Glimmering in the Dark: Modeling the Low-Mass End of the $ M {\Bullet

$Glimmering in the Dark: Modeling the Low-Mass End of the $ M {\Bullet$

Draft version July 27, 2021 Typeset using LATEX twocolumn style in AASTeX61

GLIMMERING IN THE DARK: MODELING THE LOW-MASS END OF THE M• − σ RELATION AND OF THE QUASAR LUMINOSITY FUNCTION

Fabio Pacucci,1 Abraham Loeb,2 Mar Mezcua,3, 4 and Ignacio Mart´ın-Navarro5, 6

1Yale University, Department of Physics, New Haven, CT 06511, USA 2Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA 3Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Magrans s/n, E-08193 Barcelona, Spain 4Institut d’Estudis Espacials de Catalunya (IEEC), C/ Gran Capit`a,E-08034 Barcelona, Spain 5University of California Santa Cruz, Santa Cruz, CA 95064, USA 6Max-Planck Institut f¨urAstronomie, Konigstuhl 17, D-69117 Heidelberg, Germany

ABSTRACT

The M• − σ relation establishes a connection between central black holes (BHs) and their host spheroids. Supported 5 by observations at M• & 10 M , there is limited data on its validity at lower masses. Employing a semi-analytical model to simulate the combined evolution of BHs and their host galaxies, we predict the observational consequences 5 of assuming a bimodality in the accretion efficiency of BHs, with low-mass BHs (M• . 10 M ) accreting inefficiently. 5 We predict a departure from the M• − σ relation at a transitional BH mass ∼ 10 M , with lower-mass BHs unable to reach the mass dictated by the relation and becoming disconnected from the evolution of the host galaxy. This 5 6 prediction is an alternative to previous works suggesting a flattening of the relation at ∼ 10 − 10 M . Furthermore, we predict a deficit of BHs shining at bolometric luminosities ∼ 1042 erg s−1. Joined with a detection bias, this could partly explain the scarce number of intermediate-mass BHs detected. Conversely, we predict an increase in source density at lower bolometric luminosities, < 1042 erg s−1. Because our predictions assume a bimodal population of high-redshift BH seeds, future observations of fainter BHs will be fundamental for constraining the nature of these seeds.

Keywords: galaxies: evolution — galaxies: active — black hole physics — quasars: supermassive black holes — early universe — dark ages, reionization, ﬁrst stars arXiv:1808.09452v1 [astro-ph.GA] 28 Aug 2018

Corresponding author: Fabio Pacucci [email protected] 2 Pacucci et al.

