Cosmological Hints of Modified Gravity ?
CCosmological Hints of Modified Gravity ?
Eleonora Di Valentino, Alessandro Melchiorri, and Joseph Silk
1, 3, 4, 5 Institut d’Astrophysique de Paris (UMR7095: CNRS & UPMC- Sorbonne Universities), F-75014, Paris, France Physics Department and INFN, Universit`a di Roma “La Sapienza”, Ple Aldo Moro 2, 00185, Rome, Italy AIM-Paris-Saclay, CEA/DSM/IRFU, CNRS, Univ. Paris VII, F-91191 Gif-sur-Yvette, France Department of Physics and Astronomy, The Johns Hopkins University Homewood Campus, Baltimore, MD 21218, USA BIPAC, Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
The recent measurements of Cosmic Microwave Background temperature and polarizationanisotropies made by the Planck satellite have provided impressive confirmation of the ΛCDMcosmological model. However interesting hints of slight deviations from ΛCDM have been found,including a 95% c.l. preference for a ”modified gravity” structure formation scenario. In this paperwe confirm the preference for a modified gravity scenario from Planck 2015 data, find that modifiedgravity solves the so-called A lens anomaly in the CMB angular spectrum, and constrains the am-plitude of matter density fluctuations to σ = 0 . +0 . − . , in better agreement with weak lensingconstraints. Moreover, we find a lower value for the reionization optical depth of τ = 0 . ± . τ = 0 . ± .
017 obtained in the standard scenario), moreconsistent with recent optical and UV data. We check the stability of this result by consideringpossible degeneracies with other parameters, including the neutrino effective number, the runningof the spectral index and the amount of primordial helium. The indication for modified gravity isstill present at about 95% c.l., and could become more significant if lower values of τ were to befurther confirmed by future cosmological and astrophysical data. When the CMB lensing likelihoodis included in the analysis the statistical significance for MG simply vanishes, indicating also thepossibility of a systematic effect for this MG signal. PACS numbers: 98.80.-k 95.85.Sz, 98.70.Vc, 98.80.Cq
I. INTRODUCTION
The recent measurements of Cosmic Microwave Back-ground (CMB) anisotropies by the Planck satellite ex-periment [1, 2] have fully confirmed, once again, the ex-pectations of the standard cosmological model based oncold dark matter, inflation and a cosmological constant.While the agreement is certainly impressive, somehints for deviations from the standard scenario haveemerged that certainly deserve further investigation. Inparticular, an interesting hint for ”modified gravity”(MG hereafter), i.e. a deviation of the growth of densityperturbations from that expected under General Rela-tivity (GR hereafter), has been reported in [3] using aphenomenological parametrization to characterize non-standard metric perturbations.In past years, several authors (see e.g. [3–18]) haveconstrained possible deviations of the evolution of pertur-bations with respect to the ΛCDM model, by parametriz-ing the gravitational potentials Φ and Ψ and their linearcombinations. Considering the parameter Σ, that mod-ifies the lensing/Weyl potential given by the sum of theNewtonian and curvature potentials Ψ + Φ, the analysisof [3] reported the current value of Σ − . ± . − . +0 . − . (again, see [3]).This result is clearly interesting and should be furtherinvestigated. Small systematics may still certainly be present in the data and a further analysis, expected by2016, from the Planck collaboration could solve the issue.In the meantime, it is certainly timely to independentlyreproduce the result presented in [3] and to investigateits robustness, especially in view of other anomalies andtensions currently present in cosmological data.Indeed, another anomaly seems to be suggested by thePlanck data, i.e. the amplitude of gravitational lensingin the angular spectra. This quantity, parametrized bythe lensing amplitude A lens as firstly introduced in [19],is also larger than expected at the level of two standarddeviations. The Planck+LowP analysis of [2] reports thevalue of A lens = 1 . ± .
