Scale-Invariant Dynamics in the Solar System 3

arXiv:2007.00654v1 [astro-ph.CO] 1 Jul 2020 uy2020 July 2 h ag yotcSre eecp ( Telescope Survey Synoptic Large The INTRODUCTION 1

MNRAS e oehrlast nossece navreyo scales of galax- variety holds a it on inconsistencies assuming to matter, leads dark together to ies rise give might author other several by luminous shown (e.g. their the also as to and distributions, applied galaxies matter gravity of Newtonian curves of rotation predictions actual the between cies ie fteMlyWy( ( Way experiments satel- Milky for terrestrial spheroidal dwarf searches, the on sensitive of data Fermi in lites of particles years detec- 11 matter null in dark instance continued any by of challenged tion physi is particle of interpretation model This standard parti- well-tested hypothetical the of in consist not cles to have would matter ( dark matte galaxy The dark each to surrounding attributed haloes usually are discrepancies eration is ( what astronomy of establishing mysteries work great 1970 the observational of Observatory her one Rubin still of C. Vera honour the in renamed been recently has nrnlBanik Indranil System Solar the in dynamics Scale-invariant c 2 1 hre nvriy srnmclIsiue aut fMa of Faculty Institute, Astronomical University, Charles emot-ntttf Helmholtz-Institut 00TeAuthors The 2020 ∗ Email: eadeso h yohtclpril hsc that physics particle hypothetical the of Regardless .Terwr eeldvr ag yaia discrepan- dynamical large very revealed work Their ). acc 1939 Babcock 000 [email protected] , 1 [email protected] – 5 22)Pern uy22 oplduigMRSL MNRAS using Compiled 2020 July 2 Preprint (2020) ; rSrhe n enhsk(IK) nvriyo on Nus Bonn, of University (HISKP), Kernphysik und Strahlen ur ¨ osa hsa 1972 Shostak & Rogstad 1 ∗ i ta.2017 al. et Liu ofe l 2020 al. et Hoof n ae Kroupa Pavel and fteErhMo ri sicmail ihlnrlsrrnigdt a data ranging laser lunar pred with the incompatible that is show orbit We Earth-Moon expand. the to of orbits near-Keplerian two-body ag fsae,ee fi stertclywl formulated. observ well by theoretically words: significance is Key high it at if purely falsified even is is scales, freedom c of model of not range SID degree is the extra t it Relat that the Moreover, General of conclude beyond that SID. age physics arguing in conventional new works old contains the previous similarly truly than be SID whether to older clear expected much old significantly is it be however w must make which star object would giant any red This low-mass of theory. a mass true, gravitating If the past. the that predicts SID Moreover, akmatter dark ABSTRACT uogiv22) I mle htteediscrepancies these that implies SID discrepancies dynamical 2020). observed Gueorguiev the & for explanation possible a as ehnc on–saevehicles space – Moon – mechanics srkr&Pels1973 Peebles & Ostriker PvlKroupa) (Pavel Idai Banik) (Indranil h oain cl-nain yais(I)ter a recently has theory (SID) dynamics scale-invariant covariant The ). vz´ ta.2019 al. Ivezić et n nsensitive in and ) ui Ford & Rubin .Sc accel- Such ). − rs nta rmannsadr eoiydpnetfretha force velocity-dependent non-standard a from instead arise rvtto akmte peeie srmtyadcelestia and Astrometry – ephemerides – matter dark – gravitation 1 , hmtc n hsc,VHoleˇsoviˇckách Prah V 00 Physics, CZ-180 and 2, thematics 2 cs. ). r s ) oaincreadteNwoingaiainlfield gravitational Newtonian the and curve rotation rtdb h ayncdsrbto ( distribution baryonic the by erated agn rmhnrd omlin fpres( parsecs of millions to hundreds from ranging akeeg density energy dark acceleration the between relation unique 2015 uharda ceeainrlto RR a predicted was Dynamics Newtonian (RAR) Modified (MOND, relation using earlier acceleration decades several radial a Such mass eeaindsrpnisflo oermral regulari remarkable ( some follow discrepancies celeration ya ceeaindpnec ftegaiylaw gravity the provide of instead dependence are acceleration matter an dark by to attributed usually fects uvs( curves ttoa edstrength field itational ia nrydniyo rvttoa ed(equa- field gravitational a of of 9 density tion energy sical OD(rMlrma yais introduces dynamics) Milgromian (or MOND aetlaclrto cl fntr eo hc h de- Em- the significant. becomes which pirically, dynamics below Newtonian nature from of viation scale acceleration damental aay&MGuh2012 McGaugh & Famaey eakby hsi prxmtl hr h clas- the where approximately is this Remarkably, .Telnetsuidpolmi htteglci ac- galactic the that is problem studied longest The ). M eea ta.1991 al. et Begeman rniin rmteNewtonian the from transitions a igo 1983 Milgrom g 0 1 = ees1981 Peters = ale14 sallee p . 