1. INTRODUCTION ies, will provide important constraints on the nature of It is commonly accepted that the central region of BH seeds at high redshift, which constitute the progen- all massive galaxies contains a super-massive black hole itors of the z ∼ 7 quasar population (Fan et al. 2006; 6 Natarajan & Volonteri 2012; Volonteri et al. 2016; Ri- (BH, M• & 10 M , see e.g. King & Pounds 2015). There seems to be a tight correlation between the mass carte & Natarajan 2018). of the BH and the properties of the host galaxy spheroid, 2. A BIMODAL ACCRETION MODEL such as the velocity dispersion of stars. This correlation, Pacucci et al.(2017b) suggested that accretion onto named the M − σ relation (Ferrarese & Merritt 2000; • high-z BHs may be bimodal. Accretion onto BHs lighter Gebhardt et al. 2000; Kormendy & Ho 2013; McConnell than a mass threshold M is inefficient, with largely & Ma 2013), is surprising as there is a wide separation f• sub-Eddington accretion rates and alternating quiescent between the physical scale of the bulge of a galaxy and and active phases. Depending on the parameters of the the sphere of influence of its central BH. The bulge of model, M ∼ 105 − 106 M . Previous studies already the Milky Way galaxy, for example, is ∼ 104 times larger f• proposed that lower-mass BHs accrete more inefficiently than the radius of influence of its BH. The feedback re- than higher-mass ones (e.g., Pacucci et al. 2015, 2017a; sulting from BH accretion is thought to be the driving Inayoshi et al. 2016; Park et al. 2016). The novelty of force in establishing the M −σ relation, regulating both • the proposal by Pacucci et al.(2017b) was to identify the star formation in massive host galaxies and the gas the physical conditions that allow high-efficiency accre- inflow onto the central BH (Fabian et al. 2000; Begel- tion. This identification allows to calculate the proba- man & Nath 2005; King & Pounds 2015; Mart´ın-Navarro bility that a BH seed formed with an accelerated growth et al. 2018). rate. van den Bosch(2016), employing a heterogeneous set The high-efficiency region in the two-dimensional pa- of 230 BHs with a minimum mass ∼ 4 × 105 M , found rameter space of BH mass and gas number density a relation of the form (M•, n∞) is found by combining three conditions for ef- log M• = (8.32 ± 0.04) + (5.35 ± 0.23) log σ200 , (1) ficient accretion on large (r & RB) and small (r RB) spatial scales, where r is the distance from the BH and where M• is in solar masses and σ200 is expressed in RB is its Bondi radius (Bondi 1952). Assuming that −1 units of 200 km s . Due to observational constraints, photon trapping is active in the interior part of the ac- the low-mass regime of the relation is far less explored. cretion flow, the three conditions are as follows (Pacucci 4 Currently, the lightest central BH (M• ∼ 3 × 10 M ) et al. 2015; Inayoshi et al. 2016; Begelman & Volon- is observed in a dwarf galaxy at z ∼ 0.03 (Chilingar- teri 2017). The growth efficiency on small scales is de- ian et al. 2018). Due to the paucity of the detected termined by the extent of the transition radius, above 2 6 intermediate-mass BHs (10 M . M• . 10 M , e.g. which the radiation pressure dominates the accretion Greene & Ho 2004; Reines et al. 2013; Baldassare et al. flow: 2 2015; Mezcua et al. 2015, 2016, 2018a; see the review n∞ M > 10−11 M . (2) by Mezcua 2017), it is still hard to infer whether or not • 1 cm−3 low-mass galaxies follow the extrapolation of the M• −σ By increasing n∞ the minimum seed mass required to relation (Xiao et al. 2011; Baldassare et al. 2015; Mezcua sustain efficient growth increases as well. The growth 2017; Mart´ın-Navarro & Mezcua 2018). efficiency on large scales is determined by the compar- A complete description of galaxy evolution requires a ison between RB and the extent of the ionized region better understanding of the low-mass BH regime. Star around the BH: formation and BH quenching in low-mass galaxies could n −1 M > 109 ∞ M . (3) be driven by different mechanisms, involving young stars • 1 cm−3 and supernovae instead of the central BH (Dubois et al. By increasing n the minimum seed mass required to 2015; Anglés-Alcázaret al. 2017; Habouzit et al. 2017). ∞ sustain efficient growth decreases. Finally, the infalling For masses lighter than a transition mass, the central gas needs to overcome the angular momentum barrier BH might be disentangled from the evolution of the host in order to accrete onto the BH. The condition is galaxy. −5/4 In this Letter we assume a bimodality in the accretion 19 n∞ 3 M• > 2.2 × 10 λB M . (4) efficiency of BHs (Pacucci et al. 2017b), and predict the 1 cm−3 shape of the M• −σ relation and of the luminosity func- Here, λB is the ratio of the specific angular momentum of 5 tion for BHs with M• . 10 M . Our predictions, when the gas `B to its Keplerian value, computed at the trap- compared to future observations of BHs in dwarf galax- ping radius (distance from the BH inside which photon M• − σ relation and luminosity function for low-mass black holes 3

1/2 trapping is eﬃcient): λB = `B/(GM•RB) , where G is the gravitational constant. The minimum mass of a BH to be inside the high- eﬃciency region of the (M•, n∞) parameter space is

λ 24/13 M 5 × 105 B M . (5) • & 10−1

At higher BH masses, the BH is in the high-eﬃciency region for an increasingly larger range of gas density: high-eﬃciency accretion is, thus, more likely to occur.

