Detectability of torsion gravity via galaxy clusteringand cosmic shear measurements

Stefano Camera,1,* Vincenzo F. Cardone,2 and Ninfa Radicella31CENTRA, Instituto Superior Técnico, Universidade de Lisboa,

Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal2INAF, Osservatorio Astronomico di Roma, Via Frascati 33, 00040 Monte Porzio Catone (Roma), Italy

3Dipartimento di Fisica “E.R. Caianiello,” Università di Salerno, and INFN, Sezione di Napoli,GC di Salerno, Via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy

(Received 6 November 2013; published 8 April 2014)

Alterations of the gravity Lagrangian introduced in modified torsion gravity theories—also referred to asfðTÞ gravity—allows for an accelerated expansion in a matter-dominated Universe. In this framework, thecosmic speed-up is driven by an effective “torsion fluid.” Besides the background evolution of theUniverse, structure formation is also modified because of a time-dependent effective gravitational constant.Here, we investigate the imprints of fðTÞ gravity on galaxy clustering and weak gravitational lensing to theaim of understanding whether future galaxy surveys could constrain torsion gravity and discriminatebetween it and standard general relativity. Specifically, we compute Fisher matrix forecasts for two viablefðTÞmodels to both infer the accuracy on the measurement of the model parameters and evaluate the powerthat a combined clustering and shear analysis will have as a tool for model selection. We find that with sucha combination of probes it will indeed be possible to tightly constrain fðTÞ model parameters. Moreover,the Occam’s razor provided by the Bayes factor will allow us to confirm an fðTÞ power-law extension ofthe concordance ΛCDM model, if a value larger than 0.02 of its power-law slope were measured, whereasin ΛCDM it is exactly 0.

DOI: 10.1103/PhysRevD.89.083520 PACS numbers: 98.80.-k, 95.36.+x, 98.80.Es

I. INTRODUCTION

The accelerated cosmic expansion has been confirmedup to now by a wide range of cosmological data sets, fromtype-Ia supernovae [SNeIa; Refs. [1,2]] to the cosmicmicrowave background (CMB) radiation [3], baryon acous-tic oscillations [BAOs; Ref. [4]], and the gamma-ray burstHubble diagram [5]. Although these pieces of evidence canfit the framework of general relativity (GR) if we assumethe presence of a cosmological constant term in Einstein’sfield equations, this is a deeply unsatisfactory answer froma theoretical viewpoint [e.g., Ref. [6]]. Conversely, the ideathat we may instead be disregarding some gravitationaleffect occurring on cosmological scales is rather intriguing,and it somehow follows an Einstein-inspired approach—i.e., to look for a generalization of the law of gravitywhereby data requires it.Amongst the wide class of the extended theories of

gravity, we here consider the so-called fðTÞ gravity theory.It is a generalization of the teleparallel gravity, wheretorsion, instead of curvature, is responsible for the gravi-tational interaction [7–9]. As a consequence, the torsionscalar T replaces the curvature scalar R in the Lagrangian.In this framework, the underlying Riemann-Cartan space-time is endowed with the Weitzenbock connection, whichis curvature free. In this scenario, torsion acts as a force,

allowing for the interpretation of gravity as a gaugetheory of the translation group [10]. Despite conceptualdifferences, teleparallel gravity and GR yield thoroughlyequivalent dynamics, the interpretation of the gravitationalinteraction in terms of a spacetime with curvature or torsionbeing therefore only a matter of convenience, at least at theclassical level.Nevertheless, when one generalizes teleparallel gravity

to a modified fðTÞ version, inspired by the fðRÞ extendedgravity theories [11,12], the equivalence with GR breaksdown: the two classes of models differ in facts [13,14].Differently from fðRÞ theories, that can be viewed as alow-energy limit of some fundamental theory, fðTÞgravity is just a phenomenological extension of tele-parallelism but preserves the advantage of giving equa-tions that are still second order in field derivatives,oppositely to the fourth-order equations deduced infðRÞ gravity. Thus, it would be interesting to test it asa possible alternative candidate for a theory providing anaccelerated cosmic expansion without the need of anyexotic component. However, there is a caveat, as thesemodels suffer from the lack of local Lorentz invariance. Itmeans that all 16 components of the vierbien are inde-pendent, and one cannot simply get rid of six of them byfixing a specific gauge [15].Moreover, we want to emphasize that fðTÞ gravity does

not belong to the vast family of models reproduced by theHorndeski Lagrangian, which actually includes scalar-field*stefano.camera@tecnico.ulisboa.pt

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dark energy models [16], but also modified gravity theoriessuch as fðRÞ and fðGÞ gravity [17–19], scalar-tensor(including Brans-Dicke) models [20,21], K-essence[22,23], and Galileons [24–26]. Therefore, it is worthscrutinizing generalized torsion cosmologies, since theycannot be confirmed or ruled out on the basis of an analysisperformed for Horndeski models.Motivated by these considerations, in Ref. [27] we

have analyzed two fðTÞ gravity models that present theinteresting feature of an effective equation-of-state param-eter, weffðzÞ, crossing the so-called phantom divide line,i.e., weff ¼ −1. We have shown that both models are invery good agreement with a wide set of data, includingSNIa and gamma-ray burst Hubble diagrams, BAOs atdifferent redshifts, Hubble expansion rate measurements,and the WMAP7 distance priors. Yet, that wide data setis unable to severely constrain the model parameters andhence discriminate amongst the considered fðTÞ modelsand the ΛCDM scenario. The point is that the data onlyprobe the Universe’s background expansion history. Here,we present a step forward, focusing on the subhorizonlimit, where torsion gravity leads to a rescaling of theNewton gravitational constant by a time-dependent factorthat explicitly depends upon the modified Lagrangian. Asa consequence, the growth of perturbations is differentcompared to what is predicted in the ΛCDM model, andthis can be tested by present and oncoming surveysdesigned to probe the large-scale cosmic structure, suchas the Dark Energy Survey1 (DES; Ref. [28]), Euclid2

[29,30], Pan-STARRS3 or the Large Synoptic SurveyTelescope4 (LSST; Refs. [31,32]) in the optical and nearinfrared bands, or the Square Kilometre Array5 (SKA;Ref. [33]) and its pathfinders (e.g., Refs. [34–37]) in theradio band.We calculate both the three-dimensional and the pro-

jected two-dimensional matter power spectrum from galaxyclustering and the cosmic shear signal, as predicted inviable fðTÞ cosmologies. Then, we study the constrainingpotentiality of a Euclid-like survey. First, we focus on oneof the models already tested in Ref. [27], which not onlysuccessfully passes geometrical data tests, but also showsagreement with growth data. Second, we analyze the casewhere the Universe is correctly described by ΛCDM, butthe true, underlying cosmology is actually an fðTÞ modelwhose parameter space “contains” that of ΛCDM. In thiscase, we also ask ourselves which of the competingtheoretical frameworks is preferred, given the data. Wedo so by calculating the Bayes factor [38–41], in thecontext of the model selection problem.

The layout of the paper is as follows: A summary of themain equations for fðTÞ theories is given in Sec. II, wherewe also present the models we will investigate. Theobservational probes used as input to the Fisher matrixforecasts are discussed in Sec. III, while the results obtainedwhen using each single probe separately or in combinationare given in Sec. IV. Bayesian model selection is discussedin Sec. V. A summary and future perspectives are finallygiven in Sec. VI.

