- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Thermal description of braneworld effective theories

Soham Bhattacharyya* and Soumitra SenGupta†

  • *Contact author: soham1316@gmail.com
  • †Contact author: tpssg@iacs.res.in
PDF

Phys. Rev. D 113, 064019 – Published 9 March, 2026

DOI: https://doi.org/10.1103/dyf3-h6gb

Abstract

Low-energy effective theories provide the natural description of four-dimensional physics in higher-dimensional geometries, where the imprint of the bulk appears as parameters of the lower-dimensional theory. Motivated by the recent progress in the first-order thermodynamic formulation of modified gravity theories, we investigate the thermodynamics of effective theories in braneworld scenarios and thereby the attractor mechanism toward general relativity in such theories. We consider the 2-brane Randall-Sundrum model, where the low-energy theory on either brane is of scalar-tensor nature with the extradimensional radion playing the role of the scalar. We study the thermodynamic implications of a nonvanishing gravitational contribution to the radion potential, and further explore the dynamics in the presence of a bulk stabilizing field.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (71)

  1. S. B. Giddings, S. Kachru, and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66, 106006 (2002).
  2. N. Arkani–Hamed, S. Dimopoulos, and G. Dvali, The hierarchy problem and new dimensions at a millimeter, Phys. Lett. B 429, 263 (1998).
  3. L. Randall and R. Sundrum, Large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83, 3370 (1999).
  4. S. Kanno and J. Soda, Radion and holographic brane gravity, Phys. Rev. D 66, 083506 (2002).
  5. T. Shiromizu, K. I. Maeda, and M. Sasaki, The Einstein equations on the 3-brane world, Phys. Rev. D 62, 024012 (2000).
  6. T. Chiba, Scalar-tensor gravity in a two 3-brane system, Phys. Rev. D 62, 021502(R) (2000).
  7. C. Csáki, M. L. Graesser, and G. D. Kribs, Radion dynamics and electroweak physics, Phys. Rev. D 63, 065002 (2001).
  8. J. Lesgourgues and L. Sorbo, Goldberger-wise variations: Stabilizing brane models with a bulk scalar, Phys. Rev. D 69, 084010 (2004).
  9. S. Anand, D. Choudhury, A. A. Sen, and S. SenGupta, Geometric approach to modulus stabilization, Phys. Rev. D 92, 026008 (2015).
  10. S. Das, D. Maity, and S. SenGupta, Cosmological constant, brane tension and large hierarchy in a generalized Randall-Sundrum braneworld scenario, J. High Energy Phys. 05 (2008) 042.
  11. A. Dey, D. Maity, and S. SenGupta, Critical analysis of goldberger-wise stabilization of the Randall-Sundrum braneworld scenario, Phys. Rev. D 75, 107901 (2007).
  12. A. Das, H. Mukherjee, T. Paul, and S. SenGupta, Radion stabilization in higher curvature warped spacetime, Eur. Phys. J. C 78, 108 (2018).
  13. R. Koley, J. Mitra, and S. SenGupta, Modulus stabilization of the generalized Randall-Sundrum model with a bulk scalar field, Europhys. Lett. 85, 41001 (2009).
  14. T. Paul, “Brane localized energy density” stabilizes the modulus in higher dimensional warped spacetime, arXiv:1702.03722.
  15. I. Banerjee, T. Paul, and S. SenGupta, Critical analysis of modulus stabilization in a higher dimensional f(r) gravity, Phys. Rev. D 104, 104018 (2021).
  16. I. Banerjee and S. SenGupta, Modulus stabilization in a non-flat warped braneworld scenario, Eur. Phys. J. C 77, 277 (2017).
  17. A. Das, D. Maity, T. Paul, and S. SenGupta, Bouncing cosmology from warped extra dimensional scenario, Eur. Phys. J. C 77, 813 (2017).
  18. I. Banerjee, S. Chakraborty, and S. SenGupta, Radion induced inflation on nonflat brane and modulus stabilization, Phys. Rev. D 99, 023515 (2019).
