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Inflation Theory
---------------
Inflation theory proposes a period of extremely rapid (exponential)
expansion of the universe between 10^{-36} and 10^{-32} seconds after the
Big Bang followed by a more gradual expansion. During this time the
universe expanded ~ e^{60} times.
For inflation to have occurred it is postulated that some scalar field
exists with a particular potential function such that when its vacuum
type energy density is very large that density decreases very slowly.
When its energy density decreases past a certain point, it stops
behaving in this way and starts decreasing at the same rate as ordinary
matter. Such a field is referred to as the 'INFLATON' field, φ. φ
can be represented by:
Potential
Energy
V(φ) slight negative slope
| |
| V
| A------------B
| \
| \ <-- Big Bang (starting heat energy (aka reheating)
| \
| \
| -- <- Λ (Dark energy)
------------------------------ φ
We can imagine the universe as a ball slowly rolling down the slope and
then falling over a 'cliff' where the ball literally represents the
value of the field at any particular point.
The energy of the field per unit volume is given by:
.
E = φ^{2}/2 + (∇φ)^{2} + V(φ)
For a homogeneous and isotropic field ∇φ = 0. We can write:
.
Lagrangian: L = [φ^{2}/2 - V(φ)]a^{3}
.
Euler-Lagrange: d(a^{3}φ/2)/dt = -a^{3}∂V/∂φ
.. . .
=> φ + 3(a/a)φ = -∂V/∂φ
.
The 3(a/a) term is analagous the 'friction' term in the viscosity
equation above. We can write:
.. .
φ + 3Hφ = -∂V/∂φ
This is the HUBBLE FRICTION equation.
Returning to the FRW equation:
H^{2} = 8πGρ/3
.
^{ }= 8πG(φ^{2}/2 + V(φ))/3
Assume that φ is moving very slowly so we can write:
H^{2} = 8πGV(φ)/3
.
(a/a) = √(8πGV(φ)/3)
This has a solution:
a = c'exp{√(8πGV(φ)/3)t} ~ e^{60}!
So during the inflationary period the universe expanded enormously
fast. Any 'wrinkles' got flattened out and space becomes very
homogeneous and isotropic. Consider a radiation dominated
universe with k = +1:
H^{2} = (8/3)πMG/a^{4} - k/a^{2}.
During inflation the energy density is enormously large and decreases
very slowly. If the scale factor increases extremely rapidly, the
(8/3)πMG/a^{4} will dominate the k/a^{2} term despite the fact that there
is a factor of 100 difference in their denominators. Essentially,
the k/a^{2} term becomes insignificant which is consistant with what we
observe today. Another way to look at the flatness issue is to
consider a small ballon. An ant on the surface would see a certain
amount of curvature. If that balloon is expanded to the size of the
Earth, the ant would see the surface as being flat in all directions.
Likewise any 'lumpiness' on the surface of the balloon would get
smoothed out in the process. The heights of the lumps won't change
with respect to the surface (technically not true but good enough
to illustrate the idea) but their widths will. As a result, their
inclines decrease giving the appearance of a flatter surface.
Inflation also successfully explains the absence of relic particles.
Inflation allows for magnetic monopoles to exist as long as they were
produced prior to the period of inflation or during its early stages
when temperatures were hot enough. During inflation, the density of
monopoles drops exponentially, so their abundance has dropped to
undetectable levels. It is worth noting that since inflation reduces
the particle density to virtually zero, we know the particles that
exist in the universe today must all have been produced after the
inflation epoch.
Finally, since inflation supposes a burst of exponential expansion in
the early universe, it follows that distant regions were actually much
closer together prior to inflation than they would have been with only
a standard Big Bang expansion. Thus, such regions could have been in
contact prior to inflation and could have attained a uniform
temperature. Thus, inflation also solves the 'horizon problem'.
Damped Oscillator
-------------------
The damped oscillator is described by the equation:
.. .
mx + ηx + ω^{2}x = 0
Where ω is the restoring force.
