# Friedmann Equation and a contracting universe

Consider a universe with a nonzero curvature and matter.

One can write the Friedmann Equation in this universe as such:

$$\frac{H(t)^2}{H_0^2} = \frac{\Omega_0}{a^3}+\frac{1-\Omega_0}{a^2}$$

Where $$H(t)$$ is the Hubble parameter, $$\Omega_0$$ is the density parameter for matter (since it is the only component in the universe aside curvature, I will ignore the subscript) and $$a$$ is scale factor.

Now consider a universe where $$\Omega_0$$ is greater than 1 (positive curvature), leading in a negative term of $$\frac{1-\Omega_0}{a^2}$$. It is obvious that the universe will expand first, then come to a halt when $$H(t) = 0$$.

However, when $$H(t)$$ turns negative, LHS of the equation is again positive, which is no difference from the expanding scenario. Thus I conclude this universe will expand forever, however this universe will meet its end in the Big Crunch. Which means the universe will first expand until it reach a maximum of scale factor, then contract until everything crash together.

Where did I made a mistake?

• @paulina, I conclude that the universe will expand forever, not contract. That is why I am asking where did I made a mistake. Commented Jul 2 at 0:24

The first Friedmann equation says nothing about the sign of $$H$$. During expansion, it says how fast the universe is expanding. During contraction, it says how rapidly the universe is contracting. Think of it like a conservation-of-energy expression in classical mechanics. If you know a particle's conserved energy (analogous to $$1-\Omega_0$$) and position (analogous to $$a$$), that tells you the particle's speed (analogous to $$|aH|$$) but not the direction of its velocity (analogous to $$aH$$).
• At the point where $H=0$ (so $\dot a=0$), you can use the second Friedmann equation to show that $\ddot a<0$. Then you know the sign of $\dot a$ (and hence $H$) is changing from positive to negative, and so it must be negative thereafter.
• Alternatively, you can simply integrate the second Friedmann equation from the outset. The analogy with classical particle mechanics is that the first Friedmann equation is like $E=\frac{1}{2}m\dot{\vec{x}}^2+m\phi(\vec x)$, while the second Friedmann equation is like $\ddot{\vec{x}} = -\nabla\phi(\vec x)$. The first is convenient, but all you need in principle is the second.