A WST halo model for clusters of galaxies

Erhard Scholz∗

Dec 01, 2015

Abstract

A new model for the dark halos of galaxy clusters is introduced. Itis based on the Weyl geometric scalar tensor theory of gravity (WST)with MOND-like approximation in the weak field static limit, proposedin [21]. On this basis a three component halo model for galaxy clustersis derived. It is uniquely determined by the baryonic mass distributionof hot gas and stars. The model is tested against recent observa-tional data for 19 clusters of which 2 are outliers [25], [26]. Modulo acaveat resulting from different background theories (Einstein gravityplus ΛCDM versus WST), the total mass for 15 of the outlier reducedensemble of 17 clusters seems to be predicted correctly (in the senseof overlapping 1σ error intervals). For the other two doubled errorintervals (2σ) intersect.

ContentsIntroduction 2

1 Theoretical framework 41.1 Weyl geometric scalar tensor theory of gravity (WST) . . . . . . . . 41.2 The weak field static approximation . . . . . . . . . . . . . . . . . . 81.3 WST gravity with cubic kinematic Lagrangian (WST-3L) . . . . . . 91.4 A WST approach with MOND-like phenomenology . . . . . . . . . 121.5 Comparison with usual MOND theories . . . . . . . . . . . . . . . . 141.6 Short resumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Halo model for clusters of galaxies 182.1 Cluster models for baryonic mass (hot gas and stars) . . . . . . . . 182.2 Scalar field halo of ρbar in the cluster-barycentric MOND approxi-

mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Scalar field halos of galaxies in their respective galacto-centric MOND

approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 A three-component halo model for clusters of galaxies . . . . . . . . 20

∗University of Wuppertal, Department C, Mathematics, and Interdisciplinary Centrefor History and Philosophy of Science; scholz@math.uni-wuppertal.de

1

arX

iv:1

506.

0913

8v3

[as

tro-

ph.C

O]

1 D

ec 2

015

3 A first comparison with empirical data 223.1 Theory dependence of mass data for galaxy clusters . . . . . . . . . 223.2 Determination of mass values in Einstein gravity/ΛCDM . . . . . . 223.3 Observational data . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 The WST halo model with the Coma cluster as test case . . . . . . 263.5 Halos and total mass for 17(+2) clusters of galaxies . . . . . . . . . 303.6 Comparison with TeVeS and NFW halos . . . . . . . . . . . . . . . 363.7 A side-glance at the bullet cluster . . . . . . . . . . . . . . . . . . . 37

4 Discussion 38

5 Appendix 40

Bibliography 40

Introduction

In this paper we investigate the gravitational dynamics of galaxy clustersfrom the point of view of Weyl geometric scalar tensor theory of gravity(WST) with a non-quadratic kinematic Lagrange term for the scalar field(3L), introduced in [21]. To make the paper as self-contained as possible,it starts with an outline of WST-3L (section 1). WST-3L has two (inho-mogeneous) centrally symmetric static weak field approximations: (i) theSchwarzschild-de Sitter solution with its well known Newtonian approxima-tion, which is valid if the scalar field and the WST-typical scale connectionplays a negligible role; (ii) a MOND-like approximation which is appropriateunder the constraints that the scale connection cannot be ignored but is stillsmall enough to allow for a Newtonian weak field limit of the (generalized)Einstein equation. The acceleration in the MOND approximation consistsof a Newton term and an additional acceleration of which three quarters aredue to the energy density of the scalar field and one quarter to the scaleconnection typical for Weyl geometric gravity.

In centrally symmetric constellations the scalar field energy forms a haloabout the baryonic mass concentrations. An additional phantom halo maybe ascribed to the the acceleration due to the scale connection, if the latteris expressed by a fictitious mass in Newtonian terms. The scalar field haloconsists of true energy derived from the energy-momentum tensor; it is inde-pendent of the reference system, as long as one restricts the consideration toreference systems with low (non-relativistic) relative velocities. The phan-tom halo is a symbolical construct and valid only in the chosen referencesystem (and scale gauge). For galaxy clusters we find two components of thescalar field halo, one deriving from the total baryonic mass in the MONDapproximation of the barycentric rest system of the cluster (component 1),and one arising from the superposition of all the scalar field halos formingaround each single galaxy in the MOND approximation of the latter’s rest

2

system (component 2). Because velocities of the galaxies with regard to thecluster barycenter are small (non-relativistic) and also the respective energydensities are small, the two components can be superimposed additively (lin-ear approximation). With regard to the barycentric rest system of the clustera three component halo for clusters of galaxies arises, two components beingdue to the scalar field energy and one purely phantom (section 2).

The two component scalar field halo is a distinctive feature of the Weylgeometric scalar tensor approach; it is not present in the usual MOND ap-proaches. One may pose the question whether it, and its specific transitionfunction µw(x) (predicting more “phantom” energy in the MOND sense thanthe usually assumed transition function µ2(x), or others), suffice for explain-ing the deviation of the cluster dynamics from the Newtonian expectationwithout additional dark matter. If we call the totality of the three com-ponents the transparent halo of the cluster (section 2.4), the question iswhether the (theoretically derived) transparent halo can explain the darkhalo of galaxy clusters, observationally determined in the framework of Ein-stein gravity and ΛCDM .

In section 3 we confront our model with empirical data. The transparenthalo of the present approach is derived from the baryonic content of thegalaxy cluster. Because component 2 of the scalar field is added up fromthe external halos of the individual galaxies, it is much more sensitive tothe amount of star matter than other approaches. We therefore use recentdata on the total mass (dark plus baryonic), hot gas, and the star matterfor 19 galaxy clusters, which have been determined from different raw datasources and are thus more precise than earlier ones [25], [23], [26]. 2 of the19 clusters show a surprisingly large relation of total mass to gas mass. Theyare separated as outliers from the rest of the ensemble already by the authorsof the study; so do we. 17 non-outlying clusters remain as our core referenceensemble.

In the mentioned studies total mass, gas mass and star mass are deter-mined on the background of Einstein gravity plus ΛCDM . That raises theproblem of data compatibility with the WST framework. It is discussed in3.1, 3.2 and leads to a certain caveat with regard to the observational valuesfor the total mass (M200, M500) and the reference distances r200, r500 to thecluster centers. But it does not seem to obstruct the possibility for a firstempirical check of our model (section 3.2). More refined studies are welcome.They have to use the WST framework for evaluating the observational rawdata or, at least, to analyze the transfer problem of mass data from oneframework to the other in more detail.

For the profile of the baryonic content we use a β-model with parametersrc, β given in [16] for the 19 clusters in addition to the mass data of [25],[23], [26]. Moreover, mass data at r200 from [16] are taken into account asan additional information, not as a core criterion (section 3.3).

For 15 of the 17 main reference clusters the observational and the theo-

3

retical values for the total mass agree in the sense of overlapping 1σ errorintervals. The remaining two overlap in the 2σ range. The two outliers ofthe original study do not lead to overlapping intervals even in the 4σ range(section 3.5). In the present approach the dynamics of the Coma cluster isexplained without assuming a component of particle dark matter. It is beingdiscussed in more detail than the other clusters in section 3.4.

A short comparison with the halos of R. Sanders’ µ2-MOND model withan additional neutrino core [18], and with the NFW halo [12] is given forComa (section 3.6). The paper is rounded off by a short remark on the bulletcluster (section 3.7) and a final discussion (section 4).

1 Theoretical framework

1.1 Weyl geometric scalar tensor theory of gravity (WST)

Among the family of scalar tensor theories of gravity the best known onesand closest to Einstein gravity are those with a Langrangian containing amodified Hilbert term coupled to a scalar field φ. Their Lagrangian has thegeneral form

L =1

2(ξφ)2R+ Lkin −

λ

4φ4 . . . (1)

L = L√|g| , |g| = |det g| .

Here g is an abbreviation for a 4-dimensional pseudo-Riemannian metricg = (gµν) of signature (−+++). φ is a real valued scalar field on spacetime,Lkin its kinetic term, ξ a constant coefficient, and the dots indicate matterand interaction terms. Under conformal rescaling of the metric,

gµν 7→ g′µν = Ω2 gµν (Ω a positive real valued function), (2)

the scalar field changes with weight −1, i.e. φ 7→ φ′ = Ω−1 φ. So far thisis similar to the well known Jordan-Brans-Dicke (JBD) scalar-tensor theoryof gravity.1 But here we work in a scalar-tensor theory in the framework ofWeyl’s generalization of Riemannian geometry.2

Crucial for the Weyl geometric scalar tensor approach (WST) is that thescalar curvature R and all dynamical terms involving covariant derivativesare expressed in Weyl geometric scale covariant form.3 The Lagrangian den-sity L is invariant under conformal rescaling for any value of the coefficient

1[13], [7], [10], [9].2See e.g., [2, 14, 15, 20, 21].3Fields X are scale covariant if they transform under rescaling by X 7→ X = ΩwX,

with w ∈ Q (in most cases even in Z); w is called the weight of X. Covariant derivativesof scale covariant fields are defined such that the result of covariant derivation DµX isagain scale covariant of the same weight w as X.

4

ξ2 of the modified Hilbert term, not only for ξ2 = 16 (because of the weight

−4 for L in spacetime dimension 4). For matter and interaction terms of thestandard model of elementary particles, scale invariance is naturally ensuredby the coupling to the Higgs field which has the same rescaling behaviouras the gravitational scalar field. For classical matter we expect that a betterunderstanding of the quantum to classical transition, e.g. by the decoher-ence approach, allows to consider scale invariant Lagrangian densities also.For the time being we introduce the scale invariance of matter terms in theLagrangian as a postulate. In our context its most important consequence isthe scale covariance of the Hilbert energy momentum tensor

Tµν = − 2√|g|

δLmδgµν

, (3)

which is of weight w(Tµν) = −2. That is consistent with dimensional con-siderations on a phenomenological level.4 It has been shown that the matterLagrangian of quantum matter (Dirac field, Klein Gordon field) is consistentwith test particle motion along geodesics (autoparallels) γ(τ) of the affineconnection, if the underlying Weyl geometry is integrable (see below, equ.(7)) [2]. For classical matter we assume the same.5

For the sake of consistency under rescaling we consider scale covariantgeodesics γ(τ) with scale gauge dependent parametrizations of the geodesiccurves of weight w(γ) = −1:

uλ + Γλµνuµuν − ϕµuµuλ = 0 , uµ = γµ (4)

Here the affine connection contains a ϕ-dependent term in addition to thewell-known Levi-Civita connection ϕΓ

λµν derived from gµν (see below, equ

(9). The last term on the l.h.s. of (4) takes care for the scale dependentparametrization.6

For any gauge of the Weylian metric and the scalar field, (g, ϕ, φ), anytimelike geodesic has a generalized eigentime parametrization with gµνuµuν =−1, where uµ = γµ. Happily, in the reparametrization from τ to eigentime tthe ϕµ-term on the l.h.s. of (4) cancels. This results in a differential equationwhich is formally identical to the one in Einstein gravity, but with derivativesin the sense of the Weyl geometric affine connection. Written in coordinates

4Energy density ρ has the physical dimension [ρ] = EL−3, corresponding to the scaleweight w(ρ) = −4. (Remember that we work here with geometrical/scale weights, notwith the energy weights of high energy physics with the inverse sign convention.) Thatis consistent with the weight of Tµν , the symbolic machine giving the energy-momentumdensity vector pµ = TµνX

ν for an observer with timelike tangent vector Xν of its world-line, ρX = T oνX

ν , w(Tµν) = gµλTλν = −4.5Probably it can be proven similar to Einstein gravity.6We thus work with a projective family of paths.