2 GMa × r 10 − g 6 -31 on Germany Bonn, D-53115 16, − .I hsmdl h yaia ef- dynamical the model, this In ). 0 tdistance at u 10 eoe oprbet the to comparable becomes ) Λ htcnb umrzda a as summarized be can that ) for m/s ; cag 2011 McGaugh ≡ − 2 ,g g, omnyatiue to attributed commonly omthglx rotation galaxy match to ρ N r Λ cag ta.2016 al. et McGaugh c rma sltdpoint isolated an from ≪ 2 nglxe (Maeder galaxies in rta nstandard in than er smc mle in smaller much as ahmtcl We mathematical. vt,wt several with ivity, g a GM/r htconvention- that A 0 ce expansion icted nerdfo the from inferred T . tosars a across ations enproposed been E ). tl l v3.0 file style X ,CehRepublic Czech 8, a eUniverse, he rua2012 Kroupa a 2 − 0 t ompletely a to law safun- a as > h grav- the causes t g N 200 gen- ties (1) σ ). d l . , 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 & Zlo´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). Our conclusions are given field, a consequence of the non-linear MOND equations in Section 4. (Milgrom 1986). However, including the Galactic external field did not much improve the agreement with observations (Derakhshani & Haghi 2014). MOND can also explain the 2 SOLAR SYSTEM EPHEMERIDES velocity dispersion of the ultra-diffuse galaxy Dragonfly 2 (DF2, Kroupa et al. 2018b), DF4 (Haghi et al. 2019a), and SID has some success reproducing the RAR (section 5 of DF44 (B´ılek et al. 2019), with the latter galaxy having a Maeder & Gueorguiev 2020). This comes about because of measured dispersion profile that is consistent with MOND a non-standard velocity-dependent force which causes a slow at 2.40σ (Haghi et al. 2019b). The calculations for DF2 and outwards expansion of a two-body near-Keplerian orbit (see DF4 are significantly affected by the EFE. DF44 is more their equation 26). This has always been an important part isolated than the almost gas-free DF2 (Chowdhury 2019; of the SID theory (Maeder 1978). As shown in equation 68 Sardone et al. 2019) and indeed has a higher internal velo- of Maeder & Bouvier (1979), the extra term causes a near-

MNRAS 000, 1–5 (2020) Scale-invariant dynamics in the Solar System 3

r r ˙ ′′ 2 circular orbit of radius r to expand as Model ˙tide,mm/yr ˙SID , mm/yr Ω, /century . r Standard 38 0 −25 74 r˙SID = , (3) SID 28 10 −19.42 t Observed Total = 38.05 ± 0.04 −25.80 ± 0.03 where t is the time since the Big Bang andq ˙ dq for any ≡ dt Table 1. The observed lunar recession rate is partly caused quantity q. The subscript indicates a theoretical expec- SID by tides raised on the Earth. It might contain an additional tation. In this contribution, it will be important to measure term predicted by SID, but if so, the effect of tides must be re- time from the Big Bang since the dynamical equations of SID duced since the total is tightly constrained by lunar laser ranging are not invariant under a translation of the time variable. (Folkner et al. 2014). Therefore, the scenarios predict different Since the expansion rate of the SID universe is angular deceleration rates for the lunar orbit (Equation 4). The similar to the Lambda-Cold Dark Matter (ΛCDM) observed rate comes from their section 3c. standard cosmological paradigm (section 4.1 of Maeder & Gueorguiev 2020), we assume that currently The conventional expectation is thatr ˙SID = 0 and t = 13.8 Gyr (Planck Collaboration XIII 2016). Thus, the ′′ 2 r˙tide = 38 mm/yr, implying that Ω˙ = 25.74 /century . 9.58 AU orbit of Saturn should expand atr ˙SID = 104 m/yr. − Over a decade, the expansion should easily be detectable If instead the precisely observedr ˙obs consists ofr ˙tide = 10 mm/yr andr ˙SID = 28 mm/yr, we should observe that in Cassini radio tracking data given that its accuracy is ˙ ′′ 2 32 m (Viswanathan et al. 2017). However, they found Ω = 19.42 /century . These possibilities are summarized ≈ − no significant deviation of Saturn from its conventionally in Table 1. calculated trajectory. It is difficult to see how a constant Space age observations of the Moon tell us that Ω˙ = 25.80 0.03′′/century2 (section 3c of Folkner et al. outwards expansion could be masked by e.g. changing the − ± masses of other planets. 