  19. R. K. Mishra and L. Randall, Consequences of a stabilizing field’s self-interactions for RS cosmology, J. High Energy Phys. 12 (2023) 036.
  20. K. A. Milton, S. D. Odintsov, and S. Zerbini, Bulk versus brane running couplings, Phys. Rev. D 65, 065012 (2002).
  21. I. Brevik, K. A. Milton, S. Nojiri, and S. D. Odintsov, Quantum (in)stability of a brane-world universe at nonzero temperature, Nucl. Phys. B599, 305 (2001).
  22. C. Csaki, TASI lectures on extra dimensions and branes, arXiv:hep-ph/0404096.
  23. V. A. Rubakov, Large and infinite extra dimensions, Phys. Usp. 44, 871 (2001).
  24. R. Sundrum, TASI 2004 lectures: To the fifth dimension and back, arXiv:hep-th/0508134.
  25. W. D. Goldberger and M. B. Wise, Modulus stabilization with bulk fields, Phys. Rev. Lett. 83, 4922 (1999).
  26. C. Csáki, M. L. Graesser, and G. D. Kribs, Radion dynamics and electroweak physics, Phys. Rev. D 63, 065002 (2001).
  27. S. Giardino, First-order thermodynamics of modified gravity, Ph.D. thesis, University of Heidelberg (main), 2024.
  28. V. Faraoni, Black hole interiors in the thermal view of scalar-tensor gravity, Phys. Rev. D 112, L021504 (2025).
  29. S. Giardino, V. Faraoni, and A. Giusti, First-order thermodynamics of scalar-tensor cosmology, J. Cosmol. Astropart. Phys. 04 (2022) 053.
  30. V. Faraoni, A. Giusti, S. Jose, and S. Giardino, Peculiar thermal states in the first-order thermodynamics of gravity, Phys. Rev. D 106, 024049 (2022).
  31. V. Faraoni, S. Giardino, A. Giusti, and R. Vanderwee, Scalar field as a perfect fluid: Thermodynamics of minimally coupled scalars and Einstein frame scalar-tensor gravity, Eur. Phys. J. C 83, 24 (2023).
  32. A. Giusti, S. Giardino, and V. Faraoni, Past-directed scalar field gradients and scalar-tensor thermodynamics, Gen. Relativ. Gravit. 55, 47 (2023).
  33. S. Giardino, A. Giusti, and V. Faraoni, Thermal stability of stealth and de Sitter spacetimes in scalar-tensor gravity, Eur. Phys. J. C 83, 621 (2023).
  34. S. Giardino and A. Giusti, First-order thermodynamics of scalar-tensor gravity, Ricerche di matematica 74, 43 (2025).
  35. V. Faraoni and A. Giusti, Thermodynamics of scalar-tensor gravity, Phys. Rev. D 103, L121501 (2021).
  36. V. Faraoni, A. Giusti, and A. Mentrelli, New approach to the thermodynamics of scalar-tensor gravity, Phys. Rev. D 104, 124031 (2021).
  37. N. Karolinski and V. Faraoni, Einstein gravity as the thermal equilibrium state of a nonminimally coupled scalar field geometry, Phys. Rev. D 109, 084042 (2024).
  38. J. Houle and V. Faraoni, New phenomenology in the first-order thermodynamics of scalar-tensor gravity for Bianchi universes, Phys. Rev. D 110, 024067 (2024).
  39. V. Faraoni and T. B. Françonnet, Stealth metastable state of scalar-tensor thermodynamics, Phys. Rev. D 105, 104006 (2022).
  40. V. Faraoni, P.-A. Graham, and A. Leblanc, Critical solutions of nonminimally coupled scalar field theory and first-order thermodynamics of gravity, Phys. Rev. D 106, 084008 (2022).
  41. M. Miranda, P.-A. Graham, and V. Faraoni, Effective fluid mixture of tensor-multi-scalar gravity, Eur. Phys. J. Plus 138, 387 (2023).
  42. V. Faraoni and J. Côté, Imperfect fluid description of modified gravities, Phys. Rev. D 98, 084019 (2018).
  43. M. Miranda, S. Giardino, A. Giusti, and L. Heisenberg, First-order thermodynamics of Horndeski cosmology, Phys. Rev. D 109, 124033 (2024).
  44. A. Giusti, S. Zentarra, L. Heisenberg, and V. Faraoni, First-order thermodynamics of Horndeski gravity, Phys. Rev. D 105, 124011 (2022).
