Constraining f(R) gravity with PLANCK data on galaxy cluster profiles
I. De Martino, M. De Laurentis, F. Atrio-Barandela, S. Capozziello
aa r X i v : . [ a s t r o - ph . C O ] M a y Mon. Not. R. Astron. Soc. , 000–000 (2013) Printed September 8, 2018 (MN L A TEX style file v2.2)
Constraining f(R) gravity with PLANCK data on galaxycluster profiles
I. De Martino , ⋆ , M. De Laurentis , , , F. Atrio-Barandela , S. Capozziello , F´ısica Te´orica, Universidad de Salamanca, 37008 Salamanca, Spain; email: [email protected]; [email protected] Department of Theoretical Physics, Tomsk State Pedagogical University (TSPU), pr. Komsomolsky, 75, Tomsk, 634041, Russia Dipartimento di Fisica, Universit`a di Napoli ”Federico II” INFN sez. di Napoli Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126 - Napoli, Italy
Accepted xxxx Yyyyber zz. Received xxxx Yyyymber zz; in original form xxxx Yyyyber zz
ABSTRACT
Models of f ( R ) gravity that introduce corrections to the Newtonian potential in theweak field limit are tested at the scale of galaxy clusters. These models can explainthe dynamics of spiral and elliptical galaxies without resorting to dark matter. Wecompute the pressure profiles of 579 galaxy clusters assuming that the gas is in hy-drostatic equilibrium within the potential well of the modified gravitational field. Thepredicted profiles are compared with the average profile obtained by stacking the dataof our cluster sample in the Planck foreground clean map SMICA. We find that the re-sulting profiles of these systems fit the data without requiring a dominant dark mattercomponent, with model parameters similar to those required to explain the dynamicsof galaxies. Our results do not rule out that clusters are dynamically dominated byDark Matter but support the idea that Extended Theories of Gravity could providean explanation to the dynamics of self-gravitating systems and to the present periodof accelerated expansion, alternative to the concordance cosmological model. Measurements based on Supernovae Type Ia (SNeIa)have indicated that the Universe has entered a period ofaccelerated expansion (Riess et al. 2004; Astier et al. 2006;Clocchiati 2006). Data on Cosmic Microwave Background(CMB) temperature anisotropies measured by the Wilkin-son Microwave Anisotropy Probe (WMAP) (Hinshaw et al.2013) and the Planck satellite (Planck Results XV 2013;Planck Results XVI 2013; Planck Results XX 2013;Planck Results XXI 2013), on Baryon Acoustic Oscillations(BAO) (Blake et al. 2011) and other observables, togetherwith SNeIa data, favor the concordance ΛCDM model. Inthis model, the energy component that drives the currentperiod of accelerated expansion is a cosmological constantΛ. The associated energy density is Ω Λ ≃ .
7, in units of thecritical density. The second most important component isDark Matter (DM), a matter component required to explainthe formation of galaxies and the emergence of Large ScaleStructure, with Ω DM ≃ .
