Why is the flatness problem a problem? Friedmann equation states
$$ H^2 \equiv \left( \frac{\dot{a}}{a} \right) = \frac{8\pi G \rho}{3} - \frac{k}{a^2}  $$
which describes the evolution of the expansion of the universe. Writing $\Omega(t) = \frac{8 \pi G}{3 H^2(t)}\rho(t)$ we get
$$ |\Omega(t) - 1| = \frac{|k|}{a^2H^2}.$$
Which in a radiation or matter dominated universe gives
$$ |\Omega(t) - 1| \propto t^n, \quad n\geq  2/3.$$
Thus, if the universe is flat now, it was a lot flatter right after the big bang. This is called the flatness problem, but I am having a hard time understanding why it is a problem.
My way of thinking is this: The topology of a closed (open) universe and a flat universe are very different. If the universe is flat ($k=0$) then the spatial part of universe has the topology of $\mathbb{R}^3$, but the topology of a closed universe is a 3-sphere $S^3$. Now, since we have good reasons to think that our universe is pretty flat, why is that not an argument for the universe being exactly flat? As in exactly $k=0$. If $k$ was shown to be exactly $k=10^{-100}$, I understand that we would have a fine-tuning problem. But since the topology of the entire universe would be profoundly different going from being a non-compact $\mathbb{R}^3$ ($k=0$) to being a compact 3-sphere $S^3$ ($k=10^{-100}$), I can easily accept that $k=0$ is a possibility.
Why do physicist think that $k=0$ is unrealistic? Where am I wrong? Have I misunderstood something? Please correct me if I am wrong about something.
 A: Hi Johannes:  As I understand your question, you have assumed a flat universe. There is no flatness problem with a flat universe. The problem is that if the universe is not quite flat now, it would have been a lot less flat without inflation. Inflation is not included in the Friedmann equations, so I am not sure why you are discussing them. Also, your first equation has an error. The $ \frac {\dot a} {a}$ should be squared. Also you you have omitted the cosmological constant $\Lambda$, which is needed to model the universe which is based on the most current research results to be most like the universe we probably have.
A: Good question.
The matter and energy in the universe curves space-time, the problem is that the matter density appears to have a value very close to the critical value
$\rho_{crit}=\frac{3H(z)^2}{8\pi G}$
that would make the universe flat,
$H(z)$ is the Hubble parameter and $G$ is the gravitational constant
It’s nice to think that the universe would be flat, but there was no theoretical reason to know why it is so flat (it could have a curvature).
It’s similar in Newtonian gravity to wondering why the density of the universe is such that there is this coincidence - that the radius $R$ of the observable universe ($\frac{c}{H}$) is close to
$R=\frac{GM}{c^2}\tag{1}$.
It's fine if it's true, but physicists don't like unexplained co-incidences i.e. why is it true?  That's why the flatness problem is a problem.
In the traditional expanding universe any departure from the critical density would increase with time, so measurements nowadays that show it’s close to critical density, means it was very very close in the past.
Inflation was meant to solve this problem, but some would argue that it’s an ad-hoc theory with no explanation of why it began or ended and is mainly introduced to solve the flatness problem.
Another idea here, Cosmology - an expansion of all length scales is that the expansion happens to all length scales
The total energy due to each mass $m$ is
$mc^2-\frac{GMm}{R^2}$
and it changes with time as
$(mc^2-\frac{GMm}{R^2})e^{2Ht}$
If we require conservation of energy as the universe expands
$mc^2-\frac{GMm}{R^2}=0$
so
$G=\frac{Rc^2}{M}\tag{2}$
Explaining the coincidence above and the flatness problem.
The interpretation of this is that the rate of expansion is constant, and gravity is caused by the expansion in order to conserve energy as the universe expands.
The flatness problem is important as most cosmologists think that it should be explained by a correct cosmological model.  Maybe the current concordance model is flawed - it can only explain the flatness problem by introducing inflation and another ad-hoc and unexplained quantity, dark energy.
Perhaps the flatness problem will guide us to a better cosmological model.
A: The flatness problem is akin to asking: why does the universe have these initial conditions?
This is a question that, by definition, cannot be answered in the framework of physics, because every physical theory requires initial conditions as an additional input: a physical theory cannot provide an explanation for the values of its initial conditions.
Initial conditions can be explained in the framework or another, more general theory. For example, even if Friedmann's cosmology cannot explain why $k$ has a certain value, we can introduce inflation and say that we actually started from another value $k'$, and the ln inflation flattened the universe, bringing us to the measured value of $k\sim 0$. But of course, this is just shifting the problem backwards, now we have to explain $k'$!
So, why do people say that explaining $k$ is a problem, but nobody is worried about explaining $k'$? The simple and sad truth is that, since we know very little about inflation, $k'$ cannot be presently measured in any way, and therefore there is no measured value that needs explaining.
Ok, but why is there a flatness problem and not, for instance a "fine structure constant  problem"? Why does $k$ merit such a discussion?
The reason is more phylosophical, than physical (as I said at the beginning, the value of initial conditions cannot be explaned withing physics). When a theory requires adimensional initial parameters, physicists like when those parameters assume values reasonably close to 1. In these cases, no questions asked. But if the number is very large, or very small, such as in the case of $k$, then someone will try to find an explanation.
This is partly motivated by historical reasons, because we never had a physical theory with very small, adimensional parameters that were not later explained in some more general theory. But, to be fair, we do not have many unexplained adimensional parameters at all, in modern physics.
Another reason finds roots in the belief of many scientists that nature must have a fundamental, intrinsic matemathical beauty. And that the simplest and most beautiful number that some variable can assume is, of course 1. Not 0, which looks like there was some magical cancellation coincidence, and not $10^{-100}$, which is even uglier than 0.
