# Linear waves on the expanding region of Schwarzschild-de Sitter spacetimes:
forward asymptotics and scattering from infinity

Louie Bernhardt<EMAIL_ADDRESS>School of Mathematics and
Statistics, University of Melbourne, Parkville, 3010, VIC, Australia

###### Abstract

We study solutions to the linear wave equation on the cosmological region of
Schwarzschild-de Sitter spacetimes. We show that all sufficiently regular
finite-energy solutions to the linear equation possess a particular finite-
order asymptotic expansion near the future boundary. Specifically, we prove
that several terms in this asymptotic expansion are identically zero. This is
accomplished with new weighted higher-order energy estimates that capture the
global expansion of the cosmological region. Furthermore we prove existence
and uniqueness of scattering solutions to the linear wave equation on the
expanding region. Given two pieces of scattering data at infinity, we
construct solutions that have the same asymptotics as forward solutions. The
proof involves constructing asymptotic solutions to the wave equation, as well
as a new weighted energy estimate that is suitable for the backward problem.
This scattering result extends to a large class of expanding spacetimes,
including the Kerr de Sitter family.

##  1 Introduction

In this paper we prove results relating to the asymptotic behaviour of
solutions to the linear wave equation

$\square_{g}\psi=0$ (1.1)

on $3+1$-dimensional Schwarzschild-de Sitter spacetimes. For an introduction
to the global geometry of Schwarzschild-de Sitter, see [21]. We focus on the
expanding region which we denote by $\mathcal{R}^{+}$; see Fig. 1.1. The
expanding region is bounded in the past by two cosmological horizons
$\mathcal{C}^{+},\overline{\mathcal{C}}^{+}$, and to the future by the future
boundary $\Sigma^{+}$. We introduce on $\mathcal{R}^{+}$ the standard
coordinate chart
$(r,t,\theta,\varphi)\in(r_{\mathcal{C}},\infty)\times\mathbb{R}\times(0,\pi)\times(0,2\pi)$
and metric

$g_{\Lambda,m}=-\frac{1}{\frac{\Lambda}{3}r^{2}-1+\frac{2m}{r}}\textnormal{d}r^{2}+\Big{(}\frac{\Lambda}{3}r^{2}-1+\frac{2m}{r}\Big{)}\textnormal{d}t^{2}+r^{2}\>\textnormal{d}\theta^{2}+r^{2}\sin^{2}\theta\>\textnormal{d}\varphi^{2},$

where $r_{\mathcal{C}}$ denotes the largest positive root of
$\frac{\Lambda}{3}r^{2}-1+\frac{2m}{r}$. For more properties of the expanding
region, see [27].

$\mathcal{H^{+}}$$\overline{\mathcal{H}}^{+}$$\mathcal{C}^{+}$$\overline{\mathcal{C}}^{+}$$\Sigma^{+}$$\mathcal{R}^{+}$$\mathcal{S}$$\overline{\mathcal{S}}$$\Sigma_{r}$$\iota^{+}$$\overline{\iota}^{+}$$\Sigma^{\prime}$
Figure 1.1: Geometry of the expanding region $\mathcal{R}^{+}$ of
Schwarzschild-de Sitter, shaded in gray here, which is bounded in the past by
cosmological horizons $\mathcal{C}^{+},\overline{\mathcal{C}}^{+}$, and to the
future by $\Sigma^{+}$. One can also consider the Cauchy problem for the
linear wave equation on the region
$\mathcal{S}\cup\mathcal{R}^{+}\cup\overline{\mathcal{S}}$, where one sets
initial data on a Cauchy surface like $\Sigma^{\prime}$.

A natural foliation of $\mathcal{R}^{+}$ is by the level sets of constant
radius which we denote by $\Sigma_{r}$. These are spacelike on the interior of
the expanding region. We note that the hypersurfaces $\Sigma_{r}$ and future
boundary $\Sigma^{+}$ are all diffeomorphic to
$\mathbb{R}\times\mathbb{S}^{2}$, terminating at $\iota^{+}$ on one end, and
at $\overline{\iota}^{+}$ on the other. As such we will often identify
functions on $\Sigma_{r}$ or $\Sigma^{+}$ with functions on the cylinder in
this paper. Define the following energy:111We will discuss the geometric
interpretation of this energy in Section 2.

$E[\psi](r)=\frac{1}{2}\int_{\Sigma_{r}}\phi^{-1}[(\partial_{r}\psi)^{2}+|\overline{\nabla}\psi|^{2}]\textnormal{d}\mu_{\overline{g}_{r}},$
(1.2)

where

$\phi=\frac{1}{\sqrt{\frac{\Lambda}{3}r^{2}-1+\frac{2m}{r}}}$

is the _lapse_ for the $\Sigma_{r}$-foliation, $\overline{\nabla}\psi$ is the
covariant derivative restricted to $\Sigma_{r}$, and
$\textnormal{d}\mu_{\overline{g}_{r}}$ is the induced volume form on
$\Sigma_{r}$.

We now present the two main theorems of this paper. The first theorem states
that there is a finite-order asymptotic expansion in powers of $r^{-1}$ for
forward solutions to the linear wave equation on Schwarzschild-de Sitter.

######  Theorem 1.1 (Forward asymptotics of linear waves on
$\mathcal{R}^{+}$).

Let $\psi$ be a finite energy solution to (1.1) on $\mathcal{R}^{+}$, in the
sense that on a level set $\Sigma_{r_{0}}\subset\mathcal{R}^{+}$ we have

$\sum_{k=0}^{2}E[\partial_{r}^{k}\psi](r_{0})<\infty.$ (1.3)

Then the following limits exist, where we emphasise that the limits are taken
with respect to the norm of the homogeneous Sobolev space
$\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})$:

$\psi_{0}=\lim_{r\rightarrow\infty}\psi,\quad\psi_{2}=\lim_{r\rightarrow\infty}\Big{[}r^{2}(\psi-\psi_{0})\Big{]},\quad\psi_{3}=\lim_{r\rightarrow\infty}\Big{[}r^{3}\Big{(}\psi-\psi_{0}-\frac{\psi_{2}}{r^{2}}\Big{)}\Big{]}.$
(1.4)

It immediately follows from Theorem 1.1 that such solutions $\psi$ have
asymptotics of the form

$\psi\sim\psi_{0}+\frac{\psi_{2}}{r^{2}}+\frac{\psi_{3}}{r^{3}}$ (1.5)

for large $r$.

######  Remark 1.2.

This extends a result from [27], where it was shown that $\psi$ has a nonzero
limit on $\Sigma^{+}$ with respect to the $\dot{H}^{1}$ norm, proving the
existence of just the $\psi_{0}$ term in the asymptotic expansion (1.5).
Similar asymptotic expansions to (1.5) were given in [30] for smooth solutions
to the linear wave equation on a class of de Sitter like spacetimes that
include Schwarzschild-de Sitter. In particular it was shown that on this class
of spacetimes, smooth solutions $\psi$ to the linear wave equation have an
asymptotic expansion of the form

$\psi\sim\psi_{0}+\frac{\psi_{1}}{r}+\frac{\psi_{2}}{r^{2}}+\frac{\log
r\>\psi_{3,1}}{r^{3}}+\frac{\psi_{3}}{r^{3}}.$ (1.6)

However in [30] it is not specified whether all terms in this expansion are
generically nonzero.222It is shown in [30] that if the metric of a de Sitter-
like spacetime has a Taylor expansion near $\Sigma^{+}$ that contains only
even powers of $r^{-1}$, then the $r^{-3}\log r\>\psi_{3,1}$ term is
identically zero. However this does not apply to Schwarzschild-de Sitter due
to the black hole mass, as one can see for instance that
$g_{tt}=\frac{\Lambda}{3}r^{2}-1+\frac{2m}{r}+O(r^{-2}).$ Importantly, Theorem
1.1 implies that the $r^{-1}\psi_{1}$ and $r^{-3}\log r\>\psi_{3,1}$ terms in
(1.6) are absent for linear waves on Schwarzschild-de Sitter.

######  Remark 1.3.

In the context of the Cauchy problem, where one poses data on a Cauchy surface
such as $\Sigma^{\prime}$ in Figure 1.1, the bound (1.3) holds for
sufficiently regular finite-energy solutions to the linear wave equation. This
follows in particular from the works [2, 6, 10] which studied solutions to the
linear wave equation on the _static region_ $\mathcal{S}$ of Schwarzschild-de
Sitter (and the related stationary region of Kerr-de Sitter). For more recent
work, see for example [24]. In fact, it is known that the static region
$\mathcal{S}$ is nonlinearly stable as a solution to Einstein’s vacuum
equations with positive cosmological constant. This was proved in the seminal
work [16], see also [11].

Moreover we consider the scattering problem for the linear wave equation on
Schwarzschild-de Sitter. We show that $\psi_{0},\psi_{3}$ in the asymptotic
expansion (1.5) constitute the correct notion of scattering data, so that
given the functions $\psi_{0},\psi_{3}$ there exists a unique solution to
(1.1) on $\mathcal{R}^{+}$.

######  Theorem 1.4 (Scattering of linear waves on $\mathcal{R}^{+}$).

Let $\psi_{0}\in H^{4}(\mathbb{R}\times\mathbb{S}^{2}),\psi_{3}\in
H^{2}(\mathbb{R}\times\mathbb{S}^{2})$. Then there exists a unique solution
$\psi$ to the linear wave equation on the expanding region of Schwarzschild-de
Sitter such that for all $r\geq r_{\mathcal{C}}+\epsilon$,

$E[\psi](r)\lesssim\|\psi_{0}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})}^{2}+\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}^{2}.$
(1.7)

Moreover we have the following $L^{\infty}$ decay estimate for large $r$:

$\sup_{\Sigma_{r}}\Big{|}\psi-\psi_{0}-\frac{\psi_{2}}{r^{2}}-\frac{\psi_{3}}{r^{3}}\Big{|}\lesssim\frac{1}{r^{4}}\|\psi_{0}\|_{H^{6}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{5}}\|\psi_{3}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})},$
(1.8)

provided the right hand side is finite. Here $\psi_{2}$ is a function
determined entirely by $\psi_{0}$.

######  Remark 1.5.

The fact that there are two pieces of scattering data are due to the spacelike
nature of $\Sigma^{+}$. For the corresponding Cauchy problem, one prescribes
two pieces of data on a spacelike hypersurface which are the solution and its
normal derivative restricted to $\Sigma^{+}$. In Theorem 1.10, $\psi_{0}$,
$\psi_{3}$ are the two pieces of data, with $\psi_{3}$ playing the role of the
normal derivative.333However $\psi_{3}$ is not strictly speaking the normal
derivative of the solution $\psi$, as any radial derivative $\psi$ would
contain also the term $\psi_{2}$, which is determined by $\psi_{0}$. See the
discussion in Remark 1.15 below.

######  Remark 1.6.

For related scattering results, we mention again [30], where it was shown that
given smooth functions $\psi_{0},\psi_{3}$, there exists a unique smooth
solution to the linear wave equation on a class of de Sitter-like spacetimes.
See in particular the discussion below Theorem 1.2 in [30]. In [4], a
scattering theory was established for the linear wave equation on de Sitter
spacetimes with even number of spatial dimensions.

######  Remark 1.7.

Scattering problems have also been considered for linear wave equations on
asymptotically flat black hole spacetimes, see [1, 9, 19]. The scattering
problem for Einstein’s vacuum equation with positive cosmological constant has
also been treated, see for example [15], while for scattering of Einstein’s
vacuum equations in the asymptotically flat setting, see [5].

###  1.1 Existence of solutions to the linear wave equation with prescribed
scattering data on $\Sigma^{+}$

We give a complete treatment of scattering of wave equations on Schwarzschild-
de Sitter in Section 3. We prove Theorem 1.4 by first constructing an
asymptotic solution to the wave equation of the form

$\psi_{\text{asymp}}(t,\omega)=\psi_{0}(t,\omega)+\frac{\psi_{2}(t,\omega)}{r^{2}}+\frac{\psi_{3}(t,\omega)}{r^{3}},$

where $\psi_{0}$, $\psi_{3}$ are freely chosen and $\psi_{2}$ is determined by
$\psi_{0}$.444We are not constructing smooth scattering solutions to the wave
equation by performing an infinite power series expansion in $r^{-1}$,
although if one does this it turns out that $\psi_{0}$, $\psi_{3}$ are
necessarily the leading order terms in such an expansion that can be freely
chosen. Clearly $\psi_{\text{asymp}}$ captures the asmyptotics of forward
solutions proved in Theorem 1.1, but moreover the asymptotic solution solves
the wave equation up to some error that decays quickly in $r$, specifically we
have $\square_{g}\psi_{\text{asymp}}=O(r^{-4})$.

Then we showing that there exists an actual solution to $\square_{g}\psi=0$
that converges to that asymptotic solution. The key tool we use to prove this
is the following weighted energy estimate suitable for the backward problem:

######  Proposition 1.8 (Weighted energy estimate for the backward problem).

Suppose $\psi$ is a solution to the inhomogeneous wave equation

$\square_{g}\psi=F$ (1.9)

in the expanding region $\mathcal{R^{+}}$ of Schwarzschild-de Sitter
spacetime. Then for all $r_{2}>r_{1}>r_{\mathcal{C}}$,

$r_{1}^{2}(E[\psi](r_{1}))^{1/2}\leq
r_{2}^{2}(E[\psi](r_{2}))^{1/2}+\frac{1}{\sqrt{2}}\int_{r_{1}}^{r_{2}}\Big{(}\int_{\Sigma_{r}}r^{2}\phi
F^{2}\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}\textnormal{d}r,$ (1.10)

provided the right hand side is finite.

We prove Proposition 1.8 by identifying a vectorfield multliplier
$M=r^{2}\phi^{-2}\partial_{r}$ that produces a _nonpositive_ bulk integral in
the energy estimate; see Section 3.2 for relevant definitions. We prove this
implies that the weighted energy $r^{2}E[\psi]$ is monotone decreasing
_backwards-in-time_ 555This is in contrast to the redshift estimate (1.16)
from [27], where a vectorfield was identified that produces a _nonnegative_
bulk integral, yielding an energy $E[\psi]$ that is monotonically decreasing
forwards-in-time. These two energy estimates can also be compared to the
standard energy estimate for the wave equation on Minkowski spacetime. There
one can apply the timelike Killing vectorfield multiplier
$T=\frac{\partial}{\partial t}$, which yields an energy that is conserved,
rather than just monotone..

We will apply the weighted energy estimate from Proposition 1.8 not to the
solution itself, but to a fast-decaying remainder $\psi_{\text{rem}}$ that is
obtained by subtracting from the true solution an asymptotic solution
$\psi_{\text{asymp}}$ constructed in Section 3.1. This asymptotic solution
captures the leading order asymptotics of the solution.

######  Remark 1.9.

This approach to constructing scattering solutions is inspired by [23], where
scattering problems for nonlinear wave equations on Minkowski are treated. In
that paper asymptotic solutions are constructed from a radiation field
prescribed at null infinity, and then a fast-decaying remainder is estimated
backwards-in-time via a _fractional Morawetz estimate_ , which plays the same
role as Proposition 1.8 in the present paper. For other papers where this
method is used for constructing scattering solutions to wave equations and
other dispersive equations, see for example [14, 32].

#### Extension to the cosmological horizon

While Theorem 1.4 guarantees existence of solutions to $\square_{g}\psi=0$ on
the interior of the expanding region, the energy estimate from Proposition 1.8
degenerates at the cosmological horizons, and does not guarantee that
solutions remain uniformly bounded on the cosmological horizons.

We define a nondegenerate energy on the cosmological horizons. We denote this
energy by $E^{N}[\psi]$. The full definition of $E^{N}[\psi]$ can be found in
Section 3.5, but for now we mention that on the cosmological horizons:

$E^{N}[\psi]=\frac{1}{2}\int_{\mathcal{C}^{+}\cup\overline{\mathcal{C}}^{+}}(T\psi)^{2}+|\not{\nabla}\psi|^{2},$

where $\not{\nabla}$ denotes the covariant derivative on the sphere of radius
$r$.

In order to solve all the way to the cosmological horizons
$\mathcal{C}^{+}\cup\overline{\mathcal{C}}^{+}$ in a nondegenerate energy
space, we additionally assume exponential decay of the data $\psi_{0}$,
$\psi_{3}$ along the future boundary $\Sigma^{+}$ towards $\iota^{+}$ and
$\overline{\iota}^{+}$.

######  Theorem 1.10 (Scattering up to the cosmological horizon).

Let $\psi_{0},\psi_{3}$ be scattering data defined on the future boundary
$\Sigma^{+}$ such that $\psi_{0}\in
H^{4}(\mathbb{R}\times\mathbb{S}^{2}),\psi_{3}\in
H^{2}(\mathbb{R}\times\mathbb{S}^{2})$. Moreover, assume that $\psi_{0}$,
$\psi_{3}$ decay exponentially, in that there exists some sufficiently large
constant $\beta=\beta(\Lambda,m)>0$ such that as $\tau\rightarrow\infty$ we
have

$\|\psi_{0}\|_{H^{4}((-\tau,\tau)^{c}\times\mathbb{S}^{2})},\|\psi_{3}\|_{H^{2}((-\tau,\tau)^{c}\times\mathbb{S}^{2})}\lesssim_{\Lambda,m}e^{-\beta\tau}.$
(1.11)

Then there exists a unique solution $\psi$ to (1.1) on
$\mathcal{R}^{+}\cup\mathcal{C}^{+}\cup\overline{\mathcal{C}}^{+}$ with
scattering data $\psi_{0},\psi_{3}$, and at the cosmological horizon, we have
the energy estimate

$E^{N}[\psi]\lesssim_{\Lambda,m}\|\psi_{0}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})}^{2}+\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}^{2}.$
(1.12)

######  Remark 1.11.

The infimum of all values of the constant $\beta$ for which Theorem 1.10 holds
is $\frac{1}{2}\kappa_{\mathcal{C}}$, where $\kappa_{\mathcal{C}}$ is the
_surface gravity_ of the cosmological horizons. This requirement of a
sufficient degree of exponential decay is tied to the _blueshift effect_ , a
property of Killing horizons such as cosmological horizons. The previously-
discussed redshift effect leads to boundedness of waves forwards-in-time along
Killing horizons. However for the backwards problem this is seen as a
blueshift, which leads to exponential growth of waves backwards-in-time. While
it is primarily discussed in the setting of black holes in asymptotically flat
spacetimes (see [5, 9, 26]), the blueshift effect is present for all
nondegenerate Killing horizons.

By applying the pointwise boundedness estimate found in Corollary 2.12 in
[27], it follows that the solutions constructed in Theorem 1.10 are uniformly
bounded on $\overline{\mathcal{R}}^{+}$.

######  Corollary 1.12.

Let $\psi_{0},\psi_{3},\psi$ be as in Theorem 1.4, with $\psi_{0}\in
H^{6}(\mathbb{R}\times\mathbb{S}^{2})$,$\psi_{3}\in
H^{4}(\mathbb{R}\times\mathbb{S}^{2})$.Then we have have the pointwise bound
away from the cosmological horizons:

$\sup_{\\{r\geq
r_{\mathcal{C}}+\epsilon\\}}|\psi|\lesssim_{\Lambda,m,\epsilon}\|\psi_{0}\|_{H^{6}(\mathbb{R}\times\mathbb{S}^{2})}+\|\psi_{3}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})}.$
(1.13)

If additionally $\psi_{0}$, $\psi_{3}$ decay exponentially along $\Sigma^{+}$
like in Theorem 1.10, then the solution $\psi$ is uniformly bounded up to the
horizon:

$\sup_{\overline{\mathcal{R}}^{+}}|\psi|\lesssim_{\Lambda,m}\|\psi_{0}\|_{H^{6}(\mathbb{R}\times\mathbb{S}^{2})}+\|\psi_{3}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})}.$
(1.14)

######  Remark 1.13.

Corollary 1.12 is proved by commuting energy estimates with the Killing
vectorfields that span the hypersurfaces $\Sigma_{r}$, $\mathcal{C}^{+}$ and
$\overline{\mathcal{C}}^{+}$. These are the generators of the spherical
isometries $\Omega_{i}:i=1,2,3$, and the coordinate vectorfield
$T=\frac{\partial}{\partial t}$. A Sobolev embedding on these hypersurfaces
gives pointwise bounds on $\psi$, see [27] for details.

###  1.2 Forward asymptotics for linear waves on Schwarzschild-de Sitter

We now turn our attention to the proof of Theorem 1.1, which we set up and
prove in Section 4. Theorem 1.1 relies on several new higher-order energy
estimates. These estimates capture the global expansion of $\mathcal{R}^{+}$.
We have:

######  Theorem 1.14 (Higher-order redshift estimate).

Let $\psi$ be a solution to the wave equation $\square_{g}\psi=0$ on the
expanding region of Schwarzschild-de Sitter, and let $r_{0}$ be a radius that
is slightly larger than $r_{\mathcal{C}}$. Let $X,Y$ denote the vectorfields

$X=r\phi^{-2}\partial_{r},\quad Y=r\phi^{-1}\partial_{r}.$

Then for all $r_{2}>r_{1}\geq r_{\mathcal{C}}$ we have the energy estimate

$E_{\Sigma_{r}}^{\partial_{r}}[Y(X\psi)](r_{2})+E_{\Sigma_{r}}^{\partial_{r}}[X\psi](r_{2})+E_{\Sigma_{r}}^{\partial_{r}}[\psi](r_{2})\\\
\lesssim_{\Lambda,m,r_{0}}E_{\Sigma_{r}}^{\partial_{r}}[Y(X\psi)](r_{1})+E_{\Sigma_{r}}^{\partial_{r}}[X\psi](r_{1})+E_{\Sigma_{r}}^{\partial_{r}}[\psi](r_{1}).$
(1.15)

######  Remark 1.15.

The weights in $r$ for the vectorfield commutators $X,Y$ are, to leading
order:

$X\sim r^{3}\partial_{r},\quad Y\sim r^{2}\partial_{r},$

and so applying these vectorfields to a solution $\psi$ to the wave equation
essentially identifies higher-order terms in the asymptotic expansion (1.5).
To see this, if we assume the asymptotic expansion (1.5) holds, then

$\psi\sim\psi_{0},\quad r^{3}\partial_{r}\psi\sim\psi_{2},\quad
r^{2}\partial_{r}(r^{3}\partial_{r}\psi)\sim\psi_{3}.$

######  Remark 1.16.

In [27], it was shown that the zeroth-order energy $E[\psi]$ is monotone in
$r$, so that if $\square_{g}\psi=0$, then

$E[\psi](r_{2})\leq E[\psi](r_{1})$ (1.16)

for all $r_{2}\geq r_{1}>r_{\mathcal{C}}$. In that paper a _vectorfield
multiplier_ is identified that captures the global expansion of
$\mathcal{R}^{+}$, in the sense that it produces a nonnegative bulk term in
the resulting energy identity.

