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I don't understand the reason and hypothesis behind why the quarks appeared first—as per the big bang cosmology—shortly after the strong and electroweak force separated. We don't know what the quarks or the electrons are composed of, so how can we estimate that the quarks and gluons formed first, and then the leptons followed?

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The terms quark epoch and lepton epoch do not mean that is when those particles were created. As far as we know the Standard model particles were created at the end of the inflationary epoch and both quarks and leptons would have been created at this time (though it must be emphasised that all the theories for what happened this early in the history of the universe are speculative). Instead the terms refer to the times when the behaviour of the matter in the universe was dominated by the quarks and leptons respectively.

The quark epoch refers to the time when the quarks were not bound into hadrons but instead existed as a quark gluon plasma. The leptons would have been present at this time, but it was the QGP that dominated the properties of the matter in the universe. We have managed to create a quark gluon plasma in the ALICE detector at the LHC, but even so I think it's fair to say it's early days and the QGP isn't fully understood so we need to be a little cautious about what exactly went on at this point in the evolution of the universe.

The lepton epoch is the time when the quarks had all combined into hadrons to produce the particles we see today, but the temperature was still high enough to create electron-positron pairs. These pairs were produced in such large numbers that the combined mass of the electrons and positrons was larger than the hadrons and therefore the leptons briefly dominated the universe.

The lepton epoch followed the quark epoch because it needs much more energy to separate quarks into a quark-gluon plasma than it does to create electron-positron pairs, so it happened earlier in the evolution of the universe when the temperature was higher.

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