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On Dark Energy, Weyl’s Geometry, Different Derivations of the Vacuum Energy Density and the Pioneer Anomaly

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Two different derivations of the observed vacuum energy density are presented. One is based on a class of proper and novel generalizations of the (Anti) de Sitter solutions in terms of a family of radial functions R(r) that provides an explicit formula for the cosmological constant along with a natural explanation of the ultraviolet/infrared (UV/IR) entanglement required to solve this problem. A nonvanishing value of the vacuum energy density of the order of \({10^{- 123} M_{\rm Planck}^4}\) is derived in agreement with the experimental observations. A correct lower estimate of the mass of the observable universe related to the Dirac–Eddington–Weyl’s large number N = 1080 is also obtained. The presence of the radial function R(r) is instrumental to understand why the cosmological constant is not zero and why it is so tiny. Finally, we rigorously prove why the proper use of Weyl’s Geometry within the context of Friedman–Lemaitre–Robertson–Walker cosmological models can account for both the origins and the value of the observed vacuum energy density (dark energy). The source of dark energy is just the dilaton-like Jordan–Brans–Dicke scalar field that is required to implement Weyl invariance of the most simple of all possible actions. The full theory involving the dynamics of Weyl’s gauge field Aμ is very rich and may explain also the anomalous Pioneer acceleration and the temporal variations (over cosmological scales) of the fundamental constants resulting from the expansion of the Universe. This is consistent with Dirac’s old idea of the plausible variation of the physical constants but with the advantage that it is not necessary to invoke extra dimensions.

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References

  1. L. Abrams, Can. J. Phys. 67, 919 (1989). Phys. Rev. D 20 , 2474 (1979). Phys. Rev. D 21, 2438 (1980). Phys. Rev. D 21, 2941 (1980).

  2. C. Castro, “On novel static spherically symmetric solutions of Einstein equations and the cosmological constant problem” (CTSPS preprint, May 2006).

  3. Brillouin M. (1923). J. Phys. Rad. 23, 43

    Google Scholar 

  4. Schwarzschild K. (1916). Sitzungsber. Preuss. Akad. Berlin I, 189

    Google Scholar 

  5. S. Crothers, Prog. Phys. 1, 68 (2005). Prog. Phys. 2, 3 (2005). Prog. Phys. 3, 7 (2005).

  6. P. Fiziev, “Gravitational field of massive point particle in general relativity,” [arXiv.org: gr-qc/0306088]. P. Fiziev and S. V. Dimitrov, “Point electric charge in general relativity” [arXiv.org: hep-th/0406077].

  7. D. Hilbert, Nachr. Ges. Wiss. Gottingen. Math. Phys. K 1, 53 (1917). H. Weyl, Ann. Phys. 54, 117 (1917). J. Droste, Proc. Ned. Akad. West Ser A 19, 197 (1917).

  8. M. Reuter and J. M. Schwindt, “A minimal length from cutoff modes in asymptotically safe quantum gravity” [arXiv.org: hep-th/0511021].

  9. C. Castro and J. A. Nieto, On 2 + 2 spacetimes, strings and black holes (submitted to Phys. Rev. D, September 2006).

  10. Nottale L. (1992). Fractal Spacetime and Microphysics: Towards Scale Relativity. World Scientific, Singapore

    Google Scholar 

  11. J. F. Colombeau, New Generalized Functions and Multiplication of Distributions (North Holland, Amsterdam, 1984). Elementary Introduction to Generalized Functions (North Holland, Amsterdam, 1985). M. Grosser, M. Kunzinger, M. Oberguggenberger and R. Steinbauer, Geometric Theory of Generalized Functions with Applications to Relativity (Kluwer series on Mathematics and its Applications, vol. 537, Kluwer Academic, Dordrecht, 2001). J. Heinzke and R. Steinbauer, J. Math. Phys. 43, 1493 (2002). “Remarks on the distributional Schwarzschild geometry” [arXiv.org: gr-qc/0112047].

  12. V. G. Gurzadyan and S. C. Xue, Mod. Phys. Letts. A 18, 561 (2003). G. Vereschchaigin and G. Yegorian, Phys. Lett. B 636, 150 (2006).

    Google Scholar 

  13. T. Padmanabhan, “Dark energy: mystery of the millenium” [arXiv.org: astro-ph/0603114].

  14. E. Guendelman and A. Kaganovich, Phys. Rev. D 53, 7020 (1996). Phys. Rev. D 60, 065004 (1999). Int. J. Mod. Phys. A 17, 417 (2002). E. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva, “Weyl-conformally invariant light-like p-brane theories” [arXiv.org: hep-th/0409078].

