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A&A
Volume 520, September-October 2010
Article Number A112
Number of page(s) 6
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/201014825
Published online 12 October 2010
A&A 520, A112 (2010)

Effect of the variation of the Higgs vacuum expectation value upon the deuterium binding energy and primordial abundances of D and 4He

M. E. Mosquera1,2 - O. Civitarese2

1 - Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque, 1900 La Plata, Argentina
2 - Department of Physics, University of La Plata, cc 67, 1900 La Plata, Argentina

Received 19 April 2010 / Accepted 5 July 2010

Abstract
Aims. We calculate the constraints on the time variation of the Higgs vacuum expectation value from Big Bang Nucleosynthesis.
Methods. Starting from the calculation of the deuterium binding-energy, as a function of the pion-mass and using the NN-Reid 93 potential, we calculate the abundances of primordial D and 4He by modifying Kawano's code. The Higgs vacuum expectation value (v) and the baryon to photon ratio $(\eta_{\rm B})$ enter the calculation as free parameters. By using the observational data of D and 4He, we set constraints on $\eta _{\rm B}$ and on the variation of v, relative to a constant value of $\Lambda_{\rm QCD}$.
Results. Results are consistent with null variation in v and $\epsilon_{\rm D}$ for the early universe, within 6$\sigma $.
Conclusions. We obtained a linear dependence of $\epsilon_{\rm D}$ upon v and found that the best-fit-value of the variation of v is null within 6$\sigma $.

Key words: primordial nucleosynthesis - cosmological parameters - cosmology: theory

1 Introduction

One of the most powerful tools to study the early Universe is the Big Bang nucleosynthesis (BBN). Since BBN is sensible to parameters such as the fine structure constant, the electron mass, the Higgs vacuum expectation value (v), the deuterium binding energy $(\epsilon_{\rm D})$, among others, it is an important test to set constraints on deviations from the standard cosmology, and on physical theories beyond the standard model (SM). There are some theories which allow fundamental constants to vary over cosmological times scales (Damour et al., 2002a; Gleiser & Taylor, 1985; Barr & Mohapatra, 1988; Maeda, 1988; Brax et al., 2003; Klein, 1926; Palma et al., 2003; Kaluza, 1921; Wu & Wang, 1986; Overduin & Wesson, 1997; Damour & Polyakov, 1994; Weinberg, 1983; Youm, 2001a; Damour et al., 2002b; Youm, 2001b). The time variation of fundamental constants (e.g. the fine structure constant, the electron mass, the Planck mass), was studied in Ichikawa & Kawasaki (2004); Campbell & Olive (1995); Yoo & Scherrer (2003); Chamoun et al. (2007); Cyburt et al. (2005); Bergström et al. (1999); Landau et al. (2008); Mosquera et al. (2008); Coc et al. (2007); Müller et al. (2004); Landau et al. (2006); Nollett & Lopez (2002); Ichikawa & Kawasaki (2002).

The deuterium binding energy plays a crucial role in the reaction rates involved in the formation of primordial elements during the Big Bang Nucleosynthesis (BBN). All the primordial abundances would be different from the BBN predictions if the deuterium was deeply- or weakly-bound in that epoch (e.g. the abundance of deuterium depends exponentially on $\epsilon_{\rm D}$). In Flambaum & Shuryak (2003,2002); Dmitriev & Flambaum (2003); Dmitriev et al. (2004); Berengut et al. (2010) the variation of $\epsilon_{\rm D}$ as function of the quark masses was studied and the authors applied their results to set constraints using data from cosmological epochs. In Flambaum & Wiringa (2007) the dependence of nuclear binding on hadronic mass was studied. In Yoo & Scherrer (2003) the dependence of the deuterium binding energy on the Higgs vacuum expectation value was considered using the results of Beane & Savage (2003); Epelbaum et al. (2003). In the same work, $\epsilon_{\rm D}$ was represented as a linear function of v and this dependence was used to set constraints on the variation of the Higgs vacuum expectation value during cosmological times. Dent et al. (2007) studied the dependence of the primordial abundances with several parameters such as $G_{\rm N}$, neutron decay time, $\alpha$, $m_{\rm e}$, the average nucleon mass, the neutron-proton mass difference and D, T, 3He, 4He, 6Li, 7Li, and 7Be binding energies, and found that the deuterium and lithium abundances are strongly dependent on the Higgs vacuum expectation value. However, in Dent et al. (2007), the variations of the binding energies are assumed to obey a linear dependence on the pion mass, as given by Beane & Savage (2003).

