Mass Density Of Supermassive Black Holes Research Articles

SMBH seeds from dissipative dark matter

The existence of supermassive black holes (SMBHs) with masses greater than ∼ 109 M ☉ at high redshift (z ≳ 7) is difficult to accommodate in standard astrophysical scenarios. We study the possibility that (nearly) totally dissipative self-interacting dark matter (tdSIDM)–in rare, high density dark matter fluctuations in the early Universe — produces SMBH seeds through catastrophic collapse. We use a semi-analytic model, tested and calibrated by a series of N-body simulations of isolated dark matter halos, to compute the collapse criteria and timescale of tdSIDM halos, where dark matter loses nearly all of its kinetic energy in a single collision in the center-of-momentum frame. Applying this model to halo merger trees, we empirically assign SMBH seeds to halos and trace the formation and evolution history of SMBHs. We make predictions for the quasar luminosity function, the M BH-σv * relation, and cosmic SMBH mass density at high redshift and compare them to observations. We find that a dissipative dark matter interaction cross-section of σ/m ∼ 0.05 cm2/g is sufficient to produce the SMBHs observed in the early Universe while remaining consistent with ordinary SMBHs in the late Universe.

Constraining black hole–galaxy scaling relations and radiative efficiency from galaxy clustering

The masses of supermassive black holes are observed to increase with either the total mass or the mean (random) velocity of the stars in their host galaxies. The origin of these correlations remains elusive due to observational systematics and biases that severely limit our knowledge of the local demography of supermassive black holes. Here, we show that the large-scale spatial distribution of local active galactic nuclei (AGN) can constrain the shape and normalization of the black hole–stellar mass relation, thus bypassing resolution-related observational biases. In turn, our results can set more stringent constraints on the AGN radiative efficiency, e. For currently accepted values of the AGN obscured fractions and bolometric corrections, our estimated local supermassive black hole mass density favours mean e values of ~10–20%, suggesting that the vast majority of supermassive black holes are spinning moderately to rapidly. With large-scale AGN surveys coming online, our methodology will enable even tighter constraints on the fundamental parameters that regulate the growth of supermassive black holes. The large-scale spatial distribution of local active galactic nuclei can constrain the black hole–stellar mass relation and their mean radiative efficiency to 10–20%, suggesting moderate to high spins for the vast majority of supermassive black holes.

THE BLACK HOLE MASS FUNCTION DERIVED FROM LOCAL SPIRAL GALAXIES

We present our determination of the nuclear supermassive black hole (SMBH) mass function for spiral galaxies in the Local Universe, established from a volume-limited sample consisting of a statistically complete collection of the brightest spiral galaxies in the Southern Hemisphere. Our SMBH mass function agrees well at the high-mass end with previous values given in the literature. At the low-mass end, inconsistencies exist in previous works that still need to be resolved, but our work is more in line with expectations based on modeling of SMBH evolution. This low-mass end of the spectrum is critical to our understanding of the mass function and evolution of SMBHs since the epoch of maximum quasar activity. A luminosity distance $\leq$ 25.4 $Mpc$ and an absolute B-band magnitude $\leq$ -19.12 define the sample. These limits define a sample of 140 spiral galaxies, with 128 measurable pitch angles to establish the pitch angle distribution for this sample. This pitch angle distribution function may be useful in the study of the morphology of late-type galaxies. We then use an established relationship between the pitch angle and the mass of the central SMBH in a host galaxy in order to estimate the mass of the 128 respective SMBHs in this sample. This result effectively gives us the distribution of mass for SMBHs residing in spiral galaxies over a lookback time $\leq$ 82.1 $h_{67.77}^{-1}$ $Myr$ and contained within a comoving volume of 3.37 $\times$ $10^4$ $h_{67.77}^{-3}$ $Mpc^3$. We estimate the density of SMBHs residing in spiral galaxies in the Local Universe is $5.54_{-2.73}^{+6.55}$ $\times$ $10^4$ $h_{67.77}^3$ $M_{\odot}$ $Mpc^{-3}$. Thus, our derived cosmological SMBH mass density for spiral galaxies is $\Omega_{BH} = 4.35_{-2.15}^{+5.14}$ $\times$ $10^{-7}$ $h_{67.77}$.

