Redshift Suppression of Nonlinear Scalar Fields in Accelerating FLRW Spacetimes

We study small–data solutions of a nonlinear scalar field equation on spatially flat $d$–dimensional FLRW spacetimes ($d\ge 4$). In conformal time $\tau$ the field satisfies a damped semilinear wave/Klein–Gordon equation with time–dependent coefficients determined by the scale factor $a(\tau )$ and the conformal Hubble rate $H(\tau )=\dot{a}\mathrm{/}a$. We focus on accelerated conformal expansion of the form $H(\tau )=H_0(1+\tau {)}^{-\alpha }$ with $H_0>0$ and $0\le \alpha <1$, for which $a(\tau )$ grows stretched–exponentially, and we assume a power potential $V(\phi )=-\frac{\epsilon }{m+1}\mathrm{\mid }\phi {\mathrm{\mid }}^{m+1}$. For global solutions arising from sufficiently small, spatially localized initial data, we introduce the conformal rescaling $\phi =a^{(d-2)\mathrm{/}2}\phi$, which removes the first–order Hubble damping and exposes the interaction as a \emph{time–dependent coupling}. In the rescaled equation the nonlinearity is weighted by $g(\tau )=a(\tau {)}^{\sigma }$ with $\sigma =\frac{d+2-(d-2)m}{2}$, so the conformal power $m_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}=\frac{d+2}{d-2}$ is the sharp threshold for redshift suppression: $g$ decays for $m>m_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$, is constant for $m=m_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$ (classical conformal invariance), and grows for $1<m<m_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$. For accelerated conformal expansion $H(\tau )=H_0(1+\tau {)}^{-\alpha }$ with $0\le \alpha <1$ and for superconformal interactions $m>m_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}$, we prove that $g\in L^1([0,\mathrm{\infty }))$ and deduce small–data global existence together with scattering/asymptotic linearization for $\phi$. As a complementary result in the diffusion–dominated regime $1<m<1+\frac{2}{d-1}$, we adapt a weighted energy method for variable damping to deduce explicit $L^2$ and $L^1$ decay rates. These bounds provide a quantitative PDE formulation of redshift–induced suppression of nonlinear scalar self–interactions at late conformal times.