II. MODIFIED TORSION GRAVITY

Teleparallelism promotes the vierbein field eaμðxÞ to therole of a dynamical object with components related to themetric tensor, as

gμνðxÞ ¼ ηabeaμðxÞebνðxÞ; (1)

where ηab ¼ diagð1;−1;−1;−1Þ. Notice that Latin indicesrefer to the tangent space, while Greek letters labelcoordinates on the manifold. The dynamics is thendescribed by the Lagrangian

L ¼ e16πG

½T þ fðTÞ� þ LM; (2)

where e≡ det eaμ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi− det gμν

p, LM is the matter field

Lagrangian, and the term fðTÞ originates the deviationsfrom standard GR. It is a generic function of the torsionscalar T, which is defined as

T ¼ 1

4TλμνTλμν þ

1

2TλμνTνμλ − Tμν

μTλνλ; (3)

with the torsion tensor given by

Tλμν ¼ eλað∂νeaμ − ∂μeaνÞ: (4)

By varying the action with respect to the vierbein eaμðxÞ,one gets the field equations

e−1∂μðeeρaSρμνÞð1þ f;TÞ þ eλaSρνμTρμλð1þ f;TÞ

þ eρaSρμν∂μðTÞf;TT þ 1

4eνaðT þ fÞ ¼ 4πGeμaΘν

μ; (5)

where with Θνμ we indicate the matter energy-momentum

tensor, not to create ambiguities with the torsion tensor;here, a comma denotes a derivative with respect to T.To investigate cosmology, it should be kept in mind that

two pairs of vierbein that lead to the same metric tensor arenot equivalent from the point of view of the theory. Thismeans that we are not allowed to simply insert theFriedmann-Lemaître-Robertson-Walker (FLRW) metricinto Eq. (5). Nevertheless, in the case of spatially flatmetric, a convenient choice is represented by the diagonalvierbein [42–45], i.e.,

1http://www.darkenergysurvey.org2http://www.euclid‑ec.org3http://pan‑starrs.ifa.hawaii.edu4http://www.lsst.org5http://www.skatelescope.org

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e0 ¼ dt; (6)

ei ¼ aðtÞdxi; (7)

where aðtÞ is the scale factor as a function of cosmic time t.With such a choice, the modified Friedmann equationsbecome

H2 ¼ 8πG3

ρ −1

6fðTÞ − 2H2f;TðTÞ;

ðH2Þ0 ¼ 16πGpþ 6H2 þ fðTÞ þ 12H2f;TðTÞ24H2f;TTðTÞ − 2 − 2f;TðTÞ

; (8)

with H ¼ d ln a=dt the usual Hubble parameter, and ρðtÞand pðtÞ the (background) energy density and pressure ofthe matter component, respectively. Note that hereafter, wewill denote with a prime and with a dot differentiation withrespect to ln a and t, respectively. In this case, the torsionscalar reduces to T ¼ −6H2.Equation (8) can be rewritten in the usual form by

introducing an effective “dark torsion” fluid with energydensity ρT, such as

H2 ¼ 8πG3

½ρm þ ρT �; (9)

with

ρT ¼ 2Tf;TðTÞ − fðTÞ16πG

: (10)

Since matter still minimally couples to gravity, its con-servation equation will be unaffected, so we still haveρm ∝ a−3 and ρr ∝ a−4 for the scaling laws of dust matterand radiation. Imposing the Bianchi identities, the con-servation equation for the effective torsion fluid reads

_ρT þ 3Hð1þ wTÞρT ¼ 0; (11)

having defined

wT ¼ −f=T − f;T þ 2Tf;TT þΩrðf;T þ 2Tf;TTÞ=3

ð1þ f;T þ 2Tf;TTÞðf=T − 2f;TÞ;

(12)

the equation-of-state parameter of the dark torsion fluid.Note the coupling to the radiation energy density throughthe term ΩrðaÞ ¼ 8πGρrðaÞ=3H2ðaÞ. For fðTÞ ¼ 0, onehas ρT ¼ 0, and modified teleparallel gravity goes back tothe standard GR, while the choice fðTÞ ¼ const. giveswT ¼ −1, and the ΛCDM model is recovered.Equations (10) and (12) clearly show the key role played

by the choice of the fðTÞ functional expression in deter-mining the dynamics of the Universe. Here, we shallconsider two different models. Motivated by the resultsin Ref. [27], as a first case, we set

fðTÞ ¼ αð−TÞnT ð1 − epTT0=TÞ; (13)

where a 0 subscript denotes the present-day value of aquantity, and the constant αmay be set as a function of Ωm,Ωr, and the fðTÞ parameters, nT and pT , as detailed inRef. [27] and references therein. In the following, we willrefer to this case as the “exp” fðTÞ model. Note that inRef. [27], we have also investigated a different model, butwe discard it here, since it is not in agreement with availablemeasurements of the growth rate.Although in agreement with data probing the back-

ground expansion, the fðTÞ-exp model of Eq. (13) does notreduce to ΛCDM for any particular choice of the ðnT; pTÞparameters. Albeit this is an interesting feature on its own,such a peculiarity does not allow us to investigate whetherclustering and shear data can discriminate between torsiongravity and GR. Therefore, as a second case, we considera power-law (hereafter “pl”) model given by [46]

fðTÞ ¼ αð−TÞnT ; (14)

where, again, α may be expressed as a function of Ωm, Ωr,and nT . Note that the fðTÞ-pl model exactly reduces toΛCDM for nT ¼ 0. As will see later on, we can take nT ¼ 0as a fiducial value and look at how strong the constraints onnT are, thus quantifying whether or not clustering and sheardata can discriminate between modified torsion gravityand GR.For what concerns cosmological perturbations, the same

caveat as before should be considered when perturbing themetric, and the simplest choice may lead to inconsistencies.That is, focusing on the scalar degrees of freedom, one mustperturb the vierbein with six unknown functions and thenchoose the longitudinal gauge on the perturbed metrictensor. This reduces the number of free functions to three,one degree of freedom more than in the GR case [47].Nevertheless, this term plays an important role in theevolution of perturbations at large scales. In the subhorizonlimit, this leads to an effective gravitational constant withrespect to the Newtonian constantGN , which takes the form

GeffðzÞ ¼GN

1þ f;T ½TðzÞ�(15)

so that it can be straightforwardly evaluated once themodified Friedmann equations have been solved.

III. COSMOLOGICAL OBSERVABLES

We adopt the Fisher matrix formalism [48–50] to makepredictions on the fðTÞ cosmological models presented inSec. II. By doing so, we can scrutinize to which degree ofaccuracy one of the future large-scale surveys will be ableto constrain fðTÞ model parameters, thus allowing us todiscriminate between it and ΛCDM—if the constraints aretight enough. In the assumption of a Gaussian likelihood,

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L, for the model parameters, ϑ ¼ fϑαg, the Fisher matrixapproximates the inverse of the parameter covariancematrix in a neighborhood of the likelihood peak, i.e.,

F ¼ −�∂2 lnL

∂ϑ2

�; (16)

the marginal error on parameter ϑα is thence σðϑαÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðF−1Þαα

p. Each of the cosmological probes that we study

here will then produce its own Fisher matrix, viz. Fg3D , Fg2D

and Fγ for three- and two-dimensional galaxy clusteringand cosmic shear tomography, respectively. We alsointroduce another important quantity, namely the correla-tion between the parameter pair ðϑα; ϑβÞ, which reads

rðϑα; ϑβÞ ¼ðF−1Þαβffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðF−1ÞααðF−1Þββq : (17)

This quantity tells us whether the two parameters are com-pletely uncorrelated [when rðϑα; ϑβÞ ¼ 0] or thoroughlydegenerate [if rðϑα; ϑβÞ ¼ �1].

