On the Presence of a Universal Acceleration Scale in Elliptical Galaxies

Historically, empirical laws such as Kepler's kinematic laws have played crucial roles in physics. For the study of galaxies, these include the Faber–Jackson (Faber & Jackson 1976) and Fundamental Plane (Djorgovski & Davis 1987; Dressler et al. 1987) relations for pressure-supported early-type galaxies, and the Tully–Fischer (Tully & Fisher 1977), baryonic Tully–Fisher (McGaugh et al. 2000), central density (Lelli et al. 2016), and radial acceleration relations (RARs; McGaugh et al. 2016) for rotationally supported late-type galaxies.

The RAR—a relation between the observed radial (centripetal) acceleration g_obs and the expected Newtonian acceleration g_bar due to the observed distribution of baryonic matter, for rotationally supported galaxies—is of particular interest as it exhibits a characteristic acceleration scale (denoted g_†) for dark matter phenomenology. If this scale is truly universal among all galaxies, it would naturally correspond to the critical acceleration (denoted a₀) of the Modified Newtonian dynamics (MOND) paradigm (Milgrom 1983). On the other hand, if there is an intrinsic galaxy-to-galaxy scatter in g_†, MOND would be called into question and g_† must be an emergent property of the dark matter phenomenology and/or the physics of galaxy formation.

When McGaugh et al. (2016) reported the RAR from 153 galaxies selected from the SPARC database (Lelli et al. 2016), they noted that data points from individual galaxies scattered around a universal RAR of g_† ≈ 1.2 × 10⁻¹⁰ m s⁻² with a typical scatter of ∼0.1 dex consistent with typical observational errors. Li et al. (2018) carried out Bayesian modeling of individual SPARC galaxies and noted that allowing a broad range of g_† did not improve overall fit qualities compared with the case of fixing (or imposing a narrow range of) g_†. Therefore, up to realistic uncertainties, rotation curves of all rotationally supported galaxies appeared to be consistent with a universal g_†.

This view was challenged by several authors (Rodrigues et al. 2018a, 2018b; Chang & Zhou 2019; Marra et al. 2020), who carried out Bayesian modeling using the formal uncertainties of individual rotation curves (and using uninformative priors in some cases). However, Kroupa et al. (2018) and McGaugh et al. (2018) have highlighted the uncertain nature of some formal errors and the issue of appropriate priors, and Cameron et al. (2020) further pointed out the issue of the potential model misspecification in Bayesian applications to galaxy rotation curves arising from problems in data and/or the model.

Stone & Courteau (2019) considered an order-of-magnitude larger collection of rotation curves including the SPARC galaxies as a subsample (the "PROBES" sample) and argued for an intrinsic scatter in the RAR. However, individual rotation curves of the PROBES sample are typically not as accurate and extended as those of SPARC galaxies because they largely come from one-dimensional long-slit Hα spectroscopy, which does not allow for accurate estimates of kinematic inclinations or identification of warped disks and noncircular motions. In addition, Stone & Courteau (2019) considered stellar mass rather than total baryonic mass distributions. However, unlike the SPARC mass models based on homogeneous Spitzer photometry at 3.6 μm, in which the stellar mass-to-light ratio (${M}_{\star }/L$) is approximately constant (e.g., Schombert et al. 2019), the mass models of Stone & Courteau (2019) are based on heterogeneous photometry in different bands, in which variations in M_⋆/L are a major concern (e.g. McGaugh & Schombert 2014). Therefore, it is more challenging to test the intrinsic scatter of g_† with PROBES galaxies.

To broaden the discussion of the universality or not of g_†, here we consider fitting the RAR to individual dispersion-supported elliptical galaxies, which we select from uniform integral field spectroscopic (IFS) surveys: SDSS-IV MaNGA and ATLAS^3D. This is attractive for a number of reasons. First, both kinematic and photometric data come from uniform observations while SPARC and PROBES are collections of heterogeneously derived rotation curves. This means that measurement uncertainties of stellar velocity dispersions and light distributions of our galaxies are better understood. Second, we select only nearly spherical, slow-rotator (SR) galaxies and model them with the spherical Jeans equation. Although this means we are left with relatively few objects, modeling complications and errors can be minimized because of the relative simplicity of spherical models. Third, the velocity dispersion profiles of our galaxies cover acceleration ranges that are slightly larger than g_†. Although this weakens the sensitivity to g_†, it also makes the external field effect less of a concern (see Figure 1 of Chae et al. 2020). Finally, our constraints on g_† are independent of those from rotationally supported galaxies. Thus, our constraints on the universality of g_† in elliptical galaxies can be directly compared with those from rotationally supported galaxies.

