[Paper Review] Primordial reheating in $f(R)$ cosmology by spontaneous decay of scalarons

Arun Mathew, Malay K. Nandy|arXiv (Cornell University)|Dec 27, 2020

Cosmology and Gravitation Theories4 citations

TL;DR

This paper proposes a reheating mechanism in $f(R)$ gravity where scalarons—quantum excitations of the Ricci scalar—spontaneously decay to produce particles, driven by Heisenberg's uncertainty principle. It shows that reheating begins after inflation with a preheating phase lasting $\sim 10^5\,t_{\rm P}$, followed by thermalization, reaching a reheating temperature of $T_r \sim 10^{13}$ GeV and a maximum energy density at $t_r \sim 10^{11}\,t_{\rm P}$, all governed by a single parameter: the scalaron mass $M \sim 10^{-5}\,M_{\rm P}$. The mechanism naturally explains the transition to a radiation-dominated era without an inflaton field.

ABSTRACT

We employ a viable $f(R)$ gravity model capable of giving an inflationary phase in order to study the subsequent reheating phase due to particle creation at the expense of energy in the scalaron field. Since quantum mechanics is expected to play a dominant role in particle creation, we formulate a plausible scenario of reheating obeying Heisenberg's uncertainty principle that imposes constraints on the particles created in the configuration space. We show that, so long as the energy available in the scalaron field is sufficient to populate the entire configuration space, the energy density of the particles grows, attaining a maximum value giving an efficient reheating. Beyond this maximum, the available energy becomes insufficient to populate the entire configuration space leading to a declining energy density. We further find that there is a negligible growth of energy density in the inflationary phase that lasts for $\\sim 10^7 \\, t_{\ m P}$, although particles are constantly created in this phase. The subsequent reheating phase spans for $\\sim10^{11} \\, t_{\ m P}$ and it begins with a well-defined preheating stage lasting for $\\sim 10^{5} \\, t_{\ m P}$, making a cross-over to a thermilization regime. The temperature at the beginning of the thermilization is found to be $T_{\ m th}\\sim 10^{12}$ GeV, whereas the reheating temperature is estimated as $T_{r}\\sim10^{13}$ GeV. Importantly, these estimates follow from a single parameter, the scalaron mass, $M\\sim10^{-5} \\, M_{\ m P}$.

Motivation & Objective

• To understand the reheating phase following inflation in viable $f(R)$ gravity models without a fundamental inflaton field.
• To model particle creation from scalaron decay using quantum mechanical constraints, particularly Heisenberg’s uncertainty principle.
• To determine the duration and efficiency of reheating, including the onset of thermalization and the final reheating temperature.
• To identify the physical conditions marking the end of reheating, based on energy sufficiency and configuration space population.

Proposed method

• Formulates a viable $f(R)$ gravity model that supports an inflationary phase via a scalaron field arising from higher-order curvature terms.
• Applies Heisenberg’s uncertainty principle to constrain particle creation in configuration space, ensuring energy conservation during decay.
• Uses a Schrödinger-like equation for scalaron wave function dynamics under time-varying Ricci scalar and scale factor.
• Estimates particle production rate via $\Gamma \sim R^2 / \sqrt{\epsilon}$, with $\epsilon$ related to the $R^2$ term in the action.
• Analyzes energy density evolution $\rho(t)$ using a balance between source term (particle creation) and Hubble expansion term.
• Defines the end of reheating as the point where available energy in scalarons just suffices to fully populate the configuration space, marked by $\dot{\rho} \to 0$.

Experimental results

Research questions

• RQ1 How does particle creation from scalaron decay in $f(R)$ gravity lead to efficient reheating without an inflaton field?
• RQ2 What role does Heisenberg’s uncertainty principle play in limiting and shaping the particle creation process during reheating?
• RQ3 When does the preheating phase end and thermalization begin in this scenario, and what determines the transition?
• RQ4 What is the maximum energy density and corresponding reheating temperature achievable in this model?
• RQ5 How does the scalaron mass $M \sim 10^{-5}M_{\rm P}$ determine the duration and efficiency of reheating?

Key findings

• The reheating phase spans approximately $10^{11}\,t_{\rm P}$, beginning with a preheating stage lasting $\sim 4.0572 \times 10^5\,t_{\rm P}$, during which particles are created but not in thermal equilibrium.
• The temperature at the onset of thermalization is estimated at $T_{\rm th} \sim 10^{12}$ GeV, with the final reheating temperature reaching $T_r \sim 10^{13}$ GeV.
• The maximum energy density $\rho_r$ is achieved at $t_r \sim 10^{11}\,t_{\rm P}$, when the available energy in the scalaron field exactly matches the energy required to populate the entire configuration space.
• Energy density growth is negligible during the inflationary phase due to quasi-de-Sitter expansion, despite continuous particle creation.
• The system transitions from preheating to thermalization when the collision rate $\Gamma_{\rm coll}$ exceeds the Hubble expansion rate $H$, marking the onset of thermal equilibrium.
• The final state of reheating is characterized by $\dot{\rho} \to 0$, indicating that the energy density has reached a maximum and begins to decline due to insufficient energy to sustain further population of configuration space.