-\phi\Big{(}\frac{2\Lambda}{3}r-\frac{2m}{r^{2}}\Big{)}[\partial_{r}(Y(X_{s}\psi))]^{2}+O\big{(}\frac{1}{r}\big{)}\partial_{r}(Y(X_{s}\psi))\partial_{r}(X_{s}\psi)\\\
+O\big{(}\frac{1}{r^{4}}\big{)}\partial_{r}(Y(X_{s}\psi))\partial_{t}^{2}(X_{s}\psi)+O\big{(}\frac{1}{r^{3}}\Big{)}\partial_{r}(Y(X_{s}\psi))\partial_{t}^{2}\psi.$
(4.20)

We restate that

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

and point out that the coefficients of the $[\partial_{r}(Y(X_{s}\psi))]^{2}$
terms found above and in (4.20) cancel to leading order. This means that
essentially all terms now become lower-order error terms that we may control
with lower-order energy estimates, elliptic estimates and a Grönwall-type
inequality, similar to in Theorem 4.2. Thus we have

$\phi\nabla\cdot
J^{\partial_{r}}[Y(X_{s}\psi)]=O\big{(}\frac{1}{r^{2}}\big{)}[\partial_{r}(Y(X_{s}\psi))]^{2}+O\big{(}\frac{1}{r^{6}}\big{)}[\partial_{t}(Y(X_{s}\psi))]^{2}\\\
+O\big{(}\frac{1}{r}\big{)}\partial_{r}(Y(X_{s}\psi))\partial_{r}(X_{s}\psi))\\\
+O\big{(}\frac{1}{r^{4}}\big{)}\partial_{r}(Y(X_{s}\psi))\partial_{t}^{2}(X_{s}\psi)+O\big{(}\frac{1}{r^{3}}\big{)}\partial_{r}(Y(X_{s}\psi))\partial_{t}^{2}\psi.$
(4.21)

We now bound all terms on the right hand side of (4.21) pointwise by energy
energy currents, if possible. We have

$\displaystyle\frac{1}{r^{2}}[\partial_{r}(Y(X_{s}\psi))]^{2}\leq
O\big{(}\frac{1}{r^{3}}\big{)}J^{\partial_{r}}[Y(X_{s}\psi)]\cdot n.$
$\displaystyle\frac{1}{r^{6}}[\partial_{t}(Y(X_{s}\psi))]^{2}\leq
O\big{(}\frac{1}{r^{3}}\big{)}J^{\partial_{r}}[Y(X_{s}\psi)]\cdot n.$
$\displaystyle\frac{1}{r}\Big{|}\partial_{r}(Y(X_{s}\psi))\partial_{r}(X_{s}\psi))\Big{|}\leq
O\big{(}\frac{1}{r^{2}}\big{)}J^{\partial_{r}}[Y(X_{s}\psi)]\cdot
n+O\big{(}\frac{1}{r^{2}}\big{)}J^{\partial_{r}}[X_{s}\psi]\cdot n.$
$\displaystyle\frac{1}{r^{4}}\Big{|}\partial_{r}(Y(X_{s}\psi))\partial_{t}^{2}(X_{s}\psi)\Big{|}\leq
O\big{(}\frac{1}{r^{2}}\big{)}J^{\partial_{r}}[Y(X_{s}\psi)]\cdot
n+O\big{(}\frac{1}{r^{7}}\big{)}(\partial_{t}^{2}(X_{s}\psi))^{2}.$
$\displaystyle\frac{1}{r^{3}}\Big{|}\partial_{r}(Y(X_{s}\psi))\partial_{t}^{2}\psi\Big{|}\leq
O\big{(}\frac{1}{r^{2}}\big{)}J^{\partial_{r}}[Y(X_{s}\psi)]\cdot
n+O\big{(}\frac{1}{r^{5}}\big{)}(\partial_{t}^{2}\psi)^{2}.$

It follows from (4.17) and (4.21) that

$E[Y(X_{s}\psi)](r_{2})\leq
E[Y(X_{s}\psi)](r_{1})+\int_{r_{1}}^{r_{2}}O\big{(}\frac{1}{r^{2}}\big{)}E[Y(X_{s}\psi)](r)\textnormal{d}r\\\
+\int_{r_{1}}^{r_{2}}O\big{(}\frac{1}{r^{2}}\big{)}E[X_{s}\psi](r)\Big{]}\textnormal{d}r+\int_{r_{1}}^{r_{2}}O\big{(}\frac{1}{r^{7}}\big{)}\Big{[}\int_{\Sigma_{r}}(\partial_{t}^{2}(X_{s}\psi))^{2}\textnormal{d}\mu_{\overline{g}_{r}}\Big{]}\textnormal{d}r\\\
+\int_{r_{1}}^{r_{2}}O\big{(}\frac{1}{r^{5}}\big{)}\Big{[}\int_{\Sigma_{r}}(\partial_{t}^{2}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}\Big{]}\textnormal{d}r.$
(4.22)

The final term can be controlled by $O(1/r^{2})[E[X_{s}\psi](r)+E[\psi](r)]$
via an elliptic estimate like in Theorem 4.2. For the second-to-last term,
another elliptic estimate gives

$\frac{1}{r^{7}}\int_{\Sigma_{r}}(\partial_{t}^{2}(X_{s}\psi))^{2}\textnormal{d}\mu_{\overline{g}_{r}}\leq
O\big{(}\frac{1}{r^{3}}\big{)}\int_{\Sigma_{r}}(\overline{\Delta}(X_{s}\psi))^{2}\textnormal{d}\mu_{\overline{g}_{r}}+O\big{(}\frac{1}{r^{7}}\big{)}\sum_{i=1}^{3}\int_{\Sigma_{r}}|\partial_{i}(X_{s}\psi)|^{2}\textnormal{d}\mu_{\overline{g}_{r}}.$

The final term can be controlled by $O(r^{-4})E[X_{s}\psi](r)$. Moreover,
observe that

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

and therefore

$\frac{1}{r^{3}}(\overline{\Delta}(X_{s}\psi))^{2}\leq
O\big{(}\frac{1}{r^{2}}\big{)}J^{\partial_{r}}[X_{s}\psi]\cdot
n+O\big{(}\frac{1}{r^{7}}\big{)}(\partial_{t}^{2}\psi)^{2}+O\big{(}\frac{1}{r^{4}}\big{)}J^{\partial_{r}}[Y(X_{s}\psi)]\cdot
n.$

We estimate each of these terms as before, giving

$\frac{1}{r^{3}}\int_{\Sigma_{r}}(\overline{\Delta}(X_{s}\psi))^{2}\textnormal{d}\mu_{\overline{g}_{r}}\leq
O\big{(}\frac{1}{r^{2}}\big{)}\Big{\\{}E[Y(X_{s}\psi)](r)+E[X_{s}\psi](r)+E[\psi](r)\Big{\\}}.$

Therefore

$E[Y(X_{s}\psi)](r_{2})\leq
E[Y(X_{s}\psi)](r_{1})+\int_{r_{1}}^{r_{2}}O\big{(}\frac{1}{r^{2}}\big{)}E[Y(X_{s}\psi)](r)\textnormal{d}r\\\
+\int_{r_{1}}^{r_{2}}O\big{(}\frac{1}{r^{2}}\big{)}\Big{\\{}E[X_{s}\psi](r)+E[\psi](r)\Big{\\}}\textnormal{d}r.$
(4.23)

An application of Theorem 4.2 implies

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

The result now follows by a Grönwall inequality, and the fact that the factor
$r^{-2}$ is integrable. ∎

###  4.4 Pointwise decay estimates

For sufficiently regular solutions to $\square_{g}\psi=0$, the energy
estimates from Theorem 4.2 implies pointwise decay of higher derivatives of
$\psi$. We return to the argument of Section 3.4 that takes advantage of the
Killing vectorfields of Schwarzschild-de Sitter, as well as the Sobolev
estimate from Appendix B. To this end, recall the notation that given a
multiindex $I=(I_{1},I_{2},I_{3},I_{4})$, $Z^{I}$ denotes the differential
operator

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

where $T,\Omega_{i}:i=1,2,3$ are Killing vector fields that span the tangent
space of the $\Sigma_{r}$ hypersurfaces.

