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In general the program uses neither physical nor conformal
coordinates but a run-specific set of rescalings defined as
![\begin{displaymath}
f_{pr} \equiv A a^r f;\;dt_{pr} \equiv B a^s dt;\;x_{pr} \equiv B
x.
\end{displaymath}](img169.png) |
(5.26) |
(See section 6.1.) Noting that
scales like
and
![\begin{displaymath}
\dot{a} = B a^s a';\;\ddot{a} = B^2 \left(a^{2 s} a'' + s a^{2 s -
1} a'^2\right)
\end{displaymath}](img171.png) |
(5.27) |
![\begin{displaymath}
f_k = {a^{-r} \over A} f_{k,pr};\;\dot{f}_k = B a^s f_k' = {...
...r A} \left(a^{s-r} f_{k,pr}' - r a^{s-r-1} a' f_{k,pr}\right),
\end{displaymath}](img172.png) |
(5.28) |
the equations for
,
, and
become
![\begin{displaymath}
n_k = {a^{2 - 2 r} \over A^2 B^3} {dx_{pr}^6 \over 2 L_{pr}^...
...k} \vert f_{k,pr}' + (1-r) {a' \over a}
f_{k,pr}\vert^2\right]
\end{displaymath}](img173.png) |
(5.29) |
![\begin{displaymath}
\rho_k = {a^{2 - 2 r} \over A^2 B^3} {dx_{pr}^6 \over 2 L_{p...
...^2 \vert
f_{k,pr}' + (1-r) {a' \over a} f_{k,pr}\vert^2\right]
\end{displaymath}](img174.png) |
(5.30) |
![\begin{displaymath}
\omega_k^2 = B^2\left(k_{pr}^2 + a^{2 + 2 r} {A^2 \over B^2}...
..._{pr}^2}\right> - a^{2 s + 1} a'' - (1+s) a^{2 s} a'^2\right).
\end{displaymath}](img175.png) |
(5.31) |
Using the additional definitions (sections 6.1.1
and 6.3.2)
![\begin{displaymath}
V_{pr} \equiv {A^2 \over B^2} a^{-2 s + 2 r} V;\;m_{pr}^2 \e...
...al
f_{pr}^2}\right>;\;\omega_{k,pr} \equiv {\omega_k \over B},
\end{displaymath}](img176.png) |
(5.32) |
we bring the equations to their final forms
![\begin{displaymath}
n_k = {a^{2 - 2 r} \over A^2 B^2} {dx_{pr}^6 \over 2 L_{pr}^...
...}} \vert f_{k,pr}' + (1-r) {a' \over a}
f_{k,pr}\vert^2\right]
\end{displaymath}](img177.png) |
(5.33) |
![\begin{displaymath}
\rho_k = {a^{2 - 2 r} \over A^2 B} {dx_{pr}^6 \over 2 L_{pr}...
...s} \vert
f_{k,pr}' + (1-r) {a' \over a} f_{k,pr}\vert^2\right]
\end{displaymath}](img178.png) |
(5.34) |
![\begin{displaymath}
\omega_{k,pr}^2 = k_{pr}^2 + m_{pr}^2 - a^{2 s + 1} a'' - (1+s)
a^{2 s} a'^2.
\end{displaymath}](img179.png) |
(5.35) |
Next: Number and Energy Spectra
Up: Definitions of Number and
Previous: Occupation Number and Energy
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