# What is the extreme long-term elemental composition of the universe?

In Freeman Dyson's classic 1979 Time Without End paper he points out that, if proton decay does not occur, then normal matter will spontaneously fuse to iron on a timescale of $$10^{1500}$$ years, and heavier elements will undergo fission or alpha emission, again producing iron (although he does not calculate the timescale). My question is how correct this picture is.

It is a well-known quibble that strictly speaking, $$^{56}$$Fe is not the most tightly bound nucleus: it has 8790.4 keV binding energy per particle but $$^{58}$$Fe has 8792.3 and $$^{62}$$Ni has 8794.5. More subtly, this depends on whether we consider adding the right number of nucleons or the right number of protons and neutrons. So it might appear that the long-term composition of matter would hence be nickel-dominated.

However, just like the stellar formation processes favour $$^{56}$$Fe over $$^{62}$$Ni despite the energy difference because of the relative scarcity of neutrons to bridge the gap between them, the actual decay processes in the far future also seem to lead to trapping in other isotopes. $$\alpha$$-decays require a positive Q-value for $$^{A}_{Z}X\rightarrow ^{A-4}_{Z-2}Y+\alpha+Q$$: this becomes common above $$A>102$$, but below the energy difference is too small to allow $$\alpha$$-decay. $$\beta$$-decay to the rescue! They (and electron capture) happen across the entire $$A$$ range, but only change $$Z$$ by $$\pm1$$. So this appears to lead to a lock-in where heavy element decay get stuck in the region $$62>A>102$$. In this argument I have assumed that any energetically allowed transition will eventually happen; at least some $$\beta$$-decays are presumably blocked by spin considerations.

Plot of hypothetical decays in (A,Z) plane: blue beta decay/electron capture, red alpha decays. Green isotopes do not have any energetically allowed decays. White circles mark iron-56 and nickel-62.

I seriously doubt Dyson would have made a mistake (even though this is an extremely minor part of the paper), so what is going on? Spontaneous fission reducing everything to the iron peak? Or is the long-term chemical composition of the universe a mixture of stable heavyish elements?

[ Some other minor issues: in $$\Lambda$$CDM cosmology rapid expansion also makes many atoms and molecules isolated so their fusion processes end long before iron. There are also issues with spontaneous ionization and possibly nuclear decay due to the Herzfeld "paradox" of divergent partition functions for completely isolated atoms. And, as Dyson pointed out, iron is metastable to tunnelling into neutron star or black hole states. ]

• My question is how correct this picture is. Doesn't this require us to wait about $10^{1500}$ years as otherwise we're just opinion based ? Note if the standard model requires changes for e.g. dark matter then we can't know what impact that has on the ultimate fate of matter. Likewise if dark energy proves to require changes to our understanding of fundamental processes. particularly on the cosmic scale you're describing, then this complete unknown renders any answer purely speculative. If there's some more islands of stability for nuclei - similar issue possible. – StephenG Mar 29 at 23:28
• This is a question about what our current understanding of physics predicts. That is not more speculative than asking about the dynamics of the Milky Way-Andromeda merger in a few gigayears or what how the scale factor in $\Lambda$CDM changes. – Anders Sandberg Mar 29 at 23:38

What is the extreme long-term elemental composition of the universe?

To clarify, all matter turns to iron is talking about matter in stars,

After the time ($$=10^{1500}$$ yrs) elapsed, most of the matter in the universe is in the form of ordinary low-mass stars that have settled down into white dwarf configurations and become cold spheres of pure iron.

not all matter in the universe, which still will have in the huge empty spaces hydrogen mainly but also a lot of atomic matter debris which at the temperature of zero will not meet another debris to fuse into anything. (Of course radiation will be there also). He proposes iron t so he can do some calculations.

Whether stars become iron or another element , or many elements is immaterial to his discussion which leads to the alternatives of neutron stars and black holes, which is the point of his perambulation. All this assuming that the protons do not decay, and he gives references to later theoretical papers, where proton decay is calculated.

• Yes, but my question is for a more detailled answer about the elemental composition. The iron claim, which I think is partially fallacious, is widely reported but we should be able to do better 41 years after the initial paper. – Anders Sandberg Mar 30 at 6:31
• It is not necessary for Dysons argument. it would just make it more difficult to do the calculations. – anna v Mar 30 at 8:43
• Yes, but I am interested in the stability of particular kinds of matter. That is why I want to be able to do the calculations. – Anders Sandberg Mar 30 at 12:25
• It seems to me not worth it. For example, a lot of first order effects can be computed without knowing the exact potentials, by using the harmonic oscillator potential. I think the assumption "it is all iron" is a first order approximation that allows estmates , and assuming complicated possibilities will not change much. the outcomes . – anna v Mar 30 at 13:38
• @Anders Sandberg Given the long association of iron with artifice in human society, I was surprised by Dyson's conclusion that it might be the prevalent matter in the late times of an open universe. Might the fallacy that you're imagining bear on some possible re-evolution of a universe with a more naturalistic resemblance to our own (especially given Anna's description of an iron universe as only a 1st approximation)? – Edouard Mar 31 at 1:40