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.

### A. Klainerman–Sarnak method

The Klainerman–Sarnak method consists of finding an operator $O\u0302$ that maps solutions *ϕ* of the wave equation in a given FLRW background to solutions $O\u0302\varphi $ of the conformally invariant wave equation on a simpler, ultra-static spacetime (characterized by the same spatial curvature), where a spherical means formula for $O\u0302\varphi $ is available; by inverting the operator $O\u0302$, 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 $O\u0302$ such that $O\u0302\varphi $ 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 $O\u0302\varphi $ satisfies the strong Huygens principle: the values of $O\u0302\varphi $ 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 ($a(t)=t23$ and $a(t)=t12$, 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, $t\u221223$ and $t\u221212$. 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 $t\u221232$ 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

*S*

^{3}and

*a*(

*t*) is the scale factor. The

*conformal*time coordinate is defined as

*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.

Spacetime . | Scale factor . | t(τ)
. | Large times . | Huygens . |
---|---|---|---|---|

Minkowski | a(τ) = 1 | t ∝ τ | |ϕ| ≲ t^{−1} | Yes (strong) |

Dust | a(τ) ∝ τ^{2} | t ∝ τ^{3} | |ϕ| ≲ t^{−1} | No |

Radiation | a(τ) ∝ τ | t ∝ τ^{2} | |ϕ| ≲ t^{−1} | Yes (strong) |

de Sitter | a(τ) ∝ τ^{−1} | t ∝ − log τ | |∂_{t}ϕ| ≲ 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) ∝ τ^{2} | t ∝ τ^{3} | |ϕ| ≲ t^{−1} | No |

Radiation | a(τ) ∝ τ | t ∝ τ^{2} | |ϕ| ≲ t^{−1} | Yes (strong) |

de Sitter | a(τ) ∝ τ^{−1} | t ∝ − log τ | |∂_{t}ϕ| ≲ e^{−2t} | Yes (incomplete) |

*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

**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.

Spacetime . | Scale factor . | t(τ)
. | Large times . | Huygens . |
---|---|---|---|---|

Dust | a(τ) = cosh τ − 1 | t = sinh τ − τ | $|\varphi |\u2272t\u221232$ | No |

Radiation | a(τ) = sinh τ | t = cosh τ | |ϕ| ≲ t^{−2} | Yes (strong) |

AdS | a(τ) = sech τ | $t=2arctantanh\tau 2$ | ϕ → c | Yes (incomplete) |

de Sitter | a(τ) = −cosech τ | $t=\u2212log\u2212tanh\tau 2$ | |∂_{t}ϕ| ≲ 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 | $|\varphi |\u2272t\u221232$ | No |

Radiation | a(τ) = sinh τ | t = cosh τ | |ϕ| ≲ t^{−2} | Yes (strong) |

AdS | a(τ) = sech τ | $t=2arctantanh\tau 2$ | ϕ → c | Yes (incomplete) |

de Sitter | a(τ) = −cosech τ | $t=\u2212log\u2212tanh\tau 2$ | |∂_{t}ϕ| ≲ 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.

**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).

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 | $t=2arctanhtan\tau 2$ | |∂_{t}ϕ| ≲ 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 | $t=2arctanhtan\tau 2$ | |∂_{t}ϕ| ≲ 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

*S*

^{3}. Introducing the conformal time

*K*∈ {−1, 0, 1} represents the curvature of the spatial sections we are considering, then (6) implies that $O\u0302\varphi $ satisfies the conformally invariant wave equation,

*f*and

*g*are functions to be determined. Plugging this expression into (8), we obtain the condition

## III. FLAT CASE

*K*= 0. Equation (11) holds if and only if

*τ*> 0. It can be easily checked that there are non-trivial solutions

*f*and

*g*of the constraint equations (12) 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

*j*= 0, the constraint equations (12) force us to choose

*α*= 0 so that the functions

*f*and

*g*become

*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

### B. Dust-filled flat universe

*j*= 2, conditions (12) force the functions

*f*and

*g*to be given by

*a*(

*τ*) =

*τ*

^{2}and

*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).

