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zwicky showed that there is an inconsistency between the dynamical mass and the luminous mass for the coma cluster . he postulated that there must exist dark matter at the scale of clusters of galaxies  @xcite . the observations of the rotation curves of spiral galaxies by rubin also indicated that galaxies need missing mass in the context of newtonian gravity  @xcite . the main paradigm to interpret missing mass in the universe is the theory of dark matter , supposing that the unseen matter accounts for more than @xmath7 of the universe s matter content . this hypothetical matter is made of particles ( e.g.,wimps ) that may only interact weakly with ordinary baryonic matter . ongoing experiments have been conducted to directly detect dark matter particles by sensitive particle physics detectors . so far , no positive signal has been detected  @xcite . an alternative approach for solving the problem of missing mass is to modify the law of gravity . one of the first phenomenological efforts to modify newtonian gravity was modified newtonian dynamics ( mond )  @xcite . this model is based on a non - linear poisson equation for the gravitational potential with a characteristic acceleration @xmath8 ; newtonian gravity is modified for particles with acceleration @xmath9 . the model can successfully describe the rotation curves of galaxies  @xcite , but it fails to be consistent with the temperature and gas density profiles of clusters of galaxies without invoking dark matter  @xcite . there are other modified gravity models such as the relativistic extension of mond by bekenstein ( 2004 ) , conformal gravity  @xcite and non - local gravity  @xcite . we attempt in the following to test observationally a covariant modified gravity ( mog ) theory , also called scalar tensor  vector gravity ( stvg ) at the scale of clusters of galaxies  @xcite . in this theory , the dynamics of a test particle are given by a modified equation of motion where , in addition to the conventional geodesic motion of a particle , a massive vector field couples to the fifth force charge of a particle . the fifth force charge of a particle is proportional to the inertial mass of the particle . we have shown that in the weak field approximation the potential for a matter distribution of an extended object behaves like the newtonian potential with an enhanced gravitational constant and an additional yukawa potential  @xcite . we applied the effective potential in the weak field approximation to the study of the dynamics of galaxies and demonstrated that mog can successfully explain the rotation curves of galaxies without the need for dark matter . we fixed the two free parameters of the potential and chose the stellar mass - to - light ratio as the only free parameter in calculating the dynamics of the galaxies . we extend in the following the observational tests in moffat & rahvar ( 2013 ) to one order of magnitude larger system scale . we use the observational data of relaxed clusters of galaxies in the chandra x - ray sample for this purpose . for a set of clusters of galaxies , we have the temperature profile as well as the plasma gas and galaxy distribution . these observables are sufficient to calculate the dynamical mass and compare it to the baryonic mass of clusters . in section ( [ formalism ] ) , we briefly introduce the stvg theory and review the weak field approximation limit . in section ( [ sectionboltzmann ] ) , we obtain a full relativistic version of the boltzmann equation in mog . furthermore , we derive the virial equation in the weak field approximation . in section ( [ data ] ) , the observational data from the chandra x - ray sample of clusters are examined in conjunction with the mog and finally in ( [ conc ] ) , we provide a summary and conclusions . the stvg theory is a covariant theory of gravity . the action of stvg is composed of scalar , tensor and vector fields  @xcite : @xmath10 the components of the action are ( i ) the einstein gravity action : @xmath11 where @xmath12 is the cosmological constant , ( ii ) the massive vector field @xmath13 action : @xmath14\sqrt{-g}~d^4x,\end{aligned}\ ] ] where @xmath15 is the vector field and @xmath16 , ( iii ) the action for the scalar @xmath17 and @xmath1 fields is : @xmath18\sqrt{-g}~d^4x , \label{scalar}\end{aligned}\ ] ] where @xmath19 denotes the covariant derivative with respect to the metric @xmath20 . @xmath21 denotes a dimensionless coupling constant and @xmath22 and @xmath23 denote self - interaction potentials of @xmath1 and @xmath17 fields . while in the generic formalism of mog , @xmath17 and @xmath1 are taken into account as scalar fields , for simplicity we keep these two fields as constant parameters . finally , ( iv ) @xmath24 is the matter action . the action for pressureless dust coupled to the vector field @xmath13 is given by @xmath25 where the density of the fifth force charge @xmath26 is related to the density of matter by @xmath27 where @xmath28 is a constant . for a test particle with the density @xmath29 and a fifth force charge @xmath30 , a variation of the action in eq . ( [ sm ] ) with respect to the comoving time @xmath31 , yields the equation of motion : @xmath32 where the right - hand side is a lorentz - type force . since the fifth force charge is proportional to the inertial mass , the equation of motion is independent of the mass . the main difference of the lorentz - type force with standard electromagnetism is the same sign for the charge of particles , unlike the normal electromagnetic force where there are opposite sign particles ( results in a neutral medium for an ensemble of particles in a plasma ) . a physical consequence of the same sign charged particles is a repulsive force produced by an ensemble of particles . since the vector field in mog is massive , after a characteristic range of the field given by @xmath33 , the effect of the repulsive force vanishes and eq . ( [ geo ] ) reduces to the pure metric field geodesic equation of motion .      