26. More general models assumethat the acceleration is due to an evolving form of DarkEnergy (DE) characterized by an equation of state parame-ter ω − /
3. For these models, cosmological observationsindicate that w = − . +0 . − . (Planck Results XVI 2013),fully compatible with a cosmological constant ( w = − R , to a more general function. Thesimplest extensions are f ( R ) models, where the Lagrangianis a function (possibly analytic) of the Ricci scalar. In thesemodels, the higher order gravity terms introduced in theaction are responsible for the present period of acceleratedexpansion. In some Extended Theories of Gravity (ETG),the Newtonian limit is also modified and models have beenconstructed where the dynamics of galaxies can be ex-plained without requiring a DM component. For instance,analytic f ( R ) models give rise to Yukawa-like correctionto the gravitational potential (Capozziello & De Laurentis2011, 2012) that do not require DM to explain the flat rota-tion curves of spiral galaxies (Cardone & Capozziello 2011)or the velocity dispersion of ellipticals (Napolitano et al.2012). The constraints derived from planetary dynamicsare weak since the Yukawa correction is negligible at thosescales (Capozziello & Troisi 2005; Allemandi et al. 2005;Berry & Gair 2011). ETG also modify the hydrostatic equi-librium of stars: Capozziello et al. (2011), Farinelli et al.(2013) have compared the Lan´e-Endem solution of poly-tropic gases in both f ( R ) and general relativity and foundthem to be compatible while Capozziello et al. (2012) ana-lyzed Jeans instabilities in self-gravitating systems and stud-ied star formation in f ( R ) gravity.Since the exact functional form of the Lagrangian is un-known, theoretical considerations need to be complementedwith observations. Thus, it is important to test potential c (cid:13) I. De Martino, M. De Laurentis, F. Atrio-Barandela, S. Capozziello models using all available data. At present, clusters of galax-ies are the largest virialized objects in the Universe and offerthe opportunity to test these alternative theories of gravityon scales larger than galaxy scales. Using the mass profilesof clusters of galaxies, Capozziello, De Filippis & Salzano(2009) showed that ETG provide a fit to the distributionof baryonic matter (stars+gas) derived from X-ray obser-vations in 12 clusters without requiring DM. Nevertheless,in conventional cosmological models, the non-linear evolu-tion and virialization of self-gravitating objects is studied us-ing numerical simulations. f ( R ) models have a much largernumber of degrees of freedom and the study of galaxy andcluster formation requires more complex simulations, spe-cific for each particular Lagrangian. A first attempt to con-strain ETG using cluster abundances in numerical simula-tions has been carried out by Ferraro, Schimdt & Hu (2011)and Schimdt, Vikhlinin & Hu (2009). Other numerical con-straints on f ( R ) models can be found in Song, Hu & Sawicki(2007), Sawicki & Hu (2007), Hu & Sawicki (2007a,b) andLima & Liddle (2013). More promising is the study of tem-perature fluctuations on the CMB. Galaxy clusters arereservoirs of hot gas that induces anisotropies by meansof the Sunyaev-Zeldovich (SZ) effect (Sunyaev & Zeldovich1972, 1980). Pressure profiles of galaxies can be computedin ETG assuming that the gas is in hydrostatic equilib-rium within the potential well of clusters. This is in agree-ment with the results of numerical simulations based onthe concordance cosmology that showed that gas is in hy-drostatic equilibrium in the intermediate regions of clus-ters, while in the cluster cores, the physics of baryons ismore complex and in the outer regions it is dominated bynon-equilibrium processes (Kravtsov & Borgani 2012). Re-cently, hydro-numerical simulations are being carried out tostudy the properties of galaxy clusters and groups in ETG.Arnold, Puchwein & Springel (2013) showed that the intra-cluster medium temperature increases in f ( R ) gravity in lowmass halos but the difference disappears in massive objects.Based on these results we will assume that the physics ofthe gas will be weakly dependent on the underline theory ofgravity.The SZ anisotropies generated by individual clus-ters and by the unresolved cluster population have beenmeasured by the Atacama Cosmology Telescope (ACT)(Hand et al. 2011; Hasselfield et al. 2013; Menanteau et al.2013; Sehgal et al. 2011), the South Pole Telescope(SPT) (Benson et al. 2013; Staniszewsk et al. 2009;Williamson et al. 2011; Vanderlinde et al. 2010) and thePlanck satellite (Planck Intermediate Results V 2013;Planck Intermediate Results X 2013; Planck Results XX2013; Planck Results XXIX 2013). Gas profiles based onthe Navarro-Frenk-White (hereafter NFW, Navarro et al.