Apartes de la Conferencia de la SJG del 14 y 21 de Enero de 2012: Astroparticle Physics 35_(2012)_362–370

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Authors personal copy Astroparticle Physics 35 (2012) 362–370 Contents lists available at SciVerse ScienceDirect Astroparticle Physics journal homepage: www.elsevier.com/locate/astropartHigh-energy scalarons in R2 gravity as a model for Dark Matter in galaxiesC. Corda a,⇑,1, H.J. Mosquera Cuesta a,b,c,d,2, R. Lorduy Gómez ea International Institute for Theoretical Physics and Advanced Mathematics Einstein-Galilei, Via Santa Gonda, 14, 59100 Prato, Italyb Departamento de Física, Centro de Ciências Exatas e Tecnológicas (CCET), Universidade Estadual Vale do Acaraú (UVA), Avenida Doutor Guarani, 317, CEP 62040-730,Bairro Derby, Sobral, Ceará, Brazilc Instituto de Cosmologia, Relatividade e Astrofísica (ICRA-BR), Centro Brasilero de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, CEP 22290-180, Urca, Rio de Janeiro, RJ, Brazild International Center for Relativistic Astrophysics Network (ICRANet), International Coordinating Center, Piazzale della Repubblica, 10, 065112 Pescara, Italye Grupo de Astronomia Quasar, Universidad EAFIT-Campus Principal, Carrera 49 7sur 50, Avenida Las Vegas, Medellín, Apartado Aereo 3300, Colombiaa r t i c l e i n f o a b s t r a c tArticle history: We show that in the framework of R2 gravity and in the linearized approach it is possible to obtain spher-Received 9 October 2010 ically symmetric stationary states that can be used as a model for galaxies. Such approach could representReceived in revised form 25 August 2011 a solution to the Dark Matter Problem. In fact, in the model, the Ricci curvature generates a high energyAccepted 28 August 2011 term that can in principle be identified as the dark matter field making up the galaxy. The model can alsoAvailable online 21 September 2011 help to have a better understanding on the theoretical basis of Einstein–Vlasov systems. Specifically, we discuss, in the linearized R2 gravity, the solutions of a Klein–Gordon equation for the spacetime curvature.Keywords: Such solutions describe high energy scalarons, a field that in the context of galactic dynamics can be inter-Galactic high energy scalaronsEinstein–Vlasov system preted like the no-light-emitting galactic component. That is, these particles can be figured out like wave-Dark Matter packets showing stationary solutions in the Einstein–Vlasov system. In such approximation, the energy of the particles can be thought of as the galactic dark matter component that guarantees the galaxy equi- librium. Thus, because of the high energy of such particles the coupling constant of the R2-term in the gravitational action comes to be very small with respect to the linear term R. In this way, the deviation from standard General Relativity is very weak, and in principle the theory could pass the Solar System tests. As pertinent to the issue under analysis in this paper, we present an analysis on the gravitational lensing phenomena within this framework. Although the main goal of this paper is to give a potential solution to the Dark Matter Problem within galaxies, we add a section where we show that an important property of the Bullet Cluster can in principle be explained in the scenario introduced in this work. To the end, we discuss the generic prospective to give rise to the Dark Matter component of most galaxies within extended gravity. Ó 2011 Elsevier B.V. All rights reserved.1. Introduction Background Radiation (CMBR), dim Lyman Limit Systems (LLS) and type la supernovae (SNeIa) data, a good framework to under- The accelerated expansion of the universe, that is today ob- stand the today observed Universe. However, it presents severalserved, suggests that cosmological dynamics is dominated by the shortcomings as the well known ‘‘coincidence’’ and ‘‘cosmologicalso-called Dark Energy field which provides a large negative pres- constant’’ problems [1]. An alternative approach is to change thesure. This is the standard picture, in which such new ingredient left hand side of the field equations, and check if observed cosmicis considered as a source of the right hand side of the field equa- dynamics can be achieved by extending general relativity [2–6].tions. It should be some form of non-clustered non-zero vacuum In this different context, it is not required to search candidatesenergy which, together with the clustered Dark Matter, drives for Dark Energy and Dark Matter, which till now have not beenthe global dynamics. This is the so-called ‘‘concordance model’’ found. Rather, one can only stand on the ‘‘observed’’ ingredients:(KCDM) which gives, in agreement with the Cosmic Microwave curvature and baryon matter, to account for the observations. Con- sidering this point of view, one can think of that gravity is not scale-invariant. Such an assumption opens a room for alternative ⇑ Corresponding author. theories to be introduced [7–9]. In principle, the most popular Dark E-mail address: cordac.galilei@gmail.com (C. Corda). Energy and Dark Matter models can be achieved by considering 1 CC is partially supported by a Research Grant of The R. M. Santilli Foundation f(R) theories of gravity [2–9], where R is the Ricci curvature scalar.Number RMS-TH-5735A2310. 2 HJMC is fellow of the Ceará State Foundation for the Development of Science and In this picture, even the sensitive detectors for gravitationalTechnology (FUNCAP), Fortaleza, CE, Brazil. waves like bars and interferometers (i.e. those which are currently0927-6505/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.astropartphys.2011.08.009