for the astrophysical application of mog , we need the weak field approximation of the theory . we briefly review in this section the formalism of the weak field approximation . a detailed derivation of this limit for extended distributions of matter was presented in moffat & rahvar ( 2013 ) .    for a slow moving particle , we expand the metric around minkowski space : @xmath34 the left - hand side of eq . ( [ geo ] ) can be written in terms of @xmath35 , while on the right - hand side we can ignore the magnetic - like force compared to the electric - like force due to the slow motion of the particle . ( [ geo ] ) can now be written as @xmath36 where @xmath37 and we define the effective potential as @xmath38 we substitute in the right - hand side of ( [ geo2 ] ) the solutions for @xmath35 and @xmath39 from the field equations . for the tensor component , @xmath40 where @xmath41 , and the zero component of the vector field @xmath39 satisfies the yukawa equation : @xmath42 where @xmath43 is the zero component of four dimensional current as the source of vector field and relates to density of matter as @xmath27 . substituting the solution of the poisson equation and the solution of @xmath39 from the yukawa equation into ( [ geo2 ] ) and taking into account the asymptotic solutions of effective potential at smaller distances ( i.e. @xmath44 ) to the newtonian gravity @xcite , we obtain the following effective potential in the weak field approximation : @xmath45 where @xmath17 has been replaced by @xmath46 , @xmath0 is a dimensionless parameter , @xmath47 is the newtonian gravitational constant and @xmath1 is the mass of the vector field . we have used this effective potential in moffat & rahvar ( 2013 ) and successfully fitted the rotation curves of galaxies , yielding the best values of the parameters : @xmath48 and @xmath49 kpc@xmath4 .    the first term on the right - hand side of eq . ( [ potential22 ] ) is the potential of the newtonian attractive force with the effective gravitational constant @xmath50 and the second term results from the repulsive vector force . for scales much larger than @xmath51 , the yukawa term is suppressed and the gravitational potential is enhanced by factor of @xmath52 compared to the newtonian potential . on the other hand , for smaller scales the repulsive force cancels the enhanced attractive force and the result is the standard newtonian potential at solar system scales . in this section we derive the full relativistic version of the boltzmann equation in mog and for the astrophysical applications , we calculate the boltzmann equation in the weak field approximation . the result will be used in studying the dynamics of x - ray clusters of galaxies .    for an ensemble of collisionless particles moving due to their self - gravity , the number of particles inside a differential element in the phase - space does not change and @xmath53 . this equation can be written as a sum of partial derivatives in terms of time , space and momentum . since in momentum space , we have the constraint @xmath54 , there are three degrees of freedom for @xmath55 and we keep only the spatial components of the momentum space in our calculation  @xcite . the boltzmann equation can be written as @xmath56 we can write ( [ geo ] ) in terms of the momentum of a particle : @xmath57 by substituting ( [ 3 ] ) into ( [ 1 ] ) , we obtain the boltzmann equation in curved space with the extra vector field as follows : @xmath58f = 0.\ ] ]    in order to have a relativistic euler equation , we multiply the boltzmann equation by the spatial component of four - momentum , @xmath59 , and integrate over momentum space : @xmath60 where @xmath61 is the differential volume in momentum space . in the first term , the space - time partial derivative can be moved outside the integral , and in the second term the christoffel symbol and the vector field strength can also be moved outside the integral . integrating over momentum space gives the result : @xmath62 where the bar sign denotes the average value over momentum space . for simplicity we rewrite the relativistic euler equation in terms of the four velocity @xmath63 : @xmath64 now we multiply this equation by @xmath65 and integrate over the 3-space volume and calculate this expression term by term . using the definition of the inertia tensor : @xmath66 the first term of ( [ b11 ] ) can be written in terms of partial time derivatives of @xmath67 and for the static and stationary systems it is zero . the second term represents the energy- momentum term which can be written as  @xcite : @xmath68 for simplicity , we can split the kinetic tensor into the energy - momentum tensor from the bulk velocity and the internal energy from the dispersion velocity , yielding the result : @xmath69 where @xmath70 and where @xmath71 is the dispersion tensor with @xmath72 . the integrations of the third , fourth and fifth terms in eq . ( [ b11 ] ) contain kinetic energy contributions . the newtonian gravitational part of the integrations can be interpreted as the gravitational potential of the system . we assign the last