(1997)) profile, derived from numerical simulations,have been found to be in agreement with TSZ(Atrio-Barandela et al. 2008) and X-ray observations(Arnaud et al. 2010). Nevertheless, the contributionof the unresolved cluster population in WMAP 7yrdata has been found to be smaller than expectedbased on theoretical and numerical modeling of clus-ters (Komatsu et al. 2011). For the Coma cluster, theanalysis of Planck data (Planck Intermediate Results V2013; Planck Intermediate Results X 2013) finds a normal-ization of ∼ −
15% lower compared with the parameters derived from XMM observations. These discrepancies canbe related to the existence of complex structures and sub-structures in clusters of galaxies as well as to the limitationsof the theoretical modeling (Fusco-Femiano et al. 2013),that is the approach we are going to consider here.In this article, we will compare the pressure profiles ofclusters of galaxies in f ( R ) models with Planck data. Toconstruct the pressure profiles, we will assume that the gasis in hydrostatic equilibrium within the potential well gen-erated by the cluster. At this level, our assumption can notbe applied to models not in equilibrium like the Bullet clus-ter (Clowe et al. 2006). We will restrict our analysis to f ( R )models of gravity that introduce Yukawa corrections to theNewtonian potential in order to test if the dynamics of clus-ters of galaxies can be also described without a dominantdark matter component. The paper is organized as follows:in Sec. 2, we consider the weak field limit of f ( R ) gravityderiving the gravitational potential for self-gravitating ob-jects; in Sec. 3, we present the pressure profiles based onthe NFW profile and X-ray data most commonly used andwe compute the pressure profile for f ( R ) models; in Sec. 4we describe the data used in our analysis; in Sec. 5, we dis-cuss our results and, finally, in Sec. 6 we present our mainconclusions. F ( R ) -GRAVITY In f ( R ) ETG, field equations are derived from the action A = c πG Z d x √− gf ( R ) + L m , (1)yielding f ′ ( R ) R µν − f ( R )2 g µν − f ′ ( R ) ; µν + g µν (cid:3) g f ′ ( R ) = 8 πGT µν , where f ′ ( R ) = df ( R ) /dR is the first derivative with respectto the Ricci scalar, (cid:3) g = ; σ ; σ is the d’Alembertian with co-variant derivatives, T µν = − − g ) − / δ ( √− g L m ) /δg µν isthe matter energy-momentum tensor, T its trace, g the de-terminant of the metric tensor g µν . Greek indices run from0 to 3.We search for spherically symmetric solutions of theform ds = g tt c dt − g rr dr − r d Ω , (2)where d Ω is the solid angle. Let us restrict our study tothose f ( R )-Lagrangians that can be expanded in Taylor se-ries around a fixed point R f ( R ) = X n f n ( R ) n ! ( R − R ) n ≃ f + f ′ R + f ′′ R + ... . (3)The fixed point represents the Ricci-scalar in GR for thesame mass distribution. In this case f is a cosmologicalconstant and f ′ = 1. Then, the field Eqs. (2) can be solvedat different orders in terms of the Taylor expansion. In theNewtonian limit the first correction is of order c . The metric c (cid:13) , 000–000 onstraining f ( R ) gravity with PLANCK data on galaxy cluster profiles tensor can be written as g tt ≃ grav ( r ) , (4) g rr ≃ − (1 + Φ N ( r )) , (5) g θθ ≃ − r , (6) g φφ ≃ − r sin θ, (7)whereΦ N ( r ) = − GM ( r ) r , Φ grav ( r ) = Φ N ( r )(1 + δ ) (cid:16) δe − rL (cid:17) , (8)Analytic f ( R ) models that modify the Newtonian limit canbe seen as alternative to Dark Matter. The Yukawa correc-tion to the gravitational field allows us to study the dy-namics of galaxies without requiring dark matter. The pa-rameters ( δ , L) are related to the coefficients in the Tay-lor expansion as: f ′ = 1 + δ and L = [ − f ′ / (6 f ′′ )] / ,where δ represents the deviation from GR at zero or-der and L the scale length of the self-gravitating object(Capozziello & De Laurentis 2011, 2012). In the limit δ = 0,we recover the Newtonian limit of GR, irrespective of thescale parameter L . In ETG, L depends on the scale of thesystem considered; it assumes different