Primordial nucleosynthesis with varying fundamental constants

Big Bang Nucleosynthesis (henceforth BBN) is a cornerstone of the standard particle cosmology paradigm, and a sensitive probe of physics beyond the standard model (Steigman, 2007; Iocco et al., 2009; Pitrou et al., 2018). Nevertheless, its success is limited by the well-known Lithium problem, wherein the theoretically expected abundance of Lithium-7 (given our present knowledge of astrophysics, nuclear and particle physics) exceeds the observed one by a factor of about 3.5 (Zyla et al., 2020). Although there have been many attempts to solve the problem, there is no clear known solution. Further discussion of these solutions can be found in Fields (2011), Mathews et al. (2020) and the BBN section of the latest Particle Data Group (henceforth PDG) review by Zyla et al. (2020).

Recent progress in experimental measurements of the required nuclear cross-sections has all but excluded the possibility of nuclear physics systematics (Iliadis & Coc, 2020; Mossa et al., 2020). On the astrophysics side, some degree of lithium depletion can occur in stars due to the mixing of the outer layers with the hotter interior (Sbordone et al., 2010), and some authors have suggested that a depletion by factor as large as 1.8 may have occurred (Ryan et al., 2000; Korn et al., 2006). It has been argued that this scenario might be difficult to reconcile with the existence of extremely iron-poor dwarf stars with lithium abundances very close to the Spite plateau (Aguado et al., 2019), but there is no consensus on this point. Finally, the Lithium problem could also point to new physics beyond the standard model; one such possibility is the variation of nature’s fundamental constants, which is unavoidable in many extensions of the standard particle physics and cosmological models (Damour et al., 2002)—a recent review of the topic is Martins (2017). This possibility has been recently revisited in Clara & Martins (2020) and Martins (2021) (henceforth Paper 1 and Paper 2, respectively), who show that there is a preference for a value of the fine-structure constant, $\alpha$, at the BBN epoch that is larger than the current laboratory one by a few parts per million of relative variation, even when allowing for possible changes to the most relevant cosmological parameters impacting BBN: the neutron lifetime, the number of neutrino species, the and baryon-to-photon ratio.

Stars are formed with an initial chemical composition representative of their birth environment. During their evolution, this chemical composition is modified by several processes, namely nuclear reactions and transport processes. Nuclear reactions affect the abundance profiles of some elements in the central region of stars, increasing or decreasing their abundances. Transport processes of chemical elements modify the elements distribution in the whole star. There are macroscopic transport processes as, for example, convection, which efficiently transports chemical elements and leads to an homogenized chemical composition in the convective zone. Other macroscopic transport processes are less efficient and reduce the potential internal gradient of chemical composition while transporting them deeper in stars (i.e. transport induced by the rotation: e.g. Palacios et al. 2003, and references therein; Talon 2008, and references therein; Maeder 2009, and references therein, thermohaline convection: e.g. Vauclair 2004, and references therein; Denissenkov 2010, and references therein; Brown et al. 2013, and references therein; Deal et al. 2016, and references therein). These macroscopic transport processes are in competition with microscopic transport processes, namely atomic diffusion (see Michaud et al., 2015, for a detailed description). This comes from physics first principles and is induced by the internal gradients of pressure, temperature and composition. It is efficient in radiative zones and leads to a selective transport of chemical elements, mainly driven by the competition between the gravity and radiative accelerations. Gravity moves elements towards the center of stars while radiative accelerations, a mechanism for transfer of momentum between photons and ions, move some elements toward the surface of stars (depending on their ionisation states and abundances). As a consequence, the surface abundance of elements are either larger or smaller than the initial ones. Considering all the potential processes affecting the chemical elements distribution in stars, it is expected that observed surface abundances are often different from the initial ones, especially on the main sequence where the surface convective zones are not too deep to allow surface abundance variations.