10 at 68% c.l.. This anomalypersists even when considering a significantly extendedparameter space as shown in [20]. It is therefore manda-tory to check if this deviation is in some way connectedwith the ”Σ ” anomaly performing an analysis by vary-ing both parameters at the same time. This has beensuggested but not actually done in [3].Moreover, some mild tension seems also to be presentbetween the large angular scale Planck LFI polarizationdata (that, alone, provides a constraint on the opticaldepth τ = 0 . ± .
023 [2]) and the Planck HFI small-scale temperature and polarization data that, when com-bined with large-scale LFI polarization, shifts the con-straint to τ = 0 . ± .
017 [2]. Since the Planck con-straints on τ are model-dependent, is meaningful to checkif the assumption of MG could, at least partially, resolvethe ” τ ” tension.Another tension concerns the amplitude of the r.m.s.density fluctuations on scales of 8 Mpc h − , the so-called σ parameter. The constraints on σ derived by the a r X i v : . [ a s t r o - ph . C O ] D ec Planck data under the assumption of GR and Λ-CDMare in tension with the same quantity observed by lowredshift surveys based on clusters counts, lensing andredshift-space distortions (see e.g. [21] and [2]). This ten-sion appears most dramatic when considering the weaklensing measurements provided by the CFHTLenS sur-vey (see discussion in [3]), which prefer lower values of σ with respect to those obtained by Planck. Several so-lutions to this mild tension have been proposed, includ-ing dynamical dark energy [22], decaying dark matter[23, 24], ultralight axions [25], and voids [26]. It is there-fore timely to further check if the ” σ tension” could bereconciled by assuming MG. This approach has alreadybeen suggested, for example, by [17].Finally, there are also extra parameters such as therunning of the spectral index dn S /dlnk , the neutrino ef-fective number N eff (see e.g. [27]), and the helium abun-dance Y p (see e.g. [28]) that could be varied and thatcould in principle be correlated with MG. Since the val-ues of these parameters derived under Λ-CDM (see [2])are consistent with standard expectations, it is crucialto investigate whether the inclusion of MG could changethese conclusions.This paper is organized as follows: in the next sectionwe describe the MG parametrization that we consider,while in Section III we describe the data analysis methodadopted. In Section IV, we present our results and inSection V we derive our conclusions. II. PERTURBATION EQUATIONS
Let us briefly explain here how MG is implemented inour analysis, discussing the relevant equations. Assum-ing a flat universe, we can write the line element of theFriedmann-Lemaitre-Robertson-Walker (FLRW) metricin the conformal Newtonian gauge as: ds = a ( τ ) [ − (1 + 2Ψ) dτ + (1 − dx i dx i ] , (1)where a is the scale factor, τ is the conformal time, Ψis the Newton’s gravitational potential, and Φ the spacecurvature .Given the line element of the Eq. 1, we can use a phe-nomenological parametrization of the gravitational po-tentials Ψ and Φ and their combinations. We considerthe parametrization used in the publicly available code MGCAMB [31, 32], introducing the scale-dependent function µ ( k, a ), that modifies the Poisson equation for Ψ: k Ψ = − πGa µ ( k, a ) ρ ∆ , (3) In the synchronous gauge, that is the one adopted in boltzmanncodes as
