Testing the Strong Equivalence Principle: Detection of the External Field Effect in Rotationally Supported Galaxies

The Spitzer Photometry and Accurate Rotation Curves (SPARC) database (Lelli et al. 2016) contains 175 rotationally supported galaxies in the nearby universe.⁶ These galaxies have stellar masses ranging from M_⋆ ≃ 10¹¹M_⊙ to M_⋆ ≃ 10⁷M_⊙ and cover all Hubble types of late-type, star-forming galaxies, including low-surface-brightness disk galaxies. The database provides the observed rotation velocities (V_obs) from spatially resolved H I observations and the Newtonian circular velocities from the observed distribution of stars and gas. The latter include the stellar disk contribution (V_disk) and (if present) the bulge contribution (V_bul) for a baseline mass-to-light ratio of unity, as well as the gas contribution (V_gas) for a total-to-hydrogen mass ratio of 1.33. For convenience, in this paper we redefine V_gas for a total-to-hydrogen mass ratio of unity. The reported velocity V_obs of a galaxy is tied to the reported inclination i_obs. If the inclination is changed to i, the rotation velocity becomes

$$\begin{eqnarray}&&{V}_{\mathrm{rot}}={V}_{\mathrm{obs}}\displaystyle \frac{\sin ({i}_{\mathrm{obs}})}{\sin (i)}.\end{eqnarray} \tag{ 2 }$$

The circular velocity due to the baryonic mass distribution depends on the galaxy distance D and is given by

$$\begin{eqnarray}&&{V}_{\mathrm{bar}}=\sqrt{\hat{D}({{\rm{\Upsilon }}}_{\mathrm{disk}}{V}_{\mathrm{disk}}^{2}+{{\rm{\Upsilon }}}_{\mathrm{bul}}{V}_{\mathrm{bul}}^{2}+{{\rm{\Upsilon }}}_{\mathrm{gas}}{V}_{\mathrm{gas}}| {V}_{\mathrm{gas}}| )},\end{eqnarray} \tag{ 3 }$$

where $\hat{D}\equiv D/{D}_{\mathrm{obs}}$ with D_obs being the fiducial distance. In Equation (3), ϒ_disk and ϒ_bul are the mass-to-light ratios of the disk and the bulge in units of the solar value M_⊙/L_⊙ at 3.6 μm, while ϒ_gas is the ratio of the total gas mass to the H I mass. When the SPARC database was published, this ratio was assumed to be 1.33 to account for the cosmic abundance of helium from Big Bang nucleosynthesis. Here we consider the small amounts of helium and metals formed via stellar nucleosynthesis during galaxy evolution (McGaugh et al. 2020), so that ϒ_gas = X⁻¹ where X is a function of stellar mass (M_⋆):

$$\begin{eqnarray}&&X=0.75\mbox{--}38.2{\left(\displaystyle \frac{{M}_{\star }}{{M}_{0}}\right)}^{\alpha },\end{eqnarray} \tag{ 4 }$$

with M₀ = 1.5 × 10²⁴M_⊙ and α = 0.22. We do however allow the possibility of varying ϒ_gas from X⁻¹ to consider the uncertainties in the H I flux, gas disk geometry, and the gas mass to H I mass ratio. In some cases V_gas is negative at small radii, representing the fact that the Newtonian gravitational field is not oriented toward the center when a large fraction of the gas disk lies in the outer regions. To account for the cases of negative V_gas we write ${V}_{\mathrm{gas}}| {V}_{\mathrm{gas}}|$ rather than ${V}_{\mathrm{gas}}^{2}$ in the last term of Equation (3), although this detail has negligible effects on our study.

Empirically, the observed centripetal acceleration (${g}_{\mathrm{obs}}={V}_{\mathrm{obs}}^{2}/R$) is related to the Newtonian baryonic acceleration (${g}_{\mathrm{bar}}={V}_{\mathrm{bar}}^{2}/R$) via the RAR ν₀(g_bar/g_†) of Equation (1) with a free parameter g_† (McGaugh et al. 2016; Lelli et al. 2017). In a MOND framework, g_† = a₀ is a fundamental constant of nature (Milgrom 1983) and Equation (1) can be obtained by modifying either inertia (Newton's second law of dynamics) or gravity (Poisson's equation) at the nonrelativistic level (Famaey & McGaugh 2012). In MOND modified-inertia theories Equation (1) holds exactly for any circular orbit (Milgrom 1994), while in MOND modified-gravity theories holds only for highly symmetric mass distributions (such as spheres) and represents a first-order approximation for actual disk galaxies (Brada & Milgrom 1995). In all these scenarios, however, Equation (1) is strictly valid only for isolated systems, when the EFE is negligible.