values for the vari-ous self-gravitating systems like galaxies or cluster of galax-ies while its effects are totally negligible at Solar Systemscales where GR, i.e. the Newtonian limit, is totally restored(Capozziello & De Laurentis 2012).The physical meaning of the characteristic length L deserves further discussion. As pointed out inCapozziello & De Laurentis (2011), L can be seen as an ex-tra gravitational radius similar to the Schwarzschild radius.Compared with GR, that is a second order theory, f ( R )gravity is fourth-order and contains a larger number of de-grees of freedom that, in the weak field limit, give rise toa new characteristic scale length. The paradigm can be ex-tended to (2 k +2)-order theories of gravity so any further twoderivation orders imply a new characteristic length in theNewtonian limit (see Quandt & Schmidt (1991) for details),resulting in some important implications for the theory. Firstgravity is no longer a scale invariant interaction but dependson the size of the self-gravitating systems. In other words,gravitational corrections emerge depending on scales. Sec-ond, the Gauss theorem does not hold at finite scales butonly asymptotically. This is not a problem since Bianchiidentities hold for f ( R ) as for any ETG theory and con-servation laws are fulfilled like in GR. Third, GR is totallyrestored at Solar System scales so f ( R ) theory agrees withstandard classical tests (Capozziello & Tsujikawa 2008). Fi-nally, the approach allows to represent DM effects only bygravity without requiring new ingredients at the fundamen-tal level. This fact could be considered as an astrophysicaltestbed for relativistic theories of gravity since the additionalgravitational length L introduced in this model could be ac-curately matched with observational data as we are going toshow below. F ( R ) GRAVITY
Model c α a β a γ a P Arnaud . h / Planck
Sayers β β n c, /m − r c /Mpc T e /keV2/3 3860. 0.25 6.48 f ( R ) δ L/Mpc γ -0.98 0.1 1.2 Table 1.
Parameters of the Generalized NFW, β and f ( R ) mod-els represented in Fig. 1; the Generalized NFW pressure profiledata is from Arnaud et al. (2010), Planck Intermediate Results V(2013) and Sayers et al. (2013), the β -model data corresponds tothe Coma cluster and the f ( R ) profile data is the best fit modelto Planck data (see Sec. 5). When CMB photons cross the potential wells of clus-ters of galaxies, they are scattered off by the electrons ofthe Intra-Cluster medium, inducing secondary temperatureanisotropies on the CMB of two different type by meansof the SZ effect: a thermal contribution due to the motionof the electrons within the cluster potential well and kine-matic one (KSZ) due to the motion of the cluster as a whole.The TSZ is the only SZ anisotropy that has been measuredfor individual clusters and is given by (Sunyaev & Zeldovich1972) ∆ TT = g ( ν ) k B σ T m e c Z n e T e dl, (9)where T e is the electron temperature, n e the electron densityand the integration is carried out along the line of sight l . InEq. (9) k B is the Boltzmann constant, m e c the electron an-nihilation temperature, c the speed of light, ν the frequencyof observation, σ T Thomson cross section and T the meantemperature of the CMB. Finally, g ( ν ) = x coth( x/ − x = hν/KT .To compute the TSZ anisotropy we need to specify thepressure profile n e T e of clusters. Using X-ray data and nu-merical simulations, several cluster profiles have appeared inthe literature: • The X-ray emitting region of clusters of galax-ies is well fit by the isothermal β -model profiles(Cavaliere & Fusco-Femiano 1976, 1978). In this model, theelectron density is given by: n e ( r ) = n e, [1 + ( r/r c ) ] − β/ ,where the core radius r c , the central electron density n e, ,the electron temperature and the slope β need to be deter-mined from observations. From the X-ray surface brightnessof clusters, β = 0 . − . β model parameters of the Comacluster. • Outside the central cluster regions, the β modeloverpredicts the TSZ contribution (Atrio-Barandela et al.2008). If the electrons are in hydrostatic equilibriumwithin the potential well of dark matter halos, the pres-sure profile is well describe by a Komatsu-Seljak model(Komatsu & Seljak 2002; Atrio-Barandela et al. 2008).More recently, Arnaud et al. (2010) proposed a phenomeno- c (cid:13) , 000–000 I. De Martino, M. De Laurentis, F. Atrio-Barandela, S. Capozziello
Figure 1.
