Hindawi Publishing Corporation Journal of Gravity Volume 2016, Article ID 9706704, 22 pages http://dx.doi.org/10.1155/2016/9706704

Review Article Clusters of Galaxies in a Weyl Geometric Approach to Gravity

Erhard Scholz

Faculty of Mathematics & Natural Sciences and Interdisciplinary Centre for History and Philosophy of Science, University of Wuppertal, 42119 Wuppertal, Germany

Correspondence should be addressed to Erhard Scholz; [email protected]

Received 14 March 2016; Revised 20 April 2016; Accepted 3 May 2016

Academic Editor: Jose Antonio De Freitas Pacheco

Copyright © 2016 Erhard Scholz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction to the Weyl geometric scale connection arises in the present approach. If the latter is expressed by a fictitious mass in In this paper the gravitational dynamics of galaxy clusters is Newtonian terms, a phantom halo canbeascribedtoit. investigated from the point of view of Weyl geometric scalar It indicates the amount of mass one has to assume in the tensor theory of gravity (WST) with a nonquadratic kine- framework of Newton dynamics to produce the same amount maticLagrangetermforthescalarfield(3L)similartothefirst of additional acceleration. The scalar field halo consists of relativistic MOND theory rAQUAL (“relativistic a-quadratic true energy derived from the energy momentum tensor; Lagrangian”) [1]. To make the paper as self-contained as it is independent of the reference system, as long as one possible, it starts with an outline of WST-3L (Section 2). restricts the consideration to reference systems with low WST-3L has two (inhomogeneous) centrally symmetric static (nonrelativistic) relative velocities. The phantom halo, on weak field approximations: (i) the Schwarzschild-de Sitter the other hand, is a symbolical construct and valid only in solution with its Newtonian approximation, which is valid if the chosen reference system (and scale gauge). For galaxy the scalar field and the WST-typical scale connection plays clusters we find two components of the scalar field halo, a negligible role; (ii) a MOND-like approximation which is one deriving from the total baryonic mass in the MOND appropriate under the constraints that the scale connection approximation of the barycentric rest system of the cluster cannot be ignored but is still small enough to allow for (component 1) and one arising from the superposition of a Newtonian weak field limit of the (generalized) Einstein all the scalar field halos forming around each single galaxy equation. The acceleration in the MOND approximation in the MOND approximation of the latter’s rest system consists of a Newton term and additional acceleration of (component 2). Because velocities of the galaxies with regard which three-quarters are due to the energy density of the to the cluster barycenter are small (nonrelativistic) and also scalarfieldandone-quarterisduetothescaleconnection theenergydensitiesaresmall,thetwocomponentscanbe typical for Weyl geometric gravity. superimposed additively (linear approximation). With regard In centrally symmetric constellations the scalar field to the barycentric rest system of the cluster a three-component energy forms a halo about the baryonic mass concentrations. halo for clusters of galaxies arises, two components being due Besides the acceleration derived from the (Riemannian) Levi- to the scalar field energy and one purely phantom (Section 3). Civita connection induced by the baryonic matter and the The two-component scalar field halo is a distinctive scalar field energy an additional acceleration component due feature of the Weyl geometric scalar tensor approach; it is 2 Journal of Gravity neither present in the nonrelativistic MOND approaches nor a Lagrangian containing a modified Hilbert term coupled in rAQUAL. One may pose the question whether it suffices with a scalar field 𝜙. Their Lagrangian has the general form for explaining the deviation of the cluster dynamics from 1 2 𝜆 the Newtonian expectation without additional dark matter. 𝐿= (𝜉𝜙) 𝑅+𝐿 − 𝜙4 ⋅⋅⋅ , 2 𝜙 4 If we call the totality of the three components the transparent (1) halo of the cluster (Section 3.5), the question is whether the 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 L =𝐿√󵄨𝑔󵄨, 󵄨𝑔󵄨 = 󵄨det 𝑔󵄨 . (theoretically derived) transparent halo can explain the dark 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 halo of galaxy clusters, observationally determined in the Here 𝑔 is an abbreviation for a 4-dimensional pseudo- framework of Einstein gravity and Λ𝐶𝐷𝑀. Riemannian metric 𝑔=(𝑔𝜇]) of signature (−+++). 𝜙 is a Section4containsafirsttestofthemodelbyconfronting real valued scalar field on spacetime, 𝐿𝜙 its kinetic term, it with empirical data on mass distribution available in the and 𝜉 a constant coefficient and the dots indicate matter and astronomical literature. A full-fledged test would presuppose interaction terms. Under conformal rescaling of the metric, an evaluation of raw observational data in the framework 󸀠 2 of the present approach and is beyond the scope of this 𝑔𝜇] 󳨃󳨀→ 𝑔 𝜇] =Ω𝑔𝜇] study. Here we use recently published data on the total mass (2) (Ω ), (dark plus baryonic), hot gas, and the star matter for 19 a positive real valued function galaxy clusters, which have been determined from different −1 𝜙 󳨃→𝜙󸀠 = observational data sources (and are thus more precise than the scalar field changes with weight ;thatis, Ω−1𝜙 earlier ones) [2–4]. Two of the 19 clusters show a surprisingly .SofarthisissimilartothewellknownJordan-Brans- large relation of total mass to gas mass. They are separated as Dicke (JBD) scalar tensor theory of gravity [7–10]. But here outliers from the rest of the ensemble already by the authors we work in a scalar tensor theory in the framework of Weyl’s generalization of Riemannian geometry [1, 11–14]. ofthestudyandsodowe.17nonoutlyingclustersremainas Crucial for the Weylgeometricscalartensorapproach our core reference ensemble. (WST) is that the scalar curvature 𝑅 and all dynamical In the mentioned studies total mass, gas mass, and star terms involving covariant derivatives are expressed in Weyl mass are determined on the background of Einstein gravity geometric scale covariant form. Fields 𝑋 are scale covariant Λ𝐶𝐷𝑀 𝑤 plus . That raises the problem of compatibility with if they transform under rescaling by 𝑋 󳨃→ 𝑋=Ω̃ 𝑋, the WST framework. It is discussed in Sections 4.1 and 4.2 with 𝑤∈𝑄(in most cases even in 𝑍); 𝑤 is called the and leads to a certain caveat with regard to the empirical weight of 𝑋. Covariant derivatives of scale covariant fields 𝑀 ,𝑀 values for the total mass ( 200 500) and the reference are defined such that the result of covariant derivation 𝐷𝜇𝑋 𝑟 ,𝑟 distances 200 500 to the cluster centers. But it does not is again scale covariant of the same weight 𝑤 as 𝑋.The seem to obstruct the possibility for a first empirical check of Lagrangian density L is invariant under conformal rescaling 2 our model (Section 4.2). More refined studies are welcome. foranyvalueofthecoefficient𝜉 of the modified Hilbert term. TheyhavetousetheWSTframeworkforevaluatingthe For the matter and interaction terms of the standard model of observational raw data or, at least, to analyze the transfer elementary particles, scale invariance is naturally ensured by problemofmassdatafromoneframeworktotheotherin thecouplingtotheHiggsfieldwhichhasthesamerescaling more detail. behaviour as the gravitational scalar field. Although there is For 15 of the 17 main reference clusters the empirical and no complete identity, there is a close relationship between the theoretical values for the total mass agree in the sense of scale and conformal invariance in quantum field theory [16]. overlapping 1𝜎 error intervals. The remaining two overlap in For classical matter we expect that a better understanding the 2𝜎 range. The two outliers of the original study do not lead of the quantum to classical transition, for example, by the to overlapping intervals even in the 4𝜎 range (Section 4.6). decoherence approach, allows considering scale invariant In the present approach the dynamics of the Coma cluster Lagrangian densities also. For the time being we introduce is explained without assuming a component of particle dark the scale invariance of matter terms in the Lagrangian as a postulate. matter. It is being discussed in more detail than the other In our context an important consequence is the scale clusters in Section 4.5. covariance of the Hilbert energy momentum tensor A short comparison with the halos of Sanders’ 𝜇2-MOND 2 𝛿L model with an additional neutrino core [5] and with the NFW 𝑇 =− 𝑚 , 𝜇] 󵄨 󵄨 𝜇] (3) halo[6]isgivenforComa(Section4.7).Thepaperisrounded √󵄨𝑔󵄨 𝛿𝑔 off by a short remark on the bullet cluster (Section 4.8) and a 󵄨 󵄨 final discussion (Section 5). which is of weight 𝑤(𝑇𝜇])=−2. That is consistent with dimensional considerations on a phenomenological level. It 2. Theoretical Framework has been shown that the matter Lagrangian of quantum matter (Dirac field, Klein Gordon field) is consistent with 2.1. Weyl Geometric Scalar Tensor Theory of Gravity (WST). test particle motion along geodesics (autoparallels) 𝛾(𝜏) of Among the family of scalar tensor theories of gravity the best theaffineconnection,iftheunderlyingWeylgeometryis known ones and closest ones to Einstein gravity are those with integrable (see (8)) [11]. For classical matter we assume the Journal of Gravity 3 same. It is to be expected that it can be proven similar to Having done so, the Weylian metric, given as (𝑔,̃ 𝜑)̃ ,is Einstein gravity. characterized by its Riemannian component 𝑔̃𝜇] only (and a We do not want to heap up too many technical details; vanishing scale connection 𝜑̃𝜇 =0). By obvious reasons we morecanbefoundintheliteraturegivenabove.Butwehave call this the Riemann gauge. This is the analogue of the choice to mention that a Weylian metric canbegivenbyanequiva- of Jordan frame in JBD theory. lence class of pairs (𝑔, 𝜑) consisting of a pseudo-Riemannian 𝑔 𝜇 ] In any case, the choice of representative 𝜇] also fixes metric 𝑔=𝑔𝜇]𝑑𝑥 𝑑𝑥 ,theRiemannian component of the 𝜑 𝜇 𝜇; both together define a scale gauge of the Weylian metric. Weyl metric, and a differentiable one-form 𝜑=𝜑𝜇𝑑𝑥 , Conformal rescaling of the metric is accompanied by the the scale connection (or, in Weyl’s original terminology, the gauge transformation of the scale connection (4). From a “length connection”). 𝜑𝜇 is often called the Weyl covector or mathematical point of view all the scale gauges are on an equal even the Weyl vector field. In fact, 𝜑 denotes a connection + footing, and the physical content of a WST model can be with values in the Lie algebra of the scale group (𝑅 ,⋅) and extracted, in principle, from any scale gauge. One only needs can locally be represented by a differentiable 1-form. The to form a proportion with the appropriate power of the scalar equivalence is given by rescaling the Riemannian component field. From the physical point of view there are, however, two of the Weylian metric according to (2), while 𝜑 has the particularly outstanding scale gauges. Of special importance peculiar gauge transformation behaviour of a connection, besides Riemann gauge is the gauge in which the scalar field is rather than that of an ordinary vector (or covector) field in scaled to a constant 𝜙𝑜 (scalar field gauge). For the particular a representation space of the scale group: choice of the constant value such that 𝜕 Ω 2 −1 2 󸀠 𝜇 (𝜉𝜙 ) = (8𝜋𝐺) =𝐸 , 𝜑 󳨃󳨀→ 𝜑 =𝜑 − (4) 𝑜 pl (9) 𝜇 𝜇 𝜇 Ω 𝐺 󸀠 with the Newton gravitational constant ,thisgaugeiscalled or, shorter, 𝜑 =𝜑−𝑑log Ω. 𝐸 Einstein gauge ( pl the reduced Planck energy). It is the A Weylian metric has a uniquely determined compatible analogue of Einstein frame in JBD theory. In this gauge the Γ affine connection ;inphysicaltermsitcharacterizestheiner- metrical quantities (scalar, vector, or tensor components) of tiogravitational guiding field. It can be additively composed of physical fields are directly expressed by the corresponding Γ the well known Levi-Civita connection 𝑔 of the Riemannian field or field component of the mathematical model (without component 𝑔 of any gauge (𝑔, 𝜑) and an additional expression the necessity of forming proportions). Γ 𝜑 in the scale connection; in short In Riemann gauge Γ reduces to 𝑔Γ by definition. Thus, 𝜇 in this gauge, the guiding field is given by the ordinary Γ= Γ + Γ Γ𝜇 =𝛿𝜇𝜑 +𝛿 𝜑 −𝑔 𝜑𝜇. 𝑔 𝜑 with 𝜑 ]𝜆 ] 𝜆 𝜆 ] ]𝜆 (5) expression for the Levi-Civita connection. On the other hand, in Einstein gauge the measuring behaviors of clocks are Covariant derivatives 𝐷 in the Lagrangians (1) and most immediately represented by the metric field and also (below) (28) and consequently in the expression for the other physical observables are most directly expressed by energy momentum tensor of the scalar field (equations (35) the field values in this scale. Then the expressions for the and (36)) below denote those of Weyl geometry. For a ] gravitational field and the accelerations contain contributions covariant field 𝑋 of weight 𝑤 the derivativation ∇ with regard from the Weylian scale connection. Thus a specific dynamical to Γ of (5) 𝐷 is supplemented by a term due to the scaling difference to Einstein gravity and Riemannian geometry (and weight 𝑤 of 𝑋: to JBD theory) arises even in the case of WST with its 𝐷 𝑋] =∇𝑋] +𝑤𝜑 𝑋] =𝜕𝑋] +Γ] 𝑋𝜆 +𝑤𝜑 𝑋]. integrable Weyl geometry. 𝜇 𝜇 𝜇 𝜇 𝜇𝜆 𝜇 (6) ̃ Writing the scalar field 𝜙 in Riemann gauge (𝑔,̃ 0) in ̃ 𝜔 Itturnsoutthatforthemetric 𝑔𝜇] the full covariant derivative exponential form, 𝜙=𝑒 , turns its exponent is zero: ̃ 𝜔 fl ln 𝜙 (10) 𝐷𝜆𝑔𝜇] =0←→ (7) ∇ 𝑔 +2𝜑 𝑔 =0. into a scale invariant expression for the scalar field. In the 𝜆 𝜇] 𝜆 𝜇] following we shall omit the tilde sign to simplify notation. The 𝜑=𝜑̂ This is the Weyl geometric compatibility condition between scale connection in scalar field gauge is then metric and affine connection (sometimes called “semimetric- 𝜑=−𝑑𝜔,̂ (11) ity”). ̃ 𝜔 In the low energy regime there are physical reasons to because Ω=𝜙=𝑒 is the rescaling function from Riemann constrain the scale connection to the integrable case with a gaugetoscalarfieldgauge.Formoredetailssee[1,12,13,17, closed differentiable form 𝑑𝜑;thatis, =0 𝜕𝜇𝜑] =𝜕]𝜑𝜇.Then 𝜑 18]. is a gradient (at least locally) and may be given by For the sake of consistency under rescaling we consider 𝛾(𝜏) 𝜑𝜇 =−𝜕𝜇𝜔. (8) scale covariant geodesics with scale gauge dependent parametrizations of the geodesic curves of weight 𝑤(𝛾)=−1̇ :

