Imposing Neumann boundary condition on cosmological perturbation equations and trajectories of particles

  • Hossein Shenavar
Original Article


We impose Neumann boundary condition to solve cosmological perturbation equations and we derive a modified Friedmann equation and a new lensing equation. To check the new lensing equation and the value of Neumann constant, a sample that contains ten strong lensing systems is surveyed. Except for one lens, masses of the other lenses are found to be within the constrains of the observational data. Furthermore, we argue that by using the concept of geometrodynamic clocks it is possible to modify the equation of motion of massive particles too. Also, a sample that includes 101 HSB and LSB galaxies is used to re-estimate the value of the Neumann constant and we found that this value is consistent with the prior evaluation from Friedmann and lensing equations. Finally, the growth of structure is studied by a Newtonian approach which resulted in a more rapid rate of the structure formation in matter dominated era.


Cosmological perturbation Neumann BC Lensing equation Measuring space-time Marzke-Wheeler clocks 



I would like to thank the authors Mu-Chen Chiu, Chung-Ming Ko, Yong Tian, HongSheng Zhao, David J. Bacon, Andy N. Taylor, Keith Horne, J.R. Brownstein and J.W. Moffat for providing their data freely. Also, I thank Mahmood Roshan and Koroush Javidan for very useful discussions. I acknowledge the anonymous reviewer whose comments helped to improve and clarify this manuscript. The section on the growth of structures is added due to reviewer’s suggestion. This research has made use of NASA’s Astrophysics Data System.