A. 3D galaxy clustering

Large-scale galaxy redshift surveys allow us to inves-tigate the clustering properties of galaxies through mea-surements of their correlation function and its Fouriertransform—the power spectrum, the observable we con-sider here. BAOs at the last scattering surface give rise to acharacteristic peak at the typical scale of ∼150 Mpc in thegalaxy correlation function, which translates into wigglesin the matter power spectrum. This scale may be taken as astandard ruler, fixed by the sound horizon at last scatteringand accurately measured by CMB experiments. By com-paring the BAO peak positions at the different redshifts, wecan constrain both the Hubble parameterHðzÞ, in the radialdirection, and the comoving angular diameter distancedAðzÞ, perpendicularly to the line of sight. Since theunderlying cosmology is not known a priori, the distanceto an object is hence unknown, and what galaxy surveysactually measure is the clustering in the redshift space. As aconsequence, the power spectrum also contains the imprintof the linear growth rate of structure in the form of ameasurable anisotropy due to the coherent flows of matterfrom low to high densities. When redshift is used to replacedistances, peculiar velocities of galaxies introduce distor-tions in the clustering pattern, which can be observed asanisotropies in the correlation function. At linear order,such redshift space distortions (RSDs) depend upongðzÞσ8ðzÞ, where g½zðaÞ� ¼ d lnDþ=d ln a is the growthrate, DþðzÞ is the growth factor, σ8ðzÞ ¼ σ8ðz ¼ 0ÞDþðzÞ,and σ8 is the variance of the density perturbations on thescale of 8h−1 Mpc. We can constrain σ8 via CMB mea-surements; RSDs are consequently a powerful probe of

gðzÞ, being capable of discriminating among different darkenergy models and modified gravity theories.The Fisher matrix for galaxy clustering as measured

from galaxies in a redshift bin centred on z reads [51]

Fg3Dαβ ¼

Zkmax

kmin

d3kð2πÞ3

1

2

∂ lnPobsðkÞ∂ϑα

∂ lnPobsðkÞ∂ϑβ VeffðkÞ;

(18)

where PobsðkÞ ¼ Pobsðk; μÞ is the anisotropic observedpower spectrum (with μ being the cosine of the angle withthe line of sight) and VeffðkÞ is the survey effective volume.In each redshift bin, it is given by

Veffðk; μÞ ¼ Vsurvey

�1þ

�dNðspÞ

dzPobsðk; μÞ

�−1�−2; (19)

with VsurveyðzÞ being the survey volume probed by galaxiesin the redshift bin centred on z and dNðspÞ=dzðzÞ thenumber density of galaxies with measured spectroscopicredshift in the redshift interval ½z; zþ dz�. We set kmin ¼0.001h Mpc−1 to safely remain in the subhorizon limit, andkmax ¼ 0.15h Mpc−1 to stay in the linear regime. Note thatfuture galaxy surveys will measure Pobsðk; μÞ up to muchlarger k values, but we prefer here to avoid such smallscales and neglect poorly understood nonlinear effects.Indeed, they have yet to be investigated—e.g., throughN-body simulations—in fðTÞ theories, and so a mappingfrom the linear to the nonlinear power spectrum is unavail-able at the time being. Setting kmax ¼ 0.15h Mpc−1 guar-antees that we are in the linear regime, so that we do not addfurther uncertainties or systematic error due to neglecting orincorrectly modeling nonlinearities.The observed power spectrum is a distorted representa-

tion of the underlying matter power spectrum

Pδðk; zÞ ¼ Askns ½TðkÞDþðzÞ�2; (20)

where TðkÞ is the transfer function, which we calculatefollowing the fitting formulas of Ref. [52], ns is the spectralindex, and the normalization constant As can be relatedto σ8. The redshift dependence is introduced throughthe linear growth factor DþðzÞ, which can be convenientlycomputed by integrating the growth rate, gðzÞ. In thesubhorizon limit we are interested in here, it may be obtainedas the solution of the following nonlinear differentialequation:

dgðzÞdz

þ�d lnE2ðzÞd lnð1þ zÞ þ 2þ gðzÞ

�gðzÞ1þ z

þ 3

2

Ωmð1þ zÞ2E2ðzÞ

GeffðzÞGN

¼ 0; (21)

with EðzÞ ¼ HðzÞ=H0.

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It is worth noting, however, that Eq. (15) only holds fork > kmin. On larger scales, the full set of perturbedEinstein’s equations has to be solved. For this reason,we have chosen to focus our attention on the subhorizonlimit, thus simplifying the analysis without any loss of thesurvey constraining power. Indeed, future galaxy surveyswill not typically be able to probe such extremely largescales, for which alternative techniques are more effective[e.g., Refs. [53–56]]. To give a flavor of the alterations thatthe fðTÞ models we analyze bring to the Newtonianconstant, we show in Fig. 1 the quantity Geff=GN as afunction of the scale factor a. The solid black line is theconstant value of GR, while the exp and pl fðTÞmodels aredepicted by the short-dashed blue and long-dashed redcurves, respectively. The former is calculated with thefiducial values found in Ref. [27], while for the latter, wepresent a few values of nT , specifically −0.1, −0.01, 0.01,and 0.1 from top to bottom. We remind the reader thatnT ¼ 0 recovers GR.In order to go from Pδðk; zÞ to Pobsðk; μ; zÞ, one has to

include anisotropies due to RSDs and account for thefact that the actual measurement concerns the powerspectrum of galaxies rather than that of underlyingmatter fluctuations. Moreover, since the conversion fromredshifts to distances is only possible by assuming areference cosmological model—which can be differentfrom the actual (unknown) one—, a further distortion,referred to as the Alcock-Paczynski effect [57], takesplace. The final observed power spectrum then reads[58–62]

Pobsðk; μ; zÞ ¼�HrefðzÞHðzÞ

��drefA ðzÞdAðzÞ

�2

× ½b2gðzÞ þ 2μ2bgðzÞgðzÞ þ μ4g2ðzÞ�

× exp

�−�qνcσspzHrefðzÞ

�2�Pδðq; ν; zÞ: (22)

Here,

qðk; μ; zÞ ¼ D1=2ðμ; zÞk; (23)

νðk; μ; zÞ ¼ D−1=2ðμ; zÞ½HðzÞ=HrefðzÞ�μ; (24)

and

D ¼�dAðzÞdrefA ðzÞ

�2

þ��

HrefðzÞHðzÞ

�2

−�dAðzÞdrefA ðzÞ

�2�μ2: (25)

It is worth a brief comment upon the different termsentering Eq. (22). First, the power spectrum is notevaluated directly in ðk; μÞ rather than in the shiftedvariables ðq; νÞ as a consequence of the Alcock-Paczynski effect. Indeed, when the reference cosmologyused to measure the power spectrum from the datamatches the true one, Href ¼ H and drefA ¼ dA so thatðq; νÞ ¼ ðk; μÞ, and the multiplicative bias disappearstoo. Secondly, the term in the second line is due toRSDs which have been modeled here to linear order.Here, bgðzÞ is the galaxy bias, which takes the differ-ence between the galaxy distribution and matter densityfluctuations into account. As a matter of fact, moresophisticated expressions could be used to improve theagreement with numerical simulations. However, all ofthem are very well approximated in the linear regime byour formula. Lastly, the third exponential term accountsfor errors in the spectroscopic redshift measurement,parametrized here as σspz .

