Why do the constants allow anything to exist?

A 2012 review by Luke Barnes went through the published cases in detail: the cosmological constant, the amplitude of primordial density fluctuations, the ratio of electromagnetic to gravitational strength, the neutron-proton mass difference, the strength of the strong force.

Individual cases have been argued down, and the review records it. The carbon resonance Hoyle reasoned his way to is the famous one. Weinberg has argued that even-even nuclei are known to have states of that kind, and later stellar models including helium shell flashes lessened how tuned it looks. Barnes answers that the state must also sit at the right energy, and keeps the case.

On the cosmological constant the review is firmer. Quantum field theory predicts contributions to the vacuum energy about 120 orders of magnitude larger than the observed total, and no accepted calculation removes the gap.

Barnes concludes that the universe is fine-tuned for life, and says in the same paper that he does not attempt to defend any conclusion based on it. The calculations are one thing. What follows from them is another.

An objection pressed by Victor Stenger is that the calculations change one number and hold the rest fixed. Barnes answers it: the published work varies several parameters together and still finds the life-permitting region boxed in, and a one-parameter estimate overstates rather than understates the room available.

Fred Adams, reviewing the same literature in 2019, weighs it the other way: more working universes appear when more than one parameter varies, and most of the relevant parameters can move by several orders of magnitude and still allow complex structure. The two partly measure different things. Barnes compares the life-permitting range with everything the parameter could have been; Adams measures its width in orders of magnitude around the value it has.

What stays open is the question underneath. There is no agreed probability distribution over possible values. Saying a value is improbable requires knowing what it could have been and how likely each option was, and the laws of nature tell us what is physically possible without telling us which laws are possible.

That a future theory will derive the numbers, making the question disappear. That there are many regions with different values and observers necessarily find themselves where observers are possible. That the values indicate purpose.

The first is a research programme that has not yet derived any of the constants in question. The second follows from some versions of theories proposed for other reasons and has no accepted test. The third is a claim of a different kind, and the honest thing to say about it is that it is not decided by the physics and is not refuted by it either.

In 1987, before any nonzero value had been measured, Steven Weinberg argued that if the cosmological constant varies between regions, observers will only exist where galaxies can form, which bounds how large it can be.

The bound is roughly the cosmic mass density at the earliest time galaxies form, about two hundred times the present mass density. Weinberg wrote at the time that if galaxy counts ruled out a value within one or two orders of magnitude of the bound, anthropic considerations would not explain why it is small.

The value measured eleven years later sits about two orders of magnitude under the bound. Weinberg later put the chance of finding one that small at 5 to 12 per cent, and called it a little low but not implausibly so.

It is often offered as anthropic reasoning producing a number ahead of a measurement. Hoyle's carbon resonance is not a second case, because that argument runs without the anthropic premise. Whether this one counts is argued. It is a bound rather than a prediction, and it assumes the many regions it was invoked to support.

Starkman and Trotta showed in 2006 that weighting universes by the total number of possible observations makes a value as large as ours extremely improbable. Adams notes that the bound loosens by about twenty-one orders of magnitude when the fluctuation amplitude and the baryon-to-photon ratio are allowed to vary too.

Some philosophers hold that the question needs no answer: we could not be here to ask under other values, and an improbable outcome is not by itself a thing to explain.

Others, John Leslie among them, hold that it should not be dismissed as how things just happen to be. This disagreement is about what counts as an explanation, and the physics does not settle it.