How Can Separating Variables Show Scale Factor Growth in Friedmann Equation?

[hnnhcmmngs]

Homework Statement

Consider a flat expanding universe with no cosmological constant and no curvature (k=0 in the Friedmann equation). Show that if the Universe is made of "dust", so the energy density scales like 1/R^3 , then the scale factor, R(t), grows as t^(2/3). Show if it is made of radiation (so the energy density scales as 1/R^4 -- the extra factor of R comes from the redshift), then it grows as t^(1/2). In both cases, show that for early times, the scale factor grows faster than light. Is this a problem?

Relevant Equations

(adot/a)^2 = 8*pi*G*rho/3 (because k=0)

I was shown that adot^2/a^2 = c/a^3, adot = c / √(a), then da/dt = c / √(a) . Then I was told that I have to integrate this, but I don't understand where to go from there or how this will show me that the scale factor grows as t^(2/3).

 

[TSny]

da/dt = c / √(a)

Separate the variables a and t.
https://www.mathsisfun.com/calculus/separation-variables.html