Decay of solutions of the wave equation in expanding cosmological spacetimes

Abstract

We study the decay of solutions of the wave equation in some expanding cosmological spacetimes, namely flat Friedmann-Lemaître-Robertson-Walker (FLRW) models and the cosmological region of the Reissner-Nordström-de Sitter (RNdS) solution. By introducing a partial energy and using an iteration scheme, we find that, for initial data with finite higher order energies, the decay rate of the time derivative is faster than previously existing estimates. For models undergoing accelerated expansion, our decay rate appears to be (almost) sharp.

Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

“…Theorem 2.4 (Maximal development and its domain Q for IVP C + ). The characteristic initial value problem IVP C + , with initial data as described above, has a unique classical solution (in the sense that all the partial derivatives occurring in (21)-( 27) are continuous) defined on a maximal past set Q containing a neighborhood 10 of…”

“…In this analogy, the previous formulation of cosmic no-hair would require (generic) solutions to decay to a constant at I + . However, it is well known that this is not the case [10,22,25]. To get an idea of why, note that cosmic silence is illustrated by the following construction: given any two inextendible causal curves γ i , i = 1, 2, parameterized by proper time t and reaching I + as t → ∞, there is "a late enough" (t 1) Cauchy surface Σ t such that the sets D i = J − (γ i )∩J + (Σ t ) are disjoint.…”

Critical exponent of Fujita type for semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime

“…If 𝛽 ≥ 5 3 for n = 1, 𝛽 ≥ 3 for n = 2, or 𝛽 ≥ n + 2 for n ≥ 3, by assuming data in the energy spaces with additional regularity L 1 (R n ), the global (in time) existence result for (1) was proved in D'Abbicco 8 for p > p F (n) . = 1+ 2 n , the well-known Fujita index. 9 The exponent p F (n) is critical for this model, that is, for p ≤ p F (n) and suitable, arbitrarily small data, there exists no global weak solution.…”

“…In Mathematical Physics, there are homogeneous and isotropic cosmological models represented by the wave equation in a given spacetime. In some models, an $$$ \left(n+1\right) $$$‐dimensional spacetime of interest is described by the family of the Friedmann–Lemaître–Robertson–Walker (FLRW), which is endowed with a metric written in the form $$$$ d{s}^2=-d{t}^2+{\alpha}^2(t)\left(d{x}_1^2+\dots +d{x}_n^2\right), $$$$ with an appropriate scale factor $$$ \alpha (t) $$$ (see, e.g., previous studies 1–3 ). If $$$ \frac{d}{dt}\alpha (t)>0 $$$ and $$$ \frac{d^2}{d{t}^2}\alpha (t)<0 $$$, we can say that the universe is in decelerating expansion.…”

“…with an appropriate scale factor 𝛼(t) (see, e.g., previous studies [1][2][3] ). If d dt 𝛼(t) > 0 and d 2 dt 2 𝛼(t) < 0, we can say that the universe is in decelerating expansion.…”