This constraint is part of the defining properties of WST. Then 𝜆 𝜆 𝜇 ] 𝜇 𝜆 𝜇 𝜇 it is possible to “integrate the scale connection away” [11]. 𝑢̇ +Γ𝜇]𝑢 𝑢 −𝜑𝜇𝑢 𝑢 =0, 𝑢 = 𝛾̇ . (12) 4 Journal of Gravity

Heretheaffineconnectioncontainsa𝜑-dependent term The free fall of test particles in Weyl geometric gravity follows 𝜆 in addition to the well known Levi-Civita connection 𝜑Γ𝜇] scale covariant geodesics. It is governed by a differential derived from 𝑔𝜇] (5). The last term on the l.h.s. of (12) takes equationformallyidenticaltotheoneinEinsteingravity(15). care of the scale dependent parametrization (compare (6)). In Here we look at the weak field static case for low velocities in this way we work with a projective family of paths. order to study the dynamics of stars in galaxies and galaxies in ForanygaugeoftheWeylianmetricandthescalarfield, clusters. For studies of the gas dynamics and its modification (𝑔, 𝜑,, 𝜙) any timelike geodesic has thus a generalized proper in our framework the velocity dependent terms of (15) have 𝜇 ] 𝜇 time parametrization 𝛾(𝜏) with 𝑔𝜇]𝑢 𝑢 =−1,where𝑢 = tobetakenintoaccount.Thisisnotbeingdonehere. 𝜇 𝛾̇ . Inverting the coordinate time function 𝑡(𝜏) along the Analogous to Einstein gravity, the coordinate acceleration 󸀠 󸀠 𝑎 𝑥(𝜏) geodesic by 𝜏(𝑡) we have, in abbreviated notation, 𝜏 𝑡 =1 for a low velocity motion in proper time parametriza- and thus tion is given by 𝑑2𝑥𝑖 𝑑𝜏 𝑑 𝑑𝜏 𝑑𝑥𝑖 𝑑2𝑥𝑗 = ( ) 𝑎𝑗 = ≈−Γ𝑗 . (16) 𝑑𝑡2 𝑑𝑡 𝑑𝜏 𝑑𝑡 𝑑𝜏 𝑑𝜏2 𝑜𝑜 (13) 𝑑𝜏 2 𝑑2𝑥𝑖 𝑑𝑡 −3 𝑑2𝑡 𝑑𝑥𝑖 According to (5) the total acceleration decomposes into =( ) −( ) . 𝑑𝑡 𝑑𝜏2 𝑑𝜏 𝑑𝜏2 𝑑𝜏 𝑎𝑗 =− Γ𝑗 − Γ𝑗 =𝑎𝑗 +𝑎𝑗 𝑔 𝑜𝑜 𝜑 ]𝜆 𝑅 𝜑 (17) With (12) and indices 𝑖, 𝑗, 𝑘 = 1,, 2,3 𝜇, ],... = 0,1,2,3this leads to (𝑗 = 1, 2, 3 indices of the spacelike coordinates), where 2 𝑖 2 𝜇 ] 𝑖 𝜇 𝑑 𝑥 𝑑𝜏 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑗 𝑗 =( ) (−Γ𝑖 +𝜑 ) 𝑎 =− Γ (18) 𝑑𝑡2 𝑑𝑡 𝜇] 𝑑𝜏 𝑑𝜏 𝜇 𝑑𝜏 𝑑𝜏 𝑅 𝑔 𝑜𝑜

𝑑𝜏 3 𝑑𝑥𝑖 𝑑𝑥𝜇 𝑑𝑥] 𝑑𝑡 𝑑𝑥𝜇 is the Riemannian component of the acceleration known −( ) (−Γ0 +𝜑 ) from Einstein gravity. Clearly 𝑑𝑡 𝑑𝜏 𝜇] 𝑑𝜏 𝑑𝜏 𝜇 𝑑𝜏 𝑑𝜏 (14) 𝑎𝑗 =− Γ𝑗 𝑑𝑥𝜇 𝑑𝑥] 𝑑𝑥𝑖 𝑑𝑥𝜇 𝜑 𝜑 𝑜𝑜 (19) =−Γ𝑖 +𝜑 𝜇] 𝑑𝑡 𝑑𝑡 𝜇 𝑑𝑡 𝑑𝑡 represents an additional acceleration due to the Weylian 𝜇 ] 𝑖 𝜇 𝑖 𝑔= 0 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥 scale connection. For a diagonal Riemannian metric +Γ −𝜑 . (𝑔 ,...,𝑔 ) 𝜇] 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝜇 𝑑𝑡 𝑑𝑡 diag 𝑜𝑜 33 the general expression (5) simplifies to − Γ𝑗 =−𝑔𝜑𝑗 𝜑 𝑜𝑜 𝑜𝑜 . General considerations on observable Happily, the length connection terms coming from the scale quantities and consistency with Einstein gravity show that, in covariance modification of the geodesic equation (12) cancel order to confront it with empirically measurable quantities, the equation of motion for mass points in Weyl geometric we have to take its expression in Einstein gauge if we want to gravity, parametrized in coordinate time, becomes avoid additional rescaling calculations [14, sec. 4.6]. 𝑑2𝑥𝑖 𝑑𝑥𝑖 𝑑𝑥𝑗 𝑑𝑥𝑗 𝑑𝑥𝑘 For a (diagonalized) weak field approximation in Einstein =−Γ𝑖 +Γ0 −2Γ𝑖 −Γ𝑖 gauge, 𝑑𝑡2 00 00 𝑑𝑡 0𝑗 𝑑𝑡 𝑗𝑘 𝑑𝑡 𝑑𝑡 󵄨 󵄨 (15) 󵄨 󵄨 𝑑𝑥𝑖 𝑑𝑥𝑗 𝑑𝑥𝑖 𝑑𝑥𝑗 𝑑𝑥𝑘 𝑔𝜇] =𝜂𝜇] +ℎ𝜇], 󵄨ℎ𝜇]󵄨 ≪1, (20) +2Γ0 +Γ0 . 0𝑗 𝑑𝑡 𝑑𝑡 𝑗𝑘 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝜂=𝜖 (−1, +1, +1, +1) with sig diag , the Riemannian compo- In the result the dynamics of mass points in Weyl geometric nent of the acceleration is the same as in Einstein gravity. Its gravity is governed by the guiding field (the affine connec- leading term (neglecting 2nd-order terms in ℎ)is tion), as in the semi-Riemannian case [19, equ. (9.1.2)]. Note, 1 however, that in (15) the length connection enters into the affine 𝑗 𝑗 𝑗𝑗 𝑎𝑅 = 𝑔Γ ≈ 𝜂 𝜕𝑗ℎ𝑜𝑜 (no summation over 𝑗) . (21) connection andinfluencesthedynamicsbecauseof(5). 𝑜𝑜 2 The geodesic equation thus contains terms in the scale In the limit, Φ𝑁 fl −(1/2)ℎ𝑜𝑜 behaves like a Newtonian connection 𝜑𝜇. In the low velocity, weak field regime the potential equation of motion reduces to the form well known from 𝑑2𝑥𝑗/𝑑𝑡2 =−Γ𝑗 Γ𝑗 = Γ𝑗 + Γ𝑗 Einstein gravity 00.Here 00 𝑔 00 𝜑 00 𝑎𝑅 ≈−∇Φ𝑁, (22) (𝑗 = 1, 2, 3)arethecoefficientsoftheWeylgeometric Γ𝑗 where ∇ is understood to operate in the 3-spacelike coordi- affine connection with 𝜑 00 given by (5). This is the crucial modifying term for gravity in the Weyl geometric approach nate space with Euclidean coefficients as the leading term of (low velocity case). the metric. In Einstein gauge the Weylian scale connection 𝜑̂𝜇 arises Ω=𝑒𝜔 𝜑̂ =−𝜕𝜔 2.2. The Weak Field Static Approximation. We want to under- from Riemann gauge by rescaling with , 𝜇 𝜇 𝑗 𝑗 stand the scale connection for the motion of point particles. (4), and 𝑎𝜑 ≈𝜑. In other words, the additional acceleration Journal of Gravity 5

2𝜓 duetothescaleconnection(19)isgeneratedbythescale 𝑔𝜇], 𝑔̃𝜇] =𝑒 𝑔𝜇].Itsroleisveryclosetoour𝜔 in (10). A invariant representative 𝜔 of the scalar field as its potential: corresponding scale covariant form of the Lagrangian could be 𝑎𝜑 ≈−∇𝜔. (23) 2 −1 2 −1 −2 ] 𝐿𝜙3 =(𝜉𝜙) (𝜂 𝜙) 𝑓((𝜂 𝜙) 𝜕]𝜔𝜕 𝜔) , (27) If we compare with Newton gravity, we can calculate the −1 fictitious mass density which one had to assume on the right- where 𝜂 𝜙𝑜 in Einstein gauge plays a role similar to 𝑎𝑜.The hand side of the Poisson equation, in addition to the real sign has deliberately been changed; the reasons are given masses, in order to generate the same amount of additional below (39). acceleration. Obviously here it is We are here interested in additive modifications (19) of Einstein gravity, mainly in a domain in which the effects of 𝜌 = (4𝜋𝐺)−1 ∇2𝜔. ph (24) the scale connection 𝜑] =−𝜕]𝜔, compared with those of the Riemannian component of the metric, cannot be neglected. In the terminology of the MOND literature the acceleration This will be called a regime with MOND approximation.For duetotheWeylianscaleconnectioncorrespondstoaphan- ∇𝜔 𝜌 nontimelike theLagrangian(27)thenbecomes tom energy density ph (see,e.g.,[20],p.48).Weseethat 2 −1 alreadyonthegenerallevelthedynamicsofWSTdiffers 𝐿 = (𝜉𝜙)2 (𝜂−1𝜙) (𝐷 𝜙𝐷]𝜙)3/2 from Einstein gravity. Only for trivial scalar field, 𝜔=const, 𝜙3 3 ] (28) the usual Newton limit is recovered; otherwise it is modified. 2 We shall explore how this modification relates to the usual = 𝜉2𝜂𝜙 |∇𝜔|3 , 3 MOND approaches. wherewehaveusedtheabbreviation 2.3. WST Gravity with Cubic Kinematic Lagrangian (WST- 󵄨 󵄨1/2 |∇𝜔| fl 󵄨𝜕 𝜔𝜕]𝜔󵄨 3L). The most common form of the kinetic term for the scalar 󵄨 ] 󵄨 (29) ] field is that of a Klein Gordon field, 𝐿𝜙2 = −(𝛼/2)𝐷]𝜙𝐷 𝜙, (|⋅⋅⋅| absolute value) (for timelike ∇𝜔 see [1]. Here we quadratic in the norm of the (scale covariant) gradient (in exclusively deal with the spacelike (or null) case). 𝜂 denotes a WST 𝐷] denotes the a scale covariant derivative of 𝜙, 𝐷]𝜙= 𝜕 𝜙−𝜙𝜑 constant coefficient responsible for the relative strength of the ] ]). For our form of the gravitational Lagrangian 𝜔 𝛼=−6𝜉2 cubic kinetic term and is the scale invariant representative (1) it is conformally coupled for .Inspiredby ofthescalarfieldintroducedin(10)and∇ its gradient (𝑤(𝜙) = the approach of the relativistic “a-quadratic Lagrangian” −1 and 𝑤(‖∇𝜔‖) = −1 imply the scale weight 𝑤(𝐿𝜙3)=−4,as (rAQUAL), the first relativistic attempt of a MOND theory it must be for scale invariance of L𝜙3). ofgravity[21,22],wefindthataWeylgeometricscalartensor We introduce the constant 𝑎̃𝑜 defined in Einstein gauge theory of gravity leads to a MOND-like phenomenology if we ̂ (𝑔,̂ 𝜑)̂ with constant scalar field 𝜙 š 𝜙𝑜, add a cubic term to the kinetic Lagrangian of the scalar field −1 𝑎̃𝑜 fl 𝜂 𝜙𝑜. (30) 𝐿𝜙 =𝐿𝜙2 +𝐿𝜙3. (25) Then the cubic term of the kinetic Lagrangian in Einstein The crucial difference to the early approach of rAQUAL is gauge reads the scale covariant reformulation in the framework of Weyl geometry. It results in a different behaviour of the scalar field 2 −1 3 𝐿 ≐ (8𝜋𝐺𝑎̃ ) |∇𝜔| . (31) energy density. Bekenstein/Milgrom’s model relied crucially 𝜙3 3 𝑜 on implementing a transition function 𝑓(𝑦) between the The dotted equality sign “≐” indicates that the respective Newton and the deep MOND regime into the kinetic term. equationisnotscaleinvariantbutpresupposesaspecialgauge The constraint of scale invariance of the Lagrangian reduces made clear by the context. Here, as in most cases in this the underdetermination of the Lagrangian and suggests a paper, it indicates the Einstein gauge (similar for ≈̇). 𝑎̃𝑜 has slightly different form of the kinetic term. It is still quite near thedimensionofinverselength/timeandwillplayarole to the one of the relativistic AQUAL theory. analogous to the MOND acceleration 𝑎𝑜 ≈[𝑐]𝐻,where𝐻 In the review paper [22] Bekenstein gives the rAQUAL is the Hubble parameter (at “present”) and 𝑐 is the velocity Lagrangian in the form (his equation (6)) of light. Coefficients of type [𝑐] will often be suppressed in −1 −2 2 ] the following general considerations. They will be plugged in 𝐿𝜓 =−(8𝜋𝐺𝑁) 𝐿 𝑓(𝐿 𝜕]𝜓𝜕 𝜓) , (26) only in the final step. Below we shall see that for 𝑎̃𝑜 =𝑎𝑜/16 ≈ 𝐻/100 theWSTmodelwithcubickinematicLagrangian where 𝐿 is “a constant with dimension of length introduced (WST-3L) acquires a MOND-like phenomenology in a weak for dimensional consistency” (later it is identified as the 2 −1 gravitational field in which the scalar field and the scale MOND acceleration 𝑎𝑜 via 𝑐 𝐿 =𝑎𝑜). Asymptotically connection cannot be neglected. 𝑓(𝑦) ∼ (2/3)𝑦3/2 𝑦≪1 𝑓(𝑦) ∼ for (MOND regime); similarly A reasonable choice of adaptable parameters brings 𝜉 and 𝑦 𝑦≫1 𝜓 for (Newton regime). is the logarithm of a rescaling 𝜂 to nearby orders of magnitude, 𝜂=𝛽𝜉,with function between the Jordan frame metric 𝑔̃𝜇] (called the −1 “physical” metric) and the Einstein frame (“primitive”) metric 𝛽=𝜉 𝜂, 𝛽 ∼ 100. (32) 6 Journal of Gravity

