Skip to main content

Advertisement

SpringerLink
Gravitational potential and galaxy rotation curves in multi-fractional spacetimes
Download PDF
Download PDF
  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 02 August 2022

Gravitational potential and galaxy rotation curves in multi-fractional spacetimes

  • Gianluca Calcagni  ORCID: orcid.org/0000-0003-2631-45881 &
  • Gabriele U. Varieschi2 

Journal of High Energy Physics volume 2022, Article number: 24 (2022) Cite this article

  • 73 Accesses

  • 1 Citations

  • 8 Altmetric

  • Metrics details

A preprint version of the article is available at arXiv.

Abstract

Multi-fractional theories with integer-order derivatives are models of gravitational and matter fields living in spacetimes with variable Hausdorff and spectral dimension, originally proposed as descriptions of geometries arising in quantum gravity. We derive the Poisson equation and the Newtonian potential of these theories starting from their covariant modified Einstein’s equations. In particular, in the case of the theory Tv with weighted derivatives with small fractional corrections, we find a gravitational potential that grows logarithmically at large radii when the fractional exponent takes the special value α = 4/3. This behaviour is associated with a restoration law for the Hausdorff dimension of spacetime independently found in the dark-energy sector of the same theory. As an application, we check whether this potential can serve as an alternative to dark matter for the galaxies NGC7814, NGC6503 and NGC3741 in the SPARC catalogue. We show that their rotation curves at medium-to-large radii can indeed be explained by purely geometric effects, although the Tully-Fisher relation is not reproduced well. We discuss how to fix the small-radius behaviour by lifting some approximations and how to test the model with other observables and an enlarged galaxy sample.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. D. Oriti ed., Approaches to Quantum Gravity, Cambridge University Press, Cambridge, U.K. (2009).

  2. G. F. R. Ellis, J. Murugan and A. Weltman eds., Foundations of Space and Time, Cambridge University Press, Cambridge, U.K. (2012).

  3. L. Modesto and L. Rachwał, Nonlocal quantum gravity: A review, Int. J. Mod. Phys. D 26 (2017) 1730020 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  4. G. Calcagni, Next step in gravity and cosmology: fundamental theory or data-driven models?, Front. Astron. Space Sci. 7 (2020) 52 [arXiv:2009.00846] [INSPIRE].

    Article  ADS  Google Scholar 

  5. G. ’t Hooft, Dimensional reduction in quantum gravity, in proceedings of the Conference on Highlights of Particle and Condensed Matter Physics (SALAMFEST), Trieste, Italy, 8–12 March 1993, A. Ali, J.R. Ellis and S. Randjbar-Daemi eds., World Scientific, Singapore (1993) [Conf. Proc. C 930308 (1993) 284] [gr-qc/9310026] [INSPIRE].

  6. S. Carlip, Spontaneous Dimensional Reduction in Short-Distance Quantum Gravity?, AIP Conf. Proc. 1196 (2009) 72 [arXiv:0909.3329] [INSPIRE].

    Article  ADS  Google Scholar 

  7. G. Calcagni, Fractal universe and quantum gravity, Phys. Rev. Lett. 104 (2010) 251301 [arXiv:0912.3142] [INSPIRE].

    Article  ADS  Google Scholar 

  8. G. Calcagni, Multiscale spacetimes from first principles, Phys. Rev. D 95 (2017) 064057 [arXiv:1609.02776] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  9. S. Carlip, Dimension and Dimensional Reduction in Quantum Gravity, Class. Quant. Grav. 34 (2017) 193001 [arXiv:1705.05417] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  10. J. Mielczarek and T. Trześniewski, Towards the map of quantum gravity, Gen. Rel. Grav. 50 (2018) 68 [arXiv:1708.07445] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  11. G. Calcagni, ABC of multi-fractal spacetimes and fractional sea turtles, Eur. Phys. J. C 76 (2016) 181 [Erratum ibid. 76 (2016) 459] [arXiv:1602.01470] [INSPIRE].

  12. G. Calcagni, Multifractional theories: an unconventional review, JHEP 03 (2017) 138 [Erratum JHEP 06 (2017) 020] [arXiv:1612.05632] [INSPIRE].

