Decay of solutions to the Klein-Gordon equation on some expanding cosmological spacetimes

The decay of solutions to the Klein-Gordon equation is studied in two expanding cosmological spacetimes, namely the de Sitter universe in flat Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) form, and the cosmological region of the Reissner-Nordstr\"om-de Sitter (RNdS) model. Using energy methods, for initial data with finite higher order energies, decay rates for the solution are obtained. Also, a previously established decay rate of the time derivative of the solution to the wave equation, in an expanding de Sitter universe in flat FLRW form, is improved, proving Rendall's conjecture. A similar improvement is also given for the wave equation in the cosmological region of the RNdS spacetime.


Introduction
The aim of this article is to obtain exact decay rates for solutions to the Klein-Gordon equation in a fixed background of some expanding cosmological spacetimes. The two spacetimes we will consider are the de Sitter universe in flat Friedmann-Lemaître-Robertson-Walker (FLRW) form and the cosmological region of the Reissner-Nordström-de Sitter (RNdS) model. The problem we consider is linear, in that the background is fixed. This constitutes a first step towards understanding the more complicated nonlinear coupled problem, where one also considers the effect of the energy-momentum tensor of the solution to the Klein-Gordon equation on the Einstein equation. This nonlinear coupled problem is much more complicated and usually requires, as a first step, a detailed understanding of our simpler linear problem. There are several motivations behind the interest in this question. Firstly, one may consider the linear wave equations as a proxy for the Einstein equations, with the ultimate goal of understanding the qualitative behaviour of solutions to the Einstein equations. (The vacuum Einstein equations become wave-like equations in harmonic coordinates; see for example [18, §5.4, p. 110]). After this first step, one may then proceed to consider linearised Einstein equations (which can be reduced to tensor wave-like linear equations) and, finally, the full nonlinear Einstein equations. With the addition of a positive cosmological constant to the Einstein field equations, the expectation is that the resulting accelerated expansion has a dominating effect on the decay of solutions. Precise estimates on solutions may then prove useful in formulating and proving cosmic no-hair theorems (e.g. [3,7]).
The wave equation g φ = 0 in expanding cosmological spacetimes (M, g) has been amply studied in the literature; see for example [5,8,10,23] and the references therein. It is a natural question to also study the Klein-Gordon equation g φ − m 2 φ = 0, the degenerate version of which, when m 2 = 0, is the wave equation. For example, in [23, §6], also the case of the Klein-Gordon equation in the Schwarzschild-de Sitter spacetime is considered. In [21], the asymptotic behaviour of the solutions to the Klein-Gordon equation near the Big Bang singularity is studied, while we investigate the asymptotics of the Klein-Gordon equation in the far future in the case of the de Sitter universe in flat FLRW form, and in the cosmological region of the Reissner-Nordström-de Sitter solution. Recently, in [11] (see also [26] and [22]), among other things, decay estimates for the solutions to the Klein-Gordon equation were obtained in de Sitter models (see in particular, Corollary 2.1 and the less obvious Proposition 3.1 of [11]). However, these results are proved via Fourier transformation (reminiscent of our mode calculation in Appendix A (Section 6)) and do not seem to be as sharp as our Theorem 3.1.
The wave equation in the de Sitter spacetime having flat three-dimensional spatial sections was considered in Rendall [20]. There, it was shown that the time derivative ∂ t φ =:φ decays at least as e −Ht = (a(t)) −1 , where H = Λ/3 is the Hubble constant and Λ > 0 is the cosmological constant. Moreover, it was conjectured that the decay is of the order e −2Ht = (a(t)) −2 . The almostexact conjectured decay rate of |φ| (a(t)) −2+δ (where δ > 0 can be chosen arbitrarily at the outset) follows as a corollary of a result shown recently [8,Remark 1.1]. We improve this result, to obtain full conformity with Rendall's conjecture, in our result Theorem 2.3.
Finally, from the pure mathematical perspective, analysis of linear wave equations on Lorentzian manifolds is a natural topic of study within the realm of hyperbolic partial differential equations and differential geometry; see for example [2], [25,§7,Chap. 2].
A naive heuristic indication of the effect of the accelerated expansion on the decay of the solution, based on physical energy considerations, can be obtained as follows. Considering an expanding FLRW model with flat ndimensional spatial sections of radius a(t), we have on the one hand that the energy density of a solution φ of the Klein-Gordon equation is of the order of m 2 φ 2 . On the other hand, if the wavelength of the particles associated with φ follows the expansion, then it is proportional to a(t), and so the energy varies as E 2 ∼ m 2 + p 2 ∼ A + B (a(t)) 2 , where A, B > 0 are constants. Thus, giving m 2 φ 2 ∼ (a(t)) −n (A + B (a(t)) 2 ). Asȧ 0 (expanding FLRW spacetime), the term A + B (a(t)) 2 approaches a finite positive value, and so one may expect φ ∼ (a(t)) − n 2 .
We will find out that in fact things are much more complicated: this decay rate is valid only for |m| n 2 . In order to obtain precise conjectures on the expected decay, we will consider Fourier modes for spatially periodic solutions to the Klein-Gordon equation or, equivalently, consider the expanding de Sitter universe in flat FLRW form with toroidal spatial sections. This exercise already demonstrates that the underlying decay mechanism is the cosmological expansion, as opposed to dispersion. The Fourier mode analysis, which is peripheral to the rest of the paper, is relegated to Appendix A (Section 6).
In the cosmological region of the Reissner-Nordström-de Sitter spacetimes, the expanding region is foliated by spacelike hypersurfaces of 'constant r'. One expects the decay rate with respect to r, for the solution to the Klein-Gordon equation, in the cosmological region of the Reissner-Nordström-de Sitter spacetime, to be the same as the one for the de Sitter universe in flat FLRW form, when e t is replaced by r. We show that this expectation is correct, and a suitable modification of the technique used in the case of the de Sitter universe in flat FLRW form does enable one to obtain the expected decay rates also for the case of the Reissner-Nordström-de Sitter spacetime.
Our main results are as follows: Finally, in Appendix C (Section 8), we establish the sharpness of the bound of the |m| = n 2 case of Theorem 3.1.

