Constraints From Dwarf Galaxies on Black Hole Seeding and Growth Models With Current and Future Surveys

By construction, SAMs are set up to explore a range of seeding and growth modes for a population of BHs as a function of redshift. Time slices from these SAMs can then be compared to measured BHs. In this work, we focus on the SAM developed by Ricarte & Natarajan (2018b) and explore new predictions for the low-mass end of the local BH mass function. In particular, we include the properties of the host galaxies explicitly in the scheme and do so flexibly by exploring multiple scaling relations between them and MBH properties. It is this new feature that allows us to delve more deeply into further predictions in combination with the Bayesian inference framework in this work.

Intriguingly, as first noted and discussed in detail in Ricarte & Natarajan (2018a), signatures of seeding survive to late times via the observed occupation fraction, though there is dependence on the details of their mass assembly history. There are currently other SAMs from Somerville et al. (2008) and Bower et al. (2010) and updated versions thereof, which do not focus on seeding and include recipes for star and galaxy formation as well as MBH growth, and these models inevitably have many more tuneable parameters. Here, we stick with a pared-down, phenomenologically driven model that is primarily focused on seeding signatures from nearly local observations (z ∼ 0). The Ricarte & Natarajan (2018a) SAM considers two seed models: either a massive and rare "heavy" seed typically produced from DC following the seeding prescription of Lodato & Natarajan (2006) or, Volonteri et al. (2008), or abundant "light" seeds drawn from a power-law (PL) initial mass function between 30 and 100 solar masses motivated by cosmological simulations (Hirano et al. 2015). For each seed, the SAM explored in Ricarte & Natarajan (2018b) follows three different MBH growth prescriptions:

• 1.  
  PL Model: in this model, major halo mergers trigger a burst of growth at the Eddington rate until MBHs reach the mass determined by the observed local M_•–σ_* relation. When MBHs are not in the burst mode, their accretion rates are drawn from a PL tuned to help reproduce low-redshift luminosity functions (Ricarte & Natarajan 2018b).
• 2.  
  Active Galactic Nucleus-Main Sequence (AGN-MS) Model: major halo mergers trigger burst mode growth at the Eddington rate until MBHs reach the M_•–σ_* relation. When MBHs are growing secularly, their accretion rates are set to be 10⁻³ × the star formation rate of their host. This mode is motivated by the empirical relation determined by Mullaney et al. (2012). Since star formation is not actually tracked in this SAM, scaling relations are instead used to estimate the stellar masses of galaxies as a function of halo mass and redshift (Ricarte & Natarajan 2018b).
• 3.  
  Broadline Quasars (BLQ) Model: major halo mergers (defined to be those with a halo mass ratio of at least 1:10) trigger burst mode growth. Additionally, at each time step the accretion rate is drawn from a distribution of Eddington rates observationally inferred for BLQs in SDSS (Kelly & Shen 2013). Specifically, we adopt a redshift-evolving log-normal distribution fit by Tucci & Volonteri (2017). ³  No additional steady accretion mode for mass growth is added since the AGN luminosity function is adequately predicted with this prescription (Ricarte et al. 2019). However, as discussed in this work, this assumption fails when considering deep surveys of dwarf galaxies that are not selected a priori to be AGN.

In this model, only 10% of the halo mergers lead to BH mergers. This is done to prevent the overgrowth of BHs in the most massive halos, whose BHs would otherwise grow mostly from BH–BH mergers at low redshift. We do not expect this assumption to have an impact on the dwarf galaxies considered in this work, which have much lower merger rates. In our study, we do not include satellite halos, since the SAM does not include a prescription for environmental processes such as ram-pressure/tidal stripping, which in turn are expected to affect the stellar mass as well as the growth of MBH. To increase our sample size, we compile the output from three snapshots at z = 0, z = 0.1, and z = 0.2. This yields a total of 6579 halos per seed × growth model, so the uncertainties from Poisson (small number) statistics on the predictions are small, even in the lower stellar mass bins. The scaling relations between the galaxy and MBH properties evolve very marginally in this redshift range, and therefore no further modification is applied to the z = 0.1 and 0.2 halos.

