The tight radial acceleration relation (RAR) obeyed by rotationally supported disk galaxies is one of the most successful a priori prediction of the modified Newtonian dynamics (MOND) paradigm on galaxy scales.
Another important consequence of MOND as a classical modification of gravity is that the strong equivalence principle (SEP) - which requires the dynamics of a small free-falling self-gravitating system to not depend on the external gravitational field in which it is embedded - should be broken. Multiple tentative detections of this so-called external field effect (EFE) of MOND have been made in the past, but the systems that should be most sensitive to it are galaxies with low internal gravitational accelerations residing in galaxy clusters, within a strong external field.
Here, we show that ultra-diffuse galaxies (UDGs) in the Coma cluster do lie on the RAR, and that their velocity dispersion profiles are in full agreement with isolated MOND predictions, especially when including some degree of radial anisotropy. However, including a breaking of the SEP via the EFE seriously deteriorates this agreement.
We discuss various possibilities to explain this within the context of MOND, including a combination of tidal heating and higher baryonic masses. We also speculate that our results could mean that the EFE is screened in cluster UDGs. The fact that this would happen precisely within galaxy clusters, where classical MOND fails, could be especially relevant for the nature of the residual MOND missing mass in clusters of galaxies.
This approach, known as Modified Newtonian Dynamics (MOND; see, e.g., Sanders & McGaugh 2002; Famaey & McGaugh 2012; Milgrom 2014, for reviews) postulates that the gravitational acceleration g approaches √ g(N)a(0) when the Newtonian gravitational acceleration g(N) falls below a characteristic acceleration scale a(0) ≈ 10^−10 m s^−2 , but remains Newtonian above this threshold. This allows one to directly predict the dynamics of galaxies from their baryonic mass distribution alone. This empirical modification of the gravitational law was initially proposed (Milgrom 1983a,b,c) to solve the missing mass problem in the high surface brightness galaxies known at the time, in particular their asymptotically flat circular velocity curves (e.g., Bosma 1978; Rubin et al. 1978; Faber & Gallagher 1979). It is particularly intriguing that this simple recipe has survived almost 40 years of scrutiny at galactic scales, as it has been able to predict the dynamics of a wide variety of galaxies (e.g., Begeman et al. 1991; Sanders 1996; McGaugh & de Blok 1998; Sanders & Verheijen 1998; de Blok & McGaugh 1998; Sanders & Noordermeer 2007; Gentile et al. 2007b; Swaters et al. 2010; Gentile et al. 2011; Famaey & McGaugh 2012; Milgrom 2012; McGaugh & Milgrom 2013a,b; Sanders 2019), including low surface brightness and dwarf galaxies where internal accelerations can be well below a0 such that the MOND acceleration should a priori deviate significantly from the Newtonian acceleration. This was a core prediction of the original MOND papers, and one of its most intriguing successes. MOND can also provide possible answers to various other puzzles in galaxy dynamics, such as the prevalence of bulgeless disks (Combes 2014) and of fast bars (Tiret & Combes 2007, 2008; Roshan et al. 2021), or the detailed kinematics of polar ring galaxies (Lüghausen et al. 2013). Generally speaking, there now appears to be a clear and direct connection between the baryonic mass distribution and the rotation curve in most disk galaxies, known as the radial acceleration relation (RAR; McGaugh et al. 2016; Lelli et al. 2017), and this empirical relation is actually indistinguishable from the original MOND prescription (Li et al. 2018). Evaluating whether the RAR holds for all types of galaxies and in all environments is thus of high importance to assess the viability of MOND as an alternative to particle DM in galaxies.However, it is important to note that it was also originally predicted that the non-linearity of the MOND acceleration should typically lead to a violation of the strong equivalence principle of GR, according to which the internal dynamics of a self-gravitating system embedded in a constant gravitational field should not depend on the external field strength. Within MOND, systems embedded in an external field stronger than their internal one should experience an ‘external field effect’ (EFE; Milgrom 1983c; Bekenstein & Milgrom 1984; Famaey & McGaugh 2012; McGaugh & Milgrom 2013a,b; Milgrom 2014; Wu & Kroupa 2015; Haghi et al. 2019b) whose consequence is notably that the deviations