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This answer to Why don't or (can't) stars be more than 325 or so times the mass of the sun? What limits their size? includes the following:

...The upper limit you refer to is for compositions similar to the Sun. For stars born in the distant past that were metal-poor, or even born from primordial material with no metals, the upper limit could be much higher.

Question: What was the absolute limit to the possible sizes of the first stars formed from "primordial material with no metals"? I'm primarily asking about the size that was possible to exist as a star. There may be other limits associated with the possibility of formation in the first place due to matter distribution and that would be interesting to know as well.

NOTE: There is also the well-received and as yet unanswered question Formation of the First Stars If this topic is better addressed as an answer there instead, then please do so and we can close this as a duplicate.

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    $\begingroup$ Define "star" ? $\endgroup$ – ProfRob Jul 25 '20 at 7:16
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    $\begingroup$ On the Detection of Supermassive Primordial Stars and The Hunt for Enormous Early Stars Yikes! $\endgroup$ – uhoh Jul 25 '20 at 7:32
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    $\begingroup$ I mean, is a "quasi-star" a star? Or is something that starts nuclear fusion, but is always collapsing, a star? I guess an answer would need to address this. $\endgroup$ – ProfRob Jul 25 '20 at 7:33
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    $\begingroup$ @RobJeffries Maybe something like a simply connected compact region of space(time) in which (p-p chain?) fusion occurs which either has non-trivial outflux of new fusion products (including those of other fusion chains, s-process, r-process, etc.) or (failing that) does not collapse to a black hole? Probably needs some sort of bounds on minimum and maximum timescales for the collapse bit that could confound the whole idea, though; and I think it'd technically include brown dwarfs, which might be silly...and possibly the Earth because of all these scientists and bombs... $\endgroup$ – zibadawa timmy Jul 25 '20 at 16:56
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    $\begingroup$ start of answer: radiation pressure imposes the limit to the mass of stars. Too much energy/radiation will push the star apart. The radiation pressure depends upon the composition. Different elements absorb different amounts of radiation at different wavelengths. How much of the gas is ionized depends upon its composition. The fraction of free electrons will change the radiation absorption and hence radiation pressure. $\endgroup$ – TazAstroSpacial Aug 26 '20 at 5:03
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First off, let us try to clarify a few terms:

  1. As usual in astrophysics, metal-free star means atomic number $Z \leq 3$, i.e. it only consists of the primordial elements hydrogen, helium, and lithium.
  2. Primordial star literally means original star and refers to the first star(s) (generation) formed after the big bang. It is IMHO equivalent to metal-free, and a primordial metal-free star would be a pleonasm.
  3. Star can be most generally be defined as massive, self-luminious gas sphere (see Lexikon der Astronomie, p. 412), which leads to the question what each of these four terms mean exactly. Massive would probably mean at least about $0.09 M_\odot$ as 0.09 solar masses is the weight of the smallest star observed, AB Doradus, which is undergoing nuclear fusion.
  4. A quasi-star is also worth a definition for the current question. The first paragraph of that Wiki-page summarizes it pretty neatly:

A quasi-star (also called black hole star) is a hypothetical type of extremely massive and luminous star that may have existed early in the history of the Universe. Unlike modern stars, which are powered by nuclear fusion in their cores, a quasi-star's energy would come from material falling into a black hole at its core.

ProfRob already pointed out that a key issue is the definition of star.

Or is something that starts nuclear fusion, but is always collapsing, a star?

Related is the question about the stability of a star, which is studied by Isabelle Baraffe et al. in arXiv:astro-ph/0009410 :

The stability of metal-free very massive stars ($ Z= 0; M = 120 \ldots 500 M_\odot$) is analyzed and compared with metal-enriched stars. Such zero-metal stars are unstable to nuclear-powered radial pulsations on the main sequence, but the growth time scale for these instabilities is much longer than for their metal-rich counterparts.

The metal-free stars analyzed in that manuscript would still be smaller than the quasi-stars, which require at least $1000 M_\odot$. Again a Wikipedia-quote:

Quasi stars would have had a short maximum lifespan, approximately 7 million years, during which the core black hole would have grown to about $10^3 \ldots 10^5 M_\odot$ for modern stars.

To summarize: The maximal size limit varies, depending on what we exactly regard as primordial metal-free star, and also how long such an object has to exist/ be stable. I would assume that the upper limit of primordial metal-free stars is probably larger than the upper limit of stability of $80 \ldots 100 M_\odot$ which holds for modern stars (see p. 458 in Lexikon der Astronomie).

References

  • Helmut Zimmermann, Alfred Weigert: Lexikon der Astronomie. Edition 8. Heidelberg/ Berlin 1999. ISBN 3-8274-0575-0. (in German)
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This Okhubo 2009 paper presents two fiducial models: (a) stars between Pop III-1 stars of 40 to 300 M☉ stars not affected by stellar feedback which end in core-instability SN and BHs [they speculate that some became seeds of SMBHs]; and (b) Pop III-2 stars of 40 to 60 M☉ which do include radiative feedback and explode as Type II SNs, seeding the universe with its first metals. There are other papers in this genre but this is a good introduction.

We are missing a point in this preoccupation with mass: how much ionising luminosity did these stars impart to a universe perhaps >10 times denser than today whose gas was neutral hydrogen? IOW were these classes of Pop III stars the sole contributor to reionization? Or did partial reionisation occur until Pop II stars were able to form and undergo the SN cycle and complete ionisation? The literature is not especially abundant on the subject, but it does seem that preoccupation with mass size misses the more important matter of long-term effect galactic assembly.

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