A&A 459, 43-54 (2006)
DOI: 10.1051/0004-6361:20064945
L. Miller1 - W. J. Percival2,3 - S. M. Croom4 - A. Babic1
1 - Dept. of Physics, University of Oxford,
Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
2 - Institute for Astronomy,
University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK
3 -
Institute of Cosmology and Gravitation,
University of Portsmouth,
Portsmouth, PO1 2EG, UK
4 -
Anglo-Australian Observatory, PO Box 296, Epping, NSW 2121, Australia
Received 1 February 2006 / Accepted 8 August 2006
Abstract
Aims. We investigate the cosmological growth of dark halos and follow the consequences of coeval growth for the accretion history of associated supermassive black holes.
Methods. The Press-Schechter approximation is used to obtain an analytic expression for the mean rate of growth of dark matter halos. Dark halo accretion rates are compared with numerical work and the consequences for understanding AGN evolution are described.
Results. The mean accretion rate onto dark matter halos is shown to have a simple analytic form that agrees with previous numerical work and that may easily be calculated for a wide range of halo mass, redshift and cosmological parameters. The result offers a significant improvement over published fitting formulae deduced from merger trees. We then consider the growth of associated supermassive black holes, and make a basic test of the simple hypothesis of "Pure Coeval Evolution'' (PCE) in which, on average, black hole growth tracks dark halo growth. We demonstrate that both the absolute value of the integrated AGN bolometric luminosity density and its cosmological evolution derived from hard X-ray surveys are well-reproduced by PCE. Excellent agreement is found at ,
although the observed luminosity density drops by a factor 2 compared with PCE by z=0: black hole growth appears to decouple from halo growth at low redshifts, and this may be related to the phenomenon of "cosmic downsizing''. Overall, AGN evolution appears either to be caused by or to be closely linked to the slow-down in the growth of cosmic structure. We also discuss the mean Eddington ratio averaged over all galaxies, which is predicted to show strong evolution to higher values with redshift.
Key words: accretion, accretion disks - galaxies: formation - galaxies: active - cosmology: theory
One of the great mysteries in the study of active galaxies and QSOs is the physical origin of the strong cosmological evolution in their space density at a given luminosity. For many years it has appeared that, at least in broad terms, the evolution was best described as "pure luminosity evolution'' in which QSOs appeared on average to have dimmed with time rather than changing in comoving space density (Croom et al. 2004; Marshall et al. 1983; Boyle et al. 1988). This is in accord with recent evidence that almost all galaxies with a massive spheroid component contain a supermassive black hole at the present epoch (Magorrian et al. 1998; Ferrarese & Merritt 2000; Onken et al. 2004; Tremaine et al. 2002; Gebhardt et al. 2000), the great majority of which must be largely inactive today. If these black holes were luminous in a phase of accretion and growth at higher redshift then their luminosities must indeed have declined with time.
Yet this picture is not easily in accord with modern ideas of galaxy formation, since if the matter content of the universe is dominated by cold dark matter we expect galaxies to grow hierarchically, and we expect the mass function of black holes at the centres of galaxies to increase with time - what then is the mechanism that allows the mass function to increase but the luminosity to decrease with cosmic epoch? One obvious possibility is that black holes may increase in mass but with a decreasing mean accretion rate. This picture has been incorporated into models such as those of Haehnelt & Rees (1993) and Kauffmann & Haehnelt (2000) in which simple parameterisations of such cosmological evolution in accretion rate were allowed, and it was shown that with a suitable choice of parameters the evolution in the QSO optical luminosity function could be matched. The physical origin of this variation in accretion rate has not yet been determined, however, and explanations range from a systematic depletion in available gaseous material with cosmic epoch to long-term variations in the accretion process itself: these latter explanations would of course only work if black holes were formed at high redshift and the luminosity evolution that has been observed were a reflection of the variation in accretion history of individual black holes.
More recently however it has become clear that optically luminous QSOs
are a short-lived phenomenon, at least compared with the Hubble time.
Measurement of the clustering of QSOs
in the 2dF QSO Redshift Survey (2QZ: Croom et al. 2005) has shown that
their clustering amplitude does not increase with time as would be
expected if QSOs were long-lived, given that cosmic structure is expected
still to be growing at
in a Universe with a cosmological
matter density parameter
.
This
implies that we must look for a universal variation in accretion rate
rather than anything intrinsic to an individual accretion "event''.
