# Clusters of Galaxies in a Weyl Geometric Approach to Gravity

Authors: Erhard Scholz
Journal: Journal of Gravity
 (2016)

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## Abstract

A model for the dark halos of galaxy clusters, based on the Weyl geometric scalar tensor theory of gravity (WST) with a MOND-like approximation, is proposed. It is uniquely determined by the baryonic mass distribution of hot gas and stars. A first heuristic check against empirical data for 19 clusters (2 of which are outliers), taken from the literature, shows encouraging results. Modulo a caveat resulting from different background theories (Einstein gravity plusΛCDM versus WST), the total mass for 15 of the outlier reduced ensemble of 17 clusters seems to be predicted correctly (in the sense of overlapping 1σ error intervals).


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## Body

## 1. Introduction

In this paper the gravitational dynamics of galaxy clusters is investigated from the point of view of Weyl geometric scalar tensor theory of gravity (WST) with a nonquadratic kinematic Lagrange term for the scalar field (3L) similar to the first relativistic MOND theory rAQUAL (“relativistic a-quadratic Lagrangian”) [1]. To make the paper as self-contained as possible, it starts with an outline of WST-3L (Section 2). WST-3L has two (inhomogeneous) centrally symmetric static weak field approximations: (i) the Schwarzschild-de Sitter solution with its Newtonian approximation, which is valid if the scalar field and the WST-typical scale connection plays a negligible role; (ii) a MOND-like approximation which is appropriate under the constraints that the scale connection cannot be ignored but is still small enough to allow for a Newtonian weak field limit of the (generalized) Einstein equation. The acceleration in the MOND approximation consists of a Newton term and additional acceleration of which three-quarters are due to the energy density of the scalar field and one-quarter is due to the scale connection typical for Weyl geometric gravity.In centrally symmetric constellations thescalar field energy forms ahalo about the baryonic mass concentrations. Besides the acceleration derived from the (Riemannian) Levi-Civita connection induced by the baryonic matter and the scalar field energy anadditional acceleration component due to the Weyl geometric scale connection arises in the present approach. If the latter is expressed by a fictitious mass in Newtonian terms, aphantom halo can be ascribed to it. It indicates the amount of mass one has to assume in the framework of Newton dynamics to produce the same amount of additional acceleration. The scalar field halo consists of true energy derived from the energy momentum tensor; it is independent of the reference system, as long as one restricts the consideration to reference systems with low (nonrelativistic) relative velocities. The phantom halo, on the other hand, is a symbolical construct and valid only in the chosen reference system (and scale gauge). For galaxy clusters we find two components of the scalar field halo, one deriving from the total baryonic mass in the MOND approximation of the barycentric rest system of the cluster (component 1) and one arising from the superposition of all the scalar field halos forming around each single galaxy in the MOND approximation of the latter’s rest system (component 2). Because velocities of the galaxies with regard to the cluster barycenter are small (nonrelativistic) and also the energy densities are small, the two components can be superimposed additively (linear approximation). With regard to the barycentric rest system of the cluster athree-component halo for clusters of galaxies arises, two components being due to the scalar field energy and one purely phantom (Section 3).The two-component scalar field halo is a distinctive feature of the Weyl geometric scalar tensor approach; it is neither present in the nonrelativistic MOND approaches nor in rAQUAL. One may pose the question whether it suffices for explaining the deviation of the cluster dynamics from the Newtonian expectation without additional dark matter. If we call the totality of the three components thetransparent halo of the cluster (Section 3.5), the question is whether the (theoretically derived) transparent halo can explain thedark halo of galaxy clusters, observationally determined in the framework of Einstein gravity and ΛCDM.Section4 contains a first test of the model by confronting it with empirical data on mass distribution available in the astronomical literature. A full-fledged test would presuppose an evaluation of raw observational data in the framework of the present approach and is beyond the scope of this study. Here we use recently published data on the total mass (dark plus baryonic), hot gas, and the star matter for 19 galaxy clusters, which have been determined from different observational data sources (and are thus more precise than earlier ones) [2–4]. Two of the 19 clusters show a surprisingly large relation of total mass to gas mass. They are separated as outliers from the rest of the ensemble already by the authors of the study and so do we. 17 nonoutlying clusters remain as our core reference ensemble.In the mentioned studies total mass, gas mass, and star mass are determined on the background of Einstein gravity plusΛCDM. That raises the problem of compatibility with the WST framework. It is discussed in Sections 4.1 and 4.2 and leads to a certain caveat with regard to the empirical values for the total mass (M200,M500) and the reference distances r200,r500 to the cluster centers. But it does not seem to obstruct the possibility for a first empirical check of our model (Section 4.2). More refined studies are welcome. They have to use the WST framework for evaluating the observational raw data or, at least, to analyze the transfer problem of mass data from one framework to the other in more detail.For 15 of the 17 main reference clusters the empirical and the theoretical values for the total mass agree in the sense of overlapping1σ error intervals. The remaining two overlap in the 2σ range. The two outliers of the original study do not lead to overlapping intervals even in the 4σ range (Section 4.6). In the present approach the dynamics of theComa cluster is explained without assuming a component ofparticle dark matter. It is being discussed in more detail than the other clusters in Section 4.5.A short comparison with the halos of Sanders’μ2-MOND model with an additional neutrino core [5] and with the NFW halo [6] is given for Coma (Section 4.7). The paper is rounded off by a short remark on the bullet cluster (Section 4.8) and a final discussion (Section 5).


## 2. Theoretical Framework

### 2.1. Weyl Geometric Scalar Tensor Theory of Gravity (WST)

Among the family of scalar tensor theories of gravity the best known ones and closest ones to Einstein gravity are those with a Lagrangian containing a modified Hilbert term coupled with a scalar fieldϕ. Their Lagrangian has the general form(1)L=12ξϕ2R+Lϕ-λ4ϕ4⋯,L=Lg,g=det⁡g.Here g is an abbreviation for a 4-dimensional pseudo-Riemannian metric g=(gμν) of signature (−+++). ϕ is a real valued scalar field on spacetime, Lϕ its kinetic term, and ξ a constant coefficient and the dots indicate matter and interaction terms. Under conformal rescaling of the metric,(2)gμν⟼gμν′=Ω2gμνΩapositiverealvaluedfunction,the scalar field changes with weight -1; that is, ϕ↦ϕ′=Ω-1ϕ. So far this is similar to the well known Jordan-Brans-Dicke (JBD) scalar tensor theory of gravity [7–10]. But here we work in a scalar tensor theory in the framework of Weyl’s generalization of Riemannian geometry [1, 11–14].Crucial for theWeyl geometric scalar tensor approach (WST) is that the scalar curvature R and all dynamical terms involving covariant derivatives are expressed in Weyl geometric scale covariant form. Fields X are scale covariant if they transform under rescaling by X↦X~=ΩwX, with w∈Q (in most cases even in Z); w is called the weight of X. Covariant derivatives of scale covariant fields are defined such that the result of covariant derivation DμX is again scale covariant of the same weight w as X. The Lagrangian density L isinvariant under conformal rescaling for any value of the coefficient ξ2 of the modified Hilbert term. For the matter and interaction terms of the standard model of elementary particles, scale invariance is naturally ensured by the coupling to the Higgs field which has the same rescaling behaviour as the gravitational scalar field. Although there is no complete identity, there is a close relationship between scale and conformal invariance in quantum field theory [16]. For classical matter we expect that a better understanding of the quantum to classical transition, for example, by the decoherence approach, allows considering scale invariant Lagrangian densities also. For the time being we introduce the scale invariance of matter terms in the Lagrangian as apostulate.In our context an important consequence is the scale covariance of the Hilbert energy momentum tensor(3)Tμν=-2gδLmδgμν,which is of weight w(Tμν)=-2. That is consistent with dimensional considerations on a phenomenological level. It has been shown that the matter Lagrangian of quantum matter (Dirac field, Klein Gordon field) is consistent with test particle motion along geodesics (autoparallels) γ(τ) of the affine connection, if the underlying Weyl geometry is integrable (see (8)) [11]. For classical matter we assume the same. It is to be expected that it can be proven similar to Einstein gravity.We do not want to heap up too many technical details; more can be found in the literature given above. But we have to mention that aWeylian metric can be given by an equivalence class of pairs (g,φ) consisting of a pseudo-Riemannian metric g=gμνdxμdxν, theRiemannian component of the Weyl metric, and a differentiable one-form φ=φμdxμ, thescale connection (or, in Weyl’s original terminology, the “length connection”). φμ is often called the Weyl covector or even the Weyl vector field. In fact, φ denotes a connection with values in the Lie algebra of the scale group (R+,·) and can locally be represented by a differentiable 1-form. The equivalence is given by rescaling the Riemannian component of the Weylian metric according to (2), while φ has the peculiar gauge transformation behaviour of aconnection, rather than that of an ordinary vector (or covector) field in a representation space of the scale group:(4)φμ⟼φμ′=φμ-∂μΩΩor, shorter, φ′=φ-dlog⁡Ω.A Weylian metric has a uniquely determined compatible affine connectionΓ; in physical terms it characterizes theinertiogravitational guiding field. It can be additively composed of the well known Levi-Civita connection Γg of the Riemannian component g of any gauge (g,φ) and an additional expression Γφ in the scale connection; in short(5)Γ=Γg+ΓφwithΓφνλμ=δνμφλ+δλμφν-gνλφμ.Covariant derivativesD in the Lagrangians (1) and (below) (28) and consequently in the expression for the energy momentum tensor of the scalar field (equations (35) and (36)) below denote those of Weyl geometry. For a covariant field Xν of weight w the derivativation ∇ with regard to Γ of (5) D is supplemented by a term due to the scaling weight w of X:(6)DμXν=∇μXν+wφμXν=∂μXν+ΓμλνXλ+wφμXν.It turns out that for the metric gμν the full covariant derivative is zero:(7)Dλgμν=0⟷∇λgμν+2φλgμν=0.This is the Weyl geometric compatibility condition between metric and affine connection (sometimes called “semimetricity”).In the low energy regime there are physical reasons to constrain the scale connection to theintegrable case with a closed differentiable form dφ=0; that is, ∂μφν=∂νφμ. Then φ is a gradient (at least locally) and may be given by(8)φμ=-∂μω.This constraint is part of the defining properties of WST. Then it is possible to “integrate the scale connection away” [11]. Having done so, the Weylian metric, given as (g~,φ~), is characterized by its Riemannian component g~μν only (and a vanishing scale connection φ~μ=0). By obvious reasons we call this theRiemann gauge. This is the analogue of the choice of Jordan frame in JBD theory.In any case, the choice of representativegμν also fixes φμ; both together define ascale gauge of the Weylian metric. Conformal rescaling of the metric is accompanied by the gauge transformation of the scale connection (4). From a mathematical point of view all the scale gauges are on an equal footing, and the physical content of a WST model can be extracted, in principle, from any scale gauge. One only needs to form a proportion with the appropriate power of the scalar field. From the physical point of view there are, however, two particularly outstanding scale gauges. Of special importance besides Riemann gauge is the gauge in which the scalar field is scaled to a constant ϕo (scalar field gauge). For the particular choice of the constant value such that(9)ξϕo2=8πG-1=Epl2,with the Newton gravitational constant G, this gauge is calledEinstein gauge (Epl the reduced Planck energy). It is the analogue of Einstein frame in JBD theory. In this gauge the metrical quantities (scalar, vector, or tensor components) of physical fields are directly expressed by the corresponding field or field component of the mathematical model (without the necessity of forming proportions).In Riemann gaugeΓ reduces to Γg by definition. Thus, in this gauge, the guiding field is given by the ordinary expression for the Levi-Civita connection. On the other hand, in Einstein gauge the measuring behaviors of clocks are most immediately represented by the metric field and also other physical observables are most directly expressed by the field values in this scale. Then the expressions for the gravitational field and the accelerations contain contributions from the Weylian scale connection. Thus a specific dynamical difference to Einstein gravity and Riemannian geometry (and to JBD theory) arises even in the case of WST with its integrable Weyl geometry.Writing the scalar fieldϕ~ in Riemann gauge (g~,0) in exponential form, ϕ~=eω, turns its exponent(10)ω≔ln⁡ϕ~into ascale invariant expression for the scalar field. In the following we shall omit the tilde sign to simplify notation. The scale connection φ=φ^ in scalar field gauge is then(11)φ^=-dω,because Ω=ϕ~=eω is the rescaling function from Riemann gauge to scalar field gauge. For more details see [1, 12, 13, 17, 18].For the sake of consistency under rescaling we considerscale covariant geodesicsγ(τ) with scale gauge dependent parametrizations of the geodesic curves of weight w(γ˙)=-1:(12)u˙λ+Γμνλuμuν-φμuμuλ=0,uμ=γ˙μ.Here the affine connection contains a φ-dependent term in addition to the well known Levi-Civita connection φΓμνλ derived from gμν (5). The last term on the l.h.s. of (12) takes care of the scale dependent parametrization (compare (6)). In this way we work with aprojective family of paths.For any gauge of the Weylian metric and the scalar field,(g,φ,ϕ), any timelike geodesic has thus a generalized proper time parametrization γ(τ) with gμνuμuν=-1, where uμ=γ˙μ. Inverting the coordinate time function t(τ) along the geodesic by τ(t) we have, in abbreviated notation, τ′t′=1 and thus (13)d2xidt2=dτdtddτdτdtdxidτ=dτdt2d2xidτ2-dtdτ-3d2tdτ2dxidτ.With (12) and indices i,j,k=1,2,3,  μ,ν,…=0,1,2,3 this leads to (14)d2xidt2=dτdt2-Γμνidxμdτdxνdτ+φμdxidτdxμdτ-dτdt3dxidτ-Γμν0dxμdτdxνdτ+φμdtdτdxμdτ=-Γμνidxμdtdxνdt+φμdxidtdxμdt+Γμν0dxμdtdxνdtdxidt-φμdxμdtdxidt.Happily, the length connection terms coming from the scale covariance modification of the geodesic equation (12) cancel the equation of motion for mass points in Weyl geometric gravity, parametrized in coordinate time, becomes (15)d2xidt2=-Γ00i+Γ000dxidt-2Γ0jidxjdt-Γjkidxjdtdxkdt+2Γ0j0dxidtdxjdt+Γjk0dxidtdxjdtdxkdt.In the result the dynamics of mass points in Weyl geometric gravity is governed by the guiding field (the affine connection), as in the semi-Riemannian case [19, equ. (9.1.2)]. Note, however, that in (15) thelength connection enters into the affine connection and influences the dynamics because of (5).The geodesic equation thus contains terms in the scale connectionφμ. In the low velocity, weak field regime the equation of motion reduces to the form well known from Einstein gravity d2xj/dt2=-Γ00j. Here Γ00j=Γg00j+Γφ00j (j=1,2,3) are the coefficients of the Weyl geometric affine connection with Γφ00j given by (5). This is the crucial modifying term for gravity in the Weyl geometric approach (low velocity case).


### 2.2. The Weak Field Static Approximation

We want to understand the scale connection for the motion of point particles. The free fall of test particles in Weyl geometric gravity follows scale covariant geodesics. It is governed by a differential equation formally identical to the one in Einstein gravity (15). Here we look at the weak field static case for low velocities in order to study the dynamics of stars in galaxies and galaxies in clusters. For studies of the gas dynamics and its modification in our framework the velocity dependent terms of (15) have to be taken into account. This is not being done here.Analogous to Einstein gravity, the coordinate accelerationa for a low velocity motion x(τ) in proper time parametrization is given by(16)aj=d2xjdτ2≈-Γooj.According to (5) the total acceleration decomposes into(17)aj=-Γgooj-Γφνλj=aRj+aφj(j=1,2,3 indices of the spacelike coordinates), where(18)aRj=-Γgoojis the Riemannian component of the acceleration known from Einstein gravity. Clearly(19)aφj=-Γφoojrepresents anadditional acceleration due to the Weylian scale connection. For a diagonal Riemannian metric g=diag⁡goo,…,g33 the general expression (5) simplifies to -Γφooj=-gooφj. General considerations on observable quantities and consistency with Einstein gravity show that, in order to confront it with empirically measurable quantities, we have to take its expression inEinstein gauge if we want to avoid additional rescaling calculations [14, sec. 4.6].For a (diagonalized) weak field approximation in Einstein gauge,(20)gμν=ημν+hμν,hμν≪1,with η=ϵsigdiag⁡-1,+1,+1,+1, the Riemannian component of the acceleration is the same as in Einstein gravity. Its leading term (neglecting 2nd-order terms in h) is(21)aRj=Γgooj≈12ηjj∂jhoonosummationoverj.In the limit, ΦN≔-(1/2)hoo behaves like a Newtonian potential(22)aR≈-∇ΦN,where ∇ is understood to operate in the 3-spacelike coordinate space with Euclidean coefficients as the leading term of the metric.In Einstein gauge the Weylian scale connectionφ^μ arises from Riemann gauge by rescaling with Ω=eω, φ^μ=-∂μω (4), and aφj≈φj. In other words, the additional acceleration due to the scale connection (19) is generated by the scale invariant representative ω of thescalar field as itspotential:(23)aφ≈-∇ω.If we compare with Newton gravity, we can calculate the fictitious mass density which one had to assume on the right-hand side of the Poisson equation, in addition to the real masses, in order to generate the same amount of additional acceleration. Obviously here it is(24)ρph=4πG-1∇2ω.In the terminology of the MOND literature the acceleration due to the Weylian scale connection corresponds to aphantom energy density ρph (see, e.g., [20], p. 48). We see that already on the general level the dynamics of WST differs from Einstein gravity. Only for trivial scalar field, ω=const, the usualNewton limit is recovered; otherwise it is modified. We shall explore how this modification relates to the usual MOND approaches.