As our goal is to test whether a common acceleration scale is present in elliptical galaxies, we will carry out kinematic modeling in the framework of MOND. In Section 2 we describe our methodology and data. We present our results in Section 3, and discuss and conclude in Section 4.

2.1. Galaxy Samples

As we will use the spherical model, we select galaxies that are morphologically round E0s (defined here to be b/a > 0.9 where a and b are the semimajor and semiminor axes of the light distribution) and kinematically SRs. Because reliable kinematic modeling requires accurate velocity dispersion distributions of good spatial resolution, we consider the MaNGA (Bundy et al. 2015) and ATLAS^3D (Cappellari et al. 2011) databases. The ATLAS^3D database provides velocity dispersion distributions with good spatial resolution but the sample size is small (260); the MaNGA database (not complete at the time of this writing) will eventually provide ∼10,000 galaxies but with poorer spatial resolution than ATLAS^3D (Bundy et al. 2015) because the objects are more distant.

Of the subsample of 24 disk-less (i.e., pure-bulge) ATLAS^3D galaxies selected in Chae et al. (2018, 2019a), 4 are round SRs: NGC 4486, 4636, 5846, and 6703. Their velocity dispersion maps and light distributions are shown in Chae et al. (2018) and references therein.

We select galaxies from the MaNGA DR15 catalog (Aguado et al. 2019) as follows. Based on the photometric and morphological properties presented by Fischer et al. (2019), we require (1) TType $\lt 0$ (select early-types), (2) P_S0 < 0.3 (small probability of being S0 rather than elliptical), (3) B/T > 0.7 (bulge dominates light), (4) $\varepsilon (\equiv 1-b/a)\lt 0.1$ (image is round), (5) n_Ser > 3 (light is centrally concentrated), and (6) λ_e < 0.08 + ε/4 (ensure SRs; see Fischer et al. 2019). Of 4672 MaNGA DR15 galaxies only 30 satisfy these stringent criteria.

The velocity profiles of these galaxies are computed from the kinematic measurements derived by the MaNGA Data Analysis Pipeline contained in the MAPS-VOR10-GAU-MILESHC. An example is shown in Figure 1. The velocity profile is constructed in two ways. The first one (the yellow line) is the median value of the velocity dispersion of the unmasked spaxels with signal-to-noise ratio (S/N) > 5 in circularized radial bins of equal width. A 3σ clipping is also applied to prevent outliers from affecting the profile shape. The second one (the blue line and red points) is constructed by defining radial bins with an equal number of velocity dispersions (N ≥ 11). Both agree and we use the red points with error bars for our modeling.

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A MaNGA velocity dispersion map is discarded if it has too few independent values from which to construct a velocity dispersion radial profile with reasonable estimates of statistical uncertainties, or it is obviously asymmetric and therefore inconsistent with the spherical symmetry assumption, or the constructed radial profile does not cover at least twice the radius affected by the finite fiber size (2''). Half of the selected MaNGA galaxies are discarded in this way, and thus we are left with 15 good-quality MaNGA spherical, SR galaxies. This cut based on IFU data quality will, of course, not be biased against any particular value of g_†.

The selected 15 MaNGA SR E0s are listed in Table 1. These are all luminous giant ellipticals brighter than a characteristic luminosity L_* and larger than ≈5 kpc.

Table 1.  List of MaNGA Selected Slow-rotator E0 Galaxies

IFU Plate	Mr (mag)a	a (kpc)b	nSer	$b/a$
8243-6101	−23.12	21.92	6.411	0.932
8249-6103	−22.50	9.43	5.288	0.946
8323-9101	−22.13	14.69	7.369	0.927
8459-6104	−22.47	10.99	7.245	0.901
8482-1901	−23.05	25.36	8.000	0.908
8555-6101	−23.26	11.98	4.873	0.953
8718-3704	−23.29	17.98	8.000	0.902
8726-9102	−22.96	19.10	8.000	0.928
9024-3701	−21.17	4.74	6.003	0.978
9047-6102	−22.53	13.91	6.965	0.914
9088-3701	−22.26	8.26	8.000	0.923
9485-6102	−23.32	13.29	5.498	0.928
9501-6104	−23.08	16.20	5.080	0.947
9868-3704	−22.25	6.79	5.997	0.906
9888-6102	−23.13	10.18	4.000	0.930

Notes.