Authors personal copy C. Corda et al. / Astroparticle Physics 35 (2012) 362–370 363in operation and the ones which are in a phase of planning and pro- (1). Note that in this paper we work with 8pG = 1, c = 1 and ⁄ = 1,posal stages) [10] could, in principle, be important to confirm or while the sign conventions for the line element, which generaterule out the physical consistency of general relativity or of any the sign conventions for the Riemann/Ricci tensors, are (À, +, +, +).other theory of gravitation. This is because, in the context of Ex- By varying the action (1) with respect to glm (see Refs.tended Theories of Gravity, some differences between General Rel- [11,16,17] for a parallel computation) the field equations areativity and the alternative theories can be pointed out as far as the obtained:linearized theory of gravity is concerned [11,12,23,33]. & ! 1 In the general picture of high order theories of gravity, recently Glm þ b 2R Rlm À g lm R þ À2R;l;m þ 2g lm ÃR ¼ T ðmÞ ; lm ð3Þ 4the R2 theory, which was originally proposed by Starobinski [13],has been analysed in various interesting frameworks, see [14–17] with the associated Klein–Gordon equation for the Ricci curvaturefor example. Specifically, the non-singular behavior of this class scalarof models is discussed in [14]. In [15] R2 inflation is combined withthe Dark Energy stage and in [16] an oscillating Universe, which is ÃR ¼ E2 ðR þ TÞ; ð4Þwell tuned with some cosmological observations is discussed. Fi- which is obtained by taking the trace of Eq. (3), where h is thenally, in [17] the production and potential detection of gravity- d’Alembertian operator and the energy term, E, has been introducedwaves from this particular theory has been shown. for dimensional motivations: It is also quite important to emphasize that the R2 is the sim- 1plest one among the class of viable models with Rm terms in addi- E2 ; ð5Þtion to the Einstein–Hilbert theory. In Ref. [5], it has been shown 6bthat such models may lead to the (cosmological constant or quin- thus, b has to be positive [16].tessence) acceleration of the universe as well as an early time era In the above equations T ðmÞ is the standard stress-energy tensor lmof inflation. Moreover, they seem to pass the Solar System tests, of the matter. Note that General Relativity is obtained for b = 0 ini.e. they have the acceptable newtonian limit, no instabilities and Eq. (3).no Brans–Dicke problem (decoupling of the scalar) in the scalar- Because we want to study interactions between stars at galactic tensor version. scales, the linearized theory in vacuum T ðmÞ ¼ 0 , which gives a lm In this paper it is shown that in the framework of R2 gravity, in better approximation than Newtonian theory, can be analysed,the linearized approach, it is possible to obtain spherically sym- by considering a small perturbation of the background, which is as-metric and stationary galaxy states which can be interpreted like sumed to be given by a Minkowskian background.an approximated solution to the Dark Matter Problem. In fact, in In the linearization procedure we will perform a computationthe proposed model the Ricci curvature scalar generates an energy very similar to the ones in [16,17], but with some difference thatterm that can in principle be identified as the Dark Matter content will emphasize the importance of the Ricci scalar for the purposein the galaxy. The model can also help to have a better understand- of this paper: to provide a curvature-inspired model for galaxies.ing of the physics of Einstein–Vlasov systems [18]. Putting As the observed gravitational lensing phenomena are well re- g lm ¼ glm þ hlm ; ð6Þlated to the S-type galaxies in the Hubble sequence, here we brieflydiscuss in Section 4 the implications of this theory for observations e e e to first order in hlm, calling R lmqr ; R lm and R, respectively, the line-of bending of light by foreground galaxies. arized quantity which correspond to Rlmqr, Rlm and R, the linearized The main goal of this paper is to give a potential solution to the field equations are obtained as [16,17]:Dark Matter Problem within galaxies. In other words, we do not ban e Rthe presence of a Dark Matter component in our Universe which can e e e R lm À glm ¼ À@ l @ m b R þ glm Ãb R;explain properties of clusters of galaxies which cannot be achieved 2 ð7Þby alternative gravity theories. In any case, we add a section where e e Ã R ¼ E2 R:we show that an important property of the bullet cluster can be, in e Since R lmqr and Eq. (7) are invariants for gauge transformationsprinciple, explained in the scenario given in this paper [46]. of the kind [16,17] At the end of the paper, we discuss the general possibility to 0give rise to a Dark Matter within extended gravity. hlm ! hlm ¼ hlm À @ ðl mÞ ; ð8Þ e e e b R ! b R 0 ¼ b R;2. The field equations and the linearized approach then the gauge transformation Let us consider the high order action [15–17] h e hlm hlm À glm þ glm b R ð9Þ Z 2 4 pffiffiffiffiffiffiffi 2S¼ d x Àg ðR þ bR þ Lm Þ: ð1Þ can be defined. Thus, by considering the transformation rule for the parameter lIn order to avoid confusion, in all this paper b represents the cou- Ãm ¼ @ l hlm ; ð10Þpling constant of the R2 term. The action (1) is a particular choice with respect to the well a gauge analogous to the Lorenz one for electromagnetic waves canknown canonical action characterizing General Relativity (the Ein- be chosen, which then readsstein–Hilbert action [19]), which reads Z @ l hlm ¼ 0: ð11Þ 4 pffiffiffiffiffiffiffiS¼ d x Àg ðR þ Lm Þ: ð2Þ In this way, the field equations then read like If the gravitational Lagrangian is not linear in the curvature Ãhlm ¼ 0; ð12Þinvariants, the Einstein field equations has an order higher than e e Ãb R ¼ E2 b R: ð13Þsecond. For this reason, such theories are often called higher-ordergravitational theories [2–9]. This is exactly the case of the action Solutions of Eqs. (12) and (13) are plan waves:

Authors personal copy364 C. Corda et al. / Astroparticle Physics 35 (2012) 362–370 p bhlm ¼ Alm ð~Þ expðip xb Þ þ c:c:; ð14Þ To preserve the conditions @ l hlm ¼ 0 and h ¼ 0, transformations e b likeb R ¼ að~Þ expðiq xb Þ þ c:c:; p ð15Þwhere @ l Ãl ¼ 0; ð25Þ @ l l ¼ 0;pb ðx; ~Þ; p x ¼ p j~j; p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð16Þ can be used and, by taking ~ in the z direction, a gauge in which only pqb ðxE ; ~Þ; p xE ¼ E2 þ p2 : A11, A22, and A12 = A21 are different from zero can be chosen. The condition h ¼ 0 gives A11 = ÀA22. In Eqs. (12) and (14) the equation and the solution for the tensorwaves, exactly like in general relativity, have been obtained Now, by substituting these equations in Eq. (20), one gets[16,17], while Eqs. (13) and (15) are, respectively, the equationand the solution for the third mode stemming from the curvature e hlm ðt; zÞ ¼ Aþ ðt À zÞeðþÞ þ AÂ ðt À zÞeðÂÞ þ b Rðt; zÞglm : ð26Þ lm lm(see also [16,17]). þ ðþÞ Â ðÂÞ e The term A ðt À zÞelm þ A ðt À zÞelm describes the two standard The fact that the dispersion law for the modes of the field b R is polarizations of gravitational waves obtained from general relativ-not linear has to be emphasized. The velocity of every ‘‘ordinary’’ e ity, while the term b Rðt; zÞglm represents a mode arising from the(i.e. which arises from General Relativity) mode hlm is the light curvature term associate to the R2 high order theory [16,17].speed c, but the dispersion law (the second of Eq. (16)) for the e In other words, in the R2 theory of gravity, the linearized Riccimodes of b R is that of a wave-packet [16,17]. Also, the group- e scalar is a source of a third polarization mode for gravitationalvelocity of a wave-packet of b R centered in ~ is p waves which is not present in standard general relativity. This! ~ p third mode is associated to a ‘‘curvature’’ energy E (see Eq. (4)).vG ¼ : ð17Þ We also recall that the idea of considering the Ricci scalar as an x effective scalar field (scalaron) arose from Starobinski [13]. From the second of Eqs. (16) and (17) it is simple to obtain: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 À E 2 3. Application to the Einstein–Vlasov system: stationaryvG ¼ : ð18Þ x galaxies Then, it is also written like [16,17] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Now, we are going to discuss a model of stationary, sphericallyE¼ ð1 À v 2 Þx: G ð19Þ symmetric galaxy, assuming that the dynamics of the matter, i.e. of the stars making of the galaxy, is described by the Einstein–Vlasov Now, the analysis can remain in the Lorenz gauge with transfor- system [18]. In this way, the gravitational forces between the par-mations of the type hm = 0. This gauge gives a condition of trans- ticles of the system, i.e., a galaxy, will be mediated by the thirdverse effect for the ordinary part of the field, i.e. klAlm = 0, but does mode of Eq. (26), i.e. by the spacetime curvature. Thus, the keynot give the transverse effect for the total field hlm. assumption is that in a cosmological context such a mode, which From Eq. (9) it reads is given by the (linearized) spacetime curvature, becomes domi- e nant at galactic and cosmological scales (i.e. Aþ ; AÀ ( b R). In this h ehlm ¼ hlm À glm þ glm b R: ð20Þ way the ‘‘curvature’’ energy E can be identified as the dark matter 2 content of a galaxy of typical mass-energy: E ’ 1045 g [25], in or- At this point, if one were still to remain within general relativity dinary c.g.s. units. These two assumptions constitute an adaptationone would impose [19], to modeling a galaxy of the assumptions related to the modelÃl ¼ 0; describing an oscillating Universe in [16]. Now, let us emphasize an important point. Assuming E ’ 1045 g, h ð21Þ e@ l l ¼ À þ b R; from Eq. (5) we get b ’ 10À34 cm4 in natural units. Thus, in our 2 assumption, the constant coupling of the R2 term in the gravita-which would give the total transverse effect of the field. However, tional action results infinitesimal with respect to the linear termfor the present case this is impossible. In fact, applying the d’Alem- R. In this way, the variation from standard General Relativity isbertian operator to the second of Eq. (21) and using the field Eqs. very weak, thus the theory can pass the Solar System tests. Regard-(12) and (13), it comes out that ing this important issue, it is important to provide citations of prec- edent work illustrating this and explicitly show that the bounds e@ l Ãl ¼ E2 b R; ð22Þ there agree with the preferred values of the number E. The keywhich is in contrast with the first of Eq. (21). In the same way, it is point is that as the effective scalar field arising from curvature ispossible to show that no linear relation exists at all between the very energetic, then the constant coupling of the R2 nonlinear etensor field hlm and the linearized term b R stemming from the cur- term ? 0 [30]. In this case, the Ricci curvature, which is an extravature scalar. Thus, a gauge in which hlm is purely spatial cannot be dynamical quantity in the metric formalism, must have a rangechosen (i.e. one cannot put hl0 = 0, see Eq. (20)). longer than the size of the Solar System. An important work is Nonetheless, the traceless condition to the field hlm can be writ- Ref. [31], where it is shown that this is correct if the effectiveten in the form: length of the scalar field l is much shorter than the value of 0.2 mm. In such a case, the presence of this effective scalar is hid-Ã l ¼ 0 den from Solar System and terrestrial experiments. The value of E h ð23Þ that we are assuming here guarantees the condition l ( 0.2 mm.@ l l ¼ À : Another important test concerns the deflection of light by the 2 Sun. This effect was studied in R2 gravity by calculating the Feyn- These equations imply man amplitudes for photon scattering, and it was found that, to lin- l earized order, this deflection is the same as in standard General@ hlm ¼ 0: ð24Þ Relativity [32].