### 2.3. WST Gravity with Cubic Kinematic Lagrangian (WST-3L)

The most common form of the kinetic term for the scalar field is that of a Klein Gordon field,Lϕ2=-(α/2)DνϕDνϕ, quadratic in the norm of the (scale covariant) gradient (in WST Dν denotes the a scale covariant derivative of ϕ,  Dνϕ=∂νϕ-ϕφν). For our form of the gravitational Lagrangian (1) it is conformally coupled for α=-6ξ2. Inspired by the approach of the relativistic “a-quadratic Lagrangian” (rAQUAL), the first relativistic attempt of a MOND theory of gravity [21, 22], we find that a Weyl geometric scalar tensor theory of gravity leads to a MOND-like phenomenology if we add acubic term to the kinetic Lagrangian of the scalar field(25)Lϕ=Lϕ2+Lϕ3.The crucial difference to the early approach of rAQUAL is the scale covariant reformulation in the framework of Weyl geometry. It results in a different behaviour of the scalar field energy density. Bekenstein/Milgrom’s model relied crucially on implementing a transition function f(y) between the Newton and the deep MOND regime into the kinetic term. The constraint of scale invariance of the Lagrangian reduces the underdetermination of the Lagrangian and suggests a slightly different form of the kinetic term. It is still quite near to the one of the relativistic AQUAL theory.In the review paper [22] Bekenstein gives the rAQUAL Lagrangian in the form (his equation (6))(26)Lψ=-8πGN-1L-2fL2∂νψ∂νψ,where L is “a constant with dimension of length introduced for dimensional consistency” (later it is identified as the MOND acceleration ao via c2L-1=ao). Asymptotically f(y)~(2/3)y3/2 for y≪1 (MOND regime); similarly f(y)~y for y≫1 (Newton regime). ψ is the logarithm of a rescaling function between the Jordan frame metric g~μν (called the “physical” metric) and the Einstein frame (“primitive”) metric gμν, g~μν=e2ψgμν. Its role is very close to our ω in (10). A corresponding scale covariant form of the Lagrangian could be(27)Lϕ3=ξϕ2η-1ϕ2fη-1ϕ-2∂νω∂νω,where η-1ϕo in Einstein gauge plays a role similar to ao. The sign has deliberately been changed; the reasons are given below (39).We are here interested in additive modifications (19) of Einstein gravity, mainly in a domain in which the effects of the scale connection φν=-∂νω, compared with those of the Riemannian component of the metric, cannot be neglected. This will be called a regime withMOND approximation. For nontimelike ∇ω the Lagrangian (27) then becomes(28)Lϕ3=23ξϕ2η-1ϕ-1DνϕDνϕ3/2=23ξ2ηϕ∇ω3,where we have used the abbreviation(29)∇ω≔∂νω∂νω1/2(⋯ absolute value) (for timelike ∇ω see [1]. Here we exclusively deal with the spacelike (or null) case). η denotes a constant coefficient responsible for the relative strength of the cubic kinetic term and ω is the scale invariant representative of the scalar field introduced in (10) and ∇ its gradient (w(ϕ)=-1 and w(∇ω)=-1 imply the scale weight w(Lϕ3)=-4, as it must be for scale invariance of Lϕ3).We introduce the constanta~o defined in Einstein gauge (g^,φ^) with constant scalar field ϕ^≕ϕo,(30)a~o≔η-1ϕo.Then the cubic term of the kinetic Lagrangian in Einstein gauge reads(31)Lϕ3≐238πGa~o-1∇ω3.Thedotted equality sign “≐” indicates that the respective equation is not scale invariant but presupposes a special gauge made clear by the context. Here, as in most cases in this paper, it indicates the Einstein gauge (similar for ≈˙). a~o has the dimension of inverse length/time and will play a role analogous to the MOND acceleration ao≈[c]H, where H is the Hubble parameter (at “present”) and c is the velocity of light. Coefficients of type [c] will often be suppressed in the following general considerations. They will be plugged in only in the final step. Below we shall see that for a~o=ao/16≈H/100 the WST model with cubic kinematic Lagrangian (WST-3L) acquires a MOND-like phenomenology in a weak gravitational field in which the scalar field and the scale connection cannot be neglected.A reasonable choice of adaptable parameters bringsξ and η to nearby orders of magnitude, η=βξ, with(32)β=ξ-1η,β~100.On the other hand, because of (9) and (30) the product of both coefficients is a “large number” in the sense of η·ξ=Epl/a~o=a~o-1/Lpl~1063. To the kinetic Lagrangian of the scalar field a potential term is added. It must be of order 4 to provide for scale invariance of the density (1):(33)LV4=-λ4ϕ4.Variation of the Lagrangian leads to the dynamical equations of WST, the Einstein equation, and the scalar field equation. Thescale invariant Einstein equation is (this means that not only the equation but all of its constitutive (additive) terms are scale invariant)(34)Ric-R2g=ξϕ-2Tm+Θ,where g denotes the whole collection of metrical coefficients and T(m) the energy tensor of matter (3). The scalar field contributes to the total energy momentum with two terms, Θ=Θ(I)+Θ(II), the first of which is proportional to the metric (thus formally similar to a vacuum energy tensor) (see, e.g., [23], [18, pp. 96ff.]):(35)ΘI=ϕ-2-DλDλϕ2+ξ-2LV4+Lϕg,(36)ΘμνII=ϕ-2DμDνϕ2-2ξ-2∂Lϕ∂gμν.Varying with regard toϕ gives the scalar field equation. Subtracting the trace of the Einstein equation for a conformally coupled Lϕ2 term (α=-6ξ2) strongly simplifies it and introduces the trace of the matter tensor into the scalar field equation. In Einstein gauge, with g the Riemannian component of the metric, it can be written in terms of the covariant derivative ∇g with regard to g (Levi-Civita connection in Einstein gauge) as ([1, pp. 15f., sec. 7.2, postprint version arXive v4]) (37)∇νg∇ω∂νω≐-4πGa~otrTm.If we introduce the corresponding Riemannian covariant operator(38)□Mgω=∇νg∇ω∂νω=∂ν∇ω∂νω+∇ω∇νg∂νω,the scalar field equation for a fluid with matter density ρm and pressure pm simplifies to the covariantMilgrom equation(39)□Mgω≐4πGa~oρm-3pm.In this derivation, with trT(m) entering by subtracting the trace of the Einstein equation, a sign choice like in (26) leads to the wrong sign on the r.h.s. of the Milgrom equation. This explains our sign choice in (27). By obvious reasons (38) will be called thecovariant Milgrom operator. In the static weak field static case ω does not depend on the time coordinate. Moreover with gμν≈ημν, the expression ∇ν(∇ω∂νω) turns into the nonlinear Laplace operator ∇·(∇ω∇ω) of the MOND theory with Euclidean scalar product · and norm ⋯.In the general case we have to complement (39) with the Einstein equation in Einstein gauge(40)Ric-R2g≐8πGTm+Θ,In vacuum, the trivial scalar fieldω=const is a basic solution of (39). Then WST reduces to Einstein gravity. In particular, the Schwarzschild and the Schwarzschild-de Sitter solutions of Einstein gravity are special (degenerate) solutions of WST-3L equations for λ/4=0 or λ/4≈6, respectively. In fact, they solve (40) and (39) for ϕ≐const in Riemann gauge, that is, in the case of Einstein gauge equal to Riemann gauge (g^,φ^)=(g~,0). The Riemannian component of the metric (g~=g^≕g) is given by(41)ds2=-1-2Mr-κr2dt2+1-2Mr-κr2-1dr2+r2dx22+sin2⁡x2dx32.Then Ric-(R/2)g=-3κg and Θ=Θ(I)=-(λ/4)β2a~o2g. Therefore the Einstein equation is satisfied for 3κ=(λ/4)β2a~o2; that is, κ≈2H2 for λ/4≈6 and β≈100. We see that in the case of a negligible Weylian scale connection the classical (nonhomogeneous) point symmetric solutions of Einstein gravity are valid also for the dynamics of WST. This implies thatin the case of a negligible scale connection Newton dynamics is an effective approximation for point symmetric solutions of WST (in Einstein gauge).In order to make such a type of Einstein limit compatible with our Lagrangian, a suppression of theLϕ3-term for sufficiently large accelerations aR of (18) is necessary. One may consider plugging a factor f~(ao/aR) with a function f~ such that f~(y)~1 for y>0.01 and f~(y)~0 for y≪1 into the r.h.s. expression of (28) for Lϕ3 but such a choice would have the blemish of a coordinate dependent argument of the function. A better alternative is provided by thehypothesis that the scalar field inhomogeneities are suppressed if any of the sectional curvatures κ (with respect to the Riemannian component of the metric in Einstein gauge) surpasses a certain threshold (e.g., κ≥109aoc-22). In the next section we investigate the case of a nonnegligible scale connection. A more detailed discussion of the transition between the two domains has to be left open for another occasion.


### 2.4. A WST Approach with MOND-Like Phenomenology

If the conditions for the weak field approximation (20) are given, it is possible to identify aMOND regime as a region in which the Newton acceleration aN is smaller than ao (here aN can be identified with aR in (22)). Then the scalar field equation (39) reduces, in reliable approximation, to(42)∇·∇ω∇ω≐-4πGa~otrTm,with the Euclidean ∇-operator. We call this aMOND approximation. For pressureless matter with energy density ρm we get(43)∇·∇ω∇ω≐4πGa~oρm.That is similar to the AQUAL approach [21, 22]. Note that only thematter energy momentum tensor,without thescalar field energy density, appears on the r.h.s. of (42).Straightforward verification shows that, independent of symmetry conditions, a solution of (43) is given by ω with a gradient ∇ω=-aφ such that(44)aφ=a~oaNaN=a~oaNaNaN,where aN denotes the Newton acceleration of the given mass density,(45)∇2ΦN=4πGρm,aN=-∇ΦN(calculations in the approximating Euclidean space with norm ⋯). The solution of the nonlinear Poisson equation (43) is much simpler than one might expect at a first glance. In a first step the linear Poisson equation of the Newton theory is to be solved and then an algebraic transformation of type (44) leads to the acceleration due to the solution of the nonlinear partial differential equation (43). In fact, aφ has the form of the deep MOND acceleration of the ordinary MOND theory (but with different constant a~o) (44). In the terminology of the MOND community, the MOND approximation of WST-3L behaves like a special case of a QMOND theory [20, pp. 46ff.].This raises the question of the Newtonian limit. Equation (44) implies aφ≪aN in regions, where aN≫ao(>a~o). Therefore aφ can effectively be neglected in the case of “large” values of aN derived from (45). Then, according to the observation at the end of Section 2.3, the Newton approximation is also reliable in WST gravity. That is true irrespective of the question of how to characterize the transition between the MOND and the Newton approximation. Here we shall consider the MOND approximationin an “upper transition” regime only, where roughly aN≤102ao. One might speak of theupper transition regime for ao≤aN≤100ao of theMOND regime if aN≤ao and of thedeep MOND regime for, let us say, aN≤10-2ao   [1, sec. 7.3].For centrally symmetric mass distributionsρ(r) with mass M(r) integrated up to r (where r=y denotes the Euclidean distance from the symmetry center, y=(y1,y2,y3) the coordinates of the approximating Euclidean space) this implies(46)aφ=-∇ω≈˙-GMra~oyy2,aφ=GMra~or.But this is only the most immediate modification of Newton gravity. There is also the additional term in (40) of theenergy density due to the scalar field, ρsf=(8πG)-1Θoo. It modifies the r.h.s. of the Newton limit of Einstein gravity (in contrast ρsf does not enter the r.h.s. of the scalar field equation (39) and therefore doesnot enter the r.h.s. of (45)).Neglecting contributions at the order of magnitude of cosmological terms (~H) the energy density of the scalar field in Einstein gauge simplifies to ([1, sec. 4.3]) (47)ρsf≈˙4πG-1∇2ω+Γjkj∂kω,where Latin indices j,k,… refer to space coordinates only.In the central symmetric case with Euclidean metric in spherical coordinates and a mass functionM(r) with M′(r)=0 for r>ro (for some distance ro), we find from (46) ∇2ω=a~oGM(r)/r2 and Γjkj∂kω=(2/r)(a~oGMr/r)=2∇2ω; (for ds2=dr2+r2(dθ2+sin2θdϑ2) the crucial affine connection components are Γ111=0,Γ212=Γ313=r-1, and thus(48)ρsf≈˙4πG-13∇2ω.That is three times the value of the phantom energy density corresponding to the acceleration of the scale connection (24). The total “anomalous” additive acceleration (in comparison to Newton gravity) is therefore(49)aadd=aφ+asf=4aφ.In the central symmetric case(50)aadd=4GMra~or.For consistency with the deep MOND acceleration ao we have to set(51)a~o=ao16≈10-2Hc≈6·10-10cms-2.Because of (44) the total acceleration a is then(52)a=aN+aadd=aN1+aoaN,aadd=aoaN.


### 2.5. Comparison with Usual MOND Theories

We can now compare our approach with other models of the MOND family. Simply adding a deep MOND term to the Newton acceleration of a point mass is unusual. Milgrom rather considered a multiplicative relation between the MOND accelerationa and the Newton acceleration aN by a kind of “dielectric analogy,”(53)aN=μaaoa,withμx⟶1forx⟶∞xforx⟶0,or the other way round (here μ(x)→x means μ(x)-x=O(x); i.e., (μ(x)-x)/x remains bounded for x→0. Cf. [20, pp.51f.]) (54)a=νaNaoaN,withνy⟶1fory⟶∞y-1/2fory⟶0.From this point of view our acceleration (52) is specified by well defined transition functions(55)μwx=1+1-1+4x2x,νwy=1+y-1/2.One has to keep in mind, however, that ourtransition functionsμ,νare reliable only in the MOND regime and the upper transitional regime (roughly aN≤102ao). They cannot be used for discussing the Newtonian limit. It will be important to see how they behave in the light of empirical data, in particular galactic rotation curves and cluster dynamics.In the MOND literature the amount of a (hypothetical) mass which in Newton dynamics would produce the same effects as the respective MOND correctionaadd is calledphantom massMph. For any member of the MOND family the additional acceleration can be expressed by the modified transition function ν~=ν-1 with ν as in (54)(56)aadd=ν~aNaoaN.Thephantom mass densityρph attributed to the potential Φph satisfies 4πGρph=∇2Φph and ∇Φph=-aadd. A short calculation shows that it may be expressed as(57)ρph=ν~aNaoρm+4πGao-1ν~′aNao∇aN·aN.It consists of a contribution proportional to ρm with factor ν~, which dominates in regions of ordinary matter, and a term derived from the gradient of aN dominating in the “vacuum” (where however scalar field energy is present). For the Weyl geometric model with ν~w(y)=y-1/2,ν~w′(y)=-(1/2)y-3/2ρph turns into(58)ρt=aoaN1/2ρm+8πG-1aoaN∇aN·aN,(59)ρsf=34ρt,ρph=14ρt.The first expression of (59) is compatible with (47).In our case it would be utterly wrong to consider the whole ofρt as “phantom energy.” Three-quarters of it is due to the scalar field energy density and thescalar field haloρsf and expressed atrue energy density. This energy density appears on the right-hand side of the Einstein equation (40) and the Newtonian Poisson equation as its weak field, static limit. It is decisive forlensing effects of the additional acceleration. Only one-quarter, ρph, is phantom, that is, a fictitious mass density producing the same acceleration as the Weylian scale connection (24). Only for the sake of comparison with other MOND models we may speak of ρt as some kind ofgross phantom energy, in contrast to the “net” phantom energy ρph1.We have to distinguish between the influence of the additional structure, scalar field, and scale connection, on light rays and on (low velocity) trajectories of mass particles. Bending of light rays is influenced by the scalar field halo only, the acceleration of massive particles with velocities far belowc by the scalar field haloand the scale connection.Also in another respect our theory differs from the usual MOND approaches. In MONDexternal acceleration fields of a system under consideration are difficult to handle. In WST, as in GR, a freely falling (small) system does not feel the external acceleration field if it is sufficiently small, relative to the inhomogeneities of the external gravitational field, for neglecting tidal forces. In this sense, the external acceleration problem does not arise in the WST MOND approximation (42).Another important consequence follows: the scalar field energy formed around a freely falling subsystem of a larger gravitating system, calculated in the MOND approximation of the freely falling subsystem, contributes to the r.h.s. of the Einstein equation of any other subsystem (in relative motion) and also to that of a superordinate larger system. In principle that presupposes that the whole energy momentum tensors (35) and (36) (and its system dependent representation) are considered. For slow motions and weak field approximation a superposition of energy densities as in Newton dynamics seems legitimate. This has to be taken into account for modeling the dynamics of clusters of galaxies.


### 2.6. Short Resumé

We have derived the most salient features of the Weyl geometric MOND approximation (WST MOND) and are prepared for a comparison with empirical data. Before we do so, it may be worthwhile to collect the results which are necessary for applying it to real constellations in a short survey.Consider a gravitating system which in the Newton approximation of Einstein gravity is described by the baryonic matter densityρm, the acceleration am, and potential Φm with(60)∇2Φm=4πGρm,am=-∇Φm.The modification due to WST MOND leads to additional acceleration aadd with the following features:(i)
The total accelerationa is a=am+aadd with(61)a=am1+aoam-1/2=νamaoam,whereν(y)=1+y-1/2. For am≫ao the Newton approximation applies. Equation (61) holds for am≤102ao only (“upper” transition regime). No information can be drawn from it for am larger but not yet ≫ao (the “lower” transition regime).(ii)
The “reciprocal” transformation function defined byam=μa/ama is(62)μx=1+1-1+4x2x.(iii)
aadd consists of two components aadd=aφ+asf=4aφ. The first one is derived from a potential ω satisfying the nonlinear Poisson equation(63)∇·∇ω∇ω=π4aoGρm,aφ=-∇ω.(iv)
The second one,asf, can be understood as Newton acceleration due to the energy density ρsf of a scalar field (part of the modified gravitational structure). Its potential satisfies a Newtonian Poisson equation. It satisfies(64)-∇asf=4πGρsfwith energy density(65)ρsf=34aoam1/2ρm+8πG-1∇am·amam.ρsf is part of the energy momentum tensor of the scalar field ϕ and in this sense “real” rather than phantom.(v)
aφ is formally derivable in Newton dynamics from a fictitious energy density(66)ρph=13ρsf.ρph is the net phantom energy of WST MOND. For comparison with other models of the MOND family one may like to consider ρph+ρsf=ρt as a kind of “gross phantom energy” (although the larger part of it is real). It is transparent rather than “dark” (see (75) below).(vi)
(i)–(v) are reliable approximations also for small (local) gravitating systems freely falling in a larger gravitating system, ifam⪅102ao in the local system. The subsystem can be considered as “small” with respect to the supersystem, if tidal forces of the supersystem can be neglected. In hierarchical systems like galaxy clusters the energy density contributions ρsf of the subsystems and the supersystem (calculated in different reference coordinate systems) add up to the total energy density of the scalar field, if the velocities of the subsystems relative to the barycenter of the supersystem are small. This is a crucial difference between WST MOND and ordinary MOND theories. If one likes, ρsf can be considered as the “dark matter” component of WST MOND although the energy density of the scalar field it is not constituted by the usual (hypothetical) quantum particles (WIMPs, axions, etc.). To demarcate this difference it might better be calledtransparent matter/energy of WST.(vii)
Gravitational lensing is due to the scalar field energy density only (ρsf), while the dynamics of WST corresponds to the total phantom density (ρt=ρsf+ρph). It remains to be seen whether such a difference is in agreement with observations.