SAMs provide the mass and mass accretion rates for MBHs at every snapshot. The simplest way to compute the luminosity is to convert the accretion rate into a bolometric luminosity with some efficiency _f , and then use a bolometric correction factor to infer the luminosity in some specific energy/wavelength range. Like much of the literature, we assume a radiative efficiency of accretion _f = 0.1. The SAM is known to contain too little scatter (Ricarte & Natarajan 2018a) since AGN are essentially always on. In practice, a duty cycle would increase the scatter in the observed instantaneous luminosity of the BHs. Meanwhile, we know that relatively complete surveys like AMUSE (Gallo et al. 2008; Miller et al. 2012), with a limiting luminosity of 2 × 10³⁸ erg s⁻¹ at 0.5–7.0 keV, find AGNs in ∼40% of galaxies with 10⁸ < M_*/ M_⊙ < 10¹². AMUSE is a remarkably uniform and complete survey of AGNs in nearby galaxies performed with the Chandra X-ray telescope, observing 200 galaxies—some in the field, some in the Virgo galaxy cluster—to a limiting flux of 10^38.3 erg s⁻¹, as shown in Figure 1 with a dotted line. While more recent catalogs exist, they are compiled from the Chandra archive and are thus necessarily nonuniform in exposure (e.g., Gallo & Sesana 2019). We measure the detected fraction of AGN in each SAM as:

$$\begin{eqnarray}&&{w}_{i}=\displaystyle \frac{{\rm{\Phi }}({M}_{* ,\mathrm{SAM}})}{{\rm{\Phi }}({M}_{* ,\mathrm{obs}})}\end{eqnarray} \tag{ 1 }$$

,

$$\begin{eqnarray}&&{f}_{d}=\displaystyle \frac{{\sum }_{i=0}^{{N}_{a}}{w}_{i}}{{\sum }_{j=0}^{{N}_{g}}{w}_{j}},\end{eqnarray} \tag{ 2 }$$

where w_i is the relative abundance of the host galaxy stellar mass in the SAM compared to the observations. The sum in the numerator is over N_a galaxies with L_X > 0 and in the denominator is over all N_g galaxies. The weighting corrects for the different stellar mass functions of the SAMs and the observed sample. Both the numerator and denominator include galaxies at 10⁸ < M_*/ M_⊙ < 10¹². The PL and AGN-MS growth models predict the existence of many AGNs below the AMUSE flux limit, shown as the dotted horizontal line in each panel.

Zoom In Zoom Out Reset image size

If the SAMs are taken at face value, 75%–90% of the galaxies in the AGN-MS and PL models would have had AGN detections in an AMUSE-like survey. We, therefore, add scatter to the L_X in these SAMs until the detected fraction, as defined above, is 40% for an AMUSE-like survey. These values are shown in Table 1, and the L_X –M_* relations for the SAMs with the added scatter are shown in Figure 2.

Zoom In Zoom Out Reset image size

Table 1. Scaling Relations for the PL and AGN-MS SAMs, Measured for 10⁸ < M_*/ M_⊙ < 10¹⁰ and with σ_add Adjusted So That ∼40% of the Galaxies Have AGNs with $\mathrm{log}({L}_{X}/\mathrm{erg}\ {{\rm{s}}}^{-1})\gt 38.3$, the Limiting Luminosity of the AMUSE Survey

Model	σadd	β	α	${\sigma }_{\mathrm{intr}}$
PopIII—PL	0.5	0.50	34.0	1.13
PopIII—AGN-MS	1.9	0.36	34.8	0.28
DCBH.- PL	0.6	0.6	37.3	1.28
DCBH—AGN-MS	1.8	0.35	34.7	0.29

Download table as:  ASCIITypeset image

Shen et al. (2020) measured the bolometric correction for the 0.5–2 keV and 2–10 keV X-ray bands and for the infrared B band, as a function of bolometric luminosity. This corresponds to 1/k_band = L_bol/L_band = 10–30 in each of the X-ray bands and 4–7 in the B band, each with a scatter of about 50%. For each BH, we draw the bolometric correction from a normal distribution with the mean and scatter based on its bolometric luminosity.