X-ray emission from high-redshift miniquasars: self-regulating the population of massive black holes through global warming

Observations of high-redshift quasars at z>6 imply that supermassive black holes (SMBHs) with masses over a billion solar masses were in place less than 1 Gyr after the Big Bang. If these SMBHs assembled from "seed" BHs left behind by the first stars, then they must have accreted gas at close to the Eddington limit during a large fraction (>50%) of the time. A generic problem with this scenario, however, is that the mass density in million-solar-mass SMBHs at z=6 already exceeds the locally observed SMBH mass density by several orders of magnitude; in order to avoid this overproduction, BH seed formation and growth must become significantly less efficient in less massive protogalaxies, while proceeding uninterrupted in the most massive galaxies that formed first. Using Monte-Carlo realizations of the merger and growth history of BHs, we show that X-rays from the earliest accreting BHs can provide such a feedback mechanism. Our calculations paint a self-consistent picture of black-hole-made climate change, in which the first miniquasars---among them the ancestors of the z>6 quasar SMBHs---globally warm the IGM and suppress the formation and growth of subsequent generations of BHs. We present two specific models with global miniquasar feedback that provide excellent agreement with recent estimates of the z=6 SMBH mass function. For each of these models, we estimate the rate of BH mergers at z>6 that could be detected by the proposed gravitational-wave observatory eLISA/NGO.

EXAMINING SUBGRID MODELS OF SUPERMASSIVE BLACK HOLES IN COSMOLOGICAL SIMULATION

While supermassive black holes (SMBHs) play an important role in galaxy and cluster evolution, at present they can only be included in large-scale cosmological simulation via subgrid techniques. However, these subgrid models have not been studied in a systematic fashion. Using a newly-developed fast, parallel spherical overdensity halo finder built into the simulation code FLASH, we perform a suite of dark matter-only cosmological simulations to study the effects of subgrid model choice on relations between SMBH mass and dark matter halo mass and velocity dispersion. We examine three aspects of SMBH subgrid models: the choice of initial black hole seed mass, the test for merging two black holes, and the frequency of applying the subgrid model. We also examine the role that merging can play in determining the relations, ignoring the complicating effects of SMBH-driven accretion and feedback. We find that the choice of subgrid model can dramatically affect the black hole merger rate, the cosmic SMBH mass density, and the low-redshift relations to halo properties. We also find that it is possible to reproduce observations of the low-redshift relations without accretion and feedback, depending on the choice of subgrid model.

The Millennium Galaxy Catalogue: theMbh–Lspheroidderived supermassive black hole mass function

Supermassive black hole (SMBH) mass estimates are derived for 1743 galaxies from the Millennium Galaxy Catalogue (MGC) using the recently revised empirical relation between SMBH mass and the luminosity of the host spheroid. The MGC spheroid luminosities are based on R1/n bulge plus exponential-disc decompositions. The majority of black hole masses reside between 106 M⊙ and an upper limit of 2 × 109 M⊙. Using previously determined space-density weights, we derive the SMBH mass function which we fit with a Schechter-like function. Integrating the black hole mass function over 106 < Mbh/M⊙ < 1010 gives an SMBH mass density of (3.8 ± 0.6) × 105h370 M⊙ Mpc−3 for early-type galaxies and (0.96 ± 0.2) × 105h370 M⊙ Mpc−3 for late-type galaxies. The errors are estimated from Monte Carlo simulations which include the uncertainties in the Mbh–L relation, the luminosity of the host spheroid and the intrinsic scatter of the Mbh–L relation. Assuming SMBHs form via baryonic accretion, we find that (0.008 ± 0.002) h370 per cent of the Universe's baryons are currently locked up in SMBHs. This result is consistent with our previous estimate based on the Mbh–n (Sérsic index) relation.

