Cosmic Acceleration and Dark Energy
19
2 The New Effective Equation of State Parameter
The fluid of universe has radiation, matter (both normal and dark), and dark energy in
it and let the pressure, inspired by the changing pressure parameter and the decreasing
acceleration of [8], be
P =
A − B
R 0
R
ρ c
2
(5)
where P = ω e f f ρc
2 .
R 0 is the scale factor of the universe at the shift from the phase of deceleration
to acceleration, ω e f f is the effective equation of state parameter of the total fluid
pressure of the universe, and A and B are two dimensionless constants.
Then, integrating d(R
3
ρ) + ω e f f ρd(R
3
) = 0
H =
1
R 1.5(1+A) exp
1.5B
R 0
R i
−
R 0
R
(6)
R i ∼ 10
26 is the scale factor of the universe at the end of cosmic inflation [2, 7] and
H the Hubble parameter when the scale factor is R.
And Friedmann equations are
H
2
−
8πG
3
ρ r + ρ m(n) + ρ m(d) + ρ d
= 0
( 7 )
2
¨
R
R
+ H
2
+ 8πG
A − B
R 0
R
ρ r + ρ m(n) + ρ m(d) + ρ d
= 0
( 8 )
giving
¨
R = −
4πG R
3
1 + 3
A − B
R 0
R
(ρ r + ρ m(n) + ρ m(d) + ρ d )
(9)
Friedmann equations with the equation of state and conservation of energy are quite
enough to talk about the dynamics of the universe.
3 Results
Since the present Hubble number H p ∼ 10
−18 , R p ∼ exp (B(
R 0
R i
−
R 0
R
))(H
12
1+A )
(Table 1).
Let A = −0.571 expecting R p ∼ 10
26 .
ω e f f = −0.333 at R = R 0 in Eq. ((5)) has then B = −0.238.
ω e f f =
1
3
gives
R 0
R i
= 3.798 that acceleration started when the size of the universe
was 3.798 times the initial size.