B. 2D galaxy clustering

The study of three-dimensional galaxy clustering pre-sented in Sec. III A has as a basic assumption that we canaverage the matter power spectrum within each redshift bin.However, the measured redshift is used for both estimatingdistances, through the radial comoving distance χðzÞ, andtime, since zðtÞ ¼ 1=aðtÞ − 1. In practice, in the ith redshiftslice, we reconstruct the galaxy power spectrum Pgðk; ziÞby computing correlations amongst galaxy-number densityfluctuations whose physical separation estimates are func-tions of the galaxy redshifts. We then relate the recon-structed Pgðk; ziÞ to the redshift zi (usually the center of thebin). Nonetheless, the sources contained in the volumeVsurveyðziÞ have emitted their photons at different instants inthe time interval Δt centred in ti ¼ tðziÞ. In homogenizingeverything to the central redshift value, zi, we thereforedisregard the time evolution of the underlying matter

FIG. 1 (color online). Rescaled effective gravitational constantvs the scale factor for the fiducial fðTÞ-exp model (short-dashedred curves) and in fðTÞ-pl models (long-dashed blue curves) withnT ¼ −0.1, −0.01, 0.01, and 0.1 from top to bottom.

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density field δ ¼ deltaρm=ρm. This approximation is harm-less, provided the width of the redshift slice is thin enoughso that evolution within the bin is negligible—that is to say,the growth rate is substantially constant. As we will see inSec. IV, this is indeed the case with a spectroscopic galaxysurvey, for the spectro-z error, σspz , is small, and we cansafely consider small-sized, sharp-edged redshift slices.However, in some situations there is no radial informa-

tion available, or it is poor, consequently meaning that theredshift slices are broad. For instance, this is the case ofphotometry, where the scatter between the measured andthe actual redshift may be large. In this case, we insteaddeal with projected quantities. Thus, the angular powerspectrum CgðlÞ of galaxy-number density fluctuationsreads

Cgl ¼ 4π

Zdkk½Wgðl; kÞ�2Pgðk; z ¼ 0Þ; (26)

with l being the angular wave number and Wgðl; kÞ aproper line-of-sight weight function. A widely used sim-plification is given by so-called Limber’s approximation[63,64], where l ¼ kχ. Limber’s approximation is validwhen l ≫ 1, but it has been shown that the convergence isalready good for l≳ 10 [e.g., Ref. [65]]. Therefore, it is asuitable approximation, since for larger angular scales thecosmic variance uncertainty is dominant. In this limit, andif we can further subdivide the source sample into someredshift bins, we then have

CgijðlÞ ¼

Zdχ

Wgi ðχÞWg

jðχÞχ2

Pδ

�lχ; χ

; (27)

with WgðχÞ defined by

Wg½χðzÞ� ¼ HðzÞbgðzÞdNðphÞ

dzðzÞ; (28)

where “ph” denotes photometry. This is usually referred toas redshift tomography, and the two-dimensional galaxypower spectrum is rather a tomographic matrix Cg

ijðlÞ,whose entries are the angular power spectra of each bin.Lastly, there is a further subtlety that has to be taken

into account when dealing with Limber’s approximation.Indeed, since it links the angular scale l to the physicalwave number k through the radial comoving distance χ, it isno longer possible to neatly separate linear to nonlinearscales as small or large multipoles—conversely to what onedoes with the three-dimensional Pδðk; zÞ. Therefore, wedecide to proceed as follows: We now include the nonlinearevolution of the matter power spectrum; to do so, we useHALOFIT fitting formulas. That the nonlinear evolution ofdensity fluctuations in fðTÞ cosmology follows that of theΛCDM model might be seen as a rather strong assumption.However, we believe it is acceptable for two reasons:

(i) Li, Sotiriou, and Barrow [66] have clearly demonstratedthat viable fðTÞ cosmologies differ from ΛCDM in thelargest, linear scales, otherwise recovering the GR predic-tion when approaching the nonlinear regime; and (ii) weanyway limit our analysis to a range of l’s whereby onlymildly nonlinear k’s are involved, as will be clear in thediscussion of the results.For a square patch of the sky, the Fourier transform leads

to uncorrelated modes, provided the modes are separatedby 2π=Θrad, where Θrad is the side of the square in radians.Then, the Fisher matrix is simply the sum of the Fishermatrices of each l mode [67], namely

Fg2Dαβ ¼ fsky

Xlmax

l¼lmin

2lþ 1

2Tr

�∂CgðlÞ∂ϑα

~Cgl−1 ∂CgðlÞ

∂ϑβfCgl−1�;

(29)

where fsky is the fraction of the sky covered by the surveyunder analysis, and

½fCgl�ij ¼ Cg

ijðlÞ þ1

NðiÞg

δKij (30)

is the observed (signal plus noise) galaxy angular power

spectrum, with NðiÞg being the galaxy-number density per

square arcminute in the ith bin and δK the Kronecker deltasymbol.

C. Cosmic shear

The presence of intervening matter along the path ofphotons emitted by distant sources causes gravitationallensing distortions of the high-redshift source images.The weak lensing regime occurs when lensing effectscan be evaluated on the null geodesic of the unperturbed(unlensed) photon [68]. Such distortions—directly relatedto the distribution of matter on large scales and to theUniverse’s geometry and dynamics—can be decomposedinto a convergence, κ, and a (complex) shear, γ ¼ γ1 þ iγ2[68,69]. Let us now consider a perturbed metric about theflat FLRW background in the longitudinal gauge, viz.

e0 ¼ ð1þ 2ΦÞdt; (31)

ei ¼ aðtÞð1þ 2ΨÞdxi; (32)

where Φ and Ψ are the two metric potentials. For them,Φ ¼ −Ψ holds in GR and in the absence of anisotropicstress, but this is not, in general, true in extended/modifiedtheories of gravitation. In the subhorizon regime, we knowthat matter density fluctuations δ obey the approximateevolution equation [47,66,70]

δ̈þ 2H _δ − 4πGeffρmδ≃ 0; (33)

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where Geff is given in Eq. (15). To confront our model withweak lensing observations, we have to define the so-calleddeflecting potential (e.g., Refs. [40,71–73])

ϒ ¼ Φ −Ψ2

(34)

and use its Poisson-like equation

∇2ϒ ¼ 4πGeffa2ρmδ: (35)

It relates matter density fluctuations to the combination ofmetric potentials that are responsible for weak gravitationallensing effects. Thence, we can similarly link the powerspectrum of the weak lensing source field (namely, thedeflecting potential wells) to the three-dimensional matterpower spectrum through

Pϒðk; zÞ ¼�−3

2H0

2Ωmð1þ zÞk−2 GeffðzÞGN

�2

Pδðk; zÞ:(36)

