Testing Grumiller's Modified Gravity at Galactic Scales

Physics Letters B 728 (2014) 537–542

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Physics Letters B

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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]. 2rd rd 2rd 2rd 2rd 2rd A geodesics study of time-like test particles moving in a (10) 4-dimensional spherical, symmetric background, according to Eqs. (1), (3), and (4), results in the following equations: where I and K are the modified Bessel functions. The stars’ contribution to the rotation curves is normally mul- r˙2 tiplied by the mass-to-light ratio (Υ ), that is an additional free E = + V eff, (5) 2 parameter in the mass model, introduced because we generally 2 2 can only measure the distribution of the light instead of the mass. =−M + + + V eff ar 1 , (6) When we estimate the Rindler parameters (a,n), Υ is an im- r 2r2 r2 portant source of uncertainty, because these parameters are de- where E = const. and V eff is the effective potential. generate through Eq. (11), see below. However, since stars have a When we apply the previous solution to a galactic arena, we major contribution near the center of the galaxy and the Rindler = set the cosmological constant equal to zero (Λ 0) since the acceleration contribute most at large distance, Υ does not signif- mean energy density of a galaxy is much larger than the cos- icantly affect the uncertainties of the Rindler parameters, as we mic inferred Λ.Wesetalsol = 0, to avoid an additional angular have shown in Ref. [23]. momentum to the system that in fact shall not account for the The Υ has been modeled, e.g. in Salpeter [33],Kroupa[34], kinematical deficit of rotational curves. and Bottema [35], but the precise value for an individual galaxy is Considering now the effects on rotation curves, the Rindler ac- not well known and depends on extinction, star formation history, celeration yields an additional term in the rotation’s speed (v T ): initial mass function, among others. Some assumptions have to be made respect to Υ in order to reduce the number of free param-

dφT eters in the model. In a previous work [23], one of us (JLCC) has v T (r) = r + ar, (7) dr studied the Kroupa, diet-Salpeter, and Υ as a model-independent J.L. Cervantes-Cota, J.A. Gómez-López / Physics Letters B 728 (2014) 537–542 539

Table 1 Best ﬁts for the standard Rindler model (n = 1). It is shown in column (2) the galactic type and in (3) its disk radius, in (4) the acceleration parameter, in (5) the galactic 2 disk mass, (6) the B-band mass-to-light ratio in solar units, and in (7) the χred . 2 2 B 2 Galaxy Type Rd (kpc) a (km /s kpc) Md (M ) Γ χred +70 +0.4 × 7 DDO 47 IB 0.50 820−50 1.0−0.1 10 0.14.9 ESO 116-G12 SBcd 1.70 1010 ± 50 3.0 ± 0.4 × 109 0.63.8 ± +0.2 × 10 ESO 287-G13 SBc 3.28 920 30 3.9−0.1 10 1.31.8 +20 +0.2 × 7 IC 2574 SABm 1.78 400−10 1.0−1.0 10 < 0.138.9 a ± +0.02 × 11 M31 Sb 4.50 340 60 1.68−0.01 10 8.41.6 M33 Sc 1.42 900± 30 5.0 ± 0.1 × 109 0.94.3 NGC 55 SBm 1.60 870 ± 50 5.9 ± 2.3 × 108 0.23.7 +50 ± × 9 NGC 300 Scd 1.70 790−30 2.7 0.3 10 1.20.7 NGC 1090 Sbc 3.40 580 ± 20 5.0 ± 0.2 × 1010 1.33.8 a ± +0.4 × 10 NGC 2403 Sc 2.08 900 20 1.4−0.3 10 1.74.0 NGC 3877 Sc 2.80 1500 ± 400 2.7 ± 0.6 × 1010 1.00.9 NGC 3917 Scd 3.10 1000 ± 100 1.3 ± 0.2 × 1010 1.23.6 NGC 3949 Sbc 1.70 2300 ± 1200 1.5 ± 0.5 × 1010 0.80.4 NGC 3953 SBbc 3.80 1300 ± 400 8.7 ± 1.2 × 1010 2.10.4 ± +2.1 × 9 NGC 3972 Sbc 2.00 1800 300 3.8−2.0 10 0.60.5 ± +2.8 × 9 NGC 4085 Sc 1.60 2700 600 3.4−2.7 10 0.52.0 NGC 4100 Sbc 3.37 500 ± 100 6.7 ± 0.4 × 1010 2.71.0 NGC 4157 Sb 2.60 940 ± 100 5.2 ± 0.5 × 1010 1.70.8 NGC 4183 Scd 3.20 370 ± 70 1.6 ± 0.2 × 1010 1.70.1 ± +0.5 × 10 NGC 4217 Sb 2.90 1100 200 4.6−0.6 10 2.20.7 NGC 5585 SABc 1.26 870 ± 30 1.3 ± 0.1 × 109 0.97.3 NGC 6503a Sc 1.74 610 ± 10 1.37 ± 0.03 × 1010 2.76.8 NGC 7339a SABb 1.50 2900 ± 300 1.2 ± 0.1 × 1010 1.61.4 ± +0.8 × 10 UGC 128 Sd 6.40 350 70 2.6−0.7 10 3.00.2 UGC 6399 Sm 2.40 660 ± 260 3.6 ± 2.2 × 109 2.30.3 UGC 6917 SBd 2.90 590 ± 170 1.0 ± 0.2 × 1010 2.30.2 UGC 6983 SBcd 2.70 510 ± 100 1.1 ± 0.2 × 1010 2.70.7 UGC 8017a Sab 2.10 2000 ± 150 1.58 ± 0.04 × 1011 4.06.3 UGC 10981a Sbc 5.40 430 ± 40 2.10 ± 0.02 × 1011 1.816.0 UGC 11455a Sc 5.30 2800 ± 200 1.2 ± 0.1 × 1011 2.617.7