CAMB [29], we have: ds = a ( τ ) [ − dτ + ( δ ij + h ij ) dx i dx j ] , (2)where h ij are defined as in [30]. where ρ is the dark matter energy density, ∆ is the co-moving density perturbation. Furthermore one can con-sider the function η ( k, a ), that takes into account thepresence of a non-zero anisotropic stress: η ( k, a ) = ΦΨ . (4)We can then easily introduce the function Σ( k, a ), whichmodifies the lensing/Weyl potential Φ+Ψ in the followingway: − k (Φ + Ψ) ≡ πGa Σ( k, a ) ρ ∆ , (5)and that can be obtained directly from µ ( k, a ) and η ( k, a )as Σ = µ η ) . (6)Of course, if we have GR then µ = η = Σ = 1.It is now useful to give an expression for µ and η . Fol-lowing Ref.[3], we parametrize µ and η as: µ ( k, a ) = 1 + f ( a ) 1 + c ( λH/k ) λH/k ) ; (7) η ( k, a ) = 1 + f ( a ) 1 + c ( λH/k ) λH/k ) , (8)where H = ˙ a/a is the Hubble parameter, c and c areconstants and the f i ( a ) are functions of time that char-acterize the amplitude of the deviation from GR.Again, following [3] we choose a time dependence forthese functions related to the dark energy density: f i ( a ) = E ii Ω DE ( a ) , (9)where E ii are, again, constants and Ω DE ( a ) is the darkenergy density parameter. As discussed in Ref. [3], theinclusion of scale dependence does not change signifi-cantly the results, we can therefore consider the scaleindependent parametrization, in which c = c = 1.In other words, we modify the publicly available code MGCAMB [31, 32], by substituting to the original µ and η ,the following parametrizations: µ ( k, a ) = 1 + E Ω DE ( a ) ; (10) η ( k, a ) = 1 + E Ω DE ( a ) . (11)A detection of E ii (cid:54) = 0 could therefore indicate a depar-ture of the evolution of density perturbations from GR.In order to further simplify the problem, we assume acosmological constant for the background evolution. III. METHOD
We consider flat priors listed in Table I on all the pa-rameters that we are constraining. They are: the six
Parameter PriorΩ b h [0 . , . c h [0 . , . s [0 . , τ [0 . , . n s [0 . , . A s ] [2 , E [ − , E [ − . , dn s dlnk [-1,1] N eff [0.05,10] A lens [0,10] Y P [0.1,0.5]TABLE I: External flat priors on the cosmological parametersassumed in this paper. parameters of the ΛCDM model, i.e. the Hubble con-stant H , the baryon Ω b h and cold dark matter Ω c h energy densities, the primordial amplitude and spectralindex of scalar perturbations, A s and n s respectively, (atpivot scale k = 0 . hM pc − ), and the reionization op-tical depth τ ; the constant parameters of MG, E and E ; the several extensions to ΛCDM model. In partic-ular we vary the neutrino effective number N eff (see e.g.[27]), the running of the scalar spectral index dn S /dlnk ,the primordial Helium abundance Y P and the lensing am-plitude in the angular power spectra A lens . We also varyforeground parameters following the same method of [33]and [2].We constrain these cosmological parameters by usingrecent cosmological datasets. First of all, we consider thefull Planck 2015 release on temperature and polarizationCMB angular power spectra, including the large angularscale temperature and polarization measurement by thePlanck LFI experiment and the small-scale temperatureand polarization spectra by Planck HFI. We refer to thePlanck HFI small angular scale temperature data pluslarge angular scale Planck LFI temperature and polar-ization data as Planck TT , while when we include smallangular scale polarization from Planck HFI as
Planck pol (see [33]). We also use information on CMB lensing fromPlanck trispectrum data (see [34]) and we refer to thisdataset as lensing . Finally, we consider the weak lensinggalaxy data from the CFHTlenS [35] survey with the pri-ors and conservative cuts to the data as described in [2]and we refer to this dataset as
W L .To perform the analysis, we use our modified version,according to the Eqs. 10, of the publicly available code
MGCAMB [31, 32] that modifies the original publicly code
CAMB [29] implementing the pair of functions µ ( a, k ) and η ( a, k ), as defined in [32]. This code has been developedand tested in a completely independent way to the oneused in [3].We integrate MGCAMB in the latest July 2015 version ofthe publicly available Monte Carlo Markov Chain pack-age cosmomc [36] with a convergence diagnostic based onthe Gelman and Rubin statistic. This version includes Σ − τ Planck TTPlanck pol Σ − H Planck TTPlanck pol
