The Effect of Interacting Dark Energy on Mass-Temperature Relation in Galaxy Clusters
TThe Effect of Interacting Dark Energy on Mass-Temperature Relation inGalaxy Clusters
Mahdi Naseri ∗ Department of Physics, K.N. Toosi University of Technology, P. O. Box 15875-4416, Tehran, Iran
Javad T. Firouzjaee † Department of Physics, K.N. Toosi University of Technology, P. O. Box 15875-4416, Tehran, Iran andSchool of physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran
Abstract:
There are a variety of cosmological models for dark matter and dark energy in whicha possible interaction is considered between these two significant components of the universe. Wefocus on five suggested models of interacting dark matter and dark energy and derive the modifiedvirial theorem for them by developing a previous approach. It provides an opportunity to study theevolution of this modified virial theorem with time and interacting constants for different interactingmodels. Then we use this obtained virial condition to investigate the modified mass-temperaturerelation in galaxy clusters via three various methods. It reveals that the effect of interaction betweendark matter and dark energy merely appears in the normalization factor of M ∝ T . This relationalso leads to a new constraint on the constants of interacting models, which only depends on theconcentration parameter and density profile of the cluster. Then we use five observational datasets to check some proposed figures for the constants of interaction which have been resulted fromother observational constraints. Finally, by fitting the observational results to the modified mass-temperature relation, we obtain values for interacting constants of three models and four specificcases of the two remained models. In agreement with many other observational outcomes, we findthat according to observational data for masses and temperatures of the galaxy clusters, energytransfer occurs from dark matter to dark energy in the seven investigated models. Contents
I. Introduction II. Interacting Dark Energy Models and Virial Theorem
III. Mass Temperature Relation of Galaxy Clusters
IV. Results and Discussion V. Conclusion References I. INTRODUCTION
As different observational outcomes have revealed the existence of two unfamiliar contributors to thephysics of the universe, research into the ”dark sector” has gained currency in modern cosmology. Darkmatter (DM) proposed to clarify rotation curves of spiral galaxies, and the idea behind dark energy(DE) was initially formed to explain the late-time acceleration of the universe. Eventually, the ΛCDM ∗ Electronic address: [email protected] † Electronic address: fi[email protected] a r X i v : . [ a s t r o - ph . C O ] N ov model accounted for the primary suggestion for the cosmos.In spite of gravitational evidence for DM from galaxies [1], clusters of galaxies [2], cosmic microwavebackground (CMB) anisotropies [3], cosmic shear [4], structure formation [5] and large-scale structure ofthe Universe [6], last years of direct and indirect searches of those DM particles did not give any convincedresult [7]. In addition, the accelerated expansion of the universe modeled with Λ [8] raised several prob-lems, including the ”cosmological constant finetuning problem” and the ”cosmic coincidence problem” [9].However, it could be possible to assume and investigate more elaborate alternatives in which there is afeasible non-gravitational interaction between DM and DE. The idea has extended in [10], where DM par-ticle mass is determined according to its interaction with a scalar field with the energy density of DE. Suchan assumption resembles how the Higgs field results in quark and lepton masses via interacting with them.Not only is the notion of interacting dark sector interesting, but it could also be beneficial in termsof solving some cosmological problems. By way of illustration, it may explain why the densities ofDE and DM are of the same order, despite the fact that they evolve differently with redshift, namelythe ”coincidence problem” (see e.g. [11]). The interacting dark energy model should justify the sameobservation in contrast to the ΛCDM model which modified gravity models do [12, 21].One can study the effects of modified gravity with structure formation and verified employingdark-matter-only N-body simulations [13]. Since experiments only measure photons which are emittedfrom the baryonic matter, photons properties cannot be directly calculated only from dark mattersimulations. However, hydrodynamical simulations are more appropriate in the observational aspect, asthey provide observables, such as the halo profile, the turnaround radius [14], the splashback radius [15]and the mass-temperature (M-T) relation [16].There are a wide range of observations, simulations, and theoretical research into the relationshipbetween mass and temperature of galaxy clusters which have been done heretofore. The only consensusamong all these endeavors is admit of an evident correlation between the total gravitational mass of theclusters, X-Ray luminosity, and thereby, their temperature (that is the temperature of the intraclustermedium i.e ICM). It is of significance to study this relation, owing to the fact that the cluster massesare arduous to measure directly through observation. Fundamental arguments based on virializationdensity suggest that M ∝ T , where T is the temperature of a cluster within a certain radius (e.g. thevirial radius) and M is the mass within the same radius (see [17, 18] for advanced discussion). Themass-temperature relation can be directly compared with observations. This relation has been used toput constraints on modified gravity models. For example, using the hydrodynamical simulations, [16]showed that the M-T relation obtained in modified gravity theories is different from the expectations ofthe general relativity. Nevertheless, [19] showed that the mass-temperature relation of the ΛCDM modelis similar to that of the f(R) and symmetron models.The paper is organized as follows. Section II briefly presents the interacting dark energy model andspecifically, introduces five interacting models on which we concentrate in this study. We also obtain thevirial theorem for these interacting models. Section III is devoted to the mass-temperature relation ofgalaxy clusters concerning the interaction between dark matter and dark energy. Section IV makes acomparison between observational data and obtained M-T relation to study constants of interaction inthe five models. We summarize and give our final thoughts in Section V. II. INTERACTING DARK ENERGY MODELS AND VIRIAL THEOREM
