What is the limiting abundances of elements at the end of the stelliferous era?

Is there any reputable published source on expected elemental abundances at the end of the era of stellar fusion? I am here interested in the contents of galaxies; much of intergalactic gas will be lost and remain primordial.

Adams & Laughlin mention estimates of Timms that the final chemical composition will be $X\sim 0.2, Y \sim 0.6, Z \sim 0.2$ but does not analyse the composition in much detail beyond the chemical evolution of white dwarfs (which mostly happens in the later degenerate era). Fukugita and Peebles also make use of a very approximate estimate.

It seems entirely doable to run an IMF model, add some intergalactic gas infall assumpptions, add some recycling assumptions for different mass classes, generate a composition and iterate until a limiting value. But it also seems to be a lot of work that I strongly suspect somebody has already done somewhere far better than I could do.

Another, very crude, approach might be to argue that since we know the cosmic element abundances today after 13.6 billion years of stellar activity we can rescale them to a limiting value: $A(t)=A(\infty)(1-e^{-t/\tau})$ where $\tau$ is the timeconstant of stellar elemental conversion. That would give the desired $A(\infty)=A(t_\mathrm{now})/(1-e^{-t_\mathrm{now}/\tau})$ (at least for metals; hydrogen and helium would be depleted). This assumes that the rate of conversion does not change (highly doubtful, since it will be more red dwarf conversion of hydrogen to just helium in the future than in the past where many more heavy stars formed due to the peak of SFR). If we use the estimate that $Z$ will increase by a factor of 10 from the current $Z\approx 0.02$ we get $\tau=1.3097\cdot 10^{11}$ years which seems reasonable, and of course just $A(\infty)=10A(t_\mathrm{now})$. That would for example make 10% of the mass oxygen and 4.6% carbon. But it would be underestimating the r-elements produced in neutron star collisions (since these were more rare in the past) and cosmic ray spallation elements.

• Can you be clearer about you mean by abundance ratios and the end of the stelliferous era? Do you mean in the universe as a whole? If so, the accelerating expansion of the universe will ensure that most hydrogen never gets near being inside of a star. Or do you mean a hypothetical scenario where all gas ends up in stars? If the latter then you have to assume something about stellar populations, binarity, recycling, star formation rates, mass loss, where r-process elements actually come from etc. Etc. – ProfRob Jan 28 '18 at 10:41
• @RobJeffries - Yes, I was thinking of the matter bound to galaxies, will edit the question a bit. There are many pieces to put together for a full model. For example, arxiv.org/abs/1708.07885 has a nice model of the neutron star merger rate, which might be combined .with the references in cosmic-origins.org to get r-element abundances. LiBeB follow Fe, so spallation might be straightforward. One could use Mesa and StarTrack to do stellar element evolution. I know a guy doing ISM infall. But I'd rather leave this to professionals than do it myself as an academic generalist. – Anders Sandberg Jan 28 '18 at 16:00
• Ok, it is a good question, but critically dependent on the time-dependence of the IMF. We don't understand what determines this, but do think it was different in the past. The future? Also, what do you want to assume for the star formation efficiency and does one assume no Galactic gas loss via supernovae and winds. – ProfRob Jan 28 '18 at 16:07
• @RobJeffries - I got the impression the evidence for a strongly changing IMF is relatively weak arxiv.org/abs/1001.2965 and that a high future metallicity may not have huge effects arxiv.org/abs/1102.2023 One could run with models like arxiv.org/abs/1710.04222 I am honestly not too picky; I am interested in a defensible mainstream model rather than trying to model all possible subtleties. – Anders Sandberg Jan 28 '18 at 21:25
• It is commonly believed that the mass function of stars born from very low metallicity gas was very different. We have very little to no observational information on what may happen if the H/He ratio changes significantly, or if the metallicity trebles or more. I am only making these comments because I can't provide an answer... – ProfRob Jan 28 '18 at 22:02