2.1. A mass threshold for super-Eddington rates We present here a simplified argument to show that a fundamental transition between low-efficiency and high-efficiency accretion occurs in the BH mass range Figure 1. Conditions for efficient BH growth in the 105 − 106 M , an assumption that is at the core of our (M•, M˙ B ) parameter space. The high-efficiency region is bimodal model. The simplified assumptions introduced shaded green; the Eddington rate is shown as a black line. in this section are in no way used in the actual growth model described in Sec.3. We eliminate the parameter 3.1. Description of the data n∞ from Eqs. (2) and (4) by computing the correspond- 2 2 3 We test our model against observational data of both ing Bondi rate M˙ B = 4πρG M /c , where ρ is the gas • s low-mass and high-mass galaxies for which BH mass and mass density, and cs is the sound speed. The Bondi rate is a convenient approximation to use in this simpli- stellar velocity measurements are available. We use the same sample as that of Mart´ın-Navarro & Mezcua(2018) fied model; at scales & RB is also a reasonable one for the accretion rate, as the accretion disk forms at much (blue stars in Fig.2), which includes a compilation of 127 low-mass Seyfert 1 galaxies with σ < 100 km s−1 from smaller scales. Assuming cs = σg ∼ σs (σg and σs are the velocity dispersions for gas and stars, respectively; Xiao et al.(2011) and Woo et al.(2015) drawn from the Sloan Digital Sky Survey (SDSS; Abolfathi et al. this simplified model remains valid as long as σg and σs are within the same order of magnitude) and that the 2018). The BH masses of these galaxies were estimated from the width of optical broad emission lines under the M• − σ relation (Eq.1) is valid, the constraints of Eqs. assumption that the gas is virialized (Xiao et al. 2011; (2) and (4) in the (M•, M˙ B) parameter space are shown in Fig.1. Woo et al. 2015). In the regime of massive galaxies, 5 the sample includes 205 early-type galaxies with direct Beginning the growth at M• 10 M , a BH seed can reach, at most, the Eddington rate. Growing in BH mass measurements from van den Bosch(2016) and −1 mass, it gets progressively closer to the high-efficiency σ & 100 km s . Typical errors in M• and σ are up to −1 regime, being able to enter it when the Eddington rate a factor ∼ 3 and 10 − 15 km s , respectively. crosses the angular momentum barrier shown in Fig. 1 as a green line. Once inside the high-efficiency re- 3.2. Modeling the M• − σ assuming bimodality gion, super-Eddington rates are reachable (Begelman & We employ a merger tree code (Parkinson et al. 2008; Volonteri 2017). The only condition that matters in this Dayal et al. 2017) to track the cosmological evolution of regime involves the angular momentum (Eq.4), whose a population of dark matter halos distributed following relevance is not restricted to the high-z Universe. An the Sheth-Tormen halo mass function (Sheth & Tormen important assumption in this derivation is that there is 1999). We sample logarithmically the cosmological scale always a gas supply to grow the BH at or above the factor a = (z + 1)−1 between z = 20 and z = 0.1, the Eddington limit: the growth is, thus, supply-limited. mean redshift of the sample described in Sec. 3.1. We assign a single BH seed to each z = 20 galaxy with a 3. DATA AND METHODS 7 halo mass Mh & 5 × 10 M (such that its virial tem- Next, we predict the cosmological evolution of a pop- perature is higher than the atomic cooling threshold, see ulation of BHs accreting in a bimodal regime. We com- e.g. Barkana & Loeb 2001). We further assume a ratio of 4 2 pare our predictions with a sample of ∼ 300 galaxies in high-mass (M• > 10 M ) to low-mass (M• < 10 M ) −1 the range 30 . σ(km s ) . 400. BH seeds of 1 : 100. In fact, the formation of a high- 4 Pacucci et al. mass seed is a much rarer event, because of the addi- remain qualitatively unchanged for all values of zt > 3. tional requirements (see e.g. Bromm & Loeb 2003) to We note, however, that the assumption of constant avail- prevent the fragmentation of the gas cloud. The ratio ability of gas for the central BHs is not realistic at these employed here is a proxy for the relative abundance of low z. sources in the high-luminosity and the low-luminosity Whenever a BH is active, we describe the time evo- ends of the quasar luminosity function (see e.g. the one lution of M• with two parameters: the duty cycle D for 3 < z < 5 in Masters et al. 2012). This mixture of (fraction of the active phase spent accreting) and the light and heavy seeds reproduces the z ∼ 0 quasar lu- Eddington ratio fEdd = M/˙ M˙ Edd. The first parameter 43 −1 minosity function for Lbol & 10 erg s (Hopkins et al. describes the continuity of the gas inflow, and the sec- 2007), as detailed in Sec. 4.2. We model high-mass seeds ond one quantifies the amount of mass flowing in. The as direct collapse black holes (DCBHs; e.g., Bromm & time evolution of M•, starting from M•(z0) = M•,0, is Loeb 2003) and low-mass seeds as Pop III stellar rem- obtained from the integral nants (e.g., Hirano et al. 2014). We model the initial Z z0 C(D, fEdd)dz mass function of DCBHs with a log-gaussian distribu- M•(z) = M•,0 exp , (7) tion, having a mean µ = 5.1 and a standard deviation z E(z) σ = 0.2, both in logarithm of mass. For Pop III stars where C(D, fEdd) incorporates various constants and the we employ a model with a Salpeter-like exponent and a two parameters of the model, E(z) depends on the cos- low-mass cutoff Mc: mology of choice (see Pacucci et al. 2017b for details). The values adopted for D and fEdd at each redshift de- Mc Φ(PopIII, m) ∝ m−2.35 exp − . (6) pend on the properties of the BH in the (M , n ) pa- m • ∞ rameter space. In general, D 1 and fEdd 1 in