In this work, we calculate the dependence of the deuterium binding energy with the pion-mass, using an effective nucleon-nucleon interaction. There exist several nucleon-nucleon effective potentials (Nagels et al., 1977; Reid, 1968; Nagels et al., 1975; Wiringa et al., 1995; Machleidt et al., 1987; Stoks et al., 1994; Lacombe et al., 1980); for the sake of the present calculation we have chosen the Reid 93 potential (Stoks et al., 1994). Following Berengut et al. (2010), we assume $\Lambda_{\rm QCD}$ is constant, that is, we measure all dimensions in units of $\Lambda_{\rm QCD}$. After determining the dependence of $\epsilon_{\rm D}$ on the dimensionless parameter $N=v/\Lambda_{\rm QCD}$, we concentrate on the calculation of BBN observables, like the abundances of deuterium (D) and helium (4He), to determine their sensitivity upon $\epsilon_{\rm D}$ and ${\rm N}$. Hereafter, the relative variations $\frac{\delta v}{v_0}$ and $\frac{\delta m_\pi}{\left( m_\pi \right)_0}$ might be understood as the relative variations $\frac{\delta {N}}{{N}_0}$ and $\frac{\delta {M}}{{M}_0}$, where ${M}=\frac{m_\pi}{\Lambda_{\rm QCD}}$, respectively. We actually determine BBN abundances, after calculating the D-binding energy, as a function of $v/\Lambda_{\rm QCD}$, through the variation of the pion mass. In this aspect, our attempt differs from the one of Dent et al. (2007), where the variation of the binding energies of the nuclei involved in BBN is taken in a parameter form.

The paper is organized as follows. In Sect. 2, we discuss the dependence of the deuterium binding energy with the pion-mass. In Sect. 3, we calculate the primordial abundances and obtain constraints on the variation of the deuterium binding energy and on the Higgs vacuum expectation value. Our conclusions are presented in Sect. 4. The details of the formalism, concerning the calculation of various quantities which are needed to computed BBN abundances, are presented in Appendix A.

2 Dependence of the deuterium binding energy with the pion-mass

We are interested in the effects on the deuterium-binding-energy due to the change of the pion-mass; a change which is related to the variation of v. Assuming that the pion-mass acquires different values in different epochs of the Universe, some observables, such as the primordial abundances, might differ from their values predicted by the Standard Model (Sarkar, 1996).

The variation of v produces different effects on the mass of different mesons, namely: light-mesons, like the pion, are effected more drastically than heavier mesons (Flambaum & Wiringa, 2007).

The Reid potential represents the nucleon-nucleon interaction through the one-pion exchange mechanism (OPE) and a combination of central, tensor and spin-orbit functions with cut-off parameters (non-OPE) (Stoks et al., 1994). The Reid 93 potential is the regularized version of the Reid 68 potential (Reid, 1968). The regularization is made to remove the singularities at the origin, by introducing a dipole form-factor in the Fourier transformation that leads from the momentum-space potential to the configuration-space potential (Stoks et al., 1994).

