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 ($d\ge4$).

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 $H(\tau)=\dot a/a$.

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 $V(\varphi)=-\frac{\varepsilon}{m+1}|\varphi|^{m+1}$.

For global solutions arising from sufficiently small, spatially localized initial data, we introduce the conformal rescaling $\phi=a^{(d-2)/2}\varphi$, 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 $m_{\mathrm{conf}}=\frac{d+2}{d-2}$ 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 $1<m<m_{\mathrm{conf}}$.

For accelerated conformal expansion $H(\tau)=H_0(1+\tau)^{-\alpha}$ with $0\le\alpha<1$ and for superconformal interactions $m>m_{\mathrm{conf}}$, we prove that $g\in L^1([0,\infty))$ 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.