The simple picture of QSO luminosity evolution itself has now been shown to be more complicated when active galaxies are selected at X-ray wavelengths. First, it now seems likely that the hard X-ray background is produced by X-rays emitted from active galaxies, but in that case a large fraction of those active galaxies must be highly absorbed, with equivalent X-ray absorption column densities in neutral hydrogen >1023 cm-2(Risaliti et al. 1999; Comastri et al. 2001; Ueda et al. 2003). These active galaxies would be likely to be optically obscured also. A second significant discovery from X-ray surveys however has been that low luminosity X-ray selected active galaxies, that would be classified as Seyfert galaxies were they optically selected, display cosmological evolution in their space density that appears to have a maximum at lower redshift than their high luminosity QSO counterparts (Zheng et al. 2004; Barger et al. 2005; Steffen et al. 2003; Ueda et al. 2003; Cowie et al. 2003). It may be to some extent that the identification of these weak active galaxies may be less complete at higher redshifts, but early indications are that there is a significant shift of the space density maximum to lower redshifts with decreasing luminosity.
Hence it appears that there is much still to understand both observationally and theoretically about the evolution of active galaxies. Nonetheless, the basic inference that there must have been evolution in accretion rate seems inescapable. In this paper we test the extent to which the growth of black holes may be linked to the growth of galaxies, and argue that cosmological evolution in accretion rate can be understood primarily as arising from the cosmological evolution in the rate of accretion of matter onto galaxies. In Sect. 3 we present a new calculation of the accretion rate onto dark halos using the extended Press-Schechter approximation. We find that the analytic expression agrees well with both our own and previous numerical estimation of the accretion rate calculated from merger trees. The analytic approach here circumvents the need for such numerical estimation, and we also find that it provides mathematically better behaviour than fitting functions to the numerical results that have previously been proposed in the literature.
In Sect. 4 we argue that coeval growth of black holes with their associated galaxies and dark halos leads to a simple expression for the total mass accretion rate onto all black holes. We calculate the expected integrated luminosity density arising from accretion onto black holes and show that it agrees remarkably well with observation. When expressed as the Eddington ratio (the ratio of the actual mass accretion rate to the rate required to attain the Eddington limiting luminosity) it shows significant cosmological evolution almost independent of halo mass and without depending strongly on the choice of cosmological parameters.
One hypothesis, whose consequences will be followed in this paper, is that the population of supermassive black holes, that in the nearby universe inhabit the nuclei of massive galaxies, grew coevally with their host galaxies and associated dark halos. Before launching into detailed calculation, in this section we first consider the rationale for testing this hypothesis.
First,
given present data on the black hole/bulge relationship, some degree of
coeval growth seems hard to avoid, at least for galaxies with measurable bulge
components. In all massive galaxies where both a bulge mass and a
black hole mass have been measured dynamically, there is an extremely
tight correlation between the two (Onken et al. 2004; Tremaine et al. 2002; Gebhardt et al. 2000).
At lower masses, with velocity dispersion
km s-1, the correlation is less well-established, but does
appear to extend at least down to
km s-1(Barth et al. 2005). When the bulge mass is determined from velocity
dispersion measurements, the cosmic scatter in the relation appears
smaller than the measurement errors, and probably is less than a
factor three (Onken et al. 2004; Tremaine et al. 2002). But it is thought that galaxies
have grown hierarchically over a wide range of cosmic epoch, and with
galaxies at any observed redshift having accumulated their masses at
many different times. In order for the tight present-day
black-hole/bulge relation to exist, the process of formation of black
hole and galaxy must have been interrelated, the simplest explanation
being that they grew in mass at the same time.
Second, the luminosity emitted by AGN is thought to arise from accretion,
and therefore is a signature of ongoing black hole growth at low redshifts.
Ueda et al. (2003) and Marconi et al. (2004) have shown that both the X-ray background and
the local mass density in black holes are consistent with being created by
the luminous phase of accretion visible in hard X-ray AGN surveys,
and that the bulk of that black hole growth occurs at low redshifts (z < 3).
The picture of supermassive black holes forming at high redshifts and then
remaining largely unchanged since then is not consistent with the
observed AGN luminosity density and the inferred local black hole mass density,
unless the radiative efficiency of the AGN luminosity we can see is unfeasibly
high (
,
Marconi et al. 2004). So it appears that black holes
have been continuing to form during the cosmic epochs in which dark halos
and their associated galaxies have also continued to grow.
Previous AGN models (e.g. Kauffmann & Haehnelt 2000) have assumed a relationship between dark halo mass and black hole mass. It is hard to see how the black hole could know what mass of halo it is in unless there has been some causal link between halo growth and black hole growth. The link does not need to be direct, and it may be that feedback processes have an important role in regulating the black hole/halo relationship, but in effect, these models have implicitly assumed that coeval growth has occurred. Whether coeval growth is still occurring in the present-day universe is another question, and one that we shall attempt to investigate in this paper.
The growth of galaxies by hierarchical mergers implies that
there is a contribution of mergers to black hole growth, too.
