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Demography
how many AGN in the sky?
- number counts
of normal galaxies
radio sources
optically selected AGN
X-ray selected AGN
how many AGN per cubic Mpc?
- Luminosity functions and their evolution
normal galaxies
optically selected AGN
X-ray selected AGN
QSO: probes of high z Universe
- Supermassive black hole volume density
Number counts
Flux limited sample: all sources in a given region of the sky with flux > than
some detection limit Flim.
• Consider a population of objects with the same L
• Assume Euclidean space
n(r)  space density; dN(r)  n(r)dV  n(r)r2 drd total number of sources
1/ 2
dN(r)
L
 n(r)r2 dr surface density; F 
Flux;
2
d
4 r
N(Flim ) 

dN
F  Flim  
d

dN
r  rmax  
d
 L 
rmax  

4 Flim 
F  Flim
rmax
 n(r)r dr
2
0
T otal number of sources per unit solid angle (cumulative distribut ion)
Uniform density of objects n(r) n 0
3/2


r
n
L
N(Flim )  n 0
 0 

3
3 4Flim 
3
max
 n L3 / 2  3
0
logN(Flim )  log
3 4 3 / 2 
 2 logFlim     1.5
   
Number counts
Test of evolution of a source population (e.g. radio sources). Distances of
individual sources are not required, just fluxes or magnitudes:
the number of objects increases by a factor of 100.6=4 with each magnitude.
So, for a constant space density, 80% of the sample will be within 1 mag
from the survey detection limit.
m  2.5 logFlim  so logFlim   -0.4m
3
 logFlim   0.6m  log N(m ) 0.6m
2
If the sources have some distribution in L:
n(r,L)drdL  n(r)(L)drdL
(L)dL  Luminosity Function
N(r) 
rmax (L)
 
0
n0
3/ 2
n(r,L)r drdL  4Flim   L3/ 2(L)dL
3
2
Problems with the derivation of the number counts
• Completeness of the samples.
• Eddington bias: random error on mag measurements can alter the number
counts. Since the logN-logFlim are steep, there are more sources at faint
fluxes, so random errors tend to increase the differential number counts.
If the tipical error is of 0.3 mag near the flux limit, than the correction
is 15%.
• Variability.
• Internal absorption affects “color” selection.
• SED, ‘K-correction’, redshift dependence of the flux (magnitude).
Radio sources number counts
Galaxy number counts
Optically selected AGN number counts
Z<2.2
B=22.5 100 deg-2
B=19.5 10 deg-2
z>2.2
B=22.5 50 deg-2
B=19.5 1 deg-2
B-R=0.5
X-ray AGN number counts
<X/O> OUV sel. AGN=0.3
R=22 ==> 310-15 1000deg-2
R=19 ==> 510-14 25deg-2
The surface density of X-ray selected AGN
is 2-10 times higher than OUV selected AGN
The V/Vmax test
Marteen Schmidt (1968) developed a test for evolution not sensitive to
the completeness of the sample.
Suppose we detect a source of luminosity L and flux F >Flim at a distance
r in Euclidean space:
 L 1/ 2
r  

4 F 
 L 
the same source could have been detected at a distance
rmax  

4

F

lim 
1/ 2
3
4 r 3
4 rmax
So we can define 2 spherical volumes
: V
; Vmax 
3
3
If we consider a sample of sources distributed uniformly, we expect that
half will be found in the inner half of the volume Vmax and half in the outer
half. So, on average, we expect V/Vmax=0.5

The V/Vmax test
V 


rmax
 4 r / 3n(r)r drd
3
2
0
rmax
 


n(r)r2 drd
0
6
3
4  rmax
/ 6 4  rmax


so :
3
3 rmax / 3
3 2
4 n 0
3
rmax

r 5 dr
0
rmax
n0

r 2 dr
0
V
 0.5
Vmax
In an expanding Universe the luminosity distance must be used in place
of r and rmax and the constant density assumption becomes one of
constant density per unit comuving volume .
N
Vi ( z )
V

Vmax
i 1 Vi ( z max )
Luminosity function
In most samples of AGN <V/Vmax> > 0.5. This means that the luminosity
function cannot be computed from a sample of AGN regardless of their z.
Rather we need to consider restricted z bins.
If thesources are drawn froma volumelimitedsample:
( L)l  
1
Vmax

