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Fierz , exhaustive by of an proposed theory first for modified 2012 theoretical quantum-field al. gravita- A et of Clifton models (see multiple vast tion universe exploring a the works initiated has of describe idea number to the- this able components; Modified dark be without might scales. gravity all of on ories Einsteinian is gravitation scales. largest its on universe ac- the an of triggers expansion which celerated and equations field Einstein’s into h oenAtooia oit (2015) Society Astronomical Korean The oensadr omlg uesfo w criti- two the from suffers issues: cosmology cal standard Modern ora fteKra srnmclSociety Astronomical 00 Korean the of Journal e xtceeetr atce,weesdr nryis a energy unre- to dark linked whereas be undiscovered particles, yet to elementary as exotic with assumed identified new is commonly matter dark are matter lated: energy Dark physi- dark identified. which been and for not be have components counterparts to dark universe cal exotic the al. of by et content provided Ade mass/energy (e.g., the 95% of about requires 2014) 1999) al. et Bahcall energy orsodn author: Corresponding CMcsooyi ae nteasmto that assumption the on based is cosmology ΛCDM 1 : nwihgaiaini eitdb ita bosons, virtual by mediated is gravitation which in eateto hsc n srnm,SolNtoa Univer National Seoul Astronomy, and Physics of Department Abstract: htmsiegaiycudb h ehns eidbt,dr matt dark both, behind mechanism words: the Key be param could cosmological gravity third massive the that and constant, Hubble the constant, eurdms fgaioslike gravitons of mass required hc skont eoetene o atcedr atrfo gala from matter dark particle for which constant, need Milgrom’s the acceleration, 1 (or characteristic gravity remove a of with to law demonstr comes known I Milgrom’s for problem. is provide matter which can dark gravity the massive resolving relations, for candidate a also . ∼ 1 rbe.CnnclΛD omlg (e.g., cosmology ΛCDM Canonical problem. × ,21 June 2015 4, hthv eysal(oprdt l other all to (compared small very a have that , 10 M omlgclconstant cosmological − 10 ILGROM s m asv rvt rvdsantrlslto o h akeeg pr energy dark the for solution natural a provides gravity Massive akmatter dark .I 1. rvtto omlg akmte akenergy dark — matter dark — cosmology — gravitation − 2 NTRODUCTION ntedrvto rsne ee hscntn rssnaturally arises constant this here, presented derivation the In . C .Trippe S. ONNECTS p ' S 1 / L ,tedcyo gravity of decay the Λ, rbe n the and problem WAND AW ,wihi inserted is which Λ, , eevdMrh1,21;acpe ue2 2015 2, June accepted 2015; 11, March Received a 0 G ∝ asv grav- massive LCI AND ALACTIC c √ Λ Λ ' S ∝ acaTrippe Sascha dark S cH HADOW 0 √ 1 3Ω MA,adsraems density–acceleration mass discrepancy–acceleration surface baryonic and kinematics, mass Tully–Fisher, galactic (MDA), baryonic of the Faber–Jackson, laws specifically scaling fundamental Mlrm18,19) qain 1 ,3,with 3), 2, (1, Equations 1994). a 1984, (Milgrom eln gemn ihamdfiaino etna dy- Newtonian (the of namics modification a with agreement cellent strength, with 1983a) (Milgrom eeain(rgaiainlfil strength) field gravitational (or celeration iclrobt ihradius with orbits circular in eaintksteform the dispersion takes clus- velocity relation galaxy 3D and with galaxies ters) elliptical (especially systems oethat note with n rniinfnto with function transition a ing eilyds aais is galaxies) disk pecially v tegh hnasmn etnslwo rvt.In gravity. limit field of MOND” (or law “deep acceleration Newton’s the of assuming function when a strength) becomes and unity µ httertoo yaia n uiosmse fa of luminous and the dynamical system, of stellar ratio the that c 0 Λ iy eu 5-4,Korea; 151-742, Seoul sity, nsalrsae,tednmc fglxe si ex- in is galaxies of dynamics the scales, smaller On = fsasi oainspotddnmclsses(es- systems dynamical rotation-supported in stars of igo’ law Milgrom’s ihΛ, with , C ≈ H : g/a G OSMIC 1 en etnscntn Mlrm1983a,b,c); (Milgrom constant Newton’s being . 