Explicit formulas and decay rates for the solution of the wave equation in cosmological spacetimes

Abstract

We obtain explicit formulas for the solution of the wave equation in certain Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes. Our method, pioneered by Klainerman and Sarnak, consists in finding differential operators that map solutions of the wave equation in these FLRW spacetimes to solutions of the conformally invariant wave equation in simpler, ultra-static spacetimes, for which spherical mean formulas are available. In addition to recovering the formulas for the dust-filled flat and hyperbolic FLRW spacetimes originally derived by Klainerman and Sarnak and generalizing them to the spherical case, we obtain new formulas for the radiation-filled FLRW spacetimes and also for the de Sitter, anti-de Sitter, and Milne universes. We use these formulas to study the solutions with respect to the Huygens principle and the decay rates and to formulate conjectures about the general decay rates in flat and hyperbolic FLRW spacetimes. The positive resolution of the conjecture in the flat case is seen to follow from known results in the literature.

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

“…Differently from the flat case, all scale factors under consideration here reveal the same exponentially-increasing asymptotic behaviour (when represented using the conformal time coordinate). In this context, we exhibit a new pattern of decay rates in function of the parameter w, and recover the behaviours described in [6] for the dust (w = 0) and radiation (w = 1 3 ) cases, for which an explicit formula for the solution could be obtained. We show these results in section 4.2.…”

“…Finally, in the dust-filled Universe with hyperbolic spatial sections, the numerical methods that we adopted confirm the decay rate t − 3 2 showed in [6], and thus provide a numerical counterexample to the rate of t −2 stated in [5].…”

“…Here, p m and ρ m are the pressure and the energy density, respectively, of the perfect fluid which constitutes the matter content of the FLRW Universe (see also appendix A in [6]). In this context, we evolve solutions starting from smooth, spherically symmetric initial data, and recover the results showed in [6], while explicitly describing the dynamics inside and outside the lightcone centred at the origin of the coordinate system. These results are shown in section 4.1.…”

“…We will restrict our analysis to the case Λ = 0, due to the existence of closed-form expressions for the solutions to the Friedmann equations. Recent developments on the problem can be found in [2,[5][6][7][8][9][10][11][12][13][14][15][16] and references therein, both for the case Λ = 0 and for a non-zero cosmological constant. It has been settled, in particular, that the decay of the energy associated to waves, caused by the redshift effect, is only one of the many factors that influence the propagation, together for instance with the non-compactness of the space sections and the dispersive properties of the background spacetime.…”

“…We will often refer to the companion paper [6], where explicit expressions and decay rates for the solutions of (1) in a class of FLRW spacetimes are provided by use of analytical techniques. In the present work, we provide evidence for the sharpness of the known estimates and complement the results via numerical methods.…”