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In terms of the conformal variables of the previous section it
makes sense to define occupation number as
![\begin{displaymath}
n_k \equiv {1 \over 2}\left(\omega_k \vert\tilde{F}_{k,c}\vert^2 +
{1 \over \omega_k} \vert\tilde{F}_{k,c}'\vert^2\right).
\end{displaymath}](img159.png) |
(5.18) |
Note that this quantity is adiabatically invariant, meaning it is
conserved in the limit
. Note
also that because it is defined in terms of
instead of
,
is unitless.
The energy density
is defined as
![\begin{displaymath}
\rho_k \equiv \omega_k n_k = {1 \over 2}\left(\omega_k^2
\vert\tilde{F}_{k,c}\vert^2 + \vert\tilde{F}_{k,c}'\vert^2\right).
\end{displaymath}](img162.png) |
(5.19) |
To convert these definitions back to physical coordinates note
that
![\begin{displaymath}
a' = a \dot{a};\;a'' = a^2 \ddot{a} + a
\dot{a}^2;\;\tilde{F...
...a' \tilde{F}_k =
a^2 \dot{\tilde{F}}_k + a \dot{a} \tilde{F}_k
\end{displaymath}](img163.png) |
(5.20) |
so
![\begin{displaymath}
n_k ={1 \over 2}\left( a^2 \omega_k \vert\tilde{F}_k\vert^2 ...
...dot{\tilde{F}}_k + {\dot{a} \over a}
\tilde{F}_k\vert^2\right)
\end{displaymath}](img164.png) |
(5.21) |
![\begin{displaymath}
\rho_k ={1 \over 2}\left( a^2 \omega_k^2 \vert\tilde{F}_k\ve...
...dot{\tilde{F}}_k + {\dot{a} \over a}
\tilde{F}_k\vert^2\right)
\end{displaymath}](img165.png) |
(5.22) |
![\begin{displaymath}
\omega_k^2 = k^2 + a^2 \left<{\partial^2 V \over \partial
f^2}\right> - a \ddot{a} - \dot{a}^2.
\end{displaymath}](img166.png) |
(5.23) |
Finally, in terms of the discrete Fourier transform
![\begin{displaymath}
n_k = {a^2 dx^6 \over 2 L^3} \left[\omega_k \vert f_k\vert^2...
...omega_k} \vert\dot{f}_k + {\dot{a} \over a}
f_k\vert^2\right].
\end{displaymath}](img167.png) |
(5.24) |
![\begin{displaymath}
\rho_k = {a^2 dx^6 \over 2 L^3} \left[\omega_k^2 \vert f_k\vert^2
+ a^2 \vert\dot{f}_k + {\dot{a} \over a} f_k\vert^2\right].
\end{displaymath}](img168.png) |
(5.25) |
Next: Program Variables
Up: Definitions of Number and
Previous: Conformal Coordinates
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This
documentation was generated on 2008-01-21