0 1 tr epciey ydrvto suggests derivation My respectively. eter, OW g a for 0 × = ODparadigm MOND sosrainlycntandto constrained observationally is scaling reasonable assuming that, ate denoting H 10 v g/a raddr energy. dark and er c 0 2 M n Ω and , − /r mdfidNwoindynamics”) Newtonian “modified D IS:1225-4614 pISSN: 10 0 asdiscrepancy mass ASSIVE tcdnmc.Mlrmslaw Milgrom’s dynamics. ctic YNAMICS = σ ≪ g s m v 4 N GM c 4 = = .Sc oicto implies modification a Such 1. igo’ constant Milgrom’s − = be fcsooyadis and cosmology of oblem g Λ 2 N 9 4 µ g/a dyn aual rvd o the for provide naturally , M G rpit-n O assigned DOI no - preprint rmtecosmologically the from ( en h cosmological the being M G [email protected] en h etna field Newtonian the being g/a r /r o pressure-supported For . 0 ntelmto ml ac- small of limit the in ) G 0 µ ≪ 2 0 0 a ) for 1 = and a RAVITY 0 g σ ,tecrua speed circular the 1, 0 http://jkas.kas.org h mass–velocity the , · M g IS:2288-890X eISSN: N dyn = g/a /M and , g GM expressed , 0 0 a exceeds , ≫ 0 0 /r ≈ and 1 µ 2 be- (3) (1) (2) for 2 Trippe relations, plus the asymptotic flattening of rota- the cosmic expansion history requires Tg 1/Λ, re- ≈ tion curves and the occurrence of “dark rings” in sulting in p galaxy clusters (Sanders 1994, 2010; Rhee 2004a,b; ~ mg √Λ . (8) McGaugh 2004, 2005a,b; Milgrom & Sanders 2008; ≈ c2 McGaugh 2011; Gentile et al. 2011; Cardone et al. According to the standard “cosmic triangle” formalism 2 2011; Famaey & McGaugh 2012; Trippe 2014; (e.g., Bahcall et al. 1999), Λ = 3H0 ΩΛ, resulting in Walker & Loeb 2014; Wu & Kroupa 2015; Milgrom ~ 2015; Chae & Gong 2015). Milgrom’s law eliminates mg 2 H0 3ΩΛ ; (9) particle from galactic dynamics together ≈ c p −1 −1 with its substantial difficulties (cf., e.g., Kroupa 2012, for H0 70kms Mpc and ΩΛ 0.7 (e.g., ≈ −35 −2 ≈ 2015). Ade et al. 2014), Λ 1.1 10 s and thus mg As pointed out by Trippe (2013a,b,c, 2015), massive 4 10−69 kg 2 10≈−33 eV×c−2. ≈ gravity can, at least in principle, provide Milgrom’s law ×A natural choice≈ × for a quantum-physical mass scale, together with an expression for the function µ in agree- x in our case, is the Planck mass ment with observations. Following up on this ansatz, I investigate the link of galactic dynamics to the cosmic ~ c x mP = (10) expansion history. A physical connection is established ≡ r G by the graviton mass mg √Λ. Milgrom’s constant is with G being Newton’s constant; combination of Equa- ∝ −61 given by a0 c√Λ cH0√3ΩΛ, with c,Λ, H0, and tions (9) and (10) leads to mg/mP 2 10 . The ∝ ∝ ≈ × ΩΛ being the , the , choice for the scale y is less obvious; for our purpose, we the (present-day) Hubble constant, and the (present- require a scale characteristic for gravitationally bound day) third cosmological parameter, respectively. Ac- dynamical systems. A good choice (to be discussed in cordingly, massive gravity might offer a natural alter- Section 3) is the magnitude of the gravitational poten- native explanation for the phenomena conventionally tial, associated with dark matter and . GMdyn ǫ = (11) rc2 2. CALCULATIONS (cf. also Baker et al. 2015). As ǫ is dimensionless, we Massive gravity implies (Trippe 2013a,b,c, 2015) that can supply y with the unit of a length by multiplying ǫ any luminous (“baryonic”) mass M0 is the source of a and the Planck length spherical graviton halo with mass density ~ G −2 lP = 3 (12) ρg(R)= mg ng(R)= mg ηM0 R (4) r c so that y ǫl . where m is the graviton mass, n is the graviton parti- P g g Inserting≡ Equations (8, 10, 11, 12) into Equation (6) cle density, R is the radial coordinate, and η is a scaling −2 leads to factor. The proportionality ρg M0R follows from ∝ M r2 (i) consistency with the classical force law in the New- dyn =1+4 πfc √Λ . (13) tonian limit, and (ii) the inverse-square-of-distance law M0 GMdyn of flux conservation (but see also the discussion in Sec- Using the classical expression for gravitational field tion 3). For a circular orbit of radius r around M , the 0 strength, g = GM /r2, one finds total, dynamical mass experienced by a test particle dyn follows from integrating ρg(R) from 0 to r, M c √Λ dyn =1+4 πf (14) M0 g Mdyn = M0 + Mg = M0 (1+4 πηmg r) (5) or, using the expression given by Equation (9), with Mg being the mass contributed by the graviton halo. One can bring this expression into the more in- Mdyn cH0 √3ΩΛ =1+4 πf . (15) tuitive form M0 g M m r dyn =1+4 πf g (6) Comparison of Equations (14, 15) to the expres- M0 x y sion following from the “simple µ function” (e.g., Famaey & McGaugh 2012) of MOND, where f is a dimensionless number, x is a mass scale, and y is a length scale (with η f/(xy)). Mdyn a0 ≡ =1+ , (16) With gravitons being virtual exchange particles, their M0 g effective mass can be estimated from (cf., e.g., Griffiths shows that the expressions are equivalent, with Mil- 2008) Heisenberg’s uncertainty relation for energy and grom’s constant being given by a =4πfc√Λ. Match- time, 0 2 ing this expression with the empirical value a0 1.1 mgc Tg ~ (7) −10 −2 ≈ × × ≈ 10 ms requires a (universal) value of f 0.0088 ≈ with Tg being the life time of gravitons and ~ being the (or 4πf 1/9). It is straightforward to see that in the ≈ reduced form of Planck’s constant. Consistency with limit g a0, Equation (16) leads to Equation (2). ≪ How Massive Gravity Connects Galactic and Cosmic Dynamics 3

3. DISCUSSION sive gravity – which has not been found yet. Es- pecially, it will be necessary to connect the classical As demonstrated in Section 2, massive gravity can derivation of Equation (16) with and be used to link the cosmological constant to a spe- the resulting deviations from Newtonian gravity in the cific version of Milgrom’s law, namely the one com- strong-field limit (g a0). A complete theory should prising the “simple µ function”. The transition func- also provide for the≫ basic properties of gravitons, es- tion given by Equation (16), with a0 = (1.06 0.05) pecially the absence of graviton–graviton interactions −10 −2 ± × 10 ms , is in excellent agreement with the observed which is required for the validity of the classical force mass discrepancy–acceleration relation of disk galax- law (as used in Equation (4); cf., Trippe 2013a – but ies (Trippe 2013c). I thus present a physical mecha- see also Vainshtein 1972; Babichev & Deffayet 2013; nism that provides astrophysically meaningful expres- Avilez-Lopez et al. 2015). Last but not least, a com- sions for both µ and Milgrom’s constant. My derivation plete theory of massive gravity should also comprise implies that dark matter and dark energy could be un- non-dynamical effects that are currently unexplained in derstood as phenomena arising from the same effect – the frame of general relativity; possible candidates are the non-zero mass and thus finite life time of gravitons. the “radio flux ratio anomaly” affecting the images of This mechanism also explains why a0 cH0 as already multiply gravitationally lensed quasars (e.g., Xu et al. ∼ noted by Milgrom (1983a). I also note that a simi- 2015, and references therein) and the discrepancy be- lar approach has been discussed by van Putten (2014) tween mass estimates based on galactic dynamics and in the specific context of de-Sitter space (but see also those based on gravitational lensing found recently for van Putten 2015). filaments of the Virgo cluster of galaxies (Lee et al. The result expressed in Equation (14) critically de- 2015). pends on the choice of the scales x and y. Identifying x with the Planck mass is a