3.8 Big bang nucleosynthesis

3.8.1 Overview

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 G, αW, αEM and αS 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 [117Jump To The Next Citation Point, 409Jump To The Next Citation Point] proceeds in three main steps:

  1. for T > 1 MeV, (t < 1 s) a first stage during which the neutrons, protons, electrons, positrons an neutrinos are kept in statistical equilibrium by the (rapid) weak interaction
    n ← → p + e− + ¯νe, n + νe ← → p + e− , n + e+ ← → p + ¯νe. (119)
    As long as statistical equilibrium holds, the neutron to proton ratio is
    −Qnp∕kBT (n∕p) = e (120)
    where Qnp ≡ (mn − mp )c2 = 1.29 MeV. The abundance of the other light elements is given by [409Jump To The Next Citation Point]
    ( ) [ ] ζ(3)- A−1 (3A−5)∕2 5∕2 kBT---3(A−1)∕2 A−1 Z A−Z BA∕kBT YA = gA √ π- 2 A m c2 η Yp Yn e , (121) N
    where gA is the number of degrees of freedom of the nucleus AZX, mN is the nucleon mass, η the baryon-photon ratio and BA ≡ (Zmp + (A − Z)mn − mA )c2 the binding energy.
  2. Around T ∼ 0.8 MeV (t ∼ 2 s), the weak interactions freeze out at a temperature Tf determined by the competition between the weak interaction rates and the expansion rate of the universe and thus roughly determined by Γ w(Tf) ∼ H (Tf) that is
    G2 (k T )5 ∼ ∘GN----(k T )2 (122) F B f ∗ B f
    where GF is the Fermi constant and N ∗ the number of relativistic degrees of freedom at Tf. Below Tf, the number of neutrons and protons change only from the neutron β-decay between T f to T ∼ 0.1 MeV N when p + n reactions proceed faster than their inverse dissociation.
  3. For 0.05 MeV < T < 0.6 MeV (3 s < t < 6 min), the synthesis of light elements occurs only by two-body reactions. This requires the deuteron to be synthesized (p + n → D) and the photon density must be low enough for the photo-dissociation to be negligible. This happens roughly when
    nd- 2 nγ ∼ η exp(− BD ∕TN ) ∼ 1 (123)
    with − 10 η ∼ 3 × 10. The abundance of 4He by mass, Yp, is then well estimated by
    Yp ≃ 2 --(n-∕p)N--- (124) 1 + (n ∕p)N
    (n ∕p)N = (n∕p )f exp(− tN ∕τn) (125)
    with tN ∝ G −1∕2T− 2 N and τ−1 = 1.636G2 (1 + 3g2)m5 ∕(2π3 ) n F A e, with gA ≃ 1.26 being the axial/vector coupling of the nucleon. Assuming that 2 BD ∝ αS, this gives a dependence −1∕2 2 2 tN∕τp ∝ G αSG F.
  4. The abundances of the light element abundances, Yi, are then obtained by solving a series of nuclear reactions
    ˙Yi = J − Γ Yi,
    where J and Γ are time-dependent source and sink terms.

From an observational point of view, the light elements abundances can be computed as a function of η and compared to their observed abundances. Figure 6View Image summarizes the observational constraints obtained on helium-4, helium-3, deuterium and lithium-7. On the other hand, η 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

−10 η = ηWMAP = (6.19 ± 0.15 ) × 10 .
This number being fixed, all abundances can be computed. At present, there exists a discrepancy between the predicted abundance of lithium-7 based on the WMAP results [108Jump To The Next Citation Point, 107Jump To The Next Citation Point] for η, 7Li∕H = (5.14 ± 0.50 ) × 10 −10 and its values measured in metal-poor halo stars in our galaxy [63], 7Li∕H = (1.26 ± 0.26 ) × 10 −10, which is a factor of three lower, at least [116] (see also [469]), than the predicted value. No solution to this Lithium-7 problem is known. A back of the envelope estimates shows that we can mimic a lower η parameter, just by modifying the deuterium binding energy, letting TN unchanged, since from Equation (123View Equation), one just need ΔBD ∕TN ∼ − ln9 so that the effective η parameter, assuming no variation of constant, is three times smaller than ηWMAP. This rough rule of thumb explains that the solution of the lithium-7 problem may lie in a possible variation of the fundamental constants (see below for details).

