Dark Energy coupling with electromagnetism as seen from future low-medium redshift probes
E. Calabrese, M. Martinelli, S. Pandolfi, V. F. Cardone, C. J. A. P. Martins, S. Spiro, P. E. Vielzeuf
DDark Energy coupling with electromagnetism as seen from future low-mediumredshift probes
E. Calabrese , M. Martinelli , S. Pandolfi ,V. F. Cardone , C. J. A. P. Martins , S. Spiro , P. E. Vielzeuf , Sub-department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK SISSA, Via Bonomea 265, Trieste, 34136, Italy Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen,Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark. I.N.A.F. - Osservatorio Astronomico di Roma, via Frascati 33, 00040 - Monte Porzio Catone (Roma), Italy Centro de Astrof´ısica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal I.N.A.F. - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy and Faculdade de Ciˆencias, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal (Dated: October 17, 2018)Beyond the standard cosmological model the late-time accelerated expansion of the universe canbe reproduced by the introduction of an additional dynamical scalar field. In this case, the field isexpected to be naturally coupled to the rest of the theory’s fields, unless a (still unknown) symmetrysuppresses this coupling. Therefore, this would possibly lead to some observational consequences,such as space-time variations of nature’s fundamental constants. In this paper we investigate thecoupling between a dynamical Dark Energy model and the electromagnetic field, and the corre-sponding evolution of the fine structure constant ( α ) with respect to the standard local value α . Inparticular, we derive joint constraints on two dynamical Dark Energy model parametrizations (theChevallier-Polarski-Linder and Early Dark Energy model) and on the coupling with electromag-netism ζ , forecasting future low-medium redshift observations. We combine supernovae and weaklensing measurements from the Euclid experiment with high-resolution spectroscopy measurementsof fundamental couplings and the redshift drift from the European Extremely Large Telescope,highlighting the contribution of each probe. Moreover, we also consider the case where the fielddriving the α evolution is not the one responsible for cosmic acceleration and investigate how futureobservations can constrain this scenario. PACS numbers: 98.80.-k, 95.36.+x, 97.60.Bw, 98.80.Es
I. INTRODUCTION
Since the discovery of cosmic acceleration from mea-surements of luminosity distances of type Ia Supernovae(SN) in 1998 [1, 2] and its confirmation by several otherindependent cosmological data, the nature of the compo-nent driving this acceleration, the so-called Dark Energy(DE hereafter), has been deeply debated. In the standardcosmological model, the Λ Cold Dark Matter (ΛCDM),the acceleration is produced by the cosmological constantΛ. This model is consistent with the majority of the ob-servational data, but the known theoretical problems ofthe cosmological constant led cosmologists to formulateseveral other alternative models able, from one side, torelieve the aforementioned theoretical issues and, on theother side, to explain observations.Alternative models for the DE, such as quintessence,are called (models of) dynamical dark energy and, evenif not favoured, they are currently not excluded by ob-servations [3, 4]. Several of these alternative models arecharacterized by the existence of an additional scalar fieldwhich drives the accelerated expansion of the universe. Ifthis is the case, it is expected that this additional com-ponent is coupled to the rest of the theory’s fields.In this paper we study the coupling of dynamicalDE models with the electromagnetic field: indeed, thepresence of this coupling would lead to a space-time variation of the fine-structure constant α [5]. This, inturn, would generate distinctive signatures in cosmolog-ical data, such as the Cosmic Microwave Background(CMB) (see e.g. [6–9]), but also in low and medium red-shift probes, for example in the peak of luminosity in SNor in the metal absorption lines of distant quasars (QSO).The present work aims to extend and to complete theanalysis done in [9], where constraints on the coupling ofa time-varying fine structure constant in the presence ofEarly Dark Energy were obtained with CMB data. Inthis paper we focus on low-medium redshift observables,forecasting SN and QSO data, Weak Lensing shear powerspectrum measurements (WL), and redshift-drift (RD)data. The relevance of this combination of probes is thecoverage of a wide redshift range (0 < z (cid:46)
5) which is avery powerful way to discriminate between a cosmologicalconstant and a dynamical DE model, as it makes possibleto investigate the onset of DE. In other words, given thepossibility of a dynamical field that is moving very slowly(in appropriate units), searching for deviations from acosmological constant is optimally done by maximizingthe lever arm of probed redshifts.In the present work we assume only a time-varying finestructure constant, neglecting spatial variation. Recentanalyses of CMB data [10] have shown no evidence ofa spatial variation; there is instead some evidence of aspatial variation from lower redshift QSO measurements a r X i v : . [ a s t r o - ph . C O ] F e b [11], and attempts are being made to independently con-firm it [12, 13]. For the moment we note that our methodcould in principle be extended to the more complex mod-els needed to account for such spatial variations.We consider two different classes of time-varying α models. In the first class the scalar field causing the α variation is also responsible for the accelerated expansionof the universe, and therefore observational tests of theevolution of α directly contribute to constrain dark en-ergy scenarios [14]. In the second class the additional de-gree of freedom which causes the α variation is not (or atmost is only partially) the source of the DE component.This second class is important for two reasons. Firstly,although consistency tests are available, erroneous darkenergy properties could be inferred if the α evolution isascribed to DE instead of this “external” degree of free-dom; this scenario has been discussed in [15]. Moreover,there may be a bias induced on the cosmological param-eters estimation due to a wrong assumption on the un-derlying cosmological model, i.e. selecting a dataset witha non-zero variation of α , but assuming no variation inthe analysis. We investigate this possibility here. Shouldsuch a bias be non-negligible and found in future data, itcould hint for the need of an extended underlying theo-retical model in the analysis.The paper is organized as follows. In Section II weintroduce the dynamical DE models considered in thiswork and derive the time evolution of α . Section IIIcontains the description of the different probes we exploitand we highlight the main features of each observable.Section IV details the analysis we perform and the resultsare presented in Sec. V. We then discuss our results inthe concluding Sec. VI. II. THEORETICAL MODELS FOR THEEVOLUTION OF THE FINE STRUCTURECONSTANT.
