Harmonic Modes
Key Concepts:
-
Potential fluctuations on all
scales
-
Each mode oscillates
independently
-
Modes that are half
as long oscillate
twice as fast
The final subtlety is that inflation lays down potential
fluctuations
on all scales. Mathematically we take
the potential fluctuation in space and Fourier decompose it into plane
waves of various wavelengths. Each of
these wave-modes behave independently and so we can think of each individually.
There is however a special relationship between the
temporal
behavior of modes whose wavenumbers are related
by integral multiples:
Because it takes half as long for the fluid to compress
into a potential of half the length scale, the bottom mode in the figure
oscillates exactly twice as fast
as the top mode. Mathematically the frequency of the oscillation
is equal to the wavenumber times the speed of sound: w
=
kcs.
