Diego Saez-Gomez

General Relativity: Big Bang model
R0
0 = 3ä
a , (5)
(ä
Rj i =
a + 2ȧ2
a 2 + 2K )
a 2 δj i , (6)
(ä
R = 6
a + ȧ2
a 2 + K )
a 2 , (7)
FLRW cosmologies
where a dot denotes a derivative with respect to t.
Let us consider an ideal perfect fluid as the source of
the energy momentum tensor T µ ν .Inthiscasewehave
T ν µ =Diag(−ρ,p,p,p) , (8)
B. The evolution of the universe filled with a
where ρB. andThe p are evolution theperfect energy of the density fluid universe and the filledpressure
with a
density of the fluid, respectively. perfect Then fluid Eq. (4) gives the
FLRW equations in General two Relativity independent equations
Let us consider the evolution of the universe filled with
B. The Let usevolution consider (ȧ the ofevolution the universe of the universe filled filledawith
abarotropicperfectfluidwithanequationofstate
ion of homogeneity and isotropy of the
pproximately true on large scales. The
m homogeneity at early epochs played
oleinthedynamicalhistoryofourunil
density perturbations grew via gravy
into the structure we see today in
temperature anisotropies observed in
wave Background (CMB) are believed
from quantum fluctuations generated
nary stage in the early universe. See
3, 74, 75, 76] for details on density pered
by inflationary cosmology. In this
eview the main features of the homopic
cosmology necessary for the subseic
is given by [70, 77, 78, 79]
) 2
[
dr 2
]
abarotropicperfectfluidwithanequationofstate
H 2 ≡ perfect = fluid
8πGρ − K
)
1 − Kr 2 + r2 (dθ 2 +sin 2 θdφ 2 ) ,
a 3 a 2 , (9)
w = p/ρ ,
(1)
Ḣ = −4πG(p + ρ)+ K a 2 , (10) (17)
Let us consider the evolution of the universe filled with
abarotropicperfectfluidwithanequationofstate
where w is assumed to be constant. Then by solving the
where
factorwithcosmictimet. Thecoordire
known as comoving For a perfect coordinates. fluid with
Einstein H is the equations Hubble given parameter, in Eqs. ρ(9) and and p denote (10) with the K to-=0tal
energy density and pressure of all the species present
A wean inobtain
equation we obtainof state, w = p/ρ , (17)
the universe at a given epoch.
icle comes to rest in these coordinates.
The energy momentum tensor is conserved 2 by virtue of
a purely kinematic statement. In this where w is assumed toH be=
constant. 2 Then by solving the
the Bianchi identities, leading to the continuity equation
ics is associated with the scale factor–
H = 3(1 + w)(t − t 0 ) , (18)
Einstein equations given in Eqs. (9) and
2
(10) with K =0,
ations allow us to determine the scale we obtain
a(t) ∝ (t − t 0 )
3(1+w) , (19)
2
e matter content of the universe is spect
K in the metric (1) describes the ge-
2
a(t) ˙ρ +3H(ρ ρ ∝(t a −3(1+w) −+ p) t 0 ) =0. , (11)(20)
tial section of space time, with closed, Equation (11) can H be =
where t derived from Eqs. (9) and (10),
3(1 + w)(t − t
erses corresponding to K =+1, 0, −1, which means that two of Eqs. (9), (10) 0 ) , (18)
0 is constant. We note that the above solution
is valid for w ≠ −1. The radiation and (11) are independent.
Eliminating a(t) ∝ the (t −K/a t 2 2 dominated universe
0 )
3(1+w) term from , Eqs. (9) and
which contains an initial singularity corresponds (and toperhaps w = 1/3, a future whereas one..). the dust dominated (19)
enient to write the metric (1) in the (10), universe we obtain to w
ρ=0. ∝ aInthesecaseswehave
−3(1+w) , (20)
ä
t) [ dχ 2 + fK 2 (χ)(dθ2 +sin 2 θdφ 2 ) ] universe toRadiation w =0. a = Inthesecaseswehave
: − a(t) 4πG ∝ (ρ(t +3p) − t 0 ) . (12)
where t 3
, (2)
0 is constant. We note that 1/2 , ρ ∝ a above −4 , (21)
solution
is valid Hencefor thewaccelerated ≠Dust −1.
:
The
a(t)
expansion radiation
∝ (t − t 0 ) 2/3
occurs dominated
, ρ ∝ a −3
for +3p