The amount of ^{4}He 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:

- for , () a first stage during which the neutrons, protons, electrons, positrons an neutrinos are kept in statistical equilibrium by the (rapid) weak interaction As long as statistical equilibrium holds, the neutron to proton ratio is where . The abundance of the other light elements is given by [409] where is the number of degrees of freedom of the nucleus , is the nucleon mass, the baryon-photon ratio and the binding energy.
- Around ( s), the weak interactions freeze out at a temperature determined by the competition between the weak interaction rates and the expansion rate of the universe and thus roughly determined by that is where is the Fermi constant and the number of relativistic degrees of freedom at . Below , the number of neutrons and protons change only from the neutron -decay between to when reactions proceed faster than their inverse dissociation.
- For (), the synthesis of light elements occurs only by
two-body reactions. This requires the deuteron to be synthesized () and the photon
density must be low enough for the photo-dissociation to be negligible. This happens roughly when
with . The abundance of
^{4}He by mass, , is then well estimated by with with and , with being the axial/vector coupling of the nucleon. Assuming that , this gives a dependence . - The abundances of the light element abundances, , are then obtained by solving a series of nuclear
reactions

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 [296] 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 [500]) generally focused on one of these
parameters. For instance, Kolb et al. [295] calculated the dependence of primordial ^{4}He 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 ^{4}He 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 [77] kept track of the changes in and separately
and deduced that while more recently the analysis [308] focused on and
.

Let us now see how the effect of all these parameters are now accounted for in BBN codes.

Bergström et al. [51] started to focus on the -dependence of the thermonuclear rates (see also Ref. [260]). 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

where arises from the Coulomb barrier and is given in terms of the charges and the reduced mass of the two interacting particles as The form factor has to be extrapolated from experimental nuclear data but its -dependence as well as the one of the reduced mass were neglected. Keeping all other constants fixed, assuming no exotic effects and taking a lifetime of 886.7 s for the neutron, it was deduced that . This analysis was then extended [385] to take into account the -dependence of the form factor to conclude thatThen 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 [207] 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. [364] considering the dependence on the parameters independently,

This was generalized by Landau et al. [309] 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. [104] 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

and at a 2 level, were set (assuming ). The deuterium data set the tighter constraint . Note also on Figure 6 that the lithium-7 abundance can be brought in concordance with the spectroscopic observations provided that was smaller during BBNThis analysis was extended [146] 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 [146]. These BBN parameters were then related to the same 6 “fundamental” parameters used in [364].

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} [19]. Two approaches have been
followed.

- Pion mass. A first route is to use the dependence of the binding energy on the pion
mass [188, 38], which is related to the u and d quark masses by
[364], following the computations of [426], adds an electromagnetic contribution so that

but this latter contribution has not been included in other work. - Sigma model. In the framework of the Walecka model, where the potential for the nuclear forces keeps
only the and meson exchanges,

The case of the binding energies of the other elements has been less studied. [146] 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 [104],
or that the variation of the constant is induced by a string dilaton [77]. At each step, one gets
more stringent constraints, which can reach the 10^{–4} [146] to 10^{–5} [104] 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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