The interacting dark energy model is composed of dark matter and dark energy only, as a flatFriedmann-Lemaitre-Robertson-Walker (FLRW) background metric. The dark sector interaction is mod-eled with a heat flux in the Bianchi identities between the two dark components as ∇ µ T µν ( λ ) (cid:54) = 0 , (1)where T µν ( λ ) in the energy-momentum tensor of each individual component which is no longer conserved.There are a number of interacting models which have been suggested and investigated recently. Ac-cording to [11], the balance, Raychaudhuri and FLRW equations can be written as˙ ρ b = − Hρ b , (2)˙ ρ c = − Hρ c + Q , (3)˙ ρ x = − w x ) Hρ x − Q , (4)˙ H = − πG [ ρ b + ρ c + (1 + w x ) ρ x ] , (5) H = 8 πG ρ b + ρ c + ρ x ) , (6)where H is the Hubble parameter, ρ c is the cold dark matter density, ρ b is the baryonic matter densityand ρ x represents the density of dark energy (with w x < Q describes the rate of energy density transfer between DE and DM, which is resulted from theinteraction between them. For Q >
0, it describes the transfer of energy from DE to DM, and on theother hand,
Q < b ) and photons ( γ ),are not coupled to the dark sector; therefore, Q γ and Q b considered to be equal to zero.A variety of functions have been proposed and studied for Q , including linear and non-linear combina-tions of ρ x and ρ c . In this paper, we concentrate on five various models for Q , which are rather simpleand common in literature: Model I : Q = 3 H ( α c ρ c + α x ρ x ) , Model II : Q = 3 Hξ ρ c ρ x ρ c + ρ x , Model III : Q = 3 Hξ ρ x ρ c + ρ x , Model IV : Q = 3 Hξ ρ c ρ c + ρ x , Model V : Q = 3(Γ c ρ c + Γ x ρ x ) . (7)Here, ξ , ξ , ξ , α j and Γ j are the main parameters of interacting dark sector ( j = c, x ). First four modelsare interesting, due to being coefficient with the Hubble parameter, which leads to more straightforwardcalculations. Whereas, Model V is more complicated and has a physical meaning. According to thismodel, the oscillation inflaton field decays into relativistic particles during reheating process afterinflation in early universe, and Γ j describes decay width [11]. Constant parameters in Models I toIV are dimensionless, while in Model V, Γ j has the dimension of the Hubble parameter. For furtherexplanations about these choices for Q , look at [20] and [21]. A. Virial Theorem in Interacting Models
In any theory of modified gravity, the virial theorem may significantly change from its Newtonianform. To find a virial relation in the context of general relativity, one has to use the covariant collisionlessBoltzmann equation (see [22] and reference therein). This approach has been extended to the virialtheorem in the modified gravity theories to study the dynamics of clusters of galaxies [23]. In homoge-neous and isotropic background in which gravity is not strong, the virial theorem gets the Newtonian form.Before analyzing mass-temperature relation in galaxy clusters, we have to investigate modifications tothe virial theorem with regard to interacting dark sector. In order to achieve this objective, we derivethe Layser-Irvine equation for Models I to V and then use this equation to obtain the virial condition.This equation, and hence the virial theorem, has been driven in [24] for Model I; however, we re-writecalculations so as to check it for the other four models, as well.Considering Model V, the perturbation equations for DE and DM in the subhorizon scale, which havebeen driven in [25], can be written in the real space as∆ (cid:48) c + ∇ ¯ r · v c = 3Γ x (∆ x − ∆ c ) /R , (8) v (cid:48) c + H v c = −∇ ¯ r Ψ − c + Γ x /R ) v c . (9)Here, H indicates the Hubble parameter in the conformal time, v c represents velocity of dark matterelement, ¯ r refers to conformal coordinates and the prime denotes the derivative with respect to conformaltime. Density contrasts of DM and DE are defined as ∆ c ≈ δρ c /ρ c = δ c and ∆ x ≈ δρ x /ρ x = δ x , andwe symbolize dark energy to dark matter (DM-DE) ratio by R = ρ c /ρ x . Moreover, Ψ = ψ m + ψ d is thepeculiar potential and is described by Poisson equation: ∇ ψ j = 4 πG (1 + 3 w j ) δρ j , (10)where ” j ” stands for DM or DE. Considering ∇ r = a ∇ ¯ r and defining σ c = δρ c and σ x = δρ x , Eqs. (8)and (9) can be written as ˙ σ c + 3 Hσ c + ∇ r ( ρ c v c ) = 3(Γ c σ c + Γ x σ x ) , (11) ∂∂t ( av c ) = −∇ r ( aψ c + aψ x ) − c + Γ x /R )( av c ) , (12)where a is background scale factor and H is its Hubble parameter. Following the method of [24] and [26],we multiply both sides of Eq. (12) by av c ρ c ˆ ε and then integrate them over the volume (”ˆ ε ” indicatesvolume element with criterion of expansion ∂∂t ˆ ε = 3 H ˆ ε ). For the left-hand side of Eq. (12), it is possibleto write: (cid:90) av c ∂∂t ( av c ) ρ c ˆ ε = (cid:90) av c ( ˙ av c + a ˙ v c ) ρ c ˆ ε = (cid:90) a Hρ c v c ˆ ε + (cid:90) a ρ c v c ˙ v c ˆ ε . (13)The kinetic energy ” K c ”, which stems from the movement of DM particles, is defined as: K c = 12 (cid:90) v c ρ c ˆ ε . (14)It is possible to use this definition and write: ∂∂t (cid:0) a K c (cid:1) = 2 a ˙ aK c + a ∂∂t K c = 2 a HK c + a [ (cid:90) v c ˙ v c ρ c ˆ ε + 12 (cid:90) v c ˙ ρ c ˆ ε + 12 3 H (cid:90) v c ρ c ˆ ε ] . (15)Using Eq. (15) in Eq. (13) we have: (cid:90) av c ∂∂t ( av c ) ρ c ˆ ε = ∂∂t (cid:0) a K c (cid:1) − a (cid:90) v c ˙ ρ c ˆ ε −
12 3 Ha (cid:90) v c ρ c ˆ ε . (16)Then, using Eq. (3) with Q of the Model V in the last equation gives: (cid:90) av c ∂∂t ( av c ) ρ c ˆ ε = ∂∂t (cid:0) a K c (cid:1) − a (Γ c + Γ x /R ) K c . (17)For the first term in the right-hand side of Eq. (12), integration gives: − (cid:90) av c ∇ r ( aψ c + aψ x ) ρ c ˆ ε = a (cid:90) ∇ r ( ρ c v c ) ψ c ˆ ε + a (cid:90) ∇ r ( ρ c v c ) ψ x ˆ ε . (18)With the aid of Eq. (11), it can be related to potential energy − (cid:82) av c ∇ r ( aψ c + aψ x ) ρ c ˆ ε = − a ( ˙ U cc + HU cc ) − a (cid:82) ψ x ∂∂t ( σ c ˆ ε ) + 3 a { Γ c U cx + Γ x U xc + 2Γ c U cc + 2Γ x U xx } , (19)where U αβ = (cid:82) σ α ψ β ˆ ε ; ” α ” and ” β ” stand for DM and DE, interchangeably.Eventually, integrating the second term in the right-hand side of Eq. (12) leads to − (cid:90) ( av c ) c + Γ x /R ) ρ c ˆ ε = − a (Γ c + Γ x /R ) K c . (20)Now, the Layzer-Irvine equation could be easily produced by combination of Eqs. (17), (19) and (20) as˙ K c + ˙ U cc + H (2 K c + U cc ) = − (cid:82) ψ x ∂∂t ( σ c ˆ ε ) − c + Γ x /R ) K c + 3 { Γ c U cx + Γ x U xc + 2Γ c U cc + 2Γ U xx } . (21)In