Bill Z. has brought my attention to a December 2012 nuclear physics paper that was updated 3 days ago,
The fate of carbon-based life as a function of the light quark massThey (Evgeny Epelbaum, Hermann Krebs, Timo A. Lähde, Dean Lee, Ulf-G. Meißner) try to determine the precision with which God or non-God had to fine-tune the average light quark mass – a parameter defined as \((m_{\rm up}+m_{\rm down})/2\) – in order to guarantee that there would be enough carbon, oxygen, and other elements that are crucial for the type of life that is recommended by 4 of 5 dentists.
The detailed calculations are concerned with the Hoyle state. What is it and what did the authors of the new paper conclude?
In 1954, Fred Hoyle noticed that we were lucky about a seemingly technical coincidence in nuclear physics that was apparently needed for us to exist. Note that in the baryonic (proton- and neutron-based i.e. visible) matter in the Universe around us, hydrogen and helium are the dominant elements.
It's no coincidence. It was mostly hydrogen (\(Z=1\)) and helium (\(Z=2\)) – and just some lithium (\(Z=3\)), aside from negligible trace amounts of heavier elements (\(Z\geq 4\)), that was directly produced during the Big Bang nucleosynthesis in the first three minutes after the Big Bang. One may reconstruct the temperature in the Universe during these early formative stages of our Cosmos and calculate, using the statistical methods, the ratios of the concentrations of these three light elements. The results seem to match the observations of hydrogen, helium, lithium rather impressively – and this agreement is one of the important pieces of evidence supporting the Big Bang paradigm.
Looking from a practical perspective, are the three lightest elements enough to get everything we need to be happy? Well, lithium may be helpful for some laptop and cell phone batteries and helium is useful at most for funny tricks to change your voice into the voice of the Smurfs (fine, it's also great as the gas in balloons, either for kids or adults, and as the coolant in NMR and the LHC). We could also describe helium as the main waste product of the thermonuclear reactions in the Sun and other stars if it weren't too disrespectful.
Hydrogen is useful for a huge fraction of compounds we need and love. But where is the rest? It's obvious that the three elements aren't enough to build life and the civilization as we know it. In particular, two other major heavier elements behind the miraculous project of life – carbon and oxygen – seem to be absent. Yes, these are the same two elements found in the gas that they call a pollution but we call it life.
Aside from hydrogen, carbon, and oxygen, the three major elements, nitrogen, phosphorus, and sulfur are three more elements that are paramount for life we know. Other elements such as silicon or calcium or fluorine (because I mentioned the dentists) may be helpful to shape our bodies at various moments and/or to create various intelligent gadgets but they're no longer a universal "must". Where do all these elements come from?
The heavier elements are abundantly enough produced by helium burning in stars that have gone red giants. Our carbon, oxygen, and other elements arose from the production in these red giant stars that once existed but they are no more. In 7.5 billion years or so, our beloved Sun will become a red giant, too. It will devour the Earth and other planets – that's the less dramatic part of the story – but it will also produce completely new carbon, oxygen, and other elements that may be incorporated into the bodies of a future extraterrestrial civilization.
Fine. How do the red giants produce carbon (\(Z=6\))?
Note that six is a multiple of two: we call these numbers "even". So it has the right number of protons to arise from several, namely three, nuclei of helium which seem abundant. Moreover, the ordinary carbon nuclei we need have 6 protons and 6 neutrons, the same number, so it seems appropriate to combine three helium-4 nuclei to create carbon-12:\[
\Large {}^4_2{\rm He} + {}^4_2{\rm He} + {}^4_2{\rm He} \rightarrow {}^{12}_6{\rm C}
\] It's somewhat unlikely for three helium-4 nuclei – which I will call the alpha-particles just like everyone else – to hit each other and produce the carbon nucleus directly. So the reaction actually proceeds in two steps, with an unstable level of beryllium-8 or \({}^8_4{\rm Be}\) in between. This beryllium-8 nucleus combines with another alpha-particle to get the desired carbon but this second step has too low a rate.
We wouldn't get enough carbon in this way (if it were just a generic fusion of these nuclei) and it's actually known that something special is going on. There is a resonance, a \(0^+\) state of carbon-12 known as the Hoyle state. Fred Hoyle actually predicted – using the apparent abundance of carbon as the only input – that it should be somewhere over there and indeed, the prediction was soon experimentally confirmed.
