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19 Inflation and Some Questions
Thus the demand we wish to place on the scale factor is that the horizon coordinate
distance during inflation be large, or perhaps infinite. That is, explicitly
σ 12 = c
t r
t I
a 0
a(t)
dt > σ 0 .
(19.1)
Here t I denotes the beginning of the inflation era and t r denotes the end of the inflation
era and beginning of the radiation era. The horizon at decoupling will be larger than
this and thus large enough to resolve the horizon puzzle. In the following paragraphs
we will elucidate how this can come about.
As a simple example of how to satisfy the demand in (19.1) consider an ad hoc
choice of a scale factor, a power of the cosmic time,
a = a 0
t
t 0
m
,
(19.2)
where the power m is taken to be large, say of order 10 or more. Then the integral for
σ 12 in (19.1) diverges and the horizon is infinite. This model of inflation is naturally
called power law inflation; it corresponds to a model universe filled with a fluid with
an unusual equation of state. The power law model of inflation provides an excellent
pedagogic example, but it is not presently a favored model, so we have relegated
further discussion to Appendix 1 (Peebles 1993).
As a second example consider another ad hoc choice, a de Sitter space with an
exponential scale factor, which we have already discussed in Chap. 13 and Exercise 17.5. Recall that de Sitter space describes a universe containing only vacuum
energy, or equivalently a cosmological constant. It is very important to distinguish
this vacuum energy during inflation from the present vacuum energy in the LCDM
model; the energy scale during inflation is vastly larger. This model provides an illuminating phenomenology for the basic features of inflation. We will use it to work
out the coordinate horizon σ 12 and its relation to the coordinate distance σ 0 of the
last scattering. Thus we choose during inflation
a = Ae
H I t
, H I = const.
(19.3)
A qualitative sketch is shown in Fig. 19.1. To make a rough estimate of the constant
A we equate this to the scale factor for the radiation era at its beginning as given in
(17.12) and thereby obtain.
a = a 0
t r
t 0
1/2
e
H I (t−t r )
, t ≤ t r , t 0 present time.
(19.4)
19 Inflation and Some Questions
Thus the demand we wish to place on the scale factor is that the horizon coordinate
distance during inflation be large, or perhaps infinite. That is, explicitly
σ 12 = c
t r
t I
a 0
a(t)
dt > σ 0 .
(19.1)
Here t I denotes the beginning of the inflation era and t r denotes the end of the inflation
era and beginning of the radiation era. The horizon at decoupling will be larger than
this and thus large enough to resolve the horizon puzzle. In the following paragraphs
we will elucidate how this can come about.
As a simple example of how to satisfy the demand in (19.1) consider an ad hoc
choice of a scale factor, a power of the cosmic time,
a = a 0
t
t 0
m
,
(19.2)
where the power m is taken to be large, say of order 10 or more. Then the integral for
σ 12 in (19.1) diverges and the horizon is infinite. This model of inflation is naturally
called power law inflation; it corresponds to a model universe filled with a fluid with
an unusual equation of state. The power law model of inflation provides an excellent
pedagogic example, but it is not presently a favored model, so we have relegated
further discussion to Appendix 1 (Peebles 1993).
As a second example consider another ad hoc choice, a de Sitter space with an
exponential scale factor, which we have already discussed in Chap. 13 and Exercise 17.5. Recall that de Sitter space describes a universe containing only vacuum
energy, or equivalently a cosmological constant. It is very important to distinguish
this vacuum energy during inflation from the present vacuum energy in the LCDM
model; the energy scale during inflation is vastly larger. This model provides an illuminating phenomenology for the basic features of inflation. We will use it to work
out the coordinate horizon σ 12 and its relation to the coordinate distance σ 0 of the
last scattering. Thus we choose during inflation
a = Ae
H I t
, H I = const.
(19.3)
A qualitative sketch is shown in Fig. 19.1. To make a rough estimate of the constant
A we equate this to the scale factor for the radiation era at its beginning as given in
(17.12) and thereby obtain.
a = a 0
t r
t 0
1/2
e
H I (t−t r )
, t ≤ t r , t 0 present time.
(19.4)