Light elements (such as lithium and beryllium) are especially impacted by transport processes during the evolution of stars. Atomic diffusion leads to a depletion of these elements from the surface (radiative accelerations are most of the time negligible for these elements). Moreover, these elements are destroyed by nuclear reactions at rather low temperatures (about $2.5$ and $3.5$ million K for lithium and beryllium, respectively), not very deep inside stars. When macroscopic transport processes are efficient enough and affect deep regions, lithium is then transported in regions where the temperature is large enough to be destroyed by nuclear reactions. The combination of both atomic diffusion and macroscopic transport processes is expected to lead to a discernible depletion of lithium from the surface, hence to lithium-7 surface abundances smaller than the initial one (e.g. Vauclair, 1988; Charbonnel et al., 1994; Korn et al., 2006, 2007; Gruyters et al., 2013, 2016; Dumont et al., 2020).

Population II stars are not an exception and undergo the same kind of depletion thanks to the same processes (e.g. Michaud et al., 1984; Richard et al., 2005). This is the reason why the lithium-7 surface abundances observed in population II stars (known as the Spite plateau, Spite & Spite 1982) are most likely not the initial ones, hence the discrepancy with a larger cosmological lithium-7 abundance (from BBN). It is interesting to note that a lot of stars are found to have a lithium abundance smaller than the lithium plateau (Bonifacio et al., 2007; Cayrel et al., 2008; Sbordone et al., 2010), which can also be explained by stellar models, for example, in the case of carbon enhanced metal poor stars with excess in s process elements (CEMP-s stars) (e.g. Stancliffe, 2009; Deal et al., 2021).

Here we draw on the self-consistent perturbative analysis formalism introduced in Paper 1 and extended in Paper 2 to revisit the role of stellar depletion in the Lithium problem, while simultaneously allowing for time variation of nature’s fundamental constants, in a broad class of Grand Unified theory (GUT) scenarios where all the gauge and Yukawa couplings are allowed to vary. It will be seen that the impacts of the two mechanisms can be separately constrained. We will start by assuming that the there is a depletion factor relating the cosmological and astrophysical lithium-7 abundances

Throughout our statistical analyses $\Delta$ is allowed to span the entire $[0,1]$ range, under the assumption of a uniform prior. Clearly this is a purely phenomenological parameter, and given any pair or cosmological and astrophysical lithium abundances, for which the former is larger than the latter, there will always be a choice of $\Delta$ that makes the two compatible. In this sense, the observed lithium-7 abundance simply provides a measurement of $\Delta$. However, the hypothesis underlying this assumption is that there exist stellar physics mechanisms and/or physical transport processes occurring in stars that can account for the the relevant value of $\Delta$. Thus our purpose in introducing Eq. (1) is twofold. Firstly, it will allow us to quantify the impact of astrophysical depletion in models which also allow for new physics mechanisms (specifically, in the present work, through varying fundamental constants), thereby comparing the possible roles of the two. Secondly, it provides a convenient way to assess the extent to which stellar physics mechanisms could lead to depletion.

Following the approach of Paper 1 and Paper 2, we have done our statistical likelihood analyses using three different combinations of the four abundances, as follows

• •

  The baseline case uses the abundances of helium-4, deuterium and lithium-7, which are the three available cosmological abundances. This will providing the reference values for the best-fit BBN values for $\alpha$.

• •

  The extended case adds the helium-3 abundance to the former three; this separation stems from the fact that its observed abundance is a local rather than a cosmological one. In any case, as was pointed out in Paper 1 and Paper 2, this has a negligibly small impact on the derived constraints.

• •

  The null case, which uses the helium-4, deuterium and helium-3 abundances but does not use lithium-7. The motivation for this case is that it provides a useful null test of the BBN sensitivity to the value of the fine-structure constant. In other words, if all the relevant physics is the standard one, it is expected that the standard value of $\alpha$ is recovered in this case to some degree of sensitivity that is useful to quantify and compare to other probes, It also provides an indication of the BBN constraints on $\alpha$ on the assumption that the Lithium problem has an astrophysical solution.