In this section we discuss the two broad classes of mod-els for the evolution of the fine structure constant andpresent specific examples for each class, then used in therest of the paper. In the first class, the dynamical degreeof freedom providing the α variation is also responsiblefor the observationally required dark energy, while in thesecond class the degree of freedom is not, or only par-tially, responsible for the dark energy component. Theobservational probes are affected in different ways bythese scenarios, thus leading, in principle, to constraintson DE parameters and on the coupling with electromag-netism which are specific to the particular model. A. Type I models: A single dynamical degree offreedom
In this first case we assume that there is a single ad-ditional degree of freedom (typically, a scalar field) re- sponsible for the cosmic acceleration, and coupled to theelectromagnetic sector, thus leading to the time varia-tion of the fine structure constant α . We consider twodifferent models for the DE component: a phenomeno-logical generic parametrization of the DE equation ofstate parameter, the Chevallier-Polarski-Linder (CPL)parametrization, and a more physically motivated EarlyDark Energy (EDE) model. • In the CPL model [16, 17] the DE equation of state(EoS) is written as w CPL ( z ) = w + w a z z , (1)where w is the present value of w CPL (i.e. w CPL ( z = 0) = w ) and w a is the coefficient ofthe time-dependent term of the EoS.In this model the EoS has a trend with redshiftthat is not intended to mimic a particular modelfor dark energy, but rather to allow to probe possi-ble deviations from the ΛCDM standard paradigmwithout the assumption of any underlying theory.Nevertheless, we can assume that also this kind ofDE is produced by a scalar field. • In the EDE model [18], the dark energy den-sity fraction Ω
EDE ( a ) (i.e., the fraction of energydensity of the DE component over the total en-ergy density) and equation of state w EDE ( a ) areparametrized in the following wayΩ EDE ( a ) = Ω − Ω e (cid:0) − a − w (cid:1) Ω + Ω a w +Ω e (cid:0) − a − w (cid:1) (2) w EDE ( a ) = − − Ω EDE ] d ln Ω EDE d ln a + a eq a + a eq ) (3)where a eq is the scale factor at matter-radiationequality and Ω and Ω are the current dark en-ergy and matter density, respectively. A flat uni-verse is assumed and the present value for the equa-tion of state is obtained demanding w ( a = 1) = w .The energy density Ω de ( a ) has a scaling behaviourevolving with time and going to a finite constantΩ e in the past.In this case the EoS follows the behaviour of thedominant component at each cosmic time; w EDE ≈ / w EDE ≈ w EDE ≈ − φ , and electro-magnetism stems from a gauge kinetic function B F ( φ ) L φF = − B F ( φ ) F µν F µν (4)which, to a good approximation, can be assumed linear[20, 21], B F ( φ ) = 1 − ζ √ πG ( φ − φ ) . (5)This form of the gauge kinetic function can be seenas the first term of a Taylor expansion, which is indeeda good approximation for a slowly varying field at lowredshifts, as the low-redshift constraints on couplings,obtained both directly from astrophysical measurementsand through local tests of equivalence principle viola-tions, are quite tight. For the latter category we canrefer to the conservative constraint [22, 23] | ζ local | < − . (6)In [9], the authors obtained an independent few-percentconstraint on this coupling using CMB and large-scalestructure data in combination with direct measurementsof the expansion rate of the universe.With these assumptions, the evolution of α is given by∆ αα ≡ α − α α = ζ √ πG ( φ − φ ) , (7)and, since the evolution of the putative scalar field canbe expressed in terms of the dark energy properties Ω φ and w as [21, 24] w = − √ πGφ (cid:48) ) φ , (8)where the prime denotes the derivative with respect tothe logarithm of the scale factor, we finally obtain thefollowing explicit relation for the evolution of the finestructure constant in this dynamical dark energy class ofmodels ∆ αα ( z ) = ζ (cid:90) z (cid:113) φ ( z ) [1 + w ( z )] dz (cid:48) z (cid:48) . (9)As expected, in this class of models the magnitudeof the α variation is controlled by the strength of thecoupling ζ . We also note that these two equations can bephenomenologically generalized to the case of phantomequations of state, by simply switching the sign of the(1 + w ) term [25]. Here Ω φ ( z ) is the fraction of energy density providedby the scalar field, thus it corresponds to Eq.(2) in theEDE case, while for the CPL parametrization it’s easilyfound to beΩ CPL ( z ) = Ω CP L Ω + Ω (1 + z ) − w + w a ) e (3 w a z/ z ) . (10)where Ω and Ω are, respectively, the present timeenergy densities of matter and DE. B. Type II models: Independent degrees offreedom
