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Gravity at Large Distances

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Gravity, Strings and Particles

Abstract

As discussed in the previous chapter, at small enough distances we may expect several possible modifications of classical Newton’s law describing the behavior of the gravitational force. These modifications can be attributed to the entry in a microscopic regime characterized by new symmetries and governed by new (quantum) rules, or to the presence of extra dimensions inducing additional short-range contributions to the gravitational force. None of such modifications, however, has been experimentally confirmed so far.

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Notes

  1. 1.

    Mainly thanks to the work of Vera Rubin, a young astronomer of the Carnegie Institution (Washington), who first was able to measure accurately enough the rotation velocity of the stars as a function of their distance from the center, in the cases of many spiral galaxies.

  2. 2.

    The discovery was reported in the papers by Riess et al. and Perlmutter et al. [17]. The work of both these groups has been awarded the Nobel prize in Physics in 2011.

  3. 3.

    The abbreviation stands for Modified Newtonian Dynamics, a model of classic mechanics (proposed by Milgrom in 1981) which deviates from standard Newtonian mechanics only in the limit of very small accelerations. In that limit, a force is no longer proportional to the acceleration, but tends to be proportional to the square of the acceleration.

  4. 4.

    The name of these models is due to the fact that they modify the equations of general relativity by replacing the space–time curvature—represented by the symbol R—with an arbitrary function of the curvature, denoted indeed by f(R).

  5. 5.

    The acronym AMS means Alpha Magnetic Spectrometer, an instrument used for the detection and study of the particles of antimatter present in the cosmic rays reaching the Earth. Such an instrument is installed in the outside of the International Space Station (ISS) orbiting around the Earth, and has the task of intercepting the cosmic rays before they can interact with the terrestrial atmosphere (and lose precious information about their origin).

  6. 6.

    The abbreviation DGP comes from the names of the authors, as the model was jointly proposed by Dvali, Gabadadze, and Porrati [18].

  7. 7.

    Because of the presence of the so-called ghost states, unphysical states are characterized by a negative probability, appearing in the quantum version of the DGP model.

  8. 8.

    This is what happens, for instance, in models where the behavior of the quintessence field is described by the so-called tracker solutions. See, e.g., the paper by Slatev, Wang, and Steinhardt [19].

  9. 9.

    Current observations tell us that among the total matter present in our Universe, and composed of heavy, non-relativistic particles, there is a fraction of baryonic matter corresponding to about 10 % of the total. All the rest is composed of dark matter particles, of the same type as those particles that modify the rotation velocity of the stars inside a galaxy.

  10. 10.

    See for instance the paper by Armendariz-Picon, Mukhanov, and Steinhardt [20].

  11. 11.

    More precisely, when the exponential of the dilaton field (which controls the strength of all interactions) is much smaller than one.

  12. 12.

    A concrete example has been discussed in a paper I wrote in 2001 [21].

  13. 13.

    See for instance the paper by Gasperini, Piazza, and Veneziano [22].

  14. 14.

    This occurs when the Universe crosses the so-called equality epoch, characterized by a temperature about 10,000 larger than the present temperature of 2. 7 K. We can say, more precisely, that the Universe becomes matter dominated when the radiation temperature falls below 14, 700 K (see, e.g., [2]).

  15. 15.

    As shown in a paper by Amendola, Gasperini, and Piazza [23].

  16. 16.

    The removal of the shortest wavelengths approaching zero is a procedure usually applied in many physical situations. We say, in that case, that we have introduced an “ultraviolet cutoff.”

  17. 17.

    See for instance the review paper by Weinberg [24].

  18. 18.

    As we shall see, however, there are possible exceptions due to the presence of extra spatial dimensions.

  19. 19.

    As shown for the first time by Zumino [25]. They are also supersymmetric models where the vacuum energy density is constant and negative, but they do not seem to be compatible with a supersymmetric description of all interactions based on string theory, as discussed, for instance, by Witten [26].

  20. 20.

    In order to obtain a quantitative estimate, and to compare the different results, we have to use the numerical values of the reference length scales L P , L SUSY and L H . They are given, respectively, by L P  ∼ 10−33 cm, L SUSY ∼ 10−16 cm, and L H  ∼ 1028 cm.

  21. 21.

    The proportionality factor connecting E and L has to be the inverse of a squared length, for dimensional reasons. Here, as we are mainly referring to the gravitational interactions, we have chosen the fundamental Planck length for the definition of the proportionally factor 1∕L P 2.

  22. 22.

    See, e.g., the discussion by Bousso [27].

  23. 23.

    See, e.g., the paper by Padmanabhan [28].

  24. 24.

    The statistical distribution of Poisson is characterized by the property that the variance (or mean square deviation) of a distribution coincides with its mean value.

  25. 25.

    There is also a recent proposal of computing the quantum energy of the vacuum by including, for any positive energy state, the contribution of a corresponding negative energy state, according to the so-called procedure of time-symmetric quantization [29]. The model, however, is affected by formal problems when trying to describe interacting fields.

  26. 26.

