Probing the universality of acceleration scale in the modified Newtonian dynamics with SPARC galaxies

  • We probe the universality of acceleration scale $ a_0$ in Milgrom's modified Newtonian dynamics (MOND) using the recently released rotation curve data from SPARC galaxies. We divide the SPARC data into difference subsamples according to the morphological types of galaxies, and fit the rotation curve data of each subsample with the theoretical prediction of MOND. MOND involves an arbitrary interpolation function which connects the Newtonian region and MOND region. Here we consider five different interpolation functions that are widely discussed in literatures. It is shown that the best-fitting $ a_0$ significantly depends on the interpolation functions. For a specific interpolation function, $ a_0$ also depends on the morphological types of galaxies, implying that $ a_0$ may be not a universal constant. Introducing a dipole correction on $ a_0$ can significantly improves the fits, and the dipole directions for four in five interpolation functions point towards an approximately consistent direction, but $ a_0$ still varies for differently interpolation functions.
  • 加载中
  • [1] F. Zwicky, Helvetica Physica Acta 6, 110 (1933)
    [2] S. Smith, Astrophys. J. 83, 23 (1936) doi: 10.1086/143697
    [3] V. C. Rubin and W. K. Jr. Ford, Astrophys. J. 159, 379 (1970) doi: 10.1086/150317
    [4] V. C. Rubin, W. K. Jr. Ford and N. Thonnard, Astrophys. J. 238, 471 (1980) doi: 10.1086/158003
    [5] V. C. Rubin, Scientific American 248, 96 (1983)
    [6] D. Clowe, S. W. Randall and M. Markevitch, Nucl. Phys. B: Proc. Suppl. 173, 28 (2007) doi: 10.1016/j.nuclphysbps.2007.08.150
    [7] P. A. R. Ade et al., Astron. Astrophys. 594, A13 (2016) doi: 10.1051/0004-6361/201525830
    [8] O. Adriani et al., Nature 458, 607 (2009) doi: 10.1038/nature07942
    [9] X.-Y. Cui, et al. (PandaX-Ⅱ Collaboration), Phys. Rev. Lett. 119, 181302 (2017) doi: 10.1103/PhysRevLett.119.181302
    [10] M. Milgrom, Astrophys. J. 270, 365 (1983) doi: 10.1086/161130
    [11] M. Milgrom, Astrophys. J. 270, 371 (1983) doi: 10.1086/161131
    [12] M. Milgrom, Astrophys. J. 270, 384 (1983) doi: 10.1086/161132
    [13] M. Carmeli, Int. J. Theor. Phys 37, 2621 (1998) doi: 10.1023/A:1026672604958
    [14] J. W. Moffat, J. Cosmol. Astropart. Phys. 05, 003 (2005)
    [15] D. Grumiller, Phys. Rev. Lett. 105, 211303 (2010) doi: 10.1103/PhysRevLett.105.211303
    [16] B. Famaey and S. S. McGaugh, Living Rev. Relativ. 15, 10 (2012)
    [17] R. H. Sanders, and S. S. McGaugh, Ann. Rev. Astron. Astrophys. 40, 263 (2002) doi: 10.1146/annurev.astro.40.060401.093923
    [18] S. M. Kent, Astron. J. 93, 816 (1987) doi: 10.1086/114366
    [19] K. G. Begeman, A. H. Broeils and R. H. Sanders, Mon. Not. Roy. Astron. Soc. 249, 523 (1991)
    [20] B. Famaey and J. Binney, Mon. Not. Roy. Astron. Soc. 363, 603 (2005) doi: 10.1111/j.1365-2966.2005.09474.x
    [21] G. Gentile, P. Salucci, U. Klein and G.L. Granato, Mon. Not. Roy. Astron. Soc. 375, 199 (2007) doi: 10.1111/j.1365-2966.2006.11283.x
    [22] R. H. Sanders and E. Noordermeer, Mon. Not. Roy. Astron. Soc. 379, 702 (2007) doi: 10.1111/j.1365-2966.2007.11981.x
    [23] O. Tiret and F. Combes, Astron. Astrophys. 464, 517 (2007) doi: 10.1051/0004-6361:20066446
    [24] S. S. McGaugh, Astrophys. J. 683, 137 (2008) doi: 10.1086/589148
