Issue 
A&A
Volume 654, October 2021



Article Number  A148  
Number of page(s)  14  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/202141353  
Published online  26 October 2021 
Euclid: Constraining dark energy coupled to electromagnetism using astrophysical and laboratory data^{⋆}
^{1}
Instituto de Física Teórica UAMCSIC, Campus de Cantoblanco, 28049 Madrid, Spain
email: matteo.martinelli@uam.es
^{2}
Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150762 Porto, Portugal
^{3}
Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150762 Porto, Portugal
^{4}
Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain
^{5}
Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France
^{6}
Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capitá 24, 08034 Barcelona, Spain
^{7}
INFNSezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
^{8}
Dipartimento di Fisica, Universitá degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy
^{9}
INAFOsservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese, TO, Italy
^{10}
INAFIASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy
^{11}
AIM, CEA, CNRS, Université ParisSaclay, Université de Paris, 91191 GifsurYvette, France
^{12}
Université St Joseph; UR EGFEM, Faculty of Sciences, Beirut, Lebanon
^{13}
Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK
^{14}
Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile
^{15}
Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
^{16}
INAFOsservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy
^{17}
Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany
^{18}
INFNSezione di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
^{19}
Department of Mathematics and Physics, Roma Tre University, Via della Vasca Navale 84, 00146 Rome, Italy
^{20}
Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra, Barcelona, Spain
^{21}
INAFOsservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy
^{22}
Department of Physics “E. Pancini”, University Federico II, Via Cinthia 6, 80126 Napoli, Italy
^{23}
INFN section of Naples, Via Cinthia 6, 80126 Napoli, Italy
^{24}
INAFOsservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy
^{25}
Dipartimento di Fisica e Astronomia ‘Augusto Righi’ – Alma Mater Studiorum Universitá di Bologna, Via Piero Gobetti 93/2, 40129 Bologna, Italy
^{26}
INAFOsservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy
^{27}
Institut national de physique nucléaire et de physique des particules, 3 rue MichelAnge, 75794 Paris Cédex 16, France
^{28}
Centre National d’Etudes Spatiales, Toulouse, France
^{29}
AixMarseille Univ, CNRS, CNES, LAM, Marseille, France
^{30}
Department of Astronomy, University of Geneva, ch. d’Écogia 16, 1290 Versoix, Switzerland
^{31}
Université ParisSaclay, CNRS, Institut d’astrophysique spatiale, 91405 Orsay, France
^{32}
INFNPadova, Via Marzolo 8, 35131 Padova, Italy
^{33}
Univ. Lyon, Univ. Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, 69622 Villeurbanne, France
^{34}
INAFOsservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34131 Trieste, Italy
^{35}
Istituto Nazionale di Astrofisica (INAF) – Osservatorio di Astrofisica e Scienza dello Spazio (OAS), Via Gobetti 93/3, 40127 Bologna, Italy
^{36}
Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy
^{37}
INAFOsservatorio Astronomico di Padova, Via dell’Osservatorio 5, 35122 Padova, Italy
^{38}
UniversitätsSternwarte München, Fakultät für Physik, LudwigMaximiliansUniversität München, Scheinerstrasse 1, 81679 München, Germany
^{39}
Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029, Blindern, 0315 Oslo, Norway
^{40}
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
^{41}
von Hoerner and Sulger GmbH, SchloßPlatz 8, 68723 Schwetzingen, Germany
^{42}
MaxPlanckInstitut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
^{43}
Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai ErnestAnsermet, 1211 Genève 4, Switzerland
^{44}
Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2, 00014 University of Helsinki, Finland
^{45}
NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands
^{46}
ArgelanderInstitut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
^{47}
Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK
^{48}
INFNBologna, Via Irnerio 46, 40126 Bologna, Italy
^{49}
Observatoire de Sauverny, Ecole Polytechnique Fédérale de Lau sanne, 1290 Versoix, Switzerland
^{50}
INFNSezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
^{51}
European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
^{52}
Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, 8000 Aarhus C, Denmark
^{53}
Institute of Space Science, Bucharest 077125, Romania
^{54}
Departamento de Astrofísica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain
^{55}
Instituto de Astrofísica de Canarias, Calle Vía Làctea s/n, 38204 San Cristóbal de la Laguna, Tenerife, Spain
^{56}
AixMarseille Univ, CNRS/IN2P3, CPPM, Marseille, France
^{57}
Dipartimento di Fisica e Astronomia “G. Galilei”, Universitá di Padova, Via Marzolo 8, 35131 Padova, Italy
^{58}
Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
^{59}
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349018 Lisboa, Portugal
^{60}
Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, 1749016 Lisboa, Portugal
^{61}
Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, 30202 Cartagena, Spain
^{62}
Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA
^{63}
Dipartimento di Fisica e Astronomia, Universitá di Bologna, Via Gobetti 93/2, 40129 Bologna, Italy
^{64}
European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy
^{65}
ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain
^{66}
School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, UK
^{67}
INAFIASF Bologna, Via Piero Gobetti 101, 40129 Bologna, Italy
^{68}
APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France
^{69}
Space Science Data Center, Italian Space Agency, Via del Politecnico snc, 00133 Roma, Italy
Received:
20
May
2021
Accepted:
19
August
2021
In physically realistic, scalarfieldbased dynamical dark energy models (including, e.g., quintessence), one naturally expects the scalar field to couple to the rest of the model’s degrees of freedom. In particular, a coupling to the electromagnetic sector leads to a time (redshift) dependence in the finestructure constant and a violation of the weak equivalence principle. Here we extend the previous Euclid forecast constraints on dark energy models to this enlarged (but physically more realistic) parameter space, and forecast how well Euclid, together with highresolution spectroscopic data and local experiments, can constrain these models. Our analysis combines simulated Euclid data products with astrophysical measurements of the finestructure constant, α, and local experimental constraints, and it includes both parametric and nonparametric methods. For the astrophysical measurements of α, we consider both the currently available data and a simulated dataset representative of Extremely Large Telescope measurements that are expected to be available in the 2030s. Our parametric analysis shows that in the latter case, the inclusion of astrophysical and local data improves the Euclid dark energy figure of merit by between 8% and 26%, depending on the correct fiducial model, with the improvements being larger in the null case where the fiducial coupling to the electromagnetic sector is vanishing. These improvements would be smaller with the current astrophysical data. Moreover, we illustrate how a genetic algorithms based reconstruction provides a null test for the presence of the coupling. Our results highlight the importance of complementing surveys like Euclid with external data products, in order to accurately test the wider parameter spaces of physically motivated paradigms.
Key words: cosmology: observations / cosmological parameters / space vehicles: instruments / methods: data analysis / methods: statistical / surveys
© ESO 2021
1. Introduction
The search for the physical mechanism underlying the observed lowredshift acceleration of the Universe is a pressing objective of contemporary cosmology. A first task in this endeavour is to map the behaviour of the energy density (or its equation of state parameter) of the dark energy component as a function of redshift – with the simplest case of a cosmological constant corresponding to a constant energy density. To this end, simple parametrizations are often used, commensurate with the limited constraining power of the currently available data, for instance for the dark energy equation of state parameter one has a tight constraint on its present value and a reasonable constraint on its rate of change. Even so, these parametrizations should be seen as convenient proxies for more physically realistic models, possibly containing a larger number of model parameters. While such wider parameter spaces are not significantly constrained by current data, they can in principle be constrained by future surveys.
Euclid is a mediumclass mission of the European Space Agency due for launch in 2022. Using a visible imager (Cropper et al. 2018) and a nearinfrared spectrophotometric instrument (Costille et al. 2018), it will perform a photometric and spectroscopic galaxy survey over 15 000 squared degrees of extragalactic sky, plus a deeper survey over 40 squared degrees (Laureijs et al. 2011). The main goal of Euclid is to provide measurements of the geometry of the Universe and the growth of structures up to redshift z ∼ 2, and beyond. Euclid will provide three primary cosmological probes: weak gravitational lensing, the clustering of galaxies using measurements from the photometric galaxy survey, and the clustering of galaxies of the spectroscopic survey. The latter will enable precise measurements of the baryon acoustic oscillations and redshiftspace distortions. Given the high complementarity of these largescale structure probes, we expect very precise constraints from Euclid observations, not only on the concordance cosmological constant and cold dark matter (ΛCDM) model, but also on theoretical extensions of it (see e.g., Euclid Collaboration 2020; Tutusaus et al. 2020).
In Euclid Collaboration (2020; hereafter EC20), the constraining power of Euclid on dark energy models was estimated using the common ChevallierPolarskiLinder (CPL) parameterization (Chevallier & Polarski 2001; Linder 2003) as a phenomenological proxy for generic dynamical dark energy models. However, in physically realistic, scalarfieldbased dynamical dark energy models (including, e.g. quintessence), one naturally expects the scalar field to couple to other sectors of the theory, unless unknown symmetries suppress such a coupling. Here we focus on the possible coupling of a dark energy scalar field to the electromagnetic sector, which would lead to a time (redshift) dependence of the finestructure constant, α, a violation of the Einstein equivalence principle (Carroll 1998; Dvali & Zaldarriaga 2002; Chiba & Kohri 2002), and also a violation of the distance duality relation. Forecast constraints on the latter from Euclid and contemporary surveys are discussed in Martinelli et al. (2020).
There are two immediate consequences of this. The first is that one should deal with a wider parameter space: the coupling of the scalar field to the electromagnetic sector is a further relevant parameter, all the more so because, as can be seen in what follows, it is degenerate with the parameters describing the dark energy evolution. The second consequence is that Euclid, at least with its primary probes, is not able to constrain such a coupling, as its observables are not sensitive to the variation in the finestructure constant, and therefore one needs to add astrophysical and local constraints on α and the Einstein equivalence principle to the analysis in order to test this kind of scenario. A recent review of the synergies between these astrophysical and local tests and cosmological observations is given in Martins (2017).
