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

Natário, José; Rossetti, Flavio

doi:10.1063/5.0135092

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.

I. INTRODUCTION

The aim of this article is to obtain explicit formulas and exact decay rates for the solution of the wave equation in certain Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes, taken as fixed backgrounds. Besides its intrinsic interest in modeling electromagnetic or gravitational waves, the linear wave equation may be considered as a first step to understand the qualitative behavior of solutions of the Einstein equations, which usually requires a good quantitative grasp of the linearized Einstein equations.

Explicit formulas, albeit applying only to particular FLRW models, are very useful in that they provide detailed information about the solutions and may indeed inspire conjectures about the behavior of solutions of the wave equation in other spacetimes (see Sec. VI). Moreover, they highlight special properties of the solutions of the wave equation in the particular FLRW models where they do apply, such as (some form of) the Huygens principle.

The wave equation in cosmological spacetimes has been amply studied in the literature (see, for example, Refs. 1–14 and references therein). Our own contribution is described in more detail below.

A. Klainerman–Sarnak method

The Klainerman–Sarnak method consists of finding an operator $\hat{O}$ that maps solutions ϕ of the wave equation in a given FLRW background to solutions $\hat{O} ϕ$ of the conformally invariant wave equation on a simpler, ultra-static spacetime (characterized by the same spatial curvature), where a spherical means formula for $\hat{O} ϕ$ is available; by inverting the operator $\hat{O}$ ⁠, one then obtains an explicit expression for ϕ. It should be noted, however, that such operators can only be found for very specific FLRW spacetimes.

Klainerman and Sarnak introduced this method in Ref. 8 for the dust-filled FLRW universes with flat and hyperbolic spatial sections. Abbasi and Craig extended their analysis in Ref. 1, where they elaborated on the decay rates of the solutions of the wave equation and also on the validity of the Huygens principle. Physical applications were later discussed in Refs. 15 and 16.

In this paper, we generalize the Klainerman–Sarnak method to the spherical dust-filled FLRW universe, as well as to a number of other FLRW spacetimes. More precisely, using the parameter K to denote the curvature of the spatial sections (so that K = 0, −1, 1 corresponds to flat, hyperbolic, or spherical geometries, respectively), we obtain new explicit formulas for the solution of the wave equation in radiation-filled (K = 0, −1, 1), de Sitter (K = 0, −1, 1), anti-de Sitter (K = −1), and Milne (K = −1) universes. In each case, we are able to find an operator $\hat{O}$ such that $\hat{O} ϕ$ satisfies the conformally invariant wave equation in the following:

Flat case: Minkowski’s spacetime.
Hyperbolic case: The ultra-static universe with hyperbolic spatial sections (that is the hyperbolic analog of Einstein’s universe).
Spherical case: The ultra-static universe with spherical spatial sections (that is, Einstein’s universe).

The well-known fact that Minkowski’s spacetime is conformal to a globally hyperbolic region of Einstein’s universe makes it plausible that a spherical means formula for the conformally invariant wave equation should exist in this universe. A similar argument applies to its hyperbolic analog, which can be seen to be conformal to a globally hyperbolic region of Minkowski’s spacetime.

B. Huygens principle

All solutions for which we obtain explicit formulas have the property that $\hat{O} ϕ$ satisfies the strong Huygens principle: the values of $\hat{O} ϕ$ at some spacetime point depend only on the values of the initial data at points on its light cone (that is information propagates along null geodesics). However, as noted by Abbasi and Craig in Ref. 1, this is not necessarily the case for ϕ itself. In this paper, we show that the strong Huygens principle does hold for the radiation-filled (K = 0, −1, 1) and the Milne universes. Moreover, a weaker form of the Huygens principle, dubbed the incomplete Huygens principle in Ref. 17, is satisfied by both the de Sitter (K = 0, −1, 1) and the anti-de Sitter universes.

C. Decay rates

The explicit formulas derived in this paper give very detailed information about the solutions, such as decay rates as t → +∞, where t is the physical time coordinate of the FLRW metric (see Sec. I E).

In the flat case, it is interesting to notice that the decay rates in both the dust-filled and the radiation-filled universes coincide with the decay rate t⁻¹ of waves propagating in Minkowski’s spacetime, despite the different behavior of their scale factors (⁠ $a (t) = t^{\frac{2}{3}}$ and $a (t) = t^{\frac{1}{2}}$ ⁠, respectively). These decay rates differ from those obtained in Ref. 2 for the nonzero Fourier modes in the case of toroidal spatial sections, which were, respectively, $t^{- \frac{2}{3}}$ and $t^{- \frac{1}{2}}$ ⁠. The explanation for this discrepancy is dispersion: whereas in the toroidal models, the decay is solely due to the cosmological redshift resulting from the expansion, for noncompact spatial sections, the solutions can disperse across an unbounded region. The faster the expansion, the larger the decay due to the cosmological redshift, but the smaller the decay due to dispersion (as faraway regions are moving away faster). By adapting the analysis of the damped wave equation in Ref. 18, we show that indeed there exists a regime (corresponding to a slow enough expansion of the spatial sections) where the two effects exactly balance each other to give the same decay rate as in Minkowski’s spacetime. For faster expansion, the cosmological redshift does not fully compensate for the lack of dispersion, and the decay becomes slower, until no decay occurs at all.

In the hyperbolic case, we obtain faster decay rates than in the flat case for both the dust-filled and the radiation-filled universes, as could be expected from the fact that dispersion is more effective for hyperbolic spatial geometry. Based on these two examples and also on the numerical results obtained in Ref. 19, we formulate a conjecture for the decay rates in general FLRW spacetimes with hyperbolic sections. We also correct a claim in Ref. 1 by showing that the decay rate in the dust-filled universe is $t^{- \frac{3}{2}}$ rather than t⁻².

D. Behavior at the Big Bang

Generically, solutions of the wave equation blow up at the Big Bang (and also at the Big Crunch, when there is one). Nonetheless, some solutions (e.g., constant functions) have a well-defined limit at the Big Bang. In the cases of the dust-filled (K = 0, −1, 1) and radiation-filled (K = 0, −1, 1) universes, we show how to obtain explicit expressions for these solutions. In fact, by taking limits of solutions with appropriate initial data, we give explicit expressions for solutions with any given function as its limit at the Big Bang (see also Refs. 7 and 9).

E. Summary of the main results and decay conjectures

Recall that a (3 + 1)-dimensional FLRW spacetime

(R \times Σ_{3}, g)

has metric

g = - d t^{2} + a^{2} (t) d Σ_{3}^{2},

(1)

where

d Σ_{3}^{2}

is the standard Riemannian metric for

Σ_{3} = R^{3}, H^{3}

⁠, or S³ and a(t) is the scale factor. The conformal time coordinate is defined as

τ = \int \frac{d t}{a (t)},

(2)

and we refer to t as the physical time coordinate. In what follows, we give a precise quantitative summary of the main results and conjectures derived from the exact formulas.

Flat case: The results for FLRW universes with flat spatial sections are summarized in Table I. The decay rates for the dust-filled and the de Sitter universes had already been obtained in Refs. 1 and 20, respectively. The fact that the de Sitter universe satisfies the incomplete Huygens principle was first noticed in Ref. 17.

TABLE I.

Properties of the solutions of the wave equation □_gϕ = 0 in FLRW models with flat space sections.

Spacetime	Scale factor	t(τ)	Large times	Huygens
Minkowski	a(τ) = 1	t ∝ τ	\|ϕ\| ≲ t⁻¹	Yes (strong)
Dust	a(τ) ∝ τ²	t ∝ τ³	\|ϕ\| ≲ t⁻¹	No
Radiation	a(τ) ∝ τ	t ∝ τ²	\|ϕ\| ≲ t⁻¹	Yes (strong)
de Sitter	a(τ) ∝ τ⁻¹	t ∝ − log τ	\|∂_tϕ\| ≲ e^−2t	Yes (incomplete)

Spacetime	Scale factor	t(τ)	Large times	Huygens
Minkowski	a(τ) = 1	t ∝ τ	\|ϕ\| ≲ t⁻¹	Yes (strong)
Dust	a(τ) ∝ τ²	t ∝ τ³	\|ϕ\| ≲ t⁻¹	No
Radiation	a(τ) ∝ τ	t ∝ τ²	\|ϕ\| ≲ t⁻¹	Yes (strong)
de Sitter	a(τ) ∝ τ⁻¹	t ∝ − log τ	\|∂_tϕ\| ≲ e^−2t	Yes (incomplete)

We now consider the family of flat FLRW metrics whose scale factor satisfies a(t) = t^p, with p ≥ 0. These correspond to universes filled with a perfect fluid with linear equation of state p_m = wρ_m (and zero cosmological constant), where

p = \frac{2}{3 (1 + w)}

(see Appendix A). We summarize our (proved) conjecture for the decay of solutions of the wave equations in these spacetimes in Fig. 1. The precise statement can be found in Sec. VI.

FIG. 1.

Conjectured (and proved) decay rates in the flat case. The rates for w≥−13 follow from Refs. 18 and 21 and the absence of decay for w<−13 follows from Ref. 2. The Klainerman–Sarnak method gives the precise decay for the dust-filled (w = 0) and radiation-filled (w=13) universes.

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Conjectured (and proved) decay rates in the flat case. The rates for $w \geq - \frac{1}{3}$ follow from Refs. 18 and 21 and the absence of decay for $w < - \frac{1}{3}$ follows from Ref. 2. The Klainerman–Sarnak method gives the precise decay for the dust-filled (w = 0) and radiation-filled $(w = \frac{1}{3})$ universes.