We assume Mc = 10 M and convolve this progenitor the low-efficiency region and D ∼ 1 and fEdd & 1 in mass function with the relation (Woosley et al. 2002) the high-efficiency region (Pacucci et al. 2015; Inayoshi between the mass of the remnant and the stellar mass. et al. 2016; Begelman & Volonteri 2017). The gas den- As long as there is a clear separation between the initial sity profile of the host galaxies is assumed to follow an mass functions for low-mass and high-mass seeds, their isothermal sphere. exact shape does not significantly influence our results. Once two galaxies merge, their central BHs are as- We calculate the central stellar velocity dispersion of sumed to coalesce instantaneously. The only cap im- the host from the asymptotic circular velocity vc (as posed for the growth by BH mergers is the mass of the 10 a function of total halo mass and redshift), which is a most massive BHs thus far observed (∼ 5 × 10 M , Mezcua et al. 2018b). Expressed as a function of the ve- proxy for the total√ mass of the dark matter halo of the galaxy: σ = vc/ 2. locity dispersion, the cap for the growth by accretion is −1 The fueling of the central BH seed in each galaxy is devised to match the M• −σ relation for σ & 100 km s . implemented with the following scheme. A BH is active This mass cap is formally equal to Eq. (1). The relation 5.35 whenever gas is available. In the early Universe before M• ∝ σ substantially agrees with the hypothesis of a redshift threshold z > zt, we assume that a sufficient BH growth being regulated by energy-driven wind feed- 5 2 amount of gas is always available to feed the seed: thus, back (e.g., King 2010): M• ' fgκσ /(πG c), where fg the BH is always active. Simulations (e.g., Dubois et al. is the cosmic baryon fraction and κ is the gas opacity. 2014) and analytical estimates (e.g., Wyithe & Loeb 4. OBSERVATIONAL PREDICTIONS 2012) suggest that even super-Eddington infall rates are fairly common at high redshift. Begelman & Volonteri We are now in a position to make observational pre- (2017) point out that the fraction of AGNs accreting at dictions from our model, regarding the M• − σ relation super-Eddington rates could be as high as ∼ 10−3 at and the quasar luminosity function. −2 z = 1 and ∼ 10 at z = 2. For z < zt, we instead assume that the central BH is active only when a ma- 4.1. The low-mass regime in the M• − σ jor merger occurs, defined as a merger with a mass ratio Our main results are shown in the left panel of Fig. equal or larger than 1 : 10. This criterion is meant to re- 2. The simulations are evolved to z = 0.1 to match flect the necessity of an external reservoir of gas to over- the mean redshift of the data sample in Mart´ın-Navarro come the angular momentum barrier. Whenever a ma- & Mezcua(2018). The simulations closely follow the jor merger occurs, the BH is set in the active phase for data and the theoretical model by van den Bosch(2016) a time equal to the merger time scale (Boylan-Kolchin for σ > 70 km s−1 (see also Mart´ın-Navarro & Mezcua −1 et al. 2008). We set zt = 6 and check that our results 2018; Ricarte & Natarajan 2018). At σ ∼ 70 km s M• − σ relation and luminosity function for low-mass black holes 5