Authors personal copy C. Corda et al. / Astroparticle Physics 35 (2012) 362–370 365 On the other hand, Eq. (15) guarantees that spacetime curvature In such approximation, the energy of the particle will be seen likedoes not remain too small and thus it can, in principle, become the the Dark Matter component that guarantees the galaxy’s equilib-Dark Matter component in the galaxy. rium. It is stressed that such approximation is not as precise as we The model that we shall discuss is similar to the one introduced would, but it could be a starting point for further robust analysis.by Nordstrom in [21]. The relevance of the Einstein–Vlasov model The Vlasov equation (31) implies that the function f is constantwas emphasized in the 1992 in a famous paper by Rein and Rendall on the geodesic flow over the spacetime (27). The Christoffel coef-[18]. And the following results will be obtained adapting the ideas ficients of such a spacetime are obtained from (note that as we areintroduced in [18,22,26,28]. working in the linearized approach, in the following computations e In the hypothesis Aþ ; AÀ ( b R, the spacetime of our model will e only terms up to first order in the linearized curvature R will bebe given by the conformally flat metric considered while high-order terms will be assumed equal to zero) 2 e 2 2 2 2 1 a e e 1 eds ¼ ½1 þ b Rðt; zÞðdx þ dy þ dz À dt Þ: ð27Þ Ca m ¼ l dm @ l b R þ da @ m b R À l g lm @ a b R : ð34Þ 2 e 1 þ 2b R Note: in general, conformal transformations are performed by In this way, the Vlasov equation in the spacetime defined by therescaling the line-element like [23] line-element (27) becomes " #g ab ¼ eU g ab :~ ð28Þ pa 1 a e @abR @t f þ @ xa f À e p 2ðpl @ l b RÞ 0 þ @ pa f ¼ 0: ð35Þ Here we choose the scalar field as being p0 2 p e ð1 þ 2b RÞp0 eU b R; ð29Þ Now, let us recall that two quantities are important for the Vla- sov equation in a curved spacetime [18,22,26,28]. The first is thewhich also implies current density Z eeU ¼ 1 þ b R; ð30Þ dp pffiffiffi l Nl ¼ À gp f ð36Þ p0in our linearized approach. Thus, the conformal transformation and the second is the stress-energy tensorwhich translates the analysis into the conformal frame (Einstein Z dp pffiffiffi l mframe, see [5]) is performed by the spacetime curvature. The poten- T lm ¼ À gp p f : ð37Þtial of using such conformally flat line-element in the perspective of p0describing an oscillating Universe has been discussed in [16]. Here g is the usual determinant of the metric tensor, which in The condition that the particles in the spacetime make up an the case of the line-element (27) is given byensemble with no collisions (27) is satisfied if the particle density e g ¼ 1 þ 4b R: ð38Þis a solution of the Vlasov equation [18,22,26,28] l lm pa pl pm Keep in mind that both N and T are divergence free (conser-@t f þ @ xa f À Ca m 0 @ pa f ¼ 0; l ð31Þ vation of energy): p0 pwhere Ca m are the Christoffel coefficients, f is the particle density rl Nl ¼ 0; l ð39Þand p0 is given by pa (a = 1, 2, 3) according to the relation rl T lm ¼ 0:[18,22,26,28] The mass shell conditions (32) can be rewritten asg lm pl pm ¼ À1: ð32Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p0 ¼ e ð1 þ b RÞÀ1 þ dab pa pb : ð40Þ Eq. (32) implies that the four momentum pl lies on the mass-shell of the spacetime (greek indices run from 0 to 3) [18,22,26,28]. From the Christoffel connections (34), computing the Riemann We recall that, in general, the Vlasov–Poisson system is given by tensor, Ricci tensor and Ricci scalar, the ‘‘effective’’ Einstein field[18,22,26,28] equations@ t f þ v Á rx f À rx U Á rv f ¼ 0; Glm ¼ T lm ; ð41Þr Á U ¼ 4pq; can be obtained together with the ‘‘effective’’ Klein–Gordon Z ð33Þ equationqðt; xÞ ¼ dv f ðt; x; v Þ; e Ãb R ¼ ÀT; ð42Þwhere t denotes the time and x and v the position and the velocity where T Tl is the trace of the stress-energy tensor. lof the stars. The function U = U(t, x) is the average Newtonian poten- To simplify the computations the analysis can be performed in atial generated by the stars. This system represents the non-relativ- conformal frame. Thus, rescaling the stress-energy tensor in theistic kinetic model for an ensemble of particles with no collisions forminteracting through gravitational forces which they generate collec-tively [18,22,26,28]. e T lm ¼ ð1 þ 3b RÞT lm ; ð43Þ Ã Thus, such a system can be used for a description of the motionof the stars within a galaxy, if stars are considered as pointlike par- one obtainsticles, and the relativistic effects are negligible [18,22,26,28]. In e T Ã ¼ ð1 þ 3b RÞT: ð44Þthis approach, the function f(t, x, v) in the Vlasov-Poisson system(33) is non-negative and gives the density on phase space of the Then, Eq. (42) becomesstars within the galaxy. e Ãb R ¼ ÀT Ã : ð45Þ In other words, we are going to discuss the solutions of a Klein–Gordon equation for the spacetime curvature like galactic high en- It should be noticed that the particle density is still defined onergy scalarons, i.e. particles that can be figured out like wave-pack- the mass shell of the starting line-element gab. In order to removeets, and stationary solutions in terms of an Einstein–Vlasov system even this last connection with the starting frame (Jordan Frame)will be shown. we can rescale the momentum as

