Constraining spatial variations of the fine-structure constant in symmetron models

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Physics Letters B

www.elsevier.com/locate/physletb

Constraining spatial variations of the fine-structure constant in symmetron models

A.M.M. Pinho

^a,b,c

, M. Martinelli

^d

, C.J.A.P. Martins

^a,e,∗

aCentrodeAstrofísica,UniversidadedoPorto,RuadasEstrelas,4150-762Porto,Portugal bFaculdadedeCiências,UniversidadedoPorto,RuadoCampoAlegre687,4169-007Porto,Portugal

cInstitutfürTheoretischePhysik,Ruprecht-Karls-UniversitätHeidelberg,Philosophenweg16,69120Heidelberg,Germany dInstituteLorentz,LeidenUniversity,POBox9506,Leiden2300RA,TheNetherlands

eInstitutodeAstrofísicaeCiênciasdoEspaço,CAUP,RuadasEstrelas,4150-762Porto,Portugal

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received13February2017

Receivedinrevisedform11April2017 Accepted12April2017

Availableonline14April2017 Editor: M.Trodden

Keywords:

Cosmology

Fundamentalcouplings Fine-structureconstant Astrophysicalobservations

Weintroduceamethodologytotestmodelswithspatialvariationsofthefine-structureconstantα,based onthe calculation ofthe angularpower spectrum ofthesemeasurements. Thismethodologyenables comparisonsofobservationsandtheoreticalmodelsthroughtheirpredictionsonthestatisticsoftheα

variation.Hereweapplyittothecaseofsymmetronmodels.Wefindnoindicationsofdeviationsfrom the standard behavior, with currentdata providingan upper limitto thestrength ofthe symmetron couplingtogravity(logβ²<−⁰.9)whenthisistheonlyfreeparameter,andnot abletoconstrainthe modelwhenalsothesymmetrybreakingscalefactoraS S Bisfreetovary.

©^{2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Astrophysicaltestsofthestabilityofdimensionlessfundamen- talcouplingssuchasthefine-structureconstant

α

areapowerful probeof cosmology aswell as offundamental physics [1,2]. The analysisofa datasetof293archival datameasurementsfromthe Keck and VLT telescopes by Webb et al. provided an indication of spatial variations with an amplitude of a few parts per mil- lion,withastatisticalsignificanceof4−

σ

[3].Eventhoughthere are concerns aboutpossible systematiceffects inthis dataset [4]

andthe statistical significance itself decreases when this dataset is analyzed jointly withmore recent data [5], it is important to considerthe theoreticalimplications of such results,alsobearing inmindthatforthcomingastrophysical facilitieswillenable much moreprecisetestsinthenearfuture.

Ataphenomenologicallevelitiscommontofittheastrophys- icalmeasurementswithasimpledipole,withorwithoutanaddi- tionaldependenceonredshiftorlook-backtime[3,5].Ontheother hand, from a theoretical point of view simplistic dipole models wouldrequirestrong fine-tuning toexplain such abehavior, and

*

Correspondingauthorat:CentrodeAstrofísica,UniversidadedoPorto,Ruadas Estrelas,4150-762Porto,Portugal.

E-mailaddresses:am.pinho@thphys.uni-heidelberg.de(A.M.M. Pinho), martinelli@lorentz.leidenuniv.nl(M. Martinelli),Carlos.Martins@astro.up.pt (C.J.A.P. Martins).

aphysicallymotivatedapproachwouldrelyonenvironmentalde- pendencies[6].Thisthereforecallsformorerobustmethodologies whichenableaccuratecomparisonsbetweenmodelsandobserva- tions.EarlyworkalongtheselineswasdonebyMurphyet al.,who calculatedthetwo-pointcorrelationfunctionoftheKecksubsam- pleoftheaforementionedarchivaldata,findingittobeconsistent withzero[7].Inthispaperwe movefromthetwopoint angular correlationfunction tothecalculationoftheangularpowerspec- trumofthesemeasurements.Theaimofadoptingthisapproachis tobe abletocompress thedatainformationinsucha wayto al- lowforcomparisonwiththepredictionsoftheoreticalmodels.As aproofofconcept,inthispaperweapplythismethodtothecase ofthesymmetronmodel,forwhichtheenvironmentaldependence of

α

hasbeenpreviouslystudiedusingN-bodysimulations[8].