FIG. 1: Constraints at 68% and 95% confidence levels on theΣ − τ plane (top panel) and on the Σ − H plane(bottom panel) from the Planck TT and Planck pol datasets.The 6 parameters of the ΛCDM model are varied. Noticethat Σ is different from one (dashed vertical line) at about95 % confidence level. A small degeneracy is present betweenΣ and τ : smaller optical depths are more compatible withthe data if Σ is larger than one (see top panel). Another de-generacy is present with the Hubble constant: larger values ofthe Hubble constant are more compatible with the considereddata in case of Σ different from one (bottom panel). Σ − A L Planck TTPlanck pol
FIG. 2: Constraints at 68% and 95% confidence levels onthe Σ − A lens plane from the Planck TT and Planckpol datasets. A strong degeneracy is present between Σ and A lens : larger values of A lens are more compatible with thedata if Σ is smaller than one. Planck TT Planck TT + WL Planck TT + lensing Planck pol Planck pol + WL Planck pol + lensing E . +0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . E . +1 . − . . +1 . − . . +0 . − . . +1 . − . . +1 . − . . +0 . − . µ − . +0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . η − . +0 . − . . +1 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Σ − . ± .
15 0 . +0 . − . . +0 . − . . ± .
13 0 . ± .
13 0 . +0 . − . Ω b h . ± . . ± . . ± . . ± . . ± . . ± . c h . ± . . ± . . ± . . ± . . ± . . ± . H . ± . . ± . . ± .
99 67 . ± .
71 68 . ± .
69 67 . ± . τ . ± .
021 0 . +0 . − . . ± .
019 0 . ± .
020 0 . ± .
019 0 . ± . n s . ± . . ± . . ± . . ± . . ± . . ± . σ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . ± . E and E )and varying the 6 parameters of the standard ΛCDM model. Σ − N e ff Planck TTPlanck pol
FIG. 3: Constraints at 68% and 95% confidence levels onthe Σ − N eff plane from the Planck TT and Planckpolarization datasets. Notice that Σ is different from unity(dashed vertical line) at about the 95 % confidence level. Asmall direction of degeneracy is present between Σ and N eff :larger N eff are more compatible with the data if Σ is largerthan one in case of the Planck TT dataset. the support for the Planck data release 2015 LikelihoodCode [33] (see http://cosmologist.info/cosmomc/ )and implements an efficient sampling using the fast/slowparameter decorrelations [37]. IV. RESULTS
We first report the results assuming a modified grav-ity scenario parametrized by η and µ and varying onlythe 6 parameters of the standard ΛCDM model. Theconstraints on the several parameters are reported in Ta-ble II. When comparing the first and second column ofour table, we see a complete agreement with the resultspresented in the first and third column of Table 6 of [3].Namely we find evidence at ∼
95% c.l. for Σ − , alsoslightly shifting its value towards a better compatibilitywith standard ΛCDM. We can see however that the in-clusion of small angular scale polarization does not altersubstantially the conclusions obtained when using justthe Planck TT dataset.Considering just the Planck TT dataset, it is interest-ing to note that in this modified gravity scenario, theHubble constant is constrained to be H = 68 . ± . Planck TT Planck TT + WL Planck TT + lensing Planck pol Planck pol + WL Planck pol + lensing E . +0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . E . +1 . − . . +1 . − . . +1 . − . . +1 . − . . ± . . +1 . − . µ − . +0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . η − . +0 . − . . +1 . − . . +0 . − . . +0 . − . . ± .
81 0 . +0 . − . Σ − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Ω b h . ± . . ± . . ± . . ± . . ± . . ± . c h . ± . . ± . . ± . . ± . . ± . . ± . H . ± . . ± . . ± . . ± .
73 68 . ± .
69 67 . ± . τ . +0 . − . . ± .
021 0 . ± .
021 0 . ± .