In the flat-sky approximation, the shear is expanded in itsFourier modes, and the two-dimensional angular powerspectrum CγðlÞ is thus given by

hγðlÞγ�ðl0Þi ¼ ð2πÞ2δDðl − l0ÞCγðlÞ; (37)

with l ¼ jlj being the angular wave number and δD theDirac delta function. In the case where one has distanceinformation for individual sources, we can use this infor-mation for statistical studies. A natural course of action is todivide the survey into slices at different distances andperform a study of the shear pattern on each slice [67]. Thisprocedure is the same redshift tomography introduced inSec. III B. By doing so, we can construct the tomographicshear matrix CγðlÞ, whose elements read

CγabðlÞ ¼

Zdχ

WγaðχÞWγ

bðχÞχ2

Pδ

�lχ; χ

; (38)

from Eq. (36), we have the weak lensing selection(or weight) function in the ath redshift bin:

WγaðχÞ ¼ 3

2H0

2Ωmχ

aðχÞGeffðχÞGN

Z∞

χdχ0

χ0 − χ

χ0dNðphÞ

a

dχ0;

(39)

with

dNðphÞa

dχ¼ dNðphÞ

a

dzdzdχ

(40)

being the redshift distribution of the sources. Here, as in thecase of two-dimensional galaxy clustering, dNi=dχ is

basically the probability of finding a source within theath bin, and, as such, it must have unity area. Also, we haveagain used Limber’s approximation.For cosmic shear tomography, the Fisher matrix Fγ is

functionally identical to that of two-dimensional angularclustering in Eq. (30), where now the observed (signal plusnoise) shear angular power spectrum reads

½fCγl�ab ¼ Cγ

abðlÞ þσγ

2

NaδKab; (41)

with σγ ≃ 0.3 the galaxy-intrinsic shear rms in onecomponent.

D. Galaxy shear cross correlation

Thanks to the formalism described in Sec. III B, two-dimensional galaxy clustering also enables us to estimateits cross correlation with the cosmic shear signal. It can beeasily computed through

CgγiaðlÞ ¼

Zdχ

Wgi ðχÞWγ

aðχÞχ2

Pδ

�lχ; χ: (42)

Then, the observed (signal plus noise) cross correlation is

½fCgγl �ia ¼ Cgγ

iaðlÞ; (43)

since clustering and shear noise contributions do notcorrelate.

IV. RESULTS AND DISCUSSION

First of all, we need to specify a reference survey whoseconstraining power we want to test with the Fisher matrixformalism sketched in Sec. III. For our purpose, we findthat a Euclid-like experiment [29,30] perfectly suits ourendeavor, since it will perform both (spectroscopic) galaxyclustering and (photometric) cosmic shear measurements.Euclid is an ESA medium-class space mission selected inOctober 2011 in the Cosmic Vision 2015–2025 program,and it results of the merging of the DUNE and SPACEmissions. The Euclid mission aims at understanding whythe expansion of the Universe is accelerating and what isthe nature of the source responsible for this acceleration.Therefore, it is in thorough agreement with the effort ofour work.The spectroscopic survey (hereafter Euclid-sp) will

measure galaxy redshifts in the infrared band 0.9–2 μmfor ∼65 × 106 galaxies using a slitless spectrograph relyingon the detection of emission lines in the galaxy spectra. Inthe chosen wavelength range, the most favorable line willbe the Hα line, redshifted to 0.7 ≤ z ≤ 2. We divide thisrange into equally spaced bins of width 0.1. This is muchlarger than the typical errors on spectro-z’s, which we takeas σspz ¼ 0.001ð1þ zÞ. By doing so, redshift errors aremuch smaller than the bin width, and the constraints on

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cosmological parameters from different bins are thusindependent from each other. Therefore, we can firstmarginalize over the bias in each bin and then sum theresulting Fisher matrices to get the final Fg3D . The redshiftdistribution should be computed taking into account theinstrumental setup and its efficiency coupled to a model ofthe number density of Hα emitters. Following Ref. [60–62]and the definition study report [29]—to which we refer fordetails—we use the distribution of Hα emitters of Ref. [74]and weight it according to a proper flag [75]. Thiseventually provides the dNðspÞ=dzðzÞ profile shown inthe bottom panel of Fig. 2.For what concerns the photometric measurements

(Euclid-ph), we compute our results for a 15; 000 deg2

cosmic shear experiment. The source distribution overredshifts has the form [76]

dNðphÞ

dzðzÞ ∝ z2e−ð

zz0Þ1.5 ; (44)

where z0 ¼ zm=1.41, and zm ¼ 0.9 is the median redshift ofthe survey. The number density of the sources, withestimated photometric redshift and shape, is 30 per squarearc minute. To perform the tomographic analyses outlinedin Secs. III B to III D, we divide the redshift distribution ofsources into ten redshift bins. However, the Euclid imagingsurvey will only provide photometric redshift measure-ments, which are known to be less accurate than thoseobtained from spectroscopy. The scatter between the trueredshift and the photometric estimate is assumed to be oforder 3% and to scale linearly with z, that is to sayσphz ¼ 0.03ð1þ zÞ. The top panel of Fig. 2 illustrates thetotal dNðphÞ

g =dz (solid black curve) and the ten photometricredshift bins we use (dashed red curves).

A. 3D galaxy clustering constraints

Let us start by examining the constraints on fðTÞ gravityfrom the three-dimensional galaxy power spectrumalone. First, we consider the exp model and estimate theFisher matrix with respect to the parameters ϑ ¼fΩm; h; nT; pT; ns; σ8g, and we marginalize over the biasbgðziÞ in each redshift bin. As fiducial values, we choosefΩm; h; nT; pTg ¼ f0.287; 0.731; 0.736;−0.100g, accord-ing to the results in Ref. [27], while we set fns; σ8g ¼f0.820; 0.9608g, in agreement with the WMAP9 con-straints. As done in Ref. [60–62], the fiducial bias valuesfor each bin have been set following Ref. [77].Despite the background parameters Ωm and h and the

power-spectrum-related quantities ns and σ8 being wellconstrained, confidence ranges for fðTÞ parameters arequite broad. In particular, we find σðnTÞ ¼ 3.0 andσðpTÞ ¼ 3.7. Such a result can be qualitatively explainedas follows: Over the redshift range 0.7–2.0, the termT0=T ¼ E−2ðzÞ quickly decreases so that the exponentialin Eq. (13) approaches unity. Hence, the fðTÞ term in the

Lagrangian becomes subdominant. Such a behavior holdswhatever the values of nT and pT , thus explaining why it isso difficult to constrain these parameters using three-dimensional clustering alone. However, it is worth empha-sizing that this result is mainly due to the redshift rangeinvestigated rather than the observational probe adopted—as will be clear in the following section. One could naïvelyexpect that shifting the median survey redshift to a smaller zwould improve the constraining power, since the powerspectrum would be more sensitive to the fðTÞ parameters.However, to have a lower median redshift for the survey,one should change the instrumental setup and rely ondifferent emission lines—that is to say, different kinds oftarget galaxies. As a consequence, the redshift distributionwould also change, and it is not possible a priori to inferwhether the constraints will improve or degrade.Fisher matrix forecasts depend not only on the obser-

vational probe adopted and the precision in the measure-ments, but also on the fiducial cosmological model.

FIG. 2 (color online). Normalized source redshift distributionsdN=dzðzÞ. Bottom panel: Spectroscopic galaxy survey (solidblack curve) and its 14 bins (dashed red curves). Top panel:Photometric imaging survey (solid black curve) and its ten bins(dashed red curves).