natural choice. Identification 4. SUMMARY AND CONCLUSIONS of y with the magnitude of the gravitational potential, I explore a possible physical connection between dark ǫ, (multiplied with the Planck length) is suggested by matter and dark energy. Assuming that gravitation is the properties of gravity on astronomical scales. As mediated by virtual gravitons with non-zero mass (mas- pointed out recently by Baker et al. (2015), astrophysi- sive gravity), the resulting limited life time of gravi- cal gravitating systems are characterized completely by tons provides for a decay of gravity on cosmological their location in a two-dimensional plane spanned by (i) scales and thus an accelerated expansion of the universe Kretschmann scalar the parameter ǫ and (ii) the which (“dark energy”); the graviton mass follows from the cos- corresponds to the magnitude of the Riemann 2 mological constant like mg ~√Λ/c via Heisenberg’s tensor (thus quantifying the local strength of spacetime ≈ curvature in the frame of general relativity) and which uncertainty relation. Massive gravity also implies that any luminous (“baryonic”) mass is surrounded by a √ 3 2 takes the form ξ = 48 GMdyn/(r c ) for spherical sys- (electromagnetically invisible) halo of gravitons that tems. However, ξ can be expressed in units of the cos- 2 contributes additional mass (“dark matter”). Assum- mic curvature, Λ/c ; thus the curvature (or at least a ing reasonable scaling relations for such graviton halos, specific value thereof) already appears in Equation (8), one recovers Milgrom’s law of gravity and finds that which suggests the introduction of ǫ (multiplied with Milgrom’s constant is a = 4πfc√Λ=4πfcH √3Ω the Planck length for dimensional reasons) as a new 0 0 Λ with f 0.9% (or 4πf 1/9). scaling parameter. Conveniently, my choice of x and ≈ ≈ y cancels out the constant ~ coming with the gravi- My derivation suggests that dark matter and dark energy could be interpreted as two effects arising from ton mass (Equation (8)) because mP lP = ~/c. The dimensionless factor f arises from the need to provide the same physical mechanism: massive gravity. This a normalization factor for the graviton halo density in would imply a natural connection between the dynam- Equation (4) of the form η f/(xy); I emphasize that ics of galaxies and the dynamics of the universe. f is the only free parameter≡ left in the derivation be- ACKNOWLEDGMENTS cause neither mP nor lP nor ǫ are tunable. For mass density distributions, scaling factors 0

Bahcall, N. A., Ostriker, J. P., Perlmutter, S., & Steinhardt, 121303 P. J. 1999, The Cosmic Triangle: Revealing the State of Milgrom, M. 1983, A Modification of the Newtonian Dy- the Universe, Science, 284, 1481 namics as a Possible Alternative to the Hidden Mass Hy- Babichev, E. & Deffayet, C. 2013, An Introduction to pothesis, ApJ, 270, 365 the Vainshtein Mechanism, Class. Quantum Grav., 30, Milgrom, M. 1983, A Modification of the Newtonian Dy- 184001 namics: Implications for Galaxies, ApJ, 270, 371 Baker, T., Psaltis, D., & Skordis, C. 2015, Linking Tests of Milgrom, M. 1983, A Modification of the Newtonian Dy- Gravity on All Scales: From the Strong-Field Regime to namics: Implications for Galaxy Systems, ApJ, 270, 384 Cosmology, ApJ, 802, 63 Milgrom, M. 1984, Isothermal Spheres in the Modified Dy- Cardone, V. F., Angus, G., Diaferio, A., et al. 2011, The namics, ApJ, 287, 571 Modified Newtonian Dynamics Fundamental Plane, MN- Milgrom, M. 1994, Modified Dynamics Predictions Agree RAS, 412, 2617 with Observations of the H i Kinematics in Faint Dwarf Cardone, V. F., Radicella, N., & Parisi, L. 2012, Constrain- Galaxies Contrary to the Conclusions of Lo, Sargent, and ing Massive Gravity with Recent Cosmological Data, Young, ApJ, 429, 540 Phys. Rev. D, 85, 124005 Milgrom, M. 2015, MOND Theory, Can. J. Phys. 93, 107 Chae, K.