Thus we expect the mass function of black holes to be determined
by hierarchical merging, as is the mass function of galaxies. Mergers
of black holes cannot change the integrated irreducible mass in black
holes, however, so the integrated mass in black holes should
depend on the total amount of matter accreted and not on the merger
history. The importance of mergers does not violate the argument
originally due to Soltan (1982) that black holes in the local
universe are produced by accretion at earlier cosmic epochs. The
growth of supermassive black holes through mergers does in
principle allow high black hole masses to be attained at rates faster
than the Salpeter rate without violating the Eddington limit, which
may help us to understand how supermassive black holes can exist
at high redshift,
(e.g. Willott et al. 2003).
In practice we might expect the phases of significant mass accretion in the life of a black hole to be correlated with periods of galaxy merging: it has long been suggested that mergers may drive matter into galaxy centres and trigger phases of black hole accretion (e.g. Barnes & Hernquist 1992). This picture creates an attractive link between galaxy mergers and prompt AGN activation but is not required in what follows.
If the matter content of the universe is dominated by cold dark matter, then dark matter halos associated with galaxies form hierarchically. Bond et al. (1991) have shown that a more rigorous treatment of the work of Press & Schechter (1974) ("extended Press-Schechter'', hereafter EPS) can be used to obtain information about the build-up of structure, and the resulting evolving mass functions agree with the results of N-body simulations, especially if some additional modification of EPS theory is allowed (Sheth et al. 2001; Sheth & Tormen 1999,2002).
In this picture of hierarchical galaxy formation, at any moment in time, any given overdensity is increasing in mass through the process of accretion of matter. There has been much discussion in the literature about whether one can use EPS theory to further analyse this accretion in terms of merger events (Benson et al. 2005; Lacey & Cole 1993; Cole et al. 2000; Lacey & Cole 1994) but we are not interested here in attempting such a detailed view of the build-up of galaxies. In previous papers (Percival et al. 2000; Percival & Miller 1999) the formation rate of halos within EPS theory has been calculated, and hence if the space density of halos is also known this may be converted into a mean accretion rate onto halos as a function of their mass and redshift, measuring the overall accretion of matter, not broken down into individual events.
In those papers it was shown that a joint distribution function
in mass M and
cosmic time t of new overdensities, F(M,t), could be defined that
is related to the mass function, F(M|t),
by the relation
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However, the normalisation of the joint distribution F(M,t) is problematic, as discussed by Percival et al. (2000). In Sect. 3.2 we follow a different approach and calculate the mean accretion rate analytically from the EPS conditional mass function. This is complemented in Sect. 3.3 by a numerical derivation of the rate from Monte-Carlo realisations of the EPS process.
Lacey & Cole (1993,1994) give the conditional probability that a location in a halo of mass
M at cosmic time t was previously part of a halo of lower mass M' at earlier
time t':
Hence the expectation value for the halo's increase in mass
is
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Hence we can write the mean rate of mass accretion as
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The function f(M) may be evaluated numerically for an assumed matter
power spectrum. For cold dark matter power spectra it depends only
weakly on mass and varies little with shape of the power spectrum. It
varies inversely with the normalisation parameter .
Figure 1 shows the variation of f(M) with halo mass for a
CDM power spectrum approximated as in Efstathiou et al. (1992) for
a range of values of the shape parameter
and with fixed
,
h=0.7.
The relationship between variance
and mass was
calculated assuming a spherical top-hat smoothing function, although
we should note that strictly Eq. (3) is derived assuming
smoothing with a top-hat function in k-space.
At a halo mass of
,
appropriate for the halo masses of
luminous QSOs (Croom et al. 2005), f(M) varies only by a factor 2 over the
range
for h=0.7. The variation with mass is
also weak for masses in the range appropriate for massive galaxies:
at
,
f(M) varies by a factor 2 over the
range
.
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Figure 1:
The variation of the function f(M) with log10(halo mass) as a function
of the power-spectrum shape parameter ![]() ![]() ![]() |
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For convenience we provide a fitting formula for f(M). Over the range of
and mass shown in Fig. 1, we adopt the relation
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Figure 2: Comparison of the mean mass accretion rate calculated analytically (solid curves) and measured from Monte-Carlo realisations (solid symbols), shown as a function of redshift for a variety of masses. |
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To validate the derivation presented in Sect. 3.2, we now compare the analytic instantaneous rate with the rate calculated from Monte-Carlo realisations of the EPS process. In EPS theory, the overdensity recovered if a realisation of the density field is smoothed on different scales around a fixed point can be matched to a one-dimensional Brownian random walk. The collapsed mass at a given time is given by the trajectory's first upcrossing of a barrier at the critical overdensity corresponding to the chosen time. Because we can only calculate finite representations of these trajectories, there are a number of numerical issues that complicate their use. In particular
Figure 2 shows the result from the Monte-Carlo measurements,
calculated from the average mass accreted within
,
corresponding to 0.0035 Gyr at z=7 and 0.5 Gyr at z=0. Decreasing
further does
not significantly change the recovered rate. The agreement with the
analytic instantaneous rate is extremely close. Any differences
may be revealed by
comparing f(M) with the equivalent quantity from the Monte-Carlo
realisations by factoring out the analytic linear dependence on mass
and
(Fig. 3). The maximum difference
between the analytic and numerical results is <5% and is
likely to be caused by remaining problems owing to the discrete nature
of the numerical trajectories. Note that
the curves in Fig. 2 show the immediate mass accretion rate
onto a halo which has a specified mass at a specified redshift,
not how the mean accretion rate varies with time for
an individual halo.