######  Corollary 4.6.

Let $\psi$ be a solution to the linear wave equation $\square_{g}\psi=0$, with
finite higher-order energy on an initial hypersurface $\Sigma_{r_{0}}$,

$D[\psi](r_{0})=\sum_{|I|\leq
2}\Big{\\{}E[Y(X_{s}(Z^{I}\psi))](r_{0})+E[X_{s}(Z^{I}\psi)](r_{0})+E[Z^{I}\psi](r_{0})\Big{\\}}<\infty.$

Then we have the following pointwise bounds for large $r$:

$\displaystyle\sup_{\Sigma_{r}}|\partial_{r}\psi|\lesssim\frac{1}{r^{3}}(D[\psi](r_{0}))^{1/2}.$
(4.24)
$\displaystyle\sup_{\Sigma_{r}}|\tilde{\nabla}(X_{s}\psi)|\lesssim(D[\psi](r_{0}))^{1/2}.$
(4.25)
$\displaystyle\sup_{\Sigma_{r}}|\partial_{r}(X_{s}\psi)|\lesssim\frac{1}{r^{2}}(D[\psi](r_{0}))^{1/2}.$
(4.26)
$\displaystyle\sup_{\Sigma_{r}}|\tilde{\nabla}(Y(X_{s}\psi))|\lesssim(D[\psi](r_{0}))^{1/2}.$
(4.27)

###### Proof.

Using the Sobolev estimate from Proposition B.1, we have

$\sup_{\Sigma_{r}}|X_{s}\psi|\lesssim\frac{1}{r^{3/2}}\sum_{|I|\leq
2}\|X_{s}(Z^{I}\psi)\|_{L^{2}(\Sigma_{r})},$

keeping in mind that the differential operators $Z^{I}$ commute with $X_{s}$.
An application of the fundamental theorem of calculus on $[r_{0},r]$ then
implies

$\frac{1}{r^{3/2}}\|X_{s}(Z^{I}\psi)\|_{L^{2}(\Sigma_{r})}\lesssim\|X_{s}(Z^{I}\psi)\|_{L^{2}(\Sigma_{r_{0}})}+\int_{r_{0}}^{r}\frac{1}{r^{3/2}}\|\partial_{r}(X_{s}(Z^{I}\psi))\|_{L^{2}(\Sigma_{r})}\textnormal{d}r.$

We control the last term with the higher-order energy estimate from Theorem
4.5. Since $Z^{I}$ is composed of Killing vector fields, $Z^{I}\psi$ satisfies
the homogeneous wave equation, and so by (4.5) we have

$\displaystyle\frac{1}{r^{3/2}}\|\partial_{r}(X_{s}(Z^{I}\psi))\|_{L^{2}(\Sigma_{r})}$
$\displaystyle\lesssim\frac{1}{r^{2}}\Big{(}E[X_{s}(Z^{I}\psi)](r)\Big{)}^{1/2}$
$\displaystyle\lesssim\frac{1}{r^{2}}\Big{(}E[X_{s}(Z^{I}\psi)](r_{0})+E[Z^{I}\psi](r_{0})\Big{)}^{1/2}$
$\displaystyle\lesssim\frac{1}{r^{2}}\Big{(}D[\psi](r_{0})\Big{)}^{1/2}.$

Therefore

$\displaystyle\sup_{\Sigma_{r}}|X_{s}\psi|$
$\displaystyle\lesssim\sum_{|I|\leq
2}\Big{\\{}\|X_{s}(Z^{I}\psi)\|_{L^{2}(\Sigma_{r_{0}})}+\int_{r_{0}}^{r}\frac{1}{r^{2}}\Big{(}D[\psi](r_{0})\Big{)}^{1/2}\textnormal{d}r\Big{\\}}$
$\displaystyle\lesssim\Big{(}D[\psi](r_{0})\Big{)}^{1/2}.$

Since $X_{s}\sim r^{3}\partial_{r}$, we divide the above inequality through by
$r^{3}$ and obtain (4.24). As for (4.25), we use the Sobolev estimate again
and control the resulting $L^{2}$-norm by the higher-order energy, so that

$\displaystyle\sup_{\Sigma_{r}}|\tilde{\nabla}(X_{s}\psi)|^{2}$
$\displaystyle\lesssim\frac{1}{r^{3}}\sum_{|I|\leq
2}\|\tilde{\nabla}(X_{s}\psi)\|_{L^{2}(\Sigma_{r})}^{2}$
$\displaystyle\lesssim\sum_{|I|\leq 2}E[X_{s}(Z^{I}\psi)](r),$

and finally appeal to the higher-order estimate from Theorem 4.2. The decay
estimates (4.26) and (4.27) follow in the same manner from the second-order
estimate of Theorem 1.14. ∎

###  4.5 Concluding the proof of Theorem 1.1

We conclude this section by proving the existence of the limits

$\psi_{0}=\lim_{r\rightarrow\infty}\psi,\quad\psi_{2}=-2\lim_{r\rightarrow\infty}[r^{3}\partial_{r}\psi],\quad\psi_{3}=3\lim_{r\rightarrow\infty}[r^{2}\partial_{r}(r^{3}\partial_{r}\psi)]$
(4.28)

with respect to the $\dot{H}^{1}$-norm on the future boundary $\Sigma^{+}$,
which we recall is diffeomorphic to $\mathbb{R}\times\mathbb{S}^{2}$. This is
equivalent to proving that the quantities $\psi,X_{s}\psi,Y(X_{s}\psi)$ all
have a limit on $\Sigma^{+}$. The particular constants in front of the limits
for $\psi_{2}$, $\psi_{3}$ are just normalisation factors that account for
repeated differentiation in $r$. We note that the energy controls the
$\dot{H}^{1}$-norm on the $\Sigma_{r}$ hypersurfaces, meaning there exists a
constant $C(\Lambda,m,r_{0})>0$ independent of $r$ such that

$\displaystyle\|\psi(r)\|_{\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})}^{2}\leq
CE[\psi](r),$

for all $r\geq r_{0}$. To prove the existence of the limits (4.28), we appeal
to a density argument. Let $(\varphi_{j})_{j\in\mathbb{N}}$ be a sequence of
smooth, compactly supported solutions to the linear wave equation that
converge to $\psi$, so that for some $r_{0}>r_{\mathcal{C}}$ we have

$\lim_{j\rightarrow\infty}\Big{[}\|\psi-\varphi_{j}\|_{\partial_{r},\Sigma_{r_{0}}}+\|X_{s}(\psi-\varphi_{j})\|_{\partial_{r},\Sigma_{r_{0}}}+\|Y(X_{s}(\psi-\varphi_{j}))\|_{\partial_{r},\Sigma_{r_{0}}}\Big{]}=0.$

This can be accomplished by taking a sequence of smooth compactly supported
data on $\Sigma_{r_{0}}$ that converge to the data of $\psi$ on
$\Sigma_{r_{0}}$, and then considering the corresponding solution to the wave
equation. By commuting the energy estimate from Theorem 1.14 with Killing
vectorfields, one can then show that for each $j$, the functions
$\varphi_{j},X_{s}\varphi_{j},Y(X_{s}\psi)$ each have a pointwise limit on
$\Sigma^{+}$. This pointwise limit extends to a limit with respect to the
$\dot{H}^{1}$-norm via the dominated convergence theorem. Finally, it follows
from the energy estimate of Theorem 4.2 that $\lim_{r\rightarrow\infty}\psi$,
$\lim_{r\rightarrow\infty}X_{s}\psi$, and
$\lim_{r\rightarrow\infty}Y(X_{s}\psi)$ all exist with respect to the
homogeneous Sobolev norm $\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})$.