For convenience, we will use the Baseline, Extended and Null terms to denote each of the three combinations when presenting the results in figures in the early part of the work. (This facilitates comparisons with the results in Paper 1 and Paper 2.) In the later part, and when summarizing results in the text and tables throughout the work, we will concentrate on the Baseline case, with Null case results provided for comparison when relevant.

The plan of the rest of this work is as follows. We start in Sect. 2 by phenomenologically quantifying the preferred depletion factor $\Delta$ assuming that the three key cosmological parameters—the neutron lifetime, number of neutrinos and baryon-to-photon ratio—are allowed to vary, but constrained by priors external to BBN. In Sect. 3, we report on the analogous study for GUT scenarios where all the gauge and Yukawa couplings are allowed to vary, confirming the previously reported preference for a larger value of $\alpha$ at the BBN epoch. In this case the baryon-to-photon ratio and number of neutrinos are assumed to be fixed at their standard values, while the neutron lifetime is unavoidably affected. Together, these two sections confirm a discrepancy in the values of the baryon-to-photon ratio—or equivalently the primordial Deuterium abundance—preferred by BBN and cosmic microwave background (CMB) plus baryon acoustic oscillation (BAO) data, which has been recently reported (Pitrou et al., 2021; Yeh et al., 2021). In Sect. 4 we address the Lithium problem, showing that the combination of atomic diffusion, rotation and penetrative convection can bridge the gap between the cosmological abundance and the surface abundances measured in Population II stars. In Sect. 5 we address the Deuterium discrepancy, showing that it (and not the Lithium problem) is driving the aforementioned preference for a larger value of $\alpha$ and further quantifying the model dependence of this preference. Finally, we offer some conclusions in Sect. 6.

Generically, the sensitivity of the primordial BBN abundances to the various relevant model parameters can be described as

where $C_{ij}=\partial\ln{(Y_{i})}/\partial\ln{(X_{j})}$ are the sensitivity coefficients. The perturbation is always done with respect to the predicted abundance values in some baseline theoretical model, which in our case is the one recently presented in Pitrou et al. (2021). These predicted abundances are listed in Table 1 for convenience, together with the observed abundances as recommended by the BBN review in Zyla et al. (2020). The two are compared using standard statistical likelihood methods, and theoretical and observational uncertainties being added in quadrature. We note that our fiducial model differs from the one used in Paper 1 and Paper 2, which was based on the earlier work of Pitrou et al. (2018). Therefore our present results can be approximately (but not exactly) compared with those of the earlier papers.

In this section we consider the effect of the values of the neutron lifetime $\tau_{n}$, the number of neutrino species $N_{\nu}$, and the baryon-to-photon ratio $\eta_{10}$ (where, for convenience, we have defined $\eta_{10}=\eta\times 10^{10}$) on the BBN abundances. The fiducial values of these parameters (which will be used as statistical Gaussian priors in the analysis) are, respectively

for the neutron lifetime (Zyla et al., 2020) (see also Rajan & Desai (2020) for a recent discussion of experimental measurements of this lifetime),

for the number of neutrino species, which comes from the LEP measurement (Zyla et al., 2020), and

from the combination of recent CMB (Planck 2018) and BAO data (Aghanim et al., 2020). For reference, the latter corresponds to the physical baryon density

With these assumptions our perturbative analysis for the BBN abundances has the specific form

where $t_{i}$, $v_{i}$ and $w_{i}$ are the sensitivity coefficients listed in Table 2 and previously discussed in Paper 2 and references therein. Note that Deuterium is far more sensitive to the baryon fraction than to the neutron lifetime or the number of neutrinos, while the opposite occurs for Helium-4. Additionally, we will make the assumption that the cosmological and astrophysical lithium-7 abundances are related through Eq. (1), so the depletion factor $\Delta$ becomes a fourth model parameter, together with the relative variations of the three cosmological parameters.

