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Dynamics of reheating in loop quantum cosmology

Yogesh1,2,*, Bao-Fei Li1,†, Mayukh R. Gangopadhyay3,‡, and Anzhong Wang4,§

  • *Contact author: yogesh@zjut.edu.cn, yogeshjjmi@gmail.com
  • †Contact author: libaofei@zjut.edu.cn
  • ‡Contact author: mayukhraj@gmail.com
  • §Contact author: Anzhong_Wang@baylor.edu
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APS Open Sci. 1, 000120 – Published 2 September, 2026

DOI: https://doi.org/10.1103/fz2y-fnzp

Abstract

In loop quantum cosmology (LQC), the initial singularity is replaced by a quantum bounce, leading to a universal postbounce evolution characterized by three distinct epochs: bouncing, transition, and slow-roll inflation, before the hot big-bang universe starts. While the generic nature of inflation in the presence of an inflaton field in LQC is well established, the subsequent reheating phase—the process that thermalizes the universe and marks the beginning of the hot big bang—has remained unexplored in this quantum gravitational framework. This paper presents the first comprehensive integration of the (generalized) reheating mechanism into the LQC paradigm. Using the power law plateau potential and comparing predictions with the latest Planck 2018 and ACT 2025 data, we demonstrate that the inclusion of a reheating phase with a generic equation of state is fully consistent with the cosmological constraints. In addition, using the observational data for the amplitude and spectral index of the scalar perturbations and the tensor-to-scalar ratio, we also constrain the total number of e-folds from the bounce to the present day and find a lower bound, which is less constrained than that obtained previously from the fitting of the high-l cosmic microwave background temperature power spectrum (TT), the polarization data (TT, TE, EE), and the low-l polarization data (lowP). We also show that at the horizon crossing, there exist six independent algebraic equations, which uniquely determine six physical parameters, once the amplitude and the power spectral index of the pivot mode of the scalar perturbations are provided. Three of the six parameters form a complete set of the initial conditions at the horizon crossing and uniquely determine the entire evolution of the universe, including both the pre- and postbounce phases, through the modified Friedmann and Klein-Gordon equations of LQC. This is sharply in contrast to the case without considering the reheating phase.

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References (137)

  1. A. H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D 23, 347 (1981).
  2. A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91, 99 (1980).
  3. A. D. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B 108, 389 (1982).
  4. A. H. Guth and S. Y. Pi, Fluctuations in the new inflationary universe, Phys. Rev. Lett. 49, 1110 (1982).
  5. A. D. Linde, Chaotic inflation, Phys. Lett. B 129, 177 (1983).
  6. D. Baumann, Inflation, in Theoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small (World Scientific Connect, Boulder, Colorado, 2011), pp. 523–686.
  7. E. Komatsu et al., Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmological interpretation, Astrophys. J. Suppl. Ser. 192, 18 (2011).
  8. P. A. R. Ade et al. (Planck Collaboration), Planck 2013 results. XXII. Constraints on inflation, Astron. Astrophys. 571, A22 (2014).
  9. P. A. R. Ade et al. (Planck Collaboration), Planck 2015 results, Astron. Astrophys. 594, A20 (2016).
  10. Y. Akrami et al. (Planck Collaboration), Planck 2018 results, Astron. Astrophys. 641, A10 (2020).
  11. P. A. R. Ade et al. (BICEP, Keck Collaboration), Improved constraints on primordial gravitational waves using Planck, WMAP, and BICEP/Keck observations through the 2018 observing season, Phys. Rev. Lett. 127, 151301 (2021).
  12. T. Louis et al. (ACT Collaboration), The atacama cosmology telescope: DR6 power spectra, likelihoods and ΛCDM parameters, J. Cosmol. Astropart. Phys. 11 (2025) 062.
  13. E. Calabrese et al. (ACT Collaboration), The atacama cosmology telescope: DR6 constraints on extended cosmological models, J. Cosmol. Astropart. Phys. 11 (2025) 063.
