Baryonic Tully-Fisher test of Grumiller's modified gravity model
Samrat Ghosh, Arunava Bhadra, Amitabha Mukhopadhyay, Kabita Sarkar
BBaryonic Tully-Fisher test of Grumiller’s modified gravity model
Samrat Ghosh and Arunava Bhadra
High Energy & Cosmic Ray Research Centre, University of North Bengal, Siliguri, West Bengal, India 734013
Amitabha Mukhopadhyay
Department of Physics, University of North Bengal, Siliguri, West Bengal, India 734013
Kabita Sarkar
Department of Mathematics, Swami Vivekananda Institute of Science & Technology, Dakshin Gobindapur, Kolkata-700145
We test the Grumiller’s quantum motivated modified gravity model, which at large distancesmodifies the Newtonian potential and describes the galactic rotation curves of disk galaxies interms of a Rindler acceleration term without the need of any dark matter, against the baryonicTully-Fisher feature that relates the total baryonic mass of a galaxy with flat rotation velocity ofthe galaxy. We estimate the Rindler acceleration parameter from observed baryonic mass versusrotation velocity data of a sample of sixty galaxies. Grumiller’s model is found to describe theobserved data reasonably well.
PACS numbers: 04.60.-m, 95.35.+d, 98.62.DmKeywords: Dark matter, Modified gravity model, Tully-Fisher
Preprint: This a preprint of the Materials acceptedfor publication in Gravitation and Cosmology, copyright(2021), the copyright holder indicated in the Journal
I. INTRODUCTION
Several astrophysical observations and specially theobservation of flat rotation curve of galaxies lead to thehypothesis of dark matter. However, despite several ef-forts so far there is no direct evidence of dark matterparticles, nor their existence is predicted by any stan-dard theoretical model of particle physics. Consequentlymany alternative explanations of flat rotation curve ofgalaxies exist in the literature including modification ofgravitational law at large distances [1], [3] or even modi-fication of Newton’s laws of dynamics [11].Grumiller proposed a quantum motivated theory ofgravity that aims to explain the galactic flat rotation interms of a Rindler acceleration term without the need ofany dark matter [4], [5]. Assuming spherical symmetry,Grumiller considered the most general form of metric infour dimensions ds = g αβ ( x µ ) dx α dx β + Φ ( x µ ) (cid:0) dθ + sin θdϕ (cid:1) (1) , α, β, µ = 0 , g αβ ( x µ ) is a two dimensional metric and the sur-face radius Φ ( x µ ) is a 2-dimensional dilaton field. Toobtain g αβ ( x µ ) and Φ ( x µ ) Grumiller considered themost general two dimensional renormalizable gravita-tional theory of the form S = − (cid:90) √− g [Φ R + 2 ∂ Φ − + 8 a Φ + 2] d x , (2)which contains two fundamental constants, Λ and a , thecosmological constant and a Rindler acceleration, respec- tively. The specialty of the gravitational theory drivenby the above action is that it gives a standard Newto-nian kind of potential, and the theory has no curvaturesingularities at large Φ( x µ ). The solution of the two di-mensional fields g αβ ( x µ ) and Φ( x µ ) are given by g αβ dx α dx β = − B ( r ) dt + dr B ( r ) , (3)Φ ( x µ ) = r , (4)where B ( r ) = 1 − Mr − Λ r + 2 ar, (5) M is a constant of motion. When Λ = a = 0, the abovesolution reduces to the Schwarzschild solution and for M = Λ = 0 the solution becomes the 2-dimensionalRindler metric. The above solutions are mapped intothe four dimensional world through Eq.(1).The theory has found to explain the rotation curvesof spiral galaxies well [6]. By fitting the rotationcurves of eight galaxies of The HI Nearby Galaxy Sur-vey (THINGS) [18] the Rindler acceleration term wasfound as a ∼ × − m s − [6]. When a larger sam-ple (thirty galaxies) of rotation curves were consideredthe fitting of the data by the Rindler acceleration wasfound not very good [9], [2] but the goodness of fittingwith the Grumiller’s theory was still found comparable tothat using standard Navarro–Frenk–White (NFW) pro-file [13], [14]. The fitted Rindler acceleration parameter,however, exhibit considerably large spread, at least oneorder of magnitude with mean around 3 × − m s − [2]. a r X i v : . [ g r- q c ] J a n The rotation velocity of galaxies is known to relatewith their (galaxies) luminosity [17]. The optical Tully-Fisher relation, however, shows break; the relation is notuniversal for bright and faint galaxies [10]. Instead galac-tic rotation velocity is found to exhibit universal relationwith the total baryonic mass ( M ) of the galaxy with theform M ∝ v rot [10].In the present work we have examinedt the Grumillertheory against baryonic Tully-Fisher relation and subse-quently we have estimated the Rindler acceleration pa-rameter in the framework of Grumiller’s model using ob-served total baryonic mass versus rotation velocity datafor a sample of sixty galaxies. II. ROTATION VELOCITY AS A FUNCTIONOF BARYONIC MATTER IN GRUMILLERTHEORY