We derive the higher-order redshift estimate of Theorem 1.14 by showing that
as _vectorfield commutators_ , $X$ and $Y$ satisfy good commutation
identities, once again leading to a nonnegative bulk term. These commutators
are proportional to $\partial_{r}$, and so are not Killing. This means that
$X$ and $Y$ have commutation identities containing lower-order terms that also
need to be controlled.

######  Remark 1.17.

A similar higher-order global redshift estimate has been proved in [29] for
_Weyl fields_ on expanding spacetimes. Weyl fields are trace-free solutions to
the Bianchi equations

$\nabla^{\mu}W_{\mu\nu\lambda\delta}=0.$

In that paper, the Bianchi equations are the hyperbolic differential equations
for which a higher-order redshift estimate is proved. In the context of the
local redshift effect on black hole horizons, these vectorfield commutator
estimates stem from Dafermos and Rodnianski in [8]. We also mention [12], in
which similar ideas are used to proved higher-order energy estimates for
linear waves on the black hole region of Schwarzschild spacetimes.

######  Remark 1.18.

See [18, 25] for results on forward asymptotics for the wave equation on
related cosmological spacetimes. Forward asymptotics have been studied for
solutions to Einstein’s vacuum equations. In particular, asymptotic expansions
for perturbations of Minkoswki are obtained in [22, 17].

Through a density argument presented in Section 4.5, Theorem 1.14 implies that
the quantities $\psi,X\psi,Y(X\psi)$ all have a nonzero limit on $\Sigma^{+}$
with respect to the energy norm. A simple application of the fundamental
theorem of calculus then implies the asymptotic expansion (1.5), due to the
weights in $r$ possessed by the vectorfield commutators $X,Y$.

######  Remark 1.19.

We expect that one can derive successive higher-order weighted estimates by
repeatedly commuting the existing estimate (1.15) with the commutator $Y$.
This would allow use to derive for solutions an asymptotic expansion to
arbitrary order in powers of $r^{-1}$. However, the present estimate is
sufficient for us to produce an asymptotic expansion that identifies the two
pieces of data $\psi_{0}$ ,$\psi_{3}$ for the corresponding scattering
problem.

###  1.3 Scattering on perturbations of Schwarzschild-de Sitter

The results we prove in this paper Schwarzschild-de Sitter can be suitably
generalised to a broad class of expanding spacetimes. This is because they do
not rely on fundamental solution techniques. In Section 5 we highlight this by
extending the scattering result from Theorem 1.4 to a large class of expanding
spacetimes. In particular, we show that the construction of scattering
solutions extends to Kerr-de Sitter, as well as spacetimes that do not
necessarily converge to Schwarzschild-de Sitter. The class of metrics we
consider, broadly speaking, are those with metrics that have the same leading-
order asymptotic behaviour as Schwarzschild-de Sitter in terms of growth or
decay in $r$.666This class of spacetimes is similar to those considered in
[30], with the difference being that the metric components here can be assumed
to be of finite-order regularity rather than smooth. We do not assume
closeness of the metric to Schwarzschild-de Sitter, and we make no symmetry
assumption. An explicit description of the metrics we consider can be found in
Definition 5.2

On this class of spacetimes we have the following scattering result, stated
informally:

######  Theorem 1.20 (Scattering of linear waves on perturbations of
Schwarzschild-de Sitter).

An analogue of the scattering result of Theorem 1.4 holds for solutions to the
linear wave equation on spacetimes with the same leading-order asymptotic
behaviour as Schwarzschild-de Sitter.

For a formal statement of this result, see Theorem 5.10 in Section 5.3. We
show that the two components of the proof, which are the construction of
asymptotic solutions and a weighted energy estimate suitable for the backward
problem, generalise from Schwarzschild-de Sitter to the perturbed setting.

Depending on the form of the metrics, the asymptotic solutions to the wave
equation on a given spacetime we construct are of the form

$\psi_{\text{asymp}}\sim\psi_{0}+\frac{\psi_{2}}{r^{2}}+\frac{\log
r\>\psi_{3,1}}{r^{3}}+\frac{\psi_{3}}{r^{3}}.$

Thus a $r^{3}\log r\>\psi_{3,1}$ term can be present in the expansion. We
specify sufficient conditions on the metric for which this log term is not
present. In particular these conditions are satisfied by the expanding region
of the Kerr-de Sitter family of spacetimes. This implies the following:

######  Corollary 1.21 (Scattering on Kerr-de Sitter).

Let $\psi_{0},\psi_{3}$ be sufficiently regular functions at the future
boundary of Kerr-de Sitter. Then there exists a unique, finite-energy solution
to the wave equation $\square_{g}\psi=0$ on the expanding region of Kerr-de
Sitter such that for large $r$ we have

$\psi\sim\psi_{0}+\frac{\psi_{2}}{r^{2}}+\frac{\psi_{3}}{r^{3}},$

where $\psi_{2}$ is determined by $\psi_{0}$.

###  1.4 Acknowledgements

I would like to express my gratitude to my adviser Volker Schlue for
suggesting this problem to me, and for his support and guidance throughout the
process of writing this paper. I would also like to thank the University of
Melbourne for its financial support.

##  2 Geometry of the Schwarzschild-de Sitter family of spacetimes

In this section we recall some of the relevant properties of Schwarzschild-de
Sitter spacetimes, and the energies, norms, and other notation which we use
throughout this paper. For a more complete introduction of the geometry of
Schwarzschild-de Sitter, see [21], as well as [27].

###  2.1 Schwarzschild-de Sitter spacetime

The Schwarzschild-de Sitter family of spacetimes, discovered independently in
[20] and [31], constitute a one-parameter family of solutions to Einsteins
vacuum equations with positive cosmological constant

$\text{Ric}(g)-\Lambda g=0,\quad\Lambda>0.$ (2.1)

These spacetimes are parametrised by the black hole mass, denoted by $m$. We
assume that $0<3m<1/\sqrt{\Lambda}$, which restricts the class of spacetimes
we analyse to subextremal Schwarzschild-de Sitter.

Recall the metric

$g_{\Lambda,m}=-\frac{1}{\frac{\Lambda}{3}r^{2}-1+\frac{2m}{r}}\textnormal{d}r^{2}+\Big{(}\frac{\Lambda}{3}r^{2}-1+\frac{2m}{r}\Big{)}\textnormal{d}t^{2}+r^{2}\>\textnormal{d}\theta^{2}+r^{2}\sin^{2}\theta\>\textnormal{d}\varphi^{2},$
(2.2)

defined on the expanding region. From this, we see that the coordinate vector
field $\frac{\partial}{\partial r}$ is timelike on $\mathcal{R}^{+}$, and we
can view $r$ as a time function for the expanding region. Recall also the
lapse of the $\Sigma_{r}$-foliation

$\phi=\frac{1}{\sqrt{\frac{\Lambda}{3}r^{2}-1+\frac{2m}{r}}}.$

As $\phi$ frequently appears in weighted energies in this paper, we emphasise
that for large $r$, $\phi\sim 1/r$. One can decompose the metric (2.2) so that

$g=-\phi^{2}\textnormal{d}r^{2}+\overline{g}_{r},$ (2.3)

where $\overline{g}_{r}$ is the induced metric on the hypersurfaces
$\Sigma_{r}$.

We also recall from [27] the _Kruskal coordinates_ $(u,v)$ on Schwarzschild-de
Sitter The coordinates $(r,t,\theta,\varphi)$ cover only the expanding region
$\mathcal{R}^{+}$, and the metric (2.2) becomes singular at
$r=r_{\mathcal{C}}$. In contrast the Kruskal coordinates
$(u,v,\theta,\varphi)$ cover the cosmological horizons
$\mathcal{C}^{+},\overline{\mathcal{C}}^{+}$, and are related to
$(r,t,\theta,\varphi)$ coordinates implicitly by

$\displaystyle
uv=\frac{r-r_{\mathcal{C}}}{(r-r_{\mathcal{H}})^{\alpha_{\mathcal{H}}}(r+|\overline{r_{\mathcal{C}}}|)^{\overline{\alpha_{\mathcal{C}}}}},$
$\displaystyle\log\Big{|}\frac{u}{v}\Big{|}=-\frac{\Lambda}{3}\frac{(r_{\mathcal{C}}-r_{\mathcal{H}})(r_{\mathcal{C}}+|\overline{r_{\mathcal{C}}}|)}{r_{\mathcal{C}}}t,$

where

$\alpha_{\mathcal{H}}=\frac{r_{\mathcal{H}}}{r_{\mathcal{C}}}\frac{r_{\mathcal{C}}+\overline{r_{\mathcal{C}}}}{r_{\mathcal{H}}+\overline{r_{\mathcal{C}}}},\quad\overline{\alpha_{\mathcal{C}}}=\frac{\overline{r_{\mathcal{C}}}}{r_{\mathcal{C}}}\frac{r_{\mathcal{C}}-r_{\mathcal{H}}}{r_{\mathcal{C}}+\overline{r_{\mathcal{C}}}}$

are positive constants such that
$\alpha_{\mathcal{H}}+\overline{\alpha_{\mathcal{C}}}=1$. We will also write

$\kappa_{\mathcal{C}}\doteq\frac{1}{2}\frac{\Lambda}{3}\frac{(r_{\mathcal{C}}-r_{\mathcal{H}})(r_{\mathcal{C}}+|\overline{r_{\mathcal{C}}}|)}{r_{\mathcal{C}}},$
(2.4)

and mention briefly that $\kappa_{\mathcal{C}}$ is the so-called surface
gravity of the cosmological horizons. With respect to these coordinates, the
metric is

$g=-\Omega^{2}\textnormal{d}u\textnormal{d}v+r^{2}\gamma_{\mathbb{S}^{2}},$
(2.5)

where $\Omega^{2}$ is the function

$\Omega^{2}=\frac{\Lambda}{3}\frac{1}{\kappa_{\mathcal{C}}^{2}r}(r-r_{\mathcal{H}})^{1+\alpha_{\mathcal{H}}}(r+|\overline{r_{\mathcal{C}}}|)^{1+\overline{\alpha_{\mathcal{C}}}}.$
(2.6)

We emphasise that $\Omega^{2}$ does not vanish on $\mathcal{C}^{+}$. The
Kruskal coordinates $(u,v)\in\mathbb{R}^{2}$ cover the region
$r_{\mathcal{H}}<r(u,v)<\infty$, and the expanding region $\mathcal{R}^{+}$ is
equivalent to the set $\\{(u,v):0<uv<1\\}$. Moreover, for the cosmological
horizons $\mathcal{C}^{+}=\\{(u,v):v=0,u\geq 0\\}$,
$\overline{\mathcal{C}}^{+}=\\{(u,v):u=0,v\geq 0\\}$. Finally, the future
boundary $\Sigma^{+}$ corresponds with the level set $\\{(u,v):uv=1,u,v>0\\}$.

We recall the Killing vector field $T$, which in Kruskal coordinates is given
by

$T=\kappa_{\mathcal{C}}\Big{(}u\frac{\partial}{\partial
u}-v\frac{\partial}{\partial v}\Big{)}.$ (2.7)

On $\mathcal{R}^{+}$, $T$ is precisely the coordinate vectorfield
$\frac{\partial}{\partial t}$, and is spacelike. On the cosmological horizons
$T$ is null. We also define the vectorfield

$Y|_{\mathcal{C}^{+}}=\frac{1}{\iota_{\mathcal{C}}}\frac{1}{u}\frac{\partial}{\partial
v},$ (2.8)

where

$\iota_{\mathcal{C}}=\frac{4}{\kappa_{\mathcal{C}}}\frac{1}{\Omega^{2}}\Big{|}_{\mathcal{C}^{+}}.$

One can see that $Y$ is conjugate to $T$ on the horizon, specifically we have

$g(T,Y)|_{\mathcal{C}^{+}}=-\frac{1}{2}\kappa_{\mathcal{C}}\iota_{\mathcal{C}}\Omega^{2}|_{\mathcal{C}^{+}}=-2.$

###  2.2 Wave equations and the energy method

In this paper we apply the classical vectorfield method to prove estimates on
finite-energy solutions to linear wave equations, see [8] for an introduction
to these methods in a general relativity context. Recall the standard energy-
momentum tensor $T$ for the linear wave equation, given by

$T_{\mu\nu}[\psi]=\partial_{\mu}\psi\partial_{\nu}\psi-\frac{1}{2}g_{\mu\nu}(\partial^{\alpha}\psi\partial_{\alpha}\psi),$

as well as the energy current $J^{X}$ with respect to a given vectorfield $X$,
a $1$-form defined by

$J^{X}[\psi]\cdot Y=T[\psi](X,Y).$

Given timelike vector fields $X,Y$, $J^{X}[\psi]\cdot Y$ is a positive-
definite quadratic form of $\partial\psi$ on $\mathcal{R}^{+}$, in the sense
that the following pointwise inequality holds:

$J^{X}[\psi]\cdot Y\gtrsim\sum_{|\alpha|=1}|\partial^{\alpha}\psi|^{2}.$

We will use the following notation for energy fluxes and norms induced by
these energy currents.

######  Definition 2.1.

Given a causal vectorfield $X$, and spacelike or null hypersurface $\Sigma$,
Let $E_{\Sigma}^{X}[\psi]$ denote the energy flux through $\Sigma$ with
respect to $X$, so that

$E_{\Sigma}^{X}[\psi]\doteq\int_{\Sigma}\>^{*}J^{X}[\psi],$ (2.9)

where $\>{}^{*}J^{X}[\psi]$ denotes the Hodge dual of the corresponding
vectorfield $(J^{X})^{\sharp}$. We also define the corresponding induced
energy norm

$\|\psi\|_{X,\Sigma}\doteq\Big{(}E_{\Sigma}^{X}[\psi]\Big{)}^{1/2}.$ (2.10)

If $\Sigma$ is a spacelike hypersurface, we may write

$\>{}^{*}J^{X}[\psi]|_{\Sigma}=J^{X}[\psi]\cdot
n_{\Sigma}\>\textnormal{d}\mu_{\overline{g}_{\Sigma}},$

where $n_{\Sigma}$ is the unit normal of $\Sigma$, and $\overline{g}_{\Sigma}$
is the induced metric on $\Sigma$.

######  Remark 2.2.

We point out that the energy $E[\psi]$ defined in (1.2) is equivalent to the
energy flux $E_{\Sigma_{r}}^{\partial_{r}}$:

$E[\psi](r)=E_{\Sigma_{r}}^{\partial
r}[\psi]=\int_{\Sigma_{r}}J^{\partial_{r}}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}$

To understand the weights in $r$ associated with the energy $E[\psi]$, we note
that

$\displaystyle n_{\Sigma_{r}}=\phi^{-1}\frac{\partial}{\partial r}\sim
r\frac{\partial}{\partial
r},\quad\textnormal{d}\mu_{\overline{g}_{r}}=r^{2}\phi^{-1}\>\textnormal{d}t\wedge\textnormal{d}\mu_{\mathbb{S}^{2}}\sim
r^{3}\>\textnormal{d}t\wedge\textnormal{d}\mu_{\mathbb{S}^{2}}.$

Thus for large $r$ we have

$\displaystyle E[\psi]$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}\times\mathbb{S}^{2}}r^{2}\phi^{-2}(\partial_{r}\psi)^{2}+r^{2}\phi^{2}(\partial_{t}\psi)^{2}+r^{2}|\not{\nabla}\psi|^{2}\textnormal{d}t\textnormal{d}\mu_{\mathring{\gamma}}$
$\displaystyle\sim
r^{4}\|\partial_{r}\psi(r)\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})}^{2}+\|\tilde{\nabla}\psi(r)\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})}^{2}.$

where $\not{\nabla}\psi$ is the standard covariant derivative on a sphere of
radius $r$, $\tilde{\nabla}\psi$ is the standard covariant derivative on the
cylinder $\mathbb{R}\times\mathbb{S}^{2}$, and
$\textnormal{d}\mu_{\mathring{\gamma}}$ is the standard volume form on
$\mathbb{S}^{2}$.

We will also define the following $L^{2}$-norm on the
$\Sigma_{r}$-hypersurfaces. Let

$\|\psi\|_{L^{2}(\Sigma_{r})}^{2}=\int_{\Sigma_{r}}\psi^{2}\textnormal{d}\mu_{\overline{g}_{r}}\sim
r^{3}\|\psi(r)\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})}^{2}.$

In this paper, we adopt the following notation conventions to distinguish
differential operators on relevant manifolds or submanifolds. We write
$\not{\nabla}$ to denote the covariant derivative restricted to a sphere of
radius $r$, $\overline{\nabla}$ for the covariant derivative restricted to the
level sets $\Sigma_{r}$, and $\tilde{\nabla}$ for the covariant derivative
with respect to the conformal metric at $\Sigma^{+}$:

$\tilde{g}=\lim_{r\rightarrow\infty}\Big{(}\frac{1}{r^{2}}\overline{g}_{r}\Big{)}=\frac{\Lambda}{3}dt^{2}+\gamma_{\mathbb{S}^{2}}.$

We will also use this convention for other differential operators which appear
in this paper, such as the Laplacian.

##  3 Scattering of linear waves on Schwarzschild-de Sitter

In this section we will prove existence and uniqueness of scattering solutions
to the linear wave equation on the expanding region. The main results of this
section are Theorem 1.4 which we prove in Section 3.3, and Theorem 1.10 which
we prove in Section 3.6.

First we will prove the existence and uniqueness of scattering solutions on
the interior of the expanding region, which corresponds to Theorem 1.4.
Specifically, we will show that given scattering data $\psi_{0}$, $\psi_{3}$,
there exists a unique solution to $\square_{g}\psi=0$ that is uniformly
bounded up to a $\Sigma_{r_{0}}$ hypersurface for some
$r_{0}>r_{\mathcal{C}}$. We will consider an asymptotic solution of the form

$\psi_{\text{asymp}}=\psi_{0}(t,\omega)+\frac{\psi_{2}(t,\omega)}{r^{2}}+\frac{\psi_{3}(t,\omega)}{r^{3}},$

where $\psi_{0},\psi_{3}$ are freely chosen, and $\psi_{2}$ is determined
entirely by $\psi_{0}$. We will show that that $\psi_{\text{asymp}}$
approximates a true solution, in the sense that

$|\square_{g}\psi_{\text{asymp}}|=O\big{(}\frac{1}{r^{4}}\big{)}.$

$\Sigma_{r_{0}}$$\Sigma_{R}$$\Sigma^{+}$ Figure 3.1: Spacetime region on which
we solve the wave equation on the interior of the expanding region. We solve a
sequence of finite problems, i.e. we consider a solution $\psi_{R}$ with
prescribed data on $\Sigma_{R}$, and then show that the limit
$\lim_{R\rightarrow\infty}\psi_{R}$ exists in a suitable energy space. In this
step we solve up to a $\Sigma_{r_{0}}$ hypersurface, where
$r_{0}>r_{\mathcal{C}}$.

We will prove existence of scattering solutions with scattering data
$\psi_{0},\psi_{3}$ on the interior of the expanding region $\mathcal{R}^{+}$.
We do this by taking a sequence of solutions of finite problems, and showing
that the limiting function exists and solves the scattering problem.
Specifically, we take sequence of solutions $\psi_{R}$ to (1.1) with
prescribed data on the hypersurface $\Sigma_{R}$; see Figure 3.1. We set
$\psi_{R}=\psi_{\text{asymp}}+\psi_{\text{rem}}^{R}$, where
$\psi_{\text{rem}}^{R}$ satisfies

$\square_{g}\psi_{\text{rem}}^{R}=-\square_{g}\psi_{\text{asymp}},$

and has trivial data on $\Sigma_{R}$, so that

$\psi_{\text{rem}}^{R}|_{\Sigma_{R}}=0,\quad
n_{\Sigma_{R}}\cdot\psi_{\text{rem}}^{R}|_{\Sigma_{R}}=0.$

The key technical tool we use to control the sequence of remainders
$\psi_{\text{rem}}^{R}$ is the weighted energy estimate Proposition 1.8, which
we prove in Section 3.2. Using this weighted energy estimate, we show that the
limit $\psi=\lim_{R\rightarrow\infty}\psi_{R}$ exists with respect to the
energy norms (1.2), and $\psi$ remains uniformly bounded on $\Sigma_{r_{0}}$,
proving Theorem 1.4.

We then extend the solutions constructed in Theorem 1.4 to the cosmological
horizons by solving the wave equation backwards-in-time from $\Sigma_{r_{0}}$
to the $\mathcal{C}^{+}\cup\overline{\mathcal{C}}^{+}$, proving Theorem 1.10.
For this we will assume exponential decay of the scattering data
$\psi_{0},\psi_{3}$ along $\Sigma^{+}$ toward $\iota,\overline{\iota}$. We
will consider a sequence of solutions $\psi^{T}$ to the wave equation
$\square_{g}\psi=0$. This time $\psi_{T}$ has data that is prescribed on a
truncation of the level set $\Sigma_{r_{0}}$, and on null hypersurfaces
$C_{T}^{c}$ ,$\overline{C}_{T}^{c}$ that are transversal to each cosmological
horizon; see Figure 3.2. The prescribed data on $\Sigma_{r_{0}}$ matches the
solution constructed in Section 3.3, while the prescribed data on $C_{T}^{c}$,
$\overline{C}_{T}^{c}$ is trivial. We will show that the limit
$\lim_{T\rightarrow\infty}\psi^{T}=\psi$ exists with respect to nondegenerate
energy norms, and that $\psi$ has finite nondegenerate energy on
$\mathcal{C}^{+}\cup\overline{\mathcal{C}}^{+}$.