Therefore our analysis in this work, which builds upon previous works by Calabrese et al. (2014), Martins et al. (2016), and EC20, has two main goals: firstly, to forecast how well Euclid (in combination with external data, specifically highresolution spectroscopic data and local experimental results) can constrain these models, and secondly, to quantify the change in forecast constraints on CPL parameters when the assumption of a vanishing coupling between the dark energy driving scalar field and electromagnetism is removed.
The plan of the rest of the paper is as follows. In Sect. 2 we review theoretical models relating dynamical dark energy and a varying finestructure constant. In Sect. 3 we describe the currently available data and the forecast future data used in this work, including both Euclid measurements and astrophysical and local data. In Sects. 4 and 5 we describe the analysis methods used in this study: a standard likelihood analysis for the CPL parametrization and a modelindependent reconstruction using genetic algorithms. We present the results obtained with a likelihood analysis in Sect. 6 and the results derived with the genetic algorithms in Sect. 7. We present our discussion and conclusions in Sect. 8.
2. Dynamical dark energy and varying alpha
Dynamical scalar fields in an effective fourdimensional field theory are naturally expected to couple to the rest of the theory, unless a still unknown symmetry is postulated to suppress these couplings. In particular, these couplings unavoidably exist in string theory (Taylor & Veneziano 1988; Casas et al. 1991, 1992), and their cosmological role is especially interesting in models where such a dilatontype scalar field is also responsible for the acceleration of the universe (Carroll 1998; Dvali & Zaldarriaga 2002; Chiba & Kohri 2002; Damour et al. 2002). In what follows, we assume this coupling does exist for the dynamical degree of freedom responsible for the dark energy. Specifically we are interested in the coupling between a canonical scalar field, denoted ϕ, and the electromagnetic sector, which stems from a gauge kinetic function B_{F}(ϕ)
$$\begin{array}{c}\hfill {\mathcal{L}}_{\varphi F}=\frac{1}{4}{B}_{F}(\varphi ){F}_{\mu \nu}{F}^{\mu \nu}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$(1)
Since the local behaviour of electromagnetism is extremely well known and any variation in α is constrained to be very small (as further discussed below), one can safely assume this function to be linear,
$$\begin{array}{c}\hfill {B}_{F}(\varphi )=1\zeta \kappa (\varphi {\varphi}_{0})\phantom{\rule{0.166667em}{0ex}},\end{array}$$(2)
(where we have defined κ^{2} = 8πG) since, as has been pointed out in Dvali & Zaldarriaga (2002), the absence of such a term would require the presence of a ϕ → −ϕ symmetry. Such a symmetry must be broken throughout most of the cosmological evolution, because ϕ is a timedependent field, changing (possibly very slowly) as the universe expands. In other words, the absence of such a term would require finetuning. With this definition ζ is a dimensionless coupling, which will be crucial in our subsequent discussion. As is physically clear, the relevant parameter in the cosmological evolution is the field displacement relative to its presentday value (in particular ϕ_{0} could be freely set to zero).
With these assumptions one can explicitly relate the evolution of α to that of dark energy, as in Calabrese et al. (2011), whose derivation we summarize. The evolution of α can be written as
$$\begin{array}{c}\hfill \frac{\mathrm{\Delta}\alpha}{\alpha}\equiv \frac{\alpha {\alpha}_{0}}{{\alpha}_{0}}={B}_{F}^{1}(\varphi )1=\zeta \kappa (\varphi {\varphi}_{0})\phantom{\rule{0.166667em}{0ex}}.\end{array}$$(3)
Defining the fraction of the dark energy density as
$$\begin{array}{c}\hfill {\mathrm{\Omega}}_{\varphi}(z)\equiv \frac{{\rho}_{\varphi}(z)}{{\rho}_{\mathrm{tot}}(z)}\simeq \frac{{\rho}_{\varphi}(z)}{{\rho}_{\varphi}(z)+{\rho}_{\mathrm{m}}(z)}\phantom{\rule{0.166667em}{0ex}},\end{array}$$(4)
where in the last step we have neglected the contribution from radiation (since we are interested in low redshifts, z < 5, where it is indeed negligible), the evolution of the putative scalar field can be expressed in terms of the dark energy properties Ω_{ϕ} and w_{ϕ} as (Nunes & Lidsey 2004)
$$\begin{array}{c}\hfill 1+{w}_{\varphi}=\frac{{(\kappa {\varphi}^{\prime})}^{2}}{3{\mathrm{\Omega}}_{\varphi}}\phantom{\rule{0.166667em}{0ex}},\end{array}$$(5)
with the prime denoting the derivative with respect to the logarithm of the scale factor. We finally obtain
$$\begin{array}{c}\hfill \frac{\mathrm{\Delta}\alpha}{\alpha}(z)=\zeta {\displaystyle {\int}_{0}^{z}\sqrt{3{\mathrm{\Omega}}_{\varphi}({z}^{\prime})[1+{w}_{\varphi}({z}^{\prime})]}\frac{\mathrm{d}{z}^{\prime}}{1+{z}^{\prime}}\phantom{\rule{0.166667em}{0ex}}.}\end{array}$$(6)
The above relation assumes a canonical scalar field, but the argument can be repeated for phantom fields, as discussed in Vielzeuf & Martins (2014), leading to
$$\begin{array}{c}\hfill \frac{\mathrm{\Delta}\alpha}{\alpha}(z)=\zeta {\displaystyle {\int}_{0}^{z}\sqrt{3{\mathrm{\Omega}}_{\varphi}({z}^{\prime})1+{w}_{\varphi}({z}^{\prime})}\frac{\mathrm{d}{z}^{\prime}}{1+{z}^{\prime}}\phantom{\rule{0.166667em}{0ex}}.}\end{array}$$(7)
Physically, the change of sign stems from the fact that one expects phantom fields to roll up the potential rather than down. Naturally, the two definitions match across the phantom divide (w = −1). These relations have been applied to several possible choices for the scalar field equation of state (Martins 2017). Here we choose to parameterize the w_{ϕ} function following the CPL parameterization (Chevallier & Polarski 2001; Linder 2003). In this case the dark energy equation of state has the form
$$\begin{array}{c}\hfill {w}_{\mathrm{CPL}}(z)={w}_{0}+{w}_{a}\frac{z}{1+z}\phantom{\rule{0.166667em}{0ex}},\end{array}$$(8)
while the fraction of energy density provided by the scalar field is easily found to be
$$\begin{array}{c}\hfill {\mathrm{\Omega}}_{\mathrm{CPL}}(z)=\frac{1{\mathrm{\Omega}}_{\mathrm{m}}}{1{\mathrm{\Omega}}_{\mathrm{m}}+{\mathrm{\Omega}}_{\mathrm{m}}{(1+z)}^{3({w}_{0}+{w}_{a})}{\mathrm{e}}^{3{w}_{a}z/(1+z)}}\phantom{\rule{0.166667em}{0ex}},\end{array}$$(9)
where we assumed a flat Universe, with a vanishing curvature parameter Ω_{K} = 0. When going beyond background probes, the CPL parameterization is commonly combined with the Parametrized PostFriedmann (PPF) framework for DE perturbations (Hu & Sawicki 2007; Hu 2008; Fang et al. 2008). The dependence of Eqs. (6) and (7) on w_{ϕ}(z) also makes it clear that we should expect (as discussed in Calabrese et al. 2014) degeneracies between the coupling ζ and the CPL dark energy parameters, w_{0} and w_{a}, while the correlation with the matter density should be much weaker. On the other hand, note that the α variation is independent of the Hubble constant. In Fig. 1 we show the theoretical prediction for the redshift trend of Δα/α for several values of the CPL parameters w_{0} and w_{a}. It can be seen how we also expect degeneracies between these parameters, with constant w(z) cases (w_{a} = 0) that produce redshift evolution very similar to cases where w_{a} ≠ 0.
Fig. 1. Example trends in redshift for the variation in α changing the CPL parameters. The values of the other cosmological parameters entering Eqs. (6) and (7) are Ω_{m} = 0.32, and ζ = 10^{−6}. 
A varying α violates the Einstein equivalence principle since it clearly violates local position invariance. The realization that varying fundamental couplings also induce violations of the universality of free fall goes back at least to the work of Dicke (Dicke 1964) – we refer the reader to Damour & Donoghue (2010) for a recent thorough discussion. The key point in our present context is that a light scalar field, such as the one we are considering here, inevitably couples to nucleons due to the α dependence of their masses, and therefore it mediates an isotopedependent longrange force. This can be simply quantified through the dimensionless Eötvös parameter η, which describes the level of violation of the Weak Equivalence Principle (WEP). One can show that for the class of models we are considering here, the Eötvös parameter and the dimensionless coupling ζ are simply related by (Dvali & Zaldarriaga 2002; Chiba & Kohri 2002; Damour & Donoghue 2010)
$$\begin{array}{c}\hfill \eta \approx {10}^{3}{\zeta}^{2}\phantom{\rule{0.166667em}{0ex}}\u037e\end{array}$$(10)
therefore local experimental constraints on the former can be used to constrain the latter.
We note that there is in principle an additional source term driving the evolution of the scalar field, due to a ${F}^{2}{B}_{F}^{\prime}$ term. By comparison to the standard (kinetic and potential energy) terms, the contribution of this term is subdominant, both because its average is zero for a radiation fluid and because the corresponding term for the baryonic density is constrained for the reasons discussed in the previous paragraph. For these reasons, in what follows we neglect this term, which would lead to spatial and environmental dependencies. We nevertheless note that this term can play a role in cosmological scenarios where the dominant standard term is suppressed, such as the models studied in Olive & Pospelov (2008), Silva et al. (2014), Pinho et al. (2017).
