Physical Foundations of Cosmology

64 Propagation of light and horizons

2.5.2Luminosity–redshiftrelation

A second method of recovering the expansion history is with the help of the
luminosity–redshift relation. Let us consider a source of radiation with total lum-
inosity (energy per unit time)Llocated at comoving distanceχemfrom us. The
total energy released by the source at timetemwithin a conformal time intervalη
is equal to

Eem=Ltem(η)=La(tem)η. (2.74)

All of the emitted photons are located within a shell of constant conformal width
χ=η. The radius of this shell grows with time and the frequencies of the
photons are redshifted. Therefore, when these photons reach the observer at time
t 0 ,the total energy within the shell is

Eobs=Eem

a(tem)

a 0

=L

a^2 (tem)

a 0

η. (2.75)

At this moment, the shell has surface area

Ssh(t 0 )= 4 πa 022 (χem)

and physical width

lsh=a 0 χ=a 0 η.

The shell passes the observer’s position over a time interval (measured by the
observer)tsh=lsh=a 0 η.Therefore, the measured bolometric flux (energy
per unit area per unit time) is equal to

F≡

Eobs

Ssh(t 0 )tsh

=

L

4 π^2 (χem)

a^2 (tem)

a^40

(2.76)

or, as a function of redshift,

F=

L

4 πa^202 (χem(z))( 1 +z)^2

. (2.77)

Hereχem(z)is given by (2.65). Instead ofF,astronomers often use the apparent
(bolometric) magnitude,mbol, defined as

mbol(z)≡− 2 ,5 log 10 F=5 log 10 ( 1 +z)+5 log 10 ( (χem(z)))+const, (2.78)

where const isz-independent.
Forz 1 ,we find that, irrespective of the spatial curvature and matter compo-
sition of the universe,

mbol(z)=5 log 10 z+

2. 5

ln 10

( 1 −q 0 )z+O

(

z^2

)

+const, (2.79)