MODERN COSMOLOGY

320 Highlights in modern observational cosmology

the nearby universe, where curvature terms can be neglected, are characterized
by aEuclidean slopeof− 1 .5 (or 0.6 mag). In general, at large distances,
curvature effects (cf equations (11.13) and (11.18)) cause number counts to have
slopes always shallower than the Euclidean one. However, as we will see in
section 11.3.3, evolutionary effects (φ=φ(L,z)) can counteract such a natural
behaviour and produce counts steeper than 1.5.

11.2.3.2 Redshift distribution and number counts (general case)

We now have all the ingredients to compute the expected redshift distribution,
n(z), and number counts,n(> S), for an evolving population of sources with
LFφ(L,z). Typically, on the basis of the known local LF,φ(L, 0 ), one wants
to compare theobservedredshift distribution of sources with the one expected
on an empirical evolutionary scenario, or the one predicted by some theory of
structure formation. In general, there will be some degree of degeneracy between
evolutionary parameters and cosmological paramaters (
m,
) when matching
theoretical models with observational data.
WithQ(z,
m,
)given by equation (11.19) (or (11.20)), the number of
sources per unit solid angle and redshift, in the luminosity rangeLtoL+dL,is:

d^2 N

d
dz

φ(L)dL=Q(z,
m,
)

φ∗

L∗

(

L

L∗

)−α

e−L/L∗dL. (11.22)

We now change variable,y=L/L∗, and callL 1 andL 2 the minimum and
maximum luminosity of the source population (for example, a magnitude range
within which we want to compute the redshift distribution). Thus, the surface
density of sources, per unit redshift, observed down to the fluxScan be written
as:

dN(>S,z)

d
dz

=φ∗Q(z,
m,
)

∫y 2

y 1 (z)

y−αe−ydy (11.23)

=φ∗( 1 −α)Q(z,
m,
)[P( 1 −α,y 2 )−P( 1 −α,y 1 )],

wherePis the generalized-function,y 2 =L 2 /L∗,and

y 1 (z)=max

(

L 1

L∗

,

Lmin(S,z)

L∗

)

, Lmin(S,z)=S 4 πD^2 L(z)Kz. (11.24)

Lminis the rest-frame miminum luminosity detectable at redshiftz, at the limiting
fluxS(equation (11.15)).
The numerical integration of equations (11.23) and (11.24) can also include
an evolving LF, e.g.φ∗=φ∗(z),L∗=L∗(z). The result can be directly compared
with the observed redshift distribution of sources, i.e. the number of sources per
deg^2 , in each redshift bin. The number countsn(>S)are obtained by integrating
(11.23) over all redshifts.