  14. J. Martin, C. Ringeval, and V. Vennin, Encyclopædia inflationaris, Phys. Dark Universe 5–6, 75 (2014).
  15. J. Martin and R. H. Brandenberger, Trans-Planckian problem of inflationary cosmology, Phys. Rev. D 63, 123501 (2001).
  16. R. H. Brandenberger and J. Martin, Trans-Planckian issues for inflationary cosmology, Classical Quantum Gravity 30, 113001 (2013).
  17. A. Borde, A. H. Guth, and A. Vilenkin, Inflationary spacetimes are incomplete in past directions, Phys. Rev. Lett. 90, 151301 (2003).
  18. A. Borde and A. Vilenkin, Eternal inflation and the initial singularity, Phys. Rev. Lett. 72, 3305 (1994).
  19. A. Ashtekar and P. Singh, Loop quantum cosmology: A status report, Class. Quantum Gravity 28, 213001 (2011).
  20. M. Bojowald, Quantum cosmology: A review, Rep. Prog. Phys. 78, 023901 (2015).
  21. B. Elizaga Navascués and G. A. M. Marugán, Hybrid loop quantum cosmology: An overview, Front. Astron. Space Sci. 8, 81 (2021).
  22. I. Agullo, A. Wang, and E. Wilson-Ewing, Loop quantum cosmology: Relation between theory and observations, in Handbook of Quantum Gravity, edited by C. Bambi, L. Modesto, and I. Shapiro (Springer Nature Singapore, Singapore, 2023), pp. 1–46.
  23. B.-F. Li and P. Singh, Loop quantum cosmology: Physics of singularity resolution and its implications, in Handbook of Quantum Gravity (Springer, Singapore, 2023).
  24. A. Ashtekar, W. Kaminski, and J. Lewandowski, Quantum field theory on a cosmological, quantum space-time, Phys. Rev. D 79, 064030 (2009).
  25. V. Taveras, Corrections to the Friedmann equations from loop quantum gravity for a universe with a free scalar field, Phys. Rev. D 78, 064072 (2008).
  26. M. Bojowald and A. Skirzewski, Effective equations of motion for quantum systems, Rev. Math. Phys. 18, 713 (2006).
  27. M. Bojowald, B. Sandhofer, A. Skirzewski, and A. Tsobanjan, Effective constraints for quantum systems, Rev. Math. Phys. 21, 111 (2009).
  28. I. Agullo, A. Ashtekar, and W. Nelson, Quantum gravity extension of the inflationary scenario, Phys. Rev. Lett. 109, 251301 (2012).
  29. I. Agullo, A. Ashtekar, and W. Nelson, Extension of the quantum theory of cosmological perturbations to the Planck era, Phys. Rev. D 87, 043507 (2013).
  30. I. Agullo, A. Ashtekar, and W. Nelson, The pre-inflationary dynamics of loop quantum cosmology: Confronting quantum gravity with observations, Class. Quantum Gravity 30, 085014 (2013).
  31. S. Iteanu and G. A. Mena Marugán, Mass of cosmological perturbations in the hybrid and dressed metric formalisms of loop quantum cosmology for the Starobinsky and exponential potentials, Universe 8, 463 (2022).
  32. B. Elizaga Navascués, D. Martin de Blas, and G. A. Mena Marugán, Time-dependent mass of cosmological perturbations in the hybrid and dressed metric approaches to loop quantum cosmology, Phys. Rev. D 97, 043523 (2018).
  33. L. Castelló Gomar, G. A. Mena Marugán, D. Martín De Blas, and J. Olmedo, Hybrid loop quantum cosmology and predictions for the cosmic microwave background, Phys. Rev. D 96, 103528 (2017).
  34. M. Fernandez-Mendez, G. A. Mena Marugan, and J. Olmedo, Hybrid quantization of an inflationary universe, Phys. Rev. D 86, 024003 (2012).
  35. M. Bojowald, G. M. Hossain, M. Kagan, and S. Shankaranarayanan, Gauge invariant cosmological perturbation equations with corrections from loop quantum gravity, Phys. Rev. D 79, 043505 (2009); 82, 109903(E) (2010).