three terms of eq . ( [ b11 ] ) to the potential energy tensor : @xmath73 combining all the integration terms , the conservation of the energy - momentum fluid can be written as @xmath74 for astrophysical systems such as globular clusters , elliptical galaxies and clusters of galaxies , the morphology of the systems do not change in time , in other words , these systems have been virialized and do not interact with other systems . we set the left - hand side of ( [ v1 ] ) to zero for this case . by choosing the center - of - mass of the system as the center of the coordinate system , the global velocity cancels and @xmath75 , whereby the energy - momentum equation simplifies to the so - called virial equation : @xmath76 in the next section , we will use the boltzmann equation in mog for studying clusters of galaxies . while here we derived a generic euler equation and virial condition in mog , we will use just the equilibrium condition for studying the clusters .      in this section , we study the boltzmann equation for spherically symmetric systems in the weak field approximation . in order to calculate the euler equation for a spherically symmetric system , we start with eq . ( [ b11 ] ) . let us choose @xmath77 and assume that @xmath78 is a function of radial distance @xmath79 , radial particle velocity @xmath80 and tangential particle velocity , @xmath81 . for a virialized system , the shape of the system does not change in time and terms with partial derivatives with respect to time vanish . then the jeans equation simplifies to @xmath82 by assuming a spherically symmetric metric for this structure , @xmath83 we can choose @xmath84 and assuming a perturbation @xmath85 and the definition @xmath86 , the jeans equation in the weak field approximation can be written as @xmath87 = -\rho\partial_r\phi_{eff}(r ) . \label{jeans2}\ ] ]    for an isotropic distribution @xmath88 due to rotational symmetry there is no preferred transverse direction i.e. , @xmath89 . we can still have a degree of anisotropy in the velocity distribution at each point given by @xmath90 , and then the jeans equation can be written as @xmath91=   g(r ) , \label{vir}\ ] ] where the acceleration at the right hand side , using @xmath92 , is given by @xmath93d^3{\bf r'},\ ] ] where for the scales @xmath94 , the exponential term approaches to one and by canceling the last two terms in the integrand , the gravitational acceleration reduces to newtonian gravity . on the other hand , for the case that @xmath95 , the exponential term approaches to zero and we get newtonian gravity with an enhanced gravitational constant @xmath96 . in what follows , we can set @xmath97 in eq . ( [ vir ] ) for the isotropic distribution of the velocity .    [ cols="^,^,^,^,^,^,^,^,^,^,^ " , ]         here , for a spherical symmetric structure with scales of the order of @xmath98 mpc , we can simplify our numerical calculation , noting that the length corresponding to the mass of the vector field , @xmath99 kpc , is much smaller than the size of a cluster . for these structures , the gradient of the density inside the cluster within the length scale of @xmath99 kpc is negligible and we can ignore the gravitational acceleration exerted on a test particle by the fluid surrounding it within this radius . hence , the second term of eq . ( [ potential22 ] ) for a cluster - size structure is negligible . we have numerically checked this simplification by comparing the overall effective gravitational acceleration with the simplified relation of keeping only the first term of equation ( [ potential22 ] ) . figure ( [ acc ] ) compares the relative acceleration on a test mass particle with and without considering the yukawa term . since the measurements of central parts of clusters in the chandra observations are not accurate enough and the data is scattered  @xcite , using the simplified formula for the acceleration at the outer radii of the clusters ( i.e. @xmath100 kpc ) and ignoring the yukawa term is considered to be valid . using gauss s theorem for a spherically symmetric system with a @xmath101 potential , the dynamical mass from eq . ( [ halo ] ) can be written as @xmath102 where @xmath103 is given in terms of solar mass , @xmath79 is the distance from the center of the cluster in kpc , @xmath104 is the temperature of the gas in kev . we use the value @xmath2 which has been universally fixed by the fits to the rotation curve data of galaxies  @xcite .    in figure ( [ fig2 ] ) , we compare the profile of dynamical mass @xmath105 as a function of distance from the center of the cluster with the baryonic mass of clusters composed of the mass of gas plus the mass contribution from stars , @xmath106 . the overall dynamical and baryonic masses at the distance @xmath107 are given in table ( [ tab3 ] ) . the error bar associated to the mass of baryonic component is adapted from vikhlinin et al ( 2006 ) where they reported the fraction of baryonic mass to the overall mass in the dark matter scenario . assuming the error bar corresponds to this fractional mass as a constant value throughout the cluster , we calculate the error bar of the baryonic mass being proportional to the mass . we plot the error bars for the baryonic matter of clusters as a function of distance from the center of the cluster and with the cadence of @xmath108 mpc separation . for some of clusters as cga262 and cga1991 there is an offset between the observation and theoretical mass profiles with about @xmath109 deviation . for the rest of the clusters there is consistency between the theoretical and