We finally prove the existence of the limits (1.4) by showing that

$\displaystyle\lim_{r\rightarrow\infty}\|r^{2}(\psi-\psi_{0})-\psi_{2}\|_{\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})}=0.$
(4.29)
$\displaystyle\lim_{r\rightarrow\infty}\Big{\|}r^{3}\Big{(}\psi-\psi_{0}-\frac{\psi_{2}}{r^{2}}\Big{)}-\psi_{3}\Big{\|}_{\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})}=0.$
(4.30)

To prove (4.29), we know that
$\|\psi-\psi_{0}-r^{-2}\psi_{r}\|_{\dot{H}^{1}}\rightarrow 0$ as
$r\rightarrow\infty$, and so by the fundamental theorem of calculus we have

$\Big{\|}\psi-\psi_{0}-\frac{1}{r^{2}}\psi_{2}\Big{\|}_{\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})}\leq\int_{r}^{\infty}\Big{\|}\partial_{r}\psi+\frac{2}{r^{3}}\psi_{2}\Big{\|}_{\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})}\textnormal{d}r,$

noting that
$\partial_{r}(\psi-\psi_{0}-r^{-2}\psi_{2})=\partial_{r}\psi+2r^{-3}\psi_{2}$.
We bound

$\displaystyle\int_{r}^{\infty}\Big{\|}\partial_{r}\psi+\frac{2}{r^{3}}\psi_{2}\Big{\|}_{\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})}\textnormal{d}r$
$\displaystyle\leq\int_{r}^{\infty}\frac{1}{r^{3}}\sup_{[r,\infty)}\|r^{3}\partial_{r}\psi+2\psi_{2}\|_{\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})}\textnormal{d}r$
$\displaystyle\leq\frac{1}{2r^{2}}\sup_{[r,\infty)}\|r^{3}\partial_{r}\psi+2\psi_{2}\|_{\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})}.$

This implies (4.29), as

$\|r^{2}(\psi-\psi_{0})-\psi_{2}\|_{\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})}\leq\frac{1}{2}\sup_{[r,\infty)}\|r^{3}\partial_{r}\psi+2\psi_{2}\|_{\dot{H}^{1}(\mathbb{R}\times\mathbb{S}^{2})}.$

The right hand side vanishes as $r\rightarrow\infty$, hence (4.29) holds. The
limit (4.30) follows in a similar manner.

##  5 Scattering on general expanding spacetimes

We will now generalise the scattering result Theorem 1.4 to the linear wave
equation on a wide class of expanding spacetimes. No symmetries are required
on the expanding region, and the geometry at infinity of these spacetimes can
differ significantly from Schwarzschild-de Sitter. In particular the expanding
region of Kerr-de Sitter is contained in the class of spacetimes we consider,
so we prove existence and uniqueness of scattering solutions to the linear
wave equation on Kerr-de Sitter.

######  Remark 5.1.

Wave equations on another class of perturbations of Schwarzschild-de Sitter
were also considered in [27], and the redshift estimate (1.16) for the forward
problem was shown to generalise to those perturbations. We note the perturbed
spacetimes considered in [27] are in a more restricted class compared to those
that we consider in this paper, as their metric components are assumed to
converge at infinity to their counterparts on Schwarzschild-de Sitter.

For a given spacetime, we will construct asymptotic solutions to the wave
equation which are specialised to that spacetime. We note that for the class
of perturbations of Schwarzschild-de Sitter we consider, logarithmic terms are
present in the asymptotic solution, see Remark 3.5. However we will show that
for a restriction of the perturbed class that includes Kerr-de Sitter,
logarithmic terms are not present.

We will then prove a weighted energy estimate suitable for the backward
problem, extending Proposition 1.8 to this large class of spacetimes. The
existence of scattering solutions from Theorem 1.20 then follows by repeating
the argument from Section 3.3, but in a perturbed setting.

We define the following class of spacetimes:

######  Definition 5.2.

Let $\Sigma^{+}$ denote a 3-dimensional Riemannian manifold, equipped with a
Riemannian $C^{3}$-metric $h$, and fix $\Lambda>0$. Let $M$ denote the
differentiable manifold $M=(0,\rho_{0})\times\Sigma^{+}$ for some
$\rho_{0}>0$, and let $g$ be a Lorentzian $C^{3}$-metric defined on $M$.

We say $(M,g)\in\mathcal{G}_{\Lambda}^{1}$ if the conformal metric
$\tilde{g}=\rho^{2}g$ admits a $C^{3}$-extension across
$\\{\rho=0\\}\times\Sigma^{+}$ of the form

$\tilde{g}=-\frac{3}{\Lambda}\textnormal{d}\rho^{2}+h+O(\rho).$ (5.1)

Moreover, we say $(\mathcal{M},g)\in\mathcal{G}_{\Lambda}^{2}$ if the
conformal metric $\tilde{g}$ is like

$\tilde{g}=-\frac{3}{\Lambda}\textnormal{d}\rho^{2}+h+O(\rho^{2}).$ (5.2)

######  Remark 5.3.

This class of spacetimes is very similar those considered in [30], where in
particular smooth scattering solutions are constructed from infinity. The
primary difference here is that we do not assume smoothness of the metric, but
instead $C^{3}$-regularity. This class is also interesting because of its
relation to the scattering problem for Einstein’s vacuum equations with
positive cosmological constant, see for example [15].

######  Remark 5.4.

A straightforward computation shows that de Sitter, Schwarzschild-de Sitter
and Kerr-de Sitter spacetimes are all members of $\mathcal{G}_{\Lambda}^{2}$.
For de Sitter and Schwarzschild-de Sitter, one can see this by taking the
metric in standard coordinates on the expanding region (for example take (2.2)
for Schwarzschild-de Sitter), and introducing the coordinate $\rho=r^{-1}$.
For Kerr-de Sitter this can be seen by considering the metric in _Boyer-
Lindquist_ coordinates on the expanding region, and again introducing the
coordinate $\rho=r^{-1}$.

It is simpler for us to do the computations in this section with respect to
the conformal metric $\tilde{g}$ rather than $g$. For the rest of this
section, we assume that unless explicitly stated otherwise, all quantities are
defined in terms of $\tilde{g}$ and not the original metric.

First, we note that the homogeneous wave equation $\square_{g}\psi=0$ is
equivalent to the following wave equation on the conformal metric $\tilde{g}$:

$\square_{\tilde{g}}\psi=\frac{2}{\rho}\tilde{g}^{\rho\nu}\partial_{\nu}\psi,$
(5.3)

where a subscript or superscript $\rho$ is to be treated as the component in
terms of the $\rho$-coordinate and not an abstract index. If we set
$\phi=(-\tilde{g}^{\rho\rho})^{-1/2}$, then the future-pointing timelike unit
normal $n$ of the level sets of constant $\rho$ is given by

$n^{\mu}=\phi\tilde{g}^{\rho\mu}$ (5.4)

and so we may write

$\square_{\tilde{g}}\psi=\frac{2}{\rho\phi}n\psi.$ (5.5)

Note that for the conformal metric neither the unit normal $n$ nor the lapse
$\phi$ possess weights in $\rho$, and both are uniformly continuously
differentiable up to $\Sigma^{+}$.

###  5.1 Constructing asymptotic solutions

In this section we construct asymptotic solutions to the wave equation for
metrics in $\mathcal{G}_{\Lambda}^{1}$, $\mathcal{G}_{\Lambda}^{2}$. While
these constructions are similar to their counterparts on Schwarzschild-de
Sitter, in the perturbed setting we include logarithmic terms in the
asymptotic solution. The wave equation (5.5) can be written as

$\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}\psi=\tilde{g}^{\rho\rho}\partial_{\rho}^{2}\psi+2\tilde{g}^{\rho
i}\partial_{\rho}\partial_{i}\psi+\tilde{g}^{ij}\partial_{i}\partial_{j}\psi\\\
-\tilde{g}^{\mu\nu}\Gamma_{\mu\nu}^{\rho}\partial_{\rho}\psi-\tilde{g}^{\mu\nu}\Gamma_{\mu\nu}^{i}\partial_{i}\psi-\frac{2}{\rho\phi}\tilde{g}^{\rho\rho}\partial_{\rho}\psi-\frac{2}{\rho\phi}\tilde{g}^{\rho
i}\partial_{i}\psi.$ (5.6)

If the metric belongs to $\mathcal{G}_{\Lambda}^{1}$, then a short computation
using (5.1) implies that the asymptotics of the various coefficients that
appear above are

$\displaystyle\tilde{g}^{\rho\rho}=-\frac{\Lambda}{3}+O(\rho),\quad\tilde{g}^{\rho
i}=O(\rho),\quad\tilde{g}^{ij}=h^{ij}+O(\rho),$ (5.7)
$\displaystyle\tilde{g}^{\mu\nu}\Gamma_{\mu\nu}^{\rho}=O(1),\quad\tilde{g}^{\mu\nu}\Gamma_{\mu\nu}^{i}=f_{\Sigma^{+}}^{i}+O(\rho),$
(5.8)

where the $f_{\Sigma^{+}}^{i}$ are $C^{2}$-functions on $\Sigma^{+}$. If the
metric also belongs to $\mathcal{G}_{\Lambda}^{2}$, then by (5.2) the
asymptotics are