This has solutions of the form x = Aexp(αt). Substituting this
with m = 1 gives:
(α^{2} + ηα + ω^{2}) = 0 (the exp(αt) term is ignored)
Therefore,
α = -η/2 ± √(η^{2} - 4ω^{2})/2
There are 3 cases:
Underdamped: η^{2} < 4ω^{2}
With solution:
x = Aexp{-ηt/2)exp[±i√(4ω^{2} - η^{2})/2]t}
Critically Damped: η^{2} = 4ω^{2} or η = ±2ω
Overdamped: η^{2} > 4ω^{2}
It is the underdamped and critically damped solutions that are
of the most interest.
Recall, the wave equation for a scalar field in flat spacetime
is (c = 1):
∂^{2}φ/∂t^{2} - ∂^{2}φ/∂x^{2} = -∂V/∂φ
For the inflaton field we write:
∂^{2}φ/∂t^{2} - (1/a^{2})∂^{2}φ/∂x^{2} = 0
Note the 1/a^{2} term is needed because of the stretching of space.
We can rewrite the Hubble Friction equation as:
∂^{2}φ/∂t^{2} + 3H∂φ/∂t - (1/a^{2})∂^{2}φ/∂x^{2} = 0
With solution φ = φ(t)exp(ik'x) where k' = 2πa/λ
Thus, we get:
.. .
φ + 3Hφ + (k'/a)^{2}φ = 0 (the exp(ik'x) term is ignored)
This has exactly the same form as the damped oscillator equation
where k'/a has taken the place of ω and η = 3H
So what this means is that waves that start out with different values
of k' get stretched as the universe expands. They go from underdamped
to overdamped and then 'freeze' when the restoring force, k'/a,
becomes 0. The point at which the transition between underdamped
and overdamped (critical damping) occurs when η = 2ω or
η = 2k'/a. Thus,
3H = 2k'/a
or
a = 4πa/3Hλ after substituting for k' = 2πa/λ
For this equation to be true λ must equal:
λ = 4π/3H
The distance at which they get frozen corresponds to 4π/3H - the
HUBBLE HORIZON. Thus, the shorter λ is the longer it takes for the
wave to stretch to the current horizon and freeze and vice versa.
The point is that, regardless of its λ, a wave can only stretch as
far as the current horizon before it gets frozen.
Once critical damping takes place, the restoring force starts to
disappear. The equation for the damped oscillator from before
becomes:
(α^{2} + ηα) = 0
Or,
α(α + η) = 0
Which has the solution:
A_{0} + A_{1}exp(-ηt)
Thus, the wave exponentially decays and freezes at the value A_{0}.
So what is the source of these microscopic waves or oscillations in
the inflaton field (inflaton waves) that eventually get frozen at the
horizon? The answer comes from quantum mechanics. As the universe
effectively 'rolls' down the slope of the field, there are quantum
fluctuations in the vacuum that continously generate microscopic
waves of different wavelengths. These waves superpose on top of each
other with each one eventually undergoing the process described above.
Schematically, we can look at this in the following manner. Consider
an artificial 'overhead' view.
B = top of the cliff
x |
| v
| | +
| \X + X'
| / +
| Y / + Y'
| \ +
| | +
-----------------------> φ
-----------------------> t, a
If there were no fluctuations in the vacuum, the 'zig-zag' line
would just be straight. X and Y would fall over the cliff at the
same time and their subsequent densities would decrease at the
same rate. If, however, we now add the effect of quantum
fluctuations we get a different result. Now the line is blurred
('zig-zag'). In this case, X goes over the cliff first and slides
down the hill where reheating occurs. X becomes X' and moves off
with decreasing density as the universe continues to expand. Y
follows X to form Y' but Y' is delayed in time so his density will
be different to X'. This process created inhomogenities in the
universe that eventually led to the higher density regions
condensing into the stars, galaxies, and clusters of galaxies
that we see today.