Transcript luminosity distance - INAF-Osservatorio Astronomico di Roma
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
Concordance Model ~ dark energy or vacuum energy concordanc e open 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 q 0 <1/2 with other substitutions the same expression is found
luminosity distance
valid for q 0 > 0 q 0 =1/2 (Einstein - de Sitter): for q 0 =0 we can expand the square root for small values of q 0 z q 0 =0: (this is also an approximation for small z and generic q 0 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/H o : 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
in units of c/H 0
A(z) z
look-back time
time elapsed from emission to observation we need a z-t relation, we write in differential form: for q 0 =0 or q 0 =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: 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/deg 2 at B=21 cf 1600 stars/deg 2 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 spectroscopy X-ray emission variability absence of proper motion IR luminosity only concerns radio-loud quasars id.
UV excess, later multi-band non-stellar color many objects together, identified through em. lines property shared by ~all AGNs id., requires repeated measures, function of z and L
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 10 0.6
~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=n o ): dependency on limit flux is still -3/2 this is expected for a uniform population otherwise, if slope is steeper or this is an indication that density increases with r
quasar counts in various bands
-1.7
0.85
-1.6
radio
Ryle 1968
optical
Koo 1986
X-ray
Boyle 1993
Eddington effect
effect of measurement errors near the limit flux because of errors and slope, measured count is higher than true number 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
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 thus relation between flux and luminosity becomes factor (1+z) accounts for change of frequency interval (a) for power-law spectra, we can compute the emitted spectrum at and obtain a specific expression (b) radiation observed at is emitted at this holds in general z>0 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 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 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 comparison of measured photometry with “ template ” computed photometry selection of AGN candidates classification determination of a “ photometric ” redshift spettroscopical confirmation 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) Trevese et al 1994 non variable objects 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 discarded as SNe can recovered and become useful as AGN candidates (Trevese et al 2008) COMBO-17: quasars galaxies stars spettroscopical follow up (Boutsia et al 2009) NELG quasar 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 q 0 =1/2 and α=0.7 counts are expected flatter than Euclidean. the same holds also for reasonable values of q o and α to fit the steep observed counts, it is needed a number of sources increasing with distance, and thus with redshift
V/V
max
test
or luminosity-volume test Euclidean case if sources are uniformly distributed, half are expected to be found within a source V max /2 volume V max /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)=n 0 : uniformly distributed sources volumes V are uniformly distributed between 0 and V max V V max
V/V
max
statistic uncertainty
V/V
max
cosmological case compute absolute magnitude solve for z max for which a source with absolute magnitude M would be observed at m lim element of comoving volume volume integral: compute for z ’ =z and for z ’ =z max for each quasar (i=1,2 ... N)
V/V
max
results
high z: trend inverts
luminosity function
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 M if the sample is volume-limited (all the quasars within the volume V max ) luminosity function (LF) is found by the count in each absolute magnitude interval if the sample is flux limited 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
luminosity function
Seyfert 1 extrapolation of quasars at z=0 >10 2 some possible evolutionary forms, up to 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 QSOs (LE) source function: birth and death of quasars (DE)
luminosity function
Cavaliere et al 1985 Croom et al 2004 2dF 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
cosmic downsizing
X-ray surveys position of maximum depends on luminosity: AGNs with lower L X have maximum density at lower z La Franca 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 Ueda et al 2003
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/L Edd < ~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)