**C. Inflation**

The deep issue inflation addresses is the origin
of the large-scale homogeneity of the observable universe.
In a relativistic model with positive pressure we can see distant
galaxies that have not been in causal contact with each other
since the singular start of expansion
(Sec. II.C,
Eq. [26]); they are said
to be outside each other's particle horizon. Why do apparently
causally unconnected parts of space look so
similar? ^{26}
Sato (1981a,
1981b),
Kazanas (1980), and
Guth (1981)
make the key
point: if the early universe were dominated by the energy density
of a relatively flat real scalar field (inflaton) potential
*V*() that acts
like ,
the particle horizon could spread beyond
the universe we can see. This would allow for the possibility that
microphysics during inflation could smooth inhomogeneities sufficiently
to provide an explanation of the observed large-scale
homogeneity. (We are unaware of a definitive demonstration
of this idea, however.)

In the inflation scenario the field
rolls down its
potential until eventually
*V*() steepens
enough to terminate
inflation. Energy in the scalar field is supposed to
decay to matter and radiation, heralding the usual Big Bang
expansion of the universe. With the modifications of
Guth's (1981)
scenario by
Linde (1982)
and Albrecht and
Steinhardt (1982),
the community quickly accepted this
promising and elegant way to understand the origin of our homogeneous
expanding universe. ^{27}

In Guth's (1981)
picture the inflaton kinetic energy density is
subdominant during inflation,
^{2}
<< *V*(), so
from Eqs. (30) the pressure
*p*_{} is very close to the negative of the mass density
_{}, and the
expansion of the universe approximates the de Sitter solution,
*a*
exp(*H*_{}*t*) (Eq. [27]).

For our comments on the spectrum of mass density fluctuations produced by inflation and the properties of solutions of the dark energy models in Sec. III.E we will find it useful to have another scalar field model. Lucchin and Matarrese (1985a, 1985b) consider the potential

(38) |

where *q* and *A* are parameters.
^{28} They show that the
scale factor and the
homogeneous part of the scalar field evolve in time as

(39) |

where
*N* = 2*q*(
*A*)^{1/2} / [*G*(6 - *q*)]^{1/2}.
If *q* < 2 this model inflates.
Halliwell (1987) and
Ratra and Peebles (1988)
show that the solution (39) of the homogeneous equation of
motion has the attractor property
^{29} mentioned in
connection with Eq. (31). This exponential
potential is of historical interest: it provided the first clear
illustration of an attractor solution. We return to this point in
Sec. III.E.

A signal achievement of inflation is that it offers a theory for
the origin of the departures from homogeneity. Inflation
tremendously stretches length scales, so cosmologically
significant lengths now correspond to extremely short lengths
during inflation. On these tiny length scales quantum mechanics
governs: the wavelenghts of zero-point field fluctuations generated
during inflation are stretched by the inflationary expansion,
^{30} and these fluctuations
are converted to classical density fluctuations
in the late time universe. ^{31}

The power spectrum of the fluctuations depends on the model for inflation. If the expansion rate during inflation is close to exponential (Eq. [27]), the zero-point fluctuations are frozen into primeval mass density fluctuations with power spectrum

(40) |

Here (*k*, *t*) is
the Fourier transform at wavenumber *k* of the mass density contrast
(, *t*) =
(,
*t*) /
<(*t*)>
- 1,
where is the
mass density and
<> the
mean value. After inflation, but at very large redshifts, the spectrum in
this model is *P*(*k*) *k* on all interesting length scales.
This means the curvature fluctuations produced by the mass
fluctuations diverge only as log *k*. The form
*P*(*k*)
*k* thus need not be cut off anywhere near observationally
interesting lengths, and in this sense it is
scale-invariant. ^{32} The
transfer function *T*(*k*) accounts for the effects of
radiation pressure and the dynamics of nonrelativistic matter on
the evolution of
(*k*, *t*),
computed in linear perturbation theory, at redshifts
*z*
10^{4}. The constant *A* is
determined by details of the chosen inflation model we need
not get into.