## 2.1. Weyl Geometric Scalar Tensor Theory of Gravity (WST)

Among the family of scalar tensor theories of gravity the best known ones and closest ones to Einstein gravity are those with a Lagrangian containing a modified Hilbert term coupled with a scalar fieldϕ. Their Lagrangian has the general form(1)L=12ξϕ2R+Lϕ-λ4ϕ4⋯,L=Lg,g=det⁡g.Here g is an abbreviation for a 4-dimensional pseudo-Riemannian metric g=(gμν) of signature (−+++). ϕ is a real valued scalar field on spacetime, Lϕ its kinetic term, and ξ a constant coefficient and the dots indicate matter and interaction terms. Under conformal rescaling of the metric,(2)gμν⟼gμν′=Ω2gμνΩapositiverealvaluedfunction,the scalar field changes with weight -1; that is, ϕ↦ϕ′=Ω-1ϕ. So far this is similar to the well known Jordan-Brans-Dicke (JBD) scalar tensor theory of gravity [7–10]. But here we work in a scalar tensor theory in the framework of Weyl’s generalization of Riemannian geometry [1, 11–14].Crucial for theWeyl geometric scalar tensor approach (WST) is that the scalar curvature R and all dynamical terms involving covariant derivatives are expressed in Weyl geometric scale covariant form. Fields X are scale covariant if they transform under rescaling by X↦X~=ΩwX, with w∈Q (in most cases even in Z); w is called the weight of X. Covariant derivatives of scale covariant fields are defined such that the result of covariant derivation DμX is again scale covariant of the same weight w as X. The Lagrangian density L isinvariant under conformal rescaling for any value of the coefficient ξ2 of the modified Hilbert term. For the matter and interaction terms of the standard model of elementary particles, scale invariance is naturally ensured by the coupling to the Higgs field which has the same rescaling behaviour as the gravitational scalar field. Although there is no complete identity, there is a close relationship between scale and conformal invariance in quantum field theory [16]. For classical matter we expect that a better understanding of the quantum to classical transition, for example, by the decoherence approach, allows considering scale invariant Lagrangian densities also. For the time being we introduce the scale invariance of matter terms in the Lagrangian as apostulate.In our context an important consequence is the scale covariance of the Hilbert energy momentum tensor(3)Tμν=-2gδLmδgμν,which is of weight w(Tμν)=-2. That is consistent with dimensional considerations on a phenomenological level. It has been shown that the matter Lagrangian of quantum matter (Dirac field, Klein Gordon field) is consistent with test particle motion along geodesics (autoparallels) γ(τ) of the affine connection, if the underlying Weyl geometry is integrable (see (8)) [11]. For classical matter we assume the same. It is to be expected that it can be proven similar to Einstein gravity.We do not want to heap up too many technical details; more can be found in the literature given above. But we have to mention that aWeylian metric can be given by an equivalence class of pairs (g,φ) consisting of a pseudo-Riemannian metric g=gμνdxμdxν, theRiemannian component of the Weyl metric, and a differentiable one-form φ=φμdxμ, thescale connection (or, in Weyl’s original terminology, the “length connection”). φμ is often called the Weyl covector or even the Weyl vector field. In fact, φ denotes a connection with values in the Lie algebra of the scale group (R+,·) and can locally be represented by a differentiable 1-form. The equivalence is given by rescaling the Riemannian component of the Weylian metric according to (2), while φ has the peculiar gauge transformation behaviour of aconnection, rather than that of an ordinary vector (or covector) field in a representation space of the scale group:(4)φμ⟼φμ′=φμ-∂μΩΩor, shorter, φ′=φ-dlog⁡Ω.A Weylian metric has a uniquely determined compatible affine connectionΓ; in physical terms it characterizes theinertiogravitational guiding field. It can be additively composed of the well known Levi-Civita connection Γg of the Riemannian component g of any gauge (g,φ) and an additional expression Γφ in the scale connection; in short(5)Γ=Γg+ΓφwithΓφνλμ=δνμφλ+δλμφν-gνλφμ.Covariant derivativesD in the Lagrangians (1) and (below) (28) and consequently in the expression for the energy momentum tensor of the scalar field (equations (35) and (36)) below denote those of Weyl geometry. For a covariant field Xν of weight w the derivativation ∇ with regard to Γ of (5) D is supplemented by a term due to the scaling weight w of X:(6)DμXν=∇μXν+wφμXν=∂μXν+ΓμλνXλ+wφμXν.It turns out that for the metric gμν the full covariant derivative is zero:(7)Dλgμν=0⟷∇λgμν+2φλgμν=0.This is the Weyl geometric compatibility condition between metric and affine connection (sometimes called “semimetricity”).In the low energy regime there are physical reasons to constrain the scale connection to theintegrable case with a closed differentiable form dφ=0; that is, ∂μφν=∂νφμ. Then φ is a gradient (at least locally) and may be given by(8)φμ=-∂μω.This constraint is part of the defining properties of WST. Then it is possible to “integrate the scale connection away” [11]. Having done so, the Weylian metric, given as (g~,φ~), is characterized by its Riemannian component g~μν only (and a vanishing scale connection φ~μ=0). By obvious reasons we call this theRiemann gauge. This is the analogue of the choice of Jordan frame in JBD theory.In any case, the choice of representativegμν also fixes φμ; both together define ascale gauge of the Weylian metric. Conformal rescaling of the metric is accompanied by the gauge transformation of the scale connection (4). From a mathematical point of view all the scale gauges are on an equal footing, and the physical content of a WST model can be extracted, in principle, from any scale gauge. One only needs to form a proportion with the appropriate power of the scalar field. From the physical point of view there are, however, two particularly outstanding scale gauges. Of special importance besides Riemann gauge is the gauge in which the scalar field is scaled to a constant ϕo (scalar field gauge). For the particular choice of the constant value such that(9)ξϕo2=8πG-1=Epl2,with the Newton gravitational constant G, this gauge is calledEinstein gauge (Epl the reduced Planck energy). It is the analogue of Einstein frame in JBD theory. In this gauge the metrical quantities (scalar, vector, or tensor components) of physical fields are directly expressed by the corresponding field or field component of the mathematical model (without the necessity of forming proportions).In Riemann gaugeΓ reduces to Γg by definition. Thus, in this gauge, the guiding field is given by the ordinary expression for the Levi-Civita connection. On the other hand, in Einstein gauge the measuring behaviors of clocks are most immediately represented by the metric field and also other physical observables are most directly expressed by the field values in this scale. Then the expressions for the gravitational field and the accelerations contain contributions from the Weylian scale connection. Thus a specific dynamical difference to Einstein gravity and Riemannian geometry (and to JBD theory) arises even in the case of WST with its integrable Weyl geometry.Writing the scalar fieldϕ~ in Riemann gauge (g~,0) in exponential form, ϕ~=eω, turns its exponent(10)ω≔ln⁡ϕ~into ascale invariant expression for the scalar field. In the following we shall omit the tilde sign to simplify notation. The scale connection φ=φ^ in scalar field gauge is then(11)φ^=-dω,because Ω=ϕ~=eω is the rescaling function from Riemann gauge to scalar field gauge. For more details see [1, 12, 13, 17, 18].For the sake of consistency under rescaling we considerscale covariant geodesicsγ(τ) with scale gauge dependent parametrizations of the geodesic curves of weight w(γ˙)=-1:(12)u˙λ+Γμνλuμuν-φμuμuλ=0,uμ=γ˙μ.Here the affine connection contains a φ-dependent term in addition to the well known Levi-Civita connection φΓμνλ derived from gμν (5). The last term on the l.h.s. of (12) takes care of the scale dependent parametrization (compare (6)). In this way we work with aprojective family of paths.For any gauge of the Weylian metric and the scalar field,(g,φ,ϕ), any timelike geodesic has thus a generalized proper time parametrization γ(τ) with gμνuμuν=-1, where uμ=γ˙μ. Inverting the coordinate time function t(τ) along the geodesic by τ(t) we have, in abbreviated notation, τ′t′=1 and thus (13)d2xidt2=dτdtddτdτdtdxidτ=dτdt2d2xidτ2-dtdτ-3d2tdτ2dxidτ.With (12) and indices i,j,k=1,2,3,  μ,ν,…=0,1,2,3 this leads to (14)d2xidt2=dτdt2-Γμνidxμdτdxνdτ+φμdxidτdxμdτ-dτdt3dxidτ-Γμν0dxμdτdxνdτ+φμdtdτdxμdτ=-Γμνidxμdtdxνdt+φμdxidtdxμdt+Γμν0dxμdtdxνdtdxidt-φμdxμdtdxidt.Happily, the length connection terms coming from the scale covariance modification of the geodesic equation (12) cancel the equation of motion for mass points in Weyl geometric gravity, parametrized in coordinate time, becomes (15)d2xidt2=-Γ00i+Γ000dxidt-2Γ0jidxjdt-Γjkidxjdtdxkdt+2Γ0j0dxidtdxjdt+Γjk0dxidtdxjdtdxkdt.In the result the dynamics of mass points in Weyl geometric gravity is governed by the guiding field (the affine connection), as in the semi-Riemannian case [19, equ. (9.1.2)]. Note, however, that in (15) thelength connection enters into the affine connection and influences the dynamics because of (5).The geodesic equation thus contains terms in the scale connectionφμ. In the low velocity, weak field regime the equation of motion reduces to the form well known from Einstein gravity d2xj/dt2=-Γ00j. Here Γ00j=Γg00j+Γφ00j (j=1,2,3) are the coefficients of the Weyl geometric affine connection with Γφ00j given by (5). This is the crucial modifying term for gravity in the Weyl geometric approach (low velocity case).


## 2.2. The Weak Field Static Approximation

We want to understand the scale connection for the motion of point particles. The free fall of test particles in Weyl geometric gravity follows scale covariant geodesics. It is governed by a differential equation formally identical to the one in Einstein gravity (15). Here we look at the weak field static case for low velocities in order to study the dynamics of stars in galaxies and galaxies in clusters. For studies of the gas dynamics and its modification in our framework the velocity dependent terms of (15) have to be taken into account. This is not being done here.Analogous to Einstein gravity, the coordinate accelerationa for a low velocity motion x(τ) in proper time parametrization is given by(16)aj=d2xjdτ2≈-Γooj.According to (5) the total acceleration decomposes into(17)aj=-Γgooj-Γφνλj=aRj+aφj(j=1,2,3 indices of the spacelike coordinates), where(18)aRj=-Γgoojis the Riemannian component of the acceleration known from Einstein gravity. Clearly(19)aφj=-Γφoojrepresents anadditional acceleration due to the Weylian scale connection. For a diagonal Riemannian metric g=diag⁡goo,…,g33 the general expression (5) simplifies to -Γφooj=-gooφj. General considerations on observable quantities and consistency with Einstein gravity show that, in order to confront it with empirically measurable quantities, we have to take its expression inEinstein gauge if we want to avoid additional rescaling calculations [14, sec. 4.6].For a (diagonalized) weak field approximation in Einstein gauge,(20)gμν=ημν+hμν,hμν≪1,with η=ϵsigdiag⁡-1,+1,+1,+1, the Riemannian component of the acceleration is the same as in Einstein gravity. Its leading term (neglecting 2nd-order terms in h) is(21)aRj=Γgooj≈12ηjj∂jhoonosummationoverj.In the limit, ΦN≔-(1/2)hoo behaves like a Newtonian potential(22)aR≈-∇ΦN,where ∇ is understood to operate in the 3-spacelike coordinate space with Euclidean coefficients as the leading term of the metric.In Einstein gauge the Weylian scale connectionφ^μ arises from Riemann gauge by rescaling with Ω=eω, φ^μ=-∂μω (4), and aφj≈φj. In other words, the additional acceleration due to the scale connection (19) is generated by the scale invariant representative ω of thescalar field as itspotential:(23)aφ≈-∇ω.If we compare with Newton gravity, we can calculate the fictitious mass density which one had to assume on the right-hand side of the Poisson equation, in addition to the real masses, in order to generate the same amount of additional acceleration. Obviously here it is(24)ρph=4πG-1∇2ω.In the terminology of the MOND literature the acceleration due to the Weylian scale connection corresponds to aphantom energy density ρph (see, e.g., [20], p. 48). We see that already on the general level the dynamics of WST differs from Einstein gravity. Only for trivial scalar field, ω=const, the usualNewton limit is recovered; otherwise it is modified. We shall explore how this modification relates to the usual MOND approaches.


## 2.3. WST Gravity with Cubic Kinematic Lagrangian (WST-3L)

The most common form of the kinetic term for the scalar field is that of a Klein Gordon field,Lϕ2=-(α/2)DνϕDνϕ, quadratic in the norm of the (scale covariant) gradient (in WST Dν denotes the a scale covariant derivative of ϕ,  Dνϕ=∂νϕ-ϕφν). For our form of the gravitational Lagrangian (1) it is conformally coupled for α=-6ξ2. Inspired by the approach of the relativistic “a-quadratic Lagrangian” (rAQUAL), the first relativistic attempt of a MOND theory of gravity [21, 22], we find that a Weyl geometric scalar tensor theory of gravity leads to a MOND-like phenomenology if we add acubic term to the kinetic Lagrangian of the scalar field(25)Lϕ=Lϕ2+Lϕ3.The crucial difference to the early approach of rAQUAL is the scale covariant reformulation in the framework of Weyl geometry. It results in a different behaviour of the scalar field energy density. Bekenstein/Milgrom’s model relied crucially on implementing a transition function f(y) between the Newton and the deep MOND regime into the kinetic term. The constraint of scale invariance of the Lagrangian reduces the underdetermination of the Lagrangian and suggests a slightly different form of the kinetic term. It is still quite near to the one of the relativistic AQUAL theory.In the review paper [22] Bekenstein gives the rAQUAL Lagrangian in the form (his equation (6))(26)Lψ=-8πGN-1L-2fL2∂νψ∂νψ,where L is “a constant with dimension of length introduced for dimensional consistency” (later it is identified as the MOND acceleration ao via c2L-1=ao). Asymptotically f(y)~(2/3)y3/2 for y≪1 (MOND regime); similarly f(y)~y for y≫1 (Newton regime). ψ is the logarithm of a rescaling function between the Jordan frame metric g~μν (called the “physical” metric) and the Einstein frame (“primitive”) metric gμν, g~μν=e2ψgμν. Its role is very close to our ω in (10). A corresponding scale covariant form of the Lagrangian could be(27)Lϕ3=ξϕ2η-1ϕ2fη-1ϕ-2∂νω∂νω,where η-1ϕo in Einstein gauge plays a role similar to ao. The sign has deliberately been changed; the reasons are given below (39).We are here interested in additive modifications (19) of Einstein gravity, mainly in a domain in which the effects of the scale connection φν=-∂νω, compared with those of the Riemannian component of the metric, cannot be neglected. This will be called a regime withMOND approximation. For nontimelike ∇ω the Lagrangian (27) then becomes(28)Lϕ3=23ξϕ2η-1ϕ-1DνϕDνϕ3/2=23ξ2ηϕ∇ω3,where we have used the abbreviation(29)∇ω≔∂νω∂νω1/2(⋯ absolute value) (for timelike ∇ω see [1]. Here we exclusively deal with the spacelike (or null) case). η denotes a constant coefficient responsible for the relative strength of the cubic kinetic term and ω is the scale invariant representative of the scalar field introduced in (10) and ∇ its gradient (w(ϕ)=-1 and w(∇ω)=-1 imply the scale weight w(Lϕ3)=-4, as it must be for scale invariance of Lϕ3).We introduce the constanta~o defined in Einstein gauge (g^,φ^) with constant scalar field ϕ^≕ϕo,(30)a~o≔η-1ϕo.Then the cubic term of the kinetic Lagrangian in Einstein gauge reads(31)Lϕ3≐238πGa~o-1∇ω3.Thedotted equality sign “≐” indicates that the respective equation is not scale invariant but presupposes a special gauge made clear by the context. Here, as in most cases in this paper, it indicates the Einstein gauge (similar for ≈˙). a~o has the dimension of inverse length/time and will play a role analogous to the MOND acceleration ao≈[c]H, where H is the Hubble parameter (at “present”) and c is the velocity of light. Coefficients of type [c] will often be suppressed in the following general considerations. They will be plugged in only in the final step. Below we shall see that for a~o=ao/16≈H/100 the WST model with cubic kinematic Lagrangian (WST-3L) acquires a MOND-like phenomenology in a weak gravitational field in which the scalar field and the scale connection cannot be neglected.A reasonable choice of adaptable parameters bringsξ and η to nearby orders of magnitude, η=βξ, with(32)β=ξ-1η,β~100.On the other hand, because of (9) and (30) the product of both coefficients is a “large number” in the sense of η·ξ=Epl/a~o=a~o-1/Lpl~1063. To the kinetic Lagrangian of the scalar field a potential term is added. It must be of order 4 to provide for scale invariance of the density (1):(33)LV4=-λ4ϕ4.Variation of the Lagrangian leads to the dynamical equations of WST, the Einstein equation, and the scalar field equation. Thescale invariant Einstein equation is (this means that not only the equation but all of its constitutive (additive) terms are scale invariant)(34)Ric-R2g=ξϕ-2Tm+Θ,where g denotes the whole collection of metrical coefficients and T(m) the energy tensor of matter (3). The scalar field contributes to the total energy momentum with two terms, Θ=Θ(I)+Θ(II), the first of which is proportional to the metric (thus formally similar to a vacuum energy tensor) (see, e.g., [23], [18, pp. 96ff.]):(35)ΘI=ϕ-2-DλDλϕ2+ξ-2LV4+Lϕg,(36)ΘμνII=ϕ-2DμDνϕ2-2ξ-2∂Lϕ∂gμν.Varying with regard toϕ gives the scalar field equation. Subtracting the trace of the Einstein equation for a conformally coupled Lϕ2 term (α=-6ξ2) strongly simplifies it and introduces the trace of the matter tensor into the scalar field equation. In Einstein gauge, with g the Riemannian component of the metric, it can be written in terms of the covariant derivative ∇g with regard to g (Levi-Civita connection in Einstein gauge) as ([1, pp. 15f., sec. 7.2, postprint version arXive v4]) (37)∇νg∇ω∂νω≐-4πGa~otrTm.If we introduce the corresponding Riemannian covariant operator(38)□Mgω=∇νg∇ω∂νω=∂ν∇ω∂νω+∇ω∇νg∂νω,the scalar field equation for a fluid with matter density ρm and pressure pm simplifies to the covariantMilgrom equation(39)□Mgω≐4πGa~oρm-3pm.In this derivation, with trT(m) entering by subtracting the trace of the Einstein equation, a sign choice like in (26) leads to the wrong sign on the r.h.s. of the Milgrom equation. This explains our sign choice in (27). By obvious reasons (38) will be called thecovariant Milgrom operator. In the static weak field static case ω does not depend on the time coordinate. Moreover with gμν≈ημν, the expression ∇ν(∇ω∂νω) turns into the nonlinear Laplace operator ∇·(∇ω∇ω) of the MOND theory with Euclidean scalar product · and norm ⋯.In the general case we have to complement (39) with the Einstein equation in Einstein gauge(40)Ric-R2g≐8πGTm+Θ,In vacuum, the trivial scalar fieldω=const is a basic solution of (39). Then WST reduces to Einstein gravity. In particular, the Schwarzschild and the Schwarzschild-de Sitter solutions of Einstein gravity are special (degenerate) solutions of WST-3L equations for λ/4=0 or λ/4≈6, respectively. In fact, they solve (40) and (39) for ϕ≐const in Riemann gauge, that is, in the case of Einstein gauge equal to Riemann gauge (g^,φ^)=(g~,0). The Riemannian component of the metric (g~=g^≕g) is given by(41)ds2=-1-2Mr-κr2dt2+1-2Mr-κr2-1dr2+r2dx22+sin2⁡x2dx32.Then Ric-(R/2)g=-3κg and Θ=Θ(I)=-(λ/4)β2a~o2g. Therefore the Einstein equation is satisfied for 3κ=(λ/4)β2a~o2; that is, κ≈2H2 for λ/4≈6 and β≈100. We see that in the case of a negligible Weylian scale connection the classical (nonhomogeneous) point symmetric solutions of Einstein gravity are valid also for the dynamics of WST. This implies thatin the case of a negligible scale connection Newton dynamics is an effective approximation for point symmetric solutions of WST (in Einstein gauge).In order to make such a type of Einstein limit compatible with our Lagrangian, a suppression of theLϕ3-term for sufficiently large accelerations aR of (18) is necessary. One may consider plugging a factor f~(ao/aR) with a function f~ such that f~(y)~1 for y>0.01 and f~(y)~0 for y≪1 into the r.h.s. expression of (28) for Lϕ3 but such a choice would have the blemish of a coordinate dependent argument of the function. A better alternative is provided by thehypothesis that the scalar field inhomogeneities are suppressed if any of the sectional curvatures κ (with respect to the Riemannian component of the metric in Einstein gauge) surpasses a certain threshold (e.g., κ≥109aoc-22). In the next section we investigate the case of a nonnegligible scale connection. A more detailed discussion of the transition between the two domains has to be left open for another occasion.