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A interesting example is provided here by the results for thefðTÞ-pl model. We assume nT ¼ 0, which de facto impliesa ΛCDM scenario. As expected, the constraints on thestandard parameters fΩm; h; ns; σ8g are comparable withthose obtained for the exp model and other in the literature.Moreover, the slope nT of the fðTÞ term is now wellconstrained, with σðnTÞ ¼ 0.021. This encouraging resultsuggests that three-dimensional galaxy clustering alone isable to detect torsion gravity departures from the GR basedΛCDM scenario.

B. 2D galaxy clustering constraints

Now, we analyze the results from two-dimensional(photometric) galaxy clustering. Since the redshift slicesare broader than before, we can no longer marginalizeover the bias amplitude in each bin, and then sumover the bins. Hence, in this case the parameter set—for example, for the fðTÞ-exp model—is ϑ ¼fΩm; h; nT; pT; ns; σ8; bgg; bg is a vector of nuisanceparameters which account for the bias amplitude in eachredshift bin. This is slightly different from what is donein the 3D case, where a nuisance bias parameter isincluded in each redshift-binned Fisher matrix, thenmarginalized over to eventually sum all the marginalizedFisher matrices. This happens because in the 3D case,one considers the various redshift bins as uncorrelatedvolumes of the Universe, whereas in the 2D case, one inprinciple also includes cross-correlations between bins.Anyway, we emphasize that this is somehow an over-conservative approach, because, even though we do notexactly know the Hα galaxy bias, it cannot freely vary ineach bin. Nevertheless, we decide to proceed so, also tosafely deal with our ignorance of the halo bias in fðTÞgravity. According to Ref. [29], the angular multipolesthat will be probed by Euclid are in the rangel ∈ ½5; 5000�. However, we find this assumption rathertoo optimistic for the present case: on the one hand, forl≲ 10, Limber’s approximation is less safe [69,78]; onthe other hand, at very small angular scales (large l’s),nonlinear effects—as well as feedback from baryonicphysics—became non-negligible [79,80]. Therefore, wedecide to scrutinize three different scenarios, dubbedEuclid-ph I, II, and III, where l ∈ ½10; 1000�, [10, 3000],and [5,5000], respectively.In Table II, we present the forecast 68.3% marginal

errors on fðTÞ-exp and fðTÞ-pl model parameters. It isstraightforward to notice that, as expected, the wider therange of angular multipoles, the tighter the constraints.Besides, it is interesting to verify the explanation pre-sented in Sec. IVA on the reason for why the fðTÞ-expparameters were poorly constrained by three-dimensionalgalaxy clustering. Indeed, the range of redshifts probedby the Euclid imaging survey is wider than the 0.7 ≤z ≤ 2.0 interval motivated by Hα-line spectroscopy. As aconsequence, forecast marginal errors obtained now are

3.5 to > 8 times more stringent for nT than those fromFg3D , and 3 to 6 times more stringent for pT.Regarding the fðTÞ-pl model, constraints from the

two-dimensional angular power spectrum in the mostconservative Euclid-ph I configuration are almost 1 orderof magnitude weaker than those in the three-dimensionalcase. A reason for this can be understood by looking atthe correlations amongst nT and the other parameters,namely the rðnT; ϑαÞ coefficients. They read 0.78,−0.835, 0.73, and −0.14 for Ωm, h, ns, and σ8,respectively. This means that the slope of the power-law modification to the teleparallel gravity Lagrangian isdegenerate with almost all the standard cosmologicalparameters—particularly those related to the backgroundexpansion history. This happens because the functionalform of the fðTÞ-pl model is basically a rescaled versionof the Hubble parameter. (Please recall that T ¼ −6H2.)Thus, the nonstandard parameter nT simply alters theevolution in the redshift of the torsion scalar, T, withoutintroducing any peculiar behavior, as is instead the casewith the fðTÞ-exp model. Nonetheless, things are better ifwe increase the analyzed l range. For example, alreadywith the Euclid-ph II configuration, we have a promisingσðnTÞ ¼ 0.050, and indeed the rðnT; ϑαÞ coefficients arenow 0.59, −0.53, 0.46, and 0.34.

C. Cosmic shear constraints

Let us now move on to analyze fðTÞ model parameterconstraints coming from cosmic shear alone. In Table III,we present the forecast 68.3% marginal errors on cosmo-logical parameters for both fðTÞ-exp and fðTÞ-pl models.Again, the wider the range of angular multipoles, the tighterthe constraints. Besides, we can easily see that they areoverall better than in the case of two-dimensional cluster-ing. This behavior has a straightforward reason. Indeed, theweak-lensing weight function of Eq. (39) does have afurther (and more direct) dependence upon Geff compared

TABLE I. Forecast 1σ marginal errors on fðTÞ model param-eters from three-dimensional galaxy clustering alone.

exp model pl modelnT pT nT

g3D Euclid-sp 3.0 3.7 0.021

TABLE II. Forecast 1σ marginal errors on fðTÞ model param-eters from two-dimensional galaxy clustering alone.

exp model pl modelnT pT nT

g2D Euclid-ph I 0.86 1.2 0.12g2D Euclid-ph II 0.54 0.84 0.050g2D Euclid-ph III 0.37 0.61 0.035

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to galaxy clustering. Thus, cosmic shear—and weaklensing effects more generically—is more effective indetecting modified gravity effects.

D. Combined constraints

After having analyzed the constraining power of galaxyclustering and cosmic shear singularly, having thus under-stood the most important aspects and peculiarities of thetwo probes, it is now time to look at the combination of thetwo. To better investigate the effect of modified torsiongravity on weak lensing, in Sec. IV C we have presented theresults for three different Euclid-like scenarios. However,we now restrict ourselves to the most conservative case.Indeed, with lmin ¼ 10, we are confident that Limber’sapproximation is used in its regime of validity. Moreover,we do not want our results to rely on nonlinear scales,whose dynamics and growth of perturbations have not yetbeen studied in fðTÞ cosmology. Thus, lmax ¼ 1000 bettersuits our purpose.The only consistent way to combine clustering and shear

forecasts, as discussed in Ref. [81], is by using angularpower spectra for both. By doing so, we construct a newFisher matrix containing not only all theCg andCγ spectra,but also their cross-correlations. Hence, we can build acombined tomographic matrix

~Cl ¼� fCg

lfCgγlfCgγ

lfCγl

; (45)

and its corresponding Fisher matrix takes again the sameform as Eq. (29) [82]. By doing so, we obtain theconstraints presented in Table IV.It is immediately apparent that the combination of the

two probes greatly enhances the constraining potential ofthe survey. This is due to the fact that parameter degen-eracies in Fg2D and Fγ are almost always “perpendicular”—in the sense that the correlation coefficients of galaxyclustering and cosmic shear analyses have opposite signs.Thence, all parameter errors shrink. In particular, wehave that the 1σ marginal errors on fðTÞ-exp modelparameters are σðnTÞ ¼ 0.063 and σðpTÞ ¼ 0.14. Evenmore impressively, for the fðTÞ-pl parameter, we obtainσðnTÞ ¼ 0.0097, more than twice as stringent as thatobtained with three-dimensional galaxy clustering, andalmost 12 times better compared to photometric probes