a For these galaxies we have no gas data. free parameter. It was shown that the different stellar mass models parison of these profiles see Refs. [12,13]. To extract information do not significantly change the determined value of the Rindler pa- for the Rindler parameters, as an input we will need the observa- rameters for most of the galaxies, and from the three models, the tional rotation curve and the computed gas contribution. free Υ yields best fits to rotation curves. Thus, for the purpose of this Letter, we adopt the Υ free mass model. 4. Rotation curve fits and results Gathering all contributions to the total (T ) rotation curve and including a generalized Rindler (GR) term,1 To perform the fits we employed the same method as in Ref. [23], but here it is applied to a set of thirty galaxies that has 2 = 2 + 2 + 2 v T (r) Υ v vG vGR(r), (11) been used in the past to test alternative gravity models [18].Most of galaxies possess wanted properties such as smoothness in the where we explicitly use Υ and therefore assume M with a solar d data, symmetry, and they are extended to large radii. We fit the mass-to-light in Eqs. (9), (10); the power-law generalized Rindler observational velocity curve to the theoretical model (11) using term is the χ 2 goodness-of-fit test (χ 2 test), that computes the param- 2 v2 (r) ≡ a|r|n. (12) eters’ best fits. In general the χ test statistics are of the form: GR m 2 The case n = 1 is the original model of modified gravity at large vobs − vmodel (r,a,n) χ 2 = i i , (13) distances [19,20], as the Rindler contribution in Eq. (7). The new = σi free parameters of the model of galactic rotation curves are a and i 1 n, and they have to be determined by observations. In standard where σ is the standard deviation, and m is the number of obser- 2 ≡ 2 − − dark matter profiles such as NFW [25,26],Burkert[27],pseudo- vations. One defines the reduced χred χ /(m p 1), in which isothermal, or alternative Bound Dark Matter [11] one also uses m is the number of observations and p is the number of fitted two free parameters, and therefore the number of degrees of free- parameters. The total velocity (11) defines our model – vmodel in domtofitissameasinthegeneralizedRindlermodel;foracom- Eq. (13) – and depends on the three parameters: Υ (or alterna- tively the mass of the disk Md that we actually fit), and the two Rindler parameters (a,n). 1 Notice that the squared sum in Eq. (11) is meant to add the different gravi- We firstly analyze the original Rindler model (n = 1) and pro- tational contributions of gas, stars, and Rindler, but it is not meant to represent ceed to fit the parameters a and M . Their physical units are a vectorial’s squared sum since the different velocities are not orthogonal con- d 2 2 tributions to the total. We thank Marcelo Salgado for pointing it out about this km /s kpc and M, respectively. We assume a flat prior for 7 12 commonly used notation. these parameters in the following intervals: 10 < Md < 10 and 540 J.L. Cervantes-Cota, J.A. Gómez-López / Physics Letters B 728 (2014) 537–542