In this scenario the degree of freedom responsible forthe α variation does not provide the dark energy, or atleast is constrained to provide only a fraction of it bycurrent observations. One effectively has a ΛCDM modelwith an additional (often phenomenological) degree offreedom accounting for the α variation.In this case the direct link between varying couplingsand dark energy discussed above is also lost. Neverthe-less, it is possible to observationally infer that a given α variation is not due to a Type I model, as such anassumption could lead to consequences that can be ob-servationally ruled out. This possibility has already beendiscussed in [15]. Here we will discuss this class in aslightly different context.The simplest toy model of this kind is the Bekenstein-Sandvik-Barrow-Magueijo (BSBM) model [26]. Thesetheories require some fine-tuning, even to fit purely tem-poral α variations as that of [27], but for our purposesthey are useful for parametrizing the biases introducedin cosmological parameter estimations if there is an α variation which is neglected in the analysis. For the α variation itself we can, to a good approximation, assumea simple one-parameter ( ξ ) evolution, like∆ αα = − ξ ln (1 + z ) . (11)An alternative example of this class is provided bythe string-theory inspired runaway dilaton scenario [28],where the α evolution is also relatively simple. III. OBSERVATIONAL PROBES
In this section we characterize the different observableswe will use in our analysis.
A. Supernovae Type Ia data
Type Ia Supernovae are a particular class of Super-novae, providing bright, standardizable candles, and con-straining cosmic acceleration through the Hubble dia-gram. At present, they are the most effective and matureprobe of dark energy.Moreover, as the SN peak luminosity ( L peak ) dependson photon diffusion time, which in turn depends on α through the opacity, the α variation could affect L peak [29]. The key mechanism is the energy deposition rate inthe decay chain N i → Co → F e . This leads to∆ L peak L peak ∼ − .
94 ∆ αα (12)which corresponds to∆ αα ∼ .
98 ∆ M (13)where ∆ M = M − M with M the absolute magnitudeat peak, and the subscript “0” indicates we are not ac-counting for the α variation.Decreasing alpha decreases the opacity, allowing pho-tons to escape faster, thus increasing L peak . This can betrivially translated to a change in the distance modulus µ = m − M , with m the apparent magnitude as µ ( z ) = m − M = m − ( M + ∆ M ) = µ ( z ) − .
98 ∆ αα ( z )(14)where µ ( z ) = 5 log ( d L ( z )) + 25 is function of the lu-minosity distance d L ( z ) = 1 + zH (cid:90) z dzE ( z ) . (15)The E ( z ) = H ( z ) /H expression encodes the chosen darkenergy model.We build the SN datasets following the procedure pre-sented in [30]. We use Euclid specifications [31, 32] toforecast a SN survey at low-intermediate z , containing1700 supernovae uniformly distributed in the redshiftrange 0 . < z < . B. Quasar absorption systems data
The frequencies of narrow metal absorption lines inquasar absorption systems are sensitive to α [33], andthe different transitions have different sensitivities. Ob-servationally, one expects relative velocity shifts betweentransitions in a given absorber, in a single spectrum, if α does vary; this comparison can therefore be used toobtain measurements of α in these absorption systems.Indeed a survey able to observe quasar absorption linesat different redshifts is able to reconstruct the variationof α with respect to the present value and to provide adataset corresponding to the left side of Eq. (9).Currently, there is controversial evidence [11] for aspace-time variation of α at the level of a few parts permillion, roughly in the redhsift range 1 < z <
4. Part ofthe uncertainty in these results stems from the fact thatthe large samples of spectra being used have been gath-ered for other purposes and are therefore inhomogeneous,and may be vulnerable to systematic errors which are difficult to quantify. An ongoing dedicated VLT-UVESLarge Program is trying to clarify this issue [12, 13], butthe ultimate solution is to use high-resolution ultra-stablespectrographs, for which these measurements are a keyscience driver.For representative future datasets we use the baseline(conservative) case discussed in [14]. We consider theEuropean Extremely Large Telescope (E-ELT) equippedwith a high-resolution, ultra-stable spectrograph (ELT-HIRES), for which the COsmic Dynamics Experiment(CODEX) Phase A study [34] provides a baseline refer-ence. We assume uniformly distributed measurementsin the redshift range 0 . < z < .