This comparison has shown that the results obtained from numerical realisations of the Press-Schechter process are indeed consistent with the analytic results that we have obtained. This type of realisation forms the basis of the commonly-used "merger trees'' and in the next section we compare our results with earlier numerical attempts to understand the growth of dark matter halos.
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Figure 3:
Comparison of f(M) (solid curve) with that
deduced from the Monte-Carlo realisations by factoring out the linear dependence
on M and
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Van den Bosch (2002) has calculated the average mass accretion
histories of dark matter haloes based on the extended Press-Schechter
formalism and the N-branch merger-tree construction algorithm of
Somerville & Kolatt (1999).
He provides a fit to the numerically-obtained mass accretion histories by
fitting a function of the form
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Figure 4:
Comparison of the mean accretion rate from this paper (dotted lines)
with the two alternative numerical fits proposed by Van den Bosch (2002)
(solid lines) and by Wechsler et al. (2002) with parameter values estimated
by Van den Bosch (2002) (dot-dashed lines).
See the text for more details of the
calculation of these curves, which are shown for values of
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Wechsler et al. (2002) have also discussed the growth of halos in merger trees, and
find a reasonable parameterisation of the accretion histories of
halos to have the form:
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In Fig. 4
we compare our accretion rate with those obtained from these two
alternative fits to the numerical mass accretion histories.
The solid line shows the mean accretion rate obtained by
differentiating Eq. (13) and the dot-dashed line
from differentiating Eq. (15), both with the
prescriptions for the free parameters given by Van den Bosch (2002)
that are quoted above.
The dotted line shows the mass accretion rate from Eq. (8)
(note that, as in Fig. 2,
each curve corresponds to a fixed value of mass at each redshift:
i.e. these are curves of constant M(z) and not constant M0).
The comparison is made for several values of halo mass:
.
There is excellent
agreement between the three curves over most of the range of redshifts.
However, at the lowest redshifts the Van den Bosch (2002) accretion
rate shows significant departures, and it is straightforward to see
that this arises from the choice of fitting function in Eq. (13):
in the limit
,
if calculated from
that equation, for parameter values
.
Such behaviour is unphysical,
it implies that all halos at the present day have stopped growing.
This is purely an artefact of the choice of fitting function and
leads to significant departures from the analytic calculation for z<0.5.
The result from the Wechsler et al. (2002) form consistently results in
slightly higher values of accretion rate, but predicts the same
behaviour as the analytic calculation at redshifts approaching zero.
We also find a further limitation of the numerical prescriptions,
which is that the calculated growth histories become unphysical
at high masses and redshifts, with problems arising when the present-day
mass
.
The problem is that the M(z)functions for differing masses cross at high redshift, such that a high mass
halo might be predicted to have had a lower-mass progenitor than a lower mass
halo. For example, the main progenitor
mass of a
halo is predicted by Eq. (13)
to be smaller than that of a
halo at redshifts
.
We roughly estimate that the Van den Bosch (2002) fit can be used up to
,
where the pairs of numbers denote the locus on the M0, z plane beyond which
the prescription becomes unreliable.
The same problem is found when using the parameterisation
of Eq. (15),
with the range of valid redshift and present-day halo mass moved to lower values:
.
Given the relationship between the Wechsler et al. (2002) and analytic functions, it
can be seen that the problem in this case arises from the prescription for
the value of
rather than being an intrinsic problem with the choice of numerical function.
Overall we conclude that the analytic and previous numerical results are in reasonable agreement in the regions of mass and redshift space where the numerical parameterisations are reliable, but that there exist significant regions of that space where the previous numerical fits become unusable.
Sheth & Tormen (1999) have shown, however, that the EPS mass
function may be modified to produce even closer agreement with the N-body
simulations. This work has been developed further by Sheth et al. (2001)
and Sheth & Tormen (2002) and confirmed in the Jenkins et al. (2001) simulations. The principal
modification is to reduce the critical density for collapse,
,
by
a factor
where a is determined from comparison with the simulations
to have a value
.