To build a general fitting function that approximates the EFE, we start from the nonlinear MOND modified Poisson's equation (Bekenstein & Milgrom 1984) in the one-dimensional case. If we assume a uniform external gravitational field g_ext (Famaey & McGaugh 2012) and the so-called Simple interpolating function (IF; Famaey & Binney 2005), we have

$$\begin{eqnarray}&&{g}_{\mathrm{MOND}}(R)={\nu }_{e}\left(\displaystyle \frac{{g}_{\mathrm{bar}}}{{g}_{\dagger }}\right){g}_{\mathrm{bar}}(R)\end{eqnarray} \tag{ 5 }$$

with

$$\begin{eqnarray}&&{\nu }_{e}(z)=\displaystyle \frac{1}{2}-\displaystyle \frac{{A}_{e}}{z}+\sqrt{{\left(\displaystyle \frac{1}{2}-\displaystyle \frac{{A}_{e}}{z}\right)}^{2}+\displaystyle \frac{{B}_{e}}{z}},\end{eqnarray} \tag{ 6 }$$

where z ≡ g_bar/g_†, A_e ≡ e(1 + e/2)/(1 + e), B_e ≡ (1 + e), and e ≡ g_ext/g_†. For e = 0, ν_e(z) is reduced to the Simple IF ${\nu }_{0}(z)=1/2+\sqrt{1/4+1/z}$. Equation (6) is based on the footnote to Equation (59) of Famaey & McGaugh (2012), but we corrected a small typo and rearranged it. Note here that the Simple IF allows the convenient analytic form of Equation (6) with e > 0 while it is only subtly different (Chae et al. 2019) in the (EFE-irrelevant) high acceleration limit from the function used by McGaugh et al. (2016) and Lelli et al. (2017) to fit the SPARC galaxies. Our results on the EFE detection are not affected by the choice of the Simple IF. Then, the expected circular velocity is given by

$$\begin{eqnarray}&&{V}_{\mathrm{MOND}}(R)=\sqrt{{\nu }_{e}\left(\displaystyle \frac{{g}_{\mathrm{bar}}}{{g}_{\dagger }}\right)}{V}_{\mathrm{bar}}(R).\end{eqnarray} \tag{ 7 }$$

Although ν_e(z) (Equation (6)) is based on idealized assumptions, it captures the basic feature of the EFE: a systematic downward deviation from ν₀(z) (Equation (1)) when e > 0 as $z\to 0$. Equation (6) also allows for upward deviations when e < 0, which seem unphysical but may be preferred by the data at the empirical level. These features are illustrated in Figure 1. MOND with the EFE predicts that the RAR must be a family of functions rather than a universal function. This also means that if galaxies in different environments are tried to be fitted with a single functional form of Equation (1), then there will arise some small intrinsic scatter of g_† due to the EFE. Most importantly, regardless of its MOND origin, Equation (6) may be considered a mere fitting function that improves over Equation (1) by adding the free parameter e, which has no a priori knowledge of the external gravitational field in which galaxies reside.

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In our Bayesian analysis the posterior probability of parameters ${\boldsymbol{\beta }}=\{{\beta }_{k}\}$ is defined by

$$\begin{eqnarray}&&p({\boldsymbol{\beta }})\propto \exp \left(-\displaystyle \frac{{\chi }^{2}}{2}\right)\displaystyle \prod _{k}\Pr ({\beta }_{k}),\end{eqnarray} \tag{ 8 }$$

where Pr(β_k) is the prior probability of parameter β_k and χ² is given by

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{j=1}^{N}{\left(\displaystyle \frac{{V}_{\mathrm{rot}}({R}_{j})-{V}_{\mathrm{MOND}}({\boldsymbol{\beta }};{R}_{j})}{{\sigma }_{{V}_{\mathrm{rot}}({R}_{j})}}\right)}^{2},\end{eqnarray} \tag{ 9 }$$

with ${\sigma }_{{V}_{\mathrm{rot}}({R}_{j})}={\sigma }_{{V}_{\mathrm{obs}}({R}_{j})}\sin ({i}_{\mathrm{obs}})/\sin (i)$ where ${\sigma }_{{V}_{\mathrm{obs}}({R}_{j})}$ is the reported error of V_obs(R_j) for the reported inclination i_obs. As in earlier studies of the RAR using SPARC galaxies (McGaugh et al. 2016; Lelli et al. 2017), we use only 153 galaxies with i_obs ≥ 30° and Q ≤ 2 (a quality cut on the rotation curve).