The OPE contribution to the Reid 93 potential is then written as[*] (Stoks et al., 1994)

                     $\displaystyle V_{\rm OPE}(r)$ = $\displaystyle -f_\pi^2 \left\{\left(\frac{m_{\pi^0}}{m_{\rm s}}\right)^2 m_{\pi^0} \left[\phi_{\rm T}^0(m_{\pi^0},r) S_{12}\phantom{\frac{1}{3}}\right.\right.$  
    $\displaystyle \left. \left. +\frac{1}{3} \phi_{\rm C}^0(m_{\pi^0},r) \left(\sigma_1 \sigma_2 \right) \right] \right.$  
    $\displaystyle \left.+2 \left(\frac{m_{\pi^\pm}}{m_{\rm s}}\right)^2 m_{\pi^\pm}\left[\phi_{\rm T}^0(m_{\pi^\pm},r) S_{12}\phantom{\frac{1}{3}} \right.\right.$  
    $\displaystyle \hspace*{-1.1cm}\left.\phantom{\left(\frac{m_{\pi^0}}{m_{\rm s}}\... ...\phi_{\rm C}^0(m_{\pi^\pm},r)\left(\sigma_1 \sigma_2 \right) \right] \right\} ,$  

where $m_{\pi^0}$ and $m_{\pi^\pm}$ are the mass of the neutral and charged pion respectively. The non-OPE contribution are written
                               $\displaystyle V_{\rm C}(r)$ = $\displaystyle \overline{m}_\pi \sum_{p=2}^6 \alpha_p ~ p ~ \phi_{\rm C}^0(p ~ \overline{m}_\pi,r) ,$  
$\displaystyle V_{\rm T}(r)$ = $\displaystyle 4 \overline{m}_\pi~ \beta_{4} \phi_{\rm T}^0(4 ~ \overline{m}_\pi,r)+ 6 \overline{m}_\pi~ \beta_{6}~ \phi_{\rm T}^0(6 ~ \overline{m}_\pi,r) ,$  
$\displaystyle V_{\rm LS}(r)$ = $\displaystyle 3 \overline{m}_\pi~ \gamma_{3} ~\phi_{\rm SO}^0(3~ \overline{m}_\pi,r)+ 5 \overline{m}_\pi~ \gamma_{5}~ \phi_{\rm SO}^0(5 ~ \overline{m}_\pi,r) ,$  

where $\overline{m}_\pi=(m_{\pi^0}+2m_{\pi^\pm})/3$, $\phi_{\rm C}^0(m,r)$, $ \phi_{\rm T}^0(m,r)$and $\phi_{\rm SO}^0(m,r)$ are the central, tensor and spin-orbit contribution to the potential respectively (Stoks et al., 1994).

Indeed, by multiplying the pion-mass by a constant factor (which is the same for charged and neutral pion), while keeping the scaling masses $m_{\rm s}$ and $\overline{m}_\pi$ at a fixed value (Flambaum & Wiringa, 2007), the pion-mass can be varied to affect OPE vertices of the NN potential. Although OPE is not the unique mechanism where the pion-mass appears explicitly, it is the only mechanism accounted for by the Reid 93 potential. Neither the two-pion exchange nor the heavy-meson-exchange mechanisms appear explicitly in this potential.

The effects on the potential due to the change of the pion-mass are noticeable (Flambaum & Wiringa, 2007). Therefore one might expect that both, the binding energy $\epsilon_{\rm D}$ and the D ground-state wave function $\phi(r)$would be affected by changes in $m_\pi$. The deuteron wave function can be written as a finite set of Yukawa-type functions (Krutov & Troitsky, 2007; Lacombe et al., 1981) because of the functional structure of the potential.

After modifying the Reid potential, to take into account the variation of the pion-mass (as said before affecting only the OPE terms), we calculate the deuterium wave function and the deuterium binding energy for different values of the pion-mass, by solving the corresponding radial Schrödinger equation. With the obtained wave function, for each value of the pion-mass, we have calculated the deuterium binding energy and cast the results as a function of the relative variation $\frac{\delta m_\pi}{\left( m_\pi \right)_0}$. If we call $\frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}$ the relative variation of the deuterium binding energy (quantities with subindex 0represent the actual values of the mentioned quantity), we found that the dependence of the variation of the deuterium binding energy on the variation of the pion-mass can be fitted by the straight-line $ \frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}=-3.65 ~\frac{\delta m_\pi} {\left(m_\pi \right)_0}$. To put this result in perspective, one can compare it with the values reported by Beane & Savage (2003); Flambaum & Shuryak (2002); Yoo & Scherrer (2003); Epelbaum et al. (2003), where the same dependence yields values in the interval (-18, +3). As a consequence of this effect the deuterium binding energy would be dependent on v, since $m_\pi^2 \propto v$. A comparison of the previous and our results is shown in Fig. 1.