Relation of Our Results to Previous Work
Our decay rates for the Klein-Gordon equation solutions in the case of the de Sitter universe can be retrieved from the article [26] by setting x = e −t , Y = R n therein. However, the methods used are entirely different: our proof in this case is more explicit and more elementary (relying on energy methods, rather than technical tools from microlocal analysis of partial differential operators). In the article [13], the Klein-Gordon equation is studied in the Nariai spacetime using energy methods, and en route it is also established that solutions of the Klein-Gordon equation decay exponentially in the de Sitter case (with spherical spatial sections). However, the decay rates are not given explicitly.
The article [9] contains a general discussion of redshift estimates, which we use to prove our results in the context of the Reissner-Nordström-de Sitter spacetime. Similar estimates are used in the article [23] to study the wave equation in the Schwarzschild-de Sitter spacetime, of which the Reissner-Nordström-de Sitter spacetime is a perturbation for large radius. Nevertheless, we do not appeal to these results, and instead of extracting what we need from these sources, we give a less technical, self-contained derivation for the convenience of the reader in Sect. 4.5. Here, we follow [8] (where a similar derivation was given for the wave equation).

Decay in the de Sitter Universe in Flat FLRW Form; m = 0
In [8, Theorem 1], the following result was shown: dt < +∞, • n 2, • (M, g) is an expanding FLRW spacetime with flat n-dimensional sections, given by I × R n , with the metric Then Here, the symbol is used to mean that there exists a constant C(δ), independent of , such that We also use the standard notation H k (R n ) for the Sobolev space, . .}, |α| := α 1 + · · · + α n , and ∂ α = (∂ x1 ) α1 · · · (∂ xn ) αn ; see for example [27, p. 249]  3), we will assume, for the sake of simplicity of exposition, that the solution φ to the wave/Klein-Gordon equation is smooth. However, these theorems are also true without this assumption. To see this, we note that for non-smooth solutions with initial data in H k × H k−1 , we can approximate the initial data by smooth functions in H k ×H k−1 , prove the bounds for the H k norms of the corresponding solutions, and then take limits. Since the solution of the problem with rough initial data is in C 0 (I, H k ) ∩ C 1 (I, H k−1 ), these bounds will continue to be true in the limit, and we can then use the Sobolev embedding theorem. This enables one to drop the smoothness assumption.
In Theorem 2.1, in particular, if a(t) = e Ht , where H is the Hubble constant, then since > 0 can be taken to be arbitrarily small, we obtain and this is in agreement with Rendall's conjecture up to the small quantity δ > 0. We will show below that in fact one gets the exact rate (a(t)) −2 when n > 2. There is no loss of generality in assuming that H = 1. Our result is the following. Then Proof. We proceed in several steps.
Step 1: Bound on Δφ. We will follow the preliminary steps of the proof of [8,Theorem 1] in order to obtain a bound on Δφ, which will be needed in the proof of our Theorem 2.3. We repeat this preliminary step here from [8, §2.2] for the sake of completeness and for the convenience of the reader. For a vector field X = X μ ∂ μ , it can be shown that where g := det[g μν ] is the determinant of the matrix [g μν ] describing the metric in the chart. Then, it follows that Thus, g φ = 0 can be rewritten as ∂ μ ( √ −g ∂ μ φ) = 0. With the metric for the de Sitter universe in flat FLRW form given by g = −dt 2 + (a(t)) 2 (dx 1 ) 2 + · · · + (dx n ) 2 , the wave equation can be rewritten as ∂ μ (a n ∂ μ φ) = 0, that is, We recall (see e.g. [27,Appendix E]) that the energy-momentum tensor for the wave equation is Then, it can be shown that ∇ μ T μν = 0. From (2), we have in particular that Define the vector field Then, X is future-pointing (g(X, ∂ t ) < 0) and causal (X is time-like since g(X, X) < 0). We form the current J, given by Then, (set m = 0 in [18, Ex.5.7(1), p. 116], but a justification is given below) Here, X · φ is the application of X on φ. To see ( ), note that It follows from ( ) that g(J, J) 0, so that J is causal. Also, J is pastpointing. To see this, we choose E 1 , . . . , E n orthogonal and spacelike such that {X, E 1 , . . . , E n } forms an orthogonal basis in each tangent space. Then, expressing grad φ = c 0 X + c 1 E 1 + · · · c n E n , we obtain Set N = ∂ ∂t , the future unit normal vector field. We define the energy E by We note that [X, The deformation tensor Π associated with the multiplier X is Here, we used the facts L X (a 2 ) = a 2−n ∂ ∂t a 2 = a 2−n 2aȧ and L X dx i = 0 , and so from the above expression for [X, ∂ μ ], we obtain It can be shown that ∇ μ J μ = T μν Π μν . Indeed, from the expression for the Lie derivative of the metric given in [18, Exercise 2, p. 93] and the expression for the divergence of T μν X ν given in [18, §5.2], we have So the 'bulk term' is For each R > 0, define the set Let t 1 > t 0 . We will now apply the divergence theorem to the region For preliminaries on the divergence theorem in the context of a time-oriented Lorentzian manifold, we refer the reader to [27,Appendix 7]. We have 1 That is,ċ is future-pointing.
where ∂R denotes the boundary of R, is the volume form on M induced by g, and ⌟ denotes contraction in the first index.
Since J is past-pointing, the boundary integral over the null portion C of the boundary ∂R is nonpositive. Also, because ∇ μ J μ is nonnegative, we have that the volume integral over R is nonnegative. This gives an inequality on the two boundary integrals, one over B 0 and the other over Passing the limit R → ∞ yields E(t 0 ) E(t 1 ). (We note that the radius of B 1 also goes to infinity as R → ∞.) As the choice of t 1 > t 0 was arbitrary, we have that for all t t 0 , But since each partial derivative ∂ i φ is also a solution of the wave equation, and as k 2, we obtain, by applying the above to the partial derivatives ∂ i φ, that also In fact, since k > n 2 + 2, we also obtain that for a k > n 2 , Δφ H k (R n ) 1. Finally, by the Sobolev inequality (see e.g. [14, (7.30), p. 158]), we obtain This completes Step 1 of the proof of Theorem 2.3. We note that this step loses two derivatives when we drop the smoothness assumption on φ.
Step 2: The wave equation in conformal coordinates.
The key point of departure from the earlier derivation of the estimates from [8] is the usage of 'conformal coordinates', which renders the wave equation in a form where it becomes possible to integrate, leaving essentially just the time derivative of φ with other terms (e.g. Δφ) for which we have a known bound. An application of the triangle inequality will then deliver the desired bound.
ds. Then we obtain that and With a slight abuse of notation, we write a(τ ) := a(t(τ )).
and so we obtain ∂ μ (a n+1 ∂ μ φ) = 0. Separating the partial derivative operators with respect to the τ and x coordinates, we obtain the wave equation in conformal coordinates ∂ τ (a n−1 ∂ τ φ) = a n−1 Δφ, where Δ is the usual Laplacian on R n . This completes Step 2 of the proof of Theorem 2.3.
Step 3: n > 2 and a(t) = e t . We have and so a(τ ) = 1 Hence, using the bound from (3), namely Δφ(t, ·) L ∞ (R n ) C for all t t 0 , Hence, and so This completes the proof of Theorem 2.3.
Remark 2.4. The case when n = 2 and a(t) = e t : Integrating ∂ τ (a∂ τ φ) = aΔφ from τ = 0 to τ , we obtain and so Hence, This can be viewed as an improvement to [8,Theorem 1] in the special case when a(t) = e t and n = 2, since log a(t) = t e δt = 1 + δt + · · · .
Remark 2.5. The case when a(t) = t p , p 1: One can prove an analogue of Theorem 2.3 when a(t) = t p as well. In this case, the from Theorem 2.1 can be chosen to be any number satisfying and so Theorem 2.1 gives the decay estimate where δ > 0 can be chosen arbitrarily. We can improve this to the following: The proof is the same, mutatis mutandis, as that of Theorem 2.3.
Remark 2.6. Using a similar method, one can also obtain an improvement to [8,Theorem 2]. But we will postpone this discussion until after Sect. 4, since we will need some preliminaries about the RNdS spacetime, which will be established in Sect. 4.

