284
19 Inflation and Some Questions
In summary, if the universe expands exponentially at a very early time, and the
exponential expansion continues for about 50–60 e-foldings, then the horizon is such
that all points in the sky were once in causal contact and could have been thermalized
to the same temperature: in terms of this phenomenological picture the horizon puzzle
may be thereby solved.
Presumably the era of inflation ends when its large associated energy somehow is
transformed into the particles of matter that we see in the universe today; this process
is termed reheating as we have noted in Chap. 18; clearly it is not a very descriptive
phrase since there was probably no previous heating. The mechanism for reheating
is a subject of theoretical study, but little is known (Peebles 1993).
In the next section we will discuss a flexible and popular mechanism to describe
inflation using field theory.
19.2 Inflation Via Scalar Fields
Most presently favored theoretical models for inflation behave qualitatively similar
to the above phenomenological example using de Sitter space. The vacuum energy
density in such a de Sitter space can be thought of as a constant density fluid with
the equation of state p = −ρ. It is possible to devise a scalar field which behaves
much like such a fluid, and this gives a plausible and flexible type of theory for the
material and geometry during the inflationary era. Indeed there are many versions of
such fields and models, and naturally the fields are generically called inflaton fields.
In this section we will outline one of the simplest examples of an inflaton field, but
there is much current research in this field and we will only give a general discussion
(Liddle 2003; Peebles 1993; Linde 2003).
We will assume the reader has some familiarity with the Lagrangian approach to
field theory; for the reader who is not familiar with field theory. Appendix 2 gives a
brief overview. It is now almost universal practice in field theory to use “natural units”
in which = 1 and c = 1 (Bjorken 1964). This convention makes the equations
look a great deal simpler and we will adopt it in this section and in Appendix 2.
For a self-interacting real scalar inflaton field ϕ the standard form of the
Lagrangian and action are
L =
1
2
ϕ ,μ ϕ ,ν g
μν
− V (ϕ), S =
L
−|g|d
4 x.
(19.9)
Note that in this chapter only we use a different notation convention for the metric
determinant than we did previously in Chaps. 4 and 6; there we took |g| to be the
absolute value of the determinant: here we keep the sign of |g| explicit. We do this to
clarify some algebraic manipulations involving signs, especially in Appendix 2. Here
V is a self-interaction potential function of the field, to be specified; for example the
choice of a quadratic potential describes a massive scalar field (Bjorken 1965). The
equation of motion for ϕ is then found using the standard Euler–Lagrange procedure
19 Inflation and Some Questions
In summary, if the universe expands exponentially at a very early time, and the
exponential expansion continues for about 50–60 e-foldings, then the horizon is such
that all points in the sky were once in causal contact and could have been thermalized
to the same temperature: in terms of this phenomenological picture the horizon puzzle
may be thereby solved.
Presumably the era of inflation ends when its large associated energy somehow is
transformed into the particles of matter that we see in the universe today; this process
is termed reheating as we have noted in Chap. 18; clearly it is not a very descriptive
phrase since there was probably no previous heating. The mechanism for reheating
is a subject of theoretical study, but little is known (Peebles 1993).
In the next section we will discuss a flexible and popular mechanism to describe
inflation using field theory.
19.2 Inflation Via Scalar Fields
Most presently favored theoretical models for inflation behave qualitatively similar
to the above phenomenological example using de Sitter space. The vacuum energy
density in such a de Sitter space can be thought of as a constant density fluid with
the equation of state p = −ρ. It is possible to devise a scalar field which behaves
much like such a fluid, and this gives a plausible and flexible type of theory for the
material and geometry during the inflationary era. Indeed there are many versions of
such fields and models, and naturally the fields are generically called inflaton fields.
In this section we will outline one of the simplest examples of an inflaton field, but
there is much current research in this field and we will only give a general discussion
(Liddle 2003; Peebles 1993; Linde 2003).
We will assume the reader has some familiarity with the Lagrangian approach to
field theory; for the reader who is not familiar with field theory. Appendix 2 gives a
brief overview. It is now almost universal practice in field theory to use “natural units”
in which = 1 and c = 1 (Bjorken 1964). This convention makes the equations
look a great deal simpler and we will adopt it in this section and in Appendix 2.
For a self-interacting real scalar inflaton field ϕ the standard form of the
Lagrangian and action are
L =
1
2
ϕ ,μ ϕ ,ν g
μν
− V (ϕ), S =
L
−|g|d
4 x.
(19.9)
Note that in this chapter only we use a different notation convention for the metric
determinant than we did previously in Chaps. 4 and 6; there we took |g| to be the
absolute value of the determinant: here we keep the sign of |g| explicit. We do this to
clarify some algebraic manipulations involving signs, especially in Appendix 2. Here
V is a self-interaction potential function of the field, to be specified; for example the
choice of a quadratic potential describes a massive scalar field (Bjorken 1965). The
equation of motion for ϕ is then found using the standard Euler–Lagrange procedure