Pressure profiles integrated along the line of sight forthe Coma cluster. We represent three GNFW profiles (dashed,solid and dash-dotted lines), one β = 2 / f ( R ) model (red solid line). The model parametersare given in Table 1. The angular diameter distance is that of theComa cluster ( z = 0 . logical parametrization of the electron pressure profile basedon generalized Navarro-Frenk-White (GNFW) profiles de-rived from the numerical simulations of Nagai et al. (2007).This profile has the following functional form p ( x ) ≡ P ( c x ) γ a [1 + ( c x ) α a ] ( β a − γ a ) /α a , (10)In this expression x is the radial distance in units of r ,the radius where the average density is 500 times thecritical density, and c is the concentration parameterat r . Different groups have fit the model parameters[ c , α a , β a , γ a , P ] to X-ray or CMB data; their best fit val-ues are given in Table 1.GNFW models fit the DM distribution in numericalsimulations that use newtonian gravity and therefore cannot be used to describe the dynamics in the ETG we areconsidering. Instead, baryons reside in the potential well ofclusters. The Yukawa correction to the Newtonian potentialof eq. (8) modifies the gravitational structure of clusters andthere is not longer any need to introduce dark matter to ex-plain their dynamics. In this limit, to compute the pressureprofile n e T e of Eq. (9) we assume that the gas is in hydro-static equilibrium within the (modified) potential well of thecluster dP ( r ) dr = − ρ ( r ) d Φ grav ( r ) dr , (11)and to describe the physical state of the gas we further as-sume that it follows a polytropic equation of state P ( r ) ∝ ρ γ ( r ) . (12)Eqs. (11) and (12) together with mass conservation dM ( r ) dr = 4 πρ ( r ) , (13)and the cluster gravitational potential given by eq. (8) forma close system of equations that can be solved numericallyto obtain the pressure profiles of any given cluster as a func-tion of two gravitational parameter ( δ, L ) and the polytropicindex γ . For illustration, in Fig. 1 we plot the different pro-files integrated along the line of sight with the parameters given in Table 1. We particularize the models for the Comacluster. For convenience, all distances are written in units of r and the angular scale is θ = r /d ComaA where d ComaA is the angular diameter distance of Coma. Dashed, solid anddash-dotted lines correspond to GNFW profiles with theArnaud et al. (2010), Planck Intermediate Results V (2013)and Sayers et al. (2013) parameters, respectively. The long-dashed line corresponds to the β model and the red solidline to the f ( R ) model. To constrain the ETG model described in Sec. 2, we will usethe pressure profiles of clusters of galaxies given in Sec. 3.To that purpose we shall use Planck data and a proprietarycluster catalog.
Our cluster catalogue contains 579 clusters selected fromROSAT All Sky-Survey (RASS). Those clusters are out-side the minimal Planck mask that removes a ∼
20% ofthe sky in the Plane of the Galaxy. Clusters are drawn fromthe three flux limited cluster samples: the extended Bright-est Cluster Sample (eBCS, (Ebeling et al. 1998, 2000)),the ROSAT-ESO Flux Limited X-ray catalog (REFLEX,(B¨ohringer et al. 2004)), and the Clusters in the Zoneof Avoidance (CIZA, (Ebeling et al. 2002; Kocevski et al.2007)). For each cluster, the catalog lists position, flux, andluminosity measured directly from RASS data and spectro-scopically measured redshifts. The X-ray electron temper-ature is derived from the L X − T X relation of White et al.(1997). For each cluster, the spatial profile of the X-ray emit-ting gas is fit to a β -model convolved with the RASS point-spread function to the RASS data. Due to the poor sam-pling of the surface brightness profile for all but the mostnearby clusters, β is fixed to the canonical value of β = 2 / r c and centralelectron densities n e, are derived from the data. Thus, ourcatalog provides enough information to compute the Comp-tonization parameter of the X-ray emitting region of all theclusters in our sample. In Atrio-Barandela et al. (2008), itwas found that the predicted values and those measured inWMAP 3yr data were in agreement with the β model forthe inner part of the clusters, being the discrepancy betweenthe TSZ prediction and observation below 10%. The release of WMAP 9yr data (Bennet et al. 2013) at theend of 2012 was followed by the first data release of thePlanck satellite in April 2013. Nine maps spanning a fre-quency range from 32 to 845GHz have been made publiclyavailable by the Planck Collaboration . While the WMAPteam provided foreground clean maps of all Differencing As-semblies (DA), the Planck Collaboration did not validateforeground clean maps at all frequencies. Instead, they usedcomponent separation methods to construct a map of CMB http://irsa.ipac.caltech.edu/Missions/planck.htmlc (cid:13) , 000–000 onstraining f ( R ) gravity with PLANCK data on galaxy cluster profiles Figure 2.