virial equilibrium, the first and second terms of the previous equation are equal to zero. With theassumption of homogeneous distribution of DE, σ x = 0, we get K c = − H − c H + 3Γ c + 3Γ x /R U cc . (22)In order to facilitate following calculations, we define parameter ” λ i ” and represent the virial conditionas K c = − λ i U cc . (23)Obviously, ” λ i ” is not necessarily equal to in interacting models and depends on interaction constantswithin Q . The same procedure could be undergone for Models I to IV. To sum up the results for all thefive models, λ i is ( i = I, II, III, IV, V ):Model I : λ I = 1 − α c α c + 3 α x /R , Model II : λ II = 1 − ξ R +1 ξ R +1 , Model III : λ III = 12 + ξ R ( R +1) , Model IV : λ IV = 1 − Rξ R +1 Rξ R +1 , Model V : λ V = H − c H + 3Γ c + 3Γ x /R . (24)Constant of the EOS, w j , has a similar behavior for cold dark matter (CDM) and baryonic matter, thatis w m = w c = 0. Thus, Poisson equation or Eq. (10) leads to the same potential energy for both CDMand baryonic matter. It is very common to assume that baryons can merely interact with dark sector viagravitational field. In this case, which we call ”First Possibility”, Eq. (23) results in: K = K c + K b = − λ i U G . (25)Notwithstanding such a simple assumption, interaction between CDM and baryons might be considereda bit more intricate. Although both CDM and baryonic matter have the same potential function, theymay interact separately, solely with their own type of matter. Given the circumstances, which we name”Second Possibility”, Eq. (23) gives K = − ( λ i Ω c Ω c + Ω b + 12 Ω b Ω c + Ω b ) U G (26)where Ω is relevant density parameter for each element of matter. In order to brief calculations, weintroduce parameter λ (cid:48) i and write the last equation as λ (cid:48) i = λ i Ω c Ω c + Ω b + 12 Ω b Ω c + Ω b , (27) K = − λ (cid:48) i U G . (28)Eqs. (25) and (28) are the substitutes for the classical virial condition in dynamical equilibrium withrespect to interaction between DE and DM (considering the First or the Second Possibilities). It isapparent that these equations with α j = ξ = ξ = ξ = Γ j = 0 reduce to the familiar K = − U innon-interacting models.Note that the assumption of homogeneous distribution of DE in Eq. (22) would be denied bynon-standard models of DE. As an example, detecting fewer clusters than the prediction of the primaryCMB anisotropies via the Sunyaev-Zel’dovich effect by Planck satellite [27] has given rise to the ideaof clustering DE. In this regime, DE contributes to clustering, and hence, we cannot omit DE termsin Eq. (21) whereby virial theorem changes to a more intricate form (see [28] and [29] to find out howclustering DE model alters characteristics of virialized haloes). In this work, we consider the commonstandard DE and postpone more investigations on modified virial theorem with respect to DE withnegligible sound speeds to future studies. III. MASS TEMPERATURE RELATION OF GALAXY CLUSTERS
The primary approach to form mass-temperature relation is combining the virial theorem withconservation of energy, which brings about M ∝ T ζ . While the power-law index appears to be ζ = inmost masses, a ”break” is predicted in a myriad of observations and simulations at low masses, whichgives rise to ζ > in this particular range. The physics behind this behavior has been under studyfor a while; [30] attributed it to the cooling process, and the heating process is stated in [31] to be therationale for this ”break”, to name but a few. In order to reconstruct theories concerning this ”break”,Afshordi & Cen have attributed it to the nonsphericity of the initial protoclusters in [17], and DelPopolo has taken the angular momentum acquisition by protoclusters into account in [18]. Nevertheless,more recent studies, embracing [32] and [33], revealed that there is no evidence of double slope in M-Trelation. However, the existence of this ”break” is still under discussion.We try to take a look at three different methods which have been provided by Afshordi & Cen andDel Popolo to reconstruct mass-temperature relation in galaxy clusters, considering the modified virialtheorem for interacting dark matter and dark energy. The double slope in mass-temperature relation isnot our principal focus and we neglect it, although there will be some mentions to that. A. Derivation of Mass-Temperature Relation
In this section, we develop the method used by Afshordi & Cen in [17] to rebuild M-T relation ingalaxy clusters for interacting models. They begin with a definition of the kinetic and potential energiesand pursue calculations by using velocity as a function of the gravitational potential in the perturbationtheory, Poisson equation, and Gauss’s theorem to obtain the initial energy of a protocluster (i.e. E ta or the total energy of that at turnaround radius r ta ). Since up to this point there is no indication ofinteracting dark sector, we avoid repeating calculations, and we just mention the outcome obtained in[17]: E ta = − πG ρ ta r ta B . (29)Here, B is defined as B ≡ (cid:90) ˜ δ ta (˜ r )(1 − ˜ r ) d ˜ r , (30)where ˜ r ≡ rr ta , ˜ δ ta ≡ δ ta + (Ω ta − ta and δ ta are density parameter and density contrast atturnaround time, respectively.Taking a surface pressure term into account (which is exerted at the boundary of the cluster), virialcondition gives K vir + E vir = (1 − λ i ) U vir + 3 P ext V . (31)There is the point where the impact of interacting dark sector emerges. Here, P ext denotes the pressureon the outer boundary of the virialized cluster, and V stands for the volume. It is clear that the lastequation could reduce to the classical equation (used by Afshordi & Cen), if λ i = . Another equationfor surface pressure is expressed by 3 P ext V = − νU vir , (32)where the parameter ν is a coefficient constant to indicate the considered correlation between exertedpressure and the potential energy. Combining two preceding equations gives K vir + E vir = (1 − λ i − ν ) U vir . (33)The surface pressure term also alters the relation between kinetic and potential energy after virializationto K vir = − λ i + ν U vir . (34)Inserting U vir from Eq. (34) into Eq. (33) leads to − λ i + ν − λ i − ν E vir = K vir . (35)Then, the kinetic energy of the cluster can be separated into fully ionized baryonic gas and DM as K vir = 32 M c σ v + 3 M b k B T µm p , (36)where σ v stands for the mass-weighted mean velocity dispersion of DM particles in one dimension, M b is the total baryonic mass, k B is the Boltzmann constant, T is temperature, µ = 0 .
