Redshift Suppression of Nonlinear Scalar Fields in Accelerating FLRW Spacetimes

Abstract

We study small--data solutions of a nonlinear scalar field equation on spatially flat $d$--dimensional FLRW spacetimes ($d\ge4$). In conformal time $τ$ the field satisfies a damped semilinear wave/Klein--Gordon equation with time--dependent coefficients determined by the scale factor $a(τ)$ and the conformal Hubble rate $H(τ)=\dot a/a$. We focus on accelerated conformal expansion of the form $H(τ)=H_0(1+τ)^{-α}$ with $H_0>0$ and $0\leα<1$, for which $a(τ)$ grows stretched--exponentially, and we assume a power potential $V(φ)=-\frac{\varepsilon}{m+1}|φ|^{m+1}$. For global solutions arising from sufficiently small, spatially localized initial data, we introduce the conformal rescaling $ϕ=a^{(d-2)/2}φ$, 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 $g(τ)=a(τ)^σ$ with $σ=\frac{d+2-(d-2)m}{2}$, so the conformal power $m_{\mathrm{conf}}=\frac{d+2}{d-2}$ is the sharp threshold for redshift suppression: $g$ decays for $m>m_{\mathrm{conf}}$, is constant for $m=m_{\mathrm{conf}}$ (classical conformal invariance), and grows for $1 m_{\mathrm{conf}}$, we prove that $g\in L^1([0,\infty))$ and deduce small--data global existence together with scattering/asymptotic linearization for $ϕ$. 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.