Finally, another important observable is the current drift rate of the value of α, which can easily be found to be
$$\begin{array}{c}\hfill D\equiv {\left(\frac{\dot{\alpha}}{\alpha}\right)}_{0}=\mp \zeta {H}_{0}\sqrt{3{\mathrm{\Omega}}_{\varphi 0}1+{w}_{0}}\phantom{\rule{0.166667em}{0ex}},\end{array}$$(11)
with the minus and plus signs corresponding respectively to the canonical and phantom cases. Naturally, the drift rate depends on the present value of the dark energy equation of state (and vanishes for w_{0} = −1), but it is independent of w_{a}. This observable provides a second way to constrain these models using local experiments, since the drift rate can be constrained using laboratory experiments which compare atomic clocks based on transitions with different sensitivities to α.
3. Available and future data
The purpose of this work is to constrain canonical scalarfield based dynamical dark energy models which allow for the possible variation in the finestructure constant, as detailed in Sect. 2, using both currently available data and the ones expected from future surveys. In particular, we use data from observations of quasistellar object (QSO) spectral lines from archival datasets and dedicated measurements, complemented by laboratory constraints, detailed in Sect. 3.1, as well as forecast data for the future measurements of Δα/α from the Extremely Large Telescope (ELT) as discussed in Leite et al. (2016); the assumptions made to generate mock datasets for this experiment are shown in Sect. 3.2.
However, as pointed out in Calabrese et al. (2014) and as can also be seen in Eqs. (6) and (7), the coupling parameter that drives the variation in α is significantly degenerate with other standard cosmological parameters, namely Ω_{m}, w_{0} and w_{a}. The constraining power of Euclid on these parameters is therefore crucial if one wants to constrain this kind of models.
3.1. Currently available data for α variation
Our astrophysical data consists of highresolution spectroscopy tests of the stability of α. These measurements are done in lowdensity absorption clouds along the line of sight of bright quasars, typically with wavelength resolution R = λ/Δλ ∼ 50 000 (although the exact value is different for different measurements). We use a total of 319 measurements, of which 293 come from the analysis of archival data by Webb et al. (2011) and the remaining 26 are more recent dedicated measurements (Martins 2017; Murphy & Cooksey 2017; Welsh et al. 2020; Milaković et al. 2021). The latter subset is therefore smaller than the former, but it contains more stringent measurements, so overall the archival and dedicated subsets have comparable constraining power (Martins & Vila Miñana 2019). Overall, this dataset includes measurements up to redshift z ∼ 4.18.
In our analysis we assume a homogeneous and isotropic universe, and therefore we also assume that the measurements of α depend only on redshift. That said, we note that Webb et al. (2011) has claimed a spatial variation in α, consistent with a dipolelike behaviour and with an amplitude at the parts per million level of relative variation, although this claim has been disputed by Whitmore & Murphy (2015). If these claims prove to be correct, the main implication would be ruling out the assumption of a FriedmannLemaîtreRobertsonWalker universe. Such a scenario would therefore require a separate analysis.
Additionally, the current drift rate of α is constrained by local comparison experiments between atomic clocks, with the most stringent bound being the one by Lange et al. (2021)
$$\begin{array}{c}\hfill {D}_{\mathrm{obs}}\equiv {\left(\frac{\dot{\alpha}}{\alpha}\right)}_{0}=(1.0\pm 1.1)\times {10}^{18}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.333333em}{0ex}}{\text{yr}}^{1}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$(12)
Last but not least, we also use the recent MICROSCOPE bound on the Eötvös parameter of Touboul et al. (2019)
$$\begin{array}{c}\hfill \eta =(0.1\pm 1.3)\times {10}^{14}\phantom{\rule{0.166667em}{0ex}},\end{array}$$(13)
which, as previously discussed, constrains the model’s coupling to the electromagnetic sector.
In the rest of the paper, we refer to the combination of all these data as current α data, and we show the constraints produced by such a combination. However, we note that the 293 archival data an the 26 dedicated ones are in slight tension with each other (Martins 2017; Martins & Vila Miñana 2019). Despite assuming here that they can be safely combined, we discuss this issue in more detail in Appendix A.
In our analysis, we do not include geophysical constraints on the variation in α, coming from the Oklo natural nuclear reactor (Fujii et al. 2002; Davis & Hamdan 2015) and meteoric data (Olive et al. 2004). This is motivated by the model dependence of such data, as they only provide stringent constraints on α if one assumes that only the finestructure constant can vary while the strong sector of the theory is unchanged, which is a simplistic assumption for constraints that stem from nuclearphysics processes (Martins 2017). Thus, they are less reliable than the spectroscopic and atomic clock data.
3.2. ELT forecast
Here we assume a future dataset to be put together by the highresolution ultrastable spectrograph currently known as HIRES (Marconi et al. 2020), that will operate the 39.3 m Extremely Large Telescope. For simplicity, we assume a set of 50 α measurements, uniformly spaced in the redshift range 0.7 ≤ z ≤ 3.2, each with an uncertainty of 0.03 parts per million, which is commensurate with the assumptions in Leite et al. (2016) and the TopLevel Requirements for the instrument (Liske et al. 2014).
With these specifications in hand, we generated the fiducial redshift dependence of Δα/α using Eqs. (6) and (7) with two different fiducial cosmologies, dubbed ΛCDM and ζw_{0}w_{a}CDM, shown in Table 1. These two fiducial cosmologies correspond to a standard case in which no α variation is present (ΛCDM) and to one where instead we assume the coupling parameter ζ is non vanishing, but still compatible with the laboratory constraints discussed in Sect. 3.1, and a dark energy component that does not behave as a cosmological constant. In this second case (ζw_{0}w_{a}CDM) the value of α varies in redshift and we aim at finding signatures of such variation.
Fiducial values for the two cosmologies considered here and used to obtain the mock datasets for ELT measurements.
Once the fiducial behaviour for Δα/α is obtained as discussed earlier, we then create the corresponding mock dataset drawing the data points at each redshift from a Gaussian distribution centred at the fiducial model and with σ the expected observational error of HIRES. We assume such errors to be uncorrelated, which observationally is a safe assumption since each measurement comes from a highresolution (R ∼ 100 000) signaltonoise limited spectrum of a point source along a different line of sight.
3.3. Euclid forecast methodology and Fisher matrices
As previously discussed, observations from Large Scale Structures probe are not very sensitive to variations in α and therefore they cannot significantly constrain the coupling ζ. They are however crucial to break the degeneracies between the coupling and the cosmological parameters, and can be therefore combined with the datasets discussed above. In this work we consider specifically Euclid as our probe of LSS. It is worth mentioning that, strictly speaking, LSS probes can provide some relevant constraints on variations in α. In Albareti et al. (2015), for example, constraints on Δα/α were provided using the O III doublet from BOSS DR12 quasar spectra. Following similar approaches, we could extract information on the variation in α from the future Euclid data. Furthermore, a typeIa supernovae survey using Euclid data (Astier et al. 2014) could provide some information on the variation in α, as illustrated in Calabrese et al. (2014). However, we prefer to focus here on the main Euclid probes and their constraints on the cosmological parameters.
In this work, in order to forecast the constraints from the future Euclid data, we follow the methodology presented in EC20. We consider a Fisher matrix formalism and make use of the TotallySAF^{1} code (YahiaCherif et al. 2021; Tutusaus et al. 2020) validated therein for the main Euclid probes: spectroscopic galaxy clustering (GCsp), photometric galaxy clustering (GCph), weak lensing (WL), and the crosscorrelation (XC) terms between the photometric probes. As was done in EC20, we neglect any correlation between the spectroscopic and photometric probes.
Starting with the spectroscopic probe, we build a Fisher matrix for the observed anisotropic power spectrum of Hα emitters (see Eq. (87) in EC20), accounting for a phenomenological model for nonlinearities, the AlcockPaczynski effect, redshiftspace distortions, and the FingersofGod effect. As in EC20, we consider two scenarios for these forecasts. In the optimistic case, we consider all scales up to a maximum of k_{max} = 0.30 h Mpc^{−1}, and we fix the nuisance parameters associated to nonlinearities. In the pessimistic scenario we limit our analysis to scales k < k_{max} = 0.25 h Mpc^{−1} and marginalize over the nonlinear nuisance parameters.
With respect to the photometric probes, we build a Fisher matrix for the tomographically binned projected angular power spectra. The same formalism is used for WL, GCph, and their XC terms, with the only difference being the kernels used in the projection from the power spectrum of matter perturbations to the sphericalharmonic space observable. As in EC20, we use the Limber, flatsky and spatially flat approximations (Kitching et al. 2017; Kilbinger et al. 2017; Taylor et al. 2018). We also neglect redshiftspace distortions, magnification, and other relativistic effects (Deshpande et al. 2020), but marginalize over the galaxy bias and intrinsic alignment nuisance parameters. We refer to EC20 for all the details on the modelling. As in the spectroscopic case, we consider two different scenarios in our forecasts. In the optimistic setting, we consider all multipoles between ℓ_{min} = 10 and ℓ_{max} = 3000 for GCph and the XC terms and ℓ_{max} = 5000 for WL. We then combine with the spectroscopic constraints assuming they are independent. In the pessimistic scenario, we limit the multipoles to ℓ_{max} = 750 for GCph and the XC terms and ℓ_{max} = 1500 for WL. In this case, when combining with the spectroscopic probe, we introduce a redshift cut of z < 0.9 for GCph and the XC terms, in order to remove any possible correlation with the spectroscopic sample starting at z = 0.9.
In accordance with Sect. 2 and EC20, we consider in this work a cosmological model with a dark energy equation of state parametrized with the CPL parametrization and with Ω_{K} = 0. In contrast with EC20, here we do not consider just a single ΛCDM fiducial (since under the assumptions of the present work there would be no α variation in that case), and obtain the Fisher matrices in both assumed cosmologies of Table 1. In addition to these assumed parameters, the analysis of EC20 also requires to specify the fiducial values of other cosmological parameters, namely the baryon energy density (Ω_{b} = 0.05), the primordial spectral index (n_{s} = 0.96) and the current amplitude of density perturbations (σ_{8} = 0.816). These additional parameters take the same fiducial values in both the ΛCDM and ζw_{0}w_{a}CDM cosmologies.