The parameters ${\boldsymbol{\beta }}$ in Equation (8) are given by ${\boldsymbol{\beta }}=\{{{\rm{\Upsilon }}}_{\mathrm{disk}},{{\rm{\Upsilon }}}_{\mathrm{bul}},{{\rm{\Upsilon }}}_{\mathrm{gas}},\hat{D},i,e\}$ for the case of using Equation (5) with a fixed g_† = 1.2 × 10⁻¹⁰ m s⁻². The priors on these parameters are summarized in Table 1. The mean values and standard deviations of ϒ_disk and ϒ_bul are motivated by state-of-the-art stellar population synthesis models for star-forming galaxies (Schombert et al. 2019). The mean value of ϒ_gas is given by Equation (4), while the standard deviation is motivated by the typical error on the H I flux calibration, but it could also represent variations in the assumed gas disk thickness and/or the mean gas-to-H I mass ratio. The mean values and standard deviations of $\hat{D}$ and i consider the baseline SPARC values and their fiducial errors. For e we adopt an uninformative uniform prior covering a reasonably broad range.

Table 1.  Summary of Prior Constraints on the Model Parameters

Parameter	Distribution	(μ, σ) or Range
ϒdisk	Lognormal	(${\mathrm{log}}_{10}(0.5)$, 0.1)
ϒbul	Lognormal	(${\mathrm{log}}_{10}(0.7)$, 0.1)
ϒgas	Lognormal	(${\mathrm{log}}_{10}({X}^{-1})$, 0.04)
$\hat{D}$	Lognormal	(0, ${\mathrm{log}}_{10}(1+{\sigma }_{{D}_{\mathrm{obs}}}/{D}_{\mathrm{obs}})$)
i	Gaussian	(iobs, ${\sigma }_{{i}_{\mathrm{obs}}}$)
e	Uniform	[−0.5, 0.5]

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The posterior probability density functions (PDFs) of the model parameters are derived from Markov Chain Monte Carlo (MCMC) simulations through the public code emcee (Foreman-Mackey et al. 2013). These simulations represent an extension to the previous SPARC analysis (Li et al. 2018) including the EFE parameter e. We choose N_walkers = 10,000 and N_iteration = 6000. We discard models up to N_iteration = 500 and thin the rest by a factor of 50 as the autocorrelation lengths for the parameters are <100. The posterior PDFs of $x={\mathrm{log}}_{10}{g}_{\mathrm{bar}}(R)$ and $y={{\rm{log}}}_{10}{g}_{{\rm{MOND}}}(R)$ follow from the posterior PDFs of the parameters i, ${\mathrm{log}}_{10}\hat{D}$, ${\mathrm{log}}_{10}{{\rm{\Upsilon }}}_{\mathrm{disk}}$, ${\mathrm{log}}_{10}{{\rm{\Upsilon }}}_{\mathrm{bul}}$, and ${\mathrm{log}}_{10}{{\rm{\Upsilon }}}_{\mathrm{gas}}$.

We estimate the environmental gravitational field g_env due to the large-scale distribution of matter at the positions of the SPARC galaxies. We perform this calculation within the standard ΛCDM context (Desmond et al. 2018). A similar calculation is not feasible in a MOND context due to the strong nonlinearities in the theory and the lack of a proper MOND cosmology. The ΛCDM calculation, however, is a good first-order approximation for MOND and other modified gravity theories (Desmond et al. 2018), up to some systematic uncertainty due to the unknown relation between g_env in these theories. We use g_env primarily for the purpose of picking out extreme cases with exceptionally high or low g_env (which should remain true in a relative sense in any cosmological scenario) and to check that the maximum-likelihood values of e from fitting Equation (5) are sensible in an order-of-magnitude fashion.