We focus first on how observational selection can impact the derived BHOF and explore the optimal future survey strategies that will enable potential discrimination between seeding models. We assume that BH masses are available from the full range of possible observational probes and documented empirical correlations—either from dynamical modeling of galaxy cores or other techniques. AGNs, for instance, have been identified via their optical variability (Baldassare et al. 2018, 2020) or line ratio diagnostics (Cid Fernandes et al. 2010; Kewley et al. 2013; Polimera et al. 2022); masses are then measured via reverberation mapping (e.g., Peterson et al. 2004) or direct dynamical modeling (e.g., Walsh et al. 2013; Seth et al. 2014; Nguyen et al. 2017), and then used to construct scaling relations based on easily observed quantities like emission line strengths. Therefore, we proceed with the premise that we have in hand M_BH measurements/upper limits for every single galaxy in our SAM survey sample. We first investigate the impact of the completeness of the mass measurements on our inference of the BHOF.

Figure 3 shows the occupation fraction for different values of BH mass completeness. If we assume in the best-case scenario that all BHs above 10³ M_⊙ can be detected, then the Population III seeds produce a higher occupation fraction than the DCBH models. This is particularly noticeable for galaxies with M_* < 10⁹ M_⊙. We note that this clearly reflects the efficiency and abundance of the initial seeds, as light seeds are more ubiquitous and are expected to form more efficiently compared to heavy seeds in the SAMs. If, however, we can only detect BHs more massive than 3 × 10⁴ M_⊙, then light seeds growing through the AGN-MS channel predict f_occ = 1 for all stellar masses; for the other two growth channels, the seed models become indistinguishable for M_* < 4 × 10⁸ M_⊙. If we can only find MBHs above 10⁶ M_⊙, information about seeding cannot be cleanly disentangled from accretion. These findings clearly show that the detectable present-day mass needs to be close to the initial assumed "heavy" seed mass from high redshifts, namely, 10⁴⁻⁵ M_⊙, in order to be disentangled from the confounding effects of the accretion physics.

Zoom In Zoom Out Reset image size

In summary, if the initial seeds have not grown much by any of the accretion modes or mergers, then their memory of seeding is retained in the BHOF as measured even at low redshift.

In practice, MBHs are usually detected when they are accreting and luminous. The scaling relations used to measure BH masses, regardless of wavelength deployed or method used, all suffer from large intrinsic scatter (Woo & Urry 2002) and are most reliable at X-ray wavelengths where obscuration is the lowest. Therefore, we stick to X-ray observations for comparison with the SAMs for the rest of this study. Figure 4 shows the fraction of galaxies with AGNs detected at 0.5–2.0 keV as a function of the survey detection limit. The BLQ prescription is ruled out because it predicts almost no AGNs in dwarf galaxies. A relatively shallow survey of 10⁴⁰ erg s⁻¹ would only be marginally able to distinguish between the AGN-MS and PL growth channels modes, with the former predicting 10%–20% more detections in dwarf galaxies; if all AGNs above 10³⁸ erg s⁻¹ are found, the detected fraction is identical for both of these growth channels. We need a survey that either directly detects, or can model the presence of, all AGNs with luminosities down to 10³⁶ erg s⁻¹ if the heavy and light seed models are to be clearly distinguished.

Zoom In Zoom Out Reset image size

Figures 3 and 4 reveal that most of the differences between the models are apparent at the lowest halo masses. First, this highlights the importance of using a parent sample of galaxies that has stellar masses complete down to 10⁸ M_⊙, pushing down an order of magnitude compared to current observational limits. Figure 5 therefore shows the occupation fraction of galaxies with stellar masses in the range 10⁸−10¹⁰ M_⊙ as a function of the minimum detectable BH mass. If all the information we have is the BH and stellar masses, we can only tell different seed models apart if we find every MBH at least as massive as 10⁴ M_⊙. If we can separately confirm that the growth channel is like the AGN-MS model, then we can distinguish seeds using all MBHs above 10⁵ M_⊙.

Zoom In Zoom Out Reset image size

In practice, again, we almost always detect BHs when they are luminous. Figure 6 shows the fraction of galaxies with stellar masses of 10⁸ < M_*/ M_⊙ < 10¹⁰ with detectable AGNs, as a function of survey-limiting luminosity. Heavy and light seeds can be distinguished if the survey has a limiting luminosity better than 10³⁷ erg s⁻¹.