^aSDSS r-band absolute magnitude is from a Sérsic profile truncated at 8R_e. ^bMajor axis of the ellipse enclosing one-half of the total light of the Sérsic profile assuming H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_M0 = 0.3, and Ω_Λ0 = 0.7. The circularized effective radius used for modeling is ${R}_{e}\equiv \sqrt{{ab}}$.

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2.2. Methodology

The observed two-dimensional map of line-of-sight velocity dispersions that defines a radial profile σ_los(R) provides all the dynamical constraints for our models. Figure 1 shows an example of good-quality MaNGA velocity dispersions and the range of scales that is actually used for kinematic modeling.

In our modeling we allow for velocity dispersion anisotropies and stellar mass-to-light ratio (M_⋆/L) radial gradients. We follow the modeling procedures used recently (Chae et al. 2018, 2019b). All our galaxies have negligible amounts of cold gas in the IFU-probed regions, so throughout baryons mean stars. Here we describe briefly the essential elements.

We work in the MOND framework in which the empirical⁴ (i.e., dynamical) radial acceleration g_dyn(r) experienced by stars in a spherical system is related to the Newtonian acceleration due to baryons g_bar(r) as

$$\begin{eqnarray}&&{g}_{\mathrm{dyn}}(r)=\nu \left(\displaystyle \frac{{g}_{\mathrm{bar}}(r)}{{g}_{\dagger }}\right){g}_{\mathrm{bar}}(r),\end{eqnarray} \tag{ 1 }$$

where ν(x) is known as the MOND interpolating function (IF) defining the RAR in the MOND framework, and g_† is the acceleration parameter of interest here. We consider

$$\begin{eqnarray}&&\nu (x)=\displaystyle \frac{1}{2}+\sqrt{\displaystyle \frac{1}{4}+\displaystyle \frac{1}{x}},\end{eqnarray} \tag{ 2 }$$

which is known as the Simple IF (Famaey & Binney 2005). This particular form seems to be preferred by elliptical galaxies (Chae et al. 2019b); it is similar to the form used by McGaugh et al. (2016) to describe the SPARC galaxies, differing only in a subtle way at relatively high accelerations. Because we intend to test the universality of ${g}_{\dagger }$, we do not allow the IF functional form to vary. Our goal of testing the universality of g_† does not depend on this choice of the Simple IF. An alternative choice of the SPARC IF can shift the inferred g_† by an amount smaller than the uncertainty of the global value that we will obtain.

We assume that the projected mass density of stars follows the observed surface brightness distribution with a possible radial variation of ${{\rm{\Upsilon }}}_{\star }\equiv {M}_{\star }/L$ in the inner region ≲0.8R_e:

$$\begin{eqnarray}&&{{\rm{\Upsilon }}}_{\star }(R)={{\rm{\Upsilon }}}_{\star 0}\times \max \left\{1+K\left[A-B(R/{R}_{{\rm{e}}})\right],1\right\},\end{eqnarray} \tag{ 3 }$$

with A = 2.33 and B = 3 and K is a parameter representing the strength of a gradient (Chae et al. 2018). As estimated by Bernardi et al. (2018) K = 1 corresponds to the strong gradient reported in the literature (van Dokkum et al. 2017), but here we take a mild gradient of K = 0.21, i.e., ${M}_{\star }/L(R=0)=1.5\times {M}_{\star }/L(R\geqslant 0.8{R}_{{\rm{e}}})$, with a scatter of 0.1 based on Domínguez Sánchez et al. (2019) and Bernardi et al. (2019). However, even if a broad range of 0 < K < 1 is allowed with a flat prior as in Chae et al. (2019b), it has only a minor impact on our study.

We can link the observed σ_los(R) profile with the empirical acceleration g_dyn(r) through the spherical Jeans equation (Binney & Tremaine 2008) and a velocity dispersion anisotropy profile $\beta (r)\equiv 1-{\sigma }_{{\rm{t}}}^{2}(r)/{\sigma }_{{\rm{r}}}^{2}(r)$, where σ_r(r) and σ_t(r) are, respectively, the one-dimensional radial and tangential velocity dispersions. We assume a smoothly varying anisotropy profile of

$$\begin{eqnarray}&&\beta (r)={\beta }_{0}+({\beta }_{\infty }-{\beta }_{0})\displaystyle \frac{{\left(r/{r}_{a}\right)}^{2}}{1+{\left(r/{r}_{a}\right)}^{2}},\end{eqnarray} \tag{ 4 }$$

where β₀ (${\beta }_{\infty }$) is the anisotropy at r = 0 ($\infty$) and r_a is the radius where the anisotropy is the middle between the two.

