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A comparison between the Jordan and Einstein frames in Brans-Dicke theories with torsion

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Abstract

In recent years, gravitational models motivated by quantum corrections to gravity which introduce higher order terms like \(R^{2}\) or terms in which the Riemann tensor is not symmetric have been studied by several authors in the form of a general Brans-Dicke type model containing the Ricci scalar, the Holst term and the Nieh-Yan invariant. In this paper we focus on the less explored Jordan frame of such theories and in the comparison between both this frame and the Einstein one. Furthermore, we discuss the role of the transformation of the torsion under conformal transformations and show that the transformation proposed in this paper (extended conformal transformation) contains a special case of the projective transformation of the connection used in some of the papers that motivated this work. We discuss the role and advantages of the extended conformal transformation and show that this new approach can have interesting consequences by working with different variables such as the metric and torsion. Moreover, we study the stability of the system via a dynamical analysis in the Jordan frame, this in order to analyze whether or not we have the fixed points that can be later identified as the inflationary attractor and the unstable fixed point where inflation could take place. Finally we study the scale invariant case of the general model in the Jordan frame. We find out that both the scalar spectral index and the tensor-to-scalar ratio are in agreement with the latest Planck results.

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Data Availability Statement

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Notes

  1. Recall that, in general, \(\omega _{J}\) depends on h.

  2. It is clear that the scale factor is also dynamical but we are interested in the Hubble parameter, not the scale factor.

  3. Generic potentials may not solve (5.3) at \(\chi =0\).

  4. This minimum is achieved through the Hubble friction term.

  5. After inflation and the possible reheating of the universe we need to ensure that we recover a non-accelerating cosmology and that we have the GR theory in order to be in agreement with the \(\Lambda \)CDM model.

  6. Recall that the scalar field has units of energy.

References

  1. Albert Einstein, The Foundation of the General Theory of Relativity. Annalen Phys. 49, 769–822 (1916)

    Article  ADS  Google Scholar 

  2. Albert Einstein, Einheitliche Feldtheorie von Gravitation und Elektrizitat (in German). Sitzungsb. Preuss. Akad. Wiss. 22, 414 (1925)

    MATH  Google Scholar 

  3. Albert Einstein, Auf die Riemann-Metrik und den Fern-Parallelismus gegr\(\ddot{u}\)ndete einheitliche Feldtheorie (in German). Math. Ann. 102, 1–66 (2015)

    Google Scholar 

  4. Albert. Einstein, Neue Moglichkeit fur eine einheitliche Theorie von Gravitation und Elektrizitat (in German), Sitzungsb. Preuss. Akad. Wiss. (1928) 224

  5. Albert. Einstein, Riemann-Geometrie unter Aufrechterhaltung des Begriffes des Fernparallelismus (in German), Sitzungsb. Preuss. Akad. Wiss. (1928) 217

  6. R. Aldrovandi, J.G. Pereira, Teleparallel gravity, Fund. Theor. Phys. vol 173 Springer, Dordrecht, The Netherlands (2013)

  7. Cai, Yi-Fu, Capozziello, Salvatore, De Laurentis, Mariafelicia, Saridakis, Emmanuel N., f(T) teleparallel gravity and cosmology, Rept. Prog. Phys. vol 79 106901 (2016)

  8. A. Golovnev, Introduction to teleparallel gravities, in 9th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, Belgrade, Serbia, 18-23 September 2017

  9. R.T. Hammond, Torsion gravity. Rept. Prog. Phys. 65, 599–649 (2002)

    Article  ADS  Google Scholar 

  10. J.W. Maluf, The teleparallel equivalent of general relativity. Annalen Phys. 525, 339–357 (2013)

    Article  ADS  Google Scholar 

  11. I.L. Shapiro, Physical aspects of the space-time torsion. Phys. Rept. 357, 113 (2002)

    Article  ADS  MATH  Google Scholar 

  12. J.A. Helayel-Neto, A. Penna-Firme, I.L. Shapiro, Conformal symmetry, anomaly and effective action for metric-scalar gravity with torsion. Phys. Lett. B 479, 411–420 (2000)

    Article  ADS  MATH  Google Scholar 

  13. Sami Raatikainen, Syksy Rasanen, Higgs inflation and teleparallel gravity. JCAP 12, 021 (2019)

    Article  ADS  MATH  Google Scholar 

  14. J.M. Nester, H.-J. Yo, Symmetric teleparallel general relativity. Chin. J. Phys. 37, 113 (1999)

    Google Scholar 

  15. M. Ferraris, M. Francaviglia, C. Reina, Variational formulation of general relativity from 1915to 1925 “Palatini’s method’’ discovered by Einstein in 1925. Gen. Rel. Grav. 14, 243 (1982)

    Article  ADS  MATH  Google Scholar 

  16. Miklos Långvik, Juha-Matti. Ojanperä, Sami Raatikainen, Syksy Räsänen, Higgs inflation with the Holst and the Nieh–Yan term. Phys. Rev. D 103, 083514 (2021)