-H. & Gong, I.-T. 2015, Testing Modified New- Milgrom, M. & Sanders, R. H. 2008, Rings and Shells of tonian Dynamics through Statistics of Velocity Disper- “Dark Matter” as MOND Artifacts, ApJ, 678, 131 sion Profiles in the Inner Regions of Elliptical Galaxies, Rhee, M.-H. 2004, On the Physical Basis of the Tully–Fisher arXiv:1505.02936 Relation, JKAS, 37, 15 Clifton, T., Ferreira, P. G., Padilla, A., & Skordis, C. 2012, Rhee, M.-H. 2004, Mass-to-Light Ratio and the Tully– Modified Gravity and Cosmology, Phys. Rep., 513, 1 Fisher Relation, JKAS, 37, 91 De Felice, A., G¨umr¨ukc¨uoglu, A. E., Lin, C., & Mukohyama, Sanders, R. H. 1994, A Faber–Jackson Relation for Clusters S. 2013, On the Cosmology of Massive Gravity, Class. of Galaxies: Implications for Modified Dynamics, A&A, Quantum Grav., 30, 184004 284, L31 de Rham, C. 2014, Massive Gravity, Living Rev. Relativ., Sanders, R. H. 2010, The Universal Faber–Jackson Relation, 17, 7 MNRAS, 407, 1128 Famaey, B. & McGaugh, S. S. 2012, Modified Newtonian Tasinato, G., Koyama, K., & Niz, G. 2013, Exact Solutions Dynamics (MOND): Observational Phenomenology and in Massive Gravity, Class. Quantum Grav., 30, 184002 Relativistic Extensions, Living Rev. Relativ., 15, 10 Trippe, S. 2013, A Simplified Treatment of Gravitational Fierz, M. & Pauli, W. 1939, On Relativistic Wave Equations Interaction on Galactic Scales, JKAS, 46, 41 for Arbitrary in an Electromagnetic Field, Proc. R. Trippe, S. 2013, A Derivation of Modified Newtonian Dy- Soc. London A, 173, 211 namics, JKAS, 46, 93 Gentile, G., Famaey, B., & de Blok, W. J. G. 2011, THINGS Trippe, S. 2013, Can Massive Gravity Explain the Mass about MOND, A&A, 527, A76 Discrepancy–Acceleration Relation of Disk Galaxies?, Griffiths, D. 2008, Introduction to Elementary Particles, JKAS, 46, 133 2nd edn. (Weinheim: Wiley-VCH) Trippe, S. 2014, The ‘Missing Mass Problem’ in Astronomy Hinterbichler, K. 2012, Theoretical Aspects of Massive and the Need for a Modified Law of Gravity, Z. Natur- Gravity, Rev. Mod. Phys., 84, 671 forsch. A, 69, 173 Kroupa, P. 2012, The Dark Matter Crisis: Falsification of Trippe, S. 2015, The “Graviton Picture”: a Bohr Model for the Current of Cosmology, PASA, 29, Gravitation on Galactic Scales?, Can. J. Phys., 93, 213 395 Vainshtein, A. I. 1972, To the Problem of Non-Vanishing Kroupa, P. 2015, Galaxies as Simple Dynamical Systems: Gravitation Mass, Phys. Lett. B, 39, 393 Observational Data Disfavor Dark Matter and Stochastic van Putten, M. H. P. M. 2014, Galaxy Rotation Curves in Star Formation, Can. J. Phys., 93, 169 , arXiv:1411.2665 Lee, J., Kim, S., & Rey, S.-C. 2015, A New Dynamical Mass van Putten, M. H. P. M. 2015, Accelerated Expansion from Estimate for the Virgo Cluster Using the Radial Velocity Cosmological Holography, MNRAS, 450, L48 Profile of the Filament Galaxies, arXiv:1501.07064 Volkov, M. S. 2012, Cosmological Solutions with Massive McGaugh, S. S. 2004, The Mass Discrepancy–Acceleration Gravitons in the Bigravity Theory, J. High Energy Phys., Relation: Disk Mass and the Dark Matter Distribution, 2012, 35 ApJ, 609, 652 Walker, M. G. & Loeb, A. 2014, Is the Universe Simpler McGaugh, S. S. 2005, The Baryonic Tully–Fisher Relation than ΛCDM?, Contemp. Phys., 55, 198 of Galaxies with Extended Rotation Curves and the Stel- Wu, X. & Kroupa, P. 2015, Galactic Rotation Curves, lar Mass of Rotating Galaxies, ApJ, 632, 859 the Baryon-to-Dark-Halo-Mass Relation and Space-Time McGaugh, S. S. 2005, Balance of Dark and Luminous Mass Scale Invariance, MNRAS, 446, 330 in Rotating Galaxies, Phys. Rev. Lett., 95, 171302 Xu, D., Sluse, D., Gao, L., et al. 2015, How Well Can Dark- McGaugh, S. S. 2011, Novel Test of Modified Newtonian Matter Substructures Account for the Observed Radio Dynamics with Gas Rich Galaxies, Phys. Rev. Lett., 106, Flux-Ratio Anomalies?, MNRAS, 447, 3189