Our model has the following free parameters:

$$\begin{eqnarray*}&&\vec{{\rm{\Theta }}}=\{{M}_{\star 0}(\equiv {{\rm{\Upsilon }}}_{\star 0}L),K,{\beta }_{0},{\beta }_{\infty },{r}_{a},{g}_{\dagger }\},\end{eqnarray*}$$

where L is the luminosity. We impose the following priors: a Gaussian prior with $(\mu ,\sigma )=(0.21,0.1)$ for K, and flat priors −2 < β₀ < 0.7, $-2\lt {\beta }_{\infty }\lt 0.7$, $0.1{R}_{{\rm{e}}}\lt {r}_{a}\lt 1{R}_{{\rm{e}}}$, and −15 < log₁₀g_† < −5 where R_e is the effective radius and g_† is in units of m s⁻². In general Bayesian applications, it would be preferable to impose an empirical common prior distribution on g_† estimated from elliptical galaxies themselves. In our study, we carefully selected a clean sample of galaxies with well-determined σ_los profiles. This means that g_† may be robustly estimated in each galaxy even when the prior on its value is uninformative. The prior ranges on the other parameters are discussed in Chae et al. (2018, 2019a).

We define a χ² function by

$$\begin{eqnarray}&&{\chi }^{2}=\sum _{i=1}^{{N}_{\mathrm{bin}}}{\left(\displaystyle \frac{{\sigma }_{\mathrm{los}}^{\mathrm{mod}}({R}_{i})-{\sigma }_{\mathrm{los}}^{\mathrm{obs}}({R}_{i})}{{\delta }_{i}}\right)}^{2},\end{eqnarray} \tag{ 5 }$$

where ${\sigma }_{\mathrm{los}}^{\mathrm{obs}}({R}_{i})$ and δ_i are the mean and its error in each radial bin from the velocity dispersion map (see Figure 1), and ${\sigma }_{\mathrm{los}}^{\mathrm{mod}}({R}_{i})$ is the MOND model prediction (Chae et al. 2018, 2019b). The likelihood is $\propto {e}^{-{\chi }^{2}/2}$ and the posterior probability density function (PDF) of g_† is derived from Markov Chain Monte Carlo (MCMC) simulations with the public code emcee (Foreman-Mackey et al. 2013). As was discussed in the appendix of Chae et al. (2019b), because of the complexity of the parameter space a narrow distribution of initial walkers around the maximum-likelihood estimate of the parameters can produce unrealistically narrow PDFs. However, if initial walkers are widely distributed within the prior ranges, then the PDFs returned by MCMC widen and become more similar to the results from the simple Monte Carlo simulations discussed by Chae et al. (2019b). Therefore, here we consider only MCMC simulations with widely sampled initial walkers, which are qualitatively similar to the simple Monte Carlo simulations.

Figure 2 shows MCMC results for 15 MaNGA E0 galaxies. It shows posterior probability distributions in the g_x/g_bar (where ${g}_{{\rm{x}}}\equiv {g}_{\mathrm{dyn}}-{g}_{\mathrm{bar}}$) versus g_bar plane. Note that for the supercritical (i.e., ${g}_{\mathrm{bar}}\gtrsim {10}^{-10}$ m s⁻²) acceleration regime, subtly different cases can be better distinguished in this modified RAR space (Chae et al. 2019b). Similar results for ATLAS^3D galaxies can be found in Chae et al. (2019b) and slightly revised results from this work are not shown. Figure 2 indicates that the case with ${g}_{\dagger }=1.2\times {10}^{-10}$ m s⁻² is included within the 2σ, i.e., 95%, confidence limits (CLs) of every individual galaxy displayed.

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The top panel of Figure 3 shows the individual PDFs of g_† for the 19 MaNGA and ATLAS^3D E0 SRs sorted in order of ascending g_†. As indicated in Figure 2, individual PDFs are quite broad, with 2σ CLs covering ∼3 dex. This is, in part, because uninformative priors are used for g_†, but reflects the fact that our velocity dispersion profiles probe an acceleration range that is only weakly sensitive to g_†.

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It is evident from Figure 3 that all PDFs overlap one another well. All PDFs include the global median within 2σ. Also, 12 out of 19 galaxies include the global median within 1σ, i.e., 68%, CL. Thus, our results are statistically consistent with a universal value of g_† among all E0 galaxies. The global median for our E0s ${g}_{\dagger }={1.5}_{-0.6}^{+0.9}\times {10}^{-10}$ m s⁻² is in excellent agreement with the canonical value for rotationally supported galaxies ${g}_{\dagger }=1.2\times {10}^{-10}$ m s⁻².