InSection 2we presentaconciseoverviewofthesymmetron model.Section3presentsthe methodologyusedtocompress the

α

measurements intoangularpower spectra.Insection 4wecal- culate the theoretical power spectrum for thesymmetron model andpresentouranalysis methodology,leading to theresults dis- cussedinSection5.Finally,inSection6wesummarizeourresults andtheoutlookforthismethodology.

2. Symmetronmodel

The symmetron modelis a scalar-tensormodification ofgrav- ity,introducedinordertoachieve anadditionallongrangescalar http://dx.doi.org/10.1016/j.physletb.2017.04.027

0370-2693/©^{2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

force while still satisfyinglocal gravity constraints thanks to the environment density dependence of its coupling to matter. This modificationofgravityisdescribedbytheaction[9]

S

=



dx⁴

√

−

^g



_R 2M²_pl

−

¹

2

(∂φ)

²

−

^V

(φ)

 +

^Sm

(

m

;

^gμνA²

(φ))

(1) where g=^det(gμν),M_pl=¹/√

8

π

G and S_m isthematter-action.

The conformal coupling between the scalar field and the matter fieldsmexpressedby ˜gμν=gμν A²(φ),isassumedtobethesim- plestoneconsistentwiththepotentialsymmetry,

A

(φ) =

¹

+

¹ 2

 φ

M



2

,

(2)

withM and

μ

arbitrarymassscales.Thiscouplingleadstoafifth force,whichinthenon-relativisticlimitisgivenby

−

→F_φ

≡

^{d A}

(φ)

d

φ

−

→

∇φ = φ

^→−

∇φ

M²

.

(3)

Thepotentialischosentobeofthesymmetrybreakingform

V

(φ) = −

¹

2

μ

²

φ

²

+

¹

4

λφ

⁴

.

(4)

The dynamicsof the scalarfield φ is determinedby an effective potentialwhichinthenon-relativisticlimit(relevantfortheastro- physicalmeasurements)hastheform

V_{e f f}

(φ) =

^V

(φ) +

^A

(φ) ρ

m

=

¹ 2

 ρ

m

μ

²M²

−

¹



μ

²

φ

²

+

¹

4

λφ

⁴

;

⁽⁵⁾

thismeansthatintheearlyUniverseor,ingeneral,whenthemat- terdensityis high, the effective potential hasa minimum φ=⁰ where thefield will reside. As the Universe expands, the matter densitydilutes untilitreachesa criticaldensity

ρ

S S B=

μ

²M² for whichthesymmetrybreaksandthefieldmovestooneofthetwo newminimaφ= ±φ0= ±

μ

/√

λ.

The fifth-forcebetweentwo testparticles residing in aregion ofspacewherethefieldhasthevalueφ= φlocal canbecalculated tobe[9]

F_φ

F_gravity

=

²

β

²

 φ

_local

φ

0



2

∼

²

β

²



1

− ρ

μ

²M²



,

(6)

for separations of the Compton wavelength λlocal = ¹/ V_{e f f,φφ}(φlocal), where the coupling strength to gravity is given by

β = φ

0M_pl

M² (7)

For larger separations or in the cosmological background before symmetry breaking, φlocal≈^{0 and the force is} suppressed. After symmetry breaking, the field moves towards φ= ±φ0 and the force is comparable to gravity for β=O(1). Non-linear effects inthe field-equationensure that theforce is effectivelyscreened inhigh densityregions. The symmetry breaks atthe scale factor aS S B= (

ρ

m,0/

ρ

S S B)andtherangeofthefifth-forcewhenthesym- metry is broken isgiven by λφ0=¹/(√

2

μ

), where localgravity constrains satisfy λ_φ01 Mpc/h for symmetry breaking closeto today,i.e.aS S B≈^1[10].

Sincethesymmetronscalarfieldisadynamicaldegreeoffree- dom, one naturally expects it to couple to the other degrees of freedom inthe Lagrangian,unless a newsymmetry ispostulated to suppressthesecouplings. Inparticular, we can assume that it coupleswiththeelectromagneticsectorofthetheory[8]

Fig. 1. TheoreticalpowerspectrumPδα(k,a)givenbyeq.(10)asafunctionofthe wavenumberk fora=1,β=1,λφ0=1 Mpc/handdifferentsymmetrybreaking scalefactorsaS S B= [⁰.33,0.5,0.66]^{.Notethatstrictlyspeakingeq.(10)onlyap-} pliesinthelinearregime,sothebehavior beyondthisshouldbetakenwithcare.