The results of the analysis are depicted in Figure 1. The best-fit values of our free parameters (with their one standard deviation uncertainties) in the Baseline case are

in the Null case the first three of these are the same, while $\Delta$ is obviously not constrained. (On the other hand, if one assumes that $\tau_{n}$, $N_{\nu}$ and $\eta_{10}$ are fixed at their fiducial values rather than allowed to vary and marginalized, one finds $\Delta=0.71\pm 0.07$.) These lead to the following best fit theoretical abundances

for lithium-7 we separately list the cosmological (primordial) abundance, and the corresponding astrophysical one, corrected by the preferred seventy percent depletion. We confirm the result, already mentioned by Pitrou et al. (2021), that with the latest nuclear physics cross sections and observed abundances the BBN observations prefer a slightly lower $\eta_{10}$ than the one measured by the combination of CMB and BAO observations. Alternatively, this can be phrased as as discrepancy in the Deuterium abundance. On the other hand, different reaction rates are used in Yeh et al. (2021) and in Pisanti et al. (2021), which find no such discrepancy. In any case, as illustrated in Figure 1, the discrepancy is only slightly larger than one standard deviation, so its statistical significance is limited. We revisit this point later in this work.

The approach introduced Paper 1 and Paper 2 to study BBN with varying fundamental constants is also a perturbative one. It builds upon earlier work by Muller et al. (2004) and Flambaum & Wiringa (2007) and subsequent developments by Coc et al. (2007) and Dent et al. (2007), to generically write the relative variations of other couplings as the product of some constant particle physics coefficients and the relative variation of the fine-structure constant. With some reasonable simplifying assumptions, which stem from the work of Campbell & Olive (1995), only two such dimensionless coefficients are needed, one pertaining to electroweak physics (denoted $S$) and the other to strong interactions (denoted $R$). This enables a phenomenological description of a broad range of GUT scenarios. We refer the reader to Paper 1 and Paper 2 for a more detailed description of this formalism. In what follows, astrophysical constraints on $\alpha$ are expressed in terms of a relative variation with respect to the local laboratory value,

where $\alpha_{0}=0.0072973525693(11)$ is the local laboratory value.

Under these assumptions, the sensitivity of the primordial BBN abundances to the values of the fundamental constants will therefore be expressed as a function of the phenomenological particle physics parameters $R$ and $S$ and the relative variation of $\alpha$, in other words

the corresponding sensitivity coefficients are listed in Table 3. Note that the baryon-to-photon ratio and number of neutrinos are assumed to be fixed at their standard values. On the other hand the neutron lifetime will be affected by the $\alpha$ variation, according to

but this effect has been included in the computation of these sensitivity coefficients, so the neutron lifetime does not appear as an explicit parameter. We refer the reader to Paper 2 for a more detailed discussion.

The phenomenological parameters $R$ and $S$ can in principle be taken as free parameters, to be experimentally or observationally constrained. Our current knowledge of particle physics and unification scenarios suggests that their absolute values can be anything from order unity to several hundreds, with $R$ allowed to be positive or negative (though with the former case being the more likely one), while $S$ is expected to be non-negative.

The analysis of Paper 1 and Paper 2 has identified three models that can provide a solution to the Lithium problem (at least in the sense of making all theoretical and observed abundances agree to within three standard deviations or less) for the best-fit values of $\Delta\alpha/\alpha$ listed in Table 4. Note that the fiducial model used in these papers is not the same as the one used in the present work, and that the full parameter space was different in the two papers.