Average temperature anisotropy in the SMICA (dia-monds) and NILC (triangles) maps at the position of two clustersubset selected according to luminosity and redshift (blue and redlines) and of our full sample (solid black line). temperature anisotropies combining the data at all frequen-cies (Planck Results XII 2013). The SMICA map was pro-duced by combining all nine Planck frequency maps, previ-ously upgraded to the same resolution of 5 ′ , in spherical har-monic space using different weights at different multipoles.The NILC map was constructed in needlet space given dif-ferent weights to the multipoles and to the spatial positionsof the data in the sky. These two maps were constructed us-ing different algorithms and, therefore, it is likely that theywill differ in amplitude, distribution and spatial propertiesof the foreground residuals (Planck Results XIII 2013). Wewill perform our analysis in both foreground clean maps,both with Healpix resolution N side = 2048 (Gorski et al.2005), to test for systematics.To compute the TSZ profile of the clusters in our sam-ple, we average the temperature anisotropy at the clusterpositions. At the cluster center the average is over a disc ofradius θ / θ is the angular scale subtended bythe r radius of the cluster. Outside the inner disc, we takethe average on rings of width θ /
2. The measured valueis the averaged over all clusters in our sample. The angularposition θ we associate to each data point is the mean of theangular distance to the center of the cluster of every pixel ina disc or ring. The root mean square dispersion around themean is about 0 . θ for the central disc that contains thesmallest number of pixels and is 0 . θ or smaller for therings. In Fig. 2, we present the results for the SMICA (dia-monds) and NILC (triangles) maps. We compare the resultson both maps for the full sample (solid black line) and fortwo cluster subsets, selected according to luminosity (blue)and redshift (red). The results of both maps differ by lessthan 1% in the three samples proving that the differences inthe component separation method do not distort the TSZanisotropy associated with clusters. The agreement betweenthe TSZ profiles measured in the SMICA and NILC mapsdemonstrates that systematic effects will not affect our finalresults. To each data point we associate an error bar obtainedby evaluating 1,000 times the average profiles at 579 randompositions in the SMICA and NILC maps. To avoid over-lapping real and simulated clusters, we excise a disc of 80 ′ around each cluster in our sample. The errors on both mapsare also indistinguishable. For comparison, we analyzed theW-band of WMAP 9yr data. The results were very similarto those of Planck except for larger error bars. As remarkedin Planck Results XII (2013), at high latitudes, outside theGalactic Plane, the amplitude of the foregrounds residualspresent on the SMICA map is a few µ K, smaller than thoseon the NILC map. Therefore, since NILC or WMAP do notprovide extra information and since they are more affectedby noise or foregrounds than SMICA, we will restrict ouranalysis to the latter data.
To compare cluster profiles with observations, we measurethe angle subtended by every cluster in units of θ . Foreach cluster, the radial scale r can be derived using thefollowing scaling relation B¨ohringer et al. (2007) r = 0 . h − Mpc h ( z ) × (cid:18) L X h − erg s − (cid:19) . , (14)The radius r will allow us to test if the characteristic scaleof our ETG, L , depends on the cluster properties or not. Wechecked that our results did not depend on the uncertaintiesof eq. (14) and we will not consider them any further. Sim-ilarly, we did not consider other scaling relations based ondifferent data (Piffaretti et al. 2011; Planck Early Results X2013). Eqs. (8), (11), (12) and (13) allow us to computethe pressure profile of all clusters in the data as a functionof three parameters: ( δ, L, γ ). These profiles are integratedalong the line of sight to be compared with those measuredin the SMICA map. As indicated in Sec. 2, L characterizesthe dependence of f ( R ) gravity on the size of the gravitat-ing system. We consider two parameterizations of L to testif the theory depends on the properties of the clusters: (A) L = ζr is different for each cluster but depends homo-geneously on r for the whole sample and (B) where L is the same for all clusters. In Fig. 3 we plot the pressureprofile integrated along of line of sight, convolved with agaussian beam of 5 ′ resolution, for different model parame-ters. Our models only predict the profile but not the centralanisotropy. For this reason, we normalize all our theoreticalprofiles to unity. The data is equally normalized by dividingall the averages by the mean temperature on a disc of 0 . θ radius. Error bars are computed in the same manner, renor-malizing the disc and rings at random positions on the skyby the mean on the central disc of 0 . θ . In Figs. 3a-c L isdifferent for each cluster (Model A) and in Figs. 3d-f L is thesame for all clusters (Model B). To avoid overcrowding theplots, we fixed γ = 1 .