Hyperbolic case: The results for FLRW universes with hyperbolic spatial sections are summarized in Table II. The decay rate in the dust-filled universe corrects the claim in Ref. 1.

TABLE II.

Properties of the solutions of the wave equation □_gϕ = 0 in FLRW models with hyperbolic space sections.

Spacetime	Scale factor	t(τ)	Large times	Huygens
Dust	a(τ) = cosh τ − 1	t = sinh τ − τ	$\| ϕ \| ≲ t^{- \frac{3}{2}}$	No
Radiation	a(τ) = sinh τ	t = cosh τ	\|ϕ\| ≲ t⁻²	Yes (strong)
AdS	a(τ) = sech τ	$t = 2 \arctan (\tanh (\frac{τ}{2}))$	ϕ → c	Yes (incomplete)
de Sitter	a(τ) = −cosech τ	$t = - \log (- \tanh (\frac{τ}{2}))$	\|∂_tϕ\| ≲ e^−2t	Yes (incomplete)
Milne	a(τ) = e^τ	t = e^τ	\|ϕ\| ≲ t⁻²	Yes (strong)

Spacetime	Scale factor	t(τ)	Large times	Huygens
Dust	a(τ) = cosh τ − 1	t = sinh τ − τ	$\| ϕ \| ≲ t^{- \frac{3}{2}}$	No
Radiation	a(τ) = sinh τ	t = cosh τ	\|ϕ\| ≲ t⁻²	Yes (strong)
AdS	a(τ) = sech τ	$t = 2 \arctan (\tanh (\frac{τ}{2}))$	ϕ → c	Yes (incomplete)
de Sitter	a(τ) = −cosech τ	$t = - \log (- \tanh (\frac{τ}{2}))$	\|∂_tϕ\| ≲ e^−2t	Yes (incomplete)
Milne	a(τ) = e^τ	t = e^τ	\|ϕ\| ≲ t⁻²	Yes (strong)

We now consider the family of hyperbolic FLRW universes filled with a perfect fluid with linear equation of state p_m = wρ_m (and zero cosmological constant—see Appendix A). We summarize our conjecture for the decay of solutions of the wave equations in these spacetimes in Fig. 2. The precise statement can be found in Sec. VI.

FIG. 2.

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Conjectured decay rates in the hyperbolic case. The Klainerman–Sarnak method gives the precise decay for the dust-filled (w = 0) and radiation-filled $(w = \frac{1}{3})$ universes.

Spherical case: The results for FLRW universes with spherical spatial sections are summarized in Table III. The solutions in the dust-filled and radiation-filled universes blow up for generic initial data due to the Big Crunch singularity (see Sec. I D).

TABLE III.

Properties of the solutions of the wave equation □_gϕ = 0 in FLRW models with spherical space sections.

Spacetime	Scale factor	t(τ)	Large times	Huygens
Dust	a(τ) = 1 − cos τ	t = τ − sin τ	\|ϕ\| blows upa	No
Radiation	a(τ) = sin τ	t = 1 − cos τ	\|ϕ\| blows upa	Yes (strong)
de Sitter	a(t) = sec τ	$t = 2 arctanh (\tan (\frac{τ}{2}))$	\|∂_tϕ\| ≲ e^−2t	Yes (incomplete)

Spacetime	Scale factor	t(τ)	Large times	Huygens
Dust	a(τ) = 1 − cos τ	t = τ − sin τ	\|ϕ\| blows upa	No
Radiation	a(τ) = sin τ	t = 1 − cos τ	\|ϕ\| blows upa	Yes (strong)
de Sitter	a(t) = sec τ	$t = 2 arctanh (\tan (\frac{τ}{2}))$	\|∂_tϕ\| ≲ e^−2t	Yes (incomplete)

^a

Generically (it is possible to choose initial data such that |ϕ| does not blow up).

F. A note on regularity

In this paper, we will assume that the initial data for the wave equation are smooth so that the solutions are also smooth. Moreover, we will assume that the initial data belong to any required Sobolev space or even that it has compact support. All these assumptions can be easily relaxed, but we shall refrain from doing so as this would distract us from the main objectives of this paper.

II. CONDITION FOR THE EXISTENCE OF OPERATORS

We are interested in solving the wave equation,

□_{g} ϕ = 0,

(3)

in a (3 + 1)-dimensional FLRW background

(R \times Σ_{3}, g)

⁠, with metric

g = - d t^{2} + a^{2} (t) d Σ_{3}^{2},

(4)

where

d Σ_{3}^{2}

is the standard Riemannian metric for

Σ_{3} = R^{3}, H^{3}

⁠, or S³. Introducing the conformal time

τ = \int \frac{d t}{a (t)},

(5)

we can write the wave equation in the form

\partial_{τ}^{2} ϕ + 2 \frac{a^{'}}{a} \partial_{τ} ϕ = Δ_{Σ_{3}} ϕ,

(6)

where

Δ_{Σ_{3}}

is the Laplacian operator for the metric

d Σ_{3}^{2}

and

a^{'} (τ) = \frac{d}{d τ} a (t (τ)) .

(7)

Now, let

\hat{O}

be an operator acting on the space of smooth functions that commutes with

Δ_{Σ_{3}}

⁠. If the assumption

\hat{O} (\partial_{τ}^{2} + 2 \frac{a^{'}}{a} \partial_{τ}) ϕ = \partial_{τ}^{2} \hat{O} ϕ + K \hat{O} ϕ

(8)

holds, where K ∈ {−1, 0, 1} represents the curvature of the spatial sections we are considering, then () implies that

\hat{O} ϕ

satisfies the conformally invariant wave equation,

\partial_{τ}^{2} \hat{O} ϕ = (Δ_{Σ_{3}} - K) \hat{O} ϕ .

(9)

We will search a suitable operator

\hat{O}

by taking

\hat{O} ϕ = f (τ) \partial_{τ} ϕ + g (τ) ϕ,

(10)

where f and g are functions to be determined. Plugging this expression into (), we obtain the condition

(2 \frac{a^{'}}{a} f - 2 f^{'}) \partial_{τ}^{2} ϕ + [2 \frac{a^{'}}{a} g - 2 g^{'} - f^{''} + 2 {(\frac{a^{'}}{a})}^{'} f - K f] \partial_{τ} ϕ - (g^{''} + K g) ϕ = 0 .

(11)

We will now try to satisfy this condition for each of the three different types of spatial sections.

III. FLAT CASE

Let K = 0. Equation () holds if and only if

\{\begin{cases} f (τ) = κ a (τ), \\ g (τ) = α τ + β, \\ 2 \frac{a^{'}}{a} g - 2 g^{'} - f^{″} + 2 {(\frac{a^{'}}{a})}^{'} f = 0, \end{cases}

(12)

with

α, β, κ \in R

⁠. Let us consider the case

a (τ) = τ^{j}

(13)

with τ > 0. It can be easily checked that there are non-trivial solutions f and g of the constraint equations () only for j ∈ {−1, 0, 1, 2}, provided that α, β, κ are chosen appropriately. We will analyze each of these four possibilities next.

A. Minkowski spacetime

If j = 0, the constraint equations () force us to choose α = 0 so that the functions f and g become

\{\begin{cases} f (τ) = κ, \\ g (τ) = β . \end{cases}

(14)

In this case, a(τ) ≡ 1, and so our FLRW universe is simply Minkowski’s spacetime. Indeed, if ϕ is a solution of the wave equation in Minkowski’s spacetime, then

\hat{O} ϕ = κ \partial_{τ} ϕ + β ϕ

(15)

is a solution as well.

B. Dust-filled flat universe

If j = 2, conditions () force the functions f and g to be given by

\{\begin{cases} f (τ) = κ τ^{2}, \\ g (τ) = 3 κ τ, \end{cases}

(16)

and therefore,

\hat{O} ϕ = τ^{2} \partial_{τ} ϕ + 3 τ ϕ .

(17)

In this case, a(τ) = τ² and

\frac{d t}{d τ} = a (τ) \Leftrightarrow t \propto τ^{3}

(18)

so that

a (t) \propto t^{\frac{2}{3}} .

(19)

This scale factor a(t) is a solution of the Friedmann equations when K = 0, Λ = 0, and w = 0 (see Appendix A), corresponding to a dust-filled flat FLRW model (also known as the Einstein–de Sitter universe).

Here and in what follows, we will denote the initial data for the wave equation at t = t₀ by

\{\begin{cases} ϕ (t_{0}, x) = : ϕ_{0} (x), \\ \partial_{t} ϕ (t_{0}, x) = : ϕ_{1} (x) . \end{cases}

(20)

Therefore, the initial data for

\hat{O} ϕ = τ^{2} \partial_{τ} ϕ + 3 τ ϕ = τ^{4} \partial_{t} ϕ + 3 τ ϕ

(21)

at the initial conformal time τ = τ₀ are

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = τ_{0}^{4} ϕ_{1} (x) + 3 τ_{0} ϕ_{0} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = τ_{0}^{2} Δ ϕ_{0} (x) + τ_{0}^{3} ϕ_{1} (x) + 3 ϕ_{0} (x) = : ψ_{1} (x) . \end{cases}

(22)

Here, we used Eq. () to write

\partial_{τ} \hat{O} ϕ = \partial_{τ} (τ^{2} \partial_{τ} ϕ + 3 τ ϕ) = τ^{2} Δ ϕ + τ^{3} \partial_{t} ϕ + 3 ϕ .

