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The black hole mass function of supermassive black holes describes the evolution of the distribution of black hole mass. It is one of the primary empirical tools available for mapping the growth of supermassive black holes and for constraining theoretical models of their evolution. In this paper, we discuss methods for estimating the black hole mass function, including their advantages and disadvantages. We also review the results of using these methods for estimating the mass function of both active and inactive black holes. In addition, we review current theoretical models for the growth of supermassive black holes that predict the black hole mass function. We conclude with a discussion of directions for future research which will lead to improvement in both empirical and theoretical determinations of the mass function of supermassive black holes.

Understanding how and when supermassive black holes (SMBHs) grow is currently of central importance in extragalactic astronomy. A significant amount of empirical work has established correlations between SMBH mass and host galaxy spheroidal properties, such as luminosity [

Feedback-driven “self-regulated” growth of black holes has been able to reproduce the local

The black hole mass function (BHMF) provides a complete census of the mass of SMBHs and their evolution. Because of this, the BHMF is one of the primary empirical tools available for investigating the growth of SMBHs, and for constraining theoretical models for the growth of the SMBH population. Because SMBHs and galaxies are thought to be linked in their evolution, the BHMF provides insight into the fueling mechanisms that dominate black hole growth and therefore into the role of feedback in the evolution of the host galaxy. The BHMF is also an important tool in planning future surveys, as it provides an estimate of the distribution of SMBH mass expected for the survey. This in turn is important because mass is a fundamental quantity of the black hole and therefore is an important observational quantity for empirical studies of black hole accretion physics [

In this paper, we discuss the current status of BHMF estimation and theoretical modeling. In Section

The black hole mass function, denoted as

Similar to luminosity function estimation, the BHMF may be estimated from astronomical surveys. However, while there are many well-established methods for estimating luminosity functions, there are two complications that make BHMF estimation a more difficult problem [

The second complication is the large uncertainty in SMBH mass among mass estimators. Currently, it is not possible to obtain reliable mass estimators for large numbers of SMBHs through dynamical and modeling of the stellar or gaseous components, and thus scaling relationships are employed. Masses may be estimated using scaling relationships between

The statistical uncertainty in the mass estimates can have a significant effect on the inferred BHMF. The distribution of the mass estimates is the convolution of the intrinsic BHMF with the error distribution in the mass estimates. In general, it is typically assumed that the error in the mass estimates is independent of the actual value of

Illustration of the bias in the estimated BHMF derived from mass estimates. Shown is the true mass function (thick solid black line) for a simulated sample, and the mass function derived from the mass estimateswhen the statistical error in the mass estimates is 0.3 dex (red dashed line), 0.4 dex (green dot-dashed line), and 0.5 dex (solid blue thin line). The mass function estimated from the mass estimates is biased, especially at the high-MBH end and for large statistical error.

In order to estimate the SMBH mass function in an unbiased manner, it is necessary to match the mass function with the observed distribution of the mass estimates and any additional observational quantities that the selection function (The selection function is the probability of including a source in one’s sample as a function of its measured quantities) depends on. The basic idea is to start with an assumed mass function. Then, calculate the distribution of mass estimates implied by this mass function. In addition, calculate the distribution of observational quantities that one’s sample is selected on, say, flux, that is implied by the assumed mass function. This step allows one to correct for incompleteness but requires an additional assumption about how to relate the mass function to the quantity that one’s sample is selected on. Finally, impose the selection function for the sample and compare the predicted observed distributions of mass estimates and any other observables (e.g., flux) with the actual distributions. If they are not consistent, then the data rule out the assumed mass function and relationship between

We can make the above procedure more quantitative by deriving the likelihood function for the SMBH mass function. Kelly et al. [

An alternative form of estimating the BHMF can be used when the mass estimates are derived from an observational quantity,

The observed scaling between

The scaling relationships between

Evolution in the scaling relationships is currently an area of intense study, with most groups finding evidence that the normalization of the scaling relationships increases towards higher

When the

Local BHMF. The shaded region defines the spread in estimates obtained using the

While the procedure for estimating the local BHMF is, in theory, straightforward, a number of significant systematics remain. First, there is the observational difficulty that most BHMFs derived from the ^{−1}, making it difficult to reliably measure

By employing the argument of Soltan [

Under the assumption that SMBHs grow during phases of AGN activity, AGN demographics in combination with the local BHMF may be used to compute

Most authors employing (

A variety of lightcurve models have been used when employing (

In Figure

Comparison of two recently estimated BHMFs, calculated by Shankar et al. [

In general, most of the studies that have used (

Most SMBH growth occurs in periods when the quasar is radiating near the Eddington limit.

