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The equation of motion for a scalar field in an expanding universe
is

(5.10) 
To define an adiabatic invariant occupation number for this field
we need to switch to conformal variables in which this equation
becomes a more standard oscillator equation. These variables are

(5.11) 
Using conformal time and noting that

(5.12) 
the equation of motion becomes

(5.13) 
Then switching to conformal field values note that

(5.14) 
and thus

(5.15) 
This equation can be approximated in Fourier space by

(5.16) 
where
and

(5.17) 
Next: Occupation Number and Energy
Up: Definitions of Number and
Previous: Rescaled Fourier Transforms
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