For the metric given by Equation (1) with equation(3) the expression of rotation velocity ( v rot ) of galaxiesis given by, v rot = rB (cid:48) ( r )2 B ( r ) (6)where B (cid:48) ( r ) signifies the derivative with respect to r , r is the co-ordinate distance from galactic centre. Forthe solution of B(r) given by equation (5) the rotationvelocity becomes v rot ≈ ( mr − Λ r + ar ) / (7)Because of very small magnitude of Λ we henceforth ig-nore the corresponding term in the expression of rotationvelocity. The observed rotation velocity in galaxies is ingeneral not strictly constant even at large distances butoften has some weak dependence on radial distance. Therotation velocity in Grumiller gravity is also not exactlyflat (constant) at large r but slowly increases with r. Soan obvious question is what value of rotation velocity willbe considered for testing the Tully-Fisher relation. ForGrumillers theory we consider (local) minimum value ofrotation velocity. The radial distance ( r e ) at which ro-tation velocity reaches its extremum value can be ob-tained by differentiating equation (7) with respect to r and equating it to zero which gives r e (cid:39) ma (8)Inserting it to equation (7), we get, v (cid:48) rot = 4 am, (9)where v (cid:48) ) denotes the extremum rotation velocity. Theabove expression shows that Grumiller’s theory correctly describe the baryonic Tully-Fisher relation, at least atthe theoretical level.To match with the observed rotation curve feature apower-law generalization of the Rindler modified Newto-nian potential ( − M/r + ar n ) is proposed in the literature[2]. Such a power law generalization modifies the Eq.(9)as v (cid:48) rot ∝ m nn +1 (10)In the above case baryonic mass is not strictly propor-tional to fourth power of rotation velocity but varies as m ∝ v (cid:48) n +1) / nrot . III. ESTIMATION OF RINDLERACCELERATION PARAMETER FROMOBSERVED ROTATION VELOCITY VS MASSDATA
In this section our objective is to estimate the Rindleracceleration parameter from observed rotation velocityvs baryonic mass data for a sample of disk galaxies. Weuse the compiled data of Sanders and MacGaugh [16] asgiven in Table 1 that include the early works of manygood astronomer.The major luminous matter components in a typicalspiral galaxy are stars and gas. Accordingly the totalmass of the galaxy is considered as sum of the stellarmass and gas mass. In the used sample the mass is es-timated through photometry, particularly using redderpassbands as tracer. The HI thickness method was usedfor measuring the rotation velocity. The details of thedata used and procedure of estimation of mass and ro-tating velocity are discussed in [10], [16].The equation (9) is used to estimate the Rindler ac-celeration parameter a from the observed data. We fitthe observed rotation velocity versus baryonic mass databy the Tully-Fisher relation (equation (9)) using the χ goodness-of-fit test. The fitting gives a = (3 . ± . × − ms − with reduced χ = 2 .
0. The fitted curve isshown in figure 1.The estimated values of a for individual galaxies aregiven in the last column of the Table 1. It has a smallspread, ranges from 1 . × − ms − for UGC 6446 to7 . × − ms − for NGC 3949 with mean value 3 . × − ms − and standard deviation 0 .
90. The frequencydistribution of estimated a for the sample of sixty galaxiesis shown in figure 2.We also fit the observed rotation velocity versusbaryonic mass data for the modified Tully-Fisher rela-tion (equation (10)) led by power-law generalization ofthe Rindler modified Newtonian potential using the χ goodness-of-fit test which is also depicted in figure 2. Inthis case the fitted value of the parameters are found n = 1 .
19 and a = 9 . × − ms − with reduced χ = 1 .
50 100 150 200 250 300 3500.1110
Observed data power law fit power law fit with index fixed at n=4 T o t a l ba r y on i c m a ss ( i n s o l a r m a ss ) rotation velocity (in km/s) FIG. 1: Variation of observed total baryonic mass with rota-tion velocity. The filled (blue) circle represent the observeddata, solid (black) line gives the fitting of the data for stan-dard Rindler acceleration (power index fixed at 4) and thedotted (red) line shows the fitting of the data with general-ized Rindler acceleration under Grumiller’s modified gravitymodel. F r equen cy a (in m s -2 ) FIG. 2: Frequency distribution of estimated Rindler acceler-ation parameter.