\begin{figure} \par\includegraphics[width=170 pt,angle=-90]{14825fg1.ps} \end{figure} Figure 1:

Dependence of $\frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}$ upon the relative change of the pion mass $\frac{\delta m_\pi}{\left( m_\pi \right)_0}$, from the work of Flambaum & Shuryak (2002) (grey area) and our calculated value (dotted line).

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The effect of these dependencies upon the BBN abundances will be discussed later on (see Sect. 3).

3 Big Bang nucleosynthesis

The standard model of the BBN has only one free-parameter: the baryon to photon ratio $\eta _{\rm B}$, which is determined by the comparison between observed primordial abundances and theoretical calculations, or by the analysis of the cosmic background data (Spergel et al., 2003,2007). The theoretical abundances are consistent with the observed abundance of deuterium but they are not entirely consistent with the observed abundance of 4He. In Table 1 we present the theoretical abundances of D and 4He calculated in the standard model by using Kawano's code (Kawano, 1992,1988). If the Higgs vacuum expectation value v changes with time, while $\Lambda_{\rm QCD}$ is fixed, this discrepancy might eventually be reconciled. In order to calculate the primordial abundances of D and 4He, for variable deuterium binding energy, we modify the numerical code developed by Kawano (1992,1988), as explained in Appendix A.

Table 1:   Theoretical abundances in the standard model.

To set bounds on the variation of the deuterium binding energy and on the variation of v we have used the deuterium primordial abundance reported by Levshakov et al. (2002); Pettini & Bowen (2001); Ivanchik et al. (2010); O'Meara et al. (2006); Kirkman et al. (2003); Burles & Tytler (1998b); O'Meara et al. (2001); Burles & Tytler (1998a); Pettini et al. (2008) (see Table 2). Regarding to the 4He primordial abundance, in the literature, there have been two different methods to determine it that yield quite different results (Izotov et al., 1994; Peimbert, 2002; Izotov et al., 1997; Thuan & Izotov, 2002; Izotov et al., 2006; Thuan & Izotov, 1998; Olive et al., 1997; Izotov & Thuan, 2004; Luridiana et al., 2003; Peimbert et al., 2002). Since 2007, new atomic data were incorporated to the calculations of the 4He primordial abundance, a quantity that depends on the HeI recombination coefficients. Therefore, new calculations were performed using the new atomic data, resulting into higher values of the 4He abundance (Izotov & Thuan, 2010; Izotov et al., 2007; Aver et al., 2010; Peimbert et al., 2007). In order to study the variation of $\epsilon_{\rm D}$ or v we only consider the latest 4He data, reported by Izotov & Thuan (2010), Aver et al. (2010, see Table 3#. Regarding the consistency of the data, we have followed the treatment of Yao et al. (2006) and increase the observational error by a factor $\Theta$ (see below).

Table 2:   Deuterium observational abundances $(Y_{\rm D}^{\rm obs})$.

Table 3:   4He observational abundances $(Y_{^4{\rm He}}^{\rm obs})$.

We have computed light nuclei abundances, and performed the statistical analysis using observational data, to obtain the best fit of the deuterium binding energy, the Higgs vacuum expectation value and the baryon to photon ratio. We have considered the following cases:

i)
variation of $\epsilon_{\rm D}$, by keeping $\eta _{\rm B}$ fixed at the WMAP value;
ii)
variation of $\epsilon_{\rm D}$ and $\eta _{\rm B}$;
iii)
variation of v and keeping $\eta _{\rm B}$ fixed at the WMAP value, and;
iv)
variation of both v and $\eta _{\rm B}$.