5

(xo = t, x1, x2, x3) the geodesic equation becomes:7

d2xj

dt2= −Γj00 + Γ0

00

dxj

dt− 2Γj0i

dxi

dt− Γjik

dxi

dt

dxk

dt(5)

+2Γ00i

dxj

dt

dxi

dt+ Γ0

ik

dxj

dt

dxi

dt

dxk

dt.

In the low velocity, weak field regime the equation of motion reduces tothe form well known from Einstein gravity d2xj

dt2= −Γj00. But here the Γj00

(j = 1, 2, 3) are the coefficients of the Weyl geometric affine connection (see(9)).

We do not want to heap up too many technical details; more can be foundin the literature given in fn. 2. But we have to mention that aWeylian metriccan be given by an equivalence class of pairs (gµν , ϕ) consisting of a pseudo-Riemannian metric gµν , the Riemannian component of the Weyl metric, anda differentiable one-form ϕ, the scale connection (or, in Weyl’s original termi-nology, the “length connection”). In coordinates it has the form ϕµdx

µ andis often called the Weyl covector, or even Weyl vector field. More precisely ϕdenotes a connection with values in the Lie algebra of the scale group (R+, ·);therefore it can be considered, locally, as a differentiable 1-form. The equiva-lence is given by rescaling the Riemannian component of the Weylian metricaccording to (2), while ϕ has the peculiar gauge transformation behaviourof a connection, rather than that of an ordinary vector (or covector) field ina representation space of the scale group:

ϕµ 7→ ϕ′µ = ϕµ −∂µΩ

Ω(6)

or, shorter, ϕ′ = ϕ− d log Ω.There are good physical reasons to constrain the scale connection to the

integrable case with a closed differentiable form dϕ = 0, i.e. ∂µϕν = ∂νϕµ.8

Then ϕ is a gradient (at least locally) and may be given by

ϕµ = −∂µω . (7)

This constraint is part of the defining properties of WST. Then it is possibleto “integrate the scale connection away”. Having done so the Weylian metricis characterized by its Riemannian component gµν only (and a vanishingscale connection ϕµ = 0). By obvious reasons we call this the Riemanngauge.9

The choice of a representative gµν also fixes ϕµ; both together definea scale gauge of the Weylian metric. Conformal rescaling of the metric is

7For the classical (Riemann-Einstein) case see, e.g. [22, eq. (9.1.2)], for WST [19, p.39, equ. (60)].

8[20, sec. 4.2]9It is the analogue of the choice of Jordan frame in JBD theory.

6

accompanied by the gauge transformation of the scale connection (6). Froma mathematical point of view all the scale gauges are on an equal footing, andthe physical content of a WST model can be extracted, in principle, from anyscale gauge. One only needs to form a proportion with the appropriate powerof the scalar field.10 From the physical point of view there are, however,two particularly outstanding scale gauges. Of special importance besidesRiemann gauge is the gauge in which the scalar field is scaled to a constantφo (scalar field gauge). For the particular choice of the constant value suchthat

(ξφo)2 = (8πG)−1 = E2

pl , (8)

with the Newton gravitational constant G, this gauge is called Einstein gauge(Epl the reduced Planck energy).11 In this gauge the metrical quantities(scalar, vector or tensor components) of physical fields are directly expressedby the corresponding field or field component of the mathematical model(without the necessity of forming proportions).

A Weylian metric has a uniquely determined compatible affine connec-tion Γ; in physical terms it characterizes the inertio-gravitational guidingfield. It can be additively composed by the Levi-Civita connection gΓ of theRiemannian component g of any gauge (g, ϕ) and an additional expressionϕΓ in the scale connection, in short Γ = gΓ +ϕΓ with

ϕΓµνλ = δµνϕλ + δµλϕν − gνλϕµ. (9)

By definition Γ reduces to gΓ in Riemann gauge. Thus, in this gauge, theguiding field is given by the ordinary expression of the Levi-Civita connec-tion. On the other hand, in Einstein gauge the measuring behaviour of clocksare most immediately represented by the metric field; and also other physicalobservables are most directly expressed by the field values in this scale. Thenthe expression of the gravitational field and with it the expressions for ac-celerations contain contributions from the Weylian scale connection. Thus aspecific dynamical difference to Einstein gravity and Riemannian geometry,as well as to JBD theory, arises even in the case of WST with its integrableWeyl geometry.

Writing the scalar field φ in Riemann gauge (g, 0) in exponential form,φ = eω, turns its exponent

ω := ln φ (10)

into a scale invariant expression for the scalar field. (Further below, weshall omit the tilde sign, if the context makes clear that the scale invariantexponent is meant.) The scale connection ϕ = ϕ in scalar field gauge is then

ϕ = −dω , (11)10[20, sec. 4.6]11It is the analogue of Einstein frame in JBD theory.

7

because Ω = φ is the rescaling function from Riemann gauge to scalar fieldgauge. For more details see [1, 6, 14, 15, 21].

1.2 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 covariantgeodesics. It is governed by a differential equation formally identical to theone in Einstein gravity (5). Here we want to consider the weak field staticcase for low velocities in order to study the dynamics of stars in galaxies andgalaxies in clusters. For studies of the gas dynamics and its modificationin our framework the velocity dependent terms of (5) have to be taken intoaccount. This is not being done here.

Analogous to Einstein gravity, the coordinate acceleration a for a lowvelocity motion x(τ) in eigentime parametrization is given by

aj =d2xj

dτ2≈ −Γjoo . (12)

According to (9) the total acceleration decomposes into

aj = −gΓjoo −ϕΓjνλ = ajR + ajϕ (13)

(j = 1, 2, 3 indices of the spacelike coordinates), where ajR = −gΓjoo is the Rie-mannian component of the acceleration known from Einstein gravity. Clearlyajϕ = −ϕΓjoo represents an additional acceleration due to the Weylian scaleconnection. For a diagonal Riemannian metric g = diag (goo, . . . , g33) thegeneral expression (9) simplifies to −ϕΓjoo = −gooϕj . General considerationson observable quantities and consistency with Einstein gravity show that,in order to confront it with empirically measurable quantities, we have totake its expression in Einstein gauge if we want to avoid additional rescalingcalculations.12

For a (diagonalized) weak field approximation in Einstein gauge,

gµν = ηµν + hµν , |hµν | 1 , (14)

with η = εsig diag(−1,+1,+1,+1), the Riemannian component of the accel-eration is the same as in Einstein gravity. Its leading term (neglecting 2-ndorder terms in h) is,

ajR = gΓjoo ≈

1

2ηjj∂jhoo (no summation over j) .

In the limit, ΦN := −12hoo behaves like a Newtonian potential

aR ≈ −∇ΦN , (15)12Cf. [20, sec. 4.6].

8

where ∇ is understood to operate in the 3 spacelike coordinate space withEuclidean coefficients as the leading term of the metric.

In Einstein gauge the Weylian scale connection ϕµ arises from Riemanngauge by rescaling with Ω = eω, ϕµ = −∂µω (6), and ajϕ ≈ ϕj . In otherwords, the scale connection term of the acceleration is generated by the scaleinvariant representative ω of the scalar field as its potential:

ajϕ ≈ −∇ω (16)

If we compare with Newton gravity, a (fictitious) mass density

ρph = (4πG)−1∇2ω (17)

would lead to the same acceleration effect. In the terminology of the MONDliterature the acceleration due to the Weylian scale connection correspondsto a phantom energy density ρph. We see that already on the general level thedynamics of WST differs from Einstein gravity. Only for trivial scalar field,ω = const, the usual Newton limit is recovered, otherwise it is modified. Weshall explore how this modification relates to the usual MOND approaches.

1.3 WST gravity with cubic kinematic Lagrangian (WST-3L)

The most common form of the kinetic term for the scalar field is the oneof a Klein-Gordon field, Lφ2 = −α

2DνφDνφ, quadratic in the norm of the

(scale covariant) gradient.13 Different from this approach, we show that aWeyl geometric scalar tensor theory of gravity leads to a MOND-like phe-nomenology if the kinetic term is cubic rather than quadratic in the gradientof the scalar field or, even better, if a cubic term is added to a quadratic termwith coefficent α = 6ξ2, known from conformal coupling in Riemanniangeometry.14 This is similar to an observation made in the first relativisticattempt of a MOND theory of gravity proposed in [4], later called relativisticAQUAL (the acronym of “a-quadratic Lagrangian”). The crucial differenceto this early approach is the scale covariant reformulation in the frameworkof Weyl geometry, resulting in a different behaviour of the scalar field energydensity.15 Bekenstein/Milgrom’s model relied crucially on implementing anunspecified transition function between the Newton and the deep MONDregime into the kinetic term. The constraint of scale invariant Lagrangiandensity reduces the underdetermination of the Lagrangian drastically andsuggests a different form of the kinetic term which shows some resemblenceto the one of the relativistic AQUAL theory. For non-timelike ∇ω it can bewritten in scale covariant form as

Lφ3 =2

3ξ2η φ |∇ω|3 , (18)

13In WST Dν denotes the a scale covariant derivative of φ, Dνφ = ∂νφ− φϕν .14Cf. improved postscript of [21] in v4 of http:// arxiv.org/abs/1412.0430 .15For a review in the wider context of different relativistic MOND approaches see [3].

9

where we have used the abbreviation

|∇ω| := |∂ν ω ∂ν ω|12 (19)

(| . . . | absolute value) and assume φ to be positive.16 η denotes a constantcoefficient responsible for the relative strength of the cubic kinetic term, ω isthe scale invariant representative of the scalar field introduced in (10), and∇ its gradient.17

To the total kinetic Lagrangian of the scalar field,

Lφ = Lφ2 + Lφ3 , (20)

a potential term is added. It must be of order 4 to provide for scale invarianceof the density:

LV 4 = −λ4φ4 (21)

We introduce the constant ao defined in Einstein gauge (g, ϕ) with con-stant scalar field φ =: φo,

ao := η−1φo . (22)

Then the cubic term of the kinetic Lagrangian in Einstein gauge reads

Lφ3.= −2

3(8πG ao)

−1|∇ω|3 . (23)

The dotted equality sign “ .=”, indicates that the respective equation is notscale invariant but presupposes a special gauge made clear by the context,here, as in most cases in this paper, Einstein gauge (similar for ≈). ao has thedimension of inverse length/time and will play a role analogous to the MONDacceleration ao ≈ [c]H, where H is the Hubble parameter (at “present”), cthe velocity of light. Coefficients of type [c] will often be suppressed in thefollowing general considerations and plugged in only in the final empiricalmodel. Below we shall see that for ao = ao

16 ≈H100 the WST model with cubic

kinematic Lagrangian (WST-3L) acquires a MOND-like phenomenology ina weak gravitational field in which the scalar field and the scale connectioncannot be neglected.