from Newtonian dynamics are suppressed if the external field is strong enough, and in particular if it is larger than a(0). Its influence can be important for the stability and secular evolution of galaxies even when it is weak (Banik et al. 2020), and it can create interesting features such as asymmetric tidal tails of globular clusters (Thomas et al. 2018). The EFE is an observational necessity to allow, e.g., the dynamics of wide binary stars to remain consistent with MOND (Pittordis & Sutherland 2019, Banik 2019, although see also Hernandez et al. 2021). Because of this EFE, a rotationally supported (pressure-supported) system in isolation is expected to have a higher rotational velocity (velocity dispersion) than the same system around a massive host (e.g. Wu et al. 2007; Gentile et al. 2007a; McGaugh & Milgrom 2013a,b; Pawlowski & McGaugh 2014; Pawlowski et al. 2015; McGaugh 2016; Hees et al. 2016; Haghi et al. 2016; Müller et al. 2019; Chae et al. 2020). In particular, the latter should not follow the RAR, contrary to the more isolated systems that should lie on the RAR. This breaking of the strong equivalence principle should be a smoking gun of MOND, and it is therefore important to test it for galaxies with internal gravitational accelerations lower than the external field in which they are embedded: this is the focus of the present work.The need for DM within GR is of course not limited to galaxies. Expanding MOND predictions to the cosmological regime needs a relativistic framework for the paradigm. In order to retain the success of the standard ΛCDM cosmological model on large scales, some hybrid models have for instance been proposed, where GR is retained but gravity is effectively modified in galaxies through some exotic properties of DM itself, such as in dipolar DM (Blanchet & Le Tiec 2009; Bernard & Blanchet 2015; Blanchet & Heisenberg 2015) or superfluid DM (Khoury 2015; Berezhiani & Khoury 2015; Berezhiani et al. 2018, 2019). More traditional relativistic MOND theories rely on a multi-field framework (typically with a scalar and a vector field in addition to the metric), as originally proposed by Bekenstein (2004), but adapted to pass the most recent constraints from gravitational waves (Skordis & Złosnik 2019). It has recently been shown, as a proof of concept, how the angular power spectrum of the Cosmic Microwave Background (CMB) could be reproduced in such a framework (Skordis & Złosnik 2020): the scalar field, which gives rise to the MOND behaviour in the quasi-static limit, also plays the role of DM in the time-dependent cosmological regime, thereby providing an analogue to cosmological DM for the CMB.However, the real challenge for such an approach, and for MOND in general, is to explain the mass discrepancy in galaxy clusters. It has indeed long been known that applying the MOND recipe to galaxy clusters yields a residual missing mass problem in these objects (e.g., Sanders 1999, 2003; Pointecouteau & Silk 2005; Natarajan & Zhao 2008; Angus et al. 2008). This is essentially because, contrary to the case of galaxies, there is observationally a need for DM even where the observed acceleration is larger than a(0), meaning that the MOND prescription is not enough to explain the observed discrepancy. In the central parts of clusters, the ratio of MOND dynamical mass to observed baryonic mass can reach a value of 10. This cluster missing mass problem extends to giant ellipticals residing at the center of clusters (Bílek et al. 2019b). It is also clear that this residual missing mass must be collisionless (Clowe et al. 2006; Angus et al. 2007), and it has hence been proposed that it could be made of cold, dense molecular gas clouds (Milgrom 2008) or some form of hot dark matter (HDM) such as sterile neutrinos, which would not condense on galaxy scales (Angus et al. 2010; Haslbauer et al. 2020). In such cases, the residual missing mass should be an important gravitational source contributing to the EFE acting on galaxies residing in clusters. On the other hand, if the residual MOND missing mass problem would itself be a gravitational phenomenon, it would then not necessarily contribute as a source to the EFE. Therefore, studying the dynamics of galaxies residing in galaxy clusters, and in particular whether the EFE can be detected there, should provide powerful constraints for relativistic model-building in the MOND context, as well as illuminate our understanding of scaling relations with environment in the cold dark matter (CDM) paradigm. Galaxies with a very low internal gravity, hence ultra-diffuse ones, are best suited for such a study. . . .
