On the universality of MOG weak field approximation at galaxy cluster scale
aa r X i v : . [ a s t r o - ph . C O ] M a y On the universality of MOG weak field approximation at galaxy clusterscale
Ivan De Martino a, ∗ , Mariafelicia De Laurentis b,c,d a Department of Theoretical Physics and History of Science, University of the Basque Country UPV/EHU, Faculty ofScience and Technology, Barrio Sarriena s/n, 48940 Leioa, Spain b Institute for Theoretical Physics, Goethe University, Max-von-Laue-Str. 1, D-60438 Frankfurt, Germany c Tomsk State Pedagogical University, ul. Kievskaya, 60, 634061 Tomsk, Russia d Lab.Theor.Cosmology,Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia
Abstract
In its weak field limit, Scalar-tensor-vector gravity theory introduces a Yukawa-correction to the gravitationalpotential. Such a correction depends on the two parameters, α which accounts for the modification of thegravitational constant, and µ ∗− which represents the scale length on which the scalar field propagates. Theseparameters were found to be universal when the modified gravitational potential was used to fit the galaxyrotation curves and the mass profiles of galaxy clusters, both without Dark Matter. We test the universalityof these parameters using the the temperature anisotropies due to the thermal Sunyaev-Zeldovich effect.In our model the intra-cluster gas is in hydrostatic equilibrium within the modified gravitational potentialwell and it is described by a polytropic equation of state. We predict the thermal Sunyaev-Zeldovichtemperature anisotropies produced by Coma cluster, and we compare them with those obtained using thePlanck 2013 Nominal maps. In our analysis, we find α and the scale length, respectively, to be consistentand to depart from their universal values. Our analysis points out that the assumption of the universality ofthe Yukawa-correction to the gravitational potential is ruled out at more than 3 . σ at galaxy clusters scale,while demonstrating that such a theory of gravity is capable to fit the cluster profile if the scale dependenceof the gravitational potential is restored. Keywords:PACS: ∗ Corresponding author
Email addresses: [email protected] (Ivan De Martino), [email protected] (Mariafelicia DeLaurentis)
Preprint submitted to Physic Letters B July 17, 2018 . Introduction
Scalar-Tensor-Vector Gravity theory (STVG),also known as MOdified Gravity (MOG), addsscalar, tensor and massive vector fields to the stan-dard Hilbert-Einstein action [1, 2]. In particular,the mass of the MOG vector field and its strengthare governed by two running constants, α and µ ∗ ,that are promoted to scalar fields and can be con-strained by data.Similarly to f ( R ) gravity [3–5], MOG theory in-troduces a Yukawa-like correction to the Newtoniangravitational potential in its weak field limit [2].Specifically, the modified gravitational potential is[6]:Φ eff ( ~x ) = − G N Z ρ ( ~x ′ ) | ~x − ~x ′ | h α − αe − µ ∗ | ~x − ~x ′ | i d ~x ′ , (1)where µ ∗ is the inverse of the characteristic lengthof the modified gravitational potential, that acts ata certain scale for the self-gravitating systems, and α = ( G ∞ − G N ) /G N accounts the modification ofthe Newton constant [7], where G ∞ is the effectivegravitational constant at infinity.At cosmological scales, MOG correctly predictsaccelerated expansion of the universe and the emer-gence of the Large Scale structure [8–11]. At muchsmaller scales, it is able to correctly predict theTully-Fisher relation and the galaxy rotation curves[12, 13] with α and µ ∗ being ”universal” parame-ters with constrained values α = 8 . ± .
34 and µ ∗ = 0 . ± .