] On the other hand, because of (9) and (30) the product of Moreover with 𝑔𝜇] ≈𝜂𝜇], the expression ∇](|∇𝜔|𝜕 𝜔) turns both coefficients is a “large number” in the sense of 𝜂⋅𝜉 = into the nonlinear Laplace operator ∇⋅(|∇𝜔|∇𝜔)of the 𝐸 /𝑎̃ = 𝑎̃−1/𝐿 ∼1063 ⋅ pl 𝑜 𝑜 pl . To the kinetic Lagrangian of the MOND theory with Euclidean scalar product and norm scalar field a potential term is added. It must be of order 4to |⋅⋅⋅|. provide for scale invariance of the density (1): In the general case we have to complement (39) with the Einstein equation in Einstein gauge 𝜆 4 𝐿 =− 𝜙 . (33) 𝑉4 4 𝑅 − 𝑔≐8𝜋𝐺𝑇(𝑚) +Θ, Ric 2 (40) Variation of the Lagrangian leads to the dynamical equations of WST, the Einstein equation, and the scalar field In vacuum, the trivial scalar field 𝜔=const is a basic equation. The scale invariant Einstein equation is (this means solution of (39). Then WST reduces to Einstein gravity. In that not only the equation but all of its constitutive (additive) particular, the Schwarzschild and the Schwarzschild-de Sitter terms are scale invariant) solutions of Einstein gravity are special (degenerate) solutions 𝜆/4 = 0 𝜆/4 ≈ 6 𝑅 −2 (𝑚) of WST-3L equations for or ,respectively.In − 𝑔=(𝜉𝜙) 𝑇 +Θ, 𝜙≐ Ric 2 (34) fact, they solve (40) and (39) for const in Riemann gauge, that is, in the case of Einstein gauge equal to Riemann gauge where 𝑔 denotes the whole collection of metrical coefficients (𝑔,̂ 𝜑)̂ = (𝑔,̃ 0). The Riemannian component of the metric (𝑚) and 𝑇 theenergytensorofmatter(3).Thescalarfield (𝑔=̃ 𝑔̂ š 𝑔)isgivenby contributes to the total energy momentum with two terms, (𝐼) (𝐼𝐼) 2𝑀 Θ=Θ +Θ , the first of which is proportional to the metric 𝑑𝑠2 =−(1− −𝜅𝑟2)𝑑𝑡2 (thus formally similar to a vacuum energy tensor) (see, e.g., 𝑟 [23],[18,pp.96ff.]): 2𝑀 −1 +(1− −𝜅𝑟2) 𝑑𝑟2 (41) (𝐼) −2 𝜆 2 −2 𝑟 Θ =𝜙 (−𝐷𝜆𝐷 𝜙 +𝜉 (𝐿𝑉4 +𝐿𝜙)) 𝑔, (35) 2 2 2 2 𝜕𝐿 +𝑟 (𝑑𝑥2 + sin 𝑥2𝑑𝑥3) . Θ(𝐼𝐼) =𝜙−2 (𝐷 𝐷 𝜙2 −2𝜉−2 𝜙 ). 𝜇] 𝜇 ] 𝜕𝑔𝜇] (36) (𝐼) 2 2 Then Ric − (𝑅/2)𝑔 = −3𝜅𝑔 and Θ=Θ = −(𝜆/4)𝛽 𝑎̃𝑜𝑔. Varying with regard to 𝜙 gives the scalar field equation. ThereforetheEinsteinequationissatisfiedfor3𝜅 = 2 2 2 Subtracting the trace of the Einstein equation for a confor- (𝜆/4)𝛽 𝑎̃𝑜;thatis,𝜅≈2𝐻 for 𝜆/4 ≈ 6 and 𝛽 ≈ 100.We 2 mally coupled 𝐿𝜙2 term (𝛼=−6𝜉) strongly simplifies it see that in the case of a negligible Weylian scale connection andintroducesthetraceofthemattertensorintothescalar the classical (nonhomogeneous) point symmetric solutions fieldequation.InEinsteingauge,with𝑔 the Riemannian of Einstein gravity are valid also for the dynamics of WST. componentofthemetric,itcanbewrittenintermsof This implies that in the case of a negligible scale connection the covariant derivative 𝑔∇ with regard to 𝑔 (Levi-Civita Newton dynamics is an effective approximation for point connection in Einstein gauge) as ([1, pp. 15f., sec. 7.2, postprint symmetric solutions of WST (in Einstein gauge). version arXive v4]) In order to make such a type of Einstein limit compatible with our Lagrangian, a suppression of the 𝐿𝜙3-term for ] (𝑚) 𝑎 ∇] (|∇𝜔| 𝜕 𝜔) ≐ −4𝜋𝐺𝑎̃𝑜tr𝑇 . (37) sufficiently large accelerations 𝑅 of (18) is necessary. One 𝑔 ̃ may consider plugging a factor 𝑓(𝑎𝑜/|𝑎𝑅|) with a function ̃ ̃ ̃ If we introduce the corresponding Riemannian covariant 𝑓 such that 𝑓(𝑦) ∼1 for 𝑦 > 0.01 and 𝑓(𝑦) ∼0 for operator 𝑦≪1 𝐿 into the r.h.s. expression of (28) for 𝜙3 but such ◻ 𝜔= ∇ (|∇𝜔| 𝜕]𝜔) a choice would have the blemish of a coordinate dependent 𝑔 𝑀 𝑔 ] argument of the function. A better alternative is provided (38) by the hypothesis that the scalar field inhomogeneities are =(𝜕 |∇𝜔| 𝜕]𝜔+|∇𝜔| ∇ 𝜕]𝜔) , ] 𝑔 ] suppressed if any of the sectional curvatures 𝜅 (with respect to the Riemannian component of the metric in Einstein gauge) the scalar field equation for a fluid with matter density 𝜌𝑚 and 9 −2 2 surpasses a certain threshold (e.g., |𝜅| ≥ (10 𝑎𝑜 [𝑐 ]) ). In pressure 𝑝𝑚 simplifies to the covariant Milgrom equation the next section we investigate the case of a nonnegligible 𝑔◻𝑀 𝜔≐4𝜋𝐺𝑎̃𝑜 (𝜌𝑚 −3𝑝𝑚). (39) scale connection. A more detailed discussion of the transition between the two domains has to be left open for another (𝑚) In this derivation, with tr𝑇 entering by subtracting the occasion. trace of the Einstein equation, a sign choice like in (26) leads to the wrong sign on the r.h.s. of the Milgrom equation. This 2.4. A WST Approach with MOND-Like Phenomenology. If explains our sign choice in (27). By obvious reasons (38) will the conditions for the weak field approximation (20) are be called the covariant Milgrom operator.Inthestaticweak given, it is possible to identify a MOND regime as a region field static case 𝜔 does not depend on the time coordinate. in which the Newton acceleration 𝑎𝑁 is smaller than 𝑎𝑜 (here Journal of Gravity 7

𝑎𝑁 can be identified with 𝑎𝑅 in (22)). Then the scalar field the coordinates of the approximating Euclidean space) this equation (39) reduces, in reliable approximation, to implies

(𝑚) 𝑦 ∇⋅(|∇𝜔| ∇𝜔) ≐−4𝜋𝐺𝑎̃ 𝑇 , (42) 𝑎 =−∇𝜔≈−̇ √𝐺𝑀 (𝑟) 𝑎̃ , 𝑜tr 𝜑 𝑜 󵄨 󵄨2 󵄨𝑦󵄨 (46) with the Euclidean ∇-operator. We call this a MOND approx- 󵄨 󵄨 √𝐺𝑀 (𝑟) 𝑎̃𝑜 𝜌 󵄨𝑎 󵄨 = . imation. For pressureless matter with energy density 𝑚 we 󵄨 𝜑󵄨 𝑟 get But this is only the most immediate modification of ∇⋅(|∇𝜔| ∇𝜔) ≐4𝜋𝐺𝑎̃ 𝜌 . 𝑜 𝑚 (43) Newton gravity. There is also the additional term in (40) of 𝜌 =(8𝜋𝐺)−1Θ the energy density due to the scalar field, sf 𝑜𝑜.It That is similar to the AQUAL approach [21, 22]. Note that only modifies the r.h.s. of the Newton limit of Einstein gravity (in 𝜌 the matter energy momentum tensor, without the scalar field contrast sf does not enter the r.h.s. of the scalar field equation energy density, appears on the r.h.s. of (42). (39) and therefore does not enter the r.h.s. of (45)). Straightforward verification shows that, independent of Neglecting contributions at the order of magnitude of symmetry conditions, a solution of (43) is given by 𝜔 with a cosmological terms (∼H) the energy density of the scalar field gradient ∇𝜔 = −𝑎𝜑 such that in Einstein gauge simplifies to ([1, sec. 4.3])

𝜌 ≈̇ (4𝜋𝐺)−1 (∇2𝜔+Γ𝑗 𝜕𝑘𝜔) , 𝑎̃ 󵄨 󵄨 𝑎 sf 𝑗𝑘 (47) 𝑎 = √ 𝑜 𝑎 = √𝑎̃ 󵄨𝑎 󵄨 𝑁 , 𝜑 󵄨 󵄨 𝑁 𝑜 󵄨 𝑁󵄨󵄨 󵄨 (44) 󵄨𝑎𝑁󵄨 󵄨𝑎𝑁󵄨 where Latin indices 𝑗,𝑘,...refer to space coordinates only. In the central symmetric case with Euclidean metric in where 𝑎𝑁 denotes the Newton acceleration of the given mass 󸀠 spherical coordinates and a mass function 𝑀(𝑟) with 𝑀 (𝑟) = density, 0 for 𝑟>𝑟𝑜 (for some distance 𝑟𝑜), we find from (46) 2 √̃ 2 𝑗 𝑘 √̃ 2 ∇ 𝜔= 𝑎𝑜𝐺𝑀(𝑟)/𝑟 and Γ𝑗𝑘𝜕 𝜔 = (2/𝑟)( 𝑎𝑜𝐺𝑀(𝑟)/𝑟) = ∇ Φ𝑁 =4𝜋𝐺𝜌𝑚, 2 2 2 2 2 2 2 2∇ 𝜔;(for𝑑𝑠 =𝑑𝑟 +𝑟 (𝑑𝜃 + sin 𝜃𝑑𝜗 ) the crucial affine (45) 1 2 3 −1 𝑎𝑁 =−∇Φ𝑁 connection components are Γ11 =0,Γ21 =Γ31 =𝑟 ,and thus (calculations in the approximating Euclidean space with 𝜌 ≈̇ (4𝜋𝐺)−1 3∇2𝜔. norm |⋅⋅⋅|). The solution of the nonlinear Poisson equation sf (48) (43)ismuchsimplerthanonemightexpectatafirstglance.In a first step the linear Poisson equation of the Newton theory is That is three times the value of the phantom energy density to be solved and then an algebraic transformation of type (44) corresponding to the acceleration of the scale connection leads to the acceleration due to the solution of the nonlinear (24). The total “anomalous” additive acceleration (in compar- partial differential equation (43). In fact, 𝑎𝜑 has the form of ison to Newton gravity) is therefore the deep MOND acceleration of the ordinary MOND theory 𝑎̃ 𝑎 =𝑎 +𝑎 =4𝑎 . (but with different constant 𝑜)(44).Intheterminologyof add 𝜑 sf 𝜑 (49) the MOND community, the MOND approximation of WST- 3L behaves like a special case of a QMOND theory [20, pp. In the central symmetric case 46ff.]. This raises the question of the Newtonian limit. Equation 󵄨 󵄨 √𝐺𝑀 (𝑟) 𝑎̃ |𝑎 |≪|𝑎 | |𝑎 |≫𝑎 (> 𝑎̃ ) 󵄨𝑎 󵄨 =4 𝑜 . (50) (44) implies 𝜑 𝑁 in regions, where 𝑁 𝑜 𝑜 . 󵄨 add󵄨 𝑟 Therefore 𝑎𝜑 caneffectivelybeneglectedinthecaseof“large” values of |𝑎𝑁| derived from (45). Then, according to the obser- For consistency with the deep MOND acceleration 𝑎𝑜 we have vation at the end of Section 2.3, the Newton approximation to set is also reliable in WST gravity. That is true irrespective of the question of how to characterize the transition between 𝑎 𝑎̃ = 𝑜 ≈10−2𝐻 [𝑐] ≈6⋅10−10 −2. the MOND and the Newton approximation. Here we shall 𝑜 16 cm s (51) consider the MOND approximation in an “upper transition” |𝑎 |≤102𝑎 regime only, where roughly 𝑁 𝑜.Onemightspeak Because of (44) the total acceleration 𝑎 is then of the upper transition regime for 𝑎𝑜 ≤|𝑎𝑁| ≤ 100𝑎𝑜 of the |𝑎 |≤𝑎 MOND regime if 𝑁 𝑜 and of the deep MOND regime for, 𝑎 |𝑎 |≤10−2𝑎 𝑎=𝑎 +𝑎 =𝑎 (1 + √ 𝑜 ), let us say, 𝑁 𝑜 [1, sec. 7.3]. 𝑁 add 𝑁 󵄨 󵄨 󵄨𝑎𝑁󵄨 For centrally symmetric mass distributions 𝜌(𝑟) with (52) 𝑀(𝑟) 𝑟 𝑟=|𝑦| mass integrated up to (where denotes the 󵄨 󵄨 󵄨 󵄨 󵄨𝑎 󵄨 = √𝑎 󵄨𝑎 󵄨. Euclideandistancefromthesymmetrycenter,𝑦=(𝑦1,𝑦2,𝑦3) 󵄨 add󵄨 𝑜 󵄨 𝑁󵄨 8 Journal of Gravity

2.5. Comparison with Usual MOND Theories. We can now derived from the gradient of |𝑎𝑁| dominating in the “vacuum” compare our approach with other models of the MOND (where however scalar field energy is present). For the −1/2 󸀠 family. Simply adding a deep MOND term to the Newton Weyl geometric model with ̃]𝑤(𝑦) = 𝑦 , ̃]𝑤(𝑦) = −3/2 acceleration of a point mass is unusual. Milgrom rather −(1/2)𝑦 𝜌 turns into considered a multiplicative relation between the MOND ph 𝑎 𝑎 1/2 acceleration and the Newton acceleration 𝑁 by a kind of 𝑎 𝑎 󵄨 󵄨 𝜌 =( 𝑜 ) 𝜌 + (8𝜋𝐺)−1 ( 𝑜 )∇(󵄨𝑎 󵄨) “dielectric analogy,” 𝑡 󵄨 󵄨 𝑚 󵄨 󵄨 󵄨 𝑁󵄨 󵄨𝑎𝑁󵄨 󵄨𝑎𝑁󵄨 (58) 1 𝑥󳨀→∞ 𝑎 { for ⋅𝑎𝑁, 𝑎𝑁 =𝜇( )𝑎, with 𝜇 (𝑥) 󳨀→ { (53) 𝑎𝑜 𝑥 𝑥󳨀→0, { for 3 𝜌 = 𝜌 , sf 4 𝑡 𝜇(𝑥) →𝑥 𝜇(𝑥) − 𝑥= or the other way round (here means (59) O(𝑥); i.e., (𝜇(𝑥) − 𝑥)/𝑥 remains bounded for 𝑥→0.Cf.[20, 1 𝜌 = 𝜌 . pp.51f.]) ph 4 𝑡

