Glimmering in the Dark: Modeling the Low-mass End of the M_•–σ Relation and of the Quasar Luminosity Function

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 ${10}^{5}\mbox{--}{10}^{6}\,{M}_{\odot }$, an assumption that is at the core of our bimodal model. The simplified assumptions introduced in this section are in no way used in the actual growth model described in Section 3. We eliminate the parameter n_∞ from Equations (2) and (4) by computing the corresponding Bondi rate ${\dot{M}}_{{\rm{B}}}\,=4\pi \rho {G}^{2}{M}_{\bullet }^{2}/{c}_{s}^{3}$, where ρ is the gas mass density, and c_s is the sound speed. The Bondi rate is a convenient approximation to use in this simplified model; at scales ≳R_B it is also a reasonable one for the accretion rate, as the accretion disk forms at much smaller scales. Assuming ${c}_{s}={\sigma }_{g}\sim {\sigma }_{s}$ (σ_g and σ_s are the velocity dispersions for gas and stars, respectively; this simplified model remains valid as long as σ_g and σ_s are within the same order of magnitude) and that the ${M}_{\bullet }\mbox{--}\sigma$ relation (Equation (1)) is valid, the constraints of Equations (2) and (4) in the $({M}_{\bullet },{\dot{M}}_{{\rm{B}}})$ parameter space are shown in Figure 1.

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Beginning the growth at ${M}_{\bullet }\ll {10}^{5}\,{M}_{\odot }$, a BH seed can reach, at most, the Eddington rate. Growing in mass, it gets progressively closer to the high-efficiency regime, being able to enter it when the Eddington rate crosses the angular momentum barrier shown in Figure 1 as a green line. Once inside the high-efficiency region, super-Eddington rates are reachable (Begelman & Volonteri 2017). The only condition that matters in this regime involves the angular momentum (Equation (4)), whose relevance is not restricted to the high-z universe. An important assumption in this derivation is that there is always a gas supply to grow the BH at or above the Eddington limit: the growth is, thus, supply-limited.

We test our model against observational data of both low-mass and high-mass galaxies for which BH mass and stellar velocity measurements are available. We use the same sample as that of Martín-Navarro & Mezcua (2018; blue stars in Figure 2), which includes a compilation of 127 low-mass Seyfert 1 galaxies with σ < 100 km s⁻¹ from Xiao et al. (2011) and Woo et al. (2015) drawn from the Sloan Digital Sky Survey (SDSS; Abolfathi et al. 2018). The BH masses of these galaxies were estimated from the width of optical broad emission lines under the assumption that the gas is virialized (Xiao et al. 2011; Woo et al. 2015). In the regime of massive galaxies, the sample includes 205 early-type galaxies with direct BH mass measurements from van den Bosch (2016) and σ ≳ 100 km s⁻¹. Typical errors in ${M}_{\bullet }$ and σ are up to a factor ∼3 and $10\mbox{--}15\,\mathrm{km}\,{{\rm{s}}}^{-1}$, respectively.

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We employ a merger tree code (Parkinson et al. 2008; Dayal et al. 2017) to track the cosmological evolution of a population of dark matter halos distributed following the Sheth–Tormen halo mass function (Sheth & Tormen 1999). We sample logarithmically the cosmological scale factor $a={(z+1)}^{-1}$ between z = 20 and z = 0.1, the mean redshift of the sample described in Section 3.1. We assign a single BH seed to each z = 20 galaxy with a halo mass ${M}_{h}\gtrsim 5\times {10}^{7}\,{M}_{\odot }$ (such that its virial temperature is higher than the atomic cooling threshold, see e.g., Barkana & Loeb 2001). We further assume a ratio of high-mass (${M}_{\bullet }\gt {10}^{4}\,{M}_{\odot }$) to low-mass (${M}_{\bullet }\lt {10}^{2}\,{M}_{\odot }$) BH seeds of 1:100. In fact, the formation of a high-mass seed is a much rarer event, because of the additional requirements (see e.g., Bromm & Loeb 2003) to prevent the fragmentation of the gas cloud. The ratio employed here is a proxy for the relative abundance of sources in the high-luminosity and the low-luminosity ends of the quasar luminosity function (see e.g., the one for 3 < z < 5 in Masters et al. 2012). This mixture of light and heavy seeds reproduces the z ∼ 0 quasar luminosity function for ${L}_{\mathrm{bol}}\,\gtrsim {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (Hopkins et al. 2007), as detailed in Section 4.2. We model high-mass seeds as direct collapse black holes (DCBHs; e.g., Bromm & Loeb 2003) and low-mass seeds as Pop III stellar remnants (e.g., Hirano et al. 2014). We model the initial mass function of DCBHs with a log-Gaussian distribution, having a mean μ = 5.1 and a standard deviation σ = 0.2, both in logarithm of mass. For Pop III stars we employ a model with a Salpeter-like exponent and a low-mass cutoff M_c:

$$\begin{eqnarray}&&{\rm{\Phi }}(\mathrm{Pop}\ \mathrm{III},m)\propto {m}^{-2.35}\exp \left(-\displaystyle \frac{{M}_{c}}{m}\right).\end{eqnarray} \tag{ 6 }$$