## 2.4. A WST Approach with MOND-Like Phenomenology

If the conditions for the weak field approximation (20) are given, it is possible to identify aMOND regime as a region in which the Newton acceleration aN is smaller than ao (here aN can be identified with aR in (22)). Then the scalar field equation (39) reduces, in reliable approximation, to(42)∇·∇ω∇ω≐-4πGa~otrTm,with the Euclidean ∇-operator. We call this aMOND approximation. For pressureless matter with energy density ρm we get(43)∇·∇ω∇ω≐4πGa~oρm.That is similar to the AQUAL approach [21, 22]. Note that only thematter energy momentum tensor,without thescalar field energy density, appears on the r.h.s. of (42).Straightforward verification shows that, independent of symmetry conditions, a solution of (43) is given by ω with a gradient ∇ω=-aφ such that(44)aφ=a~oaNaN=a~oaNaNaN,where aN denotes the Newton acceleration of the given mass density,(45)∇2ΦN=4πGρm,aN=-∇ΦN(calculations in the approximating Euclidean space with norm ⋯). The solution of the nonlinear Poisson equation (43) is much simpler than one might expect at a first glance. In a first step the linear Poisson equation of the Newton theory is to be solved and then an algebraic transformation of type (44) leads to the acceleration due to the solution of the nonlinear partial differential equation (43). In fact, aφ has the form of the deep MOND acceleration of the ordinary MOND theory (but with different constant a~o) (44). In the terminology of the MOND community, the MOND approximation of WST-3L behaves like a special case of a QMOND theory [20, pp. 46ff.].This raises the question of the Newtonian limit. Equation (44) implies aφ≪aN in regions, where aN≫ao(>a~o). Therefore aφ can effectively be neglected in the case of “large” values of aN derived from (45). Then, according to the observation at the end of Section 2.3, the Newton approximation is also reliable in WST gravity. That is true irrespective of the question of how to characterize the transition between the MOND and the Newton approximation. Here we shall consider the MOND approximationin an “upper transition” regime only, where roughly aN≤102ao. One might speak of theupper transition regime for ao≤aN≤100ao of theMOND regime if aN≤ao and of thedeep MOND regime for, let us say, aN≤10-2ao   [1, sec. 7.3].For centrally symmetric mass distributionsρ(r) with mass M(r) integrated up to r (where r=y denotes the Euclidean distance from the symmetry center, y=(y1,y2,y3) the coordinates of the approximating Euclidean space) this implies(46)aφ=-∇ω≈˙-GMra~oyy2,aφ=GMra~or.But this is only the most immediate modification of Newton gravity. There is also the additional term in (40) of theenergy density due to the scalar field, ρsf=(8πG)-1Θoo. It modifies the r.h.s. of the Newton limit of Einstein gravity (in contrast ρsf does not enter the r.h.s. of the scalar field equation (39) and therefore doesnot enter the r.h.s. of (45)).Neglecting contributions at the order of magnitude of cosmological terms (~H) the energy density of the scalar field in Einstein gauge simplifies to ([1, sec. 4.3]) (47)ρsf≈˙4πG-1∇2ω+Γjkj∂kω,where Latin indices j,k,… refer to space coordinates only.In the central symmetric case with Euclidean metric in spherical coordinates and a mass functionM(r) with M′(r)=0 for r>ro (for some distance ro), we find from (46) ∇2ω=a~oGM(r)/r2 and Γjkj∂kω=(2/r)(a~oGMr/r)=2∇2ω; (for ds2=dr2+r2(dθ2+sin2θdϑ2) the crucial affine connection components are Γ111=0,Γ212=Γ313=r-1, and thus(48)ρsf≈˙4πG-13∇2ω.That is three times the value of the phantom energy density corresponding to the acceleration of the scale connection (24). The total “anomalous” additive acceleration (in comparison to Newton gravity) is therefore(49)aadd=aφ+asf=4aφ.In the central symmetric case(50)aadd=4GMra~or.For consistency with the deep MOND acceleration ao we have to set(51)a~o=ao16≈10-2Hc≈6·10-10cms-2.Because of (44) the total acceleration a is then(52)a=aN+aadd=aN1+aoaN,aadd=aoaN.


## 2.5. Comparison with Usual MOND Theories

We can now compare our approach with other models of the MOND family. Simply adding a deep MOND term to the Newton acceleration of a point mass is unusual. Milgrom rather considered a multiplicative relation between the MOND accelerationa and the Newton acceleration aN by a kind of “dielectric analogy,”(53)aN=μaaoa,withμx⟶1forx⟶∞xforx⟶0,or the other way round (here μ(x)→x means μ(x)-x=O(x); i.e., (μ(x)-x)/x remains bounded for x→0. Cf. [20, pp.51f.]) (54)a=νaNaoaN,withνy⟶1fory⟶∞y-1/2fory⟶0.From this point of view our acceleration (52) is specified by well defined transition functions(55)μwx=1+1-1+4x2x,νwy=1+y-1/2.One has to keep in mind, however, that ourtransition functionsμ,νare reliable only in the MOND regime and the upper transitional regime (roughly aN≤102ao). They cannot be used for discussing the Newtonian limit. It will be important to see how they behave in the light of empirical data, in particular galactic rotation curves and cluster dynamics.In the MOND literature the amount of a (hypothetical) mass which in Newton dynamics would produce the same effects as the respective MOND correctionaadd is calledphantom massMph. For any member of the MOND family the additional acceleration can be expressed by the modified transition function ν~=ν-1 with ν as in (54)(56)aadd=ν~aNaoaN.Thephantom mass densityρph attributed to the potential Φph satisfies 4πGρph=∇2Φph and ∇Φph=-aadd. A short calculation shows that it may be expressed as(57)ρph=ν~aNaoρm+4πGao-1ν~′aNao∇aN·aN.It consists of a contribution proportional to ρm with factor ν~, which dominates in regions of ordinary matter, and a term derived from the gradient of aN dominating in the “vacuum” (where however scalar field energy is present). For the Weyl geometric model with ν~w(y)=y-1/2,ν~w′(y)=-(1/2)y-3/2ρph turns into(58)ρt=aoaN1/2ρm+8πG-1aoaN∇aN·aN,(59)ρsf=34ρt,ρph=14ρt.The first expression of (59) is compatible with (47).In our case it would be utterly wrong to consider the whole ofρt as “phantom energy.” Three-quarters of it is due to the scalar field energy density and thescalar field haloρsf and expressed atrue energy density. This energy density appears on the right-hand side of the Einstein equation (40) and the Newtonian Poisson equation as its weak field, static limit. It is decisive forlensing effects of the additional acceleration. Only one-quarter, ρph, is phantom, that is, a fictitious mass density producing the same acceleration as the Weylian scale connection (24). Only for the sake of comparison with other MOND models we may speak of ρt as some kind ofgross phantom energy, in contrast to the “net” phantom energy ρph1.We have to distinguish between the influence of the additional structure, scalar field, and scale connection, on light rays and on (low velocity) trajectories of mass particles. Bending of light rays is influenced by the scalar field halo only, the acceleration of massive particles with velocities far belowc by the scalar field haloand the scale connection.Also in another respect our theory differs from the usual MOND approaches. In MONDexternal acceleration fields of a system under consideration are difficult to handle. In WST, as in GR, a freely falling (small) system does not feel the external acceleration field if it is sufficiently small, relative to the inhomogeneities of the external gravitational field, for neglecting tidal forces. In this sense, the external acceleration problem does not arise in the WST MOND approximation (42).Another important consequence follows: the scalar field energy formed around a freely falling subsystem of a larger gravitating system, calculated in the MOND approximation of the freely falling subsystem, contributes to the r.h.s. of the Einstein equation of any other subsystem (in relative motion) and also to that of a superordinate larger system. In principle that presupposes that the whole energy momentum tensors (35) and (36) (and its system dependent representation) are considered. For slow motions and weak field approximation a superposition of energy densities as in Newton dynamics seems legitimate. This has to be taken into account for modeling the dynamics of clusters of galaxies.


## 2.6. Short Resumé

We have derived the most salient features of the Weyl geometric MOND approximation (WST MOND) and are prepared for a comparison with empirical data. Before we do so, it may be worthwhile to collect the results which are necessary for applying it to real constellations in a short survey.Consider a gravitating system which in the Newton approximation of Einstein gravity is described by the baryonic matter densityρm, the acceleration am, and potential Φm with(60)∇2Φm=4πGρm,am=-∇Φm.The modification due to WST MOND leads to additional acceleration aadd with the following features:(i)
The total accelerationa is a=am+aadd with(61)a=am1+aoam-1/2=νamaoam,whereν(y)=1+y-1/2. For am≫ao the Newton approximation applies. Equation (61) holds for am≤102ao only (“upper” transition regime). No information can be drawn from it for am larger but not yet ≫ao (the “lower” transition regime).(ii)
The “reciprocal” transformation function defined byam=μa/ama is(62)μx=1+1-1+4x2x.(iii)
aadd consists of two components aadd=aφ+asf=4aφ. The first one is derived from a potential ω satisfying the nonlinear Poisson equation(63)∇·∇ω∇ω=π4aoGρm,aφ=-∇ω.(iv)
The second one,asf, can be understood as Newton acceleration due to the energy density ρsf of a scalar field (part of the modified gravitational structure). Its potential satisfies a Newtonian Poisson equation. It satisfies(64)-∇asf=4πGρsfwith energy density(65)ρsf=34aoam1/2ρm+8πG-1∇am·amam.ρsf is part of the energy momentum tensor of the scalar field ϕ and in this sense “real” rather than phantom.(v)
aφ is formally derivable in Newton dynamics from a fictitious energy density(66)ρph=13ρsf.ρph is the net phantom energy of WST MOND. For comparison with other models of the MOND family one may like to consider ρph+ρsf=ρt as a kind of “gross phantom energy” (although the larger part of it is real). It is transparent rather than “dark” (see (75) below).(vi)
(i)–(v) are reliable approximations also for small (local) gravitating systems freely falling in a larger gravitating system, ifam⪅102ao in the local system. The subsystem can be considered as “small” with respect to the supersystem, if tidal forces of the supersystem can be neglected. In hierarchical systems like galaxy clusters the energy density contributions ρsf of the subsystems and the supersystem (calculated in different reference coordinate systems) add up to the total energy density of the scalar field, if the velocities of the subsystems relative to the barycenter of the supersystem are small. This is a crucial difference between WST MOND and ordinary MOND theories. If one likes, ρsf can be considered as the “dark matter” component of WST MOND although the energy density of the scalar field it is not constituted by the usual (hypothetical) quantum particles (WIMPs, axions, etc.). To demarcate this difference it might better be calledtransparent matter/energy of WST.(vii)
Gravitational lensing is due to the scalar field energy density only (ρsf), while the dynamics of WST corresponds to the total phantom density (ρt=ρsf+ρph). It remains to be seen whether such a difference is in agreement with observations.


## 3. Halo Model for Clusters of Galaxies

### 3.1. Cluster Models for Baryonic Mass (Hot Gas and Stars)

In the astronomical literature, the density profile of hot gas and (smeared) star/galaxy matter in a galaxy cluster is often described by a centrally symmetric profile of the following form:(67)ρr=ρo1+rrc2-3/2β.β is the ratio of the specific energies of the galaxies and the gas, ρo the central density, and rc the core radius [5, 15, 24] (rc is the distance from the cluster center at which the projected galaxy density is half the central density ρo). Equation (67) is called a β-model for the mass distribution. For our test we assume density models for the gas mass ρgas(r) and for the galaxy mass ρstar(r) with the same form parameters β and rc. We thus work with an idealized model using proportional density profiles for the hot gas and for the galaxies with parameters β,rc determined from observations of the hot gas. The central densities ρo can, in principle, be determined from mass data for gas Mgas(r1), respectively, stars Mstar(r1), at a given distance r1. The empirical determination of Mgas(r1) and Mstar(r1) from directly observable quantities is a subtle question; it will be discussed in Section 4.1.Large scale gravitational effects on the cluster level are often modelled in the Newton approximation of Einstein gravity with baryonic matter and an assumed dark matter halo which is inferred from its gravitational effects (in Einstein/Newton gravity). Another minority approach in the literature works with an evaluation of the data in a MOND limit of the most well known relativistic MOND theory TeVeS. Sanders is one of its protagonists; he concludes that in this approach a much smaller amount of unseen matter has to be assumed in addition to the baryonic mass. Its value is consistent with the hypothesis of a halo of sterile neutrinos, concentrated about the cluster center [5]. Here we want to explore the feasibility of the WST approach, in particular regarding the question of how much unseen matter has to be added to the gravitational effects of the model in order to reproduce (“predict”) the observed acceleration, respectively, their measurable effects.


### 3.2. Two Contributions to the Scalar Field Energy in Clusters of Galaxies

The mass distribution of the hot gasρgas in a galaxy cluster is described by a β-profile (67) in a locally static coordinate system with origin at the barycenter of the cluster. The averaged star mass will be represented by a continuous distribution ρstar of a β-profile with the same parameters β,rc, but with a different value of ρo. Both together form a continuity model of the baryonic mass distribution ρbar=ρgas+ρstar. Estimates show that Newtonian gravitational acceleration induced by ρbar (far away from mass concentrations stars, galaxies, and galactic centers) is below 102·ao. They are small enough for allowing working in a weak field static approximation with the scalar field equation in the MOND approximation (43). The resulting contribution to the scalar field energy will be called ρsf1. The additional acceleration of the Weyl geometrical scale connection (66) can be expressed in terms of a phantom mass density which will be called ρph1.In this first approximation the star mass is approximated on a par with the hot gas; that is, it is described by its continuously smeared out mean density. But stars are agglomerated in galaxies which form freely falling subsystems of the cluster with considerable interspaces in the supersystem (the cluster). For each subsystem a locally static coordinate system with origin at the respective galactic center can be chosen. In this system the local inhomogeneities of star mass distribution in the cluster and the resulting inhomogeneities of the gravitational field in the vicinity of the galaxy can be calculated. On the galaxy level the MOND theory has proven effective for modelling gravitational effects deviating from Einstein and Newton gravity without assuming real dark matter [20]. Although we expect that the MOND approximation of WST gravity shows similar features, this is not the point in the present investigation.Here we are interested in the neighbouring regions of galaxies as subsystems of their respective cluster.These subsystems form scalar field halos of their own which contain real energy (different from the classical MOND theory which leads to phantom halos only). In the framework of the present approach, the scalar field halos in the neighbourhood of each galaxy contribute to the energy density which adds up globally, that is, on the cluster level, to a component of scalar field energy ρsf2 which has been suppressed in the first continuity approximation of the total baryonic mass. In a second step we therefore determine this component approximately and add it to the total the scalar field halo.


### 3.3. Scalar Field and Phantom Halosρsf1 and ρph1 in the Cluster Barycentric MOND Approximation

The baryonic mass up to radiusr,(68)Mbarr=4π∫0rρbaruu2du,determines the Newton acceleration abar=G(Mr/r2) due to the total baryonic mass. The densities of the scalar field halo ρsf1 and the phantom halo of WST MOND follow from (65) and (66). They are(69)ρsf1=34aoabar1/2ρm+8πG-1∇abar·abarabar,(70)ρph1=13ρsf1.The respective masses of the halos Msf1,Mph1 arise from integration.


### 3.4. Scalar Field Halos of Galaxies in Their Respective Galactocentric MOND Approximations

As already indicated in Section3.2 (69) and (70) do not make allowance for the fact that the star matter forms a discrete structure of an ensemble of galaxieseach of which is falling freely in the inertiogravitational field of the supersystem (hot gas and other galaxies). Every galaxy possesses a local MOND approximation with regard to its own barycentric static reference system. The acceleration abar of the supersystem (with respect to the barycenter rest system of the hot gas) is transformed away in each of the local MOND approximations. The latter leads to a galacticscalar field halo whichpersists under changes of reference systems with small, that is, nonrelativistic, relative velocities. It contributes to the total energy of the scalar field, calculated in the cluster barycentric system. (Of course this isnot the case for the phantom halo of the single galaxies.) In principle, we have to add up all of these effects to a scalar field energy density ρsf2 in order to fill in these lacunae. But an exact calculation would have to solve a highly nontrivial N-body problem for the motion of the galaxies.The experience with the calculation of the combined MOND halos of stars inside galaxies shows that a resolution of the star matter inside galaxies down to individual galaxies is not necessary to achieve good results. In the outer region of galaxies a continuity model for the distribution of star matter in the galactic disk gives reliable approximations for the MOND acceleration. Similarly we want to check whether also here a continuity model for the system of galaxies alone, abstracting from the gas mass, leads to an acceptable approximation forρsf2. For this calculation, the gas mass has to be omitted because the galaxies are falling freely in the outer field of the cluster; the gravitational potential of the latter does not enter the local MOND approximation of the galaxies.Using (65) again we get for the second (inhomogeneity) component of the scalar field energy(71)ρsf2=34aoastar1/2ρstar+8πG-1∇astar·astarastar,with astarr=G(Mstarr/r2) and Mstar(r) the integral analogous to (68) for the star density ρstar.We finally arrive at a halo model for galaxy clusters constituted by the componentsρsf1,ρsf2,andρph1. All of them are determined by the two component baryonic profile of the cluster.


### 3.5. A Three-Component Halo Model for Clusters of Galaxies

In addition to the Newtonian gravitational effects of thebaryonic mass density(72)ρbar=ρgas+ρstar,modeled by a β-model of type (67), the WST MOND approach predicts acceleration generated by thescalar field halos (69) and (71). Because of their small values their combined effect can be approximated by a linear superposition in the barycentric reference system of the cluster, and because of slow (i.e., nonrelativistic) relative velocities of the galaxies the energy densities of their respective scalar field halos can be taken over to the cluster barycentric reference system:(73)ρsf≈ρsf1+ρsf2.Moreover, there arises acceleration aφ due to the scale connection in the barycentric rest system of the cluster (45). Its gravitational effects are representable by the fictitious (net)phantom haloρph1 of (70)(74)ρph1=13ρsf1.On the other hand, the phantom energies of the individual freely falling galaxies do not survive the transformation to the cluster rest system and do not play a role on the cluster level.In the usual MOND theories there is no scalar field energy; all additional effects with regard to Newton dynamics may be ascribed to a (fictitious) phantom energy density. Phantom energy densities of single galaxies do not survive the transformation to the cluster barycentric system. In MOND there is therefore no analogy toρsf2; the latter is thecrucial distinctive feature between the two approaches. For a comparison of WST-3L and usual MOND approaches with regard to galaxy clusters it isnot sufficient to evaluate the difference between the transformation functionsμ(x) (55).For an even wider comparison with other approaches it may be useful to add up the scalar field and phantom halos to a kind of “dark matter” halo or, more precisely, to a substitute for the latter. But one must not forget that in the present model there is no dark matter in the ordinary sense. Here we only find atransparent halo made up of the (real) energy density of the scalar field and the (fictitious) phantom energy density ascribed to the acceleration effects of the scale connection in Einstein gauge (with respect to the cluster barycentric rest system):(75)ρt=ρsf+ρph1.From the gravitational lensing point of view, it would be even more appropriate to consider ρsf alone as the WST equivalent of a dark matter halo, not forgetting that even the real halo ρsf is not due to fermionic particles, but to the scalar field, and thus to the extended gravitational structure of WST. From a quantum point of view, the scalar field has to be quantized if one wants to search for a (bosonic) particle content of ρt or ρsf.The totaldynamical mass of the model (up to some distance r from the center of the cluster) is(76)Mtot=Mbar+Mt,Mt=Msf+Mph1,Msf=Msf1+Msf2with Mbar=Mgas+Mstar. Thelensing mass is a bit smaller,(77)Mlens=Mbar+Msf.Mathematically, the integral of the scalar field energy density to arbitrary distances diverges. As in the dark matter approach avirial radius of a cluster may be defined, which roughly delimits the gravitational binding zone of the cluster. Comparing ρt (resp., (ρsf)) with the critical energy density ρcrit of the universe, one may, for example, choose r200 with(78)ρtr200≈200ρcritas a representative of the virial radius.Not far beyond the gravitational binding zone of the cluster, the energy density will have fallen to such a small amount that its centrally symmetric component is inconceivably stronger than the density fluctuations in the intercluster space. To continue the integration into this region and beyond has no physical meaning. In the long range the energy density of the scalar field approaches the cosmic mean energy value. A physical limit of integration has to be chosen close to the virial radius beyond which the gravitational binding structure of the cluster is fading out.


## 3.1. Cluster Models for Baryonic Mass (Hot Gas and Stars)

In the astronomical literature, the density profile of hot gas and (smeared) star/galaxy matter in a galaxy cluster is often described by a centrally symmetric profile of the following form:(67)ρr=ρo1+rrc2-3/2β.β is the ratio of the specific energies of the galaxies and the gas, ρo the central density, and rc the core radius [5, 15, 24] (rc is the distance from the cluster center at which the projected galaxy density is half the central density ρo). Equation (67) is called a β-model for the mass distribution. For our test we assume density models for the gas mass ρgas(r) and for the galaxy mass ρstar(r) with the same form parameters β and rc. We thus work with an idealized model using proportional density profiles for the hot gas and for the galaxies with parameters β,rc determined from observations of the hot gas. The central densities ρo can, in principle, be determined from mass data for gas Mgas(r1), respectively, stars Mstar(r1), at a given distance r1. The empirical determination of Mgas(r1) and Mstar(r1) from directly observable quantities is a subtle question; it will be discussed in Section 4.1.Large scale gravitational effects on the cluster level are often modelled in the Newton approximation of Einstein gravity with baryonic matter and an assumed dark matter halo which is inferred from its gravitational effects (in Einstein/Newton gravity). Another minority approach in the literature works with an evaluation of the data in a MOND limit of the most well known relativistic MOND theory TeVeS. Sanders is one of its protagonists; he concludes that in this approach a much smaller amount of unseen matter has to be assumed in addition to the baryonic mass. Its value is consistent with the hypothesis of a halo of sterile neutrinos, concentrated about the cluster center [5]. Here we want to explore the feasibility of the WST approach, in particular regarding the question of how much unseen matter has to be added to the gravitational effects of the model in order to reproduce (“predict”) the observed acceleration, respectively, their measurable effects.


