# How does the critical density decide the fate of the Universe?

Trying to teach myself cosmology. Given the Friedmann equation $$\left[\frac{1}{R}\frac{dR}{dt}\right]^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2}}{R^{2}}$$ and the critical density $$\rho_{c}\left(t\right)=\frac{3H^{2}\left(t\right)}{8\pi G}$$

I can see why, if $\rho>\rho_{c}$ then $k=+1.$ But I can't see why this leads to a collapsing universe? Conversely, if $\rho<\rho_{c}$ then $k=-1.$ But why does this lead to an expanding universe? I'm possibly confused because I've seen graphs of other FRW models with no correlation between the sign of $k$ and whether the universe is contracting or expanding. Do I need to learn about the deceleration parameter (I've heard of it and that's all) in order to understand this? Thank you

This is something I, too, found confusing when first learning about cosmology. The correspondence that you mention between the sign of $k$ and the expansion fate of the universe only holds if there is no dark energy. It turns out that in our universe, which does indeed possess dark energy, $k$ might be zero but the universe may continue to experience accelerated expansion for all time (all available cosmological data is consistent with this scenario).

To solve for $a(t)$, you need not only the Friedmann equation but also a statement of energy conservation and an equation of state for each component of $\rho$. The energy conservation equation is $$\dot{\rho} = -3 H/c^2 (\rho c^2 + p)$$ where $p$ is the pressure, related to $\rho$ through an equation of state $$p = w \rho c^2$$

where $w = 0$ for nonrelativistic matter, $w=1$ for radiation (anything traveling at or very near $c$), $w = -1$ for a cosmological constant, and $-1 < w < -1/3$ for any other type of dark energy.

You can now solve the Friedmann equation for different initial densities and values of $k$. Generally you'll need to do this numerically. However, you can get some great insight by examining simple solutions that you can solve by hand. Barbara Ryden's textbook on cosmology is great for going through many of these partial solutions.

• thanks. Ryden uses parametric equations to get her big bang to big crunch graph of a matter only universe on p87. Just out of interest, do you know why I can't set the first term on the rhs of the Friedmann equation to a constant $C$, $k=+1$, $c=1$ to obtain $$\left[\frac{1}{R}\frac{dR}{dt}\right]^{2}=C-\frac{1}{R^{2}}$$ $$\frac{dR}{dt}=\left(R^{2}-1\right)^{1/2}$$ but when I try to solve this, I don't get the nice Big Bang to Big Crunch graph that Ryden does? – Peter4075 Apr 6 '12 at 8:13
• @Peter4075 the first term on the rhs of the Friedmann equation is not constant as the universe expands or contracts. The matter density $\rho$ scales as $a^{-3}$. – kleingordon Apr 6 '12 at 20:36
• of course! Funny how some things are obvious when they're pointed out. I was getting the strangest results trying to feed my equation into the WolframAlpha differential equation calculator. Thanks. – Peter4075 Apr 7 '12 at 10:14