Using Kirchhoff’s formula for

\hat{O} ϕ

⁠, we have

\begin{align} \hat{O} ϕ (τ, x) & = \frac{1}{4 π {(τ - τ_{0})}^{2}} \int_{\partial B_{τ - τ_{0}} (x)} ψ_{0} (y) d V_{2} (y) \\ + \frac{1}{4 π (τ - τ_{0})} \int_{\partial B_{τ - τ_{0}} (x)} \nabla ψ_{0} (y) \cdot \frac{y - x}{| y - x |} d V_{2} (y) \\ + \frac{1}{4 π (τ - τ_{0})} \int_{\partial B_{τ - τ_{0}} (x)} ψ_{1} (y) d V_{2} (y) . \end{align}

(23)

An explicit expression for ϕ can be obtained by noticing that

\hat{O} ϕ = τ^{- 1} \partial_{τ} (τ^{3} ϕ)

and then integrating Kirchhoff’s formula,

(24)

This is the formula obtained by Klainerman and Sarnak in Ref. 8. Abbasi and Craig used this formula in Ref. 1 to obtain decay estimates by noticing that

|\int_{τ_{0}}^{τ} \frac{s}{4 π {(s - τ_{0})}^{2}} \int_{\partial B_{s - τ_{0}} (x)} ψ_{0} (y) d V_{2} (y) d s| ≲ ‖ ψ_{0} ‖_{\infty} + ‖ ψ_{0} ‖_{1}

(25)

[and similarly for the other integrals in ()], since

\begin{align} |\int_{τ_{0}}^{2 τ_{0}} \frac{s}{4 π {(s - τ_{0})}^{2}} \int_{\partial B_{s - τ_{0}} (x)} ψ_{0} (y) d V_{2} (y) d s| \\ = |\int_{τ_{0}}^{2 τ_{0}} \frac{s}{4 π} \int_{S^{2}} ψ_{0} (x + (s - τ_{0}) z) d V_{2} (z) d s| ≲ ‖ ψ_{0} ‖_{\infty} \end{align}

(26)

and

\frac{s}{4 π {(s - τ_{0})}^{2}} ≲ 1 for s > 2 τ_{0} .

(27)

Therefore,

| ϕ | ≲ τ^{- 3} (‖ ϕ_{0} ‖_{\infty} + ‖ ψ_{0} ‖_{\infty} + ‖ ψ_{0} ‖_{1} + ‖ \nabla ψ_{0} ‖_{\infty} + ‖ \nabla ψ_{0} ‖_{1} + ‖ ψ_{1} ‖_{\infty} + ‖ ψ_{1} ‖_{1}),

(28)

and so, provided that the relevant norms are finite, we have

| ϕ | ≲ t^{- 1} .

(29)

From (), we expect the solutions of the wave equation to diverge as t → 0, which can be traced back to the fact that the FLRW metric is singular at t = 0. However, as noted in Ref. 1 (see also Refs. 7 and 9), it is possible to construct special solutions, which have a well-defined limit as t → 0. Indeed, if we fix ϕ₀, ϕ₁ and τ and take the limit τ₀ → 0 in Eqs. () and (), we obtain

ϕ (τ, x) = \frac{3}{4 π τ^{3}} \int_{B_{τ} (x)} ϕ_{0} (y) d V_{3} (y) .

(30)

It is clear that this expression determines a solution of the wave equation satisfying

\lim_{τ \to 0} ϕ (τ, x) = ϕ_{0} (x) .

(31)

C. Radiation-filled flat universe

If j = 1, conditions () tell us that the functions f and g must be of the form

\{\begin{cases} f (τ) = κ τ, \\ g (τ) = α τ + κ . \end{cases}

(32)

In this case, a(τ) = τ and

\frac{d t}{d τ} = a (τ) \Leftrightarrow t \propto τ^{2},

(33)

which implies

a (τ) \propto t^{\frac{1}{2}} .

(34)

This scale factor is a solution of the Friedmann equations for K = 0, Λ = 0 and

w = \frac{1}{3}

(see Appendix A). Therefore, this model corresponds to a flat, radiation-filled universe. We note that in this case,

\hat{O}

can be taken as the multiplicative operator,

\hat{O} ϕ = τ ϕ .

(35)

The reason for this is that the energy–momentum tensor of a radiation fluid is traceless, and thus, the scalar curvature R of the four-dimensional spacetime vanishes. Consequently, the wave equation coincides with the conformally invariant wave equation, and so the scalar field aϕ = τϕ satisfies the wave equation in Minkowski spacetime.

The initial data for

\hat{O} ϕ = τ ϕ

at the initial conformal time τ = τ₀ are given by

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = τ_{0} ϕ_{0} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = τ_{0}^{2} ϕ_{1} (x) + ϕ_{0} (x) = : ψ_{1} (x) . \end{cases}

(36)

Therefore, if the appropriate Sobolev norms of the initial data are finite, by the standard decay of the wave equation in Minkowski spacetime (see, e.g., Ref. 22), we have, as τ → +∞,

| \hat{O} ϕ | ≲ τ^{- 1} \Leftrightarrow | ϕ | ≲ τ^{- 2} \sim t^{- 1},

(37)

the same decay rate as in Minkowski spacetime.

The explicit expression for ϕ is again given by Kirchhoff’s formula (),

(38)

As one would expect, ϕ satisfies the strong Huygens principle; a related result was obtained in Refs. 23 and 24 for gravitational waves in the Regge–Wheeler gauge.

Again, it is possible to construct solutions with a well-defined limit as t → 0 by fixing ϕ₀, ϕ₁, and τ and taking the limit τ₀ → 0 in Eqs. () and (),

ϕ (τ, x) = \frac{1}{4 π τ^{2}} \int_{\partial B_{τ} (x)} ϕ_{0} (y) d V_{2} (y) .

(39)

It is easy to check that this expression determines a solution of the wave equation satisfying

\lim_{τ \to 0} ϕ (τ, x) = ϕ_{0} (x) .

(40)

D. Flat de Sitter universe

If j = −1, we can choose α = β = 0 so that the functions f and g become

\{\begin{cases} f (τ) = κ τ^{- 1}, \\ g (τ) = 0 . \end{cases}

(41)

These lead to the operator

\hat{O} = τ^{- 1} \partial_{τ}

⁠. In this case, a(τ) = τ⁻¹, and so

\frac{d t}{d τ} = a (τ) \Leftrightarrow t = \log τ .

(42)

This yields

a (t) = e^{- t},

(43)

corresponding to the contracting flat de Sitter spacetime (which is a globally hyperbolic region of the full de Sitter universe); the expanding case can be trivially recovered by changing the time coordinate from t to −t.

The initial data for

\hat{O} ϕ = τ^{- 1} \partial_{τ} ϕ = τ^{- 2} \partial_{t} ϕ

(44)

at the corresponding conformal time τ = τ₀ is given by

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = τ_{0}^{- 2} ϕ_{1} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = τ_{0}^{- 1} Δ ϕ_{0} (x) + τ_{0}^{- 3} ϕ_{1} (x) = : ψ_{1} (x), \end{cases}

(45)

where we used () to write

\partial_{τ} \hat{O} ϕ = \partial_{τ} (τ^{- 1} \partial_{τ} ϕ) = τ^{- 1} Δ ϕ + τ^{- 3} \partial_{t} ϕ .

(46)

If the appropriate Sobolev norms of the initial data are finite, then we have, by boundedness of the solution of the wave equation in the region 0 < τ ≤ τ₀ of Minkowski’s spacetime,

| \hat{O} ϕ | ≲ 1 \Leftrightarrow | \partial_{t} ϕ | ≲ τ^{2} = e^{2 t}

(47)

as t → −∞ (that is, as τ → 0). This translates to |∂_tϕ| ≲ e^−2t as t → +∞ in the expanding case, confirming the result in Ref. 20.

An explicit expression for ϕ can be obtained by integrating Kirchhoff’s formula (),

(48)

In the particular case, when ϕ₁ ≡ 0, we obtain

\begin{align} ϕ (τ, x) & = ϕ_{0} (x) + \int_{τ_{0}}^{τ} \frac{s}{4 π τ_{0} (s - τ_{0})} \int_{\partial B_{s - τ_{0}} (x)} Δ ϕ_{0} (y) d V_{2} (y) d s \\ = ϕ_{0} (x) + \frac{1}{4 π τ_{0}} \int_{B_{τ - τ_{0}} (x)} (1 + \frac{τ_{0}}{| z - x |}) Δ ϕ_{0} (z) d V_{3} (z) . \end{align}

(49)

Using the divergence theorem and Green’s identities, this expression can be rewritten as

\begin{align} ϕ (τ, x) & = \frac{τ}{4 π τ_{0} (τ - τ_{0})} \int_{\partial B_{τ - τ_{0}} (x)} \nabla ϕ_{0} (y) \cdot \frac{y - x}{| y - x |} d V_{2} (y) \\ + \frac{1}{4 π {(τ - τ_{0})}^{2}} \int_{\partial B_{τ - τ_{0}} (x)} ϕ_{0} (y) d V_{2} (y) . \end{align}

(50)

In particular, ϕ(τ, x) depends only on the values of ϕ₀ and ∇ϕ₀ on

\partial B_{τ - τ_{0}} (x)

⁠, that is, ϕ satisfies the strong Huygens principle in the special case when ϕ₁ ≡ 0. This fact was previously noticed by Yagdjian in Ref. 17, who dubbed this property the incomplete Huygens principle.