alone. All this can be more easily seen in Fig. 3, whichshows the forecast 1σ two-parameter marginal contours onfðTÞ-exp and fðTÞ-pl model parameters in the ðϑα; ϑβÞplanes for galaxy clustering (light colors) and cosmic shear(darker colors) alone and combined (smallest and darkestellipses). The different—often substantially orthogonal—orientations of the error ellipses demonstrate how effectiveis the combination of galaxy clustering and cosmic sheartomography for our science case. This is a general trend,but it is even more useful for the modified torsion gravitynonstandard parameters.Such a spectacular behavior is due to the fact that BAOs

and RSDs are highly complementary to weak lensing,especially in the presence of uncertainties of photo-z errorsand inaccurate knowledge of galaxy clustering bias[29,30,83]. Galaxy clustering data measure the “dark fluid”equation of state at higher redshift than SNeIa, used so farfor this task Ref. [27]. Besides, clustering is a probe forthe evolution of matter fluctuations, and thus, through thePoisson equation, of the Newtonian potentialΦ—and of themodifications occurring from modified torsion gravity tothe Newtonian gravitational constant, i.e., GeffðzÞ. On theother hand, weak lensing is sensitive, through the deflectingpotential ϒ, to the sum of the two metric potentials, whichare equal in GR but not in more general gravity theories. Asa consequence, the sensitivity to beyond-GR growthparameters mostly comes from weak lensing, which pro-vides the only direct measurements of growth (withoutbiasing) [84]. In other words, constraints on modificationsto gravity mostly depend on the errors on cosmic shear—except when intrinsic parameter degeneracies wreak havocwith the weak-lensing constraining potential. Conversely,these constraints are very weakly sensitive to the BAOerrors, showing that the uncertainties are dominated by thegrowth measurements themselves rather than residualuncertainty in the expansion history.As a final remark, the next generation of large-scale

experiments aiming at understanding the nature of present-day cosmic acceleration seek much higher precision thanthose carried out to date. Therefore, the risk of beinglimited or biased by systematic errors is much higher.Conclusions about cosmic acceleration will be far moreconvincing if they are reached independently by methodswith different systematic uncertainties. Hence, measuringangular and tracer dependence of the clustering signal andtesting redshift scaling of cosmic shear as we do here is notonly more effective, but also safer.

TABLE III. Forecast 1σ marginal errors on fðTÞ modelparameters from cosmic shear tomography alone.

exp model pl modelnT pT nT

γ Euclid-ph I 0.62 0.90 0.11γ Euclid-ph II 0.40 0.59 0.096γ Euclid-ph III 0.31 0.48 0.088

TABLE IV. Forecast 1σ marginal errors on fðTÞ modelparameters from the combination of two-dimensional galaxyclustering and cosmic shear.

exp model pl modelnT pT nT

γ þ g2D Euclid-ph I 0.063 0.14 0.0097

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V. BAYESIAN MODEL SELECTION

The main purpose of this paper is to study the detect-ability of modified torsion gravity signatures by exploitingthe potential of future surveys probing the large-scalestructure of the cosmos. To this aim, we have hithertoanalyzed to which degree of accuracy the fðTÞ modelparameters can be constrained by the Euclid satellite as areference survey. Nonetheless, there is in a sense a higher-level question than parameter estimation: model selection.In the case of the fðTÞ-pl model, we can recast the presentanalysis as a comparison between the concordance ΛCDMcosmological model, where gravity is described by stan-dard GR and the present-day cosmic acceleration is causedby a cosmological constant term, and a modified tele-parallel scenario, whereby the gravity Lagrangian containsa higher-order term, αð−TÞnT , in the torsion scalar, T. Interms of model parameters, this latter perspective doesinclude the former, where the slope of the additional termvanishes, namely nT ¼ 0.When performing standard parameter estimation, we

assume a theoretical model, within which we interpret thedata. Conversely, in model selection, what wewant to knowis which theoretical framework is preferred given the data.Clearly enough, if the alternative model had more param-eters than the standard one, chi-squared analysis would beof little use, because it would always reduce if we addedmore parameters (i.e., degrees of freedom). Otherwise,Bayesian analysis provides a useful Occam’s razor, knownas the Bayes factor, B. It involves the computation of theBayesian evidence, often called marginal likelihood ormodel likelihood (cf. Ref. [85], Sec. 4.2). For a modelM, it

is defined as a marginalization over its m parametersϑ, viz.

ZðdjMÞ ¼Z

dmϑLðϑÞπðϑjMÞ; (46)

here, LðϑÞ≡ pðdjϑ;MÞ is the likelihood function of theparameters (which equals the probability for the modelparameters given the data d), and the prior πðϑjMÞencodes our status of knowledge before seeing the data.Let us now consider two competing models M1 and

M2, the former nested in the latter. That is to say, M1 issimpler, because the set of its parameters fϑα1g is containedin the M2 parameter set fϑα2g, with α1 running from 1 tom1, α2 from 1 to m2, and with m2 > m1 by definition. Insuch a situation, one can compute the Bayes factor

B ¼ pðM1jdÞpðM2jdÞ

; (47)

which is the ratio of the two corresponding posteriorevidence probabilities. The posterior probability for eachmodel Mi is given by Bayes’ theorem,

pðMijdÞ ¼ZðdjMiÞπðMiÞ

pðdÞ : (48)

If we have no a priori preferences towards one specificmodel, this will translate into the choice of noncommittalpriors πðM1Þ ¼ πðM2Þ ¼ 1=2. Hence, the ratio of theposterior evidence probabilities [Eq. (48)] reduces tothe ratio of the evidences. Ref. [39–41] showed that

FIG. 3 (color online). Marginal error contours in the two-parameter plane for fðTÞ-exp (left panel) and fðTÞ-pl (right panel) modelparameters.

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(i) in the Laplace approximation, where the expectedlikelihoods are given by multivariate Gaussians; and (ii) ifone considers hBi as the ratio of the expected values, ratherthan the expectation value of the ratio, one eventually gets

hBi ¼ffiffiffiffiffiffiffiffiffiffiffiffidetF2

pffiffiffiffiffiffiffiffiffiffiffiffidetF1

p ð2πÞ−l=2Ylq¼1

πðϑα1þqÞe−δϑ·F2·δϑ=2: (49)

Here, Fi is the Fisher matrix relative to the ith model, l ¼m2 −m1 is the number of extra parameters, and δϑ is thevector of the parameter shifts. These shifts appear because,if the correct underlying model were M2, the maximum ofthe expected likelihood would not, in principle, be atthe correct parameter values of M1 (see Fig. 1 ofRef. [39–41]). The m1 parameters of M1 shift their valuesto compensate the fact that ϑα1þ1;…; ϑα1þl are kept fixed atsome incorrect fiducial value—most times, as in our fðTÞ-pl case, simply ϑα1þ1 ¼ � � � ¼ ϑα1þl ¼ 0. The shifts can becomputed under the assumption of a multivariate Gaussiandistribution [86] and read

