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

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:

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

Matter-Lambda only:

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

Thank you for your help.

BillSaltLake

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

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