Master Thesis

4.6 The FriedmannEquations
Summing over the time- and space-like connections one obtains:
Ω 0 ä
i =
a θ 0 ∧θ i
(4.60)
Ω i k+ ˙a2
j =
a2 θ i ∧θ j . (4.61)
One ends up at the Friedmann equations by reading off the components of the curvature tensor, contracting them to the
Ricci tensor and the scalar curvature, constructing the Einstein tensor and putting it into Einstein’s field equations (4.3):
G 11=G 22=G 33=−2 ä
G 00= 3 K+˙a2
a 2 = 8πGρ (4.62)
a
− K+˙a2
a 2 = 8πGp (4.63)
where the energy-stress tensor of the comoving rest frame of the fluid ( u µ =(1, 0, 0, 0) T , T µν = diag(ρ, p, p, p) ) has
been used. The other off-diagonal terms of Einstein’s field equation vanish.
Inserting (4.62) into (4.63) yields to the acceleration equation:
4.6.1 Densityevolution
ä=− 4πG
(ρ+ 3p)a (4.64)
3
Multiplying (4.62) by a 3 , differentiating with respect to t and inserting (4.63) at the left hand side one obtains:
d
d t (ρa3 )=−p d
d t (a3 ) ⇒ ˙ρ=−3(ρ+p) ˙a
, (4.65)
a
which is called cosmic energy conservation, because it is equivalent to the first law of thermodynamics:
dU= d(ρV)=d(ρa 3 )=dQ+dW= dW=−p dV=−p d a 3
(4.66)
Using (4.65) one can obtain how the density of different matter types changes with an increasing universe. The matter
types only differ by their ratio of pressure and density p/ρ=ω. Integrating (4.65) by usingωone ends up with:
The standard matter types are:
ρ(t)=ρ(t 0) a(t 0) 3(1+ω)
Dust The non-relativistic matter has no pressure p=0. (4.67) reads then:
a(t) 3(1+ω) =ρ(t 0)(1+z) 3(1+ω) . (4.67)
ρ M(t)=ρ M(t 0) a(t 0) 3
as it should be and is well known from everyday live too.
Radiation For the relativistic matter we have p=ρ/3. (4.67) reads then:
ρ R(t)=ρ R(t 0) a(t 0) 4
Vacuum The physical vacuum has a negative pressure: p=−ρ. (4.67) yields:
a(t) 3 =ρ M(t 0)(1+z) 3 , (4.68)
a(t) 4 =ρ R(t 0)(1+z) 4 . (4.69)
ρ Λ(t)=ρ Λ(t 0)=ρ 0(1+z) 0
(4.70)
The physical vacuum remains always constant! Some theories behind the standard model predict, thatωdoes not
remain constant at−1 for vacuums. In that case one has to go back and use (4.65) or (4.67).
Curvature The curvature itself is no matter, but it behaves like matter in the Friedmann equation (4.62):
ρ K(t)=ρ K(t 0) a(t 0) 2
The “pressure” one of this curvature density would then be:ω=−1/3 .
a(t) 2 =ρ K(t 0)(1+z) 2 . (4.71)
22 4.6 The FriedmannEquations