Ch.6
Friedmann equations
metric of Robertson-Walker
basic cosmological formulae
at present time:
critical density density parameter
basic cosmological formulaesecond Friedmann equation
second equation is depending on first one, because andare also related through energy conservation equation:for non-relativistic matter constituting the present Universe:
at present epoch:
deceleration parameter:
and also:
density, deceleration, and curvature parameters are inter-related: models depend on 1 parameter
cosmological constant
dark energy or vacuum energy
concordance
EdS
closed
open
Concordance Model ~
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
for q0<1/2 with other substitutions the same expression is found
Mattig 1958
luminosity distance
valid forq0 > 0
for q0=0 we can expand the square root for small values of q0z
q0=1/2 (Einstein - de Sitter):
(this is also an approximation for small z and generic q0 values)q0=0:
there is also an alternative formula by Terrell (1977), exact, valid for every
luminosity distance,
start again from Friedmann equations, with -term
[ ]
in this case, the integral must be computed numerically
luminosity distance,
[ ]
luminosity distance is needed to compute the luminosity of the source
where A(z) is defined as dimensionless luminosity distance, in units of c/Ho:
for example, in the optical band:
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 K-correction
luminosity distance,
z
A(z)
in units of c/H0
luminosity distance
look-back time
we need a z-t relation, we write in differential form:
for q0=0 or q0=1/2 it integrates trivially
time elapsed from emission to observation
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: number of quasars per square degree, function of F and z
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
colors
low resolution slitless spectroscopyX-ray emissionvariability
absence of proper motionIR luminosity
only concerns radio-loud quasarsid.
UV excess, later multi-band non-stellar colormany objects together, identified through em. linesproperty shared by ~all AGNsid., requires repeated measures, function of z and L
countscount all the sources down to a given limit flux Seuclidean case: flat and static Universepopulation of sources with same luminosity L
number of sourcesper square degree
uniform density:
logN
logS
-3/2
counts
in the optical band magnitudes are often useddensity 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 populationotherwise, if slope is steeper orthis is an indication that density increases with r
quasar counts in various bands
radioRyle 1968
opticalKoo 1986
X-rayBoyle 1993
0.85
-1.6
-1.7
Eddington effect
differential counts A(m) andcumulative counts N(m)
effect of measurement errors near the limit flux
Gaussian random errors around true value m ’convolution
because of errorsand slope, measured count is higher than true number
to solve for A(m) we make a Taylor expansion
Eddington effect
if measurement errors are small
but if
consider counts with slope kk=0.6 for uniform euclidean case
[ m ’ -> m ]
K-correction
z=0
z>0
we know the relation between bolometric fluxes and luminosities
for monochromatic fluxes and luminosities, we must take into account how frequency transforms
thus relation between flux and luminosity becomesfactor (1+z) accounts for change of frequency interval
for power-law spectra, we can compute the emitted spectrum at and obtain a specific expression
this holds in general
figure shows two effects: (a) displacement along the spectrum (b) variation of frequency interval
(b)
(a)
radiation observed at is emitted at
K-correction
usually the opposite is done: starting from measured flux, luminosity is determinedin terms of magnitudes, factor is inserted as follows:
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 isalmost totally suppressed.this can change the probability ofdetecting a high-z quasar, comparedto 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 Surveyhttp://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
use of sequences of “template” model SEDs for various classes of astronomical objects
comparison of measured photometry with “template” computed photometry
classificationdetermination of a “photometric” redshift
limit magnitude depends on the band, e.g. 25.7 in B
some selected fields,e.g. CDFS
telescope: ESO 2.2m
comparison and calibration with spectroscopic redshifts for reference sources
selection of AGN-candidates
spettroscopical confirmation
effect of the emission lines
wavy shape of the lines of limit magnitude indicates the effect of emission lines (Cavaliere Giallongo Vagnetti 1989)
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
B=19
.8
19.2
18.2
5
UV-excess vanishes beyond z~2, due to absorption by Lyα forest
Lyα
CIV
CIII]
MgII
effect of the emission lines
U B V R I
spectra of 8000 quasars from SDSS showing position and intensity of main emission lines as a function of redshift
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: variabilitymagnitude variation must be higher than photometric error
Bershady Trevese Kron 1998Trevese et al 1994
objects with high proper motion are excluded
efficient technique also for extended objects (galaxies with low luminosity variable nuclei)
non variable objects
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”
quasars galaxies starsCOMBO-17:
synergic AGN selection by variability in SN surveysSTRESS: Southern inTermediate Redshift ESO Supernova Searchmonitoring 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)
spettroscopical follow-up(Boutsia et al 2009)
quasar
NELGgalaxy
location of stars
location of galaxies
select AGNs, specially with extended image, which would not be found on the basis of color
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
surface density of sources
Euclidean (cumulative) counts
Euclidean differential counts
( ~ S-
5/2 )
differential counts normalized to Euclidean
hypothesis: constant comoving density
logN-logS
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
example:
z
V/Vmax testor luminosity-volume test
Euclidean case
for each source, determine the maximum volume within which it could be detected, for given LV is the volume limited by the spherical surface where the source lies
volumes V are uniformly distributed between 0 and Vmax
n(r)=n0: uniformly distributed sources
source
V Vmax
Vmax/2
if sources are uniformly distributed, half are expected to be found within a volume Vmax/2, and half beyond this
V/Vmax
statistic uncertainty
V/Vmax cosmological case
compute absolute magnitude
element of comoving volumevolume 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)
solve for zmax for which a source with absolute magnitude M would be observed at mlim
Vmin
V/Vmax results
high z: trend inverts
luminosity function
interstellar absorpton
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 intervalif the sample is flux-limitedeach quasar must be weighted with the inverse of the available volume
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)
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
M
log
z
mlim
mlim
mlim
luminosity function
extrapolation ofquasars at z=0
Seyfert 1
power-law luminosity evolution (LE), e.g. Boyle et al 1991:
>102
exponential evolution (also LE), e.g. Cavaliere et al 1985:
look-back time
some possible evolutionary forms, up to z~2:
DE
LE
luminosity function
there are the two classical models of density evolution and luminosity evolution:
DE: density decreases with t L~ constLE: 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 LEinstead, beyond z~3 LF decreases, probably because quasars are formingcontinuity equation (Cavaliere et al 1971, 1983):
considers quasar population as a fluid in the unidimensional space of luminosities
change of individual QSOs (LE)
source function: birth and death of quasars (DE)
luminosity function
Cavaliere et al 1985
Croom et al 20042dF Anglo Australian Telescope
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
position of maximum depends on luminosity: AGNs with lower LX have maximum density at lower z
cosmic downsizing
Hasinger et al 2005
La Franca et al 2003
Ueda et al 2003
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
X-ray surveys
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
cosmic downsizing
galaxies
optical AGN surveys
Wolf et al. 2003 Bongiorno et al 2007
earlier studies favored a maximum independent on z.recently, downsizing has also been observed in the optical band
cosmic downsizingHopkins 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)