$\Sigma_{r_{0}}^{T}$$C_{T}^{c}$$\overline{C_{T}^{c}}$$\Sigma^{+}$$\mathcal{C}^{+}$$\overline{\mathcal{C}}^{+}$$\iota^{+}$$\overline{\iota}^{+}$
Figure 3.2: Spacetime region on which we solve the wave equation up to the
cosmological horizons. We solve a sequence of finite problems, i.e. we
consider a solution $\psi_{T}$ with prescribed data on $\Sigma_{r_{0}}^{T}\cup
C_{T}^{c}\cup\overline{C}_{T}^{c}$, and then show that the limit
$\lim_{T\rightarrow\infty}\psi_{T}$ exists in a suitable energy space.

###  3.1 The construction of asymptotic solutions

In this section we construct asymptotic solutions that satisfy the wave
equation $\square_{g}\psi_{\text{asymp}}=0$ up to some small error that decays
toward $\Sigma^{+}$. As previously stated, we will eventually set

$\psi_{\text{asymp}}=\psi_{0}+\frac{\psi_{2}}{r^{2}}+\frac{\psi_{3}}{r^{3}},$

where $\psi_{0}$ and $\psi_{3}$ are freely chosen, and $\psi_{2}$ is
determined by $\psi_{0}$. We motivate this choice by first considering a more
general finite-order power series that includes logarithmic terms, so that

$\psi_{\text{asymp}}^{(N)}(r,t,\omega)=\sum_{n=0}^{N}\frac{\psi_{n}(t,\omega)}{r^{n}}+\sum_{n=1}^{N}\frac{\log
r\>\psi_{n,1}(t,\omega)}{r^{n}},$ (3.1)

for some positive integer $N$. Here the $\psi_{n}$, $\psi_{n,1}$ are functions
on the future boundary $\Sigma^{+}$. Since $\Sigma^{+}$ is diffeomorphic to
$\mathbb{R}\times\mathbb{S}^{2}$, the $\psi_{n}$ can also be thought of as
functions on the standard cylinder.

######  Remark 3.1.

We emphasise that we will not be constructing solutions via an infinite power
series expansion. The notion of constructing solutions to the linear wave
equation on Minkowski spacetime as a power series in $1/r$ was first explored
in detail by Friedlander, see [13]. Due to the different geometry of Minkowski
spacetime, the particular asymptotic expansion is manifestly different. In
particular, solutions to $\square\psi=0$ behave to leading order near
$\mathcal{I}^{+}$ like

$\psi(t,x)\sim\frac{F(r-t,\omega)}{r},\quad r=|x|,\quad\omega=\frac{x}{|x|},$

where $F(r-t,\omega)$ is the so-called _radiation field_ , whose existence was
established by Friedlander in the aforementioned papers. See also [12] for an
example of asymtotic expansions of linear waves near a black hole singularity.

The wave equation on the expanding region of Schwarzschild-de Sitter spacetime
is expressed as

$\square_{g}\psi=\phi^{2}\partial_{t}^{2}\psi-\frac{1}{r^{2}}\partial_{r}\Big{(}\frac{r^{2}}{\phi^{2}}\partial_{r}\psi\Big{)}+\frac{1}{r^{2}}\Delta_{\mathbb{S}^{2}}\psi.$
(3.2)

We begin with a preliminary computation that we will use to compute
$\square_{g}\psi_{\text{asymp}}^{(N)}$.

######  Lemma 3.2.

Fix $r_{0}>r_{\mathcal{C}}$. Let $n$ be a nonnegative integer, and let
$\psi_{n}\in C^{2}(\mathbb{R}\times\mathbb{S}^{2})$. Then for all $r\geq
r_{0}$, we have the following two identities:

$\square_{g}\Big{(}\frac{\psi_{n}(t,\omega)}{r^{n}}\Big{)}=-\frac{\Lambda}{3}n(n-3)\frac{1}{r^{n}}\psi_{n}+\frac{1}{r^{n+2}}\Big{(}n(n-1)\psi_{n}+\tilde{\Delta}\psi_{n}\Big{)}\\\
-n^{2}\frac{2m}{r^{n+3}}\psi_{n}+O\big{(}\frac{1}{r^{n+4}})(\partial_{t}^{2}\psi_{0}).$
(3.3) $\square_{g}\Big{(}\frac{\log
r\>\psi_{n,1}}{r^{n}}\Big{)}=-\frac{\Lambda}{3}n(n-3)\frac{\log
r}{r^{n}}\psi_{n,1}+\frac{\Lambda}{3}(2n-3)\frac{1}{r^{n}}\psi_{n,1}+O\big{(}\frac{\log
r}{r^{n+2}})\big{.}$ (3.4)

###### Proof.

By (3.2), we have

$\square_{g}\Big{(}\frac{\psi_{n}}{r^{n}}\Big{)}=\frac{\phi^{2}}{r^{n}}\partial_{t}^{2}\psi_{n}-\frac{1}{r^{2}}\partial_{r}\Big{(}\frac{r^{2}}{\phi^{2}}\partial_{r}\Big{(}\frac{1}{r^{n}}\Big{)}\Big{)}\psi_{n}+\frac{1}{r^{n+2}}\Delta_{\mathbb{S}^{2}}\psi_{n}.$
(3.5)

The leading-order behaviour of $\phi^{2}$ is

$\phi^{2}=\frac{1}{\frac{\Lambda}{3}r^{2}-(1-\frac{2m}{r})}=\frac{3}{\Lambda}\frac{1}{r^{2}}+\frac{1-\frac{2m}{r}}{\frac{\Lambda}{3}r^{2}\Big{(}\frac{\Lambda}{3}r^{2}-1+\frac{2m}{r}\Big{)}}=\frac{3}{\Lambda}\frac{1}{r^{2}}+O\big{(}\frac{1}{r^{4}}\big{)},$
(3.6)

and so

$\frac{\phi^{2}}{r^{n}}\partial_{t}^{2}\psi_{n}+\frac{1}{r^{n+2}}\Delta_{\mathbb{S}^{2}}\psi_{n}=\frac{1}{r^{n+2}}\tilde{\Delta}\psi_{n}+O\big{(}\frac{1}{r^{n+4}}\big{)}.$

Meanwhile, we compute for the radial derivatives:

$\displaystyle-\frac{1}{r^{2}}\partial_{r}\Big{(}\frac{r^{2}}{\phi^{2}}\partial_{r}\Big{(}\frac{1}{r^{n}}\Big{)}\Big{)}\psi_{n}$
$\displaystyle=\frac{n}{r^{2}}\partial_{r}\Big{(}\frac{1}{\phi^{2}r^{n-1}}\Big{)}\psi_{n}$
$\displaystyle=\frac{n}{r^{2}}\partial_{r}\Big{(}\frac{\Lambda}{3}\frac{1}{r^{n-3}}-\frac{1}{r^{n-1}}+\frac{2m}{r^{n}}\Big{)}\psi_{n}$
$\displaystyle=-\frac{\Lambda}{3}\frac{n(n-3)}{r^{n}}\psi_{n}+\frac{n(n-1)}{r^{n+2}}\psi_{n}-2m\frac{n^{2}}{r^{n+3}}\psi_{n}.$

Combining this with (3.5) yields (3.3). A similar computation implies (3.4). ∎

######  Remark 3.3.

We see from Lemma 3.2 that for all but finitely many $n$, we have

$\displaystyle\square_{g}(r^{-n}\psi_{n})=O(r^{-n}),\quad\square_{g}(r^{-n}\log
r\>\psi_{n,1})=O(r^{-n}\log r).$

The three exceptions are $\psi_{0},\psi_{3},\psi_{3,1}$, where the leading
order term vanishes, giving

$\displaystyle\square_{g}\psi_{0}=O(r^{-2}),\quad\square_{g}(r^{-3}\psi_{3})=O(r^{-5}),\quad\square_{g}(r^{-3}\log
r\>\psi_{3,1})=O(r^{-3}).$

In the next Lemma we consider a general asymptotic solution
$\psi_{\text{asymp}}^{(N)}$ of the form (3.1). We will show that
$\psi_{\text{asymp}}^{(N)}$ satisfies the linear wave equation up to a
particular order in $r$ if and only if the terms $\psi_{n}$,$\psi_{n,1}$
satisfy recurrence relations. Moreover, we will show that $\psi_{0},\psi_{3}$
capture the correct notion of scattering data for the asymptotic solution.

######  Lemma 3.4.

Let $\psi_{0},\psi_{1,1},\psi_{1},\dots,\psi_{N,1},\psi_{N}$ be sufficiently
regular functions on $\Sigma^{+}$, and let $\psi_{\text{asymp}}^{(N)}$ be of
the form (3.1). Then

$\square_{g}\psi_{\text{asymp}}=O\big{(}\frac{\log r}{r^{N+1}}\big{)}$ (3.7)

if and only if $\psi_{0},\psi_{1},\psi_{1,1},\dots,\psi_{N},\psi_{N,1}$
satisfy $2N-1$ recurrence relations. These recurrence relations have two
degrees of freedom, with $\psi_{0},\psi_{3}$ being the leading-order terms
that can be freely chosen. The recurrence relations for the logarithmic terms
reduce to

$\psi_{n,1}=0,\quad n=1,\dots,N,$ (3.8)

while the recurrence relations for the other terms reduce to

$\displaystyle\psi_{1}=0,\quad\psi_{2}=-\frac{3}{\Lambda}\tilde{\Delta}\psi_{0},\quad\psi_{n}=R_{n}(\psi_{0},\psi_{3}),\quad
n\geq 4,$

where the $R_{n}$ are linear functions of $\psi_{0},\psi_{3}$ and their
derivatives.

###### Proof.

This is a straightforward application of Lemma 3.2. In general, we have

$\square_{g}\psi_{\text{asymp}}=\sum_{n=0}^{N}\frac{F_{n}}{r^{n}}+\sum_{n=1}^{N}\frac{\log
r\>F_{n,1}}{r^{n}}+O\big{(}\frac{\log r}{r^{N+1}}\big{)},$

where the coefficients $F_{n}$ are functions of $\psi_{0},\dots,\psi_{n}$,
$\psi_{1,1}\dots,\psi_{n,1}$, while the coefficients $F_{n,1}$ are functions
of $\psi_{1,1}\dots,\psi_{n,1}$ only. From this we see that (3.7) holds if and
only if $F_{0}=\dots=F_{n}=0$, and $F_{1,1}=\dots=F_{n,1}=0$, and these
generate $2N+1$ recurrence relations between the $\psi_{n},\psi_{n,1}$. We
compute these recurrence relations in order of decay, simplifying as needed.
As $\square_{g}\psi_{0}=O(r^{-2})$, the recurrence relation $F_{0}=0$ is
trivial, and the first two nontrivial recurrence relations are

$F_{1,1}=\frac{2\Lambda}{3}\psi_{1,1}=0,\quad
F_{1}=\frac{2\Lambda}{3}\psi_{1}-\frac{\Lambda}{3}\psi_{1,1}=0,$

which reduce to $\psi_{1,1}=\psi_{1}=0$. Assuming this holds, the next two
recurrence relations are

$F_{2,1}=\frac{2\Lambda}{3}\psi_{2,1}=0,\quad
F_{2}=\tilde{\Delta}\psi_{0}+\frac{2\Lambda}{3}\psi_{2}+\frac{\Lambda}{3}\psi_{2,1}=0.$

These reduce to $\psi_{2,1}=0$ and
$\psi_{2}=-\frac{3}{2\Lambda}\tilde{\Delta}\psi_{0}$. This implies that
$\psi_{0}$ can be freely chosen. By Lemma 3.2, we see that

$\square_{g}\psi_{0}=\frac{1}{r^{2}}\tilde{\Delta}\psi_{0}+O\big{(}\frac{1}{r^{4}}\big{)}\quad\square_{g}\Big{(}\frac{\psi_{2}}{r^{2}}\Big{)}=\frac{2\Lambda}{3}\frac{\psi_{2}}{r^{2}}+O\big{(}\frac{1}{r^{4}}\big{)}.$

Moreover, we have $\square_{g}(r^{-3}\psi_{3})=O(r^{-5})$,
$\square_{g}(r^{-3}\log r\>\psi_{3,1})=O(r^{-3})$, and so the recurrence
relation $F_{3,1}=0$ is trivial, while the recurrence relation for $F_{3}$ is

$F_{3,1}=\frac{\Lambda}{3}\psi_{3,1}=0.$

For $n\geq 4$, the coefficients $F_{n,1}$ are functions of
$\psi_{1,1},\dots,\psi_{n,1}$ only. Since $\square_{g}(r^{-n}\log
r\>\psi_{n,1})=O(r^{-n}\log r)$ for $n\geq 4$, the recurrence relations for
the $F_{n,1}$ reduce to $\psi_{n,1}=0$ for all $4\leq n\leq N$. Thus the
coefficients of all log terms in $\psi_{\text{asymp}}^{(N)}$ are identically
zero only if (3.7) holds. While we do not compute the remaining recurrence
relations explicitly, we note that the relation $F_{4}=0$ implies that
$\psi_{4}$ is determined entirely by $\psi_{0}$, and for $F_{5}$ we have

$F_{5}=F_{5}(\psi_{0},\psi_{2},\psi_{3},\psi_{4},\psi_{5}).$

As $\psi_{2},\psi_{4}$ are determined by $\psi_{0}$, this recurrence relation
also has a degree of freedom, with $\psi_{3}$ being the next leading-order
term that can be freely chosen. Hence the remaining recurrence relations imply
that the remaining terms are all determined by $\psi_{0},\psi_{3}$. ∎

######  Remark 3.5.

Lemma 3.4 depends heavily on the asymptotic expansion of the components of the
metric (2.2) for Schwarzschild-de Sitter. In particular, the fact that
$\psi_{3,1}$ vanishes identically relies on the fact that
$\phi^{-2}=\frac{\Lambda}{3}r^{2}+O(1)$, i.e. $\phi^{-2}$ contains no $O(r)$
term in its expansion. One can easily construct perturbations where this is
not the case, leading to nonzero logarithmic terms in
$\psi_{\text{asymp}}^{(N)}$. We will discuss this again when we construct
scattering solutions in the perturbed setting in Section 5, see in particular
Lemma 5.5. For the moment we emphasise that for Schwarzschild-de Sitter log
terms are not present in any asymptotic solution satisfying (3.7).

To prove existence and uniqueness of scattering solutions, we will consider
the approximate solution $\psi_{\text{asymp}}^{(N)}$ with $N=3$,777With Lemma
3.4, one may construct an asymptotic solution that satisfies the wave equation
up to arbitrary order in powers of $r^{-1}$ by including more terms in the
asymptotic solution. By the argument given in this paper one could use this
asymptotic solution to construct a true scattering solution to the wave
equation with more detailed asymptotics. This would require additional
regularity of the scattering data $\psi_{0}$, $\psi_{3}$ though. For the
purpose of constructing exact scattering solutions to $\square_{g}\psi=0$, we
only require decay of our asymptotic solution like
$\square_{g}\psi_{\text{asymp}}=O\big{(}\frac{1}{r^{3+\epsilon}}\big{)}$ for
some $\epsilon>0$; (3.11) in Lemma 3.6 and (3.24). and fix
$\psi_{\text{asymp}}=\psi_{\text{asymp}}(\psi_{0},\psi_{3})$ to be the
asymptotic solution

$\psi_{\text{asymp}}=\psi_{0}-\frac{3}{2\Lambda}\frac{\tilde{\Delta}\psi_{0}}{r^{2}}+\frac{\psi_{3}}{r^{3}}.$

It follows from Lemma 3.4 that $\square_{g}\psi_{\text{asymp}}=O(r^{-4}).$
More specifically we have the pointwise bound

$|\square_{g}\psi_{\text{asymp}}|\lesssim_{\Lambda,m,r_{0}}\frac{1}{r^{4}}\sum_{|\alpha|\leq
4}|\tilde{\nabla}^{\alpha}\psi_{0}|+\frac{1}{r^{5}}\sum_{|\alpha|\leq
2}|\tilde{\nabla}^{\alpha}\psi_{3}|.$ (3.9)

In the following Lemma we show that the pointwise decay (3.9) implies certain
bounds on the $L^{2}$-and energy norms of the asymptotic solution.
Specifically we estimate the energy of the asymptotic solution
$\psi_{\text{asymp}}$ with respect to the natural energy norm (1.2). We will
also show that the quantity

$\|r\phi^{1/2}\square_{g}\psi_{\text{asymp}}\|_{L^{2}(\Sigma_{r})}$

is finite and integrable in $r$, given appropriate assumptions on the
regularity and decay of the scattering data $\psi_{0},\psi_{3}$. The
importance of this property, as well as the particular weights in $r$ are
motivated by its inclusion in the weighted energy estimate that is proved in
Proposition 1.8, see (1.10) in particular.

######  Lemma 3.6.

Fix $r_{0}>r_{\mathcal{C}}$, and let $\psi_{0}$, $\psi_{3}$ be functions on
$\Sigma^{+}$ such that

$\|\psi_{0}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})},\>\>\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}<\infty.$

Then for all $r>r_{\mathcal{C}}$,
$\|\psi_{\text{asymp}}\|_{\partial_{r},\Sigma_{r}}<\infty$, and

$\|\psi_{\text{asymp}}\|_{\partial_{r},\Sigma_{r}}\lesssim_{\Lambda,m,r_{0}}\|\tilde{\nabla}\psi_{0}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{2}}\|\psi_{3}\|_{H^{1}(\mathbb{R}\times\mathbb{S}^{2})}.$
(3.10)

Moreover, we have $\square_{g}\psi_{\text{asymp}}\in L^{2}(\Sigma_{r})$, and

$\int_{r}^{\infty}\|r\phi^{1/2}\square_{g}\psi_{\text{asymp}}\|_{L^{2}(\Sigma_{r})}\>\textnormal{d}r\lesssim_{\Lambda,m,r_{0}}\frac{1}{r}\|\psi_{0}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{2}}\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}$
(3.11)

for all $r\geq r_{0}$.

###### Proof.

An application of the triangle inequality and the definition of
$\psi_{\text{asymp}}$ implies

$\|\psi_{\text{asymp}}\|_{\partial_{r},\Sigma_{r}}\lesssim\|\psi_{0}\|_{\partial_{r},\Sigma_{r}}+\|r^{-2}\tilde{\Delta}\psi_{0}\|_{\partial_{r},\Sigma_{r}}+\|r^{-3}\psi_{3}\|_{\partial_{r},\Sigma_{r}}.$

From the definitions of the $\|\cdot\|_{\partial_{r},\Sigma_{r}}$ and
$\|\cdot\|_{L^{2}(\Sigma_{r})}$ norms, we have

$\|\psi\|_{\partial_{r},\Sigma_{r}}\lesssim_{\Lambda,m,r_{0}}r^{2}\|\partial_{r}\psi_{\text{asymp}}\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})}+\|\tilde{\nabla}\psi_{\text{asymp}}\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})},$

and so

$\displaystyle\|\psi_{\text{asymp}}\|_{\partial_{r},\Sigma_{r}}\lesssim_{\Lambda,m,r_{0}}\>\|\tilde{\nabla}\psi_{0}\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r}\|\tilde{\Delta}\psi_{0}\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{2}}\|\tilde{\nabla}(\tilde{\Delta}\psi_{0})\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})}$
$\displaystyle+\frac{1}{r^{2}}\|\psi_{3}\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{3}}\|\tilde{\nabla}\psi_{3}\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})},$

which implies (3.10).

To prove (3.11), observe that by (3.9) and the fact that
$\textnormal{d}\mu_{\overline{g}_{r}}=r^{2}\phi^{-1}\textnormal{d}\mu_{\tilde{g}}$,
we have

$\|r\phi^{1/2}\square_{g}\psi_{\text{asymp}}\|_{L^{2}(\Sigma_{r})}\lesssim_{\Lambda,m,r_{0}}\frac{1}{r^{2}}\|\psi_{0}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{3}}\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}.$

Therefore
$\|r\phi^{1/2}\square_{g}\psi_{\text{asymp}}(r)\|_{L^{2}(\Sigma_{r})}$ is
integrable in $r$, and integrating in $r$ yields (3.11). ∎

###  3.2 Weighted energy estimate for the backward problem

We now prove the weighted energy estimate of Proposition 1.8 for the wave
equation on the expanding region of Schwarzschild-de Sitter. We identify a
vectorfield multiplier whose associated energy is decreasing backwards in
time, relating to a “global redshift” on the expanding region that was first
explored in [27]. In that paper however, energy estimates are constructed for
forward solutions to (1.1), and so the vectorfield multiplier in the present
paper differs from that in [27]. This energy estimate will be applied not to
solutions of (1.1), but to a small remainder, obtained by subtracting an
asymptotic solution which we constructed in Section 3.1. This remainder will
satisfy an inhomogeneous wave equation.

