Ch.6
basic cosmological formulae
metric of Robertson-Walker
Friedmann equations
at present time:
critical density
density parameter
basic cosmological formulae
second Friedmann
equation
second equation is depending on first one, because
and
are also related through energy conservation equation:
for non-relativistic matter constituting the present Universe:
and also:
at present epoch:
deceleration parameter:
density, deceleration, and curvature parameters are inter-related: models depend on 1
parameter
cosmological constant
dark energy or vacuum
energy
Concordance Model
~
open
concordanc
e
EdS
closed
Friedmann models,
Einstein-deSitter
Friedmann models,
{
{
Mattig formula
relation between radial coordinate and redshift of a given
source
depends on cosmological model
needed to compute luminosity distance
travel of photons from source position to observer:
Mattig formula
Mattig formula
Mattig 1958
for q0<1/2 with other substitutions the same expression is found
luminosity distance
valid for
q0 > 0
q0=1/2 (Einstein - de Sitter):
for q0=0 we can expand the square root for small values of
q0z
q0=0:
(this is also an approximation
for small z and generic q0
values)
there is also an alternative formula by Terrell (1977), exact, valid for
every
luminosity distance,
start again from Friedmann equations, with
term
-
[
]
luminosity distance,
[
in this case, the integral must be computed numerically
]
luminosity distance,
luminosity distance is needed to compute
the luminosity of the source
for example, in the optical band:
where A(z) is defined as
dimensionless luminosity distance, in
units of c/Ho:
these equations hold for bolometric
fluxes, luminosities and magnitudes.
for monochromatic magnitudes, or for
magnitudes in a given photometric
band, formula must be improved with Kcorrection
luminosity distance
in units of c/H0
A(z)
z
look-back time
time elapsed from emission to observation
we need a z-t relation, we write in differential
form:
for q0=0 or q0=1/2
it integrates
trivially
look-back time
http://burro.astr.cwru.edu/JavaLab/web/main.html
quasar surveys
quasars: probes of the history of the Universe
(i) properties of the quasar population as a function of redshift
(ii) cosmic time of the first appearence of quasars -> constraints on galaxy
formation
large quasar samples are needed, not affected by selection effects (unbiased)
measured quantity:
F and z
number of quasars per square degree, function of
luminosity function (LF)
number of quasars per unity luminosity
interval and per unity comoving volume
total spatial density
these counts are difficult because quasars are few and faint:
~40 quasars/deg2 at B=21 cf 1600 stars/deg2 at galactic poles
important is the adoption of selection criteria for the construction of samples of
candidate quasars (which are to be later spectroscopically confirmed)
main selection criteria
radio position
radio position + UV excess
only concerns radio-loud
quasars
id.
colors
UV excess, later multi-band non-stellar color
low resolution slitless
spectroscopy
X-ray emission
many objects together, identified through em.
