- the decay lifetime of
^{8}Be, of order 10^{–16}s, is four orders of magnitude longer than the time for two particles to scatter, so that a macroscopic amount of beryllium can be produced, which is sufficient to lead to considerable production of carbon, - an excited state of
^{12}C lies just above the energy of^{8}Be+, which allows for - the energy level of
^{16}O at 7.1197 MeV is non resonant and below the energy of^{12}C + , of order 7.1616 MeV, which ensures that most of the carbon synthesized is not destroyed by the capture of an -particle. The existence of this resonance, the -state of^{12}C was actually discovered [111] experimentally later, with an energy of [today, ], above the ground state of three -particles (see Figure 5).

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 is increased then the carbon would be rapidly processed to oxygen since the star would need to be hotter for the triple- process to start. On the other hand, if is decreased, then all -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 , which was further improved [451] to, 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 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,

- 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;
- relate these parameters to nuclear parameters. This involves the whole nuclear physics machinery;
- to relate the nuclear parameters to fundamental constants. As for the Oklo phenomenon, it requires to link QCD to nuclear physics.

A first analysis [390, 391, 451] 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 ^{12}C
state. The NN interaction was described by the Minnesota model [297, 491] and its strength was modified
by multiplying the effective NN-potential by an arbitrary number . This allows to relate the energy of
the Hoyle level relative to the triple alpha threshold, , and the gamma width, , as a function
of the parameter , the latter being almost not affected. The modified -reaction rate was then given
by

In order to compute the resonance energy of the ^{8}Be and ^{12}C a microscopic cluster model was
developed [297]. The Hamiltonian of the system is then of the form , where
is the nucleon number, the kinetic energy and 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

First, can be related to the deuterium binding energy as

which, given the discussion in Section 3.8.3, allows to relate to fundamental constants, as, e.g., in [104]. Then, the resonance energy of the This was implemented in [103, 180] to population III stars with typical masses, and with
zero metallicity, in order to compute the central abundances at the end of the core He burning. From
Figure 5, one can distinguish 4 regimes (I) the star ends the CHe burning phase with a core composed of a
mixture of ^{12}C and ^{16}O, as in the standard case; (II) if the rate is weaker, ^{12}C is produced slower, the
reaction ^{12}C^{16}O becomes efficient earlier so that the star ends the CHe burning phase
with a core composed mostly of ^{16}O; (III) for weaker rates, the ^{16}O is further processed to
^{20}Ne and then ^{24}Mg so that the star ends the CHe burning phase with a core composed of
^{24}Mg and (IV) if the rate is stronger, the ^{12}C is produced more rapidly and the star
ends the CHe burning phase with a core composed mostly of ^{12}C. Typically this imposes that

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 [224] 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, , which enters mainly on the hydrostatic equilibrium, , which enters in the Coulomb barrier penetration through the Gamow energy, and a composite parameter , which describes globally the modification of the nuclear reaction rates. The underlying idea is to assume that the power generated per unit volume, , and which determines the luminosity of the star, is proportional to the fudge factor , 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 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.

Living Rev. Relativity 14, (2011), 2
http://www.livingreviews.org/lrr-2011-2 |
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