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The Scale Factor Equation
The equation for the scale factor
is derived from the
Friedmann equations
![\begin{displaymath}
\ddot{a} = -{4 \pi a \over 3} (\rho + 3 p)
\end{displaymath}](img243.png) |
(6.19) |
![\begin{displaymath}
\left({\dot{a} \over a}\right)^2 = {8 \pi \over 3} \rho
\end{displaymath}](img244.png) |
(6.20) |
For a set of scalar fields
in an FRW universe
![\begin{displaymath}
\rho = T + G + V;\;p = T - {1 \over 3} G - V
\end{displaymath}](img245.png) |
(6.21) |
where
,
, and
are kinetic (time derivative), gradient, and
potential energy respectively with
![\begin{displaymath}
T = {1 \over 2} \dot{f}_i^2;\;G = {1 \over 2 a^2} \vert \nabla
f_i\vert^2.
\end{displaymath}](img248.png) |
(6.22) |
Equations (6.19) and (6.20) and the field evolution
equations form an overdetermined system. In principle either scale
factor equation could be used but in practice it is easiest to combine
them so as to eliminate the time derivative term
because in the
staggered leapfrog algorithm
and
are known at different
times. Eliminating
we get
![\begin{displaymath}
T = {3 \over 8 \pi}\left({\dot{a} \over a}\right)^2 - G - V
\end{displaymath}](img250.png) |
(6.23) |
![\begin{displaymath}
\ddot{a} = -{4 \pi a \over 3} (4 T - 2 V) = -2 {\dot{a}^2 \o...
...er a}\left({1 \over 3} \vert \nabla f_i\vert^2 + a^2 V\right).
\end{displaymath}](img251.png) |
(6.24) |
To convert to program variables note that
![\begin{displaymath}
\dot{a} = B a^s a';\;\ddot{a} = B^2\left(a^{2s} a'' + s a^{2s-1}
a'^2\right),
\end{displaymath}](img252.png) |
(6.25) |
so the scale factor equation becomes
![\begin{displaymath}
a'' = (-s-2){a'^2 \over a} + {8 \pi \over A^2} a^{-2s-2r-1} ...
... 3} \vert\nabla_{pr} f_{i,pr}\vert^2 + a^{2s+2} V_{pr}\right).
\end{displaymath}](img253.png) |
(6.26) |
Next: Correcting for Staggered Leapfrog
Up: Scale Factor Evolution
Previous: Scale Factor Evolution
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This
documentation was generated on 2008-01-21