020 0 . +0 . − . . ± . n s . ± . . ± . . ± . . ± . . ± . . ± . σ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . A lens . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . TABLE III: Constraints at 68% c.l. on the cosmological parameters assuming modified gravity (parametrized by E and E )and varying the 6 parameters of the standard ΛCDM model plus A lens . at 68% c.l., i.e. a value significantly larger than the H = 67 . ± .
96 at 68% c.l. reported by the Planckcollaboration assuming ΛCDM. Combining the PlanckTT dataset with the HST prior of H = 73 . ± . − . +0 . − . at68% c.l..Moreover, the amplitude of the r.m.s. mass densityfluctuations σ in our modified gravity scenario is con-strained to be σ = 0 . +0 . − . at 68% c.l., i.e. a valuesignificantly weaker (and shifted towards smaller values)than the value of σ = 0 . ± .
014 at 68% c.l. re-ported by the Planck collaboration again under ΛCDMassumption.Considering the Planck pol dataset, the value of the op-tical depth is also significantly smaller in the MG scenario( τ = 0 . ± .
020 at 68% c.l.) respect to the value ob-tained under standard ΛCDM model of τ = 0 . ± . τ ∼ .
05 is inbetter agreement with recent optical and UV astrophys-ical data (see e.g. [44–46]) and the reionization scenariospresented in [48]. A value of τ > .
07 could imply un-expected properties for high-redshift galaxies. Assumingan external gaussian prior of τ = 0 . ± .
01 (at 68 % c.l..) as in [48] that would consider in a conservative wayreionization scenarios where the star formation rate den-sity rapidly declines after redshift z ∼ − . ± .
14 at 68% c.l., i.e. furtherimproving current hints of MG. In this respect, future,improved, constraints on the value of τ from large-scalepolarization measurements as expected from the PlanckHFI experiment will obviously provide valuable informa-tion.The degeneracies between Σ , H and τ can be clearlyseen in Figure 1 where we show the constraints at 68%and 95% confidence levels on the Σ − τ plane (toppanel) and on the Σ − H plane (bottom panel)from the Planck TT and Planck pol datasets. As wecan see, a degeneracy is present between Σ − τ :smaller optical depths are more compatible with the dataif Σ is larger than one (see top panel). As discussed, asecond degeneracy is present with the Hubble constant:larger values of the Hubble constant are more compatiblewith the considered data in case of Σ different from one(Bottom Panel).As already noticed in [3] and as we will discuss in thenext paragraph, the indication for MG from the Planckdata is strictly connected with the A lens anomaly, i.e.with the fact that Planck angular spectra show ”more Planck TT Planck TT + WL Planck TT + lensing Planck pol Planck pol + WL Planck pol + lensing E . +0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . E . ± . . +1 . − . . +0 . − . . +1 . − . . +1 . − . . +0 . − . µ − . +0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . η − . ± . . +1 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Σ − . ± .
18 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Ω b h . +0 . − . . +0 . − . . +0 . − . . ± . . ± . . ± . c h . ± . . +0 . − . . ± . . ± . . ± . . ± . H . +3 . − . . +3 . − . . +2 . − . . ± . . +1 . − . . ± . τ . +0 . − . . ± .
024 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . n s . +0 . − . . +0 . − . . +0 . − . . ± . . ± .
010 0 . ± . σ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . N eff . +0 . − . . +0 . − . . +0 . − . . ± .
20 3 . ± .
