Early Universe Flatness Problem My question is about the curvature of spacetime in the early universe (Plank era)
https://en.wikipedia.org/wiki/Flatness_problem
Based on data from Wilkinson Microwave Anisotropy Probe, in order to achieve the current flatness value for $\Omega$, the density of the early universe can't have departed from its current density by more than one part in $10^{62}$.
This blows my mind, since the mass of the universe is only $10^{56}$ grams. So if there was one gram more, or less, in the universe, it would be significantly curved?
 A: The flatness problem
To understand this we need to use the Friedmann equation$$
\Omega_M+\Omega_\Lambda-1=\frac{k}{a^2H^2}
$$
Where $\Omega_M$ and $\Omega_\Lambda$ correspond to the fraction of matter and dark energy in the universe, k the curvature of the universe($k=-1,0,1$ corresponds to hyperbolic, flat, spherical geometry), $a$ the scale factor and $H^2$ the expansion rate.
we can re-write this as$$
|\Omega_{tot}-1|=\frac{|k|}{\dot{a}^2}
$$
Now when we consider a matter dominated universe, goes as $a\sim t^{2/3}$ hence $\dot{a} \sim t^{-1/3}$. When we consider a radiation dominated universe $\dot{a}\sim t^{-1/2}$.
Now in both cases the Friedmann equation scales as $$
|\Omega_{tot}-1|\sim t^{2/3}
$$
for a matter dominated universe and
$$
|\Omega_{tot}-1|\sim t
$$
for a radiation dominated universe.
The problem is that over time, the curvature must increase! Only if it's exactly $k=0$ will the curvature stay the same.(This imposes conditions for how close to flat our universe must have been in the very beginning to stay flat until today)
Inflation solution
One way to solve this problem is to consider a rapid expansion of the universe. During inflation the scale factor goes as $$a(t)=\exp\left(t\sqrt{\frac{\Lambda}{3}}\right)$$
This changes the Friedmann equation to $$
|\Omega_{tot}-1| = \frac{3|k|}{\Lambda}\exp\left(-t\sqrt{\frac{4\Lambda}{3}}\right)
$$
Now the expansion of the universe forces the curvature back to $k=0$!!