1011 1011

1010 1010

109 109

108 108 ] ] ¯ ¯

M 7 M 7 [ [

10 10 s s s s

a 6 a 6

M 10 M 10

H H B 105 B 105

4 M σ (van den Bosch 16) 4 4 10 • − 10 M σ flattened at 5 10 M 1 • − ∼ × ¯ This work (σ < 70 km s− ) due to detection bias 103 Simulations 103 Simulations Data sample Data sample 102 102 101 102 103 101 102 103 1 1 σ [km s− ] σ [km s− ] Figure 2. Theoretical predictions of the model (green points) compared with data from Mart´ın-Navarro & Mezcua(2018) 5 (blue stars). Left panel: for M• & 10 M the simulations follow closely the well-known M• − σ relation (red line). For 5 M• . 10 M the simulations fall below the line, indicating that central BHs become disconnected from the evolution of their 5 host stellar components. The van den Bosch(2016) model intercepts our predictions for M• . 10 M at transition values −1 5 σt ∼ 65 km s and M•,t ∼ 5×10 M . Right panel: current observational capabilities do not allow the detection of central BHs 5 5 6 with M• 10 M . For this reason we might be observing a ﬂattening of the M• −σ relation toward masses M• ∼ 10 −10 M . The red line indicates a smooth transition in mass between the M• −σ relation at large BH masses and the observational limit at 4 which both the BH mass and the stellar velocity dispersion are available (shown as a red star), currently set at M• ∼ 5×10 M (Baldassare et al. 2015). The observational limit is interpreted here as a line of constant BH mass.

5 there is a clear change in trend: our simulation points M•,t ∼ 5 × 10 M . At lower masses and velocity dis- tend to be under the theoretical M• −σ relation. This is persions the model clearly predicts a departure from the a direct consequence of the bimodality in the accretion M• − σ relation, with the bulk of objects being under- 5 efficiency of BHs. For masses M• & 10 M the BHs massive. are likely to be in the high-efficiency regime (see Fig. 1). These BHs grow efficiently and are able to reach 4.2. The quasar luminosity function the M − σ within a Hubble time. When they reach • The simulated data at z = 0.1 can be used to construct the M − σ, their growth is saturated by energy-driven • a luminosity function, with the abundance of host ha- winds, which deplete the central regions from the re- los retrieved from a Sheth-Tormen halo mass function maining gas, preventing further growth. At this stage, at the same redshift. We divide the simulations in 35 growth can occur by mergers only. We find the pres- logarithmic mass bins, and for each of them we com- ence of objects with masses significantly larger than the pute the average luminosity. We then compare in Fig. M − σ. Observationally, these objects can be inter- • 3 our predictions with bolometric luminosities of BHs preted as the brightest cluster galaxies, whose BHs are in dwarf galaxies, based on Hα luminosities (Greene & found to be over-massive with respect to the scaling re- Ho 2004; Reines et al. 2013; Baldassare et al. 2017). lations (Mezcua et al. 2018b). Nonetheless, these ob- The bolometric correction is from Greene & Ho(2004), jects constitute a minority population when compared L = 2.34 × 1044(L /1042)0.86 erg s−1. The luminos- to the bulk of BHs. For M 105 M , the fraction bol Hα • . ity function is in excellent agreement with observations of BHs growing efficiently is small, and only a minority for L 1043 erg s−1 (e.g., Hopkins et al. 2007). At (∼ 3%) of them are able to reach the cap imposed by the bol & low luminosities there is an evident deficit around L ∼ M − σ relation. The vast majority of them remain at bol • 1042 erg s−1, explained by our result that BHs accreting lower masses, unable to trigger sufficiently strong winds in that luminosity range have masses close to the tran- to fully halt their growth. Instead, they keep accret- sition mass between high and low efficiency. At higher ing at very low rates for most of the Hubble time. In BH masses, the likelihood of accreting close to the Ed- the low-mass regime, we predict a much steeper relation dington limit is large, while at lower masses the BH ac- (M ∼ σ11) which is only approximate, as BH and stel- • cretion is mostly sub-Eddington. The result is a paucity lar component become increasingly disconnected. The of BHs shining at L ∼ 1042 erg s−1. The scarcity of van den Bosch(2016) model intercepts our predictions bol observations of intermediate-mass BHs in this luminos- for low masses at transition values σ ∼ 65 km s−1 and t ity range could thus be a combination of a detection-bias 6 Pacucci et al.