  36. J. Mielczarek, T. Cailleteau, A. Barrau, and J. Grain, Anomaly-free vector perturbations with holonomy corrections in loop quantum cosmology, Class. Quantum Gravity 29, 085009 (2012).
  37. T. Cailleteau, J. Mielczarek, A. Barrau, and J. Grain, Anomaly-free scalar perturbations with holonomy corrections in loop quantum cosmology, Class. Quantum Gravity 29, 095010 (2012).
  38. T. Cailleteau, A. Barrau, J. Grain, and F. Vidotto, Consistency of holonomy-corrected scalar, vector, and tensor perturbations in loop quantum cosmology, Phys. Rev. D 86, 087301 (2012).
  39. T. Cailleteau, L. Linsefors, and A. Barrau, Anomaly-free perturbations with inverse-volume and holonomy corrections in loop quantum cosmology, Class. Quantum Gravity 31, 125011 (2014).
  40. A. Barrau, M. Bojowald, G. Calcagni, J. Grain, and M. Kagan, Anomaly-free cosmological perturbations in effective canonical quantum gravity, J. Cosmol. Astropart. Phys. 05 (2015) 051.
  41. M. Bojowald, G. Calcagni, and S. Tsujikawa, Observational constraints on loop quantum cosmology, Phys. Rev. Lett. 107, 211302 (2011).
  42. M. Bojowald, G. Calcagni, and S. Tsujikawa, Observational test of inflation in loop quantum cosmology, J. Cosmol. Astropart. Phys. 11 (2011) 046.
  43. T. Zhu, A. Wang, G. Cleaver, K. Kirsten, Q. Sheng, and Q. Wu, Detecting quantum gravitational effects of loop quantum cosmology in the early universe? Astrophys. J. 807, L17 (2015).
  44. T. Zhu, A. Wang, G. Cleaver, K. Kirsten, Q. Sheng, and Q. Wu, Scalar and tensor perturbations in loop quantum cosmology: High-order corrections, J. Cosmol. Astropart. Phys. 10 (2015) 052.
  45. T. Zhu, A. Wang, K. Kirsten, G. Cleaver, Q. Sheng, and Q. Wu, Inflationary spectra with inverse-volume corrections in loop quantum cosmology and their observational constraints from Planck 2015 data, J. Cosmol. Astropart. Phys. 03 (2016) 046.
  46. B. Bonga and B. Gupt, Inflation with the Starobinsky potential in loop quantum cosmology, Gen. Relativ. Gravitation 48, 71 (2016).
  47. B. Bonga and B. Gupt, Phenomenological investigation of a quantum gravity extension of inflation with the Starobinsky potential, Phys. Rev. D 93, 063513 (2016).
  48. M. Bojowald, Absence of a singularity in loop quantum cosmology, Phys. Rev. Lett. 86, 5227 (2001).
  49. A. Ashtekar, T. Pawlowski, and P. Singh, Quantum nature of the big bang, Phys. Rev. Lett. 96, 141301 (2006).
  50. A. Ashtekar, T. Pawlowski, and P. Singh, Quantum nature of the big bang: An analytical and numerical investigation, Phys. Rev. D 73, 124038 (2006).
  51. A. Ashtekar, T. Pawlowski, and P. Singh, Quantum nature of the big bang: Improved dynamics, Phys. Rev. D 74, 084003 (2006).
  52. A. Ashtekar, A. Corichi, and P. Singh, Robustness of key features of loop quantum cosmology, Phys. Rev. D 77, 024046 (2008).
  53. J. Yang, Y. Ding, and Y. Ma, Alternative quantization of the Hamiltonian in loop quantum cosmology, Phys. Lett. B 682, 1 (2009).
  54. B.-F. Li, S. Saini, and P. Singh, Primordial power spectrum from a matter-ekpyrotic bounce scenario in loop quantum cosmology, Phys. Rev. D 103, 066020 (2021).