Zoom In Zoom Out Reset image size

Figure 7 shows the AGN bolometric luminosity function for the six models, with observations from Ueda et al. (2011) in black for comparison. The uncertainties incorporate 1000 draws from the uncertainty in the bolometric corrections; the BLQ model uncertainties are not shown because, due to its low fraction of active MBHs, has error bars down to 0 at all luminosities. The AGN-MS and PL models have diverging luminosity functions above 10⁴² erg s⁻¹, with the former model requiring more growth at late times to match the M_•–σ relation. The PL model is thoroughly consistent with the observations, although the AGN-MS model cannot be entirely ruled out due to the large scatter required to bring it into an agreement with the observed active fraction in the AMUSE survey.

Zoom In Zoom Out Reset image size

As noted above, the BHOF depends on the minimum AGN luminosity that we can measure or model. Studies like Miller et al. (2015) compute the BHOF from observed AGN luminosities and upper limits, using a model where the L_X −M_* relation contains log-normal scatter. Therefore, this BHOF accounts for AGNs even below the nominal luminosity limit of the survey, which is 10³⁸ erg s⁻¹. Since they did not publish the scaling relations inferred by their Bayesian model, we remeasure them here. To recap, the original study assumed that the BHOF had the functional form of a sigmoid function:

$$\begin{eqnarray}&&{f}_{\mathrm{occ}}=\displaystyle \frac{1}{2}\left[1+\tanh \left({2.5}^{| 8.9-{\mathrm{log}}_{10}{M}_{* ,0}| }{\mathrm{log}}_{10}\displaystyle \frac{{M}_{* }}{{M}_{* ,0}}\right)\right].\end{eqnarray} \tag{ 3 }$$

The tuning parameter M_*,0 dictates how rapidly the occupation fraction jumps from 0 to 1; in other words, a higher value for this parameter means that galaxies with lower stellar masses are less likely to host super-MBHs than their more-massive counterparts. Put another way, a higher M_*,0 means a lower BHOF.

A Bayesian inference model then constrains M_*,0 simultaneously with the normalization, slope, and scatter of the L_X − M_* relation:

$$\begin{eqnarray}&&\mathrm{log}({L}_{X}/\mathrm{erg}\,{{\rm{s}}}^{-1})=\alpha +\beta \times \mathrm{log}({M}_{* }/{M}_{\odot }).\end{eqnarray} \tag{ 4 }$$

This approach thus assumes (1) that the AGN duty cycle can be described in terms of the time spent below the detection limit if the L_X –M_* scaling relation is a PL with log-normal scatter and (2) that we have no prior knowledge of this scaling relation. Again, Figure 2 shows the L_X –M_* relation for each SAM, with points below the AMUSE detection threshold shown as translucent.

Right away, we can rule out the BLQ prescription as the primary growth channel, because this predicts that only 3%–4% of the galaxies at z < 0.2 contain AGNs, far below the 40% detection fraction in AMUSE. This prescription was motivated from observations of the brightest quasars, the high-luminosity end of the Sloan Digital Sky Survey (SDSS) quasar luminosity function. Since the abundance of bright quasars drops precipitously at low redshifts, the BLQ SAM is self consistent in predicting low active fractions at low z. The fact that this differs with observations of low-mass galaxies shows that this growth channel, which is bursty and merger driven, cannot be important in dwarf galaxies. Next, we aim to distinguish between the AGN-MS and PL growth channels.

Both the PL and AGN-MS models predict very high (75%–99%) occupation fractions. From Figure 4, we know that we need to apply a luminosity cut to the SAMs before comparing them to observations. For the Miller et al. (2015) study, this is not the same as the limiting luminosity of the observations. Rather, the BHOF in that study includes all the AGNs whose luminosities fall within log-normal scatter from the mean value of the PL L_x –M_* relation. Therefore, we start by measuring this σ.

