Adapted from P. Coles, 1999, *The Routledge Critical
Dictionary of the New Cosmology*, Routledge Inc., New York. Reprinted
with the author's permission. To order this book click here:
http://www.routledge-ny.com/books.cfm?isbn=0415923549

A modern variation of the standard **Big Bang
theory** that includes a finite period of accelerated expansion
(*inflation*) in the early stages of its evolution. Inflation is the
mechanism by which various outstanding problems of the standard
**cosmological models** might be addressed, providing a possible
resolution of the **horizon problem** and the **flatness
problem**, as well as
generating the **primordial density fluctuations** that are required for
structure formation to occur and which appear to have produced the
famous **ripples** in the **cosmic microwave background radiation**.

Assuming that we accept that an epoch of inflation is desirable for
these reasons, how can we achieve the accelerated expansion
physically. when the standard **Friedmann models** are always
decelerating? The idea that lies at the foundation of most models of
inflation is that there was an epoch in the early stages of the
evolution of the Universe in which the energy density of the vacuum
state of a **scalar field** ,
perhaps associated with one of the
fundamental interactions, provided the dominant contribution to the
energy density. In the ensuing phase the scale factor *a(t)* describing
the **expansion of the Universe** grows in an accelerated fashion, and is
in fact very nearly an exponential function of time if the energy
density of the scalar field is somehow held constant. In the
inflationary epoch that follows, a microscopically small region,
perhaps no larger than the **Planck length**, can grow - or
*inflate* - to such a size that it exceeds the size of our present
observable **Universe** (see **horizon**).

There exist many different versions of the inflationary
universe. The first was formulated by Alan **Guth** in 1981, although many
of his ideas had been presented by Alexei Starobinsky in 1979. In
Guth's model inflation is assumed to occur while the Universe
undergoes a first-order **phase transition**, which is predicted to occur
in some **grand unified theories**. The next generation of inflationary
models shared the characteristics of a model called the *new
inflationary universe*, which was suggested in 1982 independently by
Andrei Linde, and by Andreas Albrecht and Paul Steinhardt. In models
of this type, inflation occurs during a phase in which the region that
will grow to include our observable patch evolves more gradually from
a `false vacuum' to a `true vacuum'. It was later realised that this
kind of inflation could also be achieved in many different contexts,
not necessarily requiring the existence of a phase transition or a
**spontaneous symmetry-breaking**. This of parameters of a particular
grand unified theory which, in the absence of any other experimental
evidence, appears a little arbitrary. This problem also arises in
other inflationary models based on theories like **supersymmetry** or
**string theories**, which are yet to receive any experimental
confirmation or, indeed, are likely to in the foreseeable future. It
is fair to say that the inflationary model has become a sort of
paradigm for resolving some of the difficulties with the standard
model, but no particular version of it has so far received any strong
physical support from particle physics theories.

Let us concentrate for a while on the physics of generic
inflationary models involving symmetry-breaking during a phase
transition. In general, **gauge theories** of elementary particle
interactions involve an *order parameter* which we can identify with the
scalar field determining the
breaking of the symmetry. The behaviour
of the scalar field is controlled by a quantity called its *Lagrangian
action*, which has two components: one (denoted by *U*) concerns
time-derivatives of and is
therefore called the kinetic term, and
the other (*V*) describes the interactions of the scalar field and is
called the potential term. A scalar field behaves roughly like a
strange form of matter with a total energy density given by *U* +
*V* and
a pressure given by *U* - *V*. Note that if *V* is much
larger than *U*, then
the density and pressure are equal but of opposite sign. This is what
is needed to generate inflation, but the way in which the potential
comes to dominate is quite complicated.

The potential function *V* changes with the temperature of the
Universe, and it is this that induces the phase transition, as it
becomes energetically favourable for the state of the field to change
when the Universe cools sufficiently. In the language of
thermodynamics, the potential *V*() plays the role of the free energy
of the system. A graph of *V*() will typically have a minimum
somewhere, and that minimum value determines the value of
which is
stable at a given temperature. Imagine an inverted parabola with its
minimum value at = 0; the
configuration of the field can be
represented as the position of a ball rolling on this curve. In the
stable configuration it nestles in the bottom of the potential well at
= 0. This might represent the
potential *V*() at very high
temperatures, way above the phase transition. The vacuum is then in
its most symmetrical state. What happens as the phase transition
proceeds is that the shape of *V*() changes so that it develops
additional minima. Initially these `false' minima may be at higher
values of *V*() than the
original, but as the temperature continues to
fall and the shape of the curve changes further, the new minima can be
at lower values of *V* than the original one. This happens at a critical
temperature *T*_{c} at which the vacuum state of the
Universe begins to
prefer one of the alternative minima to the original one.

