296
19 Inflation and Some Questions
The basic assumption is that the scale factor is equal to some power of time with
a rather large exponent m >> 1,
a = Ct
m
, C = const.
(19.29)
Then the horizon problem is trivially resolved since the integral in the horizon condition (19.1) diverges, assuming of course that the initial inflation time t I is very small
or zero.
To see what sort of fluid this scale factor might correspond to we refer to Sect. 14.4,
in particular to (14.15b) which gives the behavior of the energy density of a fluid as
a function of the scale factor as the universe expands. We rewrite (14.15b) as
ρ =
D
a 3(1+w) , D = const.
(19.30)
It is clear that for such a fluid the appropriate Friedmann equation is (14.17) with no
cosmological constant or curvature; we express it as
a
2
a 2 =
E
a 3(1+w) , E = const.
(19.31)
Finally, to relate the w of the fluid to the power m we substitute the scale factor
(19.29) into (19.31) and have
m
t
2 =
E
(At m )
3(1+w)
.
(19.32)
Equating the powers of t we get the simple relation
w = −1 +
2
3m
.
(19.33)
Thus the fluid that produces power law inflation has an equation of state w parameter
that is a little larger than −1. In this sense it behaves similarly to cosmological
constant vacuum energy, which has w = −1.
Appendix 2: Scalar Field Theory
In this appendix we will simplify notation by setting the constants c and equal to
one, as is commonly done in particle physics and inflation theory. With this choice
every quantity in the theory can be chosen to have the dimension of distance to some
power.
19 Inflation and Some Questions
The basic assumption is that the scale factor is equal to some power of time with
a rather large exponent m >> 1,
a = Ct
m
, C = const.
(19.29)
Then the horizon problem is trivially resolved since the integral in the horizon condition (19.1) diverges, assuming of course that the initial inflation time t I is very small
or zero.
To see what sort of fluid this scale factor might correspond to we refer to Sect. 14.4,
in particular to (14.15b) which gives the behavior of the energy density of a fluid as
a function of the scale factor as the universe expands. We rewrite (14.15b) as
ρ =
D
a 3(1+w) , D = const.
(19.30)
It is clear that for such a fluid the appropriate Friedmann equation is (14.17) with no
cosmological constant or curvature; we express it as
a
2
a 2 =
E
a 3(1+w) , E = const.
(19.31)
Finally, to relate the w of the fluid to the power m we substitute the scale factor
(19.29) into (19.31) and have
m
t
2 =
E
(At m )
3(1+w)
.
(19.32)
Equating the powers of t we get the simple relation
w = −1 +
2
3m
.
(19.33)
Thus the fluid that produces power law inflation has an equation of state w parameter
that is a little larger than −1. In this sense it behaves similarly to cosmological
constant vacuum energy, which has w = −1.
Appendix 2: Scalar Field Theory
In this appendix we will simplify notation by setting the constants c and equal to
one, as is commonly done in particle physics and inflation theory. With this choice
every quantity in the theory can be chosen to have the dimension of distance to some
power.