Following[8]anormalizationfactorx=⁰.06(0.5/aS S B)wasused.(Forinterpreta- tionofthereferencestocolorinthisfigurelegend,thereaderisreferredtothe webversionofthisarticle.)

S_{E M}

= −



dx⁴

√

g B_F

(φ)

¹

4F_μν²

,

(8)

where B_F is the gauge kinetic function which leads to

α

=

α

0B⁻_F¹(φ). Withthesame choiceof quadraticcoupling B⁻_F¹(φ)= 1+¹₂β_γ²

φ M

2

onegetsthefollowingvariationofthefinestructure constant

δ

α

≡  α α ⁼

α (φ) − α

0

α

0

=

^B−_F¹

(φ) −

¹

=

¹ 2

β

γ

φ

M



2

.

(9)

Consideringperturbations ofthe scalarfield inFourierspace, the power spectrum for variations of

α

in the linear regime can be connectedtothematterpowerspectrum Pm(k,a)asfollows[8]

Pδα

(

k

,

a

) =

3



mH²₀

β

_γ²

β

² a

(

k²

+

^a2m2_φ

) 

φ φ

0



2

2

Pm

(

k

,

a

),

(10)

wheremand H₀ arethepresent-daymatterdensityandHubble parameter, β_{γ is}thescalar-photon couplingrelative tothescalar- mattercoupling,k istheco-movingwavenumber,m²_φ=^Ve f f,φφ( ¯φ) is thescalarmass inthecosmologicalbackground,and(φ/φ0)is thebackgroundscalarfieldvalue.Fora≥^aS S B wecanwrite

 φ (

a

)

φ

0



2

= 

1

− 

_{aS S B} a



3

 ,

m²_φ

(

a

) =

¹

λ

²_φ0

=



1

− 

_a

S S B

a



3



.

(11)

Pδ_α isplottedinFig. 1;itisalsousefultowriteitas

P_δα

(

k

,

a

) =

0

.

33



m10⁻⁶

β

_γ²

β

² a

((

k

/

m_φ

)

²

+

^a2

)

 λ

_φ0

Mpc

/

h



2

2

Pm

(

k

,

a

).

(12)

3. Observationaldata

Currentlyavailableastrophysicalmeasurementsof

α

comefrom high-resolution spectroscopy of absorption clouds along the line

of sight of bright quasars. In addition to the 293 archival mea- surementsofWebbet al.[3]there are 20morerecentdedicated measurements discussedin[11],making a totalof 313measure- ments. From now on we refer to the formeras Webb and asAll to the combination of these with the latter. For each of them, apartfrom themeasurement of the relative variationof

α

itself, the sky coordinates and redshift of the absorber (spanning the range 0.2<z<4.2) are known with negligible uncertainty. We cancompress thisinformationinan angular powerspectrum C_, tobecomparedwithstatisticalpredictionscomingfromtheoreti- calmodels.

Inordertodothis,weobtainthetwo-pointcorrelationfunction c(ϑ)fromtheδα(θ,φ)measurements[7]

c

(ϑ) =

¹

¯

n²f_sky

< δ

_α

(θ, φ)δ

_α

(θ

, φ

) >,

(13) where the brackets < .> correspond to the average taken over all possible orthodromic separations ϑ. Since the measurements of δ_{α are} sparse on the sky (effectively point sources), the dis- creteness of the data has been taken into account following the procedure of[12]: 4

π

f_sky steradians is the assumed coverage of the sky ofthe dataset andn¯=^N/(4

π

f_sky) is the corresponding meannumberdensityover theobserved partofthe sky(withN asthe numberof sources).The data we considerhere are effec- tively sparse point sources spread over the whole sky, therefore wetake f_sky=^{1.Thisassumptionprovidesa}conservativeestimate ofthemeasurements densityn and¯ thereforeofthepower spec- trumestimation,despitealsoaffectingcosmicvariance,decreasing itsimpact ontheestimatornoise. Futuremorecompletedatasets willallowtodealproperlywiththeseaspects,exploitingalsotech- niquescommonlyusedforothercosmologicalobservables,suchas CMBorgalaxysurveys,andthereforetoobtainamoreprecisees- timateoftheerrorcontributions.