$\displaystyle\tilde{g}^{\rho\rho}=-\frac{\Lambda}{3}+O(\rho^{2}),\quad\tilde{g}^{\rho
i}=O(\rho^{2}),\quad\tilde{g}^{ij}=h^{ij}+O(\rho^{2}).$ (5.9)
$\displaystyle\tilde{g}^{\mu\nu}\Gamma_{\mu\nu}^{\rho}=O(\rho),\quad\tilde{g}^{\mu\nu}\Gamma_{\mu\nu}^{i}=f_{\Sigma^{+}}^{i}+O(\rho^{2}).$
(5.10)

######  Lemma 5.5.

Fix a spacetime $(\mathcal{M},g)\in\mathcal{G}_{\Lambda}^{1}$, and let
$\psi_{0},\psi_{3}$ be sufficiently regular functions on $\Sigma^{+}$. Then
define the asymptotic solution

$\psi_{\text{asymp}}=\psi_{0}+\rho^{2}\psi_{2}+\rho^{3}\log\rho\>\psi_{3,1}+\rho^{3}\psi_{3},$

where $\psi_{2}$,$\psi_{3,1}$ are functions determined entirely by $\psi_{0}$
and derivatives thereof. Then there exists an integer $k\geq 0$ such that

$\Big{|}\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}(\psi_{\text{asymp}})\Big{|}\lesssim_{\mathcal{M}}\rho^{2}\log\rho\sum_{|\alpha|\leq
k}|\nabla_{\Sigma^{+}}^{\alpha}\psi_{0}|+\rho^{2}\sum_{|\alpha|\leq
k}|\nabla_{\Sigma^{+}}^{\alpha}\psi_{3}|,$ (5.11)

where $\nabla_{\Sigma^{+}}$ is the covariant derivative restricted to
$\Sigma^{+}$. If additionally $(\mathcal{M},g)\in\mathcal{G}_{\Lambda}^{2}$,
then the $\psi_{3,1}$ term vanishes identically, and we have

$\Big{|}\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}(\psi_{\text{asymp}})\Big{|}\lesssim\rho^{2}\sum_{|\alpha|\leq
k}|\nabla_{\Sigma^{+}}^{\alpha}\psi_{0}|+\rho^{3}\sum_{|\alpha|\leq
k}|\nabla_{\Sigma^{+}}^{\alpha}\psi_{3}|$ (5.12)

###### Proof.

We first assume just $(\mathcal{M},g)\in\mathcal{G}_{\Lambda}^{1}$. Then we
compute term-by-term. By (5.6), (5.7), and (5.8) we have

$\displaystyle\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}(\psi_{0})=h^{ij}\partial_{i}\partial_{j}\psi_{0}-f_{\Sigma^{+}}^{i}\partial_{i}\psi+O(\rho),$
$\displaystyle\quad\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}(\rho^{2}\psi_{2})=\frac{2\Lambda}{3}\psi_{2}+O(\rho).$

For the remaining terms, we note that in (5.6) the slowest-decaying terms are
$\tilde{g}^{\rho\rho}\partial_{\rho}^{2}\psi$ and
$\frac{2}{\rho\phi}\tilde{g}^{\rho\rho}$. But for
$\rho^{3}\log\rho\>\psi_{3,1}$, $\rho^{3}\psi_{3}$ these terms cancel to
leading order, giving

$\displaystyle\Big{(}\tilde{g}^{\rho\rho}\partial_{\rho}^{2}-\frac{2}{\rho\phi}\tilde{g}^{\rho\rho}\Big{)}(\rho^{3}\log\rho\>\psi_{3,1})=\Lambda\rho+O(\rho^{2}\log\rho),$
$\displaystyle\Big{(}\tilde{g}^{\rho\rho}\partial_{\rho}^{2}-\frac{2}{\rho\phi}\tilde{g}^{\rho\rho}\Big{)}(\rho^{3}\psi_{3})=O(\rho^{2}).$

Hence we may freely choose $\psi_{0}$, $\psi_{3}$, and we set if we set the
terms in

$\square_{\tilde{g}}\psi_{\text{asymp}}=\frac{2}{\rho\phi}n\psi_{\text{asymp}}$

of order $O(1)$ and $O(\rho)$ equal to zero, then $\psi_{2}$, $\psi_{3,1}$ are
determined entirely by $\psi_{0}$ and its derivatives, which implies (5.11).

Moreover if we assume $(\mathcal{M},g)\in\mathcal{G}_{\Lambda}^{2}$, then by
(5.6), (5.9), and (5.10) we have instead for $\psi_{0}$, $\psi_{2}$,

$\displaystyle\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}(\psi_{0})=h^{ij}\partial_{i}\partial_{j}\psi_{0}-f_{\Sigma^{+}}^{i}\partial_{i}\psi+O(\rho^{2}),$
$\displaystyle\quad\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}(\rho^{2}\psi_{2})=\frac{2\Lambda}{3}\psi_{2}+O(\rho^{2}).$

Hence setting the order $O(\rho)$ term in
$\square_{\tilde{g}}\psi_{\text{asymp}}=\frac{2}{\rho\phi}n\psi_{\text{asymp}}$
to be identically zero implies that $\psi_{3,1}$ = 0, and so (5.12) holds. ∎

######  Remark 5.6.

We compare the Lemma above to the construction of asymptotic solutions on
Schwarzschild-de Sitter in Section 3.1. In the notation used in this section,
Lemma 3.4 and (3.9) the approximate solution on Schwarzschild-de Sitter is

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

and $\psi_{\text{asymp}}$ obeys the bound

$\Big{|}\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}(\psi_{\text{asymp}})\Big{|}\lesssim\rho^{2}\sum_{|\alpha|\leq
4}|\nabla_{\Sigma^{+}}^{\alpha}\psi_{0}|+\rho^{3}\sum_{|\alpha|\leq
2}|\nabla_{\Sigma^{+}}^{\alpha}\psi_{3}|,$

which is essentially (5.12). Also note that while logarithmic terms may be
present in asymptotic solutions in the perturbed setting, the term
$\rho\>\psi_{1}$ cannot be present, even for metrics in
$\mathcal{G}_{\Lambda}^{1}$. This is because

$\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}(\rho\>\psi_{1})=O\big{(}\frac{1}{\rho}\Big{)},$

so any asymptotic solution containing $\rho\psi_{1}$ would only satisfy (5.5)
up to order $\rho^{-1}$.

######  Remark 5.7.

As previously stated, Kerr-de Sitter spacetimes are in the class
$\mathcal{G}_{\Lambda}^{2}$. Thus by the previous Lemma, asymptotic solutions
to the linear wave equation on Kerr-de Sitter are of the form

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

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

###  5.2 Energy estimate for the backward problem on expanding spacetimes

We now prove a weighted backward estimate in the perturbed setting, analogous
to Proposition 1.8. We will apply this to solutions to the inhomogeneous wave
equation

$\square_{\tilde{g}}\psi-\frac{2}{\rho\phi}n\psi=F.$ (5.13)

######  Proposition 5.8.

Let $M=\frac{1}{\rho^{4}}n$, and suppose $\psi$ satisfy the inhomogeneous wave
equation (5.13). Then there exists a constant $C>0$ such that

$K^{M}[\psi]\leq CJ^{M}[\psi]\cdot n.$

on all of $\tilde{M}$. Furthermore, there exists a (different) constant $B>0$
such that for all $\rho_{1}\rho_{2}\in(0,\rho_{0})$ with $\rho_{1}<\rho_{2}$,
we have

$\int_{\Sigma_{\rho_{2}}}J^{M}[\psi]\cdot
n\>\textnormal{d}\mu_{\gamma_{\rho}}\leq
B\Big{[}\int_{\Sigma_{\rho_{1}}}J^{M}[\psi]\cdot
n\>\textnormal{d}\mu_{\gamma_{\rho}}+\int_{\rho_{1}}^{\rho_{2}}\Big{(}\int_{\Sigma_{\rho}}\frac{1}{\rho^{4}}F^{2}\Big{)}\textnormal{d}\rho\Big{]},$
(5.14)

where $\Sigma_{\rho}$ denotes the level sets of $\rho$ in
$\tilde{\mathcal{M}}$.