A reasonable choice of adaptable parameters brings ξ and η to nearbyorders of magnitude, η = β ξ, with

β = ξ−1η , β ∼ 100 . (24)

On the other hand, because of (8) and (22) the product of both coefficientsis a “large number” in the sense of η · ξ =

Eplao

= a−1oLpl∼ 1063.

16For timelike ∇ω see [21]. Here we exclusively deal with the spacelike (or zero) case.17Note that w(φ) = −1, w(‖∇ω‖ = −1 imply the scale weight w(Lkin) = −4, as it must

be for scale invariance of Lkin.

10

Variation of the Lagrangian leads to the dynamical equations of WST,the Einstein equation and the scalar field equation. The scale invariantEinstein equation is18

Ric− R

2g = (ξφ)−2T (m) + Θ , (25)

where g denotes the whole collection of metrical coefficients and T (m) theenergy tensor of matter (3). The scalar field contributes to the total energymomentum with two terms, Θ = Θ(I) + Θ(II), the first of which is propor-tional to the metric (thus formally similar to a vacuum energy tensor):

Θ(I) = φ−2(−DλD

λφ2 + ξ−2(LV 4 + Lφ))g (26)

Θ(II)µν = φ−2

(DµDνφ

2 − 2ξ−2∂Lφ∂gµν

)(27)

Varying with regard to φ gives the scalar field equation. It is simplifiedby subtracting the trace of the Einstein equation.19 In Einstein gauge (withRiemannian component of the metric g) it can be written in terms of thecovariant derivative g∇ with regard to g (Levi-Civita connection in Einsteingauge):

g∇ν(|∇ω|∂νω).= 4πG ao tr T

(m) (28)

If we introduce the corresponding Riemannian covariant operator

gM ω = g∇ν(|∇ω|∂νω) = (∂ν |∇ω| ∂νω + |∇ω| g∇ν ∂νω) , (29)

the scalar field equation for a fluid with matter density ρm and pressure pmsimplifies to

gM ω.= 4πG ao (ρm − 3pm) . (30)

We have to complement it with the Einstein equation in Einstein gauge

Ric− R

2g.= 8πGT (m) + Θ , (31)

In the static weak field static case ω does not depend on the time coordi-nate. Moreover with gµν ≈ ηµν , the expression ∇ν(|∇ω|∂νω) turns into thenonlinear Laplace operator ∇ · (|∇ω|∇ω) of MOND theory with Euclideanscalar product · and norm | . . . |. In vacuum a trivial scalar field, ω = const,is a basic solution. Then WST reduces to Einstein gravity. By obviousreasons (29) will be called the covariant Milgrom operator.

In particular, the Schwarzschild and the Schwarzschild-de Sitter solutionsof Einstein gravity are special (degenerate) solutions of WST-3L equations

18This means that not only the equation but all its constitutive (additive) terms arescale invariant. In particular the Ricci tensor Ric of Weyl geometry is scale invariantbecause the affine connection of the Weyl metric is.

19[21, pp. 15f., sec. 7.2, postprint version arXive v4]

11

for λ4 = 0 or λ

4 ≈ 6, respectively. In fact, they solve (31), (30) for φ .= const

in Riemann gauge, i.e. in the case of Einstein gauge equal to Riemann gauge(g, ϕ) = (g, 0). The Riemannian component of the metric (g = g =: g) isgiven by

ds2 = −(1− 2M

r−κ r2)dt2 + (1− 2M

r−κ r2)−1dr2 + r2(dx22 + sin2 x2 dx3)

2 .

(32)Then Ric− R

2 g = −3κ g and Θ = Θ(I) = −λ4β

2ao2 g. Therefore the Einstein

equation is satisfied for 3κ = λ4β

2 ao2, i.e. κ ≈ 2H2 for λ

4 ≈ 6 and β ≈ 100.We see that in the case of a negligible Weylian scale connection the

classical (non-homogeneous) point-symmetric solutions of Einstein gravityare valid also for the dynamics of WST. This implies that Newton dynamicsis an effective approximation for point symmetric solutions of WST in thecase of a negligible scale connection (in Einstein gauge). In the next sectionwe investigate the alternative case of a non-negligible scale connection.

1.4 A WST approach with MOND-like phenomenology

If the conditions for the weak field approximation (14) are given, it is possibleto speak of the MOND regime as the region in which the Newton accelerationaN is smaller than ao (here aN can be identified with aR in (15)). Then thescalar field equation (30) reduces, in reliable approximation, to

∇ · (|∇ω|∇ω).= −4πG ao tr T

(m) . (33)

We call this the MOND approximation. For pressure-less matter with energydensity ρm we get

∇ · (|∇ω|∇ω).= 4πG ao ρm . (34)

That is similar to the AQUAL approach, but without the transition functiontypical for the usual MOND theories.20 Note that only the trace of thematterenergy momentum tensor, not of the scalar field energy density, appears onthe r.h.s. of (33).

Straight forward verification shows that, independent of symmetry con-ditions, a solution of (34) is given by ω with a gradient ∇ω = −aϕ suchthat

aϕ =

√ao|aN |

aN =√ao|aN |

aN|aN |

, (35)

where aN denotes the corresponding Newton acceleration to the given massdensity,

∇2ΦN = 4πGρm aN = −∇ΦN (36)

(calculations in the approximating Euclidean space with norm | . . . |). Thesolution of the non-linear Poisson equation (34) is much simpler than one

20 [4], [3].

12

might expect at a first glance: In a first step the linear Poisson equation ofthe Newton theory is to be solved, then an algebraic transformation of type(35) leads to the acceleration due to the solution of the non-linear partialdifferential equation (34). In fact, aϕ has the form of the deep MONDacceleration of the ordinary MOND theory (but with a different constantao). (35).21

This raises the question of the Newtonian limit. (35) implies |aϕ| |aN |in regions where |aN | ao (> ao). Therefore aϕ can effectively be neglectedin the case of ‘large’ values of |aN | derived from (36). Then, according tothe observation at the end of section 1.3, the Newton approximation is alsoreliable in WST gravity. This is true irrespective of the answer to the moresubtle question whether there is a smooth transition between the MONDand the Newton approximations and, if so, by which transition function itcan be characterized. Here we shall consider the MOND approximation inan “upper transition” regime only, where roughly |aN | ≤ 102ao.22

For centrally symmetric mass distributions ρ(r) with mass M(r) inte-grated up to r (where r = |y| denotes the Euclidean distance from the sym-metry center, y = (y1, y2, y3) the coordinates of the approximating Euclideanspace) this implies

aϕ = −∇ω ≈ −√GM(r) ao

y

|y|2, |aϕ| =

√GM(r) ao

r. (37)

But this is only the most immediate modification of Newton gravity.There is also the additional term in (31) of the energy density due to thescalar field, ρsf = (8πG)−1Θoo. It modifies the r.h.s. of the Newton limit ofEinstein gravity.23

Neglecting contributions at the order of magnitude of cosmological terms(∼ H ) the energy density of the scalar field in Einstein gauge simplifies to24

ρsf ≈ (4πG)−1(∇2ω + Γjjk∂kω) , (38)

where latin indices j, k . . . refer to space coordinates only.In the central symmetric case with Euclidean metric in spherical coordi-

nates25 and a mass function M(r) with M ′(r) = 0 for r > ro (for some dis-

tance ro), we find from (37) ∇2ω =

√aoGM(r)

r2and Γjjk∂

kω = 2r

√aoGM(r)

r =

21In the terminology of the MOND community, the MOND approximation of WST-3Lbehaves like a special case of a QMOND theory [8, pp. 46ff.].

22One might speak of the upper transition regime for ao ≤ |aN | ≤ 100 ao, of the MONDregime if |aN | ≤ ao and of the deep MOND regime for, let us say, |aN | ≤ 10−2ao [21, sec.7.3]. We don’t claim knowledge on the “lower” transition regime with 100 ao < |aN | butnot yet ao |aN | (however may be specified).

23In contrast ρsf does not enter the r.h.s. of the scalar field equation (30), and thereforedoes not enter the r.h.s of (36).

24[21, sec. 4.3]25For ds2 = dr2 + r2(dθ2 + sin2 θ dϑ2) the crucial affine connection compoinents are

Γ111 = 0, Γ2

21 = Γ331 = r−1.

13

2∇2ω, thusρsf ≈ (4πG)−13∇2ω . (39)

That is three times the value of the phantom energy density corresponding tothe acceleration of the scale connection (17). The total “anomalous” additiveacceleration (in comparison to Newton gravity) is therefore

aadd = aϕ + asf = 4aϕ . (40)

In the central symmetric case

|aadd| = 4

√GM(r) ao

r. (41)

For consistency with the deep MOND acceleration ao we have to set

ao =ao16≈ 10−2H [c] ≈ 6 · 10−10 cms−2 . (42)

Because of (35) the total acceleration a is then

a = aN + aadd = aN

(1 +

√ao|aN |

), |aadd| =

√ao|aN | . (43)

1.5 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 pointmass is unusual. M. Milgrom rather considered a multiplicative relationbetween the MOND acceleration a and the Newton acceleration aN by akind of ‘dielectric analogy’,

aN = µ(a

ao) a , with µ(x) −→

1 for x→∞x for x→ 0 ,

(44)

or the other way round26

a = ν(aNao

) aN , with ν(y) −→

1 for y →∞y−

12 for y → 0 .

(45)

From this point of view our acceleration (43) is specified by a well definedtransition functions

µw(x) = 1 +1−√

1 + 4x

2xand νw(y) = 1 + y−

12 . (46)

One has to keep in mind, however, that our transition functions µ, ν arereliable only in the MOND regime and the upper transitional regime (roughly

26Here µ(x) → x means µ(x) − x = O(x), i.e. µ(x)−xx

remains bounded for x → 0. Cf.[8, 51f.]

14

aN ≤ 102ao). They cannot be used for discussing the Newtonian limit.27 Itwill be important to see how they behave in the light of empirical data, inparticular galactic rotation curves and cluster dynamics.