Inthisworkwearealsoneglectingtheredshiftinformationof the δα measurements; in practice we are assuming that δ_{α has} noredshift dependenceand all thedeviations fromthe standard value are brought by spatial variations. This approach is accept- able withthe current state of the data, butit will be crucial to includeredshiftinformationwhenthedatawillreachasensitivity allowingforatomographicreconstructionofδ_{α .}Moreover,includ- ing the possibility of δ_α(z) will be necessary to test theoretical modelswhichalsopredict atime evolutionforthe finestructure constant.

WecaninprincipleperformaLegendretransformoftheangu- larcorrelationin orderto obtainthe angularpowerspectrum C as[13]

C_

=



c

(ϑ)

P_

(

cos

ϑ)

d

,

(14)

whereP(cosϑ)istheLegendrepolynomialandthesolidangle.

Inpractice,wecomputethepowerspectrumestimatorCˆ_as C

ˆ

_

=

²

π ^

ϑ

c

(ϑ)

P_

(

cos

ϑ)

sin

ϑϑ

(15)

withϑ beingthe differencebetween consecutivevaluesof the angularseparation ϑ. Theexpected errorof thepower spectrum estimatorcanbeobtainedfrom[12]



²

=

²

(

2l

+

¹

)

f_sky

 σ

_f²

¯

n

+ ˆ

^C



2

(16)

whichincludesbothcontributionsoftheshotnoiseS N andcos- micvarianceC V thatcanbeexpressedas



S N

=



2

(

2l

+

¹

)

f_sky

σ

²_f

¯

n

, 

C V

=



2

(

2l

+

¹

)

f_skyC

ˆ

_

.

(17)

σ

f isobtainedfromthemeasurements’errors¹

σ

jweightingeach measurementwithafactor w²_i givenby

w²_i

=

^N

σ

_i⁻²



j

σ

_j^{−2 (18)}

whichyieldstheaforementionedquantity

σ

f as

σ

²_f

= 

^N

j

σ

⁻_j²

^.

⁽¹⁹⁾

Often there are several measurements of δ_{α at} differentred- shiftsalongthesamelineofsightasthelightfromthequasarcan gothroughmorethanoneabsorptionclouduntilitreachesEarth.

Toavoid nullangularseparations inthe computation ofEq.(13), we chooseto usetheweighted meanmeasurement formeasure- mentsinthesamelineofsight.Ourfulldatasetincludesmeasure- mentsfrom156independentlinesofsight.Beforecomputingthe correlation function,thedatasetisanalyzedandreplacedby new valuesofweightedredshift,zw,weightedδ_α,w anditscorrespond- ingweightederror,

σ

w describedby

zw

=



iwi

×

zi



iwi

,  α

α

 

w

=



iw_i

×

^_α^α_i



iwi

,

σ

_w²

₌ 

¹

iwi

(20)

where w istheweight givenby wi=¹/

σ

_i² andtheindex i runs overthemeasurementsoneachlineofsight.

Fig. 2shows theangular power spectra Cˆ_ obtainedwith the procedure described above, considering the Webb dataset (left panel) andtheAll combination (right panel).We notice how the inclusion of the new data, although limited in the number of sources, leads to an improvement of the measured power spec- trum,thankstotheincreasedsensitivity.

Fig. 3showsinsteadtheglobalerroronthemeasurementsand the contributions coming from shot noise and cosmic variance, where we can notice how the former dominates over the latter evenatthelargescalesconsidered.

4. Theoreticalpredictionsanddataanalysis

Wecannowcomparetheobservationalpowerspectrawiththe predictionsmadebythesymmetronmodel.Thisentailsexpressing Eq.(10)intheformofanangularpowerspectrum.Genericallythis can bewritten as2D projection ofthe 3Ddensityfield which in thiscaseisthelinearpowerspectrum P(k,z).

InthispaperweexploittheLimberapproximation,whichsim- plifiesthecalculationsbyavoidingintegrationsofBesselfunctions.

Wewarn thereaderthatthisapproximationcan significantlyim- pact the calculation of the power spectra [13], especially atthe angular scales considered here; however, since the sensitivity of the currentlyavailable data is farfrom allowing precision recon- structionsof P_δα,theuseofmorerefinedmethods isoutsidethe aim of this paper, but this should be taken into account when more precise measurements willbe available.In this approxima- tion,wecancomputetheangularpowerspectrumas[13]

1 If bothsystematicandstatisticalerrorsareknown,weusethecombinederror obtainedbyaddingtheminquadrature.

Fig. 2. Angular power spectrum estimationCˆas a function of the multipolewith its expected errorfor the Webb dataset, and for the All combination.