IV. HYPERBOLIC CASE

Next, we want to find suitable operators

\hat{O}

in the case K = −1. We start by noting that the constraints () require the functions f and g in operators of the form

\hat{O} = f (τ) \partial_{τ} + g (τ)

to satisfy

\{\begin{cases} f (τ) = κ a (τ), \\ g (τ) = α \cosh (τ) + β \sinh (τ), \\ 2 \frac{a^{'}}{a} g - 2 g^{'} - f^{''} + 2 {(\frac{a^{'}}{a})}^{'} f + f = 0 . \end{cases}

(51)

These conditions can be met for certain choices of scale factors a(τ) satisfying the Friedmann equations (see Appendix A). In these cases, it is possible to obtain an explicit formula for ϕ from the expression of

\hat{O} ϕ

in terms of spherical means, analogous to Kirchhoff’s formula in

R^{3}

⁠, which we now discuss.

A. Spherical means formula for K = −1

We will now obtain a spherical means formula for the solution of the conformally invariant wave equation () in the spacetime given by the metric

g = - d t^{2} + d Σ_{3}^{2},

(52)

where

d Σ_{3}^{2}

is the line element for the hyperbolic space

H^{3}

⁠. What follows is a simplified version of the more general derivation in Ref. 8.

Consider the Cauchy problem

\{\begin{cases} \partial_{t}^{2} ϕ = Δ ϕ + ϕ, \\ ϕ (0, x) = ϕ_{0} (x), \\ \partial_{t} ϕ (0, x) = ϕ_{1} (x), \end{cases}

(53)

where Δ is the Laplacian operator in

H^{3}

⁠. Using geodesic polar coordinates about some point

x \in H^{3}

⁠, we know that

d Σ_{3}^{2} = d r^{2} + \sinh^{2} (r) d Ω^{2},

(54)

where dΩ² is the line element for the unit sphere S², and also that

Δ = \frac{\partial^{2}}{\partial r^{2}} + 2 cotanh (r) \frac{\partial}{\partial r} + \frac{1}{\sinh^{2} (r)} Δ_{S^{2}},

(55)

where

Δ_{S^{2}}

is the Laplacian on S². The geodesic sphere S(x, r) about the point x is defined as

S (x, r) : = \exp_{x} ({z \in T_{x} H^{3} : ‖ z ‖ = r}) \equiv \exp_{x} (\partial B_{r} (0)),

(56)

where

\exp_{x} : T_{x} H^{3} \to H^{3}

is the geodesic exponential map and B_r(0) is the ball of radius r about 0 in

T_{x} H^{3}

⁠. We define the spherical mean of a function ϕ(t, x) on the geodesic sphere of radius r about x as

\begin{align} M_{ϕ} (t, r, x) & : = \frac{1}{4 π \sinh^{2} (r)} \int_{S (x, r)} ϕ (t, y) d V_{2} (y) \\ = \frac{1}{4 π} \int_{S^{2}} ϕ (t, \exp_{x} (r z)) d V_{2} (z), \end{align}

(57)

where S² is the unit sphere in

T_{x} H^{3}

⁠. Note that this last formula allows us to extend the spherical mean to negative values of the second argument, yielding

M_{ϕ} (t, - r, x) = M_{ϕ} (t, r, x) .

(58)

Now, if ϕ is a solution of problem (), we have

\frac{\partial^{2}}{\partial t^{2}} M_{ϕ} (t, r, x) = (\frac{\partial^{2}}{\partial r^{2}} + 2 cotanh (r) \frac{\partial}{\partial r} + 1) M_{ϕ} (t, r, x),

(59)

where we used (), (), and the divergence theorem on the sphere. From (), we have the initial data

\{\begin{cases} M_{ϕ} (0, r, x) = M_{ϕ_{0}} (r; x), \\ \partial_{t} M_{ϕ} (0, r, x) = M_{ϕ_{1}} (r; x) . \end{cases}

(60)

Using (), it can be checked that ω(t, r, x) ≔ sinh(r)M_ϕ(t, r, x) satisfies

\partial_{t}^{2} ω = \partial_{r}^{2} ω .

(61)

Given the initial data

\{\begin{cases} ω (0, r, x) = \sinh (r) M_{ϕ_{0}} (r, x) = : γ (r, x), \\ \partial_{t} ω (0, r, x) = \sinh (r) M_{ϕ_{1}} (r, x) = : ψ (r, x), \end{cases}

(62)

the solution of Eq. () is given by the d’Alembert formula,

ω (t, r, x) = \frac{1}{2} [γ (r + t, x) + γ (r - t, x) + \int_{r - t}^{r + t} ψ (s, x) d s] .

(63)

Since M_ϕ(t, r, x) → ϕ(t, x) as r → 0, we obtain

\begin{align} ϕ (t, x) & = \lim_{r \to 0} \frac{1}{2 \sinh (r)} [γ (r + t, x) + γ (r - t, x) + \int_{r - t}^{r + t} ψ (s, x) d s] \\ = \lim_{r \to 0} \frac{1}{2 \cosh (r)} [γ^{'} (r + t, x) + γ^{'} (r - t, x) + ψ (r + t, x) - ψ (r - t, x)] \\ = γ^{'} (t, x) + ψ (t, x) = \partial_{t} [\sinh (t) M_{ϕ_{0}} (t, x)] + \sinh (t) M_{ϕ_{1}} (t, x), \end{align}

(64)

where we used L’Hôpital’s rule and the fact that γ and ψ are odd functions. From (), the solution of the Cauchy problem () is then given by

(65)

with c_z(t) = exp_x(tz).

B. Dust-filled hyperbolic universe

Let us consider the case

a (τ) = \cosh (τ) - 1

(66)

with τ > 0. This corresponds to the dust-filled hyperbolic model, since a(τ) solves Friedmann’s equations for K = −1, Λ = 0 and w = 0 (see Appendix A). With this choice of the scale factor, the wave equation () becomes

\partial_{τ}^{2} ϕ + \frac{2 \sinh (τ)}{\cosh (τ) - 1} \partial_{τ} ϕ = Δ ϕ .

(67)

In this case, the constraint equations () are satisfied if we choose κ = 1, α = 0, and

β = \frac{3}{2}

⁠, that is, if we choose

\hat{O} ϕ = (\cosh (τ) - 1) \partial_{τ} ϕ + \frac{3}{2} \sinh (τ) ϕ .

(68)

We note that we can rewrite this as

\hat{O} ϕ = \frac{2}{\sinh (\frac{τ}{2})} \partial_{τ} (\sinh^{3} (\frac{τ}{2}) ϕ),

(69)

which is the expression that can be found in Ref. 8. The initial data for

\hat{O} ϕ

at the initial conformal time τ = τ₀ are then

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = (\cosh (τ_{0}) - 1) ϕ_{1} (x) + \frac{3}{2} \sinh (τ_{0}) ϕ_{0} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = \frac{1}{2} \sinh (τ_{0}) (\cosh (τ_{0}) - 1) ϕ_{1} (x) \\ + (\cosh (τ_{0}) - 1) Δ ϕ_{0} (x) + \frac{3}{2} \cosh (τ_{0}) ϕ_{0} (x) = : ψ_{1} (x), \end{cases}

(70)

where we used () to write

\partial_{τ} \hat{O} ϕ = \frac{1}{2} \sinh (τ) (\cosh (τ) - 1) \partial_{t} ϕ + (\cosh τ - 1) Δ ϕ + \frac{3}{2} \cosh (τ) ϕ .

(71)

Then, () and () give the following expression for the solution:

(72)

Let us analyze the decay of the solution for τ → +∞. The first integral, for instance, can be estimated by noting that

|\int_{τ_{0}}^{2 τ_{0}} \cosh (s - τ_{0}) \sinh (\frac{s}{2}) \int_{S^{2}} ψ_{0} (\exp_{x} ((s - τ_{0}) z)) d V_{2} (z) d s| ≲ ‖ ψ_{0} ‖_{\infty}

(73)

and that, rewriting the integral on S² as an integral over the geodesic sphere,

\begin{align} |\int_{2 τ_{0}}^{τ} \cosh (s - τ_{0}) \sinh (\frac{s}{2}) \int_{S^{2}} ψ_{0} (\exp_{x} ((s - τ_{0}) z)) d V_{2} (z) d s| \\ = |\int_{2 τ_{0}}^{τ} \frac{\cosh (s - τ_{0})}{\sinh^{2} (s - τ_{0})} \sinh (\frac{s}{2}) \int_{S (x, s - τ_{0})} ψ_{0} (y) d A (y) d s| ≲ ‖ ψ_{0} ‖_{1} . \end{align}

(74)

Similar computations apply to the other integrals in expression (). After dividing both sides by

\sinh^{3} (\frac{τ}{2})

⁠, we can write

| ϕ | ≲ e^{- \frac{3}{2} τ} (‖ ϕ_{0} ‖_{\infty} + ‖ ψ_{0} ‖_{\infty} + ‖ ψ_{0} ‖_{1} + ‖ d ψ_{0} ‖_{\infty} + ‖ d ψ_{0} ‖_{1} + ‖ ψ_{1} ‖_{\infty} + ‖ ψ_{1} ‖_{1}),

(75)

and so, provided that the relevant norms are finite, we have

| ϕ | ≲ e^{- \frac{3}{2} τ} .

(76)

We note that, in general, a faster decay rate is not possible (see Appendix B). To obtain this estimate in terms of the coordinate time t, we see from

t = \int a (τ) d τ = \sinh (τ) - τ

(77)

that, as τ → +∞,

t \sim e^{τ},

(78)

whence

| ϕ | ≲ t^{- \frac{3}{2}} .

(79)

As in the flat case, there exist solutions that have a well-defined limit at the Big Bang: if we take the limit τ₀ → 0 while keeping ϕ₀, ϕ₁, and τ fixed, we obtain from () and () the limit solution

ϕ (τ, x) = \frac{3}{16 π \sinh^{3} (\frac{τ}{2})} \int_{0}^{τ} \sinh (s) \sinh (\frac{s}{2}) \int_{S^{2}} ϕ_{0} (\exp_{x} (z s)) d V_{2} (z) d s .