*t*=

*t*

_{0}by

*τ*=

*τ*

_{0}are

*ϕ*can be obtained by noticing that $O\u0302\varphi =\tau \u22121\u2202\tau (\tau 3\varphi )$ and then integrating Kirchhoff’s formula,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

*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

*ϕ*

_{0},

*ϕ*

_{1}and

*τ*and take the limit

*τ*

_{0}→ 0 in Eqs. (22) and (24), we obtain

### C. Radiation-filled flat universe

*j*= 1, conditions (12) tell us that the functions

*f*and

*g*must be of the form

*a*(

*τ*) =

*τ*and

*K*= 0, Λ = 0 and $w=13$ (see Appendix A). Therefore, this model corresponds to a flat, radiation-filled universe. We note that in this case, $O\u0302$ can be taken as the multiplicative operator,

*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.

*τ*=

*τ*

_{0}are given by

*τ*→ +∞,

*t*→ 0 by fixing

*ϕ*

_{0},

*ϕ*

_{1}, and

*τ*and taking the limit

*τ*

_{0}→ 0 in Eqs. (36) and (38),

### D. Flat de Sitter universe

*j*= −1, we can choose

*α*=

*β*= 0 so that the functions

*f*and

*g*become

*a*(

*τ*) =

*τ*

^{−1}, and so

*t*to −

*t*.

*τ*=

*τ*

_{0}is given by

*τ*≤

*τ*

_{0}of Minkowski’s spacetime,

*t*→ −∞ (that is, as

*τ*→ 0). This translates to |

*∂*

_{t}

*ϕ*| ≲

*e*

^{−2t}as

*t*→ +∞ in the expanding case, confirming the result in Ref. 20.

*ϕ*can be obtained by integrating Kirchhoff’s formula (23),In the particular case, when

*ϕ*

_{1}≡ 0, we obtain

*ϕ*(

*τ*,

*x*) depends only on the values of

*ϕ*

_{0}and ∇

*ϕ*

_{0}on $\u2202B\tau \u2212\tau 0(x)$, that is,

*ϕ*satisfies the strong Huygens principle in the special case when

*ϕ*

_{1}≡ 0. This fact was previously noticed by Yagdjian in Ref. 17, who dubbed this property the

*incomplete Huygens*

*principle*.

## IV. HYPERBOLIC CASE

*K*= −1. We start by noting that the constraints (11) require the functions

*f*and

*g*in operators of the form $O\u0302=f(\tau )\u2202\tau +g(\tau )$ to satisfy

*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 $O\u0302\varphi $ in terms of spherical means, analogous to Kirchhoff’s formula in $R3$, which we now discuss.

### A. Spherical means formula for *K* = −1

*dΩ*

^{2}is the line element for the unit sphere

*S*

^{2}, and also that

*S*

^{2}. The geodesic sphere

*S*(

*x*,

*r*) about the point

*x*is defined as

*B*

_{r}(0) is the ball of radius

*r*about 0 in $TxH3$. We define the

*spherical mean*of a function

*ϕ*(

*t*,

*x*) on the geodesic sphere of radius

*r*about

*x*as

*S*

^{2}is the unit sphere in $TxH3$. Note that this last formula allows us to extend the spherical mean to negative values of the second argument, yielding

*ϕ*is a solution of problem (53), we have

*ω*(

*t*,

*r*,

*x*) ≔ sinh(

*r*)

*M*

_{ϕ}(

*t*,

*r*,

*x*) satisfies

*M*

_{ϕ}(

*t*,

*r*,

*x*) →

*ϕ*(

*t*,

*x*) as

*r*→ 0, we obtain

*γ*and

*ψ*are odd functions. From (57), the solution of the Cauchy problem (53) is then given bywith

*c*

_{z}(

*t*) = exp

_{x}(

*tz*).