    [25] R. A. Swaters, R. H. Sanders and S. S. McGaugh, Astrophys. J. 718, 380 (2010) doi: 10.1088/0004-637X/718/1/380
    [26] F. Iocco, M. Pato and G. Bertone, Phys. Rev. D 92, 084046 (2015) doi: 10.1103/PhysRevD.92.084046
    [27] J. D. Bekenstein, Phys. Rev. D 70, 083509 (2004) doi: 10.1103/PhysRevD.70.083509
    [28] R. H. Sanders, Mon. Not. Roy. Astron. Soc. 363, 459 (2005) doi: 10.1111/j.1365-2966.2005.09375.x
    [29] L. Blanchet and J. Novak, arXiv: 1105.5815[astro-ph.CO] (2011)
    [30] A. Hees, B. Famaey, G. W. Angus and G. Gentile, Mon. Not. Roy. Astron. Soc. 455, 449 (2016) doi: 10.1093/mnras/stv2330
    [31] S. McGaugh, Astron. J. 143, 40 (2012) doi: 10.1088/0004-6256/143/2/40
    [32] F. Lelli, S. S. McGaugh and J. M. Schombert, Astron. J. 152, 157 (2016) doi: 10.3847/0004-6256/152/6/157
    [33] S. S. McGaugh, F. Lelli and J. M. Schombert, Phys. Rev. Lett. 117, 201101 (2016) doi: 10.1103/PhysRevLett.117.201101
    [34] F. Lelli, S. S. McGaugh, J. M. Schombert and M. S. Pawlowski, Astrophys. J. 836, 152 (2017) doi: 10.3847/1538-4357/836/2/152
    [35] P.-F. Li, F. Lelli, S. S. McGaugh and J. M. Schombert, Astron. Astrophys. 615, A3 (2018) doi: 10.1051/0004-6361/201732547
    [36] G. Amir, H. Hosein and H. Z. Akram, Mon. Not. Roy. Astron. Soc. 487, 2148 (2019) doi: 10.1093/mnras/stz1272
    [37] D.C. Rodrigues, V. Marra, A. del Popolo and Z. Davari, Nat. Astron. 2, 668 (2018) doi: 10.1038/s41550-018-0498-9
    [38] Z. Chang and Y. Zhou, Mon. Not. Roy. Astron. Soc. 486, 1658 (2019) doi: 10.1093/mnras/stz961
    [39] R. B. Tully, J. R. Fisher, Astron. & Astrophys. 54, 661 (1977)
    [40] M. Milgrom, Phys. Rev. D 100, 084039 (2019) doi: 10.1103/PhysRevD.100.084039
    [41] I. Antoniou and L. Perivolaropoulos, J. Cosmol. Astropart. Phys. 12, 012 (2010)
    [42] J. A. King, J. K. Webb, M. T. Murphy, V. V. Flambaum, R. F. Carswell, M. B. Bainbridge, M. R. Wilczynska and F. E. Koch, Mon. Not. Roy. Astron. Soc. 422, 3370 (2012) doi: 10.1111/j.1365-2966.2012.20852.x
    [43] P. A. R. Ade et al., Astron. Astrophys. 571, A23 (2014) doi: 10.1051/0004-6361/201321534
    [44] P. A. R. Ade et al., Astron. Astrophys. 594, A16 (2016) doi: 10.1051/0004-6361/201526681
    [45] Y. Zhou, Z.-C. Zhao and Z. Chang, Astrophys. J. 847, 86 (2017) doi: 10.3847/1538-4357/aa8991
    [46] Z. Chang, H.-N. Lin, Z.-C. Zhao and Y. Zhou, Chine. Phys. C 42, 115103 (2018) doi: 10.1088/1674-1137/42/11/115103
    [47] H.-S. Zhao and B. Famaey, Astrophys. J. 638, L9 (2006) doi: 10.1086/500805
    [48] Z. Chang, M.-H. Li, X. Li, H.-N. Lin and S. Wang, Eur. Phys. J. C 73, 2447 (2013)
    [49] P. T. Boggs, R. H. Byrd and R. B. Schnabel, SIAM Journal on Scientific and Statistical Computing 8, 1052 (1987) doi: 10.1137/0908085
    [50] P. T. Boggs and J. E. Rogers, Contemporary Mathematics 112, 183 (1990)
    [51] H. Akaike, IEEE Trans. Automatic Control 19, 716 (1974)
    [52] G. Schwarz, Ann. Statist. 6, 461 (1978) doi: 10.1214/aos/1176344136
    [53] H. Jeffreys, The theory of probability (Oxford University Press, Oxford U.K.) (1998)
    [54] A. R. Liddle, Mon. Not. Roy. Astron. Soc. 377, L74 (2007) doi: 10.1111/j.1745-3933.2007.00306.x
    [55] H.-N. Lin, X. Li and Y. Sang, Chine. Phys. C 42, 095101 (2018) doi: 10.1088/1674-1137/42/9/095101
    [56] D. Foreman-Mackey, D. W. Hogg, D. Lang and J. Goodman, Publications of the Astronomical Society of the Pacific 125, 306 (2013) doi: 10.1086/670067
  • 加载中