## 3.2. Two Contributions to the Scalar Field Energy in Clusters of Galaxies

The mass distribution of the hot gasρgas in a galaxy cluster is described by a β-profile (67) in a locally static coordinate system with origin at the barycenter of the cluster. The averaged star mass will be represented by a continuous distribution ρstar of a β-profile with the same parameters β,rc, but with a different value of ρo. Both together form a continuity model of the baryonic mass distribution ρbar=ρgas+ρstar. Estimates show that Newtonian gravitational acceleration induced by ρbar (far away from mass concentrations stars, galaxies, and galactic centers) is below 102·ao. They are small enough for allowing working in a weak field static approximation with the scalar field equation in the MOND approximation (43). The resulting contribution to the scalar field energy will be called ρsf1. The additional acceleration of the Weyl geometrical scale connection (66) can be expressed in terms of a phantom mass density which will be called ρph1.In this first approximation the star mass is approximated on a par with the hot gas; that is, it is described by its continuously smeared out mean density. But stars are agglomerated in galaxies which form freely falling subsystems of the cluster with considerable interspaces in the supersystem (the cluster). For each subsystem a locally static coordinate system with origin at the respective galactic center can be chosen. In this system the local inhomogeneities of star mass distribution in the cluster and the resulting inhomogeneities of the gravitational field in the vicinity of the galaxy can be calculated. On the galaxy level the MOND theory has proven effective for modelling gravitational effects deviating from Einstein and Newton gravity without assuming real dark matter [20]. Although we expect that the MOND approximation of WST gravity shows similar features, this is not the point in the present investigation.Here we are interested in the neighbouring regions of galaxies as subsystems of their respective cluster.These subsystems form scalar field halos of their own which contain real energy (different from the classical MOND theory which leads to phantom halos only). In the framework of the present approach, the scalar field halos in the neighbourhood of each galaxy contribute to the energy density which adds up globally, that is, on the cluster level, to a component of scalar field energy ρsf2 which has been suppressed in the first continuity approximation of the total baryonic mass. In a second step we therefore determine this component approximately and add it to the total the scalar field halo.


## 3.3. Scalar Field and Phantom Halosρsf1 and ρph1 in the Cluster Barycentric MOND Approximation

The baryonic mass up to radiusr,(68)Mbarr=4π∫0rρbaruu2du,determines the Newton acceleration abar=G(Mr/r2) due to the total baryonic mass. The densities of the scalar field halo ρsf1 and the phantom halo of WST MOND follow from (65) and (66). They are(69)ρsf1=34aoabar1/2ρm+8πG-1∇abar·abarabar,(70)ρph1=13ρsf1.The respective masses of the halos Msf1,Mph1 arise from integration.


## 3.4. Scalar Field Halos of Galaxies in Their Respective Galactocentric MOND Approximations

As already indicated in Section3.2 (69) and (70) do not make allowance for the fact that the star matter forms a discrete structure of an ensemble of galaxieseach of which is falling freely in the inertiogravitational field of the supersystem (hot gas and other galaxies). Every galaxy possesses a local MOND approximation with regard to its own barycentric static reference system. The acceleration abar of the supersystem (with respect to the barycenter rest system of the hot gas) is transformed away in each of the local MOND approximations. The latter leads to a galacticscalar field halo whichpersists under changes of reference systems with small, that is, nonrelativistic, relative velocities. It contributes to the total energy of the scalar field, calculated in the cluster barycentric system. (Of course this isnot the case for the phantom halo of the single galaxies.) In principle, we have to add up all of these effects to a scalar field energy density ρsf2 in order to fill in these lacunae. But an exact calculation would have to solve a highly nontrivial N-body problem for the motion of the galaxies.The experience with the calculation of the combined MOND halos of stars inside galaxies shows that a resolution of the star matter inside galaxies down to individual galaxies is not necessary to achieve good results. In the outer region of galaxies a continuity model for the distribution of star matter in the galactic disk gives reliable approximations for the MOND acceleration. Similarly we want to check whether also here a continuity model for the system of galaxies alone, abstracting from the gas mass, leads to an acceptable approximation forρsf2. For this calculation, the gas mass has to be omitted because the galaxies are falling freely in the outer field of the cluster; the gravitational potential of the latter does not enter the local MOND approximation of the galaxies.Using (65) again we get for the second (inhomogeneity) component of the scalar field energy(71)ρsf2=34aoastar1/2ρstar+8πG-1∇astar·astarastar,with astarr=G(Mstarr/r2) and Mstar(r) the integral analogous to (68) for the star density ρstar.We finally arrive at a halo model for galaxy clusters constituted by the componentsρsf1,ρsf2,andρph1. All of them are determined by the two component baryonic profile of the cluster.


## 3.5. A Three-Component Halo Model for Clusters of Galaxies

In addition to the Newtonian gravitational effects of thebaryonic mass density(72)ρbar=ρgas+ρstar,modeled by a β-model of type (67), the WST MOND approach predicts acceleration generated by thescalar field halos (69) and (71). Because of their small values their combined effect can be approximated by a linear superposition in the barycentric reference system of the cluster, and because of slow (i.e., nonrelativistic) relative velocities of the galaxies the energy densities of their respective scalar field halos can be taken over to the cluster barycentric reference system:(73)ρsf≈ρsf1+ρsf2.Moreover, there arises acceleration aφ due to the scale connection in the barycentric rest system of the cluster (45). Its gravitational effects are representable by the fictitious (net)phantom haloρph1 of (70)(74)ρph1=13ρsf1.On the other hand, the phantom energies of the individual freely falling galaxies do not survive the transformation to the cluster rest system and do not play a role on the cluster level.In the usual MOND theories there is no scalar field energy; all additional effects with regard to Newton dynamics may be ascribed to a (fictitious) phantom energy density. Phantom energy densities of single galaxies do not survive the transformation to the cluster barycentric system. In MOND there is therefore no analogy toρsf2; the latter is thecrucial distinctive feature between the two approaches. For a comparison of WST-3L and usual MOND approaches with regard to galaxy clusters it isnot sufficient to evaluate the difference between the transformation functionsμ(x) (55).For an even wider comparison with other approaches it may be useful to add up the scalar field and phantom halos to a kind of “dark matter” halo or, more precisely, to a substitute for the latter. But one must not forget that in the present model there is no dark matter in the ordinary sense. Here we only find atransparent halo made up of the (real) energy density of the scalar field and the (fictitious) phantom energy density ascribed to the acceleration effects of the scale connection in Einstein gauge (with respect to the cluster barycentric rest system):(75)ρt=ρsf+ρph1.From the gravitational lensing point of view, it would be even more appropriate to consider ρsf alone as the WST equivalent of a dark matter halo, not forgetting that even the real halo ρsf is not due to fermionic particles, but to the scalar field, and thus to the extended gravitational structure of WST. From a quantum point of view, the scalar field has to be quantized if one wants to search for a (bosonic) particle content of ρt or ρsf.The totaldynamical mass of the model (up to some distance r from the center of the cluster) is(76)Mtot=Mbar+Mt,Mt=Msf+Mph1,Msf=Msf1+Msf2with Mbar=Mgas+Mstar. Thelensing mass is a bit smaller,(77)Mlens=Mbar+Msf.Mathematically, the integral of the scalar field energy density to arbitrary distances diverges. As in the dark matter approach avirial radius of a cluster may be defined, which roughly delimits the gravitational binding zone of the cluster. Comparing ρt (resp., (ρsf)) with the critical energy density ρcrit of the universe, one may, for example, choose r200 with(78)ρtr200≈200ρcritas a representative of the virial radius.Not far beyond the gravitational binding zone of the cluster, the energy density will have fallen to such a small amount that its centrally symmetric component is inconceivably stronger than the density fluctuations in the intercluster space. To continue the integration into this region and beyond has no physical meaning. In the long range the energy density of the scalar field approaches the cosmic mean energy value. A physical limit of integration has to be chosen close to the virial radius beyond which the gravitational binding structure of the cluster is fading out.


## 4. A First Comparison with Empirical Data

### 4.1. Empirical Determination of Mass Data for Galaxy Clusters

For a first empirical exploration we confront the WST cluster model with recent mass data for 19 clusters obtained on the background of Einstein/Newton gravity andΛCDM by a group of astronomers about Zhang et al. [2, 4]. We use the form parameters β,rc of the β-models of these clusters, published in an earlier study by one author of the group [15]. In the present study we have to take the mass data and the form parameters essentially at face value. Methodological questions arising from this procedure are discussed in the next subsection. There seem to be sufficient reasons for expecting that the different background theories do not principally invalidate the results thus obtained. Of course, an authoritative empirical study would presuppose an evaluation of observational data on the background of WST itself; it can be done only by astronomers, if they get interested in the present approach.The studies of Zhang et al. have the great advantage to build upon three independent observational datasets for determining the gas massMgas,500, the star mass M∗,500, and the total mass M500 (assuming a dark matter explanation for the observed gravitational effects) at the reference distance r500. The latter is determined for each cluster at the distance r500 from the cluster center at which the total gravitational acceleration indicates a total mass density 500 times the critical density.(i)
Mgas,500 has been extracted from X-ray data on the hot intracluster medium (ICM) collected by XMM Newton and ROSAT. Surface brightness data have been used to infer an ICM radial electron number density profile, and spectral analysis data gave information on the radial temperature distribution. From that a gas density distribution has been reconstructed and the gas masses at r500 by integration (an outline of the procedure and literature for more details is given in [4, p.3]).(ii)
M∗,500 has been determined from optical imaging data due to SDSS 7 in two steps. First the total luminosity of the cluster has been determined by means of a “galaxy luminosity function” (GLF); then the mass is estimated using mass-to-light ratios depending on the cluster mass. In the last step models of the star development in the respective galaxy, elliptical or spiral, enter. They depend on assumptions on an “initial mass function” (IMF). Two possibilities for the IMF (Salpeter versus Kroupa) are considered and compared in [2, 4]. According to the authors the difference of the stellar mass estimate can result in factor 2 [4, p.4].(iii)
The total cluster massM500 has been determined on the basis of the velocity dispersion of galaxies, using spectroscopic data from [25, tab.  1]. The mass estimator used is equation (2) of [26](79)Mv=Aσv103kms-13×1014h-1M⊙,whereA=1.50±0.02 and σv is the 3-dimensional velocity dispersion inside a sphere of virial radius (by convention rv=r200). Reasons for this choice are given in [26, sec. 3]. M500 was then determined from M200 by a NFW model.


### 4.2. Theory Dependence of Mass Data for Galaxy Clusters

Mass densities of the hot gas and of star matter in galaxy clusters are indirectly inferred from observable quantities; they are thustheory dependent. Even inside the same background theory they may depend on choices of models and methods of evaluation.That makes it a difficult task to compare our model with empirical data. A fine-grained judgement presupposes an evaluation of observational raw data on the background of WST gravity or, at least, a detailed estimation of systematic errors resulting from a comparison of different background theories (Einstein gravity withΛCDM and Newton approximation or alternatively TeVeS-MOND, in comparison with WST and its MOND approximation). This task has to be left to astronomers, if they become sufficiently interested in the present approach. But, taking this caveat in mind, it still seems possible to confront available data from, for example, the Einstein gravity-ΛCDM-Newton approximation framework with our model, in order to get a first impression of its potential usefulness. A comparison with mass data derived in a TeVeS-MOND background would give welcome supplementary information. This is not attempted here.Let us discuss the possibility and the problems of a confrontation of these data with the MOND approximation of WST:(1)
The mass of the hot gas (intracluster medium)Mgas,500 (at r500) has been determined in the mentioned study from X-ray data obtained byXMM Newton andROSAT. The temperature of the gas is estimated by a fit to the measured spectrum. The gas density ρ(r) is reconstructed, using model assumptions, from intensity observables and then integrated up to r500 (in this evaluation the hydrostatic assumption was corrected by taking the velocity dispersion into account [4, sec. 2.2f.]). Up to usual model dependence, the transfer of the mass data from the standard gravity background to WST seems to be relatively uncritical.(2)
Several methods for determining the stellar massM∗,500 are mentioned in [4]. In this study the star mass is gained from optical imaging data due to SDSS 7 in two steps indicated in (ii) above. According to the authors the difference of the stellar mass estimate due to different initial mass functions can result in factor 2 [4, p.4]. Another approach would be to estimate stellar masses of the individual galaxies and “to construct the stellar mass functions in order to sum the stellar masses” (ibid., p.1). Moreover, an additional component of star matter can be associated with the intracluster light. All in all, the estimate of the star mass concentrated in galaxies seems to depend more on models of galactic star evolution than on the background gravity theory. In spite of that the precision cannot be expected to be better than by factor 2 (resp., 0.5).(3)
In Einstein gravity/ΛCDM the cluster mass can, in principle, be estimated from the velocity dispersion σ of galaxies at distance R (from the center) by an estimator derived from the virial theorem M≈G-1σ2R. The additional acceleration of WST-3L aadd=aφ+asf (49) is dynamically indistinguishable from the effects of “true” Newtonian masses. So far it seems as if the estimation oftotal mass can be transferred to the MOND approximation of WST without problems. But if the radius R does not include the “whole” cluster mass (however defined) as is here the case (item (iii), Section 4.8) a surface pressure term must be taken into account. That complicates the case.In standard gravity the necessary correction is implemented by a cubic mass estimatorM~σ3 given above as (79). Moreover, the authors of our reference study [4] reconstruct M500 and r500 from these values using the NFW profile. Because of the different profile for the scalar field halo of WST this is acritical step for our exercise (an ex post comparison of the NFW halo and the WST halo for the Coma cluster is given in Figure 6). On the other hand, if the resulting systematic errors are smaller than the error intervals of Mtot (76), implied by the observational errors of the other quantities, they do not disturb a rough empirical check of the model.(4)
Finally the dependency of the data evaluation on the background cosmology has to be taken into account. The data of the 19 clusters used in the following have redshiftz<0.1. The geometrical and dynamical corrections implied by the ΛCDM cosmology are correspondingly small. An evaluation in, for example, a Lemaitre-de Sitter model (or even a nonexpanding Weyl geometric model with redshift) [1, section  4.2] would affect the data only by a minor expansion of the error intervals.The points (1), (2), and (4), in particular the estimate of stellar mass concentrated in galaxies and gas mass, are fairly insensitive against a change of the background theory from Einstein gravity to WST. The theory dependence of r500 is uncritical in our context. Any other reference radius could have been taken, as long as it is specified in astronomical distance units. The estimate for the total masses at r500 and r200 is the critical point for our purpose (item (3)). However, if the difference of the halo profiles between the scalar field energy density of WST and NFW dark matter does not push the estimates for the total masses M500,M200 outside the error intervals of our halo model (due to observational input data), we may still be able to draw first inferences from the following evaluation.


### 4.3. Empirical Data for 19 Clusters

The studies [2, 4] contain new data on the baryon content and the total gravitational mass for 19 clusters of galaxies (as it appears in an Einstein gravity, ΛCDM framework with Newton approximation) [2] contains a correction to the main paper [4]. Here, of course, we use the corrected data). The mass data M500,Mgas,500,andM∗,500 are given in columns (5), (7),  and  (8) of Table  1 in [2]. It is reproduced in our Table 1. The values for r500 are published in [3, tab.  1, col. (5)] (here Table 2). A comparison of the total cluster masses derived from the velocity dispersion with a mass estimate derived from the gas mass shows that the two clusters A2029 and A2065 are outliers, with total cluster masses considerably higher than the corresponding gas masses would let us expect. The authors therefore separate thetwo outliers from the rest of the data, with the remaining17 clusters as a reliable dataset [4, p.3]. We shall do the same.Table 1
Properties of the 19 galaxy clusters (Table1 of [2]).
Name X-ray center (J2000) Redshift M 500 M 500 , M - M g a s M g a s , 500 M ∗ , 500 Undisturbed/cool-core RA Dec 10 14  M ⊙ 10 14  M ⊙ 10 13  M ⊙ 10 12  M ⊙ A0085 00:41:50.306 −09:18:11.11 0.0556 6.37 ± 1.00 5.68 ± 0.37 8.13 ± 0.38 7.36 ± 1.00 Y/S A0400 02:57:41.349 +06:01:36.93 0.0240 1.83 ± 0.39 1.07 ± 0.07 1.36 ± 0.05 4.39 ± 1.06 N/N IIIZw54 03:41:18.729 +15:24:13.91 0.0311 1.91 ± 0.58 1.18 ± 0.08 1.45 ± 0.26 4.57 ± 0.56 Y/W A1367 11:44:44.501 +19:43:55.82 0.0216 1.76 ± 0.27 2.11 ± 0.14 2.07 ± 0.07 4.35 ± 0.74 N/N MKW4 12:04:27.660 +01:53:41.50 0.0200 0.50 ± 0.14 0.58 ± 0.04 0.47 ± 0.02 1.16 ± 0.22 Y/S ZwCl1215 12:17:40.637 +03:39:29.66 0.0750 4.93 ± 0.98 4.34 ± 0.28 6.10 ± 0.29 7.05 ± 0.83 Y/N A1650 12:58:41.885 −01:45:32.91 0.0845 3.44 ± 0.66 4.28 ± 0.27 5.09 ± 0.73 7.47 ± 1.13 Y/W Coma 12:59:45.341 +27:57:05.63 0.0232 6.55 ± 0.79 6.21 ± 0.40 8.42 ± 0.63 13.14 ± 1.80 N/N A1795 13:48:52.790 +26:35:34.36 0.0616 3.41 ± 0.63 4.46 ± 0.29 5.11 ± 0.14 6.21 ± 0.98 Y/S MKW8 14:40:42.150 +03:28:17.87 0.0270 0.62 ± 0.12 1.10 ± 0.07 0.80 ± 0.12 1.61 ± 0.23 N/N A2029 15:10:55.990 +05:44:33.64 0.0767 14.70 ± 2.61 6.82 ± 0.44 13.35 ± 0.53 9.59 ± 1.11 Y/S A2052 15:16:44.411 +07:01:12.57 0.0348 1.39 ± 0.28 2.03 ± 0.13 1.86 ± 0.10 3.53 ± 0.40 Y/S MKW3S 15:21:50.277 +07:42:11.77 0.0450 1.45 ± 0.34 2.29 ± 0.15 2.13 ± 0.09 3.90 ± 0.43 Y/S A2065 15:22:29.082 +27:43:14.39 0.0721 11.18 ± 1.78 3.35 ± 0.22 7.66 ± 1.44 7.32 ± 0.75 N/W A2142 15:58:19.776 +27:14:00.96 0.0899 7.36 ± 1.25 10.26 ± 0.66 13.76 ± 0.73 8.42 ± 0.77 Y/W A2147 16:02:16.305 +15:58:18.46 0.0351 4.44 ± 0.67 3.63 ± 0.23 5.04 ± 0.53 6.84 ± 0.90 N/N A2199 16:28:37.126 +39:32:53.29 0.0302 2.69 ± 0.42 2.64 ± 0.17 2.97 ± 0.30 4.76 ± 0.50 Y/S A2255 17:12:54.538 +64:03:51.46 0.0800 7.13 ± 1.38 4.08 ± 0.26 7.11 ± 0.33 6.74 ± 0.97 N/N A2589 23:23:56.772 +16:46:33.19 0.0416 3.03 ± 0.75 1.88 ± 0.12 2.54 ± 0.17 5.12 ± 0.56 Y/W Notes. The cluster mass, M500,M-Mgas, is derived from the M500-Mgas,500 relation and only used for comparison with the cluster mass, M500, derived from the “harmonic” velocity dispersion. “S,” “W,” and “N” denote strong cool-core, weak cool-core, and noncool-core clusters.Table 2
Dataset used for halo model (error intervals omitted).
Cluster β r c r 500 M 500 M g a s 500 M ∗ 500 M 200 r 200 M A Coma 0.654 246 1.278 6.55 8.42 13.14 13.84 2.3 12.86 A85 0.532 59.3 1.216 6.37 8.13 7.36 7.71 1.9 8.72 A400 0.534 110. 0.712 1.83 1.36 4.39 1.48 1.09 2.93 IIIZw54 0.887 206 0.731 1.91 1.45 4.57 2.81 1.35 4.51 A1367 0.695 274 0.893 1.76 2.07 4.35 4.06 1.53 5.77 MKW4 0.44 7.86 0.58 0.5 0.47 1.16 0.71 0.86 1.79 ZwCl215 0.819 308 1.098 4.93 6.1 7.05 10.37 2.09 10.65 A1650 0.704 201 1.087 3.44 5.09 7.47 11.14 2.15 11.11 A1795 0.596 55.7 1.118 3.41 5.11 6.21 10.99 2.14 11.04 MKW8 0.511 76.4 0.715 0.62 0.8 1.61 2.38 1.28 4.0 A2029 0.582 59.3 1.275 14.7 13.35 9.59 13.42 2.29 12.59 A2052 0.526 26.4 0.875 1.39 1.86 3.53 2.21 1.25 3.79 MKW3S 0.581 47 0.905 1.45 2.13 3.9 3.46 1.45 5.11 A2065 1.162 493 1.008 11.18 7.66 7.32 16.69 2.45 14.46 A2142 0.591 110 1.449 7.36 13.76 8.42 15.03 2.36 13.61 A2147 0.444 170. 1.064 4.44 5.04 6.84 3.46 1.45 5.15 A2199 0.655 99.2 0.957 2.69 2.97 4.76 4.80 1.62 6.37 A2255 0.797 423 1.072 7.13 7.11 6.74 13.32 2.27 12.52 A2589 0.596 84.3 0.848 3.03 2.54 5.12 3.58 1.47 5.24 Source:β, rc, M200, r200, and MA from [15]; r500 from [3]; M500, Mgas500, and M∗500 from [2]. Units.rc in kpc, r500, r200 in Mpc, M500 in 1014 M⊙, Mgas,500 in 1013 M⊙, and M∗,500 in 1012 M⊙.Parameters (β,rc) for the models of these galaxy clusters (as well as of many more) have been published earlier in [15, tab.  4.1]. This publication also contains mass data M200,Mgas,200 at r200 (with error intervals) and mass values MA,MgasA (without error interval) at the Abell radius, here defined as rA=2.14Mpc, but no data for star masses (evaluated for the value of Ho assumed in the later publications [2, 4],   h=0.7). In [15] the methods for determining the total mass and the gas mass were not yet as refined as in the later study. It is therefore not possible to aggregate the different datasets to one coherent ensemble (the values for M500 and Mgas,500 given here differ from the ones in [4] outside the error intervals). As no updated values for the parameters (β,rc) of the mass profiles were available to the present author, the values of [15] are here used asestimators for the form parameters of the β model.We considerM500,Mgas,500,M∗,500,andr500 from [2, 4] as ourcrucial mass data (including reference radii). Mass data at r200 (as well as r200 itself) from the older study are welcome asadditional information; but they will not be used as core criteria for the empirical test of our model. Table 2 collects the data used (for error intervals see the respective source, [2, 15]). It shows that the cluster ensemble covers an order of magnitude variation for the gas mass Mgas,500 and one and a half orders of magnitude variation in total mass M500. The selection method by intersecting the cluster sets of different raw data sources does not seem to be influenced by any particular bias. We thus may consider the collection as a reasonable dataset for testing our cluster halo model.


