10
M. V. John
These equations appear as the familiar Friedmann equations in inflationary models,
for a universe with flat (k = 0) space sections and filled with a homogeneous scalar
field φ. This model is now quite different from that of Milne, since the geometry of
its space sections and the energy density are different. The right-hand sides of the
equations imply the presence of a scalar field with kinetic energy ˙
φ
2 and potential
energy cos(2φ)/a
2 . That is, we have a new cosmological model with real scale factor
a =| ˆ
a |, whose space sections are flat and which is non-empty. Equating also the
imaginary parts, we get two supplementary equations
¨
φ + 2 ˙
φ
˙
a
a
= 0
( 5 )
and
2 ˙
φ
˙
a
a
= −
1
a 2 sin 2φ..
(6)
These shall be of help in solving the system of equations (3) and (4). However, one
can directly solve Eqs. (1) and (2) to obtain ˆ
a = (c 1 ± ct) + ic 2 . Choosing the origin
of time such that the constant of integration c 1 = 0 and relabelling c 2 ≡ a 0 , we get
the solution for ˆ
a as
ˆ
a = ±t + ia 0 .
(7)
The solution for a =| ˆ
a | can be seen to be
a =
a
2
0 + t 2
(8)
and the argument φ can be obtained as
φ = tan
−1
a 0
t
.
(9)
This is a bouncing, nonsingular evolution with a 0 as the minimum value for the scale
factor (Fig. 2). The time at which this minimum occurs for a is taken as t = 0. In
the early phase of Ozer–Taha model [13, 14], the evolution is described by the same
equation. If we take its energy density as comprised of matter and a time-variable dark
energy and do not assume any arbitrary conservation equations for these individual
components, all the cosmological problems mentioned above can be seen to vanish
in it. The energy density ρ and pressure p in the new model, as can be deduced from
Eqs. (3) and (4), have the following evolution. At the epoch near t = 0, the kinetic
energy of the field φ is dominant, leading to the nonsingular behaviour. This phase
of evolution is capable of solving cosmological problems such as that related to the
horizon. For large t, we have ρ ∝ a
−2 and ρ + 3 p ≈ 0. The contribution for ρ in the
late universe comes almost entirely from the potential energy term. It may also be
noted that its magnitude is very nearly equal to the critical density for the universe.
One can explicitly write the variation of total density and pressure with scale factor as
M. V. John
These equations appear as the familiar Friedmann equations in inflationary models,
for a universe with flat (k = 0) space sections and filled with a homogeneous scalar
field φ. This model is now quite different from that of Milne, since the geometry of
its space sections and the energy density are different. The right-hand sides of the
equations imply the presence of a scalar field with kinetic energy ˙
φ
2 and potential
energy cos(2φ)/a
2 . That is, we have a new cosmological model with real scale factor
a =| ˆ
a |, whose space sections are flat and which is non-empty. Equating also the
imaginary parts, we get two supplementary equations
¨
φ + 2 ˙
φ
˙
a
a
= 0
( 5 )
and
2 ˙
φ
˙
a
a
= −
1
a 2 sin 2φ..
(6)
These shall be of help in solving the system of equations (3) and (4). However, one
can directly solve Eqs. (1) and (2) to obtain ˆ
a = (c 1 ± ct) + ic 2 . Choosing the origin
of time such that the constant of integration c 1 = 0 and relabelling c 2 ≡ a 0 , we get
the solution for ˆ
a as
ˆ
a = ±t + ia 0 .
(7)
The solution for a =| ˆ
a | can be seen to be
a =
a
2
0 + t 2
(8)
and the argument φ can be obtained as
φ = tan
−1
a 0
t
.
(9)
This is a bouncing, nonsingular evolution with a 0 as the minimum value for the scale
factor (Fig. 2). The time at which this minimum occurs for a is taken as t = 0. In
the early phase of Ozer–Taha model [13, 14], the evolution is described by the same
equation. If we take its energy density as comprised of matter and a time-variable dark
energy and do not assume any arbitrary conservation equations for these individual
components, all the cosmological problems mentioned above can be seen to vanish
in it. The energy density ρ and pressure p in the new model, as can be deduced from
Eqs. (3) and (4), have the following evolution. At the epoch near t = 0, the kinetic
energy of the field φ is dominant, leading to the nonsingular behaviour. This phase
of evolution is capable of solving cosmological problems such as that related to the
horizon. For large t, we have ρ ∝ a
−2 and ρ + 3 p ≈ 0. The contribution for ρ in the
late universe comes almost entirely from the potential energy term. It may also be
noted that its magnitude is very nearly equal to the critical density for the universe.
One can explicitly write the variation of total density and pressure with scale factor as