######  Remark 5.9.

We point out that for $\rho_{1}<\rho_{2}$, the hypersurface
$\Sigma_{\rho_{1}}$ lies to the _future_ of $\Sigma_{\rho_{2}}$, and the
future boundary $\Sigma^{+}$ corresponds to $\\{\rho=0\\}$. So this is a
backwards-in-time estimate just like Proposition 1.8.

###### Proof.

Recall the definition (5.4) of the timelike unit normal $n$. As $M$ is a
multiple of $n$, by the product rule we have

$\>{}^{(M)}\pi^{\mu\nu}=n^{(\mu}\tilde{g}^{\nu)\lambda}\partial_{\lambda}(\rho^{-4})+\rho^{-2}\>{}^{(n)}\pi^{\mu\nu}=-\frac{4\rho^{-5}}{\phi}n^{\mu}n^{\nu}+\rho^{-4}\>{}^{(n)}\pi^{\mu\nu},$

which implies

$K^{M}[\psi]=-\frac{4}{\rho\phi}J^{M}[\psi]\cdot
n+\frac{1}{\rho^{4}}K^{n}[\psi].$

Writing the current explicitly we may bound

$J^{M}[\psi]\cdot
n=\frac{1}{2}(M\psi)(n\psi)+\frac{1}{2}\frac{\phi^{2}}{\rho^{4}}|\overline{\nabla}\psi|^{2}\geq\frac{1}{2}(M\psi)(n\psi),$

and so

$K^{M}[\psi]\leq-\frac{2}{\rho\phi}(M\psi)(n\psi)+\frac{1}{\rho^{4}}K^{n}[\psi].$

We then claim that the following bound holds on all of $\tilde{\mathcal{M}}$:

$\phi|K^{n}[\psi]|\leq CJ^{n}[\psi]\cdot n$ (5.15)

The pointwise version of this bound is always true, but the bound being
uniform follows from the fact that the components of $\tilde{g}$ and the
function $\phi$ are continuously differentiable, moreover their derivatives
are uniformly bounded across $\\{\rho=0\\}$. It follows from (5.15) that for
the divergence of $J^{M}[\psi]$ we have

$\displaystyle\phi\nabla\cdot J^{M}[\psi]$
$\displaystyle\leq-\frac{2}{\rho}(M\psi)(n\psi)+\frac{\phi}{\rho^{4}}K^{n}[\psi]+\phi(M\psi)\Big{(}\frac{2}{n\phi}\psi+F\Big{)}$
$\displaystyle\leq\frac{\phi}{\rho^{4}}CJ^{n}[\psi]\cdot n+\phi(M\psi)F\sim
CJ^{M}[\psi]\cdot n+(M\psi)F.$

We apply the Cauchy-Schwartz inequality to the $(M\psi)F$ term, so that

$(M\psi)F\leq\frac{\rho^{4}}{2}(M\psi)^{2}+\frac{1}{2\rho^{4}}F^{2}\lesssim
CJ^{M}[\psi]\cdot n+\frac{1}{\rho^{4}}F^{2}$

which yields the bound

$\phi\nabla\cdot J^{M}[\psi]\lesssim CJ^{M}[\psi]\cdot
n+\frac{1}{\rho^{4}}F^{2}.$ (5.16)

We apply the divergence theorem to the energy current $J^{M}[\psi]$ on the
region bounded in the past by $\Sigma_{\rho_{2}}$ and to the future by
$\Sigma_{\rho_{1}}$ for $0<\rho_{1}<\rho_{2}<P$. We denote this region by
$\mathcal{D}_{\rho_{1},\rho_{2}}$. This gives

$\int_{\Sigma_{\rho_{2}}}J^{M}[\psi]\cdot
n\>\textnormal{d}\mu_{\gamma_{\rho}}=\int_{\Sigma_{\rho_{1}}}J^{M}[\psi]\cdot
n\>\textnormal{d}\mu_{\gamma_{\rho}}+\int_{\mathcal{D}_{\rho_{1},\rho_{2}}}\nabla\cdot
J^{M}[\psi]\>\textnormal{d}\mu_{\tilde{g}}=,$

where $\gamma_{\rho}$ is the Riemannian metric on $\Sigma_{\rho}$ induced from
the conformal metric $\tilde{g}$. The particular signs of each integral are
due to the fact that $n\sim-\partial_{\rho}$, i.e. the normal vectorfield is
pointing in the direction of _decreasing_ $\rho$. We assert that
$|\tilde{g}|^{1/2}=\phi|\gamma_{\rho}|^{1/2}$, which implies that the volume
form $\textnormal{d}\mu_{\tilde{g}}$ satisfies

$\textnormal{d}\mu_{\tilde{g}}=\phi\>\textnormal{d}\mu_{\gamma_{\rho}}\wedge\textnormal{d}\rho,$

and so by the coarea formula we have

$\int_{\Sigma_{\rho_{2}}}J^{M}[\psi]\cdot
n\>\textnormal{d}\mu_{\gamma_{\rho}}=\int_{\Sigma_{\rho_{1}}}J^{M}[\psi]\cdot
n\>\textnormal{d}\mu_{\gamma_{\rho}}+\int_{\rho_{1}}^{\rho_{2}}\Big{(}\int_{\Sigma_{\rho}}\phi\nabla\cdot
J^{M}[\psi]\>\textnormal{d}\mu_{\gamma_{\rho}}\Big{)}\textnormal{d}\rho.$
(5.17)

Inserting (5.16) then gives

$\int_{\Sigma_{\rho_{2}}}J^{M}[\psi]\cdot
n\>\textnormal{d}\mu_{\gamma_{\rho}}\lesssim\int_{\Sigma_{\rho_{1}}}J^{M}[\psi]\cdot
n\>\textnormal{d}\mu_{\gamma_{\rho}}+\int_{\rho_{1}}^{\rho_{2}}\Big{(}\int_{\Sigma_{\rho}}\frac{1}{\rho^{4}}F^{2}\>\textnormal{d}\mu_{\gamma_{\rho}}\Big{)}\>\textnormal{d}\rho\\\
+C\int_{\rho_{1}}^{\rho_{2}}\Big{(}\int_{\Sigma_{\rho}}J^{M}[\psi]\cdot
n\>\textnormal{d}\mu_{\gamma_{\rho}}\Big{)}\>\textnormal{d}\rho.$

The result now follows through a Grönwall-type inequality. ∎

###  5.3 Existence and uniqueness of scattering solutions on expanding
spacetimes

We now state the full version of Theorem 1.20, proving existence and
uniqueness of scattering solutions to the linear wave equation in the class of
expanding spacetimes $\mathcal{G}_{\Lambda}^{1}$. Recall from (2.10) the
notation for the energy norm

$\|\psi\|_{M,\Sigma_{\rho}}=\int_{\Sigma_{\rho}}J^{M}[\psi]\cdot
n\>\textnormal{d}\mu_{\gamma_{\rho}}$

######  Theorem 5.10.

Fix $\Lambda>0$, and let $(\mathcal{M},g)\in\mathcal{G}_{\Lambda}^{1}$ be an
expanding spacetime with future boundary $\Sigma^{+}$. Suppose
$\psi_{0},\psi_{3}\in H^{k}(\Sigma^{+})$ for some $k$ sufficiently large. Then
there exists a unique solution to the linear wave equation $\square_{g}\psi=0$
such that for all $\rho\in(0,\rho_{0})$, we have

$\|\psi\|_{M,\Sigma_{\rho}}\lesssim_{\mathcal{M}}\|\psi_{0}\|_{H^{k}(\Sigma^{+})}+\|\psi_{3}\|_{H^{k}(\Sigma^{+})}.$
(5.18)

###### Proof.

Given $\psi_{0},\psi_{3}$, let $\psi_{\text{asymp}}$ be the asymptotic
solution constructed in Lemma 5.5. Let $\psi_{\text{rem}}^{(P)}$ be a solution
to the inhomogeneous wave equation

$\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}\psi_{\text{rem}}^{(P)}=-\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}\psi_{\text{asymp}},$

with trivial data on $\Sigma_{P}$:

$\psi_{\text{rem}}^{(P)}|_{\Sigma_{P}}=n\psi_{\text{rem}}^{(P)}|_{\Sigma_{P}}=0.$