For completeness, and in order to compare with the results of this work, we provide in Table 2 the baseline Euclid forecasts obtained in EC20 for the relevant parameters in our analysis. We also note that the current constraints on the dark energy equation of state parameters are w_{0} = −0.957 ± 0.080 and ${\mathit{w}}_{a}=0.{29}_{0.26}^{+0.32}$, using the combination of Planck 2018 temperature, polarization, and lensing measurements, together with typeIa supernovae and baryon acoustic oscillations observations (Planck Collaboration VI 2020).
Baseline Euclid forecast uncertainties for the relevant parameters in this work, Ω_{m}, h, w_{0}, and w_{a}, obtained in EC20.
4. Likelihood analysis for the CPL parametrization
Following the prescription for a possible α variation described in Sect. 2, we want to combine current and forecast α measurements with the information that will be brought by Euclid, thus investigating how this survey will improve our constraints on this possible deviation from the standard cosmological paradigm. While in Sect. 3.3 we discussed how Euclid constraints can be predicted using the Fisher matrix approach, the strong nonGaussian nature of the joint cosmology and fundamental physics parameter space in varying α models (Calabrese et al. 2014) makes this approach unfeasible for both the current and future measurements that we are interested in.
Therefore, we rely here on an MCMC approach, using the publicly available sampler Cobaya (Torrado & Lewis 2021), which exploits a MetropolisHastings (MH) algorithm (Lewis & Bridle 2002; Lewis 2013). We sample the matter density parameter Ω_{m}, the Hubble constant H_{0}, the two parameters of the CPL parameterization w_{0} and w_{a}, and the coupling parameter ζ that connects the dynamical dark energy scalar field to the electromagnetic sector (see Sect. 2). The posterior distribution P(θ) that we reconstruct with this method at each point θ = (Ω_{m}, H_{0}, w_{0}, w_{a}, ζ) of the parameter space contains information coming from both α measurements and the Euclid survey,
$$\begin{array}{c}\hfill P(\theta )\propto {\mathcal{L}}_{\alpha}(\theta ){\mathcal{L}}_{\mathit{Euclid}}(\theta )\phantom{\rule{0.166667em}{0ex}},\end{array}$$(14)
where ℒ_{α} and ℒ_{Euclid} are the likelihoods of the α and Euclid datasets respectively, and we assumed that the two probes are uncorrelated.
The Euclid likelihood is constructed using the Fisher matrix F described in Sect. 3.3, and it simply exploits the Gaussian assumption done to obtain these:
$$\begin{array}{c}\hfill ln{\mathcal{L}}_{\mathit{Euclid}}\propto \frac{1}{2}{(\theta {\theta}_{\mathrm{fid}})}^{T}\stackrel{\sim}{\mathsf{F}}(\theta {\theta}_{\mathrm{fid}}),\end{array}$$(15)
where $\stackrel{\sim}{\mathsf{F}}$ is the Fisher matrix F marginalized over all parameters that are not contained in our sampled parameter space, while θ_{fid} is the fiducial cosmology under examination, which can be one of the two shown in Table 1.
On the other hand, the α likelihood contains two different contributions, again assumed to be uncorrelated, with
$$\begin{array}{c}\hfill ln{\mathcal{L}}_{\alpha}\propto (ln{\mathcal{L}}_{\mathrm{QSO}}+ln{\mathcal{L}}_{\mathrm{clocks}}),\end{array}$$(16)
with the first contribution given by observations of quasar absorption systems and the second, coming from atomic clocks measurements, giving a constraint on the possible coupling ζ at present time. These two likelihoods are taken to be
$$\begin{array}{c}\hfill ln{\mathcal{L}}_{\mathrm{QSO}}\propto \frac{1}{2}{\displaystyle \sum _{i}\frac{1}{{\sigma}_{i}^{2}}{[\frac{\mathrm{\Delta}\alpha}{\alpha}{}_{\mathrm{th}}({z}_{i})\frac{\mathrm{\Delta}\alpha}{\alpha}{}_{\mathrm{obs}}({z}_{i})]}^{2}\phantom{\rule{0.166667em}{0ex}},}\end{array}$$(17)
and
$$\begin{array}{c}\hfill ln{\mathcal{L}}_{\mathrm{clocks}}\propto \frac{1}{2}\frac{{\left({D}_{\mathrm{th}}{D}_{\mathrm{obs}}\right)}^{2}}{{\sigma}_{\mathrm{D}}^{2}}\phantom{\rule{0.166667em}{0ex}},\end{array}$$(18)
where the th subscript indicates the theoretical predictions, given by Eqs. (6) and (7) for Δα/α and Eq. (11) for D, while the quantities labelled with the obs subscript and the corresponding errors are the measurements described in Sect. 3.
We therefore sample the parameter space described above and reconstruct the posterior of Eq. (14), using flat priors on all parameters except for the coupling ζ, for which a Gaussian prior centred in ζ = 0 and with variance 1.3 × 10^{−14} is used. Such prior information is derived from the MICROSCOPE experiment discussed in Sect. 3.1, which directly constrains the Eötvös parameter η, which is related to the coupling ζ via Eq. (10).
In addition, we obtain as derived parameters also the value of Δα/α in a set of equally spaced redshifts z_{i}. Obtaining the marginalized mean values of these derived parameters and their 68% confidence limit, we reconstruct the trend of the variation in α with redshift in Sect. 6.
5. Genetic algorithm analysis
In our analysis we also use a nonparametric machine learning class of stochastic optimization methods, known as genetic algorithms (GA). These emulate natural selection, by using the data as proxies for the evolutionary pressure that drives the selection of the bestfitting functions in each generation. They are characterized by the notion of grammatical evolution, as described by the genetic operations of mutation and crossover. Specifically, a set of functions will evolve over time under the pressure of the data and the influence of the stochastic operators of crossover, that is the combination of different functions to form more complicated forms (offspring) that may fit the data better, and mutation, namely a random change in an individual function.
The GA have been used extensively to test for extensions of the standard model (Akrami et al. 2010), deviations from the cosmological constant model, both at the background and the perturbations level (Nesseris & GarciaBellido 2012; Arjona & Nesseris 2020a,b), to reconstruct a plethora of cosmological data, such as type Ia supernovae or CMB Bogdanos & Nesseris (2009), Arjona (2020) or to reconstruct various null tests such as the so called Om statistic or the curvature test (Nesseris & Shafieloo 2010; Nesseris & GarciaBellido 2013a; Sapone et al. 2014).
In our analysis we fit the finestructure data with the GA, however we solely focus on the coupling ζ. From Eq. (3) it is clear that ζ should be a constant within the context of the nonminimal coupling to the Maxwell field and the linear approximation of the gauge kinetic function B_{F}(ϕ), so we use the GA to test whether this assumption is actually supported by the data. In other words, we treat the constant ζ case as a null test and use the GA to test for deviations from that behaviour. This approach has the advantage that the coupling ζ is directly related to measurable quantities, especially in the case of the local experiments, see for example Eq. (10).
In detail, the analysis of the data with the GA proceeds as follows. First, we assume that the probability that a given function in the population will produce offspring, or equivalently its ‘eproductive success’, is proportional to its fitness. We quantify this fitness via a χ^{2} statistic, which is obtained following the same likelihood computation used in Sect. 4. Second, an initial random group of functions is chosen, each representing an initial guess for the coupling, though they are allowed to be redshift dependent, that is ζ = ζ(z).
We note that as the α data extends to high redshifts, we base our GA grammar not in terms of the redshift z, but instead in terms of $1a=\frac{z}{1+z}$, something which is commonly used in other model independent methods as well, see for example Cattoen & Visser (2007), Lazkoz et al. (2013), Guimaraes & Lima (2011). This allows us to avoid any spurious reconstructions due to lack of convergence at high redshifts.
Then, in the case of the current data, Δα/α can be related directly to ζ and compared to the data using Eqs. (6) and (7), for which we need to estimate the integral in the right hand side which can be done with the information provided by Euclid. We note that this integral is by default zero when the fiducial model is exactly the cosmological constant ΛCDM model, which means that the bestfit ζ will remain indeterminate, so in our analysis we only use the Euclidζw_{0}w_{a}CDM fiducial model, as given in in Table 1. We then use 300 realizations and we calculate both the mean and variance of the integral, with the latter then included, via error propagation, in the error estimate of all reconstructed quantities. For the atomic clocks and the MICROSCOPE bound we use Eqs. (10) and (11) and the Euclidζw_{0}w_{a}CDM fiducial in a similar fashion.
After this is done, we can compare the predictions of the GA with the data and the fitness of every test function in the population can be calculated via a standard χ^{2} statistic. Subsequently, the crossover and mutation operators are applied to a subset, usually the ∼30% bestfitting functions in every generation. These are chosen with the tournament selection – see Bogdanos & Nesseris (2009) for more details. We repeat this process thousands of times in order to ensure convergence and we also test our fits with several different random seeds, so as not to bias the results.
As soon as the GA has converged, the given bestfit function is an analytic and smooth function of the redshift z that describes the possible evolution of the coupling ζ(z).
The errors on this bestfit are estimated using an analytical approach developed by Nesseris & GarciaBellido (2012, 2013a), in which the errors are estimated by a path integral over the whole functional space. This approach has been exhaustively tested by Nesseris & GarciaBellido (2012) and has been found to be in very good agreement with Monte Carlo based error estimates.
