The amount of 4He produced during the big bang nucleosynthesis is mainly determined by the neutron to proton ratio at the freeze-out of the weak interactions that interconvert neutrons and protons. The result of Big Bang nucleosynthesis (BBN) thus depends on , , and respectively through the expansion rate, the neutron to proton ratio, the neutron-proton mass difference and the nuclear reaction rates, besides the standard parameters such as, e.g., the number of neutrino families.
The standard BBN scenario [117, 409] proceeds in three main steps:
From an observational point of view, the light elements abundances can be computed as a function of and compared to their observed abundances. Figure 6 summarizes the observational constraints obtained on helium-4, helium-3, deuterium and lithium-7. On the other hand, can be determined independently from the analysis of the cosmic microwave background anisotropies and the WMAP data  have led to to the conclusion that
In complete generality, the effect of varying constants on the BBN predictions is difficult to model because of the intricate structure of QCD and its role in low energy nuclear reactions. Thus, a solution is to proceed in two steps, first by determining the dependencies of the light element abundances on the BBN parameters and then by relating those parameters to the fundamental constants.
The analysis of the previous Section 3.8.1, that was restricted to the helium-4 case, clearly shows that the abundances will depend on: (1) , which will affect the Hubble expansion rate at the time of nucleosynthesis in the same way as extra-relativistic degrees of freedom do, so that it modifies the freeze-out time . This is the only gravitational sector parameter. (2) , the neutron lifetime dictates the free neutron decay and appears in the normalization of the proton-neutron reaction rates. It is the only weak interaction parameter and it is related to the Fermi constant , or equivalently the Higgs vev. (3) , the fine-structure constant. It enters in the Coulomb barriers of the reaction rates through the Gamow factor, in all the binding energies. (4) , the neutron-proton mass difference enters in the neutron-proton ratio and we also have a dependence in (5) and and (6) the binding energies.
Clearly all these parameters are not independent but their relation is often model-dependent. If we focus on helium-4, its abundance mainly depends on , and (and hence mainly on the neutron lifetime, ). Early studies (see Section III.C.2 of FVC ) generally focused on one of these parameters. For instance, Kolb et al.  calculated the dependence of primordial 4He on , and to deduce that the helium-4 abundance was mostly sensitive in the change in and that other abundances were less sensitive to the value of , mainly because 4He has a larger binding energy; its abundances is less sensitive to the weak reaction rate and more to the parameters fixing the value of . To extract the constraint on the fine-structure constant, they decomposed as where the first term represents the electromagnetic contribution and the second part corresponds to all non-electromagnetic contributions. Assuming that and are constant and that the electromagnetic contribution is the dominant part of , they deduced that . Campbell and Olive  kept track of the changes in and separately and deduced that while more recently the analysis  focused on and .
Let us now see how the effect of all these parameters are now accounted for in BBN codes.
Bergström et al.  started to focus on the -dependence of the thermonuclear rates (see also Ref. ). In the non-relativistic limit, it is obtained as the thermal average of the product of the cross, the relative velocity and the the number densities. Charged particles must tunnel through a Coulomb barrier to react. Changing modifies these barriers and thus the reaction rates. Separating the Coulomb part, the low-energy cross section can be written as to take into account the -dependence of the form factor to conclude that
Then the focus fell on the deuterium binding energy, . Flambaum and Shuryak [207, 208, 158, 157] illustrated the sensitivity of the light element abundances on . Its value mainly sets the beginning of the nucleosynthesis, that is of since the temperature must low-enough in order for the photo-dissociation of the deuterium to be negligible (this is at the origin of the deuterium bottleneck). The importance of is easily understood by the fact that the equilibrium abundance of deuterium and the reaction rate depends exponentially on and on the fact that the deuterium is in a shallow bound state. Focusing on the -dependence, it was concluded  that .
This shows that the situation is more complex and that one cannot reduce the analysis to a single varying parameter. Many studies then tried to determinate the sensitivity to the variation of many independent parameters.
The sensitivity of the helium-4 abundance to the variation of 7 parameters was first investigated by Müller et al.  considering the dependence on the parameters independently,
This was generalized by Landau et al.  up to lithium-7 considering the parameters , assuming constant where the variation of and the variation of the masses where tied to these parameters but the effect on the binding energies were not considered.
Coc et al.  considered the effect of a variation of on the abundances of the light elements up to lithium-7, neglecting the effect of on the . Their dependence on the independent variation of each of these parameters is depicted on Figure 6. It confirmed the result of [207, 394] that the deuterium binding energy is the most sensitive parameter. From the helium-4 data alone, the bounds
This analysis was extended  to incorporate the effect of 13 independent BBN parameters including the parameters considered before plus the binding energies of deuterium, tritium, helium-3, helium-4, lithium-6, lithium-7 and beryllium-7. The sensitivity of the light element abundances to the independent variation of these parameters is summarized in Table I of . These BBN parameters were then related to the same 6 “fundamental” parameters used in .
All these analyses demonstrate that the effects of the BBN parameters on the light element abundances are now under control. They have been implemented in BBN codes and most results agree, as well as with semi-analytical estimates. As long as these parameters are assume to vary independently, no constraints sharper than 10–2 can be set. One should also not forget to take into account standard parameters of the BBN computation such as and the effective number of relativistic particle.
To reduce the number parameters, we need to relate the BBN parameters to more fundamental ones, keeping in mind that this can usually be done only in a model-dependent way. We shall describe some of the relations that have been used in many studies. They mainly concern , and .
At lowest order, all dimensional parameters of QCD, e.g., masses, nuclear energies etc., are to a good approximation simply proportional to some powers of . One needs to go beyond such a description and takes the effects of the masses of the quarks into account.
can be expressed in terms of the mass on the quarks u and d and the fine-structure constant as
The neutron lifetime can be well approximated by
Let us now turn to the binding energies, and more particularly to that, as we have seen, is a crucial parameter. This is one the better known quantities in the nuclear domain and it is experimentally measured to a precision better than 10–6 . Two approaches have been followed.
The case of the binding energies of the other elements has been less studied.  follows a route similar than for and relates them to pion mass and assumes that
These analyses allow one to reduce all the BBN parameter to the physical constants (, , , , ) and that is not affected by this discussion. This set can be further reduced, since all the masses can be expressed in terms of as , where are Yukawa couplings.
To go further, one needs to make more assumption, such as grand unification, or by relating the Yukawa coupling of the top to by assuming that weak scale is determined by dimensional transmutation , or that the variation of the constant is induced by a string dilaton . At each step, one gets more stringent constraints, which can reach the 10–4  to 10–5  level but indeed more model-dependent!
Primordial nucleosynthesis offers a possibility to test almost all fundamental constants of physics at a redshift of . This makes it very rich but indeed the effect of each constant is more difficult to disentangle. The effect of the BBN parameters has been quantified with precision and they can be constrained typically at a 10–2 level, and in particular it seems that the most sensitive parameter is the deuterium binding energy.
The link with more fundamental parameters is better understood but the dependence of the deuterium binding energy still left some uncertainties and a good description of the effect of the strange quark mass is missing.
We have not considered the variation of in this section. Its effect is disconnected from the other parameters. Let us just stress that assuming the BBN sensitivity on by just modifying its value may be misleading. In particular can vary a lot during the electron-positron annihilation so that the BBN constraints can in general not be described by an effective speed-up factor [105, 134].
Living Rev. Relativity 14, (2011), 2
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