We perform the analysis on the Higgs vacuum expectation value, for three different value of $\kappa$ $\left(\frac{\delta \epsilon_{\rm D}}{\left(\epsilon_{\rm D}\right)_0}=\kappa ~\frac{\delta m_\pi} {\left(m_\pi \right)_0}\right)$ that is: our result $\left(\kappa=-3.65\right)$, the higher limit used by Yoo & Scherrer (2003) $\left(\kappa=-10\right)$ and the lower limit of the interval found by Flambaum & Shuryak (2002) $\left(\kappa=3\right)$, and considering $\Lambda_{\rm QCD}$ fixed.

3.1 Constraints on $\epsilon _{\sf D}$

We have computed the theoretical primordial abundances for different values of the deuterium binding energy, by keeping $\eta _{\rm B}$ fixed at the WMAP value $\eta_{\rm B}=\left(6.108 \pm 0.219\right) \times 10^{-10}$ (Spergel et al., 2007). We have found the best-fit-parameter value using a $\chi^2$-test and the observational data. The results are

             $\displaystyle \frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}$ = $\displaystyle 5.60^{+0.85}_{-0.45} \times 10^{-2} ,$  
$\displaystyle \frac{\chi^2_{\min}}{N-1}$ = 1.79 , (1)

where $\chi^2_{\min}$ is the lowest value of $\chi^2$ and N is the number of data $\left(N=12\right)$. We found variation of the deuterium binding energy even at the level of six standard deviations $(6\sigma)$. The result can be explained since an increase in the deuterium binding energy leads to a larger initial abundance of deuterium. The abundance of 4He is larger since the production of this nuclei starts sooner and the final deuterium abundance is decreased (Yoo & Scherrer, 2003).

The next step was to consider the baryon to photon ratio as an extra parameter to be fixed. Therefore, we have computed the theoretical primordial abundances for different values of the deuterium binding energy and of the baryon to photon ratio. Using the data on D and 4He, we have performed a $\chi^2$-test to find the best-fit-parameter value

                $\displaystyle \frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}$ = $\displaystyle \left(7.60 \pm 1.35 \right) \times 10^{-2} ,$  
$\displaystyle \eta_{\rm B}$ = $\displaystyle 4.978^{+0.449}_{-0.360} \times 10^{-10} ,$  
$\displaystyle \frac{\chi^2_{\min}}{N-2}$ = 0.90 . (2)

The value of $\eta _{\rm B}$ agrees with the value obtained by WMAP (Spergel et al., 2007) within three standard deviation $\sigma $. For this case, we found null variation of the deuterium binding energy at the level of 6$\sigma $. The result is presented in Fig. 2, for three values of the deviation, that is at one, two and three $\sigma $. In the same Figure we show the one-dimensional likelihood, for $\eta _{\rm B}$ and $\frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}$.

\begin{figure} \par\includegraphics[width=280 pt,angle=0]{14825fg2.ps} \end{figure} Figure 2:

1$\sigma $, 2$\sigma $ and 3$\sigma $ likelihood contours for $\eta _{\rm B}$ and $\frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}$, and one-dimensional likelihood, $\left(\frac{\rm L}{\rm L_{max}}\right)$.

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3.2 Constraints on v

Next, we have studied the variation of the Higgs vacuum expectation value and of the baryon to photon ratio.

If the Higgs vacuum expectation value varies with time, the effects upon BBN are not only the ones due to the variation of the deuterium binding energy but also those due to the variation of the electron mass $m_{\rm e}$ $\left(m_{\rm e} \propto v \right)$, the neutron-proton mass-difference $\Delta m_{\rm np}$ and the Fermi constant $G_{\rm F}$ $\left(G_{\rm F} \propto v^{-2}\right)$ (see Appendix A, for details).

We have considered the baryon to photon ratio fixed at the WMAP value, and have computed the light abundances for different values of v. Once again, we performed a $\chi^2$-test to obtain the best-fit value. The results of our analysis are shown in Table 4, (where $\delta v=v_{\rm BBN}-v_0$, $v_{\rm BBN}$is the value of the binding energy during BBN, v0 is the present value of v) for $\eta _{\rm B}$ fixed at the WMAP value $(\eta_{\rm B}^{\rm WMAP}=\left(6.108 \pm 0.219\right) \times 10^{-10})$(Spergel et al., 2007), for three different values of $\kappa$.