Decay in the de Sitter Universe in Flat FLRW Form
The Klein-Gordon equation is g φ − m 2 φ = 0, that is, In the case of the de Sitter universe in flat FLRW form, we obtain In this section, we will prove Theorem 3.1. We arrive at the guesses for the specific estimates given in Theorem 3.1, based on an analysis using Fourier modes, assuming spatially periodic solutions. This Fourier mode analysis is given in 'Appendix 6'.
• (M, g) is the expanding de Sitter universe in flat FLRW form, with flat n-dimensional sections, given by I × R n , with the metric Then, for all t t 0 , we have We recall that the conformally invariant wave equation in n + 1 dimensions is where R g is the scalar curvature of the metric g; see for instance [27]. If g is a FLRW metric with flat n-dimensional spatial sections, having the form given by (1), then Thus, in de Sitter space in flat FLRW form, the conformally invariant wave equation can be interpreted as a Klein-Gordon equation, with the mass parameter satisfying which follows from using the fact that the L ∞ -norm of ψ(·, t), defined by φ = a 1− n+1 2 ψ (see [8, eq. (178)]), is uniformly bounded with respect to t. The estimate (6)

Preliminary Energy Function and Estimates
Define the energy-momentum tensor T by Then, ∇ μ T μν = 0. Also, in particular, Then, X is time-like and hence causal, and X is futurepointing. Define J by J μ = T μν X ν . Then, J is causal and past-pointing. Let N = ∂ t . Define the energy E by As Hence, We will apply the divergence theorem to the region R : Using • ∇ μ J μ 0, and • the fact that the boundary contribution on C, the null portion of ∂R, is nonpositive (since J is causal and past-pointing), we obtain the inequality Passing the limit R → ∞ yields E(t 1 ) E(t 0 ) < +∞. As t 1 > t 0 was arbitrary, we obtain From here, it follows that for all t t 0 , where k := k − 1.

The Auxiliary Function ψ and Its PDE
Motivated by the decay rate we anticipate for φ, we define the auxiliary function ψ by ψ := a κ φ, where Then, using (5), it can be shown that ψ satisfies the equation 3.3. The Case |m| > n 2 We have κ = n 2 , so that n − 2κ = 0, while κ 2 − nκ + m 2 = m 2 − n 2 4 , and thus (7) becomesψ We note that if φ ∈ H (R n ) andφ ∈ H −1 (R n ) for some , then ψ ∈ H (R n ) too, and alsoψ = n 2 a Define the new energy E, associated with the ψ-evolution, by Then, using the fact that a = e t =ȧ > 0, and also Eq. (7), we obtain Vol. 23 (2022)

Decay of Klein-Gordon Equation Solutions 2359
For a fixed t, and for a ball B(0, r) ⊂ R n , where r > 0, it follows from the divergence theorem (sinceψ and ∇ψ are smooth) that where dσ r is the surface area measure on the sphere S r = ∂B(0, r) and n is the outward-pointing unit normal. The right-hand side surface integral tends to 0 as r → +∞, by an application of Lemma 7.1, given in 'Appendix 7'. So Then, with enough regularity on φ 0 , φ 1 at the outset, that is, if φ 0 ∈ H k (R n ) and φ 1 ∈ H k−1 (R n ) for a k > n 2 + 2, and by considering (∂ x 1 ) i1 · · · (∂ x n ) in φ as a solution to the Klein-Gordon equation, we arrive at 3 where k := k − 2. As k = k − 2 > n 2 , we have, using the Sobolev inequality, that This completes the proof of Theorem 3.1 in the case when |m| > n 2 .
Integrating from t 0 to t yields e θt · E(t) e θt0 · E(t 0 ), that is, Thus, for all t t 0 , we have By considering (∂ x 1 ) i1 · · · (∂ x n ) in φ and using the Sobolev inequality, we have This completes the proof of Theorem 3.1 in the case when |m| < n 2 .

The Case |m| = n 2
We have κ = n 2 , and equation (7) becomesψ − 1 a 2 Δψ = 0. Defining the same energy as we used earlier in the case when |m| < n 2 , we obtain gives Thus, for all t t 0 , we have Hence (by considering (∂ x 1 ) i1 · · · (∂ x n ) in φ and using the Sobolev inequality), (One can show that this bound is sharp; see Appendix C (Section 8).) This completes the proof of Theorem 3.1.