Our calculation of g_env starts with the total dynamical masses of the galaxies in the all-sky 2M++ survey (Lavaux & Hudson 2011) using abundance matching. We then use N-body simulations in ΛCDM to populate the surrounding regions with halos hosting galaxies too faint to be recorded in 2M++, using statistical correlations between halo abundances and properties of the galaxy field. Finally, we add mass in long-wavelength modes of the density field according to the inferences of the BORG algorithm (Lavaux & Jasche 2016) applied to the 2M++ catalog. We use the final density field to calculate a posterior distribution for g_env at the position of each SPARC galaxy, fully propagating uncertainties in the input quantities. We define e_env ≡ g_env/g_†, and find values in the range 0.01 ≲ e_env ≲ 0.1 with a mean of 0.033 among the SPARC galaxies: typical values are in the range 0.02–0.05.

The estimated values of the environmental gravitational field strength e_env (Section 2.4) span almost one order of magnitude, thus there is sufficient dynamic range to check whether or not the rotation-curve shapes depend on the large-scale environment. Among the SPARC galaxies whose rotation curves (RCs) reach g_obs < g_†, NGC 5033 and NGC 5055 live in exceptionally dense environments with e_env ≈ 0.1, while NGC 1090 and NGC 6674 are exceptionally isolated with e_env ≈ 0.01. The former two represent "golden galaxies" for the EFE to be detected, while the latter two are control targets for the null detection.

We fit the RCs using the EFE-incorporated RAR fitting function (Equation (5)) with a free external field g_ext parameterized by e = g_ext/g_† (Section 2.2). The case of e = 0 implies flat RCs and reduces exactly to the original RAR (Equation (1)). Figure 2 shows the MCMC results for the RCs of four galaxies in the two extreme environments. The "corner" plots showing the posterior PDFs of the parameters for these galaxies can be found in Appendix A.

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For NGC 5055, the detailed shape of the RC is very well fitted with a positive e but poorly fitted with e = 0. We find e = 0.054 ± 0.005: this is an 11σ detection. Remarkably, this value is consistent within 2σ with ${e}_{\mathrm{env}}={0.094}_{-0.022}^{+0.089}$ that is independently determined from the large-scale environment. The Bayesian information criterion (BIC ≡ $-2\mathrm{ln}{L}_{\max }+k\mathrm{ln}N$ where L_max is the maximum likelihood, k is the number of free parameters, and N is the number of the fitted rotation velocities) for e = 0 relative to the free e case is ΔBIC = 144, indicating very strong evidence for e > 0 based on the conventional criterion of ΔBIC > 10 for strong evidence.

For NGC 5033, the overall fit is also improved by freeing up e since ΔBIC = 83.9. We find $e={0.104}_{-0.012}^{+0.013}$. This is an 8σ detection, in excellent agreement with ${e}_{\mathrm{env}}={0.102}_{-0.021}^{+0.086}$ from the large-scale environment. The observed properties of this galaxy, however, are not as robust as those of NGC 5055. The rotation velocities at R < 60'' (about 5 kpc) are probably underestimated due to beam-smearing effects in the H I data, although our results on e are not affected by these data points. Moreover, while the distance of NGC 5055 is robust because it is based on the tip magnitude of the red giant branch (D = 9.90 ± 0.30), that of NGC 5033 is very uncertain because it is estimated using Hubble flow models (D = 15.70 ± 4.70). Interestingly, our MCMC result for NGC 5033 predicts a relatively large distance ($D={23.5}_{-1.8}^{+2.0}$ Mpc) with e > 0 but a low one ($D={13.0}_{-0.6}^{+0.7}$ Mpc) with e = 0. Hence, future observations can provide a key independent test.

In striking contrast to the highest e_env sample, the galaxies in the lowest e_env sample show no strong evidence for e > 0 based on ΔBIC (or any other widely used statistic). These two galaxies are similar to the golden galaxies in morphology, mass, and size. The only noticeable difference is that they are unusually isolated. The fitted e values are consistent with the independent e_env values within about 2σ.

Since the EFE has subtle effects on rotation-curve shapes, positive values of e are detected with high statistical significance only in individual galaxies where e_env is exceptionally large (like NGC 5055 and NGC 5033). The EFE, however, should also imprint a statistical signature in the low-acceleration portion of the RAR (see Figure 1).