We assume ${M}_{c}=10\,{M}_{\odot }$ and convolve this progenitor mass function with the relation (Woosley et al. 2002) between the mass of the remnant and the stellar mass. As long as there is a clear separation between the initial mass functions for low-mass and high-mass seeds, their exact shape does not significantly influence our results.

We calculate the central stellar velocity dispersion of the host from the asymptotic circular velocity v_c (as a function of total halo mass and redshift), which is a proxy for the total mass of the dark matter halo of the galaxy: $\sigma ={v}_{c}/\sqrt{2}$.

The fueling of the central BH seed in each galaxy is implemented with the following scheme. A BH is active whenever gas is available. In the early universe before a redshift threshold $z\gt {z}_{t}$, we assume that a sufficient amount of gas is always available to feed the seed: thus, the BH is always active. Simulations (e.g., Dubois et al. 2014) and analytical estimates (e.g., Wyithe & Loeb 2012) suggest that even super-Eddington infall rates are fairly common at high redshift. Begelman & Volonteri (2017) point out that the fraction of AGNs accreting at super-Eddington rates could be as high as ∼10⁻³ at z = 1 and ∼10⁻² at z = 2. For z < z_t, we instead assume that the central BH is active only when a major merger occurs, defined as a merger with a mass ratio equal or larger than 1:10. This criterion is meant to reflect the necessity of an external reservoir of gas to overcome the angular momentum barrier. Whenever a major merger occurs, the BH is set in the active phase for a time equal to the merger timescale (Boylan-Kolchin et al. 2008). We set z_t = 6 and check that our results remain qualitatively unchanged for all values of z_t > 3. We note, however, that the assumption of constant availability of gas for the central BHs is not realistic at these low z.

Whenever a BH is active, we describe the time evolution of ${M}_{\bullet }$ with two parameters: the duty cycle ${ \mathcal D }$ (fraction of the active phase spent accreting) and the Eddington ratio ${f}_{\mathrm{Edd}}\,=\dot{M}/{\dot{M}}_{\mathrm{Edd}}$. The first parameter describes the continuity of the gas inflow, and the second one quantifies the amount of mass flowing in. The time evolution of M_•, starting from ${M}_{\bullet }({z}_{0})={M}_{\bullet ,0}$, is obtained from the integral

$$\begin{eqnarray}&&{M}_{\bullet }(z)={M}_{\bullet ,0}\exp \left({\int }_{z}^{{z}_{0}}\displaystyle \frac{{ \mathcal C }({ \mathcal D },{f}_{\mathrm{Edd}})d{\rm{z}}}{{ \mathcal E }(z)}\right),\end{eqnarray} \tag{ 7 }$$

where ${ \mathcal C }({ \mathcal D },{f}_{\mathrm{Edd}})$ incorporates various constants and the two parameters of the model, ${ \mathcal E }(z)$ depends on the cosmology of choice (see Pacucci et al. 2017b for details). The values adopted for ${ \mathcal D }$ and ${f}_{\mathrm{Edd}}$ at each redshift depend on the properties of the BH in the $({M}_{\bullet },{n}_{\infty })$ parameter space. In general, ${ \mathcal D }\ll 1$ and ${f}_{\mathrm{Edd}}\ll 1$ in the low-efficiency region and ${ \mathcal D }\sim 1$ and f_Edd ≳ 1 in the high-efficiency region (Pacucci et al. 2015; Inayoshi et al. 2016; Begelman & Volonteri 2017). The gas density profile of the host galaxies is assumed to follow an isothermal sphere.