The fact that in this approach we allow for a redshift dependence of the coupling ζ allows us to examine whether our assumptions, which rely on a constant coupling, are still valid or they break down. This is done with the goal of obtaining a null test for the constancy of ζ and the linear expansion of the gauge kinetic function in Sect. 2. Indeed, should a variation in α be supported by the data, but generated by a mechanism that violates our assumptions, Eqs. (6) and (7) would not be able to model its redshift trend, and our reconstruction would yield a nonconstant coupling ζ(z). Any statistically significant deviations from a constant value at any redshift will imply that our original hypothesis of a constant coupling may be violated. This is analogous (though not identical) to the reconstruction of a parameter quantifying possible distance duality violations, discussed in Martinelli et al. (2020). Effectively, our approach allows us to determine whether or not the data is consistent with a null value of the coupling, and how well this is constrained at various redshifts. This is also akin to constraining the behaviour of the coupling in different redshift bins.
In this work, the specific numerical implementation of the GA is based on the publicly available code Genetic Algorithms^{2}. For more details and how they apply to the analysis of Euclid data, see also Martinelli et al. (2020).
6. Likelihood approach results
In this Section, we show the results of the analysis presented in Sect. 4. We recall that our goal is to illustrate how the use of external data allows Euclid to constrain dynamical dark energy models including an electromagnetic sector coupling. Specifically, the external data constrain the coupling ζ, to which Euclid itself is insensitive. We first discuss the result of combining Euclid with currently available Δα/α and other current data, and focus on the analogous results when Euclid data is combined with next generation highresolution spectroscopy data, specifically that expected from the ELT. Finally, we show the results of a Bayesian evidence analysis that quantifies the possible significance of a detection of a varying finestructure constant.
6.1. Euclid and current α measurements
Figure 2 shows in purple the model parameter constraints obtained using currently available α measurements, comprising the combination of Webb archival data and the dedicated α measurements, atomic clocks constraints and the MICROSCOPE bound. The top panel shows the constraints on the free CPL and coupling parameters, while the bottom one highlights the reconstruction of the redshift trend of Δα/α. We show the results without and with Euclid data, for a ΛCDM fiducial, and for the Euclid data we show in yellow the pessimistic case and in cyan the optimistic one.
Fig. 2. Top panel: constraints on the CPL and coupling parameters using currently available data for α measurements alone (purple contours) and in combination with Euclid forecast constraints with a ΛCDM fiducial (yellow contours for the pessimistic case and cyan contours for the optimistic case). Bottom panel: reconstruction of the mean trend in redshift of Δα/α and of the allowed 68% confidence level area, obtained interpolating the marginalized means and errors of the derived parameters described in Sect. 4. The purple dashed line and purple area refer to α measurements alone, the yellow solid line and yellow area include Euclid in the pessimistic case, while the dotted green line and green contours combine the optimistic case. 
In Table 3, it is possible to notice how the constraint on the coupling ζ becomes less stringent when Euclid is included, changing from $\zeta =(0.{1}_{3.9}^{+3.5})\times {10}^{8}$ to ζ = (−0.1±8.2) × 10^{−8} in the pessimistic case, and ζ = (0.4±8.9) × 10^{−8} when the Euclid optimistic configuration is used. The reason for such loosening of the bound is due to the ability of Euclid to tightly constrain the CPL parameters around the ΛCDM limit (w_{0}, w_{a}) = (−1, 0); due to the degeneracy with ζ – see Eqs. (6) and (7) – this makes the α data less sensitive to the coupling parameter, as now a wider range of values is able to fit the data. Such a result is compatible with what was found in Calabrese et al. (2014), where other datasets able to tightly constrain the CPL parameters were considered. In the bottom panel of Fig. 2, it can be seen however how the weaker constrain on ζ does not lead to a larger spread of the allowed Δα/α reconstructions, which are instead tightly constrained around the novariation limit by the inclusion of Euclid information, exactly because of the tight constraints on the CPL parameters.
Mean values and 68% c.l. bounds obtained using current α measurements and their combination with Euclid forecasts.
The situation changes when the Euclid results obtained for the ζw_{0}w_{a}CDM fiducial cosmology are used. In Fig. 3 it is possible to notice how in this case the Euclid bound centred on a nonΛCDM value of the CPL parameters breaks the degeneracy between these and ζ. Here the loosening of the constraint on the coupling parameter is reduced, with the bound changing from $\zeta =(0.{1}_{3.9}^{+3.5})\times {10}^{8}$ to ζ = (−3.6±5.9) × 10^{−8} (pessimistic) and ζ = (−3.8±4.9) × 10^{−8} (optimistic), and the constraint on the redshift trend of Δα/α is tightened around a nonvanishing variation in the finestructure constant. The different behaviour of the bounds on ζ when Euclid is included in the analysis is due here to the fact that the LSS information constrain the w_{0} and w_{a} parameters away from the ΛCDM limit; this implies that the degeneracy shown in the previous case is broken, and the α data do not have a larger range of allowed coupling value.
Fig. 3. Same as Fig. 2, but when the fiducial used to obtain the Euclid results is ζw_{0}w_{a}CDM. Notice that here we use currently available direct measurements of α, which are not impacted by our choice of fiducial cosmology. Thus, the modified ζ fiducial value does not affect these results, as the Euclid probes considered are not sensitive to this parameter. 
While one can see that the inclusion of Euclid data helps α measurements constrain the variation in this fundamental parameter (since it provides information on the cosmological parameters), the synergy between these two datasets goes both ways: given the assumption done in Sect. 2 that the scalar field responsible for dark energy is the one that couples with the electromagnetic sector, the α measurements also improve Euclid constraints on the CPL parameters, since Euclid on its own is not sensitive to ζ. While this improvement is not extreme, it can be noticed in our results (see Table 3), where the errors on w_{0} and w_{a} are slightly improved with respect to the EC20 bounds shown in Sect. 3.3. The additional constraining power leads to an increase in the Figure of Merit (FoM), which we define as (EC20)
$$\begin{array}{c}\hfill \mathrm{FoM}=\sqrt{det{({C}_{{w}_{0},{w}_{a}})}^{1}}\phantom{\rule{0.166667em}{0ex}},\end{array}$$(19)
with C_{w0, wa} the covariance matrix, obtained from our MCMC results, marginalized over all parameters except for the CPL ones. In the ΛCDM fiducial, the combination of Euclid and α data improves the FoM by 18% (13%) with respect to the Euclid pessimistic (optimistic) value alone. When instead the fiducial for w_{0} and w_{a} is shifted from the cosmological constant limit, such improvement becomes 3% in both the pessimistic and optimistic cases. Such a result comes from our assumption that the DE field is the one responsible for the variation in α; this relation makes the astrophysical data sensitive to the DE parameters w_{0} and w_{a}, while no contributions to the FoM would be added if the α variation is not related to DE (or if one assumes a fixed vanishing coupling ζ).
6.2. Euclid and the ELT
After quantifying the impact of current constraints on α on Euclid, we now focus on the synergy between Euclid and the next generation highresolution spectrograph for the ELT.
In Fig. 4 we show the results for the ΛCDM fiducial, with ELT constraints in purple, and those with the inclusion of pessimistic and optimistic Euclid data in yellow and cyan respectively. As for the current data case, we find that the inclusion of Euclid leads to a loosening of the constraints on the coupling parameter, see Table 4, but with a tightening of the reconstruction of Δα/α around the ΛCDM limit due to the information on w_{0} and w_{a} brought by Euclid.
Fig. 4. Same as Fig. 2, but here the purple contours and lines refer to the forecast ELT data, while the yellow and cyan refer to the combination of this simulated dataset with pessimistic and optimistic Euclid results. The fiducial used here for both ELT and Euclid is the ΛCDM one shown in Table 1. 
Mean values and 68% c.l. bounds obtained using forecast α data and their combination with Euclid forecasts.
The results shown in Fig. 5 correspond to the ζw_{0}w_{a}CDM fiducial of Table 1. The values chosen for this fiducial make the precise data of ELT incompatible with a vanishing Δα/α and this leads to a multimodal posterior distribution for the ζ, w_{0} and w_{a} parameters. Looking at Eqs. (6) and (7), the symmetry of these peaks with respect to the (ζ, w_{0}, w_{a}) = (0, −1, 0) point in the parameter space appears evident, as simultaneously changing the sign of ζ and 1 + w(z) leads to the same lowredshift evolution of Δα/α. Because of this, when Euclid information is included, we find a breaking of the symmetry between the coupling and CPL parameters, with Euclid tightening the constraints on w_{0} and w_{a} around the fiducial.
Fig. 5. Same as Fig. 4, but the fiducial cosmology used in this case is the ζw_{0}w_{a}CDM of Table 1, thus a case that deviates from ΛCDM, with ζ = −5 × 10^{−8}, w_{0} = −0.94, w_{a} = 0.1. 
Also in this case, as for current α data, the inclusion of the information on Δα/α impacts the (w_{0}, w_{a}) FoM with respect to what is obtained with Euclid alone. In the ΛCDM fiducial, the FoM improves by 26% in both the pessimistic and optimistic cases, while for the ζw_{0}w_{a}CDM fiducial the improvement in the FoM becomes 8%.
6.3. Bayesian evidence
The results discussed in this Section make use of MetropolisHastings (MH) algorithm to sample the parameter space. This algorithm might fail in reconstructing the posterior shape when this is multimodal. Given the behaviour of some of our posterior distributions we compare the results obtained through MH with those from a nested sampling approach, using the public polychord code (Handley et al. 2015a,b) available in Cobaya, finding compatible results.
Given our use of nested sampling, we obtain as a byproduct of our analysis pipeline an estimate of the Bayesian evidence for each of the cases considered. This allows us to perform a model selection analysis for the different fiducial and experimental settings being considered in this work. We take as reference the ΛCDM model, thus analyzing all the different data combinations fixing ζ = 0, w_{0} = −1, and w_{a} = 0. Once the evidence Z_{ref} is computed for this model, we compare it with the evidence Z obtained when these parameters are free to vary. In all the cases considered, we assume the same priors on the cosmological parameters, and therefore their effect on the evidence calculation should cancel out. For the coupling ζ, in the case in which this is free to vary, we always use the MICROSCOPE prior discussed in Sect. 3.1.
