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For starters, it’s important to understand that the Higgs boson is important only in that creating and detecting the existence of them is way to confirm the standard model of quantum particle physics’ Higgs field feature.  The field is what, in this theory, give leptons and quarks mass – observable Higgs boson particles aren’t necessary for this mechanism to work.

 

That said, according to our best theoretical understanding of the very early, just-after-the Big Bank history of the universe, the answer to the question “was the Highs boson the first elementary particle to be generated after the big bang” is “no”.  Though the theory in this area is more esoteric than I understand, most description state that, until about 10-12 after the Big Bang, the Higgs field had zero expectation value, so quarks and leptons were massless, and it wouldn’t have been possible to create a Higgs boson.

 

We’re talking about a very brief, very weird period of the universe’s history here, though, so need to be careful to understand that our everyday-experience-based intuition isn’t very useful in understanding it, and questions like “which particle came first?” aren’t as useful as we might imagine.

 

I’ve a suspicion you ask this question, petrushkagoogol, because you’ve heard the Higgs boson referred to as “the God particle”, suggesting that, as in theistic origin-of-the-universe stories, it created the universe.  This suggestion is about as wrong as one can be.  Most physicists are appalled that “the God particle” was ever used to refer to the Higgs, and blame it on a book publisher thinking that titling a book on the subject this would improve its sales.

 

Sources: Wikipedia article "Chronology of the Universe"; Matt Strassler's The Higgs FAQ 2.0.

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