2014). This could be affected by oscillatory perturbations from other planets, but fortunately a much longer base- line is available if we also consider pre-telescopic records 2.1 Lunar laser ranging of Solar and lunar eclipses over the past 2700 years (Stephenson & Morrison 1995). This yields an estimated Ω˙ Applying Equation 3 to the Earth-Moon distance of of 26 1′′/century2, with the error derived from their r = 3.84 108 m, we get thatr ˙ = 28 mm/yr. − ± 2 × SID estimated timing uncertainty of 0.9 s/century (their sec- Maeder & Bouvier (1979) predicted that this effect should tion 5a) and the fact that changing Ω˙ by 0.5′′/century2 be observable in “sufficiently accurate observations with affects the eclipse timings by 0.46 s/century2 (their sec- laser reflectors” (see their section 7). Nowadays, the Earth- tion 2b). Impressively, this section of their work found that Moon distance is constrained to mm accuracy by lunar Ω˙ = 26.0′′/century2 fits the ancient records much better − ′′ 2 laser ranging (LLR, Adelberger et al. 2017). The Moon is than Ω˙ = 26.2 /century , correctly anticipating subse- − indeed receding from the Earth, but at a rate ofr ˙obs = quent refinements to the modern value of Ω.˙ 38.05 0.04 mm/yr (section 3c of Folkner et al. 2014). This ˙ ± Reconciling this estimate with SID requires changing Ω is caused by the tides it raises in Earth’s oceans. These by ∆Ω˙ = 6′′/century2. The SID-predicted angular deceler- tidal bulges are carried ahead of the sub-lunar point be- ation of the Earth is only 0.94′′/century2, so SID-induced cause Earth rotates much faster than the Moon orbits it. changes in its orbital period have little effect on this dis- Folkner et al. (2014) estimated that the effect of these tides crepancy. Over T = 2000 years, the discrepancy amounts to is known to an accuracy of 0.5% or 0.2 mm/yr, making 2 ˙ ≈ a shift in the angular position of the Moon by T ∆Ω/2 = this the limit to any unconventional contributions tor ˙. 1200′′. Since the Moon takes 27.3 days to orbit the Earth, Nonetheless, we consider the possibility that this calcu- it would take 36 minutes to rotate through 1200′′ . Ancient lation is seriously in error such that the tidal contribution astronomers were well capable of noticing such a large time tor ˙obs is onlyr ˙tide = 10 mm/yr, with SID contributing the difference, especially since some eclipses occurred close to remaining 28 mm/yr in line with Equation 3. The totalr ˙ sunrise or sunset. Although there is some degeneracy be- would then be consistent with ther ˙obs measured by LLR. tween Ω˙ and changes in the Earth’s rotation rate, the time of An important aspect of the SID contribution is that day provides a tight constraint on the latter. The very fact orbital velocities v do not change even though the radius that an eclipse occurred tightly constrains the lunar orbit r t (Equation 3). Since v2 = GM/r for a circular orbit ∝ while the positions of background stars constrain Earth’s around any object with mass M, its gravitational parameter orbit around the Sun. In this way, careful observations and 1 GM t in this model. Thus, in the absence of tides, SID record-keeping can break the various degeneracies involved. ∝ −1 predicts that the Moon’s orbital angular frequency Ω t ∝ Therefore, the SID theory is falsified at extremely high sig- (equation 71 of Maeder & Bouvier 1979). However, if the nificance by both modern and ancient observations. lunar orbit expands at the same rate due to tides in a conventional context, Kepler’s Third Law implies that Ω t−3/2. ∝ Combining the tidal and SID contributions, we get that 3 TIME-VARYING G AND STELLAR 3 Ω˙ r˙SID + r˙tide EVOLUTION = 2 . (4) Ω − r We have seen that SID predicts the gravitational parameter GM t. If this is caused by changes in M, then the masses ∝ 1 This partially cancels the orbit expansion due to the non- of fundamental particles would need to grow. In particular, standard velocity-dependent force, which would otherwise be changing the electron mass t would cause a similar change ∝ twice as much. in the energy of hydrogen spectral lines, i.e. the Rydberg