NL
Vmax
More often sources are drawn from flux-limited samples, and the volume
surveyed is a function of the Luminosity L. Therefore, we need to account
for the fact that more luminous objects can be detected at larger
distances and are thus over-represented in flux limited samples. This is
done by weighting each source by the reciprocal of the volume over which
it could have been found:
1
( L, z )dL  
i Vi ( z max )
Luminosity function
1/Vmax method or
maximun likelihood method
:
?
N
i1
(Li )dzdL
z2
? 
j
j

z1
dV
dz
dz
j
L lim
(z)
 (L)dL

Galaxy luminosity functions
B band rest frame
OUV selected AGN luminosity function
AAT 2dF survey
OUV selected AGN luminosity function
-3.5, -1.5
k3.5
OUV selected AGN luminosity function
 L   L   1 dL
(L,z)dL         

L (z)  L (z)  
 L (z)
L(z)  L0 (1 z) k
OUV selected AGN LF
SDSS survey
X-ray selected AGN luminosity functions
luminosity dependent density evolution
2-10 keV AGN luminosity function models
2-10keV
0.5-2keV
LDDE with variable absorbed AGN fraction
La Franca et al. 2005
Comparison with HC models
The cosmic backgrounds energy densities
Assume that the intrinsic
spectrum of the sources
making the CXB has E=1
I0=9.810-8 erg/cm2/s/sr
’=4I0/c
Optical (and soft X-ray) surveys gives values 2-3 times lower
than those obtained from the CXB
(and of the F.&M. and G. et al. estimates)
Black hole mass density
A ~ 5x1039 erg s-1Mpc-3
A (1-) LBol
.
BH ~ ——————
 c2
LX
=0.1
.
LBol/LX=40
-5 M Yr-1 Mpc-3
~
3x10
BH
Θ
BH ~ 4x105 MΘ Mpc-3
BH growth
LE 
4GMmp c
 1.3 1046 M 8 ergs/s
T
«
L = Mc 2

L
LE
«
L
M E  E2
c
« (1 ) LE (1 ) M
dM
 (1 ) M 
=
dt
c 2
 E
LE
1

Mc 2  E
Mc 2
T c
cm 2cm s2 g
E 


 0.45Gyr
3
LE
4 Gm p
cm g s
Mc 2
 Salp 
 4.5 107 yr if   0.1
LE
 t 
(1 )  t 
M(t)  M(0)exp