IV. DISCUSSION AND CONCLUSION
The Rindler parameter was estimated in [6] by fit-ting rotation curves of eight galaxies of The HI NearbyGalaxy Survey (THINGS) and the fitted mean valueof the Rindler acceleration parameter was found a ∼ × − ms − . However, when a larger sample of galax-ies were considered for analysis the spread in the value ofacceleration parameter becomes quite large and therebythe validity of the Grumiller model is questioned [2]. Incontrast the Rindler acceleration parameter as estimatedin the present work using the rotation velocity versustotal baryonic mass data of a sample of sixty galaxiesexhibits relatively small spread. The mean value is, how-ever, nearly the same to that obtained by fitting rota-tion curves [2]. As stated already the rotation velocityin Grumiller’s theory (equation (7) is not flat but slowlydiverges asymptotically which is not in accordance with TABLE I: Galaxy data
Galaxy V rot ( km s − ) M stellar (10 M (cid:12) ) M gas (10 M (cid:12) ) a (10 − )( ms − )UGC 2885 300 30.8 5 4.19NGC 2841 287 32.3 1.7 3.70NGC 5533 250 19 3 3.29NGC 6674 242 18 3.9 2.90NGC 3992 242 15.3 0.92 3.92NGC 7331 232 13.3 1.1 3.73NGC 3953 223 7.9 0.27 5.61NGC 5907 214 9.7 1.1 3.60NGC 2998 213 8.3 3 3.37NGC 801 208 10 2.9 2.69NGC 5371 208 11.5 1 2.77NGC 5033 195 8.8 0.93 2.75NGC 3893 188 4.2 0.56 4.86NGC 4157 185 4.83 0.79 3.86NGC 2903 185 5.5 0.31 3.73NGC 4217 178 4.25 0.25 4.13NGC 4013 177 4.55 0.29 3.76NGC 3521 175 6.5 0.63 2.44NGC 4088 173 3.3 0.79 4.06NGC 3877 167 3.35 0.14 4.13NGC 4100 164 4.32 0.3 2.90NGC 3949 164 1.39 0.33 7.79NGC 3726 162 2.62 0.62 3.94NGC 6946 160 2.7 2.7 2.25NGC 4051 159 3.03 0.26 3.60NGC 3198 156 2.3 0.63 3.74NGC 2683 155 3.5 0.05 3.01NGC 3917 135 1.4 0.18 3.89NGC 4085 134 1 0.13 5.28NGC 2403 134 1.1 0.47 3.80NGC 3972 134 1 0.12 5.33UGC 128 131 0.57 0.91 3.68NGC 4010 128 0.86 0.27 4.40F568-V1 124 0.66 0.34 4.38NGC 3769 122 0.8 0.53 3.08NGC 6503 121 0.83 0.24 3.71F568-3 120 0.44 0.39 4.63NGC 4183 112 0.59 0.34 3.13F563-V2 111 0.55 0.32 3.23F563-1 111 0.4 0.39 3.56NGC 1003 110 0.3 0.82 2.42UGC 6917 110 0.54 0.2 3.66UGC 6930 110 0.42 0.31 3.71M 33 107 0.48 0.13 3.98UGC 6983 107 0.57 0.29 2.82NGC 247 107 0.4 0.13 4.58NGC 7793 100 0.41 0.1 3.63NGC 300 90 0.22 0.13 3.47NGC 5585 90 0.12 0.25 3.28NGC 55 86 0.1 0.13 4.40UGC 6667 86 0.25 0.08 3.07UGC 2259 86 0.22 0.05 3.75UGC 6446 82 0.12 0.3 1.99UGC 6818 73 0.04 0.1 3.76NGC 1560 72 0.034 0.098 3.77IC 2574 66 0.01 0.067 4.56DDO 170 64 0.024 0.061 3.66NGC 3109 62 0.005 0.068 3.75DDO 154 56 0.004 0.045 3.72DDO 168 54 0.005 0.032 4.26 the observed behaviour in typical rotation curves whererotation velocity is found to decrease slowly at large ra-dial distances [15]. This seems the main reason of poordescription of rotation velocity curves by the Grumiller’smodel. While describing the observed rotation velocityversus baryoinc mass data we have considered extremavalues of rotation velocity thereby taking out the radialdependence of rotation velocity.It was found in [2] that the goodness of fits of rota-tion curves are better in the generalized Rindler acceler-ation model than that of the standard Rindler accelera-tion model. However, the power law index n was foundto vary substantially (from 0 . .
3) to describe the ob-server rotation curves [2], which is against the universal-ity of the baryonic Tully-Fisher relation as may be notedfrom equation (10). The power law generalization is thusnot suitable for Tully-Fisher feature unless power law in-dex is kept fixed and universal for all galaxies. Since theform of the Grumiller’s solution (Eqs.(3) and (5)) is thesame to the vacuum (static spherically symmetric) solu-tion of Weyl gravity [8], [7] the present findings are alsoapplicable to Weyl gravity.The criterion of the stability of orbits in conformalgravity leads to testable upper limit on the size of thegalaxies [12]. The same conclusion should be applicableto Grumiller’s modified gravity because of the similarityof space time solution. Future observations on last stableorbit in galaxies is expected to provide an important testof the Grumiller’s model/conformal gravity prediction. In conclusion we demonstrate that Grumiller’s modi-fied gravity model reproduces the baryonic Tully-Fisherrelation at theoretical level if local minimum value ofrotation velocity is considered. The fitting of the ob-served total baryonic mass versus rotation velocity datafor a sample of sixty galaxies by Grumiller’s model al-lows to estimate the value of Rindler acceleration param-eter. The mean value of so obtained Rindler parameteris found consistent with that estimated from fitting ofrotation velocity curves of disk galaxies.
Acknowledgments
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