The probability of the parameter set [α, β, σ, M_*,0] for detection is defined as:

$$\begin{eqnarray}&&\begin{array}{l}p(\mathrm{log}({L}_{X})| \mathrm{log}({M}_{* }))={f}_{\mathrm{occ}}({M}_{* },{M}_{* ,0})\times \\ N(\mathrm{log}({L}_{X})| \mu =(\alpha +\beta \times \mathrm{log}({M}_{* }),\sigma =\sigma ),\end{array}\end{eqnarray} \tag{ 5 }$$

where N(x∣μ, σ) is the normal probability distribution function centered at μ with standard deviation σ. For the nth nondetection, the probability that there is a BH is given by the latent variable I_n , which in Miller et al. (2015) depended only on the scaling relation and occupation fraction:

$$\begin{eqnarray}\begin{array}{rcl}{L}_{X,{\rm{Pred}}} & = & \alpha +\beta \times {\rm{log}}({M}_{\ast }),\\ {\rm{\Phi }} & = & C({L}_{{\rm{lim}}}-{L}_{X,{\rm{pred}}}/\sigma ),\\ {p}_{{\rm{BH}}}({L}_{{\rm{lim}}},{M}_{\ast }) & = & {\rm{\Phi }}/((1-{f}_{{\rm{occ}}}({M}_{\ast },{M}_{\ast ,0})+{f}_{{\rm{occ}}}\times {\rm{\Phi }}),\end{array}\end{eqnarray} \tag{ 6 }$$

where C is the cumulative probability of the normal distribution. This says that a nondetection means either that there is no BH, or there is a BH and it is on but the luminosity is below the detection limit ${L}_{\mathrm{lim}}$. The latent variable I_n is then set to either 1 or 0 by drawing from a binomial distribution with p_true = p_BH. For further details of the inference pipeline, we refer the reader to Section 2 of Miller et al. (2015).

As a test, we run the AMUSE sample through a Markov Chain Monte Carlo (MCMC) pipeline four times, each time using a different scaling relation between L_X –M_* in Table 1, as inferred from Figure 1. The resulting posterior on M_*,0, the stellar mass at which the occupation fraction is 50%, is shown in Figure 8. Surprisingly, these posteriors are almost indistinguishable from each other.

Zoom In Zoom Out Reset image size

The best-fit value of σ for AMUSE is 2; as seen in Table 1, this is very comparable to the total scatter required in each SAM to match the AMUSE detection fraction. Thus, the BHOF inferred assuming an L_X –M_* relation with such a large scatter would capture all the AGNs in the SAMs with both the PL and AGN-MS growth channels, and no luminosity cut needs to be applied to the SAM MBHs.

Finally, Figure 9 compares the AMUSE constraints applied to each of the four SAMs. The observations are clearly consistent with the heavy seed models growing via either the PL or AGN-MS channels. The light seed models, even with large scatter in their L_X –M_* relations, cannot be reconciled with an X-ray observed sample like AMUSE.

Zoom In Zoom Out Reset image size

A key result of our study is that distinguishing between MBH seeding and growth models relies crucially on detecting the least-massive MBH (or least-luminous AGN) in galaxy samples that are stellar mass complete down to at least 10⁸ M_⊙. We show here how such a survey could be constructed by following up on existing surveys.

The RESOLVE survey (Eckert et al. 2015) was designed to be complete in baryonic mass down to 10^9.3 M_⊙, and thus to even lower stellar masses, within z < 0.025 (d < 110 Mpc). Figure 10 shows the spatial (left) and stellar mass (right) distributions of the RESOLVE galaxies, with the latter demonstrating stellar mass completeness above 5 × 10⁸ M_⊙. The orange histogram shows only galaxies above decl. = 30°, where emission from the Milky Way is negligible, and even faint extragalactic AGNs could be detected. In principle, a survey could include galaxies at ∣decl.∣ < 30° as long as they are in the direction opposite to the Galactic Center; we are being conservative in our recommendation here to account for telescope scheduling difficulties, which may make it difficult to point at extragalactic fields at all times. The RESOLVE survey contains >10,000 such galaxies.

Zoom In Zoom Out Reset image size

Using this subset of galaxies as a parent sample, we create three different mock catalogs. Each assumes a different value of M_*,0 and thus a different BHOF. Where an MBH exists, we assign an X-ray luminosity drawn from the AMUSE observed L_X –M_* relation, including the scatter. We pass it through the Bayesian inference pipeline described above, assuming survey luminosity limits of 10³⁶ erg s⁻¹ and 10³⁸ erg s⁻¹. Figure 11 shows the posterior distributions recovered for each of the mocks. Higher occupation fractions are harder to rule out than lower ones, due to the large scatter inherent in the L_X –M_* relation. While broad, the mode of each distribution matches the input value for ${L}_{\mathrm{lim}}={10}^{36}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$. For shallower surveys, if the intrinsic BHOF is high, the survey would return a flat posterior on M_*,0 even for this high level of completeness. On the one hand, this suggests that the uncertainties obtained from the AMUSE survey may be biased low. On the other hand, it confirms that higher values of M_*,0 would have been ruled out with high confidence.