The transition does not occur instantaneously. How it proceeds depends on the shape of the potential, and this in turn determines whether the transition is first or second order. If the phase transition is second order it moves rather smoothly, and fairly large `domains' of the new phase are generated (much like the Weiss domains in a ferromagnet). One such region (bubble or domain) eventually ends up including our local patch of the Universe. If the potential is such that the transition is first order, the new phase appears as bubbles nucleating within the false vacuum background; these then grow and coalesce so as to fill space with the new phase when the transition is complete.

Inflation arises when the potential term *V* greatly exceeds the
kinetic term *U* in the action of the scalar field. In a phase
transition this usually means that the vacuum must move relatively
slowly from its original state into the final state. In fact, the
equations governing the evolution of
are mathematically identical to
those describing a ball moving under the action of the force -d*V*
/ d,
just as in standard Newtonian dynamics. But there is also a frictional
force, caused by the expansion of the Universe, that tends to slow
down the rolling of from one state
into another. This provides a
natural self-regulation of the speed of the transition. As long as the
Universe is expanding at the start, the kinetic term *U* is quickly
reduced by the action of this friction (or viscosity); the motion of
the field then resembles the behaviour of particles during
sedimentation.

In order to have inflation we must assume that, at sometime, the
Universe contains some expanding regions in **thermal equilibrium** at a
temperature *T* > *T*_{c} which can eventually cool
below *T*_{c} before they
re-collapse. Let us assume that such a vacuum phase, is sufficiently
homogeneous and isotropic to be described by a **Robertson-Walker
metric**. In this case the evolution of the patch is described by a
Friedmann model, except that the density of the Universe is not the
density of matter, but the effective density of the scalar field, i.e.
the sum *U* + *V* mentioned above. If the field is evolving slowly then
the *U* component is negligibly small. The Friedmann equation then looks
exactly like the equation describing a Friedmann model incorporating a
**cosmological constant** term but containing no matter. The cosmological
model that results is the well-known de Sitter solution in which the
scale factor *a(t)* exp
(*Ht*), with *H* (the Hubble parameter) roughly
constant at a value given by *H*^{2} = 8 *G V* / 3. Since
does not change
very much as the transition proceeds, *V* is roughly constant.

The de Sitter solution is the same as that used in the **steady state
theory**, except that the scalar field in that theory is the so-called
C-field responsible for the continuous creation of matter. In the
inflationary Universe, however, the expansion timeseale is much more
rapid than in the steady state. The inverse of the Hubble expansion
parameter, 1/*H*, is about 10^{-34} seconds in
inflation. This quantity has
to be fixed at the inverse of the present-day value of the Hubble
parameter (i.e. at the reciprocal of the **Hubble constant**,
1/*H*_{0}) in the
steady state theory, which is around 10^{17} seconds.

This rapid expansion is a way of solving some riddles which are not
explained in the standard Big Bang theory. For example, a region which
is the same order of size as the horizon before inflation, and which
might therefore be reasonably assumed to be smooth, would then become
enormously large, encompassing the entire observable Universe today.
Any inhomogeneity and anisotropy present at the initial time will be
smoothed out so that the region loses all memory of its initial
structure. Inflation therefore provides a mechanism for avoiding the
horizon problem. This effect is, in fact, a general property of
inflationary universes and it is described by the so-called *cosmic
no-hair theorem* (see also **black hole**).

Another interesting outcome of the inflationary Universe is that the
characteristic scale of the **curvature of spacetime**, which is
intimately related to the value of the *density parameter*
, is
expected to become enormously large. A balloon is perceptibly curved
because its radius of curvature is only a few centimetres, but it
would appear very flat if it were blown up to the radius of the
Earth. The same happens in inflation: the curvature scale may be very
small indeed initially, but it ends up greater than the size of our
observable Universe. The important consequence of this is that the
**density parameter**
should be very close to 1. More precisely, the
total energy density of the Universe (including the matter and any
vacuum energy associated with a cosmological constant) should be very
close to the critical density required to make a **flat universe**.

