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Scale-Invariant Dynamics in the Solar System 3 MNRAS 000, 1–5 (2020) Preprint 2 July 2020 Compiled using MNRAS LATEX style file v3.0 Scale-invariant dynamics in the Solar System Indranil Banik1∗ and Pavel Kroupa1,2 1 Helmholtz-Institut fur¨ Strahlen und Kernphysik (HISKP), University of Bonn, Nussallee 14−16, D-53115 Bonn, Germany 2Charles University, Astronomical Institute, Faculty of Mathematics and Physics, V Holeˇsoviˇck´ach 2, CZ-180 00 Praha 8, Czech Republic 2 July 2020 ABSTRACT The covariant scale-invariant dynamics (SID) theory has recently been proposed as a possible explanation for the observed dynamical discrepancies in galaxies (Maeder & Gueorguiev 2020). SID implies that these discrepancies − commonly attributed to dark matter − arise instead from a non-standard velocity-dependent force that causes two-body near-Keplerian orbits to expand. We show that the predicted expansion of the Earth-Moon orbit is incompatible with lunar laser ranging data at > 200σ. Moreover, SID predicts that the gravitating mass of any object was much smaller in the past. If true, a low-mass red giant star must be significantly older than in standard theory. This would make it much older than the conventional age of the Universe, which however is expected to be similarly old in SID. Moreover, it is not completely clear whether SID truly contains new physics beyond General Relativity, with several previous works arguing that the extra degree of freedom is purely mathematical. We conclude that the SID model is falsified at high significance by observations across a range of scales, even if it is theoretically well formulated. Key words: gravitation – dark matter – ephemerides – Astrometry and celestial mechanics – Moon – space vehicles 1 INTRODUCTION ranging from hundreds to millions of parsecs (Kroupa 2012, 2015). The longest studied problem is that the galactic ac- The Large Synoptic Survey Telescope (Ivezi´cet al. 2019) celeration discrepancies follow some remarkable regularities has recently been renamed the Vera C. Rubin Observatory (Famaey & McGaugh 2012) that can be summarized as a in honour of her observational work establishing what is unique relation between the acceleration g inferred from the still one of the great mysteries of astronomy (Rubin & Ford rotation curve and the Newtonian gravitational field g gen- 1970). Their work revealed very large dynamical discrepan- N erated by the baryonic distribution (McGaugh et al. 2016). cies between the actual rotation curves of galaxies and the Such a radial acceleration relation (RAR) was predicted predictions of Newtonian gravity applied to their luminous several decades earlier using Modified Newtonian Dynamics matter distributions, as also shown by several other authors (MOND, Milgrom 1983). In this model, the dynamical ef- (e.g. Babcock 1939; Rogstad & Shostak 1972). Such accel- arXiv:2007.00654v1 [astro-ph.CO] 1 Jul 2020 fects usually attributed to dark matter are instead provided eration discrepancies are usually attributed to dark matter by an acceleration dependence of the gravity law the grav- haloes surrounding each galaxy (Ostriker & Peebles 1973). − itational field strength g at distance r from an isolated point The dark matter would have to consist of hypothetical parti- 2 mass M transitions from the Newtonian GM/r law to cles not in the well-tested standard model of particle physics. This interpretation is challenged by continued null detec- pGMa0 g = for g,g a . (1) tion of any dark matter particles in sensitive searches, for r N ≪ 0 instance in 11 years of Fermi data on dwarf spheroidal satel- MOND (or Milgromian dynamics) introduces a as a fun- lites of the Milky Way (Hoof et al. 2020) and in sensitive 0 damental acceleration scale of nature below which the de- terrestrial experiments (Liu et al. 2017). viation from Newtonian dynamics becomes significant. Em- Regardless of the hypothetical particle physics that −10 2 pirically, a = 1.2 10 m/s to match galaxy rotation might give rise to dark matter, assuming it holds galax- 0 × curves (Begeman et al. 1991; McGaugh 2011). ies together leads to inconsistencies on a variety of scales Remarkably, this is approximately where the clas- sical energy density of a gravitational field (equa- ∗Email: [email protected] (Indranil Banik) tion 9 of Peters 1981) becomes comparable to the 2 [email protected] (Pavel Kroupa) dark energy density u ρ c that convention- Λ ≡ Λ c 2020 The Authors 2 Indranil Banik & Pavel Kroupa ally explains the accelerating expansion of the Uni- city dispersion despite a similar baryonic distribution. In a verse (Efstathiou et al. 1990; Ostriker & Steinhardt 1995; MOND context, the EFE was recently confirmed at high sig- Riess et al. 1998; Perlmutter et al. 1999). nificance based on the relative velocities of wide binary stars 2 in the Solar neighbourhood these are inconsistent with g − < uΛ g . 