Table 4:   Best fit parameter value and 1$\sigma $ errors constraints on  $\frac{\delta v}{v_0}$.

We found variation of v at the level of six standard deviations $(6\sigma)$, for all the dependencies of the deuterium binding energy with the pion-mass. The first two rows of Table 4 indicate that there is not a good fit for $\kappa =-3.65$ and $\kappa=-10$.

Finally, we have performed the calculation of the primordial abundances and found the best fit of v and $\eta _{\rm B}$simultaneously. The results are given in Table 5, for three different values of $\kappa$.

\begin{figure} \par\includegraphics[width=280pt,clip]{14825fg3.ps} \end{figure} Figure 3:

1$\sigma $, 2$\sigma $ and 3$\sigma $ likelihood contours for $\eta _{\rm B}$ and $\frac{\delta v}{v_0}$, and one-dimensional likelihood, $\left(\frac{\rm L}{\rm L_{max}}\right)$, using $\kappa =-3.65$.

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\begin{figure} \par\includegraphics[width=280pt,clip]{14825fg4.ps} \end{figure} Figure 4:

1$\sigma $, 2$\sigma $ and 3$\sigma $ likelihood contours for $\eta _{\rm B}$ and $\frac{\delta v}{v_0}$, and one-dimensional likelihood, $\left(\frac{\rm L}{\rm L_{max}}\right)$, using $\kappa =3$.

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Table 5:   Best fit parameter value and 1$\sigma $ errors constraints on $\frac{\delta v}{v_0}$ and  $\eta _{\rm B}$.

We found null variation of v at $5\sigma$, $4\sigma$and $6\sigma$ for $\kappa =-3.65$, $\kappa=-10$ and $\kappa =3$ respectively. Meanwhile, the value for $\eta _{\rm B}$ agrees with the value of WMAP at $4\sigma$, $3\sigma$and $1\sigma$ for $\kappa =-3.65$, $\kappa=-10$ and $\kappa =3$ respectively. However, there is not a good fit if $\kappa=-10$. In Figs. 3 and 4 we present the corresponding likelihood contours for $\kappa =-3.65$ and $\kappa =3$ respectively.

4 Conclusion

In the first part of this work we have studied the dependence of the deuterium binding energy as a function of the pion-mass, which is ultimately a function of the Higgs vacuum expectation value. For the analysis, we used the Reid 93 potential to represent the nucleon-nucleon interaction. It is found that the binding energy depends linearly on the pion-mass, and that the calculated value lies in the range obtained by various authors, e. g. Flambaum & Shuryak (2002). Our result for the slope of the functional dependence of $\frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}$ vs. the variation of $m_\pi$(-3.65), may reduce the uncertainties associated to it, since in other works (Beane & Savage, 2003; Yoo & Scherrer, 2003; Epelbaum et al., 2003) a domain was reported. Next, we have calculated primordial abundances of BBN and focused on the discrepancy between standard BBN estimation for 4He and D and their observational data. We found that, by allowing variations of either $\epsilon_{\rm D}$ or v, this discrepancy is not solve.

Acknowledgements
This work has been partially supported by the National Research Council (CONICET) of Argentina (PIP 5145, PIP 112-200801-00740).

Appendix A: Modifications to Kawano's code

In this Appendix we discuss the dependence on the Higgs vacuum expectation value of the different physical quantities involved in the calculation of primordial abundances.

If during BBN v acquires a value different than the value at the present time, then the electron mass, the Fermi constant, the neutron-proton mass difference and the deuterium binding energy would also take different values (Landau et al., 2008).