Fig. 3. ContributionstotheerrorcomingfromshotnoiseS N(bluelines)andcosmicvarianceC V (redlines).TheleftpanelreferstotheWebb datasetwhiletheright oneshowsthecaseoftheAll combination.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig. 4. Leftpanel:Sourcedistributionfunctioninredshiftspaceforthearchivaldatasetof[3].Rightpanel:TheoreticalpowerspectrumCforthesymmetronmodelfor differentvaluesofthescalefactorforthesymmetrybreakingaS S B.

C

≈



dzW²

(

z

)

^H

(

z

)

d²_A

(

z

)

^Pδα



k

=

^l

+

¹

/

2 r

;

z



(21)

whereW(z)isthenormalizedsourcedistributionfunctioninred- shift space, H(z) is the Hubble parameter function, d_A(z) is the angular diameter function and P(k,z) is the linear power spec- trumpreviouslyobtainedinEq.(10),withk=^l+1_r^{/2,andr isthe} comoving distance. We reconstruct the source distribution func- tionwitha 20bins histogramfromeach datasetconsidered. The exampleofthearchivaldatasetsourcedistributioncanbefoundin Fig. 4.

In order to comparethese theoretical spectrawith theobser- vational data, we use the publicly available code COSMOMC [14], modified insuch awaytocompute fromasetofparameters the theoretical power spectrum of Eq. (21), where P_δα is given by Eq.(11).

We consider here as free parameters the scale factor when thesymmetrybreaks (a_{S S B})andtheproductofthecouplingβγβ (from now on simply named β forsimplicity), while we fix the range of the fifth force when the symmetry is broken to λφ0= 1 Mpc/h: thischoicesaturates thelocalgravityconstraintof[10]

and was also used in the N-body simulations of [8]. In princi-

Fig. 5. Toppanel: 1 and2−σ confidenceregionsintheaS S B−^logβ²planeobtainedusingtheWebb (bluecontours)andtheAll (redcontours)datasets.Bottompanels:

posteriordistributionsforaS S B (leftpanel)andlogβ²(rightpanel),wherebluesolidlinesrefertotheresultsobtainedusingtheWebb datasetandreddashedones referto theAll combination.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

ple,alsothestandardcosmologicalparametersshouldbeincluded inthe analysis as they affect the calculation of the power spec- tra through Eqs. (10), (21). However, current

α

datasets would not be able to simultaneously constrain the standard and sym- metronparameters, thereforeadditionalobservablessuch asCMB orsupernovaeshouldbeincluded,providedtheimpactofaspatial variationof

α

isincludedalsointheanalysisofthese. Wedecide toleavetheseconsiderationsforafuture,moredetailed,paperon thetopic andrather take for the standard cosmologicalparame- tersthe marginalizedvalue from Planck2015[15],focusing only onconstraintsofthesymmetronparameters.

5. Results

Inthis section we highlight theresults obtainedapplying the analysisdescribedabove.Inourfirstanalysis,weallowbotha_{S S B} andlogβ² to varyassuming flatprior distributions,² while fixing λφ0=^{1 Mpc/h.Fig. 5showstheposterior}distributionforeachof thetwofreeparameterusingdifferentdatasets.

Overallwe findthat theWebb datasetis notable toconstrain thetwoparameterssimultaneouslyandnodeviationsfromavan-

2 We samplelogβ² insteadofβ² inordertobettersamplethelowcoupling limit.

ishing

α

variation are found. As the recent dedicated measure- ments areconsistentwithanon-varying

α

,whencombinedwith the largerarchival dataset,they leadagainto an agreementwith a vanishing β²,but they are still not able to putbounds on the parameters.

Wealsoperformouranalysiswithonlyonefreeparameter,fix- inglogβ²=^{1 and}λφ0=^{1 Mpc/h,againonthegroundsthatthey} wereusedintheN-bodysimulationsof[8],wefindthattheWebb datasetisnotabletoconstrainaS S B,whilefortheAll combination we finda 2−

σ

lower limitaS S B>0.43, asdisplayed inthetop panelofFig. 6.

Onthe otherhand,considering logβ² astheonly freeparam- eterwhilefixingλφ0=1 Mpc/h anddifferentvaluesoftheepoch ofsymmetrybreaking(specifically aS S B=⁰.33,0.5 and0.66), the resultsshowninFig. 6andTable 1show that,asexpected, ifthe symmetrybreaksmorerecentlyalargercouplingvalueisallowed.

