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Reviving Quadratic Inflation: Minimal Deformation for CMB Compatibility and Reheating Consistency

Khedidja Djeha

TL;DR

This work demonstrates that a minimal higher-order deformation, implemented as a noncanonical kinetic term K(φ) = 1 + γ φ^6 and a field redefinition to a canonical χ, flattens the effective potential and brings the quadratic inflation scenario into agreement with Planck constraints. The canonicalized potential Ṽ(χ) ≈ 1/2 m^2 χ^2 − 1/14 m^2 γ χ^8 produces $n_s$ near 0.965 and a suppressed tensor-to-scalar ratio around 0.036, while also modifying the reheating phase to a slightly negative equation of state, which raises the reheating temperature by a factor ~3.4 for fixed N_reh. Numerical results show the deformed model fits the CMB data well and allows a more flexible post-inflationary history, including altered end-of-inflation energies and reheating dynamics. The study highlights that controlled deformations of simple inflationary setups can reconcile observational constraints with theoretical simplicity and motivate further exploration of EFT embeddings and observational signatures in the reheating era.

Abstract

We revisit the quadratic inflationary potential by introducing a minimal higher-order correction obtained through a simple field redefinition, leading to the potential V(chi) = (1/2) m^2 * (chi - (gamma/14) * chi^7)^2. While the uncorrected quadratic model predicts n_s approximately 0.967 and r approximately 0.13, in strong tension with CMB data, the corrected potential yields n_s approximately 0.965 and r approximately 0.036, fully consistent with Planck 2018 constraints. Beyond inflationary observables, the deformation also impacts the reheating phase. In the quadratic case, reheating corresponds to a matter-like regime with w_reh = 0, whereas the corrected potential gives w_reh approximately -0.011, a slightly softer equation of state. This modification raises the reheating temperature by a factor of about 3.4 (for N_reh = 10), or equivalently extends the reheating duration at fixed temperature. Our results demonstrate that even a minimal higher-order correction is sufficient to reconcile the quadratic model with observations while providing a more consistent post-inflationary history, highlighting the relevance of controlled deformations of simple inflationary scenarios.

Reviving Quadratic Inflation: Minimal Deformation for CMB Compatibility and Reheating Consistency

TL;DR

This work demonstrates that a minimal higher-order deformation, implemented as a noncanonical kinetic term K(φ) = 1 + γ φ^6 and a field redefinition to a canonical χ, flattens the effective potential and brings the quadratic inflation scenario into agreement with Planck constraints. The canonicalized potential Ṽ(χ) ≈ 1/2 m^2 χ^2 − 1/14 m^2 γ χ^8 produces near 0.965 and a suppressed tensor-to-scalar ratio around 0.036, while also modifying the reheating phase to a slightly negative equation of state, which raises the reheating temperature by a factor ~3.4 for fixed N_reh. Numerical results show the deformed model fits the CMB data well and allows a more flexible post-inflationary history, including altered end-of-inflation energies and reheating dynamics. The study highlights that controlled deformations of simple inflationary setups can reconcile observational constraints with theoretical simplicity and motivate further exploration of EFT embeddings and observational signatures in the reheating era.

Abstract

We revisit the quadratic inflationary potential by introducing a minimal higher-order correction obtained through a simple field redefinition, leading to the potential V(chi) = (1/2) m^2 * (chi - (gamma/14) * chi^7)^2. While the uncorrected quadratic model predicts n_s approximately 0.967 and r approximately 0.13, in strong tension with CMB data, the corrected potential yields n_s approximately 0.965 and r approximately 0.036, fully consistent with Planck 2018 constraints. Beyond inflationary observables, the deformation also impacts the reheating phase. In the quadratic case, reheating corresponds to a matter-like regime with w_reh = 0, whereas the corrected potential gives w_reh approximately -0.011, a slightly softer equation of state. This modification raises the reheating temperature by a factor of about 3.4 (for N_reh = 10), or equivalently extends the reheating duration at fixed temperature. Our results demonstrate that even a minimal higher-order correction is sufficient to reconcile the quadratic model with observations while providing a more consistent post-inflationary history, highlighting the relevance of controlled deformations of simple inflationary scenarios.

Paper Structure

This paper contains 29 sections, 58 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical Setup
  3. Inflationary Dynamics and Numerical Methods
  4. Background Equations of Motion
  5. Numerical Methodology
  6. Initial Conditions and Assumptions
  7. Numerical Analysis of Cosmological Parameters
  8. Reheating Dynamics
  9. Numerical inputs and end-of-inflation energies.
  10. Scenario A (fixed $N_{\rm reh}=10$):
  11. Scenario B (fixed $T_{\rm reh}=10^{9}\ \mathrm{GeV}$):
  12. Takeaway.
  13. Comparison of Reheating Dynamics
  14. Conclusion and Outlook
  15. Detailed Derivation of the Canonical Potential from a Non-Canonical Kinetic Term
  16. ...and 14 more sections

Figures (2)

  • Figure 1: Comparison between the observational constraints from Planck, Planck+BK14, and Planck+BAO at 68% and 95% confidence levels (colored contours), and the theoretical predictions in the $n_s - r$ plane. The green points correspond to the standard quadratic model ($\gamma = 0$), which lies outside the observationally allowed region. In contrast, the blue trajectory corresponds to the modified model ($\gamma \neq 0$), which falls inside the 68% CL region and approaches the best-fit point (red star). This highlights the improved consistency of the deformed quadratic potential with current cosmological observations.
  • Figure 2: Comparison of reheating dynamics in quadratic and deformed potentials, showing the relation between $N_{\rm reh}$ and $T_{\rm reh}$.