Decay of solutions of the wave equation in cosmological spacetimes -- a numerical analysis

TL;DR

This work numerically investigates the decay of linear waves on expanding FLRW spacetimes with Λ = 0, focusing on both spatially flat and hyperbolic cases. By evolving spherically symmetric solutions using conformal time and a high-order discretization scheme, it establishes how decay rates depend on the expansion parameter p in the flat case and on the fluid parameter w in the hyperbolic case, highlighting the role of tails inside the lightcone in shaping late-time behavior. The results confirm known analytical decay bounds in flat spacetimes, reveal tail-dominated decay regimes for certain p, and provide new uniform decay rates in hyperbolic backgrounds, including a numerical counterexample to a previously claimed t^{-2} rate. These findings deepen the understanding of redshift-dispersion interplay, support potential Morawetz-type bounds, and point to hyperboloidal slicing as a promising boundary-treatment direction for future analytic and numerical work.

Abstract

We numerically evolve spherically symmetric solutions to the linear wave equation on some expanding Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes and study the respective asymptotics for large times. We find a quantitative relation between the expansion rate of the underlying background universe and the decay rate of linear waves, also in the context of spatially-hyperbolic spacetimes, for which rigorous proofs of decay rates are not generally known. A prominent role in the decay mechanism is shown to be played by tails, i.e. scattered waves propagating in the interior of the lightcone.

Figures (7)

• Figure 1: $\sup_{\mathbb{R}^3}|\phi|$ decays as $t^{\alpha}$ for $0 < p < 1$. Initial data inidata, $dt=0.01$. For $p=0.3, 0.5$: $dr=0.5$, $r \in [0, 2000]$. For $p=2/3, 0.8, 0.9$: $dr=0.05$, $r \in [0, 100]$. The value $\alpha_f$ represents the final value of each plotted line. The early-time behaviour is due to the interplay between redshift and dispersion and is explained in detail in section \ref{['resulhyperb']}, where the same phenomenon occurs in a more evident way.
• Figure 2: $\sup_{\mathbb{R}^3}|\phi|$ and $\sup_{\mathbb{R}^3}|\partial_{\tau} \phi|$ decay as $(\log t)^{\alpha}$, when $p=1$. Notice that $a(t)\partial_t \phi = \partial_{\tau} \phi$. Initial data inidata, $dt=0.01$, $dr=0.05$, $r \in [0, 100]$. The value $\alpha_f$ represents the final value of each plotted line.
• Figure 3: Inside the lightcone, flat case: $\sup_{\{2r < \tau\}} |\phi|$ decays as $t^{\alpha}$. Initial data inidata, $dt=0.01$. For $p=0.3$: $dr=0.5$, $r \in [0, 2000]$. For $p=2/3, 0.8, 0.9$: $dr=0.05$, $r \in [0, 100]$. The value $\alpha_f$ represents the final value of each plotted line.
• Figure 4: Plots of $\phi$ as a function of time and radius for the flat case. Different colors correspond to different values of the function. For the dust-filled universe ($p=\frac{2}{3}$), a non-zero contribution given by tails can be observed inside the lightcone. In the radiation case ($p=\frac{1}{2}$), the Huygens principle holds. We truncated the range of the values of $\phi$ for clarity.
• Figure 5: Inside the lightcone, flat case: behaviour of $\sup_{\{6r < \tau \}} |\phi|$ for $p=0.5$. The solution is eventually identical to zero in this region due to Huygens' principle, but some non-zero contributions will be detected depending on the machine precision. Initial data inidata, $dt=0.01$, $dr=0.5$, $r \in [0, 3000]$.