  45. V. Faraoni and J. Houle, More on the first-order thermodynamics of scalar-tensor and Horndeski gravity, Eur. Phys. J. C 83, 521 (2023).
  46. C. Eckart, The thermodynamics of irreversible processes. III. Relativistic theory of the simple fluid, Phys. Rev. 58, 919 (1940).
  47. V. Faraoni and A. Giusti, Thermal origin of the attractor-to-general-relativity in scalar-tensor gravity, Phys. Rev. Lett. 134, 211406 (2025).
  48. T. Damour and K. Nordtvedt, General relativity as a cosmological attractor of tensor-scalar theories, Phys. Rev. Lett. 70, 2217 (1993).
  49. T. Damour and K. Nordtvedt, Tensor-scalar cosmological models and their relaxation toward general relativity, Phys. Rev. D 48, 3436 (1993).
  50. A. Serna, J. Alimi, and A. Navarro, Convergence of scalar–tensor theories towards general relativity and primordial nucleosynthesis, Classical Quantum Gravity 19, 857 (2002).
  51. W. D. Goldberger and M. B. Wise, Phenomenology of a stabilized modulus, Phys. Lett. B 475, 275 (2000).
  52. L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83, 4690 (1999).
  53. J.-L. Lehners, Ekpyrotic and cyclic cosmology, Phys. Rep. 465, 223 (2008).
  54. Z. Chacko, R. K. Mishra, and D. Stolarski, Dynamics of a stabilized radion and duality, J. High Energy Phys. 09 (2013) 121.
  55. S. Bhattacharyya and S. SenGupta, Analyzing the general conditions for modulus stabilization in a warped braneworld, Eur. Phys. J. C 85, 1430 (2025).
  56. G. Gibbons, R. Kallosh, and A. Linde, Brane world sum rules, J. High Energy Phys. 01 (2001) 022.
  57. F. Leblond, R. C. Myers, and D. J. Winters, Consistency conditions for brane worlds in arbitrary dimensions, J. High Energy Phys. 07 (2001) 031.
  58. P. Breitenlohner and D. Z. Freedman, Stability in gauged extended supergravity, Ann. Phys. (N.Y.) 144, 249 (1982).
  59. I. Banerjee, T. Paul, and S. SenGupta, Bouncing cosmology in a curved braneworld, J. Cosmol. Astropart. Phys. 02 (2021) 041.
  60. I. Banerjee, T. Paul, and S. SenGupta, Critical analysis of modulus stabilization in a higher dimensional f(r) gravity, Phys. Rev. D 104, 104018 (2021).
  61. S. Nojiri and S. D. Odintsov, Modified gravity with negative and positive powers of curvature: Unification of inflation and cosmic acceleration, Phys. Rev. D 68, 123512 (2003).
  62. V. Faraoni, Matter instability in modified gravity, Phys. Rev. D 74, 104017 (2006).
  63. S. Chakraborty and S. SenGupta, Does the stability of f(R) theories imply the stability of the dual scalar-tensor theory ?, arXiv:2411.09992.
  64. O. Pujolas, I. Sawicki, and A. Vikman, The imperfect fluid behind kinetic gravity braiding, J. High Energy Phys. 11 (2011) 156.
  65. S. Chakraborty and S. SenGupta, Radion cosmology and stabilization, Eur. Phys. J. C 74, 3045 (2014).
  66. S. G. Elahi, S. S. Mandal, and S. SenGupta, Novel modulus stabilization mechanism in higher dimensional f(r) gravity, Phys. Rev. D 108, 044062 (2023).
  67. L. Cotta-Ramusino and D. Wands, Low-energy effective theory for a Randall-Sundrum scenario with a moving bulk brane, Phys. Rev. D 75, 104001 (2007).
  68. H. Cai, Radion dynamics in the multibrane Randall-Sundrum model, Phys. Rev. D 105, 075009 (2022).
  69. A. Nicolis, R. Rattazzi, and E. Trincherini, Galileon as a local modification of gravity, Phys. Rev. D 79, 064036 (2009).
  70. A. Padilla, P. M. Saffin, and S.-Y. Zhou, Bi-galileon theory I: Motivation and formulation, J. High Energy Phys. 12 (2010) 031.
  71. Data repository for this article, Figshare, 10.6084/m9.figshare.31343065 (2026).

Outline

Information

Phys. Rev. D 113, 064019– Published 9 March, 2026
  • Received 21 August 2025
  • Accepted 12 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation