• Trippe, Sascha
  • Received : 2015.03.11
  • Published : 2015.06.30


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 , 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.


gravitation;cosmology;dark matter;dark energy


  1. Trippe, S. 2013, Can Massive Gravity Explain the Mass Discrepancy–Acceleration Relation of Disk Galaxies?, JKAS, 46, 133
  2. Tasinato, G., Koyama, K., & Niz, G. 2013, Exact Solutions in Massive Gravity, Class. Quantum Grav., 30, 184002
  3. Trippe, S. 2013, A Simplified Treatment of Gravitational Interaction on Galactic Scales, JKAS, 46, 41
  4. Trippe, S. 2013, A Derivation of Modified Newtonian Dynamics, JKAS, 46, 93
  5. Trippe, S. 2014, The ‘Missing Mass Problem’ in Astronomy and the Need for a Modified Law of Gravity, Z. Naturforsch. A, 69, 173
  6. Trippe, S. 2015, The “Graviton Picture”: a Bohr Model for Gravitation on Galactic Scales?, Can. J. Phys., 93, 213
  7. Vainshtein, A. I. 1972, To the Problem of Non-Vanishing Gravitation Mass, Phys. Lett. B, 39, 393
  8. van Putten, M. H. P. M. 2014, Galaxy Rotation Curves in de Sitter Space, arXiv:1411.2665
  9. van Putten, M. H. P. M. 2015, Accelerated Expansion from Cosmological Holography, MNRAS, 450, L48
  10. Volkov, M. S. 2012, Cosmological Solutions with Massive Gravitons in the Bigravity Theory, J. High Energy Phys., 2012, 35
  11. Walker, M. G., & Loeb, A. 2014, Is the Universe Simpler than ΛCDM?, Contemp. Phys., 55, 198
  12. Wu, X., & Kroupa, P. 2015, Galactic Rotation Curves, the Baryon-to-Dark-Halo-Mass Relation and Space-Time Scale Invariance, MNRAS, 446, 330
  13. Xu, D., Sluse, D., Gao, L., et al. 2015, How Well Can Dark-Matter Substructures Account for the Observed Radio Flux-Ratio Anomalies?, MNRAS, 447, 3189
  14. Milgrom, M. 1983, A Modification of the Newtonian Dynamics: Implications for Galaxies, ApJ, 270, 371
  15. McGaugh, S. S. 2005, Balance of Dark and Luminous Mass in Rotating Galaxies, Phys. Rev. Lett., 95, 171302
  16. McGaugh, S. S. 2011, Novel Test of Modified Newtonian Dynamics with Gas Rich Galaxies, Phys. Rev. Lett., 106, 121303
  17. Milgrom, M. 1983, A Modification of the Newtonian Dynamics as a Possible Alternative to the Hidden Mass Hypothesis, ApJ, 270, 365
  18. Milgrom, M. 1983, A Modification of the Newtonian Dynamics: Implications for Galaxy Systems, ApJ, 270, 384
  19. Milgrom, M. 1984, Isothermal Spheres in the Modified Dynamics, ApJ, 287, 571
  20. Milgrom, M. 1994, Modified Dynamics Predictions Agree with Observations of the HI Kinematics in Faint Dwarf Galaxies Contrary to the Conclusions of Lo, Sargent, and Young, ApJ, 429, 540
  21. Milgrom, M. 2015, MOND Theory, Can. J. Phys. 93, 107
  22. Milgrom, M., & Sanders, R. H. 2008, Rings and Shells of “Dark Matter” as MOND Artifacts, ApJ, 678, 131
  23. Rhee, M.-H. 2004, On the Physical Basis of the Tully-Fisher Relation, JKAS, 37, 15
  24. Rhee, M.-H. 2004, Mass-to-Light Ratio and the Tully- Fisher Relation, JKAS, 37, 91
  25. Sanders, R. H. 1994, A Faber–Jackson Relation for Clusters of Galaxies: Implications for Modified Dynamics, A&A, 284, L31
  26. Sanders, R. H. 2010, The Universal Faber–Jackson Relation, MNRAS, 407, 1128
  27. Fierz, M., & Pauli, W. 1939, On Relativistic Wave Equations for Arbitrary Spin in an Electromagnetic Field, Proc. R. Soc. London A, 173, 211
  28. De Felice, A., G¨umr¨ukc¨uoglu, A. E., Lin, C., & Mukohyama, S. 2013, On the Cosmology of Massive Gravity, Class. Quantum Grav., 30, 184004
  29. de Rham, C. 2014, Massive Gravity, Living Rev. Relativ., 17, 7
  30. Famaey, B., & McGaugh, S. S. 2012, Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions, Living Rev. Relativ., 15, 10
  31. Gentile, G., Famaey, B., & de Blok, W. J. G. 2011, THINGS about MOND, A&A, 527, A76
  32. Griffiths, D. 2008, Introduction to Elementary Particles, 2nd edn. (Weinheim: Wiley-VCH)
  33. Hinterbichler, K. 2012, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys., 84, 671
  34. Kroupa, P. 2012, The Dark Matter Crisis: Falsification of the Current Standard Model of Cosmology, PASA, 29, 395
  35. Kroupa, P. 2015, Galaxies as Simple Dynamical Systems: Observational Data Disfavor Dark Matter and Stochastic Star Formation, Can. J. Phys., 93, 169
  36. Lee, J., Kim, S., & Rey, S.-C. 2015, A New Dynamical Mass Estimate for the Virgo Cluster Using the Radial Velocity Profile of the Filament Galaxies, arXiv:1501.07064
  37. McGaugh, S. S. 2004, The Mass Discrepancy–Acceleration Relation: Disk Mass and the Dark Matter Distribution, ApJ, 609, 652
  38. McGaugh, S. S. 2005, The Baryonic Tully–Fisher Relation of Galaxies with Extended Rotation Curves and the Stellar Mass of Rotating Galaxies, ApJ, 632, 859
  39. Ade, P. A. R., et al. 2014, Planck 2013 Results. XVI. Cosmological Parameters, A&A, 571, A16
  40. Avilez-Lopez, A., Padilla, A., Saffin, P. M., & Skordis, C. 2015, The Parameterized Post-Newtonian-Vainshteinian Formalism, arXiv:1501.01985
  41. Bahcall, N. A., Ostriker, J. P., Perlmutter, S., & Steinhardt, P. J. 1999, The Cosmic Triangle: Revealing the State of the Universe, Science, 284, 1481
  42. Babichev, E., & Deffay, C. 2013, An Introduction to the Vainshtein Mechanism, Class. Quantum Grav., 30, 184001
  43. Baker, T., Psaltis, D., & Skordis, C. 2015, Linking Tests of Gravity on All Scales: From the Strong-Field Regime to Cosmology, ApJ, 802, 63
  44. Cardone, V. F., Angus, G., Diaferio, A., et al. 2011, The Modified Newtonian Dynamics Fundamental Plane, MNRAS, 412, 2617
  45. Cardone, V. F., Radicella, N., & Parisi, L. 2012, Constraining Massive Gravity with Recent Cosmological Data, Phys. Rev. D, 85, 124005
  46. Chae, K.-H., & Gong, I.-T. 2015, Testing Modified Newtonian Dynamics through Statistics of Velocity Dispersion Profiles in the Inner Regions of Elliptical Galaxies, arXiv:1505.02936
  47. Clifton, T., Ferreira, P. G., Padilla, A., & Skordis, C. 2012, Modified Gravity and Cosmology, Phys. Rep., 513, 1