### 4.4. Adapting the WST Halo Model to the Data

For the construction of the model we have to work in Einstein gauge (9). Consistency with the deep MOND acceleration demands (51). Taken together these conditions fix the coefficients of the Lagrangian (1) and (28), independently of the convention chosen for ϕo. With a value for λ on the order of magnitude 10 (e.g., λ/4=6 as at the end of Section 2.3) the contribution of the LV4 term lies many orders of magnitude below the energy densities, the dominant term, in (35) and is negligible in our context.In agreement with Section3 the test of our halo model for each of the 19 galaxy clusters can now proceed as follows:(1)
Specification of theβ models (67) for gas and for star mass and parameters (β,rc) from [15], ρo determined by fitting to Mgas,500 and M∗,500 at r500, respectively [4].(2)
Determination of the Newton acceleration of the baryonic mass components due to (68).(3)
Calculation of the scalar field halo and the phantom halo of the baryonic mass by (69), and(66).(4)
Calculation of the scalar field halo of the system of freely falling galaxies (71).(5)
Aggregation of these to the total halo (75) and choice of fading out functions beyond r200 (see Appendix).(6)
Integration of the densities to the corresponding masses: scalar field energy of the galaxiesMsf2 of the gas mass Msf1, total scalar field halo Msf, net phantom energy of the baryonic mass Mph1, and finally the total transparent matter Mt and the lensing mass Mlens (76) and (77).(7)
Calculation of the error intervals of the model at selected distances (r500,r200).(8)
Comparison of the empirical value forM500 (resp., M200) with the model value Mt(r500) (resp., Mt(r200)).The results of this test are given in Section 4.6 Before we turn to the overall evaluation, we shall have a look at one cluster as an exemplary case.


### 4.5. The WST Halo Model with the Coma Cluster as Test Case

For a first check of our model we choose theComa cluster. According to item (1) of the last section we use the empirical input data from Table 2 (units given there):(80)β,rc,r500,Mgas,500,M∗,500=0.654,246,1.278,8.42,13.14,(rc in kpc, r500 in Mpc, Mgas,500 in 1013M⊙, and M∗,500 in 1012M⊙). The different components of the cluster halo integrate to masses (up to distance r) documented in Figure 1.Figure 1
Halo components of Coma cluster: transparent matter haloMt=Msf+Mph1, total scalar field (SF) halo Msf, SF halo of freely falling galaxies Msf2, and net phantom energy Mph1 (in barycentric rest system). Empirical data (violet dot, bar): galaxy and total mass (with error intervals) at r500=1280kpc.The scalar field halo (SF) of the total baryonic massρsf1 contributes the lion share to the total transparent energy. The SF of the galaxy system ρsf2 carries about as much energy as the net phantom component ρph1 in the barycentric rest system of the cluster. It surpasses the gas mass close to the reference radius r500. Figure 2 shows the fast increase of the gravitational mass of the scalar field halo of the galactic system.Figure 2
Comparison of the contribution of the scalar field halo of the galaxies with the baryonic mass for the Coma cluster (empirical dataMbar,500 violet dot).If we add all baryonic and halo contributions, the picture given in Figure3 emerges. It shows an encouraging agreement of the observed values M500 at r500=1278kpc with the prediction of the Weyl geometric halo model Mtot(r500) (in the range of the observational errors and of model errors):(81)Mtotr500=5.66+0.97-0.68×1014M⊙,M500=6.55+0.79-0.79×1014M⊙.Model error bars have been estimated by varying the input data (80) in their respective error intervals. The reconstruction of star mass from observational raw data is a particularly delicate point. Depending on the assumptions on the stellar dynamics and the resulting data evaluation model “one can obtain up to a factor of 2 fewer stars” [2, p.6]. For obtaining our model errors we allowed a variation in stellar masses by factors 0.5 and 2.Figure 3
Contribution of the baryonic massMbar, of the scalar field and phantom energies Msf,Mph1 to the transparent mass Mt and to the total mass Mtot of the Coma cluster in the WST model. Model errors indicated at r500 and r200 (black). Empirical data for Mbar (violet dot) and for total mass M500 with error bars (violet) from [2]. Additional empirical data at r200 (yellowish) from [15].At the Abell radiusrA=2.14Mpc (in the convention of [15]) we get the model values(82)MtotrA≈10.64×1014M⊙,MlensrA≈8.82×1014M⊙for the dynamical total mass and the lensing mass (77). Due to the net phantom energy the dynamical mass is about 20% higher than the lensing mass. This is an effect by which the model can, in principle, be tested empirically and discriminated from others. The baryonic masses in the β model are Mgas(rA)≈11.73×1013M⊙, M∗(rA)≈1.83×1013M⊙. Thus not only does the total mass Mtot(r500) given by our model agree with the empirical value M500 inside the error bounds but also MA and M200 are reasonably well recovered.This is the casewithout assuming any componentof particle dark matter besides the (real) energy of the scalar field and the (phantom) energy ascribed to the additional acceleration aφ induced by the Weylian scale connection. This might still be a coincidence. In order to learn more about the question whether the findings at the Coma cluster are exemplary or not, we have to consider the data of all 19 galaxy clusters, respectively, the 17 of the reliable subensemble.


### 4.6. Halos and Total Mass for 17(+2) Clusters of Galaxies

The mass values of the halo models for the 19 clusters are calculated as described in Section4.3 with the choice of fadeout functions beyond r=r200 (see Appendix). Theresults are documented in Tables 3 and 4 and Figures 4 and 5 (for Coma see Figure 3).Table 3
Empirical and model values for total mass atr500, r200.
Cluster r 500 M t o t ( r 500 ) M 500 r 200 M t o t ( r 200 ) M 200 Coma 1278 5.66 + 0.97 - 0.68 6.55 + 0.79 - 0.79 2300 14.38 + 2.71 - 1.90 13.84 + 1.49 - 1.41  A85 1216 4.95 + 0.62 - 0.43 6.37 + 1 - 1 1900 10.88 + 1.38 - 0.97 7.71 + 0.8 - 0.74  A400 712 1.38 + 0.3 - 0.19 1.83 + 0.39 - 0.39 1093 3.08 + 0.68 - 0.44 1.48 + 0.21 - 0.18  IIIZw54 731 1.45 + 0.38 - 0.28 1.91 + 0.58 - 0.58 1350 3.56 + 1.34 - 1.09 2.81 + 2.74 - 1.1  A1367 893 1.95 + 0.34 - 0.23 1.76 + 0.27 - 0.27 1529 4.67 + 0.93 - 0.62 4.06 + 0.45 - 0.4  MKW4 580 0.6 + 0.11 - 0.08 0.5 + 0.14 - 0.14 857 1.33 + 0.26 - 0.17 0.71 + 0.07 - 0.06  ZwCl215 1098 3.94 + 0.55 - 0.38 4.93 + 0.98 - 0.98 2093 9.8 + 1.64 - 1.17 10.37 + 3.51 - 2.62  A1650 1087 3.65 + 0.71 - 0.05 3.44 + 0.66 - 0.66 2150 10.48 + 2.91 - 2.41 11.14 + 5.77 - 3.46  A1795 1118 3.65 + 0.49 - 0.32 3.41 + 0.63 - 0.63 2136 10.84 + 1.47 - 0.99 10.99 + 2.26 - 2.09  MKW8 715 0.94 + 0.2 - 0.15 0.62 + 0.12 - 0.12 1279 2.80 + 0.74 - 0.6 2.38 + 1.04 - 0.59  A2029 1275 6.75 + 0.73 - 0.51 14.7 + 2.61 - 2.61 2286 17.9 + 2.01 - 1.42 13.42 + 2.43 - 2.26  A2052 875 1.78 + 0.31 - 0.21 1.39 + 0.28 - 0.28 1250 3.43 + 0.61 - 0.41 2.21 + 0.06 - 0.08  MKW3S 905 1.96 + 0.33 - 0.22 1.45 + 0.34 - 0.34 1450 4.5 + 0.78 - 0.52 3.46 + 0.36 - 0.34  A2065 1008 4.08 + 0.81 - 0.74 11.18 + 1.78 - 1.78 2450 10.83 + 5.06 - 3.34 16.69 + 21.34 - 6.73  A2142 1449 7.5 + 0.81 - 0.59 7.36 + 1.25 - 1.25 2364 16.83 + 1.89 - 1.39 15.03 + 3.9 - 2.64  A2147 1064 3.53 + 0.61 - 0.45 4.44 + 0.67 - 0.67 1450 6.39 + 1.26 - 0.99 3.46 + 1.17 - 0.74  A2199 957 2.43 + 0.44 - 0.32 2.69 + 0.42 - 0.42 1621 5.8 + 1.14 - 0.82 4.81 + 0.37 - 0.36  A2255 1072 4.13 + 0.53 - 0.37 7.13 + 1.38 - 1.38 2271 12.04 + 1.88 - 1.37 13.32 + 1.44 - 1.19  A2589 848 2.08 + 0.39 - 0.26 3.03 + 0.75 - 0.75 1471 5.44 + 1.06 - 0.73 3.58 + 3.86 - 1.54 Model valuesMtot(rN00) and empirical values MN00 in 1014 M⊙, rN00 (empirical) in kpc (N=2,5).Table 4
Model values for halo and baryonic masses atr200.
Cluster M t M s f M s f g a l M p h 1 M g a s M ∗ f ∗ f t Coma 12.32 9.91 2.66 2.42 1.78 0.278 0.16 6.9 A85 9.08 7.22 1.61 1.87 1.65 0.15 0.09 5.5 A400 2.7 2.21 0.73 0.49 0.29 0.093 0.32 9.4 IIIZw54 3.21 2.62 0.86 0.59 0.27 0.085 0.32 11.8 A1367 4.16 3.37 0.99 0.79 0.42 0.088 0.21 10. MKW4 1.2 0.97 0.3 0.22 0.1 0.0260 0.25 11.5 ZwCl215 8.54 6.82 1.66 1.72 1.14 0.131 0.12 7.5 A1650 9.16 7.36 1.94 1.81 1.15 0.169 0.15 8. A1795 9.39 7.51 1.86 1.88 1.29 0.157 0.12 7.3 MKW8 2.53 2.05 0.6 0.48 0.22 0.045 0.2 11.3 A2029 14.58 11.53 2.37 3.05 3.1 0.223 0.07 4.7 A2052 3.03 2.44 0.70 0.58 0.34 0.065 0.19 8.9 MKW3S 3.97 3.21 0.91 0.77 0.44 0.081 0.18 9. A2065 9.63 7.66 1.75 1.97 1.12 0.107 0.1 8.6 A2142 13.93 10.98 2.13 2.95 2.74 0.168 0.06 5.10 A2147 5.38 4.31 1.11 1.07 0.89 0.121 0.14 6. A2199 5.10 4.1 1.11 1. 0.61 0.098 0.16 8.4 A2255 10.4 8.27 1.88 2.13 1.51 0.143 0.09 6.9 A2589 4.74 3.83 1.11 0.91 0.59 0.119 0.2 8. Mass values in1014 M⊙, f∗=M∗/Mgas,ft=(Mt/Mgas)(r200); for r200 see Table 3.Figure 4
Halo models for clusters 2–9 in Table2: total mass Mtot (black line) with model error bars at r500,r200, transparent matter halo Mt constituted by scalar field halo Msf, and net phantom halo (in barycentric rest system of cluster) Mph1 and baryonic mass (gas and stars) Mbar. Empirical data for the total mass with error intervals at r500 (violet) from [2]. Additional empirical data at r200 (yellow) from [15].Figure 5
Halo models for clusters 10–19 in Table2. For description see Figure 4.Figure 6
Comparison of dark/transparent/phantom mass halos for Coma in NFW, WST, and TeVeS models, free parameters of halos for NFW and TeVeS (μ2 with neutrino core) adapted to mass data (black error bars) at r500=1280kpc. Red error bar indicates variability of NFW halo values at r200≈2300 kpc (fitting the NFW profile inside the margins of empirical error bars at r500); blue error bars give model errors of WST model at r500 and r200.For 15 clusters ourmodel reproduces the total mass at r500correctly, that is, inside the error margins of data and model. This is achieved without any further adjustable parameter, only on the basis of the parameters for the β-model for baryonic mass and Mgas(r500),M∗(r500). The fading out functions do not intervene below r200. Moreover, for the majority of these and paradoxically for all other four, the less precisely determined data at r200 have overlapping 1σ error intervals. This indicates a surprising agreement between the (theoretically derived) transparent halo and the empirically determined dark halo, Mt≈Mdm.For 4 clusters, MKW8, A2255, and outliers A2029 and A2065, the error intervals of empirical data and model datado not overlap. For the first two of them, MKW8 and A2255, the model predictions are consistent with the empirical data within doubled error intervals (2σ range). Only the two outliers (A2029, A2065) lie farther apart (A2029 has the surprising property that the empirical value for the total mass at r500surpasses the one at r200, M500>M200). Otherwise the model data are in good agreement with an assumption of normally distributed statistical errors and with the assumption that the evaluation bias due to the use of theNFW profile for dark matter (item (3) in Section 4.2)does not shift the mass estimates outside the error intervals.All in all, the assessment of the WST-3L halo model has surprisingly well passed, in spite of the main caveat of item(3) (Section 4.2). The outcome found for Coma seems to be typical also for the other clusters. Moreover, the good agreement of the model with the data of the reliable subensemble of 17 clusters supports the assumption stated in the last phrase (3) (Section 4.2) (no large systematic errors due to data transfer from Einstein/ΛCDM to the WST-3L framework). But we cannot exclude the possibility of cancelling between model errors and data transfer errors. Thus we have only found empirical support for theconjecture that, on the level of galaxy clusters, the observed dark matter effects encoded in Mdm may solely be due to the combined impact of the halo Msf of the scalar field and of the scale connection, Mph1: (83)Mdm≈Msf+Mph1=Mtcf.76.


### 4.7. Comparison with TeVeS and NFW Halos

It is surprising that in the WST-3L approach the total amount of observed dark matterMdm seems to be explained by the energy of the scalar field and the phantom halo, Mt=Msf+Mph1 and Mdm≈Mt. No missing mass is left. In usual relativistic MOND approaches this is not the case for clusters, although it is essentially so for galaxies [20]. Based on a study of about 40 galaxy clusters, Sanders has proposed the hypothesis of a neutrino component “between a few times 1013 and 1014M⊙,” mostly concentrated close to the center of the cluster, in a region up to twice the core radius, supplementing the baryonic mass and the phantom energy of the TeVeS model [5, p. 902]. In our approach, this hypothesis is unnecessary. Where does this difference arise from?A model calculation for the Coma cluster, evaluating (57) for the transition function μ2=x(1+x2)-1/2 (53) and the corresponding ν2 (54) used by Sanders in [5], shows that the neutrino core had to be tuned to Mν≈1.8×1014M⊙ in order to give agreement with the empirical value Mdm(r500)=M500-Mbar(r500)≈4.7×1014M⊙, where in this framework “r500” and “r200” are to be defined by a formal convention with regard to a (fictitious) NFW halo.Table4 shows the amount of scalar field energy up to a radius r≈r200 in the WST model. It varies between about 2×1013 and 9×1014M⊙, that is, roughly in the range found necessary by Sanders for the (hypothetical) neutrino halo. Moreover, a comparison of the transition functions μw(r) (55) and μ2(x) shows that the gross phantom energy ρt of the WST approach is larger than in an μ2-MOND model [1]. Roughly half of the missing mass of Sanders’ model is covered by this effect;the other half is due to the scalar field halo of the system of galaxies up to r200.A comparison of the two MOND-like approaches is given in Figure6. Here one has to keep in mind that the TeVeS-μ2 model has a free adaptable parameter (mass of the neutrino core), while the WST has not. The general profiles of the “dark matter” halos of both models are similar. The TeVeS-μ2 mass starts from a higher score because of its neutrino core; the WST transparent mass starts from a lower initial value but rises faster because of the increasing contribution of the scalar field halo of the galaxies.On the other hand, the best known profile for dark matter distribution, used in most structure formation simulations, is the NFW halo (Navarro/Frank/White). Its profile is(84)ρr=ρor/rc1+r/rc2,with density parameter ρo and core radius rc (at which the density has reduced to half the reference value). For a first comparison of the interior halos we take rc≈180h-1kpc (h=0.7), following [6] (in studies of the exterior halo of the Coma cluster, Geller et al. have found fit values 0.182+0.03-0.03,0.167+0.029-0.029,0.192+0.035-0.035 in units h-1Mpc for rc   [6]) and determine the central density parameter ρo such that the total integrated mass MNFW assumes the empirical value of [2] at r500=1289kpc (Figure 6). The error intervals for r500 give upper and lower model values for ρo and corresponding model error bars (red) for MNFW(r200). This version of the NFW halo satisfies one set of empirical data by construction (at r500); our main interest thus goes to the other empirical dataset available at r200=2300kpc. The NFW model error interval overlaps with the empirical error bar at r200 and with the error interval of the WST model.There is a conspicuous difference between the mass profiles of the NFW halo on one side and the halo profiles of WST orμ2-TeVeS on the other. More and precise empirical data of the interior halos of galaxy clusters ought to be able to discriminate between the two model classes. An empirical discrimination between the two MOND-like approaches would need more, and more precise, profile data. At the moment the Weyl geometric MOND-like model survives the comparison fairly well, even though it has no free parameter which would allow adapting it to the halo data.


