MODERN COSMOLOGY

316 Highlights in modern observational cosmology

geometry (inverse square law). Observations (i.e. fluxes, luminosities) in a given
band[ν 1 ,ν 2 ]can be related to the rest-frame band through the computation of
theK-correction,Kz, which is essentially the ratio of fluxes in the rest-frame to
the observed (redshifted) band[( 1 +z)ν 1 ,ν 2 ( 1 +z)]. In optical astronomy the
magnitude system is used (m∼− 2 .5log(S)) so that (11.15) can be written as a
relation between the apparent(m)and absolute magnitude(M)of the object:

m=M+5log

(

DL

10 pc

)

+Kz. (11.16)

If the flux spectra density is a power law, i.e.fν∼ν−α(like most of the galaxies),
then one easily obtainsKz= 2. 5 (α− 1 )log( 1 +z). Suchatermcanaddupto
several magnitudes for early type (i.e. red) galaxies atz∼1.
A low redshift expansion of (11.16) leads to the simple formula (e.g. Sandage
1995):

m=5logz+ 1. 086 ( 1 −q 0 )z+5logcH 0 −^1 +M+ 25. (11.17)

This shows that if we can recognize a class of astrophysical sources as
‘standard candles’, by measuring the dimming of these sources over a wide
range of redshifts we can measure the deceleration parameter,q 0 , and eventually
separate
mand
. The application of this fundamental test to high redshifts
Type Ia supernovae has lead to spectacular results in recent years (e.g. Perlmutter
et al1999, Schmidtet al1998).

11.2.2.3 Number densities

One of the main goal of redshift surveys is to quantify the comoving volume
density of objects as a function of redshift. A frequently used quantity is therefore
thecomoving volume elementin the redshift interval,ztoz+dz, in the solid angle
d
, which follows directly from the FRW metric (11.1), (11.2):

dV=

dM^2

( 1 +
kc−^2 H 02 d^2 M)^1 /^2

d(dM)d
. (11.18)

Using equation (11.13), and defining the functionsE(z)andA(z)as

E(z)=

∫z

0

[
k( 1 +y)^2 +
m( 1 +y)^3 +
]−^1 /^2 dy≡

∫z

0

A(y)dy,

we have:

dV

d
dz

=(cH 0 −^1 )^3 A(z)|
k|−^1 S^2 {|
k|^1 /^2 E(z)} (11.19)

=cH 0 −^1 A(z)d^2 M≡Q(z,
m,
),