Testing Grumiller's Modified Gravity at Galactic Scales
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Physics Letters B 728 (2014) 537–542 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Testing Grumiller’s modified gravity at galactic scales ✩ ∗ Jorge L. Cervantes-Cota , Jesús A. Gómez-López Depto. de Física, Instituto Nacional de Investigaciones Nucleares, Apartado Postal 18-1027 Col. Escandón, 11801 DF, Mexico article info abstract Article history: Using galactic rotation curves, we test a – quantum motivated – gravity model that at large distances Received 12 July 2013 modifies the Newtonian potential when spherical symmetry is considered. In this model one adds a Received in revised form 9 November 2013 Rindler acceleration term to the rotation curves of disk galaxies. Here we consider a standard and a Accepted 3 December 2013 power-law generalization of the Rindler modified Newtonian potential that are hypothesized to play Available online 6 December 2013 the role of dark matter in galaxies. The new, universal acceleration has to be – phenomenologically – Editor: S. Dodelson determined. Our galactic model includes the mass of the integrated gas and stars for which we consider a free mass model. We test the model by fitting rotation curves of thirty galaxies that has been employed to test other alternative gravity models. We find that the Rindler parameters do not perform a suitable fit to the rotation curves in comparison to the Burkert dark matter profile, but the models achieve a similarfitastheNFW’sprofiledoes.However,thecomputedparametersoftheRindlergravityshow some spread, posing the model to be unable to consistently explain the observed rotation curves. © 2013 The Authors. Published by Elsevier B.V. All rights reserved. 1. Introduction Recently, a model of gravity has been put forward that stems from quantum gravity corrections to General Relativity and when It is well known that General Relativity is a theory well tested one applies it to spherical symmetry and local (galactic) scales an within the solar system and scales below, inasmuch as no devia- extra Rindler acceleration appears in addition to the standard New- tions to it have been found since many years [1,2].However,new tonian formula for rotation curves [19,20]. The new Rindler term is theories/models of gravitation have been recently proposed mo- hypothesized to play the role of dark matter in galaxies. This idea tived by different theoretical and observational reasons, see Ref. [3] has been tested already in a very recent work [21], where a fit is for a review. One of the motivations is to test gravity theories be- made to eight galaxies of The HI Nearby Galaxy Survey (THINGS) yond the solar system, and to understand what constraints could [22]. They found that six of the galaxies tend to fit well to the be drawn at different length scales. On the one hand, at cosmo- data and that there is a preferred Rindler acceleration parameter logical scales different corrections apply to the standard theory of − of around a ≈ 3.0 × 10 9 cm/s2 (= 926 km2/s2 kpc); they later large scale structure alone from General Relativity [4–6] and, in fixed this acceleration parameter and found acceptable fits for five addition, new approaches have been put forward to understand galaxies, and furthermore, an additional free parameter let them the possible deviations of data to the theory [7–10]. On the other to fit two more galaxies. We have revised this idea using a greater hand, at galactic scales rotation curves provide a unique labora- sample (seventeen) of THINGS galaxies, and for the eight original tory to test kinematical deviations from theoretical expectations and in fact rotation curves are one of the reasons why dark mat- galaxies we find similar conclusions on the fits and to a conver- ter has been hypothesized. Although cold dark matter is the most gence to a similar Rindler acceleration within 1σ confidence level. popular candidate, there are other possibilities, e.g. bound dark But when one adds more galaxies to the analysis the spread in the matter [11–13], or other theoretical approaches that modify grav- acceleration blows up, and therefore we concluded that the model ity or kinematical laws such as MOND [14,15] (see however [16]) is not tenable [23]. However, THINGS rotation curves are based on or f(R)-gravity that apart from playing the role of dark energy also gas kinematics, whereas there are claims pointing out that com- intends to replace dark matter [17,18]. plex gas dynamics could not be a good tracer of gravity in spirals, and gas and stellar motion do not exactly coincide in all the cases [24]. Given this, in the present work we test again the Rindler ac- ✩ This is an open-access article distributed under the terms of the Creative Com- celeration hypothesis (and a generalized version of it) but with mons Attribution License, which permits unrestricted use, distribution, and repro- duction in any medium, provided the original author and source are credited. a different sample of galaxies that is larger (thirty galaxies) than * Corresponding author. the previous sample and has very different systematics. This set of E-mail address: [email protected] (J.L. Cervantes-Cota). galaxies has been used to test other gravity models in the past [18] 0370-2693/$ – see front matter © 2013 The Authors. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physletb.2013.12.014 538 J.L. Cervantes-Cota, J.A. Gómez-López / Physics Letters B 728 (2014) 537–542 since most of them have desired properties such as smoothness in where φT is the total gravitational potential that test particles the data, symmetry, and they are extended to large radii. (stars and gas) feel. This is the original Grumiller’s model of gravity This work is organized as follows: In Section 2 we briefly re- at large distances [19]. The new Rindler acceleration term should view Grumiller’s model of gravity at large distances, in Section 3 account for the kinematical difference of the observed and pre- we explain the rotation curve models, in Section 4 we present our dicted rotation curves. Notice that Eq. (7) diverges asymptotically, results and compare the fits to results from standard dark matter at large radius. This is not an observed behaviour in typical ro- profiles such as Navarro–Frenk–White (NFW) [25,26] and Burkert tation curves, but on the contrary they tend to slowly decrease [27]. Finally, Section 5 is devoted to conclusions. Supplementary after a few optical radii [29]. Therefore, as a generalization of the material is included to support our conclusions. previous model one may intend to determine a power-law depen- dence in the Rindler term, as suggested in Ref. [19]. The new term 2. Grumiller’s gravity model at large distances should not diverge at large distances. Accordingly, we will consider the following generalized Grumiller model: In order to have a self-consistent description of this work we briefly review the main ideas behind Grumiller’s model, for details dφ T n vc (r) = r + ar , (8) see [19]. The model starts with spherical symmetry in four dimen- T dr sions split in the following way: where there are two undetermined Rindler parameters (a,n).The 2 α β 2 2 2 2 ds = gαβ dx dx + Φ dθ + sen θ dφ , (1) case n = 1 yields acceleration units to a,butadifferentn implies 2−n γ length units; one could extract an acceleration parameter here where gαβ (x ) is a 2-dimensional metric and the surface radius time2 γ γ ={ } n n−1 Φ(x ) depend upon x t, r . The idea is to describe these fields if one defines ar ≡ anewr(r/rnew) , but we would only add an in two dimensions since the gravitational potentials gαβ and Φ extra parameter (rnew) that is completely degenerated with anew. that are intrinsically two-dimensional, and their solutions can be This could be done a posteriori, if needed. mapped into the 4-dimensional world through Eq. (1). The most general 2-dimensional gravitational theory that is 3. Rotation curve model renormalizable, that yields a standard Newtonian potential, and that avoids curvature singularities at large Φ is: In this section we closely follow the model presented in √ Ref. [23], but for the sake of completeness we present it here S =− −g Φ2 R + 2∂Φ2 − 6ΛΦ2 + 8aΦ + 2 d2x, (2) again. The galaxy model consists of gas and stars orbiting on a disk plane, and instead of dark matter we include the Rindler ac- that depends on two fundamental constants, Λ and a, the cosmo- celeration, explained in the previous section. The contribution of logical constant and a Rindler acceleration, respectively. The solu- gas is computed by integrating the surface brightness as in the tions to this action will describe the original line element, Eq. (1), standard Newtonian case by assuming an infinitely thin disk. One that will model gravity in the infrared. The solutions are: directly integrates its contribution to the rotation curve (vG ). For stars we take a standard Freeman disk [30,31]: 2 α β 2 2 dr gαβdx dx =−K dt + , (3) 2 Md −r/r K ρ (r) = e d , (9) 2M 2πr2 K 2 = 1 − − Λr2 + 2ar, (4) d r where Md is the mass of the disk and rd its radius. The rotation with K being the norm the Killing vector ∂t and M a constant of curve contribution from stars within standard Newtonian dynam- motion. Of course, if Λ = a = 0, one recovers the Schwarzschild ics, yields [32]: solution. If M = Λ = 0, it yields the 2-dimensional Rindler metric. GM r 2 r r r r Therefore, the resulting gravity theory differs from General Relativ- 2 = d − v (r) I0 K0 I1 K1 , ity only by the addition of a Rindler acceleration, see also [28].