In the MOND literature the amount of a (hypothetical) mass which inNewton dynamics would produce the same effects as the respective MONDcorrection aadd is called phantom mass Mph. For any member of the MONDfamily the additional acceleration can be expressed by the modified transitionfunction ν = ν − 1 with ν like in (45)

aadd = ν

(|aN |ao

)aN . (47)

The phantom mass density ρph attributed to the the potential Φph satisfies4πGρph = ∇2Φph and ∇Φph = −aadd. A short calculation shows that itmay be expressed as

ρph = ν

(|aN |ao

)ρm + (4πGao)

−1ν ′(|aN |ao

)(∇|aN |) · aN . (48)

It consists of a contribution proportional to ρm with factor ν, which dom-inates in regions of ordinary matter, and a term derived from the gradientof |aN | dominating in the “vacuum” (where however scalar field energy ispresent). For the Weyl geometric model with νw(y) = y−

12 , ν ′w(y) = −1

2y− 3

2

ρph turns into:

ρt =

(ao|aN |

) 12

ρm + (8πG)−1(ao|aN |

)∇(|aN |) · aN (49)

ρsf =3

4ρt , ρph =

1

4ρt (50)

The first expression of (50) is compatible with (38).In our case it would be wrong to consider the whole of ρt as “phantom

energy”. Three quarters of it are due to the scalar field energy density,the scalar field halo ρsf , and express a true energy density. This energydensity appears on the right hand side of the Einstein equation (31) andthe Newtonian Poisson equation as its weak field, static limit. It is decisivefor lensing effects of the additional acceleration. Only one quarter, ρph,is phantom, i.e. a fictitious mass density producing the same accelerationeffects as the Weylian scale connection (17). Only for the sake of comparisonwith other MOND models we may speak of ρt as some kind of gross phantomenergy.

We have to distinguish between the influence of the additional structure,scalar field and scale connection, on light rays and on (low velocity) trajec-tories of mass particles. Bending of light rays is influenced by the scalar field

27See fn. 22 and the text above it.

15

halo only, the acceleration of massive particles with velocities far below c bythe the scalar field halo and the scale connection.28

Also in another respect our theory differs from usual MOND approaches.External acceleration fields of a system under consideration generate con-siderable headache for the latter. In GR, on the other hand, Einstein’sequivalence principle allows to “transform away” acceleration fields in freelyfalling systems if they are small enough (relative to the inhomogeneities ofthe external gravitational field) to neglect tidal forces. Therefore a MONDapproximation (33) in WST of a freely falling (small) system does not “feel”the external acceleration field, just like in GR. The external accelerationproblem does not arise in WST MOND.

This has another important consequence: the scalar field energy formedaround a freely falling subsystem of a larger gravitating system, calculated inthe MOND approximation of the freely falling subsystem, contributes to ther.h.s. of the Einstein equation of any other subsystem (in relative motion)and also to that of a superordinate larger system.29 This has to be takeninto account for modeling the dynamics of clusters of galaxies.

1.6 Short resumé

We have derived the most salient features of the Weyl geometric MONDapproximation (WST MOND) and are prepared for a comparison with ob-servational data. Before we do so, it may be worthwhile to collect the resultswhich are necessary for applying it to real constellations in a short survey.

Consider a gravitating system which in the Newton approximation ofEinstein gravity is described by the baryonic matter density ρm, the accel-eration am and potential Φm with

∇2Φm = 4πGρm , am = −∇Φm. (51)

The modification due to WST MOND leads to an additional accelerationaadd with the following features:

(i) The total acceleration a is a = am + aadd with

a = am

(1 +

(ao|am|

)− 12

)= ν

(|am|ao

)am , (52)

where ν(y) = 1 + y−12 . For am ao the Newton approximation

applies. (52) holds for |am| ≤ 102ao only (“upper” transition regime).28That may look like bad news for explaining lensing at clusters and microlensing at

substructures. But the particular transition function seems to compensate much of thiseffect.

29In principle that presupposes that the whole energy momentum tensor (26, 27) (andits system dependent representation) is considered. For slow motions and weak field ap-proximation a superposition of energy densities like in Newton dynamics seems legitimate.

16

No information can be drawn from it for |am| larger but not yet ao(the “lower” transition regime).

(ii) The “reciprocal” transformation function defined by am = µ(|a||am|

)a

is

µ(x) = 1 +1−√

1 + 4x

2x. (53)

(iii) aadd consists of two components aadd = aϕ + asf = 4aϕ. The first oneis derived from a potential ω satisfying the non-linear Poisson equation

∇ · (|∇ω|∇ω) =π

4aoGρm , aϕ = −∇ω . (54)

(iv) The second one, asf , can be understood as a Newton acceleration dueto the energy density ρsf of a scalar field (part of the modified gravita-tional structure). Its potential satisfies a Newtonian Poisson equation.It satisfies

−∇asf = 4πGρsf (55)

with energy density

ρsf =3

4

(ao|am|

) 12(ρm + (8πG)−1∇(|am|) ·

am|am|

)(56)

ρsf is part of the energy-momentum tensor of the scalar field φ and inthis sense “real” rather than phantom.

(v) aϕ is formally derivable in Newton dynamics from a fictitious energydensity

ρph =1

3ρsf . (57)

ρph is the net phantom energy of WST MOND. For comparison withother 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 itis real). It is transparent rather than “dark” (see (65) below).

(vi) (i) – (v) are reliable approximations also for small (local) gravitatingsystems freely falling in a larger gravitating system, if am / 102 ao inthe local system. The subsystem can be considered as “small” withrespect to the super-system, if tidal forces of the super-system canbe neglected. In hierarchical systems like galaxy clusters the energydensity contributions ρsf of the subsystems and the super-system (cal-culated in different reference coordinate systems) add up to the totalenergy density of the scalar field, if the velocities of the subsystemsrelative to the barycenter of the super-system are small.30 If one likes,

30This is a crucial difference between WST MOND and ordinary MOND theories.

17

ρsf can be considered as the “dark matter” component of WST MOND;although as the energy density of the scalar field it is not constitutedby the usual (hypothetical) quantum particles (WIMPs, axions etc.).To demarcate this difference it might better be called transparent mat-ter/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 inagreement with observations.

2 Halo model for clusters of galaxies

2.1 Cluster models for baryonic mass (hot gas and stars)

The density profile of hot gas and (smeared) star/galaxy matter in a (galaxy)cluster may be described by the (centrally symmetric) profile of the β-modelof the astronomical literature:31

ρ(r) = ρo

(1 +

(r

rc

)2)− 3

2β

(58)

β is the ratio of the specific energies of the galaxies and the gas, ρo the centraldensity and rc is the core radius.32 For any cluster one may form densitymodels for the gas mass ρgas(r) and for the galaxy mass ρstar(r) with thesame form parameters β and rc.

In this way we work with an idealized model using proportional densityprofiles for the hot gas and for the galaxies with parameters determinedfrom the gas, and central densities adapted to mass/distance observations.The central densities ρo can, in principle, be determined from mass data forgas Mgas(r1), respectively stars Mstar(r1), at a given distance r1. Empiricaldetermination of Mgas(r1) and Mstar(r1) is a subtle question which will bediscussed in section 3.2.

Large scale gravitational effects on the cluster level may be modelled inthe Newton approximation of Einstein gravity with baryonic matter and adark matter halo, or in a MOND limit of the most well known relativisticMOND theory TeVeS. The latter has to assume a much smaller amount ofunseen matter in addition to the baryonic one [18]. Here we want to explorethe feasibility of the WST approach, in particular regarding the questionof how much unseen matter has to be added to the gravitational effectsof the model in order to reproduce (“predict”) the observed accelerations(respectively their measurable effects).

31[17, 18, 16].32rc is the distance from the cluster center at which the projected galaxy density is half

of the central density ρo.

18

2.2 Scalar field halo of ρbar in the cluster-barycentric MOND approx-imation

We start by determining the scalar field halo and the phantom halo of thetotal mean baryonic mass, gas plus stars/galaxies, in a continuity modelρbar = ρgas + ρstar according to (58). In this first approximation the starmass is approximated on a par with the hot gas, i.e. described by its con-tinuously smeared out mean density as an additive component to the gas.But stars are agglomerated in galaxies which form freely falling subsystemswith considerable inter-spaces in the super-system (the cluster). These sub-systems form scalar field halos of their own which are suppressed in the firstapproximation. In a second step we therefore add a component to the scalarfield halo which, approximately, fills in this lacuna.

The MOND approximation of the idealized total baryonic mass has to becalculated in a cluster-barycentric static reference system with a weak fieldapproximation to Euclidean/Minkowski space (14). With the model (58) forρbar(r) the baryonic mass up to radius r,

Mbar(r) = 4π

∫ r

0ρbar(u)u2du , (59)

determines the Newton acceleration abar = GM(r)r2

due to mean distributionof total baryonic mass. The densities of the scalar field halo and the phantomhalo of WST MOND follow from (56), (57)

ρsf 1 =3

4

(ao|abar|

) 12(ρm + (8πG)−1∇(|abar|) ·

abar|abar|

), (60)

ρph 1 =1

3ρsf 1 . (61)

The respective masses of the halos Msf 1, Mph 1 arise from integration.

2.3 Scalar field halos of galaxies in their respective galacto-centricMOND approximations

The galaxy-component in the calculation above does not make allowancefor the fact that the star matter forms a discrete structure of an ensembleof galaxies each of which is falling freely in the inertio-gravitational field ofthe super-system (hot gas and other galaxies). Every galaxy possesses alocal MOND approximation with regard to its own barycentric static ref-erence system. The acceleration abar of the super-system (with respect tothe barycenter rest system of the hot gas) is transformed away in each ofthe local MOND approximations. The latter leads to a galactic scalar fieldhalo which persists under changes of reference systems with small, i.e. non-relativistic, relative velocities. It contributes to the total energy of the scalar

19

field calculated in the cluster barycentric system.33 Equs. (60), (61) cannot,and do not, take this effect into account. In principle, we have to add upall these effects to a scalar field energy density ρsf 2 in order to fill in thislacunae.

An exact calculation would have to solve a highly non-trivial N -bodyproblem for the motion of the galaxies. But, luckily, the experience withthe calculation of MOND halos of galaxies allows us to expect that a secondcontinuity model for the system of galaxies alone (omitting the gas mass),gives already an acceptable approximation for ρsf 2.34 Using (56) again weget

ρsf 2 =3

4

(ao|astar|

) 12(ρstar + (8πG)−1∇(|astar|) ·

astar|astar|

), (62)

with |astar(r)| = GMstar(r)r2

and Mstar(r) the integral analogous to (59) forthe star density ρstar.

2.4 A three-component halo model for clusters of galaxies

We thus arrive at a three-component halo model for galaxy clusters. Inaddition to the Newtonian gravitational effects of the baryonic mass density

ρbar = ρgas + ρstar , (63)

modeled by the β-model, the WST MOND approach predicts accelerationsgenerated by scalar field halo of the gas and the galaxies (60), (62),35

ρsf ≈ ρsf 1 + ρsf 2 . (64)

In addition there arises another acceleration aϕ due to the scale connectionin the barycentric rest system of the cluster (36). It may be associated to afictitious (net) phantom halo ρph 1 of the baryonic matter

ρph 1 =1

3ρsf 1

like in (61). The net phantom energies of the freely falling galaxies do not sur-vive the transformation to the cluster rest system. In the cluster-barycentricrest system ρph 1 is the only component of phantom energy ascribable to theacceleration of the scale connection.