  15. N. Mavromatos, “The issue of dark energy in string theory” [arXiv.org: hep-th/0607006].

  16. Castro C. (2002). Mod. Phys. Lett. A 17: 2095

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. A. Loinger, On Black Holes and Gravitational Waves (La Goliardica Pavese, June 2002).

  18. S. Antoci and D. E. Liebscher, “Reinstating Schwarzschild’s original manifold and its singularity” [arXiv.org: gr-qc/0406090].

  19. C. Castro, Phys. Lett B 626, 209 (2005). Found. Phys 35, 971 (2005). Prog. Phys. 2, 86 (2006).

  20. J. Nieto, Phys. Lett. A 262, 274 (1999), S-duality for linearized gravity [arXiv.org: hep-th/9910049]. J. Nieto, L. Ruiz, and J. Silvas, “Thoughts on duality and the fundamental constants” [arXiv.org: hep-th/0512256].

  21. V. Kozlov and I. Volovich, “Finite action Klein Gordon solutions on Lorentzian manifolds” [arXiv.org: gr-qc/0603111].

  22. Castro C. (2005). Prog. Phys. 1, 20

    MathSciNet  Google Scholar 

  23. L. Nottale, “The Pioneer anomalous acceleration as a measurement of the cosmological constant at the scale of the solar system” [arXiv.org: gr-qc/0307042].

  24. E. Scholz, “On the geometry of cosmological model building” [arXiv.org: gr-qc/0511113].

  25. LaViolette P. (1985). Int. J. Gen. Systems 11: 329

    Google Scholar 

  26. H. Kim, “Can the Brans–Dicke gravity possibly with Λ be a theory of dark matter?” [arXiv.org: astro-ph/0604055]. M. Arik and M. Calik, “Can Brans–Dicke scalar field account for dark energy and dark matter?” [arXiv.org: gr-qc/0505035].

  27. S. Capozzielo, V. Cardone and A. Troisi, J. Cosmol. Astroparticle Phys. 8, 1 (2006). S. Capozziello, S. Nojiri and S. Odintsov, Phys. Letts. B 634, 93 (2006).

  28. B. G.Sidharth, “Discrete space time and dark Energy [arXiv.org: physics/0402007]. The mysterious dark energy” [arXiv.org: physics/0411224]. How fundamental is gravitation [arXiv.org: physics/0409088].

  29. Castro C., Pavsic M. (2005). Prog. Phys. 1: 31

    MathSciNet  Google Scholar 

  30. H. Wei and R.-G. Cai, “Cheng–Weyl vector field and cosmological application” [arXiv.org: astro-phy/0607064].

  31. J. Crooks and P. Frampton, “Conformal transformations and accelerated cosmologies” [arXiv.org: astro-ph/0601051]. M. Cardoni, “Conformal symmetry of gravity and the cosmological constant problem” [arXiv.org: hep-th/0606274].

  32. M. Israelit, Found. Phys. 29, 1303 (1999). Found. Phys. 32, 295 (2002). Found. Phys. 32, 945 (2002). The Weyl-Dirac Theory and Our Universe (Nova Science, 1999).

  33. D. Rapoport, Found. Phys 35, 1205 (2005). Found. Phys. 35, 1383 (2005).

    Google Scholar 

  34. M. Kafatos, S. Roy, and R. Amoroso, “Scaling in cosmology and the arrow of time in Studies on the Structure of Time” R. Buccheri et al., eds. (Kluwer Academic. New York, 2000). J. Glanz, Science 282, 2156 (1998).

  35. J. Rosales, “The Pioneer’s anomalous Doppler drift as a Berry’s phase” [arXiv.org: gr-qc/0401014].

  36. Einstein A. (1915). Sitzungsber Preuss Akad Berlin II, 831

    Google Scholar 

  37. Stavroulakis N. (2006). Prog. Phys. 2: 68

    MathSciNet  Google Scholar 

  38. Pavsic M. (1981). Obzornik za Matematiko in Fiziko 28, 5

    Google Scholar 

  39. M. Ibison, Private communication.

  40. C. Fronsdal, Phys. Rev. 116, 778 (1959). M. Kruskal, Phys. Rev. 119, 1743 (1960). G. Szekers, Publ. Mat. Debreca 7, 285 (1960).

  41. E. I. Guendelman and A. B. Kaganovich, “k-essence, avoidance of the Weinberg’s cosmological constant no-go theorem and other dark energy effects of two measures field theory” [arXiv.org: gr-qc/0606017].

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  1. Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA, 30314, USA

    Carlos Castro

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Correspondence to Carlos Castro.

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Dedicated to the loving memory of Rachael Bowers.

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Castro, C. On Dark Energy, Weyl’s Geometry, Different Derivations of the Vacuum Energy Density and the Pioneer Anomaly. Found Phys 37, 366–409 (2007). https://doi.org/10.1007/s10701-007-9106-z

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