δϑ ¼ −F1−1 ·G2 · δψ ; (50)

with G2 being a subset of theM2 Fisher matrix and δψ theshifts of the l extra parameters ψ. The so-called “Jeffreysscale” gives empirically calibrated levels of significance forthe strength of evidence [87]. A recent version of theJeffreys scale sets j lnBj < 1 as “inconclusive” evidence infavor of a model, 1 < j lnBj < 2.5 as “positive,” 2.5 <j lnBj < 5 as “moderate,” and j lnBj > 5 as “strong” [38].Figure 4 shows the Bayes factor, j lnBj, as a function of

the extra-ΛCDM parameter nT, for the case of the fðTÞ-plmodel. Red and blue lines, respectively, refer to the useof galaxy clustering and cosmic shear solely, while thecombination of the two yields the green curve. Horizontal,thin dotted lines indicate the boundaries of Jeffreys scaleconfidence levels. When the simpler model, M1, ispreferred by the data, the ratio in Eq. (47) is larger thanunity; j lnBj is thus positive (solid lines). Instead, if M2,the more complex model, has a larger posterior evidenceprobability, Eq. (47) is < 1, and the graph in Fig. 4 isconsequently negative (dashed lines).To make a clarifying example, if the reference Euclid-

like survey were to measure a nT value of 0.08, this wouldimply lnB ¼ −4.0, 0.96 and 1.0 for sole three- and two-dimensional clustering and shear, respectively. Occam’srazor for cosmic shear would therefore positively favorΛCDM—that is to say, the Euclid-like experiment wouldnot be able to decisively state the goodness of ΛCDM overmodified torsion gravity. Even worse, two-dimensionalclustering would be inconclusive, because of its weakconstraints on nT . Oppositely, σðnTÞ ¼ 0.021 coming fromgalaxy clustering would provide a moderate evidencetowards fðTÞ gravity. However, the complementarity ofgalaxy clustering and cosmic shear is such that the

combined forecast marginal errors are much tighter thanthose obtained by single probes. Indeed, for a measure ofnT ¼ 0.08, we would have lnB ¼ −30 (black line, dashedbranch), which falls into the strong confidence level of theJeffreys scale—a strong observational evidence in favor ofthe fðTÞ-pl model. In other words, the odds for modifiedtorsion gravity to ΛCDM would be ∼1013∶1.

VI. CONCLUSIONS

In this paper, we have analyzed cosmological modelsderived from modified torsion gravity theories, commonlyreferred to as fðTÞ cosmologies. Our aim has been toinvestigate the detectability of fðTÞ signatures via measure-ments of the growth and dynamics of the large-scale cosmicstructure. The motivation for this work effort is twofold.First, amongst the plethora of modified gravity theoriesproposed as solutions to the dark energy puzzle, fðTÞcosmologies represent an intriguing scenario, for they stillgive second-order equations in field derivatives, oppositelyto the fourth-order equations of fðRÞ gravity. Moreover,fðTÞmodels violate Lorentz invariance and do not thereforebelong to the family of Horndeski theories. Thus, they areworth being scrutinized, since constraints on HorndeskiLagrangians will not be able to confirm or rule out modifiedtorsion gravity. Secondly, a vast number of experimentsaimed at probing the properties of the Universe’s large-scalestructure are close to becoming a reality. This is a necessary

FIG. 4 (color online). j lnBj vs the fðTÞ-pl nT parameter fortwo- and three-dimensional galaxy clustering (red and bluecurves, respectively) and cosmic shear alone (red curve), andfor the full, combined two-dimensional clustering and shear(black curve). Solid (dashed) curves refer to B > 1 (B < 1).Jeffreys scale confidence levels fall between horizontal, thindotted lines.

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further step in the understanding of the cosmos, since thebackground evolution of the Universe seems to be in goodagreement with the ΛCDM paradigm. Therefore, the regimeof cosmological perturbations is the only arena where wemight detect deviations from it.For those reasons, we have focused on two viable fðTÞ

models. The first model, dubbed “exp,” has been proven tobe in good agreement with data concerning the backgroundcosmic evolution, and its model parameter values havebeen accordingly fitted [27]. The second model, baptized“pl,” is a power-law generalization of the teleparallelLagrangian and reduces to ΛCDM when its power nTvanishes; we have therefore taken nT ¼ 0 as fiducial.Regarding the experimental setup, we have chosen theESA Euclid satellite as a reference survey. By doing so, weare able to perform both galaxy clustering measurementsand cosmic shear tomography within the same experiment.This is utterly useful for our purpose, since the two probesare highly complementary, and each one helps in lifting theother’s own degeneracies.We have performed a Fisher matrix analysis to forecast

the survey constraining potential and estimate the errors onparameter measurements. To better understand the mostimportant aspects of the problem, we have firstly pursuedgalaxy clustering and cosmic shear alone. The forecast 1σmarginal errors on fðTÞ model parameters are presented inTables I, II, and III, the second and third of these for threesurveying configurations. Modified torsion gravity param-eters are quite differently constrained by clustering andweak lensing. This is an interesting and novel result, and itenables us to more deeply understand the properties of thefðTÞ models under investigation. For example, constraintson extra-ΛCDM parameters nT and pT of the fðTÞ-expmodel are poorly constrained by three-dimensional cluster-ing. This is because, over the redshift range 0.7–2.0 probedby the Euclid spectroscopic galaxy survey, the term T0=Tin the exponential in Eq. (13) quickly decreases. Hence, thefðTÞ term in the Lagrangian becomes subdominant. On thecontrary, the photometric imaging survey covers a widerredshift range, and this yields 5 to ∼10 times tighterconstraints on nT , and 4 to > 7 times on pT . This behavioris reversed when analyzing the fðTÞ-pl model: weaklensing error bounds are almost a order of magnitudebroader than those from galaxy clustering. This somehowunexpected result may be explained by the strong

degeneracies amongst the nT slope and, particularly, back-ground-related ΛCDM parameters.The primary result of this work, however, comes from

the combination of clustering and shear, presented inSec. IV D. The high complementarity of the two cosmo-logical observables yields an impressive enhancement inthe survey constraining power. This can be clearly seen inFig. 3, where we show the forecast 1σ two-parametermarginal contours on fðTÞ-exp and pl model parameters inthe ðϑα; ϑβÞ planes. Lighter (darker) ellipses refer to galaxyclustering (cosmic shear), whereas the combination of thetwo Euclid observables is depicted by the smallest, darkestellipses. As a final result, we can quote the final 68.3%marginal errors: σðnTÞ ¼ 0.063 and σðpTÞ ¼ 0.14 for theexp model, and σðnTÞ ¼ 0.0097 for the pl model.Eventually, in Sec. V we have made use of the reference

Euclid experiment as a tool for model selection. Thecalculation of the Bayes factor—the ratio of the posteriorevidence probabilities—of two competing models allowsthe (predicted) data to decide whether one model is favoredover the other. In a sense, it provides us with a usefulBayesian Occam’s razor that can assess which theoreticalframework is preferred given the data—without computingthe chi squared, for it will always reduce if we add moreparameters. Within this framework, we compared theconcordance ΛCDM model and the fðTÞ-pl model; theformer is in fact formally a subclass of the latter withnT ¼ 0. Specifically, we have found that if Euclid measuresa nonzero value for nT of a few percent, there will be astrong evidence in favor of modified torsion gravity overΛCDM (see Fig. 4).

ACKNOWLEDGMENTS

We thank Pedro G. Ferreira for valuable comments andBianca Garilli for the spectroscopic redshift distribution ofEuclid galaxies. S. C. is funded by FCT-Portugal underPostdoctoral Grant No. SFRH/BPD/80274/2011. V. F. C. isfunded by the Italian Space Agency (ASI) through ContractNo. Euclid-IC (I/0.31/10/0). V. F. C. acknowledges finan-cial contribution from Agreement No. ASI/INAF/I/023/12/0.N. R. wishes to thank Agenzia Spaziale Italiana (ASI) forpartial support. S. C. gratefully acknowledges CompositaCreative Collective for hospitality during the developmentof this project.

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