Table 2 Best ﬁts for the power-law generalized Rindler model. It is shown in column (2) the acceleration parameter, in (3) the power-law exponent, in (4) the galactic disk mass, 2 (5) the B-band mass-to-light ratio in solar units, and in (6) the χred . 2 n 2 B 2 Galaxy a (km /s kpc ) nMd (M ) Γ χred ± ± +2.3 × 7 DDO 47 420 60 1.5 0.13.3−2.1 10 0.31.2 ESO 116-G12 1100 ± 200 1.0 ± 0.12.7 ± 0.4 × 109 0.63.7 +90 ± ± × 10 ESO 287-G13 320−70 1.3 0.14.3 0.1 10 1.51.6 IC 2574 180 ± 81.6 ± 0.01.1 ± 0.2 × 108 0.12.4 +1.1 ± ± × 11 M31 0.3−0.2 3.3 0.51.8 0.0 10 8.81.1 M 33 1600 ± 100 0.8 ± 0.03.8 ± 0.1 × 109 0.73.3 ± ± +1.5 × 8 NGC 55 1000 100 0.9 0.13.9−1.8 10 0.13.5 +130 ± ± × 9 NGC 300 630−120 1.1 0.13.0 0.3 10 1.30.7 NGC 1090 1400 ± 300 0.8 ± 0.14.5 ± 0.2 × 1010 1.23.4 NGC 2403 3600 ± 200 0.5 ± 0.08.0 ± 0.4 × 109 1.02.3 +640 ± ± × 10 NGC 3877 550−340 1.4 0.43.2 0.4 10 1.20.9 +180 +0.2 ± × 10 NGC 3917 430−140 1.3−0.1 1.7 0.2 10 1.53.4 +1160 +0.8 +0.3 × 10 NGC 3949 800−590 1.5−0.6 1.8−0.2 10 0.90.4 +4500 ± +1.3 × 10 NGC 3953 7900−3100 0.5 0.26.0−1.2 10 1.50.4 +460 +0.3 +1.5 × 9 NGC 3972 880−350 1.3−0.2 6.4−1.4 10 1.00.5 +900 +0.4 +1.8 × 9 NGC 4085 1400−600 1.3−0.3 5.6−1.7 10 0.82.0 +730 +0.3 ± × 10 NGC 4100 770−440 0.9−0.2 6.6 0.4 10 2.61.0 +1700 ± ± × 10 NGC 4157 2600−1100 0.7 0.24.5 0.5 10 1.50.7 +580 ± ± × 10 NGC 4183 830−390 0.8 0.21.4 0.2 10 1.40.1 +690 ± ± × 10 NGC 4217 730−410 1.2 0.34.8 0.4 10 2.30.7 ± +0.1 ± × 9 NGC 5585 1000 100 0.9−0.0 1.1 0.1 10 0.76.9 NGC 6503 7900 ± 400 0.2 ± 0.04.3 ± 0.3 × 109 0.91.4 NGC 7339 2000 ± 300 1.2 ± 0.11.3 ± 0.1 × 1010 1.81.4 +320 ± ± × 10 UGC 128 260−180 1.1 0.32.8 0.6 10 3.20.2 +340 +0.6 +1.3 × 9 UGC 6399 240−170 1.4−0.5 5.1−1.2 10 3.20.3 +200 +0.7 ± × 10 UGC 6917 110−90 1.6−0.5 1.3 0.1 10 2.90.2 +280 ± ± × 10 UGC 6983 340−170 1.1 0.31.2 0.1 10 2.90.7 UGC 8017 8500 ± 500 0.6 ± 0.01.1 ± 0.0 × 1011 2.94.8 UGC 10981a 12000 ± 600 0.2 ± 0.01.5 ± 0.0 × 1011 1.210.4 +140 ± ± × 11 UGC 11455 650−120 1.5 0.11.5 0.0 10 3.316.9 a The upper limit for a has been extended to ﬁnd the χ 2 minimum.

