On the Evidence for the Violation of the Equivalence Principle in Disk Galaxies

Abstract

We examine the claimed observations of a gravitational external field effect (EFE) reported by Chae et al. We show that observations suggestive of the EFE can be interpreted without violating Einstein’s equivalence principle, namely from known correlations between the morphology, the environment, and dynamics of galaxies. While Chae et al.’s analysis provides a valuable attempt at a clear test of modified Newtonian dynamics, an evidently important topic, a re-analysis of the observational data does not permit us to confidently assess the presence of an EFE or to distinguish this interpretation from that proposed in this article.

1. Introduction

One of the most pressing problems in contemporary science is the origin of an apparent substantial missing mass in large gravitational systems, such as galaxies or clusters of galaxies. The prevailing explanation for this phenomenon is the presence of non-baryonic weakly interacting matter known as dark matter. However, the consensus on dark matter’s existence is not universal. This is partly due to the lack of detection of dark matter particles either directly [1] or indirectly [2]. Furthermore, advances in particle physics have ruled out the most natural extensions of the Standard Model of particle physics, which previously offered promising dark matter candidates [3]. This has fueled alternative explanations for the missing mass problem. The most well-known of these alternatives is modified Newtonian dynamics (MOND) [4]. MOND has achieved notable success in economically describing observations at the galactic scale and even making predictions that have been subsequently validated; see [5,6] for a review. One such prediction is the radial acceleration relation (RAR), a strict correlation between the measured accelerations experienced by baryonic matter within galaxies and the accelerations predicted by Newtonian gravity in the absence of dark matter. This strict correlation has been observed [7,8]. While a loose correlation may be expected within the context of dark matter and galaxy formation, they offer no uncontrived explanation for the strict correlation. MOND is foremost a phenomenological theory. If its origin is fundamental rather than being an effective theory for phenomena arising from the complexity of galaxy dynamics and/or of general relativity (GR), it would violate some of GR’s tenets. Notably, the characteristic acceleration, denoted as $a_0$, should lead to an external field effect (EFE), which contradicts GR’s strong equivalence principle. The EFE would substantially weaken the self-gravity of dwarf galaxies in a cluster environment, perhaps explaining why dwarf galaxies in the Fornax Cluster are much more susceptible to tides than expected in LCDM [9,10].

Recently, observations of the EFE have been reported by [11]. A refined analysis of the RAR data indicates that galaxies subjected to an external gravitational field, such as that emanating from neighboring galaxies, tend to display less missing mass compared to more isolated galaxies which, conversely, tend to have flatter rotation curves. This observation is naturally expected from MOND; given that the onset of MOND’s regime starts below $a_0$, a constant external field added to the galactic field delays the transition to this regime. This, in turn, keeps the galaxy in the Newtonian regime longer, reducing their missing mass.

If these observations hold, it would mark the first experimental observation of a departure from GR and, equally significant, strongly challenge the dark matter hypothesis. Thus, it is imperative to study and exhaust alternative mundane explanations for the observations in [11]. The study’s authors examine various such hypotheses and reject them. In this article, we examine the claim that the observations in [11] evince the existence of an EFE. First, we investigate a possible explanation that has not been covered in [11], one that preserves GR and its strong equivalence principle while still obviating the need for dark matter. Specifically, we show how correlations between galactic properties, rotation curves, and environment may explain the observations in [11]. The appendices provide technical details of the analyses in Section 2 and a more quantitative analysis on how well MOND, with and without the EFE, fits the observational data based on the same MCMC approach taken in [11]. In particular, we examine how the uncertainties associated with the analysis in [11] affect the claim of EFE observation.

2. Modeling Radial Acceleration Relation with General Relativity Self-Interaction

The observations in [11] can be naturally explained without jeopardizing the strong equivalence principle by considering two facts. The first is that rotation curves depend on the galaxy’s distinct morphology. This is generally expected for any non-spherical extended system, regardless of whether Newtonian physics or GR is considered. If the system is large enough, the self-interaction of the gravitational field in GR is expected to cause $V_{\infty }$ (the rotation speed at the outskirts of the galactic disk) to noticeably depend on the morphology of the galaxy, particularly on the relative size of the bulge versus that of the disk. The second fact is the relationship between the environment and galactic morphology. It has been observed that the distribution of bulge types in the nearby universe clearly depends on the environment in which the galaxies reside [12]. This is expected in current hierarchical models of galaxy formation, in which the bulges of disk galaxies grow from environmental disturbances, particularly mergers [13], and this is supported by the observation that the bulge-to-total mass ratio of nearby galaxies correlates with their number of satellite galaxies [14]. Taken together, the dependence of $V_{\infty }$ on morphology and the correlation of the latter with the environment could explain the observations in [11] without violating the strong equivalence principle of GR. We will now quantitatively investigate this possibility.