Insertion of this factor into
Eq. (3) would simply imply that f(M) is also reduced
by a factor
.
This causes a slight shift in the curves
plotted here but otherwise has no effect on the redshift or mass dependence.
We note that the merger trees of Van den Bosch (2002) have also been compared
with N-body simulations, and reasonable agreement was found there too, albeit
with a tendency for halos to form at higher redshifts which could also be
reproduced by introduction of the factor .
In the case of the
Van den Bosch (2002) simulations the best-fitting value
for
varied over the range
,
depending on halo mass.
Hence it may be that there is some
systematic and possibly mass-dependent departure of the EPS accretion rate
from the "true'' (dark-matter N-body) rate, but most likely at a
level <20% in accretion rate.
Finally, we note that
Benson et al. (2005) have also argued that the distribution of mass discontinuities
in EPS does not allow a self-consistent interpretation of these as merger
events. This casts doubt on one of the fundamental assumptions of EPS-based
merger trees, but should not affect the calculation of the mean accretion rate,
which does not rely on that assumption.
A simple extension of this hypothesis would allow some non-linear evolution
of the relation between black holes and dark halos. Equation (17)
could be modified by a factor
such that
In this section we calculate the observationally-determined AGN luminosity density and compare with the value predicted by PCE. We define the integrated luminosity density to be the luminosity emitted during the black hole accretion process per comoving cubic Mpc, summing over AGN of all luminosities, as previously calculated by a number of authors (e.g. Marconi et al. 2004). The quantity predicted by the PCE hypothesis is the bolometric luminosity density, so we will need to apply bolometric corrections to observed quantities.
Figure 5 shows the bolometric luminosity density
derived from the Ueda et al. (2003) LDDE model at 2-10 keV
integrated over the absorption-corrected luminosity range
erg s-1. The X-ray luminosity density
has been converted to a bolometric luminosity density
and a small correction for missing Compton-thick AGN has also been
applied, as described below. The luminosity density is not sensitive
to the precise limits of integration: increasing the lower limit to
erg s-1 decreases the z=0 luminosity density
only by 0.08 dex without changing the maximum value (to 0.01 dex)
or its redshift.
Decreasing the upper limit to
erg s-1
has no effect to within 0.01 dex. Most of the bolometric luminosity
density is produced by AGN with
erg s-1.
Also shown in Fig. 5 are estimates of the
uncertainty in
.
To calculate the uncertainty, we first refit the
binned data shown in Fig. 11 of Ueda et al. (2003), kindly provided by Y. Ueda,
with the same six-parameter model (see Ueda et al. 2003, for details of the
model), but with the normalisation parameterised instead as the integrated
luminosity density. We then estimate confidence intervals on the
integrated luminosity density for each of the five redshift slices
(0.015-0.2,0.2-0.4,0.4-0.8,0.8-1.6,1.6-3.0) in turn,
marginalising over the remaining five parameters of the fit. This
procedure thus includes the full statistical uncertainty in the data
and includes an element of uncertainty arising from the fact that to cover
the full luminosity range requires some extrapolation of the model,
although with the caveat that only the set of
model functions that may be parameterised by the six-parameter function
are allowed. The best-fit values for each redshift slice
and the deduced uncertainties are shown in Fig. 5.
Some points deviate somewhat from the nominal curve deduced from
Ueda et al. (2003)'s best fit: this is likely because their best fit was obtained
from a maximum likelihood fit to unbinned data whereas our points and
errors were evaluated from fits to the binned data. We nonetheless
expect that the
size of the error bars should also be a good indication of the size of the 68 percent confidence region for the likelihood-determined
function.
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Figure 5:
The bolometric luminosity density deduced from the best-fit model of
Ueda et al. (2003), integrating over the range
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It is recognised that 2-10 keV X-ray fluxes must be corrected for
absorption, and Ueda et al. (2003) derive an estimate for the
luminosity-dependent distribution of absorption column which has been incorporated
into their analysis. In addition,
it is likely that some fraction of AGN are Compton-thick, that
the Compton-thick fraction is a function of intrinsic luminosity
and that even hard X-ray surveys miss these objects (Ueda et al. 2003).
Ueda et al. (2003) assume the (very uncertain) correction, based on the results of
Risaliti et al. (1999), that there are 1.6 times as many Compton-thick AGN
as there are AGN with absorption column densities in the range
cm-2. We apply the same correction
here. Although very uncertain, the net effect is to increase the
bolometric luminosity density by only a factor 1.48 (i.e. 0.17 dex).
Varying the
Compton-thick fraction from this nominal value would have an equivalent
effect on the value of
deduced, but does not
significantly affect the conclusions of this paper.