    Article  ADS  Google Scholar 

  17. Mikhail Shaposhnikov, Andrey Shkerin, Inar Timiryasov, Sebastian Zell, Higgs inflation in Einstein-Cartan gravity. JCAP 02, 008 (2021)

    Article  ADS  MATH  Google Scholar 

  18. R. Hojman, C. Mukku, W.A. Sayed, Parity violation in metric torsion theories of gravitation. Phys. Rev. D 22, 1915–1921 (1980)

    Article  ADS  Google Scholar 

  19. Nelson, C. Philip, Gravity with propagating pseudoscalar torsion. Phys. Lett. A 79, 285 (1980)

    Article  ADS  Google Scholar 

  20. S. Holst, Barbero’s Hamiltonian derived from a generalized Hilbert-Palatini action, Phys. Rev. D 53 (1996)

  21. H.T. Nieh, M.L. Yan, An identity in Riemann-cartan geometry. J. Math. Phys. 23, 373 (1982)

    Article  ADS  MATH  Google Scholar 

  22. V. Faraoni, E. Gunzig, Einstein frame or Jordan frame? Int. J. Theor. Phys. 38, 217–225 (1999)

    Article  MATH  Google Scholar 

  23. Marieke Postma, Marco Volponi, Equivalence of the Einstein and Jordan frames. Phys. Rev. D. 90, 103516 (2014)

    Article  ADS  Google Scholar 

  24. Salvatore Capozziello, S. Nojiri, S.D. Odintsov, A. Troisi, Cosmological viability of f(R)-gravity as an ideal fluid and its compatibility with a matter dominated phase. Phys. Lett. B. 639, 135–143 (2006)

    Article  ADS  Google Scholar 

  25. Juan Garcia-Bellido, Javier Rubio, Mikhail Shaposhnikov, Daniel Zenhausern, Higgs-Dilaton cosmology: from the early to the late universe. Phys. Rev. D. 84, 123504 (2011)

    Article  ADS  Google Scholar 

  26. S. Capozziello, S. Nojiri, S.D. Odintsov, Dark energy: the equation of state description versus scalar-tensor or modified gravity. Phys. Lett. B. 634, 93–100 (2006)

    Article  ADS  Google Scholar 

  27. L.C. Garcia de Andrade, Cosmic relic torsion from inflationary cosmology. Int. J. Mod. Phys. D 8, 725–729 (1999)

    Article  ADS  Google Scholar 

  28. T. M. Guimarães, R. Lima, de C. and S. H. Pereira, Cosmological inflation driven by a scalar torsion function, Eur. Phys. J. C vol 81 271 (2021)

  29. An alternative to cosmic inflation, Popławski, Nikodem J. Cosmology with torsion. Phys. Lett. B. 694, 181–185 (2010)

    Google Scholar 

  30. Sergio Bravo Medina, Marek Nowakowski, Davide Batic, Einstein-Cartan Cosmologies. Annals Phys. 400, 64–108 (2019)

    Article  ADS  MATH  Google Scholar 

  31. Dirk Puetzfeld, Status of non-Riemannian cosmology. New Astron. Rev. 49, 59–64 (2005)

    Article  ADS  Google Scholar 

  32. Y. Akrami, others, Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. vol 641 A10 (2020)

  33. Wayne Hu, Rennan Barkana, Andrei Gruzinov, Cold and fuzzy dark matter. Phys. Rev. Lett. 85, 1158–1161 (2000)

    Article  ADS  Google Scholar 

  34. Marsh, David J. E. Axion Cosmology, Phys. Rept. vol 643 1-79 (2016)

  35. C. P. Burgess, Maxim. Pospelov, Tonnis. ter Veldhuis, The Minimal model of nonbaryonic dark matter: A Singlet scalar, Nucl. Phys. B. vol 619 709-728 (2001)

  36. Lam Hui, Jeremiah P. Ostriker, Scott Tremaine, Edward Witten, Ultralight scalars as cosmological dark matter. Phys. Rept. 95, 043541 (2017)

    ADS  Google Scholar 

  37. J.E. Marsh David, Jens C. Niemeyer, Strong constraints on fuzzy dark matter from ultrafaint Dwarf galaxy Eridanus II. Phys. Rev. Lett. 123, 051103 (2019)

    Article  ADS  Google Scholar 

  38. Edmund J. Copeland, M. Sami, Shinji Tsujikawa, Dynamics of dark energy. Int. J. Mod. Phys. D 15, 1753–1936 (2006)

    Article  ADS  MATH  Google Scholar 

  39. Ivaylo Zlatev, Li-Min. Wang, Paul J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant. Phys. Rev. Lett. 82, 896–899 (1999)

    Article  ADS  Google Scholar 

  40. Luca Amendola, Coupled quintessence. Phys. Rev. D 62, 043511 (2000)

    Article  ADS  Google Scholar 

  41. Alexander Yu Kamenshchik, Ugo Moschella, Vincent Pasquier, An Alternative to quintessence. Phys. Lett. B 511, 265–268 (2001)