Most, if not all, of the local black hole mass function can be explained as the relic of previous AGN activity, implying that mergers of SMBHs are not important for building up the local mass function.

SMBH growth is antihierarchical, with the most massive black holes growing first. This has also been termed “downsizing” of active SMBHs.

The lifetime of AGN activity is ~ a few ×10^{8} yr.

Most SMBHs have nonzero spin, as implied by inferred radiative efficiencies of

However, while (

An alternative to the methods described in Section

The BHFP is a scaling relationship between

Figure 5 from Merloni and Heinz [

Broad-line AGN BHMFs at a variety of redshifts. Shown are the local BHMF estimated by Greene and Ho [

The method of estimating the BHMF from the BHFP developed by Merloni [

Thus far, we have focused on methods for estimating the mass function of all SMBHs. In this section, we will describe methods for estimating the BHMF for those SMBHs in AGN and the results that have come from the application of these methods.

The steady improvement in reverberation mapping of AGN [

Early estimates of the mass function of SMBHs in BLAGN were obtained by binning up the virial mass estimates and applying a

Shen et al. [

Similar to the methods based on the continuity equation, investigations of the BHMF for BLAGN have found evidence for the anti-hierarchical growth of SMBHs, that is, cosmic “down-sizing” of BLAGN activity. The inferred Eddington ratio distributions are wide, and the density of SMBHs continues to increase toward Eddington ratios which are below the survey completeness limit. In addition, Kelly et al. [

Mass functions estimated from scaling relationships for BLAGN have the advantage that they are derived from estimates of

As with all methods of BHMF estimation, the virial mass estimates and the mass functions derived from them still suffer from systematics. First, there is the usual problem of calculating a bolometeric correction, although this only affects the estimated Eddington ratio distribution and not the BHMF. Second, there are a few concerns with the virial mass estimates which could introduce systematic error; some of these have been discussed by Greene and Ho [

Before broad-line mass estimates, there were two earlier attempts at estimating the BHMF for AGN, which we briefly mention here. Siemiginowska and Elvis [

Franceschini et al. [

There have been numerous theoretical models for the formation and growth of supermassive black holes, and coevolution with their host galaxies. Understanding this formation, growth, and coevolution is one of the current most important outstanding issues in extragalactic astrophysics. Because the black hole mass function provides a census of the SMBH population and its evolution, it is one of the most fundamental observational quantities available for constraining models of SMBH formation and growth. As such, many theoretical investigations have predicted a BMHF for comparison with the empirical BHMF. In this section, we review some of the models for SMBH formation and growth. There have been numerous theoretical models for SMBH growth and formation, and it is beyond the scope of this primarily empirically-focused review to review all of them; instead, we focus on those theoretical models that predict a BHMF.

Early models for the coevolution of SMHBs and galaxies linked the growth of black holes to the properties of host dark matter halos, with periods of SMBH growth occurring in quasar phases initiated by mergers. In general, early studies that predicted a BHMF used various prescriptions to relate

Granato et al. [

Hopkins et al. [

Most recently, Fanidakis et al. [

Almost all models for the cosmological coevolution of SMBHs and galaxies that predict a BHMF have been of an analytical or semi-analytical nature. An exception is the study done by Di Matteo et al. [^{−1} kpc. Instead, Di Matteo et al. [

In Figure

Compilation of BHMFs predicted by several recent models for SMBH formation and growth. Shown are the BHMFs predicted by Hopkins et al. [

Recent work has made improvement to models for the BHMF by focusing on theoretical modeling of the distribution of seed SMBHs. The discovery of quasars at

Volonteri et al. [

Natarajan and Volonteri [

Lippai et al. [

Before concluding this paper, we present a discussion of possible future empirical and theoretical work relevant to BHMF studies. These include the following.