(80)

Using L’Hôpital’s rule, one can easily check that

\lim_{τ \to 0} ϕ (τ, x) = ϕ_{0} (x) .

(81)

C. Radiation-filled hyperbolic universe

Let us now consider the case

a (τ) = \sinh (τ)

(82)

with τ > 0. This corresponds to the radiation-dominated hyperbolic universe, since a(τ) = sinh(τ) solves the Friedmann equations for K = −1, Λ = 0, and

w = \frac{1}{3}

(see Appendix A). With this choice of the scale factor, our wave equation() becomes

\partial_{τ}^{2} ϕ + 2 cotanh (τ) \partial_{τ} ϕ = Δ ϕ .

(83)

In this case, the constraint equations() are satisfied if we choose α = κ = 0 and β = 1, that is, if we choose

\hat{O} ϕ = \sinh (τ) ϕ .

(84)

Again, this operator can be taken as multiplicative because in this universe, the wave equation coincides with the conformally invariant wave equation.

The initial data for

\hat{O} ϕ

at the initial conformal time τ = τ₀ are

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = \sinh (τ_{0}) ϕ_{0} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = \cosh (τ_{0}) ϕ_{0} (x) + \sinh^{2} (τ_{0}) ϕ_{1} (x) = : ψ_{1} (x) . \end{cases}

(85)

The solution in terms of spherical means is then given by (),

(86)

As in the flat case, ϕ satisfies the strong Huygens principle.

Let us now analyze the decay of the solution as τ → +∞. We note that after dividing the formula above by sinh(τ), the first term of the solution can be estimated as

\begin{align} | ϕ | & = |\frac{\cosh (τ - τ_{0})}{4 π \sinh (τ) \sinh^{2} (τ - τ_{0})} \int_{S (x, τ - τ_{0})} ψ_{0} (y) d A (y)| \\ ≲ (‖ ψ_{0} ‖_{L^{1} (H^{3})} + ‖ d ψ_{0} ‖_{L^{1} (H^{3})}) e^{- 2 τ} . \end{align}

(87)

In fact, assuming for simplicity that the initial data have compact support, we have

\begin{align} \int_{S (x, R)} & ψ_{0} (y) d A (y) = - \int_{R}^{+ \infty} \frac{d}{d r} (\int_{S (x, r)} ψ_{0} (y) d A (y)) d r \\ = - \int_{R}^{+ \infty} 2 \sinh (r) \cosh (r) (\int_{S^{2}} ψ_{0} (\exp_{x} (r z)) d V_{2} (z)) d r \\ - \int_{R}^{+ \infty} \sinh^{2} (r) (\int_{S^{2}} d ψ_{0} ({\dot{c}}_{z} (r)) d V_{2} (z)) d r, \end{align}

(88)

where c_z(r) = exp_x(rz), whence

|\int_{S (x, R)} ψ_{0} (y) d A (y)| ≲ ‖ ψ_{0} ‖_{L^{1} (H^{3})} + ‖ d ψ_{0} ‖_{L^{1} (H^{3})}

(89)

for all R > 0. The remaining two terms have the same decay rate, which can be found after similar computations. Therefore, if the relevant Sobolev norms of the initial data are finite, we have

| ϕ | ≲ e^{- 2 τ} .

(90)

To obtain the decay rate in terms of the time variable t, we use

t = \int \sinh (τ) d τ = \cosh (τ) \sim e^{τ}

(91)

so that as t → +∞,

| ϕ | ≲ t^{- 2} .

(92)

Again there exist solutions with a well-defined limit at the Big Bang: if we take the limit τ₀ → 0 while keeping ϕ₀, ϕ₁, and τ fixed, we obtain from () and () the limit solution

ϕ (τ, x) = \frac{1}{4 π} \int_{S^{2}} ϕ_{0} (\exp_{x} (τ z)) d V_{2} (z),

(93)

which clearly converges to ϕ₀(x) as τ → 0.

D. Anti-de Sitter universe

Let us now analyze the case

a (τ) = sech (τ) .

(94)

This corresponds to a globally hyperbolic region of the anti-de Sitter universe, since a(τ) = sech(τ) is a solution for the Friedmann equations for K = −1, Λ = −3, and ρ₀ = 0 (see Appendix A). We can obtain the scale factor as a function of the physical time t by noting that

t = \int \frac{d τ}{\cosh (τ)} = 2 \arctan (\tanh (\frac{τ}{2})),

(95)

and so

a (t) = sech (2 arctanh (\tan (\frac{t}{2}))) = \cos (t),

(96)

with

t \in (- \frac{π}{2}, \frac{π}{2})

⁠.

With this choice of the scale factor, () becomes

\partial_{τ}^{2} ϕ - 2 \tanh (τ) \partial_{τ} ϕ = Δ ϕ .

(97)

We can satisfy the constraint equations () by choosing α = β = 0 and κ = 1 so that we have

\hat{O} ϕ = sech (τ) \partial_{τ} ϕ .

(98)

As before, we can express the solution in terms of spherical means. We have the following initial data for

\hat{O} ϕ

at the initial conformal time τ = τ₀,

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = {sech}^{2} (τ_{0}) ϕ_{1} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = {sech}^{2} (τ_{0}) \tanh (τ_{0}) ϕ_{1} (x) + sech (τ_{0}) Δ ϕ_{0} (x) = : ψ_{1} (x), \end{cases}

(99)

where we used () to write

\partial_{τ} \hat{O} ϕ = {sech}^{2} (τ) \tanh (τ) \partial_{t} ϕ + sech (τ) Δ ϕ .

(100)

The spherical means solution is then

(101)

In the particular case when ϕ₁ ≡ 0, we obtain

\begin{align} ϕ (τ, x) & = ϕ_{0} (x) + \int_{0}^{τ - τ_{0}} \frac{\sinh (r)}{4 π} \cosh (r + τ_{0}) \int_{S^{2}} sech (τ_{0}) Δ ϕ_{0} (\exp_{x} (r z)) d V_{2} (z) d r \\ = ϕ_{0} (x) + \frac{1}{4 π} \int_{B_{τ - τ_{0}} (x)} (\tanh (τ_{0}) + cotanh (r)) Δ ϕ_{0} (y) d V_{3} (y), \end{align}

(102)

where we defined

r (y) = {dist}_{H^{3}} (y, x)

⁠. Using the divergence theorem and Green’s identities, together with Δ cotanh(r) = 0, this expression can be rewritten as

\begin{align} ϕ (τ, x) & = \frac{\cosh (τ)}{4 π \cosh (τ_{0}) \sinh (τ - τ_{0})} \int_{\partial B_{τ - τ_{0}} (x)} \nabla ϕ_{0} (y) \cdot \nabla r (y) d V_{2} (y) \\ + \frac{1}{4 π \sinh^{2} (τ - τ_{0})} \int_{\partial B_{τ - τ_{0}} (x)} ϕ_{0} (y) d V_{2} (y) . \end{align}

(103)

Therefore, ϕ(τ, x) depends only on the values of ϕ₀ and ∇ϕ₀ on

\partial B_{τ - τ_{0}} (x)

⁠, that is, ϕ satisfies Yagdjian’s incomplete Huygens principle. In particular, if ϕ₀ is compactly supported, then we have

\lim_{τ \to + \infty} ϕ (τ, x) = 0 .

(104)

Interestingly, in the case when ϕ₀ ≡ 0, we have, for compactly supported ϕ₁,

\lim_{τ \to + \infty} ϕ (τ, x) = - \frac{{sech}^{3} (τ_{0})}{4 π} \int_{H^{3}} ϕ_{1} (y) d V_{3} (y)

(105)

(in particular, this limit does not depend on x). By the superposition principle, this result holds, in general, for compactly supported initial data; it can be interpreted by noting that for fixed x, the curve τ ↦ (τ, x) approaches a single point in the full anti-de Sitter spacetime as τ → +∞ (see Ref. 25).

To check (), we note that for ϕ₀ ≡ 0, Eq. () can be written in the form

\begin{align} ϕ (τ, x) & = - \frac{\partial}{\partial τ_{0}} \int_{0}^{τ - τ_{0}} \frac{\sinh (r)}{4 π} \cosh (r + τ_{0}) \int_{S^{2}} ψ_{0} (\exp_{x} (r z)) d V_{2} (z) d r \\ + \int_{0}^{τ - τ_{0}} \frac{\sinh (r)}{4 π} \cosh (r + τ_{0}) \int_{S^{2}} ψ_{1} (\exp_{x} (r z)) d V_{2} (z) d r . \end{align}

(106)

Discarding the boundary term, whose limit will vanish for compactly supported initial data, and using (), we obtain

\lim_{τ \to + \infty} ϕ (τ, x) = - \frac{{sech}^{3} (τ_{0})}{4 π} \int_{0}^{+ \infty} \sinh^{2} (r) \int_{S^{2}} ϕ_{1} (\exp_{x} (r z)) d V_{2} (z) d r .

(107)

E. Hyperbolic de Sitter universe

Let us now consider the case

a (τ) = - cosech (τ)

(108)

with τ < 0. This corresponds to a globally hyperbolic region of the de Sitter universe, since a(τ) = −cosech(τ) is a solution for the Friedmann equations for K = −1, Λ = −3, and ρ₀ = 0 (see Appendix A). We can obtain the scale factor as a function of the physical time t by noticing that

t = - \int \frac{d τ}{\sinh (τ)} = - \log (- \tanh (\frac{τ}{2})),

(109)

and so

a (t) = - cosech (2 arctanh (- e^{- t})) = \sinh (t),

(110)

with t ∈ (0, +∞).

