Increasing matter density in the Friedman equations? From what I understand, the Friedman equation is $H^2=\frac{8\pi G}{3} \rho$ and the $H$, the Hubble constant, is a measure of the rate of expansion of the universe. What I am confused about is that this seems to say that as the matter density of the universe increases, the rate of expansion of the universe increases. Should this not be the other way around because greater matter density means greater spacetime curvature (greater force of gravity) so everything gets pulled back more?
 A: $H$ tells us how fast the universe is expanding, relative to how much it has already expanded. It has units of inverse time. For example, if $H=0.1\ \mathrm{s}^{-1}$, then the universe is expanding by 10% every second. Suppose that the density of mass-energy in the universe was so small that deceleration was negligible, and suppose that at $t=1$ s, we have $H=0.1\ \mathrm{s}^{-1}$. Now a billion years go by. At $t=1$ billion years, clearly the universe is no longer expanding 10% per second. The value of $H$ is going to be much smaller. But this doesn't mean that the expansion has been decelerating. In fact, in this special case of the Friedmann universe, with negligible matter density, we have flat spacetime, so it makes sense to talk about the velocity of one galaxy relative to some other, cosmologically distant galaxy. This velocity hasn't changed at all, but $H$ has gone down.

But I'm still not sure why greater matter density would result in greater expansion rate

It isn't really a straightforward cause-and-effect relationship where the density causes a certain expansion rate. In the example given above, expansion has actually caused a decrease in density, while density has had no effect on the rate of expansion.
