$\displaystyle\,+(\hat{\Gamma}-\Gamma)^{l}_{jk}v^{k}v^{m}\gamma_{ml}\gamma_{ir}v^{r}\cdot
8\mathcal{F}^{\prime\prime}(\lvert v\rvert_{\gamma}^{2})$
$\displaystyle\,+\left[-\Gamma_{ij}^{k}\gamma_{kl}+\partial_{i}\gamma_{jl}-\Gamma_{il}^{k}\gamma_{jk}\right]v^{l}\cdot
2\mathcal{F}^{\prime}(\lvert v\rvert_{\gamma}^{2})$

All but the final line on the right hand side can now be estimated as above,
while for the final line, we use

$\partial_{i}\gamma_{jl}=\hat{\Gamma}_{ij}^{k}\gamma_{kl}+\hat{\Gamma}_{il}^{k}\gamma_{kj}$

to write it as

$\left[(\hat{\Gamma}-\Gamma)_{ij}^{k}\gamma_{kl}+(\hat{\Gamma}-\Gamma)^{k}_{il}\gamma_{kj}\right]v^{l}\cdot
2\mathcal{F}^{\prime}(\lvert v\rvert_{\gamma}^{2})$

which can also be estimated as before. For higher orders, one can successively
perform such commutations for each vertical derivative so that any appearance
of Christoffel symbols again reduces to the difference tensor, and then apply
the product rule. However, since we already encounter zero order curvature
terms for $K=1,L=2$ in the mixed derivative case, the highest order curvature
terms one can encounter in the general case are
$\nabla^{K-1}\left(\text{{Ric}}[G]-2\kappa G\right)$, leading to the curvature
norm in (4.10a). Again, the respective supremum norm estimates follow from the
same bounds, using the a priori bounds (4.3c) and (4.3e) to control the metric
terms.

Regarding (4.10d), we split up $\bm{X}f_{FLRW}$ into the horizontal and
vertical derivative terms and prove that both parts satisfy the claimed bound:
The terms arising from the horizontal components of $\bm{X}f$ can be bounded
directly using (4.10c) since $\mathbf{A}_{i}\xi=\underline{\nabla}_{i}\xi$.
For the remaining vertical components, we may encounter terms where only
vertical terms fall on $f_{FLRW}$. However, since we are only interested in a
uniform pointwise bound, we don’t need to ensure that all Christoffel terms
that arise from the Sasaki connection coefficients reduce to difference tensor
terms, and note that (4.3c) also implies the crude bound

$\lvert\partial^{\leq 11}\Gamma\rvert_{G}\lesssim a^{-c\sqrt{\varepsilon}}$

in a coordinate neighbourhood on which $\hat{\Gamma}$ remains bounded. The
bound then follows for the vertical parts of $\bm{X}f$ using (4.3e), (3.13)
and (4.3b) and that $M$ is compact. ∎

######  Lemma 4.8 ($C^{0}_{1,\underline{G}_{0}}$-estimate for the Vlasov
distribution function).

For $t\in(t_{Boot},t_{0}]$, one has:

(4.13)
$\displaystyle\|f-f_{FLRW}\|_{C^{0}_{1,\underline{G}_{0}}(TM_{t})}\lesssim$
$\displaystyle\,\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}$

###### Proof.

We start out by rewriting the rescaled Vlasov equation (2.21a) as an
inhomogeneous transport equation for $f-f_{FLRW}$:

$\displaystyle\,\partial_{t}\left(f-f_{FLRW}\right)-\bm{X}(f-f_{FLRW})$
$\displaystyle=$
$\displaystyle\,-a^{-1}\frac{v^{j}}{v^{0}}\mathbf{A}_{j}f_{FLRW}+a^{-1}v^{0}\nabla^{\sharp
j}N\mathbf{B}_{j}f_{FLRW}+2N\frac{\dot{a}}{a}v^{j}\mathbf{B}_{j}f_{FLRW}-2(N+1)a^{-3}\Sigma^{\sharp
j}_{i}v^{i}\mathbf{B}_{j}f_{FLRW}$

