The Dynamics of Reheating in Loop Quantum Cosmology

TL;DR

The paper addresses incorporating the reheating epoch into loop quantum cosmology (LQC), linking the quantum bounce to the hot Big Bang within a unified cosmological history. It adopts the Power Law Plateau (PLP) potential and analyzes background evolution, slow-roll inflation, and a generalized reheating phase with an adjustable equation of state, then confronts the model with Planck 2018 and ACT 2025 data. Key results include consistency of reheating in LQC with current observations and a robust lower bound on the total number of e-folds from the bounce to today, $N_T \gtrsim 130$, derived from $A_s$, $n_s$, and $r$ (distinct from prior high-$l$ CMB fits). The work provides a coherent framework tying quantum geometric effects to observable thermal history and sets the stage for extensions to other potentials and LQC variants, with implications for primordial perturbations and early-universe phenomenology.

Abstract

In loop quantum cosmology (LQC), the initial singularity is replaced by a quantum bounce, leading to a universal post-bounce 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 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-$l$ CMB temperature power spectrum (TT), the polarization data (TT, TE, EE) and the low-$l$ polarization data (lowP).

Figures (8)

• Figure 1: Numerical Evolution of the background quantities for the potential (\ref{['potential']}), for the kinetic energy dominated at bounce with $\dot \phi>0$. Fig. \ref{['atm1phidpos']} represents the evolution of the scale factor $a(t)$ for the different initial values of the field $\phi$ at the bounce. The black dotted line is the representative of the analytical solution. Fig. \ref{['wtm1phidpos']} shows the evolution of the equation of state $(\omega(\phi))$ parameter for the different initial values of the $\phi$. In Fig. \ref{['eHm1phidpos']} we show the solution for the first slow roll parameter $(\epsilon_{H})$ and in Fig. \ref{['energym1phidpos']} shows the comparison between the potential $V(\phi)$, kinetic energy density $\dot \phi^2 /2$ along with the total energy density $\rho = \dot \phi ^2 /2 + V(\phi)$, here we take the initial field value to be $\phi_B=-4$ keeping $\dot \phi>0$.
• Figure 2: Evolution of e-folds for the slow roll inflation against the different choices of $\phi_B$. In the left panel, we have considered $\dot \phi<0$ whereas in the right panel, we consider $\dot \phi>0$. Three horizontal dotted lines green, blue, and red denote the $N_{\inf}=50,60,$ and $70$ respectively.
• Figure 3: Numerical Evolution of the background quantities for the potential (\ref{['potential']}), for the kinetic energy dominated at bounce with $\dot \phi<0$. Fig \ref{['atm1phidneg']} represents the evolution of the scale factor $a(t)$ for the different initial values of the field $\phi$. The black dotted is for the analytical solution. Fig \ref{['wtm1phidneg']} shows the evolution of the equation of state $(\omega(\phi))$ parameter for the different initial values of the $\phi$. In fig \ref{['eHm1phidneg']} we show the solution for the first slow roll parameter $(\epsilon_{H})$ and in fig \ref{['energym1phidneg']} shows the comparison between the potential $V(\phi)$, kinetic energy density $\dot \phi^2 /2$ along with the total energy density $\rho = \dot \phi ^2 /2 + V(\phi)$, here we take the initial filed value to be $\phi_B=4$ keeping $\dot \phi<0$.
• Figure 4: Horizon-crossing (exit) for a given mode $k$ at $t_*$. The line of $\ln L_H \equiv \ln(1/a H)$ is the Hubble horizon. At the exit, we have $k = a(t_*) H(t_*)$. The pivot mode $k_*$ used by Planck 2018 is $k_*/a_0 = 0.05$/Mpc Planck:2018jri.
• Figure 5: Figure for $(r-n_s)$ with different numbers of e-fold during the inflation, along with the initial field value $\phi_B$ at the bounce. The two light and dark pink (green) contours are for Planck BICEP/Keck Array 2018 (ACT 2025 + Planck BICEP/Keck Array 2018) Planck:2018jriBICEP:2021xfzACT:2025fjuACT:2025tim, respectively for 1$\sigma$ and 2$\sigma$ bounds. Here the e-fold $N$ is defined as $N \equiv N_* = \ln(a_{\text{end}}/a_*)$, as defined in Eq.(\ref{['Ninf_general']}).