0, with an error σ α = 10 − . C. Redshift-drift data
QSO observations can be also used to constrain DEmodels through the so called redshift-drift of thesesources [35, 36]. The redshift-drift is the change of theredshift due to the expansion of the universe betweentwo observations of the same distant source spectrum,repeated after a given amount of (terrestrial) years. Therequired time lapse depends on the instrument used (andspecifically on its calibration stability) but is typically ofthe order of a decade with next-generation facilities.With this kind of observations one can exploit distantastrophysical sources as a probe of the expansion of theuniverse in a model independent way [37–39]. As pointedout in [15, 40] QSO are the ideal astrophysical objects toobserve the redshift variation ∆ z between two observa-tions. This ∆ z can be translated to a spectroscopic ve-locity ∆ v = c ∆ z/ (1 + z ) and connected to cosmologicalquantities through the relation∆ vc = H ∆ t (cid:20) − E ( z )1 + z (cid:21) , (16)where c is the speed of light and ∆ t is the time inter-val between two observations of the same astrophysicalsource.A CODEX-like spectrograph will have the ability todetect the cosmological redshift-drift in the Lyman α ab-sorption lines of distant (2 < z <
5) QSOs, even thoughthis is a very small signal. The E-ELT can decisivelydetect the redshift variation with a 4000 hours of inte-gration in a period of ∆ t = 20 years [41]. These may becomplemented by measurements at other redshifts usingSKA [42, 43].According to Monte Carlo simulations of the CODEXPhase A study [34], the error on the measured spectro-scopic velocity shift ∆ v that can be expressed as: σ ∆ v = 1 .
35 2370
S/N (cid:115) N QSO (cid:18)
51 + z QSO (cid:19) x cm s − , (17)where S/N is the signal to noise ratio, N QSO the numberof observed quasars, z QSO their redshift and the exponent x is equal to 1 . z ≤
4, while it becomes 0 . S/N = 3000 and a number of QSO N QSO = 30 is as-sumed to be uniformly distributed among the followingredshift bins z QSO = [2 . , . , . , . , . D. Weak lensing data
Weak gravitational lensing of distant galaxies isa powerful observable to probe the geometry of theuniverse and to map the dark matter distribution. Wedescribe the distortion of the images of distant galaxiesthrough the tensor [44] ψ ij = (cid:18) − κ − γ − γ − γ − κ + γ (cid:19) where κ is the convergence field and γ = γ + iγ is thecomplex shear field. We can rewrite these quantities asa function of the projected Newtonian potentials ψ ,ij γ = 12 ( ψ , − ψ , ) + iψ , ,κ = 12 ( ψ , − ψ , )where the commas indicate the derivatives with re-spect to the directions transverse to the line of sightand the projected potentials are given by ψ ,ij = − (1 / (cid:82) g ( z )(Ψ ,ij + Φ ,ij ) dz , i.e. integrating the gravi-tational potentials with the lensing kernel g ( z ) = (cid:90) dz (cid:48) n ( z (cid:48) ) r ( z, z (cid:48) ) r (0 , z (cid:48) )with n ( z ) the galaxy redshift distribution and r the co-moving distance r ( z, z (cid:48) ) = (cid:90) z (cid:48) z dz (cid:48)(cid:48) E ( z (cid:48)(cid:48) ) . (18)We can define the convergence power spectra in a givenredshift bin in the following way P ij ( (cid:96) ) = H (cid:90) ∞ dzE ( z ) W i ( z ) W j ( z ) P NL [ P L (cid:18) H (cid:96)r ( z ) , z (cid:19) ](19)where P NL is the non-linear matter power spectrum atredshift z , obtained correcting the linear one P L . W ( z )is a weighting function W i ( z ) = 32 Ω m (1 + z ) (cid:90) z i +1 z i dz (cid:48) n i ( z (cid:48) ) r ( z, z (cid:48) ) r (0 , z (cid:48) ) (20)with subscripts i and j indicating the redshift bin. bin z bin z − .
496 6 1 . − . . − .
654 7 1 . − . . − .
784 8 1 . − . . − .
907 9 1 . − . . − .
031 10 1 . − . The observed power spectra are affected mainly by sys-tematic uncertainties arising from the intrinsic ellipticityof galaxies γ . These uncertainties can be reduced av-eraging over a large number of sources. The observedconvergence power spectra will be hence C ij = P ij + δ ij γ ˜ n − j (21)where ˜ n j is the number of sources per steradian in the j − th bin.In this paper we simulate a weak lensing dataset ac-cording to the specifications expected for the Euclid sur-vey [31]: the mission will observe n g (cid:39)
30 gal / arcmin over an area Ω = 15000 deg , corresponding to a skyfraction f sky ∼ (cid:96) -by- (cid:96) convergencepower spectrum and the 1 σ uncertainties, computed as[45, 46] σ (cid:96) = (cid:115) (cid:96) + 1) f sky (cid:18) P ( (cid:96) ) + γ rms n gal (cid:19) . (22) E. Atomic clocks bounds
In models where the same dynamical degree of freedomis responsible for both the dark energy and the variationof α , at redshift z = 0 the atomic clock bounds [47] willalways give a constraint on the combination of a funda-mental physics parameter (e.g. the coupling of the field,which is obtained by the Equivalence Principle violation)and a cosmological parameter (usually the dark energyequation of state w , although depending on the modelother parameters may be involved too). For the modelsin subsection II A, we have (cid:113) φ (1 + w ) H ζ = ( − . ± . × − yr − , (23)and there will be analogous relations for the other mod-els. In some cases it may be possible to set such a boundat non-zero redshifts too.For II B-like models Eq. (23) simplifies to4 H ξ = ( − . ± . × − yr − . (24) IV. ANALYSIS
The cosmological parameters that we sample can bedivided in “standard parameters” quantifying the con-tent of the universe and the power spectrum of primor-dial scalar perturbations, { Ω b h ,Ω c h ,Ω Λ , n s , A s } , pecu-liar DE parameters characterizing different parametriza-tions, { w , w a } for the CPL case and { w , Ω e } for EDE,and the coupling ζ ( ξ for the BSBM model).We build simulated datasets assuming a fiducial cos-mology given by the observations of the WMAP satelliteafter 9 years of data [48] for the standard parameters: thebaryon and cold dark matter densities, Ω b h and Ω c h ,the amount of energy density given by dark energy at thepresent time Ω Λ , the optical depth to reionization τ , thescalar spectral index n s and the overall normalization ofthe spectrum A s (see Table II). We fix the DE parame-ters in such a way to mimic the ΛCDM expansion (i.e. w = − , w a = 0 in the CPL case and w = − , Ω e = 0for EDE) and a vanishing coupling ζ = 0. In all themodels and analysis we require spatial flatness of the uni-verse. Basically, this fiducial set of parameters (Set1 inTable III) represents the standard ΛCDM cosmology asmeasured by WMAP-9. Ω b h Ω c h Ω Λ τ n s A s . . .