Zoom In Zoom Out Reset image size

On the other hand, we have checked that the eFEDS survey, despite its uniform coverage overlapping with ∼500 times more galaxies than AMUSE and detection of several new AGN candidates in dwarf galaxies (Latimer et al. 2021a), yields flat posteriors on M_*,0, and that this situation is not expected to improve even with the deepest, polar regions of eRASS8. The key takeaway from this analysis is that survey depth, and the resulting completeness in stellar mass and MBH mass/luminosity, are much more powerful than survey volume when measuring the underlying BHOF.

A key challenge at low MBH masses, and therefore luminosities, is confusion with other sources such as ultra- or hyperluminous X-ray sources (ULXs/HLXs) (Swartz et al. 2004, 2011), which are likely the high Eddington ratio end of the accreting stellar BH population (Stobbart et al. 2006; Sutton et al. 2013), although some studies argue that the brightest of these may indeed be intermediate-mass BHs (IMBHs; Miller et al. 2003; Bellovary et al. 2010; Sutton et al. 2012). ULXs may be differentiated from IMBHs by a lack of radio emission or a position offset from the galaxy center (Thygesen et al. 2023), although Bellovary et al. (2021) showed that in the shallow potentials of dwarf galaxies, MBHs could live off center. Multiband X-ray observations can enable spectral studies to differentiate further between MBHs and high-accreting stellar BHs, and multiepoch observations may find different variability scales. Further contamination at the faintest ends may arise from populations of X-ray binaries (XRBs), although this can potentially be modeled via low-scatter scaling relations with the stellar mass and star formation rate (Lehmer et al. 2010, 2019).

Whether the BH selection is by mass (Figure 3) or luminosity (Figure 4), seeding models cannot be distinguished unless we find every MBH ≳10⁴ M_⊙. This finding was also independently arrived at in recent work by Burke et al. (2022) using future variability surveys. These smallest MBHs have been recently discovered using optical variability (Baldassare et al. 2018) and line diagnostics (Baldassare et al. 2015; Mezcua & Dominguez Sanchez 2020; Polimera et al. 2022). Emission line widths, such as Hα, can yield BH masses, although these can easily be confused with ongoing star formation.

MBHs are, however, notorious for the wide variety and variability of their spectral energy distributions (SEDs). This means that we do not yet have a complete understanding of how the emission from the reprocessed material feeding a BH is distributed at different wavelengths. Below, we summarize some of the opportunities and challenges at various wavelengths for future low-redshift surveys.

4.2.1. Optical and IR

4.2.2. Spectroscopic Line Ratios

4.2.3. Optical/IR Variability Selection

4.2.4. Radio Wavelengths

AGN have also been detected in nearby dwarf galaxies through their radio emission, which stand out due to the very steep spectral index (Davis et al. 2022; Sargent et al. 2022; Yang et al. 2022). Liodakis (2022) models the expected emission from jets in IMBHs in dwarf galaxies and predicts that ∼40% of them will be detectable with ngVLA, SKA, and the Rubin observatory.

Using the fundamental plane of BH activity (Merloni et al. 2003), we can use BH masses and X-ray luminosities to predict the expected radio emission for the different SAMs. The most recent measurement of this relation by Gultekin et al. (2019):

