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.
Similar content being viewed by others
Data Availability Statement
No Data associated in the manuscript
Notes
Recall that, in general, \(\omega _{J}\) depends on h.
It is clear that the scale factor is also dynamical but we are interested in the Hubble parameter, not the scale factor.
Generic potentials may not solve (5.3) at \(\chi =0\).
This minimum is achieved through the Hubble friction term.
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.
Recall that the scalar field has units of energy.
References
Albert Einstein, The Foundation of the General Theory of Relativity. Annalen Phys. 49, 769–822 (1916)
Albert Einstein, Einheitliche Feldtheorie von Gravitation und Elektrizitat (in German). Sitzungsb. Preuss. Akad. Wiss. 22, 414 (1925)
Albert Einstein, Auf die Riemann-Metrik und den Fern-Parallelismus gegr\(\ddot{u}\)ndete einheitliche Feldtheorie (in German). Math. Ann. 102, 1–66 (2015)
Albert. Einstein, Neue Moglichkeit fur eine einheitliche Theorie von Gravitation und Elektrizitat (in German), Sitzungsb. Preuss. Akad. Wiss. (1928) 224
Albert. Einstein, Riemann-Geometrie unter Aufrechterhaltung des Begriffes des Fernparallelismus (in German), Sitzungsb. Preuss. Akad. Wiss. (1928) 217
R. Aldrovandi, J.G. Pereira, Teleparallel gravity, Fund. Theor. Phys. vol 173 Springer, Dordrecht, The Netherlands (2013)
Cai, Yi-Fu, Capozziello, Salvatore, De Laurentis, Mariafelicia, Saridakis, Emmanuel N., f(T) teleparallel gravity and cosmology, Rept. Prog. Phys. vol 79 106901 (2016)
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
R.T. Hammond, Torsion gravity. Rept. Prog. Phys. 65, 599–649 (2002)
J.W. Maluf, The teleparallel equivalent of general relativity. Annalen Phys. 525, 339–357 (2013)
I.L. Shapiro, Physical aspects of the space-time torsion. Phys. Rept. 357, 113 (2002)
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)
Sami Raatikainen, Syksy Rasanen, Higgs inflation and teleparallel gravity. JCAP 12, 021 (2019)
J.M. Nester, H.-J. Yo, Symmetric teleparallel general relativity. Chin. J. Phys. 37, 113 (1999)
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)
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)
Mikhail Shaposhnikov, Andrey Shkerin, Inar Timiryasov, Sebastian Zell, Higgs inflation in Einstein-Cartan gravity. JCAP 02, 008 (2021)
R. Hojman, C. Mukku, W.A. Sayed, Parity violation in metric torsion theories of gravitation. Phys. Rev. D 22, 1915–1921 (1980)
Nelson, C. Philip, Gravity with propagating pseudoscalar torsion. Phys. Lett. A 79, 285 (1980)
S. Holst, Barbero’s Hamiltonian derived from a generalized Hilbert-Palatini action, Phys. Rev. D 53 (1996)
H.T. Nieh, M.L. Yan, An identity in Riemann-cartan geometry. J. Math. Phys. 23, 373 (1982)
V. Faraoni, E. Gunzig, Einstein frame or Jordan frame? Int. J. Theor. Phys. 38, 217–225 (1999)
Marieke Postma, Marco Volponi, Equivalence of the Einstein and Jordan frames. Phys. Rev. D. 90, 103516 (2014)
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)
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)
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)
L.C. Garcia de Andrade, Cosmic relic torsion from inflationary cosmology. Int. J. Mod. Phys. D 8, 725–729 (1999)
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)
An alternative to cosmic inflation, Popławski, Nikodem J. Cosmology with torsion. Phys. Lett. B. 694, 181–185 (2010)
