Fine-Tuning and the Myth of “One variable at a time”

A commenter over at my post “Got a cosmology question?” asks:

Someone told me “there is not a single paper which finds fine tuning that has allowed multivariation”. Can you please refute this?

Incidentally, cosmology questions are still very welcome over there.

“Multivariation” is not a word, but in this context presumably means varying more than one variable at a time. There is an objection to fine-tuning that goes like this: all the fine-tuning cases involve varying one variable only, keeping all other variables fixed at their value in our universe, and then calculating the life-permitting range on that one variable. But, if you let more than one variable vary at a time, there turns out to be a range of life-permitting universes. So the universe is not fine-tuned for life.

This is a myth. The claim quoted by our questioner is totally wrong. The vast majority of fine-tuning/anthropic papers, from the very earliest papers in the 70’s until today, vary many parameters¹. I’ve addressed these issues at length in my review paper. I’ll summarise some of that article here.

The very thing that started this whole field was physicists noting coincidences between the values of a number of different constants and the requirements for life. Carter’s classic 1974 paper “Large number coincidences and the anthropic principle in cosmology” notes that in order for the universe to have both radiative and convective stars we must have (in more modern notation to his equation 15, but it’s the same equation),

where, in Planck units, , , , and is the charge on the electron. (Interestingly, Barrow and Tipler show that the same condition must hold for stars emit photons with the right energy to power chemical reactions e.g. photosynthesis.) Similarly for cosmological cases: for the universe to live long enough for stars to live and die, we must have,

where  is related to the curvature of space and  is roughly the baryon to photon ratio.

This continues in the classic anthropic papers. Carr and Rees (1977) show that to have hydrogen to power stars left over from big bang nucleosynthesis, and to have supernovae distribute heavy elements, we must have (in Planck units, rearranging their equation 61),

where is the weak coupling constant.

Barrow and Tipler’s “The Anthropic Cosmological Principle” shows that, for carbon and larger elements to be stable, we must have:

where  is the strong force coupling constant (evaluated at , if you’re interested).

The whole point of these relations and more like them, which the early anthropic literature is entirely concerned with, is that they relate a number of different physical parameters. There are approximations in these calculations – they are order-of-magnitude – but this usually involves assuming that a dimensionless mathematical constant is approximately one. At most, a parameter may be assumed to be in a certain regime. For example, one may assume that and are small (much less than one) in order to make an approximation (e.g. that the nucleus is much heavier than the electron, and the electron orbits non-relativistically). These approximations are entirely justified in an anthropic calculation, because we have other anthropic limits that are known to (not merely assumed to) involve one variable – e.g. if is large, all solids are unstable to melting, and if  is large then all atoms are unstable. See section 4.8 of my paper for more information and references.

More modern papers almost always vary many variables. Examples abound. Below is figure 2 from my paper, which shows Figures from Barr and Khan and Tegmark, Aguirre, Rees and Wilczek. (Seriously, people … Wilczek is a Nobel prize winning particle physicist and Martin Rees is the Astronomer Royal and former president of the Royal Society. These people know what they are doing.)

The top two panels show the anthropic limits on the up-quark mass (x axis) and down-quark mass (y axis). 9 anthropic limits are shown. The life-permitting region is the green triangle in the top right plot. The lower two panels show cosmological limits on the cosmological constant (energy density) , primordial inhomogeneity Q, and the matter density per CMB photon. Tegmark et al. derive from cosmology 8 anthropic constraints on the 7 dimensional parameter space . Tegmark and Rees (1997) derive the following anthropic constraint on the primordial inhomogeneity Q:

(1)

Needless to say, there is more than one variable being investigated here. For more examples, see Figures 6, 7 (from Hogan), 8 (from Jaffe et al.) and 9 (from Tegmark) of my paper. The reason that the plots above only show two parameters at a time is because your screen is two dimensional. The equations and calculations from which these plots are constructed take into account many more variables than can be plotted on two axes.

This myth may have started because, when fine-tuning is presented to lay audiences, it is often illustrated using one-parameter limits. Martin Rees, for example, does this in his excellent book “Just Six Numbers“. Rees knows that the limits involve more than one parameter – he derived many of those limits. But equation (1) above would be far too intimidating in a popular level book.

My paper lists about 200 publications relevant to the field. I can only think of a handful that only vary one parameter. The scientific literature does not simply vary one parameter at a time when investigating life-permitting universes. This is a myth, born of (at best) complete ignorance.

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Postscript: The questioner’s discussion revolves around the article of Harnik, Kribs & Perez (2006) on a universe without weak interactions. It’s a very clever article. Their weakless universe requires “judicious parameter adjustment” and so is also fine-tuned. Remember that fine-tuning doesn’t claim that our universe is uniquely life-permitting, but rather that life-permitting universes are rare in the set of possible universe. Thus, the weakless universe is not a counterexample to fine-tuning. There are also concerns about galaxy formation and oxygen production. See the end of Section 4.8 of my paper for a discussion.

Footnotes:

1. Even if fine-tuning calculations varied only one parameter, it wouldn’t follow that fine-tuning is false. Opening up more parameter space in which life can form will also open up more parameter space in which life cannot form. As Richard Dawkins (1986) rightly said: “however many ways there may be of being alive, it is certain that there are vastly more ways of being dead, or rather not alive.” For more, see section 4.2.2 of my paper.

More of my posts on fine-tuning are here.