Quantum Fluctuations in the Very Early Universe
how the tiniest fluctuations evolved through inflation of
the universe into density perturbations
that eventually led to the
largest-scale structures we observe today
The quantum fluctuations in the early universe can be expressed as a
spatial perturbation to the scalar field:
These fluctuations can be decomposed into their Fourie components (using
the plane-wave expansion): 
,
where δk(t) is the mode amplitude, and
ak and ak† are annihilation
and creation operators, respectively.
Each Fourier mode oscillates independently of others.
These commutation relations underlie the uncertainty relations, which
in turn motivate the zero-point vacuum fluctuations to the scalar
field. The scalar field is assumed to be in the vacuum state
(corresponding to no inflaton particles) because the radiation energy
density due to the inflaton particles should be much less than the
potential for inflation to occur.
We can use the Lagrangian density: 
to come up with a scalar field equation describing
the evolution:
,
where 
is the Hubble parameter [a(t) = universal scale factor].
For a free (non-interacting) scalar field, the potential is:
.
Applying the perturbation
to the field:
.
Fourier transform of this gives the field equation for each Fourier
component:
,
where k is the comoving wavenumber vector of the fluctuation mode.
Until a few Hubble times after a given Fourier component exits the
horizon (which happens when a(t) = ck/H), the mass is negligible
(m2 << H2 during slow-roll inflation) so that the
field equation can be simplified to: [afterwards a(t) ~ e H
t]
.
With the above expansion (), the mode amplitude
δk(t) must satisfy this evolution equation:
.
Plus, δk(t) must satisfy the initial boundary
condition to be consistent with the flat-space quantum theory when
a(t) < ck/H. Approximating H to be constant during such a
short time period, the solution is:
.
A few Hubble times after the Fourier mode exited the horizon,
a(t*) > ck/H, the fluctuation amplitude becomes frozen at:
.
The power spectrum of such fluctuations would be:
.
These fluctuations in the scalar field give rise to perturbations in
the energy density: 
,
which, through gravitational effects, eventually resulted in large-scale
structures of the universe