3.8.2 Constants everywhere…

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) α G, 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 Tf. This is the only gravitational sector parameter. (2) τn, 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 GF, or equivalently the Higgs vev. (3) α EM, the fine-structure constant. It enters in the Coulomb barriers of the reaction rates through the Gamow factor, in all the binding energies. (4) Qnp, the neutron-proton mass difference enters in the neutron-proton ratio and we also have a dependence in (5) mN and me 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 Qnp, Tf and TN (and hence mainly on the neutron lifetime, τn). Early studies (see Section III.C.2 of FVC [500Jump To The Next Citation Point]) generally focused on one of these parameters. For instance, Kolb et al. [295Jump To The Next Citation Point] calculated the dependence of primordial 4He on G, GF and Qnp to deduce that the helium-4 abundance was mostly sensitive in the change in Qnp and that other abundances were less sensitive to the value of Qnp, 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 (n∕p ). To extract the constraint on the fine-structure constant, they decomposed Qnp as Q = α Q + βQ np EM α β where the first term represents the electromagnetic contribution and the second part corresponds to all non-electromagnetic contributions. Assuming that Q α and Q β are constant and that the electromagnetic contribution is the dominant part of Q, they deduced that |Δ αEM ∕αEM | < 10−2. Campbell and Olive [77Jump To The Next Citation Point] kept track of the changes in Tf and Qnp separately and deduced that ΔYp-≃ ΔTf − ΔQnp- Yp Tf Qnp while more recently the analysis [308] focused on α EM and v.

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 αEM-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 αEM modifies these barriers and thus the reaction rates. Separating the Coulomb part, the low-energy cross section can be written as

σ(E ) = S(E-)e−2πη(E ) (126 ) E
where η(E ) arises from the Coulomb barrier and is given in terms of the charges and the reduced mass Mr of the two interacting particles as
∘ ----2- η(E) = αEMZ1Z2 Mrc--. (127 ) 2E
The form factor S (E ) has to be extrapolated from experimental nuclear data but its αEM-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 |Δ αEM ∕αEM | < 2 × 10−2. This analysis was then extended [385Jump To The Next Citation Point] to take into account the α EM-dependence of the form factor to conclude that
∘ ----2- σ(E ) = --2π-η(E-)-- ≃ 2π αEMZ1Z2 Mrc-- exp−2πη(E). exp2πη(E)− 1 c2
Ref. [385] also took into account (1) the effect that when two charged particles are produced they must escape the Coulomb barrier. This effect is generally weak because the Qi-values (energy release) of the different reactions are generally larger than the Coulomb barrier at the exception of two cases, 3He (n,p)3H and 7Be (n,p)7Li. The rate of these reactions must be multiplied by a factor (1 + a Δ α ∕α ) i EM EM. (2) The radiative capture (photon emitting processes) are proportional to αEM since it is the strength of the coupling of the photon and nuclear currents. All these rates need to be multiplied by (1 + Δ αEM ∕αEM ). (3) The electromagnetic contribution to all masses was taken into account, which modify the Qi-values as Qi → Qi + qiΔ αEM ∕αEM ). For helium-4 abundance these effects are negligible since the main α EM-dependence arises from Q np. Equipped with these modifications, it was concluded that +0.010 Δ αEM ∕αEM = − 0.007− 0.017 using only deuterium and helium-4 since the lithium-7 problem was still present.

Then the focus fell on the deuterium binding energy, BD. Flambaum and Shuryak [207Jump To The Next Citation Point, 208Jump To The Next Citation Point, 158, 157] illustrated the sensitivity of the light element abundances on BD. Its value mainly sets the beginning of the nucleosynthesis, that is of T N 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 BD is easily understood by the fact that the equilibrium abundance of deuterium and the reaction rate p(n,γ)D depends exponentially on BD and on the fact that the deuterium is in a shallow bound state. Focusing on the TN-dependence, it was concluded [207Jump To The Next Citation Point] that ΔBD ∕BD < 0.075.

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.

View Image

Figure 6: (Left): variation of the light element abundances in function of η compared to the spectroscopic abundances. The vertical line depicts the constraint obtained on η from the study of the cosmic microwave background data. The lithium-7 problem lies in the fact that ηspectro < ηWMAP. From [107]. (right): Dependence of the light element abundance on the independent variation of the BBN parameters, assuming η = ηWMAP. From [105Jump To The Next Citation Point]

The sensitivity of the helium-4 abundance to the variation of 7 parameters was first investigated by Müller et al. [364Jump To The Next Citation Point] considering the dependence on the parameters {Xi} ≡ {G, αEM, v,me, τn,Qnp, BD } independently,

∑ (X ) Δ ln YHe = ci Δ ln Xi i
and assuming ΛQCD fixed (so that the seven parameters are in fact dimensionless quantities). The (X) ci are the sensitivities to the BBN parameters, assuming the six others are fixed. It was concluded that YHe ∝ α−EM0.043v2.4m0.e024τ0n.24Q −np1.8B0.D53G0.405 for independent variations. They further related (τn,Qnp, BD ) to (αEM, v,me, mN, md − mu ), as we shall discuss in the next Section 3.8.3.

This was generalized by Landau et al. [309] up to lithium-7 considering the parameters {αEM, GF, ΛQCD, Ωbh2 }, assuming G constant where the variation of τn and the variation of the masses where tied to these parameters but the effect on the binding energies were not considered.