 M(0)exp



t

E 
 growth 
z  6  969Myr
z(0) = 20  186Myr
M(t)=150 exp(15.66) 109 MSun
 growth 
t = 763Myr
 E 
 0.05Gyr if   0.1,   1
(1 )
M(0)=150Msun
CXB and SMBH census
Two seminal results:
1. The discovery of SMBH in the
most local bulges; tight
correlation between MBH and
bulge properties.
2. The BH mass density obtained
integrating the AGN L.-F. and
the CXB  that obtained from
local bulges
 most BH mass accreted during
luminous AGN phases!
Most bulges passed a phase of activity:
Complete SMBH census and full
understanding of AGN evolution
to understand galaxy evolution
Local SuperMassive Black Hole
AGN are powered by accretion on a SuperMassive Black Hole of 106-1010M,
Thus SMBH should exist in the nuclei of all galaxies that have experimented
a violently active phase.
Are all local SMBH relics of AGN activity?
Do other mechanisms, as merging, play a role?
Is possible to answer this question comparing the local BH mass
function with that of AGN
There are strong correlations between the BH mass and host galaxy
properties: bulge luminosity and mass and central stellar velocity
dispersion.
These correlations can be used to estimate the mass function of local BH
and thus their total mass density BH in the local Universe.
The mass function of local BH
-(x)dx
number of galaxies per unit of comoving volume with observable x
between x and x + dx
-logMBH = a + blogx
-(MBH)
log linear correlation between the BH mass and the
observable x
intrinsic dispersion; it is similar for all the correlations is one refers
only to galaxies with secure BH detection
-P(logMBH| logx)=(2)-1/2exp-[0.5(logMBH-a-blogx/(MBH))2]
probability that the MBH is between logMBH and logMBH + dlogMBH for a given logx
assuming a normal distribution
-(MBH, x) dMBH dx=[P(logMBH|logx)/MBHlog10] dMBH (x)dx
Number of BH with mass between MBH and MBH+dMBH, and observable between x
and x + dx
Then the local BH mass function is …
and the total mass density of local BH is
if the observable x is the bulge luminosity L
It is important to test that the MBH- and MBH-Lbul correlation derived
from a selected sample are consistent.
The selected sample is a SDSS sample of 9000 early type galaxy for
which it is possible to determine independently the velocity and
luminosity functions.
BHMF from the MBH-s correlation
The MBH- correlation from a group of selected early type of galaxy with
secure determination of BH mass is:
log MBH=(8.300.07)+(4.110.33)(log-2.3)
The assumed velocity function is that by Sheth et al. 2003 for early type
galaxy.
1000 Montecarlo realization of the BH mass function were computed by
randomly varying the input parameters .
These parameters are assumed normally distributed, and their 1
uncertainties are given by their measurement errors.
Two BHMF with (MBH)=0 and (MBH)=0.3.
BHMF from the MBH-Lbul correlation
The MBH-Lbul correlation from the group of selected early type galaxy with
secure determination of BH mass in the K band is:
log MBH=(8.210.07)+(1.130.12)(logLK,bul-10.9)
The luminosity function is that by Bernardi et al. 2003.
In the case of S0 galaxy the bulge luminosity has to be corrected for a
factor m, and is related to the total luminosity by:
bulge(m)=fS0(m-m)/(fE+fS0)
fE~0.1, fS0~0.2
The correction factor m is few dependent on the photometrical band,
thus in the computation of the bulge luminosity function is possible to
assume the B band.
The use of the MBH- correlation is more secure because it has not to be corrected
for the bulge fraction, but it is more difficult to measure
The BHMFs derived from the two
correlations
• The effect of a dispersion in
the correlation is that to
softening the decrease of the
BHMF at high mass thus
increasing the total density
•The use of the same
intrinsic dispersion provide
consistently BHMF’s with
the same mass densities BH
The BHMF for Early Type Galaxies
Can the use of luminosity functions from different galaxy survey and
photometric band affect the determination of BHMF in early type galaxy?
•Bernardi et al.: SDSS (3500-9000 A)
sample of 9000 early type galaxies;
•Marzke et al.: CfA survey (B(0)≤14.5);
the luminosity function is for morphological
type and the luminosities are in Zwicky
magnitudes
•Kochanek et al.: luminosity function in
the K band
•Nakamura et al.: SDSS sample;
luminosity function in the r* band
The different BHMF are in good agreement. Discrepancies arise only at low mass,
MBH<108M0, and are due to the extrapolation of the different functions adopted to
fit the data.
The BHMF for all Galaxy types
It has been derived using both the MBH-Lbul and the MBH- correlations.
All the BHMF’s and the BH densities BH are in agreement within the
errors.
The best estimate in the density
of local massive BH is
BH=4.6·105 M0 Mpc-3
About the 70% of this density is
given by early type galaxies.
The Mass Function of AGN Relics
I. The continuity Equation
The continuity equation links the relic BHMF N(M, t) to the AGN luminosity
function (L, z). AGN are powered by mass accretion on the central massive BH.
is the mean accretion rate
on the BH of mass M

The right term of the eq. containing the source function
is equal to zero.