Decay in the Cosmological Region of the RNdS Spacetime
The Reissner-Nordström-de Sitter (RNdS) spacetime (M, g) is a solution to the Einstein-Maxwell equations with a positive cosmological constant, and it represents a pair 4 of antipodal charged black holes in a spherical 5 universe which is undergoing accelerated expansion. The Reissner-Nordström-de Sitter metric in n + 1 dimensions is given by 4 We note that there is no solution analogous to RNdS but with only one black hole. This is analogous to (but much more complicated than, and still not fully understood) the fact that one cannot have a single electric charge on a spherical universe. (Gauss's law requires that the total charge must be zero.) In fact, the fundamental solution of the Laplace equation on the sphere gives a unit positive charge at some point and a unit negative charge at the antipodal point. One can have more than two black holes, for instance the so-called Kastor-Traschen solution [17]. 5 'Spherical' here means that the Cauchy hypersurface (that is, 'space') is an n-sphere. and dΩ 2 is the unit round metric on S n−1 . The constants M and e are proportional to the mass and the charge, respectively, of the black holes, and the cosmological constant is chosen to be Λ = n(n − 1) 2 by an appropriate choice of units.
Consider the polynomial As p(0) = −e 2 < 0 and as p(r) r→∞ −→ ∞, it follows that p will have a real root in (0, +∞), and the largest real root of p, which we denote by r c , must be positive. If r > r c , then clearly p(r) > 0, and so also V (r) > 0.
It can also be seen that p has at most three distinct positive roots. Suppose, on the contrary, that p has more than three distinct positive roots: r 1 < r 2 < r 3 < r 4 . Applying Rolle's theorem to p on [r i , r i+1 ] (i = 1, 2, 3), we conclude that p must have three distinct roots r i ∈ (r i , r i+1 ) (i = 1, 2, 3). Applying Rolle's theorem to p on [r i , r i+1 ] (i = 1, 2), we conclude that p must have two distinct roots r i ∈ (r i , r i+1 ) (i = 1, 2). But which has only one positive root, a contradiction. The 'subextremality' assumption on the RNdS spacetime made in Theorem 3.1 refers to a nondegeneracy of the positive roots of p: we assume that there are exactly three positive roots, r − , r + and r c , and 0 < r − < r + < r c . These describe the event horizon r = r + , and the Cauchy 'inner' horizon r = r − . It can be seen that the subextremality condition then implies p (r c ) > 0. (Indeed, p (r c ) cannot be negative, as otherwise p would acquire a root larger than r c since p(r) r→∞ −→ ∞. Also, if p (r c ) = 0, then Rolle's theorem implies again that p would have three positive roots, ones in (r − , r + ) and (r + , r c ), and one at r c , which is impossible, as we had seen above.) p (r c ) > 0 implies that V (r c ) > 0. We will also assume that (See [6, Appendix 6] for the range of parameters for which this is guaranteed.) Our assumptions have the following consequence, which will be used in our proof of Theorem 4.2. Proof. We have V (r) = rp (r) − (n − 1)p(r) r n = 2r n+1 + 2(3 − n)Mr + (n − 1)e 2 r n =: q(r) r n . As V (r c ) > 0, we have q(r c ) > 0. Also, V (r c ) > 0 and so V is increasing near r c . But then q(r) = r n V (r) is also increasing near r c , and in particular, q (r c ) 0. Let us suppose that there exists an r * > r c such that V (r * ) = 0, and let r * be the smallest such root. Then, q(r * ) = 0 too. We note that q = 2(n + 1)r n + 2(3 − n)M, and so q can have only one nonnegative root, namely (n−3) n+1 M 1 n 0.
If in addition q (r c ) = 0, then we arrive at a contradiction, since q then has two positive roots (at r c and at r * ), which is impossible.
If q (r c ) > 0, then we arrive at a contradiction as follows. As q is increasing near r c , and since q(r c ) > 0 = q(r * ), it follows by the intermediate value theorem that there is some r c ∈ (r c , r * ) such that q(r c ) = q(r c ). But by Rolle's theorem applied to q on [r c , r c ], there must exist an r * ∈ (r c , r c ) such that q (r * ) = 0. Again, q acquires two zeros (at r * and at r * ), which is impossible. 2 • r * is a simple root of q. But as q(r) r→∞ −→ ∞, it follows that there must be at least one more root r * * > r * of q. By Rolle's theorem applied to q on [r * , r * * ], it follows that q (r * * ) = 0 for some r * * ∈ (r * , r * * ).
If in addition q (r c ) = 0, then we arrive at a contradiction, since q then has two positive roots (at r c and at r * * ), which is impossible.
If q (r c ) > 0, then, as in the last paragraph of 1 • above, there exists an r * ∈ (r c , r c ) ⊂ (r c , r * ) such that q (r * ) = 0. Thus, q again gets two positive roots (at r * and at r * * ), which is impossible.
This shows that our assumption that V is zero beyond r c is incorrect.
We will need the previous result in Sect. 4.5 in the analysis following (12) (where in particular V appears in the denominator).
The hypersurfaces of constant r are spacelike cylinders with a futurepointing unit normal vector field N = V The global structure of a maximal spherically symmetric extension of this metric can be depicted by a conformal Penrose diagram shown below, repeated periodically; see for example [7].
We are interested in the behaviour of the solution to the Klein-Gordon equation in the cosmological region R 5 of this spacetime (see Figure 1), bounded by the cosmological horizon branches CH + 1 , CH + 2 , the future null infinity I + , and the point i + . In particular, we want to obtain estimates for the decay rate of φ as r → ∞. We guess the decay rates simply by substituting r instead of e t in the estimates we had obtained for the decay rate of φ with respect to t in the case of the de Sitter universe in flat FLRW form from the previous Sect. 3.
We will prove the following result.