Then taking the difference
$v=\psi_{\text{rem}}^{(P_{1})}-\psi_{\text{rem}}^{(P_{2})}$ for
$0<P_{1}<P_{2}$, one can repeat the proof of existence of scattering solutions
from Section 3.3. In particular, we show that for all $\rho\in(0,\rho_{0})$,
$v\rightarrow 0$ as $P_{1},P_{2}\rightarrow 0$, where the limit of $v$ is
taken with respect to the norm $\|\cdot\|_{n,\Sigma_{\rho}}$. Here we use the
bound (5.11) for the asymptotic solution, and the weighted estimate of
Proposition 5.8. The key inequality we reach is (with a constant independent
of $P_{1}$, $P_{2}$):

$\|v\|_{M,\Sigma_{\rho}}^{2}\leq
C\int_{P_{1}}^{P_{2}}\Big{(}\int_{\Sigma_{\rho}}\frac{1}{\rho^{4}}\Big{[}\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}\psi_{\text{asymp}}\Big{]}^{2}\>\textnormal{d}\mu_{\gamma_{\rho}}\Big{)}\>\textnormal{d}\rho,$

for all $\rho>P_{2}$. We compare this to the corresponding bound (3.24) in
Section 3.3. By the pointwise estimate (5.11), we have the $L^{2}$-bound

$\int_{P_{1}}^{P_{2}}\int_{\Sigma_{\rho}}\frac{1}{\rho^{4}}\Big{[}\Big{(}\square_{\tilde{g}}-\frac{2}{\rho\phi}n\Big{)}\psi_{\text{asymp}}\Big{]}^{2}\textnormal{d}\mu_{\gamma_{\rho}}\textnormal{d}\rho\lesssim\int_{P_{1}}^{P_{2}}(\log\rho)^{2}(\|\psi_{0}\|_{H^{k}(\Sigma^{+})}^{2}+\|\psi_{3}\|_{H^{k}(\Sigma^{+})}^{2})\>\textnormal{d}\rho.$

Since $(\log\rho)^{2}$ is integrable up to $\rho=0$, the right hand side above
vanishes as $P_{1},P_{2}\rightarrow 0$, and so $\lim_{P_{1},P_{2}\rightarrow
0}\|v\|_{M,\Sigma_{\rho}}=0$. Hence there exists $\psi$ such that for each
$\rho\in(0,\rho_{0})$, we have

$\lim_{P\rightarrow\infty}\|\psi-\psi_{\text{asymp}}-\psi_{\text{rem}}^{(P)}\|_{M,\Sigma_{\rho}}=0.$

The bound (5.18) from bounding

$\|\psi\|_{n,\Sigma_{\rho}}\leq\|\psi-\psi_{\text{asymp}}-\psi_{\text{rem}}^{(P)}\|_{M,\Sigma_{\rho}}+\|\psi_{\text{asymp}}\|_{n,\Sigma_{\rho}}+\|\psi_{\text{rem}}^{(P)}\|_{n,\Sigma_{\rho}}.$

By (5.11) we estimate
$\|\psi_{\text{asymp}}\|_{n,\Sigma_{\rho}}\lesssim\|\psi_{0}\|_{H^{k}(\Sigma^{+})}+\|\psi_{3}\|_{H^{k}(\Sigma^{+})}$.
To control $\|\psi_{\text{rem}}^{(P)}\|_{n,\Sigma_{\rho}}$, we apply
Proposition 5.8 to $\psi_{\text{rem}}^{(P)}$ on the spacetime slab between
$\Sigma_{P}$ and $\Sigma_{\rho}$, giving

$\|\psi_{\text{rem}}^{(P)}\|_{n,\Sigma_{\rho}}^{2}\lesssim\rho^{2}(\|\psi_{0}\|_{H^{k}(\Sigma^{+})}^{2}+\|\psi_{3}\|_{H^{k}(\Sigma^{+})}^{2})\int_{P}^{\rho}(\log
s)^{2}\>\textnormal{d}s\lesssim\|\psi_{0}\|_{H^{k}(\Sigma^{+})}^{2}+\|\psi_{3}\|_{H^{k}(\Sigma^{+})}^{2},$

where we used the fact that that
$\|\cdot\|_{n,\Sigma_{\rho}}\sim\rho^{2}\|\cdot\|_{M,\Sigma_{\rho}}$. This
implies

$\|\psi\|_{n,\Sigma_{\rho}}\leq\|\psi-\psi_{\text{asymp}}-\psi_{\text{rem}}^{(P)}\|_{M,\Sigma_{\rho}}+\|\psi_{0}\|_{H^{k}(\Sigma^{+})}+\|\psi_{3}\|_{H^{k}(\Sigma^{+})}.$

Taking the limit $P\rightarrow 0$, the result now follows. ∎

##  Appendix A Elliptic Estimates

In this appendix we will state an elliptic estimate which we use to prove the
higher-order energy estimates in Section 4. Given a 3-dimensional Riemannian
manifold $(\Sigma,\overline{g})$, we denote the metric connection by
$\overline{\nabla}$, and the Laplacian
$\overline{\nabla}^{i}\overline{\nabla}_{i}=\overline{\Delta}$. We also write

$|\overline{\nabla}^{2}\psi|^{2}=\overline{g}^{ij}\overline{g}^{kl}\overline{\nabla}_{i}\overline{\nabla}_{k}\psi\overline{\nabla}_{j}\overline{\nabla}_{l}\psi.$

We recall the following result from Section 4 of [3]; see Remark 4 therein.

######  Proposition A.1 (Elliptic estimate from [3]).

Let $\psi$ be a scalar function on the 3-dimensional Riemannian manifold
$(\Sigma,\overline{g})$. Then there exists some positive constant $C$ such
that

$\int_{\Sigma}|\overline{\nabla}^{2}\psi|^{2}\textnormal{d}\mu_{\overline{g}}\leq
C\Big{(}\int_{\Sigma}(\overline{\Delta}\psi)^{2}\textnormal{d}\mu_{\overline{g}}+\int_{\Sigma}|\emph{Ric}[\overline{g}]^{ij}\overline{\nabla}_{i}\psi\overline{\nabla}_{j}\psi|\textnormal{d}\mu_{\overline{g}}\Big{)}.$
(A.1)

As a consequence, if we consider the spacelike hypersurfaces $\Sigma_{r}$ on
Schwarzschild-de Sitter, the Ricci curvature of the induced metric
$\overline{g}_{r}$ obeys the bound

$|\text{Ric}[\overline{g}_{r}]^{ij}|\sim\frac{1}{r^{4}}.$

for sufficiently large $r$. Moreover, one may bound

$\displaystyle|\overline{\nabla}^{2}\psi|^{2}$
$\displaystyle\lesssim\sum_{i,j=1}^{3}\overline{g}_{r}^{ii}\overline{g}_{r}^{jj}(\partial_{i}\partial_{j}\psi)^{2}+\sum_{i=1}^{3}(\overline{g}_{r}^{ii})^{2}(\partial_{i}\psi)^{2}$
$\displaystyle\sim\frac{1}{r^{4}}\sum_{i,j=1}^{3}(\partial_{i}\partial_{j}\psi)^{2}+\frac{1}{r^{4}}\sum_{i=1}^{3}(\partial_{i}\psi)^{2}.$
(A.2)

We note that the final term in (A.2) can be controlled by the integrand of the
final term in (A.1), which allows us to absorb this term into the right hand
side of (A.1). Thus Proposition (A.1) implies the following Corollary:

######  Corollary A.2.

Let $(\mathcal{M}_{\Lambda,m},g)$ be a subextremal member of the
Schwarzschild-de Sitter family of spacetimes, and fix $r_{0}\geq
r_{\mathcal{C}}$, and let $\psi$ be a scalar field on the expanding region
$\mathcal{R}^{+}$. Then for all $r\geq r_{0}$, we have the following elliptic
estimate:

$\sum_{i,j=1}^{3}\int_{\Sigma_{r}}(\partial_{i}\partial_{j}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}\lesssim_{\Lambda,m,r_{0}}C\Big{(}r^{4}\int_{\Sigma_{r}}(\overline{\Delta}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}+\sum_{i=1}^{3}\int_{\Sigma_{r}}(\partial_{i}\psi)^{2}\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}.$
(A.3)

##  Appendix B Sobolev Estimates

In this appendix we prove a Sobolev estimates that we use to derive pointwise
estimates of solutions to the wave equation on Schwarzschild-de Sitter.