In Table 5 we show the difference of the logarithms of the evidence (K = log Z − log Z_{ref}); here a positive value indicates a preference for the extended model, while a negative value supports the ΛCDM model. The results shown highlight how, assuming a ΛCDM fiducial, the model comparison favours the reference model (negative values) while for ζw_{0}w_{a}CDM the extended model is supported. Thanks to the constraining power of Euclid, all the cases (pessimistic and optimistic) and for both current and forecast α data provide “decisive” evidence for one or the other model according to the Jeffrey’s scale (Jeffreys 1939). However, it has been noted in Nesseris & GarciaBellido (2013b) that the values of the Jeffrey’s scale should be interpreted with caution, especially in cases of nested models, as they may lead to biased conclusions.
Differences (K) in the logarithm of the Bayesian evidence, between the extended model allowing for α variation and the reference ΛCDM model.
We note that the extreme values of K shown in Table 5 are mainly due to the constraining power of Euclid on w_{0} and w_{a}. If the reference model is taken to be a w_{0}w_{a}CDM model, thus with ζ = 0, but with w_{0} and w_{a} free to vary, the situation changes significantly. In the ΛCDM fiducial case, the comparison between a model with varying α and the w_{0}w_{a}CDM model is always inconclusive, as expected since both of them compare similarly with respect to the favoured ΛCDM model. If we move to the ζw_{0}w_{a}CDM fiducial instead, when using current α data, the comparison of the two models is still inconclusive, thus highlighting how the decisive preference with the previous reference was totally driven by Euclid constraints on CPL parameters, and that the sensitivity of current α data do not allow to distinguish the model under examination here from a simple w_{0}w_{a}CDM. The case of forecast data combined with Euclid instead still shows a decisive evidence in favour of the varying α model, also with respect to the w_{0}w_{a}CDM reference. This clearly shows how the improvement brought by the combination of ELT and Euclid is able to distinguish a varying α model not only from the standard ΛCDM case, but also from a cosmology where the dark energy scalar field is not coupled to the electromagnetic sector.
7. Coupling null test results
In this Section we present the GA reconstruction of the coupling ζ as a null test of whether any possible redshift dependence of α could be detected through the combination of Euclid and current astrophysical data or future ELT data. We use a machine learning approach based on the GA to reconstruct ζ, as this effectively provides a null test for the constancy of the gauge kinetic term and its linear expansion given by Eq. (2).
7.1. Euclid and current α data results
First, we directly reconstruct the coupling ζ using the currently available α measurements, comprising the combination of Webb archival data and the dedicated α measurements, atomic clocks constraints and the MICROSCOPE bound, together with the Euclid forecast constraints with a ζw_{0}w_{a}CDM fiducial. In the top panel of Fig. 6 we show the GA reconstruction of the coupling ζ as a function of redshift (red line) together with the 1 σ error shaded region.
Fig. 6. Top panel: GA Reconstruction of the coupling ζ as a function of redshift using the currently available α measurements and Euclid forecast constraints with a ζw_{0}w_{a}CDM fiducial. Bottom panel: reconstruction of the relative variation in the finestructure constant Δα/α. The Webb data is shown in grey background points and the dedicated α measurements are shown in black background points. In both panels the red line corresponds to the GA reconstruction, while the shaded region is the 1σ error. 
The nominal GA reconstruction of ζ(z) is found to be
$$\begin{array}{c}\hfill {\zeta}_{\mathrm{GA}}(z)=\frac{0.011+z\phantom{\rule{0.277778em}{0ex}}(1.558+3.041z)}{0.256+z(1.011+z)}\times {10}^{6},\end{array}$$(20)
but as can be seen in the Figure, ζ is fully consistent with zero within the errors. At high redshifts the GA leads to a value for the coupling of ζ_{GA}(z ∼ 4)≃(−2.704 ± 8.293)×10^{−6}. On the other hand, at z = 0 the GA gives the value ζ_{GA}(z = 0) = (−4.285 ± 1.510)×10^{−8}, which is in very good agreement with the parametric case when the coupling is assumed to be constant.
For completeness, in the bottom panel of Fig. 6 we show a reconstruction of the relative variation in the finestructure constant Δα/α (red line), under the previously mentioned assumptions, in order to assess its redshift trend. Here the Webb archival data is shown in grey background points, while the dedicated α measurements are shown in black background points. We confirm that the allowed Δα/α is tightly constrained around zero, even at high redshifts.
7.2. Synergy between Euclid and ELT
Next, we consider the scenario of combining Euclid data with the forecast ELT data, in the case where the fiducial model has a nonzero coupling. In the top panel of Fig. 7 we show the GA reconstruction of the coupling ζ as a function of redshift (red line) using the forecast ELT α measurements, together with the Euclid forecast constraints with a ζw_{0}w_{a}CDM fiducial. We use the same atomic clocks constraints and the MICROSCOPE as in the previous subsection. We find that the GA reconstruction is consistent with the fiducial value (shown with the dotdashed line) within the 1σ bound. In particular, at high redshifts (z > 1) a deviation from zero is detected, in agreement with the fiducial value used in our mocks, highlighting the importance of the extension of the redshift lever arm provided by the ELT measurements of α.
Fig. 7. Top panel: GA reconstruction of the coupling ζ as a function of redshift (red line) using the forecast ELT α data (grey background points), the atomic clocks constraints and the MICROSCOPE bound (both at z = 0) and Euclid forecast constraints with a ζw_{0}w_{a}CDM fiducial. Bottom panel: reconstruction of the relative variation in the finestructure constant Δα/α (red line) along with binned values of the data in one (green point) and two bins (blue points). In both cases the red line corresponds to the GA reconstruction, while the shaded region is the 1σ error. 
In the bottom panel of Fig. 7 we also show a GA reconstruction of the relative variation in the finestructure constant Δα/α (red line). As a test of our approach and given the much higher sensitivity of the ELT data, we also bin the data either in one bin (green point) or in two bins (blue points) splitting the ELT data at z = 2. We find that the GA reconstruction of the relative variation in the finestructure constant is in excellent agreement with both the fiducial model, given by the dotdashed line, and with the binning of the data.
8. Discussion and outlook
When studying theoretical models beyond the concordance ΛCDM framework, many realistic extensions introduce couplings between the different degrees of freedom. In particular, the coupling of physically realistic dynamical dark energy scalar fields to the electromagnetic sector will lead to a time dependence of the finestructure constant (Carroll 1998; Dvali & Zaldarriaga 2002; Chiba & Kohri 2002), which could be detected and interpreted as a smoking gun for the existence of extra scalar fields. In this work we have studied the role of the forthcoming Euclid mission in constraining such theoretical models. Euclid will provide us with very precise cosmological information using the clustering of galaxies and the weak lensing measurements from the largescale structure of the Universe. However, these probes are not enough to constrain the full parameter space of these models. We therefore need to add astrophysical (and local) data, specifically to constrain the coupling between the dark energy scalar field and the electromagnetic sector.
In this work we have considered current astrophysical tests of the stability of the finestructure constant from quasistellar object spectral lines (both from archival data and from dedicated measurements), as well as current laboratory constraints on the presentday drift rate of α from atomic clock experiments and constraints on the Eötvös parameter from the MICROSCOPE satellite. However, at the time Euclid data will be available we expect to have even more precise astrophysical measurements of α, so we have also forecast the precision of the highresolution ultrastable spectrograph HIRES at the Extremely Large Telescope. We have used both a parametric approach, under a standard likelihood analysis, and a nonparametric machine learning class of stochastic optimization methods to forecast the constraints from the joint analysis of Euclid and astrophysical and local data.
Starting with the parametric approach, we have first considered the synergy between Euclid and current measurements of the relative variation in the finestructure constant. Our baseline scenario has a ΛCDM fiducial, meaning that the fiducial corresponds to a vanishing coupling constant and a cosmological constant as dark energy. In Sect. 6.1 we have seen that Euclid very significantly restricts the allowed values of the finestructure constant as a function of redshift compared to the constraints with astrophysical and laboratory data alone. This is due to Euclid’s ability to constrain the dark energy equation of state parameters. However, for this same reason, and because of the degeneracy between the coupling constant and w_{0} and w_{a} when they are close to the cosmological constant corresponding values, the addition of Euclid data loosens the constraints on the coupling ζ compared to the constraints from finestructure constant data alone.
We have then performed the same analysis using the ζw_{0}w_{a}CDM fiducial for the Euclid results, where the coupling fiducial is nonnull and the fiducial values of w_{0} and w_{a} no longer correspond to a cosmological constant. In this case the increase of constraining power on the evolution of α when adding Euclid data is still present. Concerning the bounds on the coupling constant, the addition of Euclid data only marginally loosens the constraints, since the fiducial is not exactly on ΛCDM and the degeneracy between the coupling constant and the dark energy parameters is partially broken. It is also worth mentioning that adding these astrophysical and local tests of the variation in the finestructure constant improves the constraints on the dark energy equation of state parameters from Euclid data alone, with the FoM for dark energy parameters improving between 3% and 18%, as long as the model considered connects the variation in α to the dark energy parameters, thus making the α data sensitive to them.
Still with the parametric approach, we have considered the synergy between Euclid and the ELT. The results obtained have been qualitatively the same, with Euclid data helping to constrain the evolution of the finestructure constant as a function of redshift while the astrophysical and local data help constraining the dark energy equation of state. However, given the fact that the ELT data is much more precise than current measurements, the contribution of Euclid on Δα/α is proportionally somewhat smaller, while the contribution of ELT on w_{0} and w_{a} is slightly larger, with the FoM now improving between 8% and 26%. Furthermore, we also performed a model comparison analysis, which highlighted how both current and forecast α data, in combination with Euclid, can significantly distinguish between ΛCDM and a model with a coupled scalar field, while only when considering ELT forecast can the latter be distinguished from a w_{0}w_{a}CDM cosmology with a vanishing ζ coupling.