3.2.1. The Systematic Trend in the Low-acceleration Portion of the RAR

We use 153 galaxies from the SPARC database (Section 2.1). Figure 3 (top panels) shows the RAR for 2696 points having accuracy in V_rot better than 10%. In the top left panel we first show the original SPARC mass models (Lelli et al. 2016) with fixed mass-to-light ratios at 3.6 μm of ϒ_disk = 0.5M_⊙/L_⊙ for the disk and ϒ_bulge = 0.7M_⊙/L_⊙ for the bulge (Lelli et al. 2017). The MCMC mass models obtained here with varied mass-to-light ratios (Section 2.3) are shown in the top right panel. We divide the data points into bins perpendicular to the best-fit curve assuming Equation (1). Each data point (x, y) is projected onto the point (x₀, y₀), so the orthogonal residual Δ_⊥ encodes any possible systematic deviation from the e = 0 case.

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The data show a small systematic deviation from Equation (1) for x₀ ≲ −11. This trend is present, though weakly, in the original SPARC mass models with fixed mass-to-light ratios for the disks and bulges. The MCMC models in the middle column show a stronger effect. The systematic deviation is weak in absolute terms (0.05–0.08 dex for the lowest x₀ bin) but at least 4 times larger than the bootstrap error of the median in the bin. This demonstrates that Equation (1) does not fully capture the trends in the RAR. Introducing e as an additional free parameter, we obtain a better fit and find e ≈ 0.02–0.04 in close agreement with the independent estimate of $\langle {e}_{\mathrm{env}}\rangle \simeq 0.033$ from an all-sky galaxy catalog (Section 2.4).

3.2.2. The Statistical Detection of the EFE

The systematic trend in the RAR also implies that the fitted individual values of e of Equation (5) will be systematically displaced from the non-EFE case e = 0. The posterior PDFs of e are quite broad with a typical standard deviation of ∼0.04 (see Appendix A for examples). Nevertheless, the statistical distribution of the fitted values will have a signature. Because e was allowed to take any value (positive or negative), this distribution provides a blind test of MOND EFEs (Section 2.2).

Figure 4 shows the distribution of the orthogonal residual Δ_⊥ and the fitted median value of e from the MCMC simulations with Equation (5). From Figure 1 it is expected that data points at high enough accelerations do not have any sensitivity to e. Indeed, for data points with −10.3 < x₀ < −9 the distribution of Δ_⊥ gives a null result. Similarly, for galaxies with $-10.3\lt \langle {x}_{0}\rangle \lt -9$, the distribution of e (median value) gives a null result.

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Data points at low enough accelerations will have sensitivity to e and distributions with nonzero mean value are expected from Figure 3. For data points with x₀ < −11.3 the distribution of Δ_⊥ has a mean of −0.061 ± 0.008 (a bootstrap error) which is statistically significant at more than 7σ. For a much larger number of data points with x₀ < −10.3, Δ_⊥ has a smaller deviation of −0.035 ± 0.003, but the statistical significance of the deviation is more than 11σ.

Figure 5 shows individual e values and their uncertainties for a subset of 113 galaxies with median $\langle {x}_{0}\rangle \lt -10.3$. Due to the large uncertainties on e, some galaxies can occasionally return negative values. However, the median value of e is 0.052 ± 0.011 (bootstrap error), which represents ≈5σ detection of positive e. This value is statistically consistent with the median environmental gravitational field for these galaxies ($\langle {e}_{\mathrm{env}}\rangle =0.034\pm 0.001$ (bootstrap error)). Furthermore, based on the robust binomial statistic with equal probabilities for e > 0 and e < 0, 78 cases of e > 0 out of 113 is 4σ away from the expected mean of 56.5 cases.

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Figure 6 further shows the distribution of the individual difference e−e_env. It has a broad distribution due to the large uncertainty in e but is clearly consistent with zero: $\langle e-{e}_{\mathrm{env}}\rangle =0.011\pm 0.013$. It is intriguing that the mere fitting parameter e returns, on average, the same value of the mean environmental gravitational field of the nearby universe, computed in a fully independent way.

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3.2.3. Statistical Properties of the Posterior Parameters of the Galaxies

Figure 7 shows the distribution of the parameters from the MCMC simulations with Equation (5) for all 153 selected galaxies. The distribution of the distances is consistent with the SPARC reported values with an rms scatter of 0.02 dex (5%). This is smaller than typical measurement uncertainties of ∼14%. The posterior inclination angles are also consistent with the SPARC reported values with an rms scatter of 21, smaller than typical measurement uncertainties of ∼4°. The distributions of the mass-to-light ratios (ϒ_disk and ϒ_bul) for the disk and the bulge are consistent with the estimates from infrared studies, i.e., ϒ_disk = 0.5 M_⊙/L_⊙ and ϒ_bul = 0.7 M_⊙/L_⊙, with an rms scatter of 0.14 dex. If anything ϒ_bul might be 0.6, a little smaller than 0.7. Finally, the distribution of ϒ_gas is in excellent agreement with X⁻¹ from Equation (4), giving a mean value of 1.38 ± 0.04, which is intermediate between a metal-poor dwarf galaxy with X⁻¹ = 1.34 and a metal-rich giant spiral with X⁻¹ = 1.42.