Following notation from [29], let
$\mathcal{D}_{r_{1},r_{2}}^{(\overline{u},\overline{v})}\subset\mathcal{R}^{+}$
denote the spacetime region

$\mathcal{D}_{r_{1},r_{2}}^{(\overline{u},\overline{v})}=\bigcup_{r_{1}\leq
r\leq
r_{2}}\Sigma_{r}\bigcap\\{u\geq\overline{u}\\}\bigcap\\{v\geq\overline{v}\\},$
(3.12)

as depicted in Figure 3.3. Here, $u,v$ are the null Kruskal coordinates
defined in Section 2. It is apparent that
$\mathcal{D}_{r_{1},r_{2}}^{(\overline{u},\overline{v})}$ is bounded by the
two spacelike hypersurfaces $\Sigma_{r_{1}}^{c}$, $\Sigma_{r_{2}}^{c}$, and
two null hypersurfaces, which we denote by
$\mathcal{F}_{\overline{u}}^{c},\mathcal{G}_{\overline{v}}^{c}$. The
superscript $c$ denotes that these hypersurfaces are ‘capped’.

$\mathcal{C}^{+}$$\overline{\mathcal{C}}^{+}$$\Sigma^{+}$$\mathcal{R}^{+}$$\Sigma_{r_{2}}^{c}$$\Sigma_{r_{1}}^{c}$$\mathcal{D}_{r_{1},r_{2}}^{(\overline{u},\overline{v})}$$\mathcal{F}_{\overline{v}}^{c}$$\mathcal{G}_{\overline{u}}^{c}$
Figure 3.3: Spacetime domain
$\mathcal{D}_{r_{1},r_{2}}^{(\overline{u},\overline{v})}$ of the backwards
weighted energy estimate.

Let $X$ be a timelike vectorfield. Given some scalar function $\psi$, we apply
the divergence theorem to the energy current $J^{X}[\psi]$ on
$\mathcal{D}_{r_{1},r_{2}}^{(\overline{u},\overline{v})}$, which gives

$\int_{\Sigma_{r_{1}}^{c}}J^{X}[\psi]\cdot
n_{\Sigma_{r}}+\int_{\mathcal{F}_{\overline{v}}^{c}}\>^{*}J^{X}[\psi]+\int_{\mathcal{G}_{\overline{u}}^{c}}\>^{*}J^{X}[\psi]\\\
=\int_{\Sigma_{r_{2}}^{c}}J^{X}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}+\int_{\mathcal{D}_{r_{1},r_{2}}^{(\overline{u},\overline{v})}}\nabla\cdot
J^{X}[\psi]\textnormal{d}\mu_{g},$ (3.13)

where $\>{}^{*}J^{M}[\psi]$ denotes the Hodge dual of the energy current
$J^{M}[\psi]$. Note that the boundary integrals in (3.13) are over spacelike
or null hypersurfaces. For a timelike future pointing vectorfield $X$, these
boundary integrals will all be nonnegative. The key to producing energy
estimates for wave equations lies in identifying vectorfield multipliers $X$
for which the divergence of the energy current $\nabla\cdot J^{X}[\psi]$ is
coercive in some sense, so that we may control the bulk integral in (3.13). By
the Leibniz product rule we have

$\nabla\cdot J^{X}[\psi]=K^{X}[\psi]+(X\psi)(\square_{g}\psi),$ (3.14)

where

$K^{X}[\psi]=\>^{(X)}\pi^{\mu\nu}T_{\mu\nu}[\psi].$ (3.15)

We define the vectorfield

$M\doteq r^{2}\phi^{-2}\frac{\partial}{\partial
r}=r^{2}\Big{(}\frac{\Lambda}{3}r^{2}-1+\frac{2m}{r}\Big{)}\frac{\partial}{\partial
r}$ (3.16)

We emphasize that $M$ is timelike, and for large $r$, $M\sim
r^{4}\partial_{r}$. In the following proposition, we will show that the
associated energy current (3.15) is nonpositive.

######  Proposition 3.7.

Let $M$ denote the vectorfield (3.16). Then everywhere on $\mathcal{R}^{+}$,
we have

$K^{M}[\psi]\leq 0.$

###### Proof.

The deformation tensor of $M$ is given by

${}^{(M)}\pi^{\mu\nu}$
$\displaystyle=\frac{1}{2}(\nabla^{\mu}M^{\nu}+\nabla^{\nu}M^{\mu})$
$\displaystyle=\frac{1}{2}\Big{(}g^{\mu\lambda}\partial_{\lambda}M^{\nu}+g^{\nu\lambda}\partial_{\lambda}M^{\mu}+g^{\mu\lambda}\Gamma_{\lambda\eta}^{\nu}M^{\eta}+g^{\nu\lambda}\Gamma_{\lambda\eta}^{\mu}M^{\eta}\Big{)}.$

From (2.2), we find that the nonzero Christoffel symbols of Schwarzschild-de
Sitter on $\mathcal{R}^{+}$ are

$\displaystyle\Gamma_{rr}^{r}=-\Big{(}\frac{\Lambda}{3}r-\frac{m}{r^{2}}\Big{)}\phi^{2},\quad\Gamma_{rt}^{t}=\Big{(}\frac{\Lambda}{3}r-\frac{m}{r^{2}}\Big{)}\phi^{2},\quad\Gamma_{tt}^{r}=\Big{(}\frac{\Lambda}{3}r-\frac{m}{r^{2}}\Big{)}\phi^{-2}.$
$\displaystyle\Gamma_{AB}^{r}=\frac{\phi^{-2}}{r}g_{AB},\quad\Gamma_{Br}^{A}=\frac{1}{r}\delta_{B}^{A}.$

Another computation reveals that the nonzero components of the deformation
tensor of $M$ are

$\>{}^{(M)}\pi^{tt}=\big{(}\frac{\Lambda}{3}r^{3}-m\big{)}\phi^{2},\quad\>^{(M)}\pi^{rr}=\frac{\frac{\Lambda}{3}r^{3}-m}{\phi^{2}}-\frac{r}{\phi^{2}}\Big{(}\frac{4\Lambda}{3}r^{2}-2+\frac{2m}{r}\Big{)}.$
$\>{}^{(M)}\pi^{\theta\theta}=\frac{1}{r\phi^{2}},\quad\>^{(M)}\pi^{\varphi\varphi}=\frac{1}{r\phi^{2}\sin^{2}\theta}.$

Then the corresponding components of the stress-energy-momentum tensor are

$\displaystyle
T_{tt}[\psi]=\frac{1}{2}\Big{(}(\partial_{t}\psi)^{2}+\frac{1}{\phi^{4}}(\partial_{r}\psi)^{2}-\frac{1}{\phi^{2}}|\not{\nabla}\psi|^{2}\Big{)}.$
$\displaystyle
T_{rr}[\psi]=\frac{1}{2}\Big{(}\phi^{4}(\partial_{t}\psi)^{2}+(\partial_{r}\psi)^{2}+\phi^{2}|\not{\nabla}\psi|^{2}\Big{)}.$
$\displaystyle
T_{\theta\theta}[\psi]=\frac{1}{2}\Big{(}-r^{2}\phi^{2}(\partial_{t}\psi)^{2}+\frac{r^{2}}{\phi^{2}}(\partial_{r}\psi)^{2}+(\partial_{\theta}\psi)^{2}-\frac{1}{\sin^{2}\theta}(\partial_{\varphi}\psi)^{2}\Big{)}.$
$\displaystyle
T_{\varphi\varphi}[\psi]=\frac{\sin^{2}\theta}{2}\Big{(}-r^{2}\phi^{2}(\partial_{t}\psi)^{2}+\frac{r^{2}}{\phi^{2}}(\partial_{r}\psi)^{2}-(\partial_{\theta}\psi)^{2}+\frac{1}{\sin^{2}\theta}(\partial_{\varphi}\psi)^{2}\Big{)}.$

We then compute the sum $\>{}^{(M)}\pi^{\mu\nu}T_{\mu\nu}[\psi]$. The sum of
the angular components are

$\>{}^{(M)}\pi^{\theta\theta}T_{\theta\theta}[\psi]+\>^{(M)}\pi^{\varphi\varphi}T_{\varphi\varphi}[\psi]=-r(\partial_{t}\psi)^{2}+\frac{r}{\phi^{4}}(\partial_{r}\psi)^{2},$

while the time and radial components sum to

$\>{}^{(M)}\pi^{tt}T_{tt}[\psi]+\>^{(M)}\pi^{rr}T_{rr}[\psi]$
$\displaystyle=\big{(}\frac{\Lambda}{3}r^{3}-m\big{)}\phi^{2}(\partial_{t}\psi)^{2}+\frac{\frac{\Lambda}{3}r^{3}-m}{\phi^{2}}(\partial_{r}\psi)^{2}$
$\displaystyle-\frac{r}{\phi^{2}}\Big{(}\frac{2\Lambda}{3}r^{2}-1+\frac{m}{r}\Big{)}\Big{(}\phi^{4}(\partial_{t}\psi)^{2}+(\partial_{r}\psi)^{2}+\phi^{2}|\not{\nabla}\psi|^{2}\Big{)}.$

Adding all terms together gives

$\>{}^{(M)}\pi^{\mu\nu}T_{\mu\nu}[\psi]$
$\displaystyle=(r-3m)\phi^{2}(\partial_{t}\psi)^{2}+\frac{r}{\phi^{2}}\Big{(}\frac{2\Lambda}{3}r^{2}-1+\frac{m}{r}\Big{)}(\partial_{r}\psi)^{2}$
$\displaystyle-\frac{r}{\phi^{2}}\Big{(}\frac{2\Lambda}{3}r^{2}-1+\frac{m}{r}\Big{)}\Big{(}\phi^{4}(\partial_{t}\psi)^{2}+(\partial_{r}\psi)^{2}+\phi^{2}|\not{\nabla}\psi|^{2}\Big{)}$
$\displaystyle=\phi^{2}\Big{(}r-3m-r\Big{(}\frac{2\Lambda}{3}r^{2}-1+\frac{m}{r}\Big{)}\Big{)}(\partial_{t}\psi)^{2}-r\Big{(}\frac{2\Lambda}{3}r^{2}-1+\frac{m}{r}\Big{)}|\not{\nabla}\psi|^{2}.$
$\displaystyle=-2r(\partial_{t}\psi)^{2}-r\Big{(}\frac{2\Lambda}{3}r^{2}-1+\frac{m}{r}\Big{)}|\not{\nabla}\psi|^{2}$

The function $\frac{2\Lambda}{3}r^{2}-1+\frac{m}{r}$ is positive on
$\mathcal{R}^{+}$, since we have

$\Big{(}\frac{2\Lambda}{3}r^{2}-1+\frac{m}{r}\Big{)}=\frac{2}{\phi^{2}}+\frac{1}{r}(r-3m),$

and $r>r_{\mathcal{C}}>3m$. Hence $K^{M}[\psi]\leq 0$. ∎

######  Remark 3.8.

Due to the identity (3.14), an immediate consequence of Proposition 3.7 is
that the divergence of the standard energy current $J^{M}[\psi]$ obeys the
bound

$\nabla\cdot J^{M}[\psi]\leq(M\psi)(\square_{g}\psi).$ (3.17)

######  Remark 3.9.

The multiplier $M$ and its associated bound (3.17) is sharp in the following
sense. Given an exponent $\alpha>0$, the vectorfield $r^{\alpha}M$ is also
timelike on $\mathcal{R}^{+}$, and possesses the property that the associated
energy current $K^{r^{\alpha}M}$ obeys the bound

$\phi K^{r^{\alpha}M}\leq-\frac{\alpha}{r}J^{r^{\alpha}M}[\psi]\cdot
n_{\Sigma_{r}}.$ (3.18)

To see this, observe that by the product rule we have

$\>{}^{(r^{\alpha}M)}\pi^{\mu\nu}$
$\displaystyle=r^{\alpha}\>{}^{(M)}\pi^{\mu\nu}+\frac{1}{2}\Big{(}g^{\mu\lambda}\partial_{\lambda}(r^{\alpha})M^{\nu}+g^{\nu\lambda}\partial_{\lambda}(r^{\alpha})M^{\mu}\Big{)}$
$\displaystyle=r^{\alpha}\>{}^{(M)}\pi^{\mu\nu}-\frac{1}{2}\alpha
r^{\alpha-1}\phi^{-2}\Big{(}\delta_{r}^{\mu}M^{\nu}+\delta_{r}^{\nu}M^{\nu}\Big{)},$

and so

$\phi K^{r^{\alpha}M}=r^{\alpha}\phi K^{M}-\alpha r^{\alpha-1}J^{M}[\psi]\cdot
n_{\Sigma_{r}}.$

As $K^{M}\leq 0$ and $r^{\alpha}J^{M}[\psi]=J^{r^{\alpha}M}[\psi]$, we have
(3.18). Proposition 3.7 corresponds to the endpoint case $\alpha=0$. Through a
Grönwall-type inequality, (3.18) implies an energy estimate similar to (1.10).
If $\square_{g}\psi=0$, the resulting energy estimate is in fact the same.
Howeve we will be applying this energy estimate to solutions of the
inhomogeneous wave equation $\square_{g}\psi=F$, and the term containing $F$
in (1.10) is multiplied by an extra factor of $r^{\alpha}$, so the estimate is
no longer sharp for $\alpha>0$.

###### Proof of Proposition 1.8.

We will apply the divergence theorem to the energy current $J^{M}[\psi]$ on
the spacetime region $\mathcal{D}_{r_{1},r_{2}}^{(\overline{u},\overline{v})}$
defined in (3.12), which implies by (3.13) that

$\int_{\Sigma_{r_{1}}^{c}}J^{M}[\psi]\cdot
n_{\Sigma_{r}}+\int_{\mathcal{F}_{\overline{v}}^{c}}\>^{*}J^{M}[\psi]+\int_{\mathcal{G}_{\overline{u}}^{c}}\>^{*}J^{M}[\psi]\\\
=\int_{\Sigma_{r_{2}}^{c}}J^{M}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}+\int_{\mathcal{D}_{r_{1},r_{2}}^{(\overline{u},\overline{v})}}\nabla\cdot
J^{M}[\psi]\textnormal{d}\mu_{g},$

We write the boundary integrals on $\Sigma_{r_{1}}^{c},\Sigma_{r_{2}}^{c}$ in
terms of the energy norms $\|\cdot\|_{M,\Sigma_{r}^{c}}$ and apply the coarea
formula to the bulk term, which implies

$\|\psi\|_{M,\Sigma_{r_{1}}^{c}}^{2}+\int_{\mathcal{F}_{\overline{v}}^{c}}\>^{*}J^{M}[\psi]+\int_{\mathcal{G}_{\overline{u}}^{c}}\>^{*}J^{M}[\psi]=\|\psi\|_{M,\Sigma_{r_{2}}^{c}}^{2}+\int_{r_{1}}^{r_{2}}\Big{(}\int_{\Sigma_{r}}\phi\nabla\cdot
J^{M}[\psi]\>\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}\textnormal{d}r.$

We then differentiate in $r_{2}$, keeping $r_{1},\overline{u},\overline{v}$
fixed so that

$\partial_{r_{2}}\Big{(}\int_{\mathcal{F}_{\overline{v}}^{c}}\>^{*}J^{M}[\psi]+\int_{\mathcal{G}_{\overline{u}}^{c}}\>^{*}J^{M}[\psi]\Big{)}=\partial_{r_{2}}(\|\psi\|_{M,\Sigma_{r_{2}})}^{2})+\int_{\Sigma_{r_{2}}}\phi\nabla\cdot
J^{M}[\psi]\textnormal{d}\mu_{\overline{g}_{r}}.$

The left hand side of the above expression is nonnegative, as
$\>{}^{*}J^{M}[\psi]$ restricted to a null hypersurface is nonnegative, and
the hypersurfaces
$\mathcal{F}_{\overline{v}}^{c},\mathcal{G}_{\overline{u}}^{c}$ are increasing
in $r_{2}$. Hence the right hand side is also nonnegative.

From Proposition 3.7 the current $K^{M}[\psi]\leq 0$, and so we bound

$\int_{\Sigma_{r_{2}}^{c}}\phi\nabla\cdot
J^{M}[\psi]\textnormal{d}\mu_{\overline{g}_{r}}\leq\int_{\Sigma_{r_{2}}^{c}}\phi(M\psi)(\square_{g}\psi)\textnormal{d}\mu_{\overline{g}_{r}}.$

We then apply the Cauchy-Schwartz inequality to the right hand side above
which implies

$\int_{\Sigma_{r_{2}}^{c}}\phi(M\psi)(\square_{g}\psi)\textnormal{d}\mu_{\overline{g}_{r}}\leq
2\Big{(}\frac{1}{2}\int_{\Sigma_{r_{2}}^{c}}\frac{\phi}{r^{2}}(M\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}^{1/2}\Big{(}\frac{1}{2}\int_{\Sigma_{r_{2}}^{c}}r^{2}\phi
F^{2}\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}^{1/2}.$

By the definition of $M$, we have the bound

$\frac{1}{2}\frac{\phi}{r^{2}}(M\psi)^{2}==\frac{1}{2}(n_{\Sigma_{r}}\psi)(M\psi)\leq
J^{M}[\psi]\cdot n_{\Sigma_{r}},$

therefore one can bound the first integral on the right hand side by
$\|\psi\|_{M,\Sigma_{r_{2}}^{c}}$, while the second integral is equivalent to
$\frac{1}{\sqrt{2}}\|r\phi^{1/2}F\|_{L^{2}(\Sigma_{r_{2}}^{c})}$. Meanwhile,

$\partial_{r_{2}}(\|\psi\|_{M,\Sigma_{r_{2}}^{c}}^{2})=2\|\psi\|_{M,\Sigma_{r_{2}}^{c}}\partial_{r_{2}}\|\psi\|_{M,\Sigma_{r_{2}}^{c}},$

and so

$0\leq
2\|\psi\|_{M,\Sigma_{r_{2}}^{c}}\partial_{r}\|\psi\|_{M,\Sigma_{r_{2}}^{c}}+\frac{2}{\sqrt{2}}\|\psi\|_{M,\Sigma_{r_{2}}^{c}}\|r\phi^{1/2}F\|_{L^{2}(\Sigma_{r_{2}}^{c})}.$

We then divide through by $2\|\psi\|_{M,\Sigma_{r_{2}}^{c}}$ and integrate in
the radial coordinate on the interval $[r_{1},r_{2}]$, yielding a localised
energy estimate,

$\|\psi\|_{M,\Sigma_{r_{1}}^{c}}\leq\|\psi\|_{M,\Sigma_{r_{2}}^{c}}+\int_{r_{1}}^{r_{2}}\frac{1}{\sqrt{2}}\|r\phi^{1/2}F\|_{L^{2}(\Sigma_{r}^{c})}\>\textnormal{d}r.$
(3.19)

The capped hypersurfaces $\Sigma_{r}^{c}$ are described precisely as

$\Sigma_{r}^{c}=\Sigma_{r}\cap\\{u\geq\overline{u}\\}\cap\\{v\geq\overline{v}\\},$

where $\overline{u},\overline{v}>0$ to ensure that the $\Sigma_{r}^{c}$ are
compact. We retrieve the global estimate (1.10) by taking limits
$\overline{u},\overline{v}\rightarrow 0$. ∎

######  Remark 3.10.

Theorem 1.8 can be used to create an array of higher order estimates. The key
observation is that the tangent spaces of the hypersurfaces $\Sigma_{r}$ are
spanned by Killing vectorfields. These consist of the Killing vectorfield $T$
that coincides in the expanding region with the coordinate vectorfield
$\partial/\partial_{t}$, and the generators of the spherical isometries
$\Omega_{i}:i=1,2,3$. The wave equation on a given Lorentzian manifold
commutes with the Killing vectorfields of that manifold, and so one can
commute the estimate Theorem (1.8) with these Killing vectorfields, yielding
estimates of the form

$\|T^{k}\Omega_{i_{1}}\cdots\Omega_{i_{l}}\psi\|_{M,\Sigma_{r_{1}}}\leq\|T^{k}\Omega_{i_{1}}\cdots\Omega_{i_{l}}\psi\|_{M,\Sigma_{r_{2}}}\\\
+\frac{1}{\sqrt{2}}\int_{r_{1}}^{r_{2}}\|r\phi^{1/2}T^{k}\Omega_{i_{1}}\cdots\Omega_{i_{l}}F\|_{L^{2}(\Sigma_{r})}\>\textnormal{d}r,$
(3.20)

for $k\in\mathbb{N},i_{1},\dots,i_{l}=1,2,3$. We will use such estimates in
Section 3.4 to obtain pointwise estimates for solutions to the backward
scattering problem.

