Disk and elliptical galaxies within renormalization group improved gravity
Abstract
The paper is about possible effects of infrared quantum contributions to General Relativity on disk and elliptical galaxies. The Renormalization Group corrected General Relativity (RGGR model) is used to parametrize these quantum effects. The new RGGR results presented here concern the elliptical galaxy NGC 4374 and the dwarf disk galaxy DDO 47. Using the effective approach to Quantum Field Theory in curved background, one can argue that the proper RG energy scale, in the weak field limit, should be related to the Newtonian potential. In the context of galaxies, this led to a remarkably small variation of the gravitational coupling G, while also capable of generating galaxy rotation and dispersion curves of similar quality to the the best dark matter profiles (i.e., the profiles that have a core).^{1}^{1}1This paper is based on a talk given by D.C. Rodrigues at the I CosmoSul meeting (Rio de Janeiro, RJ  Brazil. August, 0105, 2011).
I Introduction
In Refs. Rodrigues, Letelier, and Shapiro (2010); Rodrigues (2012); Rodrigues, Letelier, and Shapiro (2011); Farina et al. (2011); Fabris et al. (2012a) we presented new results on the application of renormalization group (RG) corrections to General Relativity in the astrophysical domain, in particular on a possible relation between RG large scale effects and dark matterlike effects in galaxies. The resulting phenomenological model was named RGGR. These developments were directly based on the RG application to gravity of Ref. Shapiro, Sola, and Stefancic (2005), and are consistent with the phenomenological consequences of diverse approaches to the subject, including the related to the asymptotic safety scenario of quantum gravity, see in particular Refs. Reuter and Weyer (2004a, b, 2006).
Currently, in the context of quantum field theory in curved space time, it is impossible to construct a formal proof that the coupling parameter is a running parameter in the infrared. However, this possibility can not be ruled out. The possibility of General Relativity being modified in the far infrared due to the renormalization group (RG) has been considered in different contexts, for instance, Goldman et al. (1992); Bertolami, Mourao, and PerezMercader (1993); Dalvit and Mazzitelli (1994); Bertolami and GarciaBellido (1996). The previous attempts to apply this picture to galaxies have considered for simplicity pointlike galaxies (e.g., Shapiro, Sola, and Stefancic (2005); Reuter and Weyer (2004a)). We extended previous considerations by identifying a proper renormalization group energy scale and by evaluating the consequences considering the observational data of disk Rodrigues, Letelier, and Shapiro (2010) and elliptical Rodrigues (2012) galaxies. We proposed in Ref. Rodrigues, Letelier, and Shapiro (2010) the existence of a relation between and the local value of the Newtonian potential (this relation was reinforced afterwards Domazet and Stefancic (2011)). With this choice, the renormalization groupbased approach (RGGR) was capable to mimic dark matter effects with great precision. Also, it is remarkable that this picture induces a very small variation on the gravitational coupling parameter , namely a variation of about of its value across a galaxy (depending on the matter distribution). We call our model RGGR, in reference to Renormalization Group corrected General Relativity.
Here we present a brief review of the RGGR achievements in galaxies and present new results on the galaxies NGC 4374 and DDO 47. The first is a giant elliptical galaxy that was first analyzed in Ref. Rodrigues (2012) in the context of RGGR and MOND, the second is a well known dwarf disk galaxy whose results within RGGR are for the first time here presented, and it is part of a larger work yet to be published Fabris et al. (2012b). It should be pointed that disk and elliptical galaxies behave as stationary systems that are stable due to different physical reasons, disk galaxies are essentially axisymmetric bodies supported by rotation, while elliptical galaxies are about spherically symmetric and mainly supported by velocity dispersions. Hence models for galaxy kinematics may in principle succeed in, say, disk galaxies but fail at the ellipticals ones. It is remarkable that the RGGR model is doing fine in both cases.
Ii A brief review on RGGR
The gravitational coupling parameter may behave as a true constant in the far IR limit, leading to standard General Relativity in such limit. Nevertheless, in the context of QFT in cuved space time, there is no proof on that. According to Refs. Shapiro, Sola, and Stefancic (2005); Farina et al. (2011), a certain logarithmic running of is a direct consequence of covariance and must hold in all loop orders. Hence the situation is as follows: either there is no new gravitational effect induced by the renormalization group in the far infrared, or there are such deviations and the gravitational coupling runs as
(1) 
Equation (1) leads to the logarithmically varying function,
(2) 
where is a reference scale introduced such that . The constant is the gravitational constant as measured in the Solar System (actually, there is no need to be very precise on where assumes the value of , due to the smallness of the variation of ). The dimensionless constant is a phenomenological parameter which depends on the details of the quantum theory leading to eq. (2). Since we have no means to compute the latter from first principles, its value should be fixed from observations. Even a small of about can lead to observational consequences at galactic scales. Note that the first possibility, namely of no new gravitational effects in the far infrared, corresponds to .
