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?
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$\begingroup$ The spacetime curvature is proportional to $H^2$. One hand wavy way to think about spacetime curvature is that it is a measure of how the local spacetime geometry is changing. So in cosmology, a large spacetime curvature means a large expansion rate (ie the spacetime is changing rapidly), which by einsteins equations (or if you prefer the Friedman equation) is driven by a large density. So there is no contradiction. $\endgroup$– AndrewCommented Oct 19, 2014 at 0:12
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$\begingroup$ @Andrew thanks for your reply! But I'm still not sure why greater matter density would result in greater expansion rate- shouldn't greater matter density mean a greater force of gravity so slower expansion rate? $\endgroup$– MeepCommented Oct 19, 2014 at 7:55
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$H$ 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.