###  3.3 Existence of scattering solutions to the wave equation on the
expanding region $\mathcal{R}^{+}$: proof of Theorem 1.4

We now prove that solutions to $\square_{g}\psi=0$ exist with prescribed
scattering data on $\Sigma^{+}$. Fix $r_{0}>r_{\mathcal{C}}$. For $R\geq
r_{0}$ let $\psi_{R}=\psi_{\text{asymp}}+\psi_{\text{rem}}^{R}$, where
$\psi_{\text{rem}}^{R}$ solves the inhomogeneous wave equation

$\square_{g}\psi_{\text{rem}}^{R}=-\square_{g}\psi_{\text{asymp}},$ (3.21)

with trivial initial data on the hypersurface $\Sigma_{R}$, so that

$\psi_{\text{rem}}^{R}|_{\Sigma_{R}}=0,\quad
n_{\Sigma_{R}}\cdot\psi_{\text{rem}}^{R}|_{\Sigma_{R}}=0.$

Let $R_{1},R_{2}\geq r_{0}$ and assume without loss of generality that
$R_{1}<R_{2}$. Then consider the difference

$v=\psi_{R_{2}}-\psi_{R_{1}}=\psi_{\text{rem}}^{R_{2}}-\psi_{\text{rem}}^{R_{1}}.$

We will prove that $v\rightarrow 0$ with respect to the weighted energy norm
$\|\cdot\|_{M,\Sigma_{r}}$ as $R_{1},R_{2}\rightarrow\infty$. Clearly $v$
satisfies the homogeneous wave equation $\square_{g}v=0$, so we may apply the
weighted energy estimate from Proposition 1.8 to $v$ on the spacetime region

$\mathcal{D}_{(r,R_{1})}=\bigcup_{r\leq\overline{r}\leq
R_{1}}\Sigma_{\overline{r}},$

which gives

$\|v\|_{M,\Sigma_{r}}\leq\|v\|_{M,\Sigma_{R_{1}}}.$ (3.22)

As $\psi_{\text{rem}}^{R_{1}}$ has trivial data on $\Sigma_{R_{1}}$, we have

$\|v\|_{M,\Sigma_{R_{1}}}=\|\psi_{\text{rem}}^{R_{2}}\|_{M,\Sigma_{R_{1}}}$

Then we apply the weighted energy estimate from Proposition 1.8 again, now to
$\psi_{\text{rem}}^{R_{2}}$ on the region $\mathcal{D}_{(R_{1},R_{2})}$. This
gives

$\|\psi_{\text{rem}}^{R_{2}}\|_{M,\Sigma_{R_{1}}}\leq\|\psi_{\text{rem}}^{R_{2}}\|_{M,\Sigma_{R_{2}}}+\frac{1}{\sqrt{2}}\int_{R_{1}}^{R_{2}}\|r\phi^{1/2}\square_{g}\psi_{\text{asymp}}\|_{L^{2}(\Sigma_{r})}\textnormal{d}r.$
(3.23)

Since $\psi_{\text{rem}}^{R_{2}}$ has vanishing data on $\Sigma_{R_{2}}$ we
have $\|\psi_{\text{rem}}^{R_{2}}\|_{M,\Sigma_{R_{2}}}=0$, and so it follows
from (3.22) and (3.23) that

$\|v\|_{M,\Sigma_{r}}\leq\frac{1}{\sqrt{2}}\int_{R_{1}}^{R_{2}}\|r\phi^{1/2}\square_{g}\psi_{\text{asymp}}\|_{L^{2}(\Sigma_{r})}\textnormal{d}r$
(3.24)

for all $r\in[r_{0},R_{1}]$. By Lemma 3.6, the integral on the right hand side
remains bounded as $R_{2}\rightarrow\infty$, and we have the bound

$\|v\|_{M,\Sigma_{r}}\lesssim_{\Lambda,m,r_{0}}\frac{1}{R_{1}}\|\tilde{\nabla}\psi_{0}\|_{H^{3}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{R_{1}^{2}}\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}.$

Thus for each $r\geq r_{0}$, $\|v\|_{M,\Sigma_{r}}\rightarrow 0$ as $R_{1}$,
$R_{2}\rightarrow 0$. Hence there exists a function $\psi$ such that

$\lim_{R\rightarrow\infty}\|\psi-\psi_{R}\|_{M,\Sigma_{r}}=0$

for each $r\geq r_{0}$.

We now derive the estimates on $\psi$, in particular we prove (1.7). By the
triangle inequality we have

$\|\psi_{\text{rem}}\|_{M,\Sigma_{r}}\leq\|\psi_{\text{rem}}-\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}}+\|\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}}.$

We apply the weighted energy estimate Proposition 1.8 to
$\psi_{\text{rem}}^{R}$ on $\mathcal{D}_{(r,R)}$, so that

$\|\psi_{\text{rem}}\|_{M,\Sigma_{r}}\leq\|\psi_{\text{rem}}-\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}}++\frac{1}{\sqrt{2}}\int_{r}^{R}\|s\phi^{1/2}(s)\square_{g}\psi_{\text{asymp}}\|_{L^{2}(\Sigma_{r})}\textnormal{d}s,$

where we used the fact that $\psi_{\text{rem}}^{R}$ has vanishing data on
$\Sigma_{R}$, so that $\|\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}}=0$. Again from
Lemma 3.6, the final integral is bounded as $R\rightarrow\infty$, and we have
the bound

$\|\psi_{\text{rem}}\|_{M,\Sigma_{r}}\lesssim_{\Lambda,m,r_{0}}\|\psi_{\text{rem}}-\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}}+\frac{1}{r}\|\tilde{\nabla}\psi_{0}\|_{H^{3}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{2}}\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}.\\\
$ (3.25)

Letting $R\rightarrow\infty$,
$\|\psi_{\text{rem}}-\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}}\rightarrow 0$, and
so it follows that for $\psi-\psi_{\text{asymp}}=\psi_{\text{rem}}$ we have.

$\displaystyle\|\psi-\psi_{\text{asymp}}\|_{\partial_{r},\Sigma_{r}}$
$\displaystyle=\|\psi_{\text{rem}}\|_{\partial_{r},\Sigma_{r}}$
$\displaystyle\lesssim_{\Lambda,m,r_{0}}\frac{1}{r^{2}}\|\psi_{\text{rem}}\|_{M,\Sigma_{r}}$
$\displaystyle\lesssim_{\Lambda,m,r_{0}}\frac{1}{r^{3}}\|\tilde{\nabla}\psi_{0}\|_{H^{3}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{4}}\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}.$
(3.26)

From this and (3.10) we also have for all $r\geq r_{0}$,

$\displaystyle\|\psi\|_{\partial_{r},\Sigma_{r}}$
$\displaystyle\lesssim_{\Lambda,m,r_{0}}r^{2}\|\psi_{\text{rem}}\|_{M,\Sigma_{r}}+\|\psi_{\text{asymp}}\|_{\partial_{r},\Sigma_{r}}$
$\displaystyle\lesssim_{\Lambda,m,r_{0}}\|\tilde{\nabla}\psi_{0}\|_{H^{3}(\mathbb{R}\times\mathbb{S}^{2})}+\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})},$

which is (1.7). Uniqueness follows immediately, since if we set
$\psi_{0}=\psi_{3}=0$, then $\psi=0$ by the above inequality.∎

######  Remark 3.11.

When defining the sequence of remainder functions $\psi_{\text{rem}}^{R}$, we
point out that the inhomogeneous wave equation (3.21) satisfied by
$\psi_{\text{rem}}^{R}$ does not contain a cutoff function in $r$, such as

$\square_{g}\psi_{\text{rem}}^{R}=\chi(r/R)\square_{g}\psi_{\text{asymp}},$

where $\chi$ is a smooth, compactly supported function such that $\chi(s)=1$
in a neighbourhood of $0$, and $\chi(s)=0$ in particular for all $s\geq 1$.
Such a cutoff function is typically employed when constructing scattering
solutions to wave equations on Minkowski in this manner, see [23] for example.
However we will use cutoffs (in a slightly different manner) when extending
solutions to the cosmological horizon, c.f. (3.41).

###  3.4 Pointwise estimates

Pointwise estimates can be obtained by repeatedly applying the Killing
vectorfields $T,\Omega_{i}:i=1,2,3$ to the solution $\psi$ constructed in the
previous section, and then using a Sobolev estimate adapted to the
hypersurfaces $\Sigma_{r}$. The proof for this Sobolev estimate can be found
in Appendix B. In particular we prove the pointwise decay estimate (1.8) in
Theorem 1.4.

First we establish the decay of the $L^{2}$-norms of the remainder
$\psi_{\text{rem}}$.

######  Lemma 3.12.

Let $\psi$ be the scattering solution constructed in Theorem 1.4 from
scattering data $\psi_{0}$, $\psi_{3}$. Then the $L^{2}$-norm of the remainder
$\psi_{\text{rem}}$ obeys the bound

$\|\psi_{\text{rem}}\|_{L^{2}(\Sigma_{r})}\lesssim_{\Lambda,m}\frac{1}{r^{5/2}}\|\psi_{0}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{7/2}}\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}$
(3.27)

for all large $r$. Moreover, $\psi\in L^{2}(\Sigma_{r})$ for all $r\geq r_{0}$
for some $r_{0}$ slightly larger than $r_{\mathcal{C}}$, and we have

$\|\psi\|_{L^{2}(\Sigma_{r})}\lesssim_{\Lambda,m,r_{0}}r^{3/2}\Big{(}\|\psi_{0}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})}+\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}\Big{)}.$
(3.28)

###### Proof.

We will give a sketch of the proof here as this is a simple modification of
Theorem 1.4. First, we observe that (3.28) follows from the decay estimate
(3.27), and the fact that

$\|\psi\|_{L^{2}(\Sigma_{r})}\sim_{r_{0}}r^{3/2}\|\psi(r)\|_{L^{2}(\mathbb{R}\times\mathbb{S}^{2})}$

for all $r\geq r_{0}$. To prove (3.27), we start with the straightforward
bound

$\|\varphi\|_{L^{2}(\Sigma_{r})}\lesssim
r^{3/2}\Big{(}\frac{1}{R^{3/2}}\|\varphi\|_{L^{2}(\Sigma_{R})}+\int_{r}^{R}\frac{1}{s^{3/2}}\|\partial_{r}\varphi\|_{L^{2}(\Sigma_{s})}\textnormal{d}s\Big{)}$

which is obtained via the fundamental theorem of calculus. We note that

$\|\partial_{r}\varphi\|_{L^{2}(\Sigma_{s})}\lesssim\frac{1}{s^{5/2}}\|\varphi\|_{M,\Sigma_{s}}.$

Through the same method of proof as that of Theorem 1.4, one can show that the
sequence of functions $\psi_{\text{rem}}^{R}$ that solve
$\square_{g}\psi_{\text{rem}}^{R}=-\square_{g}\psi_{\text{asymp}}$ with
trivial data on $\Sigma_{R}$ have a limit as $R\rightarrow\infty$. Moreover,
the uniqueness statement from Theorem 1.4 implies that this limiting function
agrees with the limiting function found in Theorem 1.4. The bound (3.27) is
obtained by estimating

$\displaystyle\|\psi_{\text{rem}}\|_{L^{2}(\Sigma_{r})}$
$\displaystyle\leq\|\psi_{\text{rem}}^{R}\|_{L^{2}(\Sigma_{r})}+\|\psi_{\text{rem}}-\psi_{\text{rem}}^{R}\|_{L^{2}(\Sigma_{r})}$
$\displaystyle\lesssim\int_{r}^{R}\frac{1}{s^{4}}\|\psi_{\text{rem}}^{R}\|_{M,\Sigma_{s}}\>\textnormal{d}s+\|\psi_{\text{rem}}-\psi_{\text{rem}}^{R}\|_{L^{2}(\Sigma_{r})}$
(3.29)

then employing the weighted energy estimate from Proposition 1.8 to bound the
integrand

$\frac{1}{r^{4}}\|\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}}\lesssim\frac{1}{r^{5}}\|\tilde{\nabla}\psi_{0}\|_{H^{3}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{6}}\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}.$

Inserting this bound into (3.29) and taking the limit $R\rightarrow\infty$
implies (3.27). ∎

###### Proof of pointwise decay (1.8).

We will apply the Sobolev estimate from Appendix B to $\psi_{\text{rem}}$. Let
$I=(I_{1},I_{2},I_{3},I_{4})$ be a multiindex, and let $Z^{I}$ denote the
differential operator

$Z^{I}=\Omega_{1}^{I_{1}}\Omega_{2}^{I_{2}}\Omega_{3}^{I_{3}}T^{I_{4}}.$

By Proposition B.1, we have

$\displaystyle\sup_{\Sigma_{r}}|\psi_{\text{rem}}|^{2}\lesssim_{\Lambda,m,r_{0}}$
$\displaystyle\frac{1}{r^{3}}\Big{\\{}\|\psi_{\text{rem}}\|_{L^{2}(\Sigma_{r})}^{2}+\|(T\psi_{\text{rem}})\|_{L^{2}(\Sigma_{r})}^{2}+\sum_{i=1}^{3}\|\Omega_{i}\psi_{\text{rem}}\|_{L^{2}(\Sigma_{r})}^{2}$
$\displaystyle\quad+\sum_{i=1}^{3}\|\Omega_{i}T\psi_{\text{rem}}\|_{L^{2}(\Sigma_{r})}^{2}+\sum_{i,j=1}^{3}\|\Omega_{i}\Omega_{j}\psi_{\text{rem}}\|_{L^{2}(\Sigma_{r})}^{2}\Big{\\}}$
$\displaystyle\lesssim_{\Lambda,m,r_{0}}$
$\displaystyle\frac{1}{r^{3}}\sum_{|I|\leq
2}\|Z^{I}\psi_{\text{rem}}\|_{L^{2}(\Sigma_{r})}^{2}.$ (3.30)

As the operators $Z^{I}$ are composed of Killing vector fields, they commute
with the wave equation, and so $Z^{I}\psi_{\text{rem}}$ solves the equation

$\square_{g}(Z^{I}\psi_{\text{rem}})=-\square_{g}(Z^{I}\psi_{\text{asymp}}).$

Moreover, the function $Z^{I}\psi_{\text{asymp}}$ remains a good asymptotic
solution, since

$Z^{I}\psi_{\text{asymp}}=Z^{I}\psi_{0}+\frac{1}{r^{2}}Z^{I}\psi_{2}+\frac{1}{r^{3}}Z^{I}\psi_{3},$

where

$Z^{I}\psi_{2}=-\frac{3}{2\Lambda}\tilde{\Delta}(Z^{I}\psi_{0}).$

By Theorem 1.4, the solution $\varphi$ to the wave equation with scattering
data $Z^{I}\psi_{0}$, $Z^{I}\psi_{3}$ is unique, and so the function
$\psi_{\text{rem}}=\varphi-Z^{I}\psi_{\text{asymp}}$ obeys the $L^{2}$-based
decay estimate (3.27) from Propostion 3.12, i.e.

$\|Z^{I}\psi_{\text{rem}}\|_{L^{2}(\Sigma_{r})}\lesssim\frac{1}{r^{5/2}}\|\tilde{\nabla}(Z^{I}\psi_{0})\|_{H^{3}(\mathbb{R}\times\mathbb{S}^{2})}+\frac{1}{r^{7/2}}\|Z^{I}\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}.$

Substituting this into (3.30), the result now follows. ∎

###  3.5 Nondegenerate energy estimates on the cosmological horizon

We now prove that the scattering solutions from Theorem 1.4 extend to the
cosmological horizon, provided the scattering data $\psi_{0},\psi_{3}$ decay
exponentially quickly along $\Sigma^{+}$.

We first prove a nondegenerate energy estimate in a neighbourhood of the
cosmological horizons. Recall the redshift vectorfield $N$ from [27] that
captures the local redshift effect on the cosmological horizons
$\mathcal{C}^{+}$. Here we slightly modify the construction of $N$ so that it
is suitable for energy estimates backward-in-time. We begin by introducing on
$\mathcal{C^{+}}$ the null frame $\\{T,Y,E_{A}:A=1,2\\}$ comprised of the
Killing vector field $T$ introduced in Section 1.2, $Y$ which is null and
conjugate to $T$, and an orthonormal frame $E_{A}:A=1,2$ that is Lie
transported by $T$ along the cosmological horizons. See (2.7), (2.8) for
formulae for $T,Y$ on $\mathcal{C}^{+}$. As in [27], we extend $Y$ off
$\mathcal{C}^{+}$ into the interior of the expanding region $\mathcal{R}^{+}$
like

$\nabla_{Y}Y=-\sigma(Y+T).$ (3.31)

This construction, capturing the local redshift effect of Killing horizons,
was initially carried out in [7, 8], and was adapted to a neighbourhood to the
future of the cosmological horizons in [27]. In our construction we choose the
constant $\sigma$ to be negative.

######  Proposition 3.13.

For $r_{0}>r_{\mathcal{C}}$, let $\mathcal{D}_{(r_{\mathcal{C}},r_{0})}$
denote the spacetime region $\\{(u,v):r_{\mathcal{C}}\leq r(u,v)\leq
r_{0}\\}$. Let $\sigma<0$ be some sufficiently negative constant, and let $Y$
denoted the vectorfield define in (2.8) and (3.31). The vectorfield multiplier
$N=Y+T$ is strictly timelike on $\mathcal{D}_{(r_{\mathcal{C}},r_{0})}$, and
has the property that there exists some constant $c=c(\Lambda,m)>0$ such that
on $\mathcal{D}_{(r_{\mathcal{C}},r_{0})}$ we have

$K^{N}[\psi]\leq\frac{c}{u}J^{N}\cdot\frac{\partial}{\partial v}.$ (3.32)

###### Proof.

From [8] and [27], we have

$K^{Y}[\psi]\Big{|}_{\mathcal{C}^{+}}=-\frac{1}{2}|\sigma|(T\psi)^{2}+\frac{1}{2}\kappa_{\mathcal{C}}(Y\psi)^{2}-\frac{1}{2}|\sigma||\not{\nabla}\psi|^{2}+\frac{2}{r_{\mathcal{C}}}(T\psi)(Y\psi).$
(3.33)

By the Cauchy-Schwartz inequality, we may bound

$\frac{2}{r}(T\psi)(Y\psi)\leq\frac{1}{2}|\sigma|(T\psi)^{2}+\frac{2}{r_{\mathcal{C}}^{2}|\sigma|}(Y\psi)^{2},$

and so by (3.33), we have

$K^{Y}[\psi]\Big{|}_{\mathcal{C}^{+}}\leq\Big{(}\frac{1}{2}\kappa_{\mathcal{C}}+\frac{2}{r_{\mathcal{C}}^{2}|\sigma|}\Big{)}(Y\psi)^{2}.$
(3.34)

A short computation reveals that the energy current $J^{N}\cdot\partial_{v}$
restricted to the cosmological horizons is

$J^{N}[\psi]\cdot\frac{\partial}{\partial
v}\Big{|}_{\mathcal{C}^{+}}=\frac{u}{\iota_{\mathcal{C}}}\Big{(}(Y\psi)^{2}+\kappa_{\mathcal{C}}^{2}|\not{\nabla}\psi|^{2}\Big{)}.$

Since $T$ is Killing everywhere, $K^{N}=K^{Y}+K^{T}=K^{Y}$, and so by (3.34),

$K^{N}[\psi]\Big{|}_{\mathcal{C}^{+}}\leq\Big{(}\frac{1}{2}\kappa_{\mathcal{C}}+\frac{2}{r_{\mathcal{C}}^{2}\sigma}\Big{)}\frac{\iota_{\mathcal{C}}}{u}J^{N}\cdot\frac{\partial}{\partial
v}\Big{|}_{\mathcal{C}^{+}}.$

By continuity this inequality can be extended to a neighbourhood of
$\mathcal{C}^{+}$, and so we may choose some $r_{0}$ sufficiently close to
$r_{\mathcal{C}}$, and some $\sigma$ sufficiently negative such that (3.32)
holds on $\mathcal{D}_{(r_{\mathcal{C}},r_{0})}$. ∎

We now introduce some notation that will be important for the following
proposition. Let $\tau$ be a function on $\mathcal{C}^{+}$ that is constant on
the spheres of symmetry $S\subset\mathcal{C}^{+}$, and

$T\cdot\tau\Big{|}_{\mathcal{C}^{+}}=1,$

and set $\tau=0$ on the sphere for which $u=1$. It follows from (2.7) that

$\frac{\textnormal{d}\tau}{\textnormal{d}u}\Big{|}_{\mathcal{C}^{+}}=\frac{1}{\kappa_{\mathcal{C}}u}.$
(3.35)

For given $\tau_{1},\tau_{2}\in\mathbb{R}$, let $\mathcal{C}_{\tau_{1}}^{+}$
be the segment of $\mathcal{C}^{+}$ where $\tau\geq\tau_{1}$, and let
$C_{\tau_{1}}$ denote the null hypersurface which lies transverse to the
cosmological horizons and intersects $\mathcal{C}^{+}$ at the sphere on which
$\tau=\tau_{1}$. For $r_{0}>r_{\mathcal{C}}$, let
$\Sigma_{r_{0}}^{(\tau_{1})}=\Sigma_{r_{0}}\cap
J^{+}(\mathcal{C}_{\tau_{1}}^{+})$. Moreover, define the compact hypersurfaces

$\displaystyle\mathcal{C}_{\tau_{1},\tau_{2}}^{+}=\mathcal{C}_{\tau_{1}}^{+}\backslash\mathcal{C}_{\tau_{2}}^{+}.$
$\displaystyle C_{\tau_{1}}^{c}=C_{\tau_{1}}\cap J^{-}(\Sigma_{r_{0}}).$
$\displaystyle\Sigma_{r_{0}}^{(\tau_{1},\tau_{2})}=\Sigma_{r_{0}}^{(\tau_{1})}\backslash\Sigma_{r_{0}}^{(\tau_{2})}.$

We note here that for any $\tau_{1},\tau_{2}\in\mathbb{R}$ with
$\tau_{1}<\tau_{2}$, and $r_{0}>r_{\mathcal{C}}$, the sets
$\mathcal{C}_{\tau_{1},\tau_{2}}^{+}$, $C_{\tau_{1}}^{c}$, $C_{\tau_{2}}^{c}$,
and $\Sigma_{r_{0}}^{(\tau_{1},\tau_{2})}$ enclose a compact region of
$\mathcal{R}^{+}$, which we denote by
$\mathcal{R}_{\tau_{1},\tau_{2}}^{(r_{0})}$; see Figure 3.4.

We will now prove a nondegenerate energy estimate at the cosmological
horizons. We refer the reader to Proposition 4.4 in [27] for the corresponding
energy estimate on the cosmological horizons proved for the forward problem.

$\mathcal{C}_{\tau_{1},\tau_{2}}^{+}$$C_{\tau_{1}}^{c}$$C_{\tau_{2}}^{c}$$\mathcal{R}_{\tau_{1},\tau_{2}}^{(r_{0})}$$\Sigma_{r_{0}}^{c}$
Figure 3.4: Spacetime domain $\mathcal{R}_{\tau_{1},\tau_{2}}^{(r_{0})}$ of
the nondegenerate energy estimate along $\mathcal{C}^{+}$.