  55. B.-F. Li, J. Olmedo, P. Singh, and A. Wang, Primordial scalar power spectrum from the hybrid approach in loop cosmologies, Phys. Rev. D 102, 126025 (2020).
  56. B.-F. Li, P. Singh, and A. Wang, Primordial power spectrum from the dressed metric approach in loop cosmologies, Phys. Rev. D 101, 086004 (2020).
  57. B.-F. Li, P. Singh, and A. Wang, Qualitative dynamics and inflationary attractors in loop cosmology, Phys. Rev. D 98, 066016 (2018).
  58. B.-F. Li, P. Singh, and A. Wang, Towards cosmological dynamics from loop quantum gravity, Phys. Rev. D 97, 084029 (2018).
  59. B.-F. Li, P. Singh, and A. Wang, Genericness of pre-inflationary dynamics and probability of the desired slow-roll inflation in modified loop quantum cosmologies, Phys. Rev. D 100, 063513 (2019).
  60. T. Zhu, A. Wang, G. Cleaver, K. Kirsten, and Q. Sheng, Pre-inflationary universe in loop quantum cosmology, Phys. Rev. D 96, 083520 (2017).
  61. T. Zhu, A. Wang, K. Kirsten, G. Cleaver, and Q. Sheng, Universal features of quantum bounce in loop quantum cosmology, Phys. Lett. B 773, 196 (2017).
  62. B.-F. Li, T. Zhu, A. Wang, K. Kirsten, G. Cleaver, and Q. Sheng, Preinflationary perturbations from the closed algebra approach in loop quantum cosmology, Phys. Rev. D 99, 103536 (2019).
  63. J. Qiao, G.-H. Ding, Q. Wu, T. Zhu, and A. Wang, Inflationary perturbation spectrum in extended effective field theory of inflation, J. Cosmol. Astropart. Phys. 09 (2019) 064.
  64. G. L. L. W. Levy and R. O. Ramos, α-attractor potentials in loop quantum cosmology, Phys. Rev. D 110, 043507 (2024).
  65. L. N. Barboza, G. L. L. W. Levy, L. L. Graef, and R. O. Ramos, Constraining the Barbero-Immirzi parameter from the duration of inflation in loop quantum cosmology, Phys. Rev. D 106, 103535 (2022).
  66. L. N. Barboza, L. L. Graef, and R. O. Ramos, Warm bounce in loop quantum cosmology and the prediction for the duration of inflation, Phys. Rev. D 102, 103521 (2020).
  67. A. Mohammadi, Exploring the pre-inflationary dynamics in loop quantum cosmology with a DBI scalar field, J. Cosmol. Astropart. Phys. 10 (2024) 062.
  68. L. L. Graef and R. O. Ramos, Probability of warm inflation in loop quantum cosmology, Phys. Rev. D 98, 023531 (2018).
  69. L. L. Graef, R. O. Ramos, and G. S. Vicente, Gravitational particle production in loop quantum cosmology, Phys. Rev. D 102, 043518 (2020).
  70. M. Gasperini and G. Veneziano, Pre-big-bang in string cosmology, Astropart. Phys. 1, 317 (1993).
  71. J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, Ekpyrotic universe: Colliding branes and the origin of the hot big bang, Phys. Rev. D 64, 123522 (2001).
  72. D. Battefeld and P. Peter, A critical review of classical bouncing cosmologies, Phys. Rep. 571, 1 (2015).
  73. R. Brandenberger and P. Peter, Bouncing cosmologies: Progress and problems, Found. Phys. 47, 797 (2017).
  74. A. Ijjas and P. J. Steinhardt, A new kind of cyclic universe, Phys. Lett. B 795, 666 (2019).
  75. A. Ijjas, P. J. Steinhardt, D. Garfinkle, and W. G. Cook, Smoothing and flattening the universe through slow contraction versus inflation, J. Cosmol. Astropart. Phys. 07 (2024) 077.