### B. Dust-filled hyperbolic universe

*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 (6) becomes

*κ*= 1,

*α*= 0, and $\beta =32$, that is, if we choose

*τ*=

*τ*

_{0}are then

*τ*→ +∞. The first integral, for instance, can be estimated by noting that

*S*

^{2}as an integral over the geodesic sphere,

*t*, we see from

*τ*→ +∞,

*τ*

_{0}→ 0 while keeping

*ϕ*

_{0},

*ϕ*

_{1}, and

*τ*fixed, we obtain from (70) and (72) the limit solution

### C. Radiation-filled hyperbolic universe

*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 $w=13$ (see Appendix A). With this choice of the scale factor, our wave equation(6) becomes

*α*=

*κ*= 0 and

*β*= 1, that is, if we choose

*τ*=

*τ*

_{0}are

*ϕ*satisfies the strong Huygens principle.

*τ*→ +∞. We note that after dividing the formula above by sinh(

*τ*), the first term of the solution can be estimated as

*c*

_{z}(

*r*) = exp

_{x}(

*rz*), whence

*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

*t*, we use

*t*→ +∞,

*τ*

_{0}→ 0 while keeping

*ϕ*

_{0},

*ϕ*

_{1}, and

*τ*fixed, we obtain from (85) and (86) the limit solution

*ϕ*

_{0}(

*x*) as

*τ*→ 0.

### D. Anti-de Sitter universe

*a*(

*τ*) = sech(

*τ*) is a solution for the Friedmann equations for

*K*= −1, Λ = −3, and

*ρ*

_{0}= 0 (see Appendix A). We can obtain the scale factor as a function of the physical time

*t*by noting that

*α*=

*β*= 0 and

*κ*= 1 so that we have

*τ*=

*τ*

_{0},

*ϕ*

_{1}≡ 0, we obtain

*r*) = 0, this expression can be rewritten as

*ϕ*(

*τ*,

*x*) depends only on the values of

*ϕ*

_{0}and ∇

*ϕ*

_{0}on $\u2202B\tau \u2212\tau 0(x)$, that is,

*ϕ*satisfies Yagdjian’s incomplete Huygens principle. In particular, if

*ϕ*

_{0}is compactly supported, then we have

*ϕ*

_{0}≡ 0, we have, for compactly supported

*ϕ*

_{1},

*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).

*ϕ*

_{0}≡ 0, Eq. (101) can be written in the form

### E. Hyperbolic de Sitter universe

*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

*ρ*

_{0}= 0 (see Appendix A). We can obtain the scale factor as a function of the physical time

*t*by noticing that

*t*∈ (0, +∞).

*α*=

*β*= 0 and

*κ*= −1 so that

*τ*=

*τ*

_{0}are

*ϕ*

_{1}≡ 0, we have

*ϕ*satisfies Yagdjian’s incomplete Huygens principle.

*t*→ +∞,

*τ*of both sides of (115), we have, as

*τ*→ 0,

*L*

^{∞}norms of

*ψ*

_{0},

*dψ*

_{0}, and

*ψ*

_{1}are finite.

### F. Milne universe

*t*,

*t*∈ (0, +∞).

*α*=

*β*= 1 and

*κ*= 0. In this case, the operator is

*τ*=

*τ*

_{0}:

*ϕ*satisfies the strong Huygens principle. Note that the decay of the solution as

*τ*→ +∞ is the same as that of the solution (86) if we assume that the appropriate norms are finite. Thus, as

*t*→ +∞, we have

*t*> |

*x*| of Minkowski’s spacetime and recalling the well-known estimate

## V. SPHERICAL CASE

*K*= 1. The constraint equations (11) now give

*a*(

*τ*), for which it is possible to obtain an explicit expression for

*ϕ*from the expression of $O\u0302\varphi $ in terms of spherical means.