0 < a < 10000. Given the large space that would require to show two orders of magnitude between the smallest and biggest values all rotation curves fits and parameters’ contour plots, we include for the Rindler acceleration. On the other hand, the power-law ex- this information as supplementary material. We gather our results ponent ranges from 0.16 ± 0.02 for NGC 6503 to 3.31 ± 0.54 for in Table 1. The uncertainties in the rotation velocity are reflected M31 or, again excluding M31, 1.60 ± 0.03 for IC 2574, which is in the uncertainties in the model parameters. One observes some an order of magnitude difference. For both parameters the uncer- +64.84 spread in the values for a, ranging from 341.26−64.46 for M31 to tainties are too small to account for the encountered differences. +293.75 Similarly as the previous Rindler model, one only has thirteen (of 2891.25−287.25 for NGC7339 to account for a difference of an order 2 of magnitude, but the uncertainties are small to account for such thirty) galaxies with χred 1, and again, for these later galaxies a difference. In addition to this discrepancy, the fits to some of the we have included distribution plots (as supplementary material) 2 that show a big spread in both parameters of the generalized galaxies present very high χred values that result in poor fittings. 2 Rindler model. Only thirteen (of thirty) galaxies had χred 1, and for these later galaxies we have included a distribution plot (as supplementary By comparing both fits (n = 1vsn = 1), the goodness of fits material) that shows a big spread in the values of the acceleration are better in the generalized model for sixteen galaxies, and for parameter. We also plot the B-band mass-to-light ratios that are fourteen both models are equally well fitted. The Rindler acceler- similar to others reported in the literature [36,18]. ation varied for the standard Rindler model one order of magni- For the generalized model (n = 1) we consider a flat prior in tude and for the generalized model two orders of magnitude. In the interval 0 < n < 10, and the same other conditions as previ- our previous work, when one of us analyzed the THINGS’ galax- 2 ous model. We determine now the two Rindler parameters (a,n) ies [23],boththeχred and Rindler acceleration values changed and the stellar disk’s mass. The results are shown in Table 2.Again more substantially: two and three orders of magnitude, respec- a spread is observed in the acceleration parameter, ranging from tively. With respect to the power-law exponent of the generalized + + 1.09 564.60 model, the present analysis results in one order of magnitude dif- 0.26−0.21 for M31 to 11605.40−555.00 for UGC 10981 resulting in a difference of four orders of magnitude. Since the a value for M31 ference, whereas for the THINGS galaxies resulted in two orders of is very small and is the only one with a value much less than magnitude. The reason to have smaller differences in the computed one hundred, and since we did not include gas data to the anal- parameters is that the present set of galaxies, while it is a larger ysis, we may exclude it for this analysis. The second lowest value collection, its uncertainties in data are bigger. Although the present +195.61 of a is 113.39−85.89 for UGC 6917. Still there is a difference of analysis soften the difference in the parameter computation, still J.L. Cervantes-Cota, J.A. Gómez-López / Physics Letters B 728 (2014) 537–542 541 such differences are not justified. On the other hand, the mass-to- Table 3 2 light ratios found do not present systematic differences. Summary of the χred values for the different dark matter profiles and gravity mod- From the set of galaxies considered in the present work and els. 2 that in Ref. [23] there is a common galaxy, NGC 2403. The data Galaxy χred considered are different and subject to different systematics. How- Burkert NFW (n = 1) (n-free) ever, we may expect some similar parameter estimation. The com- DDO 47 1.05.74.91.2 puted parameters in Ref. [23] were, for the n = 1 Rindler model, ESO 116-G12 0.92.63.83.7 + +9.8 = 97.65 = 10.2−6.4 ESO 287-G13 1.52.11.81.6 a 797.22−0.32 and M D 10 M, whereas for the present + IC 2574 2.043.538.92.4 computation Table 1 shows a = 900 ± 20 and M = 1.4 0.4 × D −0.3 M31 1.91.71.61.1 10 10 M. Clearly, both Rindler parameters and stellar disk masses M33 5.54.14.33.3 are within 1 σ . For the generalized model the previous work gives NGC 55 0.32.93.73.5 9.9±8 a = 3070 ± 16, n = 0.59 ± 0.002, and M D = 10 M, whereas NGC 300 0.40.60.70.7 in the present work we have a = 3600 ± 200, n = 0.54 ± 0.02, and NGC 1090 0.81.83.83.4 9 NGC 2403 1.51.14.02.3 M D = 8 ± 0.4 × 10 M. In this case, the Rindler parameters are NGC 3877 0.31.10.90.9 not quite different, but given the uncertainties, they are a few σ NGC 3917 0.73.93.63.4 away from each other; stellar disk masses are within 1 σ . The dis- NGC 3949 0.20.60.40.4 crepancy in Rindler parameters’ uncertainties may be due to the NGC 3953 