722 0 .
089 0 .
972 2 . · − TABLE II. Fiducial values for the six standard ΛCDM cos-mological parameters, corresponding to the marginalized bestfit values of the WMAP-9 years analysis.Fiducial w w a Ω e ζ ξ Set1 − − Set2 − .
95 0 − − × − − Set3 − . − . − × − − Set4 − − − − × − TABLE III. Fiducial values for the DE parameters and cou-plings used in the different analyses.
We also build simulated datasets with a non vanishingvariation of α assuming the same value of Table II forthe standard parameters, but different values for the onesinvolved in the α variation, listed in Table III. In order toproduce an evolving α , DE parameters must depart fromthe standard ΛCDM scenario, nevertheless we assumefiducial model values compatible with presently availableconstraints [3, 22, 23, 49, 50] . In particular, for the CPLcase we assume w = − . , w a = 0 and a coupling ζ = − × − (Set2). For the EDE case we choose adark energy described by w = − . , Ω e = 0 .
02 anda coupling ζ = − × − (Set3). We exploit these lasttwo datasets to constrain the DE parameters beyond thestandard ΛCDM model and in order to investigate thepossible bias on cosmological parameters introduced ifwe neglect the variation of α in the analysis.In the BSBM framework instead we only use one fidu-cial model (Set4) generating a non vanishing ∆ α/α witha coupling ξ = 5 × − , in order to inquire how the pos-sible presence of a scalar field not driving the acceleratedexpansion, but coupled with α , can bias the recoveredcosmological parameters. In this case the DE parame-ters are fixed to the ΛCDM values as we assume that thebackground expansion is not affected by this scalar field.We show in Fig. 1 the resulting time variation of α (top panel) and the corresponding EoS (bottom panel)for the non standard scenarios defined by Table III. −5 z ∆ α / α ( z ) CPL : Set2EDE : Set3BSBM : Set4 z w ( z ) CPL : Set2EDE : Set3BSBM : Set4
FIG. 1. Top panel: Evolution with redshift of ∆ α/α in theCPL (red solid line), EDE (blue dashed line) and BSBM(green dash-dotted line) parametrizations using the fiducialcosmology in Table III. Bottom panel: corresponding varia-tion in the DE equation of state.
In this work we rely on a MCMC technique to samplethe parameter space and we use a modified version of thepublicly available package cosmomc [51] with a conver-gence diagnostic using the Gelman and Rubin statistics.We assume flat priors on the sampled parameters.
V. RESULTS
In this section we present the most interesting resultswe obtained, discussing the impact of different observ-ables on the constraints. The complete set of constraints,resulting from using different combinations of probes, isreported in the Appendix A.
A. Vanishing ∆ α/α As stated in the previous section, the first investigationwe carry out deals with vanishing ∆ α/α mock datasets.We consider different combinations of the probes intro-duced in Section III and discuss the main features ob-tained by this analysis, exploring how the main geomet-rical probes (WL and SN) affect constraints on DE pa-rameters and on the coupling ζ .We first report the results for the CPL model. In Fig. 2we can notice how the Euclid survey will greatly narrowthe allowed parameter space for the EoS parameters w and w a , mainly thanks to the combination of the SN andWL measurements. When we consider all datasets weget σ ( w ) = 0 .