We have also used a modelindependent approach to reconstruct the coupling between the dark energy scalar field and the electromagnetic sector. This is effectively a null test reconstruction of the behaviour of the coupling ζ. Specifically, any deviation from a constant coupling would either suggest unidentified systematics in the astrophysical data or indicate that the assumptions made in Sect. 2 break down and our modelling is not accurate enough to explain the observations; in the latter case this would imply that the putative dynamical dark energy and varying α would not be due to the same underlying physical mechanism – which would in itself be a significant result. Our analysis shows how the GA are able to reconstruct the coupling function in agreement with the fiducial values assumed, and are compatible with a constant coupling.
Overall, we have found that the synergies between the main probes of Euclid and astrophysical measurements of α can tightly constrain models where the dark energy scalar field is coupled to the electromagnetic sector. Moreover, recent analyses have shown that combining Euclid with future cosmic microwave background experiments will provide very precise constraints on the cosmological parameters (Euclid Collaboration 2021). Therefore, although the cosmic microwave background sensitivity to the finestructure constant is not at the same level as astrophysical constraints, we can expect the improvement on cosmological parameter constraints to also propagate to the varying finestructure constant model considered in this work. In addition to these combinations of Euclid with external data, one must notice that other Euclid probes (such as a possible SNIa survey) can also be directly sensitive to a varying finestructure constant, and this would further improve the contribution of Euclid on tests of such coupled models.
Acknowledgments
MM has received the support of a fellowship from “la Caixa” Foundation (ID 100010434), with fellowship code LCF/BQ/PI19/11690015, and the support of the Centro de Excelencia Severo Ochoa Program SEV2016059. The work of CJM was financed by FEDER – Fundo Europeu de Desenvolvimento Regional funds through the COMPETE 2020 – Operational Programme for Competitiveness and Internationalisation (POCI), and by Portuguese funds through FCT – Fundação para a Ciência e a Tecnologia in the framework of the project POCI010145FEDER028987. IT acknowledges support from the Spanish Ministry of Science, Innovation and Universities through grant ESP201789838, and the H2020 programme of the European Commission through grant 776247. SN acknowledges support from the research project PGC2018094773BC32, the Centro de Excelencia Severo Ochoa Program SEV2016059 and the Ramón y Cajal program through Grant No. RYC201415843. The Euclid Consortium acknowledges the European Space Agency and a number of agencies and institutes that have supported the development of Euclid, in particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid Consortium, the Centre National d’Etudes Spatiales, the Deutsches Zentrum für Luft und Raumfahrt, the Danish Space Research Institute, the Fundaçáo para a Ciência e a Tecnologia, the Ministerio de Economia y Competitividad, the National Aeronautics and Space Administration, the Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the State Secretariat for Education, Research and Innovation (SERI) at the Swiss Space Office (SSO), and the United Kingdom Space Agency. A complete and detailed list is available on the Euclid web site (http://www.euclidec.org).
References
 Akrami, Y., Scott, P., Edsjo, J., Conrad, J., & Bergstrom, L. 2010, JHEP, 04, 057 [Google Scholar]
 Albareti, F. D., Comparat, J., Gutiérrez, C. M., et al. 2015, MNRAS, 452, 4153 [CrossRef] [Google Scholar]
 Arjona, R. 2020, JCAP, 08, 009 [CrossRef] [Google Scholar]
 Arjona, R., & Nesseris, S. 2020a, PRD, 101, 123525 [NASA ADS] [CrossRef] [Google Scholar]
 Arjona, R., & Nesseris, S. 2020b, JCAP, 11, 042 [NASA ADS] [CrossRef] [Google Scholar]
 Astier, P., Balland, C., Brescia, M., et al. 2014, A&A, 572, A80 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Bogdanos, C., & Nesseris, S. 2009, JCAP, 05, 006 [CrossRef] [Google Scholar]
 Calabrese, E., Menegoni, E., Martins, C. J. A. P., Melchiorri, A., & Rocha, G. 2011, PRD, 84, 023518 [NASA ADS] [CrossRef] [Google Scholar]
 Calabrese, E., Martinelli, M., Pandolfi, S., et al. 2014, PRD, 89, 083509 [NASA ADS] [CrossRef] [Google Scholar]
 Carroll, S. M. 1998, PRL, 81, 3067 [NASA ADS] [CrossRef] [Google Scholar]
 Casas, J. A., GarciaBellido, J., & Quiros, M. 1991, Nucl. Phys. B, 361, 713 [NASA ADS] [CrossRef] [Google Scholar]
 Casas, J. A., GarciaBellido, J., & Quiros, M. 1992, Class. Quant. Grav., 9, 1371 [NASA ADS] [CrossRef] [Google Scholar]
 Cattoen, C., & Visser, M. 2007, ArXiv eprints [arXiv:grqc/0703122] [Google Scholar]
 Chevallier, M., & Polarski, D. 2001, Int. J. Mod. Phys. D, 10, 213 [Google Scholar]
 Chiba, T., & Kohri, K. 2002, Prog. Theor. Phys., 107, 631 [CrossRef] [Google Scholar]
 Costille, A., Caillat, A., Rossin,, C., et al. 2018, in Space Telescopes and Instrumentation 2018: Optical, Infrared, and Millimeter Wave, eds. M. Lystrup, G. G. Fazio, et al., SPIE, 10698, 730 [Google Scholar]
 Cropper, M., Pottinger, S., Azzollini, R., et al. 2018, in Space Telescopes and Instrumentation 2018: Optical, Infrared, and Millimeter Wave, eds. M. Lystrup, H. A. MacEwen, G. G. Fazio, et al., SPIE, 10698, 709 [Google Scholar]
 Damour, T., & Donoghue, J. F. 2010, Class. Quant. Grav., 27, 202001 [NASA ADS] [CrossRef] [Google Scholar]
 Damour, T., Piazza, F., & Veneziano, G. 2002, Phys. Rev. Lett., 89, 081601 [Google Scholar]
 Davis, E. D., & Hamdan, L. 2015, Phys. Rev. C, 92, 014319 [NASA ADS] [CrossRef] [Google Scholar]
 Deshpande, A. C., Kitching, T. D., Cardone, V. F., et al. 2020, A&A, 636, A95 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Dicke, R. H. 1964, Relativité, Groupes et Topologie: Proceedings, École d’été de Physique Théorique, Session XIII, Les Houches, 165 [Google Scholar]
 Dvali, G., & Zaldarriaga, M. 2002, PRL, 88, 091303 [NASA ADS] [CrossRef] [Google Scholar]
 Euclid Collaboration (Blanchard, A., et al.) 2020, A&A, 642, A191 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Euclid Collaboration (Ilić, S., et al.) 2021, A&A, in press, https://doi.org/10.1051/00046361/202141556 [Google Scholar]
 Fang, W., Hu, W., & Lewis, A. 2008, PRD, 78, 087303 [NASA ADS] [CrossRef] [Google Scholar]
 Fujii, Y., Iwamoto, A., Fukahori, T., et al. 2002, J. Nucl. Sci. Tech., 39, 592 [CrossRef] [Google Scholar]
 Guimaraes, A. C., & Lima, J. A. S. 2011, Class. Quant. Grav., 28, 125026 [NASA ADS] [CrossRef] [Google Scholar]
 Handley, W. J., Hobson, M. P., & Lasenby, A. N. 2015a, MNRAS, 450, L61 [NASA ADS] [CrossRef] [Google Scholar]
 Handley, W. J., Hobson, M. P., & Lasenby, A. N. 2015b, MNRAS, 453, 4385 [NASA ADS] [CrossRef] [Google Scholar]
 Hu, W. 2008, PRD, 77, 103524 [NASA ADS] [CrossRef] [Google Scholar]
 Hu, W., & Sawicki, I. 2007, PRD, 76, 104043 [NASA ADS] [CrossRef] [Google Scholar]
 Jeffreys, H. 1939, The Theory of Probability, Oxford Classic Texts in the Physical Sciences [Google Scholar]
 Kilbinger, M., Heymans, C., Asgari, M., et al. 2017, MNRAS, 472, 2126 [Google Scholar]
 Kitching, T. D., Alsing, J., Heavens, A. F., et al. 2017, MNRAS, 469, 2737 [Google Scholar]
 Lange, R., Huntemann, N., Rahm, J., et al. 2021, PRL, 126, 011102 [NASA ADS] [CrossRef] [Google Scholar]
 Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, ArXiv eprints [arXiv:1110.3193] [Google Scholar]
 Lazkoz, R., Alcaniz, J., EscamillaRivera, C., Salzano, V., & Sendra, I. 2013, JCAP, 12, 005 [CrossRef] [Google Scholar]
 Leite, A., Martins, C., Molaro, P., Corre, D., & Cristiani, S. 2016, PRD, 94, 123512 [NASA ADS] [CrossRef] [Google Scholar]
 Lewis, A. 2013, PRD, 87, 103529 [NASA ADS] [CrossRef] [Google Scholar]
 Lewis, A., & Bridle, S. 2002, PRD, 66, 103511 [CrossRef] [Google Scholar]
 Linder, E. V. 2003, PRL, 90, 091301 [NASA ADS] [CrossRef] [Google Scholar]
 Liske, J., Bono, G., Cepa, J., et al. 2014, Top Level Requirements ForELTHIRES, Tech. rep., Document ESO 204697 Version 1 [Google Scholar]
 Marconi, A., Abreu, M., Adibekyan, V., et al. 2020, SPIE Conf. Ser., 11447, 1144726 [NASA ADS] [Google Scholar]
 Martinelli, M., Martins, C. J. A. P., Nesseris, S., et al. 2020, A&A, 644, A80 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Martins, C., Pinho, A., Carreira, P., et al. 2016, PRD, 93, 023506 [NASA ADS] [CrossRef] [Google Scholar]
 Martins, C. J. A. P. 2017, Rep. Prog. Phys., 80, 126902 [Google Scholar]
 Martins, C. J. A. P., & Vila Miñana, M. 2019, Phys. Dark Univ., 25, 100301 [CrossRef] [Google Scholar]
 Milaković, D., Lee, C.C., Carswell, R. F., et al. 2021, MNRAS, 500, 1 [Google Scholar]
 Murphy, M. T., & Cooksey, K. L. 2017, MNRAS, 471, 4930 [Google Scholar]
 Nesseris, S., & GarciaBellido, J. 2012, JCAP, 11, 033 [Google Scholar]
 Nesseris, S., & GarciaBellido, J. 2013a, PRD, 88, 063521 [NASA ADS] [CrossRef] [Google Scholar]
 Nesseris, S., & GarciaBellido, J. 2013b, JCAP, 08, 036 [Google Scholar]
 Nesseris, S., & Shafieloo, A. 2010, MNRAS, 408, 1879 [NASA ADS] [CrossRef] [Google Scholar]
 Nunes, N. J., & Lidsey, J. E. 2004, PRD, 69, 123511 [CrossRef] [Google Scholar]