######  Proposition B.1 (Sobolev Estimate for Schwarzschild-de Sitter).

Let $(\mathcal{M}_{\Lambda,m},g)$ be a subextremal member of the
Schwarzschild-de Sitter family of spacetimes, and fix $r_{0}\geq
r_{\mathcal{C}}$. Recall the family of Killing vectors $T$,
$\Omega_{i}:i=1,2,3$ that span the level sets $\Sigma_{r}$. Then there exists
a constant $C=C(\Lambda,m)>0$ such that for all $r\geq r_{0}$, we have

$\sup_{\Sigma_{r}}|\varphi|^{2}\leq\frac{C}{r^{3}}\int_{\Sigma_{r}}\Big{\\{}\varphi^{2}+(T\varphi)^{2}+\sum_{i=1}^{3}|\Omega_{i}\varphi|^{2}+\sum_{i=1}^{3}|\Omega_{i}T\varphi|^{2}+\sum_{i,j=1}^{3}|\Omega_{i}\Omega_{j}\psi|^{2}\Big{\\}}\textnormal{d}\mu_{\overline{g}_{r}}.$
(B.1)

###### Proof.

It was shown in [29] for a class of spacetimes that includes Schwarzschild-de
Sitter, we have for any tensor field $\theta$ that

$\sup_{t}\Big{(}\int_{S_{t}}r^{4}|\theta|^{4}\textnormal{d}\mu_{\not{g}}\Big{)}^{1/4}\lesssim_{\Lambda,m,r_{0}}\Big{(}\int_{\Sigma_{r}}|\theta|^{2}+r^{2}|\overline{\nabla}\theta|^{2}\textnormal{d}\mu_{\overline{g}_{r}}\Big{)}^{1/2},$

where $S_{t}$ is the sphere on $\Sigma_{r}$ on which the coordinate $t$ is
constant, and $\textnormal{d}\mu_{\not{g}}$ is the induced volume form on
$S_{t}$. Moreover, in [28] it was shown that

$\sup_{S_{t}}|\varphi|^{2}\lesssim_{\Lambda,m,r_{0}}\frac{1}{r}\Big{(}\int_{S_{t}}\varphi^{4}+r^{4}|\not{\nabla}\varphi|^{4}\textnormal{d}\mu_{\not{g}}\Big{)}^{1/2}.$

Combining these two estimates gives

$\displaystyle\sup_{\Sigma_{r}}|\varphi|^{2}$
$\displaystyle\lesssim\frac{1}{r}\sup_{t}\Big{(}\int_{S_{t}}\varphi^{4}+r^{4}|\not{\nabla}\varphi|^{4}\textnormal{d}\mu_{\not{g}}\Big{)}^{1/2}$
$\displaystyle\lesssim\frac{1}{r^{3}}\int_{\Sigma_{r}}\varphi^{2}+r^{2}|\overline{\nabla}\varphi|^{2}+r^{4}|\overline{\nabla}\not{\nabla}\varphi|^{2}\textnormal{d}\mu_{\overline{g}_{r}}.$

We can bound the covariant derivative restricted to the sphere and the
cylinder with spheres of radius $r$ like

$\displaystyle r^{2}|\overline{\nabla}\varphi|^{2}$
$\displaystyle\lesssim_{\Lambda,m,r_{0}}|T\varphi|^{2}+\sum_{i=1}^{3}|\Omega_{i}\varphi|^{2}$
$\displaystyle r^{2}|\not{\nabla}\varphi|^{2}$
$\displaystyle\lesssim_{\Lambda,m,r_{0}}\sum_{i=1}^{3}|\Omega_{i}\varphi|^{2},$

and therefore (B.1) holds. ∎

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# Building Open Knowledge Graph for Metal-Organic Frameworks (MOF-KG):
Challenges and Case Studies

Yuan An, Jane Greenberg, Xintong Zhao, Xiaohua Hu, Scott McCLellan, Alex
Kalinowski
College of Computing and Informatics
_Drexel University_
_Philadelphia, PA, USA_
<EMAIL_ADDRESS>
Fernando J. Uribe-Romo, Kyle Langlois, Jacob Furst
Department of Chemistry
_University of Central Florida_
_Orlando, FL, USA_
<EMAIL_ADDRESS>
Diego A. Gómez-Gualdrón, Fernando Fajardo-Rojas, Katherine Ardila
Department of Chemical and Biological Engineering
_Colorado School of Mines_
_Golden, CO, USA_
<EMAIL_ADDRESS>
###### Abstract

Metal-Organic Frameworks (MOFs) are a class of modular, porous crystalline
materials that have great potential to revolutionize applications such as gas
storage, molecular separations, chemical sensing, catalysis, and drug
delivery. The Cambridge Structural Database (CSD) reports 10,636 synthesized
MOF crystals which in addition contains ca. 114,373 MOF-like structures. The
sheer number of synthesized (plus potentially synthesizable) MOF structures
requires researchers pursue computational techniques to screen and isolate MOF
candidates. In this demo paper, we describe our effort on leveraging knowledge
graph methods to facilitate MOF prediction, discovery, and synthesis. We
present challenges and case studies about (1) construction of a MOF knowledge
graph (MOF-KG) from structured and unstructured sources and (2) leveraging the
MOF-KG for discovery of new or missing knowledge.

##  I INTRODUCTION

Metal-Organic Frameworks (MOFs) are materials that possess modular, porous
crystal structures that can be (conceptually) modified by “swapping”
constituent building blocks. MOF building blocks correspond to metal-based
clusters and organic linkers, which are interconnected in patterns described
by an underlying net (Figure 1). MOFs have great potential to revolutionize
applications such as gas storage, molecular separations, sensing, catalysis,
and drug delivery, primarily due to their usually high surface area and
exceptionally tunable properties [1]. But the combinatorics of building blocks
means that chemists have access to a (not fully explored) “material design
space” of trillions of structures. To date, there are 10,636 synthesized MOF
crystals reported in the Cambridge Structural Database (CSD) [2] which in
addition contains ca. 114,373 MOF-like structures. The sheer number has made
the identification of optimal MOFs (and subsequent) synthesis for a given
application a very challenging task. Thus, considerable efforts have been put
into developing effective computational techniques to screen and isolate
candidate MOF structures for the application of choice.

Figure 1: The Underlying Net and Framework of MOF-3

Previous efforts include the creation of large MOF databases and development
of high throughput automated workflows such as molecular simulation and
machine learning for MOF property prediction. MOF databases contains both
synthesized and “hypothesized” MOF structures [3]. When a hypothesized MOF
structure is identified as promising, it is unclear if and/or how such
structure can be realized synthetically. This is partly due to the
crystallization process leading to MOF formation not being fully understood.
Worse, a large amount of MOF synthesis data are not readily available for
computation but scattered in scientific literature. The separation between the
crystals’ structure information and their synthesis data exacerbates the
difficulty of screening MOFs.

We have undertaken a project aiming to leverage advanced knowledge graph
methods to facilitate MOF prediction, discovery, and synthesis. In this demo
paper, we describe the challenges for building such an open knowledge graph
for MOFs from structured and unstructured data. We present case studies on
addressing several challenges.

Knowledge graph (KG) represents the knowledge in a domain using a graphical
structure consisting of (typed-) vertices and (typed-) links. Despite the
growing number of materials science related databases [4], knowledge graphs
built for materials science domains are still rare [5, 6]. There is no
applicable knowledge graph for MOFs or Reticular Chemistry in general. The
_first challenge_ we face in building such a knowledge graph is to choose the
underlying graph architecture, for example, whether using RDF triple store or
labeled property graph model. Materials science data typically describe
numeric electronic, chemical, and physical properties. These properties may
connect through different relationships for guiding the design of
multifunctional materials. Additional properties can be extracted from the
literature. Considering the need of capturing the properties, we choose the
labeled property graph model for the underlying architecture and implement it
in the Neo4j platform.

Figure 2: Partial Backbone of the Data Model for the MOF-KG.

