I know that in our particular case the inflaton field expanded the volume of the universe while simultaneously maintaining a mass-energy density close to the critical density all the while, thus the flatness.
But why that number? Why didn't the inflaton field maintain a density which would have caused some other kind of curvature instead? I thought inflation was supposed to solve the fine tuning problem but I still run into the problem.
Basically, what makes inflation "choose" which energy density to maintain? And if it always picks the critical density (for flat curvature), why is that?
Answer
Have a look at http://en.wikipedia.org/wiki/Flatness_problem#Inflation
The key point is the equation:
$$(\Omega^{-1} - 1)\rho a^2 = \frac {-3kc^2}{8 \pi G}$$
In inflation the scale factor, $a$, increases enormously, but at the same time the energy density, $\rho$, remains roughly constant because it's dominated by the inflaton field, so the product $\rho a^2$ increases enormously. However the right hand side of the equation is just constant, so this means $(\Omega^{-1} - 1)$ must get vanishingly small i.e. $\Omega$ is driven towards unity. This result is quite general and doesn't depend on the fine details of how inflation works.
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