Nearby Supernova Factory: Étalonnage des données de SNIFS et ...

tel-00372504, version 1 - 1 Apr 2009
CHAPTER 1. BIG BANG COSMOLOGY
1.3.5 Cosmological observables
All the introduced cosmological parameters depend on a certain number of observables, of
which we already introduced the redshift. We will now introduce another one: distances.
Proper distance
In the comoving frame at which we are at the origin, the distance from us to an object is not
an observable. That’s because a distant object can only be observed by the light it emitted at
an earlier time, where the universe had a different scale factor, none of which observable either.
We need then to define a relationship between the physical distance and the redshift, as a
function of the dynamics and the contents of the universe.
The proper distance dP is defined so that 4πd2 P is the area of the sphere over which light from
a far emitting source spreads in the time it travels to us. It can be derived using the metric for
a null geodesic
dP
0
dr
√
1 − kr2 =
t0 dt
0
′
a =
z
0
dz ′
H(z ′ ) =
1
a
Evaluating the left hand side integral and using (1.37) we get
dP =
with Ωk = Ω0 − 1 (see footnote 7),
and
Z(z ′ ) =
H0
1
|Ωk| X
|Ωk|
z
Z(z
0
′ )dz ′
⎧
⎨ sin x if k > 0
X [x] = x
⎩
sinh x
if k = 0
if k < 0
da ′
a ′2 H(a ′ )
. (1.39)
, (1.40)
, (1.41)
Ωr(1 + z ′ ) 4 + Ωm(1 + z ′ ) 3 + Ωx(1 + z ′ ) 3(1+wx) + Ωk(1 + z ′ ) 2 − 1
2
If we now have k ∼ 0, (1.40) reduces to (cf. Fig. 1.4)
dP = 1
Luminosity distance
H0
z
0
. (1.42)
dz ′
Ωr(1 + z ′ ) 4 + Ωm(1 + z ′ ) 3 + Ωx(1 + z ′ . (1.43)
) 3(1+wx)
The main method to calculate distances to any stellar objects is to estimate its true luminosity
and compare that to the observed flux (inversely proportional to the square distance).
The measured flux f is connected to the intrinsic luminosity L by
f = L
4πd 2 L
, (1.44)
where dL is the luminosity distance. In an expanding universe, this relation does not hold: the
observed flux is reduced by (1+z) 2 due to the cosmological redshift. From (1.13) we see that one
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