The optical QSO survey that to date has most successfully probed to low intrinsic luminosity is the 2SLAQ survey (Richards et al. 2005). In Fig. 5 we plot the bolometric luminosity density obtained by integrating the pure luminosity evolution model derived from the 2SLAQ data alone, integrating over the equivalent luminosity range as for the X-ray determination, with the Marconi et al. (2004) optical bolometric correction and Ueda et al. (2003) Compton-thick fraction as above. Because of optical selection biases the valid redshift range is restricted to 0.4<z<2.1 (Richards et al. 2005).
The function shown is more sensitive to the limits of integration
than the X-ray case above, because although the large survey areas allow
optical QSO surveys
to probe to higher AGN bolometric luminosities, the range of bolometric
luminosity covered is substantially smaller than the composite X-ray surveys
discussed above (Richards et al. 2005). Increasing the lower luminosity limit
to match a value
erg s-1 decreases the z=0.4bolometric luminosity density by 0.19 dex and the maximum value
by 0.14 dex. Again, decreasing the upper luminosity limit to match a value
erg s-1 leads to no significant change in
luminosity density, to 0.01 dex.
Considering the differing selection of AGN types in optical and X-ray
surveys,
the X-ray- and optical-derived bolometric luminosity densities show
remarkably good agreement. This demonstrates that the corrections applied
are unlikely to be too far wrong, at least over the optical-X-ray part
of the spectrum (both bolometric corrections could be missing
components in other parts of the spectrum such as the far infrared - this
is discussed further by Marconi et al. 2004). The differences between
the two functions are most apparent at ,
although even here they
agree within a factor 1.6. This difference arises from the additional
low-luminosity AGN that are found by hard X-ray surveys but not by
the large-area optical QSO surveys (see
Richards et al. 2005, for further discussion).
This diagram also illustrates the limited effect of possible "cosmic downsizing'' on the luminosity density: the optical density is derived from a pure luminosity-evolution model, the X-ray from a luminosity-dependent density evolution model. We can see that the two models are measurably different, but that any "cosmic downsizing'' does not dominate the redshift evolution of the luminosity density.
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The result is shown in Fig. 5 for two cases. The first
shows what would happen if the black hole mass density did not change with
cosmic epoch: the luminosity density would then be determined entirely by
the invariant black hole mass density and the epoch-dependent accretion
rate, as shown by the dotted curve. For consistency with Ueda et al. (2003)
we assume a CDM cosmology with parameters
,
H0=73 km s-1 Mpc-3 (the observational values are also adjusted
to this value of H0) and
(Spergel et al. 2006).
The remaining parameter is the average radiative efficiency
which we have
set to a value 0.04 to approximately match data and prediction at
.
Such a value
is consistent both with the range of values
that might be expected for accretion onto black holes and with determinations
that compare the X-ray background with the local black hole mass function,
leading to a luminosity-function-dependent value for the radiative
efficiency specifically of luminous AGN of
(Marconi et al. 2004).
Note that this could
differ from the average efficiency for all black holes if a proportion
were growing by radiatively-inefficient accretion: taken literally the
agreement with the PCE model with
implies that such inefficient growth does not dominate
the overall growth of black holes at
,
although some "ADAF'' contribution is allowed. However,
the deduced value of
is degenerate with
the rather uncertain
values of the bolometric correction and black hole mass density
and with the value of
.
The local
black hole mass density is uncertain by
30 percent and the value
of
currently has an uncertainty
10 percent
(Spergel et al. 2006). The uncertainty in the bolometric correction is not
well determined. Also, introduction of non-linear coupling between
black hole and dark halo growth (Sect. 4.1)
would also modify the deduced value of
by a factor
(Eq. (18)).
Hence we should not attach too much importance to its value, provided that it
is in the range expected for accretion onto supermassive black holes
(e.g.
0.06 for a Schwarzschild black hole). However,
a significantly lower value might imply a significant contribution
to black hole growth either from radiatively inefficient accretion
or from obscured growth.
The second case shown in Fig. 5
considers also the expected evolution in the black hole
mass function: if black holes do indeed grow coevally with their host galaxies
and halos we expect their mass function to show similar evolution. The lower
solid curve shows the result expected if the integrated black hole
mass density evolves the same way as massive dark halos with
.
This evolution was calculated by integrating the
Sheth & Tormen (2002) mass function above this limit
.
We find that imposing
an upper limit has little effect, but that the amount of turn-down
with increasing redshift in the
integrated luminosity density does depend on the lower halo mass limit, with
lower masses having less high-z turn-down,
as expected from standard CDM models.
Further, the integrated luminosity density
is dominated by the contribution from AGN around the "break'' in the
luminosity function, with
erg s-1: AGN of this
luminosity typically have black hole masses around
(Fine et al. 2006)
and QSOs of this luminosity in the 2dF QSO Redshift Survey (2QZ) have
dark halo masses independently estimated from their clustering bias of
(Croom et al. 2005).