##  II DEFINING A DATA MODEL FOR THE MOF KNOWLEDGE GRAPH (MOF-KG)

The knowledge graph for MOFs (MOF-KG) should facilitate researchers in
reticular chemistry to practice their distinguished activities. The past
decades have seen an explosive increase in synthesis and characterization of
MOFs. A systematic workflow pattern in the field has emerged [7]. The pattern
is an iterative cycle consisting of 3 refinement steps: _Synthesis,
Activation_ , and _Analysis_. The workflow typically starts with a _Synthesis_
step where the researchers will screen and identify various crystalline
structures by considering many parameters including starting compounds, molar
ratios, temperature, concentration, reaction additives, modulators, solvents,
and reaction time. The researchers can apply computational techniques such as
high-throughput synthesis or screening to aid their experimental process [8].

Next at _Activation_ , the researchers will assess the permanent porosity and
architectural stability of the crystal by removing all guest molecules
(including solvent) from the pores of the framework without causing collapse
of its structure. The third step is _Analysis_ where the researchers will
identify and characterize the physical and chemical properties of the crystal.
Here, the researchers may resort to analytical, computational, and machine
learning tools to study and characterize porosity [9], gas uptake [10],
stability [11], and other important properties [12]. The workflow will cycle
through these 3 steps iteratively until successful results are obtained.

Due to the rapid development in reticular chemistry, there is no a general
agreed system of nomenclature for describing MOFs and related activities. A
couple of initiatives have worked on standardizing terminologies [13, 14],
although the diversity in the focus and the scientific inquiry has led to a
variety of terminological usages for this class of compounds. The _second
challenge_ in building the MOF-KG is to define a data model to conceptualize
MOF and its related activities corresponding to the workflow pattern.

To address this challenge, we define a data model for the MOF-KG with 4 major
areas: _synthesis, structure, atomic composition_ , and _publication_. Figure
2 illustrates a partial backbone of the data model with main concepts and
relationships. Each concept and relationship has its own set of properties.
The data model is continuously being refined and extended with newly
identified concepts and additional relationships.

Figure 3: Excerpt of the MOF-KG Showing the Connections between Journals,
Authors, the Same Solvent, and MOFs.

##  III BUILDING THE MOF-KG FROM THE CSD MOF COLLECTION

(a)

(b)

Figure 4: Extracting synthesis procedure steps from synthesis paragraph.

To instantiate the data model with instances for building the KG, we extract
MOF related entities, relationships, and properties from both structured and
unstructured sources. For structured sources, we currently focus on the
Cambridge Structural Database (CSD) curated by the Cambridge Crystallographic
Data Centre (CCDC), a world-leading organization that compiles and maintains
small-molecule organic and metal-organic crystal structures. An important
feature of the CSD is that it contains successful structures (crystals) that
have been realized in experiment, with crystal structure experimentally
measured and solved using diffraction techniques (X-rays, neutrons,
electrons). The database contains data such as crystal symmetry, atom
positions, occupancy, and displacements in the form of a CIF (crystallographic
information file) which is a highly standard format for storing
crystallographic structural data and metadata. The _third challenge_ for us is
to map the structured information in CSD to the MOF-KG described by the data
model. To address the challenge, we develop a schema mapping tool to
(semi-)automatically extract, transform, and load data from CSD to the MOF-KG.

Once the MOF-KG is populated with the instances extracted from the CSD
database, we can issue queries against the knowledge graph to explore
connections among the MOF related entities. Figure 3 shows an excerpt from the
resultant graph after issuing the following query: _“retrieve the authors and
publication journals of the MOFs that have the same solvent.”_

##  IV AUGMENTING THE MOF-KG WITH SCHOLARLY ARTICLES

While the curated databases such as CSD contain significant amounts of
information about crystal structures, scholarly articles that are rapidly
growing in quantities contain rich knowledge resources of the synthesis
procedures. However, computers cannot recognize the sequence of synthesis
actions reported in plain text (Figure 4(a)) [15]. Hence, the _fourth
challenge_ is to extract synthesis conditions from unstructured text to
augment the MOF-KG (Figure 4(b)). There is seldom a standardized way for
reporting synthesis conditions. Every researcher presents synthesis in
different manners, e.g., in main article, implicit or explicit, in appendix or
supplementary information file. Also, these conditions are often reported
incomplete, as many researchers have pre-assumptions of their readership. This
incomplete/assumed information may hinder immediate reproducibility of the
synthesis.

Some effort has been put into the use of natural language processing (NLP)
techniques [16, 17, 18, 19, 20, 21] for synthesis extraction. Existing
techniques include rule-based (e.g., regular expression, pre-defined rules)
and deep learning/machine learning based approaches [20, 21]. As a case study,
we apply a recently developed rule-based NLP approach [19] to extract the
synthesis information from 114 articles that match 114 MOFs in the CSD
Collection. There are 46 synthesis routes were recognized and they turned out
to be accurate. However, only three solvent records were extracted. We
manually examined ten synthesis procedures reported in articles. The results
indicate that solvents are indeed reported in text but not recognized by the
extraction algorithm. One possible reason is that solvent information is
described in various contexts and different ways (e.g., water, N,N-dimethyl
formamide, DMF).

Rule-based approaches are convenient to implement but brittle to the text
context. They suffer from the low recall problem. Deep learning and general
machine learning based approaches show better performance than rule-based
approaches, but they suffer from the lack of annotated training corpus.
Currently, we are developing weakly-supervised information extraction
algorithms to address this problem.

##  V USING THE MOF-KG FOR DISCOVERING NEW AND MISSING KNOWLEDGE

Knowledge graphs often suffer from incompleteness. For example, even the
state-of-the-art NLP techniques cannot extract all available information from
text. A flurry of research has been conducted on knowledge graph completion by
predicting missing links [22]. A notable approach is to learn knowledge graph
representations, that is, low-dimensional embedding vectors for downstream
classification and prediction [23]. The MOF-KG is inevitably incomplete due to
either incomplete information in the CSD database or incomplete extraction
from text. We aim to develop MOF-KG related link prediction techniques for
discovering new and missing information.

As a case study, we choose to predict the ‘HAS_SOLVENT’ links in the MOF-KG.
Solvent information is an important variable for MOF synthesis and 97% of the
solvent information is missing in the CSD MOF collection. We extract the
publication, author, atom, and bond information for the 268 MOFs that have
solvent information in the CSD collection. We apply various knowledge graph
embedding models including _TransE, ConvE, ComplEx, DistMult_ and _SimplE_ on
the data set. We randomly sample 20% as the test data and predict the
‘HAS_SOLVENT’ relation for them. The results are reported in Table I. We use
the following rank-based evaluation metrics:

  * •

MRR (Inverse Harmonic Mean Rank): higher is better, range $[0,1]$.

  * •

Hits@K (with K as one of $\\{1,5,10\\}$): higher is better, range $[0,1]$.

  * •

AMRI (Adjusted Arithmetic Mean Rank Index): higher is better, range $[-1,1]$.
AMRI=0 means the model is not better than random scoring.

KGE Model | MRR | AMRI | Hits@10 | Hits@5 | Hits@1
---|---|---|---|---|---
TransE | 0.15 | 0.97 | 0.38 | 0.25 | 0.04
ConvE | 0.12 | 0.94 | 0.21 | 0.18 | 0.07
ComplEx | 0 | -0.13 | 0 | 0 | 0
DistMult | 0.25 | 0.99 | 0.5 | 0.42 | 0.14
SimplE | 0 | 0.11 | 0 | 0 | 0
TABLE I: Apply Various Knowledge Graph Embedding (KGE) Models for Predicting
Missing Solvent information for MOFs

The results show that the DistMult KG embedding model achieved the best result
in terms of all the metrics. The AMRI is close to 1 and Hits@10 is 0.5. With
the simple information of publication and atomic element, a KG embedding can
successfully predict important missing information. Encouraged by these
results, we plan to develop and apply more sophisticated link prediction
approaches to the MOF-KG.

##  VI CONCLUSION AND NEXT STEPS

The challenges presented here show that building a knowledge graph for MOFs
requires a strong synergy among materials scientists, chemists,
informaticians, and data and computer scientists. The case studies demonstrate
the potentials of knowledge graphs for discovering information that is missing
in the original distributed and heterogeneous sources. This project is a part
of the effort undertaken by the NSF Institute for Data Driven Dynamical Design
(ID4). The next steps include enriching the MOF-KG and building user-friendly
natural language and chatbot query interfaces for domain scientists to conduct
further knowledge graph-empowered materials discovery.

## Acknowledgment

The research reported on in this paper is supported, in part, by the U.S.
National Science Foundation, Office of Advanced Cyberinfrastructure (OAC):
Grant: 1940239 and 1940307.

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