Once two galaxies merge, their central BHs are assumed to coalesce instantaneously. The only cap imposed for the growth by BH mergers is the mass of the most massive BHs thus far observed ($\sim 5\times {10}^{10}\,{M}_{\odot }$, Mezcua et al. 2018b). Expressed as a function of the velocity dispersion, the cap for the growth by accretion is devised to match the ${M}_{\bullet }\mbox{--}\sigma$ relation for $\sigma \,\gtrsim 100\,\mathrm{km}\,{{\rm{s}}}^{-1}$. This mass cap is formally equal to Equation (1). The relation ${M}_{\bullet }\propto {\sigma }^{5.35}$ substantially agrees with the hypothesis of BH growth being regulated by energy-driven wind feedback (e.g., King 2010): ${M}_{\bullet }\simeq {f}_{g}\kappa {\sigma }^{5}/(\pi {G}^{2}c)$, where f_g is the cosmic baryon fraction and κ is the gas opacity.

Our main results are shown in the left panel of Figure 2. The simulations are evolved to z = 0.1 to match the mean redshift of the data sample in Martín-Navarro & Mezcua (2018). The simulations closely follow the data and the theoretical model by van den Bosch (2016) for $\sigma \gt 70\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (see also Martin-Navarro & Mezcua 2018; Ricarte & Natarajan 2018). At $\sigma \sim 70\,\mathrm{km}\,{{\rm{s}}}^{-1}$ there is a clear change in trend: our simulation points tend to be under the theoretical ${M}_{\bullet }\mbox{--}\sigma$ relation. This is a direct consequence of the bimodality in the accretion efficiency of BHs. For masses ${M}_{\bullet }\gtrsim {10}^{5}\,{M}_{\odot }$ the BHs are likely to be in the high-efficiency regime (see Figure 1). These BHs grow efficiently and are able to reach the ${M}_{\bullet }\mbox{--}\sigma$ within a Hubble time. When they reach the ${M}_{\bullet }\mbox{--}\sigma$, their growth is saturated by energy-driven winds, which deplete the central regions from the remaining gas, preventing further growth. At this stage, growth can occur by mergers only. We find the presence of objects with masses significantly larger than the ${M}_{\bullet }\mbox{--}\sigma$. Observationally, these objects can be interpreted as the brightest cluster galaxies, whose BHs are found to be over-massive with respect to the scaling relations (Mezcua et al. 2018b). Nonetheless, these objects constitute a minority population when compared to the bulk of BHs. For ${M}_{\bullet }\lesssim {10}^{5}\,{M}_{\odot }$, the fraction of BHs growing efficiently is small, and only a minority (∼3%) of them are able to reach the cap imposed by the ${M}_{\bullet }\mbox{--}\sigma$ relation. The vast majority of them remain at lower masses, unable to trigger sufficiently strong winds to fully halt their growth. Instead, they keep accreting at very low rates for most of the Hubble time. In the low-mass regime, we predict a much steeper relation (${M}_{\bullet }\sim {\sigma }^{11}$) which is only approximate, as BH and stellar component become increasingly disconnected. The van den Bosch (2016) model intercepts our predictions for low masses at transition values ${\sigma }_{t}\sim 65\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${M}_{\bullet ,t}\sim 5\,\times {10}^{5}\,{M}_{\odot }$. At lower masses and velocity dispersions the model clearly predicts a departure from the ${M}_{\bullet }\mbox{--}\sigma$ relation, with the bulk of objects being under-massive.

The simulated data at z = 0.1 can be used to construct a luminosity function, with the abundance of host halos retrieved from a Sheth–Tormen halo mass function at the same redshift. We divide the simulations in 35 logarithmic mass bins, and for each of them we compute the average luminosity. We then compare in Figure 3 our predictions with bolometric luminosities of BHs in dwarf galaxies, based on Hα luminosities (Greene & Ho 2004; Reines et al. 2013; Baldassare et al. 2017). The bolometric correction is from Greene & Ho (2004), ${L}_{\mathrm{bol}}=2.34\times {10}^{44}{({L}_{{\rm{H}}\alpha }{10}^{42})}^{0.86}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. The luminosity function is in excellent agreement with observations for ${L}_{\mathrm{bol}}\gtrsim {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (e.g., Hopkins et al. 2007). At low luminosities there is an evident deficit around ${L}_{\mathrm{bol}}\,\sim {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, explained by our result that BHs accreting in that luminosity range have masses close to the transition mass between high and low efficiency. At higher BH masses, the likelihood of accreting close to the Eddington limit is large, while at lower masses the BH accretion is mostly sub-Eddington. The result is a paucity of BHs shining at ${L}_{\mathrm{bol}}\sim {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. The scarcity of observations of intermediate-mass BHs in this luminosity range could thus be a combination of a detection bias with an intrinsically low probability of observing sources in that luminosity range. The importance of a detection bias in the observation of intermediate-mass BHs has been thoroughly investigated. For example, Mezcua et al. (2016) showed, by means of X-ray stacking, that a population of faintly accreting intermediate-mass BHs (with X-ray luminosity ${L}_{{\rm{x}}}\sim {10}^{38}\mbox{--}{10}^{40}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$) should be present in dwarf galaxies; however, their detection is challenging due to their faintness and mild obscuration. Figure 3 also suggests that the abundance of sources with ${L}_{\mathrm{bol}}\lt {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ should rise again at lower luminosities. We point out, though, that the detection of these abundant faint sources is challenging, because they enter the luminosity regime of stellar X-ray binaries (Mezcua et al. 2018a).

