at the initial hypersurface $\Sigma_{t_{0}}$. We also assume this rescaled
initial data is close to that of the FLRW reference solution (see (2.1) and
(2.2)) in the sense that

(8.1)
$\mathcal{H}(t_{0})\color[rgb]{0,0,0}+\mathcal{H}_{top}(t_{0})\color[rgb]{0,0,0}+\mathcal{C}(t_{0})\color[rgb]{0,0,0}\leq\color[rgb]{0,0,0}\varepsilon^{2}$

is satisfied (with $\mathcal{H}$ and $\mathcal{C}$ as in Definition 3.6).
131313Essentially, this translates to smallness in
$H_{\gamma}^{\color[rgb]{0,0,0}19\color[rgb]{0,0,0}}$ and $C_{\gamma}^{17}$ .
For $\varepsilon=0$, the solution is the FLRW reference solution.

Then, the past maximal globally hyperbolic development $((0,t_{0}]\times
M,\overline{g},\phi)$ of this data within the Einstein scalar-field system
(1.1a)-(1.1c) in CMC gauge (2.10) with zero shift is foliated by the CMC
hypersurfaces $\Sigma_{s}=t^{-1}(\\{s\\})$, and one has

(8.2)
$\mathcal{H}(t)+\mathcal{C}(t)+\mathcal{C}_{\gamma}(t)\lesssim\color[rgb]{0,0,0}\varepsilon^{\frac{7}{4}}\color[rgb]{0,0,0}a(t)^{-c\varepsilon^{\frac{1}{8}}}$

for some $c>0$ and any $t\in(0,t_{0}]$. In particular, this implies the
following statements:

Asymptotic behaviour of solution variables: We denote the solution metric as
$\overline{g}=-n^{2}dt^{2}+g$, the second fundamental form (viewed as a
$(1,1)$-tensor) with respect to $\Sigma_{t}$ as $k$ and the volume form with
regards to $g$ on $\Sigma_{t}$ by ${\text{{vol}}}_{g}$. There exist a smooth
function $\Psi_{Bang}\in C_{\gamma}^{15}(M)$, a $(1,1)$-tensor field
$K_{Bang}\in C^{15}_{\gamma}(M)$ and a volume form ${\text{{vol}}}_{Bang}\in
C^{15}_{\gamma}(M)$ such that the following estimates hold for any
$t\in(0,t_{0}]$:

(8.3a) $\displaystyle\|n-1\|_{C^{l}_{\gamma}(\Sigma_{t})}\lesssim$
$\displaystyle\,\begin{cases}\varepsilon
a(t)^{4-c\varepsilon^{\frac{1}{8}}}&l\leq 14\\\ \varepsilon
a(t)^{2-c\varepsilon^{\frac{1}{8}}}&l=15\end{cases}$ (8.3b)
$\displaystyle\left\|a^{-3}{\text{{vol}}}_{g}-{\text{{vol}}}_{Bang}\right\|_{C^{l}_{\gamma}(\Sigma_{t})}\lesssim$
$\displaystyle\,\begin{cases}\varepsilon
a(t)^{4-c\varepsilon^{\frac{1}{8}}}&l\leq 14\\\ \varepsilon
a(t)^{2-c\varepsilon^{\frac{1}{8}}}&l=15\end{cases}$ (8.3c)
$\displaystyle\left\|a^{3}\partial_{t}\phi-(\Psi_{Bang}+C)\right\|_{C^{l}_{\gamma}(\Sigma_{t})}\lesssim$
$\displaystyle\,\begin{cases}\varepsilon
a(t)^{4-c\varepsilon^{\frac{1}{8}}}&l\leq 14\\\ \varepsilon
a(t)^{2-c\varepsilon^{\frac{1}{8}}}&l=15\end{cases}$ (8.3d)
$\displaystyle\left\|\phi-\int_{t}^{t_{0}}a(s)^{-3}\,ds\cdot(\Psi_{Bang}+C)\right\|_{\dot{C}^{l}_{\gamma}(\Sigma_{t})}\lesssim$
$\displaystyle\,\begin{cases}\varepsilon
a(t)^{4-c\varepsilon^{\frac{1}{8}}}&1\leq l\leq 14\\\ \varepsilon
a(t)^{2-c\varepsilon^{\frac{1}{8}}}&l=15\end{cases}$ (8.3e)
$\displaystyle\left\|a^{3}k-K_{Bang}\right\|_{C^{l}_{\gamma}(\Sigma_{t})}\lesssim$
$\displaystyle\,\begin{cases}\varepsilon
a(t)^{4-c\varepsilon^{\frac{1}{8}}}&l\leq 14\\\ \varepsilon
a(t)^{2-c\varepsilon^{\frac{1}{8}}}&l=15\end{cases}$

Further, these footprint states satisfy the equations141414These are precisely
the (generalized) Kasner relations, see Section 1.2.3.

(8.4a) $\displaystyle{(K_{Bang})^{a}}_{a}=$ $\displaystyle\,-\sqrt{12\pi}C\,,$
(8.4b) $\displaystyle
8\pi(\Psi_{Bang}+C)^{2}+{(K_{Bang})^{a}}_{b}{(K_{Bang})^{b}}_{a}=$
$\displaystyle\,12\pi C^{2}$

and remain close to the data of the reference solution in the following sense,
where $\mathbb{I}$ denotes the Kronecker symbol:

(8.5a)
$\displaystyle\left\|{\text{{vol}}}_{\gamma}-{\text{{vol}}}_{Bang}\right\|_{C^{15}_{\gamma}(M)}\lesssim$
$\displaystyle\,\varepsilon$ (8.5b)
$\displaystyle\left\|\Psi_{Bang}\right\|_{C^{15}_{\gamma}(M)}\lesssim$
$\displaystyle\,\varepsilon$ (8.5c)
$\displaystyle\left\|K_{Bang}+\sqrt{\frac{4\pi}{3}}C\mathbb{I}\right\|_{C^{15}_{\gamma}(M)}\lesssim$
$\displaystyle\,\varepsilon\,$

Additionally, there exists a $(0,2)$-tensor field $M_{Bang}\in
C^{15}_{\gamma}(M)$ satisfying

(8.6) $\left\|M_{Bang}-\gamma\right\|_{C^{15}_{\gamma}(M)}\lesssim\varepsilon$

and, with $\odot$ and $\exp$ meant in the matrix product and exponential sense
respectively, one has

(8.7)
$\left\|g\odot\exp\left[\left(\color[rgb]{0,0,0}-2\color[rgb]{0,0,0}\int_{t}^{t_{0}}a(s)^{-3}\,ds\right)\cdot
K_{Bang}\right]-M_{Bang}\right\|_{C^{l}_{\gamma}(\Sigma_{t})}\lesssim\begin{cases}\varepsilon
a(t)^{4-c\varepsilon^{\frac{1}{8}}}&l\leq 14\\\ \varepsilon
a(t)^{2-c\varepsilon^{\frac{1}{8}}}&l=15\,.\end{cases}$

Moreover, the Bel-Robinson variables $E$ and $B$ satisfy the following
estimates:

(8.8a) $\displaystyle\|E\|_{C^{16}_{\gamma}(\Sigma_{t})}\lesssim$
$\displaystyle\,\varepsilon a^{-4-c\varepsilon^{\frac{1}{8}}}$ (8.8b)
$\displaystyle\|B\|_{C^{l}_{\gamma}(\Sigma_{t})}\lesssim$
$\displaystyle\,\begin{cases}\varepsilon
a^{-2-c{\varepsilon}^{\frac{1}{8}}}&l\leq 15\\\ \varepsilon
a^{-4-c\varepsilon^{\frac{1}{8}}}&l\leq 16\end{cases}$

Causal disconnectedness: Let $\bm{\alpha}$ be a past directed causal curve on
$(\color[rgb]{0,0,0}(0,t]\color[rgb]{0,0,0}\times M,\overline{g})$ for $t\leq
t_{0}$ with domain $[s_{1},s_{max})$ such that
$\bm{\alpha}(s_{1})\in\color[rgb]{0,0,0}\Sigma_{t}\color[rgb]{0,0,0}$ and
$s_{max}$ is maximal. Then, there exists a constant $\mathcal{K}>0$ that does
not depend on $\bm{\alpha}$ such that one has

(8.9)
$L[\bm{\alpha}]=\int_{s_{1}}^{s_{max}}\sqrt{(\gamma_{ab})_{\bm{\alpha}(s)}\dot{\bm{\alpha}}^{a}(s)\dot{\bm{\alpha}}^{b}(s)}\,ds\leq\mathcal{K}a(t)^{2-c\varepsilon^{\frac{1}{8}}}\,,$

where $\gamma$ is the negative Einstein spatial reference metric on $M$ (see
Definition 2.1).

Geodesic incompleteness: Let $\bm{\alpha}$ be a past directed, affinely
parametrized causal geodesic emanating from $\Sigma_{t_{0}}$ with parameter
time $\mathcal{A}$ normalized to $\mathcal{A}(t_{0})=0$. Then,

(8.10)
$\mathcal{A}(0)\leq\mathcal{K}_{1}\cdot\lvert\mathcal{A}^{\prime}(t_{0})\rvert\cdot
a(t_{0})^{1+K_{2}\varepsilon}\int_{0}^{t_{0}}a(s)^{-1-\mathcal{K}_{2}\varepsilon}\,ds<\infty\,,$

holds for suitable constants $\mathcal{K}_{1},\mathcal{K}_{2}>0$ that are
independent of $\bm{\alpha}$, and thus any such geodesic crashes into the Big
Bang hypersurface in finite affine parameter time. Hence, for sufficiently
points $p,q\in\Sigma_{t}$ with
$\text{dist}_{\gamma}(p,q)>2\mathcal{K}a(t)^{2-c\varepsilon^{\frac{1}{8}}}$,
the causal pasts of $p$ and $q$ cannot intersect.

Blow-up:

The norm $\lvert k\rvert_{g}$ behaves toward the Big Bang hypersurface as
follows: (8.11a) $\left\|a^{6}\lvert
k\rvert_{g}^{2}-(K_{Bang})^{i}_{j}(K_{Bang})^{j}_{i}\right\|_{C^{0}_{\gamma}(\Sigma_{t})}\lesssim\varepsilon
a^{4-c{\varepsilon}^{\frac{1}{8}}}$ Further, with $W[\overline{g}]$ denoting
the Weyl curvature and
$P[\overline{g}]=\text{{Riem}}[\overline{g}]-W[\overline{g}]$, (8.11b)
$\left\|a^{12}P_{\alpha\beta\gamma\delta}P^{\alpha\beta\gamma\delta}-\frac{5}{3}\cdot(8\pi)^{2}(\Psi_{Bang}+C)^{4}\right\|_{C^{0}_{\gamma}(M)}\lesssim\varepsilon
a^{4-c{\varepsilon}^{\frac{1}{8}}}\,$ is satisfied, whereas there exists a
scalar footprint $W_{Bang}\in C^{15}_{\gamma}(M)$ such that one has (8.11c)
$\left\|a^{12}W_{\alpha\beta\gamma\delta}W^{\alpha\beta\gamma\delta}-W_{Bang}\right\|_{C^{0}(M)}\lesssim\varepsilon
a^{2-c{\varepsilon}^{\frac{1}{8}}}\,.$ Here, $W_{Bang}$ is a fourth order
polynomial in $\hat{K}_{Bang}=K_{Bang}+\sqrt{\frac{4\pi}{3}}C\mathbb{I}$ and
$\Psi_{Bang}$ and satisfies (8.11d)
$\|W_{Bang}\|_{C^{15}_{\gamma}(M)}\lesssim\varepsilon\,.$ Finally, the scalar
curvature $R[\overline{g}]$ and the Ricci curvature invariant
$\text{{Ric}}[\overline{g}]_{\alpha\beta}\text{{Ric}}[\overline{g}]^{\alpha\beta}$
blow-up with the asymptotics (8.11e)
$\displaystyle\|a^{6}R[\overline{g}]-8\pi(\Psi_{Bang}+C)^{2}\|_{C^{0}(M)}\lesssim$
$\displaystyle\,\varepsilon a^{4-c\varepsilon^{\frac{1}{8}}}\,\text{and}$
(8.11f)
$\displaystyle\|a^{12}\text{{Ric}}[\overline{g}]_{\alpha\beta}\text{{Ric}}[\overline{g}]^{\alpha\beta}-(8\pi)^{2}(\Psi_{Bang}+C)^{4}\|\lesssim$
$\displaystyle\,\varepsilon a^{4-c\varepsilon^{\frac{1}{8}}}\,,$ and the
Kretschmann scalar
$\mathcal{K}=\text{{Riem}}[\overline{g}]_{\alpha\beta\gamma\delta}\text{{Riem}}[\overline{g}]^{\alpha\beta\gamma\delta}$
exhibits stable blow-up in the following sense: (8.11g)
$\color[rgb]{0,0,0}\left\|a^{12}\mathcal{K}-\frac{5}{3}\cdot(8\pi)^{2}(\Psi_{Bang}+C)^{4}-W_{Bang}\right\|_{C^{0}(M)}\lesssim\varepsilon
a^{2-c\varepsilon^{\frac{1}{8}}}\color[rgb]{0,0,0}$

######  Remark 8.3 (The solution variables exhibit AVTD behaviour).

The estimates (8.3a)-(8.3e) and (8.7) imply that the solution is
asymptotically velocity term dominated (AVTD) in the sense that, toward the
Big Bang singularity, they behave at leading order like solutions to the
(formal) velocity term dominated equations. These arise by dropping any terms
containing spatial derivatives in the decomposed Einstein system, i.e. in
(2.15a), (2.15b), (2.17a) and (2.18).

###### Proof.

As argued at the end of Section 3.4, we can assume without loss of generality
that our initial data is sufficiently regular. Hence, the local existence
statement in Lemma 3.14 and the initial data requirements (8.1) ensure that
there exists a local solution to the Einstein scalar-field system on
$[t_{1},t_{0}]\times M$ and that the bootstrap assumption (see Assumption
3.16) holds on $[t_{1},t_{0}]\times M$ with $t_{1}\in(0,t_{0})$ and
$\sigma=\varepsilon^{\frac{1}{16}}$Let $\mathfrak{t}\in(0,t_{0})$ be such that
$(\mathfrak{t},t_{0}]\times M$ is the maximal domain on which the solution
variables exist and satisfy the bootstrap assumptions. For contradiction, we
now assume that $\mathfrak{t}>0$ were to hold.

Due to Corollary 7.6, there exist (summarizing all updates) constants
$c_{1},K_{1}>0$ such that, for any $t\in(\mathfrak{t},t_{0}]$,

(8.12) $\mathcal{C}(t)\leq
K_{1}\color[rgb]{0,0,0}\varepsilon^{\frac{7}{2}}\color[rgb]{0,0,0}a(t)^{-c_{1}\varepsilon^{\frac{1}{8}}}\,$

If $\varepsilon$ is small enough such that
$K_{1}\varepsilon^{\frac{1}{8}}<K_{0}$ and
$c_{1}\varepsilon^{\frac{1}{8}}<c_{0}\sigma$ hold, this is a strict
improvement of the bootstrap assumption. Furthermore, argued exactly as in the
proof of [RS18b, Theorem 15.1], above improvement ensures none of the blow-up
criteria of Lemma 3.14 are satisfied if $\mathfrak{t}>0$ were to hold,
essentially as a direct consequence of (8.12). Hence, the solution could be
classically extended to a CMC hypersurface $\Sigma_{\mathfrak{t}}$
diffeomorphic to $M$ while satisfying the improved estimates by continuity,
and further to an interval $(\mathfrak{t}^{\prime},t_{0}]$ for some
$0<\mathfrak{t}^{\prime}<\mathfrak{t}$ on which the bootstrap assumptions must
then be satisfied, also by continuity. This contradicts the maximality of
$(\mathfrak{t},t_{0}]$.

Thus, the rescaled solution variables induce a unique solution to the Einstein
scalar-field system on $(0,t_{0}]\times M$ such that (8.12) is satisfied for
any $t\in(0,t_{0}]$. The core estimate (8.2) follows since Corollary 7.3 and
Corollary 7.6 now hold on $(0,t_{0}]$.

From (8.2), the asymptotic behaviour in (8.3a)-(8.3e) and (8.7) is established
as in [RS18b, Theorem 15.1], which we briefly outline: First, we note that
(8.3a) follows directly from (8.2). For the remaining estimates, the arguments
are similar, so consider for example $\partial_{t}\phi$: By the rescaled wave
equation (2.32a) and (8.3a), we have that

$\left\|\partial_{t}\Psi\right\|_{C^{l}_{\gamma}(\Sigma_{t})}\lesssim\begin{cases}\varepsilon
a^{1-c\varepsilon^{\frac{1}{8}}}&l\leq 14\\\ \varepsilon
a^{-1-c\varepsilon^{\frac{1}{8}}}&l=15\end{cases}$

Hence, for an arbitrary decreasing sequence $(t_{m})_{m\in\mathbb{N}},$ on
$(0,t_{0}]$ that converges to zero, we have

$\|\Psi(t_{m_{1}},\cdot)-\Psi(t_{m_{2}},\cdot)\|_{C^{l}_{\gamma}(M)}\lesssim\begin{cases}\varepsilon
a(t_{m_{1}})^{4-c\varepsilon^{\frac{1}{8}}}&l\leq 14\\\ \varepsilon
a(t_{m_{1}})^{2-c\varepsilon^{\frac{1}{8}}}&l=15\end{cases}$

for any $m_{1},m_{2}\in\mathbb{N},\,m_{1}<m_{2}$ by (2.6). This shows that
$\Psi(t_{m_{1}},\cdot)$ is a Cauchy sequence in $C_{\gamma}^{15}(M)$ and hence
there exists a limit function ${\Psi}_{Bang}\in C_{\gamma}^{15}(M)$ that
satisfies

$\|\Psi(t,\cdot)-({\Psi}_{Bang}-C)\|_{C^{l}_{\gamma}(M)}\lesssim\begin{cases}\varepsilon
a(t)^{4-c\varepsilon^{\frac{1}{8}}}&l\leq 14\\\ \varepsilon
a(t)^{2-c\varepsilon^{\frac{1}{8}}}&l=15\end{cases}$

for any $t\in(0,t_{0}]$. Since $\Psi=a^{3}n^{-1}\partial_{t}\phi-C$ holds by
definition, (8.3c) now follows by examining the Taylor expansion of $n^{-1}-1$
at $0$ using (8.3a).

(8.4a) follows directly from the CMC condition (2.10), the asymptotic
behaviour (8.3e) of $a^{3}k$ and the Friedman equation (2.3). (8.4b) follows
from asymptotic limit of the Hamiltonian constraint (2.16a) with (2.3),
(8.3a), (8.3c) and (8.3e) as well as (8.2) for lower order terms. (8.7)
follows exactly as in [RS18b, Theorem 15.1], and (8.5a)-(8.6) are a direct
result of the initial data assumptions and applying the respective asymptotic
estimates to $t=t_{0}$.

For the first estimate in (8.8b), we apply the momentum constraint (2.29d) to
get

$\lvert\nabla^{J}B\rvert_{G}=\color[rgb]{0,0,0}a^{-4}\lvert\nabla^{J}\bm{B}\rvert_{G}\color[rgb]{0,0,0}=\color[rgb]{0,0,0}a^{-2}\lvert\nabla^{J}\text{{curl}}_{G}\Sigma\rvert_{G}\color[rgb]{0,0,0}\lesssim
a^{-2}\lvert\nabla^{J+1}\Sigma\rvert_{G}$

and consequently, with Lemma 7.4 as well as (4.4g) and (8.2),

$\displaystyle\|B\|_{C^{15}_{\gamma}(\Sigma_{t})}\lesssim$
$\displaystyle\,a^{-c\sqrt{\varepsilon}}\|B\|_{C^{15}_{G}(\Sigma_{t})}+\varepsilon
a^{-2-c\sqrt{\varepsilon}}\cdot
P_{{15}}(\|G-\gamma\|_{C^{15}_{\gamma}(\Sigma_{t})})$ $\displaystyle\lesssim$
$\displaystyle\,a^{-2-c\sqrt{\varepsilon}}\|\Sigma\|_{C^{16}_{G}(\Sigma_{t})}+\varepsilon
a^{-2-c\sqrt{\varepsilon}}\cdot
P_{15}(\|G-\gamma\|_{C^{15}_{\gamma}(\Sigma_{t})})$ $\displaystyle\lesssim$
$\displaystyle\,\varepsilon a^{-2-c\varepsilon^{\frac{1}{8}}}\,.$

The remaining estimates in (8.8a) and (8.8b) are contained in (8.2). The
results (8.9) and (8.10) follow as in the proofs of (15.6) and (15.7) in
[RS18b, Theorem 15.1] from the asymptotic behaviour of the solution variables
in (8.3a)-(8.3e) and (8.7). We briefly sketch the proof of (8.10): Consider a
geodesic $\bm{\alpha}$ affinely parametrized by $\mathcal{A}$ as in the
statement. The geodesic equations then lead to the following estimate for some
suitable $\mathcal{K}>0$:

$\lvert\mathcal{A}^{\prime\prime}\rvert\leq\frac{\dot{a}}{a}\lvert\mathcal{A}^{\prime}\rvert+\mathcal{K}\left[\frac{\dot{a}}{a}\lvert
N\rvert+n^{-1}\lvert\partial_{t}N\rvert+n^{-1}\lvert\nabla
N\rvert_{g}+n\lvert\hat{k}\rvert_{g}\right]\lvert\mathcal{A}^{\prime}\rvert\,.$

The leading term is hereby arises from the mean curvature condition. Arguing
as with the elliptic estimates in Section 5, one can show that
$\lvert\partial_{t}N\rvert\lesssim\varepsilon
a^{-1-c\varepsilon^{\frac{1}{8}}}$. Thus, along with the other pointwise
bounds on $n$, $g$ and $\hat{k}$, one obtains

$\lvert\mathcal{A}^{\prime\prime}\rvert\leq\frac{\dot{a}}{a}\left(1+c\varepsilon\right)\lvert\mathcal{A}^{\prime}\rvert$

and consequently
$\lvert\mathcal{A}^{\prime}(t)\rvert\leq\lvert\mathcal{A}^{\prime}(t_{0})\rvert
a^{-1-c\varepsilon}$ by the Gronwall lemma. (8.10) follows by integrating.