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3.2.4. Analysis of Possible Systematic Effects

One may wonder whether the systematic deviations from Equation (1) are due to some systematic uncertainties. There are three main observational effects that may systematically affect the low-acceleration portion of the RAR: galaxy distances, the thickness of the gas disk, and possible variations of M_⋆/L in the stellar disk with radius. To mitigate the first two uncertainties, the left panel of Figure 8 considers data points from galaxies with accurate distances based on the tip magnitude of the red giant branch, Cepheids, or supernovae (Lelli et al. 2016), as well as low gas contributions (f_gas = M_gas/M_bar < 0.4). Compared with Figure 3, it is clear that the scatter is smaller and the median trend is consistent with the full data set.

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The thickness of the gas disk is a concern because the EFE is detected in the galaxy outskirts, where the gas contribution becomes nonnegligible or even dominating in some cases. Recent studies (Bacchini et al. 2019) suggest that gas disks may become thicker at large radii: this would systematically decrease V_gas, hence g_bar, moving points to the left of the RAR. Therefore, we repeat the MCMC fits considering gas disks that are 3 times thicker than assumed in the SPARC database. This is a very extreme scenario because not all galaxies will have such thick gas disks. Our goal is simply to provide an upper bound on the possible impact of this effect. Figure 8 (middle panel) shows that there is still a significant systematic deviation from Equation (1) even when we consider very thick gas disks.

Negative gradients of M_⋆/L with R could also systematically decrease V_disk, hence g_bar, moving points to the left of the RAR. While we are treating the bulge separately in the most massive spirals (Sa to Sb), the stellar disk may potentially display a radial variation of its stellar populations. At 3.6 μm these variations have a relatively weak effect (Schombert et al. 2019), but we nevertheless repeat the MCMC fits considering a linear decrease in ϒ_disk by a factor of 2 from the center to the outermost observed radius. Again, this is an extreme scenario since most stellar disk are likely not showing such strong radial gradients in ϒ_disk. Figure 8 (right panel) shows that there is still a significant systematic deviation from Equation (1).

Only a few attempts have been made so far to detect the EFE from the RCs of galaxies (Wu & Kroupa 2015; Haghi et al. 2016). In particular, Haghi et al. (2016) considered the RCs of 18 galaxies taken from the literature available at that time. These galaxies are known to have relatively nearby massive neighbors. Eleven of them are also included in our sample of 153 galaxies studied here. They are DDO 154, IC 2574, NGC 2998, NGC 3198, NGC 3521, NGC 3769, NGC 4100, NGC 4183, NGC 5033, NGC 5055, and NGC 5371.

Haghi et al. (2016) obtained values of e ranging from about 0.1 to 0.6 with a median of ∼0.3 and a typical uncertainty of ∼0.1 for these 11 galaxies. Their values are systematically higher than our values ranging from about −0.1 to 0.3 with a median of ∼0.075 and a typical uncertainty of ∼0.04. This is primarily due to the fact that the disk models of Haghi et al. (2016) are based on a baryonic mass profile that declines more slowly than observed at large radii, requiring a larger EFE in the MOND context (a deficit of DM in the ΛCDM context).

There have also been indications of the EFE in pressure-supported galaxies (McGaugh & Milgrom 2013a, 2013b; Famaey et al. 2018; Kroupa et al. 2018). Pressure-supported galaxies are analyzed through their observed line-of-sight velocity dispersions. Because their stellar orbits are complex and not observed directly, a robust kinematic analysis to infer the EFE is challenging. However, McGaugh & Milgrom (2013a, 2013b) have found that the observed velocity dispersions of the dwarf galaxies of the Andromeda galaxy are consistent with a MOND theory with EFE. More recently, galaxies that appeared to have too low observed velocity dispersions and thus lack dark matter in the ΛCDM context (van Dokkum et al. 2018, 2019) may well be explained by the MOND EFE (Famaey et al. 2018; Kroupa et al. 2018; Haghi et al. 2019; Müller et al. 2019).