The exponential potential model in Eq. (38) produces
the power spectrum ^{33}

(41) |

When *n* 1(*q*
0) the power spectrum is
said to be tilted. This
offers a parameter *n* to be adjusted to fit the observations of
large-scale structure, though as we will discuss the simple
scale-invariant case *n* = 1 is close to the best fit to the
observations.

The mass fluctuations in these inflation models are
said to be adiabatic, because they are what you get by
adiabatically compressing or decompressing parts of an exactly
homogeneous universe. This means the initial conditions for the
mass distribution are described by one function of position,
(, *t*). This
function is a realization of a
spatially stationary random Gaussian process, because it
is frozen out of almost free quantum field fluctuations.
Thus the single function of position is
statistically prescribed by its power spectrum, as in
Eqs. (40) and (41).
More complicated models for inflation produce density
fluctuations that are not Gaussian, or do not have simple
power law spectra, or have parts that break adiabaticity, as
gravitational waves
(Rubakov, Sazhin, and
Veryaskin, 1982)
or magnetic fields
(Turner and Widrow,
1988;
Ratra, 1992b)
or new hypothetical fields. All these extra features may be invoked to
fit the observations, if needed. It may be significant that none
seem to be needed to fit the main cosmological structure constraints
we have now.

**2. Inflation in a low density universe**

We do need an adjustment from the simplest case -- an
Einstein-de Sitter cosmology -- to account for
the measurements of the mean mass density. In the two models that
lead to Eqs. (40) and (41)
the enormous expansion factor during inflation suppresses the
curvature of space sections, making
_{K0}
negligibly small. If
= 0,
this fits the Einstein-de Sitter model (Eq. [35]),
which in the absence of data clearly is the elegant choice. But
the high mass density in this model was already seriously
challenged by the data available in 1983, on the low streaming
flow of the nearby galaxies toward the nearest known large mass
concentration, in the Virgo cluster of galaxies,
and the small relative velocities of galaxies outside the rich
clusters of galaxies. ^{34}
A striking and long familiar example of the latter is that the galaxies
immediately outside the Local Group of galaxies, at distances of a
few megaparsecs, are moving away from us in a good approximation
to Hubble's homogeneous flow, despite the very clumpy
distribution of galaxies on this scale.
^{35} The options (within
general relativity) are that the mass density is low, so its
clumpy distribution has
little gravitational effect, or the mass density is high and the
mass is more smoothly distributed than the galaxies. We comment
on the first option here, and the second in connection with the
cold dark matter model for structure formation in
Sec. III.D.

Under the first option we have two choices: introduce a
cosmological constant, or space curvature, or maybe even both.
In the conventional inflation picture space curvature is
unacceptable, but there is another line of thought that leads to
a universe with open space sections.
Gott's (1982)
scenario commences with a
large energy density in an inflaton at the top of its
potential. This behaves as Einstein's cosmological
constant and produces a near de Sitter universe expanding
as *a*
exp(*H*_{} *t*), with sufficient inflation to
allow for a microphysical explanation of the large-scale
homogeneity of the observed universe. As the
inflaton gradually rolls down the potential it reaches a point
where there is a small bump in the potential. The inflaton
tunnels through this bump by nucleating a bubble. Symmetry
forces the interior of the bubble to have open spatial sections
(Coleman and De Luccia,
1980),
and the continuing presence of a non-zero
*V*() inside the
bubble acts like
, resulting in an
open inflating universe. The potential
is supposed to steepen, bringing the second limited epoch of
inflation to an end before space curvature has been completely
redshifted away. The region inside the open bubble at the end of
inflation is a radiation-dominated Friedmann-Lemaître open model,
with 0 <
_{K0}
< 1 (Eq. [16]). This can fit the dynamical evidence for low
_{M0} with
= 0.
^{36}

The decision on which scenario, spatially-flat or open, is
elegant, if either, depends ultimately on which Nature has
chosen, if either. ^{37}
But it is natural to make judgments in advance of the evidence.
Since the early 1980s there have been occasional explorations
of the open case, but the community generally has favored the flat case,
_{K0} = 0,
without or, more recently, with a cosmological
constant, and indeed the evidence now is that space sections are
close to flat. The earlier preference for the Einstein-de Sitter
case with
_{K0} = 0
and _{0} =
0 led to considerable interest in the
picture of biased galaxy formation in the cold dark matter model, as
we now describe.