Coc et al. [104Jump To The Next Citation Point] considered the effect of a variation of (Qnp, BD, τn,me ) on the abundances of the light elements up to lithium-7, neglecting the effect of αEM on the . Their dependence on the independent variation of each of these parameters is depicted on Figure 6View Image. It confirmed the result of [207Jump To The Next Citation Point, 394] that the deuterium binding energy is the most sensitive parameter. From the helium-4 data alone, the bounds

−2 Δτn- −2 −2 ΔQnp-- − 2 − 8.2 × 10 ≲ τ ≲ 6 × 10 , − 4 × 10 ≲ Q ≲ 2.7 × 10 , (128 ) n np
− 7.5 × 10− 2 ≲ ΔBD---≲ 6.5 × 10−2, (129 ) BD
at a 2σ level, were set (assuming η WMAP). The deuterium data set the tighter constraint −2 − 2 − 4 × 10 ≲ Δ ln BD ≲ 3 × 10. Note also on Figure 6View Image that the lithium-7 abundance can be brought in concordance with the spectroscopic observations provided that BD was smaller during BBN
− 2 ΔBD--- −2 − 7.5 × 10 ≲ BD ≲ − 4 × 10 ,
so that BD may be the most important parameter to resolve the lithium-7 problem. The effect of the quark mass on the binding energies was described in [49]. They then concluded that a variation of Δmq ∕mq = 0.013 ± 0.002 allows to reconcile the abundance of lithium-7 and the value of η deduced from WMAP.

This analysis was extended [146Jump To The Next Citation Point] 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 [146Jump To The Next Citation Point]. These BBN parameters were then related to the same 6 “fundamental” parameters used in [364Jump To The Next Citation Point].

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 η and the effective number of relativistic particle.

3.8.3 From BBN parameters to fundamental constants

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 Qnp, τn and BD.

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 ΛQCD. One needs to go beyond such a description and takes the effects of the masses of the quarks into account.

Qnp can be expressed in terms of the mass on the quarks u and d and the fine-structure constant as

Qnp = aαEM ΛQCD + (md − mu ),
where the electromagnetic contribution today is (aαEM ΛQCD )0 = − 0.76 MeV and therefore the quark mass contribution today is (m − m ) = 2.05 d u [230] so that
ΔQnp Δ αEM Δ(md − mu ) -Q----= − 0.59-α---- + 1.59 (m---−-m--)-. (130 ) np EM d u
All the analyses cited above agree on this dependence.

The neutron lifetime can be well approximated by

2 [ ( )] −1 1-+-3gA- 2 5 ∘ -2---- 4 2 ∘ -2---- τn = 120π3 GFm e q − 1(2q − 9q − 8) + 15 ln q + q − 1 ,
with q ≡ Qnp ∕me and √ -- 2 GF = 1∕ 2v. Using the former expression for Qnp we can express τn in terms of αEM, v and the u, d and electron masses. It follows
Δ τn Δ αEM Δv Δme Δ (md − mu ) ----= 3.86------ + 4----+ 1.52 -----− 10.4-------------. (131 ) τn αEM v me (md − mu )
Again, all the analyses cited above agree on this dependence.

Let us now turn to the binding energies, and more particularly to B D 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.

The case of the binding energies of the other elements has been less studied. [146Jump To The Next Citation Point] follows a route similar than for BD and relates them to pion mass and assumes that

∂B B ----i= fi(Ai − 1)-D-r ≃ − 0.13fi(Ai − 1), ∂m π m π
where fi are unknown coefficients assumed to be of order unity and Ai is the number of nucleons. No other estimates has been performed. Other nuclear potentials (such as Reid 93 potential, Nijmegen potential, Argonne v18 potential and Bonn potential) have been used in [101] to determine the dependence of BD on v and agree with previous studies.

These analyses allow one to reduce all the BBN parameter to the physical constants (αEM, v, me, md − mu, mq) and G that is not affected by this discussion. This set can be further reduced, since all the masses can be expressed in terms of v as m = h v i i, where h i 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 v by assuming that weak scale is determined by dimensional transmutation [104Jump To The Next Citation Point], or that the variation of the constant is induced by a string dilaton [77Jump To The Next Citation Point]. At each step, one gets more stringent constraints, which can reach the 10–4 [146Jump To The Next Citation Point] to 10–5 [104Jump To The Next Citation Point] level but indeed more model-dependent!

3.8.4 Conclusion

Primordial nucleosynthesis offers a possibility to test almost all fundamental constants of physics at a redshift of z ∼ 108. 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 G in this section. Its effect is disconnected from the other parameters. Let us just stress that assuming the BBN sensitivity on G by just modifying its value may be misleading. In particular G 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 [105Jump To The Next Citation Point, 134Jump To The Next Citation Point].

  Go to previous page Go up Go to next page