To illustrate our model, we generated a set of 40,000 simulated disk galaxies with bulges sampling the correlations derived from observations in the manner previously described in detail in [15].

We account for GR self-interaction (GR-SI) in a disk galaxy by modeling it following [15]: the galaxy has two components, a disk and a bulge made solely of baryonic matter. The model does not require dark matter, since for galaxies large enough, field self-interaction constrains the gravitational field lines to remain within the disk, effectively confining gravity to approximately the two dimensions of the disk. This phenomenon, however, is suppressed in the bulge because its three-dimensional volume presents no particular direction or plane into which the field lines can collapse [16]. This expectation has been verified by observing that in elliptical galaxies, the larger the ellipticity, the more the dynamics of baryonic matter departs from the Newtonian expectation [17,18].

The model in [15] entails that the total acceleration within the galaxy ($g_{\mathrm{SI}}$) is the sum of the Newtonian acceleration ($g_{3\mathrm{D}}$) from the bulge and that of gravity is constrained to propagate in 2D ($g_{2\mathrm{D}}$). We present an improved model:

$$\begin{array}{ccc}\hfill g_{\mathrm{SI}}\left(r\right)& \equiv & g_{3\mathrm{D}}\left(r\right)+g_{2\mathrm{D}}\left(r\right)=G{\displaystyle \frac{M_{\mathrm{B}}\left(r\right)}{r^2}}+\sqrt{G\alpha }{\displaystyle \frac{\sqrt{M_{\mathrm{D}}\left(r\right)}}{r}},\hfill \end{array}$$

(1)

with $\sqrt{G\alpha }$ being the effective coupling constant of gravity in two dimensions. $M_{\mathrm{B}}\left(r\right)$ and $M_{\mathrm{D}}\left(r\right)$ are the masses of the baryonic bulge and disk, respectively, contained within a sphere of radius r. We improved the model mainly with the handling of the factor $\alpha$ (discussed in detail in Appendix A) such that it satisfies the Tully–Fisher relation (detailed in Appendix B). The expression of the 2D part relates to the effective potential emerging from the solution found in [19] for a system with cylindrical symmetry in GR, but it is extrapolated to an extended source with a given mass profile here. It is worthwhile to emphasize that this and previous GR-based analyses of approximate 2D systems [16,20,21] positions our approach strictly within the framework of GR and its approximations and the Standard Model of particle physics, in contrast to MOND and $\Lambda$CDM. The resulting $g_{\mathrm{SI}}$ is evaluated numerically (cf. Appendix B) and plotted versus $g_{3\mathrm{D}}$ in Figure 1. As the evaluation depends on the ratio of disk and bulge mass, or equivalently the bulge-to-total mass ratio $\mu \equiv M_{\mathrm{B}}^{\mathrm{tot}}/(M_{\mathrm{B}}^{\mathrm{tot}}+M_{\mathrm{D}}^{\mathrm{tot}})$, where $M_{\mathrm{B}}^{\mathrm{tot}}$ and $M_{\mathrm{D}}^{\mathrm{tot}}$ are total masses of the bulge and the disk, respectively, the distribution of this parameter in the numerical evaluation is plotted for reference. Clearly, at low accelerations, galaxies with large bulges (red color in the figure) are closer to the Newtonian expectation (diagonal line) than those with a small bulge (blue color). Equation (1) can be re-expressed as the ratio between the total acceleration to that from Newton’s gravity:

$${\displaystyle \frac{g_{\mathrm{SI}}}{g_{3\mathrm{D}}}}=1+\sqrt{{\displaystyle \frac{\alpha }{G}}}{\displaystyle \frac{\sqrt{M_{\mathrm{D}}\left(r\right)}}{M_{\mathrm{B}}\left(r\right)}}r,$$

(2)

which can then be compared to those proposed in [7,11] in the context of a violation of the strong equivalence principle; see Appendix B.

