**4.5. The Inflationary Paradigm and
= 1**

It hass been 30 years since Penzias and Wilson discovered the CMB. This
observation is the very foundation of the Hot Big Bang cosmological
model. In its early formulation, this model lead to a low density,
baryon dominated Universe of age
18 billion years
(e.g., Sandage 1982).
However, with the evidence presented above about the existence of
dark matter, a fundamental alteration of this baryon-dominated
cosmological model might be required. In addition, when looked at
in detail, the Hot Big Bang model does not naturally predict some
aspects of the large scale nature of the Universe. These predictive
problems are enumerated
below. In an effort to develop a model which more predictive power,
a radical new model was proposed fifteen years ago by Alan Guth
and colleagues (see Guth 1980). This alteration is known as the
inflationary model. In brief, this model postulated the existence of
a phase-transition in the earliest moments of the Universe which
lead to a brief period in which universal expansion was exponential
in nature. This exponential expansion epoch is called "inflation"
because, among other things, any initial curvature in the early
Universe would be inflated away by the rapid expansion thus producing
a **spatially flat** Universe. Spatial flatness requires the
curvature term in the Robertson-Walker metric (K in
Chapter 1) to be
zero. This in turn requires =
1 and a dark matter
dominated Universe. Hence, if inflation indeed is the correct paradigm
for the physics of the early Universe, then it makes a fairly definition
prediction regarding the mass distribution.

An excellent and detailed overview of the physics of the early
Universe under this model is offered in the book *The Early
Universe* by Turner and Kolb. Besides being a possible "natural"
result of a phase-transition, inflation also solves three problems
that that are not adequately addressed by the standard Big Bang
model that has no inflationary epoch. These problems are:

1. The Flatness Problem: Equations 1.23 and 1.28 can be combined to yield

(22a) |

Since is time dependent (its decreasing with expansion) then is also time dependent and we an write

(22b) |

where

(22c) |

At early times
was enormous and hence *f(t)* is very
small and is very close to
unity and the Universe is
spatially flat. This condition is independent of the present epoch value of
(_{0}). The observation that
_{0} is in the range
0.01 -1, despite the fact that the scale factor has increased enormously
has lead many to suggest that
_{0} = 1 or else
the Universe would have either re-collapsed long ago or be curvature
dominated (radius of curvature
the Hubble length) at the present
epoch. A specific value of
_{0} in the range 0.01-1
would imply that the conditions
at the present epoch are somehow imprinted on the initial conditions
that determine the expansion.

2. The Horizon Problem: In
an expanding universe there are
particle horizons. The size of these horizons to first order is set
by the speed of light and the expansion rate so that *r*_{hor} ~
*cT*_{exp}. As the universe ages (expands), the particle
horizon increases
and more material can come into causal contact with that particle.
At early times, individual particle horizons could encompass
only a fraction of the volume of the Universe. It can be shown (see
the derivation in Kolb and Turner) that the horizon size at any redshift
epoch is proportional to the entropy (basically the number of photons)
within that horizon volume. For the matter dominated Universe this
can be expressed as

(23) |

At the present epoch (*z* = 0), we have *S*_{HOR}
10^{88} (which corresponds
to a CMB photon density of about 400 cm^{-3}). At the time of
recombination, *z*
1000, we have *S*_{HOR}
10^{83} so that
the Universe at *z* = 0 consists of
10^{5} causally
disconnected regions.
At the time of recombination, the angular size of a particle horizon was
1-2 degrees and yet over 360 degrees of sky, the CMB photon density
is the same. Consideration of the observed abundances of light elements
only exacerbates this issue. At the time of cosmological nucleosynthesis,
*S*_{HOR}
10^{63} yet the observed abundances of light elements today
show no variation. Thus, over 10^{25} causally disconnected regions,
the Universe shows homogeneity. This demands that the initial conditions
of the Big Bang were homogeneous and that would be a very improbable state.