Again therecentmeasurementsimprovethe constraintsfromthe archivalmeasurements and,alsointhiscase,we findthat there- sultsareconsistentwithavanishingβ.

6. Conclusionsandoutlook

Inthiswork wehaveintroduced anewmethodology toaccu- ratelytestmodelswithspatialand/orenvironmental(localdensity

Fig. 6. Toppanel: posteriordistributionfortheaS S B parameter,usingtheWebb dataset(bluesolidlines)andtheAll combination(reddashedlines).Herewehavefixed log(β²)=^{1 and}λφ0=^1.Bottompanels: posteriordistributionforthelogβ²parameterwithdifferentvaluesofaS S B.OntheleftpanelweusetheWebb dataset,andonthe rightpaneltheAll combination.Inallbothcaseswekeepλφ0=1 fixed.

Table 1

Two-σ constraints on the symmetron parameter logβ² givenbytheWebb datasetandtheAll combination,fordif- ferentfixedvaluesofaS S B;λφ0=^{1 Mpc/hwasalsoused} throughout.

aS S B=0.33 aS S B=0.50 aS S B=0.66 Webb <−0.5 <0.2 <1.2 All <−0.9 <−0.2 <0.7

dependent)variations of thefine-structure constant

α

. Theseare basedon thecalculation oftheangular powerspectrum ofthese measurements,whicharestandardinothercosmologicalcontexts.

Forconcreteness we have alsoapplied thesetools to thecase of

α

variations in symmetron models. We findthat currently avail- abledataarenotabletoconstrainthesymmetronparametersaS S B andlogβ² whenthey are both considered as free parameters. If instead the only free parameter is the strength of the coupling to gravity β², we find that thedata do not show anydeviations fromthestandardbehaviorandratherprovideanupperlimit for thiscoupling,whichislogβ²<−⁰.9 inthemostconstrainingcase consideredhere.

Ourresultshighlightthefactthatarelativelysmallnumberof stringentmeasurements—therecentdedicated measurementsdis-

cussed in [11]—lead to stronger constraints when they are com- bined with the larger dataset of earlier measurements. The cur- rentbestconstraintsontheparameterβ comefrompulsartiming constraints on Brans–Dicke type scalar tensor theories (of which symmetronsareanexample),whichcorrespondtoβ^{10−2 [10].}

While our constraint is weaker, it comes from

α

measurements alone. Combining this withother cosmological datasets will lead to morestringentconstraints;we leavethisextendedanalysisfor subsequentwork.

Ourresultsshould beseenasaproof ofconcept,inthesense that theyarelimitedbytheuncertaintiesoftheavailable

α

mea- surements. Future high-resolution ultra-stable spectrographs, in particularESPRESSO[16](dueforcommissioningatthecombined CoundéfocusoftheVLTin2017)andELT-HIRES[17](fortheEu- ropeanExtremelyLargeTelescope,whosefirstlightisforeseenfor 2024), both ofwhich have thesemeasurements asa key science and design driver, will lead to significantly more sensitive mea- surements,both interms ofstatisticaluncertainties andinterms of control over possible systematics.As discussed above, the use ofmoresensitivedataandcompletesurveyswillrequireafurther step in the analysis, revisingthe assumptions made herefor the calculation ofboth thetheoretical andobservational powerspec- tra. On the other hand,these will enable more detailed studies,

including a tomographic analysis (dividing the data into several differentredshiftbins)and,shouldvariationsbeconfirmed,model selectionstudiescomparingvariouspossibletheoreticalparadigms.

Adiscussionofthesepossibilitiesisleftforsubsequentwork.

Acknowledgements

We are grateful to Luca Amendola, Micol Bolzonella and Adi Nusserfor several discussions anduseful suggestions during the variousstagesofthiswork.

ThisworkwasdoneinthecontextofprojectPTDC/FIS/111725/

2009 (FCT, Portugal). CJM is also supported by an FCT Re- searchProfessorship, contractreferenceIF/00064/2012,funded by FCT/MCTES (Portugal) and POPH/FSE. MM is supported by the Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organization for Scientific Research/Ministry of Sci- enceandEducation (NWO/OCW). MM andCJM acknowledge ad- ditionalsupportfromtheCOSTAction CA1511(CANTATA),funded byCOST(EuropeanCooperationinScienceandTechnology).

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