007 and σ ( w a ) = 0 . ζ . Thisresult is however easily explained considering the chosenfiducial cosmological model. Eq.(9) in fact implies thata vanishing ∆ α/α can be obtained in two ways: either ζ = 0 and/or w ( z ) = −
1. This leads to the fact thatwhen w and w a are poorly constrained (i.e. when WLand SN are removed from the analysis) the QSO fore-casted measurements require a coupling ζ close to zero.On the contrary when WL and SN impose tight inde-pendent constraints on DE parameters and the recov-ered w ( z ) is close to −
1, a larger range of ζ values is inagreement with the QSO measurements. We can inter-pret this result considering that, as our chosen fiducialcosmology is the standard ΛCDM universe, our probestightly constrain the Dark Energy to be close to a cos-mological constant, thus a non dynamical field (or onerolling down the potential extremely slowly), and there-fore a vanishing ∆ α/α is reproduced for every choice ofthe coupling. This effect is displayed in Fig. 3 wherewe report the recovered 1-dimensional posterior distri-butions for the coupling and the DE parameters. Thesolid red curves show the combination of all observableswith very tight constraints on DE parameters and thelarger distribution for ζ ; the dotted cyan curves are ob-tained removing SN, the constraints on w - w a are slightly −1.2 −1.1 −1 −0.9 −0.8−0.3−0.2−0.100.10.20.3 w w a allall−SNall−WLall−WL−SN FIG. 2. 2-dimensional contours at 68% and 95% confidencelevels for the w - w a parameters. The solid red contours showthe combination of all observables; dotted cyan curves de-scribe the degradation of the constraints when removing SN;blue dot-dashed contours broaden because of the exclusionof WL; the green dashed regions are obtained removing bothWL and SN measurements. Ω Λ −1.1 −1.0 −0.9 w w a−5 0 5 10x 10 −7 ζ allall−SNall−WLall−WL−SN FIG. 3. Marginalized 1-dimensional posterior distributionsfor the DE parameters w , w a , Ω Λ and the coupling ζ , fordifferent combinations of probes. broader and the coupling is slightly better constrained;the blue dot-dashed lines exclude WL: DE parametersare still measured by SN but the constraints are largelybroadened allowing for a tighter measurement of ζ ; thegreen dashed lines show the constraints on parameterswhen removing both WL and SN: in this case we get themost stringent constraint on the coupling because of theunmeasured w - w a parameters. In Fig. 4 we show the 2-dimensional contours at 68% and 95% confidence levels inthe ζ - w and ζ - w a planes only for the two extreme cases:the combination of all probes and the analysis excludingWL and SN. Again we can see that when DE parametersare constrained thanks to WL and SN, the coupling canlie in a larger region, while it is tightly constrained whenloose bounds on w - w a are obtained.In the EDE case the considered low redshift combi-nation of probes leads to very tight constraints on themodel parameters, narrowing the parameter space in a w ζ −1.2 −1.1 −1 −0.9 −0.8−6−4−20246 x 10 −7 allall−WL−SN −0.3 −0.2 −0.1 0 0.1 0.2 0.3−6−4−20246 x 10 −7 w a ζ allall−WL−SN FIG. 4. 2-dimensional contours at 68% and 95% confidencelevels showing ζ versus w / w a with (closed blue contours)and without (open red contours) the inclusion of WL and SNobservations. competitive way with respect to present high redshift re-sults on this kind of models (see [49],[50] for latest re-sults). We obtain w < − .
992 and Ω e < . α variation is driven by this kind of darkenergy parametrization: the more datasets we consider,the broader the constraints on the coupling are. −3 −2 −1 log(1+w ) Ω e allall−SNall−WLall−WL−SN FIG. 5. Same as Fig. 2 for the EDE parameters w -Ω e . Herewe plot log (1 + w ) to better show the w ∼ − Ω Λ −1 −0.95 −0.90 w Ω e−5 0 5 x 10 −7 ζ allall−SNall−WLall−WL−SN FIG. 6. Same as Fig. 3 for EDE parameters. w ζ −0.96 −0.95 −0.94−3.4−3.2−3−2.8 x 10 −5 w a ζ −0.01 −0.005 0 0.005 0.01−3.4−3.2−3−2.8 x 10 −5 w ζ −0.96 −0.95 −0.94−2.2−2.1−2−1.9 x 10 −5 Ω e ζ −5 FIG. 7.
Top panels : 2-dimensional contours at 68% and 95% confidence levels showing ζ versus w / w a for the CPL modelwhen a Set2 fiducial cosmology is assumed in the data building. Bottom panels : same as top panel showing ζ versus w /Ω e EDE model parameters. The black crosses show the chosen fiducial values.
B. Non-Vanishing ∆ α/α In a second step of our analysis we select fiducial mod-els (Set2, Set3, and Set4) where ∆ α/α is not vanishingand the DE parameters move from the standard ΛCDMscenario. We report constraints on DE parameters forboth the CPL and EDE parametrizations, as well as forthe coupling arising in a BSBM model.In this case, the peculiar w − ζ behaviour mentionedabove, due to the ζ = 0 fiducial value, is not presentand the degeneracies between these parameters show upclearly, as we report in Fig. 7 for both CPL and EDEmodels.We also notice that probing a different fiducial cos-mology will give different constraints on the parameters.For the CPL parametrization we recover the input fidu-cial values and we obtain σ ( w ) = 0 . σ ( w a ) = 0 . σ ( ζ ) = 1 . × − . The constraint on w improvesby a factor of about two and the measurement of w a becomes about one order of magnitude better: moving the fiducial region away from the special point ( ζ = 0, w = −
1) prevents the loss of constraining power becauseof the pathological degeneracies described in Fig. 4 andtherefore all the observables can fully contribute in con-straining the cosmological parameters. In particular, inthese non standard scenarios, the QSO contribution willbe non vanishing. Even though QSO data have a muchlower constraining power than other dark energy observ-ables, in Fig. 8 it is possible to notice how this datasetcan provide independent (and almost orthogonal) limitson dark energy parameters and can be used to break de-generacies between w and w a .The same behaviour is observed in the EDE analy-sis where we find σ ( w ) = 0 . σ (Ω e ) = 0 .