This choice of the scale factor in () leads to

\partial_{τ}^{2} ϕ - 2 cotanh (τ) \partial_{τ} ϕ = Δ ϕ .

(111)

We can satisfy the constraint equations () by choosing α = β = 0 and κ = −1 so that

\hat{O} ϕ = - cosech (τ) \partial_{τ} ϕ .

(112)

Again we can repeat the same steps as in the previous cases to obtain the solution in terms of spherical means. The initial data for

\hat{O} ϕ

at the initial conformal time τ = τ₀ are

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = {cosech}^{2} (τ_{0}) ϕ_{1} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = {cosech}^{2} (τ_{0}) cotanh (τ_{0}) ϕ_{1} (x) - cosech (τ_{0}) Δ ϕ_{0} (x) = : ψ_{1} (x), \end{cases}

(113)

where we used () to write

\partial_{τ} \hat{O} ϕ = {cosech}^{2} (τ) cotanh (τ) \partial_{t} ϕ - cosech (τ) Δ ϕ .

(114)

The spherical means solution is then

(115)

Repeating the same steps as was done for the anti-de Sitter solution, one can show that when ϕ₁ ≡ 0, we have

\begin{align} ϕ (τ, x) & = \frac{\sinh (τ)}{4 π \sinh (τ_{0}) \sinh (τ - τ_{0})} \int_{\partial B_{τ - τ_{0}} (x)} \nabla ϕ_{0} (y) \cdot \nabla r (y) d V_{2} (y) \\ + \frac{1}{4 π \sinh^{2} (τ - τ_{0})} \int_{\partial B_{τ - τ_{0}} (x)} ϕ_{0} (y) d V_{2} (y) . \end{align}

(116)

Therefore, again ϕ satisfies Yagdjian’s incomplete Huygens principle.

Similarly to what happens in the flat de Sitter universe, the time derivative of the solution decays exponentially as t → +∞,

| \partial_{t} ϕ | ≲ e^{- 2 t} .

(117)

Indeed, if we take the partial derivative with respect to τ of both sides of (), we have, as τ → 0,

| \sinh (t) \partial_{t} ϕ | = | \partial_{τ} ϕ | ≲ - \sinh τ = \frac{1}{\sinh (t)},

(118)

where we are assuming that the L^∞ norms of ψ₀, dψ₀, and ψ₁ are finite.

F. Milne universe

We will now work with the scale factor

a (τ) = e^{τ} .

(119)

This model represents the Milne universe, as we can see by writing the scale factor as a function of the physical variable t,

t = \int e^{τ} d τ = e^{τ},

(120)

whence

a (t) = t

(121)

for t ∈ (0, +∞).

With this choice, our wave equation () becomes

\partial_{τ}^{2} ϕ + 2 \partial_{τ} ϕ = Δ ϕ .

(122)

To satisfy the constraint equations (), we can pick α = β = 1 and κ = 0. In this case, the operator is

\hat{O} ϕ = e^{τ} ϕ .

(123)

To get an explicit expression for solutions, we consider the following initial data for

\hat{O} ϕ

at the initial conformal time τ = τ₀:

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = e^{τ_{0}} ϕ_{0} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = e^{τ_{0}} ϕ_{0} (x) + e^{2 τ_{0}} ϕ_{1} (x) = : ψ_{1} (x) . \end{cases}

(124)

The solution in terms of spherical means is given by (),

(125)

As could be expected from the fact that the Milne universe is a globally hyperbolic region of Minkowski’s spacetime, ϕ satisfies the strong Huygens principle. Note that the decay of the solution as τ → +∞ is the same as that of the solution () if we assume that the appropriate norms are finite. Thus, as t → +∞, we have

| ϕ | ≲ e^{- 2 τ} = t^{- 2} .

(126)

This decay rate can be understood by viewing the Milne universe as the region t > |x| of Minkowski’s spacetime and recalling the well-known estimate

| ϕ (x, t) | ≲ \frac{1}{(1 + t + | x |) \sqrt{1 + | t - | x ‖}}

(127)

for solutions of the wave equation with compactly supported initial data (see, for instance, Ref. 26).

V. SPHERICAL CASE

Finally, we consider the case K = 1. The constraint equations () now give

\{\begin{cases} f (τ) = κ a, \\ g (τ) = α \cos (τ) + β \sin (τ), \\ 2 \frac{a^{'}}{a} g - 2 g^{'} - f^{''} + 2 {(\frac{a^{'}}{a})}^{'} f - f = 0 . \end{cases}

(128)

Again these conditions are satisfied for certain choices of a(τ), for which it is possible to obtain an explicit expression for ϕ from the expression of

\hat{O} ϕ

in terms of spherical means.

A. Spherical means formula for K = 1

Let us find the solution of the conformally invariant wave equation () in the Einstein universe, that is, in the spacetime given by

g = - d t^{2} + d Σ_{3}^{2},

(129)

where

d Σ_{3}^{2}

is the line element for the three-sphere S³. We therefore consider the Cauchy problem

\{\begin{cases} \partial_{t}^{2} ϕ = Δ ϕ - ϕ, \\ ϕ (0, x) = ϕ_{0} (x), \\ \partial_{t} ϕ (0, x) = ϕ_{1} (x), \end{cases}

(130)

where Δ is the Laplacian operator in S³. Following the exact same steps as in the hyperbolic case but replacing the hyperbolic functions by their trigonometric counterparts, we arrive at the Kirchhoff-like formula,

(131)

where again c_z(t) = exp_x(tz). It is interesting to note that ϕ is periodic in t with period 2π.

B. Dust-filled spherical universe

Let us consider the case

a (τ) = 1 - \cos (τ),

(132)

with τ ∈ (0, 2π). This corresponds to the dust-filled spherical model, since a(τ) solves Friedmann’s equations for K = 1, Λ = 0, and w = 0 (see Appendix A). With this choice of the scale factor, the wave equation () becomes

\partial_{τ}^{2} ϕ + \frac{2 \sin (τ)}{1 - \cos (τ)} \partial_{τ} ϕ = Δ ϕ .

(133)

The constraint equations() are satisfied if we choose κ = 1, α = 0, and

β = \frac{3}{2}

⁠, that is, if we choose

\hat{O} ϕ = (1 - \cos (τ)) \partial_{τ} ϕ + \frac{3}{2} \sin (τ) ϕ,

(134)

which can also be written as

\hat{O} ϕ = \frac{2}{\sin (\frac{τ}{2})} \partial_{τ} (\sin^{3} (\frac{τ}{2}) ϕ),

(135)

as remarked in Ref. 8. The initial data for

\hat{O} ϕ

at the initial conformal time τ = τ₀ are given by

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = \frac{3}{2} \sin (τ_{0}) ϕ_{0} (x) + (1 - \cos (τ_{0})) ϕ_{1} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = \frac{1}{2} \sin (τ_{0}) (1 - \cos (τ_{0})) ϕ_{1} (x) \\ + (1 - \cos (τ_{0})) Δ ϕ_{0} (x) + \frac{3}{2} \cos (τ_{0}) ϕ_{0} (x) = : ψ_{1} (x), \end{cases}

(136)

where we used () to write

\partial_{τ} \hat{O} ϕ = \frac{1}{2} \sin (τ) (1 - \cos (τ)) \partial_{t} ϕ + (1 - \cos (τ)) Δ ϕ + \frac{3}{2} \cos (τ) ϕ .

(137)

Finally, the spherical means formulas () and () give

(138)

In general, these solutions will diverge as τ → 0 and as τ → 2π. As in the flat case, there exist solutions that have a well-defined limit at the Big Bang: if we take the limit τ₀ → 0 while keeping ϕ₀, ϕ₁, and τ fixed, we obtain from () and () the limit solution

ϕ (τ, x) = \frac{3}{16 π \sin^{3} (\frac{τ}{2})} \int_{0}^{τ} \sin (s) \sin (\frac{s}{2}) \int_{S^{2}} ϕ_{0} (\exp_{x} (z s)) d V_{2} (z) d s .

(139)

Using L’Hôpital’s rule, one can easily check that

\lim_{τ \to 0} ϕ (τ, x) = ϕ_{0} (x) .

(140)

Similarly, we can find solutions with a well-defined limit at the Big Crunch by taking the limit τ₀ → 2π while keeping ϕ₀, ϕ₁, and τ fixed. In this case, we obtain the limit solution

ϕ (τ, x) = \frac{3}{16 π \sin^{3} (\frac{τ}{2})} \int_{2 π}^{τ} \sin (s) \sin (\frac{s}{2}) \int_{S^{2}} ϕ_{0} (\exp_{x} (z s)) d V_{2} (z) d s,

(141)

which, again using L’Hôpital’s rule, satisfies

\lim_{τ \to 2 π} ϕ (τ, x) = ϕ_{0} (x) .

(142)

Interestingly, if ϕ₀ satisfies

\int_{0}^{2 π} \sin (s) \sin (\frac{s}{2}) \int_{S^{2}} ϕ_{0} (\exp_{x} (z s)) d V_{2} (z) d s = 0

(143)

for all x ∈ S³, then the two limit solutions coincide, yielding a solution with the same limit at the Big Bang and at the Big Crunch (this includes, of course, the constant solutions).

C. Radiation-filled spherical universe

Next, let us take the scale factor

a (τ) = \sin (τ)

(144)

with τ ∈ (0, π). This corresponds to a radiation-dominated spherical universe, since a(τ) = sin(τ) solves the Friedmann equations for K = 1, Λ = 0, and

w = \frac{1}{3}

(see Appendix A). With this choice of the scale factor, our wave equation () becomes

\partial_{τ}^{2} ϕ + 2 cotan (τ) \partial_{τ} ϕ = Δ ϕ .

