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

[ck99]

Homework Statement

r₁ = _t∫^t1 1/a(t) dt

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

Homework Equations

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

The Attempt at a Solution

Start with Hubble parameter definition, and rearrange to find dt

aH = da/dt

dt = 1/aH da

Substitute this into the original integral to get

r₁ = _a∫¹ (1/a) (1/aH) da

Simplify, because

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

So r₁ = _a∫¹ 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!

 

[mighty2000]

Your approach is correct so far! To integrate the expression with respect to a, you can use the substitution method. Let u = aa'. Then du = a'da. Substituting this into the integral, we get:

r1 = ∫1 1/u du

Integrating, we get:

r1 = ln(u) + C

Substituting back for u, we get:

r1 = ln(aa') + C

Since we know that a(t1) = 1, we can substitute this in to get:

r1 = ln(aa') - ln(1) = ln(aa')

To solve for a, we can take the exponential of both sides:

e^r1 = aa'

a = e^r1 / a'

This gives us the final expression for r1 in terms of a, using the Hubble parameter definition.

Hope this helps!