    Current observations tell us that the mean curvature of the three-dimensional space, at the cosmic level, is equal to zero with an accuracy of one percent. See for instance the official site of the Particle Data Group at http://pdg.lbl.gov, containing an updated compilation of all relevant data for particle physics, astrophysics, and cosmology.

  27. 27.

    This effect has been discussed in a paper I wrote recently [30].

  28. 28.

    See for instance the paper by Antoniadis, Dudas, and Sagnotti [31].

  29. 29.

    To obtain this result we must insert the numerical value of the Planck mass, M P  ∼ 1019 GeV, and recall that 1 TeV = 103 GeV.

References

  1. R. Durrer, The Cosmic Microwave Background (Cambridge University Press, Cambridge, 2008)

    Book  Google Scholar 

  2. S. Weinberg, Cosmology (Oxford University Press, Oxford, 2008)

    MATH  Google Scholar 

  3. M. Gasperini, Lezioni di Cosmologia Teorica (Spriger, Milano, 2012)

    Book  MATH  Google Scholar 

  4. E.G. Adelberger, B.R. Heckel, A.E. Nelson, Ann. Rev. Nucl. Part. Sci. 53, 77 (2003)

    Article  ADS  Google Scholar 

  5. E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, S.H. Aronson, Phys. Rev. Lett. 56, 3 (1986)

    Article  ADS  Google Scholar 

  6. R. Barbieri, S. Cecotti, Z. Phys. C 33, 255 (1986)

    Google Scholar 

  7. M. Gasperini, Phys. Rev. D 40, 325 (1989)

    Google Scholar 

  8. M. Gasperini, Phys. Rev. D 63, 047301 (2001)

    Google Scholar 

  9. T. Taylor, G. Veneziano, Phys. Lett. B 213, 459 (1988)

    Google Scholar 

  10. M. Gasperini, Phys. Lett. B 327, 314 (1994); M. Gasperini, G. Veneziano, Phys. Rev. D 50, 2519 (1994)

    Google Scholar 

  11. J. Khoury, A. Weltman, Phys. Rev. D 69, 044026 (2004)

    Google Scholar 

  12. R. Sundrum, Phys. Rev. D 69, 044014 (2004)

    Google Scholar 

  13. T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin 1921, 966 (1921); O. Klein, Z. Phys. 37, 895 (1926)

    Google Scholar 

  14. N. Arkani Hamed, S. Dimopoulos, G.R. Dvali, Phys. Lett. B 429, 263 (1998)

    Google Scholar 

  15. I. Antoniadis, Phys. Lett. B 246, 377 (1990)

    Google Scholar 

  16. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 4960 (1999)

    Google Scholar 

  17. A.G. Riess et al., Astron. J. 116, 1009 (1998); S. Perlmutter et al., Astrophys. J 517, 565 (1999)

    Google Scholar 

  18. G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485, 208 (2000)

    Google Scholar 

  19. I. Slatev, L. Wang, P.J. Steinhardt, Phys. Rev. Lett. 82, 869 (1999)

    Google Scholar 

  20. C. Armendariz-Picon, V. Mukhanov, P.J. Steinhardt, Phys. Rev. Lett. 85, 4438 (2000)

    Article  ADS  Google Scholar 

  21. M. Gasperini, Phys. Rev. D 64, 043510 (2001)

    Google Scholar 

  22. M. Gasperini, F. Piazza, G. Veneziano, Phys. Rev. D 65, 023508 (2001)

    Google Scholar 

  23. L. Amendola, M. Gasperini, F. Piazza, JCAP 09, 014 (2004); Phys. Rev. D 4, 127302 (2006)

    Google Scholar 

  24. S. Weinberg, Rev. Mod. Phys. 61, 1 (1989)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. B. Zumino, Nucl. Phys. B 89, 535 (1975)

    Google Scholar 

  26. E. Witten, in Sources and Detection of Dark Matter and Dark Energy in the Universe, ed. by D.B. Cline (Springer, Berlin, 2001), p. 27

    Chapter  Google Scholar 

  27. R. Bousso, Gen. Rel. Grav. 40, 607 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. T. Padmanabhan, Gen. Rel. Grav. 40, 529 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. A. Kobakhidze, N.L. Rodd, Int. J. Theor. Phys. 52, 2636 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Gasperini, JHEP 06, 009 (2008)

    Article  ADS  Google Scholar 

  31. I. Antoniadis, E. Dudas, A. Sagnotti, Phys. Lett. B 464, 38 (1999)

    Google Scholar 

  32. G.F.R. Ellis, Spacetime and the Passage of Time, arXiv:1208.2611, published in “Springer Hanbook of Spacetime”, ed. by A. Ashtekar, V. Petkov (Springer, Berlin, 2014)

