3.5 Stellar constraints

Stars start to accumulate helium produced by the pp-reaction and the CNO cycle in their core. Furthermore, the products of further nuclear reactions of helium with either helium or hydrogen lead to isotopes with A = 5 or A = 8, which are highly unstable. In order to produce elements heavier than A > 7 by fusion of lighter isotopes, the stars need to reach high temperatures and densities. In these conditions, newly produced 12C would almost immediately be fused further to form heavier elements so that one expects only a tiny amount of 12C to be produced, in contradiction with the observed abundances. This led Hoyle [257] to conclude that a then unknown excited state of the 12C with an energy close to the 3α-threshold should exist since such a resonance would increase the probability that 8Be captures an α-particle. It follows that the production of 12C in stars relies on the three conditions:
View Image

Figure 5: Left: Level scheme of nuclei participating to the 4He(α α,γ)12C reaction. Right: Central abundances at the end of the CHe burning as a function of δNN for a 60M ⊙ star with Z = 0. From [103Jump To The Next Citation Point].

The variation of any constant that would modify the energy of this resonance would also endanger the stellar nucleosynthesis of carbon, so that the possibility for carbon production has often been used in anthropic arguments. Qualitatively, if E + 02 is increased then the carbon would be rapidly processed to oxygen since the star would need to be hotter for the triple-α process to start. On the other hand, if E0+2 is decreased, then all α-particles would produce carbon so that no oxygen would be synthesized. It was estimated [334] that the carbon production in intermediate and massive stars is suppressed if the various of the energy of the resonance is outside the range − 250 keV ≲ ΔE + ≲ 60 keV 02, which was further improved [451Jump To The Next Citation Point] to, − 5 keV ≲ ΔE + ≲ 50 keV 02 in order for the C/O ratio to be larger than the error in the standard yields by more than 50%. Indeed, in such an analysis, the energy of the resonance was changed by hand. However, we expect that if E0+ 2 is modified due to the variation of a constant other quantities, such as the resonance of the oxygen, the binding energies and the cross sections will also be modified in a complex way.

In practice, to draw a constraint on the variation of the fundamental constants from the stellar production of carbon, one needs to go through different steps, any of them involving assumptions,

  1. to determine the effective parameters, e.g., cross sections, which affects the stellar evolution. The simplest choice is to modify only the energy of the resonance but it may not be realistic since all cross sections and binding energies should also be affected. This requires one to use a stellar evolutionary model;
  2. relate these parameters to nuclear parameters. This involves the whole nuclear physics machinery;
  3. to relate the nuclear parameters to fundamental constants. As for the Oklo phenomenon, it requires to link QCD to nuclear physics.

A first analysis [390Jump To The Next Citation Point, 391, 451Jump To The Next Citation Point] used a model that treats the carbon nucleus by solving the 12-nucleon Schrödinger equation using a three-cluster wavefunction representing the three-body dynamics of the 12C state. The NN interaction was described by the Minnesota model [297Jump To The Next Citation Point, 491Jump To The Next Citation Point] and its strength was modified by multiplying the effective NN-potential by an arbitrary number p. This allows to relate the energy of the Hoyle level relative to the triple alpha threshold, 𝜀 ≡ Q ααα, and the gamma width, Γ γ, as a function of the parameter p, the latter being almost not affected. The modified 3 α-reaction rate was then given by

( ) [ ] 3∕2 3 2π ℏ2 3 Γ 𝜀(p ) rα = 3 N α M--k-T-- -ℏ exp − k--T- , (106 ) α B B
where M α and N α are the mass and number density of the α-particle, The resonance width Γ = Γ αΓ γ∕(Γ α + Γ γ) ∼ Γ γ. This was included in a stellar code and ran for red giant stars with 1.3, 5 and 20M ⊙ with solar metallicity up to thermally pulsating asymptotic giant branch [390] and in low, intermediate and high mass (1.3,5,15,25M ⊙) with solar metallicity also up to TP-AGB [451] to conclude that outside a window of respectively 0.5% and 4% of the values of the strong and electromagnetic forces, the stellar production of carbon or oxygen will be reduced by a factor 30 to 1000.