lines
property shared by ~all AGNs
variability
id., requires repeated measures, function of z
and L
absence of proper motion
IR luminosity
counts
count all the sources down to a given limit flux S
euclidean case: flat and static Universe
population of sources with same luminosity L
number of sources
per square degree
uniform density:
logN
-3/2
logS
counts
in the optical band magnitudes are often used
density increases by 100.6~4 for each
magnitude: 80% of the sources lie within 1
mag from limit flux
if sources do not have all the same L, but assuming that luminosity
distribution is the same at each distance r, then we can separate
dependency on L and r, and we have (still assuming n=no):
dependency on limit flux is still -3/2
this is expected for a uniform population
otherwise, if slope is steeper
this is an indication that density increases with r
or
quasar counts in various bands
-1.6
-1.7
0.85
radio
Ryle 1968
optical
Koo 1986
X-ray
Boyle 1993
Eddington effect
effect of measurement errors near the limit flux
differential counts A(m)
and
cumulative counts N(m)
Gaussian random errors
around true value m’
convolution
to solve for A(m) we make a
Taylor expansion
because of errors
and slope, measured
count is higher than true
number
Eddington effect
[ m’ -> m ]
consider counts with slope k
k=0.6 for uniform euclidean case
if measurement errors are small
but if
K-correction
we know the relation between
bolometric fluxes and
luminosities
for monochromatic fluxes and
luminosities, we must take into
account how frequency transforms
radiation
observed at
is emitted at
thus relation between flux
and luminosity becomes
factor (1+z) accounts for
change of frequency interval
this holds in
general
(a)
z>0
(b)
for power-law spectra, we can
compute the emitted spectrum at
and obtain a specific expression
z=0
figure shows two effects: (a) displacement along the spectrum (b) variation of frequency
interval
K-correction
usually the opposite is done:
starting from measured flux,
luminosity is determined
in terms of magnitudes, factor
follows:
is inserted as
the expression for absolute magnitude becomes:
this for the power-law case,
otherwise it is used the more general form
with the choice of an appropriate SED shape
K-correction must be applied not only to AGNs, but also to
galaxies, and every other source at non negligible redshift
K-correction
K-corrections in UBV bands
computed for a realistic
spectrum, the average quasar
spectrum here shown (arbitrarily
translated in ordinate)
K-correction in B compared with
model power-law K-correction
problems and difficulties
euclidean counts:
we have assumed
has same shape everywhere. but it is not so:
quasar LF at z~0 is very different from what it was at z~2. this is a critical
problem for quasars, which span a wide L interval, so that at a given flux
they contribute to the counts for a large interval of distances
completeness: in principle, all sources with flux greater than the limit S must
be detected. probability of losing sources increases toward the limit flux,
mimicking the effect of a distribution decreasing with distance. completeness
tests are not that rigorous, usually only a comparison with previous surveys
is done. it is important to perform surveys with different selection criteria in
the same sky area to compensate merits and defects of the different
techniques
(e.g. Selected Area 57, color/proper motion/variability)
variability: as luminosities vary, sources near the limit flux can happen to be
above or below the detection threshold in different epochs. this alters the
counts similarly to Eddington effect, with the possible addition of a
dipendency of variability on L and z
problems and difficulties
prominence of emission lines: equivalent widths vary significantly among
different quasars. those surveys that rely critically on line prominence detect
more easily strong-lined objects, and can lose weak-lined ones. it is possible to
estimate and correct incompleteness if sensibility of the survey to EW can be
quantified, and EW distribution is approximately known
absorption lines: spectra of high redshift quasars show absorption lines due to
intervening matter along the line of sight, in particular at wavelengths below
(
-forest), where continuum is
almost totally suppressed.
this can change the probability of
detecting a high-z quasar, compared
to a non-absorbed quasar
internal absorption: dust either in the emission line regions, or in the disk of the
host galaxy. in rest-frame UV, extinction can be as high as ~0.8 mag, so
reducing detection probability for a quasar with z> ~2. or in a torus, as that
invoked for unified schemes, and this can completely remove obscured
quasars from traditional surveys. it’s the so-called quasar-2, for which
favorable bands are hard X-rays and IR
color selection
initially it was simply the UV-excess:
most famous survey of this kind is Palomar Bright
Quasar Survey by Schmidt and Green 1983, which
provided the PG (Palomar Green) quasar sample, 114
quasars at magnitude ~16 over ~10000 square
degrees
then this technique improved with the use of
more photometric bands to search, in a two- (or
many-) color diagram, objects with at least one
color index different from stars
e.g. Warren et al 1991
here, small circles are low-redshift quasars, and
big circles are high-redshift quasars
color selection
Koo Kron and Cudworth use U, J, F, N bands, and complement selection