Currently, the local BHMF is estimated from the distribution of host galaxy properties assuming that

Most studies that have invoked the continuity equation to link the local BHMF to the AGN luminosity function have assumed a single radiative efficiency, which is equivalent to assuming a single black hole spin, and a universal AGN lightcurve. Neither of these assumptions are likely to be true, and improvements to this type of modeling should include a distribution of SMBH spin and AGN lightcurves. In addition, we need to better characterize the bolometeric corrections, which remain a significant source of systematic uncertainty. The continuity equation techniques should also be extended to map the evolution of the full joint 3-dimensional distribution of black hole mass, accretion rate, and spin. While this will not necessarily have a direct effect on estimating the BHMF, it will provide insight into the dominant accretion modes experienced by active SMBH and into the dominant fueling mechanism for AGN activity, as the spin distribution traces the SMBH fueling history [

The dominant scaling relationship for estimating

There is also the need to better characterize the black hole fundamental plane. Because the BHFP describes how the emission mechanisms responsible for the radio and X-ray flux scale with

Finally, there has recently been the discovery of scaling relationships involving

From the theoretical point of view, it is clear that high-redshift scaling relations (or the lack thereof) between SMBH and their hosts provide unique and powerful constraints to models for AGN feeding and feedback, which cannot be otherwise distinguished (see, e.g., Merloni et al. [

In practical terms, a better understanding of the evolution of scaling relations may also be very advantageous for BHMF studies. As we discussed above, current technique for BH mass estimation at

From the technical point of view, a lot of work of course is needed to better understand how reliable these estimators are. Another big “technical" challenge of all studies of the evolution of scaling relations is the fact that they require a thorough assessment of the many observational biases one encounters in studying high redshift AGN and their hosts [

Many theoretical models for the BHMF do not include recoiling effects caused by the merger of two black holes. However, recent theoretical work on black hole recoils suggests that black holes can spend a significantly large enough amount of time offset from the central region of the host galaxy to alter their growth, thereby increasing the scatter about the scaling relationships and decreasing the final black hole mass [

Most current theoretical models for SMBH growth involve AGN feedback and assume a single efficiency for coupling feedback energy to the gas; this feedback efficiency is usually treated as a free parameter. An improved physical understanding of AGN feedback will improve theoretical models for the BHMF, as the feedback efficiency affects the dynamics of the SMBH’s fuel supply and therefore the amount that the SMBH accretes as a function of redshift. Recent high-resolution hydrodynamic simulations in one dimension [

On the observational side, future X-ray observations should provide considerable improvement in our understanding of AGN feedback. X-ray spectra are needed in order to determine the total column density of the gas, and thus its kinetic energy flux, which can be compared to the energy output of the SMBH. Current X-ray observations from

Full hydrodynamic cosmological simulations offer the most promising avenue for providing a physically motivated BHMF without free parameters and for unambiguously identifying the relevant physical processes in building up the BHMF. However, they currently cannot resolve scales relevant to the accretion flow onto the SMBH. Numerical codes based on

One way of improving current sub-resolution models may be to implement the results on AGN feedback based on the type of work described in the previous bullet point. Another improvement is in modifying the sub-resolution model for the SMBH accretion flow. Current methods assume the Bondi rate combined with a correction factor to account for the fact that the temperature and density of the gas are not resolved at the Bondi radius. Not surprisingly, the growth of the SMBH is sensitive to how this correction factor is modeled [

In this paper we have reviewed current estimates of the SMBH mass function, as well as theoretical models for the BHMF. As discussed above, each of the methods for estimating the BHMF has their own set of systematics. In Figure

Comparison of empirical estimates of the BHMF (grey shaded region) with BHMFs predicted by theoretical models for SMBH formation and growth (red shaded region), in both the local universe (a) and at

Despite the differences in the methods for estimating the BHMF and the theoretical models, they have lead to a number of common conclusions. In particular, the empirical results have presented a picture whereby SMBHs grow primarily via accretion in active phases (Eddington ratios

The authors would like to thank Tommaso Treu, Priya Natarajan, Marta Volonteri, Aneta Siemiginowska, Xiaohui Fan, Marianne Vestergaard, Alister Graham, and Zoltán Haiman for helpful comments. In addition, they would like to thank two anonymous referees whose comments improved our paper. B. C. Kelly acknowledges support by NASA through Hubble Fellowship Grant no. HF-51243.01 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under Contract NAS 5-26555.

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