A Non-empty Bouncing Milne Model for the Universe
9
sities of individual cosmic fluids. The authors of the former have stated that various
constituents of the total energy can adjust their relative densities via particle–particle
interactions [30] to get evolution equations that satisfy p = −(1/3) ρ. However,
no specific time evolution for the densities of these constituents is suggested in the
R h = ct papers. The authors excuse themselves by stating that when the evolution of
individual components is needed, several conservation laws and reasonable assumptions delimit their behaviour [34]. Hence, there is no model here to compare with the
eternal coasting model.
3 Complex Extension of the Milne Model
In this paper, we also show that the Milne model can lead to a bouncing non-empty,
coasting cosmological model. This is done by a complex extension of scale factor in
the Milne-type empty Friedmann model. The new model is realistic and nonsingular
and is always coasting after the Planck epoch. As a next step, a quantum cosmological
treatment of this model is made, which provides the result that the minimum radius a 0
in it is of the order of the Planck length. The complexified Milne model has Euclidean
(+ + ++) signature at t = 0, but it changes to the usual Lorentzian one (+ − − −)
immediately after the Planck epoch. We find that this leads to the widely speculated
‘signature change’ [35, 36] in the early universe.
Let us now extend the scale factor of the Milne model to the complex plane and
denote it as ˆ
a. We can now show that this results in a coasting cosmology with real
scale factor a ≡| ˆ
a |. One can write the Friedmann equations corresponding to the
empty Milne model (with units in which the speed of light c = 1) as
˙ ˆ
a
ˆ
a
2
−
1
ˆ
a 2 = 0
( 1 )
and
2
¨ ˆ
a
ˆ
a
+
˙ ˆ
a
ˆ
a
2
−
1
ˆ
a 2 = 0.
(2)
Using the polar form ˆ
a = ae
iφ in these equations and then equating their real parts,
we get
˙
a
2
a 2 = ˙
φ
2
+
1
a 2 cos 2φ,
(3)
and
2
¨
a
a
+
˙
a
2
a 2 = 3 ˙
φ
2
+
1
a 2 cos 2φ.
(4)
9
sities of individual cosmic fluids. The authors of the former have stated that various
constituents of the total energy can adjust their relative densities via particle–particle
interactions [30] to get evolution equations that satisfy p = −(1/3) ρ. However,
no specific time evolution for the densities of these constituents is suggested in the
R h = ct papers. The authors excuse themselves by stating that when the evolution of
individual components is needed, several conservation laws and reasonable assumptions delimit their behaviour [34]. Hence, there is no model here to compare with the
eternal coasting model.
3 Complex Extension of the Milne Model
In this paper, we also show that the Milne model can lead to a bouncing non-empty,
coasting cosmological model. This is done by a complex extension of scale factor in
the Milne-type empty Friedmann model. The new model is realistic and nonsingular
and is always coasting after the Planck epoch. As a next step, a quantum cosmological
treatment of this model is made, which provides the result that the minimum radius a 0
in it is of the order of the Planck length. The complexified Milne model has Euclidean
(+ + ++) signature at t = 0, but it changes to the usual Lorentzian one (+ − − −)
immediately after the Planck epoch. We find that this leads to the widely speculated
‘signature change’ [35, 36] in the early universe.
Let us now extend the scale factor of the Milne model to the complex plane and
denote it as ˆ
a. We can now show that this results in a coasting cosmology with real
scale factor a ≡| ˆ
a |. One can write the Friedmann equations corresponding to the
empty Milne model (with units in which the speed of light c = 1) as
˙ ˆ
a
ˆ
a
2
−
1
ˆ
a 2 = 0
( 1 )
and
2
¨ ˆ
a
ˆ
a
+
˙ ˆ
a
ˆ
a
2
−
1
ˆ
a 2 = 0.
(2)
Using the polar form ˆ
a = ae
iφ in these equations and then equating their real parts,
we get
˙
a
2
a 2 = ˙
φ
2
+
1
a 2 cos 2φ,
(3)
and
2
¨
a
a
+
˙
a
2
a 2 = 3 ˙
φ
2
+
1
a 2 cos 2φ.
(4)