Turning to the blow-up behaviour of geometric invariants, observe (8.11a) is a
direct consequence of (8.3e). Regarding (8.11c), we first compute using (2.19)
and standard algebraic manipulations that

$a^{12}W_{\alpha\beta\gamma\delta}W^{\alpha\beta\gamma\delta}=a^{12}\left(\color[rgb]{0,0,0}8\color[rgb]{0,0,0}\lvert
E\rvert_{g}^{2}+8\lvert
B\rvert_{g}^{2}\right)=\color[rgb]{0,0,0}8\color[rgb]{0,0,0}\lvert\bm{E}\rvert_{G}^{2}+8\lvert\bm{B}\rvert_{G}^{2}\,.$

By the rescaled constraint equation (2.29c), we have

$\bm{E}_{ij}=-\dot{a}a^{2}\Sigma_{ij}+(\Sigma\odot\Sigma)_{ij}-\left[\frac{8\pi}{3}\Psi^{2}+\frac{16\pi}{3}C\Psi\right]G_{ij}+\mathcal{O}\left(\varepsilon
a^{4-c\varepsilon^{\frac{1}{8}}}\right)\,,$

for $t\downarrow 0$. Further, by expanding (2.3) around $a=0$, we have
$\dot{a}a^{2}=\sqrt{\frac{4\pi}{3}}C+\mathcal{O}\left(a^{2}\right)$. Since
$\Sigma^{\sharp}$ and $\Psi$ converge to footprint states
$\hat{K}_{Bang}=K_{Bang}+\sqrt{\frac{4\pi C}{3}}\mathbb{I}$ and $\Psi_{Bang}$
in $C^{15}_{\gamma}(M)$ respectively, this shows that $W_{Bang}\in
C^{15}_{\gamma}(M)$ exists, is a fourth order polynomial in $\hat{K}_{Bang}$
and $\Psi_{Bang}$, and satisfies

$\left\|\lvert\bm{E}\rvert_{G}^{2}-\color[rgb]{0,0,0}\frac{1}{8}\color[rgb]{0,0,0}W_{Bang}\right\|_{C^{0}(M)}\lesssim\varepsilon
a^{2-c\varepsilon^{\frac{1}{8}}}$

as well as where (8.11d). Due to (8.8b), the $\lvert\bm{B}\rvert_{G}^{2}$-term
in the Weyl curvature scalar is negligible in comparison, and thus (8.11c)
immediately follows.
Furthermore, one has

$\displaystyle P_{\alpha\beta\gamma\delta}P^{\alpha\beta\gamma\delta}=$
$\displaystyle\,2\text{{Ric}}[\overline{g}]_{\alpha\beta}\text{{Ric}}[\overline{g}]^{\alpha\beta}-\frac{2}{9}R[\overline{g}]^{2},\,$

and the statement follows once more with (8.3c), (8.3a) as well as (8.2) for
lower orders. ∎

##  9\. Future stability

The goal of this section is to show the following theorem:

######  Theorem 9.1 (Future stability of Milne spacetime).

Let the rescaled initial data $(\bm{g},\bm{k},\nabla\phi,\phi^{\prime})$ on
$M$ be sufficiently close to $(\gamma,\frac{1}{3}\gamma,0,0)$ in $H^{5}\times
H^{4}\times H^{4}\times H^{4}$ on some initial hypersurface
$\Sigma_{\tau=\tau_{0}}$ (see Definition 9.4 and Assumption 9.7). Then, its
maximal globally hyperbolic development $(\overline{M},\overline{g},\phi)$
within the Einstein scalar-field system in CMCSH gauge is foliated by the CMC
Cauchy hypersurfaces $(\Sigma_{\tau})_{\tau\in[\tau_{0},0)}$, is future
(causally) complete and exhibits the following asymptotic behaviour:

$(\bm{g},\bm{k},\phi^{\prime},\nabla\phi)(\tau)\longrightarrow(\gamma,\frac{1}{3}\gamma,0,0)\text{
as }\tau\uparrow 0$

Since the control of geometric perturbations uses the same arguments as in
[AF20], the focus in this section will lie on dealing with the the scalar
field. The key idea is controlling decay of the scalar field using an
indefinite corrective term on top of the canonical energy (see Definition
9.6).

###  9.1. Preliminaries

####  9.1.1. Notation, gauge and spatial reference geometry

Within this section, we will decompose the Lorentzian metric as follows:

(9.1a) $\overline{g}=-n^{2}dt^{2}+g_{ab}(dx^{a}+X^{a})(dx^{b}+X^{b}dt)$ We
impose CMCSH gauge (see [AM04]) via (9.1b)
$t=\tau,\,g^{ij}(\Gamma^{a}_{ij}-\hat{\Gamma}^{a}_{ij})=0\,,$

where $\hat{\Gamma}$ refers to the Christoffel symbols with regards to the
spatial reference metric $\gamma$.
We extend the notation from the Big Bang stability analysis regarding
foliations, derivatives, indices and schematic term notation to this setting
(see Section 2.1). In particular, $\Sigma_{T}$ and $\Sigma_{\tau}$ will refer
to spatial hypersurfaces along which the logarithmic time $T$ (see (9.2c)) and
the mean curvature $\tau$ are constant (see (9.2c) on why these are
interchangeable), and we will write for example $\Sigma_{T=0}$ when inserting
a specific value to avoid potential ambiguity. We use similar notation for
scalar functions and tensors that depend on $T$ or, respectively, $\tau$.

For the extent of the future stability analysis, we have to introduce an
additional condition for the spatial geometry beyond Definition 2.1:

######  Definition 9.2 (Spectral condition for the Laplacian of the spatial
reference manifold).

Let $\mu_{0}(\gamma)$ to be the smallest positive eigenvalue of the Laplace
operator $-\Delta_{\gamma}=(\gamma^{-1})^{ab}\nabla_{a}\nabla_{b}$ acting on
scalar functions, where $(M,\gamma)$ is as in Definition 2.1. $(M,\gamma)$
additionally is assumed to satisfy

$\mu_{0}(\gamma)>\frac{1}{9}\,.$

######  Remark 9.3 (Manifolds that satisfy Definition 9.2).

The existing literature on spectra of $\Delta_{\gamma}$ usually considers
hyperbolic manifolds with sectional curvature $\kappa=-1$, and thus one needs
to verify that $\mu_{0}$ is strictly greater than $1$ to verify the analogue
of Definition 9.2 after rescaling.

Numerical work, e.g., [CS99, Ino01], provides evidence for over 250 compact
hyperbolic $3$-manifolds to satisfy this spectral bound, many of which are
closed. In particular, both [CS99]151515These results have to be interpreted
cautiously since the numerical method cannot detect eigenvalues below $1$. and
[Ino01] consider the smallest closed orientable hyperbolic $3$-manifold, the
Weeks space m003(-3,1), and compute that it falls under Definition 9.2 with
$\mu_{0}\approx 27,8$ in [CS99, Table IV] and $26\lessapprox\mu_{0}\lessapprox
27,8$ in [Ino01, p. 639, Table 2]. Moreover, as demonstrated in [Ino01, p.
642, Figure 6], many manifolds with small enough diameter $d$ satisfy this
condition. In fact, this is ensured by, for example, the analytical bound

$\mu_{0}\geq\max\left\\{\frac{\pi^{2}}{2d^{2}}-\frac{1}{2},\sqrt{\frac{\pi^{4}}{d^{4}}+\frac{1}{4}}-\frac{3}{2},\frac{\pi^{2}}{d^{2}}e^{-d}\right\\}\,\,$

(cf. [CZ95, Theorem 1.1-1.2] with $L=2$) which implies $\mu_{0}>10$ for Weeks
space, which has diameter $d\approx 0.843$. Furthermore, [Ino01] finds no
closed hyperbolic manifold violating this bound. We also note that it is
conjectured that, for any arithmetic hyperbolic 3-manifold, one at least has
$\mu_{0}\geq 1$ (see [Ber03, Conjecture 2.3]). In fact, this is tied to the
Ramanujan conjecture for automorphic forms. Finally, one can construct compact
hyperbolic manifolds where $\mu_{0}$ becomes arbitrarily small, see [Cal94,
Corollary 4.4].

####  9.1.2. Rescaled variables and Einstein equations

We will use the standard rescaling of the solution variables by $\tau$:

######  Definition 9.4 (Rescaled variables for future stability).

(9.2a)
$\displaystyle\bm{g}_{ij}=\tau^{2}g_{ij}\,,(\bm{g}^{-1})^{ij}=\tau^{-2}g^{ij}\,,\
\bm{\Sigma}_{ij}=\tau\hat{k}_{ij}$ (9.2b) $\displaystyle\bm{n}=\tau^{2}n\,,\
\hat{\bm{n}}=\frac{\bm{n}}{3}-1\,,\ \bm{X}^{a}=\tau X^{a}$ Furthermore, we
introduce the logarithmic time (9.2c)
$T=-\log\left(\frac{\tau}{\tau_{0}}\right)\,\Leftrightarrow\,\tau=\tau_{0}e^{-T}$
which satisfies $\partial_{T}=-\tau\partial_{\tau}$. Toward the future, $\tau$
increases from $\tau_{0}$ toward $0$, and thus $T$ increases from $0$ to
$\infty$. We additionally introduce: (9.2d)
$\displaystyle\widetilde{\partial_{0}}=\partial_{T}+\mathcal{L}_{\bm{X}}=-\tau(\partial_{\tau}-\mathcal{L}_{X})$
(9.2e)
$\displaystyle\phi^{\prime}=\bm{n}^{-1}\widetilde{\partial_{0}}\phi=n^{-1}(-\tau)^{-1}(\partial_{\tau}-\mathcal{L}_{X})\phi$
Moreover, for any scalar function $\zeta$, we denote by $\overline{\zeta}$ the
mean integral with regards to $(\Sigma_{T},\bm{g}_{T})$.

For symmetric $(0,2)$-tensors $h$, we define the perturbed Lichnerowicz
Laplacian

(9.3)
$\mathcal{L}_{\bm{g},\gamma}h_{ab}=-\frac{1}{\mu_{\bm{g}}}\hat{\nabla}_{k}\left((\bm{g}^{-1})^{kl}\mu_{\bm{g}}\hat{\nabla}_{l}h_{ab}\right)-2\text{{Riem}}[\gamma]_{akbl}(\bm{g}^{-1})^{kk^{\prime}}(\bm{g}^{-1})^{ll^{\prime}}h_{k^{\prime}l^{\prime}}\,.$

This operator satisfies

(9.4)
$\left(\text{{Ric}}[\bm{g}]-\text{{Ric}}[\gamma]\right)_{ij}=\frac{1}{2}\mathcal{L}_{\bm{g},\gamma}(\bm{g}-\gamma)_{ij}+J_{ij},\quad\|J_{ij}\|_{H^{l-1}}\lesssim\|\bm{g}-\gamma\|_{H^{l}}\,,$

see [AM03, Pf. of Theorem 3.1]. Under our conditions for the reference
geometry, [Krö15] implies that the smallest positive eigenvalue of
$\mathcal{L}_{\gamma,\gamma}$, denoted by $\lambda_{0}$, satisfies
$\lambda_{0}\geq\frac{1}{9}$, and that $\mathcal{L}_{\gamma,\gamma}$ has
trivial kernel. The spectral condition in Definition 9.2 is not necessary for
this to hold true.

We now collect the 3+1-decomposition of the Einstein scalar-field equations in
CMCSH gauge with the help of [AF20, (2.13)-(2.18)]:

######  Lemma 9.5 (Rescaled CMCSH equations).

The rescaled CMCSH Einstein scalar-field equations take the following form:

The constraint equations (9.5a)
$R[\bm{g}]-\lvert\bm{\Sigma}\rvert_{\bm{g}}^{2}-\frac{2}{3}=8\pi\left[\lvert\phi^{\prime}\rvert^{2}+\lvert\nabla\phi\rvert_{\bm{g}}^{2}\right],\
\text{{div}}_{\bm{g}}\bm{\Sigma}_{a}=8\pi\tau^{3}\phi^{\prime}\nabla_{b}\phi,$
the elliptic lapse and shift equations (9.5b)
$\displaystyle\left(\Delta_{\bm{g}}-\frac{1}{3}\right)\bm{n}=$
$\displaystyle\,\bm{n}\left(\lvert\bm{\Sigma}\rvert_{\bm{g}}^{2}+4\pi\left[\lvert\phi^{\prime}\rvert^{2}+\lvert\nabla\phi\rvert_{\bm{g}}^{2}\right]\right)-1,$
(9.5c)
$\displaystyle\Delta_{\bm{g}}\bm{X}^{a}+(\bm{g}^{-1})^{ab}\text{{Ric}}[\bm{g}]_{bm}\bm{X}^{m}=$
$\displaystyle\,2(\bm{g}^{-1})^{am}(\bm{g}^{-1})^{bn}\nabla_{b}\bm{n}\cdot\bm{\Sigma}_{mn}-(\bm{g}^{-1})^{ab}\nabla_{b}\hat{\bm{n}}+8\pi\bm{n}\tau^{3}\phi^{\prime}\nabla_{b}\phi$
$\displaystyle\,-2(\bm{g}^{-1})^{bk}((\bm{g}^{-1})^{cl}\bm{n}\cdot\bm{\Sigma}_{bc}-\nabla_{b}\bm{X}^{l})(\Gamma^{a}_{kl}-\hat{\Gamma}^{a}_{kl})\,,$
the geometric evolution equations (9.5d)
$\displaystyle\widetilde{\partial_{0}}\bm{g}_{ab}=$
$\displaystyle\,2\bm{n}\bm{\Sigma}_{ab}+2\hat{\bm{n}}\bm{g}_{ab}\,\,,$ (9.5e)
$\displaystyle\widetilde{\partial_{0}}(\bm{g}^{-1})^{ab}=$
$\displaystyle\,-2\bm{n}(\bm{g}^{-1})^{ac}(\bm{g}^{-1})^{bd}\bm{\Sigma}_{cd}-2\hat{\bm{n}}(\bm{g}^{-1})^{ab}\,,$
(9.5f) $\displaystyle\widetilde{\partial_{0}}\bm{\Sigma}_{ab}=$
$\displaystyle\,-2\bm{\Sigma}_{ab}-\bm{n}\left(\text{{Ric}}[\bm{g}]_{ab}+\frac{2}{9}\bm{g}_{ab}\right)+\nabla_{a}\nabla_{b}\bm{n}$
$\displaystyle\,+2\bm{n}\cdot(\bm{g}^{-1})^{mn}\bm{\Sigma}_{am}\bm{\Sigma}_{bn}-\frac{1}{3}\hat{\bm{n}}\bm{g}_{ab}-\hat{\bm{n}}\Sigma_{ab}-8\pi\bm{n}\nabla_{a}\phi\nabla_{b}\phi$
and the wave equation (9.5g)
$\widetilde{\partial_{0}}\phi^{\prime}=\langle\nabla\bm{n},\nabla\phi\rangle_{\bm{g}}+\bm{n}\Delta_{\bm{g}}\phi+(1-\bm{n})\phi^{\prime}\,.$

####  9.1.3. Energies and data assumptions

The proof will rely on the following corrected energy quantities:

######  Definition 9.6 (Energies for future stability).

(9.6a) $\displaystyle\mathbb{E}_{SF}^{(l)}=$
$\displaystyle\,(-1)^{l}\int_{M}\left[\phi^{\prime}\Delta_{\bm{g}}^{l}\phi^{\prime}-\phi\Delta_{\bm{g}}^{l+1}\phi\right]{\text{{vol}}}_{\bm{g}},\qquad\mathcal{C}_{SF}^{(l)}=(-1)^{l}\int_{M}(\phi-\overline{\phi})\Delta_{\bm{g}}^{l}\phi^{\prime}\,{\text{{vol}}}_{\bm{g}}$
(9.6b) $\displaystyle E_{SF}=$
$\displaystyle\,\sum_{m=0}^{4}\left(\mathbb{E}_{SF}^{(m)}+\frac{2}{3}\mathcal{C}_{SF}^{(m)}\right)$
(9.6c) $\displaystyle E_{\text{geom}}=$
$\displaystyle\,\sum_{m=1}^{5}\biggr{(}\frac{9}{2}\int_{M}\langle\bm{g}-\gamma,\mathcal{L}_{\bm{g},\gamma}^{m}(\bm{g}-\gamma)\rangle_{\bm{g}}{\text{{vol}}}_{\bm{g}}+\frac{1}{2}\int_{M}\langle
6\bm{\Sigma},\mathcal{L}_{\bm{g},\gamma}^{m-1}(6\bm{\Sigma})\rangle_{\bm{g}}{\text{{vol}}}_{\bm{g}}$
$\displaystyle\,\phantom{\sum_{m=1}^{l}}+c_{E}\int_{M}\langle
6\bm{\Sigma},\mathcal{L}_{\bm{g},\gamma}^{m-1}(\bm{g}-\gamma)\rangle_{\bm{g}}{\text{{vol}}}_{\bm{g}}\biggr{)}$

The constant $c_{E}$ is given by

(9.7) $c_{E}=\begin{cases}1&\lambda_{0}>\frac{1}{9}\\\
9(\lambda_{0}-\delta^{\prime})&\lambda_{0}=\frac{1}{9}\,,\end{cases}$

where $\delta^{\prime}>0$ is chosen to be small enough within the argument.

The Sobolev norms $H_{\bm{g}}^{l}$ and $C_{\bm{g}}^{l}$ are defined
analogously to Definition 3.3 and 3.4, with similar conventions on surpressing
time dependence in notation whereever possible. Since norms with respect to
$\bm{g}$ and $\gamma$ are equivalent under the bootstrap assumption (and
consequently throughout the entire argument), we will simply denote the norms
by $H^{l}$ and $C^{l}$ throughout unless the specific metric is crucial.

######  Assumption 9.7 (Initial data assumption).

The initial data on the spatial hypersurface $\Sigma_{T=0}$ is assumed to be
small in the following sense:

(9.8)
$\displaystyle\color[rgb]{0,0,0}\|\bm{g}-\gamma\|_{C^{3}}+\|\bm{\Sigma}\|_{C^{2}}+\|\hat{\bm{n}}\|_{C^{4}}+\|\bm{X}\|_{C^{4}}+\|\phi^{\prime}\|_{C^{2}}+\|\nabla\phi\|_{C^{2}}$
$\displaystyle\color[rgb]{0,0,0}+\|\bm{g}-\gamma\|_{H^{5}}+\|\bm{\Sigma}\|_{H^{4}}+\|\hat{\bm{n}}\|_{H^{6}}+\|\bm{X}\|_{H^{6}}+\|\phi^{\prime}\|_{H^{4}}+\|\nabla\phi\|_{H^{4}}\color[rgb]{0,0,0}\color[rgb]{0,0,0}$
$\displaystyle\,\leq\delta^{2}$

######  Remark 9.8 (Local well-posedness toward the future).

Under the above initial data assumption, local well-posedness is satisfied by
analogizing the arguments for local well-posedness in the vacuum setting (see
[AM03, Theorem 3.1]) with the matter coupling added. Since this only consists
of adding another wave equation to the hyperbolic system, the argument is
structurally unchanged given appropriate smallness assumptions on
$\phi^{\prime}$ and $\nabla\phi$ (where $\phi$ itself does not enter into the
Einstein system). As before, we can without loss of generality assume that the
initial is sufficiently regular to ensure that
$E_{geom},\,\mathcal{E}^{(l)}_{SF}$ and $\mathcal{C}_{SF}^{(l)}$ initially are
continuously differentiable (in time) for any $l\leq 4$.

######  Assumption 9.9 (Bootstrap assumption).

We assume that, on the bootstrap interval $T\in[0,T_{Boot})$, one has

(9.9)
$\displaystyle\color[rgb]{0,0,0}\|\bm{g}-\gamma\|_{C^{3}}+\|\bm{\Sigma}\|_{C^{2}}+\|\hat{\bm{n}}\|_{C^{4}}+\|\bm{X}\|_{C^{4}}+\|\phi^{\prime}\|_{\color[rgb]{0,0,0}C^{2}\color[rgb]{0,0,0}}+\|\nabla\phi\|_{C^{2}}\color[rgb]{0,0,0}$
$\displaystyle\color[rgb]{0,0,0}+\|\bm{g}-\gamma\|_{H^{5}}+\|\bm{\Sigma}\|_{H^{4}}+\|\hat{\bm{n}}\|_{H^{6}}+\|\bm{X}\|_{H^{6}}+\|\phi^{\prime}\|_{H^{4}}+\|\nabla\phi\|_{H^{4}}\color[rgb]{0,0,0}$
$\displaystyle\color[rgb]{0,0,0}\,\leq\delta
e^{-\frac{T}{2}}\,.\color[rgb]{0,0,0}$

We only choose not to use “$\lesssim$”-notation in the above assumptions for
notational convenience in some technical computations. As before, $\delta$ can
be chosen to have been sufficiently small for the following estimates to hold
and for the decay estimates we derive from the bootstrap assumptions to be
strict improvements. Moreover, note that (9.9) is satisfied since all of the
norms are continuous in time (see Remark 9.8)

Before moving on to the energy estimates, we quickly collect the following
immediate consequence of the bootstrap assumptions:

######  Lemma 9.10 (Sobolev estimate for the curvature).

The following estimate holds for any $l\in\mathbb{N}_{0}$:

(9.10a)
$\left\|\text{{Ric}}[\bm{g}]+\frac{2}{9}\bm{g}\right\|_{H^{l}}\lesssim\|\bm{g}-\gamma\|_{H^{l+2}}+\|\bm{g}-\gamma\|_{H^{l+1}}^{2}$
Under the bootstrap assumptions, this implies (9.10b)
$\left\|\text{{Ric}}[\bm{g}]+\frac{2}{9}\bm{g}\right\|_{C^{1}}+\left\|\text{{Ric}}[\bm{g}]+\frac{2}{9}\bm{g}\right\|_{H^{3}}\lesssim\delta
e^{-\frac{T}{2}}$

###### Proof.

By (9.4), one has

$\displaystyle\left\|\text{{Ric}}[\bm{g}]+\frac{2}{9}\bm{g}\right\|_{H^{l}}\leq$
$\displaystyle\,\frac{1}{2}\|\mathcal{L}_{\bm{g},\gamma}(\bm{g}-\gamma)\|_{H^{l}}+K\|\bm{g}-\gamma\|_{H^{l+1}}^{2}\,$

for some suitably large $K>0$ and that $\mathcal{L}_{\bm{g},\gamma}$ is
elliptic. This implies the first inequality, while the latter follows from
directly from the bootstrap assumption (9.9) and by applying the standard
Sobolev embedding. ∎

###  9.2. Elliptic estimates

We briefly collect the elliptic estimates for lapse and shift:

######  Lemma 9.11 (Elliptic estimates for lapse and shift).

Let $l\in\\{3,4,5,6\\}$. Then, one has $\bm{n}\in(0,3)$ (thus
$\hat{\bm{n}}\in(-1,0)$) and the following estimates hold:

(9.11a) $\displaystyle\|\hat{\bm{n}}\|_{H^{l}}\lesssim$
$\displaystyle\color[rgb]{0,0,0}\,\delta
e^{-\frac{T}{2}}\|\bm{\Sigma}\|_{H^{l-2}}+\delta^{2}e^{-T}\|\bm{g}-\gamma\|_{H^{l-2}}+\delta
e^{-\frac{T}{2}}\left[\|\phi^{\prime}\|_{H^{l-2}}+\|\nabla\phi\|_{H^{l-2}}\right]\color[rgb]{0,0,0}$
(9.11b) $\displaystyle\|\bm{X}\|_{H^{l}}\lesssim$
$\displaystyle\,\color[rgb]{0,0,0}\delta
e^{-\frac{T}{2}}\|\bm{\Sigma}\|_{H^{l-2}}+\delta
e^{-\frac{T}{2}}\|\bm{g}-\gamma\|_{H^{l-1}}+\delta
e^{-\frac{T}{2}}\left[\|\phi^{\prime}\|_{H^{l-2}}+\|\nabla\phi\|_{H^{l-2}}\right]\color[rgb]{0,0,0}$

###### Proof.

The pointwise bounds on $\bm{n}$ follow via (9.5b) and the maximum principle
as in Lemma 4.1. For the remaining estimates, applying elliptic regularity
theory to (9.5b) and (9.5c) implies:

$\displaystyle\|\hat{\bm{n}}\|_{H^{l}}\lesssim$
$\displaystyle\,\color[rgb]{0,0,0}\|\bm{\Sigma}\|_{C^{\lfloor\frac{l-2}{2}\rfloor}}\|\bm{\Sigma}\|_{H^{l-2}}+\|\nabla\phi\|_{C^{2}}^{2}\|\bm{g}-\gamma\|_{H^{l-2}}\color[rgb]{0,0,0}$
$\displaystyle\,\color[rgb]{0,0,0}+\left[\|\nabla\phi\|_{C^{2}}\left(1+\|\bm{g}-\gamma\|_{C^{2}}\right)+\|\phi^{\prime}\|_{C^{2}}\right]\left[\|\phi^{\prime}\|_{H^{l-2}}+\|\nabla\phi\|_{H^{l-2}}\right]\color[rgb]{0,0,0}$
$\displaystyle\|\bm{X}\|_{H^{l}}\lesssim$
$\displaystyle\,\color[rgb]{0,0,0}\|\bm{\Sigma}\|_{C^{\lfloor\frac{l-2}{2}\rfloor}}\|\bm{\Sigma}\|_{H^{l-2}}+\|\bm{g}-\gamma\|_{H^{l-1}}^{2}+\|\nabla\phi\|_{C^{1}}\|\bm{g}-\gamma\|_{H^{l-3}}\color[rgb]{0,0,0}$
$\displaystyle\,\color[rgb]{0,0,0}+\left[\|\nabla\phi\|_{C^{2}}^{2}\left(1+\|\bm{g}-\gamma\|_{C^{2}})+\|\phi^{\prime}\|_{C^{2}}\right)\right]\left[1+\|\hat{\bm{n}}\|_{C^{2}}\right]\left[\|\phi^{\prime}\|_{H^{l-2}}+\|\nabla\phi\|_{H^{l-2}}\right]\color[rgb]{0,0,0}$

The statement then follows by inserting (9.9). ∎

###  9.3. Scalar field energy estimates

####  9.3.1. Near-coercivity of $E_{SF}$

We will be able to prove a decay estimate via a Gronwall argument only for the
corrected energy $E_{SF}$ . Hence, we first need to verify that this energy
controls the solution norms, for which we first show that it controls the
“canonical” scalar field energies:

######  Lemma 9.12 (Positivity of corrected scalar field energies).

Let

$Q=\frac{\sqrt{1+9q}-1}{\sqrt{1+9q}}\ \text{with}\
q=\frac{1}{2}\left(\mu_{0}(\gamma)-\frac{1}{9}\right)\,.$

Then, for any $l\in\\{0,1,2,3,4\\}$ and $\delta>0$ small enough, one has

(9.12)
$Q\mathbb{E}_{SF}^{(l)}\leq\mathbb{E}_{SF}^{(l)}+\frac{2}{3}\mathcal{C}_{SF}^{(l)},\quad\text{hence}\quad
Q\sum_{m=0}^{4}\mathbb{E}_{SF}^{(l)}\leq E_{SF}$

###### Proof.

We denote the smallest positive eigenvalue of $\Delta_{\bm{g}}$ acting on
scalar functions on $\Sigma_{T}$ as $\mu_{0}(\bm{g}_{T})$. By the bootstrap
assumption (9.9) and since $\mu_{0}$ depends continuously on the metric, we
obtain the following for small enough $\delta>0$:

$\mu_{0}(\bm{g}_{T})\geq\mu_{0}(\gamma)-\frac{1}{2}\left(\mu_{0}(\gamma)-\frac{1}{9}\right)\geq\frac{1}{9}+q$

By the Poincaré inequality applied on $(\Sigma_{T},\bm{g}_{T})$ (see [CBM01,
p.1037]), above spectral bound implies the following for any $\zeta\in
H^{1}(\Sigma_{T})$:

(9.13)
$\|\zeta-\overline{\zeta}\|_{L^{2}_{\bm{g}}(\Sigma_{T})}^{2}\leq\mu_{0}(\bm{g}_{T})^{-1}\|\nabla\zeta\|_{L^{2}_{\bm{g}}(\Sigma_{T})}^{2}\leq\left(\frac{1}{9}+q\right)^{-1}\|\nabla\zeta\|_{L^{2}_{\bm{g}}(\Sigma_{T})}^{2}$

For $l=0$, this means

$\displaystyle\mathbb{E}_{SF}^{(0)}+\frac{2}{3}\mathcal{C}_{SF}^{(0)}\geq$
$\displaystyle\,\|\phi^{\prime}\|_{L^{2}_{\bm{g}}}^{2}+\|\nabla\phi\|_{L^{2}_{\bm{g}}}^{2}-\frac{2}{3}\|\phi-\overline{\phi}\|_{L^{2}_{\bm{g}}}\|\phi^{\prime}\|_{L^{2}_{\bm{g}}}$
$\displaystyle\geq$
$\displaystyle\,\|\phi^{\prime}\|_{L^{2}_{\bm{g}}}^{2}+\|\nabla\phi\|_{L^{2}_{\bm{g}}}^{2}-2\left(1+9q\right)^{-\frac{1}{2}}\|\nabla\phi\|_{L^{2}_{\bm{g}}}\|\phi^{\prime}\|_{L^{2}_{\bm{g}}}$
$\displaystyle\geq$
$\displaystyle\,\frac{\sqrt{1+9q}-1}{\sqrt{1+9q}}\mathbb{E}_{SF}^{(0)}\,.$

For $l=1$, notice that we can rewrite $\mathcal{C}_{SF}^{(1)}$ as follows:

$\mathcal{C}_{SF}^{(1)}=\int_{M}\langle\nabla\phi,\nabla\phi^{\prime}\rangle_{\bm{g}}\,{\text{{vol}}}_{\bm{g}}=\int_{M}\left\langle\nabla\phi,\nabla\left(\phi^{\prime}-\overline{\phi^{\prime}}\right)\right\rangle_{\bm{g}}\,{\text{{vol}}}_{\bm{g}}=-\int_{M}\left(\phi^{\prime}-\overline{\phi^{\prime}}\right)\Delta_{\bm{g}}\phi\,{\text{{vol}}}_{\bm{g}}$

Hence, applying (9.13) to $\zeta=\phi^{\prime}$ yields

$\mathbb{E}_{SF}^{(1)}+\frac{2}{3}\mathcal{C}_{SF}^{(1)}\geq\mathbb{E}_{SF}^{(1)}-2\left(1+9q\right)^{-\frac{1}{2}}\|\nabla\phi^{\prime}\|_{L^{2}_{\bm{g}}}\|\Delta_{\bm{g}}\phi\|_{L^{2}_{\bm{g}}}\geq\frac{\sqrt{1+9q}-1}{\sqrt{1+9q}}\mathbb{E}_{SF}^{(1)}\,.$

For $l=2,3,4$, notice
$\overline{\Delta_{\bm{g}}\phi}=\overline{\Delta_{\bm{g}}\phi^{\prime}}=\overline{\Delta_{\bm{g}}^{2}\phi}=0$
holds due to the divergence theorem, hence the argument proceeds as in
$l=0,1$. ∎

######  Lemma 9.13 (Near-coercivity of corrected scalar field energy).

For any scalar function $\zeta$ and $k\in\\{1,2\\}$, one has the following
under the bootstrap assumptions:

$\displaystyle\int_{M}\lvert\nabla^{2}\zeta\rvert_{\bm{g}}^{2}\,{\text{{vol}}}_{\bm{g}}\lesssim$
$\displaystyle\,\int_{M}\lvert\Delta_{\bm{g}}\zeta\rvert_{\bm{g}}^{2}+\lvert\nabla\zeta\rvert_{\bm{g}}^{2}\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle\|\zeta\|_{\dot{H}^{2k}}^{2}\lesssim$
$\displaystyle\,\|\Delta_{\bm{g}}^{k}\zeta\|_{L^{2}}^{2}+\left(\|\zeta\|_{\dot{H}^{2k-1}}^{2}+\|\zeta\|_{\dot{H}^{2k-2}}^{2}\right)+\|\nabla\zeta\|_{C^{1}}^{2}\left\|\text{{Ric}}[\bm{g}]+\frac{2}{9}\bm{g}\right\|_{H^{2k-2}}^{2}$
$\displaystyle\|\nabla\zeta\|_{\dot{H}^{2k}}^{2}\lesssim$
$\displaystyle\,\|\nabla\Delta_{\bm{g}}^{k}\zeta\|_{L^{2}}^{2}+\left(\|\nabla\zeta\|_{\dot{H}^{2k-1}}^{2}+\|\nabla\zeta\|_{\dot{H}^{2k-2}}^{2}\right)+\|\nabla\zeta\|_{C^{2}}^{2}\left\|\text{{Ric}}[\bm{g}]+\frac{2}{9}\bm{g}\right\|_{H^{2k-2}}^{2}$

Consequently, the following estimate holds:

(9.14)
$\displaystyle\|\phi^{\prime}\|_{H^{4}}^{2}+\|\nabla\phi\|_{H^{4}}^{2}\lesssim$
$\displaystyle\,E_{SF}^{(4)}+\left(\|\phi^{\prime}\|_{C^{2}}^{2}+\|\nabla\phi\|_{C^{2}}^{2}\right)\left\|\text{{Ric}}[\bm{g}]+\frac{2}{9}\bm{g}\right\|_{H^{2}}^{2}$

###### Proof.

The inequalities for $\zeta$ follows from the same arguments as Lemma 4.5,
except that we have $\|\text{{Ric}}[\bm{g}]\|_{C^{1}_{\bm{g}}}\lesssim
1+\delta\lesssim 1$ by Lemma 9.10. The final estimate then follows by applying
these estimates to $\zeta=\phi^{\prime}$ and $\zeta=\phi$ and applying Lemma
9.12. ∎

####  9.3.2. Preparations for energy estimates

Before proving the energy estimate, we need to establish two technical lemmas:
First, we collect a formula to differentiate integrals, and then some
estimates needed to deal with the mean value of $\phi$ in the base level
correction term.

######  Lemma 9.14 (Differentiation of integrals, future stability version).

For any diffentiable function $\zeta$, one has

(9.15)
$\partial_{T}\int_{M}\zeta{\text{{vol}}}_{\bm{g}}=\int_{M}\left(\widetilde{\partial_{0}}\zeta+3\hat{\bm{n}}\zeta\right)\,{\text{{vol}}}_{\bm{g}}\,.$

###### Proof.

As in the proof of (4.12), we obtain

$\displaystyle\partial_{T}\int_{M}\zeta{\text{{vol}}}_{\bm{g}}=$
$\displaystyle\,\int_{M}\partial_{T}\zeta+\frac{\partial_{T}\mu_{\bm{g}}}{\mu_{\bm{g}}}\zeta\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle=$
$\displaystyle\,\int_{M}\partial_{T}\zeta+3\hat{\bm{n}}\zeta-\frac{1}{2}(\bm{g}^{-1})^{ab}\mathcal{L}_{\bm{X}}\bm{g}_{ab}\zeta\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle=$
$\displaystyle\,\int_{M}\partial_{T}\zeta+3\hat{\bm{n}}\zeta-\text{{div}}_{\bm{g}}\bm{X}\cdot\zeta\,{\text{{vol}}}_{\bm{g}}$

The statement now follows by applying Stokes’ theorem to the final term and
rearranging. ∎

######  Lemma 9.15 (Decay estimate for the integrated time derivative).

For any $T>0$, we have

(9.16)
$\int_{\Sigma_{T}}\phi^{\prime}\,{\text{{vol}}}_{\bm{g}}=\left(\int_{\Sigma_{T=0}}\phi^{\prime}\,{\text{{vol}}}_{\bm{g}}\right)\cdot
e^{-2T}\,.$

Consequently, the bootstrap assumptions imply

(9.17)
$\left\lvert\int_{\Sigma_{T}}\widetilde{\partial_{0}}\overline{\phi}\cdot\phi^{\prime}\,{\text{{vol}}}_{\bm{g}}\right\rvert\lesssim\,\delta^{3}e^{-\color[rgb]{0,0,0}\frac{5}{2}\color[rgb]{0,0,0}T}$

for $\delta>0$ small enough.

###### Proof.

Using that the integral of $\text{{div}}_{\bm{g}}(\bm{n}\nabla\phi)$ vanishes,
we compute:

$\partial_{T}\left(\int_{M}\phi^{\prime}\,{\text{{vol}}}_{\bm{g}}\right)=\int_{M}\left(\widetilde{\partial_{0}}\phi^{\prime}+3\hat{\bm{n}}\phi^{\prime}\right)\,{\text{{vol}}}_{\bm{g}}=\int_{M}\left[(1-\bm{n})\phi^{\prime}+(\bm{n}-3)\phi^{\prime}\right]\,{\text{{vol}}}_{\bm{g}}=-2\left(\int_{M}\phi^{\prime}\,{\text{{vol}}}_{\bm{g}}\right)$

Hence, (9.16) precisely describes the solution to this ODE ($f^{\prime}=-2f$)
with prescribed initial value at $T=0$, and the initial data assumption (9.8)
implies

$\left\lvert\int_{M}\phi^{\prime}\,{\text{{vol}}}_{\bm{g}}\right\rvert\leq\|\phi^{\prime}\|_{C^{0}(\Sigma_{T=0})}{\text{{vol}}}_{\bm{g}}(\Sigma_{T=0})e^{-2T}\leq\delta^{2}e^{-2T}\,.$

Furthermore, one has by (9.15) that

(9.18)
$\color[rgb]{0,0,0}\partial_{T}{\text{{vol}}}_{\bm{g}}\left(\Sigma_{T}\right)=\int_{\Sigma_{T}}3\hat{\bm{n}}{\text{{vol}}}_{\bm{g}}\color[rgb]{0,0,0}$

Consequently, one has

$\widetilde{\partial_{0}}\overline{\phi}=\left[-\frac{\partial_{T}{\text{{vol}}}_{\bm{g}}\left(\Sigma_{T}\right)}{{\text{{vol}}}_{\bm{g}}\left(\Sigma_{T}\right)}\cdot\overline{\phi}+\frac{1}{{\text{{vol}}}_{\bm{g}}\left(\Sigma_{T}\right)}\int_{M}\left(\widetilde{\partial_{0}}\phi+3\hat{\bm{n}}\phi\right)\,{\text{{vol}}}_{\bm{g}}\right]=\int_{M}\left(\bm{n}\phi^{\prime}+3\hat{\bm{n}}(\phi-\overline{\phi})\right)\,{\text{{vol}}}_{\bm{g}}\,.$

By applying $\lvert\bm{n}\rvert<3$, the adapted Poincare inequality (9.13) and
the bootstrap assumptions (9.9), this implies

$\lvert\widetilde{\partial_{0}}\overline{\phi}\rvert\lesssim\|\phi^{\prime}\|_{L^{2}}+\|\nabla\phi\|_{L^{2}}\|\hat{\bm{n}}\|_{L^{2}_{\bm{g}}}\lesssim\delta
e^{-\frac{T}{2}}\,.$

(9.17) now follows by combining this with (9.16). ∎

####  9.3.3. Energy estimates

Now, we can collect the following estimates for the corrected scalar field
energies:

######  Lemma 9.16 (Base level estimate for the corrected scalar field
energy).

Under the bootstrap assumptions, the following estimate holds for some $K>0$:

(9.19) $\partial_{T}E_{SF}^{(0)}\leq\color[rgb]{0,0,0}-2+E_{SF}^{(0)}+K\delta
e^{-T}\sqrt{\mathcal{E}_{SF}^{(0)}}\sqrt{\left[\|\Sigma\|_{L^{2}}^{2}+\|\bm{g}-\gamma\|_{L^{2}}^{2}\right]}+K\delta^{3}e^{-\frac{5}{2}T}\color[rgb]{0,0,0}$

###### Proof.

We compute, using $[\widetilde{\partial_{0}},\nabla]\phi=0$,
$\widetilde{\partial_{0}}\phi=\bm{n}\phi^{\prime}$ and the rescaled wave
equation (9.5g):

$\displaystyle\partial_{T}\mathbb{E}_{SF}^{(0)}=$
$\displaystyle\,\int_{M}\left[2\widetilde{\partial_{0}}\phi^{\prime}\cdot\phi^{\prime}+2\left\langle\nabla\phi,\nabla\widetilde{\partial_{0}}\phi\right\rangle_{\bm{g}}+\left(\widetilde{\partial_{0}}\bm{g}^{-1}\right)^{ab}\nabla_{a}\phi\nabla_{b}\phi+3\hat{\bm{n}}\left(\lvert\phi^{\prime}\rvert^{2}+\lvert\nabla\phi\rvert_{\bm{g}}^{2}\right)\right]\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle=$
$\displaystyle\,\int_{M}\biggr{[}2\left(\langle\nabla\bm{n},\nabla\phi\rangle_{\bm{g}}+\bm{n}\Delta_{\bm{g}}\phi+(1-\bm{n})\phi^{\prime}\right)\phi^{\prime}-2(\bm{n}\phi^{\prime})\cdot\Delta_{\bm{g}}\phi$
$\displaystyle\,\,\phantom{\int_{M}}-2\bm{n}\langle\bm{\Sigma},\nabla\phi\nabla\phi\rangle_{\bm{g}}+3\hat{\bm{n}}\lvert\phi^{\prime}\rvert^{2}+\hat{\bm{n}}\lvert\nabla\phi\rvert_{\bm{g}}^{2}\biggr{]}\,{\text{{vol}}}_{\bm{g}}$

With $2(1-\bm{n})=-4-6\hat{\bm{n}}$, integration by parts and using the
bootstrap assumption (9.9) on $C$-norms, we get for some constant $K>0$ that
we update from line to line:

$\displaystyle\partial_{T}\mathbb{E}_{SF}^{(0)}\leq$
$\displaystyle\,\int_{M}-4\lvert\phi^{\prime}\rvert_{\bm{g}}^{2}\,{\text{{vol}}}_{\bm{g}}+K\left[\|\nabla\phi\|_{C^{0}}\|\hat{\bm{n}}\|_{H^{1}}\sqrt{\mathbb{E}_{SF}^{(0)}}+\left(\|\bm{\Sigma}\|_{C^{0}}+\|\hat{\bm{n}}\|_{C^{0}}\right)\mathbb{E}_{SF}^{(0)}\right]$
$\displaystyle\leq$
$\displaystyle\int_{M}-4\lvert\phi^{\prime}\rvert^{2}\,{\text{{vol}}}_{\bm{g}}+K\delta
e^{-\frac{T}{2}}\color[rgb]{0,0,0}\sqrt{\mathbb{E}_{SF}^{(0)}}\color[rgb]{0,0,0}\sqrt{\|\hat{\bm{n}}\|_{H^{1}}}$

Similarly and using the same evolution equations, we obtain:

$\displaystyle\partial_{T}\mathcal{C}_{SF}^{(0)}=$
$\displaystyle\,\int_{M}\left[\widetilde{\partial_{0}}\phi\cdot\phi^{\prime}-\widetilde{\partial_{0}}\overline{\phi}\cdot\phi^{\prime}+(\phi-\overline{\phi})\widetilde{\partial_{0}}\phi^{\prime}+3\hat{\bm{n}}(\phi-\overline{\phi})\phi^{\prime}\right]\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle=$
$\displaystyle\,\int_{M}\left[3\lvert\phi^{\prime}\rvert^{2}+3\hat{\bm{n}}\lvert\phi^{\prime}\rvert^{2}+\left(\phi-\overline{\phi}\right)\cdot\text{{div}}_{\bm{g}}\left(\bm{n}\nabla\phi\right)-2\left(\phi-\overline{\phi}\right)\phi^{\prime}-\widetilde{\partial_{0}}\overline{\phi}\cdot\phi^{\prime}\right]\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle\leq$
$\displaystyle\,-2\mathcal{C}_{SF}^{(0)}+\int_{M}3\left[\lvert\phi^{\prime}\rvert^{2}-3\lvert\nabla\phi\rvert_{\bm{g}}^{2}\right]\,{\text{{vol}}}_{\bm{g}}+3\|\hat{\bm{n}}\|_{C^{0}}\mathbb{E}_{SF}^{(0)}-\int_{M}\left(\widetilde{\partial_{0}}\overline{\phi}\cdot\phi^{\prime}\right)\,{\text{{vol}}}_{\bm{g}}$

Applying Lemma 9.15 to the last term, we get:

$\displaystyle\partial_{T}\mathcal{C}_{SF}^{(0)}\leq$
$\displaystyle-2\mathcal{C}_{SF}^{(0)}+K\delta
e^{-\frac{T}{2}}\mathbb{E}_{SF}^{(0)}+\color[rgb]{0,0,0}K\delta^{3}e^{-\frac{5}{2}T}\color[rgb]{0,0,0}$

Combining these two estimates, inserting (9.11a) and (9.12) as well as
updating $K$ yields:

$\displaystyle\partial_{T}E_{SF}^{(0)}=$
$\displaystyle\,\partial_{T}\mathbb{E}_{SF}^{(0)}+\frac{2}{3}\partial_{T}\mathcal{C}_{SF}^{(0)}$
$\displaystyle=$ $\displaystyle\,\int_{M}\left[\left(-4+\frac{2}{3}\cdot
3\right)\lvert\phi^{\prime}\rvert^{2}-\frac{2}{3}\cdot
3\lvert\nabla\phi\rvert_{\bm{g}}^{2}\right]\,{\text{{vol}}}_{\bm{g}}-2\cdot\frac{2}{3}\mathcal{C}_{SF}^{0}$
$\displaystyle\,+\color[rgb]{0,0,0}K\delta
e^{-\frac{T}{2}}\sqrt{\mathbb{E}_{SF}^{(0)}}\sqrt{\|\hat{\bm{n}}\|_{H^{1}}}+K\delta^{3}e^{-\frac{5}{2}T}\color[rgb]{0,0,0}$
$\displaystyle\leq$ $\displaystyle\color[rgb]{0,0,0}\,-2E_{SF}^{(0)}+K\delta
e^{-T}\sqrt{E^{(0)}_{SF}}\left(\|\Sigma\|_{L^{2}}+\|\bm{g}-\gamma\|_{L^{2}}+\sqrt{E^{(0)}_{SF}}\right)+\color[rgb]{0,0,0}K\delta^{3}e^{-\frac{5}{2}T}\color[rgb]{0,0,0}\color[rgb]{0,0,0}$

∎

######  Lemma 9.17 (Higher order estimates for the corrected scalar field
energy).

For any $l\in\\{1,\dots,4\\}$, the following estimate holds:

$\displaystyle\partial_{T}\left(\mathbb{E}_{SF}^{(l)}+\frac{2}{3}\mathcal{C}_{SF}^{(l)}\right)\leq$
$\displaystyle\,-2\left(\mathbb{E}_{SF}^{(l)}+\frac{2}{3}\mathcal{C}_{SF}^{(l)}\right)+K\delta
e^{-\frac{T}{2}}\left(\sum_{m=0}^{l}\sqrt{\mathbb{E}_{SF}^{(m)}}\right)\cdot$
$\displaystyle\,\qquad\cdot\color[rgb]{0,0,0}\left(\|\phi^{\prime}\|_{H^{l}}+\|\nabla\phi\|_{H^{l}}+\|\Sigma\|_{H^{l}}+\|\bm{g}-\gamma\|_{H^{l}}\right)\color[rgb]{0,0,0}$

###### Proof.

Starting with $l=2k,\,k\in\\{1,2\\}$, one calculates:

(9.20a) $\displaystyle\partial_{T}\mathbb{E}_{SF}^{(2k)}=$
$\displaystyle\,\int_{M}\biggr{[}2\Delta_{\bm{g}}^{k}\widetilde{\partial_{0}}\phi^{\prime}\cdot\Delta_{\bm{g}}^{k}\phi^{\prime}+2\langle\nabla\Delta_{\bm{g}}^{k}\phi,\nabla\Delta_{\bm{g}}^{k}\widetilde{\partial_{0}}\phi\rangle_{\bm{g}}$
(9.20b)
$\displaystyle\,\phantom{\int_{M}}+(\widetilde{\partial_{0}}{\bm{g}}^{-1})^{ab}\cdot\nabla_{a}\Delta_{\bm{g}}^{k}\phi\cdot\nabla_{b}\Delta_{\bm{g}}^{k}\phi+3\hat{\bm{n}}\left(\lvert\Delta^{k}\phi^{\prime}\rvert_{\bm{g}}^{2}+\lvert\nabla\Delta^{k}\phi\rvert_{\bm{g}}^{2}\right)$
(9.20c)
$\displaystyle\,\phantom{\int_{M}}+2[\widetilde{\partial_{0}},\Delta_{\bm{g}}^{k}]\phi^{\prime}\cdot\Delta_{\bm{g}}^{k}\phi^{\prime}+2\left\langle[\widetilde{\partial_{0}},\nabla\Delta_{\bm{g}}^{k}]\phi,\nabla\Delta_{\bm{g}}^{k}\phi\right\rangle_{\bm{g}}\biggr{]}\,{\text{{vol}}}_{\bm{g}}$
We insert the rescaled wave equation (9.5g) and
$\widetilde{\partial_{0}}\phi=\bm{n}\phi^{\prime}$ into the right hand side of
(9.20a) and obtain for some constant $K>0$:
$\displaystyle\eqref{eq:fESF-1}\leq$
$\displaystyle\,\int_{M}\biggr{[}-4\lvert\Delta_{\bm{g}}^{k}\phi^{\prime}\rvert^{2}-6\hat{\bm{n}}\lvert\Delta_{\bm{g}}^{k}\phi^{\prime}\rvert^{2}+\bm{n}\Delta_{\bm{g}}^{k+1}\phi\cdot\Delta_{\bm{g}}^{k}\phi^{\prime}\biggr{]}\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle\,+K\|\Delta^{k}\phi^{\prime}\|_{L^{2}}\left(\|\hat{\bm{n}}\|_{H^{2k+1}}\|\nabla\phi\|_{C^{0}}+\|\nabla\phi\|_{H^{2k}}\|\hat{\bm{n}}\|_{C^{2k}}\right)$
$\displaystyle\,+\int_{M}\left[-\bm{n}\Delta_{\bm{g}}^{k}\phi^{\prime}\cdot\Delta_{\bm{g}}^{k+1}\nabla\phi-3\langle\nabla\hat{\bm{n}},\nabla\Delta_{\bm{g}}^{k}\phi\rangle_{\bm{g}}\cdot\Delta_{\bm{g}}^{k}\phi^{\prime}\right]\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle\,+K\|\nabla\Delta_{\bm{g}}^{k}\phi\|_{L^{2}}\left(\|\nabla\phi\|_{H^{2k+1}}\|\phi^{\prime}\|_{C^{0}}+\|\hat{\bm{n}}\|_{C^{2k}}\|\phi^{\prime}\|_{H^{2k}}\right)$
$\displaystyle\leq$
$\displaystyle\,\int_{M}-4\lvert\Delta_{\bm{g}}^{k}\phi^{\prime}\rvert^{2}\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle\,+K\sqrt{\mathbb{E}_{SF}^{(2k)}}\cdot\Bigr{[}\left(\|\nabla\phi\|_{C^{0}}+\|\phi^{\prime}\|_{C^{0}}\right)\cdot\|\hat{\bm{n}}\|_{H^{2k+1}}+\left(\|\nabla\phi\|_{H^{2k}}+\|\phi^{\prime}\|_{H^{2k}}\right)\cdot\|\hat{\bm{n}}\|_{C^{2k}}\Bigr{]}$

For (9.20b), we use (9.5e) and the bootstrap assumption (9.9) to bound it by
$K\delta e^{-\frac{T}{2}}\mathbb{E}_{SF}^{(2k)}$. Regarding (9.20c), the
commutator formulas (12.1a)-(12.1b) imply the following:

$\displaystyle\|[\widetilde{\partial_{0}},\Delta_{\bm{g}}^{k}]\phi^{\prime}\|_{L^{2}}\lesssim$
$\displaystyle\,\color[rgb]{0,0,0}\|\bm{n}\|_{C^{2k-1}}\left(\|\phi^{\prime}\|_{C^{1}}\|\bm{\Sigma}\|_{\dot{H}^{2k-1}}+\|\bm{\Sigma}\|_{C^{2k-2}}\|\phi^{\prime}\|_{H^{2k}}\right)\color[rgb]{0,0,0}$
$\displaystyle\|[\widetilde{\partial_{0}},\nabla\Delta_{\bm{g}}^{k}]\phi\|_{L^{2}}\lesssim$
$\displaystyle\,\color[rgb]{0,0,0}\|\bm{n}\|_{C^{2k}}\left(\|\nabla\phi\|_{C^{1}}\|\bm{\Sigma}\|_{{H}^{2k}}+\|\bm{\Sigma}\|_{C^{2k-2}}\|\nabla\phi\|_{H^{2k}}\right)\color[rgb]{0,0,0}$

Summarizing, inserting the $C$-norm bounds from the bootstrap assumption (9.9)
and updating $K$, this implies

$\displaystyle\partial_{T}\mathbb{E}_{SF}^{(2k)}\leq$
$\displaystyle\,\int_{M}-4\lvert\Delta_{\bm{g}}^{k}\phi^{\prime}\rvert^{2}\,{\text{{vol}}}_{\bm{g}}+K\delta
e^{-\frac{T}{2}}\mathbb{E}_{SF}^{(2k)}$ $\displaystyle\,+K\delta
e^{-\frac{T}{2}}\sqrt{\mathbb{E}_{SF}^{(2k)}}\left(\|\phi^{\prime}\|_{H^{2k}}+\|\nabla\phi\|_{H^{2k}}\right)$
$\displaystyle\,+K\delta
e^{-\frac{T}{2}}\sqrt{\mathbb{E}_{SF}^{(2k)}}\left(\color[rgb]{0,0,0}\|\hat{\bm{n}}\|_{H^{2k+1}}+\|\Sigma\|_{H^{2k}}\color[rgb]{0,0,0}\right)$

Moving on to the corrective term, we compute:

(9.21a) $\displaystyle\partial_{T}\mathcal{C}_{SF}^{(2k)}=$
$\displaystyle\,\int_{M}\biggr{[}\Delta_{\bm{g}}^{k}\widetilde{\partial_{0}}\phi\cdot\Delta_{\bm{g}}^{k}\phi^{\prime}+\Delta_{\bm{g}}^{k}\phi\cdot\Delta_{\bm{g}}^{k}\widetilde{\partial_{0}}\phi^{\prime}$
(9.21b)
$\displaystyle\,\phantom{\int_{M}}+3\hat{\bm{n}}\cdot\Delta_{\bm{g}}^{k}\phi\cdot\Delta_{\bm{g}}^{k}\phi^{\prime}+[\widetilde{\partial_{0}},\Delta_{\bm{g}}^{k}]\phi\cdot\Delta_{\bm{g}}\phi^{\prime}+\Delta_{\bm{g}}^{k}\phi\cdot[\widetilde{\partial_{0}},\Delta_{\bm{g}}^{k}]\phi^{\prime}\biggr{]}\,{\text{{vol}}}_{\bm{g}}$

Inserting the evolution equations into the right hand side of (9.21a), we can
bound that line by

$\displaystyle\leq$
$\displaystyle\,\int_{M}\left[3\lvert\Delta_{\bm{g}}^{k}\phi^{\prime}\rvert^{2}+3\hat{\bm{n}}\lvert\Delta_{\bm{g}}^{k}\phi^{\prime}\rvert^{2}\right]{\text{{vol}}}_{\bm{g}}+K\|\hat{\bm{n}}\|_{C^{2k}}\|\phi^{\prime}\|_{H^{2k-1}}\|\Delta^{k}\phi^{\prime}\|_{L^{2}}$
$\displaystyle+\int_{M}\left[-2\Delta_{\bm{g}}^{k}\phi\cdot\Delta_{\bm{g}}^{k}\phi^{\prime}+3\hat{\bm{n}}\Delta_{\bm{g}}^{k}\phi\cdot\Delta_{\bm{g}}^{k}\phi^{\prime}+3\Delta_{\bm{g}}^{k}\phi\cdot\Delta_{\bm{g}}^{k+1}\phi+3\hat{\bm{n}}\Delta_{\bm{g}}^{k}\phi\cdot\Delta_{\bm{g}}^{k+1}\phi\right]\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle\,+K\left[\|\hat{\bm{n}}\|_{C^{2k}}\left(\|\nabla\phi\|_{H^{2k}}+\|\phi^{\prime}\|_{H^{2k-1}}\right)+\left(\|\nabla\phi\|_{C^{0}}+\|\phi^{\prime}\|_{C^{0}}\right)\|\hat{\bm{n}}\|_{H^{2k+1}}\right]\|\Delta_{\bm{g}}^{k}\phi\|_{L^{2}}$

Note that, after integrating by parts, the last two terms in the second line
can be bounded by

$\int_{M}-3\lvert\nabla\Delta_{\bm{g}}\phi\rvert_{\bm{g}}^{2}\,{\text{{vol}}}_{\bm{g}}+\|\hat{\bm{n}}\|_{C^{1}}(\|\nabla\Delta^{k}\phi\|_{L^{2}}+\|\Delta^{k}\phi\|_{L^{2}})\|\nabla\Delta^{k}\phi\|_{L^{2}}\,.$

For the terms in (9.21b), notice that the first term can be bounded by $\delta
e^{-\frac{T}{2}}\|\nabla\phi\|_{H^{2k-1}}\sqrt{\mathbb{E}_{SF}^{(2k)}}$, while
the commutator terms can be estimated as before, with

$\displaystyle\|[\widetilde{\partial_{0}},\Delta_{\bm{g}}^{k}]\phi\|_{L^{2}}\lesssim$
$\displaystyle\,\color[rgb]{0,0,0}\|\nabla\phi\|_{C^{0}}\|\bm{n}\|_{C^{0}}\|\bm{\Sigma}\|_{\dot{H}^{2k-1}}\color[rgb]{0,0,0}\color[rgb]{0,0,0}+\|\bm{n}\|_{C^{2k}}\|\bm{\Sigma}\|_{C^{2k-2}}\color[rgb]{0,0,0}\color[rgb]{0,0,0}\|\nabla\phi\|_{H^{2k-1}}\color[rgb]{0,0,0}$

Combining all of the above, we get

$\displaystyle\partial_{T}\mathcal{C}_{SF}^{(2k)}\leq$
$\displaystyle\,-2\mathcal{C}_{SF}^{(2k)}+\int_{M}\left[3\lvert\Delta_{\bm{g}}^{k}\phi^{\prime}\rvert-3\lvert\nabla\Delta_{\bm{g}}^{k}\phi\rvert_{\bm{g}}\right]\,{\text{{vol}}}_{\bm{g}}$
$\displaystyle\,+K\delta
e^{-\frac{T}{2}}\color[rgb]{0,0,0}\left[\|\phi^{\prime}\|_{H^{2k}}+\|\nabla\phi\|_{H^{2k}}+\|\hat{\bm{n}}\|_{H^{2k+1}}+\|\bm{\Sigma}\|_{H^{2k}}\right]\color[rgb]{0,0,0}$
$\displaystyle\,\phantom{+K}\cdot\left(\sqrt{\mathbb{E}_{SF}^{(2k)}}+\sqrt{\mathbb{E}_{SF}^{(2k-1)}}\right)$

Finally, combining both differential estimates yields the statement for
$l=2k$. For $l=2k-1,\,k\in\\{1,2\\}$, the argument is completely analogous and
hence omitted. ∎

###  9.4. Geometric variables

We can take the following results from prior literature, where we additionally
apply the elliptic estimates in Lemma 9.11:

######  Lemma 9.18 (Coercivity of geometric energies, [AM11, Lemma 7.4]).

For sufficiently small $\delta>0$, the following estimate holds:

(9.22) $\|\bm{g}-\gamma\|_{H^{5}}^{2}+\|\Sigma\|_{H^{4}}^{2}\lesssim
E_{\text{geom}}$

######  Lemma 9.19 (Geometric energy estimate, [AF20, Lemma 20]).

Let $\delta>0$ be chosen appropriately small, and let

(9.23) $\alpha=\begin{cases}1&\lambda_{0}>\frac{1}{9}\\\
1-3\sqrt{\delta^{\prime}}&\lambda_{0}=\frac{1}{9}\,,\end{cases}$

where $\delta^{\prime}>0$ is the same as in (9.7), in particular suitably
small. Then, there exists some constant $K>0$ such that the following estimate
holds:

(9.24) $\displaystyle\partial_{T}E_{\text{geom}}\leq$ $\displaystyle\,-2\alpha
E_{\text{geom}}+KE_{\text{geom}}^{\frac{3}{2}}+\color[rgb]{0,0,0}K\delta
e^{-\frac{T}{2}}\color[rgb]{0,0,0}\sqrt{E_{geom}}\left[\|\phi^{\prime}\|_{H^{4}}+\|\nabla\phi\|_{H^{4}}\right]$

###  9.5. Closing the bootstrap

Now, we can collect our estimates to improve the bootstrap assumptions:

######  Proposition 9.20 (Improved bounds for future stability).

Let the bootstrap assumption (see Assumption 9.9) be satisfied for
$T\in[0,T_{Boot})$ and assume the initial data holds at $T=0$ (see Assumption
9.7). For $\delta>0$ sufficiently small and $\alpha$ as in (9.23) with
$\delta^{\prime}>0$ sufficiently small, the following estimates hold:

(9.25a)
$\displaystyle\|\phi^{\prime}\|_{C^{2}}+\|\nabla\phi\|_{C^{2}}+\|\phi^{\prime}\|_{H^{4}}+\|\nabla\phi\|_{H^{4}}\lesssim$
$\displaystyle\,\delta^{\frac{3}{2}}\color[rgb]{0,0,0}e^{-\alpha
T}\color[rgb]{0,0,0}$ (9.25b)
$\displaystyle\|\bm{g}-\gamma\|_{C^{3}}+\|\Sigma\|_{C^{2}}+\|\bm{g}-\gamma\|_{H^{5}}+\|\Sigma\|_{H^{4}}\lesssim$
$\displaystyle\,\delta^{\frac{3}{2}}e^{-\alpha T}$ (9.25c)
$\displaystyle\|\hat{\bm{n}}\|_{C^{4}}+\|\bm{X}\|_{C^{4}}+\|\hat{\bm{n}}\|_{H^{6}}+\|\bm{X}\|_{H^{6}}\lesssim$
$\displaystyle\,\delta^{3}e^{-2\alpha T}$

###### Proof.

In the following, the positive constant $K$ that may be updated from line to
line.
Combining the estimate from Lemma 9.16 as well as those from Lemma 9.17 at
each level with Lemma 9.19 and applying the (near)-coercivity estimates (9.14)
and (9.22) to the right hand sides, we obtain:

$\displaystyle\partial_{T}\left(E^{(4)}_{SF}+E_{geom}\right)\leq$
$\displaystyle\,-2E_{SF}+K\delta
e^{-\frac{T}{2}}\sqrt{E_{SF}}\left(\sqrt{E_{SF}+\delta^{2}e^{-T}\left\|\text{{Ric}}[\bm{g}]+\frac{2}{9}\bm{g}\right\|_{H^{2}}^{2}}+\sqrt{E_{geom}}\right)$
$\displaystyle\,+K\delta^{3}e^{-\frac{5}{2}T}$ $\displaystyle\,-2\alpha
E_{geom}+KE^{\frac{3}{2}}_{geom}+K\delta
e^{-\frac{T}{2}}\sqrt{E_{geom}}\sqrt{E_{SF}+\delta^{2}e^{-T}\left\|\text{{Ric}}[\bm{g}]+\frac{2}{9}\bm{g}\right\|_{H^{2}}^{2}}$

Applying (9.10a) to the curvature norms, as well as (9.22) to the resulting
norms on $\bm{g}-\gamma$ and (9.9) (which implies
$\sqrt{E_{geom}}\lesssim\delta e^{-\frac{T}{2}}$), this becomes

$\partial_{T}\left(E^{(4)}_{SF}+E_{geom}\right)\leq-2\alpha(E_{SF}^{(4)}+E_{geom})+K\delta
e^{-\frac{T}{2}}\left(E_{geom}+E_{SF}^{4}\right)+K\delta^{3}e^{-\frac{5}{2}T}\,.$

and consequently, since $\alpha\leq 1$,

$\partial_{T}\left[e^{2\alpha
T}\left(E^{(4)}_{SF}+E_{geom}\right)\right]\lesssim\delta
e^{-\frac{T}{2}}\cdot e^{2\alpha
T}\left(E^{(4)}_{SF}+E_{geom}\right)+\delta^{3}e^{-\frac{T}{2}}$

The Gronwall lemma, along with the initial data assumption (9.8), now implies

(9.26) $E^{(4)}_{SF}+E_{geom}\lesssim\delta^{3}e^{-2\alpha T}\,.$

Lemma 9.18 and the standard Sobolev embedding then imply (9.25b). In
particular, this means

(9.27)
$\left\|\text{{Ric}}[\bm{g}]+\frac{2}{9}\bm{g}\right\|_{H^{2}}\color[rgb]{0,0,0}\lesssim\color[rgb]{0,0,0}\delta^{\frac{3}{2}}e^{-\alpha
T}$

due to Lemma 9.10, and for $\delta^{\prime}>0$ small enough, inserting (9.26)
and (9.27) into (9.14) shows (9.25a). This in particular improves the
$C$-norms we just used the bootstrap assumption for, and reinserting this
improvement into (9.14) improves the bound to (9.26). Moreover, (9.25c)
follows directly from Lemma 9.11 and the already obtained improvements. ∎

###### Proof of Theorem 9.1.

The problem is locally well-posed as outlined in Remark 9.8. There then is
some maximal interval $[0,T_{Boot})$ for the logarithmic time $T$ – or,
equivalently, some maximal time interval $[\tau_{0},\tau_{Boot})$ – on which
the solution exists and the bootstrap assumptions (see Assumption 9.9) are
satisfied. By the analogous argument to the proof of Theorem 8.2, the decay
estimates in Proposition 9.20 are strictly stronger than the bootstrap
assumptions for small enough $\delta,\delta^{\prime}>0$. This implies
$T_{Boot}=\infty$ (resp. $\tau_{Boot}=0$) since we could else extend the
solution strictly beyond $T_{Boot}$ while also satisfying the bootstrap
assumptions. This proves the convergence statement in Theorem 9.1.

Finally, the decay estimates imply that $\lvert\nabla n\rvert_{g}$,
respectively $\lvert k\rvert_{g}$, are bounded by $\tau^{\alpha-1}$,
respectively $\tau^{\alpha+1}$, up to constant on $[\tau_{0},\tau)$. Since
$\alpha$ is at worst slightly smaller than $1$, both functions are integrable
on $[\tau_{0},0)$ for suitably small $\delta^{\prime}>0$ . By [CBC02], this
means the spacetime is future complete. ∎

##  10\. Global stability

To prove Theorem 1.2, what still needs to be shown is that initial data as in
Theorem 1.1 develops from $\Sigma_{t_{0}}$ to some hypersurface
$\Sigma_{t_{1}}\equiv\Sigma_{\tau(t_{1})}$ in its future such that the data in
$\Sigma_{t_{1}}$ is near-Milne in the sense of Assumption 9.7 and in CMCSH
gauge. From there, near-Milne stability yields the behaviour in the future of
$\Sigma_{\tau(t_{1})}$, and hence future stability of near-FLRW spacetimes as
in Theorem 1.2.

###### Proof of Theorem 1.2.

Within this proof, $t$ will denote the “physical” time coordinate used
throughout the Big Bang stability analysis, while $\tau$ denotes the mean
curvature time used within CMCSH gauge.

Consider initial data $(g,k,\nabla\phi,\partial_{0}\phi)$ induced on the CMC
hypersurface $\Sigma_{t_{0}}$ within $\overline{M}$ such that the rescaled
variables are close to FLRW reference data in the sense of Theorem 8.2.
Moreover, let $(\mathring{g},\mathring{k},\mathring{\pi},\mathring{\psi})$ be
the geometric initial data on $M$ that induce it via the embedding
$\iota:M\hookrightarrow\overline{M}$.

Notice that

$P:H^{20}_{\gamma}(M)\rightarrow
H^{18}_{\gamma}(M),\,Y^{i}\mapsto\Delta_{\gamma}Y^{i}+(\gamma^{-1})^{il}{\text{{Ric}}[\gamma]}_{lj}Y^{j}=\Delta_{\gamma}Y^{i}-\frac{2}{9}Y^{i}=0$

is an isomorphism since $\Delta_{\gamma}$ has no positive eigenvalues. Hence,
using [FK20, Theorem 2.5, Remark 2.6], there is an metric
$\mathring{g}^{\prime}$ isometric to $\mathring{g}$ that remains
$H^{18}_{\gamma}(M)$-close to $\gamma$ and satisfies

$((\mathring{g}^{\prime})^{-1})^{ij}\left(\Gamma[\mathring{g}^{\prime}]^{k}_{ij}-\hat{\Gamma}[\gamma]^{k}_{ij}\right)=0.$

Let $\theta\in\text{Diff}(M)$ be the diffeomorphism such that
$\theta^{\ast}\mathring{g}=\mathring{g}^{\prime}$, then the proof of [FK20,
Theorem 2.5] implies that $\theta$ can be chosen close to the identity map
within $H^{18}(\text{Diff}(M))$, and consequently that
$\theta^{\ast}\mathring{k}=\mathring{k}^{\prime},\,\theta^{\ast}\mathring{\pi}=\mathring{\pi}^{\prime}$
and $\theta^{\ast}\mathring{\psi}=\mathring{\psi}^{\prime}$ remain close to
$\frac{1}{3}\gamma,0$ and $C$ in $H^{18}_{\gamma}(M)$. By the same argument as
in Remark 8.1, we can now evolve this data locally and obtain a new initial
hypersurface $\Sigma^{\prime}$ close to $\Sigma_{t_{0}}$ that is in CMCSH
gauge and that $(g,k,\nabla\phi,\partial_{0}\phi)$ is close to the reference
data in the sense of Assumption 3.10, exchanging the initial time $t_{0}$ by
some close time $t_{0}^{\prime}$.