  76. A. Ashtekar and D. Sloan, Loop quantum cosmology and slow roll inflation, Phys. Lett. B 694, 108 (2010).
  77. A. Ashtekar and D. Sloan, Probability of inflation in loop quantum cosmology, Gen. Relativ. Gravitation 43, 3619 (2011).
  78. I. Agullo and P. Singh, Loop quantum cosmology, in Loop Quantum Gravity: The First 30 Years, edited by A. Ashtekar and J. Pullin (WSP, Singapore, 2017), pp. 183–240.
  79. A. Ashtekar and J. Pullin, The overview chapter in loop quantum gravity: The first 30 years, in Loop Quantum Gravity: The First 30 Years (100 Years of General Relativity) (World Scientific, Singapore, 2017).
  80. This is also true in modified LQCs [53, 59, 134, 135]. For the latter, we refer readers to Refs. [131, 136] and references therein.
  81. M. Shahalam, M. Sharma, Q. Wu, and A. Wang, Preinflationary dynamics in loop quantum cosmology: Power-law potentials, Phys. Rev. D 96, 123533 (2017).
  82. M. Shahalam, M. Sami, and A. Wang, Preinflationary dynamics of an α-attractor in loop quantum cosmology, Phys. Rev. D 98, 043524 (2018).
  83. M. Sharma, M. Shahalam, Q. Wu, and A. Wang, Preinflationary dynamics in loop quantum cosmology: Monodromy potential, J. Cosmol. Astropart. Phys. 11 (2018) 003.
  84. M. Sharma, T. Zhu, and A. Wang, Background dynamics of pre-inflationary scenario in Brans-Dicke loop quantum cosmology, Commun. Theor. Phys. 71, 1205 (2019).
  85. M. Shahalam, M. Al Ajmi, R. Myrzakulov, and A. Wang, Revisiting pre-inflationary universe of family of α-attractor in loop quantum cosmology, Class. Quantum Gravity 37, 195026 (2020).
  86. A. Ashtekar and B. Gupt, Quantum gravity in the sky: Interplay between fundamental theory and observations, Class. Quantum Gravity 34, 014002 (2017).
  87. P. Singh, K. Vandersloot, and G. V. Vereshchagin, Nonsingular bouncing universes in loop quantum cosmology, Phys. Rev. D 74, 043510 (2006).
  88. J. Mielczarek, T. Cailleteau, J. Grain, and A. Barrau, Inflation in loop quantum cosmology: Dynamics and spectrum of gravitational waves, Phys. Rev. D 81, 104049 (2010).
  89. X. Zhang and Y. Ling, Inflationary universe in loop quantum cosmology, J. Cosmol. Astropart. Phys. 08 (2007) 012.
  90. L. Chen and J.-Y. Zhu, Loop quantum cosmology: The horizon problem and the probability of inflation, Phys. Rev. D 92, 084063 (2015).
  91. K. Dimopoulos and C. Owen, Modelling inflation with a power-law approach to the inflationary plateau, Phys. Rev. D 94, 063518 (2016).
  92. A. R. Liddle, P. Parsons, and J. D. Barrow, Formalizing the slow-roll approximation in inflation, Phys. Rev. D 50, 7222 (1994).
  93. A. Riotto, Inflation and the theory of cosmological perturbations, ICTP Lect. Notes Ser. 14, 317 (2003).
  94. The physical wave number of the pivot mode at the bounce as labeled by k*/aB is uniquely fixed by the total number of e-foldings from the bounce point to the present. While NT has a lower bound at around 130 as given in Eq.  (35), it can also attain some higher values as shown in Fig.  8, which are able to make the physical wave number of the pivot mode very trans-Planckian at the bounce.
  95. A. D. Dolgov and A. D. Linde, Baryon asymmetry in the inflationary universe, Phys. Lett. B 116, 329 (1982).
  96. L. F. Abbott, E. Farhi, and M. B. Wise, Particle production in the new inflationary cosmology, Phys. Lett. B 117, 29 (1982).