with variability and proper motion criteria
color selection
Sloan Digital Sky Survey
http://www.sdss.org/
locations of stars (black)
and of extended sources
(orange) in two-color
diagrams within ugriz
system
color selection
it is possible to simulate quasar colors assuming an SED and
parametrically modeling emission lines. the tracks so found show color
change as a function of z, and possible intersection with the location of
stars. remedy is to add more photometric bands
Giallongo and Trevese 1990
multi-band color selection
COMBO-17 survey
http://www.mpia-hd.mpg.de/COMBO/combo_index.html
5 broad bands (~UBVRI)
+12 narrow bands
=17 bands in total
limit magnitude depends
on the band, e.g. 25.7 in
B
some selected fields,
e.g. CDFS
telescope: ESO 2.2m
use of sequences of
“template” model
SEDs for various
classes of
astronomical objects
classification
selection of AGNcandidates
spettroscopical
confirmation
comparison of
measured photometry
with “template”
computed photometry
determination of a
“photometric” redshift
comparison and
calibration with
spectroscopic
redshifts for reference
sources
effect of the emission lines
emission lines can increase quasar luminosity so that it
can become detectable (where it would be undetectable
for continuum only)
and/or, they can increase
UV-excess because of
K-correction, favoring
selection of a quasar if a
strong emission line is
present in the U band
UV-excess
vanishes beyond
z~2, due to
absorption by Lyα
forest
wavy shape of the lines of limit
magnitude indicates the effect of
emission lines (Cavaliere
Giallongo Vagnetti 1989)
effect of the emission lines
spectra of 8000 quasars from
SDSS showing position and
intensity of main emission lines as
a function of redshift
U
B
V
R
I
slitless spectroscopy
it consists in making the spectrum of a wide sky area with a dispersing element
in front of the telescope, an objective-prism, or a ”grism” (prism with one side
ruled as a grating)
useful for z> ~2 because Lyα and CIV are shifted in the optical
however, it depends not much on z, because of the wider observed band
compared to photometry
integration times are longer, compared to photometric measures, but the
advantage is that many spectra are simultaneously observed
problems:
- higher limit flux
- uncertain determination of the limit flux, affected by emission lines
- some redshift intervals with few lines
- strong-line objects favored (and low-luminosity objects because of Baldwin
effect)
other selection criteria: variability
magnitude variation must be higher than photometric
error
objects with high
proper motion
are excluded
efficient technique also for
extended objects (galaxies with
low luminosity variable nuclei)
non variable objects
Trevese et al 1994
Bershady Trevese Kron 1998
variability
•variability increases with redshift, so it is more probable to select high redshift objects
•probability increases also with sampling interval and with the number of observation
epochs
Green et al 2006, simulation for the Large Synoptic Survey Telescope, a telescope with
8.4m diameter to be used for imaging surveys in the time domain (www.lsst.org) in
project to be operating in 2020:
“Good probability of detection is achieved after only 2 epochs, and after 12 epochs in a
year, almost all the AGNs to i<24 will be detected as variable”
synergic AGN selection by variability in SN surveys
STRESS: Southern inTermediate Redshift ESO Supernova
Search
monitoring of Chandra deep field South (CDFS), 8 epochs in 3 years: variable objects
discarded as SNe can recovered and become useful as AGN candidates (Trevese et
al 2008)
COMBO-17:
quasars
galaxies
stars
spettroscopical followup
(Boutsia et al 2009)
quasar
NELG
galaxy
location of stars
select AGNs, specially with
extended image, which would
not be found on the basis of
color
location of galaxies
main quasar surveys and counts
Hartwick & Shade
1990
logN-logS test for a non-Euclidean Universe
K-correction
number of sources in the
volume between r and
r+dr
relation between geometric
distance and luminosity
distance
relation between comoving
radial coordinate r and redshift
volume element
(
)
logN-logS
hypothesis: constant comoving density
surface density of sources
Euclidean (cumulative) counts
Euclidean differential counts
differential counts
normalized to
Euclidean
( ~ S-5/2 )
logN-logS
example:
z
for q0=1/2 and α=0.7 counts are expected flatter than Euclidean. the same
holds also for reasonable values of qo and α
to fit the steep observed counts, it is needed a number of sources
increasing with distance, and thus with redshift
V/Vmax test
or luminosity-volume test
Euclidean case
source
if sources are uniformly distributed,
half are expected to be found within a
volume Vmax/2, and half beyond this
for each source,
determine the maximum
volume within which it
could be detected, for
given L
V is the volume limited by
the spherical surface
where the source lies
n(r)=n0: uniformly
distributed sources
volumes V are uniformly
distributed between 0 and Vmax
Vmax/2
V
Vmax
V/Vmax
statistic uncertainty
V/Vmax
cosmological case
compute absolute magnitude
solve for zmax for which a source with absolute
magnitude M would be observed at mlim
element of comoving volume
volume integral:
compute for z’=z and for
z’=zmax for each quasar
(i=1,2 ... N)
<V/Vmax>: if distribution is
uniform, it must 1/2
if there is also a lower limit to z
because of the selection criterion (e.g.