The first of these is a ‘typical’ unification scenario, for which one has (Coc et al., 2007; Langacker et al., 2002)

we refer to this as the Unification model. The second is the dilaton-type model discussed by Nakashima et al. (2010), for which

we refer to this as the Dilaton model. Finally the Clocks model denotes the general case where the parameters $R$ and $S$ are allowed to vary and are then marginalised, in this case with uniform priors in the range $R=[0,+500]$, $S=[0,+1000]$, together with an additional prior coming from local experiments with atomic clocks (Ferreira et al., 2012)

This is therefore a more phenomenological model than the previous two (and has a larger number of free parameters than them), but serves the purpose of illustrating the range of behaviours that might be found in GUT models—although we should mention that, unsurprisingly, the results in this case will also depend on the choice of priors for $R$ and $S$. This last point has been discussed in Paper 1.

We now repeat the statistical likelihood analysis, allowing for both a variation of fundamental couplings and lithium-7 depletion, for these three models. In this section we assume that the values of the number of neutrino species and the baryon-to-photon ratio are fixed at their standard fiducial values, but the neutron lifetime is implicitly allowed to vary, as pointed out above. The results of this analysis are depicted in Figure 2 and also summarized in Table 5 for the Baseline case; naturally, in the Null case the depletion fraction is unconstrained, but for comparison we also list the value of $\Delta\alpha/\alpha$ in that case.

The observed BBN abundances allow the separate quantification of the role of both effects. We see that the preferred value of the depletion parameter decreases with respect to the one obtained with the standard value of $\alpha$ in Section 2, by about one standard deviation. Moreover, there is still a preference for a positive value of $\Delta\alpha/\alpha$. Comparison with Table 4 shows that for the Unification and Dilaton models both the best-fit values and the statistical significance of the preference for the Baseline case are decreased with respect to the values found in Paper 1 and Paper 2, while for the Null case they are slightly increased. On the other hand, for the Clocks model the best-fit value for $\alpha$ is in agreement with the one found in Paper 1 and Paper 2 in both the Null and the Baseline cases. In this case the effect of the additional depletion mechanism is manifest in the best-fit values for the phenomenological particle physics parameters $R$ and $S$. Here we find

whereas in Paper 2 the best-fit values were

Note that although $R$ and $S$ are independent parameters in the likelihood analysis, the local prior from atomic clock experiments—cf. Equation (23)—effectively correlates them. The difference between the two sets of values reflects the fact that in Paper 2 there was no allowance for a depletion factor (effectively, $\Delta=0$ was used). Moreover, bearing in mind the aforementioned slight reduction of the depletion factor when a non-standard value of $\alpha$ is allowed, it also shows that although a value of $\Delta\alpha/\alpha>0$ does alleviate the lithium problem, it does not completely solve it. Instead, a significant depletion factor is still statistically preferred.

The more noteworthy aspect is that, unlike the results of Paper 1 and Paper 2, the preferred values for $\Delta\alpha/\alpha$ are almost identical in the Baseline and Null cases, meaning that the preference for a positive value of $\Delta\alpha/\alpha$ is not driven by the lithium-7 problem. In other words, even without including lithium-7 in the analysis, there is a mild statistical preference for a variation of $\Delta\alpha/\alpha$ at the few parts-per-million level. Instead, it is driven by the small discrepancy, already mentioned in the previous section, between the baryon-to-photon ratios inferred from BBN and the corresponding CMB value; in other words, it is driven by the helium-4 and deuterium data. We note that within the standard paradigm the baryon-to-photon ratio is a constant, having the same value at the BBN and CMB epochs; a different value at the two epochs necessarily implies the presence of new physics.

When adding lithium-7 to the analysis without allowing for the possibility of a depletion mechanism (as was done in Paper 1 and Paper 2) the preferred value of $\Delta\alpha/\alpha$ increases very significantly with respect to the Null case, since a larger value of $\alpha$ is needed solve the lithium problem per se. On the other hand, if a depletion mechanism is allowed (as in the present work) then this mechanism provides the dominant contribution to address the lithium problem, and the value of $\Delta\alpha/\alpha$ is very slightly changed with the respect to the null case. Nevertheless the inclusion of $\Delta\alpha/\alpha$ is still significant, which is manifest in the fact that the preferred depletion fraction changes with respect to the case discussed in the previous section by about one standard deviation.