In order to compute the resonance energy of the 8Be and 12C a microscopic cluster model was developed [297]. The Hamiltonian of the system is then of the form ∑A ∑A H = i T (ri + j<i V(rij), where A is the nucleon number, T the kinetic energy and V the NN interaction potential. In order to implement the variation of the strength of the nuclear interaction with respect to the electromagnetic interaction, it was taken as

V (rij) = VC(rij) + (1 + δNN )VN (rij),
where δNN is a dimensionless parameter that describes the change of the nuclear interaction, VN being described in [491]. When A > 4 no exact solution can be found and approximate solutions in which the wave function of the 8Be and 12C are described by clusters of respectively 2 and 3 α-particle is well adapted.

First, δNN can be related to the deuterium binding energy as

ΔBD ∕BD = 5.7701 × δNN , (107 )
which, given the discussion in Section 3.8.3, allows to relate δNN to fundamental constants, as, e.g., in [104Jump To The Next Citation Point]. Then, the resonance energy of the 8Be and 12C scale as
8 12 ER ( Be ) = (0.09208 − 12.208 × δNN ) Mev, ER ( C ) = (0.2877 − 20.412 × δNN ) Mev, (108 )
so that the energy of the Hoyle level relative to the triple alpha threshold is Q ααα = ER (8Be) + ER (12C ).

This was implemented in [103, 180] to population III stars with typical masses, 15 and 60M ⊙ with zero metallicity, in order to compute the central abundances at the end of the core He burning. From Figure 5View Image, one can distinguish 4 regimes (I) the star ends the CHe burning phase with a core composed of a mixture of 12C and 16O, as in the standard case; (II) if the 3α rate is weaker, 12C is produced slower, the reaction 12C(α, γ)16O becomes efficient earlier so that the star ends the CHe burning phase with a core composed mostly of 16O; (III) for weaker rates, the 16O is further processed to 20Ne and then 24Mg so that the star ends the CHe burning phase with a core composed of 24Mg and (IV) if the 3α rate is stronger, the 12C is produced more rapidly and the star ends the CHe burning phase with a core composed mostly of 12C. Typically this imposes that

− 5 × 10−4 < δ < 1.5 × 10−3, − 3 × 10− 4 < ΔB ∕B < 9 × 10−3, (109 ) NN D D
at a redshift of order z ∼ 15, to ensure the ratio C/O to be of order unity.

To finish, a recent study [3] focus on the existence of stars themselves, by revisiting the stellar equilibrium when the values of some constants are modified. In some sense, it can be seen as a generalization of the work by Gamow [224Jump To The Next Citation Point] to constrain the Dirac model of a varying gravitational constant by estimating its effect on the lifetime of the Sun. In this semi-analytical stellar structure model, the effect of the fundamental constants was reduced phenomenologically to 3 parameters, G, which enters mainly on the hydrostatic equilibrium, αEM, which enters in the Coulomb barrier penetration through the Gamow energy, and a composite parameter 𝒞, which describes globally the modification of the nuclear reaction rates. The underlying idea is to assume that the power generated per unit volume, 𝜀(r), and which determines the luminosity of the star, is proportional to the fudge factor 𝒞, which would arise from a modification of the nuclear fusion factor, or equivalently of the cross section. Thus, it assumes that all cross sections are affected is a similar way. The parameter space for which stars can form and for which stable nuclear configurations exist was determined, showing that no fine-tuning seems to be required.

This new system is very promising and will provide new information on the fundamental constants at redshifts smaller than z ∼ 15 where no constraints exist at the moment, even though drawing a robust constraint seems to be difficult at the moment. In particular, an underlying limitation arises from the fact that the composition of the interstellar media is a mixture of ejecta from stars with different masses and it is not clear which type of stars contribute the most the carbon and oxygen production. Besides, one would need to include rotation and mass loss [181]. As for the Oklo phenomenon, another limitation arises from the complexity of nuclear physics.

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