  13. G. Calcagni, Multifractional theories: an updated review, Mod. Phys. Lett. A 36 (2021) 2140006 [arXiv:2103.06557] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  14. P. D. Mannheim, Alternatives to dark matter and dark energy, Prog. Part. Nucl. Phys. 56 (2006) 340 [astro-ph/0505266] [INSPIRE].

    Article  ADS  Google Scholar 

  15. Q. Li and L. Modesto, Galactic Rotation Curves in Conformal Scalar-Tensor Gravity, Grav. Cosmol. 26 (2020) 99 [arXiv:1906.05185] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  16. L. Modesto, T. Zhou and Q. Li, Geometric origin of the galaxies’ dark side, arXiv:2112.04116 [INSPIRE].

  17. S. Capozziello and M. De Laurentis, The dark matter problem from f (R) gravity viewpoint, Annalen Phys. 524 (2012) 545 [INSPIRE].

    Article  ADS  Google Scholar 

  18. R. H. Sanders and S. S. McGaugh, Modified Newtonian dynamics as an alternative to dark matter, Ann. Rev. Astron. Astrophys. 40 (2002) 263 [astro-ph/0204521] [INSPIRE].

    Article  ADS  Google Scholar 

  19. J. D. Bekenstein, Relativistic gravitation theory for the MOND paradigm, Phys. Rev. D 70 (2004) 083509 [Erratum ibid. 71 (2005) 069901] [astro-ph/0403694] [INSPIRE].

  20. B. Famaey and S. S. McGaugh, Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions, Living Rev. Rel. 15 (2012) 10 [arXiv:1112.3960] [INSPIRE].

    Article  Google Scholar 

  21. A. Giusti, MOND-like Fractional Laplacian Theory, Phys. Rev. D 101 (2020) 124029 [arXiv:2002.07133] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  22. A. Giusti, R. Garrappa and G. Vachon, On the Kuzmin model in fractional Newtonian gravity, Eur. Phys. J. Plus 135 (2020) 798 [arXiv:2009.04335] [INSPIRE].

    Article  Google Scholar 

  23. G. U. Varieschi, Newtonian Fractional-Dimension Gravity and MOND, Found. Phys. 50 (2020) 1608 [Erratum ibid. 51 (2021) 41] [arXiv:2003.05784] [INSPIRE].

  24. G. U. Varieschi, Newtonian Fractional-Dimension Gravity and Disk Galaxies, Eur. Phys. J. Plus 136 (2021) 183 [arXiv:2008.04737] [INSPIRE].

    Article  Google Scholar 

  25. G. U. Varieschi, Newtonian Fractional-Dimension Gravity and Rotationally Supported Galaxies, Mon. Not. Roy. Astron. Soc. 503 (2021) 1915 [arXiv:2011.04911] [INSPIRE].

    Article  ADS  Google Scholar 

  26. G. U. Varieschi, Relativistic Fractional-Dimension Gravity, Universe 7 (2021) 387 [arXiv:2109.02855] [INSPIRE].

    Article  ADS  Google Scholar 

  27. S. S. McGaugh, F. Lelli and J. M. Schombert, Radial Acceleration Relation in Rotationally Supported Galaxies, Phys. Rev. Lett. 117 (2016) 201101 [arXiv:1609.05917] [INSPIRE].

    Article  ADS  Google Scholar 

  28. F. Lelli, S. S. McGaugh and J. M. Schombert, SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves, Astron. J. 152 (2016) 157 [arXiv:1606.09251] [INSPIRE].

    Article  ADS  Google Scholar 

  29. G. Calcagni and A. De Felice, Dark energy in multifractional spacetimes, Phys. Rev. D 102 (2020) 103529 [arXiv:2004.02896] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  30. G. Calcagni, Geometry and field theory in multi-fractional spacetime, JHEP 01 (2012) 065 [arXiv:1107.5041] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  31. G. Calcagni and G. Nardelli, Momentum transforms and Laplacians in fractional spaces, Adv. Theor. Math. Phys. 16 (2012) 1315 [arXiv:1202.5383] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  32. G. Calcagni, Quantum field theory, gravity and cosmology in a fractal universe, JHEP 03 (2010) 120 [arXiv:1001.0571] [INSPIRE].