A summary plot of the standard theoretical and observed abundances, together with those predicted for the best-fit values of the various models discussed both in Section 2 and in the present one, can be found in Figure 3.

It is interesting to note that although the Unification and Dilaton models lead to very similar lithium-7 abundances (a consequence of the fact that this nuclide drives the statistical analysis), they lead to significantly different ones for the other nucildes, especially helium-4 and helium-3. Specifically, the Dilaton model leads to a significantly higher helium-4 abundance and a significantly lower helium-3 abundance, as compared to the Unification model. This is due to the fact that in the Dilaton model, where $R$ is comparatively large, there are large positive and negative sensitivity coefficients, respectively for helium-4 and helium-3. The comparison between the three models also confirms the well-known result that, for the observationally relevant ratios of the baryon-to-photon ratio, the deuterium and lithium-7 abundances are anticorrelated (Olive et al., 2012; Coc et al., 2015), so a reduction of the latter abundance leads to an increase of the former. Similarly, one can notice an anticorrelation between the helium-4 and helium-3 abundances, highlighting the fact that an accurate determination of the primordial cosmological abundance would provide a sensitive consistency test of BBN.

In this section we assess how realistic is the depletion factor $\Delta$, determined in the previous Section, according to stellar evolution theory. We compare $\Delta$ to predicted lithium-7 depletion in stellar models, including different transport processes of chemical elements.

Having previously identified the small discrepancy between the baryon-to-photon ratios inferred from BBN and the CMB value as the origin of the preference for a non-standard value of $\alpha$, here we assess the relative contributions of the helium-4 and deuterium data to this result.

To this end, we repeat the analysis both for the case where only the deuterium abundance is used and for the case where both deuterium and helium-4 are used. For each of these we further consider two sub-cases, ether keeping the cosmological parameters (i.e., the neutron lifetime, number of neutrinos and baryon-to-photon ratio) fixed at their best-fit values introduced in Sect. 2, or letting these parameters vary (with the previously described priors) and then marginalizing them.

In the former case, the parameter space is summarized by Eq. 19 and the relevant sensitivity coefficients are given in Table 3. In the latter case the parameter space is wider:

As previously mentioned, a variation of $\alpha$ itself impacts the neutron lifetime; unlike in Sect. 3, here we explicitly separate this effect from the rest of the effects of the $\alpha$ variation. This means that the $\alpha$-related sensitivity coefficients will change, and we have highlighted this point by adding the $n$ subscript in the previously defined sensitivity coefficients $x$, $y$ and $z$. For completeness, all these sensitivity coefficients are reproduced in Table 7.

The results of our analysis with the deuterium and helium-4 abundances are summarized in Figure 7 and Table 8; the latter can be usefully compared with the Baseline and Null cases reported in Table 5.

First and foremost, we confirm that the preference for $\Delta\alpha/\alpha>0$ is driven by the Deuterium results, and the top panels of Figure 7 also makes it clear that this is due to the positive correlation between $\Delta\alpha/\alpha$ and the baryon-to-photon ratio $\eta$, though the strength of this correlation is model-dependent. Secondly, we find that the Helium-4 abundance plays a relatively minor role, though one that again depends on the assumed model.

Specifically, for the Unification and Clocks model, the results in Table 8 are fully consistent with the ones for the previously discussed Baseline and Null cases. The positive correlation between $\Delta\alpha/\alpha$ and $\eta$ is relatively mild, In these circumstances helium-4 does not noticeably affect the constraints on $\Delta\alpha/\alpha$, while allowing the cosmological parameters to vary and marginalizing them simply increases the statistical uncertainties, and therefore decreases the significance of the $\Delta\alpha/\alpha>0$ preference.