    Article  ADS  MATH  Google Scholar 

  42. Bharat Ratra, P.J.E. Peebles, Cosmological consequences of a rolling homogeneous scalar field. Phys. Rev. D 37, 3406 (1998)

    Article  ADS  Google Scholar 

  43. C. Armendariz-Picon, T. Damour, Viatcheslav F. Mukhanov, k - inflation. Phys. Lett. B 458, 209–218 (1999)

    Article  ADS  MATH  Google Scholar 

  44. Andrei D. Linde, Chaotic Inflation, Phys. Lett. B 129, 177–181 (1983)

    Article  ADS  Google Scholar 

  45. Andrei D. Linde, Hybrid inflation. Phys. Rev. D 49, 748–754 (1994)

    Article  ADS  Google Scholar 

  46. Georges. Aad, others, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B vol 716 1-29 (2012)

  47. Fedor L. Bezrukov, Mikhail Shaposhnikov, The standard model Higgs boson as the inflaton. Phys. Lett. B 659, 703–706 (2008)

    Article  ADS  Google Scholar 

  48. Carl H. Brans, Jordan-Brans-Dicke. Theory, Scholarpedia 9, 31358 (2014)

    Article  ADS  Google Scholar 

  49. A. A. Starobinsky, in Quantum Gravity, Proceedings of the 2nd Seminar on Quantum Gravity, Moscow, 1981(INR Press, Moscow, 1982), pp. 58–72

  50. A. A. Starobinsky, reprinted inM. A. Markov and P. C. West eds., Quantum Gravity(Plenum Press, New York, 1984), pp. 103–128

  51. A.A. Starobinsky, Phys. Lett. 91B, 99102 (1980)

    Google Scholar 

  52. I.L. Buchbinder, I.L. Shapiro, On the renormalization of models of quantum field theory in an external gravitational field with torsion. Phys. Lett. B 151, 263–266 (1985)

    Article  ADS  Google Scholar 

  53. G. German, Brans-Dicke type models with torsion. Phys. Rev. D 32, 3307–3308 (1985)

    Article  ADS  Google Scholar 

  54. Matteo Piani, Javier Rubio, Higgs-Dilaton inflation in Einstein-Cartan gravity. JCAP 05, 009 (2022)

    Article  ADS  MATH  Google Scholar 

  55. Albert. Einstein, The Meaning of Relativity, 5th ed. (Princeton Univ. , Princeton, N. J., 1955)

  56. Ioannis D. Gialamas, Alexandros Karam, Thomas D. Pappas, Antonio Racioppi, Vassilis C. Spanos, Scale-invariance, dynamically induced Planck scale and inflation in the Palatini formulation. J. Phys. Conf. Ser. 2105, 012005 (2021)

    Article  Google Scholar 

  57. Ioannis D. Gialamas, Alexandros Karam, Antonio Racioppi, Dynamically induced Planck scale and inflation in the Palatini formulation. JCAP 11, 014 (2020)

    Article  ADS  MATH  Google Scholar 

  58. Giovanni Tambalo, Massimiliano Rinaldi, Inflation and reheating in scale-invariant scalar-tensor gravity. Gen. Rel. Grav. 49, 52 (2017)

    Article  ADS  MATH  Google Scholar 

  59. Massimiliano Rinaldi, Luciano Vanzo, Inflation and reheating in theories with spontaneous scale invariance symmetry breaking. Phys. Rev. D 94, 024009 (2016)

    Article  ADS  Google Scholar 

  60. Pedro G. Ferreira, Christopher T. Hill, Johannes Noller, Graham G. Ross, Scale-independent \(R^2\) inflation. Phys. Rev. D 100, 123516 (2019)

    Article  ADS  Google Scholar 

  61. Ioannis D. Gialamas, Alexandros Karam, Thomas D. Pappas, Antonio Racioppi, Vassilis C. Spanos, Scale-invariance, dynamically induced Planck scale and inflation in the Palatini formulation. J. Phys. Conf. Ser 2105, 012005 (2021)

    Article  Google Scholar 

  62. van de Bruck, Carsten and Longden, Chris, Higgs Inflation with a Gauss-Bonnet term in the Jordan Frame. Phys. Rev. D 93, 063519 (2016)

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Acknowledgments

The authors would like to thank the National Council of Science and Technology (CONACyT) for its funding and support and an anonymous referee for his/her suggestions and comments.

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Authors and Affiliations

  1. Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad S/N, 62251, Cuernavaca, Morelos, Mexico

    R. Gonzalez Quaglia & Gabriel Germán

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Correspondence to R. Gonzalez Quaglia.

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Gonzalez Quaglia, R., Germán, G. A comparison between the Jordan and Einstein frames in Brans-Dicke theories with torsion. Eur. Phys. J. Plus 138, 93 (2023). https://doi.org/10.1140/epjp/s13360-023-03725-8

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