2. In each panel we show the variationof the pressure profile with L . Notice that in Model B, when L > L is the scale length of the Yukawa correction,that becomes negligible for large values of L . For illustra-tive purposes we overplot the SMICA data shown in Fig. 2,normalized to unity, with their corresponding error bars. c (cid:13) , 000–000 I. De Martino, M. De Laurentis, F. Atrio-Barandela, S. Capozziello
Figure 3.
Pressure profiles of clusters in f ( R ) gravity and SMICA data. Panels a-c correspond to the parametrization L = ζr (ModelA), while d-f correspond to the same scale L for all clusters (Model B). To determine the model parameters that best fit the SMICAdata we generate pressure profiles for different values ofthe parameters ( δ, L, γ ), integrated along the line of sightand convolved with a Gaussian beam with the same resolu-tion of the SMICA map to compare them with the data.On physical grounds, we fix our parameter space to be δ = [ − . , .
0] since if δ < − δ = −
1. In the parametrization L = ζr we take ζ = [0 . , L is the same for all clusters, wefix the interval to be L = [0 . , γ = [1 . , . L = − χ / χ ( p ) = Σ Ni,j =0 ( y ( p, x i ) − d ( x i )) C − ij ( y ( p, x j ) − d ( x j )) (15)where N = 7 is the number of data points. The mean profile y ( p, x i ) of all the clusters in our sample depends on three parameters: p = ( δ, L, γ ). In eq. (15), d ( x i ) is the SMICAaverage profile and C i,j is the correlation function betweenbins. To compute the correlation function we choose 579random positions outside the locations of known clusters andcompute the average temperature anisotropy on discs andrings of size θ , different for each of the random clusters.The process is repeated 1,000 times and C ij is the averagecorrelation between bins of any given cluster, averaged overall clusters and all simulations.The value of the model A and B parameters that max-imize the likelihood are given in Table 2. In Fig. 4 weplot the 68% and 95% confidence contours for pairs of pa-rameters of Model A. Fig. 5 shows the same contours forthe Model B. Since the models are very similar to eachother, the likelihood function is flat close to the maximum.The 1 σ contours are cut by our physical boundaries on δ and γ . Consequently, 2D contours of the marginalized like-lihoods of pairs of parameters of these Figs are not closed,and only lower or upper limits to the parameters can bederived from their marginalize 1D likelihoods. At the 68%and 95% confidence levels those limits are δ < − . , − . ζ < . , . γ > . , .