21 2 . ± . E and E )and varying the 6 parameters of the standard ΛCDM model plus N eff . lensing” than expected in the standard scenario. It istherefore not a surprise that when the Planck lensingdata (obtained from a trispectrum analysis) that is on thecontrary fully compatible with the standard expectationsis included in the analysis the indication for modifiedgravity is significantly reduced to less than one standarddeviation, as we can see from the third column of TableII. On the other hand, when weak lensing data from theWL dataset is included, the indication for MG increases,with Σ − A lens (Table III), the neutrino ef-fective number N eff (Table IV), the running of the scalarspectral index dn S /dlnk (Table V) and, finally, the He-lium abundance Y P (Table VI).As expected, there is a main degeneracy between the A lens parameter and Σ , as we can clearly see in Figure2 where we report the 2D posteriors at 68% and 95%c.l. in the Σ − A lens plane from the Planck TTand Planck pol datasets. In practice, the main effect ofa modified gravity model is to enhance the lensing signalin the angular power spectrum. The same effect can beobtained by increasing A lens and some form of degeneracyis clearly expected between the two parameters. As we see from the results in Table III, the value of the A lens parameter, when MG is considered, is A lens = 1 . +0 . − . ,fully consistent with 1, while for the standard ΛCDMthe constraint is A lens = 1 . +0 . − . at 68% c.l.. Whenalso varying A lens we found that the Planck pol datasetsconstraint the optical depth to τ = 0 . ± .
020 at 68%c.l.On the other hand, by looking at the results in Ta-bles IV, V, VI we do not see a significant degeneracybetween the MG parameters and the new extra param-eters. A small degeneracy is however present betweenΣ and the effective neutrino number N eff . We see fromTable IV that Planck TT data provides the constraint N eff = 3 . +0 . − . at 68% c.l. that should be comparedwith N eff = 3 . +0 . − . at 68% c.l. from the same datasetbut assuming the standard ΛCDM model. While the pos-sibility of an unknown ”dark radiation” component (i.e. N eff > . − N eff planes arereported in Figure 3.We also consider the possibility of a running of thescalar spectral index dn S /dlnk . Results are reported in Planck TT Planck TT + WL Planck TT + lensing Planck pol Planck pol + WL Planck pol + lensing E . +0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . E . +1 . − . . +1 . − . . +0 . − . . +1 . − . . +1 . − . . +0 . − . µ − . +0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . η − . +1 . − . . +1 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Σ − . ± .
18 0 . +0 . − . . +0 . − . . +0 . − . . ± .
13 0 . +0 . − . Ω b h . +0 . − . . +0 . − . . ± . . ± . . ± . . ± . c h . ± . . ± . . ± . . ± . . ± . . ± . H . +1 . − . . +1 . − . . ± . . ± .
72 68 . ± .
70 67 . ± . τ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . ± . n s . ± . . ± . . ± . . ± . . ± . . ± . σ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . dn s dlnk − . +0 . − . − . +0 . − . . ± . − . ± . − . ± . . ± . E and E )and varying the 6 parameters of the standard ΛCDM model plus dn S /dlnk . Table V and we find no degeneracy with MG parameters.The Planck TT constraint of dn S /dlnk = − . +0 . − . at 68% c.l. is almost identical to the value dn S /dlnk = − . ± . Y P since it affects small angular scaleanisotropies. Our results are in Table VI. The PlanckTT constraint is found to be Y P = 0 . ± .
023 at 68%c.l., slightly larger than the standard ΛCDM value of Y p = 0 . ± .
021 at 68% c.l. obtained using the samedataset. While a larger helium abundance is in betteragreement with recent primordial helium measurementsof [41], it is important to stress that the inclusion of po-larization yields a constraint that is almost identical tothe one obtained under ΛCDM. The constraints at 68%and 95% c.l. in the Σ − dn S /dlnk and Σ − Y P planes are reported in Figure 4. V. CONCLUSIONS
In this paper, we have further investigated the currenthints for a ”modified gravity” scenario from the recentPlanck 2015 data release. We have confirmed that thestatistical evidence for these hints, assuming the conser- vative dataset of Planck TT, is, at most, at ∼
95% c.l.,i.e. not extremely significant. The statistical significanceincreases when combining the Planck datasets with theWL cosmic shear dataset. Indeed, the Planck datasetseems to provide lower values for the σ parameter withrespect to those derived under the assumption of GR andΛ-CDM.If future astrophysical or cosmological measurementswill point towards a lower value of the optical depth of τ ∼ .