There exists a state of carbon-12 whose mass/energy is equal to the mass/energy of three free alpha-particles plus \(\varepsilon=397.47(18)\keV\); we say that the state is \(\varepsilon\) above the three-alpha threshold (a threshold, in general, is the minimum mass/energy of an unstable/composite object that allows the corresponding state to decay to particular products without violating the energy conservation law).
In the relevant region of the parameter space, the reaction rate for the carbon-12 production – via the Hoyle resonance – may be approximated by \[
r_{3\alpha} = \Gamma_\gamma (N_\alpha/k_B T)^3 \exp(-\varepsilon/k_B T).
\] You see that up to an overall normalization constant, this is equal to the third power of the number (density) of alpha-particles per unit volume (because three of them have to meet) and a Boltzmannian factor that exponentially decreases with energy. This \(\varepsilon\) shouldn't be too high because the exponential suppression could be severe. It shouldn't be too small, either, for other reasons. In the past, it was argued that a 15% deviation of \(\varepsilon\) from the known value could still allow enough carbon etc. for life.
Now, what have they found about the dependence of this accident on the light quark mass?
They use a novel numerical method, nuclear lattice simulations, to calculate the dependence. It's something like lattice QCD except that it seems to work with some composite pion fields and the low-energy emergent nuclear physics mess instead of the fundamental QCD fields. The light quark mass is translated to the mass of the pion \(M_\pi\) and they discuss the dependence of the energies of several energy levels on either the light quark mass or the pion mass which is almost the same dependence.
I don't want to bore you with all the details – you may read the original paper, it's just 4 pages long. Instead, let me repost a graph summarizing some partial results of their analysis:
On the \(x\)-axis, they depicted the relative change of the binding energy of the alpha-particle; on the \(y\)-axis, you see the corresponding (much larger) relative change of the three energies related to the helium-4, beryllium-8, and carbon-18 nuclei, namely of\[
\eq{
\Delta E_h &= E_{12}^* - E_8 - E_4\\
\Delta E_b &= E_8 - 2E_4\\
\varepsilon &= E_{12}^* - 3E_4 = \Delta E_h + \Delta E_b
}
\] Because of the final relationship for \(\varepsilon\), it's not surprising that the yellow curve is in between the other two. But what may be surprising is that these two curves – and therefore all three curves – have pretty much the same slope. What does it mean? It means that the several fine-tunings are actually not independent from each other.
You could think that for the nuclear factory to work and produce the elements, you may need several miracles – several anthropic conditions, in this case three – and therefore God has to be even greater than the size He would adopt if there were just one miracle. God's omnipotence seems like the third power of a generic god's power (or three times? It depends whether His omnipotence is quantified on the log scale).
However, the new paper shows that this ain't the case. The three coincidences aren't independent from each other. Pretty much because of mathematical identities, they're more or less equivalent to one coincidence only. If one identity for the nuclear level energies holds, the other two will probably hold as well, with a highly acceptable accuracy. It means that one miracle is enough and life is much more likely than what you would expect if you thought that the three conditions are independent of each other.
Now, you could claim that we still need meta-God to explain the mathematical "metamiracle" that the three conditions are actually almost equivalent to each other. If you did so, you would cover these questions by lots of exciting religious fog. But the matter of fact is that they can actually explain this "metamiracle" – at least in a preliminary way – in terms of completely non-mysterious, irreligious arguments, too. The same slopes kind of follow from the alpha-cluster structure of beryllium-8 and carbon-12 nuclei. I won't present this derivation in its full glory but the alpha-particle-based compositeness of the two nuclei sort of rationally explains why the two slopes are almost the same.
Even though several levels and energy differences are involved, the authors de facto show that there is only one "miracle" we need for a sufficient production of the heavier elements behind life. Moreover, the tolerated error for \(\alpha_{\rm elmg}\) as well as \(m_q\) could be around 2 percent or so: the fine-tuning isn't extreme.
Although this very topic may make you "wish" that there is some evidence for the Intelligent Design and/or a stunningly convincing role for the anthropic principle, and the very fact that a paper about this metaphysical and mysterious question was written could manipulate you into a more spiritual thinking, I would say that the actual results of their analysis imply exactly the opposite conclusion. Different "miracles" aren't really independent from each other and they're not "terribly unlikely miracles", anyway. Good luck at the 1-in-50 level seems to be enough for the amount of carbon to be just fine. At most, you may need two such 1-in-50 fine-tunings – one for the fine-structure constant and one for the light quark mass – except that I think that only some combination of them will have a high enough impact on the essential processes needed for the elements of life to arise.