    Article  ADS  Google Scholar 

  33. G. Calcagni, Complex dimensions and their observability, Phys. Rev. D 96 (2017) 046001 [arXiv:1705.01619] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  34. G. Calcagni, D. Rodríguez Fernández and M. Ronco, Black holes in multi-fractional and Lorentz-violating models, Eur. Phys. J. C 77 (2017) 335 [arXiv:1703.07811] [INSPIRE].

    Article  ADS  Google Scholar 

  35. G. Calcagni and M. Ronco, Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime, Nucl. Phys. B 923 (2017) 144 [arXiv:1706.02159] [INSPIRE].

    Article  ADS  Google Scholar 

  36. G. Calcagni, Multi-scale gravity and cosmology, JCAP 12 (2013) 041 [arXiv:1307.6382] [INSPIRE].

    Article  ADS  Google Scholar 

  37. R. B. Tully and J. R. Fisher, A New method of determining distances to galaxies, Astron. Astrophys. 54 (1977) 661 [INSPIRE].

    ADS  Google Scholar 

  38. S. S. McGaugh, J. M. Schombert, G. D. Bothun and W. J. G. de Blok, The Baryonic Tully-Fisher relation, Astrophys. J. Lett. 533 (2000) L99 [astro-ph/0003001] [INSPIRE].

    Article  ADS  Google Scholar 

  39. S. S. McGaugh, The Baryonic Tully-Fisher Relation of Gas Rich Galaxies as a Test of ΛLCDM and MOND, Astron. J. 143 (2012) 40 [arXiv:1107.2934] [INSPIRE].

    Article  ADS  Google Scholar 

  40. G. Calcagni, G. Nardelli and D. Rodríguez-Fernández, Standard Model in multiscale theories and observational constraints, Phys. Rev. D 94 (2016) 045018 [arXiv:1512.06858] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  41. G. Calcagni, Classical and quantum gravity with fractional operators, Class. Quant. Grav. 38 (2021) 165005 [Erratum ibid. 38 (2021) 169601] [arXiv:2106.15430] [INSPIRE].

  42. K.-H. Chae, F. Lelli, H. Desmond, S. S. McGaugh, P. Li and J. M. Schombert, Testing the Strong Equivalence Principle: Detection of the External Field Effect in Rotationally Supported Galaxies, Astrophys. J. 904 (2020) 51 [Erratum ibid. 910 (2021) 81] [arXiv:2009.11525] [INSPIRE].

  43. D. Clowe et al., A direct empirical proof of the existence of dark matter, Astrophys. J. Lett. 648 (2006) L109 [astro-ph/0608407] [INSPIRE].

  44. S. W. Allen, A. E. Evrard and A. B. Mantz, Cosmological Parameters from Observations of Galaxy Clusters, Ann. Rev. Astron. Astrophys. 49 (2011) 409 [arXiv:1103.4829] [INSPIRE].

    Article  ADS  Google Scholar 

  45. R. Massey, T. Kitching and J. Richard, The dark matter of gravitational lensing, Rept. Prog. Phys. 73 (2010) 086901 [arXiv:1001.1739] [INSPIRE].

    Article  ADS  Google Scholar 

  46. DES collaboration, Dark Energy Survey Year 3 results: curved-sky weak lensing mass map reconstruction, Mon. Not. Roy. Astron. Soc. 505 (2021) 4626 [arXiv:2105.13539] [INSPIRE].

  47. Planck collaboration, Planck 2018 results. Part VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6 [Erratum ibid. 652 (2021) C4] [arXiv:1807.06209] [INSPIRE].

  48. J. W. Moffat, Scalar-tensor-vector gravity theory, JCAP 03 (2006) 004 [gr-qc/0506021] [INSPIRE].

  49. J. R. Brownstein and J. W. Moffat, The Bullet Cluster 1E0657-558 evidence shows Modified Gravity in the absence of Dark Matter, Mon. Not. Roy. Astron. Soc. 382 (2007) 29 [astro-ph/0702146] [INSPIRE].

  50. J. W. Moffat and V. T. Toth, Can Modified Gravity (MOG) explain the speeding Bullet (Cluster)?, arXiv:1005.2685 [INSPIRE].