THE COSMOLOGICAL CONSEQUENCE OF AN OBSCURED AGN POPULATION ON THE RADIATION EFFICIENCY

In light of recent indications for a large population of obscured active galactic nuclei (AGNs), we revisit the mean radiation efficiency from accretion onto supermassive black holes (SMBHs), epsilon, applying a bayesian approach. We use the integrated comoving energy density emitted by AGNs and compare it to the mass density of local SMBHs. When considering only optically-selected unobscured AGNs, we derive log_10[epsilon]=-1.77+0.16-0.11 or epsilon=1.7+0.8-0.4%. Using the AGNs selected using hard X-rays, which include unabsorbed and Compton-thin absorbed AGNs, we find log_10[epsilon]=-1.20+0.15-0.10 or epsilon=6.4+2.6-1.3%. Using optically-selected AGNs, and correcting for the obscured population, we inferr log_10[epsilon]=-1.17+0.11-0.08 or epsilon=6.7+1.9-1.1%, which we consider our best estimate. We also repeat this calculation for intrinsically luminous AGNs (M_B<-23, quasars), comparing to the SMBH mass density in local elliptical galaxies, and find log_10[epsilon]=-1.27+0.15-0.11 or epsilon=5.4+2.2-1.2%. We discuss how our results can be used to estimate the mean spin of accreting SMBHs, both assuming the standard thin-disk model of accretion onto black holes and also comparing to more recent simulations. Our results can rule out maximally rotating SMBHs (<a>=0.998 Gm_bh/c^2) at the >=98% confidence level, as well as high rotation values (<a> >=0.9 Gm_bh/c^2) with >=92% confidence. Our preferred values of <a> are ~0.25-0.60 Gm_bh/c^2, although even these might be overestimated. Hence, we find that on average, SMBHs are not rapidly spinning during accretion. Finally, using an independent measurement of Eddington ratios, we estimate the SMBH e-folding time for the brightest AGNs (quasars) to be <tau> =100+151-60 Myr.

A synthesis model for AGN evolution: supermassive black holes growth and feedback modes

We present a comprehensive synthesis model for the active galactic nuclei (AGN) evolution and the growth of supermassive black hole (SMBH) in the Universe. We assume that black holes accrete in just three distinct physical states, or 'modes': at low Eddington ratio, only a radiatively inefficient, kinetically dominated mode is allowed [low kinetic (LK)]; at high Eddington ratio, instead, AGN may display both a purely radiative [radio quiet, high radiative (HR)] and a kinetic [radio loud, high kinetic (HK)] mode. We solve the continuity equation for the black hole mass function using the locally determined one as a boundary condition, and the hard X-ray luminosity function as tracer of the AGN growth rate distribution, supplemented with a luminosity-dependent bolometric correction and an absorbing column distribution. Differently from most previous semi-analytic and numerical models for black hole growth, we do not assume any specific distribution of Eddington ratios, rather we determine it empirically by coupling the mass and luminosity functions and a set of fundamental relations between observables in the three accretion modes. SMBH always show a very broad accretion rate distribution, and we discuss the profound consequences of this fact for our understanding of observed AGN fractions in galaxies, as well as for the empirical determination of SMBH mass functions with large surveys. We confirm previous results and clearly demonstrate that, at least for z≤ 1.5, SMBH mass function evolves antihierarchically, i.e. the most massive holes grew earlier and faster than less massive ones. For the first time, we find hints of a reversal of such a downsizing behaviour at redshifts above the peak of the black hole accretion rate density (z ≈2). We also derive tight constraints on the (mass-weighted) average radiative efficiency of AGN: under the simplifying assumption that the mass density of both high redshift (z ∼5) and 'wandering' black holes ejected from galactic nuclei after merger events are negligible compared to the local mass density, we find that 0.065 0.07, where ξ 0 is the local SMBH mass density in units of 4.3 x 10 5 M ⊙ Mpc -3 . We trace the cosmological evolution of the kinetic luminosity function of AGN, and find that the overall efficiency of SMBH in converting accreted rest mass energy into kinetic power, e kin , ranges between e kin ≃ (3-5) x 10 -3 , depending on the choice of the radio core luminosity function. Such a 'kinetic efficiency' varies however strongly with SMBH mass and redshift, being maximal for very massive holes at late times, as required for the AGN feedback by many galaxy formation models in cosmological contexts.

Piecing together the X-ray background: bolometric corrections for active galactic nuclei

(Abridged) The X-ray background can be used to constrain the accretion history of Supermassive Black Holes (SMBHs) in Active Galactic Nuclei (AGN). A knowledge of the hard X-ray bolometric correction, \kappa_{2-10keV} is a vital input into these studies. Variations in the disk emission in the UV have not previously been taken into account in calculating \kappa_{2-10keV}; we show that such variations are important by constructing optical--to--X-ray SEDs for 54 AGN. In particular, we use FUSE UV and X-ray data from the literature to constrain the disk emission as well as possible. Previous work has suggested a dependence of \kappa_{2-10keV} on AGN luminosity, but we find significant spread in \kappa_{2-10keV} with no simple dependence on luminosity. Populations such as Narrow-Line Seyfert 1 nuclei (NLS1s), Radio Loud and X-ray Weak AGN may have values of \kappa_{2-10keV} differing systematically from the rest of the AGN population. Other sources of uncertainty include intrinsic extinction in the optical--UV, X-ray and UV variability and uncertainties in SMBH mass estimates. Our results suggest a more well-defined relationship between \kappa_{2-10keV} and Eddington ratio in AGN, with a transitional region at an Eddington ratio of ~0.1, below which the bolometric correction is typically 15 - 25, and above which it is typically 40 - 70. We consider the potential implied parallels with the low/hard and high/soft states in Galactic Black Hole (GBH) accretion, and present bolometric corrections for the GBH binary GX 339-4 for comparison. Our findings reinforce previous studies proposing a multi-state description of AGN accretion analogous to that for GBH binaries. Future calculations of the SMBH mass density may need to account for the possible dependence of \kappa_{2-10keV} on Eddington ratio.

The local supermassive black hole mass density: corrections for dependencies on the Hubble constant

Abstract We have investigated past measurements of the local supermassive black hole mass density, correcting for hitherto unknown dependencies on the Hubble constant, which, in some cases, had led to an underestimation of the mass density by factors of ∼2. Correcting for this, we note that the majority of (but not all) past studies yield a local supermassive black hole mass density that is consistent with the range 4.4–5.9 × 105f(H0) M⊙ Mpc−3 (when using H0= 70 km s−1 Mpc−1). In addition, we address a number of ways in which these past estimates can be further developed. In particular, we tabulate realistic bulge-to-total flux ratios which can be used to estimate the luminosity of bulges and subsequently their central black hole masses.

The Millennium Galaxy Catalogue: the local supermassive black hole mass function in early- and late-type galaxies

We provide a new estimate of the local supermassive black hole mass function using (i) the empirical relation between supermassive black hole mass and the Sersic index of the host spheroidal stellar system and (ii) the measured (spheroid) Sersic indices drawn from 10k galaxies in the Millennium Galaxy Catalogue. The observational simplicity of our approach, and the direct measurements of the black hole predictor quantity, i.e. the Sersic index, for both elliptical galaxies and the bulges of disc galaxies makes it straightforward to estimate accurate black hole masses in early- and late-type galaxies alike. We have parameterised the supermassive black hole mass function with a Schechter function and find, at the low-mass end, a logarithmic slope (1+alpha) of ~0.7 for the full galaxy sample and ~1.0 for the early-type galaxy sample. Considering spheroidal stellar systems brighter than M_B = -18 mag, and integrating down to black hole masses of 10^6 M_sun, we find that the local mass density of supermassive black holes in early-type galaxies rho_{bh, early-type} = (3.5+/-1.2) x 10^5 h^3_{70} M_sun Mpc^{-3}, and in late-type galaxies rho_{bh, late-type} = (1.0+/-0.5) x 10^5 h^3_{70} M_sun Mpc^{-3}. The uncertainties are derived from Monte Carlo simulations which include uncertainties in the M_bh-n relation, the catalogue of Sersic indices, the galaxy weights and Malmquist bias. The combined, cosmological, supermassive black hole mass density is thus Omega_{bh, total} = (3.2+/-1.2) x 10^{-6} h_70. That is, using a new and independent method, we conclude that (0.007+/-0.003) h^3_{70} per cent of the universe's baryons are presently locked up in supermassive black holes at the centres of galaxies.

Spin, Accretion, and the Cosmological Growth of Supermassive Black Holes

If supermassive black holes (SMBHs) are the energy sources that power quasars and active galactic nuclei (AGNs), then QSO SDSS 1148+5251, the quasar with the highest redshift (zQSO = 6.43), hosts an SMBH formed within ~0.9 Gyr after the big bang. This requirement places constraints on the cosmological formation of SMBHs, believed to grow from smaller initial seeds by a combination of accretion and mergers. We focus on gas accretion onto seeds produced by the collapse of Population III stars at high redshift. We incorporate the results of recent relativistic, MHD simulations of disk accretion onto Kerr black holes to track the coupled evolution of the masses and spins of the holes. We allow for an additional amplification of ~104 in the mass of a typical seed due to mergers, consistent with recent Monte Carlo simulations of hierarchical mergers of cold dark matter halos containing black hole seeds. We find that the growth of Population III black hole remnants to ~109 M☉ by zQSO ≳ 6.43 favors MHD accretion disks over standard thin disks. MHD disks tend to drive the holes to a submaximal equilibrium spin rate a/M ~ 0.95 and radiation efficiency ϵM ~ 0.2, while standard thin disks drive them to maximal spin (a/M = 1) and efficiency (ϵM = 0.42). This small difference in efficiency results in a huge difference in mass amplification by accretion at the Eddington limit. The MHD equilibrium efficiency is consistent with the observed ratio of the QSO plus AGN luminosity density to the local SMBH mass density. Our prototype analysis is designed to stimulate the incorporation of results from relativistic stellar collapse and MHD accretion simulations in future Monte Carlo simulations of hierarchical structure formation to better determine the cosmological role of SMBHs and their mass and spin distributions.