Thus the higher-redshift evolution of the
luminosity density is entirely in accord with the expectation
that it is dominated by accretion onto black holes contained
within massive galaxy halos.
At z<0.5 the luminosity density falls off faster than predicted, although even at z=0 the predicted and observed values agree to a factor two. The implication is that although halos and black holes grow coevally at higher redshifts, in the more nearby universe black hole growth may have decoupled from halo growth. There is growing evidence that accretion rate in the low-redshift universe depends on host galaxy mass, with lower mass galaxies having black holes that at low redshift are preferentially growing with respect to their higher-mass counterparts (e.g. Heckman et al. 2004) - so-called "cosmic downsizing''. In detail, to explain cosmic downsizing we may need to recognise that in the late universe higher mass galaxies have slowed down their growth more than lower-mass galaxies, perhaps as a result of the environment they find themselves in at late epochs, or perhaps as a result of feedback. An alternative explanation is that the mean radiative efficiency may fall if AGN accreting at lower Eddington ratios switch to an ADAF mode of accretion (e.g. Beckert & Duschl 2002).
What we can learn from the PCE model however is that these processes are secondary to the primary cause of AGN evolution: the cosmic build-up of structure and its rate of change. We reiterate that the success of the PCE hypothesis does not mean that complex physics is not operating: it simply means that averaged across all massive galaxies, there is a mean net effect which corresponds to black holes growing at about the same rate as their host dark halos. Individually we expect black holes to go through periods of inactivity punctuated by bursts of accretion probably associated with merger events between galaxies, remembering that the galaxy mass function itself is also built up hierarchically by mergers between less massive progenitors. But just as the average effect on the galaxy mass function is a steady build-up of mass, so there is a steady build-up of mass in the black hole population.
A final consistency check is to ask whether the inferred local black hole mass
density could indeed have been built up during the process of luminous
accretion. This question has largely been answered by Marconi et al. (2004),
who have shown that the Ueda et al. (2003) luminosity function correctly
predicts not only the hard extragalactic X-ray background spectrum but
also the local black hole mass density, as already discussed above. The
implication is that not only are the estimates of bolometric luminosity
density consistent with the relic black hole population, but that there
cannot have been any substantially larger amount of "hidden'' (Compton-thick
or radiatively-inefficient) black hole growth, unless the observed
AGN have an uncomfortably high radiative efficiency (
).
![]() |
= | ![]() |
|
= | ![]() |
(20) |
So what significance can we attach to the mean Eddington ratio?
If
has a high value, it implies
that, on average, there is plenty of matter available to fuel luminous
accreting black holes at or close to their Eddington rate.
In fact, values significantly higher than unity would imply that there
is actually too much accreting matter available: we should expect
that black hole growth by accretion would be limited to the Salpeter rate
but that overall growth by mergers might be important.
Conversely, a low value implies that, on average, there is
insufficient new material accreting onto massive galaxies to maintain
Eddington-limited black hole growth.
Figure 6 shows
as a function of redshift for a flat,
CDM cosmology,
assuming
,
H0 = 73 km s-1 Mpc-1,
,
,
radiative efficiency
and without non-linear modification (Sect. 4.1).
![]() |
Figure 6:
Ratio of mean accretion rate to the limiting Eddington rate,
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
The result plotted is remarkable in two respects. As already
discussed,
depends only weakly
on the mass of dark matter
halo and on the shape of the dark matter power spectrum,
with no dependence on the
fraction of a halo's total mass (dark matter plus baryons), that ends
up in the black hole. Even the dependence on cosmology is weak, with
a near power-law dependence on 1+z with exponent
at z<1 for the
CDM
cosmology compared with n = 2.5 for Einstein-de Sitter.
Second, in principle there was no reason why the growth of dark matter
halos should have any interesting relationship to the Eddington limit,
except that both are governed by gravity. Yet it turns out
that at low redshifts the mean ratio
is consistent with the range of
observationally determined
values
(determined as the ratio of bolometric luminosity to
estimated Eddington luminosity)
for galaxy samples with a mean value
(Ho 2004) as discussed below.
At moderate redshifts the mean ratio increases to a value
at
,
implying that
at that epoch every galaxy has an actively accreting nuclear
black hole (although typically with lower mass than today's black holes).
The above analysis has only enabled us to calculate the mean evolution
in luminosity density and
in
.