### A. Spherical means formula for *K* = 1

*S*

^{3}. We therefore consider the Cauchy problem

*S*

^{3}. 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,where again

*c*

_{z}(

*t*) = exp

_{x}(

*tz*). It is interesting to note that

*ϕ*is periodic in

*t*with period 2

*π*.

### B. Dust-filled spherical universe

*τ*∈ (0, 2

*π*). 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 (6) becomes

*κ*= 1,

*α*= 0, and $\beta =32$, that is, if we choose

*τ*=

*τ*

_{0}are given by

*τ*→ 0 and as

*τ*→ 2

*π*. As in the flat case, there exist solutions that have a well-defined limit at the Big Bang: if we take the limit

*τ*

_{0}→ 0 while keeping

*ϕ*

_{0},

*ϕ*

_{1}, and

*τ*fixed, we obtain from (136) and (138) the limit solution

*τ*

_{0}→ 2

*π*while keeping

*ϕ*

_{0},

*ϕ*

_{1}, and

*τ*fixed. In this case, we obtain the limit solution

*ϕ*

_{0}satisfies

*x*∈

*S*

^{3}, 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

*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 $w=13$ (see Appendix A). With this choice of the scale factor, our wave equation (6) becomes

*α*=

*κ*= 0 and

*β*= 1, that is, if we choose

*τ*=

*τ*

_{0}is given by

*ϕ*satisfies the strong Huygens principle.

*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

*τ*

_{0}→ 0 while keeping

*ϕ*

_{0},

*ϕ*

_{1}, and

*τ*fixed, we obtain from (147) and (148) the limit solution

*ϕ*

_{0}(

*x*) as

*τ*→ 0. Similarly, we can find solutions with a well-defined limit at the Big Crunch by taking the limit

*τ*

_{0}→

*π*while keeping

*ϕ*

_{0},

*ϕ*

_{1}, and

*τ*fixed. In this case, we obtain the limit solution

*ϕ*

_{0}(

*x*) as

*τ*→

*π*. Interestingly, if

*ϕ*

_{0}is an even function on the sphere

*S*

^{3}, 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

*a*(

*τ*) = sec(

*τ*) is a solution for the Friedmann equations for

*K*= 1, Λ = 3, and

*ρ*

_{0}= 0 (see Appendix A). We can obtain the scale factor as a function of the physical time

*t*by noting that

*α*=

*β*= 0 and

*κ*= 1 so that we have

*τ*=

*τ*

_{0}:

*ϕ*

_{1}≡ 0, we obtain

*r*) = 0, this expression can be rewritten as

*ϕ*(

*τ*,

*x*) depends only on the values of

*ϕ*

_{0}and ∇

*ϕ*

_{0}on $\u2202B\tau \u2212\tau 0(x)$, that is,

*ϕ*satisfies Yagdjian’s incomplete Huygens principle.

*t*→ +∞,

*τ*of both sides of (158), we have, as $\tau \u2192\pi 2$,

*t*→ −∞,

## 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 $p=23$ [dust-filled universe, see (29)] and when $p=12$ [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.

*(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*

*ϕ*

_{0}(

*x*) ≔

*ϕ*(

*t*

_{0},

*x*)

*and*

*ϕ*

_{1}(

*x*) ≔

*∂*

_{t}

*ϕ*(

*t*

_{0},

*x*)

*are sufficiently regular and belong to appropriate Sobolev spaces. Then,*

*or, equivalently,*

*Moreover, we have*

*and there is no decay for*

*p*> 1

*(i.e.,*$\u22121<w<\u221213$

*).*

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:

*Conjecture 6.1 holds for* 0 ≤ *p* ≤ 1 *(i.e.,* $w\u2265\u221213$*).*

*p*= 0 corresponds to the wave equation in Minkowski’s spacetime. For 0 <

*p*< 1, the original wave equation (6) (written in conformal time) can be seen as a damped wave equation in Minkowski’s spacetime,

*L*

^{1}–

*L*

^{∞}case, as stressed in Remark 3.3), where it was proved that

*p*= 1, the respective damped equation has constant coefficients. By a simple rescaling, $\varphi \u0303(\tau ,x):=\varphi \tau 2,x2$, we have

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.

