Solving expansion rate for a variant of the Friedmann equation

[Kyrios]

Homework Statement

For the equation H^2 = \frac{8 \pi G \rho_m}{3} + \frac{H}{r_c} how do I find the value of H for scale factor a \rightarrow \infty, and show that H acts as though dominated by \Lambda (cosmological constant) ?

Homework Equations

\rho_m \propto \frac{1}{a^3}
H > 0

The Attempt at a Solution

I'm not sure how to show that H is driven by \Lambda, but have tried to sub in the scale factor in place of matter density and make the scale factor go to infinity.
As in,
H^2 = \frac{8 \pi G }{3 a^3} + \frac{H}{r_c}
This gets rid of the \frac{8 \pi G \rho_m}{3} leaving H = \frac{1}{r_c}

 

[George Jones]

What is $r_c$? Is it constant?

 

[Kyrios]

Yes r_c is a constant which is called the cross over scale. I don't think we need to know the value of it

 

[George Jones]

Okay. Now, solve

$$H = \frac{1}{r_c}$$

for the scale factor $a$.

 

[George Jones]

Alternatively, what is $H$ for a universe that has a non-zero cosmological constant $\Lambda$, and that is otherwise empty, i.e., that has no matter or radiation content?

 

[Kyrios]

ok, so H = \frac{1}{r_c}
H = \frac{1}{a}\frac{da}{dt} = \frac{1}{r_c}
\int_{0}^{a} \frac{1}{a} da = \int_{0}^{t} \frac{1}{r_c} dt
ln(a) = \frac{1}{r_c} t
a = \exp(\frac{t}{r_c})

Which is akin to dark energy domination a \propto \exp(Hr_c) ?