How can we prove that the universe doesn't have a uniform net electric charge density? Physics students are often taught that the universe cannot have significant net electric charge density because the gravitational force appears to dominate at large length scales. According to this argument, the fraction of excess protons or electrons in the universe must be much less than $10^{-20}$.
One might mount the following counterargument: suppose the universe really did have, say, $1 + 10^{-15}$ electrons for every 1 proton, and those excess electrons were uniformly spatially distributed; how would we know? Naively, it seems that the electromagnetic force would be strong enough to maintain the uniform distribution, since gravity would be far too weak to do anything that would tend to make them cluster together, and if this distribution filled the entire universe, there would be nowhere for them to spread out to (other than simply decreasing in density as the universe expands). As a result, the electric field from this excess charge density would be zero by symmetry, and gravity would continue to dominate dynamics at large length scales.
I am pretty sure that this scenario is impossible because the gravitational effects of the electrostatic potential energy on the evolution of the universe would not be consistent with observation (assuming GR is (approximately) correct), but I don't know how to work through the math myself. I'm also interested in any other evidence that disproves this scenario.
 A: It's not possible to prove that the universe doesn't have a net electric charge.
It would only be possible to put limits on the charge that the universe has.
If the charge were big enough to affect the orbits of planets, so that the modelling of planetary orbits couldn't match existing models (that can vary the planets masses), then perhaps a specific charge that is equally distributed could be introduced to make the models match observations.
There would then be the problem of separating this effect from a tiny variation in $G$, and that's only known to a few significant figures.
So unless someone knows a way, it seems that it's very difficult to put a low limit on the net charge of the universe.
If some limit were found for the solar system, there is also the problem of whether the same limit applies to other regions of the universe.

The limit would be difficult to get lower than this:
By dividing the electrical force between two masses by the gravitational force between them (assuming a charge density of $k$ Coulombs per Kg) we get $$\frac{k^2}{4\pi\epsilon_0 G}$$
setting this equal to $2\times 10^{-5}$, the approximate uncertainty in $G$, gives $k = 4 \times 10^{-13}C/Kg$.  This number matches the $10^{-20}$ in your question, although in reality it might even be difficult to disprove the $10^{-15}$ figure.