Since $\tau$ is strictly increasing, $t\equiv t(\tau)$ exists and we can
interchangeably view $a$ as a function in $t$ or $\tau$ with some abuse of
notation. The Friedman equation (2.3) implies $\partial_{t}a\geq\frac{1}{9}$
and thus $a(t)\geq\frac{1}{9}t$ on $(0,\infty)$, as well as

$-\tau=3\frac{\dot{a}}{a}=\frac{1}{a}+\langle\text{lower order terms}\rangle\
\text{as}\ t\to\infty\ (\text{resp. }\tau\to 0)\,.$

We choose $t_{1}>\max\\{1,t_{0}^{\prime}\\}$ large enough (resp.
$\tau(t_{1})\equiv\tau_{0}$ small enough) that the following estimates hold
for some small $\chi\in(0,\frac{1}{2})$ that depends only on $\delta$:

(10.1) $\displaystyle Ca(t_{1})^{-3}\tau(t_{1})^{-1}\leq$
$\displaystyle\,\chi$ (10.2) $\displaystyle-\tau(t_{1})\cdot{a(t_{1})}\in$
$\displaystyle\,[1-\chi,1+\chi]$

As the solution is Cauchy stable, i.e., it and its maximal time of existence
depend continuously upon the initial data,161616For the argument for Einstein
vacuum in CMCSH gauge, see [AM04, Theorem 3.1]. As with local existence, the
argument in the Einstein scalar-field system is largely identical since the
only difference amounts to coupling the hyperbolic parts of the system with a
further hyperbolic one. one can choose $\varepsilon>0$ in the analogue of
Assumption 3.10 small enough to ensure the following: The solution exists
until $t_{1}>t_{0}^{\prime}$ and
$(a^{-2}g,a\hat{k},\nabla\phi,a^{3}\widetilde{\partial_{0}}\phi)$ remain
$K\varepsilon$-close to $(\gamma,0,0,C)$ in $H_{\gamma}^{6}\times
H_{\gamma}^{5}\times H_{\gamma}^{5}\times H_{\gamma}^{5}$ for some suitable
$K>0$ along the slab $\cup_{s\in[t_{0}^{\prime},t_{1}]}\Sigma_{s}$. What now
remains to be shown is that this implies Assumption 9.7 in the sense that, if
$\varepsilon$ is small enough, $\delta$ can be made as small as necessary for
Theorem 9.1 to apply.

Note that the scalings in Definition 9.4 can be rewritten as follows:

$\displaystyle\bm{g}-\gamma=(\tau\cdot
a)^{2}\cdot(a^{-2}g-\gamma)+(\tau^{2}\cdot
a^{2}-1)\gamma,\quad\bm{\Sigma}=\frac{\tau}{a}(a\hat{k})$
$\displaystyle\phi^{\prime}=C\left(-\tau^{-1}\cdot
a^{-3}\right)+\left(-\tau^{-1}\cdot
a^{-3}\right)\cdot(a^{3}n^{-1}(\partial_{\tau}-\mathcal{L}_{X})\phi-C)$

Since (10.2) implies $\tau\cdot a$ is close to $-1$ at $t_{1}$, $\|(\tau\cdot
a)^{2}(a^{-2}g-\gamma)\|_{H^{6}}$ can be bounded by $\frac{\delta^{3}}{2}$ for
small enough $\varepsilon$. Choosing $\chi<\frac{\delta^{3}}{2}$ then implies
$\|\bm{g}-\gamma\|_{H^{6}(\Sigma_{\tau_{0}})}<\delta^{3}$. That
$\|\bm{\Sigma}\|_{H^{5}}$ can be made smaller than $\delta^{3}$ for small
enough $\varepsilon>0$ follows since $\frac{\tau}{a}$ behaves like
$\frac{1}{a^{2}}$ up to a constant by (10.2).
For the normal derivative of the wave, notice that $\lvert
C\left(-\tau^{-1}\cdot a^{-3}\right)\rvert$ is bounded by $\chi$ due to
(10.1), and that $-\tau^{-1}a^{-3}$ is equivalent to $a^{-2}$ by (10.2).
Hence, we can similarly ensure that $\phi^{\prime}$ is bounded in $H^{5}$ by
$\delta^{3}$. Since $\nabla\phi$ is not changed in either rescaling, and
bounds on lapse and shift (up to constant) follow from the elliptic estimates
in Lemma 9.11, it follows each individual norm in Assumption 9.7 can be
bounded by $\delta^{3}$ up to constants that depend only on $\gamma$, and
hence the initial data assumption itself can be satisfied for suitably small
$\delta>0$.
This proves that we can develop from initial data for the Big Bang stability
proof to near-Milne initial data within a CMCSH foliation, and thus we obtain
Theorem 1.2 from Theorem 1.1 and Theorem 9.1. ∎

##  11\. Appendix – Big Bang Stability

###  11.1. Basic formulas and estimates

####  11.1.1. Tools from elementary calculus

######  Lemma 11.1 (A Gronwall lemma).

Let $f,\chi,\xi:[a,b]\longrightarrow\mathbb{R}$ be continuous functions such
that $\chi\geq 0$, $\xi$ is decreasing and, for any $s\in[a,b]$,

$f(s)\leq\int_{s}^{b}\chi(r)f(r)\,dr+\xi(s)$

is satisfied. Then, for any $t\in[a,b]$, we have

$f(t)\leq\xi(t)\exp\left(\int_{t}^{b}\chi(r)\,dr\right)\,.$

###### Proof.

This follows by standard arguments as in [Dra03, Corollary 2-3].∎

######  Lemma 11.2 (A weak fundamental theorem of calculus for square roots).

Let $f:(0,t_{0}]\longrightarrow\mathbb{R}^{+}_{0}$ be a $C^{1}$-function.
Then, we have for any $t\in(0,t_{0}]$:

(11.1) $\sqrt{f(t)}\leq\sqrt{f(t_{0})}+\int_{t}^{t_{0}}\frac{\lvert
f^{\prime}(s)\rvert}{2\sqrt{f(s)}}\,ds$

###### Proof.

This follows from a straightforward application of the monotone convergence
theorem to $g_{n}=\sqrt{f+\frac{1}{n}}$. ∎

####  11.1.2. Levi-Civita tensor identites

Herein, we collect some basic identities for the Levi-Civita tensor
$\bm{\varepsilon}[g]$: Firstly, it satisfies the contraction identities, where
$\mathbb{I}^{a}_{b}$ denotes the Kronecker-symbol:

(11.2a)
$\displaystyle\bm{\varepsilon}^{ai_{1}i_{2}}\bm{\varepsilon}_{aj_{1}j_{2}}=$
$\displaystyle\,\mathbb{I}^{i_{1}}_{j_{1}}\mathbb{I}^{i_{2}}_{j_{2}}-\mathbb{I}^{i_{1}}_{j_{2}}\mathbb{I}^{i_{2}}_{j_{1}}$
(11.2b) $\displaystyle\bm{\varepsilon}^{abi}\bm{\varepsilon}_{abj_{2}}=$
$\displaystyle\,2\mathbb{I}^{i}_{j}$ (11.2c)
$\displaystyle\bm{\varepsilon}^{abc}\bm{\varepsilon}_{abc}=$
$\displaystyle\,6$ (11.2d) $\displaystyle\nabla\bm{\varepsilon}=$
$\displaystyle\,0$

The analogous formulas hold for $\bm{\varepsilon}[G]$ when raising indices
with regards to $G$ instead of $g$.
For a tracefree and symmetric $\Sigma_{t}$-tangent $(0,2)$-tensor
$\mathfrak{T}$ and a $\Sigma_{t}$-tangent $(0,2)$-tensor $\mathfrak{A}$, the
following simplified identites hold:

(11.3a) $\displaystyle(\mathfrak{T}\times\mathfrak{A})_{ij}$
$\displaystyle={\bm{\varepsilon}_{i}}^{ab}{\bm{\varepsilon}_{j}}^{pq}\mathfrak{T}_{ap}\mathfrak{A}_{bq}+\frac{1}{3}(\mathfrak{T}\cdot\mathfrak{A})g_{ij}$
(11.3b) $\displaystyle(\mathfrak{T}\times g)_{ij}$
$\displaystyle=-\mathfrak{T}_{ij}$ (11.3c) $\displaystyle(\mathfrak{T}\times
k)_{ij}$
$\displaystyle=-\frac{\tau}{3}\mathfrak{T}_{ij}+(\mathfrak{T}\times\hat{k})_{ij}$
Further, note the following formulas (for $\tilde{\mathfrak{T}}$ as
$\mathfrak{T}$, $\tilde{\mathfrak{A}}$ as $\mathfrak{A}$ and any
$\Sigma_{t}$-tangent $(0,1)$-tensor $\xi$) (see [AM04, p.30]): (11.3d)
$\displaystyle\text{{div}}_{g}(\mathfrak{A}\wedge\tilde{\mathfrak{A}})$
$\displaystyle=-\text{{curl}}\mathfrak{A}\cdot\tilde{\mathfrak{A}}+\mathfrak{A}\cdot\text{{curl}}\tilde{\mathfrak{A}}$
(11.3e) $\displaystyle\mathfrak{A}\cdot(\xi\wedge\tilde{\mathfrak{A}})$
$\displaystyle=-2\xi\cdot(\mathfrak{A}\wedge\tilde{\mathfrak{A}})$ (11.3f)
$\displaystyle\mathfrak{T}\cdot(\mathfrak{A}\times\tilde{\mathfrak{T}})$
$\displaystyle=(\mathfrak{T}\times\mathfrak{A})\cdot\tilde{\mathfrak{T}}$

####  11.1.3. Estimates on contracted tensors

######  Lemma 11.3.

Let $\mathfrak{S},\mathfrak{T}$ be traceless and symmetric
$\Sigma_{t}$-tangent $(0,2)$-tensors, $\mathfrak{M},\mathfrak{N}$ symmetric
$\Sigma_{t}$-tangent $(0,2)$-tensors and $\xi$ a $\Sigma_{t}$-tangent
$(0,1)$-tensor. We define $G,G^{-1}$ and $\lvert\cdot\rvert_{G}$ via (2.27a)).
Then:

(11.4a)
$\displaystyle\lvert\mathfrak{M}\odot_{G}\mathfrak{N}\rvert_{G}\leq\lvert\mathfrak{M}\rvert_{G}\lvert\mathfrak{N}\rvert_{G},$
$\displaystyle\quad\mathfrak{M}\odot_{g}\mathfrak{N}=a^{-2}\mathfrak{M}\odot_{G}\mathfrak{N}$
(11.4b)
$\displaystyle\lvert\mathfrak{S}\times_{G}\mathfrak{T}\rvert_{G}\lesssim\lvert\mathfrak{S}\rvert_{G}\lvert\mathfrak{T}\rvert_{G},$
$\displaystyle\quad(\mathfrak{S}\times\mathfrak{T})_{ij}=a^{-3}(\mathfrak{S}\times_{G}\mathfrak{T})_{ij}$
(11.4c)
$\displaystyle\lvert\mathfrak{S}\wedge_{G}\mathfrak{T}\rvert_{G}\leq\lvert\mathfrak{S}\rvert_{G}\lvert\mathfrak{T}\rvert_{G},$
$\displaystyle\quad\left(\mathfrak{S}\wedge\mathfrak{T}\right)_{l}=a^{-3}\left(\mathfrak{S}\wedge_{G}\mathfrak{T}\right)$
(11.4d)
$\displaystyle\lvert\xi\wedge_{G}\mathfrak{T}\rvert_{G}\leq\lvert\xi\rvert_{G}\lvert\mathfrak{T}\rvert_{G},$
$\displaystyle\quad\left(\xi\wedge\mathfrak{T}\right)_{ij}=a^{-1}(\xi\wedge_{G}\mathfrak{T})_{ij}$
(11.4e)
$\displaystyle\lvert\text{{curl}}_{G}\mathfrak{M}\rvert_{G}\lesssim\lvert\nabla\mathfrak{M}\rvert_{G},$
$\displaystyle\quad\text{{curl}}\mathfrak{M}_{ij}=a^{-1}\text{{curl}}_{G}\mathfrak{M}_{ij}$

###### Proof.

The estimates with respect to the unrescaled metric are direct consequences of
the contraction identities (11.2a)-(11.2c) replacing $g$ with $G$, and the
scalings follow simply by tracking the effects of the rescaling in Definition
2.9. In particular, note

(11.5)
${\bm{\varepsilon}[g]_{i}}^{cd}=g^{cj}g^{dk}\bm{\varepsilon}[g]_{ijk}=\left(a^{-2}(G^{-1})^{cj}\right)\left(a^{-2}(G^{-1})^{dk}\right)a^{3}\bm{\varepsilon}[G]_{ijk}=a^{-1}{\varepsilon[G]_{i}}^{\sharp
cd}\,.$

∎

###  11.2. Commutators

Herein, we collect a variety of commutators of spatial derivative operators
with each other as well as with time derivatives. While these mostly follow by
standard computations, we use the fact that our spatial hypersurfaces are
three-dimensional to significantly simplify the spatial commutator formulas,
and need to apply the rescaled equations from Proposition 2.10 for the time
derivative formulas.

For higher order commutators, we denote by $\mathfrak{J}$ terms within the
commutator formula that contribute junk terms at any point where this
commutator formula is used. Furthermore, in the following, $\zeta$ denotes a
scalar function on $\overline{M}$ and $\mathfrak{T}$ denotes a
$\Sigma_{t}$-tangent, symmetric $(0,2)$-tensor, always with sufficient
regularity for the equations to make sense. Moreover, recall the schematic
$\ast$-notation as introduced in subsection 2.1.8.

######  Corollary 11.4 (Schematic first order spatial commutators).

For $\zeta$ and $\mathfrak{T}$ as above, the following identities hold:

(11.6a) $\displaystyle[\Delta,\nabla]\zeta=$
$\displaystyle\,\text{{Ric}}[G]\ast\color[rgb]{0,0,0}\nabla\zeta\color[rgb]{0,0,0}$
(11.6b) $\displaystyle[\Delta,\nabla^{2}]\zeta=$
$\displaystyle\,\text{{Ric}}[G]\ast\nabla^{2}\zeta+\nabla\text{{Ric}}[G]\ast\nabla\zeta$
(11.6c) $\displaystyle[\Delta,\nabla]\mathfrak{T}=$
$\displaystyle\,\text{{Ric}}[G]\ast\nabla\mathfrak{T}+\nabla\text{{Ric}}[G]\ast\mathfrak{T}$
(11.6d) $\displaystyle[\Delta,\nabla^{2}]\mathfrak{T}=$
$\displaystyle\,\text{{Ric}}[G]\ast\nabla^{2}\mathfrak{T}+\nabla\text{{Ric}}[G]\ast\nabla\mathfrak{T}+\nabla^{2}\text{{Ric}}[G]\ast\mathfrak{T}$
(11.6e) $\displaystyle[\Delta,\text{{div}}_{G}]\mathfrak{T}=$
$\displaystyle\,\text{{Ric}}[G]\ast\nabla\mathfrak{T}+\nabla\text{{Ric}}[G]\ast\mathfrak{T}$
(11.6f)
$\displaystyle\color[rgb]{0,0,0}[\Delta,\text{{curl}}_{G}]\color[rgb]{0,0,0}=$
$\displaystyle\,\color[rgb]{0,0,0}\bm{\varepsilon}[G]\ast\left(\text{{Ric}}[G]\ast\nabla\mathfrak{T}+\nabla\text{{Ric}}[G]\ast\mathfrak{T}\right)\color[rgb]{0,0,0}$

###### Proof.

Since we are working in three spatial dimensions, the following identity
holds:

$\displaystyle\text{{Riem}}[G]_{ijkl}=$
$\displaystyle\,G_{ik}\text{{Ric}}[G]_{jl}-G_{il}\text{{Ric}}[G]_{jk}+G_{jl}\text{{Ric}}[G]_{ik}-G_{jk}\text{{Ric}}[G]_{il}$
$\displaystyle\,-\frac{1}{2}(G^{-1})^{mn}\text{{Ric}}[G]_{mn}(G_{ik}G_{jl}-G_{il}G_{jk})$

Hence, for any $I\in\mathbb{N}_{0}$, any $\nabla^{I}\text{{Riem}}[G]$-term
reduces to a sum of products and contractions of $\nabla^{I}\text{{Ric}}[G]$
with various metric tensors that are all surpressed in schematic notation.
With this in mind, the above statements are simply direct consequences of
standard commutation cormulas and (11.5). ∎

######  Lemma 11.5 (Higher order spatial commutators).

For $l\in\mathbb{N},\,l\geq 2$, the following formulas hold (and extend to
$l=1$ when dropping any term involving $\Delta^{l-2}$):

(11.7a) $\displaystyle[\Delta^{l},\nabla]\zeta=$
$\displaystyle\,\Delta^{l-1}\text{{Ric}}[G]\ast\nabla\zeta+\nabla\Delta^{l-2}\text{{Ric}}[G]\ast\nabla^{2}\zeta+\mathfrak{J}([\Delta^{l},\nabla]\zeta)$
(11.7b) $\displaystyle[\Delta^{l},\nabla^{2}]\zeta=$
$\displaystyle\,\nabla\Delta^{l-1}\text{{Ric}}[G]\ast\nabla\zeta+\nabla^{2}\Delta^{l-2}\text{{Ric}}[G]\ast\nabla^{2}\zeta+\mathfrak{J}([\Delta^{l},\nabla^{2}]\zeta)$
(11.7c) $\displaystyle[\Delta^{l},\nabla]\mathfrak{T}=$
$\displaystyle\,\nabla\Delta^{l-1}\text{{Ric}}[G]\ast\mathfrak{T}+\nabla^{2}\Delta^{l-2}\text{{Ric}}[G]\ast\nabla\mathfrak{T}+\mathfrak{J}([\Delta^{l},\nabla]\mathfrak{T})\,,$
(11.7d) $\displaystyle[\Delta^{l},\nabla^{2}]\mathfrak{T}=$
$\displaystyle\,\nabla^{2}\Delta^{l-1}\text{{Ric}}[G]\ast\mathfrak{T}+\nabla^{3}\Delta^{l-2}\text{{Ric}}[G]\ast\nabla\mathfrak{T}+\mathfrak{J}([\Delta^{l},\nabla^{2}]\mathfrak{T})\,,$
(11.7e) $\displaystyle[\Delta^{l},\text{{div}}_{G}]\mathfrak{T}=$
$\displaystyle\,\nabla\Delta^{l-1}\text{{Ric}}[G]\ast\mathfrak{T}+\nabla^{2}\Delta^{l-2}\text{{Ric}}[G]\ast\nabla\mathfrak{T}+\mathfrak{J}([\Delta^{l},\text{{div}}_{G}]\mathfrak{T})\,,$
(11.7f)
$\displaystyle\color[rgb]{0,0,0}[\Delta^{l},\text{{curl}}_{G}]\mathfrak{T}=$
$\displaystyle\color[rgb]{0,0,0}\,\bm{\varepsilon}[G]\ast\left(\nabla\Delta^{l-1}\text{{Ric}}[G]\ast\mathfrak{T}+\nabla^{2}\Delta^{l-2}\text{{Ric}}[G]\ast\nabla\mathfrak{T}\right)+\mathfrak{J}([\Delta^{l},\text{{curl}}_{G}]\mathfrak{T})\color[rgb]{0,0,0}\color[rgb]{0,0,0}$
with junk terms $\displaystyle\mathfrak{J}([\Delta^{l},\nabla]\zeta)=$
$\displaystyle\,\sum_{\begin{subarray}{c}I_{1}+I_{\zeta}=2(l-1),\\\
\,I_{\zeta}\geq
2\end{subarray}}\nabla^{I_{1}}\text{{Ric}}[G]\ast\nabla^{I_{\zeta}+1}\zeta+\sum_{m=0}^{l-2}\sum_{I_{1}+\dots+I_{l-m}+I_{\zeta}=2m}\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m}}\text{{Ric}}[G]\ast\nabla^{I_{\zeta}+1}\zeta$
$\displaystyle\mathfrak{J}([\Delta^{l},\nabla^{2}]\zeta)=$
$\displaystyle\,\sum_{\begin{subarray}{c}I_{1}+I_{\zeta}=2(l-1)+1,\\\
I_{1},I_{\zeta}\geq
2\end{subarray}}\nabla^{I_{1}}\text{{Ric}}[G]\ast\nabla^{I_{\zeta}+1}\zeta+\sum_{m=0}^{l-2}\sum_{I_{1}+\dots+I_{l-m}+I_{\zeta}=2m+1}\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m}}\text{{Ric}}[G]\ast\nabla^{I_{\zeta}+1}\zeta$
$\displaystyle\mathfrak{J}([\Delta^{l},\nabla]\mathfrak{T})=$
$\displaystyle\,\sum_{\begin{subarray}{c}I_{1}+I_{\mathfrak{T}}=2(l-1)+1,\\\
I_{\mathfrak{T}}\geq
2\end{subarray}}\nabla^{I_{1}}\text{{Ric}}[G]\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}+\sum_{m=0}^{l-2}\sum_{I_{1}+\dots+I_{l-m}+I_{\mathfrak{T}}=2m+1}\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m}}\text{{Ric}}[G]\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}$
$\displaystyle\mathfrak{J}([\Delta^{l},\nabla^{2}]\mathfrak{T})=$
$\displaystyle\,\sum_{\begin{subarray}{c}I_{1}+I_{\mathfrak{T}}=2l,\\\
I_{\mathfrak{T}}\geq
2\end{subarray}}\nabla^{I_{1}}\text{{Ric}}[G]\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}+\sum_{m=0}^{l-2}\sum_{I_{1}+\dots+I_{l-m}+I_{\mathfrak{T}}=2m+2}\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m}}\text{{Ric}}[G]\ast\nabla^{I_{\mathfrak{T}}}T$
$\displaystyle\mathfrak{J}([\Delta^{l},\text{{div}}_{G}]\mathfrak{T})=$
$\displaystyle\,\sum_{\begin{subarray}{c}I_{1}+I_{\mathfrak{T}}=2(l-1)+1,\\\
I_{\mathfrak{T}}\geq
2\end{subarray}}\nabla^{I_{1}}\text{{Ric}}[G]\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}+\sum_{m=0}^{l-2}\sum_{I_{1}+\dots+I_{l-m}+I_{\mathfrak{T}}=2m+1}\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m}}\text{{Ric}}[G]\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}$
$\displaystyle\color[rgb]{0,0,0}\mathfrak{J}([\Delta^{l},\text{{curl}}_{G}]\mathfrak{T})\color[rgb]{0,0,0}=$
$\displaystyle\color[rgb]{0,0,0}\bm{\varepsilon}[G]\ast\left[\sum_{\begin{subarray}{c}I_{1}+I_{\mathfrak{T}}=2(l-1)+1,\\\
I_{\mathfrak{T}}\geq
2\end{subarray}}\nabla^{I_{1}}\text{{Ric}}[G]\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}+\sum_{m=0}^{l-2}\sum_{\begin{subarray}{c}I_{1}+\dots+I_{l-m}+\\\
+I_{\mathfrak{T}}=2m+1\end{subarray}}\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m}}\text{{Ric}}[G]\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}\right]\color[rgb]{0,0,0}$

###### Proof.

The formulas follow by applying the formulas from Lemma 11.4 inductively. ∎

######  Lemma 11.6 (Time derivative commutators).

With respect to a solution to the Einstein scalar field system as in
Proposition 2.6, the following commutator formulas hold:

(11.8a) $\displaystyle[\partial_{t},\nabla_{i}]\zeta=$ $\displaystyle\,0$
(11.8b) $\displaystyle[\partial_{t},\nabla^{\sharp i}]\zeta=$
$\displaystyle\,2(N+1)a^{-3}\Sigma^{\sharp
ij}\nabla_{j}\zeta-2N\frac{\dot{a}}{a}\nabla^{\sharp i}\zeta$ (11.8c)
$\displaystyle[\partial_{t},\Delta]\zeta=$
$\displaystyle\,2(N+1)a^{-3}\langle\Sigma,\nabla^{2}\zeta\rangle_{G}-2N\frac{\dot{a}}{a}\Delta\zeta$
$\displaystyle\,-2(N+1)a^{-3}\langle\text{{div}}_{G}\Sigma,\nabla\zeta\rangle_{G}-2a^{-3}\langle\Sigma,\nabla
N\nabla\zeta\rangle_{G}+\frac{\dot{a}}{a}\langle\nabla
N,\nabla\zeta\rangle_{G}$ (11.8d)
$\displaystyle[\partial_{t},\nabla_{i}]\mathfrak{T}_{kl}=$
$\displaystyle\,\left\\{2(N+1)a^{-3}\left[\nabla_{i}\Sigma_{(k}^{m}+\nabla_{(k}\Sigma_{i}^{\sharp
m}-\nabla^{m}\Sigma_{i(k}\right]\right.$ $\displaystyle\
+2a^{-3}\left.\left[\nabla_{i}N\Sigma^{\sharp
m}_{(k}+\color[rgb]{0,0,0}\Sigma^{\sharp
m}_{i}\nabla_{(k}N\color[rgb]{0,0,0}-\nabla^{\sharp
m}N\Sigma_{i(k}\right]\right\\}\color[rgb]{0,0,0}\mathfrak{T}_{l)m}\color[rgb]{0,0,0}$
$\displaystyle-\frac{\dot{a}}{a}\left[\nabla_{i}N\mathfrak{T}_{kl}+\nabla_{(k}N\color[rgb]{0,0,0}\mathfrak{T}_{l)i}\color[rgb]{0,0,0}-\nabla^{\sharp
m}NG_{i(k}\color[rgb]{0,0,0}\mathfrak{T}_{l)m}\color[rgb]{0,0,0}\right]$
(11.8e) $\displaystyle[\partial_{t},\Delta]\mathfrak{T}_{kl}=$
$\displaystyle\,a^{-3}(N+1)\Sigma\ast\nabla^{2}\mathfrak{T}+\frac{\dot{a}}{a}N\color[rgb]{0,0,0}\Delta\mathfrak{T}\color[rgb]{0,0,0}+a^{-3}\nabla((N+1)\Sigma)\ast\nabla\mathfrak{T}+\frac{\dot{a}}{a}\nabla
N\ast\nabla\mathfrak{T}$
$\displaystyle\,+a^{-3}\nabla^{2}((N+1)\Sigma)\ast\mathfrak{T}-\frac{\dot{a}}{a}\nabla^{2}N\ast\mathfrak{T}$

###### Proof.

(11.8a) is simply that coordinate derivatives commute, and (11.8b) follows by
applying (2.28b) and the product rule.
For the commutators (11.8c), (11.8d) and (11.8e), we write out the covariant
derivatives in local coordinates, apply the product rule, and then the
evolution equations (2.28b) and (2.34) for the inverse metric and Christoffel
symbols. ∎

######  Lemma 11.7 (High order time derivative commutators).

For $l\in\mathbb{N},\,l\geq 2$, the time derivative commutators take the
following form:

(11.9a) $\displaystyle[\partial_{t},\Delta^{l}]\zeta=$
$\displaystyle\,2a^{-3}(N+1)\langle\Sigma,\nabla^{2}\Delta^{l-1}\zeta\rangle_{G}\color[rgb]{0,0,0}+a^{-3}\nabla\Sigma\ast\nabla^{3}\Delta^{l-2}\zeta\color[rgb]{0,0,0}$
$\displaystyle\color[rgb]{0,0,0}\,-2(N+1)a^{-3}\langle\text{{div}}_{G}\Delta^{l-1}\Sigma,\nabla\zeta\rangle_{G}+(N+1)a^{-3}\nabla^{2l-3}\text{{Ric}}\ast\Sigma\ast\nabla\zeta+\mathfrak{J}([\partial_{t},\Delta^{l}]\zeta),\color[rgb]{0,0,0}$
(11.9b) $\displaystyle[\partial_{t},\nabla\Delta^{l}]\zeta=$
$\displaystyle\,2a^{-3}(N+1)\langle\Sigma,\nabla^{3}\Delta^{l-1}\zeta\rangle_{G}+a^{-3}(N+1)\nabla\Sigma\ast\nabla^{2l}\zeta$
$\displaystyle\,-2(N+1)a^{-3}\langle\nabla\text{{div}}_{G}\Delta^{l-1}\Sigma,\nabla\zeta\rangle_{G}$
$\displaystyle\,+\frac{\dot{a}}{a}\langle\nabla^{2}\Delta^{l-1}N,\nabla\zeta\rangle_{G}+(N+1)a^{-3}\nabla^{2l-2}\text{{Ric}}[G]\ast\Sigma\ast\nabla\zeta$
$\displaystyle\,+\mathfrak{J}([\partial_{t},\nabla\Delta^{l}]\zeta)$ (11.9c)
$\displaystyle[\partial_{t},\Delta^{l}]\mathfrak{T}=$
$\displaystyle\,a^{-3}\left(\Sigma\ast\nabla^{2}\Delta^{l-1}\mathfrak{T}+\nabla\Sigma\ast\nabla^{3}\Delta^{l-2}\mathfrak{T}+\nabla\mathfrak{T}\ast\nabla\Delta^{l-1}\Sigma+\mathfrak{T}\ast\Delta^{l}\Sigma\right)$
$\displaystyle\,+a^{-3}\left((N+1)\Sigma\ast\mathfrak{T}\ast\nabla^{2}\Delta^{l-2}\text{{Ric}}[G]+\nabla((N+1)\Sigma\ast\mathfrak{T})\ast\nabla^{2l-3}\text{{Ric}}[G]\right)$
$\displaystyle\,+\frac{\dot{a}}{a}\Delta^{l}N\cdot\mathfrak{T}+\frac{\dot{a}}{a}\nabla\Delta^{l-1}N\ast\nabla\mathfrak{T}+\mathfrak{J}([\partial_{t},\Delta^{l}])\mathfrak{T}\,,$
(11.9d) $\displaystyle[\partial_{t},\nabla\Delta^{l}]\mathfrak{T}=$
$\displaystyle\,a^{-3}\nabla\Sigma\ast\Delta^{l}\mathfrak{T}+a^{-3}(N+1)\Sigma\ast\nabla^{3}\Delta^{l-1}\mathfrak{T}+a^{-3}(N+1)\mathfrak{T}\ast\nabla\Delta^{l}\Sigma$
$\displaystyle+\frac{\dot{a}}{a}\nabla\Delta^{l}N\ast\mathfrak{T}+\frac{\dot{a}}{a}\nabla^{2}\Delta^{l-1}N\ast\nabla\mathfrak{T}+a^{-3}(N+1)\Sigma\ast\nabla^{3}\Delta^{l-2}\text{{Ric}}[G]\ast\mathfrak{T}$
$\displaystyle\,+\mathfrak{J}([\partial_{t},\nabla\Delta^{l}]\mathfrak{T})\,,$
where the junk terms take are as follows: (11.9e)
$\displaystyle\mathfrak{J}([\partial_{t},\Delta^{l}]\zeta)=$
$\displaystyle\,a^{-3}\sum_{\begin{subarray}{c}I_{N}+I_{\Sigma}+I_{\zeta}=2(l-1)\\\
I_{\zeta}\leq
2(l-2)\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{\zeta}+2}\zeta$
$\displaystyle\,+\frac{\dot{a}}{a}\sum_{\begin{subarray}{c}I_{N}+I_{\zeta}=2l\\\
I_{\zeta}\geq 2\end{subarray}}\nabla^{I_{N}}N\ast\nabla^{I_{\zeta}}\zeta$
$\displaystyle\,+a^{-3}\sum_{m=0}^{l-2}\sum_{\begin{subarray}{c}I_{N}+I_{\Sigma}+I_{\zeta}\\\
+\sum_{i=1}^{l-m-1}I_{i}=2m\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m-1}}\text{{Ric}}[G]\ast\nabla^{I_{\zeta}+2}\zeta$
$\displaystyle\,+a^{-3}\sum_{m=0}^{l-2}\sum_{\begin{subarray}{c}I_{N}+I_{\Sigma}+I_{\zeta}\\\
+\sum_{i=1}^{l-m-1}I_{i}=2m\\\ I_{1}\neq
2l-4\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{1}+1}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m-1}}\text{{Ric}}[G]\ast\nabla^{I_{\zeta}+1}\zeta$
$\displaystyle\,+\frac{\dot{a}}{a}\sum_{m=0}^{l-1}\sum_{\begin{subarray}{c}I_{N}+\sum_{i=1}^{l-m-1}I_{i}+I_{\zeta}=2m-1\\\
\,I_{\zeta}\neq
2(l-1)\end{subarray}}\nabla^{I_{N}}N\ast\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m-1}}\text{{Ric}}[G]\ast\nabla^{I_{\zeta}+1}\zeta$
(11.9f) $\displaystyle\mathfrak{J}([\partial_{t},\nabla\Delta^{l}]\zeta)=$
$\displaystyle\,\frac{\dot{a}}{a}\sum_{\begin{subarray}{c}I_{N}+I_{\zeta}=2l\\\
\,I_{\zeta}\neq
0\end{subarray}}\nabla^{I_{N}}N\ast\nabla^{I_{\zeta}+1}\zeta+a^{-3}\sum_{\begin{subarray}{c}I_{N}+I_{\Sigma}+I_{\zeta}=2(l-1)+1\\\
(I_{\Sigma},I_{\zeta})\neq(0,2(l-1)+1),(1,2(l-1))\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{\zeta}+1}\zeta$
$\displaystyle\,+a^{-3}\sum_{m=0}^{l-2}\sum_{\begin{subarray}{c}I_{N}+I_{\Sigma}+I_{\zeta}\\\
+\sum_{i=1}^{l-m-1}I_{i}=2m+1\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m-1}}\text{{Ric}}[G]\ast\nabla^{I_{\zeta}+2}\zeta$
$\displaystyle\,+a^{-3}\sum_{m=0}^{l-2}\sum_{\begin{subarray}{c}I_{N}+I_{\Sigma}+I_{\zeta}\\\
+\sum_{i=1}^{l-m-1}I_{i}=2m+1\\\ I_{1}\neq
2l-3\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{1}+1}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m-1}}\text{{Ric}}[G]\ast\nabla^{I_{\zeta}+1}\zeta$
$\displaystyle\,+\frac{\dot{a}}{a}\sum_{m=0}^{l-1}\sum_{\begin{subarray}{c}I_{N}+\sum_{i=1}^{l-m-1}I_{i}+I_{\zeta}=2m\\\
\,I_{\zeta}\neq
2(l-1)\end{subarray}}\nabla^{I_{N}}N\ast\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m-1}}\text{{Ric}}[G]\ast\nabla^{I_{\zeta}+1}\zeta$
(11.9g)
$\displaystyle\color[rgb]{0,0,0}\mathfrak{J}([\partial_{t},\Delta^{l}])\mathfrak{T}=$
$\displaystyle\,a^{-3}\sum_{I_{N}+I_{\Sigma}+I_{\mathfrak{T}}=2l}\nabla^{I_{N}}N\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}+a^{-3}\sum_{\begin{subarray}{c}I_{\Sigma}+I_{\mathfrak{T}}=2l\\\
I_{\Sigma},I_{\mathfrak{T}}\geq
2\end{subarray}}\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}$
$\displaystyle\,+a^{-3}\sum_{m=0}^{l-1}\sum_{\begin{subarray}{c}I_{N}+I_{\Sigma}+I_{\mathfrak{T}}\\\
+\sum_{i=1}^{l-m-1}I_{i}=2m\\\
I_{1}<2l-3\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{l-m-1}}\text{{Ric}}[G]\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}$
$\displaystyle\,+\frac{\dot{a}}{a}\sum_{I_{N}+I_{\mathfrak{T}}=2l,\,I_{\mathfrak{T}}\geq
2}\nabla^{I_{N}}N\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}$ (11.9h)
$\displaystyle\mathfrak{J}([\partial_{t},\nabla\Delta^{l}]\mathfrak{T})=$
$\displaystyle\,a^{-3}\Sigma\ast\nabla
N\ast\Delta^{l}\text{{Ric}}[G]+a^{-3}N\ast\nabla\Sigma\ast\Delta^{l}\mathfrak{T}+\frac{\dot{a}}{a}\nabla
N\ast\Delta^{l}\mathfrak{T}+\nabla\mathfrak{J}([\partial_{t},\Delta^{l}]\mathfrak{T})$

We can extend the formulas to $l=1$ by dropping any term which would contain
negative powers of $\Delta$ or a multiindex of negative order.

###### Proof.

This follows by iteratively applying the commutators in Lemma 11.6. ∎

While all of the above commutators will be essential for the mainline
argument, the a priori estimates require the following commutators:

######  Lemma 11.8 (Auxiliary commutators).

Let $J\in\mathbb{N}$. Then, we have:

(11.10a) $\displaystyle[\partial_{t},\nabla^{J}]\zeta=$
$\displaystyle\,a^{-3}\sum_{I_{N}+I_{\Sigma}+I_{\zeta}=l-1,\,I_{\zeta}<J-1}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{\zeta}+1}\zeta$
$\displaystyle\,+\frac{\dot{a}}{a}\sum_{I_{N}+I_{\zeta}=J-1,I_{N}>0}\nabla^{I_{N}}N\ast\nabla^{I_{\zeta}+1}\zeta$
(11.10b) $\displaystyle[\partial_{t},\nabla^{J}]\mathfrak{T}=$
$\displaystyle\,a^{-3}\sum_{I_{N}+I_{\Sigma}+I_{\mathfrak{T}}=J,\,I_{\mathfrak{T}}<J}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}$
$\displaystyle\,+\frac{\dot{a}}{a}\sum_{I_{N}+I_{\mathfrak{T}}=J,I_{N}>0}\nabla^{I_{N}}N\ast\nabla^{I_{\mathfrak{T}}}\mathfrak{T}$

###### Proof.

For $J=1$, this has already been shown in (11.8a) and (11.8d). For higher
orders, the formulas follow from a straightforward induction argument using
that, in local coordinates, we schematically have

$\displaystyle[\partial_{t},\nabla^{J}]\zeta=[\partial_{t},\nabla]\nabla^{J-1}\zeta+\nabla[\partial_{t},\nabla^{J-1}]\zeta=(\partial_{t}\Gamma[G])\ast\nabla^{J-1}\zeta+\nabla[\partial_{t},\nabla^{J-1}]\zeta$

and analogously replacing $\zeta$ with $\mathfrak{T}$. ∎

###  11.3. Borderline and junk terms

######  Definition 11.9 (Error terms).

Let $L\in 2\mathbb{N},\,L\geq 2$. Then, the error terms in the Laplace
commuted equations stated in Lemma 2.11 take the following form:

For the constraint equations, we have

(11.11a) $\displaystyle\mathfrak{M}_{L,Junk}=$
$\displaystyle\,-8\pi(\Psi+C)\nabla\Delta^{\frac{L}{2}-2}\text{{Ric}}[G]\ast\nabla^{2}\phi+\nabla^{L-2}\text{{Ric}}[G]\ast\nabla\Sigma+\underbrace{\nabla^{L-3}\text{{Ric}}[G]\ast\nabla^{2}\Sigma}_{\text{if
}L\neq 2}$ $\displaystyle\,+\sum_{I_{\Psi}+I_{\phi}=L,\,I_{\Psi}\neq
0}\nabla^{I_{\Psi}}\Psi\ast\nabla^{I_{\phi}+1}\phi+8\pi(\Psi+C)\mathfrak{J}([\Delta^{\frac{L}{2}},\nabla]\phi)-\mathfrak{J}([\Delta^{\frac{L}{2}},\text{{div}}_{G}]\Sigma)$
(11.11b)
$\displaystyle\color[rgb]{0,0,0}\tilde{\mathfrak{M}}_{L,Junk}=\color[rgb]{0,0,0}$
$\displaystyle\color[rgb]{0,0,0}\,\underbrace{-\bm{\varepsilon}[G]\ast\nabla^{L-3}\text{{Ric}}[G]\ast\nabla\Sigma}_{\text{if
}L\neq
2}-\mathfrak{J}([\Delta^{\frac{L}{2}},\text{{curl}}_{G}]\Sigma)\color[rgb]{0,0,0}$
(11.11c) $\displaystyle\mathfrak{H}_{L,Border}=$
$\displaystyle\,a^{-4}\left[\Sigma\ast\Delta^{\frac{L}{2}}\Sigma+\nabla\Sigma\ast\nabla^{L-1}\Sigma\right]$
(11.11d) $\displaystyle\mathfrak{H}_{L,Junk}=$
$\displaystyle\,\sum_{I_{1}+I_{2}=L}\nabla^{I_{1}+1}\phi\ast\nabla^{I_{2}+1}\phi+a^{-4}\sum_{I_{1}+I_{2}=L,I_{i}\geq
2}\nabla^{I_{1}}\Sigma\ast\nabla^{I_{2}}\Sigma$
$\displaystyle\,+\Delta^{\frac{L}{2}}\left[\frac{4\pi}{3}\lvert\nabla\phi\rvert_{G}^{2}+\frac{8\pi}{3}a^{-4}\Psi^{2}+\frac{16\pi}{3}Ca^{-4}\Psi\right]\cdot
G$

The lapse equation error terms are

(11.12a) $\displaystyle\mathfrak{N}_{L,Border}=$
$\displaystyle\,a^{-4}(N+1)\left(\Sigma\ast\Delta^{\frac{L}{2}}\Sigma+\nabla\Sigma\ast\nabla^{L-1}\Sigma+\Psi\ast\Delta^{\frac{L}{2}}\Psi+\nabla\Psi\ast\nabla^{L-1}\Psi\right)$
$\displaystyle\,+a^{-4}\left[\lvert\Sigma\rvert_{G}^{2}+\Psi^{2}+\Psi\right]\ast\Delta^{\frac{L}{2}}N+a^{-4}\nabla\left[\lvert\Sigma\rvert_{G}^{2}+\Psi^{2}+\Psi\right]\ast\nabla^{L-1}N\,$
(11.12b) $\displaystyle\mathfrak{N}_{L,Junk}=$
$\displaystyle\,a^{-4}\sum_{\begin{subarray}{c}I_{N}+I_{1}+I_{2}=L;\\\
I_{N}\leq L-2;\,I_{N}>0\text{ or }I_{1}\leq I_{2}\leq
L-2\end{subarray}}\nabla^{I_{N}}(N+1)\ast\left(\nabla^{I_{1}}\Sigma\ast\nabla^{I_{2}}\Sigma+\nabla^{I_{1}}\Psi\ast\nabla^{I_{2}}\Psi\right)$
$\displaystyle\,+a^{-4}N\ast\Delta^{\frac{L}{2}}\Psi+a^{-4}\sum_{I_{N}+I_{\Psi}=L;\,I_{\Psi}\geq
2,\,I_{N}\geq 1}\nabla^{I_{N}}N\ast\nabla^{I_{\Psi}}\Psi\,,$ as well as
(11.12c) $\displaystyle\mathfrak{N}_{L+1,Border}=$
$\displaystyle\,a^{-4}(N+1)\Bigr{(}\Sigma\ast\nabla\Delta^{\frac{L}{2}}\Sigma+\nabla\Sigma\ast\nabla^{L}\Sigma+\nabla^{2}\Sigma\ast\nabla^{L-1}\Sigma+\Psi\ast\nabla\Delta^{\frac{L}{2}}\Psi$
$\displaystyle\,+\nabla\Psi\ast\nabla^{L}\Psi+\nabla^{2}\Psi\ast\nabla^{L-1}\Psi\Bigr{)}+a^{-4}\left[\lvert\Sigma\rvert_{G}^{2}+\Psi^{2}+\Psi\right]\ast\nabla\Delta^{\frac{L}{2}}N$
$\displaystyle\,+\nabla\Psi\ast\nabla^{L}N+\nabla^{2}\Psi\ast\nabla^{L-1}\Psi\,$
(11.12d) $\displaystyle\mathfrak{N}_{L+1,Junk}=$
$\displaystyle\,a^{-4}\sum_{\begin{subarray}{c}I_{N}+I_{1}+I_{2}=L;\\\
I_{N}<L+1;\,I_{N}>0\text{ or }I_{1}\geq
I_{2}>2\end{subarray}}\nabla^{I_{N}}(N+1)\ast\left(\nabla^{I_{1}}\Sigma\ast\nabla^{I_{2}}\Sigma+\nabla^{I_{1}}\Psi\ast\nabla^{I_{2}}\Psi\right)$
$\displaystyle\,+a^{-4}N\ast\nabla\Delta^{\frac{L}{2}}\Psi+a^{-4}\sum_{I_{N}+I_{\Psi}=L+1;\,I_{N},I_{\Psi}>2}\nabla^{I_{N}}N\ast\nabla^{I_{\Psi}}\Psi$