Because of the large expansion, a small initial patch also becomes
practically devoid of matter in the form of **elementary particles**. This
also solves problems associated with the formation of monopoles and
other **topological defects** in the early Universe, because any defects
formed during the transition will be drastically diluted as the
Universe expands, so that their present density will be negligible.

After the slow rolling phase is complete, the field falls rapidly
into its chosen minimum and then undergoes oscillations about the
minimum value. While this happens there is a rapid liberation of
energy which was trapped in the potential term *V* while the transition
was in progress. This energy is basically the *latent heat* of the
transition. The oscillations are damped by the creation of elementary
particles coupled to the scalar field, and the liberation of the
latent heat raises the temperature again - a phenomenon called
*reheating*. The small patch of the Universe we have been talking about
thus acquires virtually all the energy and entropy that originally
resided in the quantum vacuum by this process of particle
creation. Once reheating has occurred, the evolution of the patch
again takes on the character of the usual Friedmann models, but this
time it has the normal form of matter. If *V*(_{0}) = 0, then the vacuum
energy remaining after inflation is zero and there will be no
remaining cosmological constant .

One of the problems left unsolved by inflation is that there is no
real reason to suppose that the minimum of *V* is exactly at zero, so we
would expect a nonzero -term
to appear at the end. Attempts to
calculate the size of the cosmological constant
induced by phase
transitions in this way produce enormously large values. It is
important that the inflationary model should predict a reheating
temperature sufficiently high that processes which violate the
conservation of baryon number can take place so as to allow the
creation of an asymmetry between matter and **antimatter** (see
**baryogenesis**).

As far as its overall properties are concerned, our Universe was reborn into a new life after reheating. Even if before it was highly lumpy and curved, it was now highly homogeneous, and had negligible curvature. This latter prediction may be a problem because, as we have seen, there is little strong evidence that the density parameter is indeed close to 1, as required.

Another general property of inflationary models, which we shall not go into here, is that fluctuations in the quantum scalar field driving inflation can, in principle, generate a spectrum of primordial density fluctuations capable of initiating the process of cosmological structure formation. They may also produce an observable spectrum of primordial gravitational waves.

There are many versions of the basic inflationary model which are based on slightly different assumptions about the nature of the scalar field and the form of the phase transition. Some of the most important are described below.

*Old inflation.*

*New inflation.*
*spinodal decomposition*,
usually leaves larger coherent domains, providing a natural way out of
the problem of old inflation. The problem with new inflation is that
it suffers from severe fine-tuning difficulties. One is that the
potential must be very flat near the origin to produce enough
inflation and to avoid excessive fluctuations due to the quantum
field. Another is that the field
is assumed to be in thermal
equilibrium with the other matter fields before the onset of
inflation; this requires to be
coupled fairly strongly to the other
fields that might exist at this time. But this coupling would induce
corrections to the potential which would violate the previous
constraint. It seems unlikely, therefore, that thermal equilibrium can
be attained in a self-consistent way before inflation starts and under
the conditions necessary for inflation to happen.

*Chaotic inflation.*
*V*, a patch of the
Universe in which is large,
uniform and relatively static will
automatically lead to inflation. In chaotic inflation we simply assume
that at some initial time, perhaps as early as the Planck time, the
field varied from place to place in an arbitrary chaotic manner. If
any region is uniform and static, it will inflate and eventually
encompass our observable Universe. While the end result of chaotic
inflation is locally flat and homogeneous in our observable patch, on
scales larger than the horizon the Universe is highly curved and
inhomogeneous. Chaotic inflation is therefore very different from
both the old and new inflationary models. This difference is
reinforced by the fact that no mention of supersymmetry or grand
unified theories appears in the description. The field that
describes chaotic inflation at the Planck time is completely decoupled
from all other physics.