Authors personal copy366 C. Corda et al. / Astroparticle Physics 35 (2012) 362–370 ! e bR l perturbed by the scalaron. The collective set of equations can inpl ¼ Ã 1þ p ð46Þ principle be solved for given initial data. 2and we can define the particle density in the Einstein frame as 4. The gravitational lensing ! ! bRefÃ ðt; x; pÃ Þ ¼ f t; x; 1 À pÃ : ð47Þ The attentive reader notices that, in principle, as the metric is 2 conformally flat, this could mean that the scalaron Dark Matter in galaxies will, in and of itself, result in no gravitational lensing Hence, we can write our adaptation of the Vlasov system in the of light rays. This could be a serious problem for the model. Thisfollowing form is because there exists substantial evidence for gravitational lens- Z ing by the large scale structure of a galaxy beyond that due to e e dpÃÃb R ¼ ð1 þ 2b RÞ fÃ ðt; x; pÃ Þ; ð48Þ the constituent baryons alone. For instance, the observed absence p0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ã of lensing by the effective dark matter in a class of metric formula-p0 ¼ 1 þ dab pa pb : Ã Ã Ã ð49Þ tions of Modified Newtonian Dynamics formed the basis for a tenta- pa 1h e e i tive no-go theorem for such theories [29].@ t fÃ þ 0 @ xa fÃ À 0 pl @ l b Rpa þ @ a b R @ pa fÃ ¼ 0: Ã Ã Ã ð50Þ Actually, we show that in the proposed model the gravitational pÃ pÃ Ã lensing can be, in principle, obtained like an effect of spacetime Because we want to restrict ourselves to spherical symmetry in curvature, i.e. due by the Ricci curvature scalar.the present approximation, the line-element (27) can be rewritten In our linearized approach, gravitational lensing can be de-as scribed in a local Lorentz frame perturbed by the first order post- h i Newtonian potential [27]. Calling V such a potential one can define 2 e 2 2ds ¼ 1 þ b Rðt; rÞ ðdr À dt Þ: ð51Þ a refractive index [27] In this equation r is the radial coordinate. Thus, in spherical n 1 þ 2jVj: ð58Þcoordinates, Eqs. (48)–(50) can be written as Some clarifications are needed concerning this point. In the 2e usual Geometrical Optics, the condition n > 1 implies that the light d bR 1 d d e 2 eÀ 2 þ 2 b Rr ¼ ð1 þ 2b RÞlðt; rÞ; ð52Þ in a medium is slower than in vacuum [34]. Then, the effective dt r dr dr Z speed of light in a gravitational field is expressed by [34] dplðt; rÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðt; x; pÞ; ð53Þ 1 1 þ p2 " ! v¼ % 1 À 2jVj: ð59Þ n p d e xÁp 1 d e@ t f þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Á @ x f À b R þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi b R pþ Thus, one can obtain the Shapiro delay [35] by integrating over 1 þ p2 dt 1 þ p2 r dr # the optical path between the source and the observer: x 1 d e Z obser v erpffiffiffiffiffiffiffiffiffiffiffiffiffiffi b R Á @ p f ¼ 0; ð54Þ 2jVjdl: ð60Þ 1 þ p2 r dr sourcewhere the suffix ⁄ has been removed for the sake of simplicity, and The situation is analogous to the prism [34].we have denoted by p the vector p = (p1, p2, p3) with p2 = jpj2, and Now, we recall that, in the weak field approximation, the con-also defined x for the vector xi = (x1, x2, x3). nection between the post-Newtonian potential and the linearized Then, in searching for stationary states, and following [16], we theory is given by the g00 component of the line-element [19]:can call k the wavelength of the ‘‘galactic’’ gravitational wave (26), g 00 % 1 þ 2V: ð61Þi.e. the characteristic length of our gravitational perturbation, andassume that k ) d, where d is the galactic scale, i.e. d $ 105 light- Thus, from Eq. (27) one obtainsyears [25]. In this way, the gravitational wave is ‘‘frozen-in’’ with e b Rðt; zÞ À 1respect to the galactic scale. V% ð62Þ Thus, the system of equations which defines the stationary 2solutions of Eqs. (52)–(54), for our galaxy model is and 1 d d e 2 e b Rr ¼ ð1 þ 2b RÞlðrÞ;e ð55Þ n % 1 þ 2b Rðt; zÞ À 1; ð63Þr 2 dr dr Z is the equation which can, in principle, be used to discuss the grav- dplðrÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðx; pÞ; ð56Þ itational lensing in our model. Then, in this case, the gravitational 1 þ p2 lensing is performed directly by spacetime curvature, i.e. by the Ric- 1 d e ci scalar.p Á @xf À b R½ðp Á xÞp þ x Á @ p f ¼ 0; ð57Þ r dr For a sake of completeness, we discuss the gravitational lensingIn other words, the idea is that the spin-zero degree of freedom from another point of view.introduced via the addition of an R2 term in the gravitational The condition k ) d, on the characteristic length of the gravita-Lagrangian may be a candidate for the dark matter. This new degree tional perturbation, guarantees that the Ricci curvature remainsof freedom is termed the scalaron [13]. ‘‘frozen, i.e. constant’’, with respect to the galactic characteristic dis- Thus, we assume that within a galaxy the dominant contribu- tance scale. Thus, we puttion to the curvature comes from the new degree of freedom (as e bR ¼ K ða constant!Þ: ð64Þit is also the case to first approximation in the general relativ-ity + cold dark matter case). We deduce this contribution via the We can search deviations from standard General Relativity withinscalaron field equation which itself has a baryonic source term. the galaxy by considering a spherically symmetric Schwarzschild-The baryons themselves are then taken to evolve according to a like metric generated by the ordinary (barion) galaxy mass M withcollisionless Boltzmann equation, propagating on a background the corrections that are generated by curvature [36]:

Authors personal copy C. Corda et al. / Astroparticle Physics 35 (2012) 362–370 367 2 2 2 2ds ¼ À expðÀkrÞ dt þ expðkrÞ dr þ r 2 ðdh2 þ sin h du2 Þ ð65Þ the innermost point at which deviations from general relativity are suppressed in will occur at the scale radius rs. Considering Eq. (70) Then, following [36], the Ricci scalar is given by: togheter with the NFW density profile on sees that halos with 4k0 2 2 masses which satisfyR ¼ expðÀkrÞ k00 À k02 þ À 2 þ 2; ð66Þ r r r K > qc dc ðMÞ: ð74Þwhere 0 stands for the derivative with respect to r. As dc(M) decreases with decreasing M [39], an upper limit is Putting the condition (64) in Eq. (66) one gets [36]: setted to the mass of halos which can be screened given K, i.e. given a b K 2 the value of the Ricci curvature, see Eq. (64). For masses below thiskðrÞ ¼ À ln À À r ð67Þ r r 2 12b threshold the theory strongly deviates from general relativity and strong lensing around these halos is permitted. We plan to furtherand, by choosing a = À2M, b = 0 in analogy with the standard develop in this direction of research.Schwarzschild metric, one gets [19,36] À1 2 2M K 2 2 2M K 2 2 5. Can R2 gravity explain the Bullet Cluster dynamics?ds ¼ À 1 À À r dt þ 1 À þ r dr r 12b r 12b 2 þ r 2 ðdh2 þ sin h du2 Þ: ð68Þ The scalaron has two aspects as a field (like classical electric or magnetic field) and a particle (like quantized photon). We treated The physical interpretation implies that within some radius rmin the scalaron as a field in the Vlasov equation (see [42] for a recentdeviation from General Relativity are highly suppressed and we say cosmological application) whose scenario is in some sense similarthat the object is screened [37–39]. This is because we are assuming to that of the MOND [43]. In such a scenario, it seems to be difficultthat, within the radius rmin, the spacetime curvature which is due to explain the Bullet Cluster [44] although the Bullet Cluster couldby the ordinary, i.e. barion, mass of the galaxy dominates with re- be naturally explained if the scalaron could be a heavy particle.spect to the intrinsic constant curvature R = K. On the other hand, Aside from this, we emphasize that the main goal of this paperfor values r > rmin, the spacetime curvature which is due by the is to give a potential solution to the Dark Matter Problem withinintrinsic curvature dominates with respect to the spacetime curva- galaxies. Therefore, we do not ban the presence of a Dark Matterture due by the barion mass of the galaxy. component in our Universe which can explain properties of clus- In order to clarify the issue of the bending of light in this R2 sce- ters of galaxies which cannot be achieved by current alternativenario, in what follows, and as matter of illustration, we will repro- gravity theories [44]. In any case, in this Section we show that anduce a piece of the discussion presented by Smith in Ref. [39], important property of the Bullet Cluster can in principle be ex-which, in comparison with our discussion at the beginning of Sec- plained in the scenario proposed in this paper.tion 2, appears as a generalization of that digression (see AppendixA). R 4 pffiffiffiffiffiffiffi 5.1. A note on the Bullet Cluster 1E0657-56 Thus, for a general action S ¼ d x Àg ðR þ f ðRÞ þ Lm Þ screen-ing occurs within a radius rmin implicitly given by [37–39] Since 2006, the argument ‘‘But the Bullet Cluster . . .’’ has comejf 0 ðRmax Þj < qðr min Þr 2 ; min ð69Þ into scene in most discussions about the Dark Matter Problem in astrophysics. It relates to the observations of the post-collisionwhere q is the local value of the density. clusters of galaxies in the source 1E0657–56 (known as the Bullet In our case it is f0 (R) = 2bR and Rmax ¼ K . Then, Eq. (69) becomes b Cluster) [44]. It is seen that the X-ray emission is a mid the couple 1 of colliding clusters, and thus is shifted with respect to the bulk ofK< qðrmin Þr2 : min ð70Þ matter making up each group of galaxies, which resides in the out- 2 skirts. In most studies of X-ray emission from galaxy clusters the Now, following [39], we can use Eq. (70) to further discuss the gas usually sits at its center. This offset can be understood bygravitational lensing. recalling that as the gas pass each other it interacts electromagnet- Let us consider a galactic Navarro, Frank and White (NFW) halo ically, heats up, ionize to emit X-rays, and hence is slowed-downof the form [38,39] during the collision. This may explain why the gas appears a bit be- r3 s hind in the heading direction of each of the group of galaxies in-qðrÞ ¼ qc dc ; ð71Þ rðr þ rs Þ2 volved. Meanwhile, non-interacting Dark Matter components do only interact through gravity and thus pass each other unhinderedwhere rs is the scale radius and qc the critical density of the universe without being slowed-down like the gas. In the image, the field[38,39]. purported to represent the vast of mass in each cluster, as com- By assuming the mass-concentration relation [39–41] pared to the X-ray emitting counterpart, is inferred from gravita- ! tional lensing analysis of images of background galaxies, under 9 Mc¼ ; ð72Þ the assumption that general relativity is the theory of gravity 1 þ z 8:12 Â 1012 hÀ1 M } (but be aware that most modified gravity theories describe cor-where z is the redshift, M} the mass of the Sun and h is the Hubble rectly the phenomenon of gravitational lensing, see Appendix A).parameter in units of 100 km/(sMpc), the amplitude dc, that relates It follows the distribution of galaxies, not the gas. As the most mas-the concentration to the virial radius with an overdensity D = 119 is sive component of the galaxy cluster is not centered at the gas, it isgiven by [39–41] argued, upon that lens analysis, that the large part of matter in the cluster is close to, and around each galaxy group. And as the visible Dc 3 mass in the galaxies is not enough to account for the velocity dis-dc ¼ h i : ð73Þ c 3 lnð1 þ cÞ À 1þc persion of a galaxy cluster, it is suggested that there is Dark Matter, too. As noninteracting component, the distribution of the Dark Eq. (72) reduces the NFW halo profile to a one-parameter family Matter can pass through each other just like the galaxies, and thatwhich is taken to be dependent on the virial mass, M [39]. For r < rs is why the bulk of mass should be found close to the galaxies in aqNFW scales like rÀ1, while for r > rs qNFW scales like rÀ4 [39]. Thus, cluster collision like this.