  97. A. Albrecht, P. J. Steinhardt, M. S. Turner, and F. Wilczek, Reheating an inflationary universe, Phys. Rev. Lett. 48, 1437 (1982).
  98. J. Martin, C. Ringeval, and V. Vennin, Observing inflationary reheating, Phys. Rev. Lett. 114, 081303 (2015).
  99. B. A. Bassett, S. Tsujikawa, and D. Wands, Inflation dynamics and reheating, Rev. Mod. Phys. 78, 537 (2006).
  100. T. Rehagen and G. B. Gelmini, Low reheating temperatures in monomial and binomial inflationary models, J. Cosmol. Astropart. Phys. 06 (2015) 039.
  101. M. A. G. Garcia, K. Kaneta, Y. Mambrini, and K. A. Olive, Inflaton oscillations and post-inflationary reheating, J. Cosmol. Astropart. Phys. 04 (2021) 012.
  102. G. N. Felder, L. Kofman, and A. D. Linde, Tachyonic instability and dynamics of spontaneous symmetry breaking, Phys. Rev. D 64, 123517 (2001).
  103. J. H. Traschen and R. H. Brandenberger, Particle production during out-of-equilibrium phase transitions, Phys. Rev. D 42, 2491 (1990).
  104. Y. Shtanov, J. H. Traschen, and R. H. Brandenberger, Universe reheating after inflation, Phys. Rev. D 51, 5438 (1995).
  105. L. Kofman, A. D. Linde, and A. A. Starobinsky, Reheating after inflation, Phys. Rev. Lett. 73, 3195 (1994).
  106. L. Kofman, A. D. Linde, and A. A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D 56, 3258 (1997).
  107. L. Kofman, Reheating and preheating after inflation, in 3rd RESCEU International Symposium on Particle Cosmology (Universal Academy Press Inc., Tokyo, Japan, 1998), pp. 1–8.
  108. R. Allahverdi, R. Brandenberger, F.-Y. Cyr-Racine, and A. Mazumdar, Reheating in inflationary cosmology: Theory and applications, Annu. Rev. Nucl. Part. Sci. 60, 27 (2010).
  109. M. A. Amin, M. P. Hertzberg, D. I. Kaiser, and J. Karouby, Nonperturbative dynamics of reheating after inflation: A review, Int. J. Mod. Phys. D 24, 1530003 (2015).
  110. K. D. Lozanov, Lectures on reheating after inflation, arXiv:1907.04402.
  111. T. Zhu, Q. Wu, and A. Wang, An analytical approach to the field amplification and particle production by parametric resonance during inflation and reheating, Phys. Dark Universe 26, 100373 (2019).
  112. F. W. J. Olver, Introduction to Asymptotics and Special Functions (Acedemic Press, Cambridge, 1974).
  113. S. Habib, K. Heitmann, G. Jungman, and C. Molina-Paris, The inflationary perturbation spectrum, Phys. Rev. Lett. 89, 281301 (2002).
  114. A. Wang, Vector and tensor perturbations in Horava-Lifshitz cosmology, Phys. Rev. D 82, 124063 (2010).
  115. T. Zhu, A. Wang, G. Cleaver, K. Kirsten, and Q. Sheng, Constructing analytical solutions of linear perturbations of inflation with modified dispersion relations, Int. J. Mod. Phys. A 29, 1450142 (2014).
  116. T. Zhu, A. Wang, G. Cleaver, K. Kirsten, and Q. Sheng, Inflationary cosmology with nonlinear dispersion relations, Phys. Rev. D 89, 043507 (2014).
  117. T. Zhu and A. Wang, Gravitational quantum effects in light of BICEP2 results, Phys. Rev. D 90, 027304 (2014).
  118. T. Zhu, A. Wang, G. Cleaver, K. Kirsten, and Q. Sheng, Gravitational quantum effects on power spectra and spectral indices with higher-order corrections, Phys. Rev. D 90, 063503 (2014).
  119. T. Zhu, A. Wang, G. Cleaver, K. Kirsten, and Q. Sheng, Power spectra and spectral indices of k-inflation: High-order corrections, Phys. Rev. D 90, 103517 (2014).