(145)

In this case, the constraint equation () are satisfied if we choose α = κ = 0 and β = 1, that is, if we choose

\hat{O} ϕ = \sin (τ) ϕ .

(146)

Again, this operator can be taken as multiplicative because in this universe, the wave equation coincides with the conformally invariant wave equation.

The initial data for

\hat{O} ϕ

at the initial conformal time τ = τ₀ is given by

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = \sin (τ_{0}) ϕ_{0} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = \cos (τ_{0}) ϕ_{0} (x) + \sin^{2} (τ_{0}) ϕ_{1} (x) = : ψ_{1} (x) . \end{cases}

(147)

The spherical means solution is then given by (),

(148)

As in the flat and hyperbolic cases, ϕ satisfies the strong Huygens principle.

Again, these solutions will, in general, diverge as τ → 0 and as τ → π, but there exist solutions that have a well-defined limit at the Big Bang: if we take the limit τ₀ → 0 while keeping ϕ₀, ϕ₁, and τ fixed, we obtain from () and () the limit solution

ϕ (τ, x) = \frac{1}{4 π} \int_{S^{2}} ϕ_{0} (\exp_{x} (τ z)) d V_{2} (z),

(149)

which clearly converges to ϕ₀(x) as τ → 0. Similarly, we can find solutions with a well-defined limit at the Big Crunch by taking the limit τ₀ → π while keeping ϕ₀, ϕ₁, and τ fixed. In this case, we obtain the limit solution

ϕ (τ, x) = \frac{1}{4 π} \int_{S^{2}} ϕ_{0} (\exp_{x} ((τ - π) z)) d V_{2} (z),

(150)

which converges to ϕ₀(x) as τ → π. Interestingly, if ϕ₀ is an even function on the sphere S³, then the two limit solutions coincide, yielding a solution with the same limit at the Big Bang and at the Big Crunch (this includes, of course, the constant solutions).

D. Spherical de Sitter universe

Finally, we consider the case

a (τ) = \sec (τ),

(151)

with

τ \in (- \frac{π}{2}, \frac{π}{2})

⁠. This corresponds to the spherical (full) de Sitter universe, since a(τ) = sec(τ) is a solution for the Friedmann equations for K = 1, Λ = 3, and ρ₀ = 0 (see Appendix A). We can obtain the scale factor as a function of the physical time t by noting that

t = \int \frac{d τ}{\cos (τ)} = 2 arctanh (\tan (\frac{τ}{2}))

(152)

and so

a (t) = \sec (2 \arctan (\tanh (\frac{t}{2}))) = \cosh (t) .

(153)

With this choice of the scale factor, () becomes

\partial_{τ}^{2} ϕ + 2 \tan (τ) \partial_{τ} ϕ = Δ ϕ .

(154)

We can satisfy the constraint equation () by choosing α = β = 0 and κ = 1 so that we have

\hat{O} ϕ = \sec (τ) \partial_{τ} ϕ .

(155)

As before, we can express the solution in terms of spherical means. We have the following initial data for

\hat{O} ϕ

at the initial conformal time τ = τ₀:

\{\begin{cases} \hat{O} ϕ (τ_{0}, x) = \sec^{2} (τ_{0}) ϕ_{1} (x) = : ψ_{0} (x), \\ \partial_{τ} \hat{O} ϕ (τ_{0}, x) = \sec (τ_{0}) Δ ϕ_{0} (x) - \sec^{2} (τ_{0}) \tan (τ_{0}) ϕ_{1} (x) = : ψ_{1} (x), \end{cases}

(156)

where we used () to write

\partial_{τ} \hat{O} ϕ = \sec (τ) Δ ϕ - \sec^{2} (τ) \tan (τ) \partial_{t} ϕ .

(157)

The spherical means solution is then

(158)

In the particular case when ϕ₁ ≡ 0, we obtain

\begin{align} ϕ (τ, x) & = ϕ_{0} (x) + \int_{0}^{τ - τ_{0}} \frac{\sin (r)}{4 π} \cos (r + τ_{0}) \int_{S^{2}} \sec (τ_{0}) Δ ϕ_{0} (\exp_{x} (r z)) d V_{2} (z) d r \\ = ϕ_{0} (x) + \frac{1}{4 π} \int_{B_{τ - τ_{0}} (x)} (cotan (r) - \tan (τ_{0})) Δ ϕ_{0} (y) d V_{3} (y), \end{align}

(159)

where we defined

r (y) = {dist}_{S^{3}} (y, x)

⁠. Using the divergence theorem and Green’s identities, together with Δcotan(r) = 0, this expression can be rewritten as

\begin{align} ϕ (τ, x) & = \frac{\cos (τ)}{4 π \cos (τ_{0}) \sin (τ - τ_{0})} \int_{\partial B_{τ - τ_{0}} (x)} \nabla ϕ_{0} (y) \cdot \nabla r (y) d V_{2} (y) \\ + \frac{1}{4 π \sin^{2} (τ - τ_{0})} \int_{\partial B_{τ - τ_{0}} (x)} ϕ_{0} (y) d V_{2} (y) . \end{align}

(160)

Therefore, ϕ(τ, x) depends only on the values of ϕ₀ and ∇ϕ₀ on

\partial B_{τ - τ_{0}} (x)

⁠, that is, ϕ satisfies Yagdjian’s incomplete Huygens principle.

As expected by analogy with the flat and hyperbolic de Sitter universes, the time derivative of the solution decays exponentially as t → +∞,

| \partial_{t} ϕ | ≲ e^{- 2 t} .

(161)

Indeed, if we take the partial derivative with respect to τ of both sides of (), we have, as

τ \to \frac{π}{2}

⁠,

| \cosh (t) \partial_{t} ϕ | = | \partial_{τ} ϕ | ≲ \cos τ = \frac{1}{\cosh (t)} .

(162)

Similarly, one can show that the time derivative of the solution decays exponentially as t → −∞,

| \partial_{t} ϕ | ≲ e^{2 t} .

(163)

VI. DECAY RATES IN FLAT AND HYPERBOLIC FLRW UNIVERSES

Let us consider the family of flat FLRW universes described by the scale factor a(t) = t^p, with p ≥ 0. In this context, we obtained the Minkowski spacetime decay rate of t⁻¹ for the wave equation when $p = \frac{2}{3}$ [dust-filled universe, see (29)] and when $p = \frac{1}{2}$ [radiation-filled universe, see (37)]. The independence of these results on p (as well as the numerical results in Ref. 19) suggests a wider range of validity for this decay.

Conjecture 6.1

(decay in the flat case). Let ϕ be a solution of the wave equation in a flat FLRW universe with scale factor a(t) = t^p, where p ≥ 0. Assume that the initial data ϕ₀(x) ≔ ϕ(t₀, x) and ϕ₁(x) ≔ ∂_tϕ(t₀, x) are sufficiently regular and belong to appropriate Sobolev spaces. Then,

‖ ϕ (t, \cdot) ‖_{L^{\infty} (R^{3})} ≲ \{\begin{cases} {(1 + t)}^{- 1}, & 0 \leq p \leq \frac{2}{3}, \\ {(1 + t)}^{- 3 (1 - p)}, & \frac{2}{3} \leq p < 1, \end{cases}

(164)

or, equivalently,

‖ ϕ (t, \cdot) ‖_{L^{\infty} (R^{3})} ≲ \{\begin{cases} {(1 + t)}^{- 1}, & w \geq 0, \\ {(1 + t)}^{- \frac{3 w + 1}{w + 1}}, & - \frac{1}{3} < w \leq 0 . \end{cases}

(165)

Moreover, we have

‖ ϕ (t, \cdot) ‖_{L^{\infty} (R^{3})} ≲ {(\log t)}^{- \frac{3}{2}} for p = 1 (i.e. w = - \frac{1}{3}),

and there is no decay for p > 1 (i.e.,

- 1 < w < - \frac{1}{3}

).

The absence of decay for p > 1 (but not for p = 1) follows from the results in Ref. 2. For 0 ≤ p ≤ 1, we have the following result:

Theorem 6.2.

Conjecture 6.1 holds for 0 ≤ p ≤ 1 (i.e., $w \geq - \frac{1}{3}$ ).

Proof.

The case p = 0 corresponds to the wave equation in Minkowski’s spacetime. For 0 < p < 1, the original wave equation () (written in conformal time) can be seen as a damped wave equation in Minkowski’s spacetime,

□ ϕ - \frac{ν}{τ} \partial_{τ} ϕ = 0,

(166)

where

□ = - \partial_{τ}^{2} + Δ

and

ν : = \frac{2 p}{1 - p} .

(167)

This equation was studied in Ref. 18, Theorem 3.5 (which can be extended to the L¹–L^∞ case, as stressed in Remark 3.3), where it was proved that

‖ ϕ (τ, \cdot) ‖_{L^{\infty} (R^{3})} ≲ \{\begin{cases} {(1 + τ)}^{- 1 - \frac{ν}{2}} \sim {(1 + t)}^{- 1}, & 0 < p \leq \frac{2}{3}, \\ {(1 + τ)}^{- 3} \sim {(1 + t)}^{- 3 (1 - p)}, & \frac{2}{3} \leq p < 1 . \end{cases}

When p = 1, the respective damped equation has constant coefficients. By a simple rescaling,

\tilde{ϕ} (τ, x) : = ϕ (\frac{τ}{2}, \frac{x}{2})

⁠, we have

□ \tilde{ϕ} - \partial_{τ} \tilde{ϕ} = 0 .