59 is mean molecularweight and m p represents the proton mass. To simplify the previous equation, ˜ β spec is defined as˜ β spec = β spec [1 + ( f β − spec −
1) Ω b Ω b + Ω c ] . (37)Here, f is the fraction of baryonic matter in hot gas and β spec ≡ σ v / ( k B T /µm p ). This definition assiststo obtain from Eq. (36): K vir = 3 ˜ β spec M k B T µm p . (38)Now, using Eqs. (29) and (38) in Eq. (35), with respect to conservation of energy ( E ta = E vir ), we find: k B T = 5 µm p π ˜ β spec ( 2 λ i + ν − λ i − ν ) H ta r ta B . (39)In order to find an expression for H ta r ta , parameter e is defined to be the energy of a test particle withunit mass at r ta , therefore, we can write it as e = v ta − GMr ta . (40)We also have collapse time (or dynamical time scale) as t = 2 πGM ( − e ) . (41)With the assumption that this time is approximately equal to the required time for virialization, andusing the Friedmann equations, one can obtain − e = 54 π H ta r ta A = ( 2 πGMt ) , (42) A ≡ (cid:90) ˜ δ i (˜ r ) d ˜ r = 25 ( 3 π t Gρ ta ) . (43)Using last two equations together with Eq. (39), the mass-temperature relation can be obtained k B T = ( µm p β spec )( 2 λ i + ν − λ i − ν )( 2 πGMt ) ( BA ) . (44)By inserting numerical values, this relation can be written as k B T = (6 . keV ) ˜ Q ( M h − M (cid:12) ) / , (45)where the dimensionless factor ˜ Q is defined:˜ Q ≡ ( ˜ β spec . − ( 2 λ i + ν − λ i − ν )( BA )( Ht ) − / . (46)Eq. (45) is the mass-temperature relation in galaxy clusters, regarding interaction between DE and DM.It is noticeable that the effect of interacting dark sector is merely appeared in factor ˜ Q . Afshordi & Cenhave extensively discussed this factor in [17]. Overall, ˜ β spec is a function of the ratio of the kinetic energyper unit mass of DM to the thermal energy of gas particles ( β spec ), the fraction of baryonic matter in hotgas ( f ) and the ratio of baryonic matter to DM in the sphere. According to different simulations andobservations, these three parameters vary slightly whereby the final value for ˜ β spec does not face dramaticchanges and is close to 0.9, hence we fix it by this figure in our calculations. The second variable, ν ,depends on density profile f ( ω ) and concentration parameter c , which is given by ν ( c, f ( ω )) ≡ − P ext VU = c (cid:82) ∞ c f ( ω ) g ( ω ) ω − dω (cid:82) c f ( ω ) g ( ω ) ωdω , (47)where: g ( ω ) = (cid:90) ω f ( ω ) ω dω . (48)For density profile, we may choose NFW profile as: f NF W ( ω ) = 1( ω ) (1 + ω ) , (49)where ω = rr s and r s is the scale radius given in [37]. This profile is proposed by Navarro, Frenk andWhite and has been widely used and studied in literature. However, there have been some objections tothat, as some recent observations have revealed a cored density profile in the inner region of the haloes.Several density profiles have been proposed to include the cored central region, including Burkert profile[34], which is expressed by f Burkert ( ω ) = 1(1 + ω ) (1 + ω ) . (50)Clearly, considering each of these profiles may affect M-T relation, as well as the other properties ofclusters.Concentration parameter c is defined as the ratio of virial radius to scale radius, that is r vir r s . Thedensity profile is exclusively described by c . In case there is not any observational data, the followingrelation (from [35]) may be used to find the value of the concentration parameter: c = 8 . M M (cid:12) ) − . , (51)where M is the mass enclosed by the radius in which the average density is 200 times the criticaldensity of the universe. Meanwhile, mass-concentration relation has extensively been under study and itwould have minuscule differences in various works, such as [36].In Eq. (46), parameter ( BA ) plays the prominent role in the ”break” of mass-temperature relation inlow masses. In spite of the fact that both A and B are proportional to scale factor, AB remains constant.Considering an initial density profile with multiple peaks (rather than a homogeneous distribution ofdensity, or a profile with one central peak), Afshordi & Cen obtain < BA > = 4(1 − n )( n − n −
2) [1 − n ( n + 3)10(1 − n ) (1 − Ω c − Ω b − Ω Λ )( Htπ (Ω c + Ω b ) ) ] , (52)where n is the index of the density power spectrum. Choosing an initial density profile with multiplepeaks would be more comprehensive and rational because, in hierarchical structure formation models,mass gradually accumulates in several regions of the initial cluster and not solely around the center.Taking nonsphericity in the geometry of the collapsing protocluster into account, which has a notablesign in low masses, Afshordi & Cen write some equations for dispersion of factor AB , or ∆ BA . Itreveals more dispersion in low masses and consequently, leads to a so-called ”break” in M-T relation.However, as we have mentioned before, not only is there no agreement on the existence of this doubleslope, but there is also no sign of interacting dark sector in this parameter, thus, we neglect it for our study.Furthermore, another parameter is introduced in [17] as: y = BA ( Ht ) . (53)This definition changes Eq. (46) to a more straightforward form. It can be written as a function ofdensity profile and concentration parameter y ( c, f ) = ∆ / (2 − λ i − ν ) c (cid:82) c f ( ω ) g ( ω ) ωdω π / g ( c ) , (54)where ∆ is the overdensity of the sphere and for a virialized cluster is somewhere in the region of∆ = 200, meaning that the cluster has an average density of 200 times as much as critical density of theuniverse. Last relation is driven in [17] regarding virial theorem and the definition of ν ; meanwhile, ow-ing to modification of virial theorem, the factor (1 − ν ) has changed to (2 − λ i − ν ) for interacting models.Both the mass and temperature of a cluster have to be positive to result in a genuine outcome.Combining this principle with Eq. (46) shows a constraint on the possible values for λ i . As all contributorsin Eq. (46) are positive quantities, the ratio ( λ i + ν − λ i − ν ) should be positive. As a result, we should whetherhave − ν < λ i < − ν , (55)0or 2 − ν < λ i < − ν . (56)Due to the fact that ν is always a positive parameter, Eq. (56) necessitates a negative λ i . Taking Eq. (25)into account, a negative λ i does not have any physical meaning; thus, just Eq. (55) could be acceptableas a criterion for the value of λ i , and its more accurate form is0 < λ i < − ν . (57)Note that Eq. (46) is derived for our ”First Possibility”. It is self-evident that by replacing ” λ i ” with” λ (cid:48) i ”, we would also be able to study the ”Second Possibility”. B. Reforming the Top-Hat Model
In order to form the ”break” in M-T relation, Del Popolo takes angular momentum acquisition of thecollapsing protoclusters into consideration in [18], and later, reinforces this method by adding anotherterm for dynamical friction in [19]. The angular momentum is acquired by interacting with neighboringprotoclusters. Del Popolo suggests two approaches to formulate M-T relation. The first approach isbased upon the development of the top-hat model and we investigate it in this section, with an additionalassumption of the interacting dark sector.To start this method, an ensemble of gravitationally growing mass concentrations is assumed and thenwith the assistance of the Liouville’s Theorem, Del Popolo obtains the radial acceleration of a particle asd v r d t = − GMr + L ( r ) M r + Λ3 r − η d r d t , (58)where η is the dynamical friction coefficient and L ( r ) denotes the acquired angular momentum in radius r from the center of the cluster. L ( r ) has a very complicated relation which can be found in [38] and [39].Integrating the previous equation leads to12 (cid:18) d r d t (cid:19) = GMr + (cid:90) r L M r d r + Λ6 r − (cid:90) r η d r d t + (cid:15) , (59)Here, (cid:15) is the specific binding energy of the shell and can be determined by condition of d r d t = 0 at r ta . Thepreceding equation represents four forms of potential energy; using them in the modified virial conditionfor interacting dark sector, we have (cid:104) K (cid:105) = − λ i (cid:104) U G (cid:105) − (cid:104) U L (cid:105) + (cid:104) U Λ (cid:105) + (cid:104) U η (cid:105) . (60)Here, (cid:104)(cid:105) indicates time averaged value of any quantity. By using Eq. (32) and (33) in the previousequation, we get (cid:104) K (cid:105) = (2 λ i + ν )( − (cid:104) U G (cid:105) − (cid:104) U L (cid:105) + (cid:104) U Λ (cid:105) + (cid:104) U η (cid:105) ) . (61)Defining r eff as the time averaged radius of mass shell, Eq. (61) can be written as (cid:104) K (cid:105) = − (cid:0) λ i + ν (cid:1) U G (cid:104) U L U G − U Λ U G − U η U G (cid:105) = (cid:0) λ i + ν (cid:1) GMr eff (cid:104) r eff GM (cid:82) r eff L ( r ) r dr − Λ r GM − r eff GM (cid:82) r eff η d r d t (cid:105) . (62)Ratio of r eff to r ta is defined by ψ = r eff r ta ; then we have M = 4 πρ b x / ,χ = r ta /x , Ω = 8 πGρ b H ; (63)1and as a result: r eff = ψχ (cid:18) GM Ω H (cid:19) / . (64)Then, putting (cid:104) K (cid:105) from Eq. (38) into Eq. (62) results in the M-T relation as k B T keV = 1 .
58 ( λ i + ν ) µβ spec ψχ Ω / (cid:18) M M (cid:12) h − (cid:19) / (1 + z ta ) × (cid:34) (cid:18) π (cid:19) / ψχρ / , ta H Ω b , M / (1 + z ta ) × (cid:90) r eff L r d r −
23 ΛΩ b , H (1 + z ta ) ( ψχ ) − / / π / (cid:18) ψχ Ω b , H (cid:19) (cid:16) ρ b , M (cid:17) /
11 + z ta × (cid:90) η d r d t d r (cid:21) . (65)Conservation of energy should be used in order to determine the value of ψ , or r eff as (cid:104) E (cid:105) = (cid:104) K (cid:105) + (cid:104) U G (cid:105) + (cid:104) U Λ (cid:105) + (cid:104) U L (cid:105) + (cid:104) U η (cid:105) = U G , ta + U Λ , ta + U L , ta + U η, ta . (66)Using Eq. (61) in this equation, we find − λ i − ν + 22 (cid:104) U G (cid:105) − (2 λ i + ν − (cid:104) U L (cid:105) + (2 λ i + ν + 1)( (cid:104) U Λ (cid:105) + (cid:104) U η (cid:105) ) = U G , ta + U Λ , ta + U L , ta + U η, ta , (67)and with the aid of the method provided by [40] for the last equation, the cubic equation below is obtained( − λ − ν + 2) + ( χψ ) (2 λ i + ν + 1) Υ − ψ (cid:0) χ (cid:1) − χ ψρ π Gr (cid:20) (2 λ i + ν − (cid:90) r eff L ( r ) r d r + (cid:90) r ta L ( r ) r d r − π λ i + ν + 1) ρ r × (cid:18)(cid:90) r eff η d r d t d r − λ i + ν + 1 (cid:90) r ta η d r d t d r (cid:19)(cid:21) = 0 , with Υ = Λ4 πGρ ta = Λ r GM = 2Ω Λ Ω (cid:18) ρ ta ρ ta , b (cid:19) − (1 + z ta ) − . (68)Then it is possible to find ψ , or r eff by solving the above equation. Note that M-T relation or Eq. (65)can be expressed in terms of r vir as k B T keV = 0 .