Figures(5) / Tables(3)

Get Citation
Xin Li, Su-Ping Zhao, Hai-Nan Lin and Yong Zhou. Probing the universality of acceleration scale in the modified Newtonian dynamics with SPARC galaxies[J]. Chinese Physics C.
Xin Li, Su-Ping Zhao, Hai-Nan Lin and Yong Zhou. Probing the universality of acceleration scale in the modified Newtonian dynamics with SPARC galaxies[J]. Chinese Physics C. shu
Milestone
Received: 2020-02-02
Article Metric

Article Views(5)
PDF Downloads(0)
Cited by(0)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

Probing the universality of acceleration scale in the modified Newtonian dynamics with SPARC galaxies

    Corresponding author: Xin Li, lixin1981@cqu.edu.cn
    Corresponding author: Su-Ping Zhao, zhaosp@cqu.edu.cn
  • 1. Department of Physics, Chongqing University, Chongqing 401331, China
  • 2. CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China

Abstract: We probe the universality of acceleration scale $ a_0$ in Milgrom's modified Newtonian dynamics (MOND) using the recently released rotation curve data from SPARC galaxies. We divide the SPARC data into difference subsamples according to the morphological types of galaxies, and fit the rotation curve data of each subsample with the theoretical prediction of MOND. MOND involves an arbitrary interpolation function which connects the Newtonian region and MOND region. Here we consider five different interpolation functions that are widely discussed in literatures. It is shown that the best-fitting $ a_0$ significantly depends on the interpolation functions. For a specific interpolation function, $ a_0$ also depends on the morphological types of galaxies, implying that $ a_0$ may be not a universal constant. Introducing a dipole correction on $ a_0$ can significantly improves the fits, and the dipole directions for four in five interpolation functions point towards an approximately consistent direction, but $ a_0$ still varies for differently interpolation functions.

    HTML

    I.   INTRODUCTION
    • In the 1930s, Zwicky [1] and Smith [2] found that the velocity dispersion of galaxies in cluster is so high that the galaxies cannot be bounded to cluster just by the gravitational force generated by visible matter. Hence it was concluded that there must be a large amount of invisible matter in cluster, which is now called dark matter (DM). Decades later, Rubin and his collaborators [35] found strong evidence for the existence of DM from the observations of galaxy rotation curves. Besides, there are other observational evidences for the existence of DM, such as the strong and weak gravitational lensing of Bullet Cluster 1E 0657-558 [6], the observation of cosmic microwave background radiation from Planck satellite [7], the excess of positron abundance in cosmic rays [8], etc. In recent decades, extensive efforts have been done to search for the DM particles, but have not succeeded yet [9]. Meanwhile, various alternative models were proposed to explain the extra gravitational force generated by DM, such as the modified Newtonian dynamics (MOND) [1012] and the modified gravities [1315].

      The MOND theory was originally proposed to account for the rotation curves of spiral galaxies, see e.g. Refs. [16, 17] for recent review. According to MOND, the Newton's second law loses its efficacy if the acceleration is bellow a critical value $ a_{0} $. MOND theory has achieved great success in interpreting the rotation curves of spiral galaxies [1826]. For a long time, MOND was a non-relativistic theory until Bekenstein [27] constructed its relativistic form. The relativistic form of MOND, i.e., the tensor-vector-scalar (TeVeS) theory [27, 28], was discussed in the weak field regime, and was tested in the Solar system [29, 30]. Moreover, MOND is consistent with the baryonic Tully-Fisher relation observed in spiral galaxies [31]. MOND theory introduces a free parameter $ a_0 $, a characteristic acceleration scale bellow which the Newtonian dynamics breaks. If MOND theory is a fundamental theory, $ a_0 $ must be a universal constant.

      To study the characteristic acceleration of MOND theory, we need to match with the observations. Recently, $ Spitzer $ Photometry and Accurate Rotation Curves (SPARC) [32] has been released, which is extensively used to investigate the radial acceleration relation [3335]. Based on the SPARC data, some debates on MOND as a fundamental theory have attracted extensive attention [3638]. Using Bayesian inference with flat priors, Rodrigues et al. [37] concluded that MOND was excluded as a fundamental theory at more than $ 10\sigma $ based on 193 high-quality disk galaxies from SPARC and The HI Nearby Galaxy Survey (THINGS) databases. Later on, Chang et al. [38] fitted the rotation curves of 175 SPARC galaxies by using similar method but with Gaussian priors, and came to the similar conclusion that MOND theory cannot hold as a fundamental theory. As the representative and currently largest collection of the rotating curved galaxies, SPARC data contains galaxies of different morphological types, luminosity and effective surface brightness. In the previous works [3338], the qualified SPARC data was used as a whole. It is meaningful to test if the acceleration scale is dependent on the galaxy morphological types or not. In our work, we will classify SPARC dataset depending on the galaxy morphological types, and test the possible galaxy-dependence of the characteristic acceleration scale.