######  Proposition 3.14.

Let $\psi$ be a solution to (1.1) with finite energy on $\Sigma_{r_{0}}$ for
some $r_{0}$ sufficiently close to $r_{\mathcal{C}}$. Set

$f(\tau)=\int_{\mathcal{C}_{\tau}^{\prime}}\>^{*}J^{N}[\psi].$

The following energy estimate holds for all solutions to $\square_{g}=0$ on
$\mathcal{R}^{+}\cup\mathcal{C}^{+}$:

$\displaystyle
f(\tau_{1})+\int_{\mathcal{C}_{\tau_{1},\tau_{2}}^{+}}\>^{*}J^{N}[\psi]\leq\>$
$\displaystyle
e^{c(\tau_{2}-\tau_{1})}f(\tau_{2})+\int_{\Sigma_{r_{0}}^{(\tau_{1},\tau_{2})}}J^{N}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}$
$\displaystyle+c\int_{\tau_{1}}^{\tau_{2}}e^{c(\tau-\tau_{1})}\Big{(}\int_{\Sigma_{r_{0}}^{(\tau,\tau_{2})}}J^{N}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}\textnormal{d}\tau$

for all $\tau_{1}<\tau_{2}$, where $c>0$ is any constant such that
$c>\kappa_{\mathcal{C}}$.

###### Proof.

Applying the divergence theorem to the energy current $J^{N}[\psi]$ on
$\mathcal{R}_{\tau_{1},\tau_{2}}^{(r_{0})}$, we get

$\int_{C_{\tau_{2}}^{c}}\>^{*}J^{N}[\psi]+\int_{\Sigma_{r_{0}}^{(\tau_{1},\tau_{2})}}J^{N}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}+\int_{\mathcal{R}_{\tau_{1},\tau_{2}}^{(r_{0})}}K^{N}[\psi]\>\textnormal{d}\mu_{g}\\\
=\int_{C_{\tau_{1}}^{c}}\>^{*}J^{N}[\psi]+\int_{\mathcal{C}_{\tau_{1},\tau_{2}}^{+}}\>^{*}J^{N}[\psi].$
(3.36)

In the Kruskal coordinates defined in Section 1.2, we have

$\textnormal{d}\mu_{g}=\frac{1}{2}\Omega^{2}r^{2}\textnormal{d}u\wedge\textnormal{d}v\wedge\textnormal{d}\mu_{\mathring{\gamma}},$

From this and Proposition 3.13, it follows that there exists some constant
$c>0$ that depends on $\kappa_{\mathcal{C}}$ such that

$\int_{\mathcal{R}_{\tau_{1},\tau_{2}}^{(r_{0})}}K^{N}[\psi]\>\textnormal{d}\mu_{g}\leq
c\int_{\tau_{1}}^{\tau_{2}}\Big{(}\int\Big{(}J^{N}\cdot\frac{\partial}{\partial
v}\Big{)}\textnormal{d}v\textnormal{d}\mu_{\mathring{\gamma}}\Big{)}\textnormal{d}\tau,$

where we used the relation (3.35) between $u$ and $\tau$. On the other hand,
we can write the integral of $\>{}^{*}J^{N}$ on $C_{\tau}^{c}$ as

$\int_{C_{\tau}^{c}}\>^{*}J^{N}=\int\Big{(}J^{N}\cdot\frac{\partial}{\partial
v}\Big{)}r^{2}\textnormal{d}v\textnormal{d}\mu_{\mathring{\gamma}},$

Therefore we may write

$\int_{\mathcal{R}_{\tau_{1},\tau_{2}}^{(r_{0})}}K^{N}[\psi]\>\textnormal{d}\mu_{g}\leq
c\int_{\tau_{1}}^{\tau_{2}}\Big{(}\int_{C_{\tau}^{c}}\>^{*}J^{N}[\psi]\Big{)}\textnormal{d}\tau.$

It follows from (3.36) that

$f(\tau_{1})+\int_{\mathcal{C}_{\tau_{1},\tau_{2}}^{+}}\>^{*}J^{N}[\psi]\leq
f(\tau_{2})+c\int_{\tau_{1}}^{\tau_{2}}f(\tau)\textnormal{d}\tau+\int_{\Sigma_{r_{0}}^{(\tau_{1},\tau_{2})}}J^{N}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}.$ (3.37)

This is a standard Grönwall-type inequality, from which the result follows by
integrating over $[\tau_{1},\tau_{2}]$ the inequality

$-\partial_{\tau}\Big{[}e^{c\tau}F(\tau,\tau_{2})\Big{]}=e^{c\tau}\Big{[}f(\tau)-cF(\tau,\tau_{2})\Big{]}\leq
e^{c\tau}\Big{[}f(\tau_{2})+\int_{\Sigma_{r_{0}}^{(\tau,\tau_{2})}}J^{N}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\Big{]},$

and applying the inequality (3.37) again. ∎

######  Remark 3.15.

While Proposition 3.14 was only proven on one segment of the cosmological
horizons $\mathcal{C}^{+}$, one can easily derive the analogous estimate for
$\overline{\mathcal{C}}^{+}$. The expanding region and horizons
$\mathcal{R}^{+}\cup\mathcal{C}^{+}\cup\overline{\mathcal{C}}^{+}$ are
symmetric with respect to the null coordinates $(u,v)$, and so precisely the
same argument holds by applying the isometry $(u,v)\mapsto(v,u)$ to
$\mathcal{R}^{+}\cup\mathcal{C}^{+}\cup\overline{\mathcal{C}}^{+}$.

###  3.6 Existence of scattering solutions to the wave equation up to the
cosmological horizons: proof of Theorem 1.10

We first prove that for the scattering solutions constructed in Theorem 1.4,
exponential decay of the scattering data $\psi_{0},\psi_{3}$ is propagated to
exponential decay of the solution $\psi$ along each $\Sigma_{r}$ hypersurface.
Let $\Sigma_{r}^{\tau}$ denote the subset of $\Sigma_{r}$ that lies in the
future domain of dependence of $\mathcal{C}_{\tau}^{+}$, and let
$\overline{\Sigma}_{r}^{\tau}$ denote the analagous subset of $\Sigma_{r}$
that lies in the future domain of dependence of
$\overline{\mathcal{C}}_{\tau}^{+}$. These should be thought of as the “ends”
of the $\Sigma_{r}$, which we recall are diffeomorphic to the standard
cylinder $\mathbb{R}\times\mathbb{S}^{2}$.

######  Proposition 3.16.

Let $\psi_{0}\in H^{4}(\mathbb{R}\times\mathbb{S}^{2})$ ,$\psi_{3}\in
H^{2}(\mathbb{R}\times\mathbb{S}^{2})$ be scattering data, and let $\psi$ be
the corresponding finite-energy solution on the expanding region
$\mathcal{R}^{+}$. In addition, assume that $\psi_{0}$, $\psi_{3}$ decay
exponentially, in the sense that as $\tau\rightarrow\infty$ we have

$\|\tilde{\nabla}\psi_{0}\|_{H^{3}((-\tau,\tau)^{c}\times\mathbb{S}^{2})},\|\psi_{3}\|_{H^{2}((-\tau,\tau)^{c}\times\mathbb{S}^{2})}\lesssim_{\Lambda,m}e^{-\beta\tau}$

for some constant $\beta>0$. Then for each $r>r_{\mathcal{C}}$, the energy
flux of $\psi$ through $\Sigma_{r}$ decays exponentially toward $\iota$ and
$\overline{\iota}$, so that

$\|\psi\|_{M,\Sigma_{r}^{\tau}},\|\psi\|_{M,\overline{\Sigma}_{r}^{\tau}}\lesssim_{\Lambda,m,r}e^{-\beta\tau}.$
(3.38)

The $L^{2}$-norm of $\psi$ also decays exponentially,

$\|\psi\|_{L^{2}(\Sigma_{r}^{\tau})},\|\psi\|_{L^{2}(\overline{\Sigma}_{r}^{\tau})}\lesssim_{\Lambda,m,r}e^{-\beta\tau}$
(3.39)

###### Proof.

This follows directly from the localised version of the weighted energy
estimate from Proposition 1.8, namely (3.19). We will only prove the estimate
for the energy flux of $\psi$ through $\Sigma_{r}^{\tau}$, the proof for
$\overline{\Sigma}_{r}^{\tau}$ is the same. We estimate

$\displaystyle\|\psi\|_{M,\Sigma_{r}^{\tau}}$
$\displaystyle\leq\|\psi_{\text{asymp}}\|_{M,\Sigma_{r}^{\tau}}+\|\psi_{\text{rem}}\|_{M,\Sigma_{r}^{\tau}}$
$\displaystyle\leq\|\psi_{\text{asymp}}\|_{M,\Sigma_{r}^{\tau}}+\|\psi_{\text{rem}}-\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}^{\tau}}+\|\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}^{\tau}}$
(3.40)

As a consequence of the localised energy estimate (3.19), we can bound the
last term like

$\|\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}^{\tau}}\leq\int_{r}^{R}\|r\phi^{1/2}\square_{g}\psi_{\text{asymp}}\|_{L^{2}(\Sigma_{r}^{\tau})}\textnormal{d}r.$

This follows from taking the limit $\overline{v}\rightarrow 0$, while still
keeping $\overline{u}$ fixed in (3.19). From Lemma 3.6 we bound the energy and
$L^{2}$-norms of the asymptotic solution

$\displaystyle\|\psi_{\text{asymp}}\|_{M,\Sigma_{r}^{\tau}}\lesssim_{\Lambda,m,r}\|\tilde{\nabla}\psi_{0}\|_{H^{2}((\tau,\infty)\times\mathbb{S}^{2})}+\|\psi_{3}\|_{H^{1}((\tau,\infty)\times\mathbb{S}^{2})},$
$\displaystyle\int_{r}^{\infty}\|r\phi^{1/2}\square_{g}\psi_{\text{asymp}}\|_{L^{2}(\Sigma_{r}^{\tau})}\>\textnormal{d}r\lesssim_{\Lambda,m,r}\|\tilde{\nabla}\psi_{0}\|_{H^{3}((\tau,\infty)\times\mathbb{S}^{2})}+\frac{1}{r^{2}}\|\psi_{3}\|_{H^{2}((\tau,\infty)\times\mathbb{S}^{2})}.$

It follows from (3.40) and the assumption of exponential decay of
$\psi_{0},\psi_{3}$ that

$\displaystyle\|\psi\|_{M,\Sigma_{r}^{\tau}}$
$\displaystyle\lesssim_{\Lambda,m,r}\|\tilde{\nabla}\psi_{0}\|_{H^{3}((\tau,\infty)\times\mathbb{S}^{2})}+\frac{1}{r^{2}}\|\psi_{3}\|_{H^{2}((\tau,\infty)\times\mathbb{S}^{2})}+\|\psi_{\text{rem}}-\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}^{\tau}}$
$\displaystyle\lesssim
e^{-\beta\tau}+\|\psi_{\text{rem}}-\psi_{\text{rem}}^{R}\|_{M,\Sigma_{r}^{\tau}}.$

The estimate (3.38) now follows after taking the limit $R\rightarrow\infty$.
The proof of (3.39) follows analogously through a localised version of the
$L^{2}$ estimate (3.28) from Proposition 3.12. ∎

######  Remark 3.17.

Proposition 3.16 holds for any exponential decay rate, i.e. one can choose any
$\beta>0$. However for the following proof of Theorem 1.10, one must assume a
sufficiently high degree of exponential decay, see in particular (3.47) below.
This is a manifestation of the blueshift effect discussed in Section 1.1. One
can show that Theorem 1.10 follows as long as $\beta>\kappa_{\mathcal{C}}/2$.
This is accomplished by carefully tracking constants throughout the proof and
choosing $r_{0}$ sufficiently close to $r_{\mathcal{C}}$.

###### Proof of Theorem 1.10.

Let $T>0$ be some constant, consider the hypersurfaces
$C_{T}^{c},\overline{C_{T}^{c}}$, where $C_{T}^{c}$ is the outgoing null
hypersurface that intersects $\mathcal{C}^{+}$ at the point $\tau=T$, and
$\overline{C_{T}^{c}}$ is the corresponding hypersurface that intersects
$\overline{\mathcal{C}}^{+}$. Let $\psi^{\prime}$ be the solution on
$\mathcal{R}^{+}$ with scattering data $\psi_{0},\psi_{3}$. By Theorem 1.4,
$\psi^{\prime}$ has finite energy on $\Sigma_{r_{0}}$ for all
$r_{0}>r_{\mathcal{C}}$. Let $T^{\prime}>0$ denote the value of the coordinate
$t$ on the sphere of intersection of the hypersurfaces $\Sigma_{r_{0}}$ and
$C_{T}^{c}$. We will consider a solution $\psi_{T}$ that satisfies the
homogeneous wave equation $\square_{g}\psi^{T}=0$, with initial data on
$\Sigma_{r_{0}}$ given as

$\psi_{T}|_{\Sigma_{r_{0}}}=\chi(t/T^{\prime})\psi^{\prime}|_{\Sigma_{r_{0}}},\quad
n_{\Sigma_{r}}\cdot\psi_{T}|_{\Sigma_{r_{0}}}=\chi(t/T^{\prime})n_{\Sigma_{r}}\cdot\psi^{\prime}|_{\Sigma_{r_{0}}},$

where $\chi\in C_{0}^{\infty}(\mathbb{R})$ is a cutoff function such that
$0\leq\chi\leq 1$, $\chi(s)=1$ for $|s|\leq 1/2$, and $\chi(s)=0$ for $|s|\geq
1$. By the localised weighted energy estimate (3.19) from Proposition 1.8, it
follows that $\psi=0$ on the entire future domain of dependence of
$\mathcal{C}_{T}^{+}\cup\overline{\mathcal{C}_{T}^{+}}$ up to
$\Sigma_{r_{0}}$. In particular, we have

$\psi_{T}|_{C_{T}^{c}}=\psi_{T}|_{\overline{C_{T}^{c}}}=0,\quad\psi_{T}|_{\mathcal{C}_{T}^{+}}=\psi_{T}|_{\overline{\mathcal{C}}_{T}^{+}}=0.$

Now consider the difference $v=\psi_{T_{2}}-\psi_{T_{1}}$, for $T_{2}>T_{1}$.
Clearly $v$ satisfies the wave equation, and on $\Sigma_{r_{0}}$, we have

$\displaystyle
v|_{\Sigma_{r_{0}}}=(\chi(t/T_{2}^{\prime})-\chi(t/T_{1}^{\prime}))\psi^{\prime}|_{\Sigma_{r_{0}}}$
$\displaystyle n_{\Sigma_{r}}\cdot
v|_{\Sigma_{r_{0}}}=(\chi(t/T_{2}^{\prime})-\chi(t/T_{1}^{\prime}))n_{\Sigma_{r}}\cdot\psi^{\prime}|_{\Sigma_{r_{0}}}.$
(3.41)

Note that the initial data $v|_{\Sigma_{r}}$ and $n_{\Sigma_{r}}\cdot
v|_{\Sigma_{r}}$ are supported only on the compact set

$\Sigma_{r_{0}}^{(T_{1}/2,T_{2})}\cup\overline{\Sigma}_{r_{0}}^{(T_{1}/2,T_{2})},$

i.e. $v=0$, $n_{\Sigma_{r}}\cdot v=0$ on $\Sigma_{r_{0}}$ for all $|t|\leq
T_{1}^{\prime}/2$ and for all $|t|\geq T_{2}^{\prime}$. We apply the
nondegenerate energy estimate of Proposition 3.14 to $v$ on the domain
$\mathcal{R}_{\tau,T_{1}}^{(r_{0})}$ near $\mathcal{C}^{+}$ so that for
$\tau<T_{1}$, $v$ satisfies the estimate

$\int_{C_{\tau}^{c}}\>^{*}J^{N}[v]+\int_{\mathcal{C}_{\tau,T_{1}}^{+}}\>^{*}J^{N}[v]\leq
e^{c(T_{1}-\tau)}\int_{C_{T_{1}}^{c}}\>^{*}J^{N}[v]+\int_{\Sigma_{r_{0}}^{(\tau,\tau_{1})}}J^{N}[v]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\\\
+c\int_{\tau}^{T_{1}}e^{c(s-\tau)}\Big{(}\int_{\Sigma_{r_{0}}^{(s,T_{1})}}J^{N}[v]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}\textnormal{d}s.$
(3.42)

Since $v=\psi_{T_{2}}$ on $C_{T_{1}}^{c}$, we have

$\int_{C_{T_{1}}^{c}}\>^{*}J^{N}[v]=\int_{C_{T_{1}}^{c}}\>^{*}J^{N}[\psi_{T_{2}}].$

We then apply Proposition 3.14 again, this time to $\psi_{T_{2}}$ on the
domain $\mathcal{R}_{T_{1},T_{2}}^{(r_{0})}$ so that

$\int_{C_{T_{1}}^{c}}\>^{*}J^{N}[\psi_{T_{2}}]+\int_{\mathcal{C}_{T_{1},T_{2}}^{+}}\>^{*}J^{N}[\psi_{T_{2}}]\leq\int_{\Sigma_{r_{0}}^{(T_{1},T_{2})}}J^{N}[\psi_{T_{2}}]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\\\
+c\int_{T_{1}}^{T_{2}}e^{c(s-T_{1})}\Big{(}\int_{\Sigma_{r_{0}}^{(s,T_{2})}}J^{N}[\psi_{T_{2}}]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}\textnormal{d}s,$
(3.43)

where we used that $\psi_{T_{2}}=0$ on $C_{T_{2}}^{c}$, and so the
corresponding energy flux through $C_{T_{2}}^{c}$ vanishes. Also, note that
the energy flux of $\psi_{T_{1}}=0$ through $\mathcal{C}_{T_{1}}^{+}$
vanishes, and therefore

$\int_{\mathcal{C}_{T_{1},T_{2}}^{+}}\>^{*}J^{N}[\psi_{T_{2}}]=\int_{\mathcal{C}_{T_{1},T_{2}}^{+}}\>^{*}J^{N}[v]=\int_{\mathcal{C}_{T_{1}}^{+}}\>^{*}J^{N}[v].$
(3.44)

Because $M$ and $N$ are both strictly timelike on $\Sigma_{r_{0}}$, it follows
that $J^{N}\cdot n\sim J^{M}\cdot n$ on $\Sigma_{r_{0}}$, and so

$\int_{\Sigma_{r_{0}}^{(T_{1},T_{2})}}J^{N}\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\sim_{\Lambda,m,r_{0}}\int_{\Sigma_{r_{0}}^{(T_{1},T_{2})}}J^{M}\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}.$

Combining this with (3.42), (3.43) and (3.44), we get

$\int_{C_{\tau}^{c}}\>^{*}J^{N}[v]+\int_{\mathcal{C}_{\tau}^{+}}\>^{*}J^{N}[v]\leq\|v\|_{M,\Sigma_{r_{0}}^{(\tau,T_{1})}}^{2}+c\int_{\tau}^{T_{1}}e^{c(s-\tau)}\|v\|_{M,\Sigma_{r_{0}}^{(s,T_{1})}}^{2}\textnormal{d}s\\\
+e^{c(T_{1}-\tau)}\Big{(}\|\psi_{T_{2}}\|_{M,\Sigma_{r_{0}}^{(T_{1},T_{2})}}^{2}+c\int_{T_{1}}^{T_{2}}e^{c(s-T_{1})}\|\psi_{T_{2}}\|_{M,\Sigma_{r_{0}}^{(s,T_{2})}}^{2}\textnormal{d}s\Big{)}.$
(3.45)

We will appeal to Proposition 3.16 to show that the right hand side of (3.45)
vanishes as $T_{1},T_{2}\rightarrow\infty$. Since $v=0$ and
$n_{\Sigma_{r_{0}}}\cdot v=0$ on $\Sigma_{r_{0}}$ for all $t\leq
T_{1}^{\prime}/2$, along with the fact that
$\Sigma_{r_{0}}^{(\tau,T_{1})}\subset\Sigma_{r_{0}}^{\tau}$, the energy flux
of $v$ through $\Sigma_{r_{0}}^{(\tau,T_{1})}$ can be estimated like

$\|v\|_{M,\Sigma_{r_{0}}^{(\tau,T_{1})}}^{2}\lesssim\begin{cases}\|\psi\|_{M,\Sigma_{r_{0}}^{T_{1}/2}}^{2}+\|\psi\|_{L^{2}(\Sigma_{r_{0}}^{T_{1}/2})}^{2},\quad&\tau\leq
T_{1}/2.\\\
\|\psi\|_{M,\Sigma_{r_{0}}^{\tau}}^{2}+\|\psi\|_{L^{2}(\Sigma_{r_{0}}^{\tau})}^{2},\quad&T_{1}/2\leq\tau\leq
T_{1}.\end{cases}$

Similarly we can bound

$\|\psi_{T_{2}}\|_{M,\Sigma_{r_{0}}^{(\tau,T_{2})}}^{2}\lesssim\|\psi\|_{M,\Sigma_{r_{0}}^{\tau}}^{2}+\|\psi\|_{L^{2}(\Sigma_{r_{0}}^{\tau})}^{2}.$

We now use the result from Proposition 3.16, which implies the exponential
decay of the energy and $L^{2}$-norm of $\psi$ along $\Sigma_{r_{0}}$. This
implies the bounds

$\|v\|_{M,\Sigma_{r_{0}}^{(\tau,T_{1})}}^{2}\lesssim\begin{cases}e^{-\beta
T_{1}},\quad&\tau\leq T_{1}/2.\\\ e^{-2\beta\tau},\quad&T_{1}/2\leq\tau\leq
T_{1}.\end{cases}.$
$\|\psi_{T_{2}}\|_{M,\Sigma_{r_{0}}^{(\tau,T_{2})}}^{2}\lesssim
e^{-2\beta\tau}.$

By these estimates, it follows from (3.45) that

$\int_{C_{\tau}^{c}}\>^{*}J^{N}[v]+\int_{\mathcal{C}_{\tau}^{+}}\>^{*}J^{N}[v]\lesssim
e^{-\beta T_{1}}+c\int_{\tau}^{T_{1}/2}e^{c(s-\tau)}e^{-\beta
T_{1}}\textnormal{d}s+c\int_{T_{1}/2}^{T_{1}}e^{c(s-\tau)}e^{-2\beta
s}\textnormal{d}s\\\ +e^{c(T_{1}-\tau)}\Big{(}e^{-2\beta
T_{1}}+c\int_{T_{1}}^{T_{2}}e^{c(s-T_{1})}e^{-2\beta
s}\textnormal{d}s\Big{)}.$ (3.46)

