Next: Power-Law Expansion
Up: Scale Factor Evolution
Previous: The Scale Factor Equation
Correcting for Staggered Leapfrog
In practice the program uses a staggered leapfrog algorithm so in
solving for
the value of
is known at
where
is the time step. See section 6.4 for more details. The
solution to this problem is to use the two equations
![\begin{displaymath}
a_+' \approx a_-' + d a'';\;a' \approx {1 \over 2}\left(a_+' +
a_-'\right)
\end{displaymath}](img257.png) |
(6.27) |
where
and
refer to the values of
at
and
respectively and all other variables are evaluated at time
. Take the evolution equation to be
![\begin{displaymath}
a'' = -C_1 {a'^2 \over a} + C_2.
\end{displaymath}](img261.png) |
(6.28) |
Plugging this form into equation (6.27) and eliminating
gives
![\begin{displaymath}
a_+' \approx a_-' + d \left(-{C_1 \over 4 a} \left(a_+' +
a_...
...\over
2 a} a_+' a_-' - {d C_1 \over 4 a} a_-'^2 + d C_2 + a_-'
\end{displaymath}](img262.png) |
(6.29) |
To determine whether to use the plus or minus sign in equation
(6.31) consider the limit as
. In this limit
![\begin{displaymath}
a_+' \approx -a_-' - {2 a \over d C_1} \pm {2 a \over d C_1}...
...{2 a \over d C_1} \pm \left({2
a \over d C_1} + 2 a_-'\right).
\end{displaymath}](img269.png) |
(6.32) |
This suggests that the plus sign must be used in order to reduce to
the limit
. Hence
![\begin{displaymath}
a_+' \approx -a_-' - {2 a \over d C_1}\left(1 - \sqrt{1 + {2 d C_1
\over a} a_-' + {d^2 C_1 C_2 \over a}}\right).
\end{displaymath}](img271.png) |
(6.33) |
In the program it's useful to calculate
, which is roughly
, so
![\begin{displaymath}
a'' \approx {1 \over d} \left[-2 a_-' - {2 a \over d C_1}\le...
...{2 d C_1 a_-' \over a} + {d^2 C_1 C_2 \over
a}}\right)\right].
\end{displaymath}](img274.png) |
(6.34) |
Thus equation (6.26) becomes
![\begin{displaymath}
a'' \approx {1 \over d}\left\{-2 a_-' - {2 a \over d C_1} \l...
...a_{pr} f_{i,pr}\vert^2 +
a^{C_4} V_{pr}\right)}\right]\right\}
\end{displaymath}](img275.png) |
(6.35) |
where
![\begin{displaymath}
C_1 = s+2;\;C_3 = 2s+2r+2;\;C_4 = 2s+2
\end{displaymath}](img276.png) |
(6.36) |
Next: Power-Law Expansion
Up: Scale Factor Evolution
Previous: The Scale Factor Equation
Go to The
LATTICEEASY Home Page
Go to Gary Felder's Home
Page
Send email to Gary Felder at gfelder@email.smith.edu
Send
email to Igor Tkachev at Igor.Tkachev@cern.ch
This
documentation was generated on 2008-01-21