# Using Hubble parameter defn to change integration variable from t to a

1. Feb 19, 2013

### ck99

1. The problem statement, all variables and given/known data

r1 = tt1 1/a(t) dt

Use the Hubble parameter definition to change from t to a, if a(t) = a and a(t1) = 1

2. Relevant equations

Hubble parameter H = a' / a where a' = da/dt

3. The attempt at a solution

aH = da/dt

dt = 1/aH da

Substitute this in to the original integral to get

r1 = a1 (1/a) (1/aH) da

Simplify, because

(1/aH)(1/a) = 1/(a2H) = 1/(aa') (from Hubble parameter defn)

So r1 = a1 1/(aa') da

I'm not sure if this is the right approach, and if it is, how do I integrate this expression with respect to a when it has an a' term in it? I think my effort is all wrong, because it does not help me at all with the second part of the question :(

Any help much appreciated!