𝑎𝑁 The first expression of (59) is compatible with (47). 𝑎=] ( )𝑎𝑁, 𝑎𝑜 In our case it would be utterly wrong to consider the whole of 𝜌𝑡 as “phantom energy.” Three-quarters of it is due to (54) 𝜌 {1 for 𝑦󳨀→∞ the scalar field energy density and the scalar field halo sf and with ] (𝑦) 󳨀→ { expressed a true energy density. This energy density appears 𝑦−1/2 𝑦󳨀→0. { for on the right-hand side of the Einstein equation (40) and the Newtonian Poisson equation as its weak field, static limit. It From this point of view our acceleration (52) is specified by is decisive for lensing effects of the additional acceleration. well defined transition functions 𝜌 Only one-quarter, ph, is phantom, that is, a fictitious mass density producing the same acceleration as the Weylian scale 1−√1+4𝑥 𝜇 (𝑥) =1+ , connection (24). Only for the sake of comparison with other 𝑤 2𝑥 (55) MOND models we may speak of 𝜌𝑡 as some kind of gross ] (𝑦) = 1 + 𝑦−1/2. phantom energy, in contrast to the “net” phantom energy 𝑤 𝜌 ph1. One has to keep in mind, however, that our transition func- We have to distinguish between the influence of the tions 𝜇, ] are reliable only in the MOND regime and the upper additional structure, scalar field, and scale connection, on 2 light rays and on (low velocity) trajectories of mass particles. transitional regime (roughly 𝑎𝑁 ≤10𝑎𝑜). They cannot be used for discussing the Newtonian limit. It will be important Bending of light rays is influenced by the scalar field halo only, to see how they behave in the light of empirical data, in the acceleration of massive particles with velocities far below 𝑐 particular galactic rotation curves and cluster dynamics. by the scalar field halo and the scale connection. In the MOND literature the amount of a (hypothetical) Also in another respect our theory differs from the usual mass which in Newton dynamics would produce the same MOND approaches. In MOND external acceleration fields of 𝑎 effects as the respective MOND correction add is called a system under consideration are difficult to handle. In WST, 𝑀 phantom mass ph. For any member of the MOND family as in GR, a freely falling (small) system does not feel the the additional acceleration can be expressed by the modified external acceleration field if it is sufficiently small, relative transition function ̃] = ] −1with ] as in (54) to the inhomogeneities of the external gravitational field, for 󵄨 󵄨 neglecting tidal forces. In this sense, the external acceleration 󵄨𝑎 󵄨 𝑎 = ̃] (󵄨 𝑁󵄨)𝑎 . problemdoesnotariseintheWSTMONDapproximation add 𝑁 (56) 𝑎𝑜 (42). Another important consequence follows: the scalar field 𝜌 The phantom mass density ph attributed to the potential energy formed around a freely falling subsystem of a larger Φ 4𝜋𝐺𝜌 =∇2Φ ∇Φ =−𝑎 ph satisfies ph ph and ph add.Ashort gravitating system, calculated in the MOND approximation calculation shows that it may be expressed as of the freely falling subsystem, contributes to the r.h.s. of the Einstein equation of any other subsystem (in relative motion) 󵄨 󵄨 󵄨 󵄨 󵄨𝑎 󵄨 −1 󵄨𝑎 󵄨 󵄨 󵄨 andalsotothatofasuperordinatelargersystem.Inprinciple 𝜌 = ̃] (󵄨 𝑁󵄨)𝜌 +(4𝜋𝐺𝑎) ̃]󸀠 (󵄨 𝑁󵄨)(∇󵄨𝑎 󵄨) ph 𝑚 𝑜 󵄨 𝑁󵄨 that presupposes that the whole energy momentum tensors 𝑎𝑜 𝑎𝑜 (57) (35) and (36) (and its system dependent representation) are ⋅𝑎𝑁. considered. For slow motions and weak field approximation a superposition of energy densities as in Newton dynamics It consists of a contribution proportional to 𝜌𝑚 with factor ̃], seems legitimate. This has to be taken into account for which dominates in regions of ordinary matter, and a term modeling the dynamics of clusters of galaxies. Journal of Gravity 9

2.6. Short Resume.´ We have derived the most salient features (v) 𝑎𝜑 is formally derivable in Newton dynamics from a of the Weyl geometric MOND approximation (WST MOND) fictitious energy density and are prepared for a comparison with empirical data. Before 1 we do so, it may be worthwhile to collect the results which 𝜌 = 𝜌 . (66) arenecessaryforapplyingittorealconstellationsinashort ph 3 sf survey. 𝜌 Consider a gravitating system which in the Newton ph is the net phantom energy of WST MOND. For approximation of Einstein gravity is described by the bary- comparisonwithothermodelsoftheMONDfamily 𝜌 +𝜌 =𝜌 onic matter density 𝜌𝑚, the acceleration 𝑎𝑚,andpotentialΦ𝑚 one may like to consider ph sf 𝑡 as a kind of with “gross phantom energy” (although the larger part of it is real). It is transparent rather than “dark” (see (75) ∇2Φ =4𝜋𝐺𝜌 , 𝑚 𝑚 below). (60) 𝑎𝑚 =−∇Φ𝑚. (vi) (i)–(v) are reliable approximations also for small (local) gravitating systems freely falling in a larger 2 The modification due to WST MOND leads to additional gravitating system, if 𝑎𝑚 ⪅10𝑎𝑜 in the local system. 𝑎 acceleration add with the following features: The subsystem can be considered as “small” with 𝑎 𝑎=𝑎 +𝑎 respect to the supersystem, if tidal forces of the super- (i) The total acceleration is 𝑚 add with system can be neglected. In hierarchical systems like −1/2 󵄨 󵄨 𝜌 𝑎 󵄨𝑎 󵄨 galaxy clusters the energy density contributions sf 𝑎=𝑎 (1+( 𝑜 ) ) = ] (󵄨 𝑚󵄨) 𝑎 , of the subsystems and the supersystem (calculated in 𝑚 󵄨 󵄨 𝑚 (61) 󵄨𝑎𝑚󵄨 𝑎𝑜 different reference coordinate systems) add up to the total energy density of the scalar field, if the velocities −1/2 of the subsystems relative to the barycenter of the where ](𝑦) = 1 + 𝑦 .For𝑎𝑚 ≫𝑎𝑜 the Newton approximation applies. Equation (61) holds supersystem are small. This is a crucial difference 2 between WST MOND and ordinary MOND theories. for |𝑎𝑚|≤10𝑎𝑜 only (“upper” transition regime). No If one likes, 𝜌 can be considered as the “dark matter” information can be drawn from it for |𝑎𝑚| larger but sf component of WST MOND although the energy not yet ≫𝑎𝑜 (the “lower” transition regime). density of the scalar field it is not constituted by (ii) The “reciprocal” transformation function defined by the usual (hypothetical) quantum particles (WIMPs, 𝑎 = 𝜇(|𝑎|/|𝑎 |) 𝑎 𝑚 𝑚 is axions, etc.). To demarcate this difference it might 1−√1+4𝑥 better be called transparent matter/energy of WST. 𝜇 (𝑥) =1+ . (62) 2𝑥 (vii) Gravitational lensing is due to the scalar field energy 𝜌 density only ( sf ), while the dynamics of WST corre- 𝜌 =𝜌 +𝜌 𝑎 𝑎 =𝑎 +𝑎 =4𝑎 sponds to the total phantom density ( 𝑡 sf ph). (iii) add consists of two components add 𝜑 sf 𝜑. The first one is derived from a potential 𝜔 satisfying It remains to be seen whether such a difference is in the nonlinear Poisson equation agreement with observations. 𝜋 ∇⋅(|∇𝜔| ∇𝜔) = 𝑎 𝐺𝜌 , 4 𝑜 𝑚 3. Halo Model for Clusters of Galaxies (63) 3.1. Cluster Models for Baryonic Mass (Hot Gas and Stars). 𝑎𝜑 =−∇𝜔. In the astronomical literature, the density profile of hot gas 𝑎 and (smeared) star/galaxy matter in a galaxy cluster is often (iv) The second one, sf , can be understood as Newton 𝜌 described by a centrally symmetric profile of the following acceleration due to the energy density sf of a scalar form: field (part of the modified gravitational structure). Its −(3/2)𝛽 potential satisfies a Newtonian Poisson equation. It 𝑟 2 satisfies 𝜌 (𝑟) =𝜌𝑜 (1 + ( ) ) . (67) 𝑟𝑐 −∇𝑎 =4𝜋𝐺𝜌 (64) sf sf 𝛽 is the ratio of the specific energies of the galaxies and the gas, 𝜌𝑜 the central density, and 𝑟𝑐 the core radius [5, 15, 24] (𝑟𝑐 with energy density is the distance from the cluster center at which the projected 1/2 galaxy density is half the central density 𝜌𝑜). Equation (67) 3 𝑎𝑜 −1 󵄨 󵄨 𝑎𝑚 𝛽 𝜌 = (󵄨 󵄨) (𝜌𝑚 + (8𝜋𝐺) ∇(󵄨𝑎𝑚󵄨)⋅󵄨 󵄨). (65) is called a -model for the mass distribution. For our test we sf 4 󵄨𝑎 󵄨 󵄨𝑎 󵄨 𝜌 (𝑟) 󵄨 𝑚󵄨 󵄨 𝑚󵄨 assume density models for the gas mass gas and for the 𝜌 (𝑟) 𝛽 𝑟 galaxy mass star with the same form parameters and 𝑐. 𝜌 sf is part of the energy momentum tensor of the We thus work with an idealized model using proportional scalar field 𝜙 and in this sense “real” rather than density profiles for the hot gas and for the galaxies with phantom. parameters 𝛽,𝑐 𝑟 determined from observations of the hot gas. 10 Journal of Gravity

The central densities 𝜌𝑜 can, in principle, be determined from Hereweareinterestedintheneighbouringregions 𝑀 (𝑟 ) 𝑀 (𝑟 ) mass data for gas gas 1 ,respectively,stars star 1 ,ata of galaxies as subsystems of their respective cluster. These 𝑟 𝑀 (𝑟 ) given distance 1. The empirical determination of gas 1 subsystems form scalar field halos of their own which contain 𝑀 (𝑟 ) and star 1 from directly observable quantities is a subtle real energy (different from the classical MOND theory which question; it will be discussed in Section 4.1. leads to phantom halos only). In the framework of the present Large scale gravitational effects on the cluster level are approach, the scalar field halos in the neighbourhood of often modelled in the Newton approximation of Einstein each galaxy contribute to the energy density which adds up globally, that is, on the cluster level, to a component of gravity with baryonic matter and an assumed dark matter 𝜌 halo which is inferred from its gravitational effects (in scalar field energy sf2 which has been suppressed in the first continuity approximation of the total baryonic mass. Einstein/Newton gravity). Another minority approach in the In a second step we therefore determine this component literature works with an evaluation of the data in a MOND approximately and add it to the total the scalar field halo. limit of the most well known relativistic MOND theory TeVeS. Sanders is one of its protagonists; he concludes that in 𝜌 𝜌 this approach a much smaller amount of unseen matter has 3.3. Scalar Field and Phantom Halos sf1 and ph1 in the Cluster to be assumed in addition to the baryonic mass. Its value is Barycentric MOND Approximation. The baryonic mass up to radius 𝑟, consistent with the hypothesis of a halo of sterile neutrinos, concentrated about the cluster center [5]. Here we want to 𝑟 𝑀 (𝑟) =4𝜋∫ 𝜌 (𝑢) 𝑢2𝑑𝑢, bar bar (68) explore the feasibility of the WST approach, in particular 0 regarding the question of how much unseen matter has to 𝑎 = 𝐺(𝑀(𝑟)/𝑟2) be added to the gravitational effects of the model in order to determines the Newton acceleration bar due reproduce (“predict”) the observed acceleration, respectively, to the total baryonic mass. The densities of the scalar field 𝜌 their measurable effects. halo sf1 and the phantom halo of WST MOND follow from (65) and (66). They are 3.2. Two Contributions to the Scalar Field Energy in Clusters 𝜌 𝜌 1 of Galaxies. The mass distribution of the hot gas gas in a sf 𝛽 galaxy cluster is described by a -profile (67) in a locally 1/2 (69) 3 𝑎𝑜 −1 󵄨 󵄨 𝑎 static coordinate system with origin at the barycenter of = (󵄨 󵄨) (𝜌 + (8𝜋𝐺) ∇(󵄨𝑎 󵄨)⋅󵄨 bar󵄨), 4 󵄨𝑎 󵄨 𝑚 󵄨 bar󵄨 󵄨𝑎 󵄨 the cluster. The averaged star mass will be represented by 󵄨 bar󵄨 󵄨 bar󵄨 𝜌 𝛽 a continuous distribution star of a -profile with the same 1 𝛽, 𝑟 𝜌 𝜌 = 𝜌 . parameters 𝑐, but with a different value of 𝑜.Both ph1 3 sf1 (70) together form a continuity model of the baryonic mass 𝜌 =𝜌 +𝜌 distribution bar gas star. Estimates show that Newtonian The respective masses of the halos 𝑀 1,𝑀 1 arise from 𝜌 sf ph gravitational acceleration induced by bar (far away from mass integration. concentrations stars, galaxies, and galactic centers) is below 102 ⋅𝑎 𝑜. They are small enough for allowing working in a 3.4. Scalar Field Halos of Galaxies in Their Respective Galac- weak field static approximation with the scalar field equation tocentric MOND Approximations. As already indicated in in the MOND approximation (43). The resulting contribution Section 3.2 (69) and (70) do not make allowance for the fact 𝜌 tothescalarfieldenergywillbecalled sf1. The additional that the star matter forms a discrete structure of an ensemble acceleration of the Weyl geometrical scale connection (66) of galaxies each of which is falling freely in the inertiogravita- can be expressed in terms of a phantom mass density which tional field of the supersystem (hot gas and other galaxies). 𝜌 will be called ph1. Every galaxy possesses a local MOND approximation with In this first approximation the star mass is approximated regard to its own barycentric static reference system. The 𝑎 on a par with the hot gas; that is, it is described by its acceleration bar of the supersystem (with respect to the continuously smeared out mean density. But stars are agglom- barycenter rest system of the hot gas) is transformed away erated in galaxies which form freely falling subsystems of in each of the local MOND approximations. The latter leads the cluster with considerable interspaces in the supersystem to a galactic scalar field halo which persists under changes of (the cluster). For each subsystem a locally static coordinate reference systems with small, that is, nonrelativistic, relative system with origin at the respective galactic center can be velocities. It contributes to the total energy of the scalar field, chosen. In this system the local inhomogeneities of star mass calculated in the cluster barycentric system. (Of course this distribution in the cluster and the resulting inhomogeneities is not the case for the phantom halo of the single galaxies.) of the gravitational field in the vicinity of the galaxy can Inprinciple,wehavetoaddupalloftheseeffectstoascalar 𝜌 be calculated. On the galaxy level the MOND theory has field energy density sf2 in order to fill in these lacunae. But proven effective for modelling gravitational effects deviating an exact calculation would have to solve a highly nontrivial from Einstein and Newton gravity without assuming real 𝑁-body problem for the motion of the galaxies. dark matter [20]. Although we expect that the MOND The experience with the calculation of the combined approximation of WST gravity shows similar features, this is MOND halos of stars inside galaxies shows that a resolution not the point in the present investigation. of the star matter inside galaxies down to individual galaxies Journal of Gravity 11 is not necessary to achieve good results. In the outer region the transformation to the cluster barycentric system. In 𝜌 of galaxies a continuity model for the distribution of star MOND there is therefore no analogy to sf2;thelatteristhe matter in the galactic disk gives reliable approximations for crucial distinctive feature between the two approaches. For a the MOND acceleration. Similarly we want to check whether comparison of WST-3L and usual MOND approaches with also here a continuity model for the system of galaxies regard to galaxy clusters it is not sufficient to evaluate the alone, abstracting from the gas mass, leads to an acceptable difference between the transformation functions 𝜇(𝑥) (55). 𝜌 approximation for sf2. For this calculation, the gas mass has Foranevenwidercomparisonwithotherapproaches to be omitted because the galaxies are falling freely in the it may be useful to add up the scalar field and phantom outer field of the cluster; the gravitational potential of the halos to a kind of “dark matter” halo or, more precisely, to latter does not enter the local MOND approximation of the a substitute for the latter. But one must not forget that in galaxies. the present model there is no dark matter in the ordinary Using (65) again we get for the second (inhomogeneity) sense. Here we only find a transparent halo made up of the component of the scalar field energy (real) energy density of the scalar field and the (fictitious) phantom energy density ascribed to the acceleration effects 1/2 3 𝑎𝑜 of the scale connection in Einstein gauge (with respect to the 𝜌 = (󵄨 󵄨) sf2 4 󵄨𝑎 󵄨 cluster barycentric rest system): 󵄨 star󵄨 (71) 𝜌 =𝜌 +𝜌 . −1 󵄨 󵄨 𝑎 𝑡 sf ph1 (75) ⋅(𝜌 + (8𝜋𝐺) ∇(󵄨𝑎 󵄨)⋅󵄨 star󵄨), star 󵄨 star󵄨 󵄨𝑎 󵄨 󵄨 star󵄨 From the gravitational lensing point of view, it would be even 𝜌 |𝑎 (𝑟)| = 𝐺(𝑀 (𝑟)/𝑟2) 𝑀 (𝑟) more appropriate to consider sf alone as the WST equivalent with star star and star the integral 𝜌 𝜌 ofadarkmatterhalo,notforgettingthateventherealhalo sf analogous to (68) for the star density star. is not due to fermionic particles, but to the scalar field, and We finally arrive at a halo model for galaxy clusters 𝜌 ,𝜌 , 𝜌 thus to the extended gravitational structure of WST. From a constituted by the components sf1 sf2 and ph1.Allof quantum point of view, the scalar field has to be quantized if them are determined by the two component baryonic profile one wants to search for a (bosonic) particle content of 𝜌𝑡 or of the cluster. 𝜌 sf . The total dynamical mass of the model (up to some 3.5. A Three-Component Halo Model for Clusters of Galaxies. distance 𝑟 from the center of the cluster) is In addition to the Newtonian gravitational effects of the 𝑀 =𝑀 +𝑀, baryonic mass density tot bar 𝑡