94 (2 λ i + ν ) µβ spec (cid:18) r ta r vir (cid:19) (cid:18) ρ ta ρ b , ta (cid:19) / Ω / (cid:18) M M (cid:12) h − (cid:19) / (1 + z ta ) × (cid:20) r vir ρ b , ta π H Ω ρ r (1 + z ta ) (cid:90) r vir L ( r )d rr −
23 Λ H Ω (cid:18) r vir r ta (cid:19) (cid:18) ρ b , ta ρ ta (cid:19) z ta ) − / π / r vir r ta (cid:18) ρ b , ta ρ ta (cid:19) / (cid:16) ρ b , M (cid:17) /
11 + z ta × λ − µ ( δ ) (cid:21) , (69)where µ ( δ ) and λ are parameters related to dynamical friction and are given in [41].The previous equation has obtained for the mass-temperature relation of galaxy clusters, consideringthe effects of angular momentum acquisition (in [18]), dynamical friction (in [19]) and eventually, theimpact of interacting dark sector, in this paper. As can be seen, λ i plays a more profound role in thisapproach, in comparison with Afshordi & Cen’s method, owing to its contribution to both Eqs. (68)and (69). Similar to the preceding model, ” λ (cid:48) i ” could be substituted for ” λ i ” to create the ”Second2Possibility” in all equations.This model is based on the assumption of cluster formation with the evolution of a spherical top-hatdensity perturbation, and the ”late-formation approximation”. The latter approximation states that anycluster at redshift z is just reached its virialization. Although it is a good assumption in some cases,including the critical case of Ω = 1 (where the cluster formation is rapid), it constructs impediments toother cosmological models. C. Continuous Formation Model
After a discussion on limitations and disadvantages to the former model in [18], Del Popolo derivesM-T relation concerning the continuous formation model, which had been used in [42] before. In thismodel, cluster formation occurs gradually, instead of instantaneously. The effects of angular momentumand dynamical friction with respect to this approach have been studied in [18] and [19], respectively.Now, we are going to study how interacting dark sector makes a difference in M-T relation in terms ofthis procedure.By integrating Eq. (58), Del Popolo obtains an expression for the ratio of the total energy of a virializedcluster to its mass or EM . We avoid iterating calculations, so the result is EM = 3 m m − (cid:18) πGt Ω (cid:19) M (cid:34) m + (cid:18) t Ω t (cid:19) / + K ( m, x )( M/M ) / + λ − µ ( δ ) + Λ χ H Ω b , (cid:21) , (70)where t Ω = π Ω H o (1 − Ω − Ω Λ ) ,K ( m, x ) = ( m − F x
LerchPhi ( x, , m/ − ( m − F LerchPhi ( x, , m/ ,LerchP hi ( x (cid:48) , y (cid:48) , z (cid:48) ) = (cid:80) ∞ n =0 x (cid:48) n ( z (cid:48) + n ) y (cid:48) ,F = / π / χρ / / H Ω (cid:82) r L ( r ) drr ,x = 1 + ( t Ω t ) / . (71)Meanwhile, M = M x − m/ and M is given in [42].Combining Eqs. (70) and (38) with the virial theorem results in k B T = 43 ˜ a µm p β spec EM , (72)and afterwards k B T keV = 25 ˜ a µm p β spec mm − (cid:18) πGt Ω (cid:19) / M / × (cid:34) m + (cid:18) t Ω t (cid:19) / + K ( m, x )( M/M ) / + λ − µ ( δ ) + Λ χ H Ω b , (cid:21) . (73)Here, the parameter ˜ a is the ratio of the kinetic to total energy of the cluster, and according to Eq. (35)we have ˜ a = 2 λ i + ν − λ i − ν . (74)3Thus, we can see that the trace of interacting dark sector emerges in a factor in M-T relation. Like-wise the previous procedures, putting ” λ (cid:48) i ” instead of ” λ i ” gives the equation for the ”Second Possibility”. IV. RESULTS AND DISCUSSION
We use five different sets of observational data to determine constants of the interacting dark sectorfor Models I to V, with the aid of M-T relation (Eq. (45)). These observational data sets are providedin [43], [44], [45], [46] and [47]. The first set provides details of mass and temperature for 32 clusters(hereafter Obs. 1999). The second source of data is used by Afshordi & Cen in [17] and consists of 39clusters (hereafter Obs. 2001). The third data set includes Chandra’s observations for 10 low-redshiftclusters (hereafter Obs. 2006) and details of 49 low-redshift clusters from Chandra are collected in thefourth data set (hereafter Obs. 2009). Finally, the last resource comprises 20 clusters from XMM-Newtonobservations (hereafter Obs. 2015).Measurements of temperature are generally based on X-ray observations, hence the temperatures givenin the mentioned catalogs are X-ray temperature and could be different than density-weighted temper-ature in Eq. (45), which is averaged over the whole cluster. The reason lies within the fact that X-raytemperature ( T X ) is exclusively measured over the central brighter portion of the cluster. To convertX-ray temperature to T in Eq. (45), we use the relation below from [48] : T = T X [1 + (0 . ± .
05) log T X ( keV ) − (0 . ± . . (75)As it has been mentioned before, it is prevalent to consider the overdensity of the virialized clusters tobe about 200 times the critical density of the universe. Therefore, M is considered to be the clustermass after virialization. The masses given in Obs. 1999 to 2015 have been obtained with respect todifferent methods and none of them incorporates M . In order to convert these masses to M (e.g. M to M ), we use the relation M δ ∝ δ − . from [49], where δ = M ( 36 and α c = − . α c = − . SN e Ia + H + CC + BAO ”, is illustrated with red lines in Fig. (1). Likewise, the constraintof ” P lanck T T ” has given α c = − . × − and its result in M-T relation is shown with black lines.For both predictions, solid lines are related to the ”First Possibility” of the NFW density profile, whiledotted lines are attributed to the ”Second Possibility” for the same density profile. The results of theBurkert density profile are presented by dashed lines (for the ”First Possibility”) and dash-dot lines (forthe ”Second Possibility”). Note that the differences between first and second possibilities are very subtlein this model whereby solid and dotted black lines are almost indistinguishable. Data sets and theirfitted curves for Obs. 1999, 2001, 2006, 2009, and 2015 are demonstrated with colors cyan, magenta,4blue, green, and brown, respectively (the fitted lines for Obs. 1999 and 2001 are virtually coincident).We immediately infer that for the case of α x = 0 in Model I, a negative α c has to be very close to zeroto not violate the constraint of Eq. (57). However, these values are not consistent with any observationaldata set. FIG. 1: The behavior of the mass-temperature relation in interacting Model I, in special case of α x = 0. Red linesindicate the outcome of ” SNe Ia + H + CC + BAO ” observations for α c and black lines display the predictionrelated to ” P lanck T T ” observations for this parameter. The other five colors denote five observational data setsfrom 1999 to 2015 (Obs. 1999: cyan; Obs. 2001: magenta; Obs. 2006: blue; Obs. 2009: green; Obs. 2015:brown). Solid and dotted lines show the ”First” and ”Second” possibilities for NFW density profile, while dashedand dash-dot lines illustrate these two possibilities for Burkert profile, respectively. Fig. (2) indicates M-T relation for another special case for Model I, which is α c = 0. Chosen colorsand types of lines are the same as Fig. (1) and again, results of ” SN e Ia + H ” (with α x = − . 26) and” SN e Ia + H + CC ” (with α c = − . 