cosmology - When did the first carbon nucleus in the Universe come into existence?

I am a chemist with a passion for astrophysics and particle physics, and one of the most marvellous things I have learned in my life is the process of stellar nucleosynthesis. It saddens me how my colleagues point to the periodic table and draw compound structures so often, yet never seem to be curious about where the elements on which they rely came from, and completely miss the beauty underneath.

As far as I understand, there are several types of nucleosynthesis conditions (big bang, stellar, supernova, black hole accretion disk, artificial, pycnonuclear, and perhaps others), the first three of which are most important when discussing the composition of the Universe, but I wish to focus on an aspect of big bang nucleosynthesis.

The Universe became extremely hot after inflation (or so it is thought), and while cooling down there was a very brief period in which the temperatures and densities were adequate for protons and neutrons to exist and fuse into heavier elements. However, this period of nucleosynthesis was so brief that most of the nucleons formed in baryogenesis didn't even have time to fuse, ending up as hydrogen. Of the amount that did manage to fuse, almost all of it stopped at helium, helped by the exceptional stability of the $^4He$ nucleus relative to its neighbours. Only a tiny amount of lithium is said to have been made, and I've rarely heard anyone discussing elements heavier than that. It is often said that "metals" (in the astronomical sense) only really appeared after stellar nucleosynthesis began, but there was already a lot of matter around after baryogenesis ended (about the same amount as the $10^{80}$ baryons present now?), so even the tiniest atomic fraction of an element could correspond to galactic masses' worth of it.

It would be really neat if it turned out that carbon, the very stuff of life as we know it, didn't exist at all until over a hundred million years after the big bang, when the first generation of stars began to light up. However, I don't know if this is the case, as confirmation or refutation relies on precise quantitative modelling of the big bang nucleosynthesis. Is it possible to figure out when the first carbon nuclei came into existence? I tried using a search engine, but my specific query is completely drowned in links covering the more general aspects of nucleosynthesis.

Note: I suppose that if one were to consider the matter outside the observable Universe, then there probably were carbon nuclei even the tiniest timestep after nucleosynthesis began, because the sheer ridiculous amount of space expected to be out there could well overwhelm the nigh-impossible odds of carbon formation. Thus, the calculation is more interesting if limited to our observable volume.

Answer

The article The path to metallicity: synthesis of CNO elements in standard BBN attempts to quantify the amount of carbon produced during big bang nucleosynthesis.

It concludes that the ratio of carbon-12 formed to hydrogen was $~4 \times 10^{-16}$

with lesser amounts of carbon-13 and carbon-14.