Foundations of Physics

, Volume 49, Issue 1, pp 63–82 | Cite as

Cosmological Density Perturbations in Newtonian- and MONDian Gravity Scenario: A Symmetry-Based Approach

  • Amitava ChoudhuriEmail author
  • Aritra Ganguly


We investigate the evolution of linear density contrasts obtained with respect to a homogeneous spatially flat Friedman-Lemaître–Robertson–Walker (FLRW) background by solving the density contrast equations governed by Newtonian and MONDian force laws using symmetry-based approach. We find eight-parameter Lie group symmetries for the linear order density perturbation equation for the Newtonian case whereas the density contrast equation follows only one parameter Lie group symmetry in MONDian case. We use Lie symmetries to find the group invariant solutions from invariant curve condition. The physical features of the evolution for various mode of density contrast with respect to the global cosmic background density in homogeneous isotropic cosmological models have been investigated using analytical group invariant solutions along with their numerical solutions. An account for cosmological density contrast and mass fluctuation also have been provided. We also have shown that the MONDian force law generates higher amplitudes in the density fluctuation, results in a more rapid structure formation that cannot be possible under the Newtonian force law.


Cosmological density perturbation Newtonian- and MONDian gravity Lie group symmetries Structure formation 



AC acknowledges UGC, The Government of India, for financial support through Project No. F.30-302/2016(BSR).


  1. 1.
    Spergel, D.N., Verde, L., Peiris, H.V., Komatsu, E., Nolta, M.R., Bennett, C.L., Halpern, M., Hinshaw, G., Jarosik, N., Kogut, A., Limon, M., Meyer, S.S., Page, L., Tucker, G.S., Weiland, J.L., Wollack, E., Wright, E.L.: First year Wilkinson Microwave Probe (WMAP) observations: determination of cosmological parameters. Astrophys. J. Suppl. Ser. 148, 175 (2003)ADSCrossRefGoogle Scholar
  2. 2.
    Sachs, R.K., Wolfe, A.M.: Perturbations of a cosmological model and angular variations of the microwave background. Astrophys. J. 147, 73 (1967)ADSCrossRefGoogle Scholar
  3. 3.
    Jeans, J.H.: The stability of spiral nebula. Philos. Trans. 199A, 49 (1902)Google Scholar
  4. 4.
    Jeans, J.: Astronomy and Cosmogony. Cambridge University Press, Cambridge (1929)zbMATHGoogle Scholar
  5. 5.
    Lifshitz, E.M.: On the gravitational instability of the expanding universe. JETP 16, 987 (1946)zbMATHGoogle Scholar
  6. 6.
    Weinberg, S.: Cosmology. Oxford University Press Inc., New York (2008)zbMATHGoogle Scholar
  7. 7.
    Peebles, P.J.E.: The Large-Scale Structure of the Universe. Princeton Series in Physics. Princeton University Press, Princeton (1980)Google Scholar
  8. 8.
    Zel’dovich, Ya B., Novikov, I.D.: Relativistic astrophysics. I. Usp. Fiz. Nauk 84, 377 (1965)Google Scholar
  9. 9.
    Sakharov, A.D.: The initial stage of an Expanding Universe and Appearance of a Nonuniform Distribution of Matter. ZhETF 49, 345 (1965); translation in JETP Lett. 22, 241 (1966)Google Scholar
  10. 10.
    Guth, A.H.: In: Freedman, W.L. (ed.) Measuring and Modeling the Universe. Carnegie Observatories Astrophysics Series, vol. 2. Cambridge University Press, Cambridge (2004)Google Scholar
  11. 11.
    Moffat, J.W.: Scalar–tensor–vector gravity theory. JCAP 0603, 004 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Skordis, C., Mota, D.F., Ferreira, P.G., Boehm, C.: Large scale structure in Bekensteins theory of relativistic modified newtonian dynamics. Phys. Rev. Lett. 96, 011301 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    McGaugh, S.S.: A tale of two paradigms: the mutual incommensurability of \(\Lambda CDM\) and MOND. Can. J. Phys. 93, 250 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    Milgrom, M.: A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. J. Astrophys. 270, 365 (1983)ADSCrossRefGoogle Scholar
  15. 15.
    Sanders, R.H., McGaugh, S.S.: Modified Newtonian dynamics as an alternative to dark matter. Annu. Rev. Astron. Astrophys. 40, 263 (2002)ADSCrossRefGoogle Scholar
  16. 16.
    McGaugh, S.S., de Blok, E.: High-resolution rotation curves of low surface brightness galaxies. I. Data. Astrophys. J. 499, 66 (1998)ADSCrossRefGoogle Scholar
  17. 17.
    Sanders, R.H.: Clusters of galaxies with modified Newtonian dynamics. Mon. Not. R. Astron. Soc. 342, 901 (2003)ADSCrossRefGoogle Scholar
  18. 18.
    Pointecouteau, E., Silk, J.: New constraints on modified Newtonian dynamics from galaxy clusters. Mon. Not. R. Astron. Soc. 364, 654 (2005)ADSCrossRefGoogle Scholar
  19. 19.
    Fabris, J.C., Velten, H.E.S.: MOND virial theorem applied to a galaxy cluster. Br. J. Phys. 39, 592 (2009)ADSCrossRefGoogle Scholar
  20. 20.
    Clowe, D.: A direct empirical proof of the existence of dark matter. Astrophys. J. Lett 648, L109 (2006)ADSCrossRefGoogle Scholar
  21. 21.
    Milgrom, M.: MOND Particularly as Modified Inertia. (2011). arXiv:1101.5122v1
  22. 22.
    Calmet, X., Kuntz, I.: What is modified gravity and how to differentiate it from particle dark matter? (2017). arXiv:1702.03832v2
  23. 23.
    Milgrom, M.: MOND theory. (2014). arXiv:1404.7661v2
  24. 24.
    Scarpa, R.: Modified Newtonian Dynamics, an Introductory Review. astro-ph/0601478 (2006)Google Scholar
  25. 25.
    Nusser, A.: Modified Newtonian dynamics of large-scale structure. Mon. Not. R. Astron. Soc. 331, 909 (2002)ADSCrossRefGoogle Scholar
  26. 26.
    Nusser, A., Pointecouteau, E.: Modeling the formation of galaxy clusters in MOND. Mon. Not. R. Astron. Soc. 366, 969 (2006)ADSCrossRefGoogle Scholar
  27. 27.
    Olver, P.J.: Applications of Lie Groups to Differential equations. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  28. 28.
    Weinberg, S.: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley & Sons, Inc., New York (1972)Google Scholar
  29. 29.
    Bonnor, W.B.: Jeans’ formula for gravitational instability. Mon. Not. R. Astron. Soc. 117, 104 (1957)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sanders, R. H.: Cluster of galaxies with modified Newtonian dynamics (MOND). (2002). arXiv:astro-ph/0212293v1
  31. 31.
    Famaey, B., McGaugh, S.S.: Modified Newtonian dynamics (MOND): observational phenomenology and relativistic extension. Living Rev. Relativ. 15, 10 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Milgrom, M.: New physics at low accelerations (MOND): an alternative to dark matter. (2010). arXiv:0912.2678v2

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsThe University of BurdwanBardhamanIndia

Personalised recommendations