observational mass profiles , except for the central regions of clusters . we note that observational data at the central parts of clusters is poor . moreover , this part might not completely virialized due to cooling flow . for most of the clusters at scales larger than @xmath110 kpc , the theoretical profiles are in good agreement with the observational data . from the mass profiles in figure ( [ fig2 ] ) , the outer regions of clusters at @xmath111 kpc contain almost @xmath112 of the mass of the clusters and for these regions the mog theory prediction follows the observational data . for the overall mass of the clusters at @xmath107 , we compare the mass profiles for the data and theory . the error bar associated with the observed mass is obtained from the uncertainty in the observational data , while for the theoretical profile , we obtain the upper and lower values for the mass resulting from the uncertainties in @xmath1 and @xmath0 , reported in  @xcite . figure ( [ fig3 ] ) compares the overall theoretical baryonic mass of clusters with the observational data . following our previous paper  @xcite , in this work we have investigated further the weak field approximation of stvg to test the dynamics of clusters of galaxies . the theory of gravity in the weak field regime is composed of an enhanced newtonian gravity term with the gravitational constant @xmath113 and a yukawa repulsive term with the mass @xmath114 kpc . at the smaller scales compared to @xmath51 , e.g. , solar system scales , standard newtonian gravity can be recovered . on the other hand , at larger scales we get newtonian gravity with a stronger effective gravitational constant . our analysis of a sample of galaxies in the previous work showed that with the fixed universal parameters @xmath0 and @xmath1 in the effective gravitational potential  @xcite , we can successfully describe the dynamics of a large set of galaxies with only their baryonic matter content .    in order to test this universality , we have analyzed a sample of nearby chandra x - ray data in which the gas density profiles and the temperatures have been measured  @xcite . our analysis shows that , using the universal values for the parameters @xmath0 and @xmath1 in the effective mog potential , we get consistent mass profiles and overall values for the dynamical and baryonic mass of clusters , except for the central parts of clusters where the observational data is scattered . these results along with the analyses of the rotation curves of galaxies demonstrate that with a unique weak field mog effective potential , we can describe the dynamics of astrophysical systems from the solar system to @xmath115mpc scale systems without exotic dark matter .    finally , we outline further questions in mog yet to be answered , ( i ) large scale structure formation in the universe . the structure formation growth has to be described by mog without using dark a matter component and using cmb boundary conditions , ( ii ) interpretation of observational data of gravitational lensing at large scales , especially the case of the asymmetric bullet cluster without invoking dark matter for the lensing system . a previous published investigation of the bullet cluster using a version of mog  @xcite , showed compatibility between the predictions of mog and the available data at the time . however , this work was based on a spherically symmetric point - like solution of the field equations , which has to be extended to continuous matter distributions using the methods of the present paper . the john templeton foundation is thanked for its generous support of this research . the research was also supported by the perimeter institute for theoretical physics . the perimeter institute was supported by the government of canada through industry canada and by the province of ontario through the ministry of economic development and innovation . we thank niayesh afshordi , bahram mashhoon , martin green and viktor toth for helpful discussions and comments . finally , we would like to thank david weinberg `` as referee of this work '' for useful comments and suggestions for improving this work .
we apply the weak field approximation limit of the covariant scalar - tensor - vector gravity ( stvg ) theory , so - called modified gravity ( mog ) , to the dynamics of clusters of galaxies by using only baryonic matter . 
 the mog effective gravitational potential in the weak field approximation is composed of an attractive newtonian term and a repulsive yukawa term with two parameters @xmath0 and @xmath1 . 
 the numerical values of these parameters have been obtained by fitting the predicted rotation curves of galaxies to observational data , yielding the best fit result : @xmath2 and @xmath3 kpc@xmath4  @xcite . 
 we extend the observational test of this theory to clusters of galaxies , using data for the ionized gas and the temperature profile of nearby clusters obtained by the chandra x - ray telescope . using the mog virial theorem for clusters , 
 we compare the mass profiles of clusters from observation and theory for eleven clusters . 
 the theoretical mass profiles for the inner parts of clusters exceed the observational data . however , the observational data for the inner parts of clusters ( i.e. , @xmath5 ) is scattered , but at distances larger than @xmath6 kpc , the observed and predicted mass profiles converge . 
 our results indicate that mog as a theory of modified gravity is compatible with the observational data from the the solar system to mega parsec scales without invoking dark matter .