27) violate the constraint of Eq. (57) and consequently, cannot bepresented. Whereas, observations of ” SN e Ia + H + CC + BAO ” (with α x = − . P lanck T T ”(with α x = − . λ i explains the ratio of kinetic to potential energy after virialization and plays the mostvital role in our calculations. Fig. (3) reveals how λ and λ (cid:48) change as a function of α c or α x , in twomentioned cases of Model I, which are more simple. The blue lines are related to Model I with α x = 0and the red lines describe the same model with α c = 0. Therefore, the horizontal axis is attributedto α c in the former case, and to α x in the latter one. Moreover, solid lines are shown as the symbolof the ”First Possibility”, and the dotted lines denote the ”Second Possibility”, mutually. The blackdashed line is drawn with respect to the obtained value of λ I for Obs. 1999 (”First Possibility”); andthe dash-dot line shows the same value, but regarding the ”Second Possibility”. We do not display theoutcomes of the other four observational data sets to avoid an overcrowded graph.For both situations of Model I, large negative values of α j are too far away from the observationalresults. As the interacting constants are declining, both cases reach to observational outcomes justbefore the zero points. Although Model I with α x = 0 almost keeps its slope for positive values, the caseof α c = 0 remains stable and would be rather comparable with observational results if λ and λ (cid:48) wereless than 0.5, even for higher values of α x . According to the definition of Q for Model I, it means thatif the transfer of energy from DE to DM primarily stemmed from the density of DE, different valuesfor interacting constant would not lead to considerable changes in the virial condition. In other words,5 FIG. 2: Comparison between observational data and the predictions of ” SNe Ia + H + CC + BAO ” and” P lanck T T ” observations for the case of α c = 0 in Model I. Colors and types of lines are chosen the sameas Fig. (1) whether the protocluster consists of a dense region of DE or not, there would be merely negligibledifferences. However, it does not have great practical importance, since we initially assumed that thedistribution of DE is unchanged through the interior and exterior of the collapsing sphere. On thecontrary, if the energy transfer between DE and DM were mostly affected by the density of DM, thevirial theorem would gradually change with interacting constant. FIG. 3: The behavior of λ I (or λ (cid:48) I ) as a function of interacting constant for two simple cases of Model I. Theblack dashed line represents the result of Obs. 1999 for the ”First Possibility” and the dash-dot line shows thisfor the ”Second Possibility”. The blue lines are related to the case of α x = 0 and the red lines indicate ModelI with α c = 0. Here, the solid lines describe the ”First Possibility”, while the dotted lines are attributed to the”Second Possibility”. Description of Model I in general (without any zero constant) is more elaborate. Nonetheless, several6constraints have been yet derived. For example, [11] obtains four constraints between α c and α x . In ourstudy, Eq. (57) gives rise to another constraint for these two parameters:0 < − α c α c + 3 α x /R < − ν . (76)Fig. (4) illustrates how different inputs of α c and α x give different amounts of λ I , for a small rangefrom − . . λ I for each given α c and α x . The redline also constrains acceptable choices for these two parameters, according to Eq. (76). Here, we chosethe value of c = 5 for a typical cluster and used NFW density profile to calculate ν . All the points inthe left-bottom corner of the figure (below the red line) are unacceptable and have no physical meaningdue to our recent constraint. In this specific region, which has been deliberately chosen to be close tonon-interacting models, every couple with α c = − α x gives approximately the same value for λ I , while α c = α x results in very different numbers. FIG. 4: Different combinations of α c and α x in the range between − . . λ I from justless than 0.2 to over 1.6, as it is illustrated in this figure. Colors stand for the given value of λ I for any givencouple of α c and α x , according to the guide strip in the right side. The red line specifies the obtained constraint,which confines real physical choices. For Models II, III and IV, there is only one interacting constant. For Model II, Fig. (5) makes acomparison between observational data and the outcome of obtained values for ξ in [21]. Similar tothe previous cases, the result of ” SN e Ia + H ”, which has given ξ = − . 53, violates Eq. (57) andleads to negative temperatures. Despite Model I, ” SN e Ia + H + CC ” (with ξ = − . 07) resultsin an allowable prediction for M-T relation, which is represented with the purple lines in Fig. (5).The characteristics of the other lines are selected similar to Figs. (1) and (2); with ξ = − . SN e Ia + H + CC + BAO ” and ξ = − . 010 for ” P lanck T T ”. In this model, predictions of” SN e Ia + H + CC ” and ” SN e Ia + H + CC + BAO ” are close to some observational data. Forexample, ” SN e Ia + H + CC + BAO ” result of the ”Second Possibility” in NFW density profile andalso the outcome of ” SN e Ia + H + CC ” for the ”First Possibility” in Burkert density profile areapproximately in agreement with Obs. 1999 and Obs. 2001.Similarly, Fig. (6) shows M-T relation for three allowable values of ξ in Model III and compares themwith fitted curves of the five observational data sets. Here, the values have been proposed as: ξ = − . SN e Ia + H ” (unacceptable), ξ = − . 04 for ” SN e Ia + H + CC ” (purple lines), ξ = − . 08 for” SN e Ia + H + CC + BAO ” (red lines) and ξ = − . P lanck T T ” (black lines). It is clearthat merely, the results of ” SN e Ia + H + CC + BAO ” for the NFW density profile are almost close toObs. 1999 and Obs. 2001 and again, the other predictions show higher masses than data sets.7 FIG. 5: The M-T diagram of galaxy clusters based on Model II. The features are identical to Figs. (1) and (2),except for the purple lines which are emerged because the predicted value from ” SNe Ia + H + CC ” observationsis allowable in this model.FIG. 6: The M-T diagram for Model III; all chosen colors and types of lines are analogous to Fig. (5) For Model IV, the result of ” SN e Ia + H + CC ” ( ξ = − . 27) is impossible to indicate an actualillustration of M-T relation, while the outcomes of ” SN e Ia + H ” ( ξ = − . SN e Ia + H + CC + BAO ” ( ξ = − . P lanck T T ”( ξ = − . × − ) are credible. Fig. (7) represents these three predictions and compares them withobservational data sets. In this graph, the prediction of ” SN e Ia + H ” is displayed by yellow lines andits ”First Possibility” of NFW density profile is virtually consistent with Obs. 2015.The evolution of λ as a function of ξ i (with i = II, III, IV ) for Models II, III, and IV are presentedin Fig. (8). The brown, green, and magenta lines are related to Models II, III, and IV, respectively.Likewise Fig. (3), black lines describe the obtained value from Obs. 1999 in which the solid lines aredrawn for the ”First Possibility” and dotted lines show the ”Second Possibility”. It demonstrates thatwhile the λ gradually decreases with the growth of ξ i in Models III and IV, it sharply falls for Model II.8 FIG. 7: The behavior of M-T relation for the predicted values of Model IV. Yellow lines represent the observationsof ” SNe Ia + H ” and the other colors and types of lines are chosen completely the same as the previous M-Tgraphs.FIG. 8: The changes of λ or λ (cid:48) as a function of interacting constant in Models II, III, and IV. The black dashedline and black dash-dot line display the ”First” and the ”Second” possibilities for Obs. 1999, respectively. Thebehavior of the three mentioned models are shown with brown (Model II), green (Model III), and magenta (ModelIV) lines. Model V is the most complicated one. In addition to the fact that there are two interacting parameters,there is also an important dependency on H (and therefore redshift z ), which means that λ evolves withtime. Although [21] does not investigate model V, [11] claims that Γ x and Γ c should have opposite signs.As a second condition, it is possible to use Eq. (57) to constrain interacting constants. Fig. (9) showsthe evolution of λ with time, for the simple cases of Γ x = 0 or Γ c = 0. The horizontal axis indicates H ( z ) H from the present time to approximately z = 0 . 