33Of course this is not the case for the phantom halo of the single galaxies.34The gas mass has to be omitted here, because its gravitational potential does not

enter the local MOND approximation.35This presupposes small values of the scalar field energies, enabling us to use a lin-

ear superposition of the contributions. The two energy densities can be related to thestatic cluster barycentric reference system because the galaxies move slowly (i.e., non-relativistically with regard to the barycenter).

20

In the usual MOND theories there is no scalar field energy; all additionaleffects with regard to Newton dynamics may be ascribed to a (fictitious)phantom energy density. Phantom energy densities of single galaxies do notsurvive the transformation to the cluster barycentric system. In MOND thereis therefore no analogy to ρsf 2; the latter is the crucial distinctive featurebetween the approaches. For a comparison of WST-3L and usual MONDapproaches with regard to galaxy clusters it is not sufficient to evaluate thedifference between the transformation functions µ(x) (46) alone.

For an even wider comparison with other approaches it may be useful toadd up the scalar field and phantom halos to a kind of “dark matter” halo.However, one must not forget that in the present model there is no darkmatter in the ordinary sense but only a transparent halo made up of the(real) energy density of the scalar field and the (fictitious) phantom energydensity ascribed to the acceleration effects of the scale connection in Einsteingauge (and with respect to the cluster barycentric rest system):

ρt = ρsf + ρph 1 (65)

From the gravitational lensing point of view, it would be more appropriateto consider ρsf alone as the WST equivalent of a dark matter halo. Howevereven 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 quantumpoint of view, the scalar field has to be quantized if one wants to search fora (bosonic) particle content of ρt.

The total dynamical mass of the model (up to some distance r from thecenter of the cluster) is now

Mtot = Mbar +Mt, Mt = Msf +Mph 1 (66)

with Mbar = Mgas +Mstar. The lensing mass is a bit smaller,

Mlens = Mbar +Msf . (67)

Mathematically, the integral of the scalar field energy density to arbitrarydistances diverges. But like in the dark matter approach a virial radius of acluster can be defined, which roughly delimits the gravitational binding zoneof the cluster. Comparing ρt (respectively (ρsf )) with the critical energydensity ρcrit of the universe, one sets, e.g., r200 such that

ρt(r200) ≈ 200 ρcrit (68)

Not far beyond the gravitational binding zone of the cluster, the energydensity will have fallen to such a small amount that its centrally symmet-ric component is inconceivably stronger than the density fluctuations in theinter-cluster space. To continue the integration into this region, and beyond,has no physical meaning. In the long range the energy density of the scalar

21

field approaches the cosmic mean energy value. A physical limit of integra-tion has to be chosen close to the virial radius beyond which the gravitationalbinding structure of the cluster is fading out.36

3 A first comparison with empirical data

3.1 Theory dependence of mass data for galaxy clusters

The determination of mass data for galaxy clusters is quite involved. Massdensities are not directly observable; they are indirectly inferred from ob-servable quantities like mass-velocity dispersion relations of stellar systems(presupposing some gravitational theory), luminosities (presupposing a stel-lar mass-to-light ratio that depends on the mass of the cluster), X-ray data(usually, although not always, presupposing hydrostatical equilibrium), orgravitational lensing (usually presupposing Einstein gravity). As a result,mass data of galaxy clusters are theory dependent, and even inside the samebackground theory they depend on choices of models and methods of evalu-ation.

That makes it a difficult task to compare our model with empirical data.A fine-grained judgement presupposes an evaluation of observational rawdata on the background of WST gravity or, at least, a detailed estimationof systematic errors resulting from a comparison of different backgroundtheories (Einstein gravity with ΛCDM and Newton approximation, or al-ternatively TeVeS-MOND, in comparison with WST and its MOND approx-imation). This task has to be left to astronomers, if they become sufficientlyinterested in the present approach. But, taking this caveat in mind, it stillseems possible to confront available data from, e.g., the Einstein gravity-ΛCDM -Newton approximation framework with our model, in order to geta first impression of its potential usefulness. A comparison with mass dataderived in a TeVeS-MOND background would give welcome supplementaryinformation. This is not attempted here.

3.2 Determination of mass values in Einstein gravity/ΛCDM

Here we refer to the mass data given in recent studies [25], [26] based onEinstein gravity with ΛCDM . These studies contain new data on the baryoncontent and the total gravitational mass (as it appears in classical gravity)

36Alternatively, or perhaps even better, one may model the fading out of the centrallysymmetric energy densities (63) and (65) by factors (fading out functions) which fallsmoothly from 1 to close to 0 (see end of section 3.3, item (5), and appendix). In theend, we have to distinguish between the halos internal to the cluster (up to virial radiusrvir), which are completely determined by the baryonic energy distribution in the cluster,and the external halo for r > rvir which is subject to field and matter disturbances of thewider region and can only be approximated after choosing a fading out function.

22

for 19 clusters of galaxies.37 We discuss the possibility and the problems ofa confrontation of these data with the MOND-approximation of WST.

(1) In Einstein gravity/ΛCDM the cluster mass can, in principle, be esti-mated from the velocity dispersion σ of galaxies at distance R (from thecenter) by an estimator derived from the virial theoremM ≈ G−1σ2R.The additional acceleration of WST-3L aadd = aϕ+asf (40) is dynam-ically indistinguishable from the effects of “true” Newtonian masses.So far it seems as if the estimation of total mass can be transferred tothe MOND approximation of WST without problems.

(2) But if the radius R does not include the “whole” cluster mass (howeverdefined) a surface pressure term must be taken into account, whichcomplicates the case. In standard gravity the necessary correctionis usually implemented by a modified cubic mass estimator M ∼ σ3

given in [5, eq. (2)]. Its calculation presupposes a dark matter halowith Navarro-Frenk-White (NFW) profile. An estimator of this typeis being used by our reference study [25] for determining M200 at R =r200 from optical spectroscopic data collected in [24]. The authorsreconstruct M500 and r500 from these values using again the NFWprofile (with parameters determined in the last step). Because of thedifferent profile for the scalar field halo of WST this is a critical stepfor our exercise. But if the resulting systematic errors are smaller thanthe error intervals of Mtot (66), implied by the observational errors ofthe other quantities, they do not disturb a rough empirical check ofthe model.

(3) The mass of the hot gas (intracluster medium) Mgas 500 (at r500) hasbeen determined in the mentioned study from X-ray data obtained byXMM-Newton and ROSAT. The temperature of the gas is estimated bya fit to the measured spectrum. The gas density ρ(r) is reconstructed,using model assumptions, from intensity observables and then inte-grated up to r500.38

(4) Several methods for determining the stellar massM∗ 500 are mentionedin [26]. In this study the star mass is gained from optical imaging datadue to SDSS 7 in two steps. First the total luminosity of the cluster isdetermined by means of a “galaxy luminosity function” (GLF), then themass is estimated using mass-to-light ratios depending on the clustermass. In the last step models of the star development in the respectivegalaxy, elliptical or spiral, enter. They depend on assumptions on an“initial mass function” (IMF). Two possibilities for the IMF (Salpeter

37I thank T. Reiprich for his hint at the mentioned observational studies.38In this evaluation the hydrostatic assumption was corrected by taking the velocity

dispersion into account [26, sec. 2.2f.].

23

versus Kroupa) are considered and compared in [25], [26]. Accordingto the authors the difference of the stellar mass estimate can result ina factor 2 [25, p.4 ]. Another approach would be to estimate stellarmasses of the individual galaxies and “to construct the stellar massfunctions in order to sum the stellar masses” (ibid, 1).39 All in all,the estimate of the star mass concentrated in galaxies depends moreon models of galactic star evolution than on the background gravitytheory. In spite of that the precision cannot be expected to be betterthan by a factor 2 (respectively 0.5).

(5) Finally the dependency of the data evaluation on the background cos-mology has to be taken into account. The data of the 19 clusters usedin the following have redshift z < 0.1. The geometrical and dynami-cal corrections implied by the ΛCDM cosmology are correspondinglysmall. An evaluation in, e.g., a Lemaitre-de Sitter model (or even anon-expanding Weyl geometric model with redshift)40 would affect thedata only by a minor expansion of the error intervals.

The points (1), (3) and (4), in particular the estimate of stellar mass concen-trated in galaxies and gas mass, are fairly insensitive against a change of thebackground theory from Einstein gravity to WST.41 The estimate for thetotal masses at r500 and r200 is the critical point for our purpose (item (2)).However, if the difference of the halo profiles between the scalar field energydensity of WST and NFW dark matter does not push the estimates for thetotal massesM500, M200 outside the error intervals of our halo model (due toobservational input data), we may still be able to draw first inferences fromthe following evaluation. The same can be said with regard to item (5).

3.3 Observational data

The studies [25], [26]. contain new data on the baryon content and the totalgravitational mass for 19 clusters of galaxies (as it appears in an Einstein–ΛCDM framework with Newton approximation).42 As described, indepen-dent raw data sources have been used to determine the different observationalparameters of the cluster. This has improved the precision of the data incomparison with earlier studies. The ensemble of 19 clusters is the intersec-tion of all cluster sets appearing in the different raw data sources used bythe authors ([24], XMM-Newton, ROSAT, SDSS DR7). The mass data are

39Moreover, an additional component of star matter can be associated to the intraclusterlight.

40Cf. [21, section 4.2]41The 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.42[26] corrects a methodological inconsistency in the first paper. In the following, refer-

ence is made to the corrected data.

24

given in columns (5), (7), (8) of [26, tab. 1]. It is reproduced in our fig. 1.The values for r500 are published in [23, tab. 1, col. (5)] (here table 1).

A comparison of the total cluster masses derived from the velocity dis-persion with a mass estimate derived from the gas mass shows that thetwo clusters A2029 and A2065 are outliers, with total cluster masses con-siderably higher than the corresponding gas masses would let expect. Theauthors therefore separate the two outliers from the rest of the data, withthe remaining 17 clusters as a reliable data set [25, p. 3]. We shall do thesame.

A&A 544, C3 (2012)DOI: 10.1051/0004-6361/201116803ec© ESO 2012

Astronomy&

Astrophysics

Star-formation efficiency and metal enrichment of the intraclustermedium in local massive clusters of galaxies

(Corrigendum)

Y.-Y. Zhang1,2, T. F. Laganá3,1, D. Pierini4,, E. Puchwein5,6, P. Schneider1, and T. H. Reiprich1

1 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germanye-mail: yyzhang@astro.uni-bonn.de

2 National Astronomical Observatories, Chinese Academy of Sciences, 100012 Beijing, PR China3 Universidade de São Paulo, Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Departamento de Astronomia,

Rua do Matão 1226, Cidade Universitária, CEP: 05508-090, São Paulo, SP, Brasil4 Max-Planck-Institut für extraterrestrische Physik, Giessenbachstraße, 85748 Garching, Germany5 Heidelberger Institut für Theoretische Studien, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany6 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85741 Garching, Germany

A&A, 535, A78 (2011), DOI: 10.1051/0004-6361/201116803

Key words. cosmology: observations – galaxies: clusters: general – methods: data analysis – surveys – X-rays: galaxies: clusters –errata, addenda

Table 1. Properties of the 19 galaxy clusters.