      The original MOND theory [1012] only requires two limit conditions. One is Newton region, that is the Newton's second law should valid while $ a_0\rightarrow0 $. The other is deep MOND region, that is the Newton's second law should be modified to satisfy the Tully-Fisher relation [39]. To match the MOND theory with the astronomical observations, an interpolation function is needed. If MOND theory is a fundamental theory, the acceleration scale $ a_0 $ should not depend on the specific choices of the interpolation function. In this paper, we will consider five popular interpolation functions to match the SPARC data.

      It has been noted that the best-fitting value $ a_0\sim1.2\times10^{-10}\; m/s^2 $, obtained from fitting to a large sample of spiral galaxies, is numerically close to $ cH_{0}/2\pi $ [40], where c is the speed of light and $ H_0 $ is the Hubble constant [10]. It is unclear if this correlation has some physical implications or is just a coincidence. If the former is true, then $ a_0 $ must be correlated with the evolution history of the universe. Some astronomical and cosmological observations, such as the luminosity of type-Ia supernovae [41], the spatial variation of the fine-structure constant [42], the cosmic microwave background radiation [43, 44], etc., show that the universe may be anisotropic. Hence it is necessary to investigate if the acceleration scale $ a_0 $ is directional dependent or not. In fact, Zhou et al. [45] have already studied the possible anisotropy of $ a_0 $ using the SPARC data, and found that the direction of maximum anisotropy is close to the direction of the “Australia dipole” for the fine-structure constant. In their following work, Chang et al. [46] have studied a dipole correction for $ a_0 $, and found a similar dipole direction. Inspired by this, here we continue to investigate the possible anisotropy of the acceleration scale. We will focus on testing if different interpolation functions have some influences on the anisotropy.

      The rest of this paper is organized as follows. In Section II, we briefly review the MOND theory and some commonly discussed interpolation functions, and introduce the SPARC data set that is used in our studies. In Section III, we fit the SPARC data to the MOND theory, taking into account five different interpolation functions and different galaxy types, as well as the sky direction of the galaxies. Finally, a short summary is given in Section IV.

    II.   MOND THEORY AND THE SPARC SAMPLE
    • It is well-known that the MOND theory [10, 11] is initially proposed by Milgrom to account for the mass-missing problems in rotationally supported galaxies to avoid the exotic dark matter. The main idea of MOND is that there is a critical acceleration $ a_0 $ bellow which the Newton's second laws no longer valid (the MOND region), while above $ a_0 $ the Newton's second law still holds (the Newton region). A interpolation function $ \mu(x) $ is used to join these two regions. Phenomenologically, the dynamics in MOND theory can be written as

      $ \mu\left(\frac{g}{a_{0}}\right)g = g_{N}, $

      (1)

      where $ {g}_{N}\equiv GM/r^{2} $ is the Newtonian acceleration, $ a_{0} $ is the critical acceleration, $ \mu(x) $ is a smooth and monotonically increasing function of x. To be accordance with the asymptotically flat feature of the observed galaxy rotation curves, $ \mu(x) $ must satisfy the asymptotical conditions $ \mu(x)\approx x $ when $ x\rightarrow0 $. And to recover Newtonian dynamics, $ \mu(x) $ should be equal to unity when $ x\rightarrow\infty $. In the deep-MOND limit, namely, $ g\ll a_{0} $, the effective acceleration becomes $ g = \sqrt{a_{0}g_{N}} $.

      Given the matter distribution in a galaxy, it is easy to calculate the Newtonian acceleration $ g_N $ by solving the Poisson equation, then the MOND acceleration g can be solved from equation (1). It is convenient to rewrite equation (1) to the following form,

      $ \nu\left(\frac{g_{N}}{a_{0}}\right)g_{N} = g. $

      (2)

      Noted that $ y\nu(y) $ is the inverse of $ x\mu(x) $, and it satisfies $ \nu(y)\approx1 $ when $ y\rightarrow\infty $, and $ \nu(y)\approx y^{-1/2} $ when $ y\rightarrow0 $. To realize the fitting of the observations with the theoretical predictions, it is necessary to clarify the form of interpolation function $ \mu(x) $ or $ \nu(y) $. However, the MOND theory could not give any hints on the concrete form of interpolation function. Any smooth and monotonic function satisfying the above asymptotical conditions is an underlying choice. In general, there are two families of interpolation functions, namely, the $ \mu $-function family and $ \nu $-function family. Any $ \mu $-function has a corresponding $ \nu $-function, although in some cases it is not easy to find an analytical expression. Different forms of interpolation functions have been discussed in the review paper [16]. In our work, five common interpolation functions are considered, including four $ \mu $-functions and one $ \nu $-function.