*Stochastic inflation.*
*eternal inflation*; as with chaotic inflation, the Universe is
extremely inhomogeneous overall, but quantum fluctuations during the
evolution of are taken into
account. In stochastic inflation the
universe will at any time contain regions which are just entering an
inflationary phase. We can picture the Universe as a continuous
`branching' process in which new `mini-universes' expand to produce
locally smooth patches within a highly chaotic background
Universe. This model is like a Big Bang on the scale of each
mini-universe, but overall it is reminiscent of the steady state
model. The continual birth and rebirth of these mini-universes is
often called, rather poetically, the *phoenix universe*. This model has
the interesting feature that the laws of physics may be different in
different mini-universe, which brings the **anthropic principle** very
much into play.

*Modified gravity.*
**Brans-Dicke theory** of gravity. The crucial point with this kind of
model is that the scalar field does not generate an exponential
expansion, but one in which the expansion is some power of time: *a(t)*
*t*^{a}. This
modification even allows old inflation to succeed: the
bubbles that nucleate the new phase can be made to merge and fill
space if inflation proceeds as a power law in time rather than an
exponential. Theories based on Brans-Dicke modified gravity are
usually called *extended inflation*.

There are many other possibilities: models with more than one scalar field, models with modified gravity and a scalar field, models based on more complicated potentials, models based on supersymmetry, on grand unified theories, and so on. Indeed, inflation has led to an almost exponential increase in the number of cosmological models!

It should now be clear that the inflationary Universe model provides
a conceptual explanation of the horizon problem and the flatness
problem. It also may rescue grand unified theories which predict a
large present-day abundance of monopoles or other topological defects.
Inflationary models have gradually evolved to avoid problems with
earlier versions. Some models are intrinsically flawed (e.g. old
inflation) but can be salvaged in some modified form (e.g. extended
inflation). The magnitude of the primordial density fluctuations and
gravitational waves they produce may also be too high for some
particular models. There are, however, much more serious problems
associated with these scenarios. Perhaps the most important is
intimately connected with one of the successes. Most inflationary
models predict that the observable Universe at the present epoch
should be almost flat. In the absence of a cosmological constant this
means that
1. While this possibility is
not excluded by
observations, it cannot be said that there is compelling evidence for
it and, if anything, observations of dark matter in galaxies and
clusters of galaxies favour an open universe with a lower density than
this. It is possible to produce a low-density universe after
inflation, but it requires very particular models. To engineer an
inflationary model that produces 0.2 at the
present epoch requires
a considerable degree of unpleasant fine-tuning of the conditions
before inflation. On the other hand, we could reconcile a low-density
universe with apparently more natural inflationary models by appealing
to a relic cosmological constant: the requirement that space should be
(almost) flat simply translates into _{0} + (*c*^{2} / 3*H*_{0}^{2})
1. It has been
suggested that this is a potentially successful model of structure
formation, but recent developments in **classical cosmology** put pressure
on this alternative.

The status of inflation as a physical theory is also of some concern. To what extent is inflation predictive? Is it testable? We could argue that inflation does predict that we live in a flat universe. This may be true, but a flat universe might emerge naturally at the very beginning if some process connected with quantum gravity can arrange it. Likewise, an open universe appears to be possible either with or without inflation. Inflationary models also produce primordial density fluctuations and gravitational waves. Observations showing that these phenomena had the correct properties may eventually constitute a test of inflation, but this is not possible at the present. All we can say is that the observed properties of fluctuations in the cosmic microwave background radiation indeed seem to be consistent with the usual inflationary models. At the moment, therefore, inflation has a status somewhere between a theory and a paradigm, but we are still far from sure that inflation ever took place.

FURTHER READING:

Guth, A.H., `Inflationary Universe: A possible solution to the horizon
and flatness problems', *Physical Review* D, 1981, **23**, 347.
Albrecht, A. and Steinhardt, P.J., `Cosmology for grand unified
theories with radiatively induced symmetry breaking', *Physical Review
Letters*, 1982, **48**, 1220.
Linde, A.D., `Scalar field fluctuations in the expanding Universe and
the new inflationary Universe scenario', *Physics Letters* B, 1982,
**116**, 335.
Narlikar, J.V. and Padmanabhan, T., `Inflation for astronomers',
*Annual Reviews of Astronomy and Astrophysics*, 1991, **29**, 325.