˙ ˙ Log BH mass (M = MEdd) [M ] tion would be the M• − M? relation, where M? is the 0 2 4 6 ¯ 8 101 80 stellar mass of the galaxy (Reines & Volonteri 2015). We

0 note, however, that many dwarf galaxies do have bulges 10 70 (e.g., NGC 4395, POX52, RGG 118, see Baldassare et al. 10-1 60 2015) and that the deﬁnition of σ can always be inter- Data for 10-2 preted as the velocity dispersion of stars within some ] IMBHs only 50 3

− -3 c 10 eﬀective radius from the center of mass of the system. p

M 40

3 -4 In this Letter we chose to focus on the M• −σ because it

h 10 [ seems to provide a tighter relation (e.g., Shankar et al.

Φ 30 10-5 2016), indicating a more fundamental connection. As a Number of objects 20 10-6 test, we performed our simulations also in the (M•,M?) Data sample for 44 1 parameter space, using the theoretical relation presented 10-7 Lbol < 10 erg s− 10 Simulations in Reines & Volonteri(2015). We confirm, also in the 10-8 0 38 40 42 44 46 48 M• −M? space, the presence of the same departure from Log BH bolometric luminosity [erg s 1] 5 − the theoretical relation, occurring at M• ∼ 10 M . Be- low we discuss the consequences of our results for BH Figure 3. Luminosity function of galactic BHs in our sim- seed models at high redshift. ulations (red symbols), obtained from sources categorized in 35 logarithmic mass bins. The upper scale in BH mass as- sumes that all BHs are accreting at the Eddington rate and 5.1. Observational predictions for BH seeding models it is for reference only. There is a clear deficit of sources 42 −1 Our main prediction is that central BHs and their with predicted luminosities Lbol ∼ 10 erg s . The data sample of intermediate-mass BHs from Greene & Ho(2004); host galaxies depart from the M• −σ relation for masses 5 Reines et al.(2013); Baldassare et al.(2017) also suggests M• . 10 M , becoming under-massive with respect to this deficit, which might be a combination of observational the extrapolation of the M• − σ to lower masses. The 5 bias and intrinsic scarcity of sources. M• − σ relation reflects, for M• & 10 M , the connection between galaxy evolution and BH growth. The with an intrinsically low probability of observing sources link is driven by the outflows generated by the BH en- in that luminosity range. The importance of a detec- ergy and momentum output. Smaller BHs grow ineffi- tion bias in the observation of intermediate-mass BHs ciently and are unable to generate strong outflows trig- has been thoroughly investigated. For example, Mezcua gering the growth-regulation process. For this reason, 5 et al.(2016) showed, by means of X-ray stacking, that BHs with M• . 10 M fail to reach the mass dictated a population of faintly accreting intermediate-mass BHs by their velocity dispersion and become under-massive. 38 40 −1 (with X-ray luminosity Lx ∼ 10 − 10 erg s ) should Previous observations (e.g., Mart´ın-Navarro et al. 2018; be present in dwarf galaxies; however, their detection is Mart´ın-Navarro & Mezcua 2018) and simulations (e.g., challenging due to their faintness and mild obscuration. Anglés-Alcázaret al. 2017; Habouzit et al. 2017) al- Figure3 also suggests that the abundance of sources ready suggested that feedback is driven by BH activity 42 −1 5 with Lbol < 10 erg s should rise again at lower lu- for M• & 10 M and by supernova-driven winds for 5 minosities. We point out, though, that the detection M• . 