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•; n1) 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 n1 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 n1 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 1 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; Angl´es-Alc´azaret al. 2017; Habouzit et al. 2017). 1 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 n1 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•; n1) 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 n1 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.