05 or of the r.m.s. amplitude of mass fluctuationsof σ ∼ .
78 then the current hints for modified gravitycould be further strenghtened.However it also important to stress that when the CMBlensing likelihood is included in the analysis the statisti-cal significance for MG simply vanishes.We also investigated possible degeneracies with extra,non-standard parameters as the neutrino effective num-ber, the running of the scalar spectral index and the pri-mordial helium abundance showing that the results onthese parameters assuming ΛCDM are slightly changedwhen considering the Planck TT dataset. Namely, undermodified gravity we have larger values for the neutrinoeffective number, N eff = 3 . +0 . − . at 68% c.l., and forthe helium abundance, Y p = 0 . ± . Planck TT Planck TT + WL Planck TT + lensing Planck pol Planck pol + WL Planck pol + lensing E . +0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . E . ± . . +1 . − . . +0 . − . . +0 . − . . +1 . − . . +0 . − . µ − . +0 . − . − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . . +0 . − . η − . +1 . − . . +1 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Σ − . ± .
16 0 . +0 . − . . +0 . − . . +0 . − . . ± .
13 0 . +0 . − . Ω b h . +0 . − . . ± . . ± . . +0 . − . . ± . . ± . c h . ± . . ± . . ± . . ± . . ± . . ± . H . +1 . − . . +1 . − . . +1 . − . . +0 . − . . ± .
81 67 . ± . τ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . ± . n s . +0 . − . . ± .
015 0 . ± .
012 0 . ± . . ± . . ± . σ . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . ± . Y P . ± .
023 0 . ± .
023 0 . ± .
021 0 . ± .
014 0 . ± .
013 0 . ± . E and E )and varying the 6 parameters of the standard ΛCDM model plus Y P . the Planck HFI polarization data.We have clearly shown that the slight Planck hintsof MG are strongly degenerate with the anomalouslensing amplitude in the Planck CMB angular spectraparametrized by the A lens parameter. Indeed, the A lens anomaly disappears when MG is considered. Clearly, un-detected small experimental systematics could be the ori-gin of this anomaly. However our conclusions are thatmodified gravity could provide a physical explanation,albeit exotic, for this anomaly that has been pointed outalready in pre-Planck CMB datasets [49], was present inthe Planck 2013 data release [50] and seems still to bealive in the recent Planck 2015 release [2] .An extra parameter we have not investigated here isthe neutrino absolute mass scale Σ m ν . Since MG is de-generate with the A lens we expect that in a MG scenariocurrent constraints on the neutrino mass from CMB an-gular power spectra should be weaker. However a moredetailed computation is needed and we plan to investi-gate it in a future paper ([52]).During the submission process of our paper, anotherpaper appeared [53], claiming an indication for MG from cosmological data. The dataset used in that paper iscompletely independent from the one used here and theMG parametrization is also different. Clearly a possi-ble connection between the two results deserves futureinvestigation. VI. ACKNOWLEDGMENTS
It is a pleasure to thank Noemi Frusciante, MatteoMartinelli and Marco Raveri for useful discussions. JSand EdV acknowledge support by ERC project 267117(DARK) hosted by UPMC, and JS for support at JHU byNational Science Foundation grant OIA-1124403 and bythe Templeton Foundation. EdV has been supported inpart by the Institute Lagrange de Paris. AM acknowledgesupport by the research grant Theoretical AstroparticlePhysics number 2012CPPYP7 under the program PRIN2012 funded by MIUR and by TASP, iniziativa specificaINFN. [1] R. Adam et al. [Planck Collaboration], arXiv:1502.01582[astro-ph.CO]. [2] P. A. R. Ade et al. [Planck Collaboration], Σ − d n s / d l n k Planck TTPlanck pol Σ − Y P Planck TTPlanck pol
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