The action for this model is simply the EinsteinHilbert one in which appears inside the integral, namely,^{2}^{2}2We use the spacetime signature.
(3) 
In the above, should be understood as an external scalar field that satisfies (2). Since for the problem of galaxy rotation curves the cosmological constant effects are negligible, we have not written the term above. Nevertheless, for a complete cosmological picture, is necessary and it also runs covariantly with the RG flow of Shapiro, Sola, and Stefancic (2005); Reuter and Weyer (2006).
There is a simple procedure to map the solutions from the Einstein equations with the gravitational constant into RGGR solutions. In this review, we will proceed to find RGGR solutions via a conformal transformation of the EinsteinHilbert action, and to this end first we write
(4) 
and we assume , which will be justified latter. Introducing the conformally related metric
(5) 
the RGGR action can be written as
(6) 
where is the EinsteinHilbert action with as the gravitational constant. The above suggest that the RGGR solutions can be generated from the Einstein equations solutions via the conformal transformation (5). Indeed, within a good approximation, one can check that this relation persists when comparing the RGGR equations of motion to the Einstein equations even in the presence of matter Rodrigues, Letelier, and Shapiro (2010).
In the context of galaxy kinematics, standard General Relativity gives essentially the same predictions of Newtonian gravity. In the weak field limit and for velocities much lower than that of light, the gravitational dynamics can be derived from the Newtonian potential, which is related to the metric by
(7) 
Hence, using eq. (5), the effective RGGR potential is given by
(8) 
An equivalent result can also be found from the geodesics of a test particleRodrigues, Letelier, and Shapiro (2010). For weak gravitational fields (with at spatial infinity), hence even if eq. (8) can lead to a significant departure from Newtonian gravity.
In order to derive a test particle acceleration, we have to specify the proper energy scale for the problem setting in question, which is a timeindependent gravitational phenomena in the weak field limit. This is a recent area of exploration of the renormalization group application, where the usual procedures for high energy scattering of particles cannot be applied straightforwardly. Previously to Rodrigues, Letelier, and Shapiro (2010) the selection of , where is the distance from a massive point, was repeatedly used, e.g. Reuter and Weyer (2004b); Dalvit and Mazzitelli (1994); Bertolami, Mourao, and PerezMercader (1993); Goldman et al. (1992); Shapiro, Sola, and Stefancic (2005). This identification adds a constant velocity proportional to to any rotation curve. Although it was pointed as an advantage due to the generation of “flat rotation curves” for galaxies, it introduced difficulties with the TullyFisher law Tully and Fisher (1977), the Newtonian limit, and the behavior of the galaxy rotation curve close to the galactic center, since there the behavior is closer to the expected one without dark matter. In Rodrigues, Letelier, and Shapiro (2010) we introduced a identification that seems better justified both from the theoretical and observational points of view. The characteristic weakfield gravitational energy scale does not comes from the geometric scaling , but should be found from the Newtonian potential , the latter is the field that characterizes gravity in such limit. Therefore,
(9) 
If would be a complicated function with dependence on diverse constants, that would lead to a theory with small (or null) prediction power. The simplest assumption, , leads to in the large limit; which is unsatisfactory on observational grounds (bad Newtonian limit and correspondence to the TullyFisher law). One way to recover the Newtonian limit is to impose a suitable cutoff, but this rough procedure does not solves the TullyFisher issues Shapiro, Sola, and Stefancic (2005). Another one is to use Rodrigues, Letelier, and Shapiro (2010)
(10) 
where and are constants. Apart from the condition , in order to guarantee , the precise value of is largely irrelevant for the dynamics, since does not depends on . The relevant parameter is , which will be commented below. The above energy scale setting (10) was recently reobtained from a more fundamental perspective Domazet and Stefancic (2011), where a renormalization group scalesetting formalism is employed.
The parameter is a phenomenological parameter that needs to depend on the mass of the system, and it must go to zero when the mass of the system goes to zero. This is necessary to have a good Newtonian limit. From the TullyFisher law, it is expected to increase monotonically with the increase of the mass of disk galaxies. In a recent paper, an upper bound on in the Solar System was derived Farina et al. (2011). In galaxy systems, , while for the Solar System, whose mass is about of that of a galaxy, . It shows that a linear increase on with the mass (ignoring possible dependences on the mass distribution) is sufficient to satisfy both the current upper bound from the Solar System and the results from galaxies. Actually, in Ref. Rodrigues (2012) it is shown that a closetolinear dependence on the mass can also be found for elliptical galaxies by using the fundamental plane.
