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This should hopefully be a quick one.

Is there any difference between the Big Bang Theory and the Hot Big Bang Theory?

Around Cambridge I hear everyone using "Hot Big Bang Theory", for example ther first line here of the pages of the Centre for Theoretical Cosmology.

In my undergraduate university it was referred to as the "Big Bang Theory".

I have tried searching online, and there does not seem to be much more mention of a "Hot Big Bang Theory" theory anywhere else on the internet. For example, wikipedia returns no results for the term.

I am almost certain that they are one and the same theory, and that this is simply a case of nomenclature, with Cambridge trying to be different as usual, but I just want to make sure. It's terribely unnecessiarly confusing!

So can someone confirm that there is no difference between these two theories for me please?

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They are one and the same, and the opposing model is the Cold Big Bang.

Cold big bang is a designation used in cosmology to denote an absolute zero temperature at the beginning of the Universe, instead of a (hot) Big Bang.

Also see the Big Bang page of wiki, it also has clues to why it may be called hot.

Cold Big Bang

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Today "hot big bang" and big bang are essentially considered the same. That is because the "cold big bang" has been ruled out once the detection of the acoustic peaks were made (CMB).

apart from that according to this article, it could not explain the expansion of the universe, nor the abundance of light elements.

So they probably mean the same exact paradigm you are used to.

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Of course Hot big bang just seems to make more sense because when the universe was much smaller the energy content of all the universe came together. That means all the universe's energy and matter content got squeezed together and formed a singularity that was extremely hot. Of course the hot big bang and regular big bang are the same because it is what physicists now agree happened at the beginning of the universe.

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I am quoting from some related research literature (https://www.researchgate.net/publication/279420332) -

Friedmann equations help explain what drives the accelerated expansion of the early universe. For a spatially flat universe with Friedmann-Robertson-Walker (FRW) metric, inflation requires a source of negative pressure p and an energy density ρ which dilutes very slowly, while allowing for an exit into the standard Big Bang cosmology at later times - which means having a mechanism which maintains a near-constant value of "change in energy density" (potential) during the inflationary period. There is a wide array of mechanisms for obtaining this near-constant "change in energy density" (potential) during inflation. Two basic approaches include (i) postulating a nearly flat potential, OR (ii) postulating an effective action for potential which contains strong self-interactions which slow the field’s evolution down a steep potential. The spacetime experiences accelerated expansion if and only if the potential energy dominates over its kinetic energy.

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For inflation to successfully address the Big Bang problems, one must simply ensure that the inflationary process produces a sufficient number of ‘e-folds’ of accelerated expansion. After a sufficient number of e-folds have been achieved, the process must terminate. The potential ("change in energy density") descends towards the minimum and "reheats" the universe, with inflaton particles decaying into radiation, and so initiating the HOT BIG BANG.

I am quoting the above in response - because I too reached your question looking for an answer. So yes "Hot Big Bang" does seem to be defined distinctly.

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