  1. Anderson, J.L.: Phys. Rev. Lett. 75, 3602 (1995) ADSCrossRefGoogle Scholar
  2. Arfken, G.B., Weber, H.-J., Harris, F.E.: Mathematical Methods for Physicists: A Comprehensive Guide. Academic Press, San Diego (2012) zbMATHGoogle Scholar
  3. Bamba, K., Capozziello, S., Nojiri, S., Odintsov, S.D.: Astrophys. Space Sci. 342(1), 155–228 (2012) ADSCrossRefGoogle Scholar
  4. Bardeen, J.M.: Phys. Rev. D 22, 1882–1905 (1980) ADSMathSciNetCrossRefGoogle Scholar
  5. Begeman, K.G., Broeils, A.H., Sanders, R.H.: Mon. Not. R. Astron. Soc. 249, 523 (1991) ADSCrossRefGoogle Scholar
  6. Behar, S., Carmeli, M.: Int. J. Theor. Phys. 39, 1397–1404 (2000). arXiv:astro-ph/9907244 MathSciNetCrossRefGoogle Scholar
  7. Bonnor, W.B.: Class. Quantum Gravity 16, 1313 (1999) ADSMathSciNetCrossRefGoogle Scholar
  8. Brahm, D.E., Gruber, R.P.: Gen. Relativ. Gravit. 24(3), 297–303 (1992) ADSMathSciNetCrossRefGoogle Scholar
  9. Brownstein, J.R., Moffat, J.W.: Astrophys. J. 636(2), 721–741 (2006) ADSCrossRefGoogle Scholar
  10. Carmeli, M.: In: Nuclear Physics B Proceedings Supplements, vol. 124, pp. 258–263 (2003). arXiv:astro-ph/0205113 Google Scholar
  11. Carroll, S.M.: Spacetime and Geometry. Addison-Wesley, Reading (2004) zbMATHGoogle Scholar
  12. Castagnino, M.A.: J. Math. Phys. 12, 2203 (1971) ADSMathSciNetCrossRefGoogle Scholar
  13. CASTLES Survey (2015).
  14. Chabrier, G.: Publ. Astron. Soc. Pac. 115(809), 763–795 (2003) ADSCrossRefGoogle Scholar
  15. Chiu, M.-C., Ko, C.-M., Tian, Y., Sheng Zhao, H.: Phys. Rev. D 83, 063523 (2011) ADSGoogle Scholar
  16. Cooperstock, F.I., Faraoni, V., Vollick, D.N.: Astrophys. J. 503, 61 (1998) ADSCrossRefGoogle Scholar
  17. de Blok, W.J.G., McGaugh, S.S.: Astrophys. J. 469, L89 (1996) ADSCrossRefGoogle Scholar
  18. Desloge, E.A.: Gen. Relativ. Gravit. 21(7), 677–681 (1989) ADSMathSciNetCrossRefGoogle Scholar
  19. Dicke, R.H., Peebles, P.J.E.: Phys. Rev. Lett. 12, 435 (1964) ADSCrossRefGoogle Scholar
  20. Einstein, A., Straus, E.G.: Rev. Mod. Phys. 17, 120 (1945) ADSMathSciNetCrossRefGoogle Scholar
  21. Famaey, B., McGaugh, S.S.: Modified Newtonian dynamics (MOND): observational phenomenology and relativistic extensions. Living Rev. Relativ. 15, 10 (2012). ADSCrossRefGoogle Scholar
  22. Faraoni, V., Jacques, A.: Phys. Rev. D 76, 063510 (2007) ADSMathSciNetCrossRefGoogle Scholar
  23. Farrar, G.R., Rosen, R.A.: Phys. Rev. Lett. 98, 171302 (2007). doi: 10.1103/PhysRevLett.98.171302 ADSCrossRefGoogle Scholar
  24. Hartnett, J.G.: Int. J. Theor. Phys. 44(3), 359–372 (2005). arXiv:gr-qc/0407082 MathSciNetCrossRefGoogle Scholar
  25. Hubble, E.: Proc. Natl. Acad. Sci. USA 15(3), 168–173 (1929) ADSCrossRefGoogle Scholar
  26. Jacobs, M.W., Linder, E.V., Wagoner, R.V.: Phys. Rev. D 45, 10 (1992) MathSciNetCrossRefGoogle Scholar
  27. Marzke, R.F., Wheeler, J.A.: Gravitation as geometry-I. In: Chiu, H.Y., Hoffman, W.F. (eds.) Gravitation and Relativity. Benjamin, New York (1964) Google Scholar
  28. McVittie, G.C.: Mon. Not. R. Astron. Soc. 93, 325 (1933) ADSCrossRefGoogle Scholar
  29. Milgrom, M.: Astrophys. J. 270, 365–370 (1983a) ADSCrossRefGoogle Scholar
  30. Milgrom, M.: Astrophys. J. 270, 371–383 (1983b) ADSCrossRefGoogle Scholar
  31. Milgrom, M.: Astrophys. J. 270, 384–389 (1983c) ADSCrossRefGoogle Scholar
  32. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, New York (1973) Google Scholar
  33. Mukhanov, V.: Physical Foundations of Cosmology. Cambridge University Press, Cambridge (2005) CrossRefzbMATHGoogle Scholar
  34. OB́rien, J.G., Mannheim, P.D.: Mon. Not. R. Astron. Soc. 421, 1273–1282 (2012) ADSCrossRefGoogle Scholar
  35. Ohanian, H.C.: Gravitation and Spacetime. Norton, New York (1976) zbMATHGoogle Scholar
  36. Pachner, J.: Phys. Rev. 132, 1837 (1963) ADSMathSciNetCrossRefGoogle Scholar
  37. Pachner, J.: Phys. Rev. B 137, 1379 (1965) ADSMathSciNetCrossRefGoogle Scholar
  38. Pyne, T., Birkinshaw, M.: Astrophys. J. 458, 46–56 (1996) ADSCrossRefGoogle Scholar
  39. Romanowsky, A.J., Douglas, N.D., Arnaboldi, M., Kuijken, K., Merrifield, M.R., Napolitano, N.R., Capaccioli, M., Freeman, K.C.: Science 301, 1696 (2003) ADSCrossRefGoogle Scholar
  40. Salpeter, E.E.: Astrophys. J. 121, 161 (1955) ADSCrossRefGoogle Scholar
  41. Sanders, R.H.: Astrophys. J. 473, 117 (1996) ADSCrossRefGoogle Scholar
  42. Shenavar, H.: (2016, in preparation) Google Scholar
  43. Sofue, Y.: Astrophys. J. 458, 120 (1996) ADSCrossRefGoogle Scholar
  44. Spergel, D.N., Flauger, R., Hložek, R.: Phys. Rev. D 91, 023518 (2015) ADSCrossRefGoogle Scholar
  45. Verheijen, M.A.W., Sancisi, R.: Astron. Astrophys. 370, 765 (2001) ADSCrossRefGoogle Scholar
  46. Zhao, H., Bacon, D., Taylor, A., Horne, K.: Mon. Not. R. Astron. Soc. 368(1), 171–186 (2006) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of PhysicsFerdowsi University of MashhadMashhadIran

Personalised recommendations