Once the identification is set, it is straightforward to find the rotation velocity for a static gravitational system sustained by its centripetal acceleration Rodrigues, Letelier, and Shapiro (2010),
(11) 
Contrary to Newtonian gravity, the value of the Newtonian potential at a given point does play a significant role in this approach. This sounds odd from the perspective of Newtonian gravity, but this is not so from the General Relativity viewpoint, since the latter has no free zero point of energy. In particular, the Schwarzschild solution is not invariant under a constant shift of the potential.
Equation (11) was essential for the derivation of the RGGR galaxy rotation curves. Since elliptical galaxies are mainly supported by velocity dispersions (VD), the main equation for galaxy kinematics in this case will not be eq.(11), but the following expression for the projected (lineofsight) VD (see also Mamon and Lokas (2005)),
(12) 
where
is the incomplete beta function, is the Gamma function, stands for the deprojected (spherical) radius, for the projected (line of sight) radius, is the luminosity intensity (its integral on an infinity surface gives the total luminosity of the galaxy), is the luminosity density (found from the deprojection of ), is the anisotropy parameter (assumed constant inside each galaxy, it is zero if the galaxy has an isotropic VD profile) and is the total (effective) mass of the system at the radius . See Ref. Rodrigues (2012) and references therein for additional details. In the case of Newtonian gravity without dark matter would stand, within a good approximation, as the internal stellar mass to radius , i.e., .
For RGGR without dark matter, the total mass inside the radius is also the baryonic mass . Nevertheless, the nonNewtonian gravitational effects of RGGR for spherically symmetric systems can be understood from the Newtonian perspective as if the total mass was given by the following total effective mass Rodrigues (2012)
(14) 
with
(15) 
Iii NGC 4374 and DDO 47
iii.1 Ngc 4374
NGC 4374 is a giant elliptical recently analyzed within RGGR Rodrigues (2012). Here we present a variation of the analysis presented in that reference, namely we here do not directly use the photometric data with the Sérsic extension to model the stellar mass, instead we here only use the Sérsic model parameters that best fit the surface brightness of this galaxy. See Table 1 and Fig. 1. This is relevant to display that our results presented in Rodrigues (2012) are sufficiently robust to small changes on the baryonic mass content.
NGC 4374  
RGGR without dark matter  
Stellar model  
Full Sérsic+  0  20.1  0.96  
Full Sérsic+  18.5  0.92  
Full Sérsic+K.IMF+  0  21.9  0.99  
Full Sérsic+K.IMF+  21.2  1.0  

In Ref. Rodrigues (2012) it is shown hat MOND fits better the NGC 4374 observational data than Newtonian gravity without dark matter. However, it is still a poor fit, in particular since: ) There is a significant tendency towards a lower VD curve at large radii, tendency which is strongly enhanced once the fits consider the expected ; ) if the expected is not used, the best fit is achieved for tangential anisotropy with . Other issues of MOND with the giant ellipticals can be found for instance in Ref.Gerhard et al. (2001). It was also shown that RGGR fit to the data is a satisfactory one and outperforms MOND in all the points above.
iii.2 Ddo 47
Here we present part of a new result from a working in progressFabris et al. (2012b). The dwarf disk galaxy DDO 47 is cited as a paradigmatic galaxy for testing dark matter effects Walter and Brinks (2001); Salucci, Walter, and Borriello (2003); Gentile et al. (2005), and in particular its fits using the NFW profile vary from bad (using the two NFW halo parameters as free parameters) to a disaster (if one of the parameters is fixed in accordance with Nbody simulations expectations) Gentile et al. (2005).
Here we use the same data and conventions used in Ref.Gentile et al. (2005) for the baryonic matter and observed rotation curve, except for the gas surface density. For the latter, we directly use the gas surface density provided by Ref. Walter and Brinks (2001), which leads to a gas rotation curve less smooth than the one presented in Ref.Gentile et al. (2005). The results are not significantly sensitive to either of the gas profiles and are shown in Fig. 2.
Iv Conclusions
In summary, RGGR is a model based on the theoretical possibility that the beta function of the gravitational coupling parameter may not be zero in the far infrared. Currently, there is no way to directly deduce this behavior from first principles, nevertheless eq. (1) is a natural, if not unique, possibility that has appeared many times before in the context of Quantum Field Theory on curved spacetime. This equation depends on a universal free parameter that can be constrained from experiments and observations, with corresponding to standard General Relativity. The eq. (1) also depends on an energy scale , which should be related to the symmetries and physical interactions that are being evaluated. Considering gravitation effects in stationary systems with weak Newtonian potential () and slow particle velocities (), it is natural to use a relation of the type (9). In Ref. Rodrigues, Letelier, and Shapiro (2010) the relation (10) was first proposed.
