Why is the flatness problem called the "flatness" problem? What is its connection to geometry? My understanding of the flatness problem is that it says that if we leave out dark energy and inflation, then the density parameter $\Omega(t)$ tends to $\infty$ or $0$ unless we have $\Omega(t) = 1$ exactly. Thus, $\Omega(t) = 1$ is an unstable equilibrium point, making it very strange to observe $\Omega(t_{0})\approx 1$ today.
My question is, why is this called the "flatness problem?" I don't see the connection to geometry or curvature.
I understand that if $\Omega(t_{0})$ is close to $1$, then $\Omega_{K}(t_{0})\equiv 1-\Omega(t_{0})$ would be close to zero, but how does this relate to the actual curvature value $K$? In particular, isn't $K$ supposed to be constant (so the deviation from flatness is fixed)?
 A: Relation between curvature $k$ and density parameter $\Omega$ can be described with 1st Friedmann equation.
$$(\frac{\dot{a}}{a})^2 +\frac{kc^2}{a^2} = \frac{ 8\pi G }{3}\rho$$
Define Hubble parameter be $H = \dot{a}/a$, and density parameter be $\Omega = 8\pi G \rho/3H^2$, then comparison between $\Omega$ and 1 has same meaning with comparison between $k$ and 0.
$$\frac{kc^2}{a^2 H^2} = \Omega-1 $$
It says $|\Omega-1| \propto 1/\dot{a}^2$. If there is no inflation, $\dot{a}^2$ will decrease, and $\Omega$ increases. Our current observations said $\Omega \simeq 1 $, so density parameter has to be closer to 1 in the early universe stage. It is called flatness problem.
A: As you pointed out, as we go back in time, $\Omega_{\rm total}-1$ needs to be very small (i.e., $|\Omega_{\rm total}-1| \propto 10^{-61}$)
This situation brings to mind the following question;
Why should the universe have started from such a unique situation?
In other words, why would the universe initially started from $|\Omega_{\rm total} -1| = 0.00000...000001$ when it could have taken different $\Omega_{\rm total}$ values?
The situation can be viewed from the following perspective;
If the universe took any other value, we wouldn't be here, so it had to take that value (anthropic principle). But physicists don't like the anthropic principle.  So the only solution is to come up with the idea that for any initial $\Omega_{\rm total}$ value, the universe would end up with $\Omega_{\rm total} \approx 1$. The inflation mechanism provides that.
A: If the universe is finite, then as it gets larger its curvature gets smaller. I am confused why this fact is not discussed in the context of "the flatness problem". The idea that the "density parameter has to be closer to 1 in the early universe stage" seems backward.