𝜌 =𝜌 +𝜌 , 𝑀𝑡 =𝑀 +𝑀 1, bar gas star (72) sf ph (76) 𝑀 =𝑀 +𝑀 modeled by a 𝛽-model of type (67), the WST MOND sf sf1 sf2 approach predicts acceleration generated by the scalar field 𝑀 =𝑀 +𝑀 halos (69) and (71). Because of their small values their com- with bar gas star.Thelensing mass is a bit smaller, bined effect can be approximated by a linear superposition in 𝑀 =𝑀 +𝑀 . thebarycentricreferencesystemofthecluster,andbecause lens bar sf (77) of slow (i.e., nonrelativistic) relative velocities of the galaxies the energy densities of their respective scalar field halos can Mathematically, the integral of the scalar field energy be taken over to the cluster barycentric reference system: density to arbitrary distances diverges. As in the dark matter approach a virial radius of a cluster may be defined, which 𝜌 ≈𝜌 1 +𝜌 2. (73) roughly delimits the gravitational binding zone of the cluster. sf sf sf 𝜌 𝜌 Comparing 𝑡 (resp., ( sf )) with the critical energy density 𝜌 𝑟 Moreover, there arises acceleration 𝑎𝜑 due to the scale crit of the universe, one may, for example, choose 200 with connection in the barycentric rest system of the cluster (45). 𝜌 (𝑟 ) ≈ 200𝜌 Its gravitational effects are representable by the fictitious (net) 𝑡 200 crit (78) phantom halo 𝜌 1 of (70) ph as a representative of the virial radius. 1 Not far beyond the gravitational binding zone of the 𝜌 = 𝜌 . ph1 3 sf1 (74) cluster, the energy density will have fallen to such a small amount that its centrally symmetric component is inconceiv- On the other hand, the phantom energies of the individual ably stronger than the density fluctuations in the intercluster freely falling galaxies do not survive the transformation to the space. Tocontinue the integration into this region and beyond cluster rest system and do not play a role on the cluster level. has no physical meaning. In the long range the energy density In the usual MOND theories there is no scalar field of the scalar field approaches the cosmic mean energy value. energy; all additional effects with regard to Newton dynamics A physical limit of integration has to be chosen close to the may be ascribed to a (fictitious) phantom energy density. virial radius beyond which the gravitational binding structure Phantom energy densities of single galaxies do not survive of the cluster is fading out. 12 Journal of Gravity

4. A First Comparison with Empirical Data where 𝐴 = 1.50 ± 0.02 and 𝜎V is the 3-dimensional velocity dispersion inside a sphere of virial radius (by 4.1. Empirical Determination of Mass Data for Galaxy Clusters. convention 𝑟V =𝑟200). Reasons for this choice are For a first empirical exploration we confront the WST cluster given in [26, sec. 3]. 𝑀500 was then determined from model with recent mass data for 19 clusters obtained on 𝑀200 by a NFW model. the background of Einstein/Newton gravity and ΛCDM by a group of astronomers about Zhang et al. [2, 4]. We 4.2. Theory Dependence of Mass Data for Galaxy Clusters. use the form parameters 𝛽,𝑐 𝑟 of the 𝛽-models of these Mass densities of the hot gas and of star matter in galaxy clus- clusters, published in an earlier study by one author of ters are indirectly inferred from observable quantities; they the group [15]. In the present study we have to take the are thus theory dependent. Even inside the same background mass data and the form parameters essentially at face value. theory they may depend on choices of models and methods Methodological questions arising from this procedure are of evaluation. discussed in the next subsection. There seem to be sufficient That makes it a difficult task to compare our model with reasons for expecting that the different background theories empirical data. A fine-grained judgement presupposes an do not principally invalidate the results thus obtained. Of evaluation of observational raw data on the background of course, an authoritative empirical study would presuppose an WST gravity or, at least, a detailed estimation of systematic evaluation of observational data on the background of WST errors resulting from a comparison of different background itself; it can be done only by astronomers, if they get interested theories (Einstein gravity with Λ𝐶𝐷𝑀 and Newton approx- in the present approach. imation or alternatively TeVeS-MOND, in comparison with ThestudiesofZhangetal.havethegreatadvantage WST and its MOND approximation). This task has to be left to build upon three independent observational datasets for to astronomers, if they become sufficiently interested in the determining the gas mass 𝑀 ,500, the star mass 𝑀∗,500,and gas present approach. But, taking this caveat in mind, it still seems the total mass 𝑀500 (assumingadarkmatterexplanationfor possible to confront available data from, for example, the the observed gravitational effects) at the reference distance Einstein gravity-Λ𝐶𝐷𝑀-Newton approximation framework 𝑟500. The latter is determined for each cluster at the distance with our model, in order to get a first impression of its 𝑟500 from the cluster center at which the total gravitational potential usefulness. A comparison with mass data derived acceleration indicates a total mass density 500 times the in a TeVeS-MOND background would give welcome supple- critical density. mentary information. This is not attempted here. 𝑀 Let us discuss the possibility and the problems of a (i) gas,500 has been extracted from X-ray data on the hot intracluster medium (ICM) collected by XMM New- confrontation of these data with the MOND approximation tonandROSAT.Surfacebrightnessdatahavebeen of WST: used to infer an ICM radial electron number density (1)Themassofthehotgas(intraclustermedium)𝑀 ,500 profile, and spectral analysis data gave information on gas (at 𝑟500) has been determined in the mentioned study the radial temperature distribution. From that a gas from X-ray data obtained by XMM Newton and density distribution has been reconstructed and the 𝑟 ROSAT. The temperature of the gas is estimated by a gasmassesat 500 by integration (an outline of the fit to the measured spectrum. The gas density 𝜌(𝑟) is procedureandliteratureformoredetailsisgivenin reconstructed, using model assumptions, from inten- [4, p.3]). sity observables and then integrated up to 𝑟500 (in this (ii) 𝑀∗,500 has been determined from optical imaging evaluation the hydrostatic assumption was corrected dataduetoSDSS7intwosteps.Firstthetotallumi- by taking the velocity dispersion into account [4, sec. nosity of the cluster has been determined by means of 2.2f.]). Up to usual model dependence, the transfer of a “galaxy luminosity function” (GLF); then the mass the mass data from the standard gravity background is estimated using mass-to-light ratios depending on to WST seems to be relatively uncritical. the cluster mass. In the last step models of the star (2) Several methods for determining the stellar mass development in the respective galaxy, elliptical or 𝑀∗,500 are mentioned in [4]. In this study the star spiral, enter. They depend on assumptions on an mass is gained from optical imaging data due to SDSS “initial mass function” (IMF). Two possibilities for 7 in two steps indicated in (ii) above. According to the IMF (Salpeter versus Kroupa) are considered and the authors the difference of the stellar mass estimate compared in [2, 4]. According to the authors the due to different initial mass functions can result in difference of the stellar mass estimate can result in factor 2 [4, p.4]. Another approach would be to factor 2 [4, p.4]. estimate stellar masses of the individual galaxies and (iii) The total cluster mass 𝑀500 has been determined “to construct the stellar mass functions in order to on the basis of the velocity dispersion of galaxies, sum the stellar masses” (ibid., p.1). Moreover, an using spectroscopic data from [25, tab. 1]. The mass additionalcomponentofstarmattercanbeassociated estimator used is equation (2) of [26] with the intracluster light. All in all, the estimate of the star mass concentrated in galaxies seems to depend 𝜎 3 V 14 −1 more on models of galactic star evolution than on 𝑀V =𝐴( ) ×10 h 𝑀⊙, (79) 103 km s−1 the background gravity theory. In spite of that the Journal of Gravity 13

precision cannot be expected to be better than by approximation) [2] contains a correction to the main paper factor 2 (resp., 0.5). [4]. Here, of course, we use the corrected data). The mass 𝑀 ,𝑀 , 𝑀 (5) (3)InEinsteingravity/Λ𝐶𝐷𝑀 the cluster mass can, in data 500 gas,500 and ∗,500 are given in columns , (7) (8) principle, be estimated from the velocity dispersion 𝜎 ,and of Table 1 in [2]. It is reproduced in our Table 1. 𝑟 (5) of galaxies at distance 𝑅 (from the center) by an esti- The values for 500 arepublishedin[3,tab.1,col. ](here −1 2 mator derived from the virial theorem 𝑀≈𝐺 𝜎 𝑅. Table 2). A comparison of the total cluster masses derived 𝑎 =𝑎 +𝑎 from the velocity dispersion with a mass estimate derived The additional acceleration of WST-3L add 𝜑 sf (49) is dynamically indistinguishable from the effects from the gas mass shows that the two clusters A2029 and of “true” Newtonian masses. So far it seems as if the A2065 are outliers, with total cluster masses considerably estimation of total mass can be transferred to the higher than the corresponding gas masses would let us MOND approximation of WST without problems. expect. The authors therefore separate the two outliers from But if the radius 𝑅 does not include the “whole” the rest of the data, with the remaining 17 clusters as a reliable dataset [4,p.3].Weshalldothesame. cluster mass (however defined) as is here the case 𝛽, 𝑟 (item (iii), Section 4.8) a surface pressure term must Parameters ( 𝑐) for the models of these galaxy clusters betakenintoaccount.Thatcomplicatesthecase. (as well as of many more) have been published earlier in [15, tab. 4.1]. This publication also contains mass data In standard gravity the necessary correction is imple- 𝑀 ,𝑀 𝑟 200 gas,200 at 200 (with error intervals) and mass values 𝑀∼𝜎3 𝑀 ,𝑀 mented by a cubic mass estimator given 𝐴 gas𝐴 (without error interval) at the Abell radius, here above as (79). Moreover, the authors of our reference defined as 𝑟𝐴 = 2.14 Mpc, but no data for star masses (eval- 𝑀 𝑟 study [4] reconstruct 500 and 500 from these values uated for the value of 𝐻𝑜 assumed in the later publications using the NFW profile. Because of the different profile [2, 4], ℎ = 0.7). In [15] the methods for determining the total for the scalar field halo of WST this isa critical step massandthegasmasswerenotyetasrefinedasinthelater for our exercise (an ex post comparison of the NFW study. It is therefore not possible to aggregate the different haloandtheWSThalofortheComaclusterisgiven datasets to one coherent ensemble (the values for 𝑀500 and 𝑀 in Figure 6). On the other hand, if the resulting gas,500 given here differ from the ones in [4] outside the systematic errors are smaller than the error intervals error intervals). As no updated values for the parameters 𝑀 of tot (76), implied by the observational errors of the (𝛽,𝑐 𝑟 )ofthemassprofileswereavailabletothepresentauthor, otherquantities,theydonotdisturbaroughempirical the values of [15] are here used as estimators for the form check of the model. parameters of the 𝛽 model. 𝑀 ,𝑀 ,𝑀 , 𝑟 (4) Finally the dependency of the data evaluation on the We consider 500 gas,500 ∗,500 and 500 from [2, 4] background cosmology has to be taken into account. as our crucial mass data (including reference radii). Mass The data of the 19 clusters used in the following have data at 𝑟200 (as well as 𝑟200 itself) from the older study are redshift 𝑧 < 0.1.Thegeometricalanddynamical welcome as additional information;buttheywillnotbeused corrections implied by the Λ𝐶𝐷𝑀 cosmology are as core criteria for the empirical test of our model. Table 2 correspondingly small. An evaluation in, for example, collects the data used (for error intervals see the respective a Lemaitre-de Sitter model (or even a nonexpanding source, [2, 15]). It shows that the cluster ensemble covers an 𝑀 Weyl geometric model with redshift) [1, section 4.2] order of magnitude variation for the gas mass gas,500 and would affect the data only by a minor expansion of the one and a half orders of magnitude variation in total mass error intervals. 𝑀500. The selection method by intersecting the cluster sets of different raw data sources does not seem to be influenced by (1) (2) (4) The points , ,and , in particular the estimate of any particular bias. We thus may consider the collection as a stellar mass concentrated in galaxies and gas mass, are fairly reasonable dataset for testing our cluster halo model. insensitive against a change of the background theory from Einstein gravity to WST. The theory dependence of 𝑟500 is uncritical in our context. Any other reference radius could 4.4. Adapting the WST Halo Model to the Data. For the have been taken, as long as it is specified in astronomical construction of the model we have to work in Einstein gauge distance units. The estimate for the total masses at 𝑟500 and (9). Consistency with the deep MOND acceleration demands 𝑟200 is the critical point for our purpose (item (3)). However, (51). Taken together these conditions fix the coefficients of if the difference of the halo profiles between the scalar field the Lagrangian (1) and (28), independently of the convention 𝜙 𝜆 energy density of WST and NFW dark matter does not push chosen for 𝑜.Withavaluefor on the order of magnitude 10 𝜆/4 = 6 the estimates for the total masses 𝑀500,𝑀200 outside the error (e.g., as at the end of Section 2.3) the contribution of 𝐿 intervals of our halo model (due to observational input data), the 𝑉4 term lies many orders of magnitude below the energy we may still be able to draw first inferences from the following densities, the dominant term, in (35) and is negligible in our evaluation. context. In agreement with Section 3 the test of our halo model for 4.3. Empirical Data for 19 Clusters. The studies [2, 4] each of the 19 galaxy clusters can now proceed as follows: contain new data on the baryon content and the total gravitational mass for 19 clusters of galaxies (as it appears (1) Specification of the 𝛽 models (67) for gas and in an Einstein gravity, Λ𝐶𝐷𝑀 framework with Newton for star mass and parameters (𝛽,𝑐 𝑟 )from[15],𝜌𝑜 14 Journal of Gravity

Table 1: Properties of the 19 galaxy clusters (Table 1 of [2]).