001 and σ ( ζ ) = 5 . × − at 68% c.l.; the EDE parameters willbe detected with high significance in this scenario.Set4 defines the non vanishing ∆ α/α fiducial modelused to forecast the coupling between the electromag-netic sector and the BSBM scalar field which, as ex-plained above, does not affect the background expansionof the universe. This implies that probes which do not0 w w a −1.00 −0.95 −0.90 −0.85 −0.80−0.10−0.0500.05 QSOall−QSOall FIG. 8. QSO contribution to the w - w a constraints. Wereport contour plots at 68% and 95 % confidence levels asobtained from QSO data only (dashed green line), all probesexcept QSO (dash-dotted red line) and all probes (solid purpleline). The black cross shows the fiducial input values. directly depend on α will constrain cosmological param-eters but will not be sensitive to the coupling ξ in anycase, given that Eq.(11) relies only on ξ as free param-eter. Therefore in this analysis ξ is constrained only byQSO and SN data, the latter through the shift a variationof α produces on the distance modulus. We constrain σ ( ξ ) = 2 . × − (see Fig. 9). −8 ξ all FIG. 9. Marginalized 1-dimensional posterior distribution forthe coupling parameter ξ between the BSBM scalar field and α . This result refers to the combination of all the considereddatasets. As a last investigation we analyse the non vanishing α data fixing the coupling parameter to zero in the cos-mological parameter estimation. This assumption willforce the analysis to fit datasets where ∆ α/α is redshiftdependent with theoretical spectra unable to reproducethis trend. Should this translate into a bias in the recov-ered cosmological parameters we will be able to quantifythe impact of a wrong assumption on ζ on cosmologicalresults.Among the observables we considered in this work,only QSO and SN are directly affected by the α evolu-tion, and in particular only SN can produce a shift in theestimated value of the cosmological parameters. ζ = 0will in fact always produce a vanishing ∆ α/α in Eq.(9).Thus, whatever value the cosmological parameters as-sume, the whole parameter sets will not give a good fitto the QSO dataset which directly probe the quantity∆ α/α . On the contrary, SN datasets generated with∆ α/α (cid:54) = 0 are shifted with respect to the Set1 dataset(see Eq.(14)), and require a shift in the cosmology affect-ing µ ( z ) to compensate this artefact. We better showthis effect in Fig.10 where we plot the relative differenceof the distance modulus µ ( z ) for different coupling val-ues with respect to the case ζ = 0 for the CPL model.We see that the greater is the departure from ζ = 0, thegreater the shift in µ ( z ) will be. −7 −6 −5 −4 −3 redshift µ ( z , ζ ) / µ ( z , ζ = ) − ζ =−3x10 −5 ζ =−3x10 −4 ζ =−3x10 −3 ζ =−3x10 −2 ζ =−3x10 −1 FIG. 10. Distance modulus µ ( z ) produced in CPL cosmologyfor different values of ζ , compared to the fiducial case with ζ = 0. We see that the greater is the departure from ζ = 0,the greater the shift in µ ( z ) will be. Nevertheless we find that, assuming Set2, Set3 andSet4 fiducial values, this bias is too small to be observedwith the considered SN survey in both types of mod-els. We do not find any significant shift in the cosmo-logical parameters induced by wrong assumptions on thecoupling, suggesting that a more sensitive and deep SNsurvey will be needed to detect this effect. Indeed theE-ELT (plus JWST [52]) is expected to find SN up to1 z ∼ z where the shift in µ ( z ) is slightly increasing. A greater value of ζ mighthave an effect as well however, as stated previously, werestrict our analyses to a parameter region in agreementwith current observations, i. e. | ζ (cid:46) − | . VI. CONCLUSIONS
In this paper we focused on the possible coupling be-tween a scalar field driven dark energy, parametrizedhere with the CPL and EDE formalisms, and electromag-netism, which can in principle bring to a time evolutionof the fine structure constant α . We have shown how thetwo sectors are connected by a coupling ζ and we inves-tigated the ability of future low-medium redshift surveysto constrain this coupling. In particular, we consideredtwo different scenarios, a standard ΛCDM one (without α variations) and dynamical dark energy where a ζ (cid:54) = 0produces a redshift evolution for the fine structure con-stant. We forecasted observables for these two fiducialcosmologies from several upcoming surveys and we anal-ysed these simulated datasets using MCMC techniques.In the vanishing ∆ α/α case we obtained constraints onthe sampled parameters, showing how, as expected, darkenergy parameters will greatly benefit from weak lensingand supernova data coming from the Euclid satellite: wefind ( σ ( w ) = 0 . σ ( w a ) = 0 .