Authors personal copy368 C. Corda et al. / Astroparticle Physics 35 (2012) 362–370 The analysis of the overimposed optical and X-ray images of the server reduces to a local Lorentz frame [19]. As the origin of thecluster can be interpreted as an endorsement on the existence of coordinates is located within the main cluster, the proper distancesDark Matter. This issue is based on a number of assumptions which between the two clusters are the coordinates of the ‘‘bullet’’. Theare not totally accepted by the astrophysics community because line element is [19]the large set of problems for the Dark Matter hypothesis are, in 2 0 i j a ds ¼ þðdx Þ2 À d dx dx À Oðjxj j2 Þ dx dx : bany case, independent from the observations of the Bullet Cluster. ð75ÞIn a different context, the collision velocity of the Bullet Cluster The effect of the gravitational force on test masses is describedseems to be in disagreement with the standard concordance model by the equationof cosmology [46] (the relative velocity of the clusters in the colli-sion is an issue being addressed next in the framework of this grav- € e xi ¼ À R i0k0 xk ; ð76Þity theory). At the same time, alternative theories of gravitation, e which is the equation for geodesic deviation in this frame [19]. R i0k0while often said to fail in explaining dynamics of galaxy clusters,can account for them rather naturally. is the linearized Riemann tensor and xi are the coordinates of the On the other hand, even if the Bullet Cluster can only be ex- ‘‘bullet’’ which represent the separation vector between the twoplained by invoking Dark Matter, the problems on small scales per- test masses [19]. e To study the effect of the scalaron on the two clusters, R i0k0 hassist. The Bullet Cluster does not improve our understanding of theLocal Group of Galaxies, nor on the dynamics of individual galaxies, to be computed in the proper reference frame of the main cluster. e But, as the linearized Riemann tensor R lmqr is invariant under(as is on focus in the present paper). The two arguments are abso-lutely independent. A conclusion could be that the lensing mass gauge transformations [19], it can be directly computed from Eq.and the hot gas have a spatial shift, and hence the visible, hot (27).gas cannot make up the bulk of mass in the system. In any case, From [19] we getwhether the cause of this phenomenon is a nonstandard law of 1 e R lmab ¼ f@ l @ b ham þ @ m @ a hlb À @ a @ b hlm À @ l @ m hab g; ð77Þgravitation or missing mass is not that easy to unveil. 2 that, in the case Eq. (27), gives5.2. Scalaron and the Bullet Cluster dynamics e bn a e e e e o R ac0 ¼ 0 @ @ 0 R g0c þ @ 0 @ c Rda À @ a @ c R g00 À @ 0 @ 0 Rda : 0 c ð78Þ As pointed out above, it is a common opinion that alternative 2gravity theories cannot explain all the properties of the bullet clus- The different elements are (only the non-zero ones will be writ-ter [44], even if scalar-vector-tensor theory could, in principle, take ten down explicitly)into account the apparent discrepancy between the gravitational 8 9lensing and X-ray maps of the colliding clusters, without requiring e < @2 R for a¼c¼0 = e tcollisionless Dark Matter [45,48]. @ a @ 0 R g0c ¼ ; ð79Þ : Meanwhile, hydrodynamic simulations of the colliding galaxy e À@ z @ t R for a ¼ 3; c ¼ 0 ;clusters 1E0657–06 show that at a distance of 4.6 Mpc between 8 9the two clusters, an infall velocity of 3000 km/s is required in order e < @2 R for a¼c¼0 = e a tto explain the observed X-ray brightness and morphology of the @ 0 @ c Rd0 ¼ ; ð80Þ :cluster [45,46]. It has also been argued that such a high infall veloc- e @t @z R for a ¼ 0; c ¼ 3 ;ity is incompatible with predictions of the standard cosmology (the 8 9concordance KCDM model) [45,47]. e > À@ 2 R for a¼c¼0 > > > t > > Next, we extend our scalaron model in order to show that the > > > > >infall velocity of the two clusters can in principle be explained in e e < @2 R z e for a¼c¼3 > = À@ a @ c R g00 ¼ @ a @ c R ¼ ; ð81Þ >the framework introduced in the present paper. To rigorously show > > e > À@ t @ z R for a ¼ 0; c ¼ 3 > > > >that the scalaron gravity can cope with the Bullet Cluster dynamics > > > > : e ;we need first to modify our assumptions. In particular, we will as- @z@t R for a ¼ 3; c ¼ 0sume that the characteristic length of the scalaron gravitationalperturbation is much longer not only than the galactic scale, but e e À@ 0 @ 0 Rda ¼ À@ 2 R for a ¼ c: ð82Þ c tthan the distance between the colliding clusters too. In otherwords, the gravitational wave is ‘‘frozen-in’’ with respect to the Bul- By inserting these results in Eq. (78) we obtainlet Cluster’s scale. In this way, we can consider the colliding clus- be e €ters (the main cluster and the ‘‘bullet’’) like test particles in the R 1 ¼ À R; 010 2gravitational field of the scalaron. In the linearized approach of this paper, the coordinate system e be€ R 2 ¼ À R; 010 ð83Þin which the space–time is locally flat has to be used and the dis- 2tance between any two points (the colliding clusters) is given sim- e b 2e e R3 ¼ 030 @z R À @2 R : tply by the difference in their coordinates in the sense of Newtonian 2physics [19]. This frame is the proper reference frame of a local ob- At the Bullet Cluster’s scale we have homogeneity and isotropyserver, which we assume to be located within the main cluster. In e [20], which imply @ 2 R ¼ 0. Therefore, Eq. (83) become zthis frame gravitational signals manifest them-self by exerting ti-dal forces on the test masses. In other words, we assume that the e be€ R 1 ¼ À R; 010space–time within the main cluster is locally flat with respect to 2the global curvature generated by the scalaron which is described be€ e R 2 ¼ À R; ð84Þby the line element (27). By using the proper reference frame of a 010 2local observer the time coordinate x0 is the proper time of the ob- e be€server O and the spatial axes are centered in O. In the special case R 3 ¼ À R: 030of zero acceleration and zero rotation the spatial coordinates xj are 2the proper distances along the axes and the frame of the local ob- By using Eq. (76), we obtain