All processes, such as merging, that can create or destroy
a BH are neglected; the
rate is very uncertain and strongly depends on the model adopted.
The intrinsic AGN luminosity is directly related to the BH accretion.
(L, t) dLog L=(M, t) N(M, t) dM
M=BH with mass M active at t
During the BH accretion the AGN luminosity is L=LEdd and the mass is
converted in energy with efficiency :
A fraction  of the mass is converted in energy and escape
from the BH
with constant  and 
Integration with initial condition [M, t(zs)]=1, i.e. all
the BH are active at the starting redshift zs.
Integration on the mass M gives the density of AGN relics:
with
II. The Bolometric Corrections
•AGN luminosity is determined in a limited energy band b; a suitable
bolometric correction, f bol, b=L/Lb is required.
•The observed luminosity is given by the integral of the observed
Spectral Energy Distribution. The IR radiation is reprocessed, thus a
correction is required.
•A template spectrum is constructed.
Optical-UV band: broken power-law
1=-0.44
1m-1300 A
2=-1.76
1200-500 A
(L~)
X-ray band: simple power law + reflection component
=1.9 , Eb=500 keV
•The spectrum, and thus the bolometric corrections, are assumed to be
independent on the redshift.
III. The Luminosity Function of AGN
AGN surveys are performed in limited spectral bands. The LF found in literature
describe only a fraction of the AGN population, i.e. a fraction of the local BHMF.
•Boyle et al. 2000: B band, all the population of red quasar is missed
•Soft X-ray (0.5-2 keV): all sources with important absorption are missed, NH>1022 cm-2
•Hard X-ray (2-10 keV): the most of AGNs; object with NH>1024 cm-2 are missed
The first two LFs function are in good agreement at high luminosity.
The third LF samples a larger fraction of AGN population at all luminosities.
Differential comoving energy density
•Objects with L>1012L0 provide ~ 50 %
of the total energy
•High and low luminosity objects have
similar redshift distributions
U=1.5·10-15 erg cm-2
BH=2.2·105 M0 Mpc-3
Integration of the continuity equation with =1, =0.1 and zs=3 give ….
•The hard X-ray LF gives the greater
number of AGN relics
• The number of relic at z=0 is greater
than the number of relic at zs.
•BH growth mainly at z ≤ zs. At higher
z BH have too little time to growth.
•The relic BHMF is substantially
independent on zs, if zs >2.5.
Solid line z=0, Dotted line z=zs, Dashed line only AGN with L >1012L0
The higher mass BH today growth during the quasar phase (L>1012L0).
If <1, the BH can accrete more mass. The number of relic with high
mass increases.
Comparison between local BH’s and AGN relics
No correction for missing
AGN population
Relic BHMF corrected using hard
X-ray LF and accounting for
Compton Thick AGN’s
EXCELLENT AGREEMENT!
•Local BH’s are AGN relics mainly grown during active phases of host galaxy
•Agreement obtained with ε = 0.1 and λ = 1
•Merging processes not important (at least for z < 3)
Constraints from the X-ray Background
It is possible to estimate the expected mass density of
relic BH’s from XRB
where
The average redshift of X-ray sources emitting the XRB is…
Perfectly consistent with the local estimate for  = 0.1
(consistency requires 0.07<  < 0.27)
Accretion efficiency and Eddington ratio
Consider the average square
deviation between the logarithms
of local and relic BHMF’s
Acceptance
region (k2 ≤ 1)
Best values (k2 = k2min):
 = 0.08,  = 0.5
 = 0.42, maximally
rotating Kerr BH
 = 0.054, non-rotating
Schwarzschild BH
k2 ≤ 0.72 which
corresponds to the
“canonical” case
(ε = 0.1, λ = 1)
BH’s should
be slowly
rotating
0.1 < λ < 1.7
BH’s grow during
luminous accretion
phases close to the
Eddington limit
Growth and accretion history of massive BH’s
Redshift dependence of ρBH
Cosmic BH’s accretion rate
BH’s accretion proportional to star
formation rate + feedback from AGN’s
explain the observed correlations MBH - 
and MBH - Lbul with host galaxies
High mass BH’s grow earlier than low
mass BH’s
The lifetime of active BH’s
The active time is
AGN’s which leave smaller
relic masses need longer
active phases
 ~ 1.5·108 yr MBH > 109 Mo
 ~ 4.5·108 yr MBH < 108 Mo
 ~ 109 yr
smaller  and 
Results in agreement with
upper limit of 109 yr set by
variability timescale
Summary
Local BHMF
Relic BHMF
Consistent and
in agreement
with XRB
constraints
•Local BH’s are AGN relics mainly grown during active phases of host
galaxy, in which accreting matter was converted into radiation with
efficiencies ε = 0.04 – 0.16 at a fraction λ = 0.1 – 1.7 of the Eddington
luminosity
•Merging processes are not important at redshift z < 3
•BH’s growth is anti-hierarchical
•The average total lifetime of AGN’s active phases ranges between 108
and 109 yr depending on the BH mass