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Boundedness and decay of waves on spatially flat decelerated FLRW spacetimes

Mahdi Haghshenas

TL;DR

This work analyzes the linear wave equation on spatially flat decelerated FLRW spacetimes, capturing how cosmological expansion influences energy and decay. By deploying twisted $t$-weighted multipliers and a Dafermos–Rodnianski $r^p$-method, the authors derive uniform energy bounds, integrated local energy decay, and a hierarchical array of $r^p$-weighted estimates across the entire decelerated regime, yielding both energy and pointwise decay. The results distinguish the radiation case $q=\tfrac{1}{2}$—where the problem exhibits a near-Minkowskian structure—and delineate how decay rates degrade as $q$ moves away from this threshold, with explicit rates depending on $\sigma_q$ and related quantities. The findings significantly advance understanding of wave dispersion in expanding spacetimes and lay groundwork for nonlinear extensions and stability analyses in cosmological contexts.

Abstract

We study the linear wave equation on a class of spatially homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes in the decelerated regime with spatial topology $\mathbb{R}^3$. Employing twisted $t$-weighted multiplier vector fields, we establish uniform energy bounds and derive integrated local energy decay estimates across the entire range of the decelerated expansion regime. Furthermore, we obtain a hierarchy of $r^p$-weighted energy estimates à la the Dafermos-Rodnianski $r^p$-method, which leads to energy decay estimates. As a consequence, we demonstrate pointwise decay estimates for solutions and their derivatives. In the wave zone, this pointwise decay is optimal in the "radiation" and "sub-radiation" cases, and almost optimal around the radiation case.

Boundedness and decay of waves on spatially flat decelerated FLRW spacetimes

TL;DR

This work analyzes the linear wave equation on spatially flat decelerated FLRW spacetimes, capturing how cosmological expansion influences energy and decay. By deploying twisted -weighted multipliers and a Dafermos–Rodnianski -method, the authors derive uniform energy bounds, integrated local energy decay, and a hierarchical array of -weighted estimates across the entire decelerated regime, yielding both energy and pointwise decay. The results distinguish the radiation case —where the problem exhibits a near-Minkowskian structure—and delineate how decay rates degrade as moves away from this threshold, with explicit rates depending on and related quantities. The findings significantly advance understanding of wave dispersion in expanding spacetimes and lay groundwork for nonlinear extensions and stability analyses in cosmological contexts.

Abstract

We study the linear wave equation on a class of spatially homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes in the decelerated regime with spatial topology . Employing twisted -weighted multiplier vector fields, we establish uniform energy bounds and derive integrated local energy decay estimates across the entire range of the decelerated expansion regime. Furthermore, we obtain a hierarchy of -weighted energy estimates à la the Dafermos-Rodnianski -method, which leads to energy decay estimates. As a consequence, we demonstrate pointwise decay estimates for solutions and their derivatives. In the wave zone, this pointwise decay is optimal in the "radiation" and "sub-radiation" cases, and almost optimal around the radiation case.

Paper Structure

This paper contains 28 sections, 21 theorems, 245 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Initial value problem
  3. Main results
  4. Related works
  5. Explicit representation formulae via transformation
  6. Conformal weakly dissipative wave equation
  7. Numerical works
  8. Non-linear stability and instability results
  9. Overview of proofs of main results
  10. Waves on Minkowski space
  11. The radiation case $q=\frac{1}{2}$
  12. The case $q\neq\frac{1}{2}$
  13. Organisation of the paper
  14. Preliminaries
  15. The twisted energy-momentum tensor
  16. ...and 13 more sections

Key Result

Theorem 1.1

Suppose $0<q<1$, and $\psi$ solves the initial value problem eq:intro:IVP-wave. Then for any $\tau_2>\tau_1\geq \tau_0$, we have Moreover, for any $\varepsilon>0$, we also have $\mathcal{E}_q^\varepsilon[\psi] (\tau_2) \lesssim \mathcal{E}_q^\varepsilon[\psi] (\tau_1)\,.$

Figures (2)

  • Figure 1: Penrose diagram of FLRW spacetimes \ref{['eq:intro:Metric-x']} for $0<q<1$, with the spacelike hypersurface $\{t=1\}$, the hypersurface $\Sigma_\tau$, and the spacelike singularity at $\{t=0\}$.
  • Figure 2: The future domain of dependence of the initial hypersurface $\Sigma_{\tau_0}$ is covered by the regions $\mathcal{U}^+$, $\mathcal{U}^\epsilon_{u_0}$, and the compact set $\mathcal{U}_\text{cmp}$.

Theorems & Definitions (44)

  • Theorem 1.1: Energy boundedness of waves on FLRW
  • Theorem 1.2: Integrated local energy decay for waves on FLRW
  • Theorem 1.3: $r^p$-estimates for waves on FLRW
  • Corollary 1.4: Decay of waves on FLRW
  • Remark 1.5: The case $q=\frac{1}{3}$
  • Remark 1.6: Definition of the energy flux
  • Remark 1.7: Non-sharp decay from energy boundedness
  • Remark 1.8: Waves on FLRW with $\mathbb{T}^3$ spatial topology
  • Lemma 2.1: Divergence theorem
  • Lemma 2.2
  • ...and 34 more