*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*

*ϕ*

_{0}(

*x*) ≔

*ϕ*(

*t*

_{0},

*x*)

*and*

*ϕ*

_{1}(

*x*) ≔

*∂*

_{t}

*ϕ*(

*t*

_{0},

*x*)

*are sufficiently regular and belongs to appropriate Sobolev spaces. Then,*

*Moreover, the decay is slower than*$(1+t)\u22123(w+1)2$

*for*$\u221213<w<0$

*, and there is no decay for*$\u22121\u2264w<\u221213$

*.*

## 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

*a*(

*t*) in the FLRW metric (4) 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),

*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

*w*is a constant (the square of the fluid’s speed of sound), then (A1) can be rewritten as

*ρ*

_{0}≥ 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.

w
. | Fluid type . |
---|---|

w < 0 | Unphysical (imaginary speed of sound) |

w = 0 | Dust |

$w=13$ | Radiation |

w = 1 | Stiff fluid |

w > 1 | Unphysical (superluminal speed of sound) |

w
. | Fluid type . |
---|---|

w < 0 | Unphysical (imaginary speed of sound) |

w = 0 | Dust |

$w=13$ | Radiation |

w = 1 | Stiff fluid |

w > 1 | Unphysical (superluminal speed of sound) |

*ρ*

_{0}> 0, Λ = 0,

*K*= 0, and $w\u2260\u221213$, the solutions of (A3) are given by

*ρ*

_{0}(amounting to a choice of units). In this work, we consider the following scale factors:

*ρ*

_{0}= 0 and Λ > 0 (here, we chose units such that Λ = 3).

*ρ*

_{0}> 0, Λ = 0,

*K*= −1, and $w\u2260\u221213$, the solutions of (A3) are given by

*ρ*

_{0}. In particular, we consider the following scale factors:

*ρ*

_{0}= 0 and Λ = −3, Λ = 3, and Λ = 0, respectively.

*ρ*

_{0}> 0, Λ = 0,

*K*= 1, and $w\u2260\u221213$, the solutions of (A3) are given by

*ρ*

_{0}. In particular, we consider

*ρ*

_{0}= 0 and Λ = 3.

### APPENDIX B: DECAY FOR THE DUST-FILLED HYPERBOLIC UNIVERSE

*τ*

_{0}> 0, we choose the functions

*ϕ*

_{0}and

*ϕ*

_{1}such that

*ψ*

_{0}≡ 0. From (70), this implies that

*B*= cosh(

*τ*

_{0}) − 1. Next, we choose the radially symmetric function $\varphi 0(x)=\phi (distH3(x,x\u0304))$, where

*N*> 0 is a free parameter. Note that

*φ*is smooth in (0, +∞) and moreover that its behavior at

*r*= 0 implies that

*ϕ*

_{0}is of class

*C*

^{2}. Plugging this choice of

*ϕ*

_{0}into the right-hand side of the spherical means solution (72) evaluated at $x\u0304$, we obtain

*r*=

*s*−

*τ*

_{0}. By (B2), the above integral is constant for

*τ*>

*τ*

_{0}+ 1. We can choose

*τ*

_{0}such that this constant is non-zero (for

*τ*

_{0}= 1,

*τ*= 2 and

*N*= 50, the evaluation of the integral in (B3) using Mathematica gives 0.151503, and this value grows linearly with

*N*), and so, after this choice of

*ϕ*

_{0}and

*ϕ*

_{1}, (72) implies

*C*

^{2}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

*ϕ*

_{0}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).

## REFERENCES

*u*= 0 on the Friedmann-Robertson-Walker space-times

*L*

^{p}–

*L*

^{q}estimates of solutions to the damped wave equation in 3-dimensional space and their application

*Lectures on Nonlinear Hyperbolic Differential Equations*

*An Introduction to Mathematical Relativity*