Journal of the Korean Astronomical Society preprint - no DOI assigned 00: 1 ∼ 4, 2015 June pISSN: 1225-4614 · eISSN: 2288-890X The Korean Astronomical Society (2015) http://jkas.kas.org MILGROM'S LAW AND Λ'S SHADOW: HOW MASSIVE GRAVITY CONNECTS GALACTIC AND COSMIC DYNAMICS Sascha Trippe Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Korea; [email protected] Received March 11, 2015; accepted June 2, 2015 Abstract: Massive gravity provides a natural solution for the dark energy problem of cosmology and is also a candidate for resolving the dark matter problem. I demonstrate that, assuming reasonable scaling relations, massive gravity can provide for Milgrom’s law of gravity (or “modified Newtonian dynamics”) which is known to remove the need for particle dark matter from galactic dynamics. Milgrom’s law comes with a characteristic acceleration, Milgrom’s constant, which is observationally constrained to a0 1.1 10−10 ms−2. In the derivation presented here, this constant arises naturally from the cosmologically≈ × required mass of gravitons like a0 c√Λ cH0√3ΩΛ, with Λ, H0, and ΩΛ being the cosmological constant, the Hubble constant, and the∝ third cosmological∝ parameter, respectively. My derivation suggests that massive gravity could be the mechanism behind both, dark matter and dark energy. Key words: gravitation — cosmology — dark matter — dark energy 1. INTRODUCTION On smaller scales, the dynamics of galaxies is in ex- cellent agreement with a modification of Newtonian dy- Modern standard cosmology suffers from two criti- namics (the MOND paradigm) in the limit of small ac- cal issues: the dark matter problem and the dark celeration (or gravitational field strength) g, expressed energy problem. Canonical ΛCDM cosmology (e.g., in Milgrom’s law Bahcall et al. 1999) requires about 95% (e.g., Ade et al. 2014) of the mass/energy content of the universe to be g = µ(g/a ) g (1) provided by exotic dark components for which physi- N 0 cal counterparts have not been identified. Dark matter (Milgrom 1983a) with gN being the Newtonian field and dark energy are commonly assumed to be unre- strength, a0 denoting Milgrom’s constant, and µ be- lated: dark matter is identified with as yet undiscovered ing a transition function with µ = 1 for g/a0 1 and new exotic elementary particles, whereas dark energy is ≫ µ = g/a0 for g/a0 1. Such a modification implies linked to a cosmological constant, Λ, which is inserted that the ratio of dynamical≪ and luminous masses of a into Einstein’s field equations and which triggers an ac- stellar system, the mass discrepancy Mdyn/M0, exceeds celerated expansion of the universe on its largest scales. unity and becomes a function of acceleration (or field ΛCDM cosmology is based on the assumption that strength) when assuming Newton’s law of gravity. In gravitation is Einsteinian on all scales. Modified the- the “deep MOND” limit g/a0 1, the circular speed ories of gravity might be able to describe the universe ≪ vc of stars in rotation-supported dynamical systems (es- without dark components; this idea has initiated a vast pecially disk galaxies) is number of works exploring multiple models of gravita- tion (see Clifton et al. 2012 for an exhaustive review). 4 vc = GM0 a0 (2) A quantum-field theoretical modified theory of gravity, first proposed by Fierz & Pauli (1939), is massive grav- with G being Newton’s constant (Milgrom 1983a,b,c); arXiv:1506.00730v1 [astro-ph.CO] 2 Jun 2015 ity in which gravitation is mediated by virtual bosons, 2 2 2 note that g = vc /r = GMdyn/r and gN = GM0/r for gravitons, that have a very small (compared to all other circular orbits with radius r. For pressure-supported elementary particles) non-zero mass. Massive gravi- systems (especially elliptical galaxies and galaxy clus- tons, being virtual exchange particles, have a limited ters) with 3D velocity dispersion σ, the mass–velocity life time governed by Heisenberg’s uncertainty relation relation takes the form for energy and time. If the mass of the graviton is 4 4 such that its life time is 1/Λ, the decay of gravity σ = GM0 a0 (3) induces an accelerated expansionp of the universe like 9 the one actually observed – thus providing an elegant (Milgrom 1984, 1994). Equations (1, 2, 3), with resolution of the dark energy problem (recently, e.g., −10 −2 a0 1.1 10 ms , naturally provide for the Cardone et al. 2012; Clifton et al. 2012; Hinterbichler ≈ × 2012; Volkov 2012; Tasinato et al. 2013; De Felice et al. fundamental scaling laws of galactic kinematics, 2013; de Rham 2014). specifically the baryonic Tully–Fisher, baryonic Faber–Jackson, mass discrepancy–acceleration Corresponding author: S. Trippe (MDA), and surface mass density–acceleration 1 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 dark matter 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 speed of light, the cosmological constant, 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 dark energy. 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.