1.3. UDGs in MONDUDGs in clusters provide a testing ground for MOND and the EFE given the singularly low internal accelerations stemming from their low surface brightness and the strong external field. The small velocity dispersion observed in the two group UDGs NGC 1052-DF2 and NGC 1052-DF4, inferring dynamical masses close to their stellar masses, was initially interpreted as a challenge for MOND (van Dokkum et al. 2018, 2019a). Indeed, the dynamical effect attributed to DM in the CDM model, and to a modification of the gravitational law within MOND in isolation, would be absent. But taking the EFE into account removes or significantly lessens the tension (Famaey et al. 2018; Kroupa et al. 2018; Müller et al. 2019; Haghi et al. 2019b). On the other hand, the large velocity dispersion of the Coma cluster UDG DF44 (van Dokkum et al. 2016, 2019b) and its relative agreement with the isolated MOND prediction without EFE has been used to place constraints on its distance from the cluster center within MOND, or on a potential need for an additional baryonic mass (Bílek et al. 2019a; Haghi et al. 2019a).Different approaches have been used to take the EFE into account in this context: Kroupa et al. (2018) and Haghi et al. (2019b) use fitting functions for the one-dimensional line-of-sight velocity dispersion in an external field stemming from MOND N-body simulations by Haghi et al. (2009); Famaey et al. (2018) and Müller et al. (2019) use the one-dimensional analytical expression for the acceleration field in the presence of an external field from Famaey & McGaugh (2012), Eq. (59), together with the Wolf et al. (2010) relation for the line-of-sight velocity dispersion. Bílek et al. (2019a) and Haghi et al. (2019a) do not quantitatively assess the EFE on the velocity dispersion for DF44, since the data required as little of it as possible. We propose here to examine more quantitatively the question of the EFE in this galaxy, and to expand the study to a larger sample of UDGs.
We discuss hereafter different possible interpretations for the tension between the measurements and the MOND prediction with EFE, constraining either MOND itself or the formation and evolution of UDGs within this theory:(1) the observed UDGs are further away from the cluster center than they seem, have fallen inside the cluster relatively recently, and/or are disrupted by tides in the cluster environment;(2) they have higher stellar mass-to-light ratios than assumed here or are surrounded by additional baryonic dark matter haloes;(3) the EFE varies from one galaxy to another depending on its individual history;(4) the characteristic acceleration scale of MOND varies with the environment, being higher in clusters;(5) the cluster environment shuts down the EFE within the parent relativistic theory of MOND.Alternatively, MOND being an effective dark matter scaling relation of course also remains a serious possibility: in that context, the fact that cluster UDGs obey the same scaling relation as field spirals, despite their very different environments and likely different formation scenarios, is still particularly intriguing, irrespective of the underlying theoretical framework.
5.5. Screening the EFE in galaxy clusters?We note that the apparent absence of EFE happens precisely within galaxy clusters, where classical MOND fails to explain the overall dynamics of the cluster, and we conjecture here that these two facts might possibly be related. In the case of EMOND, this would be explained by an effective increase of the MOND acceleration constant, not by a screening of the EFE itself. But another possibility is that the EFE is severely damped in galaxy clusters.
In a theory like that of Skordis & Złosnik (2020), the action harbours a free-function, playing the role of the MOND interpolating function, depending both on the spatial gradient squared of the scalar field |∇ϕ| 2 (with a 3/2 exponent, characteristic of MOND actions) and on its temporal derivative having a non-zero minimum leading to gravitating “dust". It is this time dependent term which allows to reproduce a reasonable angular power spectrum for the CMB, and one could therefore speculate that it can possibly also give rise to additional gravitating “dust" inside galaxy clusters, to explain the residual missing mass of MOND. However, it is not clear that, if the scalar field is dominated by this “dust" component inside the cluster itself, it would couple to the scalar field within the UDG in the same way as in the fully quasi-static limit. Therefore, one could imagine that, precisely because the residual missing mass in galaxy clusters would be caused by the same scalar field as that creating the MOND effect inside the UDG, the EFE could be effectively screened within clusters. Note that this is especially relevant for any model that would try to explain away the residual MOND missing mass in clusters of galaxies, as such an explanation would not work if the residual missing mass is made of additional hot DM like light sterile neutrinos.
