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Baryogenesis is the physical process(es) that produced baryon antibaryon asymmetry in the early universe. That means, the laws that governed the bigbang was baryon-antibaryon symmetric.

On the other hand it is usually said that LHC is probing the early universe by creating similar conditions of the early universe.

My question is, by using $p\bar{p}$ collisions, instead of $pp$ collisions, we are not exactly reproducing the early conditions of the universe because the conditions we created are not baryon antibaryon symmetric, are we?

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Recreating the conditions of the early universe is a popular explanation, not an exact statement of what is happening at the LHC. It was never the primary goal of the LHC to produce anything similar to conditions in the early universe. To whatever extent it does so, that's just a side benefit of the searches for the Higgs boson and supersymmetry.

Besides, at the level of detail needed to analyze an LHC collision, a proton is nearly half antimatter anyway. If you're trying to achieve any sort of equality between matter and antimatter, it doesn't make that much of a difference whether you use proton-proton collisions or proton-antiproton collisions; there will be plenty of individual interactions between quarks and antiquarks in either case.

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The LHC depends in large measure on it's huge luminosity for its usefulness. Making anti-proton beams is hard, and making them at high luminosity is harder still.

While there are advantages to $p\bar{p}$, they are not overwhelming and the luminosity issue drives all.

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