The Cosmic Evolution of Hard X-Ray-selected Active Galactic Nuclei

We use highly spectroscopically complete deep and wide-area Chandra surveys to determine the cosmic evolution of hard X-ray–selected active galactic nuclei (AGNs). For the deep fields, we supplement the spectroscopic redshifts with photometric redshifts to assess where the unidentified sources are likely to lie. We find that the median redshifts are fairly constant with X-ray flux at z ~ 1. We classify the optical spectra and measure the FWHM line widths. Most of the broad-line AGNs show essentially no visible absorption in X-rays, whereas the sources without broad lines (FWHM < 2000 km s-1; AGNs) show a wide range of absorbing column densities. We determine hard X-ray luminosity functions for all spectral types with LX ≥ 1042 ergs s-1 and for broad-line AGNs alone. At z < 1.2, both are well described by pure luminosity evolution, with L* evolving as (1 + z)3.2±0.8 for all spectral types and as (1 + z)3.0±1.0 for broad-line AGNs alone. Thus, all AGNs drop in luminosity by almost an order of magnitude over this redshift range. We show that this observed drop is due to AGN downsizing rather than to an evolution in the accretion rates onto the supermassive black holes. We directly compare our broad-line AGN hard X-ray luminosity functions with the optical QSO luminosity functions and find that at the bright end they agree extremely well at all redshifts. However, the optical QSO luminosity functions do not probe faint enough to see the downturn in the broad-line AGN hard X-ray luminosity functions and even appear to be missing some sources at the lowest luminosities they probe. We find that broad-line AGNs dominate the number densities at the higher X-ray luminosities, while optically narrow AGNs dominate at the lower X-ray luminosities. We rule out galaxy dilution as a partial explanation for this effect by measuring the nuclear UV/optical properties of the Chandra sources using the Hubble Space Telescope Advanced Camera for Surveys GOODS-North data. The UV/optical nuclei of the optically narrow AGNs are much weaker than expected if the optically narrow AGNs were similar to the broad-line AGNs. We therefore postulate the need for a luminosity-dependent unified model. An alternative possibility is that the broad-line AGNs and the optically narrow AGNs are intrinsically different source populations. We cover both interpretations by constructing composite spectral energy distributions—including long-wavelength data from the mid-infrared to the submillimeter—by spectral type and by X-ray luminosity. We use these spectral energy distributions to infer the bolometric corrections (from hard X-ray luminosities to bolometric luminosities) needed to map the accretion history. We determine the accreted supermassive black hole mass density for all spectral types and for broad-line AGNs alone, using the observed evolution of the hard X-ray energy density production rate and our inferred bolometric corrections. We find that only about one-half to one-quarter of the supermassive black hole mass density was fabricated in broad-line AGNs. Using either recent optical QSO luminosity function determinations or our broad-line AGN hard X-ray luminosity function determinations, we measure an accreted supermassive black hole mass density that is a factor of almost 2 lower than that measured by previous work, assuming = 0.1. This leaves room for obscured accretion when compared with the local supermassive black hole mass density. In fact, we find reasonable agreement between the accreted supermassive black hole mass density from all spectral types and the local supermassive black hole mass density, assuming ≈ 0.1–0.2. However, there is little room for further obscured sources or for low-efficiency accretion periods.