.2. Suppose that
• (M, g) is the (n + 1)-dimensional subextremal Reissner-Nordström-de Sitter solution given by the metric  [4].) Then, there exists a r 0 large enough so that for all r r 0 ,
We will also use the following notation: / ∇φ gradient of φ on S n−1 with respect to the unit round metric, |/ ∇φ| norm with respect to the unit round metric, / Δφ Laplacian of φ on S n−1 with respect to the unit round metric, / g determinant of the unit round metric.
Suppose that φ satisfies the Klein-Gordon equation g φ − m 2 φ = 0. Recall that the energy-momentum tensor associated with φ is given by Thus, We define the energy

The Auxiliary Function ψ and Its PDE
The Klein-Gordon equation g φ − m 2 φ = 0 can be rewritten as: This becomes Then, using the PDE for φ, it can be shown that where θ :=

The Case |m| > n 2
Then, κ = n 2 , and (10) becomes where θ := − n 2 We will use an energy function to obtain the required decay of ψ for large r, and in order to do so, we will need to keep careful track of the limiting behaviour of the various functions appearing in the expression for θ and the coefficients of the PDE (11). We will do this step-by-step in a sequence of lemmas.

Lemma 4.3.
Given any > 0, there exists an r 0 large enough so that for all r r 0 , Proof. This follows immediately from

Lemma 4.4.
There exists r 0 large enough so that for r r 0 , we have θ > 0.
(We note that the proof uses the fact that |m| > n 2 , and so this result is specific to this subsection.)

Proof. As lim
there exists a r 0 such that for r r 0 . Also, by the previous lemma, there exists a r 0 > r 0 such that r for all r r 0 . Then, we have for r > r 0 that where δ := m 2 − n 2 4 > 0. Taking at the outset small enough so as to satisfy 0 < < δ δ+ n 2 + n 2 4 , we see that θ > 0 for r r 0 .
Define the energy (We assume for the moment that this is finite for a sufficiently large r 0 . Later on, in the subsection on redshift estimates, we will see how our initial finiteness of Sobolev norms of φ on the two branches CH + 1 , CH + 2 of the cosmological horizon guarantees this.) We now proceed to find an expression for E (r), and to simplify it, we will use (11), and the divergence theorem, to get rid of the terms involvingψ and/ Δψ, the spherical Laplacian of ψ: We note that in the above, getting rid of the spherical Laplacian by using the divergence theorem is allowed because the compact sphere S n−1 has no boundary. For the second time derivative, however, there is a boundary at infinity (with two connected components), namely which can be seen to be equal to 0, by Lemma 7.3 from 'Appendix B'. Thus, Let > 0 be given. Then, there exists an r 0 large enough such that: Hence, using (a)-(d) above, we obtain Using Grönwall's inequality (see e.g. [12, Appendix 7(j)]), we obtain Thus, Hence, ψ(r, ·) L 2 (R×S n−1 ) r . Consequently, Recall that S n−1 admits n(n−1) 2 independent Killing vectors, given by for i < j (under the usual embedding S n−1 ⊂ R n ). As ∂ ∂t and L ij are Killing vector fields, it follows thatφ and L ij · φ are also solutions to g φ − m 2 φ = 0. Commuting with the Killing vector fields ∂ ∂t and L ij , if we assume at the moment 6 that at r 0 we have φ(r 0 , ·) H k ({r=r0}) < +∞, then we also obtain for all r By the Sobolev inequality, 7 φ(r, ·) L ∞ (R×S n−1 ) r − n 2 + . This completes the proof of Theorem 4.2 in the case when |m| > n 2 (provided we show the aforementioned finiteness of energy, which will be carried out in Sect. 4.5 on redshift estimates).

