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How did the Higgs interaction differ due to the densities, temperatures, pressures, etc. in the moments following the big bang? I am aware that around $10^{-43}$ seconds all the fundamental forces were thought to be unified into a single force.

With that being said what was the progression of the Higgs interaction during this time and the following moments.

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Here is a timeline after the big bang:

timelin bb

As far as I know, the Higgs mechanism ( not interaction)concept would be inactive after the breaking off of gravity from a unified symmetric model: the standard mode particles in that soup would have zero mass until the time of electroweak breaking in the plot, about $10^{-12}$ seconds after the BB. It is at that time that the energy of the universe reaches the electroweak breaking energy connected with the vacuum expectation value of the higgs field, It is then that the particles acquire mass, including the Higgs boson. In the interval before it follows the structure given in this link.

The Higgs field, through the interactions specified (summarized, represented, or even simulated) by its potential, induces spontaneous breaking of three out of the four generators ("directions") of the gauge group U(2). This is often written as SU(2) × U(1).

However, after symmetry breaking, these three of the four degrees of freedom in the Higgs field mix with the three W and Z bosons ( W+,W−and ), and are only observable as components of these weak bosons, which are made massive by their inclusion; only the single remaining degree of freedom becomes a new scalar particle: the Higgs boson

So it is a complicated mathematical model, maybe this talk will help in understanding (written before the discovery of the Higgs boson, one has to distinguish between the Higgs field and the boson)

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