004 kpc − , respectively [6]. De-spite its successes at galactic scale, it is not clearif the assumption of the universality of those pa-rameters holds at the scale of galaxy clusters. Ingeneral these parameters depend on the mass of thesource of the gravitational potential and, therefore,they should depend on the scale length of the self-gravitating system as their analogue in f ( R ) gravity[4]. Nevertheless, the universal parameters seem tobe able to predict the dynamical mass of galaxyclusters [14, 15].In this letter we propose an alternative test toprobe the universality of the α and µ ∗ parameters atgalaxy cluster scale. We use Planck α, µ ∗ ) of the modified gravitational potential. Wefocus our analysis on the Coma cluster since it islocated close to the galactic pole where the fore-ground emission is comparatively low. The letter is organized as follows: in Sec. 2 we briefly describethe data; in Sec. 3 we illustrate the methodologyused to fit the profile to the data; in Sec. 4 we dis-cuss the results of our analysis; in Sec. V we pointout the limitation of our analysis and the futureperspectives in this field; and finally, in Sec. 6 wegive our main conclusions.
2. Data
In 2015 the Planck Collaboration made publiclyavailable the Compton Y-maps [16] that was ob-tained applying a component separation algorithmto the high frequency channels (100-857 GHz) of thePlanck mission. This technique extracts a signalwhen its frequency dependence is specified. Let usnote that in order to specify the TSZ frequency de-pendence, one can not include any relativistic effect(due to the electron temperature) to the frequencydependence, and have to fix the CMB temperature-redshift relation to be (adiabatic): T CMB ( z ) = T CMB (0)(1+ z ). However, the intra-cluster mediumof Coma cluster has a temperature T e ∼ ∼
10% to the totalTSZ emission. Moreover, it is well known that al-ternative theories of gravity could produce a depar-ture from the adiabatic expansion since they couldchange the evolution of cosmological backgroundand its density perturbations.Therefore, although the Planck Y-maps allowsto measure the SZ cluster profiles with few percentaccuracy within its virial radius, due to a lack ofinformation about the effect of MOG theory at cos-mological scales, we prefer to be more conservativeand test the underlying theory of gravity by mea-suring the TSZ profile on the
Planck
Nominal maps[22, 23]. However, to reliably detect the TSZ tem-perature anisotropies induced by a galaxy clusterwe need to reduce the contaminations due to fore-ground emissions such as galactic dust, CO lines,synchrotron radiation, point and extended infraredsources, and the cosmological CMB signal. Forthat purpose, we applied the cleaning proceduredescribed in [20, 21] to the high frequency chan-nels. Briefly, the main steps are the following: (i) maps were brought to a common 10 arcminutes res-olution corresponding to the angular resolution ofthe 100 GHz channel; (ii)
CO lines were removedusing the CO maps released by the Planck Col-laboration [24]; (iii) intrinsic CMB signal and kSZwere removed using an LGMCA template [25, 26]; (iv) the dust emission were removed by using the2 igure 1: Patches centered at the position of A1656 (Comacluster) at 100-353 GHz. Patches are 4 ◦ × ◦ . First, second,and third rows illustrate the view of Coma cluster in PlanckNominal, foreground cleaned, and Planck Compton maps,respectively. highest frequency channel as a template. Finally,we measure the TSZ temperature anisotropies pro-duced by Coma cluster at 100, 143, and 353 GHzchannels while we discard the 217 and 545 GHzchannels: the first channel does not provide usefulinformation since the TSZ signal is greatly reduced( ∼ ◦ × ◦ patch centered atthe position of Coma cluster for the 100, 143, 217,353 GHz channels. The first row shows the view ofComa in the Planck Nominal maps; the second rowshows the results of our cleaning procedure and thelast row shows the view of the galaxy cluster in theY-map released by the Planck Collaboration. Thelatter, to be compared to our cleaned data, has beenmultiplied by the frequency dependence of the TSZeffect. Our cleaning procedure produces a noisiermap with more residuals. This is reflected in ourerror bars that are larger than the one obtainedfrom the Planck Collaboration especially at largerradii. However, we prefer to be more conservativeand use our own data to test MOG theory for thereasons explained above.To compute the error bars, we carried out 1 , C ij ) between different apertures, and weused the latter to compute the chi-square in ourstatistical analysis.