      The first interpolation function is the so-called “standard” function initially proposed by Milgrom [11], which takes the form

      $ \mu_{\rm{sta}}(x) = \frac{x}{\sqrt{1+x^{2}}}. $

      (3)

      This function was widely used to analyse the galaxy rotation curves at the beginning of MOND theory [18, 19]. However, Famaey et al. [20] found that the “simple” function, another widely used interpolation function, performs a better fitting to the terminal velocity curve of the Milky Way than the “standard” function, and it is given as

      $ \mu_{\rm{sim}}(x) = \frac{x}{1+x}. $

      (4)

      Some researches show that the “simple” function in some cases performs better than the “standard” function [22, 47].

      The initial MOND is a phenomenal model and is non-relativistic, until Bekenstein [27] constructed the relativistic form, namely, the TeVeS theory. There is still a freedom in choosing the specific form of the scalar field action in TeVeS theory. The scalar action proposed in Ref. [27] gives rise to the following “toy” interpolation function,

      $ \mu_{\rm{toy}}(x) = \frac{\sqrt{1+4x}-1}{\sqrt{1+4x}+1}. $

      (5)

      This interpolation function performs a much slower transition from MOND region to Newtonian region than the “standard” and “simple” functions. The Finslerian model proposed in Ref.[48] also leads to the “toy” interpolation function.

      The fourth interpolation functions we considered is the so-called “exponential” function [11], which reads

      $ \mu_{\rm{exp}}(x) = 1-e^{-x}. $

      (6)

      This interpolation function behaves similar with the “standard” function.

      Finally, we also consider a $ \nu $-function, which is inspired by the radial acceleration relation (RAR-inspired) found from the SPARC data by Ref.[33]. It takes the form

      $ \nu_{\rm{rar}}(y) = \frac{1}{1-e^{-\sqrt{y}}}. $

      (7)

      It was shown that this $ \nu $-function can make a good fit to the rotation curves of the Milky way [24]. Recently, this function is used by McGaugh to fit the radial acceleration relation observed from a large sample of rotationally supported galaxies [33]. The $ \mu $-function corresponding to this $ \nu $-function can be obtained numerically, which is plotted, together with the other four $ \mu $-functions in Fig. 1. The “standard” and “exponential” functions show a sharp jump from MOND region to Newtonian region, while the “toy” function changes slowly from MOND region to Newtonian region, and the “simple” and “RAR” functions fall in between.

      Figure 1.  The plot of different interpolation functions.

      To investigate the universality of acceleration scale, we fit the above five functions with the recently released SPARC dataset [32]. This sample contains 175 disk galaxies with high quality observations on near-infrared (3.6 $ \mu $m) and 21 cm, so both the baryon distribution and velocity field can be obtained. This dataset has already been used to study the radial acceleration relation [33, 34], i.e., the relation between the observed acceleration ($ g_{\rm{obs}} $) and that expected from the baryon matter ($ g_{\rm{bar}} $, which is equivalent to $ g_N $ in equation (1)) in the framework of Newtonian dynamics. In our work, the same selection criteria are adopted as Refs. [33, 34]: the 12 objects with asymmetric rotation curves which do not trace the equilibrium gravitational potential (quality flag Q = 3) and 10 face-on galaxies with $ i<30^{\circ} $ have been excluded, which leaves a sample of 153 galaxies. Besides, the condition that the relative uncertainty of the observed velocity ($ \delta V_{obs}/V_{obs} $) should be less than 10% has been considered. Therefore, the finally data set used in our work contains 2693 data points in 147 galaxies.

      To fit theoretical prediction with the observational data, the orthogonal-distance-regression algorithm [49, 50] is adopted, which considers errors on both the horizontal and vertical axes. The best-fitting parameters are obtained by minimizing the following $ \chi^{2} $,

      $ \chi^{2} = \sum\limits_{i = 1}^{N}\frac{[g_{th}(g_{bar,i}+\delta_{i})-g_{obs,i}]^{2}}{\sigma_{obs,i}^{2}}+\frac{\delta_{i}^{2}}{\sigma_{bar,i}^{2}}, $

      (8)

      where $ g_{obs} $ is observed acceleration, $ g_{bar} $ is the acceleration contributes by the baryonic matter, calculated in the framework of Newtonian dynamics, $ g_{th} $ is the theoretical acceleration predicted by MOND, $ \sigma_{obs} $ and $ \sigma_{bar} $ are the uncertainty of $ g_{obs} $ and $ g_{bar} $, respectively. $ \delta $ is an auxiliary parameter for determining the weighted orthogonal (shortest) distance from the best-fitting curve, which can be obtained interactively in the fitting procedure. The only one free parameter is the acceleration scale $ a_0 $.