Let $(X,V,Z)$ be the corresponding characteristics satisfying (4.5) emanating
from $(x,v)\in TM_{t}$ with $Z(t)=(f-f_{FLRW})(t,x,v)$. Then, we have

$\displaystyle\frac{dZ}{dt}=$
$\displaystyle\,a^{-1}\frac{V^{j}}{V^{0}}\mathbf{A}_{j}f_{FLRW}-a^{-1}V^{0}\nabla^{\sharp
j}N\mathbf{B}_{j}f_{FLRW}$
$\displaystyle\,-2N\frac{\dot{a}}{a}V^{j}\mathbf{B}_{j}f_{FLRW}+2(N+1)a^{-3}\Sigma^{\sharp
j}_{i}V^{i}\mathbf{B}_{j}f_{FLRW}\,.$

Note that, on the support of $(f-f_{FLRW})(t,x,\cdot)$, $V^{0}$ can be
controlled by $a^{-c\sqrt{\varepsilon}}$ using (4.3c) and (4.6). We obtain in
total:

(4.14)
$\left\lvert\frac{dZ}{dt}\right\rvert\lesssim{\varepsilon}a^{-1-c\sigma}+\varepsilon
a^{1-c\sigma}+\varepsilon a^{-1-c\sigma}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\lesssim\varepsilon
a^{-3-c\sqrt{\varepsilon}}+\sqrt{\varepsilon}a^{-1-c\sqrt{\varepsilon}}$

Using the initial data assumption (3.5) and again (4.6) and (4.3c) to control
$v^{0}$ on the support of $f-f_{FLRW}$, this implies

$\displaystyle\lvert v^{0}(f-f_{FLRW})(t,x,v)\rvert\leq$
$\displaystyle\,v^{0}\lvert(f-f_{FLRW})(t,X(t),V(t))-(f-f_{FLRW})(t_{0},X(t_{0}),V(t_{0}))\rvert$
$\displaystyle\,+\lvert v^{0}(f-f_{FLRW})(t_{0},X(t_{0}),V(t_{0}))\rvert$
$\displaystyle\leq$
$\displaystyle\,v^{0}\left(\chi_{\mathrm{supp}(f-f_{FLRW})(t,x,\cdot)}(v)\right)\lvert
Z(t)-Z(t_{0})\rvert+\|f-f_{FLRW}\|_{C^{0}_{1,\underline{G}_{0}}(TM_{t_{0}})}$
$\displaystyle\lesssim$
$\displaystyle\,a(t)^{-c\sqrt{\varepsilon}}\left[\int_{t}^{t_{0}}\left(\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}}+\sqrt{\varepsilon}a(s)^{-1-c\sqrt{\varepsilon}}\right)\,ds+\varepsilon^{2}\right]$

The statement now follows by (2.6). ∎

######  Lemma 4.9 (Strong $C^{l}_{1,\underline{G}_{0}}$-norm estimates for the
Vlasov distribution function).

Let $t\in(t_{Boot},t_{0}]$. Then, the following holds:

(4.15)
$\displaystyle\|f-f_{FLRW}\|_{C^{11}_{1,\underline{G}_{0}}(TM_{t})}\lesssim\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}$

Note that, for $\varepsilon>0$ suitably small, (4.15) is a bootstrap
improvement for (3.12b).

