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

tel-00372504, version 1 - 1 Apr 2009
CHAPTER 1. BIG BANG COSMOLOGY
We can thus rewrite the Friedmann equation as
with (for the i components, radiation, matter and dark energy)
Ω − 1 = k
a2 , (1.29)
H2 Ω =
Ωi =
i
i
8πGρi
3H 2 . (1.30)
It is interesting to note the direct relation given by (1.29) between energy content and
curvature of the universe: Ω {> 1, < 1, = 1} gives us respectively closed, open or spatially flat
universes. Specifically stating the different contributions to the energy density today, we can
write the “cosmic sum” which determines the overall sign of the curvature 7
Deceleration parameter
Ω0 = Ωr + Ωm + Ωx = k
H 2 0
+ 1 . (1.31)
Looking now at the second Friedmann equation (1.6), we can define a new parameter, the
so-called deceleration 8 parameter
q0 ≡ − ä
a0H 2 0
= 1
2
i
= Ωr + Ωm
2
Ω0i (1 + 3wi) (1.32)
+ Ωx
2 (1 + 3wx) , (1.33)
which relates the dynamics of the universe with its contents. q0 = 0 would represent a static
universe, whether a positive or negative value means the expansion is decelerating or accelerating.
for the latter case to happen.
For a flat universe, wx has to be < − 1
3
Hubble parameter
The (time dependent) Hubble parameter already introduced in § 1.2.4 can be related to its
present day value (Hubble’s constant). Substituting (1.28) on (1.18) we obtain
ρi = ρcΩ0a −3(1+wi) , (1.34)
which applied with (1.27) and (1.31) on the Friedmann equation (1.5), gives us the a-dependent
Hubble parameter
H(a) = H0 Ω(a) + (1 − Ω0)a−2 , (1.35)
with
Ω(a) =
i
Ω0i a−3(1+wi)
= Ωra −4 + Ωma −3 + Ωxa −3(1+wx) . (1.36)
An equivalent redshift-dependent Hubble parameter can easily be found using (1.14)
H(z) = H0 Ω(z) + (1 − Ω0)(1 + z) 2 . (1.37)
7
Some authors also define a curvature “density” parameter, as Ωk ≡ − k
H2 0
, and thus Ω0 + Ωk = 1.
8 The reason for the name is historical, since it was believed that the universe should be decelerating.
14