    III.   BEST-FITTING RESULTS

      A.   Results of galaxy-dependence

    • The SPARC data used here contains 147 galaxies of different Hubble types. Most of them are spiral galaxies from Sa to Sm, and it also includes some lenticular galaxies (S0), irregular galaxies (Im) and blue compact dwarf galaxies (BCD). To study the possible dependence of $ a_0 $ on the galaxy morphology, we intend to classify SPARC data set into several subsets according to the galaxy Hubble types. The Hubble types of the galaxies are labelled from 0 to 11: 0 = S0, 1 = Sa, 2 = Sab, 3 = Sb, 4 = Sbc, 5 = Sc, 6 = Scd, 7 = Sd, 8 = Sdm, 9 = Sm, 10 = Im, 11 = BCD. The number of galaxies in each Hubble type is shown in the histogram in Fig. 2. First, we divide the galaxies into two subsets: the Spiral galaxies (from 1 to 9) and the Other galaxies (including 0, 10 and 11). There are 2433 data points in 118 Spiral galaxies and 260 data points in 29 Other galaxies, respectively. Due to the significant difference in the number of data points between Spiral and Other galaxies, we further divide the Spiral galaxies into two subsets, with approximately equal data points in each subset: Spiral-1 galaxies (from 1 to 4) and Spiral-2 galaxies (from 5 to 9). There are 1201 data points in 41 Spiral-1 galaxies and 1232 data points in 77 Spiral-2 galaxies, respectively. The details of the galaxy classification are shown in Table I.

      FullSpiralOtherSpiral-1Spiral-2
      Hubble type0-111-90,10,111-45-9
      Number of galaxies147118294177
      Number of data points2693243326012011232

      Table 1.  The details of galaxy classification.

      Figure 2.  The number of galaxies in each Hubble type.

      First, we use the Full dataset contains 2693 data points to fit MOND theory with different interpolation functions. The best-fitting $ a_0 $ values are listed in Table II, and the corresponding fitting curves between the Newtonian acceleration $ g_{N} $ and the MOND acceleration g are shown in Fig. 3(a). We can see that the best-fitting $ a_0 $ varies between different interpolation functions. The “toy” function seems to have the smallest $ \chi^2 $, but this function also leads to the smallest $ a_{0} $. This is because the “toy” function evolves much slower than the rest four functions, see Fig. 1. The “standard” function has the largest $ a_0 $, but also has the largest $ \chi^2 $, since this function shows the sharpest jump from MOND region to Newtonian region. The $ a_0 $ values from “simple” and “RAR-inspired” functions are close to each other, because of their similar evolution behaviours.

      $\mu_{\rm{sta}}(x)$$\mu_{\rm{sim}}(x)$$\mu_{\rm{toy}}(x)$$\mu_{\rm{exp}}(x)$$\nu_{\rm{rar}}(y)$
      Full$a_{0}$$1.39\pm0.02$$1.00\pm0.01$$0.69\pm0.01$$1.24\pm0.02$$1.02\pm0.02$
      $\chi^{2}/dof$$1.73$$1.48$$1.47$$1.66$$1.49$

      Spiral$a_{0}$$1.50\pm0.02$$1.07\pm0.02$$0.73\pm0.01$$1.34\pm0.04$$1.09\pm0.02$
      $\chi^{2}/dof$$1.46$$1.26$$1.28$$1.39$$1.27$

      Other$a_{0}$$0.86\pm0.04$$0.70\pm0.04$$0.55\pm0.04$$0.79\pm0.04$$0.71\pm0.04$
      $\chi^{2}/dof$$3.06$$2.99$$3.00$$3.08$$3.01$

      Spiral-1$a_{0}$$1.70\pm0.03$$1.08\pm0.03$$0.61\pm0.02$$1.48\pm0.03$$1.12\pm0.03$
      $\chi^{2}/dof$$1.14$$0.986$$1.04$$1.10$$0.99$

      Spiral-2$a_{0}$$1.43\pm0.02$$1.06\pm0.02$$0.78\pm0.02$$1.28\pm0.02$$1.08\pm0.02$
      $\chi^{2}/dof$$1.73$$1.52$$1.48$$1.65$$1.53$

      Table 2.  The best-fitting $a_0$ values for five different interpolation functions and different datasets. The unit of $a_{0}$ is $10^{-10}$ m s$^{-2}$.

      Figure 3.  The best-fitting curves of the radial acceleration relation. The observational data points are shown by blue dots with error bars. The green solid line is the line of unity.

      Then, we divide the Full sample into Spiral and Other subsamples, and fit each subsample to the MOND theory. The best-fitting results are reported in Table II, and the best-fitting curves are plotted in Figs. 3(b) and 3(c), respectively. For both subsamples, the “simple” function has the smallest $ \chi^2 $, but the best-fitting $ a_0 $ differs significantly, namely, $ a_{0} = (1.07\pm0.02)\times10^{-10} $ m s$ ^{-2} $ for the Spiral sample and $ a_{0} = (0.70\pm0.04)\times10^{-10} $ m s$ ^{-2} $ for the Other sample. For a fixed interpolation function, the Other sample has a much smaller $ a_0 $ than the Spiral sample, and $ a_0 $ of the Full sample falls in between. This implies that the Spiral galaxies and Other galaxies have very different $ a_0 $. For both the Spiral and Other subsamples, the best-fitting $ a_0 $ is also dependent on the choice of interpolation function.