3. Methodology
The TSZ temperature anisotropies are producedwhen CMB photons are scattered off by the highenergy electrons in the Intra-Cluster Medium. Suchanisotropies are usually expressed as∆ T T SZ (ˆ n ) T = G (˜ ν ) σ T mc Z l P e ( l ) dl. (2)where P e ( l ) is pressure profile along the line of sight l , T = 2 . ± . G (˜ ν )is the spectral frequency dependence where ˜ ν = hν ( z ) /k B T ( z ) is the reduced frequency. In the nonrelativistic limit (electron temperature about fewkeV), G (˜ ν ) = ˜ ν coth(˜ ν/ −
4. Relativistic correc-tions in the electron temperature up to fourth orderhave been included to improve the model [29–31].To predict the TSZ temperature anisotropies,the pressure profile must be specified. Following[32, 33], we considered the gas in hydrostatic equi-librium within the modified potential well of thegalaxy cluster d P ( r ) dr = − ρ gas ( r ) d Φ eff ( x ) dr , (3)and well described by a polytropic equation of state P ( r ) ∝ ρ γgas ( r ) . (4)The system of equations is closed with the conser-vation of the mass dM ( r ) dr = 4 πρ gas ( r ) . (5)Let us remark that the model does not include anyDark Matter component. Thus, the pressure profile P e ( r ) = P c P ( r ) depends by the two MOG parame-ters ( α, µ ∗ ) the polytropic index γ , and the centralpressure P c . Finally, to predict the TSZ tempera-ture anisotropies, the profile was integrated along3 arameter Priors References P c / [10 − cm − keV] [0 . , .
0] [33] γ [1 . , /
3] [33] µ ∗ − /[Mpc] [0 . , .
0] [6, 18] α [0 . , .
0] [6, 18]
Table 1: Parameter space explored by the MCMC
Parameter Results P c / [10 − cm − keV] 0 . ± . γ . +0 . − . µ ∗ − /[Mpc] 4 . +0 . − . α . +3 . − . Table 2: Results from the MCMC. the line of sight and convolved with the 10 arcmin-utes antenna beam of the
Planck data.To test the universality of the MOG weakfield approximation we predicted the TSZ pro-file from 5 to 100 arcminutes (in rings of 5 ar-cminutes width), and we fit them to the data atthe same apertures carrying out a Monte CarloMarkov Chain (MCMC) analysis employing theMetropolis-Hastings sampling algorithm and theGelman-Rubin convergence criteria [34–36]. Werun four independent chains, each one composedby 25,000 steps, with randomly set starting points.The parameter space explored by our pipeline isgiven in Table 1.
4. Results and Discussion.
Once the MCMC algorithm has reach the conver-gence [36], we merged the four chains and computedthe marginalized likelihood to constrain the modelparameters. All results are summarized in Table 2,while in Fig. 2 we show the goodness of our fittingprocedure.The Table summarizes same important results:first, the parameter α is compatible at 68% CL withits universal value α ≃ .
89 [6, 18]. Second, the uni-versal value of scale length µ ∗ − is ruled out at morethan 3 . σ . Therefore, the assumption that the pa-rameters of the Yukawa-potential can be assumedscale independent is also ruled out. Third, we findthe polytropic index γ = 1 . +0 . − . to be consis-tent at 1 . σ level with the value γ ∼ . α and µ ∗ − to their universal values (red dot-dashedline). Panels (a-c) correspond to the three differentfrequencies, while the χ per d.o.f, given in eachpanel, refers to the best fit model with all param-eter free to vary. For the ”universal” MOG profilewe constrained: P c = (0 . ± . × − cm − keV, and γ = 1 . +0 . − . ; it only fits the central re-gion ( .