10 M . of these abundant faint sources is challenging, because Some papers (e.g., Greene & Ho 2006; Mezcua 2017; they enter the luminosity regime of stellar X-ray binaries Mart´ın-Navarro & Mezcua 2018) suggest an alternative (Mezcua et al. 2018a). scenario for the low-mass regime of the M• − σ rela- 5 6 tion, predicting a flattening at masses ∼ 10 − 10 M . 5. DISCUSSION AND CONCLUSIONS Mart´ın-Navarro & Mezcua(2018) explained this puta- We have employed a semi-analytical model to inves- tive flattening with a weaker coupling between baryonic tigate the low-mass regime of the M• − σ relation and cooling and BH feedback, disconnecting the BH from of the quasar luminosity function. Our model is based the evolution of the host spheroid. Alternatively, the on two main assumptions: (i) low-mass and high-mass flattening could be explained with the prevalence of a seed populations form at z ∼ 20; (ii) the accretion pro- high-mass formation channel for early seeds (Volonteri 5 cess is bimodal, with BHs with M• . 10 M accreting 2010). These high-mass seeds would fail to grow and just 5 6 inefficiently. accumulate around their original mass, ∼ 10 − 10 M . The M• − σ relation is somewhat tricky for low-mass In this Letter, we envisage that a flattening toward 5 6 galaxies, as σ is defined as the velocity dispersion of ∼ 10 − 10 M would be due to an observational bias. 5 stars inside bulges. An alternative to the M• − σ rela- Observing BHs with M• . 10 M is currently chal- M• − σ relation and luminosity function for low-mass black holes 7 lenging, and the predicted paucity of BHs shining at its host galaxy, but will ultimately provide important 42 −1 Lbol ∼ 10 erg s (Sec. 4.2) could add to this ef- constraints on the seed formation mechanisms active in fect. Instead of a flattening, our model clearly predicts the high-redshift Universe. The observation in the local a downward departure from the M• − σ relation. A Universe of intermediate-mass BHs, as well as other pro- 5 detection for M• . 10 M of a relation of the form posed techniques (e.g., deriving the local super-massive α M• ∼ σ with α & 7 would be an important indicator of BH occupation fraction, Miller et al. 2015) will help us the existence of a bimodal population of BH seeds. to understand processes occurred early in the history of 4 Pushing the detection limit to M• . 10 M (Baldas- the Universe. sare et al. 2017; Chilingarian et al. 2018) will enable to determine the relevance of the BH - galaxy connection F.P. acknowledges support from the NASA Chandra for lower-mass galaxies, and whether or not a departure award No. AR8-19021A, and enlightening discussions with Vivienne Baldassare and Elena Gallo. M.M. ac- from the M• − σ relation occurs. A major role in this observational challenge will be played by future observa- knowledges support from the Spanish Juan de la Cierva tories, both in the electromagnetic (e.g., Lynx; see Ben- program (IJCI-2015-23944). I.M.N. acknowledges sup- Ami et al. 2018) and in the gravitational (e.g., LISA; see port from the EU Marie Curie Global Fellowships. This Amaro-Seoane et al. 2017) realms. This effort will not work was supported in part by the Black Hole Initiative only shed light on the interconnection between BH and at Harvard University, which is funded by a JTF grant.