  51. N. S. Israel and J. W. Moffat, The Train Wreck Cluster Abell 520 and the Bullet Cluster 1E0657-558 in a Generalized Theory of Gravitation, Galaxies 6 (2018) 41 [arXiv:1606.09128] [INSPIRE].

    Article  ADS  Google Scholar 

  52. C. Lage and G. R. Farrar, The Bullet Cluster is not a Cosmological Anomaly, JCAP 02 (2015) 038 [arXiv:1406.6703] [INSPIRE].

    Article  ADS  Google Scholar 

  53. M. Lisanti, M. Moschella, N. J. Outmezguine and O. Slone, Testing Dark Matter and Modifications to Gravity using Local Milky Way Observables, Phys. Rev. D 100 (2019) 083009 [arXiv:1812.08169] [INSPIRE].

    Article  ADS  Google Scholar 

  54. C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, Freeman, New York, NY, U.S.A. (1973).

  55. P. C. van der Kruit, The three-dimensional distribution of light and mass in disks of spiral galaxies, Astron. Astrophys. 192 (1988) 117.

  56. M. A. Bershady, M. A. W. Verheijen, K. B. Westfall, D. R. Andersen, R. A. Swaters and T. Martinsson, The DiskMass Survey. Part II. Error Budget, Astrophys. J. 716 (2010) 234 [arXiv:1004.5043] [INSPIRE].

  57. K.-H. Chae, H. Desmond, F. Lelli, S. S. McGaugh and J. M. Schombert, Testing the Strong Equivalence Principle. Part II. Relating the External Field Effect in Galaxy Rotation Curves to the Large-scale Structure of the Universe, Astrophys. J. 921 (2021) 104 [arXiv:2109.04745] [INSPIRE].

  58. J. Binney and S. Tremaine, Galactic Dynamics, 2nd edition, Princeton University Press, Princeton, NJ, U.S.A. (2008).

  59. H. S. Cohl and J. E. Tohline, A compact cylindrical Green’s function expansion for the solution of potential problems, Astrophys. J. 527 (1999) 86.

  60. H. S. Cohl, A. R. P. Rau, J. E. Tohline, D. A. Browne, J. E. Cazes and E. I. Barnes, Useful alternative to the multipole expansion of 1/r potentials, Phys. Rev. A 64 (2001) 052509 [physics/0101086].

  61. H. S. Cohl, Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems, SIGMA 9 (2013) 042 [arXiv:1209.6047].

  62. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th edition, Academic Press, London, U.K. (2007).

  63. J. D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York, NY, U.S.A. (1998).

  64. H. S. Cohl and E. G. Kalnins, Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry, J. Phys. A 45 (2012) 145206 [arXiv:1105.0386].

  65. H. S. Cohl, On a generalization of the generating function for Gegenbauer polynomials, Integr. Transf. Special Func. 24 (2013) 807 [arXiv:1105.2735].

Download references

Author information

Authors and Affiliations

  1. Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006, Madrid, Spain

    Gianluca Calcagni

  2. Department of Physics, Loyola Marymount University, 1 LMU Drive, Los Angeles, CA, 90045, USA

    Gabriele U. Varieschi

Authors
  1. Gianluca Calcagni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriele U. Varieschi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gianluca Calcagni.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2112.13103

Rights and permissions

Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Calcagni, G., Varieschi, G.U. Gravitational potential and galaxy rotation curves in multi-fractional spacetimes. J. High Energ. Phys. 2022, 24 (2022). https://doi.org/10.1007/JHEP08(2022)024

Download citation

  • Received: 14 January 2022

  • Revised: 27 May 2022

  • Accepted: 23 June 2022

  • Published: 02 August 2022

  • DOI: https://doi.org/10.1007/JHEP08(2022)024

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Classical Theories of Gravity
  • Models for Dark Matter
  • Models of Quantum Gravity
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Over 10 million scientific documents at your fingertips

Switch Edition
  • Academic Edition
  • Corporate Edition
  • Home
  • Impressum
  • Legal information
  • Privacy statement
  • California Privacy Statement
  • How we use cookies
  • Manage cookies/Do not sell my data
  • Accessibility
  • FAQ
  • Contact us
  • Affiliate program

Not affiliated

Springer Nature

© 2023 Springer Nature Switzerland AG. Part of Springer Nature.