      Finally, we further divide the Spiral sample into two subsamples denoted by Spiral-1 and Spiral-2, and use these two subsamples to fit with the MOND theory. The best-fitting parameters are presented in the last four rows in Table II, and the best-fitting curves are plotted in Figs. 3(d) and 3(e), respectively. The “simple” function is the best one for Spiral-1 subsample, with the best-fitting value $ a_{0} = (1.08\pm0.03)\times10^{-10} $ m s$ ^{-2} $. While for the Spiral-2 subsample, the “toy” function fits the data best, with the best-fitting value $ a_{0} = (0.78\pm0.02)\times10^{-10} $ m s$ ^{-2} $. Similarly, for a fixed interpolation function, the Spiral-1 and Spiral-2 subsamples may lead to different values of $ a_{0} $, but the difference seems to be not as significant as that between the Spiral and Other subsamples. Especially, for the “simple” and “RAR-inspired” functions, the values of $ a_{0} $ for the Spiral-1 and Spiral-2 subsamples are consistent with each other within $ 1\sigma $ uncertainty.

      As a summary, we plot the best-fitting $ a_0 $ values with $ 1\sigma $ error bars for different interpolation functions and different data samples in Fig. 4. From this figure, we can draw the following conclusions. For a fixed interpolation function, the best-fitting $ a_0 $ is in general dependent on the galaxy types. The Other subsample always has a much smaller $ a_0 $ than the Spiral subsample, regardless of which interpolation function is chosen. Due to the similar behaviour between the “simple” and “RAR-inspired” functions, the fitting results of these two interpolation functions are very similar. For these two functions, the Spiral-1 and Spiral-2 subsamples have consistent $ a_0 $ values, while for the rest three functions, the $ a_0 $ values for Spiral-1 and Spiral-2 subsamples differ significantly. For a fixed data sample, the best-fitting $ a_0 $ depends on the choice of interpolation functions. The “toy” function always results a much smaller $ a_0 $ than the rest four functions due to its slow change from MOND region to Newtonian region. On the contrary, the “standard” function always results a large $ a_0 $ since it has a sharp jump from MOND region to Newtonian region. In conclusion, the best-fitting $ a_0 $ depends on both the galaxy morphology and interpolation function. For the spiral galaxies, $ a_0 $ may be a universal constant, but for the other types of galaxies there must be a different $ a_0 $ value.

      Figure 4.  The best-fitting $ a_0 $ values for different interpolation functions and different data samples

    • B.   Results of direction-dependence

    • The MOND theory assumes that the acceleration scale $ a_0 $ is a universal constant. However, if $ a_0 $ is correlated with the evolution of the universe, we may expect that $ a_0 $ depends on the sky position. This is because, as was mentioned in the introduction, some observations hint that the universe may be anisotropic. Therefore, we investigate if it is necessary to make a dipole correction to the acceleration scale. We write the direction-dependent acceleration scale in the following form,

      $ a = a_{0}(1+D\hat{n}\cdot\hat{p}), $

      (9)

      where D is the dipole amplitude, $ a_0 $ is a constant, $ \hat{n} $ and $ \hat{p} $ are the unit vectors, pointing towards the dipole direction and galaxy position, respectively.

      To investigate how much the dipole correction on acceleration scale can improve the fits, the Akaike information criterion (AIC) [51] and Bayesian information criterion (BIC) [52] are employed to make a model selection. The AIC and BIC of a model are defined by

      $ {\rm{AIC}} = \chi_{min}^{2}+2k, $

      (10)

      $ {\rm{BIC}} = \chi_{min}^{2}+k\ln N, $

      (11)

      where $ \chi_{min}^{2} $ is the minimum of $ \chi^{2} $, k is the number of the free parameters, N is the total number of data points. The model with a smaller IC (either AIC or BIC) is better than the one with a larger IC. In general, one can choose a reference model and calculate the difference of IC with respect to the reference model,

      $ \Delta {\rm{IC}}_{\rm{model}} = {\rm{IC}}_{\rm{model}}-{\rm{IC}}_{\rm{ref.-model}}. $

      (12)

      According to the Jeffreys' scale [53, 54], a model with $ \Delta $IC$ >5 $ or $ \Delta $IC$ >10 $ means that there is “strongly” or “decisive” evidence against this model with respect to the reference model [46, 55]. Here, the MOND without dipole correction is chosen as the reference model.