Now, you could still argue that 1-in-50 is a low chance. The probability \(p=0.02\) or so is pretty small, some of you could say, and this strengthens some arguments in favor of God, Intelligent Design, the anthropic principle, or something along these lines. Well, perhaps. I don't think it's a right way to think about this coincidence. Why?
First, \(p=0.02\) is equivalent to a "bump just a little bit larger than a 2-sigma bump". To make extraordinary claims about God or the anthropic principle – and one really doesn't know which of these (or other metaphysical) explanations "follows" from the "miracle" – and justify them by not-so-extraordinary evidence such as 2-sigma bumps seems to betray the lack of evidence. Extraordinary claims require extraordinary evidence and this ain't one.
Second, this not-so-extreme probability \(p=0.02\) is the \(p\)-value before the look-elsewhere effect of a particular type is included. What I want to say is that we're computing the probability that a particular system of nuclear furnaces will be able to produce a particular type of life (determined by its elements etc.). However, there could very well be other types of life – perhaps \({\mathcal O}(50)\) types of life – that may arise in the same parameter space which means that the probability that at least one of these types of life will be allowed for a "random" choice of the values of parameters may approach 100 percent.
These observations of coincidences that are needed for life are intriguing but we shouldn't get carried away for two basic reasons. First, as argued above, the probabilities that we get a tolerable value of the parameters (values compatible with life) aren't extremely tiny and we should treat these "modestly suggestive" low probability just like any other 2-sigma bumps in physics. They're not enough to settle a question, they're not enough for a paradigm shift.
Second, it's pretty much guaranteed that if we calculate the odds that some "conditions constraining parameters that makes the theory friendly to life as we know it" are obeyed, it's unsurprising that the answer will probably be Yes because our type of life does exist, after all. The proposition that "conditions apparently needed for this life to arise with a significant probability were satisfied" is pretty much tautologicaly true. These conditions simply aren't independent of some empirical known facts. We're just measuring the answer to the question "Does life exist?" using a different, perhaps more contrived, procedure. But the fact that many such questions have "Yes" answers isn't a miracle; it's pretty much tautologically guaranteed because these questions were cherry-picked by their equivalence to the existence of life (or some of the aspects of this existence).
If the probabilities arising in similar anthropic coincidences were much tinier, i.e. much more extreme than 2-sigma bumps, I could be impressed. The tiny cosmological constant could be an indication of this sort. However, we may only argue that the "probability that the cosmological constant is below \(10^{-120}m_{\rm Planck}^4\) is of order \(10^{-120}\)" if we adopt a uniform probability distribution for cosmological constants in the interval comparable to \((0,m_{\rm Planck}^4)\).
While some plausible models that make the uniform distribution look natural exist, they're not "inevitably true" and it's still easy to imagine that this uniform probability distribution is a completely naive, wrong expectation. If we replace it by another one – one that follows from a slightly sophisticated mechanism and one that gives tiny values of the cosmological constant with far higher probabilities – the "unavoidably impressive" miracle goes away once again. If you wanted to convince me that there is a miracle that harbors strong evidence in favor of the anthropic reasoning or God or anything like that, you would have to show me a coincidence that has a tiny probability according to the right calculation of probabilities (a calculation which takes all mathematically guaranteed correlations such as those above into account) and a nicely justifiable probability distribution for the parameters.
If you used a quasi-uniform one, you would have to convince me that it's reasonable to expect that the distribution is quasi-uniform for that situation. It would have to be so reasonable – almost inevitable – that, in fact, I would find your anthropic principle or God more likely as an explanation than the mundane possibility that there simply exists a better argument or "better theory" telling you that a non-uniform distribution is actually a much more sensible (and likely) one. Or a better theory that simply allows you to calculate the observed value. How strong evidence is needed to prefer God over the "better theory" depends on subjective prior probabilities but be sure that 2-sigma or even 3-sigma bumps are way too small for people like me to pick God or His best pal, the anthropic anti-God, instead of a "better, so far unknown, theory".
If you can't show me such a thing, I would keep on insisting that there doesn't exist any tangible evidence to believe the anthropic/religious paradigm and because these things aren't mathematically elegant or explaining any true pre-existing mysteries in physics, they don't really deserve to become a part of physics, at least at this moment.
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