The Case |m| n 2
Let > 0 be given. Define We now proceed to find an expression for E (r), and we will simplify it using (10) and the divergence theorem, in order to get rid of the terms involvingψ and the spherical Laplacian of ψ: This will be proved later in the subsection on redshift estimates. 7 The part of the Sobolev embedding theorem concerning inclusion in Hölder spaces holds for a complete Riemannian manifold with a positive injectivity radius and a bounded sectional curvature; see e.g. [ Again, for getting rid of the spherical Laplacian, we use the divergence theorem, noting that the sphere S n−1 has no boundary. For handling the second time derivative, as before, we note that there is a boundary at infinity (with two connected components), which can be seen to be equal to 0, by Lemma 7.3 from Appendix B (Section 7). Thus, Now, there exists an r 0 large enough such that for all r r 0 , we have: Using (i), (ii) and (iii), it can be seen that As V r 2 r→∞ −→ 1 and V V r r→∞ −→ 2, it follows that Thus, given > 0, there exists an r 0 large enough such that for r r 0 , |r 2 θ| < , that is, |θ| < r 2 . So The Cauchy-Schwarz inequality applied to the last integral gives So we obtain

Application of Grönwall's inequality yields
Thus, ψ(r, ·) L 2 (R×S n−1 ) 2 E(r 0 ) 1 Given > 0, arbitrarily small, we can choose = ( ) > 0 small enough so that + 2 √ < at the outset, so that φ(r, ·) L 2 (R×S n−1 ) r −κ+ . Again assuming at the moment that at r 0 we have φ(r 0 , ·) H k ({r=r0}) < +∞, and by commuting with the Killing vector fields ∂ ∂t and L ij , then we also obtain for all r r 0 that By the Sobolev inequality, this yields This completes the proof of Theorem 4.2 in the case when |m| n 2 (provided we show the finiteness of energy, which will be carried out in the subsection on redshift estimates below).

Redshift Estimates
The last step is to use redshift estimates to transfer finiteness of the energies along the branches CH + 1 and CH + 2 of the cosmological horizon to finiteness at r = r 0 , justifying the finiteness of the energies E(r 0 ) and E(r 0 ) assumed in the previous two subsections. Here, we do not include all the details of the computations, since they are analogous to the corresponding estimates given in [8, §3.6], and the interested reader can also find the details for our case spelt out in the arxiv version of our paper [19].
dr, where r * > r c is arbitrary, but fixed. Then, The Reissner-Nordström-de Sitter metric can be rewritten using the coordinates (u, r, . . .), instead of the old (t, r, . . .)-coordinates, as follows: . So E(r) 'loses control' of the transverse and angular derivatives as r → r c . To remedy this problem, we define a new energy E, by adding Y to X, obtaining We now have so that using E instead of E allows some control of the angular derivatives as r → r c . Note that E(r c ) is equivalent to φ 2 . We will now compute the deformation tensor Ξ corresponding to the multiplier Y .
Set C 2 (r c ) := C 2 (r c ) + C 2 (r c ). We have For r 1 ∈ (r c , r 0 ), and T > 0, define D = {r = r 1 } ∩ {−T t T }. We now apply the divergence theorem, with the current J corresponding to the multiplier X + Y , in the region T = D + (D) ∩ {r r 0 }. Noticing that the flux across the future null boundaries is less than or equal to 0, we obtain, after passing the limit T → ∞, that But We have Using the above three estimates, it follows from (14) that where k(r 0 , r c ) := C(r 0 , r c )(2r n−1 0 + 2r 2 0 + 2 m 2 ). Now suppose r 2 is such that r c < r 1 < r 2 < r 0 . If we redo all of the above steps in order to obtain (15), but with r 2 replacing r 0 , we obtain where k(r 2 , r c ) = C(r 2 , r c )(2r n−1 As C 1 (r 2 )= As r 2 r 0 , we get E(r 0 ) E(r 1 )e k(r0,rc)·(r0−r1) . This holds for all r 1 ∈ (r c , r 0 ). Passing the limit as r 1 r c , we obtain E(r 0 ) E(r c )e k(r0,rc)·(r0−rc) .
Commuting with the Killing vector fields ∂ ∂t and L ij , we see that the hypothesis from Theorem 4.2, namely φ H k (CH + 1 ) < +∞ and φ H k (CH + 2 ) < +∞, for some k > n 2 +2, yields also φ H k ({r=r0}) φ H k (CH + 1 ) + φ H k (CH + 2 ) < +∞. We now show that this justifies the assumption used in the previous two subsections. For simplicity, we only consider just one of the energies (The proof of the finiteness of E(r 0 ) is entirely analogous.) As ψ = r κ φ, we obtain finiteness of the last summand, namely We have , and as φ(r 0 , ·) ∈ H 1 ({r = r 0 }), we have ψ(r 0 , ·) ∈ L 2 (R × S n−1 ), that is, We also have Finally, Thus, each summand in the expression for E(r 0 ) is finite. This completes the proof of Theorem 4.2.