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Our main prediction is that central BHs and their host galaxies depart from the ${M}_{\bullet }\mbox{--}\sigma$ relation for masses ${M}_{\bullet }\,\lesssim {10}^{5}\,{M}_{\odot }$, becoming under-massive with respect to the extrapolation of the ${M}_{\bullet }\mbox{--}\sigma$ to lower masses. The ${M}_{\bullet }\mbox{--}\sigma$ relation reflects, for ${M}_{\bullet }\gtrsim {10}^{5}\,{M}_{\odot }$, the connection between galaxy evolution and BH growth. The link is driven by the outflows generated by the BH energy and momentum output. Smaller BHs grow inefficiently and are unable to generate strong outflows triggering the growth-regulation process. For this reason, BHs with ${M}_{\bullet }\lesssim {10}^{5}\,{M}_{\odot }$ fail to reach the mass dictated by their velocity dispersion and become under-massive. Previous observations (e.g., Martín-Navarro et al. 2018; Martín-Navarro & Mezcua 2018) and simulations (e.g., Anglés-Alcázar et al. 2017; Habouzit et al. 2017) already suggested that feedback is driven by BH activity for ${M}_{\bullet }\gtrsim {10}^{5}\,{M}_{\odot }$ and by supernova-driven winds for ${M}_{\bullet }\lesssim {10}^{5}\,{M}_{\odot }$.

Some papers (e.g., Greene & Ho 2006; Mezcua 2017; Martín-Navarro & Mezcua 2018) suggest an alternative scenario for the low-mass regime of the ${M}_{\bullet }\mbox{--}\sigma$ relation, predicting a flattening at masses $\sim {10}^{5}\mbox{--}{10}^{6}\,{M}_{\odot }$. Martín-Navarro & Mezcua (2018) explained this putative flattening with a weaker coupling between baryonic cooling and BH feedback, disconnecting the BH from the evolution of the host spheroid. Alternatively, the flattening could be explained with the prevalence of a high-mass formation channel for early seeds (Volonteri 2010). These high-mass seeds would fail to grow and just accumulate around their original mass, $\sim {10}^{5}\mbox{--}{10}^{6}\,{M}_{\odot }$. In this Letter, we envisage that a flattening toward $\sim {10}^{5}\mbox{--}{10}^{6}\,{M}_{\odot }$ would be due to an observational bias. Observing BHs with ${M}_{\bullet }\lesssim {10}^{5}\,{M}_{\odot }$ is currently challenging, and the predicted paucity of BHs shining at ${L}_{\mathrm{bol}}\sim {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (Section 4.2) could add to this effect. Instead of a flattening, our model clearly predicts a downward departure from the ${M}_{\bullet }\mbox{--}\sigma$ relation. A detection for ${M}_{\bullet }\lesssim {10}^{5}\,{M}_{\odot }$ of a relation of the form ${M}_{\bullet }\sim {\sigma }^{\alpha }$ with α ≳ 7 would be an important indicator of the existence of a bimodal population of BH seeds.

Pushing the detection limit to ${M}_{\bullet }\lesssim {10}^{4}\,{M}_{\odot }$ (Baldassare et al. 2017; Chilingarian et al. 2018) will enable to determine the relevance of the BH—galaxy connection for lower-mass galaxies, and whether or not a departure from the ${M}_{\bullet }\mbox{--}\sigma$ relation occurs. A major role in this observational challenge will be played by future observatories, both in the electromagnetic (e.g., Lynx; see Ben-Ami et al. 2018) and in the gravitational (e.g., LISA; see Amaro-Seoane et al. 2017) realms. This effort will not only shed light on the interconnection between BH and its host galaxy, but will ultimately provide important constraints on the seed formation mechanisms active in the high-redshift universe. The observation in the local universe of intermediate-mass BHs, as well as other proposed techniques (e.g., deriving the local supermassive BH occupation fraction, Miller et al. 2015) will help us to understand processes occurred early in the history of the universe.