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I have read that, in smaller stars, such as our Sun, the fusion reaction that takes place is a proton-proton chain, or PP chain for short.

From what I have learned, in larger stars, a different process takes place, known as the CNO cycle, in which carbon, nitrogen and oxygen are utilized as a catalyst.

The first stars were many times more massive than the Sun, but pretty much the only elements they had to make use of were hydrogen and a little bit of helium. So, what types of fusion reactions went on in the first stars?

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I think you already know the answer...

Pop III stars, by definition, are born from primordial gas that is basically Hydrogen, Helium with trace amounts of deuterium, tritium, lithium and beryllium; they initially contain almost no C, N, or O. Therefore the primary fusion in massive Pop III stars has to be (well, initially the deuterium is burned but this is not energetically important) the pp chain, which produces more helium from hydrogen. The lower temperature dependence of the pp chain means that the core can get hotter and denser than would be usual in more metal-rich massive stars.

It is not until He burning is initiated that C, N, and O can form, but this actually takes place before the red giant tip is reached in Pop. III stars.

Your recent questions all have a theme. You might find this presentation I came across interesting.

EDIT: Further to this: Current thinking is that CNO burning will take place on the main sequence for Pop. III stars more massive than $20 M_{\odot}$. This is because they cannot be supported by the pp chain alone; they contract and heat up sufficiently to start the triple alpha He burning phase that produce carbon; once the carbon concentration builds up to about $10^{-10}$ of hydrogen then the CNO cycle takes over (Ekstrom et al. 2008; Yoon et al. 2012).

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