03) at 68% c.l. and( w < − . e < . ζ : thechosen fiducial cosmology in fact implies that the betterdark energy parameters are constrained, the larger therange of allowed values for ζ is. When all observables areconsidered we get σ ( ζ CPL ) = 1 . × − and σ ( ζ EDE ) =1 . × − at 68% c.l..This trend disappears when the second fiducial modelis considered, as we move away from the peculiar point[ ζ, w ( z )] = [0 , −
1] of the parameter space. In the non-ΛCDM fiducial cosmology, we have shown the constrain-ing power of the considered observables on the sampledparameters, as well as the degeneracies between darkenergy parameters and ζ both for the CPL and EDEmodels, highlighting how these degeneracies affect con-straints. In particular we showed for the CPL model howthe contribution from QSOs, combined with orthogonalconstraints from Euclid observables, will improve the es-timate by a factor of 2 for w and by one order of mag-nitude for w a , finding ( σ ( w ) = 0 . σ ( w a ) = 0 . σ ( w ) = 0 . σ (Ω e ) = 0 . σ ( ζ CPL ) = 1 . × − and σ ( ζ EDE ) = 5 . × − at 68% c.l..Furthermore, we analysed this last fiducial cosmologykeeping ζ fixed to a value different from the one in in-put in order to find out if wrong assumptions on thecosmological model could produce an observable bias on parameters. We discovered this is not the case as only SNcan highlight this shift and the survey considered here isnot sensitive enough to show this small effect. A futurepaper may investigate which are the specifications (suchas the number of SN and redshift range) needed by afuture survey to detect this bias.Finally, we also considered a BSBM model, where thescalar field coupled to electromagnetism is not the onedriving the accelerated expansion of the universe. Weanalysed this model using datasets forecasted with afiducial cosmology producing a non vanishing ∆ α/α andobtained constraints on the coupling of this model withelectromagnetism, obtaining σ ( ξ ) = 2 . × − . Alsoin this case we investigated the possible existence of abias due to wrong cosmological assumptions, finding thesame results obtained for the CPL and EDE models. AKNOWLEDGMENTS
We would like to thank Silvia Galli for useful discus-sions. We acknowledge useful comments and suggestionsfrom Isobel Hook and other members of the Euclid Cos-mology Theory SWG. EC acknowledges funding fromERC grant 259505. MM acknowledges partial supportfrom the PD51 INFN grant. CJM and PEV acknowl-edge funding from the project PTDC/FIS/111725/2009(FCT, Portugal). CJM is also supported by an FCT Re-search Professorship, contract reference IF/00064/2012,funded by FCT/MCTES (Portugal) and POPH/FSE(EC). The Dark Cosmology Centre is funded by the Dan-ish National Research Foundation. SS acknowledges thesupport of ASI contract n. I/023/12/0. VFC is fundedby Italian Space Agency (ASI) through contract Euclid-IC (I/031/10/0) and acknowledges financial contributionfrom the agreement ASI/INAF/I/023/12/0.
Appendix A: Recovered Parameters
While in Section V we focused only on the key resultsof our analyses, in this appendix we list in Section A 1the constraints on all the parameters as determined fromdifferent combinations of probes, and in Section A 2 wereport the 1-dimensional posteriors and the constraintsfor the sampled parameters when a non-standard fiducialmodel is assumed.
1. Vanishing ∆ α/α For vanishing ∆ α/α datasets we performed severalanalyses, excluding each time one of the observables pre-sented in Section III. In this way we could explore andhighlight the contribution of each observable to the con-straints. In Table IV we report the 68% confidence levelerrors on relevant cosmological parameters and in Fig. 112
CPLall all-WL all-SN all-RD all-QSOCL σ (Ω b h ) 5 . × − < . < .
025 5 . × − . × − σ (Ω c h ) 6 . × − . × − . × − . × − . × − σ ( H ) 1 . × − . × − . . × − . × − σ (Ω Λ ) 6 . × − . × − . × − . × − . × − σ ( w ) 6 . × − . × − . × − . × − . × − σ ( w a ) 2 . × − . × − . × − . × − . × − σ ( ζ ) 1 . × − . × − . × − . × − . < × − EDEall all-WL all-SN all-RD all-QSOCL σ (Ω b h ) 5 . × − < . < .
025 5 . × − . × − σ (Ω c h ) 6 . × − < . < .
12 6 . × − . × − σ ( H ) 1 . × − . × − . . × − . × − σ (Ω Λ ) 3 . × − . × − . × − . × − . × − σ ( w ) < − . < − . < − . < − . < − . σ (Ω e ) < . × − < . × − < . × − < . × − < . × − σ ( ζ ) 1 . × − . × − . × − . × − < . × − TABLE IV. 68% c.l. constraints on relevant cosmological parameters when the DE equation of state is parametrized through theCPL (top) or EDE (bottom) formalism for different combinations of probes: all includes all datasets described in Section III, all-WL excludes weak lensing data, all-SN excludes supernovae data, all-RD excludes redshift drift, and all-QSOCL excludesquasars and atomic clocks bounds. we show the 1-dimensional posteriors recovered for bothCPL and EDE models. We can notice how removingQSO and atomic clocks from the analysis we lose, triv-ially, all the constraining power for the coupling ζ , whileas expected removing WL and/or SN opens the DE pa-rameters.
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