12 for the Model A parametersand δ < − . , − . L < ,
19 Mpc and γ > . , . c (cid:13) , 000–000 onstraining f ( R ) gravity with PLANCK data on galaxy cluster profiles constrained. In particular, the polytropic index constraintsdominated by the physical boundary on this parameter.The characteristic scale length L is similar in both models,whether it scales with r or is identical for all clusters. Inretrospect, this explains why the results of model A did notdepend on the uncertainties in the scaling relation of r ,given in eq. (14). But even if the parameters are weakly con-strained, let us remark that in both models, A and B, thevalue δ = 0 is excluded at more than a 95% confidence level.Since δ ≃ δ = −
1. The limitation stems from the useof first order perturbations with respect to a backgroundmodel. The contours show that at the 1 σ level L is compat-ible with zero. Physically, at L ≃ M ′ = M/ (1+ δ ). As δ +1 ≃ M ′ ≫ M and the gravita-tional field is that of a system that contains a large fractionof DM distributed like the baryonic gas. Briefly, while ourresults show that cluster TSZ profiles in ETG are compati-ble with the data, they do not rule out that clusters couldcontain a significant fraction of DM. In summary, in orderto fit the TSZ data, clusters are either dominated by DM orthe Newtonian potential includes a Yukawa correction.Comparison of Figs. 4 and 5 also shows that thedata does not have enough statistical power to discrim-inate between Models A and B. Importantly, the resultsare consistent with those obtained by Sanders (1984) andNapolitano et al. (2012) using spiral and elliptical galax-ies, respectively. In model A we find the same correlationbetween the gravity parameters L and δ that in the caseof galaxies: to accommodate the data, larger values of L require lower values of δ , while the behavior is the oppo-site in Model B. This different scaling suggests that ModelA is in better agreement with the dynamics of galaxiesthan Model B. Also, conceptually is the preferable modelsince L scales with the size of the self-gravitating system.The agreement of the central values of δ and L with thoseof galaxies, that correspond to a different linear scale, isvery reassuring; the dynamics of galaxies and clusters canbe equally described by ETG, without requiring DM. Inother words, DM and alternative gravity models are equiv-alent descriptions that could be discriminated only by somesignature at fundamental scales, i.e. the discovery of newparticles non-interacting at electromagnetic level, or theclear evidence of some new gravitational mode not relatedto GR (Capozziello & De Laurentis (2011), Bogdanos et al.(2010)).For comparison, we also compute the likelihood ofeach of the models given in Table 1 and their χ per de-gree of freedom are given in Table 2. For the β modelwe generate the profile of each cluster using the data ofour catalog. The β model does not produce a good fitto the data, in agreement with our previous results us-ing WMAP data (Atrio-Barandela et al. 2008), since thismodel only fits the X-ray emitting regions of the innerparts of clusters. Comparing the three GNFW parameters,the Arnaud et al. (2010) parameters, derived using the X-ray data of 33 clusters, performs better that either the Planck Intermediate Results V (2013) or the Sayers et al.(2013) parameters, that were obtained from TSZ observa-tions. These discrepancies are not relevant since we did notexplore the parameter space to find the best fit values ofGNFW models to the SMICA data. Nevertheless, the factthat our f ( R ) profiles fit significantly better than any othermodel is a clear indication that our assumption of a poly-tropic gas in hydrostatic equilibrium in the cluster potentialwell is supported by the data. We have constructed cluster pressure profiles based on theYukawa-like correction to the Newtonian potential obtainedin the weak field approximation of f ( R ) gravity. These mod-els do not require large fractions of DM and they have beenshown to describe well the dynamics of spiral and ellipti-cal galaxies. By fitting the pressure profiles measured in theforeground clean SMICA map released by the Planck Collab-oration, we have found that clusters can also be accuratelydescribed in these models. We have used a proprietary cata-log of 579 clusters, and have determined the parameter spacethat best fits data. Our results are predicated on the bary-onic gas being in hydrostatic equilibrium in the potentialwells of clusters. This hypothesis can only be tested usinghydrodynamical simulations and if the gas turn out not tobe in equilibrium, our conclusion will be severely weaken.Models based on f ( R )-gravity that do not require DMhalos appear as a viable alternative to generalized NFWmodels. Due to foreground contamination, we cannot usesingle frequency maps. For instance, the 217GHz channelcould be used to remove the intrinsic CMB component andthe signal at other frequencies could be fit to the profile ofeach individual clusters. Lacking frequency information in-creases our error bars and makes our final contours widerthan what they would be otherwise. Then, the constraintsfrom pressure profiles could be further improve by using fre-quency information, by carrying out the analysis in fore-ground clean maps, using the 217GHz map to remove thecosmological CMB signal and fitting the profile of each in-dividual cluster to the data. The conclusion of this and sim-ilar studies (Cardone & Capozziello 2011; Napolitano et al.2012) is that large amounts of DM are not required to de-scribe self-gravitating systems, if we relax the hypothesisthat gravity is strictly scale independent above the scale ofSolar System. ACKNOWLEDGMENTS
IDM and FAB acknowledge financial support from theSpanish Ministerio de Educaci´on y Ciencia (grant FIS2012-30926). SC and MDL acknowledge INFN (iniziative speci-fiche NA12 and OG51).
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