MNRAS 000, 1–5 (2020) 4 Indranil Banik & Pavel Kroupa constant would change. For much heavier elements, the ex- Looking beyond the Earth-Moon system, SID predicts pected shift is not exactly the same due to special relativistic that the orbit of Saturn expands at 104 m/yr (Section 2). corrections that become more important for atoms orbiting This probably violates constraints from the Cassini mission a more charged nucleus. This allowed Karshenboim & Peik since the ranging accuracy is 32 m and the orbiter func- ≈ (2008) to place stringent constraints on possible time evo- tioned for over a decade (table 11 of Viswanathan et al. lution of the Rydberg constant, completely ruling out the 2017). Unlike oscillatory perturbations from other planets, possibility that it grows t. SID predicts a constant outwards expansion that is difficult ∝ Thus, the SID model requires G t, implying a present to mask by e.g. changing the mass of Jupiter. −11 ∝ value of G/G˙ = 7.2 10 /yr. This very slow change In addition to predicting that orbits expand, SID also × is in significant tension with the precision pulsar timing implies significant time variation of any object’s gravita- constraint from PSR J0437-4715 (G/G˙ = ( 0.5 1.8) tional parameter GM (Section 3). Since changing the masses −11 − ± × 10 /yr, Verbiest et al. 2008). of fundamental particles would violate precise laboratory Over a Hubble time, the much larger expected change constraints, the most plausible interpretation is that the in G would significantly alter stellar evolution since the core change occurs solely in the gravitational constant G. This pressure and temperature depend sensitively on G. As a re- would substantially affect stellar evolution because any an- sult, its expected variation is in strong tension with astroseis- cient star would have fused hydrogen at a much lower rate mic observations of the ancient star KIC 7970740 (G/G˙ = in the early universe, implying low-mass red giants must be (1.2 2.6) 10−12/yr, Bellinger & Christensen-Dalsgaard significantly older than in standard theory. However, the SID ± × 2019). model also implies the age of the universe is similar to that More generally, since SID does not much change the age in ΛCDM (section 4.1 of Maeder & Gueorguiev 2020). of the universe, it would require significant revision to our In addition to these observational difficulties, it is not understanding of main sequence turnoffs in the oldest glob- completely clear that SID is in fact distinct from classi- ular clusters (VandenBerg et al. 2013; Correnti et al. 2018). cal General Relativity the extra degree of freedom may − We consider this unlikely because the nuclear processes in be a mathematical artefact without any physical effects stars are rather well understood and the turnoff stars have (Tsamis & Woodard 1986; Jackiw & Pi 2015). In any case, a mass of 0.8M⊙, making them only slightly less massive SID is unable to satisfy Solar System constraints and con- ≈ than the Sun. tradicts stellar astrophysics to a substantial extent. There- Any time variation of G would also change the Chan- fore, some other way should be found to explain the im- drasekhar mass G−3/2 (Gaztañaga et al. 2002). This pressive successes of Milgromian dynamics in a covariant ∝ would affect the luminosities of Type Ia supernovae, allowing framework. The recently proposed relativistic MOND theory observations of them to set limits on G˙ . Based on this idea, of Skordis & Zlo´snik (2019) may be a significant step in this the analysis of Mould & Uddin (2014) constrained G/G˙ to direction. the range ( 3, 7.3) 10−11/yr, marginally consistent with − × −11 the SID expectation of 7.2 10 /yr. Future improvements × to this technique could yield more stringent constraints. ACKNOWLEDGEMENTS IB is supported by an Alexander von Humboldt postdoc- toral research fellowship. The authors are grateful to Andre 4 CONCLUSIONS Maeder for carefully presenting his work at the Bonn Grav- The SID cosmological model claims to successfully repro- ity2019 conference. They also thank Richard Woodard and duce the acceleration-dependent pattern of dynamical dis- Moti Milgrom for helpful discussions, and the referee for crepancies in galaxies (the RAR, Maeder & Gueorguiev comments which helped to clarify this letter. 2020). This is due to an extra term in the equations, which in the Solar System implies that the Moon’s orbit around the Earth should expand at a rate ofr ˙SID = 28 mm/yr DATA AVAILABILITY (Section 2.1). Such an expansion of two-body orbits has The data underlying this article are available in the article. always been part of SID (Maeder 1978; Maeder & Bouvier 1979), with the latter work explicitly suggesting that this prediction should be tested with laser ranging. 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