Tracing the cosmological assembly of stars and supermassive black holes in galaxies

We examine possible phenomenological constraints for the joint evolution of supermassive black holes (SMBHs) and their host spheroids. We compare all the available observational data on the redshift evolution of the total stellar mass and star formation rate density in the Universe with the mass and accretion rate density evolution of SMBHs, estimated from the hard X-ray selected luminosity function of quasars and active galactic nuclei for a given radiative efficiency, ∈. We assume that the ratio of the stellar mass in spheroids to the black hole mass density evolves as (1 + z) -α , while the ratio of the stellar mass in discs + irregulars to that in spheroids evolves as ( 1 + z) -β , and we derive constraints on α, β and ∈. We find that a > 0 at more than the 4a level, implying a larger black hole mass at higher redshift for a given spheroid stellar mass. The favoured values for β are typically negative, suggesting that the fraction of stellar mass in spheroids decreases with increasing redshift. This is consistent with recent determinations that show that the mass density at high redshift is dominated by galaxies with irregular morphology. In agreement with earlier work, we constrain ∈ to be between 0.04 and 0.11, depending on the exact value of the local SMBH mass density, but almost independently of a and β.

Detection of Gravitational Waves from the Coalescence of Population III Remnants with Advanced LIGO

The comoving mass density of massive black hole (MBH) remnants from pre-galactic star formation could have been similar in magnitude to the mass-density of supermassive black holes (SMBHs) in the present-day universe. We show that the fraction of MBHs that coalesce during the assembly of SMBHs can be extracted from the rate of ring-down gravitational waves that are detectable by Advanced LIGO. Based on the SMBH formation history inferred from the evolution of the quasar luminosity function, we show that an observed event rate of 1 per year will constrain the SMBH mass fraction that was contributed by MBHs coalescence down to a level of ~10^-6 for 20 solar mass MBH remnants (or ~10^-4 for 260 solar mass remnants).

Supermassive black hole demography: the match between the local and accreted mass functions

We have performed a detailed analysis of the local super-massive black-hole (SMBH) mass function based on both kinematic and photometric data and derived an accurate analytical fit in the range 10^6 <= (M_BH/M_sun) <= 5*10^9. We find a total SMBH mass density of (4.2+/-1.1)*10^5 M_sun/Mpc^3, about 25% of which is contributed by SMBHs residing in bulges of late type galaxies. Exploiting up-to-date luminosity functions of hard X-ray and optically selected AGNs, we have studied the accretion history of the SMBH population. If most of the accretion happens at constant \dot{M_BH}/M_BH the local SMBH mass function is fully accounted for by mass accreted by X-ray selected AGNs, with bolometric corrections indicated by current observations and a standard mass-to-light conversion efficiency \epsilon ~10%. The analysis of the accretion history highlights that the most massive BHs accreted their mass faster and at higher redshifts (z>1.5), while the lower mass BHs responsible for most of the hard X-ray background have mostly grown at z<1.5. The accreted mass function matches the local SMBH mass function if \epsilon ~0.09(+0.04,-0.03) and the Eddington ratio \lambda=L/L_Edd \~0.3(+0.3,-0.1) (68% confidence errors). The visibility time, during which AGNs are luminous enough to be detected by the currently available X-ray surveys, ranges from ~0.1 Gyr for present day BH masses M_BH(z=0) ~10^6 M_sun to ~0.3 Gyr for M_BH(z=0) >= 10^9 M_sun. The mass accreted during luminous phases is >= 25-30% even if we assume extreme values of \epsilon (\epsilon \~0.3-0.4). An unlikely fine tuning of the parameters would be required to account for the local SMBH mass function accomodating a dominant contribution from 'dark' BH growth (due, e.g., to BH coalescence).

Cosmological Growth History of Supermassive Black Holes and Demographics in High-zUniverse

We study the demographics of supermassive black holes (SMBHs) in the local and high-z universe with hard X-ray luminosity functions (HXLFs) of AGNs. First, we compare the mass accretion history at z > 0 with optical luminosity functions (OLFs) and that with HXLFs. We consider the constraints on parameters of mass accretion (energy conversion efficiency and Eddington ratio) and conclude that the constraints based on HXLFs are more adequate rather than that based on OLFs. The sub-Eddington case is allowed only when we use HXLFs. Next, we estimate the upper limit of the cumulative mass density of SMBHs at any redshifts. For an application, we examine if Lyman-Break galaxies (LBGs) at z = 3 already have SMBHs in their centers which are suggested by recent observations. We show that the special conditions should be met in order that a large part of LBGs have SMBHs (for example, large energy conversion efficiency and frequent mergers and/or direct formations at z > 3). The possibility that nearly all LBGs have SMBHs with large mass ratio, such as MBH/M s t e l l a r > 0.005, is reliably ruled out.