In this context of EFE screening, one could imagine two possibilities: one where the EFE would be solely produced by the baryonic mass of the cluster, and one where it would be almost fully screened, the UDG living in its MOND bubble effectively decorrelated from the dynamics of the cluster itself. We test here the first hypothesis by redoing the analysis of Section 4 using only the Coma cluster hot gas mass distribution M(gas) (Eq. (33)) derived from the β-model of Eq. (16) as a source of EFE, instead of the mass distribution MC(r) inferred from hydrostatic equilibrium. This mass is about 1 dex below MC at a distance of 1 Mpc but reaches MC at 10 Mpc. As a consequence, the resulting velocity dispersions at distances smaller than 10 Mpc are higher than in Fig. 9, as shown in Fig. E.1. However, the difference is not sufficient to significantly alter our conclusion on the mismatch between the observed velocity dispersions and the predictions with EFE at the average d(mean).
This means that, to explain our results with the nominal values of the stellar mass-to-light ratios, the EFE should be almost fully screened for the UDGs residing inside clusters. This is actually also the case in some hybrid versions of MOND such as the superfluid DM theory (Khoury 2015; Berezhiani & Khoury 2015; Berezhiani et al. 2018, 2019). As discussed in detail in Sect. IX.B of Berezhiani et al. (2018), the superfluid core would be rather small in galaxy clusters (of the order of a few hundreds kpc at most) and no EFE would be expected for cluster UDGs, contrary to the case of satellite galaxies orbiting within the superfluid core of their host, where the EFE would be expected to be similar to the MOND case.
Isolated Point Masses
For two significant point masses with nothing else nearby, self-interactions cause the system to reduce from a three dimensional one to a flux tube causing the force between them to remain nearly constant without regard to distance.
If the mass is confined to a disk, the self-interactions cause the system to reduce from a three dimensional one to a two dimensional one, causing the force to have a 1/r form that we see in the MONDian regime of spiral galaxies.
In the geometries where Deur's approach approximate's MOND, the following formula approximate's the self-interaction term:
FG = GNM/r2 + c2(aπGNM)1/2/(2√2)r
where FG is the effective gravitational force, GN is Newton's constant, c is the speed of light, M is ordinary baryonic mass of the gravitational source, r is the distance between the source mass and the place that the gravitational force is measured, and a is a physical constant that is the counterpart of a0 in MOND (that should in principle be possible to derive from Newton's constant) which is equal to 4*10−44 m−3s2.
Thus, the self-interaction term that modifies is proportionate to (GNM)1/2/r. So, it is initially much smaller that the first order Newtonian gravity term, but it declines more slowly than the Newtonian term until it is predominant.Spherically Symmetric MassesIf the mass is spherically symmetric, the self-interactions cancel out and the system remains three dimensional causing the force to have the 1/r2 form that we associate with Newtonian gravity.
Why do galactic clusters have so much more apparent dark matter than spiral galaxies?Because geometrically, they are closer to the two point particle scenario, in which galaxies within the cluster are the point particles that exert a distance independent force upon each other (analogous to flux tubes in QCD), rather than being spherically symmetric or disk-like.Why does the Bullet Cluster behave as it does?Since gas dominates the visible mass of a cluster, the observation that most of the total (dark) mass did not stay with the gas appears to rule out modifications of gravity as an alternative to dark matter. But, actually, this isn't the case in a self-interacting graviton scenario.Because it has a gaseous component that is more or less spherically symmetric, that component has little apparent dark matter, while the galaxy components, which come close to the two point mass flux tube paradigm which is equivalent to a great amount of inferred dark matter. So, the gaseous portion and the core galaxy components are offset from each other. The apparent dark matter tracks the galaxy cores and not the interstellar gas medium between them.
A good place to review this analysis from the source is A. Deur, “Implications of Graviton-Graviton Interaction to Dark Matter” (May 6, 2009) (published at 676 Phys. Lett. B 21 (2009)).
