undergo rapid core-collapse leading to the formation of very massive stars (VMSs) with ~1000 Msun which directly collapse into black holes with similar masses. Star clusters in halos of > 10^9 Msun experience type-II supernovae before the formation of VMSs due to long core-collapse time scales. We also model the subsequent gro…
Recent observations suggested that star formation quenching in galaxies is related to galaxy structure. Here we propose a new mechanism to explain the physical origin of this correlation. We assume that while quiescent galaxies are maintained quenched by a feedback mechanism, cooling flows in the hot halo gas can still develop intermittently. We study cooling flows in a large suite of around 90 hydrodynamic simulations of an isolated galaxy group, and find that the flow development depends significantly on the gravitational potential well in the central galaxy. If the galaxy's gravity is not s…
We present a combined morphological and X-ray analysis of Was 49, an isolated, dual AGN system notable for the presence of a dominant AGN Was 49b in the disk of the primary galaxy Was 49a, at a projected radial distance of 8 kpc from the nucleus. Using X-ray data from Chandra, NuSTAR, and Swift, we find that this AGN has a bolometric luminosity of L_bol ~ 2 x 10^45 erg/s, with a black hole mass of M_BH=1.3^{+2.9}_{-0.9} x 10^8 M_Sol. Despite its large mass, our analysis of optical data from the Discovery Channel Telescope shows that the supermassive black hole is hosted by a stellar counterpar…
NGC6240 is a well-studied nearby galaxy system in the process of merging. Based on optical, X-ray, and radio observations, it is thought to harbor two active nuclei We carried out a detailed optical 3D spectroscopic study to investigate the inner region of this system in connection with existing MERLIN and VLBA data. We observed NGC6240 with very high spatial resolution using the MUSE instrument in the Narrow-Field Mode with the four-laser GALACSI adaptive optics system on the ESO VLT under seeing conditions of 0.49 arcsec. Our 3D spectra cover the wavelength range from 4725 to 9350 AA at a sp…
During galaxy mergers the gas falls to the center, triggers star formation, and feeds the rapid growth of supermassive black holes (SMBHs). SMBHs respond to this fueling by supplying energy back to the ambient gas. Numerical studies suggest that this feedback is necessary to explain why the properties of SMBHs and the formation of bulges are closely related. This intimate link between the SMBH's mass and the large scale dynamics and luminosity of the host has proven to be a difficult issue to tackle with simulations due to the inability to resolve all the relevant length scales simultaneously.…
Galactic nuclei harbouring a central supermassive black hole (SMBH), possibly surrounded by a dense nuclear cluster (NC), represent extreme environments which house a complex interplay of many physical processes that uniquely affect stellar formation, evolution, and dynamics. The discovery of gravitational waves (GW) emitted by merging black holes (BHs) and neutron stars (NSs), funnelled a huge amount of work focused on understanding how compact object binaries (COBs) can pair-up and merge together. Here, we review from a theoretical standpoint how different mechanisms concur to the formation,…
Mergers of galaxies are extremely violent events shaping their evolution. Such events are thought to trigger starbursts and, possibly, black hole accretion. Nonetheless, it is still not clear how to know the fate of a galaxy pair from the data available at a given time, limiting our ability to constrain the exact role of mergers. In this paper we use the lightcone of the Horizon-AGN simulation, for which we know the fate of each pair, to test three selection processes aiming at identifying true merging pairs. We find that the simplest one (selecting objects within two thresholds on projected d…
Disks of gas accreting onto supermassive black holes are thought to power active galactic nuclei (AGN). Stars may form in gravitationally unstable regions of these disks, or may be captured from nuclear star clusters. Because of the dense gas environment, the evolution of such embedded stars can diverge dramatically from those in the interstellar medium. This work extends previous studies of stellar evolution in AGN disks by exploring a variety of ways that accretion onto stars in AGN disks may differ from Bondi accretion. We find that tidal effects from the supermassive black hole significant…
The formation of massive black hole binaries (MBHBs) is an unavoidable outcome of galaxy evolution via successive mergers. However, the mechanism that drives their orbital evolution from parsec separations down to the gravitational wave (GW) dominated regime is poorly understood, and their final fate is still unclear. If such binaries are embedded in gas-rich and turbulent environments, as observed in remnants of galaxy mergers, the interaction with gas clumps (such as molecular clouds) may efficiently drive their orbital evolution. Using numerical simulations, we test this hypothesis by study…
We introduce a force correction term to better model the dynamical friction (DF) experienced by a supermassive black hole (SMBH) as it orbits within its host galaxy. This new approach accurately follows the orbital decay of a SMBH and drastically improves over commonly used advection methods. The force correction introduced here naturally scales with the force resolution of the simulation and converges as resolution is increased. In controlled experiments we show how the orbital decay of the SMBH closely follows analytical predictions when particle masses are significantly smaller than that of…