for slitless spectroscopy), then the
available volume is used (Avni and
Bahcall 1980)
test is efficient also in presence of multiple selection criteria: e.g. Vmax (R,O)
Vmin
V/Vmax results
high z:
trend
inverts
large samples are needed to count significant
numbers of sources in bins of M and z.
more than one sample is needed, otherwise a
ficticious M-z correlation would be found (most
objects lie near the limit magnitude)
interstellar absorpton
log z
luminosity function
M
if the sample is volume-limited (all the quasars
within the volume Vmax) luminosity function (LF)
is found by the count in each absolute magnitude
interval
if the sample is fluxlimited
each quasar must be weighted with
the inverse of the available volume
count also in z because LF depends strongly
on z
result is a LF with double-power-law shape
with a break for a particular value of L
Seyfert 1
luminosity function
extrapolation of
quasars at z=0
>102
some possible evolutionary forms, up to
z~2:
power-law
luminosity evolution (LE), e.g. Boyle et al
1991:
exponential evolution (also LE), e.g. Cavaliere et al
1985:
look-back time
luminosity function
DE
LE
there are the two classical models of density evolution and luminosity evolution:
DE:
density decreases with t L~ const
LE:
density ~ const
L decreases with t (i.e. with decreasing z)
quasars more numerous and/or more luminous in the past
up to z~2.5 data ~ agree with LE
instead, beyond z~3 LF decreases, probably because quasars are forming
continuity equation (Cavaliere et al 1971, 1983):
considers quasar population as a fluid in the
unidimensional space of luminosities
change of individual source function: birth
QSOs (LE)
and death of quasars
(DE)
luminosity function
Croom et al 2004
2dF Anglo Australian Telescope
Cavaliere et al 1985
space density of quasars
- space density of optically-selected quasars has a maximum at z~2-3
- in X-rays instead, position of maximum depends on luminosity
cosmic downsizing
X-ray surveys position of maximum depends on luminosity: AGNs
with lower LX have maximum density at lower z
La Franca et al 2003
Ueda et al 2003
Hasinger et al 2005
this behavior is called “AGN cosmic downsizing”, luminous AGNs and
quasars have a strong activity at high z, then turn off rapidly, low
luminosity AGNs are active in more recent epochs. it is a trend contrary
to hierarchical clustering, where small structures form first and cluster
later in larger structures
cosmic downsizing
galaxies
also for galaxies, there is evidence of “downsizing” (Cowie et al. 1996): massive galaxies are
characterized by a star formation rate with a maximum at high redshift, while galaxies of small mass are
typically younger systems
optical AGN surveys
earlier studies favored a maximum independent on z.
recently, downsizing has also been observed in the optical band
Wolf et al. 2003
Bongiorno et al 2007
cosmic downsizing
Hopkins et al 2007 compute models of the bolometric LF which fit data by a large number of surveys in
many different bands, and show downsizing in many of them
cosmic downsizing
two explanations:
1) SMBH downsizing : most massive BHs preferentially stop accreting at high z, while at
low z small mass BHs dominate (Heckman et al 2004, Merloni et al 2004, Barger et al
2005)
2) accretion rate downsizing : the average accretion rate decreases, and at low z L/LEdd <
~0.01 (Babic et al 2007, Fanidakis 2010)
consistent with a bimodality in the growth of BHs: at high z by merging, at low z by
stochastic slow accretion of cold gas (Hopkins and Hernquist 2006)