whereas the scalar field error terms read

(11.13a) $\displaystyle\mathfrak{P}_{L,Border}=$
$\displaystyle\,-3\Psi\frac{\dot{a}}{a}\Delta^{\frac{L}{2}}N+\frac{\dot{a}}{a}\nabla\Psi\ast\nabla^{L-1}N+2a^{-3}(N+1)\langle\Sigma,\nabla^{2}\Delta^{\frac{L}{2}-1}\Psi\rangle_{G}+2a^{-3}(N+1)\nabla^{L-3}\text{{Ric}}\ast\Sigma\ast\nabla\Psi$
$\displaystyle\,\color[rgb]{0,0,0}-2a^{-3}(N+1)\langle\text{{div}}_{G}\Delta^{\frac{L}{2}-1}\Sigma,\nabla\Psi\rangle_{G}+a^{-3}(N+1)\nabla\Sigma\ast\nabla^{3}\Delta^{\frac{L}{2}-2}\Psi\color[rgb]{0,0,0}$
(11.13b) $\displaystyle\mathfrak{P}_{L,Junk}=$
$\displaystyle\,\frac{\dot{a}}{a}\sum_{I_{N}+I_{\Psi}=L,\,I_{\Psi}\geq
2}\nabla^{I_{N}}N\ast\nabla^{I_{\Psi}}\Psi+a\sum_{I_{N}+I_{\phi}=L+1,\,I_{N},I_{\phi}\neq
0}\nabla^{I_{N}}N\ast\nabla^{I_{\phi}+1}\phi+\mathfrak{J}([\partial_{t},\Delta^{\frac{L}{2}}]\Psi)$
(11.13c) $\displaystyle\mathfrak{Q}_{L,Border}=$
$\displaystyle\,a^{-3}\Psi\nabla\Delta^{\frac{L}{2}}N+a^{-3}(N+1)\Sigma\ast\nabla^{3}\Delta^{\frac{L}{2}-1}\phi\color[rgb]{0,0,0}+a^{-3}(N+1)\nabla^{L}\phi\ast\nabla\Sigma\color[rgb]{0,0,0}$
(11.13d) $\displaystyle\mathfrak{Q}_{L,Junk}=$
$\displaystyle\,a^{-3}\sum_{I_{N}+I_{\Psi}=L+1,\,I_{N},I_{\Psi}\neq
0}\nabla^{I_{N}}N\ast\nabla^{I_{\Psi}}\Psi+a^{-3}\nabla\Delta^{\frac{L}{2}-1}N\ast\nabla^{2}\phi\ast\Sigma$
$\displaystyle\,\color[rgb]{0,0,0}+a^{-3}(N+1)\nabla^{2}\Delta^{\frac{L}{2}-1}\Sigma\ast\nabla\phi\color[rgb]{0,0,0}+(N+1)a^{-3}\nabla\Delta^{\frac{L}{2}-1}\text{{Ric}}[G]\ast\Sigma\ast\nabla\phi$
$\displaystyle\,+a^{-3}\nabla^{L-2}\text{{Ric}}[G]\ast\left((N+1)\ast\Sigma\ast\nabla\phi\right)+\frac{\dot{a}}{a}\langle\nabla^{2}\Delta^{\frac{L}{2}-1}N,\nabla\phi\rangle_{G}\color[rgb]{0,0,0}+\mathfrak{J}([\partial_{t},\nabla\Delta^{\frac{L}{2}}]\phi)\color[rgb]{0,0,0}$
and (11.13e) $\displaystyle\mathfrak{P}_{L+1,Border}=$
$\displaystyle\,-3\Psi\frac{\dot{a}}{a}\nabla\Delta^{\frac{L}{2}}N+\frac{\dot{a}}{a}\nabla\Psi\ast\nabla^{2}\Delta^{\frac{L}{2}-1}N+2a^{-3}\langle\Sigma,\nabla^{3}\Delta^{\frac{L}{2}-1}\Psi\rangle_{G}$
$\displaystyle\,+a^{-3}(N+1)\nabla\Sigma\ast\nabla^{L}\Psi+2a^{-3}\nabla^{L-2}\text{{Ric}}\ast\Sigma\ast\nabla\Psi$
$\displaystyle\,+a^{-3}(N+1)\nabla^{2}\Delta^{\frac{L}{2}-1}\Sigma\ast\nabla\Psi\color[rgb]{0,0,0}$
(11.13f) $\displaystyle\mathfrak{P}_{L+1,Junk}=$
$\displaystyle\,\frac{\dot{a}}{a}\sum_{I_{N}+I_{\Psi}=L+1,\,I_{\Psi}\geq
2}\nabla^{I_{N}}N\ast\nabla^{I_{\Psi}}\Psi+a\sum_{I_{N}+I_{\phi}=L+2,\,I_{N},I_{\phi}\neq
0}\nabla^{I_{N}}N\ast\nabla^{I_{\phi}+1}\phi$
$\displaystyle\,\color[rgb]{0,0,0}+\mathfrak{J}([\partial_{t},\nabla\Delta^{\frac{L}{2}}]\Psi)\color[rgb]{0,0,0}$
(11.13g) $\displaystyle\mathfrak{Q}_{L+1,Border}=$
$\displaystyle\,a^{-3}\Psi\Delta^{\frac{L}{2}+1}N+a^{-3}\nabla\Psi\ast\nabla\Delta^{\frac{L}{2}}N+a^{-3}(N+1)\Sigma\ast\nabla^{2}\Delta^{\frac{L}{2}}\phi+\frac{\dot{a}}{a}\nabla\Delta^{\frac{L}{2}}N\ast\nabla\phi$
$\displaystyle\color[rgb]{0,0,0}+a^{-3}(N+1)\nabla\Sigma\ast\nabla^{2}\Delta^{\frac{L}{2}-1}\phi\color[rgb]{0,0,0}$
$\displaystyle\mathfrak{Q}_{L+1,Junk}=$
$\displaystyle\,a^{-3}\sum_{I_{N}+I_{\Psi}=L+2,\,2\leq I_{\Psi}\leq
L+1}\nabla^{I_{N}}N\ast\nabla^{I_{\Psi}}\Psi+a^{-3}(N+1)\nabla^{L-2}\text{{Ric}}[G]\ast\Sigma\ast\nabla\phi$
(11.13h)
$\displaystyle\,\color[rgb]{0,0,0}+a^{-3}(N+1)\nabla^{2}\Delta^{\frac{L}{2}-1}\Sigma\ast\nabla\phi+\mathfrak{J}([\partial_{t},\Delta^{\frac{L}{2}+1}]\phi)\color[rgb]{0,0,0}$
as well as (11.13i) $\displaystyle\mathfrak{Q}_{1,Border}=$
$\displaystyle\,a^{-3}\Psi\Delta N+a^{-3}(N+1)\Sigma\ast\nabla^{2}\phi$
(11.13j) $\displaystyle\mathfrak{Q}_{1,Junk}=$
$\displaystyle\,a^{-3}\nabla\Psi\ast\nabla
N\color[rgb]{0,0,0}+a^{-3}(N+1)\nabla\Sigma\ast\nabla\phi+\mathfrak{J}([\partial_{t},\Delta]\phi)\color[rgb]{0,0,0}\,.$

The commuted rescaled evolution equation for $\Sigma$ has the error terms

(11.14a) $\displaystyle\mathfrak{S}_{L,Border}=$
$\displaystyle\,a^{-3}(N+1)\left(\Sigma\ast\nabla^{2}\Delta^{\frac{L}{2}-1}\Sigma+\nabla\Sigma\ast\nabla^{3}\Delta^{\frac{L}{2}-2}\Sigma\right)+a^{-3}\left(\Delta^{\frac{L}{2}}N\cdot(\Sigma\ast\Sigma)+\nabla\Delta^{\frac{L}{2}-1}N\ast\nabla\Sigma\ast\Sigma\right)$
$\displaystyle\,+a^{-3}(N+1)\Sigma\ast\Sigma\ast\nabla^{2}\Delta^{\frac{L}{2}-2}\text{{Ric}}[G]+\frac{\dot{a}}{a}\Delta^{\frac{L}{2}}N\ast\Sigma+\frac{\dot{a}}{a}\nabla\Delta^{\frac{L}{2}-1}N\ast\nabla\Sigma$
$\displaystyle\,+\underbrace{a^{-3}[(N+1)\nabla\Sigma\ast\Sigma+\nabla
N\ast\Sigma\ast\Sigma]\ast\nabla^{L-3}\text{{Ric}}[G]}_{\text{not present for
}L=2}$ (11.14b) $\displaystyle\mathfrak{S}_{L,Junk}=$
$\displaystyle\,-a[\Delta^{\frac{L}{2}},\nabla^{2}]N+a\sum_{I_{N}+I_{\text{{Ric}}}=L,I_{N}\neq
0}\nabla^{I_{N}}N\ast\nabla^{I_{\text{{Ric}}}}\text{{Ric}}[G]+\frac{\dot{a}}{a}\sum_{I_{N}+I_{\Sigma}=L}\nabla^{I_{N}}N\ast\nabla^{I_{\Sigma}}\Sigma$
$\displaystyle\,+a^{-3}\sum_{I_{1}+I_{2}=L,\,I_{i}>0}\nabla^{I_{1}}\Sigma\ast\nabla^{I_{2}}\Sigma+a^{-3}\sum_{I_{N}+I_{1}+I_{2}=L,\,I_{N}<L}\nabla^{I_{N}}N\ast\nabla^{I_{1}}\Sigma\ast\nabla^{I_{2}}\Sigma$
$\displaystyle\,+a\sum_{I_{N}+I_{1}+I_{2}=L}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{1}+1}\phi\ast\nabla^{I_{2}+1}\phi+\left(4\pi
C^{2}a^{-3}+\frac{1}{3}a\right)\Delta^{\frac{L}{2}}N\cdot
G+\mathfrak{J}([\partial_{t},\Delta^{\frac{L}{2}}]\Sigma)$

while the commuted Ricci tensor evolution equations have error terms

(11.15a) $\displaystyle\mathfrak{R}_{L,Border}=$
$\displaystyle\,a^{-3}\left[\nabla^{L+2}N\cdot\Sigma+\nabla^{L+1}N\ast\nabla\Sigma+\Sigma\ast\nabla^{2}\Delta^{\frac{L}{2}-1}\text{{Ric}}[G]+\nabla\Sigma\ast\nabla^{L-1}\text{{Ric}}[G]\right]$
(11.15b) $\displaystyle\mathfrak{R}_{L+1,Border}=$
$\displaystyle\,a^{-3}\left[\nabla^{L+3}N\cdot\Sigma+\nabla^{L+2}N\ast\nabla\Sigma+\Sigma\ast\nabla^{3}\Delta^{\frac{L}{2}-1}\text{{Ric}}[G]+\nabla\Sigma\ast\nabla^{L}\text{{Ric}}[G]\right]$
(11.15c) $\displaystyle\mathfrak{R}_{L,Junk}=$
$\displaystyle\,a^{-3}\sum_{I_{N}+I_{\Sigma}=L+2,\,I_{\Sigma}\geq
2}\nabla^{I_{N}}N\ast\nabla^{I_{\Sigma}}\Sigma$
$\displaystyle+a^{-3}\sum_{\begin{subarray}{c}I_{N}+I_{\Sigma}+I_{\text{{Ric}}}=L\\\
(I_{\Sigma},I_{\text{{Ric}}})\neq(0,L),(1,L-1)\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{\text{{Ric}}}}\text{{Ric}}[G]$
$\displaystyle\,+a^{-3}\sum_{m=0}^{\frac{L}{2}-1}\sum_{{I_{N}+I_{\Sigma}+\sum_{i=1}^{\nicefrac{{L}}{{2}}-m+1}I_{i}=2m}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\ast\nabla^{I_{\frac{L}{2}-m+1}}\text{{Ric}}[G]$
$\displaystyle\,+\frac{\dot{a}}{a}\left([\Delta^{\frac{L}{2}},\nabla^{2}]N+\Delta^{\frac{L}{2}}N\ast\text{{Ric}}[G]+\nabla^{L-1}N\ast\nabla\text{{Ric}}[G]\right)+\mathfrak{J}([\partial_{t},\Delta^{\frac{L}{2}}]\text{{Ric}}[G])$
(11.15d) $\displaystyle\mathfrak{R}_{L+1,Junk}=$
$\displaystyle\,a^{-3}\sum_{I_{N}+I_{\Sigma}=L+3,\,I_{\Sigma}\geq
2}\nabla^{I_{N}}N\ast\nabla^{I_{\Sigma}}\Sigma$
$\displaystyle+a^{-3}\sum_{\begin{subarray}{c}I_{N}+I_{\Sigma}+I_{\text{{Ric}}}=L\\\
(I_{\Sigma},I_{\text{{Ric}}})\neq(0,L+1),(1,L)\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{\text{{Ric}}}}\text{{Ric}}[G]$
$\displaystyle\,+a^{-3}\sum_{m=0}^{\frac{L}{2}-1}\sum_{{I_{N}+I_{\Sigma}+\sum_{i=1}^{\nicefrac{{L}}{{2}}-m+1}I_{i}=2m+1}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{1}}\text{{Ric}}[G]\ast\dots\nabla^{I_{\frac{L}{2}-m+1}}\text{{Ric}}[G]$
$\displaystyle\,+\frac{\dot{a}}{a}\left(\nabla[\Delta^{\frac{L}{2}},\nabla^{2}]N+\nabla\Delta^{\frac{L}{2}}N\ast\text{{Ric}}[G]+\nabla^{2}\Delta^{\frac{L}{2}-1}N\ast\nabla\text{{Ric}}[G]\right)+\mathfrak{J}([\partial_{t},\nabla\Delta^{\frac{L}{2}}]\text{{Ric}}[G])$

Finally, the Bel-Robinson evolution error terms are

(11.16a) $\displaystyle\mathfrak{E}_{L,Border}=$
$\displaystyle\,\frac{\tau}{3}\left(\Delta^{\frac{L}{2}}N\cdot\bm{E}+\nabla^{L-1}N\ast\nabla\bm{E}\right)-a^{-1}\left(\Delta^{\frac{L}{2}}\bm{E}\times\Sigma+\bm{E}\times\Delta^{\frac{L}{2}}\Sigma\right)$
$\displaystyle\,+a^{-3}\bm{\varepsilon}[G]\ast\bm{\varepsilon}[G]\ast\left(\nabla^{L-1}\bm{E}\ast\nabla\Sigma+\nabla\bm{E}\ast\nabla^{L-1}\Sigma\right)$
$\displaystyle\,+a^{-3}\Delta^{\frac{L}{2}}N\cdot(\bm{E}\ast\Sigma)+a^{-3}\nabla\Delta^{\frac{L}{2}-1}N\ast[\nabla\bm{E}\ast\Sigma+\bm{E}\ast\nabla\Sigma]$
$\displaystyle\,+a^{-3}\left(\Sigma\ast\nabla^{2}\Delta^{\frac{L}{2}-1}\bm{E}+\nabla\Sigma\ast\nabla^{3}\Delta^{\frac{L}{2}-2}\bm{E}+\nabla\bm{E}\ast\nabla\Delta^{\frac{L}{2}-1}\Sigma+\bm{E}\ast\Delta^{\frac{L}{2}}\Sigma\right)$
$\displaystyle\,+a^{-3}\left[(N+1)\Sigma\ast\bm{E}\ast\nabla^{2}\Delta^{\frac{L}{2}-2}\text{{Ric}}[G]+\underbrace{\nabla\left((N+1)\ast\Sigma\ast\bm{E}\right)\ast\nabla^{L-3}\text{{Ric}}[G]}_{\text{if
}L\neq 2}\right]$ $\displaystyle\,+4\pi
a^{-3}(\Psi+C)^{2}\Delta^{\frac{L}{2}}N\cdot\Sigma\,+4\pi
a^{-3}\nabla^{L-1}N\ast\left[(\Psi+C)^{2}\nabla\Sigma+2(\Psi+C)\ast\nabla\Psi\ast\Sigma\right]$
$\displaystyle\,+4\pi
a^{-3}(N+1)\left[(\Psi^{2}+2C\Psi)\Delta^{\frac{L}{2}}\Sigma+2(\Psi+C)\Delta^{\frac{L}{2}}\Psi\cdot\Sigma\right]$
$\displaystyle\,+4\pi a^{-3}\nabla^{L-1}\Sigma\ast\left[(\Psi+C)^{2}\nabla
N+2(N+1)(\Psi+C)\nabla\Psi\right]$ $\displaystyle\,+4\pi
a^{-3}(\Psi+C)\nabla^{L-1}\Psi\ast\left[(N+1)\nabla\Sigma+\nabla
N\ast\Sigma\right]$ (11.16b) $\displaystyle\mathfrak{E}_{L,top}=$
$\displaystyle\,a^{-1}(N+1)\bm{\varepsilon}[G]\ast\bm{B}\ast\nabla\Delta^{\frac{L}{2}-1}\text{{Ric}}[G]+a(N+1)(\Psi+C)\nabla\Delta^{\frac{L}{2}-1}\text{{Ric}}[G]\ast\nabla\phi$
(11.16c) $\displaystyle\mathfrak{E}_{L,Junk}=$
$\displaystyle\,\frac{\dot{a}}{a}\sum_{I_{N}+I_{\bm{E}}=L,\,I_{N}\leq
L-2}\nabla^{I_{N}}N\ast\nabla^{I_{\bm{E}}}\bm{E}$
$\displaystyle\,+a^{-1}\bm{\varepsilon}[G]\ast\left[\sum_{I_{N}+I_{\bm{B}}=L+1,\,I_{N},I_{\bm{B}}\leq
L}\nabla^{I_{N}}N\ast\nabla^{I_{\bm{B}}}\bm{B}+(N+1)\nabla^{2}\Delta^{\frac{L}{2}-2}\text{{Ric}}[G]\ast\nabla\bm{B}\right]$
$\displaystyle\,+a^{-3}\bm{\varepsilon}[G]\ast\bm{\varepsilon}[G]\ast\sum_{\begin{subarray}{c}I_{N}+I_{\bm{E}}+I_{\Sigma}=L,\\\
I_{N}\leq L-2;\,I_{N}>0\text{ or }I_{\bm{E}},I_{\Sigma}\geq
2\end{subarray}}\nabla^{I_{N}}N\ast\nabla^{I_{\bm{E}}}\bm{E}\ast\nabla^{I_{\Sigma}}\Sigma$
$\displaystyle\,+a\sum_{\begin{subarray}{c}I_{N}+I_{\Psi}+I_{\phi}=L+1\\\
I_{N},I_{\Psi},I_{\phi}\neq
L+1\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Psi}}(\Psi+C)\ast\nabla^{I_{\phi}+1}\phi$
$\displaystyle\,+a^{-3}\sum_{\begin{subarray}{c}I_{N}+I_{\Sigma}+I_{1}+I_{2}=L\\\
I_{N},I_{\Sigma},I_{i}\leq
L-2\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{1}}(\Psi+C)\ast\nabla^{I_{2}}(\Psi+C)$
$\displaystyle\,+\dot{a}a^{3}\sum_{I_{N}+I_{1}+I_{2}=L}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{1}+1}\phi\ast\nabla^{I_{2}+1}\phi$
$\displaystyle\,+a\sum_{I_{N}+I_{\Sigma}+I_{1}+I_{2}=L}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Sigma}}\Sigma\ast\nabla^{I_{1}+1}\phi\ast\nabla^{I_{2}+1}\phi$
$\displaystyle\,+a^{-1}\varepsilon[G]\ast\bm{B}\ast[\Delta^{\frac{L}{2}},\nabla]N+4\pi
a(N+1)(\Psi+C)\left[\nabla^{L-2}\text{{Ric}}[G]\ast\nabla^{2}\phi+\mathfrak{J}([\Delta^{\frac{L}{2}},\nabla^{2}]\phi)\right]$
$\displaystyle\,+a\left\\{(\Psi+C)[\Delta^{\frac{L}{2}},\nabla]N+(N+1)[\Delta^{\frac{L}{2}},\nabla]\Psi\right\\}\ast\nabla\phi+\color[rgb]{0,0,0}(N+1)a^{-1}\mathfrak{J}([\Delta^{\frac{L}{2}},\text{{curl}}_{G}]\bm{B})\color[rgb]{0,0,0}+\mathfrak{J}([\partial_{t},\Delta^{\frac{L}{2}}]\bm{E})$
$\displaystyle\,+\Delta^{\frac{L}{2}}\left[a^{-3}(N+1)\bm{E}\ast\Sigma+\frac{2\pi}{3}a^{6}(N+1)\left(\partial_{0}\left(a^{-6}(\Psi+C)^{2}+a^{-2}\lvert\nabla\phi\rvert_{G}^{2}\right)+4\pi\frac{\dot{a}}{a}(\Psi+C)^{2}\right)\right]\cdot
G$ (11.16d) $\displaystyle\mathfrak{B}_{L,Border}=$
$\displaystyle\,\frac{\tau}{3}\left(\Delta^{\frac{L}{2}}N\cdot\bm{B}+\nabla^{L-1}N\ast\nabla\bm{B}\right)-a^{-1}\left(\Delta^{\frac{L}{2}}\bm{B}\times\Sigma+\bm{B}\ast\Delta^{\frac{L}{2}}\Sigma\right)$
$\displaystyle\,+a^{-3}\bm{\varepsilon}[G]\ast\bm{\varepsilon}[G]\ast\left(\nabla^{L-1}\bm{B}\ast\nabla\Sigma+\nabla\bm{B}\ast\nabla^{L-1}\Sigma\right)$
$\displaystyle\,+a^{-3}\Delta^{\frac{L}{2}}N\cdot(\bm{B}\ast\Sigma)+a^{-3}\nabla\Delta^{\frac{L}{2}-1}N\cdot\left[\nabla\bm{B}\ast\Sigma+\bm{B}\ast\nabla\Sigma\right]$
$\displaystyle\,+a^{-3}\left(\Sigma\ast\nabla^{2}\Delta^{\frac{L}{2}-1}\bm{B}+\nabla\Sigma\ast\nabla^{3}\Delta^{\frac{L}{2}-2}\bm{B}+\nabla\bm{B}\ast\nabla\Delta^{\frac{L}{2}-1}\Sigma+\bm{B}\ast\Delta^{\frac{L}{2}}\Sigma\right)$
$\displaystyle\,+a^{-3}\left[(N+1)\Sigma\ast\bm{B}\ast\nabla^{L-2}\text{{Ric}}[G]+\underbrace{\nabla\left((N+1)\ast\Sigma\ast\bm{B}\right)\ast\nabla^{L-3}\text{{Ric}}[G]}_{\text{if
}L\neq 2}\right]$
$\displaystyle\,+a^{-1}(N+1)(\Psi+C)\cdot\bm{\varepsilon}[G]\ast\nabla\Delta^{\frac{L}{2}}\phi\ast\Sigma+a^{-1}\bm{\varepsilon}[G]\ast\nabla^{2}\nabla^{L}\phi\ast\nabla\left((N+1)(\Psi+C)\Sigma\right)$
(11.16e) $\displaystyle\mathfrak{B}_{L,top}=$
$\displaystyle\,a^{3}(N+1)\bm{\varepsilon}[G]\ast\nabla\Delta^{\frac{L}{2}-1}\text{{Ric}}[G]\ast\nabla\phi\ast\nabla\phi+a^{-1}\bm{\varepsilon}[G]\ast\bm{E}\ast\nabla\Delta^{\frac{L}{2}-1}\text{{Ric}}[G]$
(11.16f) $\displaystyle\mathfrak{B}_{L,Junk}=$
$\displaystyle\,\frac{\dot{a}}{a}\sum_{I_{N}+I_{\bm{B}}=L,I_{N}\leq
L-2}\nabla^{I_{N}}N\ast\nabla^{I_{\bm{B}}}\bm{B}$
$\displaystyle\,+a^{-1}\bm{\varepsilon}[G]\ast\left[\sum_{I_{N}+I_{\bm{E}}=L+1,I_{N},I_{\bm{E}}\leq
L}\nabla^{I_{N}}N\ast\nabla^{I_{\bm{E}}}\bm{E}+\nabla^{2}\Delta^{\frac{L}{2}-2}\text{{Ric}}[G]\ast\nabla\bm{E}\right]$
$\displaystyle\,+a^{-3}\bm{\varepsilon}[G]\ast\bm{\varepsilon}[G]\ast\sum_{\begin{subarray}{c}I_{N}+I_{\bm{E}}+I_{\Sigma}=L,\\\
I_{N}\leq L-2;\,I_{N}>0\text{ or }I_{\bm{B}},I_{\Sigma}\geq
2\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\bm{B}}}\bm{B}\ast\nabla^{I_{\Sigma}}\Sigma$
$\displaystyle\,+a^{3}\bm{\varepsilon}[G]\ast\sum_{\begin{subarray}{c}I_{N}+I_{1}+I_{2}=L,\\\
\,I_{N}>0\text{ or
}I_{2}<L\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{1}+1}\phi\ast\nabla^{I_{2}+2}\phi$
$\displaystyle\,+a^{-1}\bm{\varepsilon}[G]\ast\sum_{\begin{subarray}{c}I_{N}+I_{\Psi}+I_{\phi}+I_{\Sigma}=L\\\
I_{\phi}\leq
L-2\end{subarray}}\nabla^{I_{N}}(N+1)\ast\nabla^{I_{\Psi}}(\Psi+C)\ast\nabla^{I_{\phi}+1}\phi\ast\nabla^{I_{\Sigma}}\Sigma$
$\displaystyle\,+a^{-1}\varepsilon[G]\ast\bm{E}\ast[\Delta^{\frac{L}{2}},\nabla]N$
$\displaystyle\,+a^{-1}(N+1)(\Psi+C)\cdot\bm{\varepsilon}[G]\ast\Sigma\ast\left[\Delta^{\frac{L}{2}},\nabla\right]\phi$
$\displaystyle\,+a^{3}(N+1)\bm{\varepsilon}[G]\ast\nabla\phi\ast\left(\nabla^{2}\Delta^{\frac{L}{2}-2}\text{{Ric}}[G]\ast\nabla^{2}\phi+\mathfrak{J}([\Delta^{\frac{L}{2}},\nabla]\phi)\right)$
$\displaystyle\,\color[rgb]{0,0,0}-a^{-1}(N+1)\mathfrak{J}([\Delta^{\frac{L}{2}},\text{{curl}}_{G}]\bm{E})\color[rgb]{0,0,0}+\mathfrak{J}([\partial_{t},\Delta^{\frac{L}{2}}]\bm{B})+\Delta^{\frac{L}{2}}\left[a^{-3}(N+1)\bm{B}\ast\Sigma\right]\cdot
G$ $\displaystyle\,+\Delta^{\frac{L}{2}}\left[4\pi a^{2}\nabla^{\sharp
m}\phi(\Psi+C)+\frac{2\pi}{3}a^{5}\Delta^{\frac{L}{2}}\nabla^{\sharp
m}\left(a^{-6}(\Psi+C)^{2}+a^{-2}\lvert\nabla\phi\rvert_{G}^{2}\right)\right]\bm{\varepsilon}[G]_{(\cdot)m(\cdot)}$

###  11.4. $L^{2}_{G}$ error term estimates

In this subsection, we collect how the error terms can be controlled in terms
of energies as well as homogenous Sobolev norms of $\phi$. We don’t claim that
these estimates are optimal – in particular, we note that at low order (like
$L=2$), many of the curvature errors that appear in the estimates below could
be avoided entirely: These arise as a result of applying the general estimates
in Lemma 4.5 where the Ricci tensor doesn’t naturally occur in the respective
equations, and can be avoided at low orders by applying (4.4f) on all
curvature terms that occur.
Instead of optimality, we try to keep both notation and form of the error term
estimates as simple as possible and the energy estimates between base and top
level as unified as possible. In particular, we track the “worst” curvature
energy occurring at high orders for all estimates below, even if these terms
are added in artificially for low orders.