REFERENCES

Abolfathi B., et al., 2018, ApJS, 235, 42 Greene J. E., Ho L. C., 2006, ApJ, 641, L21 Amaro-Seoane P., et al., 2017, preprint, p. Habouzit M., Volonteri M., Dubois Y., 2017, MNRAS, 468, arXiv:1702.00786( arXiv:1702.00786) 3935 Anglés-AlcázarD., Faucher-GiguèreC.-A., Quataert E., Hirano S., Hosokawa T., Yoshida N., Umeda H., Omukai Hopkins P. F., Feldmann R., Torrey P., Wetzel A., Kereˇs K., Chiaki G., Yorke H. W., 2014, ApJ, 781 D., 2017, MNRAS, 472, L109 Hopkins P. F., Richards G. T., Hernquist L., 2007, ApJ, Baldassare V. F., Reines A. E., Gallo E., Greene J. E., 654, 731 2015, ApJ, 809 Inayoshi K., Haiman Z., Ostriker J. P., 2016, MNRAS, 459, Baldassare V. F., Reines A. E., Gallo E., Greene J. E., 3738 2017, ApJ, 836, 20 King A. R., 2010, MNRAS, 408, L95 Barkana R., Loeb A., 2001, PhR, 349, 125 King A., Pounds K., 2015, Annual Review of Astronomy Begelman M. C., Nath B. B., 2005, MNRAS, 361, 1387 and Astrophysics, 53, 115 Begelman M. C., Volonteri M., 2017, MNRAS, 464, 1102 Kormendy J., Ho L. C., 2013, Annual Review of Astronomy Ben-Ami S., Vikhlinin A., Loeb A., 2018, ApJ, 854, 4 and Astrophysics, 51, 511 Bondi H., 1952, MNRAS, 112, 195 Mart´ın-Navarro I., Mezcua M., 2018, ApJL, 855, L20 Boylan-Kolchin M., Ma C.-P., Quataert E., 2008, MNRAS, Mart´ın-Navarro I., Brodie J. P., Romanowsky A. J., 383, 93 Ruiz-Lara T., van de Ven G., 2018, Nature, 553, 307 Bromm V., Loeb A., 2003, ApJ, 596, 34 Masters D., et al., 2012, ApJ, 755, 169 Chilingarian I. V., Katkov I. Y., Zolotukhin I. Y., Grishin McConnell N. J., Ma C.-P., 2013, ApJ, 764, 184 K. A., Beletsky Y., Boutsia K., Osip D. J., 2018, preprint, (arXiv:1805.01467) Mezcua M., 2017, International Journal of Modern Physics Dayal P., Choudhury T. R., Bromm V., Pacucci F., 2017, D, 26 ApJ, 836 Mezcua M., Roberts T. P., Lobanov A. P., Sutton A. D., Dubois Y., Volonteri M., Silk J., 2014, MNRAS, 440, 1590 2015, MNRAS, 448, 1893 Dubois Y., Volonteri M., Silk J., Devriendt J., Slyz A., Mezcua M., Civano F., Fabbiano G., Miyaji T., Marchesi Teyssier R., 2015, MNRAS, 452, 1502 S., 2016, ApJ, 817, 20 Fabian A. C., et al., 2000, MNRAS, 318, L65 Mezcua M., Civano F., Marchesi S., Suh H., Fabbiano G., Fan X., et al., 2006,AJ, 131, 1203 Volonteri M., 2018a, preprint, (arXiv:1802.01567) Ferrarese L., Merritt D., 2000, ApJL, 539, L9 Mezcua M., Hlavacek-Larrondo J., Lucey J. R., Hogan Gebhardt K., et al., 2000, ApJL, 539, L13 M. T., Edge A. C., McNamara B. R., 2018b, MNRAS, Greene J. E., Ho L. C., 2004, ApJ, 610, 722 474, 1342 8 Pacucci et al.

Miller B. P., Gallo E., Greene J. E., Kelly B. C., Treu T., Sheth R. K., Tormen G., 1999, MNRAS, 308, 119 Woo J.-H., Baldassare V., 2015, ApJ, 799 Volonteri M., 2010, Astronomy and Astrophysics Review, Natarajan P., Volonteri M., 2012, MNRAS, 422, 2051 18, 279 Pacucci F., Volonteri M., Ferrara A., 2015, MNRAS, 452, Volonteri M., Habouzit M., Pacucci F., Tremmel M., 2016, in Kaviraj S., ed., IAU Symposium Vol. 319, Galaxies at 1922 High Redshift and Their Evolution Over Cosmic Time. Pacucci F., Natarajan P., Ferrara A., 2017a, ApJL, 835, L36 pp 72–79 (arXiv:1511.02588), Pacucci F., Natarajan P., Volonteri M., Cappelluti N., Urry doi:10.1017/S1743921315010005 C. M., 2017b, ApJ, 850 Woo J.-H., Yoon Y., Park S., Park D., Kim S. C., 2015, Park K., Ricotti M., Natarajan P., Bogdanovi´cT., Wise ApJ, 801, 38 J. H., 2016, ApJ, 818 Woosley S. E., Heger A., Weaver T. A., 2002, Reviews of Parkinson H., Cole S., Helly J., 2008, MNRAS, 383, 557 Modern Physics, 74, 1015 Reines A. E., Volonteri M., 2015, ApJ, 813 Wyithe J. S. B., Loeb A., 2012, MNRAS, 425, 2892 Reines A. E., Greene J. E., Geha M., 2013, ApJ, 775 Xiao T., Barth A. J., Greene J. E., Ho L. C., Bentz M. C., Ricarte A., Natarajan P., 2018, MNRAS, 474, 1995 Ludwig R. R., Jiang Y., 2011, ApJ, 739, 28 van den Bosch R. C. E., 2016, ApJ, 831 Shankar F., et al., 2016, MNRAS, 460, 3119