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}({M}_{\bullet }/\,{M}_{\odot })={\mu }_{0} & + & {\xi }_{\mu R}\mathrm{log}({L}_{R}/{10}^{38}\mathrm{erg}\,{{\rm{s}}}^{-1})\\ & + & {\xi }_{\mu X}\mathrm{log}({L}_{X}/{10}^{40}\mathrm{erg}\,{{\rm{s}}}^{-1}),\end{array}\end{eqnarray} \tag{ 7 }$$

where the L_R is the radio continuum luminosity at 5 GHz and L_X is the X-ray luminosity integrated over 2–10 keV. The best fit to their observations was μ₀ = 0.55 ± 0.22, ξ_{μ R} = 1.09 ± 0.10, and ξ_{μ L} = − 0.59 ± 0.15, with an intrinsic scatter of −0.04 in log(M_•). Figure 12 shows the predictions derived using this observed fit for the expected L_R for each of the SAMs. The heavy seed models predict slightly higher radio emission for dwarf AGNs, but almost no sources in galaxies with M_* < 10⁷ M_⊙. Additionally, in the heavy seed scenario, the PL growth channel shows larger scatter in radio luminosities for the faint AGNs in dwarf galaxies. This allows an additional way to distinguish between growth channels, other than the high-end luminosity function mentioned above, without having to assume that the same growth channels dominate for MBHs of all masses/luminosities.

Zoom In Zoom Out Reset image size

Combining optical, infrared, and X-ray observations would allow a self-consistent determination of the bolometric luminosity, removing the need for the uncertain bolometric correction factors that are currently adopted (Thornton et al. 2008; Koliopanos et al. 2017). Such a follow-up campaign to measure the masses of all the BHs detected via the range of methods above would greatly improve measurements of the BHOF.

In this idealized study, we have examined in detail the effects of seed and growth channels in isolation. It is likely that BHs may grow form via multiple channels. Spinoso et al. (2022) used a SAM to populate halos from the Millennium-II Simulation with both heavy and light seeds, and evolved their growth to z = 0 using a two-phase prescription, corresponding to the observed quasar and radio modes of AGNs. They find that with such a multiple-channel seeding model, the slope of the BH–stellar mass relation is flatter for dwarf galaxies than their more-massive counterparts. We see this in Figure 1 in the models that grow with a PL Eddington ratio distribution function (ERDF), irrespective of the seeding model, although the difference is starker for heavy seeds. Interestingly, this heavy seed PL ERDF model is also the one that is most consistent with the AMUSE observations. This is a promising sign for future X-ray surveys, suggesting that we may find more dwarf AGNs by simply pushing to slightly lower luminosities.

The most uncertain part of our model is the connection between the MBH mass and its X-ray luminosity. Observationally, we know that the MBH population is diverse—there are Type I (Richards et al. 2006) and Type II AGN, Seyfert galaxies, FR I and FR II jets, blazars (Giommi et al. 2009; Bottcher et al. 2017), BLQs, narrowline quasars (NLQs), low-luminosity AGNs (Almeida et al. 2021), and a variety of radio sources. A large part of this diversity has to do with dust obscuration, which affects X-ray wavelengths far less than optical/IR. However, there is expected to be intrinsic variability in the SEDs at different BH accretion rates, depending on what process is driving the luminosity on BH event horizon scales.

We adopted the bolometric correction to the 0.5–2.0 keV band from Shen et al. (2020), which is a function of the bolometric luminosity; they found a similar scaling for the 2–10 keV band. Their measurements agree very well with Duras et al. (2020); for the earlier Lusso et al. (2012) study, the corrections agree for L_bol ≳ 10⁴³ erg s⁻¹, but diverge significantly for fainter sources. Using the results of Lusso et al. (2012) would thus yield much more pessimistic predictions for the X-ray surveys; we note, however, that these scaling relations were computed for AGNs with L_bol ≳ 10⁴³ erg s⁻¹, far above the dwarf-AGN regime. Vasudevan & Fabian (2007) found that the bolometric correction to the 2–10 keV band is better correlated to the Eddington ratio. Woo & Urry (2002) went so far as to argue that there was no correlation between BH luminosity and mass other than due to circular reasoning in inferring the former from the latter. A relationship between the two is crucial for inferring BH demographics unless we rely on dynamical measurements of the BH mass, which are only possible for the nearest galaxies, or reverberation mapping, which is possible only for Type I AGN with an unobscured BL region.

We do not know whether the Eddington distribution—the ratio of a BH's luminosity to its Eddington luminosity—is the same as for more-massive galaxies, or if there is a "downsizing" effect where less-massive galaxies host, on average, more-active BHs. There is then a degeneracy between the M_*–L_BH,X scaling relation and the occupation fraction. The most robust constraints must jointly model these two effects, using an unbiased sample of host galaxies (Gallo et al. 2010).