3. The Smoothness Problem:
On a large scale the Universe is
extraordinarily smooth, as evidence by the low anisotropy measured
by COBE. Yet in this smooth Universe, there exists galaxies which
are local density enhancements of order 10^{4}. It will be shown in
the next chapter that density enhancements
( /
) grow
linearly, due to gravitational effects, with scale factor. A natural
gravitational timescale is known as the Planck time
which is 10^{-43} seconds. Galaxy formation commenced when
the Universe was
10^{15} seconds old and hence,
there is potentially a 10^{58} scale in the growth of density
fluctuations. If fluctuations are allowed to grow over this scale
then the observed structure in the Universe could have formed out of
a very initially smooth state. However, in the early Universe
purely baryonic density fluctuations are not allowed to be amplified linearly
The problem in the early Universe is
due to the high radiation pressure of the photon field to which
they are coupled. Baryonic density fluctuations can then only grow
linearly after recombination at 10^{5} seconds has occurred. In
this sense, structure formation is greatly aided if there is some
form of matter that is able to "gravitate" but which is not affected
by radiation drag.
The elegance of the inflationary paradigm lies in its simultaneous
solution of these three problems. The root of the paradigm is
a brief period in the early Universe where it expanded exponentially
instead of linearly. This exponential expansion caused
the Universe to increase in scale by a factor of
10^{50}. The
trigger for this inflationary epoch is a competition in the early
Universe between vacuum energy density (which acts as a source of
negative pressure) and the kinetic energy density which is essentially
an entropy field that currently drives the uniform expansion and
cooling of the Universe. It is possible to define a particular form
for the scalar field that incorporates a potential energy function,
usually expressed as *V*().
The scalar field,
is only weakly
coupled to other fields that may be present. If
is everywhere
the same (e.g., spatially homogeneous) then it is possible to express
the energy density and pressure of this fluid to first order as

(24a) | |

(24b) |

Equation 24 is a partial solution to the stress-energy tensor
use to describe the field. In the
derivation of these equations,
the early Universe is assumed to conform to the Robinson-Walker metric
described in chapter 1 and is assumed to obey
Einstein's field equations.
The term ^{2} / 2 is the
kinetic energy density. If
*V*()
varies slowly with and if the time
derivative of
is
also small then the early Universe can be characterized by
^{2} / 2 << *V*. In
this case, =
-*p* and the Universe
acts exactly as if a Cosmological Constant dominated the stress-energy
tensor.

This source of negative pressure causes the Universe to
undergo an exponential expansion of its scale factor. Presumably
this inflationary epoch must end whenever a condition is achieved such
that *V*() is at a minimum and
the potential energy of the field
is then converted into kinetic energy density. In practice,
*V*()
will oscillate around this minimum. As the scalar field is weakly
coupled to other fields, this oscillation will quickly cause the enormous
amount of vacuum energy to be dumped into the kinetic energy field
which effectively re-heats the Universe and fixes its entropy. After
this time, the Universe expands and cools.

The actual physical conditions which determine the turn-on and turn-off
phases of inflation remain obscure. Most physicists believe that
inflation is like a symmetry breaking phase-transition. The dominance
of the vacuum energy field may then be a response to symmetry breaking
at the GUTs energy scale. The end of the inflationary epoch is thought to
be caused by another phase transition in which the symmetry between the weak
nuclear force and the electrostatic force was broken. This occurs
at an energy scale of
500 Gev or a time scale of
10^{-15} seconds.
At this time, the potential has achieved a minimum meaning that the
Cosmological Constant has decayed to some minimum value and remained
constant since then. This minimum value need not have been zero.

This period of exponential expansion have inflated out any initial
curvature and
hence inflation directly solves the flatness problem. In particular,
inflation predicts a spatially flat Universe (to 1 part in 10^{50}).
The horizon problem is also directly solved as
the initial conditions could have been quite heterogeneous. A
tiny region of that heterogeneous mixture (and this tiny region was
homogeneous) inflated to produce our observable, homogeneous Universe.
Inflation allows for the
existence of other inflated Universes which occupy different domains.
Theorists enjoy speculating that boundaries between these inflated
domains, called domain walls, may have been present in our Universe with
observable consequences.

As a consequence of predicting that space must be spatially flat, inflation
then demands a Universe which is dark matter dominated. Interestingly,
this provides a solution to the smoothness problem. If dark matter
dominates the potential of structures which have trapped baryons, then
we can allow it to be of a form that does not interact with the radiation
field in the early Universe. This allows density fluctuations to begin
growing at very early times. Thus, a galaxy size density fluctuation
could grow from an initial dark matter density enhancement by a factor
of 10^{30} if we
start the growth at
the time of electroweak symmetry breaking. A universe at this time which
was smooth to one part in 10^{30} is quite consistent with the CMB
observations.

In sum, the inflationary paradigm, while operating via some unknown
but clearly fundamental
physics, provides some elegant solutions to the problems encountered
in the standard big bang model which is baryon-mass dominated.
Lately, inflation has come under fire because most observations do
not indicate that _{0}
= 1, as predicted. However, inflation
only predicts a spatially flat Universe and the curvature of the Universe
is determined by both and
. Hence, if we believe
the observations that
1 and that the inflationary
paradigm must hold, we are again driven to considerations of a non-zero
. In addition, there are some
esoteric but opaque "inflationary"
models that do predict significant spatial curvature. Since we don't
understand those, they are not considered here (but see references in
Kolb and Turner).