RGGR without dark matter is a model with one phenomenological free parameter () which is capable of dealing with the kinematics of diverse galaxies. The relation to other physical parameters we aim to understand soon Fabris et al. (2012b).
Acknowledgements
DCR thanks the I CosmoSul organizers for the invitation. We thank Nicola Napolitano for a valuable remark on the NGC 4374 stellar mass model that we use here, and Paolo Salucci for kindly providing the data of DDO 47. DCR and JCF thank CNPq and FAPES for partial financial support. PLO thanks CAPES for financial support. The work of I.Sh. has been partially supported by CNPq, FAPEMIG and ICTP.
References
 Rodrigues, Letelier, and Shapiro (2010) D. C. Rodrigues, P. S. Letelier, and I. L. Shapiro, JCAP 1004, 020 (2010), arXiv:0911.4967 [astroph.CO] .
 Rodrigues (2012) D. C. Rodrigues, To appear in JCAP (2012), arXiv:1203.2286 [astroph.CO] .
 Rodrigues, Letelier, and Shapiro (2011) D. C. Rodrigues, P. S. Letelier, and I. L. Shapiro, (2011), 1102.2188 [astroph.CO] .
 Farina et al. (2011) C. Farina, W. KortKamp, S. Mauro, and I. L. Shapiro, Phys.Rev. D83, 124037 (2011), arXiv:1101.5611 [grqc] .
 Fabris et al. (2012a) J. C. Fabris, P. L. de Oliveira, D. C. Rodrigues, A. M. VelasquezToribio, and I. L. Shapiro, Int.J.Mod.Phys. A27, 1260006 (2012a), arXiv:1203.2695 [astroph.CO] .
 Shapiro, Sola, and Stefancic (2005) I. L. Shapiro, J. Sola, and H. Stefancic, JCAP 0501, 012 (2005), arXiv:hepph/0410095 .
 Reuter and Weyer (2004a) M. Reuter and H. Weyer, JCAP 0412, 001 (2004a), arXiv:hepth/0410119 .
 Reuter and Weyer (2004b) M. Reuter and H. Weyer, Phys. Rev. D70, 124028 (2004b), arXiv:hepth/0410117 .
 Reuter and Weyer (2006) M. Reuter and H. Weyer, Int. J. Mod. Phys. D15, 2011 (2006), arXiv:hepth/0702051 .
 Goldman et al. (1992) J. T. Goldman, J. PerezMercader, F. Cooper, and M. M. Nieto, Phys. Lett. B281, 219 (1992).
 Bertolami, Mourao, and PerezMercader (1993) O. Bertolami, J. M. Mourao, and J. PerezMercader, Phys. Lett. B311, 27 (1993).
 Dalvit and Mazzitelli (1994) D. A. R. Dalvit and F. D. Mazzitelli, Phys. Rev. D50, 1001 (1994), arXiv:grqc/9402003 .
 Bertolami and GarciaBellido (1996) O. Bertolami and J. GarciaBellido, Int. J. Mod. Phys. D5, 363 (1996), arXiv:astroph/9502010 .
 Domazet and Stefancic (2011) S. Domazet and H. Stefancic, Phys. Lett. B703, 1 (2011), arXiv:1010.3585 [grqc] .
 Fabris et al. (2012b) J. C. Fabris, P. L. de Oliveira, D. C. Rodrigues, and I. L. Shapiro, Work in progress (2012b).
 (16) We use the spacetime signature.
 Tully and Fisher (1977) R. Tully and J. Fisher, Astron.Astrophys. 54, 661 (1977).
 Mamon and Lokas (2005) G. A. Mamon and E. L. Lokas, Mon.Not.Roy.Astron.Soc. 363, 705 (2005), arXiv:astroph/0405491 [astroph] .
 Gerhard et al. (2001) O. Gerhard, A. Kronawitter, R. Saglia, and R. Bender, Astron.J. 121, 1936 (2001), 21pp, 19 figs, latex. AJ, in press (April 2001) Reportno: Astr.Inst.Basel Prep.Ser.128, arXiv:astroph/0012381 [astroph] .
 Walter and Brinks (2001) F. Walter and E. Brinks, Astron. J. 121, 3026 (2001).
 Salucci, Walter, and Borriello (2003) P. Salucci, F. Walter, and A. Borriello, Astronomy Astrophysics 409, 53 (2003), arXiv:astroph/0206304 .
 Gentile et al. (2005) G. Gentile, A. Burkert, P. Salucci, U. Klein, and F. Walter, Astrophys. J. Letters 634, L145 (2005), arXiv:astroph/0506538 .