A change in the electron mass affects the sum of the electron and positron energy densities, the sum of the electron and positron pressures and the difference of the electron and positron number densities. These quantities are calculated in Kawano's code (Kawano, 1992,1988) as:

$\displaystyle \rho_{{\rm e}^-}+ \rho_{{\rm e}^+}= \frac{2}{\pi^2} \frac{\left(m... ...bar c \right)^3} \sum_n (-1)^{n+1} {\rm cosh} \left(n\phi_{\rm e}\right) M(nz),$      

$\displaystyle \frac{p_{{\rm e}^-}+ p_{{\rm e}^+}}{c^2}=\frac{2}{\pi^2} \frac{\l... ...)^3} \sum_n \frac{(-1)^{n+1}}{nz} {\rm cosh} \left(n\phi_{\rm e}\right) N(nz) ,$      

$\displaystyle \frac{\pi^2}{2}\left[\frac{\hbar c^3}{m_{\rm e} c^2}\right]^3 z^3\left(n_{{\rm e}^-}-n_{{\rm e}^+}\right)$ = $\displaystyle z^3 \sum_n (-1)^{n+1}$     
    $\displaystyle \times\;{\rm sinh}\left(n\phi_{\rm e}\right) L(nz),$       

where $z=\frac{m_{\rm e} c 2}{k_{\rm B} T_\gamma}$, $\phi_{\rm e}$is the electron chemical potential and L(z), M(z) and N(z) are combinations of the modified Bessel function Ki(z) (Kawano, 1992,1988). In order to include the variation in $m_{\rm e}$ we replace, in all the equations, $m_{\rm e}$ by $\left(m_{\rm e}\right)_0 \left(1+\frac{\delta m_{\rm e}}{\left(m_{\rm e}\right)_0} \right)$, and consider $\frac{\delta m_{\rm e}}{\left(m_{\rm e}\right)_0}=\frac{\delta v}{v_0}\cdot$

The $n\leftrightarrow p$ reaction rates and the weak decay rates of heavy nuclei are also modified if the electron mass varies with time. The $n\leftrightarrow p$ reaction rate is calculated by

                                 $\displaystyle \lambda_{n\to p}$ = $\displaystyle \lambda_0 \int_{m_{\rm e} c^2}^{\infty} {\rm d}E_{\rm e} ~ \frac{... ...rm e}^{-\left(E_{\rm e}+ \Delta m_{\rm np} c^2\right) /k_{\rm B} T_\nu -\xi_l}}$  
    $\displaystyle + \lambda_0 \int_{m_{\rm e} c^2}^{\infty} {\rm d}E_{\rm e} ~ \fra... ...\rm e}^{\left(E_{\rm e}-\Delta m_{\rm np}c^2\right) /k_{\rm B} T_\nu +\xi_l}} ,$ (A.1)  

where $\lambda_0$ is a normalization constant proportional to $G_{\rm F}^2$, $E_{\rm e}$and $p_{\rm e}$ are the electron energy and momentum respectively $\left(E_{\rm e}=\sqrt{p_{\rm e}^2 c^2+m_{\rm e}^2 c^4}\right)$, $T_\gamma$ and $T_\nu$ are the photon and neutrino temperature and $\xi_l$is the ratio between the neutrino chemical potential and the neutrino temperature. This normalization constant is obtained at a very low temperature and for no variation of v.

The Fermi constant is proportional to v-2 (Dixit & Sher, 1988), affecting the $n\leftrightarrow p$ reaction rate, since $\lambda_0 \sim G_{\rm F}^2$.

The neutron-proton mass difference changes by (Christiansen et al., 1991)

$\displaystyle \frac{\delta \Delta m_{\rm np}}{\left(\Delta m_{\rm np}\right)_0}$ = $\displaystyle 1.587 \frac{\delta v}{v_0} ,$ (A.2)

affecting $n\leftrightarrow p$ reaction rates (see Eq. (A.1)), Q-values of several reaction rates (e.g. $^3{\rm He}(\rm n,~ p)^3{\rm H}$, $^7{\rm Be}(\rm n,~ p)^7{\rm Li}$) and the initial neutrons and protons abundances:
                $\displaystyle Y_{\rm n}$ = $\displaystyle \frac{1}{1+{\rm e}^{\Delta m_{\rm np} c^2/k_{\rm B} T_9 +\xi}} ,$  
$\displaystyle Y_{\rm p}$ = $\displaystyle \frac{1}{1+{\rm e}^{-\Delta m_{\rm np} c^2/ k_{\rm B} T_9 -\xi}} ,$ (A.3)

