Complex Inflaton Potentials with Nonminimal Coupling: Robust Inflation and Geometric Reheating

Abstract

We investigate an inflationary scenario driven by a complex scalar field nonminimally coupled to gravity and subject to a non-symmetric complex potential. The real part of the potential controls the cosmological background and realizes a plateau-type inflation compatible with $α$-attractor $\mathrm{T}$-models, while the imaginary part acts as an effective non-Hermitian deformation encoding dissipative effects. Working in the Jordan frame and imposing ghost-free conditions on the effective Planck mass, we derive the background equations and define a complex equation-of-state parameter whose real part governs the expansion and whose imaginary part quantifies departures from conservative dynamics. Numerical integration shows that the duration of inflation is primarily controlled by the nonminimal coupling $ζ$, whereas the complex asymmetry parameter $Δ\varepsilon$ has a negligible impact on the real background: the real energy density and pressure vary by less than $10^{-5}$ as $Δ\varepsilon$ is scanned over its allowed range. Mapping the two-field dynamics to an effective single-field description in the Einstein frame, we obtain a spectral index $n_s\simeq 0.968-0.971$ and a tensor-to-scalar ratio $r<10^{-3}$, fully consistent with Planck 2018 bounds. We introduce a relevance parameter and show that non-Hermitian effects remain strongly suppressed during slow roll but grow to $\mathcal{O}(1)$ near the end of inflation, triggering an efficient reheating phase without additional fields or {\it ad hoc} friction terms. In this sense, the imaginary sector behaves as an effective $\mathcal{PT}$-symmetric channel for energy transfer, providing a geometrical mechanism for inflation and its exit within a non-Hermitian scalar-tensor framework.

Figures (5)

• Figure 1: Spectral index $n_s$ versus tensor-to-scalar ratio $r$ for the three best-fit parameter sets listed in Table \ref{['tab:comparison']} The points lie in the region $n_s \simeq 0.961-0.969$ and $r<0.06$, well inside the 1$\sigma$ and 2$\sigma$ Planck 2018 confidence contours (blue regions), showing that the complex inflaton model with nonminimal coupling is consistent with current CMB constraints.
• Figure 2: Number of e-folds $N_{\text{end}}(\zeta,\Delta\varepsilon)$ as a function of the nonminimal coupling $\zeta$ and the asymmetry parameter $\Delta\varepsilon$. Left: $\mu = 0.1100$ with $\zeta \in [0.1600,0.1670]$ near the conformal value. Right: $\mu = 0.1000$ with $\zeta \in [-0.0060,0.0050]$. In both panels, $\Delta\varepsilon \in [-1.0,1.0]$. The maps show that $\zeta$ controls the duration of inflation, while $\Delta\varepsilon$ has a negligible impact on the real background evolution.
• Figure 3: $S_{\rho}^{\max}$ remains extremely small across the scanned parameter space, confirming that the real background evolution is essentially insensitive to the non-Hermitian deformation of the potential.
• Figure 4: Relevance parameter $\mathcal{R}(N)$ given by equation \ref{['eq:relevance']} as a function of the number of e-folds $N = \ln a$ for the parameter choices of Table \ref{['tab:comparison']}. For small $N$, the imaginary sector contributes a sizeable fraction of the total energy density ($R \sim \mathcal{O}(1)$), but its relevance decreases as inflation proceeds and becomes negligible near and after $N_{\text{end}}$, where reheating takes place.
• Figure 5: Absolute values of the real and imaginary parts of the effective equation of state, $|w_R(N)|$ and $|w_I(N)|$, as functions of the number of e-folds $N = \ln a$. The plots illustrate that during slow roll the Universe undergoes a quasi--de Sitter phase with $w_R \simeq -1$ and $|w_I| \ll 1$, while near the end of inflation $|w_I|$ grows and transiently enhances dissipation, initiating the reheating phase.