A Non-empty Bouncing Milne Model for the Universe
11
ρ =
3
8π G
1
a 2 −
a
2
0
a 4
(10)
and
p = −
1
8π G
1
a 2 +
a
2
0
a 4
,
(11)
so that
ρ + 3 p = −
3
4π G
a
2
0
a 4 .
(12)
The term that causes the bounce to happen corresponds to a negative energy density,
which can now be separated as
ρ − = −
3
8π G
a
2
0
a 4 .
(13)
The pressure due to this is given by p − = (1/3)ρ − , which is an equation of state
characteristic of relativistic energy densities. This energy density becomes negligible
for a a 0 , when compared to the rest of the energy densities (Fig. 2).
Thus, the real model is flat, with the total energy density and pressure obeying
ρ + 3 p ≈ 0 for a a 0 . After this initial epoch, if the potential energy of the field
comprises energy corresponding to radiation/matter and a time-variable dark energy,
we can write ρ = ρ m + ρ d.e. . Taking p m = wρ m and p d.e. = −ρ d.e. , one obtains
Fig. 2 The variation of scale factor with time in the non-empty bouncing Milne model is shown.
The bounce occurs in between the coasting contraction and the coasting expansion phases and lasts
only for 10 −43 s, the Planck time