# Integrating Friedmann Equation of Multi-component universe respect to a and t

1. Dec 28, 2012

### 4everphysics

I am having a trouble finding relationship between 'a' and 't' by integrating friedmann equation in a multi-component universe.

It would be very helpful if you can help me with just
matter-curvature only universe and matter-lambda only universe.

The two integrals looks like following.

Matter-curvature only:

$$H_0 t = ∫_0^a \frac{da}{[Ω_0/a + (1-Ω_0)]^{1/2}}$$

Matter-Lambda only:

$$H_0 t = ∫_0^a \frac{da}{[Ω_0/a + (1-Ω_0)a^2]^{1/2}}$$

2. Dec 28, 2012

### BillSaltLake

Try substituting x = 1/a and then use a table of integrals.

3. Dec 31, 2012

### akhtarphysic

with matter lambda the result is
a(t)=(ro_matter/ro_lambda)^(1/3)*[sinh[(6*Pi*ro_lambda*G)^(1/2)*t]^(2/3)
Where ro_x/ro_critical=omega_0x