      The MOND with a dipole corrected acceleration scale is used to fit the Full data sample. In this time, the acceleration scale $ a_0 $ in equations (1) and (2) is replaced by the direction-dependent acceleration a given by equation (9). The affine-invariant Markov chain Monte Carlo sampler $ {\textsf{emcee}}$ [56] is used to calculate the posterior probability density distribution of free parameters ($ a_0,D,l,b $). The mean values and $ 1\sigma $ uncertainties of free parameters are reported in Table III, and the corresponding error contours are shown in Fig. 5. We also list the $ \Delta {\rm{AIC}} $ and $ \Delta {\rm{BIC}} $ values in the last two columns in Table III. It is seen that, adding the dipole correction can significantly improve the fits, regardless of which interpolation function is chosen. The dipole directions of four interpolation functions (except for the “toy” function) are consistent with each other within $ 2\sigma $ uncertainty, with an average direction centering on $ (l,b) = (165^{\circ},-14^{\circ}) $. The “toy” function has the smallest $ \chi^2 $, but the largest $ \Delta {\rm{IC}} $. Especially, $ \Delta{\rm{BIC_{toy}} }= -4.30 $, implies that the dipole correction is only moderately favoured for the “toy” function.

      $a_{0}$Dlb$\chi^{2}$$\Delta$AIC$\Delta$BIC
      $\mu_{\rm{sta}}$$1.29\pm0.02$$0.32\pm0.03$$155.97^{\circ}\pm5.58^{\circ}$$-12.65^{\circ}\pm3.27^{\circ}$$4541.87$$-116.10$$-98.41$
      $\mu_{\rm{sim}}$$0.96\pm0.02$$0.26\pm0.04$$172.53^{\circ}\pm7.38^{\circ}$$-15.76^{\circ}\pm4.92^{\circ}$$3926.12$$-46.99$$-29.30$
      $\mu_{\rm{toy}}$$0.69\pm0.02$$0.22\pm0.05$$198.83^{\circ}\pm10.22^{\circ}$$-19.46^{\circ}\pm7.21^{\circ}$$3916.84$$-21.99$$-4.30$
      $\mu_{\rm{exp}}$$1.16\pm0.02$$0.31\pm0.04$$160.90^{\circ}\pm5.70^{\circ}$$-13.13^{\circ}\pm3.52^{\circ}$$4359.68$$-94.44$$-76.71$
      $\nu_{\rm{rar}}$$0.98\pm0.02$$0.27\pm0.04$$171.42^{\circ}\pm7.21^{\circ}$$-15.44^{\circ}\pm4.62^{\circ}$$3962.02$$-52.39$$-34.70$

      Table 3.  The best-fitting values for five different interpolation functions with the dipole correction and the Full SPARC data. The unit of $a_{0}$ is $10^{-10}$ m s$^{-2}$.

      Figure 5.  The marginalized posterior probability density function and 2-dimensional marginalized contours for the parameter space $ (a_0,D, l, b) $ for different interpolation functions.

    IV.   SUMMARY
    • In this paper, we have probed the universality of the acceleration scale $ a_{0} $ in MOND theory using the recently released SPARC rotation curve data, which contains in total 175 galaxies of different Hubble types. To study the possible galaxy-dependence of $ a_{0} $, we divided the full SPARC dataset into several subsets depending on the galaxy Hubble types. To test the effect of interpolation function on $ a_{0} $, we consider five different interpolation functions that are commonly discussed in literatures. The subsamples are used to fit with the MOND theory by using different interpolation functions. Results show that the acceleration scale $ a_0 $ not only depends on the galaxy Hubble type, but also depends on the choice of interpolation function. This implies that $ a_0 $ may be not a universal constant. To test if $ a_0 $ is universal in spiral galaxies, we further divided the spiral galaxies into two subset of approximately equal data points in each subset. It is found that, for these two subsets of spiral galaxies, $ a_0 $ is consistent for the “simple” and “toy” functions. But for the rest three interpolation functions, $ a_0 $ still varies. Besides, we also check the possible direction-dependence of $ a_0 $. We found that the dipole correction of $ a_0 $ can significantly improve the fits, and the dipole directions are consistent for four interpolation functions, except for the “toy” function.

      Our results show that, although MOND theory could quantitatively account for the SPARC rotation curve data, it couldn't be a fundamental theory. If our universe does admit a fundamental theory other than dark matter, it means that MOND theory may be an approximation of the fundamental theory which could account for the astronomical observations of dark matter effects. The better performance of the interpolation functions $ \mu_{\rm{sim}}(x) $ and $ \nu_{\rm{rar}}(y) $ may give a clue of construction of the fundamental theory. The dependence of galaxy types hints that alternative effective theory of MOND theory may not be a spherical symmetric solution of the fundamental theory. And the result of the dipole correction of $ a_0 $ means that possible anisotropy effect of the universe should be considered in dealing with rotation curve data before matching the fundamental theory with the astronomical observations, since the rotation curve data are obtained by using the standard cosmological model.

Reference (56)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return