As a self-consistency check of our MCMC modeling we have considered other objects for which the spherical model is less well motivated or inadequate. We first consider an extended sample that includes all 16 SRs from the ATLAS^3D photometric pure-bulge sample (Chae et al. 2018; i.e., we relax the criterion on the circularity of the light distribution). The bottom panel of Figure 3 shows the individual PDFs for all 31 SRs. Evidently, even non-E0 (i.e., not perfectly round) SRs are consistent with a universal value of g_†. This may imply that the spherical model with velocity dispersion anisotropy can adequately describe nearly spherical SRs. If we include eight fast-rotator (FR) ATLAS^3D galaxies from the sample defined by Chae et al. (2018), then we find two individual results that deviate by more than 3σ from the universal g_† value. Of course, this is not surprising because the spherical model is bound to fail for FRs; rather, it demonstrates self-consistency of our MCMC modeling.

Figure 4 shows 387 data points from the MCMC analysis of the velocity dispersion profiles of the 19 E0s in the top panel of Figure 3 in a format that is similar to the RAR for rotationally supported SPARC galaxies (McGaugh et al. 2016; Lelli et al. 2017). Two tracks from MaNGA IFU 8243-6101 and NGC 4636 appear offset toward larger g_†, but these points are actually consistent with the RAR curve within their 2σ uncertainties. These galaxies are not otherwise unusual. Overall, the inset shows that all 387 points are statistically consistent with a universal RAR, confirming the results shown in Figures 2 and 3.

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In light of the recent debate on the presence of a universal acceleration scale involving dark matter phenomena in rotationally supported galaxies (Kroupa et al. 2018; McGaugh et al. 2018; Rodrigues et al. 2018a, 2018b; Chang & Zhou 2019; Marra et al. 2020), we have carried out kinematic analyses of a well-defined sample of E0 SR galaxies. Our Bayesian inference analysis yields individual posterior PDFs of g_† for 19 E0s that are consistent with a universal value of ${g}_{\dagger }={1.5}_{-0.6}^{+0.9}\times {10}^{-10}$ m s⁻². This value agrees well with that reported by McGaugh et al. (2016) for 153 rotationally supported galaxies from the SPARC database: ${g}_{\dagger }=(1.2\pm 0.2)\times {10}^{-10}$ m s⁻² (including systematic error).

Two crucial aspects of our Bayesian methodology are (1) the selection and use of reliable kinematic data and (2) robust and wide searches of the parameter space within the prior ranges. When these requirements are met, the presence of a common parameter may be tested for statistically. When the estimated error models are uncertain and/or the kinematic model is inappropriate, Bayesian inference is not suitable for statistically testing a common parameter (see also the discussion in Cameron et al. 2020).

To test the universality of g_† in generic early-type galaxy samples, one must use accurate nonspherical models. The larger parameter space of such models, and the greater likelihood of parameter degeneracies, will make this challenging. We believe that our results for (very nearly) spherical galaxies, which support the presence of a universal g_† for dark matter phenomena in galaxies, motivate future efforts to test the universality of g_† in generic galaxy samples, however challenging.

The presence of a universal acceleration scale fits naturally into the MOND paradigm: the empirical g_† becomes Milgrom's critical acceleration a₀. We would have falsified MOND if we had not found a universal value, or if this value had been different from that for rotating galaxies. In this respect, we note that MOND analyses were carried out recently based on MaNGA (Durazo et al. 2018) and ATLAS^3D (Tortora et al. 2014) galaxies, although our sample is more carefully defined and our analysis, which accounts for M_⋆/L radial gradients, is more general. Currently available ΛCDM simulations of galaxy formation and evolution predict intrinsic scatter of ∼0.06–0.08 dex in the value of g_† around the empirical mean value (see Stone & Courteau 2019 and references therein, in particular, Dutton et al. 2019). As this is substantially smaller than our current error bars, larger samples are needed to test if this is consistent with the data.

We thank F. Lelli, S. McGaugh, and P. Li for discussion and useful comments on the manuscript. We thank the anonymous reviewer for useful comments that helped us improve the presentation. This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1F1A1062477). M.B. acknowledges support from the NFS grant AST-1816330. H.D.S. acknowledges support from the Centro Superior de Investigaciones Científicas PIE2018-50E099.