Power-Law Expansion

LATTICEEASY is designed to self-consistently solve for the evolution of scalar fields $f$ and the scale factor $a$ 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 power-law 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 $C_i$ to denote constants of the equations. The $C_i$ 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

$$a = C_1 (t-t_0)^\gamma.$$ (6.37)

The program time is rescaled as

$$t_{pr} \equiv B \int a^s dt = C_1^s B \int (t-t_0)^{\gamma s} dt = {C_1^s B \over \gamma s + 1} (t-t_0)^{\gamma s + 1} + C_2,$$ (6.38)

which can be inverted to give

$$t-t_0 = \left({\gamma s + 1 \over C_1^s B} (t_{pr}-C_2)\right)^{1 \over \gamma s + 1}$$ (6.39)

and thus

$$a = (C_3 t_{pr}-C_4)^{\gamma \over \gamma s + 1}.$$ (6.40)

To solve for the parameters $C_i$ we want to match the values of $a$ and $H$ at the beginning of the simulation, $t_{pr}=0$. The scale factor itself has an arbitrary scaling and is set to $1$ initially, while the Hubble constant has some well defined initial value $H_0$. The first constraint trivially gives $C_4=-1$. The second constraint is most easily defined in terms of the program value of the Hubble constant,

$$H_{pr} \equiv {a' \over a} = {\gamma \over \gamma s + 1} {C_3 \over C_3 t_{pr} + 1}.$$ (6.41)

Let $H_{pr,0}$ be the value of $H_{pr}$ when $t_{pr}=0$

$$C_3 = {\gamma s + 1 \over \gamma} H_{pr,0}.$$ (6.42)

So

$$a = \left({1 \over G} H_{pr,0} t_{pr} + 1\right)^G$$ (6.43)

where

$$G \equiv {\gamma \over \gamma s + 1}.$$ (6.44)

The program value $H_{pr,0}$ is derived in section 6.3.6 and is automatically calculated by the program. The rescaling variable $s$ should be defined for your model, so all you need for a power-law expansion is to specify the value of $\gamma$, which is declared in parameters.h with the variable name expansion_power. Note that if you know the equation of state $p=\alpha \rho$ that you want the corresponding power-law expansion will be given by

$$\gamma={2 \over 3(1+\alpha)}.$$ (6.45)

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

$$f(t_{pr}) \equiv {1 \over G} H_{pr,0} t_{pr} + 1.$$ (6.46)

then the final form of the power-law expansion equations is

$$a = f^G$$ (6.47)

$$a' = H_{pr,0} f^{G-1} = {H_{pr,0} \over f} a$$ (6.48)

$$a'' = {G-1 \over G} H_{pr,0}^2 f^{G-2} = {(G-1) H_{pr,0}^2 \over G f^2} a.$$ (6.49)

The parameters $f$ and $G$ are called sfbase and sfexponent respectively in the program.