Hyperboloidal approach for linear and non-linear wave equations in FLRW spacetimes

TL;DR

This work numerically analyzes linear and nonlinear wave equations on decelerating FLRW spacetimes using a hyperboloidal foliation with compactified slices that reach future null infinity $\mathrsfs{I}^+$. By formulating both conformal-time and physical-time variants for flat and hyperbolic spatial geometries, the authors confirm sharp linear decay rates and provide evidence for small-data global existence under a generalized null condition in decelerated expansion, while identifying finite-time blow-up for Fritz John-type nonlinearities in slow expansion regimes. The hyperboloidal approach eliminates boundary contamination and enables long-time evolution, offering a robust framework to explore wave propagation in cosmological backgrounds and guiding future analytical and numerical investigations.

Abstract

In this numerical work, we deal with two distinct problems concerning the propagation of waves in cosmological backgrounds. In both cases, we employ a spacetime foliation given in terms of compactified hyperboloidal slices. These slices intersect future null infinity, so our method is well-suited to study the long-time behaviour of waves. Moreover, our construction is adapted to the presence of the time--dependent scale factor that describes the underlying spacetime expansion. First, we investigate decay rates for solutions to the linear wave equation in a large class of expanding FLRW spacetimes, whose non--compact spatial sections have either zero or negative curvature. By means of a hyperboloidal foliation, we provide new numerical evidence for the sharpness of decay--in--time estimates for linear waves propagating in such spacetimes. Then, in the spatially-flat case, we present numerical results in support of small data global existence of solutions to semi-linear wave equations in FLRW spacetimes having a decelerated expansion, provided that a generalized null condition holds. In absence of this null condition and in the specific case of $ \square_g φ= (\partial_t φ)^2 $ (Fritz John's choice), the results we obtain suggest that, when the spacetime expansion is sufficiently slow, solutions diverge in finite time for every choice of initial data.

Figures (9)

• Figure 1: Conformal Carter-Penrose diagrams depicting the hyperboloidal slices considered in the spatially flat case when the height function is time--dependent. In the horizontal axis $R = (V-U)/2$ is plotted and $T = (V+U)/2$ in the vertical one, with the common choice $U=\arctan(\tilde{u})$, $V=\arctan(\tilde{v})$ and $\tilde{u} = \tilde{\tau}-\tilde{r}$, $\tilde{v} = \tilde{\tau}+\tilde{r}$. For more details on the construction of Penrose diagrams for hyperboloidal slices see e.g. Vano-Vinuales:2023pum. The black lines correspond to level sets of the hyperboloidal time $\tau$ given by (\ref{['tautrafo']}) with the CMC height function (\ref{['hfuncflat']}), with $\kappa=1$ used in all cases. The dashed lines correspond to foliations given by hyperboloidal time $t$ following the transformation (\ref{['ttrafo']}) substituted into $\tilde{\tau} = \tilde{t}^{1-p}/(1-p)$ and using the time-dependent height function (\ref{['tdepheightfunc']}). Each color corresponds to a different choice of $C(t)$ as indicated in the legend.
• Figure 2: Schematic representation of how $\sup_{\mathbb{R}^3}|\phi|(\tilde{T}, \cdot)$ is recovered from data from the simulations on hyperboloidal slices. Given an initial radius $\tilde{r}_0$, there exists a unique $\tilde{T}$--constant level set that intersects the first hyperboloidal slice at $\tilde{r}_0$. Let it correspond to the unrescaled time $\tilde{T}_0$. Then, it is possible to recover the intersections between $\{\tilde{T} = \tilde{T}_0\}$ and the remaining hyperboloidal slices. If we run the numerical experiment with high enough accuracy, this gives a reliable approximation of the values of $\phi$ in $\{\tilde{T}_0\}\times \mathbb{R}^3$. Therefore, we can store the $L^{\infty}$ norm of $\phi$ at $\tilde{T}=\tilde{T}_0$. Then, we repeat the procedure starting from the radius $\tilde{r}_1 > \tilde{r}_0$ along the first hyperboloidal slice, and so on.
• Figure 3: Decay rates for $\phi$ in flat FLRW spacetimes with scale factor $a(t)=t^p$. Here $dt = 0.00025$, $dr=0.00125$, $r \in [0, 1]$. We obtain the expected decay rates, see also Rossetti:2023igb. The quantity $f$ measures the final value of the plotted line.
• Figure 4: Decay rates for $\partial_{\tilde{t}} \phi$ in flat FLRW spacetimes with scale factor $a(t)=t^p$. Here $dt = 0.00025$, $dr=0.00125$, $r \in [0, 1]$. The quantity $f$ measures the final value of the plotted line.
• Figure 5: Decay rates for $\phi$ in hyperbolic FLRW spacetimes with scale factor as in (\ref{['scalefactor_hyp']}) when $0 < w < \frac{1}{3}$. The $L^{\infty}$ decay rate we obtain along the $\tilde{t}$--foliation comes from the slow decay at the origin. Notice that, at the origin, the value of $\tilde{t}$ coincides with that of the hyperboloidal time $t$. Here $dt = 0.00025$, $dr=0.00125$, $r \in [0, 1]$. The quantity $f$ measures the final value of the plotted line.