###### Proof.

The following proof will require us to choose a spatial coordinate frame to
express horizontal frame derivatives, so we must first establish suitable
coordinate systems to cover the entire evolution: Since $M$ is compact, there
exists some sufficiently small $r>0$ such that an open ball $B_{r}(x)$ of
radius $r>0$ around any $x\in M$ is a coordinate neighbourhood, and there
exist $x_{1},\dots,x_{m}\in M$ such that
$\\{B_{\frac{r}{2}}(x_{i})\\}_{i=1,\dots,m}$ covers $M$. Furthermore, for
(4.5a), $\lvert\frac{d}{dt}X\rvert_{G}\leq a^{-1}$ holds, which is integrable
on $[0,t_{0}]$ (see (2.6)), and (4.5a) will describe the spatial
characteristics for all systems considered in this proof. In particular, there
exists a finite partition of $(t_{Boot},t_{0}]$ by intervals $(t_{j},t_{j-1}]$
such that $\int_{t_{j+1}}^{t_{j}}a(s)^{-1}\,ds<\frac{r}{2}$ holds. Thus, for
any $y\in B_{\frac{r}{2}}(x_{i})$, the characteristic $X$ emanating from $y$
at time $t_{j}$ is contained in the coordinate neighbourhood $B_{r}(x_{i})$ on
$(t_{j},t_{j-1}]$. Thus, we can prove (4.15) for any $t\in(t_{1},t_{0}]$ using
local coordinates by having found a choice of coordinates that remains valid
throughout the evolution, and since we can cover $M$ in a finite amount of
such coordinate neighbourhoods. Finally, the bounds then extend to $t_{1}$ by
continuity, and thus we can prove the statement iteratively on the entire
bootstrap interval by proving the bound on $(t_{j},t_{j-1}]$ from data
emanating from $M_{t_{j-1}}$.

Having now justified that we can essentially perform this proof in local
coordinates, we move on to the proof itself: First, we commute (2.21a) with
$\mathbf{B}_{i}$ to obtain

$\partial_{t}\mathbf{B}_{i}(f-f_{FLRW})=\mathbf{B}_{i}\partial_{t}f=\bm{X}\mathbf{B}_{i}(f-f_{FLRW})+\mathbf{B}_{i}\bm{X}f_{FLRW}+(\mathcal{K}_{1,0})_{i}$

with

$\displaystyle(\mathcal{K}_{1,0})_{i}(\cdot,\cdot,v)=$
$\displaystyle\,[\mathbf{B}_{i},\bm{X}](f-f_{FLRW})(\cdot,\cdot,v)$

(see (2.27g)). Moving to the corresponding characteristic system
$(X,V,Z_{i})$, $X$ and $V$ again satisfy (4.5) emanating from $(x,v)\in
TM_{t}$, while $Z_{i}(t)=\mathbf{B}_{i}(f-f_{FLRW})(t,X(t),V(t))$ satisfies

$\frac{dZ_{i}}{dt}=(\mathcal{K}_{1,0})_{i}(t,X(t),V(t))+\mathbf{B}_{i}\bm{X}f_{FLRW}(t,X(t),V(t))\,.$

The horizontal term derivative term in (2.27g) can be bounded with respect to
$\gamma$ by $\varepsilon^{\frac{1}{4}}a^{-c\sigma}$ using (3.12b) and (3.12a).
Applying the $C^{0}_{\gamma}$-bound in (4.1b) as well as (3.13) to the
remaining terms, we then obtain

$\displaystyle
V^{0}_{\gamma}\lvert\mathcal{K}_{1,0}(\cdot,X,V)\rvert_{\gamma}\lesssim$
$\displaystyle\,{\varepsilon}^{\frac{1}{4}}a^{-1-c\sigma}+\varepsilon
a^{-3}V^{0}_{\gamma}\lvert Z\rvert_{\gamma}\,.$

Due to (4.10d) and (4.3c) and since $f_{FLRW}$ as compact momentum support
with respect to $\gamma$, we also have

$\displaystyle
V^{0}_{\gamma}\lvert\mathbf{B}\bm{X}f_{FLRW}\rvert_{\gamma}\lesssim$
$\displaystyle\,\varepsilon a^{-3-c\sqrt{\varepsilon}}\,.$

