Historically, Fred Hoyle's coincidence was the first to be discovered. For the anthropic community it is an exemplar. Hoyle made a prediction which was, in short order, confirmed experimentally. This was that carbon-12 must have a resonance of just the right energy to permit rapid capture of unstable beryllium-8 nuclei. The importance of this can hardly be overstated. By this means the A = 8 stability gap is bridged. The fact that carbon-12 has a correctly placed resonance permits its formation via the triple-alpha reaction. And the formation of carbon opens the way for the creation of the heavier elements.
The rate of a resonant reaction depends exponentially on the
resonance energy with respect to the reaction threshold. Not only is the reaction rate very sensitive to the resonance energy, but the resonance energy is also very sensitive to the strength of the N-N force (the strong nuclear force). A change in the N-N interaction strength of less than one percent is sufficient to reduce either the carbon or the oxygen abundance to less than 1% of its value in this universe. However, there is a partial get-out clause.
A change in the N-N interaction strength by ~0.4% (probably an increase) would lead to Beryllium-8 being stable. This would have a profound effect on subsequent stellar nucleosynthesis since it bridges the A = 8 stability gap. This would lead to an alternative reaction pathway to carbon, oxygen and nitrogen, via direct Be8 + Be8 reactions. This would appear to be simpler than the route for nucleosynthesis that nature has actually chosen. Thus, one could argue that, rather than being fine tuned, our universe is actually rather badly tuned.
However, there still appears to be a one-sided fine tuning. If a change in the nuclear force led to higher Be8 and carbon resonance energies, the rate of the triple-alpha reaction would reduce dramatically. This could reduce carbon production to an apparently arbitrary degree. The effect would be exacerbated if the oxygen resonances also increased in energy. This would cause the rate of oxygen formation from carbon to become faster, and ultimately to be become resonant. This could potentially happen for only a small fraction of a percent change in the strong force. A faster oxygen formation reaction would further deplete the abundance of carbon, with the attendant consequences for biochemistry.
In the present state of knowledge about nuclear physics, it is not possible to definitively state whether an increase in the strength of the nuclear force would lead to an increase or a decrease in the energy difference between two nuclear species. It is such differences in energy, e.g. between a product nucleus and the reaction threshold, which determines the reaction rate. Consequently it is essentially unknown whether an increase in the nuclear force strength leads to an increase or a decrease in, say, the triple alpha reaction. It is pointed out that existing attempts to address this question are likely to be subject to over-whelming numerical error. However, this observation merely serves to re-emphasise the extreme sensitivity of reaction rates, and whether the reactions occur at all, on very small changes in the strength of the nuclear force. In particular, it is not good enough to argue that the Hoyle coincidence is merely a consequence of the weak binding of cluster states. This qualitative observation, though correct, is simply not precise enough to ensure that the triple alpha reaction is sufficiently fast.
In summary, we demote the Hoyle coincidence to a one-sided coincidence. If the strength of the nuclear force were changed so as to reduce the Be8 ground state energy with respect to the two-alpha threshold, the abundance of carbon and oxygen is probably not prejudiced fatally up to the point at which Be8 becomes stable. At this point, a new, and simpler, reaction pathway for element synthesis opens up and the sensitivity to the nuclear force reduces. However, if the nuclear force is changed so as to increase the energy of all contributing nuclear species, the carbon abundance of the universe would fall dramatically. This may well be fatal to normal biochemistry.