  120. Q. Wu, T. Zhu, and A. Wang, Primordial spectra of slow-roll inflation at second-order with the Gauss-Bonnet correction, Phys. Rev. D 97, 103502 (2018).
  121. B.-F. Li, T. Zhu, and A. Wang, Langer modification, quantization condition and barrier penetration in quantum mechanics, Universe 6, 90 (2020).
  122. G.-H. Ding, J. Qiao, Q. Wu, T. Zhu, and A. Wang, Inflationary perturbation spectra at next-to-leading slow-roll order in effective field theory of inflation, Eur. Phys. J. C 79, 976 (2019).
  123. J. Qiao, T. Zhu, W. Zhao, and A. Wang, Polarized primordial gravitational waves in the ghost-free parity-violating gravity, Phys. Rev. D 101, 043528 (2020).
  124. R. Pan, J. J. Marchetta, J. Saeed, G. Cleaver, B.-F. Li, A. Wang, and T. Zhu, Uniform asymptotic approximation method with Pöschl–Teller potential, Universe 9, 471 (2023).
  125. R. Pan, J. Saeed, and A. Wang, Stable initial conditions and analytically approximate solutions of cosmological perturbations in a modified loop quantum cosmology, J. Cosmol. Astropart. Phys. 06 (2026) 030.
  126. J. L. Cook, E. Dimastrogiovanni, D. A. Easson, and L. M. Krauss, Reheating predictions in single field inflation, J. Cosmol. Astropart. Phys. 04 (2015) 047.
  127. H. A. Khan and Yogesh, Study of Goldstone inflation in the domain of Einstein-Gauss-Bonnet gravity, Phys. Rev. D 105, 063526 (2022).
  128. M. R. Gangopadhyay, H. A. Khan, and Yogesh, A case study of small field inflationary dynamics in the Einstein–Gauss–Bonnet framework in the light of GW170817, Phys. Dark Universe 40, 101177 (2023).
  129. R. Adhikari, M. R. Gangopadhyay, and Yogesh, Lower tensor to scalar ratio in a SUGRA motivated inflationary potential, Gravitation Cosmol. 28, 1 (2022).
  130. R. Kallosh and A. Linde, On the present status of inflationary cosmology, Gen. Relativ. Gravitation 57, 135 (2025).
  131. B.-F. Li, P. Singh, and A. Wang, Phenomenological implications of modified loop cosmologies: An overview, Front. Astron. Space Sci. 8, 701417 (2021).
  132. Note that the parameter ηH used in this paper is the same as δ introduced in Ref. [30] but has a sign difference from η introduced in Ref.  [6], i.e., ηH=δ=−η.
  133. In all the analysis given in the Appendix, we assume that the potential V(ϕ) depends only on one free parameter, V0. This is the case for various viable potentials [10]. However, there are cases in which V(ϕ) depends on several parameters. In these cases, to determine the free parameters of the potentials, additional physics needs to be taken into account [137].
  134. A. Dapor and K. Liegener, Cosmological effective Hamiltonian from full loop quantum gravity dynamics, Phys. Lett. B 785, 506 (2018).
  135. M. Assanioussi, A. Dapor, K. Liegener, and T. Pawłowski, Emergent de Sitter epoch of the quantum cosmos from loop quantum cosmology, Phys. Rev. Lett. 121, 081303 (2018).
  136. J. Saeed, R. Pan, C. Brown, G. Cleaver, G. Clevear, and A. Wang, Universal properties of the evolution of the universe in modified loop quantum cosmology, Universe 10, 397 (2024).
  137. S. Bhattacharya, K. Das, and M. R. Gangopadhyay, Probing the era of reheating for reconstructed inflationary potential in the RS II braneworld, Class. Quantum Gravity 37, 215009 (2020).

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APS Open Sci. 1, 000120– Published 2 September, 2026
Vol. 1
  • Received 13 August 2024
  • Revised 12 July 2026
  • Accepted 19 August 2026
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