Theorem 1. Assume that a cooling primordial soup after the Big Bang favors creation of new baryons, that is, baryons become more abundant when the soup cools.
Assume that there exists an unknown attractive force which pulls together matter baryons, and pulls together also antimatter baryons.
Then there will form areas where matter baryons dominate and areas where antimatter baryons dominate. This is spontaneous symmetry breaking, or in a sense, crystallization of the soup.
Proof. Because of the unknown attractive force, distributions where matter baryons and antimatter baryons are separated have a lower potential than mixed distributions. When the soup cools it favors a lower potential.
It is like cooling of water where two types of ice can form, but the types cannot coexist close to each other. When the water cools, ice must form. Since the ice cannot be mixed, the two types of ice have to appear as separate "crystals". QED.
We do not know how to calculate the crystallization process. We seem to live inside one huge crystal.
Crystallization takes a long time. Is this compatible with a fast Big Bang? If we assume extremely heavy particles which can decay into either matter or antimatter, the crystallization might happen very quickly. The visible universe is then the decay product of one or more such superheavy particles.
Actually, we do not need Theorem 1 at all if we assume a superheavy particle which can decay into an entire visible universe.
https://en.m.wikipedia.org/wiki/Baryogenesis
Andrei Sakharov in 1967 wrote three "necessary" conditions for baryogenesis. It turns out the first two conditions are not necessary. There is no need for baryon number violation or any symmetry violation. The crystallization process causes apparent symmetry breaking.
The third condition of Sakharov is that interactions are out of thermal equilibrium. The third condition is trivial: if something is "born", then the system must have been out of equilibrium.
Lumping together even when there is no attraction
Even if there is no unknown attraction between matter baryons, there will be some lumping together of matter. The following toy model clarifies the thing.
Suppose that we have n baskets and a random flux of baryons and antibaryons which fall in them. We assume that in each basket, antiparticles immediately annihilate. In each basket, the number
N_baryons - N_antibaryons
will perform a random walk. After n particles, its distribution is
2 * B(n, 1/2) - n,
where B is the binomial distribution.
The typical number of particles in a basket grows like sqrt(n) after n particles have fallen into a basket.
We had to assume that individual particles choose their basket at random. If a produced pair baryon-antibaryon would always fall in the same basket, then there would be no lumping together. Since antiparticles attract each other, the cross section for annihilation is bigger for slow particles. If the particles in a pair are born with a high kinetic energy, they will often fall in different baskets.
Lumping together is reduced by the motion of particles from a basket to an adjacent basket.
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