######  Lemma 11.10 (Estimates for borderline error terms).

Let $L\in 2\mathbb{Z}_{+},\,L\leq 20$. Then, the following estimates hold:

(11.17a) $\displaystyle\|\mathfrak{H}_{L,Border}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{-4}\sqrt{\mathcal{E}^{(L)}(\Sigma,\cdot)}+\varepsilon
a^{-4-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\Sigma,\cdot)}+\varepsilon^{2}a^{-4-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}$ (11.17b)
$\displaystyle\|\mathfrak{N}_{L,Border}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{-4}\left[\sqrt{\mathcal{E}^{(L)}(\phi,\cdot)}+\sqrt{\mathcal{E}^{(L)}(\Sigma,\cdot)}\right]+\varepsilon
a^{-4}\sqrt{\mathcal{E}^{(L)}(N,\cdot)}$ $\displaystyle\,+\varepsilon
a^{-4-c\sqrt{\varepsilon}}\left[\sqrt{\mathcal{E}^{(\leq
L-2)}(\phi,\cdot)}+\sqrt{\mathcal{E}^{(\leq
L-2)}(\Sigma,\cdot)}+\sqrt{\mathcal{E}^{(\leq L-2)}(N,\cdot)}\right]$
$\displaystyle\,+\underbrace{\varepsilon^{2}a^{-4}\sqrt{\mathcal{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}+\varepsilon
a^{-4-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-4)}(\text{{Ric}},\cdot)}}_{\text{not present for }L=2}$ (11.17c)
$\displaystyle\|\mathfrak{N}_{L+1,Border}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{-6}\left[\sqrt{a^{4}\mathcal{E}^{(L+1)}(\phi,\cdot)}+\sqrt{a^{4}\mathcal{E}^{(L+1)}(\Sigma,\cdot)}\right]+\varepsilon
a^{-6}\sqrt{a^{4}\mathcal{E}^{(L+1)}(N,\cdot)}$ $\displaystyle\,+\varepsilon
a^{-4}\left[\sqrt{\mathcal{E}^{(L)}(\phi,\cdot)}+\sqrt{\mathcal{E}^{(L)}(\Sigma,\cdot)}+\sqrt{\mathcal{E}^{(L)}(N,\cdot)}\right]$
$\displaystyle\,+\varepsilon
a^{-4-c\sqrt{\varepsilon}}\left[\sqrt{\mathcal{E}^{(\leq
L-2)}(\phi,\cdot)}+\sqrt{\mathcal{E}^{(\leq
L-2)}(\Sigma,\cdot)}+\sqrt{\mathcal{E}^{(\leq L-2)}(N,\cdot)}\right]$
$\displaystyle\,+\varepsilon^{2}a^{-4-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}$ (11.17d)
$\displaystyle\|\mathfrak{P}_{L,Border}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\color[rgb]{0,0,0}\varepsilon\color[rgb]{0,0,0}a^{-3}\sqrt{\mathcal{E}^{(L)}(\phi,\cdot)}+\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L)}(N,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-2)}(N,\cdot)}$
$\displaystyle\,+\color[rgb]{0,0,0}\varepsilon\color[rgb]{0,0,0}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\phi,\cdot)}\color[rgb]{0,0,0}+\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L)}(\Sigma,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(L-2)}(\Sigma,\cdot)}\color[rgb]{0,0,0}$
$\displaystyle\,+\underbrace{\varepsilon^{2}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-3)}(\text{{Ric}},\cdot)}}_{\text{not present for }L=2}$ (11.17e)
$\displaystyle\|\mathfrak{Q}_{L,Border}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L+1)}(N,\cdot)}+\color[rgb]{0,0,0}\varepsilon\color[rgb]{0,0,0}a^{-3}\sqrt{a^{-4}\mathcal{E}^{(L)}(\phi,\cdot)}$
$\displaystyle\,+\color[rgb]{0,0,0}\varepsilon\color[rgb]{0,0,0}a^{-3-c\sqrt{\varepsilon}}\sqrt{a^{-4}\mathcal{E}^{(\leq
L-2)}(\phi,\cdot)}$ (11.17f)
$\displaystyle\|\mathfrak{P}_{L+1,Border}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\color[rgb]{0,0,0}\varepsilon\color[rgb]{0,0,0}a^{-3}\sqrt{\mathcal{E}^{(L+1)}(\phi,\cdot)}+\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L+1)}(N,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-1)}(N,\cdot)}$
$\displaystyle\,+\color[rgb]{0,0,0}\varepsilon\color[rgb]{0,0,0}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-1)}(\phi,\cdot)}+\color[rgb]{0,0,0}\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L+1)}(\Sigma,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(L-1)}(\Sigma,\cdot)}\color[rgb]{0,0,0}$
$\displaystyle\,+\varepsilon^{2}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}$ (11.17g)
$\displaystyle\|\mathfrak{Q}_{L+1,Border}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L+2)}(N,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(L+1)}(N,\cdot)}$
$\displaystyle\,+\color[rgb]{0,0,0}\varepsilon\color[rgb]{0,0,0}a^{-3}\sqrt{a^{-4}\mathcal{E}^{(L+1)}(\phi,\cdot)}+\color[rgb]{0,0,0}\varepsilon\color[rgb]{0,0,0}a^{-3-c\sqrt{\varepsilon}}\sqrt{a^{-4}\mathcal{E}^{(\leq
L-1)}(\phi,\cdot)}$ (11.17h)
$\displaystyle\|\mathfrak{S}_{L,Border}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L)}(\Sigma,\cdot)}+\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L)}(N,\cdot)}+\varepsilon^{2}a^{-3}\sqrt{\mathcal{E}^{(L-2)}(\text{{Ric}},\cdot)}$
$\displaystyle\,+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\Sigma,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-2)}(N,\cdot)}$
$\displaystyle\,+\underbrace{\varepsilon^{2}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-4)}(\text{{Ric}},\cdot)}}_{\text{not present for }L=2}$ (11.17i)
$\displaystyle\|\mathfrak{R}_{L,Border}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L+2)}(N,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L)}(N,\cdot)}$
$\displaystyle\,+\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L)}(\text{{Ric}},\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-2)}(\text{{Ric}},\cdot)}$
(11.17j) $\displaystyle\|\mathfrak{R}_{L+1,Border}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L+3)}(N,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L+1)}(N,\cdot)}$
$\displaystyle\,+\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L+1)}(\text{{Ric}},\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-1)}(\text{{Ric}},\cdot)}$
(11.17k)
$\displaystyle\,\|\mathfrak{E}_{L,Border}\|_{L^{2}_{G}}+\|\mathfrak{B}_{L,Border}\|_{L^{2}_{G}}$
$\displaystyle\lesssim$ $\displaystyle\,\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L)}(\phi,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-2)}(\phi,\cdot)}$
$\displaystyle\,+\varepsilon
a^{-3}\left(\sqrt{\mathcal{E}^{(L)}(N,\cdot)}+\sqrt{\mathcal{E}^{(L)}(\Sigma,\cdot)}\right)+\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L)}(W,\cdot)}$ $\displaystyle\,+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\left(\sqrt{\mathcal{E}^{(\leq
L-2)}(N,\cdot)}+\sqrt{\mathcal{E}^{(\leq
L-2)}(\Sigma,\cdot)}+\sqrt{\mathcal{E}^{(\leq L-2)}(W,\cdot)}\right)$
$\displaystyle\,+\varepsilon^{2}a^{-3}\sqrt{\mathcal{E}^{(L-2)}(\text{{Ric}},\cdot)}+\underbrace{\varepsilon^{2}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-4)}(\text{{Ric}},\cdot)}}_{\text{not present for }L=2}$

###### Proof.

All of these estimates follow from applying $L^{2}_{G}$-$L^{\infty}_{G}$-type
Hölder estimates to the individual nonlinear terms. The lower order terms are
either controlled by the zero order estimates in subsection 4.1 or the a
priori estimates in Lemma 4.3. Furthermore, we apply Lemma 4.5, along with
again Lemma 4.3, to translate $L^{2}_{G}$-norms into energies up to additional
curvature energy terms. For the sake of simplicity, we always estimate
$\frac{\dot{a}}{a}$ by $a^{-3}$ up to constant (see (2.3)), and liberally
apply (3.8) to deal with odd order energies and to distribute
$a^{-c\sqrt{\varepsilon}}$ factors to lower orders while updating $c>0$
whereever this is convenient. ∎

######  Lemma 11.11 (Estimates for top order error terms).

(11.18a) $\displaystyle\|\mathfrak{E}_{L,top}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\sqrt{\varepsilon}a^{1-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(L-1)}(\text{{Ric}},\cdot)}=\sqrt{\varepsilon}a^{-1-c\sqrt{\varepsilon}}\sqrt{a^{4}\mathcal{E}^{(L-1)}(\text{{Ric}},\cdot)}$
(11.18b) $\displaystyle\|\mathfrak{B}_{L,top}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{-1}\sqrt{\mathcal{E}^{(L-1)}(\text{{Ric}},\cdot)}=\color[rgb]{0,0,0}\varepsilon
a^{-3}\sqrt{a^{4}\mathcal{E}^{(L-1)}(\text{{Ric}},\cdot)}\color[rgb]{0,0,0}$

###### Proof.

This follows directly using (4.2c) and (4.4g) for the Bel-Robinson terms as
well as (4.4e). ∎

######  Lemma 11.12 (Junk terms).

Recalling $\parallel$-notation from Remark 2.12, the following holds:

(11.19a) $\displaystyle\|\mathfrak{M}_{L,Junk}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon a^{-2-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-1)}(\phi,\cdot)}+a^{-2-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\phi,\cdot)}$
$\displaystyle\,+a^{-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-1)}(\Sigma,\cdot)}+\sqrt{\varepsilon}a^{-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}$ (11.19b)
$\displaystyle\|\tilde{\mathfrak{M}}_{L,Junk}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}+a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-1)}(\Sigma,\cdot)}$ (11.19c)
$\displaystyle\|\mathfrak{H}_{L,Junk}^{\parallel}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon a^{-4-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\Sigma,\cdot)}+\sqrt{\varepsilon}a^{-2-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L)}(\phi,\cdot)}$ $\displaystyle\,+\varepsilon
a^{-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-2)}(\text{{Ric}},\cdot)}$
(11.19d) $\displaystyle\|\mathfrak{N}_{L,Junk}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon a^{-4-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(N,\cdot)}+\varepsilon a^{-c\sigma}\sqrt{\mathcal{E}^{(L)}(\phi,\cdot)}$
$\displaystyle\,+\varepsilon
a^{-4-c\sqrt{\varepsilon}}\left[\sqrt{\mathcal{E}^{(\leq
L-2)}(\phi,\cdot)}+\sqrt{\mathcal{E}^{(\leq L-2)}(\Sigma,\cdot)}\right]$
$\displaystyle\,+\underbrace{\varepsilon a^{-4}\sqrt{\mathcal{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}+\varepsilon
a^{-4-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-4)}(\text{{Ric}},\cdot)}}_{\text{not present for }L=2}$ (11.19e)
$\displaystyle\|\mathfrak{N}_{L+1,Junk}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon a^{-4-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-1)}(N,\cdot)}+\varepsilon
a^{-c\sigma}\left(\sqrt{\mathcal{E}^{(L+1)}(\phi,\cdot)}+\sqrt{\mathcal{E}^{(L+1)}(\Sigma,\cdot)}\right)$
$\displaystyle\,+\varepsilon
a^{-4-c\sqrt{\varepsilon}}\left[\sqrt{\mathcal{E}^{(\leq
L-1)}(\phi,\cdot)}+\sqrt{\mathcal{E}^{(\leq L-1)}(\Sigma,\cdot)}\right]$
$\displaystyle\,+\varepsilon^{2}a^{-4}\sqrt{\mathcal{E}^{(\leq
L-1)}(\text{{Ric}},\cdot)}+\underbrace{\varepsilon^{2}a^{-4-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-3)}(\text{{Ric}},\cdot)}}_{\text{not present for }L=2}$ (11.19f)
$\displaystyle\|\mathfrak{P}_{L,Junk}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{1-c\sigma}\sqrt{\mathcal{E}^{(L)}(\phi,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-2)}(\phi,\cdot)}$
$\displaystyle\,+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\Sigma,\cdot)}+\sqrt{\varepsilon}a^{1-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(L)}(N,\cdot)}$
$\displaystyle\,+\left[\varepsilon
a^{-3-c\sqrt{\varepsilon}}+\sqrt{\varepsilon}a^{-1-c\sqrt{\varepsilon}}\right]\sqrt{\mathcal{E}^{(\leq
L-2)}(N,\cdot)}$
$\displaystyle\,+\underbrace{\varepsilon^{2}a^{1-c\sigma}\sqrt{\mathcal{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)+\varepsilon^{2}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-3)}(\text{{Ric}},\cdot)}}}_{\text{not present for }L=2}$ (11.19g)
$\displaystyle\|\mathfrak{P}_{L+1,Junk}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{1-c\sigma}\sqrt{\mathcal{E}^{(L+1)}(\phi,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-1)}(\phi,\cdot)}$
$\displaystyle\,+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-1)}(\Sigma,\cdot)}+\sqrt{\varepsilon}a^{1-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(L+1)}(N,\cdot)}$
$\displaystyle\,+\left[\varepsilon
a^{-3-c\sqrt{\varepsilon}}+\sqrt{\varepsilon}a^{-1-c\sqrt{\varepsilon}}\right]\sqrt{\mathcal{E}^{(\leq
L-1)}(N,\cdot)}$
$\displaystyle\,+\varepsilon^{2}a^{-1-c\sigma}\sqrt{a^{4}\mathcal{E}^{(L-1)}(\text{{Ric}},\cdot)}$
$\displaystyle\,+\left(\varepsilon^{2}a^{-1-c\sigma}+\varepsilon^{2}a^{-3-c\sqrt{\varepsilon}}\right)\sqrt{\mathcal{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}$ (11.19h)
$\displaystyle\|\mathfrak{Q}_{L,Junk}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\color[rgb]{0,0,0}\varepsilon
a^{-1-c\sqrt{\varepsilon}}\sqrt{a^{-4}\mathcal{E}^{(L)}(\phi,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\mathcal{E}^{(\leq
L-2)}(\phi,\cdot)\color[rgb]{0,0,0}$
$\displaystyle\,+\color[rgb]{0,0,0}\sqrt{\varepsilon}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L)}(\Sigma,\cdot)}\color[rgb]{0,0,0}+\sqrt{\varepsilon}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L)}(N,\cdot)}$ $\displaystyle\,+\underbrace{\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}}_{\text{not present for }L=2}$ (11.19i)
$\displaystyle\|\mathfrak{Q}_{1,Junk}\|\lesssim$ $\displaystyle\,\varepsilon
a^{-3-c\sqrt{\varepsilon}}\mathcal{E}^{(1)}(N,\cdot)+\varepsilon^{\frac{3}{2}}a^{-3-c\sqrt{\varepsilon}}\|\nabla\phi\|_{L^{2}_{G}}+\color[rgb]{0,0,0}\sqrt{\varepsilon}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
1)}(\Sigma,\cdot)}\color[rgb]{0,0,0}$ (11.19j)
$\displaystyle\|\mathfrak{Q}_{L+1,Junk}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\color[rgb]{0,0,0}\varepsilon
a^{-1-c\sqrt{\varepsilon}}\sqrt{a^{-4}\mathcal{E}^{(L+1)}(\phi,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\mathcal{E}^{(\leq
L-1)}(\phi,\cdot)\color[rgb]{0,0,0}$
$\displaystyle\,\color[rgb]{0,0,0}+\sqrt{\varepsilon}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L+1)}(\Sigma,\cdot)}\color[rgb]{0,0,0}+\sqrt{\varepsilon}a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L+1)}(N,\cdot)}$ $\displaystyle\,+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-1)}(\text{{Ric}},\cdot)}$
(11.19k)
$\displaystyle\|\mathfrak{S}_{L,Junk}^{\parallel}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon
a^{1-c\sigma}\sqrt{\mathcal{E}^{(L)}(\Sigma,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(\Sigma,\cdot)}+\sqrt{\varepsilon}a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(L)}(\phi,\cdot)}$
$\displaystyle\,+\left(\varepsilon
a^{-3}+a^{1-c\sqrt{\varepsilon}}\right)\sqrt{\mathcal{E}^{(\leq
L)}(N,\cdot)}+\varepsilon a^{5-c\sigma}\sqrt{\mathcal{E}^{(\leq
L-1)}(\text{{Ric}},\cdot)}$ $\displaystyle\,+\underbrace{\varepsilon
a^{-3}\mathcal{E}^{(L-2)}(\text{{Ric}},\cdot)+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-4)}(\text{{Ric}},\cdot)}}_{\text{not present for }L=2}$ (11.19l)
$\displaystyle\|\mathfrak{R}_{L,Junk}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon^{2}a^{1-c\sigma}\sqrt{\mathcal{E}^{(\leq
L-1)}(\text{{Ric}},\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-2)}(\text{{Ric}},\cdot)}$
$\displaystyle\,+\varepsilon a^{1-c\sigma}\sqrt{\mathcal{E}^{(\leq
L+2)}(\Sigma,\cdot)}+a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L)}(\Sigma,\cdot)}$
$\displaystyle\,+a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L)}(N,\cdot)}$ (11.19m)
$\displaystyle\|\mathfrak{R}_{L+1,Junk}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon^{2}a^{1-c\sigma}\sqrt{\mathcal{E}^{(\leq
L)}(\text{{Ric}},\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-1)}(\text{{Ric}},\cdot)}$
$\displaystyle\,+\varepsilon a^{1-c\sigma}\sqrt{\mathcal{E}^{(\leq
L+3)}(\Sigma,\cdot)}+a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L+1)}(\Sigma,\cdot)}$
$\displaystyle\,+a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L+1)}(N,\cdot)}$ (11.19n)
$\displaystyle\|\mathfrak{E}_{L,Junk}^{\parallel}\|_{L^{2}_{G}}+\|\mathfrak{B}_{L,Junk}^{\parallel}\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,\varepsilon a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L)}(W,\cdot)}+\varepsilon a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(W,\cdot)}$ $\displaystyle\,+\varepsilon
a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(L)}(\phi,\cdot)}+\left(\varepsilon
a^{-3-c\sqrt{\varepsilon}}+a^{-1-c\sqrt{\varepsilon}}\right)\sqrt{\mathcal{E}^{(\leq
L-2)}(\phi,\cdot)}$
$\displaystyle\,+\sqrt{\varepsilon}a^{1-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L)}(N,\cdot)}+\varepsilon a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq
L-2)}(N,\cdot)}$ $\displaystyle\,+\varepsilon
a^{-1-c\sigma}\sqrt{\mathcal{E}^{(L)}(\Sigma,\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathcal{E}^{(\leq L-2)}(\Sigma,\cdot)}$
$\displaystyle\,+\varepsilon
a^{-3}\sqrt{\mathcal{E}^{(L-2)}(\text{{Ric}},\cdot)}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\underbrace{\sqrt{\mathcal{E}^{(\leq
L-4)}(\text{{Ric}},\cdot)}}_{\text{not present for }L=2}$

###### Proof.

Once again, this follows by applying the a priori estimates from subsection
4.1 and Lemma 4.3, as well as the bootstrap assumption (3.17h) for the lapse,
to deal with the lower order terms in the nonlinearities, and then applying
Lemma 4.5 as well as (3.8) whereever this is necessary. Further, especially in
(11.19n), it is often more convenient to use the bootstrap assumption for
$\|\nabla\phi\|_{C_{G}}$ instead of the a priori estimate (4.4e) to gain
higher powers of $\varepsilon$ in prefactors.
Recognizing that every low order curvature term can be estimated up to
constant by $a^{-c\sqrt{\varepsilon}}$ at worst (see (4.4f)), we also note
that any of the highly nonlinear curvature terms in $\mathfrak{J}$-expressions
turn out to be negligible after updating $c$ compared to Ricci energies
arising from applying Lemma 4.5 or compared to junk terms in which
$\text{{Ric}}[G]$ is tracked explicitly. ∎

##  12\. Appendix – Future stability

Here, we collect the commutators in CMCSH gauge necessary to study the
commuted scalar-field equations:

######  Lemma 12.1 (Commutator formulas for future stabilty).

Let $\zeta$ be a scalar function on $\Sigma_{T}$. Then, the following formulas
hold:

$\displaystyle[\widetilde{\partial_{0}},\nabla]\zeta=$ $\displaystyle\,0$
$\displaystyle[\widetilde{\partial_{0}},\Delta_{\bm{g}}]\zeta=$
$\displaystyle\,(\widetilde{\partial_{0}}(\bm{g}^{-1})^{ab})\nabla_{a}\nabla_{b}\zeta-2(\bm{g}^{-1})^{ab}\left(\text{{div}}_{\bm{g}}(\bm{n}\bm{\Sigma})_{a}-2\nabla_{a}\bm{n}\right)\nabla_{b}\zeta$

Schematically, for $k\in\mathbb{N}$, this implies

(12.1a) $\displaystyle[\widetilde{\partial_{0}},\Delta_{\bm{g}}^{k}]\zeta=$
$\displaystyle\,\sum_{I_{\bm{n}}+I_{\bm{\Sigma}}+I_{\zeta}=2k-1}\nabla^{I_{\bm{n}}}\bm{n}\ast_{\bm{g}}\nabla^{I_{\bm{\Sigma}}}\bm{\Sigma}\ast_{\bm{g}}\nabla^{I_{\zeta}+1}\zeta$
(12.1b)
$\displaystyle[\widetilde{\partial_{0}},\nabla\Delta_{\bm{g}}^{k}]\zeta=$
$\displaystyle\,\sum_{I_{\bm{n}}+I_{\bm{\Sigma}}+I_{\zeta}=2k}\nabla^{I_{\bm{n}}}\bm{n}\ast_{\bm{g}}\nabla^{I_{\bm{\Sigma}}}\bm{\Sigma}\ast_{\bm{g}}\nabla^{I_{\zeta}+1}\zeta$

###### Proof.

This follows from straightfoward computations, similar to Lemma 11.6 for the
low order commutators and to Lemma 11.7 for higher orders. ∎

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