Since accretion events are discrete, we expect
there to be a distribution of
values at any given cosmic
epoch. In principle one could either attempt to calculate that
distribution from EPS theory (e.g. Cole et al. 2000) or
from numerical simulation (e.g. Hopkins et al. 2006; Di Matteo et al. 2005),
or one could look at the
observational constraints on the distribution. Ho (2004) has
estimated the distribution of
for Palomar galaxies, with
bolometric luminosities estimated from H
and black hole mass
estimated from applying the Tremaine et al. (2002) relation to measured
stellar velocity dispersions. The distribution at essentially
zero redshift is broad, with a mean
,
and although there
are large uncertainties associated with the determination of
the agreement to a factor about 2 with the value
obtained at z=0 for
(shown in Fig. 6) is encouraging. The
Ho (2004) values have a large dispersion, which results in there
being a small number of galaxies with
that are
visible as AGN (Onken et al. 2004), and the vast majority of galaxies
accreting at rates closer to
.
We again emphasise that the mean value
calculated here is the mean for all galaxies and is not the mean
value expected for samples of AGN. By definition AGN are selected because
they are actively accreting and they must therefore be biased to higher
values than the mean. Using the results from this
paper to predict the distribution and evolution of AGN requires
knowledge of or assumptions about the distribution of
and will be discussed
further by Babic et al. (in preparation).
As seen in Sect. 4.2, the absolute value and the evolution
of either
the observed space or luminosity density appears to be
a combination of two competing but inter-related effects:
the build-up of massive halos and hence black holes with decreasing
redshift, and the decreasing accretion rate with decreasing redshift
calculated in this paper. Croom et al. (2005) have shown that luminous
QSOs inhabit dark matter halos of mass
:
at low redshifts (z<1) the halo mass function, and hence the black hole mass
function, is changing little at this mass, and AGN evolution is
dominated by the evolution in
.
In this case we expect the
evolution of the population to be dominated by apparent luminosity
evolution: on average black holes at lower redshift accrete at a lower
rate than black holes at higher redshift. Broadly speaking, this
is what is observed (Boyle et al. 2000), although it's not the whole
story (Zheng et al. 2004; Barger et al. 2005; Steffen et al. 2003; Ueda et al. 2003; Cowie et al. 2003). At higher redshifts (z>2)
the growth of the black hole mass function is significant and
probably dominates the evolution, causing a decline in space
density with redshift (Fig. 5).
At
approaches the values found for the
most luminous QSOs at
lower z: at this epoch in cosmic history the average dark matter
halo is accreting at a rate sufficient to supply close to the
Eddington luminosity of its nuclear black hole: the majority of
galaxies with such a black hole would be active. At higher redshifts
still the rate of accretion of matter onto dark halos exceeds that
required to supply the Eddington rate. In this case it is unlikely
that super-Eddington accretion onto black holes would occur: it is
more likely that nuclear outflows would limit the accretion process
of individual black holes to
about the Eddington rate and it is also possible that the
black-hole/bulge relation may be built up during this phase of galaxy
and black hole growth (King 2003; King & Pounds 2003).
Numerical simulations have also been used to try to understand the link between black hole growth and galaxy mergers, and in particular to understand the role of feedback in forming the well-defined relation between galaxy velocity dispersion and black hole mass (e.g. Hopkins et al. 2005; Di Matteo et al. 2005; Hopkins et al. 2006; Robertson et al. 2006). These approaches are directly complementary to the question addressed in this paper, namely to understand analytically the effect that the cosmic slowdown in halo growth has on the typical black hole accretion rate. It may be that combining the two approaches will finally lead us to a detailed understanding of AGN evolution.
Acknowledgements
We are grateful to Y. Ueda for supplying data from Ueda et al. (2003). A.B. acknowledges support from the Clarendon Fund.
In this section we discuss the calculation of
and approximations
to it for cosmologies with a cosmological constant (the analysis could be extended
to include more general dark energy cosmologies following the approach of
Percival (2005) but this is beyond the scope of this appendix).
We consider quantities that evolve as a
function of scale factor
a=(1+z)-1. If no dependence is quoted, the quantities
should be assumed to be calculated at a=1 (z=0).
The critical overdensity for collapse
in the
spherical top-hat collapse model is the linear overdensity, extrapolated
to the present day, that leads to the collapse
of a homogeneous spherical region
to a singularity at scale factor a.
This is used
in EPS theory to link overdensities with their predicted collapse
times. However, two alternative formalisms are often
considered for the collapse of perturbations in EPS theory.
![]() |
(A.1) |
![]() |
(A.2) | ||
![]() |
(A.3) |
![]() |
Figure A.2:
The percentage error in approximating the time
derivative of
![]() ![]() |
The first picture of EPS theory as corresponding to a growing field
does not follow from the spherical top-hat collapse model. However,
Fig. A.1 shows that the evolution of
along
lines of evolving (
)
is small. In the
remainder of this appendix we consider the error introduced in
by assuming that the evolution in
is
governed solely by the linear growth factor. The linear growth factor
D(a) is given by