Name X-ray center (J2000) Redshift M500 M500,M−Mgas Mgas,500 M∗,500 UndisturbedRA Dec 1014 M 1014 M 1013 M 1012 M /cool core

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/SA0400 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/NIIIZw54 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/WA1367 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/NMKW4 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/SZwCl1215 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/NA1650 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/WComa 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/NA1795 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/SMKW8 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/NA2029 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/SA2052 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/SMKW3S 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/SA2065 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/WA2142 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/WA2147 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/NA2199 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/SA2255 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/NA2589 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, derivedfrom the “harmonic” velocity dispersion. “S”, “W”, and “N” denote strong cool-core, weak cool-core, and noncool-core clusters.

In our original paper, we inadvertently tabulated the stellarmasses from a Kroupa (2001) initial mass function in Table 1,which are not consistent with the remaining content of the paper.We note that this does not affect any of our results. Nevertheless,we regret this error and provide the actual stellar masses inTable 1 from a Salpeter (1955) initial mass function.

Guest astronomer at the MPE.

Acknowledgements. The authors would like to thank Anthony Gonzalez forpointing out this error in our manuscript.

ReferencesKroupa, P. 2001, MNRAS, 322, 231Salpeter, E. E. 1955, ApJ, 121, 161

Article published by EDP Sciences C3, page 1 of 1

Figure 1: Tab.1 of [26]. Properties of the 19 galaxy clusters

Parameters (β, rc) for the models of these galaxy clusters (as well asof many more) have been published earlier by one of the authors [16, tab.4.1]. This publication also contains mass data M200, Mgas 200 at r200 (witherror intervals), mass valuesMA, MgasA (without error interval) at the Abellradius, here defined as rA = 2.14Mpc, but no data for star masses.43 In [16]the methods for determining the total mass and the gas mass were not yetas refined as in the later study. It is therefore not possible to aggregatethe different data sets to one coherent ensemble.44 On the other hand, noupdated values for the parameters (β, rc) of the mass profiles are available.We therefore use the parameter values of [16] as estimators for the formparameters of the β model and adapt the central densities ρo to the valuesof gas mass Mgas500 in [26]; similarly for the star mass.

43Evaluated for the value of Ho assumed in the later publications [25], [26], h = 0.7.44The values for M500 and Mgas500 given here differ from the ones in [25] far outside

the error intervals.

25

Table 1. Data set used for halo model (error intervals omitted)

Cluster β rc r500 M500 Mgas 500 M∗ 500 M200 r200 MA

Coma 0.654 246 1.278 6.55 8.42 13.14 13.84 2.3 12.86A85 0.532 59.3 1.216 6.37 8.13 7.36 7.71 1.9 8.72A400 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 4.51A1367 0.695 274 0.893 1.76 2.07 4.35 4.06 1.53 5.77MKW4 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 10.65A1650 0.704 201 1.087 3.44 5.09 7.47 11.14 2.15 11.11A1795 0.596 55.7 1.118 3.41 5.11 6.21 10.99 2.14 11.04MKW8 0.511 76.4 0.715 0.62 0.8 1.61 2.38 1.28 4.0A2029 0.582 59.3 1.275 14.7 13.35 9.59 13.42 2.29 12.59A2052 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 5.11A2065 1.162 493 1.008 11.18 7.66 7.32 16.69 2.45 14.46A2142 0.591 110 1.449 7.36 13.76 8.42 15.03 2.36 13.61A2147 0.444 170. 1.064 4.44 5.04 6.84 3.46 1.45 5.15A2199 0.655 99.2 0.957 2.69 2.97 4.76 4.80 1.62 6.37A2255 0.797 423 1.072 7.13 7.11 6.74 13.32 2.27 12.52A2589 0.596 84.3 0.848 3.03 2.54 5.12 3.58 1.47 5.24

Source: first and last block [16], middle block [26], r500 [23]Units: rc in kpc, r500, r200 in Mpc, M500 in 1014 M, Mgas 500 in 1013 M,M∗ 500 in 1012 M,

We considerM500, Mgas 500, M∗ 500, r500 from [25, 26] as our crucial massdata (including reference radii). Mass data at r200 (as well as r200 itself) fromthe older study are welcome as additional information; but they will not beused as core criteria for the empirical test of our model.

Table 1 collects the data used.45 It shows that the cluster ensemble coversan order of magnitude variation for the gas massMgas 500 and one and a halforders of magnitude variation in total mass M500. The selection method byintersecting the cluster sets of different raw data sources does not seem tobe influenced by any particular bias. We thus may consider the collection asa reasonable data set for testing our cluster halo model.

3.4 The WST halo model with the Coma cluster as test case

For the construction of the model we have to work in Einstein gauge (8).Consistency with the deep MOND acceleration demands (42). Togetherthese conditions fix the coefficients of the Lagrangian (1) and (18), indepen-

45For error intervals see the respective source, [26], [16].

26

dently of the convention chosen for φo.46 With a value for λ on the order ofmagnitude 10 (e.g., λ4 = 6 like at the end of section 1.3) the contribution ofthe LV 4 term lies many orders of magnitude below the energy densities thedominant term in (26) and is negligible in our context.

In agreement with section 2 the test of our halo model for each of the 19galaxy clusters can now proceed as follows:

(1) Specification of the β models (58) for gas and for star mass; parameters(β, rc) from [16], ρo determined by fitting toMgas 500 andM∗ 500 at r500respectively [26].47

(2) Determination of the Newton accelerations of the baryonic mass com-ponents.

(3) Calculation of the scalar field halo and the phantom halo of the bary-onic mass (60), (57).

(4) Calculation of the scalar field halo of the system of freely falling galaxies(62).

(5) Aggregation of these to the total halo (65); choice of fading out func-tions beyond r200 (see appendix).

(6) Integration of the densitities to the corresponding masses: scalar fieldenergy of the galaxies Msf 2, of the gas mass Msf 1, total scalar fieldhalo Msf , net phantom energy of the baryonic mass Mph 1, finally thetotal transparent matter Mt and the lensing mass Mlens (66), (67).

(7) Calculation of the error intervals of the model at selected distances(r500, r200).

For a first check of our model we choose the Coma cluster. Accordingto item (1) of the last section we use the empirical input data from table 1(units given there):

(β, rc, r500,Mgas 500,M∗ 500) = (0.654, 246, 1.278, 8.42, 13.14) , (69)

(rc in kpc, r500 in Mpc, Mgas 500 in 1013M, M∗ 500 in 1012M). The dif-ferent components of the cluster halo integrate to masses (up to distance r)documented in fig. 2.

The scalar field halo (SF) of the total baryonic mass ρsf 1 contributes thelion share to the total transparent energy. The SF of the galaxy system ρsf 2

46For example the convention given in (24).47If one wants to investigate the external halo (beyond r200) one has to choose a cutoff at

r200 or fading out functions. The main part of of our investigation deals with the internalhalo; only in the final discussion, section 4, questions of the external halo come into theplay. This remark applies, mutatis mutandis, to items (3), (4) below

27

1000 1500 2000 2500r in kpc

2×1014

4×1014

6×1014

8×1014

1×1015

Coma M(r) in (1M)

M_t

M_sf

M_sf,bar

M_sf,gal

M_ph

Figure 2: Halo components of Coma cluster: transparent matter halo Mt =Msf+Mph 1, total scalar field (SF) haloMsf , SF of baryonic mass in barycen-tric rest system Msf 1, SF halo of freely falling galaxies Msf 2, net phantomenergy Mph 1 (in barycentric rest system).Observational data: total mass (with error intervals) at r500 = 1280 kpc.

carries about as much energy as the net phantom component ρph 1 in thebarycentric rest system of the cluster. It surpasses the gas mass close to thereference radius r500. Figure 3 shows the fast increase of the gravitationalmass of the scalar field halo of the galactic system.

If we add all baryonic and halo contributions, the picture given in fig.4 emerges. It shows an encouraging agreement of the observed values M500

at r500 = 1278 kpc with the prediction of the Weyl geometric halo modelMtot(r500) (in the range of the observational errors and of model errors).

Mtot(r500) = 5.66+0.97−0.68 , M500 = 6.55

+0.79−0.79 × 1014M (70)

Model error bars have been estimated by varying the input data (69) in theirrespective error intervals:48

At the Abell radius rA = 2.14Mpc (in the convention of [16]) we get themodel values

Mtot(rA) ≈ 10.64 , Mlens(rA) ≈ 8.82 × 1014M (71)

for the dynamical total mass and the lensing mass (67).49 Due to the netphantom energy the dynamical mass is about 20 % higher than the lensing

48The reconstruction of star mass from observational raw data is a particularly delicatepoint. Depending on the assumptions on the stellar dynamics and the resulting dataevaluation model “one can obtain up to a factor of 2 fewer stars” [26, p. 6]. For obtainingour model errors we allowed a variation in stellar masses by factors 0.5 and 2.

49The total (dynamical) transparent matter contribution is Mt(rA) = 9.30, of whichMsf 2(rA) ≈ 2.01 are due to the scalar field halo of the galaxy system. The net phantomenergy amounts to Mph 1(rA) ≈ 1.82 (all values in units of 1014M).

28

500 1000 1500 2000r in kpc

5.0×1013

1.0×1014

1.5×1014

ComaM(r) in (1M)

M_sf,gal

M_bar

M_gas

M_star

Figure 3: Comparison of the contribution of the scalar field halo of thegalaxies with the baryonic mass for the Coma cluster (observational dataMbar 500, Mgas 500, M∗ 500 violet dots).

1000 1500 2000 2500r in kpc

2.0×1014

4.0×1014

6.0×1014

8.0×1014

1.0×1015

1.2×1015

1.4×1015

1.6×1015Coma M(r) in (1M)

M_tot

M_t

M_sf

M_ph

M_bar

Figure 4: Contribution of the baryonic mass Mbar, of the scalar field andphantom energies Msf , Mph 1 to the transparent mass Mt and to the totalmass Mtot of the Coma cluster in the WST model. Model errors indicatedat r500, r200 (black). Empirical data for Mbar (violet dot) and for total massM500 with error bars (violet) from [26]. Additional empirical data at r200(yellowish) from [16].

29

mass. This is an effect by which the model can, in principle, be testedempirically and discriminated from others.50 The baryonic masses in the βmodel areMgas(rA) ≈ 11.73×1013M,M∗(rA) ≈ 1.83×1013M. Thus notonly the total mass Mtot(r500) given by our model agrees with the empiricalvalue M500 inside the error bounds; also MA and M200 are reasonably wellrecovered.

This is the case without assuming any component of particle dark matterbesides the (real) energy of the scalar field and the (phantom) energy ascribedto the additional acceleration aϕ induced by the Weylian scale connection.But that might be a coincidence. In order to learn more about the questionwhether the findings at the Coma cluster are exemplary or not, we have toconsider the data of all 19 galaxy clusters, respectively the 17 of the reliablesub-ensemble.