On the other hand, for the Dilaton model the results are more sensitive to the assumptions underlying the analysis. On the one hand, the stronger positive correlation between $\Delta\alpha/\alpha$ and $\eta$ means that whether or not $\eta$ is allowed to vary has a mild but discernible impact even in the case where only deuterium is used in the analysis. On the other hand, the inclusion of helium-4 significantly lowers the preferred value of $\Delta\alpha/\alpha$, the reason being that this model tends to overpredict the helium-4 abundance, as can bee seen in Figure 3. Again, this shows the potential of BBN as a precision test of these theoretical scenarios.

We have provided an updated analysis of BBN constraints in the framework of a wide class of Grand Unified Theory scenarios (Clara & Martins, 2020; Martins, 2021), in particular addressing the long-standing Lithium problem (Fields, 2011; Mathews et al., 2020) and the more recently noticed Deuterium discrepancy (Pitrou et al., 2021). For the former we have highlighted and quantified the astrophysical mechanisms that can provide a solution, while for the latter we have identified a possible hint of new physics, specifically a mild (two to three standard deviations) preference for $\Delta\alpha/\alpha>0$, at the parts per million level. Such variations would be consistent with all other current $\alpha$ constraints (Martins, 2017).

We note that the Deuterium discrepancy identified by Pitrou et al. (2021) is not found by other authors (Yeh et al., 2021; Pisanti et al., 2021), who rely on different reaction rates. From a statistical point of view, the main difference is that the uncertainties in the theoretical abundances obtained by Pitrou et al. (2021) are smaller than those of other analyses, although there are also small differences in the preferred values of these abundances. While we have not repeated our analysis with these different reaction rates (and the corresponding theoretical abundances inferred from them) we would expect that the propagation of the larger uncertainties on the theoretical side would lead to larger uncertainties on the fine-structure constant measurements, reducing the statistical significance of the preference for a non-standard value.

We have quantitatively determined the amount a lithium depletion needed to solve the lithium problem and searched for depletion mechanisms occurring during the evolution of stars, namely, the transport processes of chemical elements. We have shown that taking into account, in stellar models, atomic diffusion, rotational-induced mixing and some amount of penetrative convection (consistent with the amount determined from solar observation, Christensen-Dalsgaard et al. 2011), allows us to reproduce the lithium surface abundances of Population II stars. More realistic modelling of some of the processes (especially the penetrative convection, and the missing transport of angular momentum) are required to draw a more robust conclusion, but this work shows the necessity of including the stellar contribution to the lithium problem. Transport processes of chemical elements are most likely the dominant contribution to the solution to this problem.

The stellar models were computed with the best fit value for the initial primordial lithium abundance derived in this study, in the context of the GUT scenarios under consideration, of approximately (⁷Li/H)$\times 10^{10}=4.45$. Considering stellar models with a larger initial primordial lithium, as obtained in the standard BBN model (i.e. (⁷Li/H)$\times 10^{10}=5.46$) would not strongly impact our conclusions. This difference would only lead to the need of a slightly larger amount of penetrative convection to explain the extra depletion. This also shows that from the stellar physics point of view, the accuracy of the primordial lithium abundance is crucial to constrain the transport of chemical elements in metal poor stars.

It is worthy of note that currently there are only two primordial (cosmological) abundances that are well known, those of Deuterium and Helium-4. It is therefore highly desirable to close the loop through a cosmological measurement of the Helium-3 abundance, especially given the anticorrelation between the helium-4 and helium-3 abundances. This would enable a key consistency test of the underlying physics, which will be particularly crucial should evidence for new physics be confirmed.

Our analysis highlights the role of BBN as consistency test of the standard cosmological paradigm, and as a sensitive probe of new physics. For the broad phenomenological but physically motivated class of models we have considered, improving the observed abundances of Deuterium and Helium-4 by a factor of 2-3 will provide a stringent test, and in particular will definitively confirm or rule out the present tentative evidence for the $\alpha$ variation. This is a highly compelling science case for the forthcoming Extremely Large Telescope (Liske et al., 2014; Marconi et al., 2020), provided it has an efficient blue wavelength coverage.