𝑀500 𝑀500,𝑀−𝑀 𝑀 ,500 𝑀∗,500 X-ray center (J2000) gas gas Name Redshift 14 14 13 12 Undisturbed/cool-core RA Dec 10 𝑀⊙ 10 𝑀⊙ 10 𝑀⊙ 10 𝑀⊙ A0085 00:41:50.306 −09:18:11.11 0.0556 6.37 ± 1.00 5.68 ± 0.37 8.13 ± 0.38 7.36 ± 1.00 Y/S A0400 02:57:41.349 +06:01:36.93 0.0240 1.83 ± 0.39 1.07 ± 0.07 1.36 ± 0.05 4.39 ± 1.06 N/N IIIZw54 03:41:18.729 +15:24:13.91 0.0311 1.91 ± 0.58 1.18 ± 0.08 1.45 ± 0.26 4.57 ± 0.56 Y/W A1367 11:44:44.501 +19:43:55.82 0.0216 1.76 ± 0.27 2.11 ± 0.14 2.07 ± 0.07 4.35 ± 0.74 N/N MKW4 12:04:27.660 +01:53:41.50 0.0200 0.50 ± 0.14 0.58 ± 0.04 0.47 ± 0.02 1.16 ± 0.22 Y/S ZwCl1215 12:17:40.637 +03:39:29.66 0.0750 4.93 ± 0.98 4.34 ± 0.28 6.10 ± 0.29 7.05 ± 0.83 Y/N A1650 12:58:41.885 −01:45:32.91 0.0845 3.44 ± 0.66 4.28 ± 0.27 5.09 ± 0.73 7.47 ± 1.13 Y/W Coma 12:59:45.341 +27:57:05.63 0.0232 6.55 ± 0.79 6.21 ± 0.40 8.42 ± 0.63 13.14 ± 1.80 N/N A1795 13:48:52.790 +26:35:34.36 0.0616 3.41 ± 0.63 4.46 ± 0.29 5.11 ± 0.14 6.21 ± 0.98 Y/S MKW8 14:40:42.150 +03:28:17.87 0.0270 0.62 ± 0.12 1.10 ± 0.07 0.80 ± 0.12 1.61 ± 0.23 N/N A2029 15:10:55.990 +05:44:33.64 0.0767 14.70 ± 2.61 6.82 ± 0.44 13.35 ± 0.53 9.59 ± 1.11 Y/S A2052 15:16:44.411 +07:01:12.57 0.0348 1.39 ± 0.28 2.03 ± 0.13 1.86 ± 0.10 3.53 ± 0.40 Y/S MKW3S 15:21:50.277 +07:42:11.77 0.0450 1.45 ± 0.34 2.29 ± 0.15 2.13 ± 0.09 3.90 ± 0.43 Y/S A2065 15:22:29.082 +27:43:14.39 0.0721 11.18 ± 1.78 3.35 ± 0.22 7.66 ± 1.44 7.32 ± 0.75 N/W A2142 15:58:19.776 +27:14:00.96 0.0899 7.36 ± 1.25 10.26 ± 0.66 13.76 ± 0.73 8.42 ± 0.77 Y/W A2147 16:02:16.305 +15:58:18.46 0.0351 4.44 ± 0.67 3.63 ± 0.23 5.04 ± 0.53 6.84 ± 0.90 N/N A2199 16:28:37.126 +39:32:53.29 0.0302 2.69 ± 0.42 2.64 ± 0.17 2.97 ± 0.30 4.76 ± 0.50 Y/S A2255 17:12:54.538 +64:03:51.46 0.0800 7.13 ± 1.38 4.08 ± 0.26 7.11 ± 0.33 6.74 ± 0.97 N/N A2589 23:23:56.772 +16:46:33.19 0.0416 3.03 ± 0.75 1.88 ± 0.12 2.54 ± 0.17 5.12 ± 0.56 Y/W

𝑀500,𝑀−𝑀 𝑀500 −𝑀 ,500 𝑀500 Notes. The cluster mass, gas ,isderivedfromthe gas relation and only used for comparison with the cluster mass, ,derivedfromthe “harmonic” velocity dispersion. “S,” “W,” and “N” denote strong cool-core, weak cool-core, and noncool-core clusters.

𝑀 𝑀 𝑟 determined by fitting to gas,500 and ∗,500 at 500, 4.5.TheWSTHaloModelwiththeComaClusterasTestCase. respectively [4]. ForafirstcheckofourmodelwechoosetheComa cluster. According to item (1) of the last section we use the empirical (2) Determination of the Newton acceleration of the input data from Table 2 (units given there): baryonic mass components due to (68). (𝛽, 𝑟 ,𝑟 ,𝑀 ,𝑀 ) 𝑐 500 gas,500 ∗,500 (3) Calculation of the scalar field halo and the phantom (80) halo of the baryonic mass by (69), and(66). = (0.654, 246, 1.278, 8.42, 13.14) ,

(4) Calculation of the scalar field halo of the system of 𝑟 𝑟 𝑀 1013𝑀 𝑀 ( 𝑐 in kpc, 500 in Mpc, gas,500 in ⊙,and ∗,500 freely falling galaxies (71). 12 in 10 𝑀⊙). The different components of the cluster halo (5) Aggregation of these to the total halo (75) and choice integrate to masses (up to distance 𝑟)documentedinFigure1. 𝑟 𝜌 of fading out functions beyond 200 (see Appendix). The scalar field halo (SF) of the total baryonic mass sf1 contributes the lion share to the total transparent energy. The (6) Integration of the densities to the corresponding 𝜌 SF of the galaxy system sf2 carries about as much energy masses: scalar field energy of the galaxies 𝑀 2 of the sf as the net phantom component 𝜌 1 in the barycentric rest 𝑀 𝑀 ph gas mass sf1, total scalar field halo sf ,netphantom system of the cluster. It surpasses the gas mass close to the energy of the baryonic mass M 1, and finally the total ph reference radius 𝑟500. Figure 2 shows the fast increase of 𝑀 𝑀 transparent matter 𝑡 and the lensing mass lens (76) the gravitational mass of the scalar field halo of the galactic and (77). system. (7) Calculation of the error intervals of the model at If we add all baryonic and halo contributions, the picture given in Figure 3 emerges. It shows an encouraging agreement selected distances (𝑟500,𝑟200). of the observed values 𝑀500 at 𝑟500 = 1278 kpc with the 𝑀 (𝑟 ) (8) Comparison of the empirical value for 𝑀500 (resp., prediction of the Weyl geometric halo model tot 500 (in 𝑀200)withthemodelvalue𝑀𝑡(𝑟500) (resp., 𝑀𝑡(𝑟200)). therangeoftheobservationalerrorsandofmodelerrors): +0.97 𝑀 (𝑟 ) = 5.66 −0.68 ×1014𝑀 , The results of this test are given in Section 4.6 Before we turn tot 500 ⊙ (81) to the overall evaluation, we shall have a look at one cluster as +0.79 −0.79 14 an exemplary case. 𝑀500 =6.55 ×10 𝑀⊙. Journal of Gravity 15

Table 2: Dataset used for halo model (error intervals omitted). 2.0 × 1014

𝛽𝑟𝑐 𝑟500 𝑀500 𝑀 500 𝑀∗500 𝑀200 𝑟200 𝑀𝐴 ) Cluster gas ⊙ 14

Coma 0.654 246 1.278 6.55 8.42 13.14 13.84 2.3 12.86 1M 1.5 × 10

A85 0.532 59.3 1.216 6.37 8.13 7.36 7.71 1.9 8.72 in ( 14 A400 0.534 110. 0.712 1.83 1.36 4.39 1.48 1.09 2.93 1.0 × 10 M(r) IIIZw54 0.887 206 0.731 1.91 1.45 4.57 2.81 1.35 4.51 5.0 × 1013 A1367 0.695 274 0.893 1.76 2.07 4.35 4.06 1.53 5.77 Coma MKW4 0.44 7.86 0.58 0.5 0.47 1.16 0.71 0.86 1.79 ZwCl215 0.819 308 1.098 4.93 6.1 7.05 10.37 2.09 10.65 500 1000 1500 2000 A1650 0.704 201 1.087 3.44 5.09 7.47 11.14 2.15 11.11 r (kpc) M M A1795 0.596 55.7 1.118 3.41 5.11 6.21 10.99 2.14 11.04 sf2 gas M M MKW8 0.511 76.4 0.715 0.62 0.8 1.61 2.38 1.28 4.0 bar star A2029 0.582 59.3 1.275 14.7 13.35 9.59 13.42 2.29 12.59 Figure 2: Comparison of the contribution of the scalar field halo of A2052 0.526 26.4 0.875 1.39 1.86 3.53 2.21 1.25 3.79 the galaxies with the baryonic mass for the Coma cluster (empirical 𝑀 MKW3S 0.581 47 0.905 1.45 2.13 3.9 3.46 1.45 5.11 data bar,500 violet dot). A2065 1.162 493 1.008 11.18 7.66 7.32 16.69 2.45 14.46 A2142 0.591 110 1.449 7.36 13.76 8.42 15.03 2.36 13.61 1.5 × 1015

A2147 0.444 170. 1.064 4.44 5.04 6.84 3.46 1.45 5.15 ) ⊙ A2199 0.655 99.2 0.957 2.69 2.97 4.76 4.80 1.62 6.37 1M 0.797 423 1.072 7.13 7.11 6.74 13.32 2.27 12.52 1.0 × 1015 A2255 in ( A2589 0.596 84.3 0.848 3.03 2.54 5.12 3.58 1.47 5.24 𝛽 𝑟 𝑀 𝑟 𝑀 𝑟 𝑀 𝑀 M(r) Source: , 𝑐, 200, 200,and 𝐴 from [15]; 500 from [3]; 500, gas500,and 5.0 × 1014 𝑀∗500 from [2]. 14 13 𝑟 𝑟 𝑟 𝑀 10 𝑀 𝑀 10 𝑀 Coma Units. 𝑐 in kpc, 500, 200 in Mpc, 500 in ⊙, gas,500 in ⊙,and 12 𝑀∗,500 in 10 𝑀⊙. 1000 1500 2000 2500 r (kpc)

15 M M 1.4 × 10 tot ph1 M M 15 t bar ) 1.2 × 10 M ⊙ sf 1.0 × 1015 1M Figure 3: Contribution of the baryonic mass 𝑀 ,ofthescalar 14 bar in ( 8.0 × 10 𝑀 ,𝑀 𝑀 field and phantom energies sf ph1 to the transparent mass 𝑡 14 𝑀 6.0 × 10 andtothetotalmass tot of the Coma cluster in the WST model. M(r) 𝑟 𝑟 14 Model errors indicated at 500 and 200 (black). Empirical data for 4.0 × 10 𝑀 𝑀 bar (violet dot) and for total mass 500 with error bars (violet) from Coma 14 2.0 × 10 [2]. Additional empirical data at 𝑟200 (yellowish) from [15].

1000 1500 2000 2500 r (kpc) our model errors we allowed a variation in stellar masses by M M factors 0.5 and 2. t sf2 M M 𝑟 = 2.14 sf ph1 At the Abell radius 𝐴 Mpc (in the convention of [15]) we get the model values Figure 1: Halo components of Coma cluster: transparent matter 𝑀 =𝑀 +𝑀 𝑀 14 halo 𝑡 sf ph1, total scalar field (SF) halo sf ,SFhalo 𝑀 (𝑟𝐴) ≈ 10.64 × 10 𝑀⊙, 𝑀 𝑀 tot of freely falling galaxies sf2, and net phantom energy ph1 (in (82) barycentric rest system). Empirical data (violet dot, bar): galaxy and 𝑀 (𝑟 ) ≈ 8.82 × 1014𝑀 lens 𝐴 ⊙ total mass (with error intervals) at 𝑟500 = 1280 kpc. for the dynamical total mass and the lensing mass (77). Due to the net phantom energy the dynamical mass is about Model error bars have been estimated by varying the input 20% higher than the lensing mass. This is an effect by data (80) in their respective error intervals. The reconstruc- which the model can, in principle, be tested empirically and 𝛽 tion of star mass from observational raw data is a particularly discriminated from others. The baryonic masses in the 𝑀 (𝑟 ) ≈ 11.73 × 1013𝑀 𝑀 (𝑟 ) ≈ 1.83 × delicate point. Depending on the assumptions on the stellar model are gas 𝐴 ⊙, ∗ 𝐴 1013𝑀 𝑀 (𝑟 ) dynamics and the resulting data evaluation model “one can ⊙.Thusnotonlydoesthetotalmass tot 500 given obtain up to a factor of 2 fewer stars” [2, p.6]. For obtaining by our model agree with the empirical value 𝑀500 inside 16 Journal of Gravity