^{26} Early discussions of this question
are reviewed by
Rindler (1956);
more recent examples are
Misner (1969),
Dicke and Peebles
(1979),
and Zee (1980).
Back.

^{27} Aspects of the present state of the
subject are reviewed by
Guth (1997),
Brandenberger (2001), and
Lazarides (2002).
Back.

^{28} Similar exponential potentials appear
in some higher-dimensional Kaluza-Klein models. For an early discussion see
Shafi and Wetterich
(1985).
Back.

^{29} Ratra
(1989,
1992a)
shows that spatial inhomogeneities do not destroy
this property, that is, for *q* < 2
the spatially inhomogeneous scalar field perturbation has no growing mode.
Back.

^{30} The strong curvature of spacetime
during inflation makes the
vacuum state quite different from that of Minkowski spacetime
(Ratra 1985).
This is somewhat analogous to how the Casimir metal
plates modify the usual Minkowski spacetime vacuum state.
Back.

^{31} For the development of these ideas see
Hawking (1982),
Starobinsky (1982),
Guth and Pi (1982),
Bardeen, Steinhardt, and
Turner (1983),
and Fischler, Ratra, and
Susskind (1985).
Back.

^{32} The virtues of a spectrum that is
scale-invariant in this sense were noted before inflation, by
Harrison (1970),
Peebles and Yu (1970),
and Zel'dovich (1972).
Back.

^{33} This is discussed by
Abbott and Wise (1984),
Lucchin and Matarrese
(1985a,
1985b),
and Ratra (1989,
1992a).
Back.

^{34} This is discussed in Davis and Peebles
(1983a,
1983b)
and Peebles (1986).
Relative velocities of galaxies in rich clusters are large, but the masses
in clusters are known to add up to a modest mean mass density. Thus most
of the Einstein-de Sitter mass would have to be outside the dense parts of
the clusters, where the relative velocities are small.
Back.

^{35} The situation a half century ago is
illustrated by the compilation of galaxy redshifts in
Humason, Mayall, and
Sandage (1956).
In this sample of 806 galaxies, 14 have negative redshifts (after
correction
for the rotation of the Milky Way galaxy and for the motion of the
Milky Way toward the other large galaxy in the Local Group, the
Andromeda Nebula), indicating motion toward us. Nine are members
of the Local Group, at distances
1 Mpc. Four are
in the direction of the Virgo cluster, at redshift
~ 1200 km s^{-1} and distance ~ 20 Mpc. Subsequent
measurements indicate two of these four really have negative redshifts,
and plausibly are members of the Virgo cluster on the tail of the
distribution of peculiar velocities of the cluster members.
(Astronomers use the term peculiar velocity to denote the deviation
from the uniform Hubble expansion velocity.) The last of the 14,
NGC 3077, is in the M 81 group of galaxies at 3 Mpc
distance. It is now known to have a small positive redshift.
Back.

^{36} Gott's scenario is resurrected by
Ratra and Peebles (1994,
1995).
See Bucher and Turok
(1995),
Yamamoto, Sasaki, and
Tanaka (1995),
and Gott (1997), for
further discussions of this model. In this case spatial curvature provides
a second cosmologically-relevant length scale (in addition to that set
by the Hubble radius *H*^{-1}), so there is no natural
preference for a power law power spectrum
(Ratra, 1994;
Ratra and Peebles,
1995).
Back.

^{37} At present, high energy physics
considerations do not provide
a compelling specific inflation model, but there are strong indications
that inflation happens in a broad range of models, so it might not
be unreasonable to think that future advances in high energy physics
could give us a compelling and observationally successful model of
inflation, that will determine whether it is flat or open.
Back.