2πa0 . (2) MOND without the Galactic EFE (Pittordis & Sutherland 8πG ⇔ 2019). The more relevant case with the EFE is quite similar In regions of space with such a weak gravitational field, the to the Newtonian case, though a more careful analysis could dominant contribution to the energy density is uΛ , which distinguish them in the near future (Banik & Zhao 2018; is often identified with the quantum-mechanical zero point Banik 2019). Recent progress with hydrodynamic MOND energy density of the vacuum. If this is correct, poorly un- simulations indicates that it can naturally form exponential derstood quantum gravity effects could well be rather im- disc galaxies out of a collapsing gas cloud (Wittenburg et al. portant to such regions but these are totally neglected − 2020) and produce realistic morphologies for the interacting in General Relativity. MOND could be an empirical way to Antennae galaxies (Renaud et al. 2016). include such effects, as suggested by the coincidence of scales Most of these observations only became apparent in Equation 2 (e.g. Milgrom 1999; Pazy 2013; Verlinde 2016; long after the MOND field equation was first published Smolin 2017). (Bekenstein & Milgrom 1984), making these achievements Regardless of its underlying microphysical explanation, successful a priori predictions. It is difficult to explain the MOND can accurately match the rotation curves of a wide success of these predictions in a conventional gravity con- variety of spiral galaxies across a vast range in mass, surface text, even with the observational facts in hand (Desmond brightness, and gas fraction using only the distribution of 2017a,b; Ghari et al. 2019). In particular, section 4.2 of the luminous matter (figure 5 of Lelli et al. 2017). Fits to in- latter work showed that it is still rather difficult to get a dividual rotation curves show that intrinsic scatter about tight RAR despite a diversity of rotation curve shapes at its predictions must be < 13% and is consistent with 0 fixed peak velocity (Oman et al. 2015). (Li et al. 2018). A few discrepant galaxies were claimed by Despite its successes, the theoretical underpinnings of Rodrigues et al. (2018), but it was later shown that the ac- MOND remain unclear. An important clue may be that in tual discrepancies are either very mild or arise when the the deep-MOND limit, the dynamics of gravitational sys- distance is particularly uncertain and could plausibly be tems become scale invariant scaling the spacetime co- − outside the range they allow (Kroupa et al. 2018a). MOND ordinates by some factor λ yields another valid solution can also explain the rotation curves of elliptical galaxies, at to the equations of motion (Milgrom 2009). In principle, least when these can be measured in a sub-dominant rotating λ could be time-dependent. This is at the basis of the fully disc of neutral hydrogen (figure 8 of Lelli et al. 2017). relativistic scale invariant dynamics theory (SID, Maeder The successes of MOND extend beyond near-circular 1978; Maeder & Bouvier 1979). SID applies the Weyl Inte- orbital motion in the non-relativistic regime. A rela- grable Geometry (Canuto et al. 1977) and reproduces some tivistic version of MOND has recently been developed of the MOND phenomenology, in particular a tight RAR in which gravitational waves propagate at the speed (Maeder & Gueorguiev 2020). There are some differences in its predicted form when g . 0.01a while MOND pre- of light (Skordis & Z lo´snik 2019), consistent with the N 0 − dicts that g √gN a0 , SID predicts that g flattens out at near-simultaneous detection of the gravitational wave → 0.1a (see their figure 1). This leads to potentially testable event GW170817 and its electromagnetic counterpart ≈ 0 (Virgo & LIGO Collaborations 2017). Galactic globular consequences in low surface brightness dwarf galaxies. clusters can be used to test MOND in dispersion-dominated In this contribution, we avoid discussing possible the- systems (Baumgardt et al. 2005; Haghi et al. 2011). In this oretical issues with SID, which may in fact be identical to regard, NGC 2419 was claimed to be problematic for MOND classical General Relativity as the extra degree of freedom (Ibata et al. 2011a,b). It was later shown that the disagree- may be a mathematical artefact (Tsamis & Woodard 1986; ment could be caused by plausible systematic uncertain- Jackiw & Pi 2015). We assume that there is genuinely new ties like a mild departure from spherical symmetry, rota- physics in SID and focus on its high-acceleration limit, where tion within the sky plane, or a radially-dependent poly- SID has some unusual consequences. tropic index (Sanders 2012a,b). In MOND, we also ex- After introducing the SID model and its context (Sec- pect NGC 2419 to be affected by the ‘external field effect’ tion 1), we discuss how it can be constrained by Solar System (EFE) whereby the internal gravity binding a system is ephemerides (Section 2). We also consider what SID implies weakened when it is embedded in an external gravitational for stellar evolution (Section 3).
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