The Makeup Of The Coma Cluster
Applying Deur's analysis to the Coma Cluster requires more of an understanding of its geometry that is commonly considered relevant. Here is what Wikipedia has to say about the Coma Cluster (emphasis mine):
The Coma Cluster (Abell 1656) is a large cluster of galaxies that contains over 1,000 identified galaxies. Along with the Leo Cluster (Abell 1367), it is one of the two major clusters comprising the Coma Supercluster. It is located in and takes its name from the constellation Coma Berenices.The cluster's mean distance from Earth is 99 Mpc (321 million light years). Its ten brightest spiral galaxies have apparent magnitudes of 12–14 that are observable with amateur telescopes larger than 20 cm. The central region is dominated by two supergiant elliptical galaxies: NGC 4874 and NGC 4889. The cluster is within a few degrees of the north galactic pole on the sky. Most of the galaxies that inhabit the central portion of the Coma Cluster are ellipticals. Both dwarf and giant ellipticals are found in abundance in the Coma Cluster.As is usual for clusters of this richness, the galaxies are overwhelmingly elliptical and S0 galaxies, with only a few spirals of younger age, and many of them probably near the outskirts of the cluster.The full extent of the cluster was not understood until it was more thoroughly studied in the 1950s by astronomers at Mount Palomar Observatory, although many of the individual galaxies in the cluster had been identified previously.The Coma Cluster is one of the first places where observed gravitational anomalies were considered to be indicative of unobserved mass. In 1933 Fritz Zwicky showed that the galaxies of the Coma Cluster were moving too fast for the cluster to be bound together by the visible matter of its galaxies. Though the idea of dark matter would not be accepted for another fifty years, Zwicky wrote that the galaxies must be held together by "...some dunkle Materie."About 90% of the mass of the Coma cluster is believed to be in the form of dark matter. The distribution of dark matter throughout the cluster, however, is poorly constrained. . . .
The Coma cluster contains about 800 galaxies within a 100 x 100 arc-min area of the celestial sphere.
NASA in an explanation of a Hubble image of the Coma cluster has more to say (emphasis mine):
The Hubble's Advanced Camera for Surveys viewed a large portion of the cluster, spanning several million light-years across. The entire cluster contains thousands of galaxies in a spherical shape more than 20 million light-years in diameter.Also known as "Abell 1656," the Coma Cluster is more than 300 million light-years away. The cluster, named after its parent constellation Coma Berenices, is near the Milky Way's north pole. This places the Coma Cluster in an area unobscured by dust and gas from the plane of the Milky Way, and easily visible by Earth viewers.Most of the galaxies that inhabit the central portion of the Coma Cluster are ellipticals. These featureless "fuzz-balls" are pale goldish brown in color and contain populations of old stars. Both dwarf, as well as giant ellipticals, are found in abundance in the Coma Cluster.Farther out from the center of the cluster are several spiral galaxies. These galaxies have clouds of cold gas that are giving birth to new stars. Spiral arms and dust lanes "accessorize" these bright bluish-white galaxies that show a distinctive disk structure.In between the ellipticals and spirals is a morphological class of objects known as S0 (S-zero) galaxies. They are made up of older stars and show little evidence of recent star formation, however, they do show some assemblage of structure -- perhaps a bar or a ring, which may give rise to a more disk-like feature.
The Coma cluster has a volume of about 4.2*10^21 cubic light years, for a mean density of about one galaxy per 4.2*10^18 light years (although much higher towards the core and lower at the fringes), suggesting a mean separation of on the order of one to a few million light years from each other (by comparison the Milky Way galaxy has a radius of about 50,000 light years).
Applying Deur's Analysis of the Coma cluster
If Deur's analysis is right, the predominantly elliptical galaxies of the Coma cluster have little apparently internal dark matter (which MOND would also predict) but unlike MOND wouldn't be producing abnormally large external to the galaxy fields either.
The external diffuse gravitational fields of the 1000 or so galaxies in the Coma cluster may also largely cancel each other out since they are approximately spherically symmetric and have so many sources. So, there may indeed, be less of a net external gravitational field on ultra diffuse galaxies in clusters than one might expect (particularly in a MOND regime).
Instead, the apparent dark matter phenomena in clusters like the Coma cluster might largely arise from flux tubes of enhanced gravity in a spiderweb of enhanced gravitational lines between galaxies that are close to each other within the cluster.
We explore the Baryonic Tully-Fisher Relation in the Local Group. Rotationally supported Local Group galaxies adhere precisely to the relation defined by more distant galaxies. For pressure supported dwarf galaxies, we determine the scaling factor that relates their observed velocity dispersion to the equivalent circular velocity of rotationally supported galaxies of the same mass such that . For a typical mass-to-light ratio in the -band, we find that . More generally, . This provides a common kinematic scale relating pressure and rotationally supported dwarf galaxies.