75, when H ( z ) H = 1 . 5. The blue lines are related to the caseof Γ x = 0, and the red lines describe the case of Γ c = 0. Solid lines and dotted lines denote the first and9the second possibilities, respectively. As an observational example, we used the result from Obs. 2001,regarding the first (black dashed line) and the second (black dash-dot line) possibilities. According tothis graph, the further the cluster is located, the more noticeable difference between the cases of Γ x = 0and Γ c = 0 can be seen. All the lines are consistent with observational data in a low-redshift, since wefixed the value of interacting constants with regard to this observational data set itself, so it is not aninteresting point. In addition, the figure clearly reveals that the constant of the virial condition wasmuch lower than its present value in the past. It means that further clusters in interacting Model Vmust behave more similarly to the non-interacting model. FIG. 9: The figure demonstrates how λ and λ (cid:48) evolve with time, considering Model V. Red and blue lines arerelated to Γ x = 0 and Γ c = 0, respectively, and black lines denote the result of Obs. 2001 (with dashed lineand solid lines representing the ”First Possibility”, and dash-dot line and dotted lines standing for the ”SecondPossibility”). The core of our work is to determine interacting constants with respect to observational data sets formass and temperature of galaxy clusters. As it has been mentioned before, we tried to find λ i (and λ (cid:48) i ) ina way that the M-T relation could accurately fit the observational curves. Tables (I) and (II) summarizethe information which has been obtained for all situations, including NFW and Burkert density profiles,the first and the second possibilities, five observational data sets and seven preferred and discussed casesof Models I to V. As far as the constants are concerned, we obtained negative values for all of them. Ourresults are in agreement with [21] in terms of obtaining negative values for these constants. It meansthat energy transfer occurs from DM to DE.In Model V, it is common to define the dimensionless constants γ j = Γ j H and write λ V as: λ V = H ( z ) H − γ c H ( z ) H + 3 γ c + 3 γ x /R . (77)Therefore, we calculated the constants γ j rather than Γ j .From the calculated constants, it can be concluded that more negative values are needed for a coreddensity profile (Burkert) than a cuspy profile (NFW) to be consistent with each observational data set.Note that even fine differences among observational results may play considerable roles in the calculatedconstants. In fact, in our method, every input value in Eq. (46) contributes to measuring interactingconstants. However, we strove to incorporate as many various assumptions as possible (embracing differ-ent density profiles, different possibilities, and different observational data sets) in order to compensatefor the inaccuracies of parameters within Q .0 TABLE I: The calculated constants of interacting models regarding the ”First Possibility”, based on makingcomparison with observational data sets of mass and temperature in galaxy clusters. α c ( α x = 0) α x ( α c = 0) ξ ξ ξ γ c ( γ x = 0) γ x ( γ c = 0)NFW Obs. 1999 − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . α c ( α x = 0) α x ( α c = 0) ξ ξ ξ γ c ( γ x = 0) γ x ( γ c = 0)NFW Obs. 1999 − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . V. CONCLUSION We investigated the mass-temperature relation of galaxy clusters for a number of interacting modelsof dark matter and dark energy, which are summarized in Eq. (7). First of all, we expanded themethod provided in [25] to derive the modified virial theorem for all these models of the interactingdark sector in Section II. It immediately suggested that there might be two different possibilities for thiscondition, regarding two plausible behaviors of dark matter through baryonic matter. Then we used themodified virial condition to obtain M-T relation with respect to three different procedures in Section III.It revealed that the effect of interaction only emerges within the normalization factor of the M-T relation.The M-T relation led to a new constraint on interacting constants, which totally depends on theconcentration parameter and density profile of the clusters (Eq. (57)). This constraint is used tocheck the suggested constants of interacting and showed that many of those suggested values are notacceptable, due to resulting in negative masses for given temperatures.To analyze the obtained M-T relation, we focused on five different observational data sets andcompared their fitted lines with many suggested values for interacting constants. We consideredtwo outstanding density profiles, which are NFW and Burkert, and managed to calculate interactingconstants for seven cases of the five interacting models. Overall, it appears that according to theseobservational data sets, energy transfer should occur from DM to DE, which leads to negative values forinteracting constants. It is completely consistent with the results of [21], which has investigated manyother observational constraints to obtain numerical values for interacting constants. Although differentobservations result in minuscule differences in the figures, the figures are usually near zero. Furthermore,the positive constants can solely be obtained for Models I and V, if both constants have non-zero values.It also appears that for a cored density profile, more negative constants are obtained in comparison witha cuspy profile.1In the meantime the M-T relation and interacting constants were being studied, we also allocatedsome parts of this paper to discuss how the ratio of kinetic to the potential energy of a virialized clusterbehaves as a function of interacting constants or redshift, for many of our interacting models. Fig. (3)and Fig. (8) show that various models of interaction cause different behaviors of λ as a function ofinteracting constant, although all of them lead to decreasing functions. The graphs also indicated thatfor Model V, the value of λ grows with time, resulting in the fact that more distant clusters must betheoretically more consistent with non-interacting models. Two specific cases of this model (Γ x = 0and Γ c = 0) are also more distinguishable from each other when the cluster is located in a higher redshift.Finally, we emphasized that the obtained values could be extremely affected by the other parametersin the normalization factor of the M-T relation, which we have fixed with particular values for ourresearch. However, considering a variety of possibilities might have compensated for these unwantederrors and impacts to some extend.We should also mention that future observations of cluster masses and temperatures may assist toobtain more exact numerical values for interacting constants. To this purpose, cluster masses shouldbe determined via the other methods of mass measurements, such as gravitational lensing, instead ofobtaining the mass from X-ray temperature. Euclid satellite and LSST are two upcoming projectswhich would provide improved mass data through gravitational lensing observations. 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