15 arcminutes) of the galaxy cluster, whileit overestimates the TSZ emission at larger aper-tures: at θ ∼
5. Further considerations on the universalnature of ( α, µ ∗ )-parameters: limitationand future perspective of the analysis Our results show a good agreement of α with itsuniversal value but a 3 . σ discrepancy in the scalelength µ ∗ − . The fact that µ ∗ − does not agreewith its universal fit could be interpreted as theconsequence of the scale dependence of the modi-fied potential: Φ eff ( r ≫ µ ∗ − ) becomes Newtonianwith an enhanced value of gravitational constant.When assuming the universal MOG parameters onefixes µ ∗ − ∼ kpc while the typical scale length fora galaxy cluster is ∼ Mpc, thus only α plays an im-portant rule in to describe the gravitational interac-tion and it fails to predict the TSZ profile at largerradii. Therefore, our results demonstrate that scaledependence of the MOG parameters play an im-portant role at galaxy cluster scale and can not beneglected.Another point of discussion is the assumptionsof hydrostatic equilibrium and spherical symme-try that are in our model. Although it has alsobeen demonstrated that in the intermediate regions,4 igure 2: Predicted and measured TSZ profile of the Comacluster at different frequencies. For each channel, the MOGbest fit model has been convolved with the antenna beam.The solid line represents the predicted model with the bestfit values in Table 2, while the red dot-dashed line showsthe fitted profile with α and µ ∗ − fixed to their universalvalues. Finally, the blue dashed line show the theoreticalprofile based on the Navarro-Frenk-White halo model withbest fit parameter from [37]. where we are testing the model, both assumptionshold [38–41], it is well known that the presenceof substructures, turbulences, heating and coolingprocesses in the cluster core, and the departurefrom the spherical symmetry [42–50] affect both theinnermost and outermost regions of Coma cluster.The effect of such phenomena determines the phys-ical state of the gas, and the degeneracy with theunderlying theory of gravity. Actually, in our anal-ysis we found a degeneracy between the gravita-tional parameter α and the polytropic index. Thisresults is illustrated in Fig. 3 where we plot the 2Dmarginalized contours obtained from our MCMCanalysis. A way of studying the α − γ degeneracy isincluding a non-thermal term in the pressure. Theproper strategy of doing this is to carry out hy-drodynamical N-body simulations of each specificMOG model, and compare the theoretical predic-tions to higher resolution data that allow to resolvethe cluster core region ( < ∼
6. Conclusions
We proposed an alternative test to probe the as-sumption of the universality of MOG weak fieldapproximation [6, 18]. Despite the fact that, un-der this assumption, MOG theory is able to ex-plain the phenomenology at galactic scale, it is notclear if it is also able to describe the galaxy cluster.Therefore, there is an important need to constrainthe modified gravitational potential in eq. (1) atgalaxy cluster scales in order to investigate if itsscale dependence can be neglected or must be con-sidered. Thus, we used the
Planck igure 3: 2D marginalized contour of the pair of parameters ( α, γ ) obtained from the MCMC analysis. For the pair ofparameters the 68% (dark gray) and 95% CL (light gray), the marginalized likelihood distributions are shown. α to be consistent atthe 68% CL with its universal value [6], while thescale length, µ ∗− , was not compatible with suchassumption at more than 3 . σ . This latter result in-dicates a breakdown of the universality of the MOGweak field approximation demonstrating that, in or-der to fit the TSZ temperature anisotropies of theComa cluster, the scale dependence of the MOGparameters can not be neglected. Acknowledgements
We warmly acknowledge John Moffat for fruit-ful discussion and his valuable comments. I.D.Macknowledge financial supports from University ofthe Basque Country UPV/EHU under the pro-gram ”Convocatoria de contrataci´on para la espe-cializaci´on de personal investigador doctor en laUPV/EHU 2015”, from the Spanish Ministerio deEconom´ıa y Competitividad through the researchproject FIS2010-15492, and from the Basque Gov-ernment through the research project IT-956-16.M. D. L. is supported by the ERC Synergy Grant“BlackHoleCam” – Imaging the Event Horizon ofBlack Holes (Grant No. 610058). M.D.L. acknowl-edge INFN Sez. di Napoli (Iniziative SpecificheQGSKY and TEONGRAV). This article is basedupon work from COST Action CA1511 Cosmol-ogy and Astrophysics Network for Theoretical Ad-vances and Training Actions (CANTATA), sup-ported by COST (European Cooperation in Scienceand Technology).
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