Authors personal copy C. Corda et al. / Astroparticle Physics 35 (2012) 362–370 369 be€ We recall that the main goal of this paper is to give a poten-€¼x Rx; ð85Þ 2 tial solution to the Dark Matter Problem within galaxies. Thus, be€ we do not ban the presence of a Dark Matter component in€y ¼ Ry ð86Þ 2 our Universe which can explain properties of clusters of galaxies which cannot be achieved by alternative gravity theories. In anyand case, in Section 5 we showed that an important property of the be€ bullet cluster, i.e. an infall velocity of 3000 km/s between the€¼z Rz: ð87Þ 2 two clusters, can be, in principle, explained in the scenario given in this paper. These are three perfectly symmetric oscillations of the scalaron(wave-packet). If one re-introduces the radial distance r, Eqs. (85)– Acknowledgement(87) are summarized by be€ The authors thank an unknown referee for precious advices and€¼r Rr: ð88Þ 2 comments which permitted to improve this paper. Christian Corda thanks the R. M. Santilli Foundation for partially supporting this re- Eq. (88) can be solved with the perturbation method [19] ! search (Research Grant No. RMS-TH-5735A2310). Herman J. Mos- be quera Cuesta thanks Fundação Cearense para o DesenvolvimentorðtÞwrð0Þ 1 þ RðtÞ : ð89Þ 2 Científico e Tecnológico (FUNCAP), Ceará, Brazil for financial support. Assuming that the wave-packet is in the contraction phase weget the relative infall velocity as Appendix A. Dynamics in f(R) theories of gravity and d be_ gravitational lensingv infall ¼ rðtÞ w rð0Þ RðtÞ: ð90Þ dt 2 Therefore, in order to have consistence with the observations We emphasize that hereafter we closely follow the paper bywe need Smith [39], and the set of references quoted there. A generic branch of scalar-tensor theories in which the Brans–Dicke parameter be_ x 0! bring in unique predictions: they feature a peculiar effectrð0Þ RðtÞ w 3000 km=s: ð91Þ 2 known as the chameleon mechanism, by which the modifications to general relativity get quickly suppressed nearby sufficientlyBeing r(0) w 4.6 Mpc, the model will be consistent with observa- massive objects, such as galaxies. This rapid decay in the theory’stions if gravitational effects present a unique size-dependent lensing foot-be_ 3000 km À1 print which is discussed next. RðtÞ w s w 2 Ã 10À17 sÀ1 : ð92Þ2 4:6 Mpc The action for f(R)-theories (functional of the Ricci scalar R) The assumption that the characteristic length of the scalaron Z 1 4 pffiffiffiffiffiffi ffigravitational perturbation is much longer than the distance be- S¼ d x Àg ½R þ f ðRÞ þ Sm ; ð93Þ 2jtween the colliding clusters implies that the frequency of thewave-packet has to be xE ( 10À14 sÀ1. For example, for a value where Sm action for the matter fields. The field equations turn out toxE $ 10À16 sÀ1 Eq. (92) guarantees theoretical consistence if bebe2 R $ 10À1 . 1 Â 0 Ã ½1 þ f 0 ðRÞGlm þ g lm Rf ðRÞ À f þ 2Ãf 0 ðRÞ À rl rl f 0 ðRÞ ¼ jT lm : 26. Final discussion and closing remarks ð94Þ In this class of theories the Ricci scalar becomes a dynamical It has been shown that in the framework of R2 gravity and in the quantity whose equation of motion is determined by the trace oflinearized approach, it is possible to obtain spherically symmetric the field equation,and stationary galaxy states which can be interpreted like anapproximated solution of the Dark Matter problem. In fact, in the 1 Ãf 0 ðRÞ ¼ ðjT þ R½1 À f 0 ðRÞ þ 2f Þ: ð95Þmodel the Ricci curvature generates a high energy term that can 3in principle be identified as the dark matter content of most the In the limit where f ? 0, Eq. (95) reduces to the standard rela-galaxies. The model can also help to gain a better understanding tion R = ÀjT of general relativity. Here T is the trace of theof the basics of the Einstein–Vlasov system. stress-energy tensor. In other words, we have discussed, in the linearized R2 gravity, Using the trace equation to rewrite the gravitational equation ofthe solutions of a Klein–Gordon equation for the spacetime curva- motion one obtainsture like galactic high energy scalarons, i.e. particles that has beenanalysed like wave-packets, and we have shown stationary solu- 1 1 0 T eff ¼ ½jT þ R þ ½2Rf ðRÞ À f : ð96Þtions in terms of the Einstein–Vlasov system. 3j 3j In this approximation, the energy of the galaxy-particle has Solutions to the Eq. (95) determine the lensing predictions forbeen seen like the dark matter component which guarantees the the theory, and can be understood by rewriting the trace of theequilibrium of galaxies. field equation in the form An important point is that, because of the high energy of such dVparticle-galaxies, the constant coupling of the R2 in the gravita- Ãf 0 ðRÞ ¼ À ; ð97Þtional action comes out to be infinitesimal with respect to the lin- df 0 ðRÞear term R. In this way, the deviation from standard general whererelativity is very weak, and, in principle, the theory could passthe Solar System tests. An analysis on the gravitational lensing dV 1 ðjT þ R½1 À f 0 ðRÞ þ 2f Þ: ð98Þhas been also performed in Section 4. df 0 ðRÞ 3

Authors personal copy370 C. Corda et al. / Astroparticle Physics 35 (2012) 362–370 Notice that for functions f(R) which reproduce the observed [11] S. Capozziello, C. Corda, Int. J. Mod. Phys. D 15 (2006) 119–1150; C. Corda, J. Cosmol. Astropart. Phys. JCAP 04009 (2007);expansion history of the universe, the minimum of this potential C. Corda, APh 28 (2007) 247–250.yields the general relativistic relationship between R and T, i.e., [12] M.E. Tobar, T. Suzuki, K. Kuroda, PhRvD 59 (1999) 102002;R = ÀjT. C. Corda, PhRvD 83 (2011) 062002. Hence, within a given object, a typical galaxy; for instance, far [13] A.A. Starobinsky, PhLB 91 (1980) 99. [14] G.F.R. Ellis, J. Murugan, C.G. Tsagas, Class. Quant. Grav. 21 (2004) 233–250.away from the center the scalar curvature starts off nearly at its [15] S. Nojiri, S.D. Odintsov, PhRvD 68 (2003) 123512.asymptotic value, Rmax, and evolves with radial distance from the [16] C. Corda, Gen. Rel. Grav. 40 (10) (2008) 2201–2212.center. 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