Table 3: Empirical and model values for total mass at 𝑟500, 𝑟200. Table 4: Model values for halo and baryonic masses at 𝑟200. 𝑟 𝑀 (𝑟 )𝑀 𝑟 𝑀 (𝑟 )𝑀 𝑀 𝑀 𝑀 𝑀 𝑀 𝑀 𝑓 𝑓 Cluster 500 tot 500 500 200 tot 200 200 Cluster 𝑡 sf sfgal ph1 gas ∗ ∗ 𝑡 +0.97 +0.79 +2.71 +1.49 Coma 1278 5.66 −0.68 6.55 −0.79 2300 14.38 −1.90 13.84 −1.41 Coma 12.32 9.91 2.66 2.42 1.78 0.278 0.16 6.9 +0.62 +1 +1.38 +0.8 A85 9.08 7.22 1.61 1.87 1.65 0.15 0.09 5.5 A85 1216 4.95 −0.43 6.37 −1 1900 10.88 −0.97 7.71 −0.74 +0.3 +0.39 +0.68 +0.21 A400 2.7 2.21 0.73 0.49 0.29 0.093 0.32 9.4 A400 712 1.38 −0.19 1.83 −0.39 1093 3.08 −0.44 1.48 −0.18 +0.38 +0.58 +1.34 +2.74 IIIZw54 3.21 2.62 0.86 0.59 0.27 0.085 0.32 11.8 IIIZw54 731 1.45 −0.28 1.91 −0.58 1350 3.56 −1.09 2.81 −1.1 +0.34 +0.27 +0.93 +0.45 A1367 4.16 3.37 0.99 0.79 0.42 0.088 0.21 10. A1367 893 1.95 −0.23 1.76 −0.27 1529 4.67 −0.62 4.06 −0.4 +0.11 +0.14 +0.26 +0.07 MKW4 1.2 0.97 0.3 0.22 0.1 0.0260 0.25 11.5 0.6 −0.08 0.5 −0.14 1.33 −0.17 0.71 −0.06 MKW4 580 857 ZwCl215 8.54 6.82 1.66 1.72 1.14 0.131 0.12 7.5 +0.55 +0.98 +1.64 +3.51 ZwCl215 1098 3.94 −0.38 4.93 −0.98 2093 9.8 −1.17 10.37 −2.62 A1650 9.16 7.36 1.94 1.81 1.15 0.169 0.15 8. +0.71 +0.66 +2.91 +5.77 A1650 1087 3.65 −0.05 3.44 −0.66 2150 10.48 −2.41 11.14 −3.46 A1795 9.39 7.51 1.86 1.88 1.29 0.157 0.12 7.3 +0.49 +0.63 +1.47 +2.26 A1795 1118 3.65 −0.32 3.41 −0.63 2136 10.84 −0.99 10.99 −2.09 MKW8 2.53 2.05 0.6 0.48 0.22 0.045 0.2 11.3 +0.2 +0.12 +0.74 +1.04 MKW8 715 0.94 −0.15 0.62 −0.12 1279 2.80 −0.6 2.38 −0.59 A2029 14.58 11.53 2.37 3.05 3.1 0.223 0.07 4.7 +0.73 +2.61 +2.01 +2.43 A2052 3.03 2.44 0.70 0.58 0.34 0.065 0.19 8.9 A2029 1275 6.75 −0.51 14.7 −2.61 2286 17.9 −1.42 13.42 −2.26 +0.31 +0.28 +0.61 +0.06 MKW3S 3.97 3.21 0.91 0.77 0.44 0.081 0.18 9. A2052 875 1.78 −0.21 1.39 −0.28 1250 3.43 −0.41 2.21 −0.08 +0.33 +0.34 +0.78 +0.36 A2065 9.63 7.66 1.75 1.97 1.12 0.107 0.1 8.6 MKW3S 905 1.96 −0.22 1.45 −0.34 1450 4.5 −0.52 3.46 −0.34 +0.81 +1.78 +5.06 +21.34 A2142 13.93 10.98 2.13 2.95 2.74 0.168 0.06 5.10 4.08 −0.74 11.18 −1.78 10.83 −3.34 16.69 −6.73 A2065 1008 2450 A2147 5.38 4.31 1.11 1.07 0.89 0.121 0.14 6. +0.81 +1.25 +1.89 +3.9 A2142 1449 7.5 −0.59 7.36 −1.25 2364 16.83 −1.39 15.03 −2.64 A2199 5.10 4.1 1.11 1. 0.61 0.098 0.16 8.4 +0.61 +0.67 +1.26 +1.17 A2147 1064 3.53 −0.45 4.44 −0.67 1450 6.39 −0.99 3.46 −0.74 A2255 10.4 8.27 1.88 2.13 1.51 0.143 0.09 6.9 +0.44 +0.42 +1.14 +0.37 A2199 957 2.43 −0.32 2.69 −0.42 1621 5.8 −0.82 4.81 −0.36 A2589 4.74 3.83 1.11 0.91 0.59 0.119 0.2 8. +0.53 +1.38 +1.88 +1.44 14 Mass values in 10 𝑀⊙, 𝑓∗ =𝑀∗/𝑀 ,𝑓𝑡 =(𝑀𝑡/𝑀 )(𝑟200);for𝑟200 see A2255 1072 4.13 −0.37 7.13 −1.38 2271 12.04 −1.37 13.32 −1.19 gas gas +0.39 +0.75 +1.06 +3.86 Table 3. A2589 848 2.08 −0.26 3.03 −0.75 1471 5.44 −0.73 3.58 −1.54 𝑀 (𝑟 ) 𝑀 1014 𝑀 𝑟 Model values tot 𝑁00 and empirical values 𝑁00 in ⊙, 𝑁00 𝑁=2,5 (empirical) in kpc ( ). intervals. This indicates a surprising agreement between the (theoretically derived) transparent halo and the empirically 𝑀 ≈𝑀 𝑀 𝑀 determined dark halo, 𝑡 dm. the error bounds but also 𝐴 and 200 are reasonably well For 4 clusters, MKW8, A2255, and outliers A2029 and recovered. A2065, the error intervals of empirical data and model data This is the case without assuming any component of do not overlap. For the first two of them, MKW8 and A2255, particle dark matter besides the (real) energy of the scalar the model predictions are consistent with the empirical data fieldandthe(phantom)energyascribedtotheadditional 2𝜎 𝑎 within doubled error intervals ( range). Only the two acceleration 𝜑 induced by the Weylian scale connection. This outliers (A2029, A2065) lie farther apart (A2029 has the might still be a coincidence. In order to learn more about surprising property that the empirical value for the total the question whether the findings at the Coma cluster are mass at 𝑟500 surpasses the one at 𝑟200, 𝑀500 >𝑀200). exemplary or not, we have to consider the data of all 19 galaxy Otherwise the model data are in good agreement with an clusters, respectively, the 17 of the reliable subensemble. assumption of normally distributed statistical errors and with the assumption that the evaluation bias due to the use of the 4.6. Halos and Total Mass for 17(+2) Clusters of Galaxies. NFW profile for dark matter (item (3) in Section 4.2) does not The mass values of the halo models for the 19 clusters are shift the mass estimates outside the error intervals. calculated as described in Section 4.3 with the choice of Allinall,theassessmentoftheWST-3Lhalomodel fadeout functions beyond 𝑟=𝑟200 (see Appendix). The results has surprisingly well passed, in spite of the main caveat aredocumentedinTables3and4andFigures4and5(for of item (3) (Section 4.2). The outcome found for Coma ComaseeFigure3). seems to be typical also for the other clusters. Moreover, the For 15 clusters our model reproduces the total mass good agreement of the model with the data of the reliable at 𝑟500 correctly, that is, inside the error margins of data subensemble of 17 clusters supports the assumption stated in and model. This is achieved without any further adjustable the last phrase (3) (Section 4.2) (no large systematic errors parameter, only on the basis of the parameters for the 𝛽- due to data transfer from Einstein/Λ𝐶𝐷𝑀 to the WST- 𝑀 (𝑟 ), 𝑀 (𝑟 ) model for baryonic mass and gas 500 ∗ 500 . The fading 3L framework). But we cannot exclude the possibility of outfunctionsdonotintervenebelow𝑟200.Moreover,forthe cancelling between model errors and data transfer errors. majority of these and paradoxically for all other four, the less Thuswehaveonlyfoundempiricalsupportfortheconjecture precisely determined data at 𝑟200 have overlapping 1𝜎 error that,onthelevelofgalaxyclusters,theobserveddarkmatter Journal of Gravity 17

7×1014 1.5 × 1015 ) 14

⊙ 6×10 ) ⊙

1M 5×1014

1M 15 1.0 × 10 in ( 4×1014 in ( 14 M(r) 3×10 M(r) 5.0 × 1014 2×1014 A400 A85 1×1014

1000 1500 2000 2500 1000 1500 2000 2500 r (kpc) r (kpc)

1×1015 )

14 ) ⊙ 6×10 ⊙ 8×1014 1M 1M

in ( 14 4×1014 in ( 6×10

M(r) 14 M(r) 4×10 2×1014 2×1014 A1367 IIIZw54

1000 1500 2000 2500 1000 1500 2000 2500 r (kpc) r (kpc) 4×1014 1.2 × 1015 ) ) ⊙

⊙ 15 3×1014 1.0 × 10 1M 1M 8.0 × 1014 in ( in ( 2×1014 6.0 × 1014 M(r)

M(r) 14

215 4.0 × 10 1×1014 14

MKW4 2.0 × 10 ZwCl

1000 1500 2000 2500 1000 1500 2000 2500 r (kpc) r (kpc) 1.4 × 1015 15 1.2 × 10 1.2 × 1015 ) ) ⊙ 15 ⊙ 1.0 × 10 1.0 × 1015 1M 1M 14 14

in ( 8.0 × 10 in ( 8.0 × 10 6.0 × 1014 6.0 × 1014 M(r) M(r) 4.0 × 1014 4.0 × 1014 A1650 2.0 × 1014 A1795 2.0 × 1014

1000 1500 2000 2500 1000 1500 2000 2500 r (kpc) r (kpc) M M M M tot ph1 tot ph1 M M M M t bar t bar M M sf sf 𝑀 𝑟 ,𝑟 𝑀 Figure 4: Halo models for clusters 2–9 in Table 2: total mass tot (blackline)withmodelerrorbarsat 500 200, transparent matter halo 𝑡 𝑀 𝑀 𝑀 constitutedbyscalarfieldhalo sf , and net phantom halo (in barycentric rest system of cluster) ph1 and baryonic mass (gas and stars) bar. Empirical data for the total mass with error intervals at 𝑟500 (violet) from [2]. Additional empirical data at 𝑟200 (yellow) from [15]. 18 Journal of Gravity

8×1014 ) ) ⊙ ⊙ 15 6×1014 1.5 × 10 1M 1M in ( in ( 15 4×1014 1.0 × 10 M(r) M(r) 2×1014 5.0 × 1014 A2029 MKW8

1000 1500 2000 2500 1000 1500 2000 2500 r (kpc) r (kpc) 15 8×1014 1×10 ) ) ⊙ ⊙ 8×1014 14 1M

1M 6×10 14 in (

in ( 6×10 4×1014 14 M(r)

M(r) 4×10 2×1014 2×1014 A2052 MKW3S

1000 1500 2000 2500 1000 1500 2000 2500 r (kpc) r (kpc) 1.2 × 1015 ) ⊙ ) 15 15 ⊙ 1.0 × 10 1.5 × 10 1M

1M 8.0 × 1014 in ( 15 in ( 1.0 × 10 6.0 × 1014 M(r)

M(r) 4.0 × 1014 5.0 × 1014 14 2.0 × 10 A2142 A2065

1000 1500 2000 2500 1000 1500 2000 2500 r (kpc) r (kpc) 1.5 × 1015 1×1015 ) ) ⊙ ⊙ 8×1014 1M 1M 1.0 × 1015 in ( in ( 6×1014

14 M(r) M(r) 5.0 × 1014 4×10 2×1014 A2199 A2147

1000 1500 2000 2500 1000 1500 2000 2500 r (kpc) r (kpc) 1.5 × 1015 1×1015 ) ) ⊙ ⊙ 8×1014 1M 1M 1.0 × 1015 in ( in ( 6×1014

14 M(r) M(r) 5.0 × 1014 4×10 2×1014 A2589 A2255

1000 1500 2000 2500 1000 1500 2000 2500 r (kpc) r (kpc) M M M M tot ph1 tot ph1 M M M M t bar t bar M M sf sf Figure 5: Halo models for clusters 10–19 in Table 2. For description see Figure 4. Journal of Gravity 19

𝑀 effects encoded in dm maysolelybeduetothecombined 15 𝑀 1.4 × 10 impact of the halo sf of the scalar field and of the scale 𝑀 1.2 × 1015 connection, ph1: ) 15 ⊙ 1.0 × 10 𝑀 ≈𝑀 +𝑀 =𝑀 ( . (76)) .

dm sf ph1 𝑡 cf (83) 1M 8.0 × 1014 in ( 14 4.7. Comparison with TeVeS and NFW Halos. It is surprising dm 6.0 × 10 M that in the WST-3L approach the total amount of observed 4.0 × 1014 𝑀 dark matter dm seems to be explained by the energy 14 𝑀 =𝑀 + 2.0 × 10 of the scalar field and the phantom halo, 𝑡 sf 𝑀 𝑀 ≈𝑀 ph1 and dm 𝑡. No missing mass is left. In usual 0 500 1000 1500 2000 2500 relativistic MOND approaches this is not the case for clusters, r (kpc) although it is essentially so for galaxies [20]. Based on a M t WST study of about 40 galaxy clusters, Sanders has proposed M dm NFW the hypothesis of a neutrino component “between a few M 13 14 phm TeVeS times 10 and 10 𝑀⊙,” mostly concentrated close to the center of the cluster, in a region up to twice the core radius, Figure 6: Comparison of dark/transparent/phantom mass halos for supplementing the baryonic mass and the phantom energy of Coma in NFW,WST, and TeVeS models, free parameters of halos for the TeVeS model [5, p. 902]. In our approach, this hypothesis NFW and TeVeS (𝜇2 with neutrino core) adapted to mass data (black 𝑟 = 1280 is unnecessary. Where does this difference arise from? error bars) at 500 kpc. Red error bar indicates variability 𝑟 ≈ 2300 AmodelcalculationfortheComacluster,evaluating(57) of NFW halo values at 200 kpc (fitting the NFW profile 2 −1/2 inside the margins of empirical error bars at 𝑟500); blue error bars for the transition function 𝜇2 =𝑥(1+𝑥) (53) and the give model errors of WST model at 𝑟500 and 𝑟200. corresponding ]2 (54) used by Sanders in [5], shows that the 14 neutrino core had to be tuned to 𝑀] ≈ 1.8 × 10 𝑀⊙ in 𝑀 (𝑟 )= order to give agreement with the empirical value dm 500 14 +0.03 +0.029 +0.035 𝑀 −𝑀 (𝑟 ) ≈ 4.7 × 10 𝑀 −0.03 −0.029 −0.035 −1 500 bar 500 ⊙,whereinthisframework 0.182 , 0.167 , 0.192 in units h Mpc for 𝑟𝑐 “𝑟500”and“𝑟200” are to be defined by a formal convention with [6]) and determine the central density parameter 𝜌𝑜 such that 𝑀 regard to a (fictitious) NFW halo. the total integrated mass NFW assumes the empirical value Table 4 shows the amount of scalar field energy up to a of [2] at 𝑟500 = 1289 kpc (Figure 6). The error intervals for 𝑟500 radius 𝑟≈𝑟200 in the WST model. It varies between about give upper and lower model values for 𝜌𝑜 and corresponding 2×1013 9×1014 𝑀 𝑀 (𝑟 ) and ⊙,thatis,roughlyintherangefound model error bars (red) for NFW 200 .Thisversionofthe necessary by Sanders for the (hypothetical) neutrino halo. NFW halo satisfies one set of empirical data by construction Moreover, a comparison of the transition functions 𝜇𝑤(𝑟) (55) (at 𝑟500); our main interest thus goes to the other empirical and 𝜇2(𝑥) shows that the gross phantom energy 𝜌𝑡 of the WST dataset available at 𝑟200 = 2300 kpc. The NFW model error approach is larger than in an 𝜇2-MOND model [1]. Roughly interval overlaps with the empirical error bar at 𝑟200 and with half of the missing mass of Sanders’ model is covered by this the error interval of the WST model. effect; the other half is due to the scalar field halo of the system There is a conspicuous difference between the mass of galaxies up to 𝑟200. profiles of the NFW halo on one side and the halo profiles A comparison of the two MOND-like approaches is given of WST or 𝜇2-TeVeS on the other. More and precise empirical in Figure 6. Here one has to keep in mind that the TeVeS-𝜇2 data of the interior halos of galaxy clusters ought to be able model has a free adaptable parameter (mass of the neutrino to discriminate between the two model classes. An empirical core), while the WST has not. The general profiles of the “dark discrimination between the two MOND-like approaches matter” halos of both models are similar. The TeVeS-𝜇2 mass would need more, and more precise, profile data. At the starts from a higher score because of its neutrino core; the moment the Weyl geometric MOND-like model survives the WST transparent mass starts from a lower initial value but comparison fairly well, even though it has no free parameter rises faster because of the increasing contribution of the scalar which would allow adapting it to the halo data. field halo of the galaxies. On the other hand, the best known profile for dark matter 4.8. A Side-Glance at the Bullet Cluster. At the end of distribution, used in most structure formation simulations, is this section let us shed a side-glance at the bullet cluster the NFW halo (Navarro/Frank/White). Its profile is 1E0657-56. It is often claimed that the latter provides direct 𝜌 𝜌 (𝑟) = 𝑜 , evidence in favour of particle dark matter and rules out 2 (84) (𝑟/𝑟𝑐) (1 + 𝑟/𝑟𝑐) alternative gravity approaches. Our considerations show that this argument is not compelling. The energy content of the with density parameter 𝜌𝑜 andcoreradius𝑟𝑐 (at which the scalar field halos of the colliding clusters endows them with density has reduced to half the reference value). For a first inertia of their own. The shock of the colliding gas exerts −1 comparison of the interior halos we take 𝑟𝑐 ≈ 180 h kpc dynamicalforcesonthegasmassesonly,notdirectlyon (ℎ = 0.7), following [6] (in studies of the exterior halo the scalar field halos. During the encounter the halos will of the Coma cluster, Geller et al. have found fit values roughly follow the inertial trajectories of their respective 20 Journal of Gravity