3.5 Halos and total mass for 17(+2) clusters of galaxies

The mass values of the halo models for the 19 clusters are calculated asdescribed in section 3.3 with the choice of fadeout functions beyond r = r200(see appendix). The results are documented in tables 2, 3 and figs. 5, 6 (forComa see fig. 4).

For 15 clusters our model reproduces the total mass at r500 correctly,i.e. inside the error margins of data and model. This is achieved withoutany further adjustable parameter, only on the basis of the parameters forthe β-model for baryonic mass and Mgas(r500),M∗(r500).51 Moreover, forthe majority of these, and paradoxically for all other four, the less preciselydetermined data at r200 have overlapping 1σ error intervals. This indicatesa surprising agreement between the (theoretically derived) transparent haloand the observationally determined dark halo, Mt ≈Mdm.

For 4 clusters, MKW8, A2255 and the outliers A2029, A2065, the errorintervals of observational data and model data do not overlap. For the firsttwo of them, MKW8 and A2255, the model predictions are consistent withthe observational data within doubled error intervals (2σ range). Only thetwo outliers (A2029, A2065) lie farther apart.52 Otherwise the model dataare in good agreement with an assumption of normally distributed statisticalerrors and with the assumption that the evaluation bias due to the use ofthe NFW profile for dark matter (item (2) in section 3.2) does not shift themass estimates outside the error intervals.

50The lensing data of [11] seem consistent with such an observation, Mlens(r1) =

6.1+12.1−3.5 × 1014M at r1 = 2.5

+0.8−0.5 Mpc, compared with MA ≈ 12.9 × 1014M at

rA ≈ 2.14Mpc in [16]. But because of their exorbitant large error intervals these data arefar from significant for our question.

51The fading out functions do not intervene below r200.52A2029 has the surprising property that the observational values for the total mass at

r500 surpasses the one at r200, M500 > M200.

30

All in all, the assessment of the WST-3L halo model has surprisinglywell passed, in spite of the main caveat of item (2), section 3.2. The outcomefound for Coma seems to be typical also for the other clusters. Moreover,the good agreement of the model with the data of the reliable sub-ensembleof 17 clusters supports the assumption stated in the last phrase (2), section3.2 (no large systematic errors due to data transfer from Einstein/ΛCDMto the WST-3L framework). But we cannot exclude the possibility of can-celling between model errors and data transfer errors. Thus we have onlyfound empirical support for the conjecture that, on the level of galaxy clus-ters, the observed dark matter effects encoded in Mdm may solely be dueto the combined impact of the halo Msf of the scalar field and of the scaleconnection, Mph 1:

Mdm ≈Msf +Mph 1 = Mt (cf. (66)) (72)

31

Table 2. Observational and model values for total mass at r500, r200Cluster r500 Mtot(r500) M500 r200 Mtot(r200) M200

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+0.71−0.55 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 values Mtot(rN00) and observational values MN00 in 1014M,rN00 (observational) in kpc (N = 1, 2)

32

Table 3. Model values for halo and baryonic masses at r200Cluster Mt Msf Msf gal Mph 1 Mgas M∗ f∗ ft

Coma 12.32 9.91 2.66 2.42 1.78 0.278 0.16 6.9A85 9.08 7.22 1.61 1.87 1.65 0.15 0.09 5.5A400 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 11.8A1367 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 7.5A1650 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 7.3MKW8 2.53 2.05 0.6 0.48 0.22 0.045 0.2 11.3A2029 14.58 11.53 2.37 3.05 3.1 0.223 0.07 4.7A2052 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 8.6A2142 13.93 10.98 2.13 2.95 2.74 0.168 0.06 5.10A2147 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 8.4A2255 10.4 8.27 1.88 2.13 1.51 0.143 0.09 6.9A2589 4.74 3.83 1.11 0.91 0.59 0.119 0.2 8.

Mass values in 1014M, f∗ = M∗Mgas

, ft = MtMgas

(r200),r200 see tab. 2

33

1000 1500 2000 2500r in kpc

5.0×1014

1.0×1015

1.5×1015

A85 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

1×1014

2×1014

3×1014

4×1014

5×1014

6×1014

7×1014

A400 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

2×1014

4×1014

6×1014

IIIZw54 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

2×1014

4×1014

6×1014

8×1014

1×1015

A1367 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

1×1014

2×1014

3×1014

4×1014

MKW4 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

2.0×1014

4.0×1014

6.0×1014

8.0×1014

1.0×1015

1.2×1015

M(r) in ZwCl215 (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

2.0×1014

4.0×1014

6.0×1014

8.0×1014

1.0×1015

1.2×1015

A1650 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

2.0×1014

4.0×1014

6.0×1014

8.0×1014

1.0×1015

1.2×1015

1.4×1015A1795 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

Figure 5: Halo models for clusters 2 – 9 in tab. 1.: total mass Mtot (blackline) with model error bars at r500, r200, transparent matter halo Mt con-stituted by scalar field halo Msf 2 and net phantom halo (in barycentric restsystem of cluster) Mph 1 and baryonic mass (gas and stars) Mbar. Obser-vational data for total mass with error intervals at r500 (violet) from [26].Additional empirical data at r200 (yellow) from [16].

34

1000 1500 2000 2500r in kpc

2×1014

4×1014

6×1014

8×1014

MKW8 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

5.0×1014

1.0×1015

1.5×1015

A2029 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

2×1014

4×1014

6×1014

8×1014

A2052 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

2×1014

4×1014

6×1014

8×1014

1×1015MKW3S M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

2.0×1014

4.0×1014

6.0×1014

8.0×1014

1.0×1015

1.2×1015

A2065 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

5.0×1014

1.0×1015

1.5×1015

A2142 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

5.0×1014

1.0×1015

1.5×1015

A2147 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

2×1014

4×1014

6×1014

8×1014

1×1015

A2199 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

5.0×1014

1.0×1015

1.5×1015

A2255 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

1000 1500 2000 2500r in kpc

2×1014

4×1014

6×1014

8×1014

1×1015

A2589 M(r) in (1M)

M_tot

M_dm

M_sf

M_ph

M_bar

Figure 6: Halo models for clusters 10 – 19 in tab. 1. Description see fig. 5

35

3.6 Comparison with TeVeS and NFW halos

It is surprising that in the WST-3L approach the total amount of observeddark matter Mdm seems to be explained by the energy of the scalar fieldand the phantom halo, Mt = Msf +Mph 1, Mdm ≈ Mt. No missing mass isleft. In usual relativistic MOND approaches this is not the case for clusters,although it is essentially so for galaxies [8]. Based on a study of about40 galaxy clusters, R Sanders has proposed the hypothesis of a neutrinocomponent “between a few times 1013 and 1014M”, mostly concentratedclose 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 TeVeSmodel [18, p. 902]. In our approach, this hypothesis is unnecessary. Wheredoes this difference arise from?

A model calculation for the Coma cluster, evaluating (48) for the tran-sition function µ2 = x(1 + x2)−

12 (44) and the corresponding ν2 (45) used

by Sanders in [18], shows that the neutrino core had to be tuned to Mν ≈1.8 × 1014M in order to give agreement with the observational valueMdm(r500) = M500 −Mbar(r500) ≈ 4.7 × 1014M, where in this framework“r500”, “r200” are to be defined by a formal convention with regard to a(fictitious) NFW halo.

Table 3 shows the amount of scalar field energy up to a radius r ≈ r200in the WST model. It varies between about 2× 1013 and 9× 1014M, i.e.,roughly in the range found necessary by R. Sanders for the (hypothetical)neutrino halo. Moreover, a comparison of the transition functions µw(r) (46)and µ2(x) shows that the gross phantom energy ρt of the WST approach islarger than in an µ2-MOND model.53 Roughly half of the missing mass ofSanders’ model is covered by this effect, the other half is due to the scalarfield halo of the system of galaxies up to r200.

A comparison of the two MOND-like approaches is given in fig. 7. Hereone has to keep in mind that the TeVeS-µ2 model has a free adaptable pa-rameter (mass of the neutrino core), while the WST has not. The generalprofiles of the “dark matter” halos of both models are similar. The TeVeS-µ2mass starts from a higher socle because of its neutrino core; the WST trans-parent mass starts from a lower initial value but rises faster because of theincreasing 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

ρ(r) =ρo

rrc

(1 + rrc

)2, (73)

with density parameter ρo and core radius rc (at which the density hasreduced to half the reference value). For a first comparison of the interior

53[21]

36

500 1000 1500 2000 2500r in kpc

2.0×1014

4.0×1014

6.0×1014

8.0×1014

1.0×1015

1.2×1015

1.4×1015

M_dm in (1M)

M_dm WST

M_dm NFW

M_dm TeVeS

Figure 7: Comparison of dark/transparent/phantom mass halos for Comain NFW, WST and TeVeS models, free parameters of halos for NFW andTeVeS (µ2 with neutrino core) adapted to mass data (black error bars) atr500 = 1280 kpc

halos we take rc ≈ 180h−1 kpc (h = 0.7), following [12],54 and determinethe central density parameter ρo such that the total integrated mass MNFW

assumes the empirical value of [25] at r500 = 1289 kpc (fig. 7). The errorintervals for r500 give upper and lower model values for ρo and correspondingmodel error bars (red) for MNFW (r200). This version of the NFW halosatisfies one set of empirical data by construction (at r500); our main interestthus goes to the other empirical data set available at r200 = 2300 kpc. TheNFW model error interval overlaps with the empirical error bar at r200 andwith the error interval of the WST model.

There is a conspicuous difference between the mass profiles of the NFWhalo on one side and the halo profiles of WST or µ2-TeVeS on the other.More and precise empirical data of the interior halos of galaxy clusters oughtto be able to discriminate between the two model classes. An empiricaldiscrimination between the two MOND-like approaches would need more,and more precise, profile data. At the moment the Weyl MONDlike modelsurvives the comparison fairly well, even though it has no free parameterwhich would allow to adapt it to the halo data.

3.7 A side-glance at the bullet cluster

At the end of this section let us shed a side-glance at the bullet cluster1E0657-56. It is often claimed that the latter provides direct evidence infavour of particle dark matter and rules out alternative gravity approaches.Our considerations show that this argument is not compelling. The energycontent of the scalar field halos of the colliding clusters endows them with

54In studies of the exterior halo of the Coma cluster, Geller e.a. have found fit values0.182

+0.03−0.03 , 0.167

+0.029−0.029 , 0.192

+0.035−0.035 in units h−1Mpc for rc [12].

37

inertia of their own. The shock of the colliding gas exerts dynamical forceson the gas masses only, not directly on the scalar field halos. During theencounter the halos will roughly follow the inertial trajectories of their re-spective clusters before collision, and they will continue to do so for a while.It will take time before a re-adaptation of the mass systems and the respec-tive scalar field halos has taken place. Clearly the MOND-approximationis unable to cover such violent dynamical processes. It describes only therelatively stable states before collision and – in some distant future – af-ter collision. But a separation of halos and gas masses for a (cosmically“short”) period is to be expected, just like in the case of a particle halo withappropriate clustering properties.