0.50.50.40.4 fact that the THINGS sample include more data and are more pre- NGC 3972 0.10.60.50.5 cise. On the other hand, we do not expect a big influence of a NGC 4085 0.52.42.02.0 small bulge, that was taken into account in Ref. [23], on the com- NGC 4100 0.60.91.01.0 puted values of the Rindler parameters, since the main influence NGC 4157 0.50.60.80.7 NGC 4183 0.10.10.10.1 of the modified gravity is in the outer parts of the galaxies, where NGC 4217 0.30.70.70.7 the bulge, or even the disk, counts less. In our present work, we NGC 5585 0.44.97.36.9 have not taken into account bulge contributions, since it is known NGC 6503 2.15.36.81.4 the present set has negligible bulges [18]. NGC 7339 1.21.51.41.4 We now compare the Rindler models with standard dark mat- UGC 128 0.00.20.20.2 ter profiles, such as NFW [25,26] and Burkert [27].Theformeris UGC 6399 0.00.40.30.3 an example of a cuspy dark matter profile, whereas the later is UGC 6917 0.40.30.20.2 UGC 6983 0.60.70.70.7 shallow. The explicit computations are as those done in our previ- UGC 8017 3.13.76.34.8 ous works [13,23] and are included as supplementary material. To UGC 10981 7.26.616.010.4 compare among the different models we constructed Table 3 with UGC 11455 13.019.317.716.9 2 = the χred values for NFW, Burkert, standard Rindler with n 1, and generalized Rindler (n-free), for the free stellar mass model. The results are as follows: to some of these galaxies has been reported in the literature [37], as well as further analysis on the NFW fits in Ref. [38]. • As already mentioned, the Rindler model with two free parameters (a, n) fits equally well or better than the model with 5. Conclusions a single parameter (a, n = 1) for all galaxies. Using a collection of rotation curves of thirty galaxies, we have • The standard Rindler model (n = 1) fits worst than Burkert’s tested the standard (n = 1) and power-law generalized (n-free) profile, but similarly well as NFW. The standard Rindler model Grumiller’s model of modified gravity at large distances. The cor- achieves an equally well or a better fit than both NFW and responding gravitational potential implies a new (Rindler) acceler- Burkert only for six galaxies (M 31, NGC 3949, NGC 3953, NGC ation constant in nature that affects the rotation curve as v2 (r) = 4183, NGC 4217, and UGC 6917) and, in addition, it fits better T Υ v2 + v2 + a|r|n, where the last term would replace the contribu- than Burkert for one galaxy (M 33), and it fits equally well or G tion of the dark matter profile. better than NFW for eleven galaxies (DDO 47, ESO 287-G13, IC The results of the fits are in Tables 1 and 2, and a comparison 2574, NGC 3877, NGC 3917, NGC 3972, NGC 4085, NGC 7339, of the goodness-of-fit to NFW’s and Burkert’s profiles is presented UGC 6399, UGC 6983, and UGC 11455). In summary, the NFW’s in Table 3. Our results show that: i) the standard Rindler model profile fits equally well or better for 16 galaxies (out of 30) (n = 1) does not achieve good fits since only thirteen (of thirty) and Burkert’s profile achieves an equally well or a better fit galaxies had χ 2 1, and these best-fitted galaxies also show a for 26 galaxies (out of 30) than the standard Rindler model. red big spread in the acceleration parameter; ii) the power-law gen- • The power-law generalized Rindler model (n-free) fits worst eralized model (n = 1) does achieve an equally well (14/30) or a than Burkert’s profile, but slightly better than NFW. This better fit (16/30) than the standard Rindler’s model for all galax- Rindler model fits equally well or better than both NFW and 2 ies, but again only thirteen (of 30) galaxies had χred 1, and Burkert models for six galaxies (M 31, M 33, NGC 3953, NGC also these best-fitted galaxies show big spreads in the Rindler 4183, NGC 6503, and UGC 6917) and, in addition, it fits equally parameters; iii) the comparison of these modified gravity models well or better than NFW for fourteen galaxies (DDO 47, ESO with standard dark matter profiles yields that the standard Rindler 287-G13, IC 2574, NGC 3877, NGC 3917, NGC 3949, NGC 3972, model (n = 1) fits worst Burkert’s profiles, but it fits equally well NGC 4085, NGC 4217, NGC 7339, UGC 128, UGC 6399, UGC as NFW. The generalized model achieves better fits than NFW’s 6983, and UGC 11455). In summary, the NFW profile fits profile, but much poorer fits than Burkert’s profile. equally well or better for 14 galaxies (out of 30) and Burkert’s The main problem, however, is that both Rindler parameters profile achieves an equally well or better fit for 25 galaxies (a,n) show at least one order of magnitude spread that cannot (out of 30) than the generalized Rindler model. The fact that be explained by the corresponding uncertainties, not pointing to Burkert’s shallow profile fits better than the cuspy NFW profile single universal values. 542 J.L. Cervantes-Cota, J.A. Gómez-López / Physics Letters B 728 (2014) 537–542

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