If we assume $\beta>c/2$, then we have

$\int_{T_{1}/2}^{T_{1}}e^{c(s-\tau)}e^{-2\beta
s}\textnormal{d}s=-\frac{e^{-c\tau}}{2\beta-c}\Big{(}e^{-(2\beta-c)T_{1}}-e^{-(2\beta-c)T_{1}/2}\Big{)}\leq\frac{e^{-c\tau}}{2\beta-c}e^{-\beta
T_{1}/2}.$ (3.47)

Bounding the other terms similarly we find that

$\int_{C_{\tau}^{c}}\>^{*}J^{N}[v]+\int_{\mathcal{C}_{\tau}^{+}}\>^{*}J^{N}[v]\lesssim(1+e^{-c\tau})e^{-\beta
T_{1}/2}.$

This holds for all $T_{2}>T_{1}$, Repeating the same argument near
$\overline{\mathcal{C}}^{+}$, it follows that

$\int_{C_{\tau}^{c}}\>^{*}J^{N}[v]+\int_{\mathcal{C}_{\tau}^{+}}\>^{*}J^{N}[v]+\int_{\overline{C_{\tau}^{c}}}\>^{*}J^{N}[v]+\int_{\overline{\mathcal{C}}_{\tau}^{+}}\>^{*}J^{N}[v]\lesssim(1+e^{-c\tau})e^{-\beta
T_{1}/2}.$

which vanishes as $T_{1},T_{2}\rightarrow\infty$. Hence there exists a
function $\psi$ on
$\mathcal{R}^{+}\cup\mathcal{C}^{+}\cup\overline{\mathcal{C}}^{+}$ such that
for each $\tau\in(-\infty,\infty)$, we have

$\displaystyle\lim_{T\rightarrow\infty}\|\psi-\psi_{T}\|_{N,C_{\tau}^{c}}=\lim_{T\rightarrow\infty}\|\psi-\psi_{T}\|_{N,\overline{C_{\tau}^{c}}}=0,$
$\displaystyle\lim_{T\rightarrow\infty}\|\psi-\psi_{T}\|_{N,\mathcal{C}_{\tau}^{+}}=\lim_{T\rightarrow\infty}\|\psi-\psi_{T}\|_{N,\overline{\mathcal{C}}_{\tau}^{+}}=0.$

This gives uniform boundedness of the energy of $\psi$ outside a small
neighbourhood of the bifurcation sphere
$\mathcal{C}^{+}\cap\overline{\mathcal{C}}^{+}$; this is simply because the
vectorfield $N$ blows up there. With any regular, strictly timelike
vectorfield $N^{\prime}$, one can show $\psi$ remains bounded at the
bifurcation sphere through a standard Grönwall-type inequality on a compact
set. Without changing notation, we modify $N$ to be regular and timelike in a
neighbourhood of $\mathcal{C}^{+}\cap\overline{\mathcal{C}}^{+}$, and set

$E^{N}[\psi]=\int_{\mathcal{C}^{+}\cap\overline{\mathcal{C}}^{+}}\>^{*}J^{N}[\psi].$

Then we have $E^{N}[\psi]\lesssim\|\psi\|_{\partial_{r},\Sigma_{r_{0}}}^{2}$,
hence $\psi$ is uniformly bounded on
$\mathcal{C}^{+}\cup\overline{\mathcal{C}}^{+}$. Moreover we obtain (1.12), as
by (1.7) we have
$\|\psi\|_{\partial_{r},\Sigma_{r_{0}}}^{2}\lesssim\|\psi_{0}\|_{H^{4}(\mathbb{R}\times\mathbb{S}^{2})}^{2}+\|\psi_{3}\|_{H^{2}(\mathbb{R}\times\mathbb{S}^{2})}^{2}$.
This proves Theorem 1.10. ∎

##  4 Asymptotics of forward solutions to the wave equation on Schwarzschild-
de Sitter

In this section we prove the main asymptotics result Theorem 1.1 for solutions
to the linear wave equation on Schwarzschild-de Sitter. We emphasise the
absence of an $O(r^{-1})$ term in this expansion, and additionally the absence
of log terms. In particular we prove the existence of the limits (1.4) in
Section 4.5.

We establish these asymptotics by employing several higher-order energy
estimates, which we also prove in this section. These estimates are obtained
by commuting the wave equation with suitably chosen vectorfield commutators.
These commutators are not the Killing vectorfields $T,\Omega_{i}$ that commute
with the wave operator $\square_{g}$, but are instead proportional to the
radial coordinate vectorfield $\frac{\partial}{\partial r}$. The main energy
estimate, found in Theorem 1.14, is proved in Section 4.3.

###  4.1 Higher order energy estimates with vectorfield commutators

In order to prove the subsequent energy estimates throughout this section, we
will be returning to a common argument for generating energy estimates on
$\mathcal{R}^{+}$. We apply the divergence theorem to the energy current
$J^{\partial_{r}}[X\psi]$ on the spacetime region bound in the past by
$\Sigma_{r_{1}}$, and to the future by $\Sigma_{r_{2}}$, where $r_{2}>r_{1}\gg
r_{\mathcal{C}}$, which yields the inequality

$\int_{\Sigma_{r_{2}}}J^{\partial_{r}}[X\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}+\int_{r_{1}}^{r_{2}}\Big{(}\int_{\Sigma_{r}}\phi\nabla\cdot
J^{\partial_{r}}[X\psi]\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}\textnormal{d}r\leq\int_{\Sigma_{r_{1}}}J^{\partial_{r}}[X\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}},$ (4.1)

where we applied the coarea formula to the bulk term. The strategy for proving
the higher-order energy estimates in this section is to identify vectorfield
commutators $X$ for which the divergence of $J^{\partial_{r}}[X\psi]$
possesses key sign and decay properties. Recall that by the product rule we
have

$\nabla\cdot
J^{\partial_{r}}[X\psi]=K^{\partial_{r}}[X\psi]+\partial_{r}(X\psi)\square_{g}(X\psi)$
(4.2)

It was computed in [27] that

$K^{\partial_{r}}[X\psi]=\Big{(}\frac{2\Lambda}{3}r-\frac{1}{r}+\frac{m}{r}\Big{)}(\partial_{r}(X\psi))^{2}+\Big{(}\frac{\Lambda}{3}r-\frac{m}{r}\Big{)}\phi^{4}(\partial_{t}(X\psi))^{2}.$
(4.3)

One can see from this that (to leading order in $r$), the energy current
$K^{\partial_{r}}$ is nonnegative. Indeed this nonnegativity is what implies
the global redshift estimate of [27]. For higher-order estimates, however, one
must consider contributions from the second term
$\partial_{r}(X\psi)\square_{g}(X\psi)$. We will restrict our attention to
commutation vectorfields of the form $X=f(r)\partial_{r}$. We begin by
deriving an identity for $\square_{g}(X\psi)$.

######  Lemma 4.1.

Let $f\in C^{2}(r_{\mathcal{C}},\infty)$ with $f(r)>0$ for all
$r>r_{\mathcal{C}}$, and let $X=f(r)\partial_{r}$. Then

$\square_{g}(X\psi)=X(\square_{g}\psi)+\frac{f^{2}}{r^{2}}\partial_{r}\Big{(}\frac{r^{2}\phi^{-2}}{f^{2}}\Big{)}\partial_{r}(X\psi)+\frac{f^{2}}{r^{2}}\partial_{r}^{2}\Big{(}\frac{r^{2}\phi^{-2}}{f}\Big{)}\partial_{r}\psi\\\
+2f\phi^{4}\Big{(}\frac{1}{r}-\frac{3m}{r}\Big{)}\partial_{t}^{2}\psi+\frac{2f}{r}\square_{g}\psi.$
(4.4)

###### Proof.

By (3.2), we can write

$[\square_{g},X]\psi=-\frac{1}{r^{2}}\partial_{r}\Big{(}r^{2}\phi^{-2}\partial_{r}(f\partial_{r}\psi)\Big{)}+f\partial_{r}\Big{(}\frac{1}{r^{2}}\partial_{r}(r^{2}\phi^{-2}\partial_{r}\psi)\Big{)}+[\overline{\Delta},X]\psi.$

Computing the final term, we have

$\displaystyle[\overline{\Delta},X]\psi$
$\displaystyle=\phi^{2}\partial_{t}^{2}(f\partial_{r}\psi)-f\partial_{r}(\phi^{2}\partial_{t}^{2}\psi)+\frac{1}{r^{2}}\Delta_{\mathbb{S}^{2}}(f\partial_{r}\psi)-f\partial_{r}\Big{(}\frac{1}{r^{2}}\Delta_{\mathbb{S}^{2}}\psi\Big{)}$
$\displaystyle=-f\partial_{r}(\phi^{2})\partial_{t}^{2}\psi-f\partial_{r}\Big{(}\frac{1}{r^{2}}\Big{)}\Delta_{\mathbb{S}^{2}}\psi$
$\displaystyle=\frac{2f}{r}\overline{\Delta}\psi+2\Big{(}\frac{1}{r}-\frac{3m}{r}\Big{)}\phi^{4}\partial_{t}^{2}\psi,$

where we used in the last line the fact that

$\displaystyle\partial_{r}(\phi^{2})$
$\displaystyle=-\Big{(}\frac{2\Lambda}{3}r-\frac{2m}{r^{2}}\Big{)}\phi^{4}$
$\displaystyle=-\frac{2}{r}\phi^{2}-2\Big{(}\frac{1}{r}-\frac{3m}{r^{3}}\Big{)}\phi^{4}.$

Substituting (3.2), it follows that

$[\square_{g},X]\psi=-\frac{1}{r^{2}}\partial_{r}\Big{(}r^{2}\phi^{-2}\partial_{r}(f\partial_{r}\psi)\Big{)}+f\partial_{r}\Big{(}\frac{1}{r^{2}}\partial_{r}(r^{2}\phi^{-2}\partial_{r}\psi)\Big{)}\\\
+\frac{2f}{r^{3}}\partial_{r}\Big{(}r^{2}\phi^{-2}\partial_{r}\psi\Big{)}+\frac{2f}{r}\square_{g}\psi+2\Big{(}\frac{1}{r}-\frac{3m}{r}\Big{)}\phi^{4}\partial_{t}^{2}\psi$

As for the radial derivatives, we have

$-\frac{1}{r^{2}}\partial_{r}\Big{(}r^{2}\phi^{-2}\partial_{r}(f\partial_{r}\psi)\Big{)}+f\partial_{r}\Big{(}\frac{1}{r^{2}}\partial_{r}(r^{2}\phi^{-2}\partial_{r}\psi)\Big{)}+\frac{2f}{r^{3}}\partial_{r}\Big{(}r^{2}\phi^{-2}\partial_{r}\psi\Big{)}\\\
=\Big{[}\frac{2\phi^{-2}}{r}f+\partial_{r}(\phi^{-2})f-2\phi^{-2}\partial_{r}f\Big{]}\partial_{r}^{2}\psi\\\
+\Big{[}\partial_{r}^{2}(\phi^{-2})f+\frac{4\partial_{r}(\phi^{-2})}{r}f+\frac{2\phi^{-2}}{r^{2}}f\\\
\quad\quad\quad\quad-\partial_{r}(\phi^{-2})\partial_{r}f-\frac{2\phi^{-2}}{r}\partial_{r}f-\phi^{-2}\partial_{r}^{2}f\Big{]}\partial_{r}\psi.$

By using that

$\partial_{r}^{2}\psi=\frac{1}{f}\partial_{r}(X\psi)-\frac{\partial_{r}f}{f}\partial_{r}\psi,$

it follows that

$\square_{g}(f\partial_{r}\psi)=f\partial_{r}(\square_{g}\psi)+\frac{2f}{r}\square_{g}\psi+2\Big{(}\frac{1}{r}-\frac{3m}{r}\Big{)}\phi^{4}\partial_{t}^{2}\psi\\\
+\Big{[}\frac{2\phi^{-2}}{r}+\partial_{r}(\phi^{-2})-2\phi^{-2}\frac{\partial_{r}f}{f}\Big{]}\partial_{r}(f\partial_{r}\psi)\\\
+\Big{[}\partial_{r}^{2}(\phi^{-2})f+\frac{2\partial_{r}(\phi^{-2})}{r}f+\frac{2\phi^{-2}}{r^{2}}f\\\
\quad\quad\quad-2\partial_{r}(\phi^{-2})\partial_{r}f-\frac{2\phi^{-2}}{r}\partial_{r}f-2\phi^{-2}\frac{(\partial_{r}f)^{2}}{f}-\phi^{-2}\partial_{r}^{2}f\Big{]}\partial_{r}\psi.\\\
$

A brief computation shows that the coefficients of the radial derivatives are
given by

$\frac{f^{2}}{r^{2}}\partial_{r}\Big{(}\frac{r^{2}\phi^{-2}}{f^{2}}\Big{)}=\Big{[}\frac{2\phi^{-2}}{r}+\partial_{r}(\phi^{-2})-2\phi^{-2}\frac{\partial_{r}f}{f}\Big{]}.$

and

$\frac{f^{2}}{r^{2}}\partial_{r}^{2}\Big{(}\frac{r^{2}\phi^{-2}}{f}\Big{)}=\Big{[}\partial_{r}^{2}(\phi^{-2})f+\frac{2\partial_{r}(\phi^{-2})}{r}f+\frac{2\phi^{-2}}{r^{2}}f-2\partial_{r}(\phi^{-2})\partial_{r}f\\\
-\frac{2\phi^{-2}}{r}\partial_{r}f-2\phi^{-2}\frac{(\partial_{r}f)^{2}}{f}-\phi^{-2}\partial_{r}^{2}f\Big{]}.$

i.e. they match those in (3.2). ∎

We now focus on the $\partial_{r}\psi$ term in (4.4). In the energy estimates
that follow, we will see that it is crucial that this term vanishes at much
higher order. There is a family of nonzero functions $f(r)$ for which the
coefficient of the $\partial_{r}\psi$ term vanishes completely, given by

$f(r)=\frac{r^{2}\phi^{-2}}{c_{1}+c_{2}r},$

for parameters $(c_{1},c_{2})\in\mathbb{R}^{2}\backslash\\{(0,0)\\}$.
Consequently we define the vector fields

$X_{s}=r\phi^{-2},\quad X_{w}=r^{2}\phi^{-2},$

which correspond with choosing $(c_{1},c_{2})=(0,1)$ and $(c_{1},c_{2})=(1,0)$
respectively. Throughout this section, we will frequently be considering only
the leading-order behaviour of terms that appear in the energy estimates, and
so we remind the reader that $\phi\sim r^{-1}$ for large $r$, and therefore
$X_{s}\sim r^{3}\partial_{r}$, $X_{w}\sim r^{4}\partial_{r}$. The subscripts
of $X_{s},X_{w}$ refer to the fact that $X_{s}$ will be used to prove a sharp
energy estimate, while $X_{w}$ will be used to prove an auxillary estimate
that yields weaker decay of certain second derivatives present in both energy
fluxes, but is nevertheless necessary to prove the sharp estimate.

###  4.2 First-order weighted energy estimates

We now state a weighted energy estimate that captures the first-order
behaviour of forward solutions to the wave equation.

######  Theorem 4.2.

Fix $r_{0}>r_{\mathcal{C}}$, and let $\psi$ be a solution to (1.1) on
$\mathcal{R}^{+}$. For $r_{2}>r_{1}\geq r_{0}$, the following higher-order
estimate of $\psi$ holds:

$\int_{\Sigma_{r_{2}}}J^{\partial_{r}}[X_{s}\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\lesssim_{\Lambda,m,r_{0}}\int_{\Sigma_{r_{1}}}J^{\partial_{r}}[X_{s}\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}+\int_{\Sigma_{r_{1}}}J^{\partial_{r}}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}.$ (4.5)

###### Proof.

We will denote the energy flux of $\psi$ through the hypersurface $\Sigma_{r}$
by

$\int_{\Sigma_{r}}J^{\partial_{r}}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}=E[\psi](r),$

and so we rewrite (4.1) as

$E[X_{s}\psi](r_{2})+\int_{r_{1}}^{r_{2}}\Big{(}\int_{\Sigma_{r}}\phi\nabla\cdot
J^{\partial_{r}}[X_{s}\psi]\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}\textnormal{d}r\leq
E[X_{s}\psi](r_{1}).$ (4.6)

for all $r_{2}>r_{1}>r_{\mathcal{C}}$. We focus on the bulk term. With the
choice $f(r)=r\phi^{-2}$, we compute some of the the terms that appear in the
commutation identity (4.4):

$\displaystyle\frac{f^{2}}{r^{2}}\partial_{r}\Big{(}\frac{r^{2}\phi^{-2}}{f^{2}}\Big{)}=-\Big{(}\frac{2\Lambda}{3}r-\frac{2m}{r^{2}}\Big{)}.$
$\displaystyle\frac{f^{2}}{r^{2}}\partial_{r}^{2}\Big{(}\frac{r^{2}\phi^{-2}}{f}\Big{)}=0.$
$\displaystyle
2f\phi^{4}\Big{(}\frac{1}{r}-\frac{3m}{r^{2}}\Big{)}=2\phi^{2}\Big{(}1-\frac{3m}{r}\Big{)}.$

It follows from Lemma 4.1 that

$\phi\>\partial_{r}(X_{s}\psi)\square_{g}(X_{s}\psi)=-\phi\Big{(}\frac{2\Lambda}{3}r-\frac{2m}{r^{2}}\Big{)}(\partial_{r}(X_{s}\psi))^{2}+2\phi^{3}\Big{(}1-\frac{3m}{r}\Big{)}\partial_{t}^{2}\psi\partial_{r}(X_{s}\psi).$
(4.7)

Moreover, from (4.3) we have

$\phi\>K^{\partial_{r}}[X_{s}\psi]=\phi\Big{(}\frac{2\Lambda}{3}r-\frac{1}{r}+\frac{m}{r}\Big{)}(\partial_{r}(X_{s}\psi))^{2}+\phi^{5}\Big{(}\frac{\Lambda}{3}r-\frac{m}{r}\Big{)}(\partial_{t}(X\psi))^{2}.$
(4.8)

We point out that to leading order in $r$, the $(\partial_{r}(X_{s}\psi))^{2}$
term on the right hand side of (4.7) and (4.8) cancel, and there is no term
that depends on $\partial_{r}\psi$. Summing these yields $\phi\nabla\cdot
J^{\partial_{r}}[\psi]$, so that

$\phi\>\nabla\cdot
J^{\partial_{r}}[X_{s}\psi]=-\phi\Big{(}\frac{1}{r}-\frac{3m}{r^{2}}\Big{)}(\partial_{r}(X_{s}\psi))^{2}\\\
+\phi^{5}\Big{(}\frac{1}{r}-\frac{3m}{r^{2}}\Big{)}(\partial_{t}(X_{s}\psi))^{2}+2\phi^{3}\Big{(}1-\frac{3m}{r}\Big{)}\partial_{t}^{2}\psi\partial_{r}(X_{s}\psi).$
(4.9)

We now wish to bound each term on the right hand side of (4.9) by the energies
$J^{\partial_{r}}[\psi]\cdot n$ and $J^{\partial_{r}}[X_{s}\psi]\cdot n$. For
reference, the energy flux density through $\Sigma_{r}$ with respect to the
vector field $\partial/\partial_{r}$ of an arbitrary function $\varphi$ is

$J^{\partial_{r}}[\varphi]\cdot
n=\frac{1}{2}\Big{(}\phi^{-1}(\partial_{r}\varphi)^{2}+\phi^{3}(\partial_{t}\varphi)^{2}+\phi|\not{\nabla}\varphi|^{2}\Big{)}.$

Therefore we may bound simply

$\displaystyle\Big{|}\phi\Big{(}\frac{1}{r}-\frac{3m}{r^{2}}\Big{)}(\partial_{r}(X_{s}\psi))^{2}\Big{|}\leq
O\big{(}\frac{1}{r^{3}}\big{)}J^{\partial_{r}}[X_{s}\psi]\cdot n.$
$\displaystyle\Big{|}\phi^{5}\Big{(}\frac{1}{r}-\frac{3m}{r^{2}}\Big{)}(\partial_{t}(X_{s}\psi))^{2}\Big{|}\leq
O\big{(}\frac{1}{r^{3}}\big{)}J^{\partial_{r}}[X_{s}\psi]\cdot n.$

The final term in (4.9) is more difficult to bound, as terms like
$\partial_{t}^{2}\psi$ are not controlled pointwise by the energy densities
$J^{\partial_{r}}[\psi]$, $J^{\partial_{r}}[X_{s}\psi]$. An application of
Cauchy-Schwartz implies

$\Big{|}2\phi^{3}\Big{(}1-\frac{3m}{r}\Big{)}\partial_{t}^{2}\psi\>\partial_{r}(X_{s}\psi)\Big{|}\leq
O\Big{(}\frac{1}{r^{2}}\Big{)}J^{\partial_{r}}[X_{s}\psi]\cdot
n+O\Big{(}\frac{1}{r^{5}}\Big{)}(\partial_{t}^{2}\psi)^{2}.$

It follows that

$\Big{|}\phi\nabla\cdot J^{\partial_{r}}[X_{s}\psi]\Big{|}\leq
O\big{(}\frac{1}{r^{2}}\big{)}J^{\partial_{r}}[X_{s}\psi]\cdot
n_{\Sigma_{r}}+O\big{(}\frac{1}{r^{5}}\big{)}(\partial_{t}^{2}\psi)^{2},$

and so by (4.6) we have

$E[X_{s}\psi](r_{2})\leq
E[X_{s}\psi](r_{1})+\int_{r_{1}}^{r_{2}}O\big{(}\frac{1}{r^{2}}\big{)}E[X_{s}\psi](r)\textnormal{d}r\\\
+\int_{r_{1}}^{r_{2}}\Big{(}O\big{(}\frac{1}{r^{5}}\big{)}\int_{\Sigma_{r}}(\partial_{t}^{2}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}\textnormal{d}r.$
(4.10)