 Olive, K. A., & Pospelov, M. 2008, Phys. Rev. D, 77, 043524 [NASA ADS] [CrossRef] [Google Scholar]
 Olive, K. A., Pospelov, M., Qian, Y.Z., et al. 2004, Phys. Rev. D, 69, 027701 [NASA ADS] [CrossRef] [Google Scholar]
 Pinho, A. M. M., Martinelli, M., & Martins, C. J. A. P. 2017, Phys. Lett. B, 769, 491 [NASA ADS] [CrossRef] [Google Scholar]
 Planck Collaboration VI. 2020, A&A, 641, A6 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Sapone, D., Majerotto, E., & Nesseris, S. 2014, PRD, 90, 023012 [NASA ADS] [CrossRef] [Google Scholar]
 Silva, M. F., Winther, H. A., Mota, D. F., & Martins, C. J. A. P. 2014, PRD, 89, 024025 [NASA ADS] [CrossRef] [Google Scholar]
 Taylor, P. L., Kitching, T. D., McEwen, J. D., & Tram, T. 2018, PRD, 98, 023522 [NASA ADS] [CrossRef] [Google Scholar]
 Taylor, T. R., & Veneziano, G. 1988, Phys. Lett. B, 213, 450 [NASA ADS] [CrossRef] [Google Scholar]
 Torrado, J., & Lewis, A. 2021, JCAP, 05, 057 [Google Scholar]
 Touboul, P., Métris, G., Rodrigues, M., et al. 2019, Class. Quant. Grav., 36, 225006 [NASA ADS] [CrossRef] [Google Scholar]
 Tutusaus, I., Martinelli, M., Cardone, V. F., et al. 2020, A&A, 643, A70 [EDP Sciences] [Google Scholar]
 Vielzeuf, P. E., & Martins, C. J. A. P. 2014, Mem. Soc. Ast. It., 85, 155 [NASA ADS] [Google Scholar]
 Webb, J., King, J., Murphy, M., et al. 2011, PRL, 107, 191101 [NASA ADS] [CrossRef] [Google Scholar]
 Welsh, L., Cooke, R., Fumagalli, M., & Pettini, M. 2020, MNRAS, 494, 1411 [NASA ADS] [CrossRef] [Google Scholar]
 Whitmore, J. B., & Murphy, M. T. 2015, MNRAS, 447, 446 [Google Scholar]
 YahiaCherif, S., Blanchard, A., Camera, S., et al. 2021, A&A, 649, A52 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
Appendix A: Current data compatibility
As pointed out in Martins (2017), Martins & Vila Miñana (2019), the currently available astrophysical tests of the stability of the finestructure constant are in slight tension with each other; the weighted mean of Δα/α obtained through the Webb archival data is in fact in tension of ≈2σ with the one obtained through the recent dedicated measurements. In order to be able to combine these datasets, as we did throughout our paper, we must assess the significance of such tension.
In the parametric approach, we can estimate the concordance of the datasets by computing the Bayesian ratio
$$\begin{array}{c}\hfill K=\frac{Z(\mathrm{Webb}+\mathrm{recent})}{Z(\mathrm{Webb})Z(\mathrm{recent})}\phantom{\rule{0.166667em}{0ex}},\end{array}$$(A.1)
where Z(Webb) is the evidence when using Webb data alone, Z(recent) when only recent data are considered and Z(Webb + recent) the case when the two are considered in combination. Such a Bayesian ratio can be used for model comparison between the case in which the two datasets are used to fit the same set of cosmological parameter, Z(Webb + recent), and the case in which these might differ for the two datasets, Z(Webb)Z(recent).
We obtain these values using polychord to sample our free parameters and, when considering only astrophysical α measurements, we find that lnK ≈ 0.2, a value that is not able to provide any conclusive evidence for either the concordance or discordance of the data. When instead we include the information brought by local measurements, we obtain lnK ≈ 1.6, thus an improved significance towards the concordance of the data. Such an increase in the value of lnK comes from the fact that the local measurements dominate the constraints obtained through the parametric approach, thus hiding any possible discordance of the two sets of astrophysical data.
This does not apply in our null test reconstruction of ζ(z); here, the local measurements only contribute to the very low redshift reconstruction, while astrophysical data dominate at higher redshift. Because of this tension between the Webb archival data and the recent dedicated measurement, the reconstruction performed with the GA attempts to be in agreement with both datasets, and this can potentially increase the error on the reconstructed function. One thus expects that removing one of the datasets, the error on the reconstruction will decrease.
In particular, in Fig. A.1 we show the GA reconstruction of the relative variation in the finestructure constant Δα/α as a function of redshift. The green line corresponds to the dedicated measurements, the orange line to the Webb archival data and the magenta line to the combination of the two, while in all cases the shaded region is the 1σ error. As expected, the reconstruction of the dedicated measurements has a smaller error (green shaded region), but when we combine them with the Webb archival data then the combined error region not only does not decrease, but instead increases due to the tension, as observed in the magenta shaded region. Equivalently, we see that when we remove the Webb data from the combination of the two, counterintuitively the error of the reconstruction decreases. However, overall the reconstructions are still compatible with each other and with zero.
Fig. A.1. GA reconstruction of the relative variation in the finestructure constant Δα/α as a function of redshift. The green line corresponds to the dedicated measurements, the orange line to the Webb archival data and the magenta line to the combination of the two, while in all cases the shaded region is the 1σ error. 
All Tables
Fiducial values for the two cosmologies considered here and used to obtain the mock datasets for ELT measurements.
Baseline Euclid forecast uncertainties for the relevant parameters in this work, Ω_{m}, h, w_{0}, and w_{a}, obtained in EC20.
Mean values and 68% c.l. bounds obtained using current α measurements and their combination with Euclid forecasts.
Mean values and 68% c.l. bounds obtained using forecast α data and their combination with Euclid forecasts.
Differences (K) in the logarithm of the Bayesian evidence, between the extended model allowing for α variation and the reference ΛCDM model.
All Figures
Fig. 1. Example trends in redshift for the variation in α changing the CPL parameters. The values of the other cosmological parameters entering Eqs. (6) and (7) are Ω_{m} = 0.32, and ζ = 10^{−6}. 

In the text 
Fig. 2. Top panel: constraints on the CPL and coupling parameters using currently available data for α measurements alone (purple contours) and in combination with Euclid forecast constraints with a ΛCDM fiducial (yellow contours for the pessimistic case and cyan contours for the optimistic case). Bottom panel: reconstruction of the mean trend in redshift of Δα/α and of the allowed 68% confidence level area, obtained interpolating the marginalized means and errors of the derived parameters described in Sect. 4. The purple dashed line and purple area refer to α measurements alone, the yellow solid line and yellow area include Euclid in the pessimistic case, while the dotted green line and green contours combine the optimistic case. 

In the text 
Fig. 3. Same as Fig. 2, but when the fiducial used to obtain the Euclid results is ζw_{0}w_{a}CDM. Notice that here we use currently available direct measurements of α, which are not impacted by our choice of fiducial cosmology. Thus, the modified ζ fiducial value does not affect these results, as the Euclid probes considered are not sensitive to this parameter. 

In the text 
Fig. 4. Same as Fig. 2, but here the purple contours and lines refer to the forecast ELT data, while the yellow and cyan refer to the combination of this simulated dataset with pessimistic and optimistic Euclid results. The fiducial used here for both ELT and Euclid is the ΛCDM one shown in Table 1. 

In the text 
Fig. 5. Same as Fig. 4, but the fiducial cosmology used in this case is the ζw_{0}w_{a}CDM of Table 1, thus a case that deviates from ΛCDM, with ζ = −5 × 10^{−8}, w_{0} = −0.94, w_{a} = 0.1. 

In the text 
Fig. 6. Top panel: GA Reconstruction of the coupling ζ as a function of redshift using the currently available α measurements and Euclid forecast constraints with a ζw_{0}w_{a}CDM fiducial. Bottom panel: reconstruction of the relative variation in the finestructure constant Δα/α. The Webb data is shown in grey background points and the dedicated α measurements are shown in black background points. In both panels the red line corresponds to the GA reconstruction, while the shaded region is the 1σ error. 

In the text 
Fig. 7. Top panel: GA reconstruction of the coupling ζ as a function of redshift (red line) using the forecast ELT α data (grey background points), the atomic clocks constraints and the MICROSCOPE bound (both at z = 0) and Euclid forecast constraints with a ζw_{0}w_{a}CDM fiducial. Bottom panel: reconstruction of the relative variation in the finestructure constant Δα/α (red line) along with binned values of the data in one (green point) and two bins (blue points). In both cases the red line corresponds to the GA reconstruction, while the shaded region is the 1σ error. 

In the text 
Fig. A.1. GA reconstruction of the relative variation in the finestructure constant Δα/α as a function of redshift. The green line corresponds to the dedicated measurements, the orange line to the Webb archival data and the magenta line to the combination of the two, while in all cases the shaded region is the 1σ error. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.