Redshift Academy


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 ~ e⁶⁰ 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 = φ²/2 + (∇φ)² + V(φ) 

For a homogeneous and isotropic field ∇φ = 0.  We can write:
                 .
Lagrangian: L = [φ²/2 - V(φ)]a³ 
                     .
Euler-Lagrange:  d(a³φ/2)/dt = -a³∂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:

H² = 8πGρ/3
          .
  =  8πG(φ²/2 + V(φ))/3

Assume that φ is moving very slowly so we can write:

H² = 8πGV(φ)/3
 .
(a/a) = √(8πGV(φ)/3)

This has a solution:

a = c'exp{√(8πGV(φ)/3)t} ~ e⁶⁰!

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:

H² = (8/3)πMG/a⁴ - k/a².  

During inflation the energy density is enormously large and decreases
very slowly.  If the scale factor increases extremely rapidly, the 
(8/3)πMG/a⁴ will dominate the k/a² term despite the fact that there 
is a factor of 100 difference in their denominators. Essentially, 
the k/a² 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 + ω²x = 0

Where ω is the restoring force.

This has solutions of the form x = Aexp(αt).  Substituting this 
with m = 1 gives:

(α² + ηα + ω²) = 0    (the exp(αt) term is ignored)

Therefore,

α = -η/2 ± √(η² - 4ω²)/2

There are 3 cases:



Underdamped:  η² < 4ω² 

With solution:

x = Aexp{-ηt/2)exp[±i√(4ω² - η²)/2]t}

Critically Damped:  η² = 4ω² or η = ±2ω

Overdamped:  η² > 4ω²

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):

∂²φ/∂t² - ∂²φ/∂x² = -∂V/∂φ

For the inflaton field we write:

∂²φ/∂t² - (1/a²)∂²φ/∂x² = 0

Note the 1/a² term is needed because of the stretching of space.

We can rewrite the Hubble Friction equation as:

∂²φ/∂t² + 3H∂φ/∂t - (1/a²)∂²φ/∂x² = 0

With solution φ = φ(t)exp(ik'x) where k' = 2πa/λ

Thus, we get:
..    .
φ + 3Hφ + (k'/a)²φ = 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:

(α² + ηα) = 0

Or,

α(α + η) = 0

Which has the solution:

A₀ + A₁exp(-ηt)

Thus, the wave exponentially decays and freezes at the value A₀.

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.