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Up: Scale Factor Evolution
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PowerLaw Expansion
LATTICEEASY is designed to selfconsistently solve for the evolution
of scalar fields and the scale factor in an expanding
universe. In some cases, however, you may wish to solve for the
behavior of a set of fields in a universe dominated by other forms of
energy, e.g. pure matter or radiation. In this case you can tell the
program to impose a fixed powerlaw expansion and evolve the fields in
this background. In this section we derive the equations for such an
expansion in program variables. Note that we use the variables
to denote constants of the equations. The in this section have
no relation to the ones in the previous (or any other) section.
For a general constant equation of state the scale factor evolution is
given by

(6.37) 
The program time is rescaled as

(6.38) 
which can be inverted to give

(6.39) 
and thus

(6.40) 
To solve for the parameters we want to match the values of
and at the beginning of the simulation, . The scale
factor itself has an arbitrary scaling and is set to initially,
while the Hubble constant has some well defined initial value
. The first constraint trivially gives . The second
constraint is most easily defined in terms of the program value of the
Hubble constant,

(6.41) 
Let be the value of when

(6.42) 
So

(6.43) 
where

(6.44) 
The program value is derived in section 6.3.6
and is automatically calculated by the program. The rescaling variable
should be defined for your model, so all you need for a powerlaw
expansion is to specify the value of , which is declared in
parameters.h with the variable name
expansion_power. Note that if you know the equation of
state that you want the corresponding powerlaw
expansion will be given by

(6.45) 
(See for example [4].) If we let

(6.46) 
then the final form of the powerlaw expansion equations is

(6.47) 

(6.48) 

(6.49) 
The parameters and are called sfbase and
sfexponent respectively in the program.
Next: Initial Conditions on the
Up: Scale Factor Evolution
Previous: Correcting for Staggered Leapfrog
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This
documentation was generated on 20080121