To control the integral of $(\partial_{t}^{2}\psi)^{2}$, we employ the
elliptic estimate of Corollary A.2 from Appendix A, which implies

$O\big{(}\frac{1}{r^{5}}\big{)}\int_{\Sigma_{r}}(\partial_{t}^{2}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}\leq
O\big{(}\frac{1}{r}\big{)}\int_{\Sigma_{r}}(\overline{\Delta}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}+O\big{(}\frac{1}{r^{5}}\big{)}\int_{\Sigma_{r}}\sum_{i=1}^{3}(\partial_{i}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}.$
(4.11)

Since $\psi$ satisfies the wave equation, we have by (3.2) that

$\overline{\Delta}\psi=\square_{g}\psi+\frac{1}{r^{2}}\partial_{r}(r^{2}\phi^{-2}\partial_{r}\psi)=\frac{1}{r^{2}}\partial_{r}(X_{w}\psi),$

and therefore

$\int_{\Sigma_{r}}(\overline{\Delta}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}=\frac{1}{r^{4}}\int_{\Sigma_{r}}(\partial_{r}(X_{w}\psi))^{2}\textnormal{d}\mu_{\overline{g}_{r}}\leq
O\Big{(}\frac{1}{r^{5}}\Big{)}E[X_{w}\psi](r).$ (4.12)

Note that we still cannot bound this solely by the higher-order energy flux
$E[X_{s}\psi]$. Moreover, if we were to naively bound it by both
$E[X_{s}\psi]$ and the standard energy flux $E[\psi]$ via the product rule,
the resulting terms would possess weights in $r$ that are too large to close
the argument through a Grönwall-type inequality. Instead, this is where we
employ a standalone, auxillary energy estimate that arises from the
alternative choice of vectorfield commutator $X_{w}$. We claim that for any
solution $\psi$ to the wave equation on $\mathcal{R}^{+}$, the following
estimate holds for all $r_{2}>r_{1}\geq r_{0}$:

$\frac{1}{r_{2}^{4}}\int_{\Sigma_{r_{2}}}J^{\partial_{r}}[X_{w}\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\lesssim_{\Lambda,m,r_{0}}\frac{1}{r_{1}^{4}}\int_{\Sigma_{r_{1}}}J^{\partial_{r}}[X_{w}\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}+\int_{\Sigma_{r_{1}}}J^{\partial_{r}}[\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}.$ (4.13)

We will delay the proof of this estimate until later in this section.
Combining the elliptic estimate (4.11), (4.12), as well as the auxillary
energy estimate (4.13), we bound

$\displaystyle
O\big{(}\frac{1}{r^{5}}\big{)}\int_{\Sigma_{r}}(\partial_{t}^{2}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}$
$\displaystyle\leq
O\big{(}\frac{1}{r^{6}}\big{)}E[X_{w}](r)+O\big{(}\frac{1}{r^{2}}\big{)}E[\psi](r)$
$\displaystyle=O\big{(}\frac{1}{r^{2}}\big{)}\Big{[}\frac{1}{r^{4}}E[X_{w}\psi](r)+E[\psi](r)\Big{]}$
$\displaystyle\lesssim
O\big{(}\frac{1}{r^{2}}\big{)}\Big{[}E[X_{w}\psi](r_{1})+E[\psi](r_{1})\Big{]},$

and so by (4.10) we have

$E[X_{s}\psi](r_{2})\lesssim
E[X_{s}\psi](r_{1})+\int_{r_{1}}^{r_{2}}O\big{(}\frac{1}{r^{2}}\big{)}E[X_{s}\psi](r)\textnormal{d}r\\\
+\Big{(}\frac{1}{r_{1}^{4}}E[X_{w}\psi](r_{1})+E[\psi](r_{1})\Big{)}\int_{r_{1}}^{r_{2}}\frac{1}{r^{2}}\textnormal{d}r.$

As $r^{-2}$ is integrable, it follows that

$E[X_{s}\psi](r_{2})\lesssim
E[X_{s}\psi](r_{1})+E[X_{w}\psi](r_{1})+E[\psi](r_{1})+\int_{r_{1}}^{r_{2}}O\big{(}\frac{1}{r^{2}}\big{)}E[X_{s}\psi](r)\textnormal{d}r.$

An application of the standard Grönwall inequality yields

$\displaystyle E[X_{s}\psi](r_{2})$
$\displaystyle\lesssim_{r_{0}}\Big{[}E[X_{s}\psi](r_{1})+E[X_{w}\psi](r_{1})+E[\psi](r_{1})\Big{]}\exp\Big{[}C\int_{r_{1}}^{r_{2}}\frac{1}{r^{2}}\textnormal{d}r\Big{]}$
$\displaystyle\lesssim E[X_{s}\psi](r_{1})+E[X_{w}\psi](r_{1})+E[\psi](r_{1})$

for all $r_{2}\geq r_{1}$. By the product rule we may simply bound the energy
fluxes through $\Sigma_{r_{1}}$ like

$E[X_{w}\psi](r_{1})\lesssim_{r_{0}}E[X_{s}\psi](r_{1})+E[\psi](r_{1}),$

which implies (4.5). ∎

######  Remark 4.3.

To highlight the associated weights in $r$ for terms that appear in the energy
estimate (4.5), we note that for large $r$ we have

$\int_{\Sigma_{r}}J^{\partial_{r}}[X_{s}\psi]\cdot
n\>\textnormal{d}\mu_{\overline{g}_{r}}\sim\int_{\mathbb{R}\times\mathbb{S}^{2}}r^{4}(\partial_{r}(r\phi^{-2}\partial_{r}\psi))^{2}+r^{6}(\partial_{t}\partial_{r}\psi)^{2}+r^{8}|\not{\nabla}\partial_{r}\psi|^{2}\textnormal{d}t\textnormal{d}\mu_{\mathring{\gamma}}.$
(4.14)

As was shown in [27], solutions to the wave equation on the expanding region
of Schwarzschild-de Sitter are bounded with respect to the energy (1.2). The
boundedness of the weighted higher-order energy already eliminates the
possibility of an $O(r^{-1})$-weighted term being present in the asymptotic
expansion of solutions, as we have

$\displaystyle\int_{\Sigma_{r}}J^{\partial_{r}}\Big{[}X_{s}\Big{(}\psi_{0}+\frac{\psi_{1}}{r}\Big{)}\Big{]}\cdot
n\>\textnormal{d}\mu_{\overline{g}_{r}}$
$\displaystyle=\int_{\Sigma_{r}}J^{\partial_{r}}\Big{[}X_{s}\Big{(}\frac{\psi_{1}}{r}\Big{)}\Big{]}\cdot
n\>\textnormal{d}\mu_{\overline{g}_{r}}$
$\displaystyle\sim\int_{\mathbb{R}\times\mathbb{S}^{2}}r^{6}\psi_{1}^{2}+r^{2}|\tilde{\nabla}\psi_{1}|^{2}\textnormal{d}t\textnormal{d}\mu_{\mathring{\gamma}},$

Therefore if a solution $\psi$ was like $\psi\sim\psi_{0}+\psi_{1}/r$ for
large $r$, this higher-order energy would not be bounded.

######  Remark 4.4.

A related energy estimate was

###### Proof of the auxillary energy estimate (4.13).

The proof of (4.13) is very similar to that of the sharp estimate in Theorem
4.2. We begin by noting by Lemma 4.1

$\phi\>\partial_{r}(X_{w}\psi)\square_{g}(X_{w}\psi)=-\phi\>\Big{(}\frac{4\Lambda}{3}r-\frac{2}{r}+\frac{2m}{r^{2}}\Big{)}(\partial_{r}(X_{w}\psi))^{2}+2\phi^{3}(r-3m)\partial_{t}^{2}\psi\partial_{r}(X_{w}\psi),$
(4.15)

and summing this with $\phi K^{\partial_{r}}[X_{w}\psi]$ gives by (4.3):

$\phi\>\nabla\cdot
J^{\partial_{r}}[X_{w}\psi]=-\phi\Big{(}\frac{2\Lambda}{3}r-\frac{1}{r}+\frac{m}{r^{2}}\Big{)}(\partial_{r}(X_{w}\psi))^{2}+\phi^{2}\Big{(}\frac{\Lambda}{3}r-\frac{m}{r}\Big{)}(\partial_{t}(X_{w}\psi))^{2}\\\
+2\phi^{3}(r-3m)\partial_{t}^{2}\psi\>\partial_{r}(X_{w}\psi).$ (4.16)

In contrast to (4.9) for the commutator $X_{s}$, the
$(\partial_{r}(X_{w}\psi))^{2}$ term in (4.16) does not vanish to leading
order, we will soon see the negative sign of this term leads to a loss of
decay in the resulting energy estimate. For now, we may bound each term like

$\displaystyle\Big{|}\phi\Big{(}\frac{2\Lambda}{3}r-\frac{1}{r}+\frac{m}{r^{2}}\Big{)}(\partial_{r}(X_{w}\psi))^{2}\Big{|}\leq\frac{4}{r}J^{\partial_{r}}[X_{w}\psi]\cdot
n+O\big{(}\frac{1}{r^{3}}\big{)}J^{\partial_{r}}[X_{w}\psi]\cdot n.$
$\displaystyle\Big{|}\phi^{5}\Big{(}\frac{1}{r}-\frac{3m}{r^{2}}\Big{)}(\partial_{t}(X_{w}\psi))^{2}\Big{|}\leq
O\big{(}\frac{1}{r^{3}}\big{)}J^{\partial_{r}}[X_{w}\psi]\cdot n.$
$\displaystyle\Big{|}2\phi^{3}(r-3m)\partial_{t}^{2}\psi\>\partial_{r}(X_{w}\psi)\Big{|}\leq
O\Big{(}\frac{1}{r^{2}}\Big{)}J^{\partial_{r}}[X_{w}\psi]\cdot
n+O\Big{(}\frac{1}{r^{3}}\Big{)}(\partial_{t}^{2}\psi)^{2}.$

It follows that

$\Big{|}\phi\nabla\cdot
J^{\partial_{r}}[X_{w}\psi]\Big{|}\leq\frac{4}{r}J^{\partial_{r}}[X_{w}\psi]\cdot
n+O\big{(}\frac{1}{r^{2}}\big{)}J^{\partial_{r}}[X_{w}\psi]\cdot
n+O\big{(}\frac{1}{r^{3}}\big{)}(\partial_{t}^{2}\psi)^{2},$

and so by (4.6) we have

$E[X_{w}\psi](r_{2})\leq
E[X_{w}\psi](r_{1})+\int_{r_{1}}^{r_{2}}\Big{[}\frac{4}{r}+O\big{(}\frac{1}{r^{2}}\big{)}\Big{]}E[X_{w}\psi](r)\textnormal{d}r\\\
+\int_{r_{1}}^{r_{2}}O\big{(}\frac{1}{r^{3}}\big{)}\int_{\Sigma_{r}}(\partial_{t}^{2}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}\textnormal{d}r.$

An application of the elliptic estimate from Corollary A.2 along with (4.12)
yields

$O\big{(}\frac{1}{r^{3}}\big{)}\int_{\Sigma_{r}}(\partial_{t}^{2}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}\lesssim
O\big{(}\frac{1}{r^{4}}\big{)}E[X_{w}\psi](r)+O\big{(}1)E[\psi](r),$

and therefore there exists a constant $C>0$ such that

$E[X_{w}\psi](r_{2})\leq
E[X_{w}\psi](r_{1})+C\int_{r_{1}}^{r_{2}}E[\psi](r)\textnormal{d}r+\int_{r_{1}}^{r_{2}}\beta(r)E[X_{w}\psi](r)\textnormal{d}r,$

where

$\beta(r)=\frac{4}{r}+\frac{C}{r^{2}}$

for some large constant $C$. We then use the fact from [27] that the natural
energy $E[\psi]$ is monotonically decreasing in $r$ to bound

$\int_{r_{1}}^{r_{2}}E[\psi](r)\textnormal{d}r\leq r_{2}E[\psi](r_{1}).$

If we set $\alpha(r)=E[X_{w}\psi](r_{1})+CrE[\psi](r_{1})$, then we have for
all $r_{2}\geq r_{1}$ that

$E[X_{w}\psi](r_{2})\leq\alpha(r_{2})+\int_{r_{1}}^{r_{2}}\beta(r)E[X_{w}\psi](r)\textnormal{d}r,$

which is a Grönwall-type inequality, and implies

$E[X_{w}\psi](r_{2})\leq\alpha(r_{2})+\int_{r_{1}}^{r_{2}}\alpha(s)\beta(s)\exp\Big{[}\int_{s}^{r_{2}}\beta(r)\textnormal{d}r\Big{]}\textnormal{d}s.$

Finally we bound

$\exp\Big{[}\int_{s}^{r_{2}}\frac{4}{r}+\frac{C}{r^{2}}\Big{]}\textnormal{d}r=\exp\Big{[}4\log(r_{2})-4\log(s)\Big{]}\exp\Big{[}\int_{s}^{r_{2}}\frac{C}{r^{2}}\textnormal{d}r\Big{]}\lesssim\frac{r_{2}^{4}}{s^{4}},$

which implies

$\displaystyle\int_{r_{1}}^{r_{2}}\alpha(s)\beta(s)\exp\Big{[}\int_{s}^{r_{2}}\beta(r)\textnormal{d}r\Big{]}\textnormal{d}s$
$\displaystyle\lesssim
r_{2}^{4}\int_{r_{1}}^{r_{2}}\frac{E[X_{w}\psi](r_{1})}{s^{5}}+\frac{E[\psi](r_{1})}{s^{4}}\textnormal{d}s$
$\displaystyle\lesssim
r_{2}^{4}\Big{[}E[X_{w}\psi](r_{1})+E[\psi](r_{1})\Big{]}.$

This implies (4.13) after dividing through by $r_{2}^{4}$. ∎

######  Remark 4.5.

The auxillary energy estimate (4.13) is weaker than that given in Theorem 4.2
in the following way. Writing the commuted energy from (4.13) more explicitly,
we have

$\int_{\Sigma_{r}}J^{\partial_{r}}[X_{w}\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\sim\int_{\mathbb{R}\times\mathbb{S}^{2}}r^{4}(\partial_{r}(r^{2}\phi^{-2}\partial_{r}\psi))^{2}+r^{8}|\tilde{\nabla}\partial_{r}\psi|^{2}\>\textnormal{d}t\>\textnormal{d}\mu_{\mathring{\gamma}},$

and so the auxillary estimate implies that the quantity

$\frac{1}{r^{4}}\int_{\Sigma_{r}}J^{\partial_{r}}[X_{w}\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\sim\int_{\mathbb{R}\times\mathbb{S}^{2}}(\partial_{r}(r^{2}\phi^{-2}\partial_{r}\psi))^{2}+r^{4}|\tilde{\nabla}\partial_{r}\psi|^{2}\>\textnormal{d}t\>\textnormal{d}\mu_{\mathring{\gamma}}$

is bounded. On the other hand, it follows from Theorem 4.2 that

$\int_{\Sigma_{r}}J^{\partial_{r}}[X_{s}\psi]\cdot
n_{\Sigma_{r}}\textnormal{d}\mu_{\overline{g}_{r}}\sim\int_{\mathbb{R}\times\mathbb{S}^{2}}(\partial_{r}(r\phi^{-2}\partial_{r}\psi))^{2}+r^{6}|\tilde{\nabla}\partial_{r}\psi|^{2}\>\textnormal{d}t\>\textnormal{d}\mu_{\mathring{\gamma}}$

is bounded, which yields stronger decay of $\partial_{r}\psi$ with respect to
the homogeneous Sobolev norm $\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})$.

###  4.3 Second-order energy estimate

We now prove the second-order estimate of Theorem 1.14. This estimate follows
from a second commutation with the vectorfield of the form
$Y=r\phi^{-1}\partial_{r}$, after already commuting once with $X_{s}$. The
commutator $Y$ once again creates cancellations in various terms that are
produced when computing $\square_{g}(Y(X_{s}\psi))$.

###### Proof of Theorem 1.14.

We begin with the inequality

$E[Y(X_{s}\psi)](r_{2})+\int_{r_{1}}^{r_{2}}\Big{(}\int_{\Sigma_{r}}\phi\nabla\cdot
J^{\partial_{r}}[Y(X_{s}\psi)]\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}\textnormal{d}r\leq
E[Y(X_{s}\psi)](r_{1}).$ (4.17)

By Lemma 4.1, for some scalar function $\varphi$ we have

$\square_{g}(Y\varphi)=\phi^{-2}\partial_{r}^{2}(r\phi^{-1})\partial_{r}\varphi+2\phi^{3}\Big{(}1-\frac{3m}{r}\Big{)}\partial_{t}^{2}\varphi+2\phi^{-1}\square_{g}\varphi+Y(\square_{g}\varphi).$

Setting $\varphi=X_{s}\psi$, it follows that

$\square_{g}(Y(X_{s}\psi))=\phi^{-2}\partial_{r}^{2}(r\phi^{-1})(X_{s}\psi)+2\phi^{3}\Big{(}1-\frac{3m}{r}\Big{)}\partial_{t}^{2}(X_{s}\psi)\\\
+2\phi^{-1}\Big{[}-\Big{(}\frac{2\Lambda}{3}r-\frac{2m}{r^{2}}\Big{)}\partial_{r}(X_{s}\psi)+2\phi^{2}\Big{(}1-\frac{3m}{r}\Big{)}\partial_{t}^{2}\psi\Big{]}\\\
+Y\Big{[}-\Big{(}\frac{2\Lambda}{3}r-\frac{2m}{r^{2}}\Big{)}\partial_{r}(X_{s}\psi)+2\phi^{2}\Big{(}1-\frac{3m}{r}\Big{)}\partial_{t}^{2}\psi\Big{]},$
(4.18)

where we used the commutation identity (4.7) for the commutator $X_{s}$. We
expand the final term giving

$Y\Big{[}-\Big{(}\frac{2\Lambda}{3}r-\frac{2m}{r^{2}}\Big{)}\partial_{r}(X_{s}\psi)+2\phi^{2}\Big{(}1-\frac{3m}{r}\Big{)}\partial_{t}^{2}\psi\Big{]}\\\
=-\phi^{-1}\Big{(}\frac{2\Lambda}{3}r^{2}-\frac{2m}{r}\Big{)}\partial_{r}^{2}(X_{s}\psi)-\phi^{-1}\Big{(}\frac{2\Lambda}{3}r-\frac{4m}{r^{2}}\Big{)}\partial_{r}(X_{s}\psi)\\\
+2\phi^{3}\Big{(}1-\frac{3m}{r}\Big{)}\partial_{t}^{2}(X_{s}\psi)+2r\phi^{-1}\partial_{r}\Big{[}\phi^{2}\Big{(}1-\frac{3m}{r}\Big{)}\Big{]}\partial_{t}^{2}\psi.$

Since $Y=r\phi^{-1}\partial_{r}$, we have

$\partial_{r}^{2}(X_{s}\psi)=\frac{\phi}{r}\partial_{r}(Y(X_{s}\psi))-\frac{\phi}{r}\partial_{r}(r\phi^{-1})\partial_{r}(X_{s}\psi).$

It follows from (4.18) that

$\square_{g}(Y(X_{s}\psi))=-\Big{(}\frac{2\Lambda}{3}r-\frac{2m}{r^{2}}\Big{)}\partial_{r}(Y(X_{s}\psi))+4\phi^{3}\Big{(}1-\frac{3m}{r}\Big{)}\partial_{t}^{2}(X_{s}\psi)\\\
+\Big{[}2r\phi^{-1}\partial_{r}\Big{[}\phi^{2}\Big{(}1-\frac{3m}{r}\Big{)}\Big{]}+2\phi\Big{(}1-\frac{3m}{r}\Big{)}\Big{]}\partial_{t}^{2}\psi\\\
+\Big{[}\phi^{-2}\partial_{r}^{2}(r\phi^{-1})+\partial_{r}(r\phi^{-1})\Big{(}\frac{2\Lambda}{3}r-\frac{2m}{r^{2}}\Big{)}-\phi^{-1}\Big{(}\frac{6\Lambda}{3}r-\frac{8m}{r^{2}}\Big{)}\Big{]}\partial_{r}(X_{s}\psi).$
(4.19)

We now observe that the coefficient of the $\partial_{r}(X_{s}\psi)$ in (4.19)
vanishes to leading order, as

$\displaystyle\phi^{-2}\partial_{r}^{2}(r\phi^{-1})=\frac{2}{r\phi^{3}}+O(1).$
$\displaystyle\partial_{r}(r\phi^{-1})\Big{(}\frac{2\Lambda}{3}r-\frac{2m}{r^{2}}\Big{)}=\frac{4}{r\phi^{3}}+O(1).$
$\displaystyle-\phi^{-1}\Big{(}\frac{6\Lambda}{3}r-\frac{8m}{r^{2}}\Big{)}=-\frac{6}{r\phi^{3}}+O(1).$

Therefore the coefficient of the $\partial_{r}(X_{s}\psi)$ term is of order
$O(1)$. This is also true for the coefficient of the $\partial_{t}^{2}\psi$
term, as

$\displaystyle
2r\phi^{-1}\partial_{r}\Big{[}\phi^{2}\Big{(}1-\frac{3m}{r}\Big{)}\Big{]}=-2\phi+O\big{(}\frac{1}{r^{2}}\big{)}$
$\displaystyle
2\phi\Big{(}1-\frac{3m}{r}\Big{)}=2\phi+O\big{(}\frac{1}{r^{2}}\big{)},$

and so the coefficient of the $\partial_{t}^{2}\psi$ term is of order
$O(r^{-2})$. If we keep track of the leading-order behaviour of the
coefficients of the other terms as well, we get

$\phi\partial_{r}(Y(X_{s}\psi))\square_{g}(Y(X_{s}\psi))=\\\