Matter and radiation are gravitationally attractive, so
in a maximally symmetric spacetime filled with matter, the gravitational
force will inevitably cause any lumpiness in the matter to grow and
condense. That's how hydrogen gas turned into galaxies and stars. But
vacuum energy comes with a high vacuum pressure, and that high vacuum
pressure resists gravitational collapse as a kind of repulsive gravitational
force. The pressure of the vacuum energy flattens out the lumpiness,
and makes space get flatter, not lumpier, as it expands.
So one possible solution to the flatness problem would
be if our Universe went through a phase where the only energy density
present was a uniform vacuum energy. If this phase occurred before the
radiation-dominated era, then the Universe could evolve to be extraordinarily
flat when the radiation-dominated era began, so extraordinarily flat
that the lumpy evolution of the radiation- and matter-dominated periods
would be consistent with the high degree of remaining flatness that
is observed today.
This type of solution to the flatness problem was proposed
in the 1980s by cosmologist Alan Guth. The model is called the Inflationary
Universe. In the Inflation model, our Universe starts out as
a rapidly expanding bubble of pure vacuum energy, with no matter or
radiation. After a period of rapid expansion, or inflation, and rapid
cooling, the potential energy in the vacuum is converted through particle
physics processes into the kinetic energy of matter and radiation. The
Universe heats up again and we get the standard Big Bang.
So an inflationary phase before
the Big Bang could explain how the Big
Bang started with such extraordinary spatial flatness that it
is still so close to being flat today.
Inflationary models also solve the horizon problem. The
vacuum pressure accelerates the expansion of space in time so that a
photon can traverse much more of space than it could in a spacetime
filled with matter. To put it another way, the attractive force of matter
on light in some sense slows the light down by slowing down the expansion
of space itself. In an inflationary phase, the expansion of space is
accelerated by vacuum pressure from the cosmological constant, and light
gets farther faster because space is expanding faster.
If there were an inflationary phase of our Universe before
the radiation-dominated era of the Big Bang, then by the end of the
inflationary period, light could have crossed the whole Universe. And
so the isotropy of the radiation from the Big Bang would no longer be
inconsistent with the finiteness of the speed of light.
The inflationary model also solves the magnetic monopole
problem, because in the particle physics that underlies the inflationary
idea, there would only be one magnetic monopole per vacuum energy bubble.
That means only one magnetic monopole per Universe.
That's why the inflationary universe theory is still the
favored pre-Big Bang cosmology among cosmologists.

But how does Inflation work?

The vacuum energy that drives the rapid expansion
in an inflationary cosmology comes from a scalar field that is part
of the spontaneous symmetry breaking dynamics of some unified theory
particle theory, say, a Grand Unified Theory or string theory.
This field is sometimes called the inflaton.
The average value of the inflaton at temperature T is the value at the
minimum of its potential energy at that temperature. The location of
this minimum changes with temperature, as is shown in the animation
to the right.
For temperatures T above some critical temperature T_{crit},
the minimum of the potential is at zero. But as the temperature cools,
the potential changes and a second minimum develops in the potential
at a nonzero value. This signals something called a phase transition,
like when steam cools and condenses into water. For water the critical
temperature T_{crit} where this phase transition happens is
100°C, or 373°K.
The two minima in the potential represent the two possible
phases of the inflaton field, and of the Universe, at the critical temperature.
One phase has the minimum of the field f=0,
and the other phase represents the vacuum energy if the ground state
has f=f_{0}.
According to the inflationary model, at the critical temperature,
spacetime starts to under go this phase transition from one minimum
to the other. But it doesn't do it smoothly, it stays in the old "false"
vacuum too long. This is called supercooling. This region of false vacuum
expands exponentially fast, and the vacuum energy of this false vacuum
is the cosmological constant for the expansion. It is this process that
is called Inflation and solves the flatness, horizon and monopole problems.
This region of false vacuum expands until bubbles of the
new broken symmetry phase with f=f_{0
}form and collide, and eventually end the inflationary phase. The
potential energy of the vacuum is converted through to kinetic energy
of matter and radiation, and the Universe expands according to the Big
Bang model already outlined.

A testable prediction?

It's always good to have testable predictions from a theory
of physics, and the inflation theory has a distinct prediction about
the density variations in the cosmic microwave background. A bubble
of inflation consists of accelerating vacuum. In this accelerating vacuum,
a scalar field will have very small thermal fluctuations that are nearly
the same at every scale, and the fluctuations will be have a Gaussian
distribution. This prediction fits current observations and will be
tested with greater precision by future measurements of the cosmic microwave
background.

So are all the problems solved?

Despite the prediction above, inflation as described above
is far from an ideal theory. It's too hard to stop the inflationary
phase, and the monopole problem has other ways of resurfacing in the
physics. Many of the assumptions that go into the model, such as an
initial high temperature phase and a single inflating bubble have been
questioned and alternative models have been developed.
Today's inflation models have evolved beyond the original
assumption of a single inflation event giving birth to a single Universe,
and feature scenarios where universes nucleate and inflate out of other
universes in the process called eternal inflation.
There is also another attempt to solve the problems of
Big Bang cosmology using a scalar field that never goes through an inflationary
period at all, but evolves so slowly so that we observe it as being
constant during our own era. This model is called quintessence,
after the ancient spiritual belief in the Quinta Essentia, the spiritual
matter from which the four forms of physical matter are made.
So where does string theory fit in all of this? That's
the next topic.