Decay in RNdS When m = 0, the Wave Equation
In [8, Theorem 2], the following result was shown: • (M, g) is the (n + 1)-dimensional subextremal Reissner-Nordström-de Sitter solution given by the metric and dΩ 2 is the metric of the unit (n − 1)-dimensional sphere S n−1 , • k > n 2 + 2, and • φ is a smooth solution to g φ = 0 such that Then, there exists a r 0 large enough so that for all r r 0 , Using a method similar to the one we used to show Rendall's conjecture in Theorem 2.3, we can improve the almost-exact bound of r −3+δ to r −3 .
Remark 5.2. As observed in [8,Remark 1.4], this decay rate bound of r −3 for ∂ r φ is in fact the decay rate one would expect in the light of Rendall's conjecture. Indeed, for freely falling observers in the cosmological region, one has where τ is the proper time, and a(τ ) is the radius of a comparable de Sitter universe in flat FLRW form, giving Thus, our improved version of Theorem 5.1 is the following result.
Sitter solution given by the metric where V = r 2 + 2M r n−2 − e 2 r n−1 − 1, and dΩ 2 is the metric of the unit (n − 1)-dimensional sphere S n−1 , • k > n 2 + 2, and • φ is a smooth solution to g φ = 0 such that Proof.
We will follow [8, §3.2] in order to obtain the bounds above, which will be needed in Step 2 of our proof below. We repeat this preliminary step here from [8, §3.2] for the sake of completeness and for the convenience of the reader. Suppose φ satisfies the wave equation g φ = 0. The energy-momentum tensor associated with φ is given by The current J is given by J μ := T μν X ν . We define the energy The deformation tensor Π associated with the multiplier X is given by It can be shown that .
We have

Consequently, the full bulk term is
Using the expression for V , we compute Hence, For each T < 0, define the set C := {r = r 0 } ∩ {−T t T }. Also, consider the region S := D + (C) ∩ {r r 1 }.
We will apply the divergence theorem to the current J on the region S. As the flux across the future null boundaries is nonpositive, we have R×S n−1 C r 3 |/ ∇φ| 2 dtdΩdr. We have Commuting with the Killing vector fields ∂ ∂t and L ij , we obtain (after the transferral of the finiteness of the energies along the branches CH + 1 and CH + 2 of the cosmological horizon to finiteness at r = r 0 , and an application of Sobolev's inequality) that for all r r 0 , φ (r, ·) L ∞ (R,S n−1 ) 1, and / Δφ(r, ·) L ∞ (R,S n−1 ) 1.
Step 2: In this step, we will write the wave equation in new coordinates which 'equalises' the magnitude of the coefficient weights for the r and t coordinates in the matrix of the metric.
Integrating from ρ 0 := ρ(r 0 ) = 0 to ρ = ρ(r), we obtain Using the fact that V ∼ r 2 for r r 0 , with r 0 large enough, and the estimates from Step 1 above, we obtain ∂ r φ(r, ·) L ∞ (R×S n−1 ) A r n+1 + Recalling that n > 2, we have This completes the proof of Theorem 5.3.
Using these, as τ 0 or t → ∞, |c k | = C e −( n 2 −Re(ν))t + O(e − n+1 2 t ). Thus, we expect φ to satisfy φ(t, ·) L ∞ (R n ) a − n So we will construct a subsequence (r km ) m of (r k ) k and a sequence (δ m ) m of positive numbers such that r k1 < r k1 + δ 1 < r k2 < r k2 + δ 2 < r k3 < · · · , and such that for the 'annuli' A m := {x : r km < |x| < r km + δ m }, we have f 2 H 1 (Am) > . We will need to keep track of the constants in the trace theorems on our annuli A m , and we will use the following [15, p. 41].) Theorem 7.2. Let Ω be a bounded open subset of R n with a Lipschitz boundary Γ. Then, for f ∈ H 1 (Ω) and for all ∈ (0, 1), where μ ∈ C 1 (Ω, R n ) is such that μ · n δ on ∂Ω, and n is the outer normal vector.
If Ω is an annulus A = {x : r < x < R} (which is clearly bounded, open, and also it has the Lipschitz boundaries which are the two spheres S r and S R ), then with μ(x) = x, we have μ · n = x = R on S R , r on S r r =: δ.