For the time being, the cluster 1E0657-56 does not help to decide be-tween the overarching alternative research strategies, particle dark matteror alternative gravity. It may be able to do so, once the dynamics of gas andof the halos has been modeled with sufficient precision in both approaches.Then a proper comparison can be made. But that is an overtly complicatedtask. It seems more likely that other types of observational evidence willoffer a simpler path to a differential evaluation of the two strategies and helpclarifying the alternative.

4 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 thebarycentric rest system of a cluster (under abstraction of the discretestructure of the star matter clustering in freely falling galaxies) (60),with integrated mass equivalent Msf 1,

(ii) an additional contribution to the scalar field energy, forming aroundthe freely falling galaxies (62), integrating to Msf 2,

(iii) and the phantom energy of the total baryonic mass in the barycentricrest system of the cluster, due to the additional acceleration of theWeylian scale connection in Einstein gauge (61) with mass equivalentMph 1.

The first two components add up to a real energy content of the scalar fieldwith mass equivalent Msf = Msf 1 +Msf 2, the third one, Mph 1, arises froma theoretical attribution in a Newtonian perspective and has fictitious char-acter. The mass equivalent of the integrated energy components combinesto a total dark-matter-like quantity Mt = Msf 1 +Msf 2 +Mph 1 (64).

All three components arise from gravitational effects of the cluster’s bary-onic mass Mbar = Mgas+Mstar in the framework of a Weyl geometric scalar

38

tensor theory of gravity (1), with its scale connection as the specific differenceto Riemannian geometry (7), (6).55 A second speciality of the theoreticalframework is the cubic kinetic term of the Lagrangian (18), analogue to theAQUAL approach but in scale covariant form. Observable quantities aredirectly given by the model in Einstein gauge (8).

The total dynamical mass of the model, Mtot = Mbar + Mt (66) hasbeen confronted with the observational values for of 17+2 galaxy clustersgiven in [23], [25], [26], complemented by data from [16]. The problem ofdata transfer between different theoretical frameworks (in particular betweenEinstein gravity /ΛCDM and WST-3L) has been discussed. It leads to acertain caveat with regard to an uncorrected taking over of the values for thetotal mass. But it does not seem to obstruct a meaningful first comparisonof the WST halo model with available mass data of clusters collected in theEinstein/ΛCDM framework.

The result of this comparison shows a surprisingly good agreement ofthe total mass predicted by the model Mtot, on the basis of data for thebaryonic mass components, with the observed total massM500 (measured atthe main reference distance r500). Moreover the model shows an acceptablygood agreement with additional observational values at the distance r200given in [16] (determined on a slightly less refined data basis and evaluationmethod). For 15 clusters the model predicts values for Mtot(r500) with errorintervals (due to the observational errors for the baryonic data) which overlapwith the observational error intervals of M500. The Coma cluster is amongthem. Two clusters have overlapping error intervals in the 2σ range. Theremaining two are outliers and have been identified as such already in [25].

In the result we have found striking empirical support for the conjecturethat the observed dark matter at galaxy cluster level may be due to the trans-parent halo of the scalar field and the phantom halo of the scale connectionof WST (72).

Details for the constitution of the total transparent matter halo fromits specific components (i), (ii), (iii) have been investigated for the Comacluster (section 3.4). They seem to be exemplary for the whole collection ofgalaxy clusters. A particular feature of the model is the scalar field energyformed in the inter-spaces between the galaxies. Its integrated energy contri-bution surpasses the baryonic mass between 1 and 1.5 Mpc (see fig. 3 andtable 3, col. 4). It is crucial for this model’s capacity to explain the totaldynamic mass on purely gravitational grounds, without any additional darkmatter component. A comparison with R. Sanders’ TeVeS-MOND model forgalaxy clusters and the NFW halo is given in section 3.6.

55In Weyl geometric scalar tensor theory the scalar field φ is the new dynamical variable,while the scale curvature dϕ = f vanishes. The latter would be an even more striking,and even irritating, difference to Riemannian geometry; in integrable Weyl geometry itplays no role.

39

At the moment the Weyl geometric scalar tensor model with cubic La-grangian fares well in all the mentioned respects. It would be very helpfulif astronomers decided to evaluate old or new raw data in the framework ofWST. That could lead to an empirical discrimination of the different mod-els. But already independent of the outcome of such a revision, the scalarfield φ and the scale connection ϕµ of WST have a remarkable propertyfrom 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 a proper energy-momentum tensor (26), (27). Theenergy momentum of the scalar field plays a crucial role for the constitutionof the transparent matter halo in the present approach. It seems to expressthe self-energy of the extended gravitational structure.56

5 Appendix

Remarks on the numerical implementation

The calculations in sections 3.2ff. have been implemented in Mathematica10 and run on a PC. Integrations of the mass values have been realized bynumerical interpolation routines in distance intervals of 100 kpc. A compar-ison with refined distance intervals 10 kpc showed differences at the order ofmagnitude 10−4 of the respective values, thus below the rounding precision.

The fading out for the scalar field and phantom halos beyond r200 hasbeen modeled by the cubic expression:

f(x,A,B) = χ(x;−∞, A) +

(1

1 + x−AB

)3

χ(x;A,∞) , (74)

with χ(x; a, b) the characteristic function of the interval [a, b]. The fadingout of f(x,A,B) starts at A and declines to 1

2 at A+B. In our case we startthe fading out close to the virial radius, A = 1.1 r200 and set B = 0.5 r200.

2000 2500 3000 3500 4000 4500 5000kpc

0.2

0.4

0.6

0.8

1.0

Figure 8: Fading out function f(x, 1.1 r200, 0.5 r200) for r200 = 2300 kpc

56Cf. [21, last section].

40

References[1] Adler, Ronald; Bazin, Maurice; Schiffer Menahem. 1975. Introduction to Gen-

eral Relativity. New York etc.: Mc-Graw-Hill. 2nd edition.

[2] Audretsch, Jürgen; Gähler, Franz; Straumann Norbert. 1984. “Wave fields inWeyl spaces and conditions for the existence of a preferred pseudo-riemannianstructure.” Communications in Mathematical Physics 95:41–51.

[3] Bekenstein, Jacob. 2004. “Relativistic gravitation theory for the modified New-tonian dynamics paradigm.” Physical Review D 70(083509).

[4] Bekenstein, Jacob; Milgrom, Mordechai. 1984. “Does the missing mass problemsignal the breakdown of Newtonian gravity?” Astrophysical Journal 286:7–14.

[5] Biviano, A.; Murante, G.; Borgani S. e.a. 2006. “On the efficiency and re-liability of cluster mass estimates based on member galaxies.” Astronomy &Astrophysics 456:23–26.

[6] Blagojević, Milutin. 2002. Gravitation and Gauge Symmetries. Bris-tol/Philadelphia: Institute of Physics Publishing.

[7] Brans, Carl; Dicke, Robert H. 1961. “Mach’s principle and a relativistic theoryof gravitation.” Physical Review 124:925–935.

[8] Famaey, Benoît; McGaugh, Stacy. 2012. “Modified Newtonian dynamics(MOND): Observational phenomenology and relativistic extensions.” LivingReviews in Relativity 15(10):1–159.

[9] Faraoni, Valerio; Gunzig, Edgard. 1999. “Einstein frame or Jordan frame.”International Journal of Theoretical Physics 38:217–225.

[10] Fujii, Yasunori; Maeda, Kei-Chi. 2003. The Scalar-Tensor Theory of Gravita-tion. Cambridge: University Press.

[11] Gavazzi, R.; Adami, C. e.a. 2009. “A weak lensing study of the Coma Cluster.”Astronomy & Astrophysics 498(2):L33–L36. arXiv:0904.0220.

[12] Geller, M.J.; Diaferio, Antonaldo; Kurtz M.J. 1999. “The mass profile of theComa galaxy cluster.” Astrophysical Journal 517:L23–L26.

[13] Jordan, Pascual. 1952. Schwerkraft und Weltall. Braunschweig: Vieweg. 2ndrevised edtion 1955.

[14] Poulis, Felipe P.; Salim, J.M. 2011. “Weyl geometry as a characterization ofspace-time.” International Journal of Modern Physics: Conference Series V3:87–97. arXiv:1106.3031.

[15] Quiros, Israel. 2014. “Scale invariant theory of gravity and the standard modelof particles.” Preprint Guadalajara. arXiv:1401.2643.

[16] Reiprich, Thomas. 2001. “Cosmological Implications and Physical Propertiesof an X-Ray Flux-Limited Sample of Galaxy Clusters.” Dissertation UniversityMunich.

41

[17] Sanders, Robert. 1999. “The virial discrepancy in clusters of galaxies in thecontext of modified Newtonian dynamics.” Astrophysical Journal 512:L23–L26.

[18] Sanders, Robert. 2003. “Clusters of galaxies with modified Newtonian dynam-ics.” Monthly Notices Royal Astronomical Society 342:901–908.

[19] Scholz, Erhard. 2005. “On the geometry of cosmological model building.”Preprint Wuppertal, arXiv:gr-qc/0511113.

[20] Scholz, Erhard. 2014. Paving the way for transitions – a case for Weyl geom-etry. To appear in Towards a Theory of Spacetime Theories, ed. D. Lehmkuhle.a. Basel: Birkhäuser (Springer). arXiv:1206.1559.

[21] Scholz, Erhard. 2015. “MOND-like acceleration in integrable Weyl geometricgravity.” Foundations of Physics, online first, Nov. 2015, arXiv:1412.0430.

[22] Weinberg, Stephen. 1972. Gravitation and Cosmology: Principles and Appli-cations of the General Theory of Relativity. New York etc.: Wiley.

[23] Zhang, Yu-Ying; Laganá, Tatiana; Reiprich Thomas; Schneider Peter. 2011.“XMM-Newton/Sloan Digital Sky Survey: Star formation efficiency in galaxyclusters and constraints on the matter density parameter.” Astronomical Jour-nal 743(13):11 pages.

[24] Zhang, Y.-Y.; Andernach, H.; Caretta C.A. 2011. “HIFLUGCS: Galaxy clus-ter scaling relations between X-ray luminosity, gas mass, cluster radius, andvelocity dispersion.” Astronomy & Astrophysics 526(A105):38pp.

[25] Zhang, Y.-Y.; Laganá, T.F.; Pierini D.; Puchwein E.; Schneider P.;Reiprich T.H. 2011. “Star-formation efficiency and metal enrichment of theintracluster medium in local massive clusters of galaxies.” Astronomy & As-trophysics 535(A78):11 pages.

[26] Zhang, Y.-Y., Laganá T.F.; Pierini D.; Puchwein E.; Schneider P.;Reiprich T.H. 2012. “Corrigendum to star-formation efficiency and metal en-richment of the intracluster medium in local massive clusters of galaxies.”Astronomy & Astrophysics 544(C3):1 page.

42