We present measurements of the radial gravitational acceleration around isolated galaxies, comparing the expected gravitational acceleration given the baryonic matter with the observed gravitational acceleration, using weak lensing measurements from the fourth data release of the Kilo-Degree Survey.
These measurements extend the radial acceleration relation (RAR) by 2 decades into the low-acceleration regime beyond the outskirts of the observable galaxy. We compare our RAR measurements to the predictions of two modified gravity (MG) theories: MOND and Verlinde's emergent gravity.
We find that the measured RAR agrees well with the MG predictions.
In addition, we find a difference of at least 6σ between the RARs of early- and late-type galaxies (split by Sérsic index and u−r colour) with the same stellar mass. Current MG theories involve a gravity modification that is independent of other galaxy properties, which would be unable to explain this behaviour. The difference might be explained if only the early-type galaxies have significant (Mgas≈M∗) circumgalactic gaseous haloes. The observed behaviour is also expected in ΛCDM models where the galaxy-to-halo mass relation depends on the galaxy formation history.
We find that MICE, a ΛCDM simulation with hybrid halo occupation distribution modelling and abundance matching, reproduces the observed RAR but significantly differs from BAHAMAS, a hydrodynamical cosmological galaxy formation simulation. Our results are sensitive to the amount of circumgalactic gas; current observational constraints indicate that the resulting corrections are likely moderate. Measurements of the lensing RAR with future cosmological surveys will be able to further distinguish between MG and ΛCDM models if systematic uncertainties in the baryonic mass distribution around galaxies are reduced.
Using a sample of 67 galaxies from the MIGHTEE Survey Early Science data we study the HI-based baryonic Tully-Fisher relation (bTFr), covering a period of ∼one billion years (0≤z≤0.081). We consider the bTFr based on two different rotational velocity measures: the width of the global HI profile and V(out), measured as the outermost rotational velocity from the resolved HI rotation curves.
Both relations exhibit very low intrinsic scatter orthogonal to the best-fit relation (σ⊥=0.07±0.01), comparable to the SPARC sample at z≃0. The slopes of the relations are similar and consistent with the z≃0 studies (3.66+0.35−0.29 for W50 and 3.47+0.37−0.30 for V(out)).
We find no evidence that the bTFr has evolved over the last billion years, and all galaxies in our sample are consistent with the same relation independent of redshift and the rotational velocity measure. Our results set up a reference for all future studies of the HI-based bTFr as a function of redshift that will be conducted with the ongoing deep SKA pathfinders surveys.
We present a novel 2D flux density model for observed HI emission lines combined with a Bayesian stacking technique to measure the baryonic Tully-Fisher relation below the nominal detection threshold. We simulate a galaxy catalogue, which includes HI lines described either with Gaussian or busy function profiles, and HI data cubes with a range of noise and survey areas similar to the MeerKAT International Giga-Hertz Tiered Extragalactic Exploration (MIGHTEE) survey.
With prior knowledge of redshifts, stellar masses and inclinations of spiral galaxies, we find that our model can reconstruct the input baryonic Tully-Fisher parameters (slope and zero point) most accurately in a relatively broad redshift range from the local Universe to z=0.3 for all the considered levels of noise and survey areas, and up to z=0.55 for a nominal noise of 90μJy/channel over 5 deg2. Our model can also determine the M(HI)−M⋆ relation for spiral galaxies beyond the local Universe, and account for the detailed shape of the HI emission line, which is crucial for understanding the dynamics of spiral galaxies. Thus, we have developed a Bayesian stacking technique for measuring the baryonic Tully-Fisher relation for galaxies at low stellar and/or HI masses and/or those at high redshift, where the direct detection of HI requires prohibitive exposure times.
We carry out a test of the radial acceleration relation (RAR) for a sample of 10 dynamically relaxed and cool-core galaxy clusters imaged by the Chandra X-ray telescope, which was studied in Giles et al. For this sample, we observe that the best-fit RAR shows a very tight residual scatter equal to 0.09 dex. We obtain an acceleration scale of 1.59×10^−9m/s^2, which is about an order of magnitude higher than that obtained for galaxies. Furthermore, the best-fit RAR parameters differ from those estimated from some of the previously analyzed cluster samples, which indicates that the acceleration scale found from the RAR could be of an emergent nature, instead of a fundamental universal scale.