Next: PowerLaw 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

(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

(6.28) 
Plugging this form into equation (6.27) and eliminating
gives

(6.29) 
To determine whether to use the plus or minus sign in equation
(6.31) consider the limit as
. In this limit

(6.32) 
This suggests that the plus sign must be used in order to reduce to
the limit
. Hence

(6.33) 
In the program it's useful to calculate , which is roughly
, so

(6.34) 
Thus equation (6.26) becomes

(6.35) 
where

(6.36) 
Next: PowerLaw 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 20080121