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Question

While the standard model of cosmology points to a beginning—often associated with the Big Bang—does modern physics strictly prohibit the idea that matter (or energy) has always existed? Or is there still, however slim, a physically coherent possibility that matter is eternal in some form?

Moreover, have any serious physicists explicitly endorsed or explored the notion of eternal matter within the bounds of established or speculative models?

I’m not asking about metaphysics or theology, but whether this idea finds any legitimate foothold—however marginal—in physical theory.

Answer 1

There are serious physicists who study alternatives to the Big Bang, and in this sense it is possible that the universe (and its contents) exist for more time than we currently believe. Roger Penrose famously discusses something known as conformal cyclic cosmology, and I believe there is a wider class of models involving a "big bounce". For example, instead of the universe starting from a single point, it existed before, contracted a lot until a very small scale, and "bounced" back into expansion.

While the standard model of cosmology points to a beginning—often associated with the Big Bang—does modern physics strictly prohibit the idea that matter (or energy) has always existed?

No. It is possible the universe did not start at a specific event (the "big bang" as usually interpreted). We do have a lot of evidence that something known as the "hot big bang" existed, but this is the modern jargon for an era in which the universe was very hot and much smaller than today. It is not necessarily an initial singularity (I discussed this a bit more in this answer).

Or is there still, however slim, a physically coherent possibility that matter is eternal in some form?

Depends on what you mean by matter being eternal. You can create and annihilate electrons and positrons from photons, for example. Hawking radiation also gives you a different way in which you can create particles from gravity. But it is possible there wasn't an initial singularity.

Moreover, have any serious physicists explicitly endorsed or explored the notion of eternal matter within the bounds of established or speculative models?

Depends on what you mean by eternal matter, but there is serious research on the possibility that the universe didn't start at a specific point in time.

Summary

• We are not fully sure whether there was initial singularity at which the universe started. We are extremely sure that an era known as the "hot big bang" existed, meaning a stage in which the universe was far smaller and hotter. It is possible something happened before that era (e.g. inflation).
• We can create electrons and positrons from photons, and other phenomena. Furthermore, energy does not need to be conserved at universal scales. This makes it difficult to argue what "eternal matter" could mean.