Sergio Bravo Medina, Marek Nowakowski, Davide Batic, Einstein-Cartan Cosmologies. Annals Phys. 400, 64–108 (2019)
Dirk Puetzfeld, Status of non-Riemannian cosmology. New Astron. Rev. 49, 59–64 (2005)
Y. Akrami, others, Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. vol 641 A10 (2020)
Wayne Hu, Rennan Barkana, Andrei Gruzinov, Cold and fuzzy dark matter. Phys. Rev. Lett. 85, 1158–1161 (2000)
Marsh, David J. E. Axion Cosmology, Phys. Rept. vol 643 1-79 (2016)
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)
Lam Hui, Jeremiah P. Ostriker, Scott Tremaine, Edward Witten, Ultralight scalars as cosmological dark matter. Phys. Rept. 95, 043541 (2017)
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)
Edmund J. Copeland, M. Sami, Shinji Tsujikawa, Dynamics of dark energy. Int. J. Mod. Phys. D 15, 1753–1936 (2006)
Ivaylo Zlatev, Li-Min. Wang, Paul J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant. Phys. Rev. Lett. 82, 896–899 (1999)
Luca Amendola, Coupled quintessence. Phys. Rev. D 62, 043511 (2000)
Alexander Yu Kamenshchik, Ugo Moschella, Vincent Pasquier, An Alternative to quintessence. Phys. Lett. B 511, 265–268 (2001)
Bharat Ratra, P.J.E. Peebles, Cosmological consequences of a rolling homogeneous scalar field. Phys. Rev. D 37, 3406 (1998)
C. Armendariz-Picon, T. Damour, Viatcheslav F. Mukhanov, k - inflation. Phys. Lett. B 458, 209–218 (1999)
Andrei D. Linde, Chaotic Inflation, Phys. Lett. B 129, 177–181 (1983)
Andrei D. Linde, Hybrid inflation. Phys. Rev. D 49, 748–754 (1994)
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)
Fedor L. Bezrukov, Mikhail Shaposhnikov, The standard model Higgs boson as the inflaton. Phys. Lett. B 659, 703–706 (2008)
Carl H. Brans, Jordan-Brans-Dicke. Theory, Scholarpedia 9, 31358 (2014)
A. A. Starobinsky, in Quantum Gravity, Proceedings of the 2nd Seminar on Quantum Gravity, Moscow, 1981(INR Press, Moscow, 1982), pp. 58–72
A. A. Starobinsky, reprinted inM. A. Markov and P. C. West eds., Quantum Gravity(Plenum Press, New York, 1984), pp. 103–128
A.A. Starobinsky, Phys. Lett. 91B, 99102 (1980)
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)
G. German, Brans-Dicke type models with torsion. Phys. Rev. D 32, 3307–3308 (1985)
Matteo Piani, Javier Rubio, Higgs-Dilaton inflation in Einstein-Cartan gravity. JCAP 05, 009 (2022)
Albert. Einstein, The Meaning of Relativity, 5th ed. (Princeton Univ. , Princeton, N. J., 1955)
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)
Ioannis D. Gialamas, Alexandros Karam, Antonio Racioppi, Dynamically induced Planck scale and inflation in the Palatini formulation. JCAP 11, 014 (2020)
Giovanni Tambalo, Massimiliano Rinaldi, Inflation and reheating in scale-invariant scalar-tensor gravity. Gen. Rel. Grav. 49, 52 (2017)
Massimiliano Rinaldi, Luciano Vanzo, Inflation and reheating in theories with spontaneous scale invariance symmetry breaking. Phys. Rev. D 94, 024009 (2016)
Pedro G. Ferreira, Christopher T. Hill, Johannes Noller, Graham G. Ross, Scale-independent \(R^2\) inflation. Phys. Rev. D 100, 123516 (2019)
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)
van de Bruck, Carsten and Longden, Chris, Higgs Inflation with a Gauss-Bonnet term in the Jordan Frame. Phys. Rev. D 93, 063519 (2016)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-023-03725-8