# Does Friedmann equation allow for complex scale factor?

1. Apr 24, 2012

### Piano man

Looking at the Friedmann equation
$$H^2=\left[\frac{\dot{a}}{a}\right]^2=\frac{8\pi G\rho}{3}-\frac{kc^2}{a^2}$$

and considering positive curvature, then for the limit where the second term dominates, we're left with
$$\left[\frac{\dot{a}}{a}\right]^2=-\frac{kc^2}{a^2}$$

This implies a complex scale factor, does it not?

2. Apr 24, 2012

### Ich

Either that, or it implies that the curvature in this limit is negative. Which is the case.
IMHO, the equation makes more sense if you rearrange it:
$$\frac{kc^2}{a^2}=\left[\frac{\dot{a}}{a}\right]^2=\frac{8\pi G\rho}{3}-H^2$$
Read: Curvature = positive contribution from energy density - negative contribution from expansion.

3. Apr 24, 2012

### Piano man

Okay, so the curvature has a time dependence k=k(t)?

4. Apr 24, 2012

### Ich

Right. For example, in a closed universe, it could be its radius.

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