### 4.8. A Side-Glance at the Bullet Cluster

At the end of this section let us shed a side-glance at the bullet cluster 1E0657-56. It is often claimed that the latter provides direct evidence in favour of particle dark matter and rules out alternative gravity approaches. Our considerations show that this argument is not compelling. The energy content of the scalar field halos of the colliding clusters endows them with inertia of their own. The shock of the colliding gas exerts dynamical forces on the gas masses only, not directly on the scalar field halos. During the encounter the halos will roughly follow the inertial trajectories of their respective clusters before collision, and they will continue to do so for a while. It will take time before a readaptation of the mass systems and the respective scalar field halos has taken place. Clearly the MOND approximation is unable to cover such violent dynamical processes. It describes only the relatively stable states before collision and in some distant future after collision. But a separation of halos and gas masses for a (cosmically “short”) period is to be expected, just as in the case of a particle halo with appropriate clustering properties.For the time being, cluster 1E0657-56 does not help to decide between the overarching alternative research strategies, particle dark matter, or alternative gravity. It may be able to do so, once the dynamics of gas and of the halos has been modeled with sufficient precision in both approaches. Then a proper comparison can be made. But that is an overtly complicated task. It seems more likely that other types of empirical evidence will offer a simpler path to a differential evaluation of the two strategies and help clarifying the alternative.


## 4.1. Empirical Determination of Mass Data for Galaxy Clusters

For a first empirical exploration we confront the WST cluster model with recent mass data for 19 clusters obtained on the background of Einstein/Newton gravity andΛCDM by a group of astronomers about Zhang et al. [2, 4]. We use the form parameters β,rc of the β-models of these clusters, published in an earlier study by one author of the group [15]. In the present study we have to take the mass data and the form parameters essentially at face value. Methodological questions arising from this procedure are discussed in the next subsection. There seem to be sufficient reasons for expecting that the different background theories do not principally invalidate the results thus obtained. Of course, an authoritative empirical study would presuppose an evaluation of observational data on the background of WST itself; it can be done only by astronomers, if they get interested in the present approach.The studies of Zhang et al. have the great advantage to build upon three independent observational datasets for determining the gas massMgas,500, the star mass M∗,500, and the total mass M500 (assuming a dark matter explanation for the observed gravitational effects) at the reference distance r500. The latter is determined for each cluster at the distance r500 from the cluster center at which the total gravitational acceleration indicates a total mass density 500 times the critical density.(i)
Mgas,500 has been extracted from X-ray data on the hot intracluster medium (ICM) collected by XMM Newton and ROSAT. Surface brightness data have been used to infer an ICM radial electron number density profile, and spectral analysis data gave information on the radial temperature distribution. From that a gas density distribution has been reconstructed and the gas masses at r500 by integration (an outline of the procedure and literature for more details is given in [4, p.3]).(ii)
M∗,500 has been determined from optical imaging data due to SDSS 7 in two steps. First the total luminosity of the cluster has been determined by means of a “galaxy luminosity function” (GLF); then the mass is estimated using mass-to-light ratios depending on the cluster mass. In the last step models of the star development in the respective galaxy, elliptical or spiral, enter. They depend on assumptions on an “initial mass function” (IMF). Two possibilities for the IMF (Salpeter versus Kroupa) are considered and compared in [2, 4]. According to the authors the difference of the stellar mass estimate can result in factor 2 [4, p.4].(iii)
The total cluster massM500 has been determined on the basis of the velocity dispersion of galaxies, using spectroscopic data from [25, tab.  1]. The mass estimator used is equation (2) of [26](79)Mv=Aσv103kms-13×1014h-1M⊙,whereA=1.50±0.02 and σv is the 3-dimensional velocity dispersion inside a sphere of virial radius (by convention rv=r200). Reasons for this choice are given in [26, sec. 3]. M500 was then determined from M200 by a NFW model.


## 4.2. Theory Dependence of Mass Data for Galaxy Clusters

Mass densities of the hot gas and of star matter in galaxy clusters are indirectly inferred from observable quantities; they are thustheory dependent. Even inside the same background theory they may depend on choices of models and methods of evaluation.That makes it a difficult task to compare our model with empirical data. A fine-grained judgement presupposes an evaluation of observational raw data on the background of WST gravity or, at least, a detailed estimation of systematic errors resulting from a comparison of different background theories (Einstein gravity withΛCDM and Newton approximation or alternatively TeVeS-MOND, in comparison with WST and its MOND approximation). This task has to be left to astronomers, if they become sufficiently interested in the present approach. But, taking this caveat in mind, it still seems possible to confront available data from, for example, the Einstein gravity-ΛCDM-Newton approximation framework with our model, in order to get a first impression of its potential usefulness. A comparison with mass data derived in a TeVeS-MOND background would give welcome supplementary information. This is not attempted here.Let us discuss the possibility and the problems of a confrontation of these data with the MOND approximation of WST:(1)
The mass of the hot gas (intracluster medium)Mgas,500 (at r500) has been determined in the mentioned study from X-ray data obtained byXMM Newton andROSAT. The temperature of the gas is estimated by a fit to the measured spectrum. The gas density ρ(r) is reconstructed, using model assumptions, from intensity observables and then integrated up to r500 (in this evaluation the hydrostatic assumption was corrected by taking the velocity dispersion into account [4, sec. 2.2f.]). Up to usual model dependence, the transfer of the mass data from the standard gravity background to WST seems to be relatively uncritical.(2)
Several methods for determining the stellar massM∗,500 are mentioned in [4]. In this study the star mass is gained from optical imaging data due to SDSS 7 in two steps indicated in (ii) above. According to the authors the difference of the stellar mass estimate due to different initial mass functions can result in factor 2 [4, p.4]. Another approach would be to estimate stellar masses of the individual galaxies and “to construct the stellar mass functions in order to sum the stellar masses” (ibid., p.1). Moreover, an additional component of star matter can be associated with the intracluster light. All in all, the estimate of the star mass concentrated in galaxies seems to depend more on models of galactic star evolution than on the background gravity theory. In spite of that the precision cannot be expected to be better than by factor 2 (resp., 0.5).(3)
In Einstein gravity/ΛCDM the cluster mass can, in principle, be estimated from the velocity dispersion σ of galaxies at distance R (from the center) by an estimator derived from the virial theorem M≈G-1σ2R. The additional acceleration of WST-3L aadd=aφ+asf (49) is dynamically indistinguishable from the effects of “true” Newtonian masses. So far it seems as if the estimation oftotal mass can be transferred to the MOND approximation of WST without problems. But if the radius R does not include the “whole” cluster mass (however defined) as is here the case (item (iii), Section 4.8) a surface pressure term must be taken into account. That complicates the case.In standard gravity the necessary correction is implemented by a cubic mass estimatorM~σ3 given above as (79). Moreover, the authors of our reference study [4] reconstruct M500 and r500 from these values using the NFW profile. Because of the different profile for the scalar field halo of WST this is acritical step for our exercise (an ex post comparison of the NFW halo and the WST halo for the Coma cluster is given in Figure 6). On the other hand, if the resulting systematic errors are smaller than the error intervals of Mtot (76), implied by the observational errors of the other quantities, they do not disturb a rough empirical check of the model.(4)
Finally the dependency of the data evaluation on the background cosmology has to be taken into account. The data of the 19 clusters used in the following have redshiftz<0.1. The geometrical and dynamical corrections implied by the ΛCDM cosmology are correspondingly small. An evaluation in, for example, a Lemaitre-de Sitter model (or even a nonexpanding Weyl geometric model with redshift) [1, section  4.2] would affect the data only by a minor expansion of the error intervals.The points (1), (2), and (4), in particular the estimate of stellar mass concentrated in galaxies and gas mass, are fairly insensitive against a change of the background theory from Einstein gravity to WST. The theory dependence of r500 is uncritical in our context. Any other reference radius could have been taken, as long as it is specified in astronomical distance units. The estimate for the total masses at r500 and r200 is the critical point for our purpose (item (3)). However, if the difference of the halo profiles between the scalar field energy density of WST and NFW dark matter does not push the estimates for the total masses M500,M200 outside the error intervals of our halo model (due to observational input data), we may still be able to draw first inferences from the following evaluation.


## 4.3. Empirical Data for 19 Clusters

The studies [2, 4] contain new data on the baryon content and the total gravitational mass for 19 clusters of galaxies (as it appears in an Einstein gravity, ΛCDM framework with Newton approximation) [2] contains a correction to the main paper [4]. Here, of course, we use the corrected data). The mass data M500,Mgas,500,andM∗,500 are given in columns (5), (7),  and  (8) of Table  1 in [2]. It is reproduced in our Table 1. The values for r500 are published in [3, tab.  1, col. (5)] (here Table 2). A comparison of the total cluster masses derived from the velocity dispersion with a mass estimate derived from the gas mass shows that the two clusters A2029 and A2065 are outliers, with total cluster masses considerably higher than the corresponding gas masses would let us expect. The authors therefore separate thetwo outliers from the rest of the data, with the remaining17 clusters as a reliable dataset [4, p.3]. We shall do the same.Table 1
Properties of the 19 galaxy clusters (Table1 of [2]).
Name X-ray center (J2000) Redshift M 500 M 500 , M - M g a s M g a s , 500 M ∗ , 500 Undisturbed/cool-core RA Dec 10 14  M ⊙ 10 14  M ⊙ 10 13  M ⊙ 10 12  M ⊙ A0085 00:41:50.306 −09:18:11.11 0.0556 6.37 ± 1.00 5.68 ± 0.37 8.13 ± 0.38 7.36 ± 1.00 Y/S A0400 02:57:41.349 +06:01:36.93 0.0240 1.83 ± 0.39 1.07 ± 0.07 1.36 ± 0.05 4.39 ± 1.06 N/N IIIZw54 03:41:18.729 +15:24:13.91 0.0311 1.91 ± 0.58 1.18 ± 0.08 1.45 ± 0.26 4.57 ± 0.56 Y/W A1367 11:44:44.501 +19:43:55.82 0.0216 1.76 ± 0.27 2.11 ± 0.14 2.07 ± 0.07 4.35 ± 0.74 N/N MKW4 12:04:27.660 +01:53:41.50 0.0200 0.50 ± 0.14 0.58 ± 0.04 0.47 ± 0.02 1.16 ± 0.22 Y/S ZwCl1215 12:17:40.637 +03:39:29.66 0.0750 4.93 ± 0.98 4.34 ± 0.28 6.10 ± 0.29 7.05 ± 0.83 Y/N A1650 12:58:41.885 −01:45:32.91 0.0845 3.44 ± 0.66 4.28 ± 0.27 5.09 ± 0.73 7.47 ± 1.13 Y/W Coma 12:59:45.341 +27:57:05.63 0.0232 6.55 ± 0.79 6.21 ± 0.40 8.42 ± 0.63 13.14 ± 1.80 N/N A1795 13:48:52.790 +26:35:34.36 0.0616 3.41 ± 0.63 4.46 ± 0.29 5.11 ± 0.14 6.21 ± 0.98 Y/S MKW8 14:40:42.150 +03:28:17.87 0.0270 0.62 ± 0.12 1.10 ± 0.07 0.80 ± 0.12 1.61 ± 0.23 N/N A2029 15:10:55.990 +05:44:33.64 0.0767 14.70 ± 2.61 6.82 ± 0.44 13.35 ± 0.53 9.59 ± 1.11 Y/S A2052 15:16:44.411 +07:01:12.57 0.0348 1.39 ± 0.28 2.03 ± 0.13 1.86 ± 0.10 3.53 ± 0.40 Y/S MKW3S 15:21:50.277 +07:42:11.77 0.0450 1.45 ± 0.34 2.29 ± 0.15 2.13 ± 0.09 3.90 ± 0.43 Y/S A2065 15:22:29.082 +27:43:14.39 0.0721 11.18 ± 1.78 3.35 ± 0.22 7.66 ± 1.44 7.32 ± 0.75 N/W A2142 15:58:19.776 +27:14:00.96 0.0899 7.36 ± 1.25 10.26 ± 0.66 13.76 ± 0.73 8.42 ± 0.77 Y/W A2147 16:02:16.305 +15:58:18.46 0.0351 4.44 ± 0.67 3.63 ± 0.23 5.04 ± 0.53 6.84 ± 0.90 N/N A2199 16:28:37.126 +39:32:53.29 0.0302 2.69 ± 0.42 2.64 ± 0.17 2.97 ± 0.30 4.76 ± 0.50 Y/S A2255 17:12:54.538 +64:03:51.46 0.0800 7.13 ± 1.38 4.08 ± 0.26 7.11 ± 0.33 6.74 ± 0.97 N/N A2589 23:23:56.772 +16:46:33.19 0.0416 3.03 ± 0.75 1.88 ± 0.12 2.54 ± 0.17 5.12 ± 0.56 Y/W Notes. The cluster mass, M500,M-Mgas, is derived from the M500-Mgas,500 relation and only used for comparison with the cluster mass, M500, derived from the “harmonic” velocity dispersion. “S,” “W,” and “N” denote strong cool-core, weak cool-core, and noncool-core clusters.Table 2
Dataset used for halo model (error intervals omitted).
Cluster β r c r 500 M 500 M g a s 500 M ∗ 500 M 200 r 200 M A Coma 0.654 246 1.278 6.55 8.42 13.14 13.84 2.3 12.86 A85 0.532 59.3 1.216 6.37 8.13 7.36 7.71 1.9 8.72 A400 0.534 110. 0.712 1.83 1.36 4.39 1.48 1.09 2.93 IIIZw54 0.887 206 0.731 1.91 1.45 4.57 2.81 1.35 4.51 A1367 0.695 274 0.893 1.76 2.07 4.35 4.06 1.53 5.77 MKW4 0.44 7.86 0.58 0.5 0.47 1.16 0.71 0.86 1.79 ZwCl215 0.819 308 1.098 4.93 6.1 7.05 10.37 2.09 10.65 A1650 0.704 201 1.087 3.44 5.09 7.47 11.14 2.15 11.11 A1795 0.596 55.7 1.118 3.41 5.11 6.21 10.99 2.14 11.04 MKW8 0.511 76.4 0.715 0.62 0.8 1.61 2.38 1.28 4.0 A2029 0.582 59.3 1.275 14.7 13.35 9.59 13.42 2.29 12.59 A2052 0.526 26.4 0.875 1.39 1.86 3.53 2.21 1.25 3.79 MKW3S 0.581 47 0.905 1.45 2.13 3.9 3.46 1.45 5.11 A2065 1.162 493 1.008 11.18 7.66 7.32 16.69 2.45 14.46 A2142 0.591 110 1.449 7.36 13.76 8.42 15.03 2.36 13.61 A2147 0.444 170. 1.064 4.44 5.04 6.84 3.46 1.45 5.15 A2199 0.655 99.2 0.957 2.69 2.97 4.76 4.80 1.62 6.37 A2255 0.797 423 1.072 7.13 7.11 6.74 13.32 2.27 12.52 A2589 0.596 84.3 0.848 3.03 2.54 5.12 3.58 1.47 5.24 Source:β, rc, M200, r200, and MA from [15]; r500 from [3]; M500, Mgas500, and M∗500 from [2]. Units.rc in kpc, r500, r200 in Mpc, M500 in 1014 M⊙, Mgas,500 in 1013 M⊙, and M∗,500 in 1012 M⊙.Parameters (β,rc) for the models of these galaxy clusters (as well as of many more) have been published earlier in [15, tab.  4.1]. This publication also contains mass data M200,Mgas,200 at r200 (with error intervals) and mass values MA,MgasA (without error interval) at the Abell radius, here defined as rA=2.14Mpc, but no data for star masses (evaluated for the value of Ho assumed in the later publications [2, 4],   h=0.7). In [15] the methods for determining the total mass and the gas mass were not yet as refined as in the later study. It is therefore not possible to aggregate the different datasets to one coherent ensemble (the values for M500 and Mgas,500 given here differ from the ones in [4] outside the error intervals). As no updated values for the parameters (β,rc) of the mass profiles were available to the present author, the values of [15] are here used asestimators for the form parameters of the β model.We considerM500,Mgas,500,M∗,500,andr500 from [2, 4] as ourcrucial mass data (including reference radii). Mass data at r200 (as well as r200 itself) from the older study are welcome asadditional information; but they will not be used as core criteria for the empirical test of our model. Table 2 collects the data used (for error intervals see the respective source, [2, 15]). It shows that the cluster ensemble covers an order of magnitude variation for the gas mass Mgas,500 and one and a half orders of magnitude variation in total mass M500. The selection method by intersecting the cluster sets of different raw data sources does not seem to be influenced by any particular bias. We thus may consider the collection as a reasonable dataset for testing our cluster halo model.


## 4.4. Adapting the WST Halo Model to the Data

For the construction of the model we have to work in Einstein gauge (9). Consistency with the deep MOND acceleration demands (51). Taken together these conditions fix the coefficients of the Lagrangian (1) and (28), independently of the convention chosen for ϕo. With a value for λ on the order of magnitude 10 (e.g., λ/4=6 as at the end of Section 2.3) the contribution of the LV4 term lies many orders of magnitude below the energy densities, the dominant term, in (35) and is negligible in our context.In agreement with Section3 the test of our halo model for each of the 19 galaxy clusters can now proceed as follows:(1)
Specification of theβ models (67) for gas and for star mass and parameters (β,rc) from [15], ρo determined by fitting to Mgas,500 and M∗,500 at r500, respectively [4].(2)
Determination of the Newton acceleration of the baryonic mass components due to (68).(3)
Calculation of the scalar field halo and the phantom halo of the baryonic mass by (69), and(66).(4)
Calculation of the scalar field halo of the system of freely falling galaxies (71).(5)
Aggregation of these to the total halo (75) and choice of fading out functions beyond r200 (see Appendix).(6)
Integration of the densities to the corresponding masses: scalar field energy of the galaxiesMsf2 of the gas mass Msf1, total scalar field halo Msf, net phantom energy of the baryonic mass Mph1, and finally the total transparent matter Mt and the lensing mass Mlens (76) and (77).(7)
Calculation of the error intervals of the model at selected distances (r500,r200).(8)
Comparison of the empirical value forM500 (resp., M200) with the model value Mt(r500) (resp., Mt(r200)).The results of this test are given in Section 4.6 Before we turn to the overall evaluation, we shall have a look at one cluster as an exemplary case.


