Ch.6. Friedmann equations metric of Robertson-Walker basic cosmological formulae at present time:...

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)