𝑀 =𝑀 +𝑀 clusters before collision, and they will continue to do so for The total dynamical mass of the model, tot bar 𝑡 a while. It will take time before a readaptation of the mass (76), has been heuristically confronted with the empirical val- systems and the respective scalar field halos has taken place. ues for of 17 +2 galaxy clusters given in [2, 4], complemented Clearly the MOND approximation is unable to cover such by data from [15]. The problem of data transfer between dif- violent dynamical processes. It describes only the relatively ferent theoretical frameworks (in particular between Einstein stable states before collision and in some distant future after gravity/Λ𝐶𝐷𝑀 and WST-3L) leads to a certain caveat with collision. But a separation of halos and gas masses for a regard to uncorrected taking-over of the values for the total (cosmically “short”) period is to be expected, just as in the mass. But it does not seem to obstruct a meaningful first case of a particle halo with appropriate clustering properties. comparison of the WST halo model with available mass data For the time being, cluster 1E0657-56 does not help to of clusters collected in the Einstein/Λ𝐶𝐷𝑀 framework. decide between the overarching alternative research strate- The result of this comparison shows a surprisingly good 𝑀 gies, particle dark matter, or alternative gravity. It may be agreement of the total mass predicted by the model tot, able to do so, once the dynamics of gas and of the halos has on the basis of data for the baryonic mass components, been modeled with sufficient precision in both approaches. with the empirically determined total mass 𝑀500 (at the Thenapropercomparisoncanbemade.Butthatisan main reference distance 𝑟500). Moreover the model shows an overtly complicated task. It seems more likely that other acceptably good agreement with additional empirical values types of empirical evidence will offer a simpler path to a at the distance 𝑟200 givenin[15](determinedonaslightly differential evaluation of the two strategies and help clarifying less refined data basis and evaluation method). For 15 clusters the alternative. 𝑀 (𝑟 ) the model predicts values for tot 500 with error intervals (due to the observational errors for the baryonic data) which 5. Discussion overlap with the empirical error intervals of 𝑀500.TheComa cluster is among them. Two clusters have overlapping error We have analyzed a three-component halo model for clusters intervals in the 2𝜎 range. The remaining two are outliers and of galaxies, consisting of have been identified as such already in [4]. (i) the scalar field energy induced by the overall baryonic Intheresultwehavefoundempiricalsupportforcon- matter in the barycentric rest system of a cluster jecturing that the observed dark matter at galaxy cluster level (under abstraction of the discrete structure of the star maybeduetothetransparenthaloofthescalarfieldandthe matter clustering in freely falling galaxies) (69), with phantom halo of the scale connection of WST (83). 𝑀 Details for the constitution of the total transparent matter integrated mass equivalent sf1, halo from its specific components (i), (ii),and(iii) have been (ii) an additional contribution to the scalar field energy, investigated for the Coma cluster (Section 4.5). They seem forming around the freely falling galaxies (71), inte- to be exemplary for the whole collection of galaxy clusters. grating to 𝑀 2, sf A particular feature of the model is the scalar field energy (iii) the phantom energy of the total baryonic mass in the formed in the interspaces between the galaxies. Its integrated barycentric rest system of the cluster, due to the addi- energy contribution surpasses the baryonic mass between tional acceleration of the Weylian scale connection in 1and1.5 Mpc (see Figure 2 and Table 4, column 4). It is 𝑀 Einstein gauge (70) with mass equivalent ph1. crucial for this model’s capacity to explain the total dynamic The first two components add up to a real energy content mass on purely gravitational grounds, without any additional 𝑀 = +𝑀 dark matter component. An overall comparison with Sanders’ of the scalar field with mass equivalent sf Msf1 sf2; 𝑀 TeVeS-MOND model for galaxy clusters and the NFW halo the third one, ph1, arises from a theoretical attribution in a Newtonian perspective and has fictitious character. The mass is given in Section 4.7. equivalent of the integrated energy components combines to At the moment the Weyl geometric scalar tensor model 𝑀 =𝑀 +𝑀 +𝑀 a total dark matter-like quantity 𝑡 sf1 sf2 ph1 with a cubic term in the kinetic Lagrangian of the scalar (73). field fares well in all the mentioned respects. It would All of the three components arise from gravitational be very helpful if astronomers decided to evaluate old or 𝑀 =𝑀 +𝑀 effects of the cluster’s baryonic mass bar gas star in new raw data in the framework of WST. That could lead the framework of a Weyl geometric scalar tensor theory of to empirical discrimination of the different models. But gravity (1), with its scale connection as the specific difference already independent of the outcome of such a revision, the to Riemannian geometry (8), (4). In Weyl geometric scalar scalar field 𝜙 and the scale connection 𝜑𝜇 of WST have a tensor theory the scalar field 𝜙 is the new dynamical variable, remarkable property from a theoretical point of view. They while the scale curvature 𝑑𝜑 =𝑓 vanishes. The latter would be complement the classical Einstein-Riemannian expression even more striking and even irritating difference to Rieman- for the gravitational structure, the metric field 𝑔𝜇],bya nian geometry (see [27]); in integrable Weyl geometry it plays feature which carries proper energy momentum (35) and no role. A second speciality of the theoretical framework is (36). The energy momentum of the scalar field plays a crucial the cubic kinetic term of the Lagrangian (28), analogue to role in the constitution of the transparent matter halo. In the the AQUAL approach but in scale covariant form. Observ- present approach it seems to express the self-energy of the able quantities are directly given by the model in Einstein extended gravitational structure. It remains to be seen whether gauge (9). this is more than a model artefact. Journal of Gravity 21

1.0 [3]T.F.Lagana,Y.-Y.Zhang,T.H.Reiprich,andP.Schneider,´ “XMM-Newton/Sloan digital sky survey: star formation effi- 0.8 ciency in galaxy clusters and constraints on the matter-density parameter,” Astrophysical Journal,vol.743,no.1,article13,11 0.6 pages, 2011. [4] Y.-Y. Zhang, T. F. Lagana,´ D. Pierini, E. Puchwein, P. Schnei- 0.4 der, and T. H. Reiprich, “Star-formation efficiency and metal enrichment of the intracluster medium in local massive clusters 0.2 of galaxies,” Astronomy & Astrophysics,vol.535,articleA78,11 pages, 2011. 2000 2500 3000 3500 4000 4500 5000 [5] R. H. Sanders, “Clusters of galaxies with modified Newtonian (kpc) dynamics,” Monthly Notices of the Royal Astronomical Society, vol. 342, no. 3, pp. 901–908, 2003. 𝑓(𝑥, 1.1𝑟 , 0.5𝑟 ) 𝑟 = Figure 7: Fading out function 200 200 for 200 [6] M. J. Geller, A. Diaferio, and M. J. Kurtz, “The mass profile of 2300 kpc. the Coma galaxy cluster,” Astrophysical Journal,vol.517,no.1, pp.L23–L26,1999. [7] P. Jordan, Schwerkraft und Weltall, Vieweg, Braunschweig, Appendix Germany, 1952, 2nd edition 1955. [8] C. Brans and R. H. Dicke, “Mach’s principle and a relativistic Remarks on the Numerical Implementation theory of gravitation,” Physical Review,vol.124,no.3,pp.925– 935, 1961. The calculations described in Sections 3.3 to 3.5 and 4.4 [9] Y. Fujii and K.-C. Maeda, The Scalar-Tensor Theory of Gravita- have been implemented in Mathematica 10 and run on a tion, Cambridge University Press, Cambridge, UK, 2003. PC. Integrations of the mass values have been realized by [10] V. Faraoni and E. Gunzig, “Einstein frame or Jordan frame?” numerical interpolation routines in distance intervals of International Journal of Theoretical Physics,vol.38,no.1,pp.217– 100 kpc. A comparison with refined distance intervals 10 kpc 225, 1999. −4 showed differences at the order of magnitude 10 of the [11] J. Audretsch, F. Gahler,¨ and N. Straumann, “Wave fields in respective values, thus below the rounding precision. Weyl spaces and conditions for the existence of a preferred pseudo-Riemannian structure,” Communications in Mathemat- If one wants to investigate the external halo (beyond 𝑟200) one has to choose fading out functions. The main part of ical Physics,vol.95,no.1,pp.41–51,1984. our investigation deals with the internal halo. Only in the [12] F. P. Poulis and J. M. Salim, “Weyl geometry as characterization Discussion, Section 5, questions of the external halo come of space-time,” International Journal of Modern Physics: Confer- ence Series,vol.3,pp.87–97,2011. into the play. The fading out for the scalar field and phantom [13] I. Quiros, “Scale invariant theory of gravity and the standard halos beyond 𝑟200 has been modeled by the cubic expression: model of particles,” http://arxiv.org/abs/1401.2643. 𝑓 (𝑥, 𝐴,) 𝐵 =𝜒(𝑥; −∞,) 𝐴 [14] E. Scholz, “Paving the way for transitions—a case for Weyl geometry,” in Towards a Theory of Spacetime Theories, D.Lehmkuhl,Ed.,Birkhauser,¨ Basel, Switzerland, 2016, 1 3 (A.1) +( ) 𝜒 (𝑥; 𝐴,) ∞ , http://arxiv.org/abs/1206.1559. 1+(𝑥−𝐴) /𝐵 [15] T. H. Reiprich, Cosmological implications and physical properties of an X-ray flux-limited sample of galaxy clusters [Ph.D. thesis], with 𝜒(𝑥; 𝑎, 𝑏) the characteristic function of the interval [𝑎, 𝑏]. LMU Munich, Munich, Germany, 2001. The fading out of 𝑓(𝑥, 𝐴, 𝐵) starts at 𝐴 and declines to 1/2 at [16] Y. Nakayama, “Scale invariance vs conformal invariance,” 𝐴+𝐵.Inourcasewestartthefadingoutclosetothevirial http://arxiv.org/abs/1302.0884. radius, 𝐴 = 1.1𝑟200,andset𝐵 = 0.5𝑟200 (see Figure 7). [17] R. Adler, M. Bazin, and S. Menahem, Introduction to General Relativity, McGraw-Hill, New York, NY,USA, 2nd edition, 1975. [18] M. Blagojevic, Gravitation and Gauge Symmetries,Instituteof Competing Interests Physics, Bristol, UK, 2002. The author declares no competing interests. [19] S. Weinberg, Gravitation and Cosmology: Principles and Appli- cations of the General Theory of Relativity,JohnWiley&Sons, New York, NY, USA, 1972. References [20] B. Famaey and S. S. McGaugh, “Modified Newtonian dynamics (MOND): observational phenomenology and relativistic exten- [1] E. Scholz, “MOND-like acceleration in integrable Weyl geomet- sions,” Living Reviews in Relativity,vol.15,article10,pp.1–159, ric gravity,” Foundations of Physics,vol.46,no.2,pp.176–208, 2012. 2016. [21] J. Bekenstein and M. Milgrom, “Does the missing mass problem [2] Y.-Y. Zhang, T. F. Lagana,´ D. Pierini, E. Puchwein, P. Schneider, signal the breakdown of Newtonian gravity?” Astrophysical and T. H. Reiprich, “Corrigendum to star-formation efficiency Journal,vol.286,pp.7–14,1984. and metal enrichment of the intracluster medium in local [22] J. D. Bekenstein, “Relativistic gravitation theory for the mod- massive clusters of galaxies,” Astronomy & Astrophysics,vol.544, ified Newtonian dynamics paradigm,” Physical Review D,vol. article C3, 1 page, 2012. 70, no. 8, Article ID 083509, 2004. 22 Journal of Gravity

[23] W. Drechsler and H. Tann, “Broken Weyl invariance and the origin of mass,” Foundations of Physics,vol.29,no.7,pp.1023– 1064, 1999. [24] R. H. Sanders, “The virial discrepancy in clusters of galaxies in the context of modified Newtonian dynamics,” Astrophysical Journal,vol.512,no.1,pp.L23–L26,1999. [25]Y.-Y.Zhang,H.Andernach,C.A.Carettaetal.,“HIFLUGCS: galaxy cluster scaling relations between X-ray luminosity, gas mass, cluster radius, and velocity dispersion,” Astronomy & Astrophysics,vol.526,articleA105,p.38,2011. [26] A. Biviano, G. Murante, S. Borgani, A. Diaferio, K. Dolag, and M. Girardi, “On the efficiency and reliability of cluster mass estimates based on member galaxies,” Astronomy & Astrophysics,vol.456,no.1,pp.23–36,2006. [27] H. C. Ohanian, “Weyl gauge-vector and complex dilaton scalar for conformal symmetry and its breaking,” General Relativity and Gravitation,vol.48,no.3,article25,pp.1–17,2016. Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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