where T9 is the temperature in units of $10^9 ~{\rm K}$. In order to include these effects we replace $\Delta m_{\rm np}$ by $\Delta m_{\rm np}\left(1+\frac{\delta \Delta m_{\rm np}}{\left(\Delta m_{\rm np}\right)_0}\right)$. We have also modified the masses of the light nuclei (Flambaum & Wiringa, 2007) affecting the Q-values and the reverse coefficient of the reactions that involve neutrons.

The deuterium binding energy must be corrected by

$\displaystyle \frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}$ = $\displaystyle \frac{\kappa}{2} \frac{\delta v}{v_0} ,$ (A.4)

where $\kappa$ is a model dependent constant. In the present work this constant is found to be $\kappa =-3.65$. This correction affects the initial value of the deuterium abundance
                        $\displaystyle Y_{\rm d}$ = $\displaystyle \frac{Y_{\rm n} Y_{\rm p} ~ {\rm e}^{\epsilon_{\rm D}/k_{\rm B} T_9}}{0.471\times 10^{-10}\left(k_{\rm B} T_9\right)^{3/2}} ,$ (A.5)

where $\epsilon_{\rm D}$ is in MeV, and the Q-values of several reactions, such as d $(\gamma, n)p$ from its reverse reaction. Once again we replace $\epsilon_{\rm D}$ by $\epsilon_{\rm D} \left(1+\frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0} \right)$ in order to modify the code.

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Footnotes

... as[*]
We adopt natural units $(\hbar=c=1)$ through the text, unless indicated.

All Tables

Table 1:   Theoretical abundances in the standard model.

Table 2:   Deuterium observational abundances $(Y_{\rm D}^{\rm obs})$.

Table 3:   4He observational abundances $(Y_{^4{\rm He}}^{\rm obs})$.

Table 4:   Best fit parameter value and 1$\sigma $ errors constraints on  $\frac{\delta v}{v_0}$.

Table 5:   Best fit parameter value and 1$\sigma $ errors constraints on $\frac{\delta v}{v_0}$ and  $\eta _{\rm B}$.

All Figures

  \begin{figure} \par\includegraphics[width=170 pt,angle=-90]{14825fg1.ps} \end{figure} Figure 1:

Dependence of $\frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}$ upon the relative change of the pion mass $\frac{\delta m_\pi}{\left( m_\pi \right)_0}$, from the work of Flambaum & Shuryak (2002) (grey area) and our calculated value (dotted line).

Open with DEXTER
In the text

  \begin{figure} \par\includegraphics[width=280 pt,angle=0]{14825fg2.ps} \end{figure} Figure 2:

1$\sigma $, 2$\sigma $ and 3$\sigma $ likelihood contours for $\eta _{\rm B}$ and $\frac{\delta \epsilon_{\rm D}} {\left(\epsilon_{\rm D}\right)_0}$, and one-dimensional likelihood, $\left(\frac{\rm L}{\rm L_{max}}\right)$.

Open with DEXTER
In the text

  \begin{figure} \par\includegraphics[width=280pt,clip]{14825fg3.ps} \end{figure} Figure 3:

1$\sigma $, 2$\sigma $ and 3$\sigma $ likelihood contours for $\eta _{\rm B}$ and $\frac{\delta v}{v_0}$, and one-dimensional likelihood, $\left(\frac{\rm L}{\rm L_{max}}\right)$, using $\kappa =-3.65$.

Open with DEXTER
In the text

  \begin{figure} \par\includegraphics[width=280pt,clip]{14825fg4.ps} \end{figure} Figure 4:

1$\sigma $, 2$\sigma $ and 3$\sigma $ likelihood contours for $\eta _{\rm B}$ and $\frac{\delta v}{v_0}$, and one-dimensional likelihood, $\left(\frac{\rm L}{\rm L_{max}}\right)$, using $\kappa =3$.

Open with DEXTER
In the text

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