Finally, by the same arguments as in (4.8),
$-\frac{dV^{0}_{\gamma}}{dt}\lesssim(\varepsilon
a^{-3}+a^{-1-c\sigma})(V^{0}_{\gamma})$ holds due to (4.5b), and thus
altogether

$-\frac{1}{2V^{0}_{\gamma}\lvert
Z\rvert_{\gamma}}\frac{d}{dt}\left[(V^{0}_{\gamma}\lvert
Z\rvert_{\gamma})^{2}\right]\lesssim\left(\varepsilon
a^{-3}+a^{-1-c\sigma}\right)V^{0}_{\gamma}\lvert Z\rvert_{\gamma}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}+{\varepsilon}^{\frac{1}{4}}a^{-1-c\sigma}$

which, after integrating and applying the Gronwall lemma as well as the scale
factor integral bounds from Lemma 2.2 and the initial data assumption, yields

(4.16)
$\lvert\underline{\nabla}_{vert}(f-f_{FLRW})(t,X(t),V(t))\rvert_{\underline{\gamma}_{0}}\lesssim\varepsilon^{\frac{1}{4}}+\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}\lesssim\varepsilon^{\frac{1}{4}}a(t)^{-c\sqrt{\varepsilon}}\,.$

Repeating the same procedure for $\mathbf{A}_{i}$, we have

$\partial_{t}\mathbf{A}_{i}(f-f_{FLRW})=\bm{X}\mathbf{A}_{i}(f-f_{FLRW})+\mathbf{A}_{i}\bm{X}f_{FLRW}+(\mathcal{K}_{1,1})_{i}$

with

$\displaystyle(\mathcal{K}_{1,1})_{i}=$
$\displaystyle\,[\mathbf{A}_{i},\bm{X}](f-f_{FLRW})+[\partial_{t},\mathbf{A}_{i}](f-f_{FLRW})\,,$

where (2.27f) and (2.27d) imply, using (3.12a) and (4.1b):

$\displaystyle\lvert[\bm{X},\mathbf{A}_{(\cdot)}](f-f_{FLRW})\rvert_{\gamma}\lesssim$
$\displaystyle\,a^{-1-c\sigma}\lvert\mathbf{A}_{(\cdot)}(f-f_{FLRW})\rvert_{\gamma}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\lvert\underline{\nabla}_{vert}(f-f_{FLRW})\rvert_{\underline{\gamma}_{0}}$
$\displaystyle\lvert[\partial_{t},\mathbf{A}_{(\cdot)}](f-f_{FLRW})\rvert_{\gamma}\lesssim$
$\displaystyle\,\varepsilon
a^{-3-c\sqrt{\varepsilon}}\lvert\underline{\nabla}_{vert}\left(f-f_{FLRW}\right)\rvert_{\underline{\gamma}_{0}}$

Using (4.16), it follows that

$\lvert\mathcal{K}_{1,1}\rvert_{\gamma}\lesssim
a^{-1-c\sigma}\lvert\mathbf{A}_{(\cdot)}(f-f_{FLRW})\rvert_{\gamma}+\varepsilon^{\frac{5}{4}}a^{-3-c\sqrt{\varepsilon}}$

With (4.10d), one also computes

$\lvert\mathbf{A}_{(\cdot)}\bm{X}f_{FLRW}\rvert_{\gamma}\lesssim\varepsilon
a^{-1-c\sigma}+\varepsilon a^{-3-c\sqrt{\varepsilon}}\,,$

and now obtains that

(4.17)
$\lvert\mathbf{A}_{(\cdot)}(f-f_{FLRW})(t,X(t),V(t))\rvert_{\gamma}\lesssim\sqrt{\varepsilon}a^{-c\sqrt{\varepsilon}}$

holds. Finally, we can redo the proof of (4.16), replacing (3.12b) with the
improved bound (4.17), which improves the bound in (4.16) to
$\sqrt{\varepsilon}a^{-c\sqrt{\varepsilon}}$ after updating constants.
Altogether, one now concludes as in the proof of Lemma 4.8 that