    Google Scholar 

  33. P.C.W. Davies, Sci. Am. (Special Edition) 21, 8 (2012)

    Google Scholar 

  34. J.B. Barbour, The End of Time: The Next Revolution in Physics (Oxford University Press, Oxford, 1999)

    Google Scholar 

  35. J.D. Barrow, D.J. Shaw, Phys. Rev. Lett. 106, 101302 (2011)

    Article  ADS  Google Scholar 

  36. N. Kaloper, K.A. Olive, Phys. Rev. D 57, 811 (1998)

    Google Scholar 

  37. E. Caianiello, La Rivista del Nuovo Cimento 15, 1 (1992)

    Article  Google Scholar 

  38. M. Gasperini, Int. J. Mod. Phys. D 13, 2267 (2004)

    Google Scholar 

  39. B. Zwiebach, A First Course in String Theory (Cambridge University Press, Cambridge, 2009)

    Book  MATH  Google Scholar 

  40. M.B. Green, J. Schwartz, E. Witten, Superstring Theory (Cambridge University Press, Cambridge, 1987)

    MATH  Google Scholar 

  41. J. Polchinski, String Theory (Cambridge University Press, Cambridge, 1998)

    Book  Google Scholar 

  42. M. Gasperini, Elements of String Cosmology (Cambridge University Press, Cambridge, 2007)

    Book  MATH  Google Scholar 

  43. M.A. Virasoro, Phys. Rev. D 1, 2933 (1970)

    Google Scholar 

  44. P. Ramond, Phys. Rev. D 3, 2415 (1971)

    Google Scholar 

  45. A. Neveau, J.H. Schwarz, Nucl. Phys. B 31, 86 (1971)

    Google Scholar 

  46. F. Gliozzi, J. Scherk, A. Olive, Nucl. Phys. B 122, 253 (1977)

    Google Scholar 

  47. A. Sagnotti, J. Phys. A 46, 214006 (2013)

    Google Scholar 

  48. K. Kikkawa, M.Y. Yamasaki, Phys. Lett. B 149, 357 (1984)

    Google Scholar 

  49. N. Sakai, I. Senda, Prog. Theor. Phys. 75, 692 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  50. A.A. Tseytlin, Mod. Phys. Lett. A 6, 1721 (1991)

    Google Scholar 

  51. G. Veneziano, Phys. Lett. B 265, 287 (1991)

    Google Scholar 

  52. E. Witten, Nucl. Phys. B 443, 85 (1995)

    Google Scholar 

  53. R. Bousso, J. Polchinski, Sci. Am. 291, 60 (2004)

    Article  Google Scholar 

  54. R. Brandenberger, C. Vafa, Nucl. Phys. B 316, 391 (1989)

    Google Scholar 

  55. A.A. Tseytlin, C. Vafa, Nucl. Phys. B 372, 443 (1992)

    Google Scholar 

  56. S. Alexander, R. Brandenberger, D. Easson, Phys. Rev. D 62, 103509 (2000)

    Google Scholar 

  57. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972)

    Google Scholar 

  58. A.A. Penzias, R.W. Wilson, Astrophys. J. 142, 419 (1965)

    Article  ADS  Google Scholar 

  59. J. Smooth et al., Astrophys. J. 396, L1 (1992)

    Article  ADS  Google Scholar 

  60. A. Guth, Phys. Rev. D 23, 347 (1981)

    Google Scholar 

  61. E.W. Kolb, M.S. Turner, The Early Universe (Addison Wesley, Redwood City, 1990)

    MATH  Google Scholar 

  62. A. Borde, A. Guth, A. Vilenkin, Phys. Rev. Lett. 90, 151301 (2003)

    Article  ADS  Google Scholar 

  63. M. Gasperini, G. Veneziano, Astropart. Phys. 1, 317 (1993); Phys. Rep. 373, 1 (2003)

    Google Scholar 

  64. M. Gasperini, The Universe Before the Big Bang: Cosmology and String Theory (Springer, Berlin, 2008)

    Google Scholar 

  65. J. Khoury, B.A. Ovrut, P.J. Steinhardt, N. Turok, Phys. Rev. D 64, 123522 (2001)

    Google Scholar 

  66. P.J. Steinhardt, N. Turok, Phys. Rev. D 65, 126003 (2002)

    Google Scholar 

  67. N. Goheer, M. Kleban, L. Susskind, JHEP 0307, 056 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  68. C. Burgess et al., JHEP 0107, 047 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  69. S. Kachru et al., JCAP 0310, 013 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  70. R. Penrose, Cycles of Time: An Extraordinary New View of the Universe (Bodley Head, London, 2010)

    Google Scholar 

  71. V.G. Gurzadyan, R. Penrose, Eur. Phys. J. Plus 128, 22 (2013)

    Article  Google Scholar 

  72. M. Gasperini, M. Giovannini, Phys. Lett. B 282, 36 (1992); Phys. Rev. D 47, 1519 (1993)

    Google Scholar 

  73. R. Brustein, M. Gasperini, G. Venenziano, Phys. Lett. B 361, 45 (1995)

    Google Scholar 

  74. M. Maggiore, Gravitational Waves (Oxford University Press, Oxford, 2007)

    Book  Google Scholar 

Download references

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  1. Dipartimento di Fisica, Università di Bari, Bari, Italy

    Maurizio Gasperini

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Gasperini, M. (2014). Gravity at Large Distances. In: Gravity, Strings and Particles. Springer, Cham. https://doi.org/10.1007/978-3-319-00599-7_3

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