The diffusive structure of the above partial differential equation (PDE) was explicitly described in Ref. 21, where the decay

‖ \tilde{ϕ} ‖_{L^{\infty} (R^{3})} ≲ τ^{- \frac{3}{2}} \sim {(\log t)}^{- \frac{3}{2}}

was established.□

A related conjecture for the hyperbolic case is motivated by the decay rates obtained in Secs. IV B and IV C and by the numerical results in Ref. 19.

Conjecture 6.3

(Decay in the hyperbolic case). Let ϕ be a solution of the wave equation in a hyperbolic FLRW universe, with the scale factor satisfying the Friedmann equation (A1) with zero cosmological constant and equation of state (A2). Assume that the initial data ϕ₀(x) ≔ ϕ(t₀, x) and ϕ₁(x) ≔ ∂_tϕ(t₀, x) are sufficiently regular and belongs to appropriate Sobolev spaces. Then,

‖ ϕ (t, \cdot) ‖_{L^{\infty} (H^{3})} ≲ \{\begin{cases} {(1 + t)}^{- 2}, & w \geq \frac{1}{3}, \\ {(1 + t)}^{- \frac{3 (w + 1)}{2}}, & 0 \leq w \leq \frac{1}{3} . \end{cases}

(168)

Moreover, the decay is slower than

{(1 + t)}^{- \frac{3 (w + 1)}{2}}

for

- \frac{1}{3} < w < 0

, and there is no decay for

- 1 \leq w < - \frac{1}{3}

.

ACKNOWLEDGMENTS

We thank João Costa and Alex Vañó-Viñuales for many useful discussions. This work was partially supported by FCT/Portugal through CAMGSD, IST-ID (Project Nos. UIDB/04459/2020 and UIDP/04459/2020). Flavio Rossetti was supported by FCT/Portugal through the Ph.D. scholarship (Grant No. UI/BD/152068/2021).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

José Natário: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Flavio Rossetti: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

APPENDIX A: THE FRIEDMANN EQUATIONS

Meaningful choices of the scale factor a(t) in the FLRW metric () are obtained by solving the Friedmann equations, which result from the Einstein equations with an ideal fluid source. In three spatial dimensions, these are given by (see, for instance, Ref. 27),

{(\frac{\dot{a}}{a})}^{2} = \frac{8 π}{3} ρ_{m} + \frac{Λ}{3} - \frac{K}{a^{2}} and \frac{\ddot{a}}{a} = - \frac{4 π}{3} (ρ_{m} + 3 p_{m}) + \frac{Λ}{3},

(A1)

where K is the curvature of the spatial sections, ρ_m is the fluid’s energy density, p_m is the fluid’s pressure, and Λ is the cosmological constant. If we assume that the fluid satisfies the linear equation of state

p_{m} = w ρ_{m},

(A2)

where w is a constant (the square of the fluid’s speed of sound), then () can be rewritten as

{(\frac{a^{'}}{a^{2}})}^{2} = \frac{8 π}{3} ρ_{0} a^{- 3 (1 + w)} + \frac{Λ}{3} - \frac{K}{a^{2}},

(A3)

where ρ₀ ≥ 0 is an integration constant and the derivatives are now taken with respect to the conformal time. Interesting values of w are given in Table IV.

TABLE IV.

Fluid types for different values of w.

w	Fluid type
w < 0	Unphysical (imaginary speed of sound)
w = 0	Dust
$w = \frac{1}{3}$	Radiation
w = 1	Stiff fluid
w > 1	Unphysical (superluminal speed of sound)

If ρ₀ > 0, Λ = 0, K = 0, and

w \neq - \frac{1}{3}

⁠, the solutions of () are given by

a (τ) = τ^{\frac{2}{1 + 3 w}}

(A4)

for an appropriate choice of ρ₀ (amounting to a choice of units). In this work, we consider the following scale factors:

a (τ) = \{\begin{cases} τ^{2} & (dust), \\ τ & (radiation), \\ τ^{- 1} & (de Sitter) . \end{cases}

(A5)

The de Sitter universe corresponds to ρ₀ = 0 and Λ > 0 (here, we chose units such that Λ = 3).

If ρ₀ > 0, Λ = 0, K = −1, and

w \neq - \frac{1}{3}

⁠, the solutions of () are given by

a (τ) = {[\sinh (\frac{1}{2} (3 w + 1) τ)]}^{\frac{2}{3 w + 1}}

(A6)

for an appropriate choice of ρ₀. In particular, we consider the following scale factors:

a (τ) = \{\begin{cases} \cosh (τ) - 1 & (dust), \\ \sinh (τ) & (radiation), \\ sech (τ) & (anti–de Sitter), \\ - cosech (τ) & (de Sitter), \\ e^{τ} & (Milne universe) . \end{cases}

(A7)

The anti-de Sitter, de Sitter, and Milne universes correspond to the cases where ρ₀ = 0 and Λ = −3, Λ = 3, and Λ = 0, respectively.

Finally, if ρ₀ > 0, Λ = 0, K = 1, and

w \neq - \frac{1}{3}

⁠, the solutions of () are given by

a (τ) = {[\sin (\frac{1}{2} (3 w + 1) τ)]}^{\frac{2}{3 w + 1}}

(A8)

for an appropriate choice of ρ₀. In particular, we consider

a (τ) = \{\begin{cases} 1 - \cos (τ) & (dust), \\ \sin (τ) & (radiation), \\ \sec (τ) & (de Sitter) . \end{cases}

(A9)

Again, the de Sitter universe corresponds to ρ₀ = 0 and Λ = 3.

APPENDIX B: DECAY FOR THE DUST-FILLED HYPERBOLIC UNIVERSE

Let us show that the decay () is optimal by fixing a point

\bar{x} \in H^{3}

and choosing initial data such that

| ϕ (τ, \bar{x}) | \sim e^{- \frac{3}{2} τ}

⁠. This can be done as follows: given τ₀ > 0, we choose the functions ϕ₀ and ϕ₁ such that ψ₀ ≡ 0. From (), this implies that

ψ_{1} (x) = A ϕ_{0} (x) + B Δ ϕ_{0} (x),

(B1)

with

A = \frac{3}{2} (\cosh (τ_{0}) - \frac{1}{2} \sinh^{2} (τ_{0}))

and B = cosh(τ₀) − 1. Next, we choose the radially symmetric function

ϕ_{0} (x) = φ ({dist}_{H^{3}} (x, \bar{x}))

⁠, where

φ (r) : = \{\begin{cases} N r^{3} \exp (\frac{1}{r - 1}), & 0 \leq r < 1, \\ 0, & r \geq 1, \end{cases}

(B2)

and N > 0 is a free parameter. Note that φ is smooth in (0, +∞) and moreover that its behavior at r = 0 implies that ϕ₀ is of class C². Plugging this choice of ϕ₀ into the right-hand side of the spherical means solution () evaluated at

\bar{x}

⁠, we obtain

\begin{align} \int_{τ_{0}}^{τ} \frac{\sinh (s - τ_{0})}{8 π} \sinh (\frac{s}{2}) \int_{S^{2}} ψ_{1} (\exp_{\bar{x}} ((s - τ_{0}) z)) d V_{2} (z) d s \\ = \frac{1}{2} \int_{0}^{τ - τ_{0}} \sinh (r) \sinh (\frac{r + τ_{0}}{2}) (A φ (r) + B φ^{″} (r) + 2 B cotanh (r) φ^{'} (r)) d r, \end{align}

(B3)

where we used () and the substitution r = s − τ₀. By (), the above integral is constant for τ > τ₀ + 1. We can choose τ₀ such that this constant is non-zero (for τ₀ = 1, τ = 2 and N = 50, the evaluation of the integral in () using Mathematica gives 0.151503, and this value grows linearly with N), and so, after this choice of ϕ₀ and ϕ₁, () implies

ϕ (τ, \bar{x}) = C \sinh^{- 3} (\frac{τ}{2}) \sim e^{- \frac{3}{2} τ} as τ \to + \infty .

(B4)

Since the C² functions on a manifold can be approximated by smooth functions (Whitney approximation theorem, see, e.g., Ref. 28), we can choose smooth initial data that approximates the chosen ϕ₀ and extend this construction to smooth initial data. The above reasoning disproves the decay rate of e^−2τ in the conformal time variable stated by Abbasi and Craig in Ref. 1. Indeed, the inequality proved in Theorem 3.1 of their article seems to hold specifically for the term they choose, but not for the last terms in their solutions (3.9) and (3.10).

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Published open access through an agreement with Universidade de Lisboa Instituto Superior Tecnico

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

I. INTRODUCTION

A. Klainerman–Sarnak method

B. Huygens principle

C. Decay rates

D. Behavior at the Big Bang

E. Summary of the main results and decay conjectures

F. A note on regularity

II. CONDITION FOR THE EXISTENCE OF OPERATORS

III. FLAT CASE

A. Minkowski spacetime

B. Dust-filled flat universe

C. Radiation-filled flat universe

D. Flat de Sitter universe

IV. HYPERBOLIC CASE

A. Spherical means formula for K = −1

B. Dust-filled hyperbolic universe

C. Radiation-filled hyperbolic universe

D. Anti-de Sitter universe

E. Hyperbolic de Sitter universe

F. Milne universe

V. SPHERICAL CASE

A. Spherical means formula for K = 1

B. Dust-filled spherical universe

C. Radiation-filled spherical universe

D. Spherical de Sitter universe

VI. DECAY RATES IN FLAT AND HYPERBOLIC FLRW UNIVERSES

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: THE FRIEDMANN EQUATIONS

APPENDIX B: DECAY FOR THE DUST-FILLED HYPERBOLIC UNIVERSE

REFERENCES

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