## 4.5. The WST Halo Model with the Coma Cluster as Test Case

For a first check of our model we choose theComa cluster. According to item (1) of the last section we use the empirical input data from Table 2 (units given there):(80)β,rc,r500,Mgas,500,M∗,500=0.654,246,1.278,8.42,13.14,(rc in kpc, r500 in Mpc, Mgas,500 in 1013M⊙, and M∗,500 in 1012M⊙). The different components of the cluster halo integrate to masses (up to distance r) documented in Figure 1.Figure 1
Halo components of Coma cluster: transparent matter haloMt=Msf+Mph1, total scalar field (SF) halo Msf, SF halo of freely falling galaxies Msf2, and net phantom energy Mph1 (in barycentric rest system). Empirical data (violet dot, bar): galaxy and total mass (with error intervals) at r500=1280kpc.The scalar field halo (SF) of the total baryonic massρsf1 contributes the lion share to the total transparent energy. The SF of the galaxy system ρsf2 carries about as much energy as the net phantom component ρph1 in the barycentric rest system of the cluster. It surpasses the gas mass close to the reference radius r500. Figure 2 shows the fast increase of the gravitational mass of the scalar field halo of the galactic system.Figure 2
Comparison of the contribution of the scalar field halo of the galaxies with the baryonic mass for the Coma cluster (empirical dataMbar,500 violet dot).If we add all baryonic and halo contributions, the picture given in Figure3 emerges. It shows an encouraging agreement of the observed values M500 at r500=1278kpc with the prediction of the Weyl geometric halo model Mtot(r500) (in the range of the observational errors and of model errors):(81)Mtotr500=5.66+0.97-0.68×1014M⊙,M500=6.55+0.79-0.79×1014M⊙.Model error bars have been estimated by varying the input data (80) in their respective error intervals. The reconstruction of star mass from observational raw data is a particularly delicate point. Depending on the assumptions on the stellar dynamics and the resulting data evaluation model “one can obtain up to a factor of 2 fewer stars” [2, p.6]. For obtaining our model errors we allowed a variation in stellar masses by factors 0.5 and 2.Figure 3
Contribution of the baryonic massMbar, of the scalar field and phantom energies Msf,Mph1 to the transparent mass Mt and to the total mass Mtot of the Coma cluster in the WST model. Model errors indicated at r500 and r200 (black). Empirical data for Mbar (violet dot) and for total mass M500 with error bars (violet) from [2]. Additional empirical data at r200 (yellowish) from [15].At the Abell radiusrA=2.14Mpc (in the convention of [15]) we get the model values(82)MtotrA≈10.64×1014M⊙,MlensrA≈8.82×1014M⊙for the dynamical total mass and the lensing mass (77). Due to the net phantom energy the dynamical mass is about 20% higher than the lensing mass. This is an effect by which the model can, in principle, be tested empirically and discriminated from others. The baryonic masses in the β model are Mgas(rA)≈11.73×1013M⊙, M∗(rA)≈1.83×1013M⊙. Thus not only does the total mass Mtot(r500) given by our model agree with the empirical value M500 inside the error bounds but also MA and M200 are reasonably well recovered.This is the casewithout assuming any componentof particle dark matter besides the (real) energy of the scalar field and the (phantom) energy ascribed to the additional acceleration aφ induced by the Weylian scale connection. This might still be a coincidence. In order to learn more about the question whether the findings at the Coma cluster are exemplary or not, we have to consider the data of all 19 galaxy clusters, respectively, the 17 of the reliable subensemble.


## 4.6. Halos and Total Mass for 17(+2) Clusters of Galaxies

The mass values of the halo models for the 19 clusters are calculated as described in Section4.3 with the choice of fadeout functions beyond r=r200 (see Appendix). Theresults are documented in Tables 3 and 4 and Figures 4 and 5 (for Coma see Figure 3).Table 3
Empirical and model values for total mass atr500, r200.
Cluster r 500 M t o t ( r 500 ) M 500 r 200 M t o t ( r 200 ) M 200 Coma 1278 5.66 + 0.97 - 0.68 6.55 + 0.79 - 0.79 2300 14.38 + 2.71 - 1.90 13.84 + 1.49 - 1.41  A85 1216 4.95 + 0.62 - 0.43 6.37 + 1 - 1 1900 10.88 + 1.38 - 0.97 7.71 + 0.8 - 0.74  A400 712 1.38 + 0.3 - 0.19 1.83 + 0.39 - 0.39 1093 3.08 + 0.68 - 0.44 1.48 + 0.21 - 0.18  IIIZw54 731 1.45 + 0.38 - 0.28 1.91 + 0.58 - 0.58 1350 3.56 + 1.34 - 1.09 2.81 + 2.74 - 1.1  A1367 893 1.95 + 0.34 - 0.23 1.76 + 0.27 - 0.27 1529 4.67 + 0.93 - 0.62 4.06 + 0.45 - 0.4  MKW4 580 0.6 + 0.11 - 0.08 0.5 + 0.14 - 0.14 857 1.33 + 0.26 - 0.17 0.71 + 0.07 - 0.06  ZwCl215 1098 3.94 + 0.55 - 0.38 4.93 + 0.98 - 0.98 2093 9.8 + 1.64 - 1.17 10.37 + 3.51 - 2.62  A1650 1087 3.65 + 0.71 - 0.05 3.44 + 0.66 - 0.66 2150 10.48 + 2.91 - 2.41 11.14 + 5.77 - 3.46  A1795 1118 3.65 + 0.49 - 0.32 3.41 + 0.63 - 0.63 2136 10.84 + 1.47 - 0.99 10.99 + 2.26 - 2.09  MKW8 715 0.94 + 0.2 - 0.15 0.62 + 0.12 - 0.12 1279 2.80 + 0.74 - 0.6 2.38 + 1.04 - 0.59  A2029 1275 6.75 + 0.73 - 0.51 14.7 + 2.61 - 2.61 2286 17.9 + 2.01 - 1.42 13.42 + 2.43 - 2.26  A2052 875 1.78 + 0.31 - 0.21 1.39 + 0.28 - 0.28 1250 3.43 + 0.61 - 0.41 2.21 + 0.06 - 0.08  MKW3S 905 1.96 + 0.33 - 0.22 1.45 + 0.34 - 0.34 1450 4.5 + 0.78 - 0.52 3.46 + 0.36 - 0.34  A2065 1008 4.08 + 0.81 - 0.74 11.18 + 1.78 - 1.78 2450 10.83 + 5.06 - 3.34 16.69 + 21.34 - 6.73  A2142 1449 7.5 + 0.81 - 0.59 7.36 + 1.25 - 1.25 2364 16.83 + 1.89 - 1.39 15.03 + 3.9 - 2.64  A2147 1064 3.53 + 0.61 - 0.45 4.44 + 0.67 - 0.67 1450 6.39 + 1.26 - 0.99 3.46 + 1.17 - 0.74  A2199 957 2.43 + 0.44 - 0.32 2.69 + 0.42 - 0.42 1621 5.8 + 1.14 - 0.82 4.81 + 0.37 - 0.36  A2255 1072 4.13 + 0.53 - 0.37 7.13 + 1.38 - 1.38 2271 12.04 + 1.88 - 1.37 13.32 + 1.44 - 1.19  A2589 848 2.08 + 0.39 - 0.26 3.03 + 0.75 - 0.75 1471 5.44 + 1.06 - 0.73 3.58 + 3.86 - 1.54 Model valuesMtot(rN00) and empirical values MN00 in 1014 M⊙, rN00 (empirical) in kpc (N=2,5).Table 4
Model values for halo and baryonic masses atr200.
Cluster M t M s f M s f g a l M p h 1 M g a s M ∗ f ∗ f t Coma 12.32 9.91 2.66 2.42 1.78 0.278 0.16 6.9 A85 9.08 7.22 1.61 1.87 1.65 0.15 0.09 5.5 A400 2.7 2.21 0.73 0.49 0.29 0.093 0.32 9.4 IIIZw54 3.21 2.62 0.86 0.59 0.27 0.085 0.32 11.8 A1367 4.16 3.37 0.99 0.79 0.42 0.088 0.21 10. MKW4 1.2 0.97 0.3 0.22 0.1 0.0260 0.25 11.5 ZwCl215 8.54 6.82 1.66 1.72 1.14 0.131 0.12 7.5 A1650 9.16 7.36 1.94 1.81 1.15 0.169 0.15 8. A1795 9.39 7.51 1.86 1.88 1.29 0.157 0.12 7.3 MKW8 2.53 2.05 0.6 0.48 0.22 0.045 0.2 11.3 A2029 14.58 11.53 2.37 3.05 3.1 0.223 0.07 4.7 A2052 3.03 2.44 0.70 0.58 0.34 0.065 0.19 8.9 MKW3S 3.97 3.21 0.91 0.77 0.44 0.081 0.18 9. A2065 9.63 7.66 1.75 1.97 1.12 0.107 0.1 8.6 A2142 13.93 10.98 2.13 2.95 2.74 0.168 0.06 5.10 A2147 5.38 4.31 1.11 1.07 0.89 0.121 0.14 6. A2199 5.10 4.1 1.11 1. 0.61 0.098 0.16 8.4 A2255 10.4 8.27 1.88 2.13 1.51 0.143 0.09 6.9 A2589 4.74 3.83 1.11 0.91 0.59 0.119 0.2 8. Mass values in1014 M⊙, f∗=M∗/Mgas,ft=(Mt/Mgas)(r200); for r200 see Table 3.Figure 4
Halo models for clusters 2–9 in Table2: total mass Mtot (black line) with model error bars at r500,r200, transparent matter halo Mt constituted by scalar field halo Msf, and net phantom halo (in barycentric rest system of cluster) Mph1 and baryonic mass (gas and stars) Mbar. Empirical data for the total mass with error intervals at r500 (violet) from [2]. Additional empirical data at r200 (yellow) from [15].Figure 5
Halo models for clusters 10–19 in Table2. For description see Figure 4.Figure 6
Comparison of dark/transparent/phantom mass halos for Coma in NFW, WST, and TeVeS models, free parameters of halos for NFW and TeVeS (μ2 with neutrino core) adapted to mass data (black error bars) at r500=1280kpc. Red error bar indicates variability of NFW halo values at r200≈2300 kpc (fitting the NFW profile inside the margins of empirical error bars at r500); blue error bars give model errors of WST model at r500 and r200.For 15 clusters ourmodel reproduces the total mass at r500correctly, that is, inside the error margins of data and model. This is achieved without any further adjustable parameter, only on the basis of the parameters for the β-model for baryonic mass and Mgas(r500),M∗(r500). The fading out functions do not intervene below r200. Moreover, for the majority of these and paradoxically for all other four, the less precisely determined data at r200 have overlapping 1σ error intervals. This indicates a surprising agreement between the (theoretically derived) transparent halo and the empirically determined dark halo, Mt≈Mdm.For 4 clusters, MKW8, A2255, and outliers A2029 and A2065, the error intervals of empirical data and model datado not overlap. For the first two of them, MKW8 and A2255, the model predictions are consistent with the empirical data within doubled error intervals (2σ range). Only the two outliers (A2029, A2065) lie farther apart (A2029 has the surprising property that the empirical value for the total mass at r500surpasses the one at r200, M500>M200). Otherwise the model data are in good agreement with an assumption of normally distributed statistical errors and with the assumption that the evaluation bias due to the use of theNFW profile for dark matter (item (3) in Section 4.2)does not shift the mass estimates outside the error intervals.All in all, the assessment of the WST-3L halo model has surprisingly well passed, in spite of the main caveat of item(3) (Section 4.2). The outcome found for Coma seems to be typical also for the other clusters. Moreover, the good agreement of the model with the data of the reliable subensemble of 17 clusters supports the assumption stated in the last phrase (3) (Section 4.2) (no large systematic errors due to data transfer from Einstein/ΛCDM to the WST-3L framework). But we cannot exclude the possibility of cancelling between model errors and data transfer errors. Thus we have only found empirical support for theconjecture that, on the level of galaxy clusters, the observed dark matter effects encoded in Mdm may solely be due to the combined impact of the halo Msf of the scalar field and of the scale connection, Mph1: (83)Mdm≈Msf+Mph1=Mtcf.76.


## 4.7. Comparison with TeVeS and NFW Halos

It is surprising that in the WST-3L approach the total amount of observed dark matterMdm seems to be explained by the energy of the scalar field and the phantom halo, Mt=Msf+Mph1 and Mdm≈Mt. No missing mass is left. In usual relativistic MOND approaches this is not the case for clusters, although it is essentially so for galaxies [20]. Based on a study of about 40 galaxy clusters, Sanders has proposed the hypothesis of a neutrino component “between a few times 1013 and 1014M⊙,” mostly concentrated close to the center of the cluster, in a region up to twice the core radius, supplementing the baryonic mass and the phantom energy of the TeVeS model [5, p. 902]. In our approach, this hypothesis is unnecessary. Where does this difference arise from?A model calculation for the Coma cluster, evaluating (57) for the transition function μ2=x(1+x2)-1/2 (53) and the corresponding ν2 (54) used by Sanders in [5], shows that the neutrino core had to be tuned to Mν≈1.8×1014M⊙ in order to give agreement with the empirical value Mdm(r500)=M500-Mbar(r500)≈4.7×1014M⊙, where in this framework “r500” and “r200” are to be defined by a formal convention with regard to a (fictitious) NFW halo.Table4 shows the amount of scalar field energy up to a radius r≈r200 in the WST model. It varies between about 2×1013 and 9×1014M⊙, that is, roughly in the range found necessary by Sanders for the (hypothetical) neutrino halo. Moreover, a comparison of the transition functions μw(r) (55) and μ2(x) shows that the gross phantom energy ρt of the WST approach is larger than in an μ2-MOND model [1]. Roughly half of the missing mass of Sanders’ model is covered by this effect;the other half is due to the scalar field halo of the system of galaxies up to r200.A comparison of the two MOND-like approaches is given in Figure6. Here one has to keep in mind that the TeVeS-μ2 model has a free adaptable parameter (mass of the neutrino core), while the WST has not. The general profiles of the “dark matter” halos of both models are similar. The TeVeS-μ2 mass starts from a higher score because of its neutrino core; the WST transparent mass starts from a lower initial value but rises faster because of the increasing contribution of the scalar field halo of the galaxies.On the other hand, the best known profile for dark matter distribution, used in most structure formation simulations, is the NFW halo (Navarro/Frank/White). Its profile is(84)ρr=ρor/rc1+r/rc2,with density parameter ρo and core radius rc (at which the density has reduced to half the reference value). For a first comparison of the interior halos we take rc≈180h-1kpc (h=0.7), following [6] (in studies of the exterior halo of the Coma cluster, Geller et al. have found fit values 0.182+0.03-0.03,0.167+0.029-0.029,0.192+0.035-0.035 in units h-1Mpc for rc   [6]) and determine the central density parameter ρo such that the total integrated mass MNFW assumes the empirical value of [2] at r500=1289kpc (Figure 6). The error intervals for r500 give upper and lower model values for ρo and corresponding model error bars (red) for MNFW(r200). This version of the NFW halo satisfies one set of empirical data by construction (at r500); our main interest thus goes to the other empirical dataset available at r200=2300kpc. The NFW model error interval overlaps with the empirical error bar at r200 and with the error interval of the WST model.There is a conspicuous difference between the mass profiles of the NFW halo on one side and the halo profiles of WST orμ2-TeVeS on the other. More and precise empirical data of the interior halos of galaxy clusters ought to be able to discriminate between the two model classes. An empirical discrimination between the two MOND-like approaches would need more, and more precise, profile data. At the moment the Weyl geometric MOND-like model survives the comparison fairly well, even though it has no free parameter which would allow adapting it to the halo data.


## 4.8. A Side-Glance at the Bullet Cluster

At the end of this section let us shed a side-glance at the bullet cluster 1E0657-56. It is often claimed that the latter provides direct evidence in favour of particle dark matter and rules out alternative gravity approaches. Our considerations show that this argument is not compelling. The energy content of the scalar field halos of the colliding clusters endows them with inertia of their own. The shock of the colliding gas exerts dynamical forces on the gas masses only, not directly on the scalar field halos. During the encounter the halos will roughly follow the inertial trajectories of their respective clusters before collision, and they will continue to do so for a while. It will take time before a readaptation of the mass systems and the respective scalar field halos has taken place. Clearly the MOND approximation is unable to cover such violent dynamical processes. It describes only the relatively stable states before collision and in some distant future after collision. But a separation of halos and gas masses for a (cosmically “short”) period is to be expected, just as in the case of a particle halo with appropriate clustering properties.For the time being, cluster 1E0657-56 does not help to decide between the overarching alternative research strategies, particle dark matter, or alternative gravity. It may be able to do so, once the dynamics of gas and of the halos has been modeled with sufficient precision in both approaches. Then a proper comparison can be made. But that is an overtly complicated task. It seems more likely that other types of empirical evidence will offer a simpler path to a differential evaluation of the two strategies and help clarifying the alternative.


## 5. Discussion

We have analyzed a three-component halo model for clusters of galaxies, consisting of(i)
the scalar field energy induced by the overall baryonic matter in the barycentric rest system of a cluster (under abstraction of the discrete structure of the star matter clustering in freely falling galaxies) (69), with integrated mass equivalent Msf1,(ii)
an additional contribution to the scalar field energy, forming around the freely falling galaxies (71), integrating to Msf2,(iii)
the phantom energy of the total baryonic mass in the barycentric rest system of the cluster, due to the additional acceleration of the Weylian scale connection in Einstein gauge (70) with mass equivalent Mph1.The first two components add up to a real energy content of the scalar field with mass equivalent Msf=Msf1+Msf2; the third one, Mph1, arises from a theoretical attribution in a Newtonian perspective and has fictitious character. The mass equivalent of the integrated energy components combines to a total dark matter-like quantity Mt=Msf1+Msf2+Mph1 (73).All of the three components arise from gravitational effects of the cluster’s baryonic massMbar=Mgas+Mstar in the framework of a Weyl geometric scalar tensor theory of gravity (1), with its scale connection as the specific difference to Riemannian geometry (8), (4). In Weyl geometric scalar tensor theory the scalar field ϕ is the new dynamical variable, while thescale curvaturedφ=f vanishes. The latter would be even more striking and even irritating difference to Riemannian geometry (see [27]); in integrable Weyl geometry it plays no role. A second speciality of the theoretical framework is the cubic kinetic term of the Lagrangian (28), analogue to the AQUAL approach but in scale covariant form. Observable quantities are directly given by the model in Einstein gauge (9).The total dynamical mass of the model,Mtot=Mbar+Mt (76), has been heuristically confronted with the empirical values for of 17+2 galaxy clusters given in [2, 4], complemented by data from [15]. The problem of data transfer between different theoretical frameworks (in particular between Einstein gravity/ΛCDM and WST-3L) leads to a certain caveat with regard to uncorrected taking-over of the values for the total mass. But it does not seem to obstruct a meaningful first comparison of the WST halo model with available mass data of clusters collected in the Einstein/ΛCDM framework.The result of this comparison shows a surprisingly good agreement of the total mass predicted by the modelMtot, on the basis of data for the baryonic mass components, with the empirically determined total mass M500 (at the main reference distance r500). Moreover the model shows an acceptably good agreement with additional empirical values at the distance r200 given in [15] (determined on a slightly less refined data basis and evaluation method). For 15 clusters the model predicts values for Mtot(r500) with error intervals (due to the observational errors for the baryonic data) which overlap with the empirical error intervals of M500. The Coma cluster is among them. Two clusters have overlapping error intervals in the 2σ range. The remaining two are outliers and have been identified as such already in [4].In the result we have found empirical support forconjecturing that theobserved dark matter at galaxy cluster levelmay be due to the transparent halo of the scalar field and the phantom halo of the scale connection of WST (83).Details for the constitution of the total transparent matter halo from its specific components(i), (ii), and (iii) have been investigated for the Coma cluster (Section 4.5). They seem to be exemplary for the whole collection of galaxy clusters. A particular feature of the model is thescalar field energy formed in the interspaces between the galaxies. Its integrated energy contribution surpasses the baryonic mass between 1 and 1.5Mpc (see Figure 2 and Table 4, column 4). It is crucial for this model’s capacity to explain the total dynamic mass on purely gravitational grounds, without any additional dark matter component. An overall comparison with Sanders’ TeVeS-MOND model for galaxy clusters and the NFW halo is given in Section 4.7.At the moment the Weyl geometric scalar tensor model with a cubic term in the kinetic Lagrangian of the scalar field fares well in all the mentioned respects. It would be very helpful if astronomers decided to evaluate old or new raw data in the framework of WST. That could lead to empirical discrimination of the different models. But already independent of the outcome of such a revision, the scalar fieldϕ and the scale connection φμ of WST have a remarkable property from a theoretical point of view. They complement the classical Einstein-Riemannian expression for the gravitational structure, the metric field gμν, by a feature which carries proper energy momentum (35) and (36). Theenergy momentum of the scalar field plays a crucial role in the constitution of the transparent matter halo. In the present approach it seems to express theself-energy of the extended gravitational structure. It remains to be seen whether this is more than a model artefact.



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