$\|\underline{\nabla}(f-f_{FLRW})\|_{{C}^{0}_{1,\underline{\gamma}_{0}}(TU_{t})}\lesssim\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}$

holds for suitably chosen coordinate neighbourhoods as outlined at the start
of the proof, and thus the same bounds on $TM_{t}$. With (4.3c), this also
implies

$\displaystyle\|f-f_{FLRW}\|_{\dot{C}^{1}_{1,\underline{G}_{0}}(TM_{t})}\lesssim$
$\displaystyle\,\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}\,.$

We now sketch how to extend this procedure to higher orders inductively, since
the line of argumentation is almost identical: Assume the improved bound has
been proven up to $C^{\ell-1}_{1,\underline{G}_{0}}$ for $\ell\leq 11$.
Commuting (2.21a) with $\mathbf{B}_{i_{1}}\dots\mathbf{B}_{i_{\ell}}$ implies
the following transport system:

$(\partial_{t}-\bm{X})\mathbf{B}_{i_{1}}\dots\mathbf{B}_{i_{\ell}}(f-f_{FLRW})=(\mathcal{K}_{\ell,0})_{i_{1}\dots
i_{\ell}}+\mathbf{B}_{i_{1}}\dots\mathbf{B}_{i_{\ell}}\bm{X}f_{FLRW}$

with

$\displaystyle(\mathcal{K}_{\ell,0})_{i_{1}\dots i_{\ell}}=$
$\displaystyle\,[\mathbf{B}_{i_{1}}\dots\mathbf{B}_{i_{\ell}},\bm{X}](f-f_{FLRW})$
$\displaystyle=$
$\displaystyle\,\mathbf{B}_{i_{1}}\dots\mathbf{B}_{i_{\ell-1}}[\mathbf{B}_{i_{\ell}},\bm{X}](f-f_{FLRW})+\dots+[\mathbf{B}_{i_{1}},\bm{X}]\mathbf{B}_{i_{2}}\dots\mathbf{B}_{i_{\ell}}(f-f_{FLRW})$

Once, again the characteristics $(X,V)$ are as in (4.5), while one has

$\frac{dZ_{i_{1}\dots i_{\ell}}}{dt}=(\mathcal{K}_{\ell,0})_{i_{1}\dots
i_{\ell}}+\mathbf{B}_{i_{1}}\dots\mathbf{B}_{i_{\ell}}\bm{X}f_{FLRW}$

with

$(V^{0}_{\gamma})^{\ell}\lvert\mathcal{K}_{\ell,0}\rvert_{\gamma}\lesssim\varepsilon
a^{-3}(V^{0}_{\gamma})^{\ell}\lvert
Z\rvert_{\gamma}+\varepsilon^{\frac{1}{4}}a^{-1-c\sigma}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\|f-f_{FLRW}\|_{C_{1,\underline{G}_{0}}^{\ell-1}}$

and

$(V^{0}_{\gamma})^{\ell}\left\lvert\mathbf{B}^{\ell}\bm{X}f_{FLRW}\right\rvert_{\gamma}\lesssim\varepsilon
a^{-3-c\sqrt{\varepsilon}}$

This yields

$\left(V^{0}_{\gamma}(t)\right)^{\ell}\left\lvert\mathbf{B}^{\ell}(f-f_{FLRW})(X(t),V(t))\right\rvert_{{\gamma}}\lesssim\varepsilon^{\frac{1}{4}}a(t)^{-c\sqrt{\varepsilon}}$

as with $\ell=0$.
From here, we can repeat this procedure for
$\mathbf{B}^{\ell-r}\mathbf{A}^{r}$ by iterating over increasing $r$ for
$r=1,\dots,\ell$, where we will have proven