We now utilize the observed correlation reported in [14], wherein a correlation was found between the bulge-to-total mass ratio of galaxies with their number of satellite galaxies. This relationship is given by

$$N^{200}=30.2(\pm 6.2)\mu +1.7(\pm 1.3),$$

(3)

where $N^{200}$ is the number of observed satellites within a distance of 200 kpc from the main galaxy. This correlation serves here as a measure of the density of the environment in which the main galaxies reside, and therefore as a measure of the external gravitational field to which the galaxies are subjected to. This approach differs from that in [11], which estimated the external field within the standard $\Lambda$CDM context using an N-body simulation and the 2M++ survey [22] to populate observed SPARC galaxies. We find our approach to be appropriate for this study due to the difficulties $\Lambda$CDM has with estimating satellite populations [23] and its challenges in explaining strong correlations between bulge mass and satellite populations [24]. The absence of accurate SPARC bulge-to-total ratios also makes a direct comparison to [11] difficult. We also acknowledge that [14] did not aim to present an exact formula for N vs. $\mu$. Instead, we use Equation (3) for the purpose of better understanding the trend presented by the data. In doing so, we note that the complex histories of individual galaxies play the most important role in determining morphology and corresponding isotropies. These complex histories present a problem when attempting to analyze individual galaxies. Therefore, analyzing individual rotation curves, while often useful, is not necessary in our context.

Reformulating Equation (2) as shown in Appendix B and combining Equations (A10) and (3) yields the observation that galaxies from a dense environment tend to be closer to the Newtonian expectation than isolated galaxies:

$${log}_{10}{g}_{\mathrm{SI}}={\displaystyle \frac{1}{2}}{log}_{10}{g}_{3\mathrm{D}}+{\displaystyle \frac{1}{2}}{log}_{10}\alpha +{\displaystyle \frac{1}{2}}{log}_{10}\left({\displaystyle \frac{30.2(\pm 6.2)}{N^{200}-1.7(\pm 1.3)}}-1\right).$$

(4)

The third term on the right-hand side of Equation (4) will result in the radial acceleration curve getting closer to the 45° diagonal as $N^{200}$ increases, as shown in Figure 2.

Using this result, we find we can thus account for the correlation observed in [11] without invoking an EFE. This is achieved by combining the verified GR-SI prediction that the departure from a Newtonian behavior of a large system is suppressed by the degree of isotropy of the system [17,18], with the observed correlations between the morphology of disk galaxies and satellite galaxy populations [12,13,14]. In short, if the dynamics of spiral galaxies are dependent on galactic morphology in the way predicted by GR-SI, and the morphology of galaxies are correlated with their number of satellite galaxies (and more generally, their environment), then we expect to see a correlation between the dynamics of galaxies and their external fields. This expected correlation is in direct contrast to the explanation given in [11] wherein external fields directly affect the dynamics of spiral galaxies via the suppression of the MOND effect.

For a quantitative comparison, the MCMC analysis in [11] should be repeated with our model. For technical details, see Appendix C. We find that the significance of the analysis in [11] is not as high as claimed. Re-conducting their analysis, we find considerably different values and large error margins. Within these errors, our results are consistent with the values in [11], but our analysis does not provide further insights aside from highlighting important instabilities in the MCMC analysis and possible insufficient convergence. This suggests that [11] might not have found a global maximum of the likelihood in the parameter space. In particular, Chae et al. [11] stops after a given number of iterations and does not demonstrate convergence of the MCMC analysis. To quantitatively determine which model is preferred would require a prohibitively expensive investigation of the parameter space to ensure proper convergence of the MCMC algorithm, which is beyond the scope of this paper. At this point, we can only state that the MCMC analysis in [11] suffers from insufficient convergence, which, in addition to the alternative explanation presented here, further weakens claims for an EFE.

3. Conclusions

We examined the claim of observations of an external field effect (EFE) in gravitation reported in [11], based on extensive data on disk galaxy rotation [7]. First, we showed with an example that there exist other plausible explanations not requiring an EFE nor contrived models tailored to explain this phenomenon. To do so, we demonstrated that an apparent deviation from Einstein’s strong equivalence principle could be explained instead by examining the known correlations between galactic morphology, the galactic environment (specifically, satellite galaxy populations), and galaxy dynamics. More precisely, it is expected from the non-linear character of general relativity that galaxies with larger bulges exhibit more Newtonian dynamics, just like how rounder elliptical galaxies exhibit a smaller missing mass problem than flatter ones [17,18]. This, together with the observation that galaxies with a higher number of satellites, which is indicative of a denser environment (and incidentally a greater external field), tend to have larger bulges [14], explain the apparent EFE. Furthermore, while re-analyzing the data following the approach in [11], we found that, given the analysis method used in [11] and in this study, observational data do not permit us to confidently assess the the existence of an EFE or to distinguish between the possible various interpretations of these data.