$\left(V^{0}_{\gamma}(t)\right)^{\ell-\tilde{r}}\left\lvert\mathbf{B}^{\ell-\tilde{r}}\mathbf{A}^{\tilde{r}}(f-f_{FLRW})(t,X(t),V(t))\right\rvert_{{\gamma}}\lesssim{\varepsilon}^{\frac{1}{4}}a(t)^{-c\sqrt{\varepsilon}}$

for $0\leq\tilde{r}<r$ in the previous steps. Additionally, the induction
hypothesis allows us to use (4.15) up to order $\ell-1$. We use all of these
bounds to control Vlasov terms of lower order and of order $\ell$ with less
horizontal derivaitves in $\mathcal{K}_{\ell,r}$, along with(4.1b),(4.3b) and
(3.12a) to control shear, metric and lapse as well as (4.6) and (4.3c) to
control superfluous momentum terms. Additionally, for $r<\ell$, we use the
bootstrap assumption (3.12b) to bound the term

(4.18)
$-a^{-1}\left(\frac{1}{v^{0}}\mathbb{I}^{j}_{i_{K}}-\frac{v^{\flat}_{i_{K}}v^{j}}{(v^{0})^{3}}\right)\mathbf{B}_{i_{1}}\dots\mathbf{B}_{i_{K-1}}\mathbf{B}_{i_{K+1}}\dots\mathbf{B}_{i_{\ell-r}}\mathbf{A}_{i_{\ell-r+1}}\dots\mathbf{A}_{i_{\ell}}\mathbf{A}_{j}(f-f_{FLRW})\,,$

which occurs when applying the product rule to the horizontal derivative term
in $\bm{X}$ , by ${\varepsilon}^{\frac{1}{4}}a^{-1-c\sigma}$. In total, this
then implies

(4.19)
$\lvert\mathcal{K}_{\ell,r}\rvert_{\gamma}\lesssim\begin{cases}(\varepsilon
a^{-3}+a^{-1-c\sigma})\lvert
Z\rvert_{\gamma}+\varepsilon^{\frac{1}{4}}a^{-1-c\sigma}+\varepsilon^{\frac{5}{4}}a^{-3-c\sqrt{\varepsilon}}&r<\ell\\\
(\varepsilon a^{-3}+a^{-1-c\sigma})\lvert
Z\rvert_{\gamma}+\varepsilon^{\frac{5}{4}}a^{-3-c\sqrt{\varepsilon}}&r=\ell\end{cases}\,.$

We again use Lemma 4.7 to control the reference terms. In total, for $m=\ell$,
we use strong low order bounds on

$\|\text{{Ric}}[G]-2\kappa G\|_{C_{G}^{\ell-1}(M_{t})},\
\|\Gamma-\hat{\Gamma}\|_{C^{\ell-1}_{G}(M_{t})}\text{ and
}\|\Sigma\|_{C^{\ell}_{G}(M_{t})}\,$

and thus this procedure can only be performed until $\ell=11$. Altogether, one
obtains

$\displaystyle\sum_{r=0}^{\ell}(V^{0}_{\gamma}(t))^{\ell-r}\left\lvert\mathbf{B}^{\ell-r}\mathbf{A}^{r}(f-f_{FLRW})(t,X(t),V(t))\right\rvert_{\gamma}\lesssim$
$\displaystyle\,\varepsilon^{\frac{1}{4}}a(t)^{-c\sqrt{\varepsilon}}\,,$

and, since (4.18) does not occur for $r=l$, also

$\left\lvert\mathbf{A}^{\ell}(f-f_{FLRW})(t,X(t),V(t))\right\rvert_{\gamma}\lesssim\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}$

just like for $\ell=r=1$. Now, we iterate over decreasing $r$: Assume that one
has proven, for any $\tilde{r}>r\geq 0$,

$\left(V^{0}_{\gamma}(t)\right)^{\ell-\tilde{r}}\left\lvert\mathbf{B}^{\ell-\tilde{r}}\mathbf{A}^{\tilde{r}}(f-f_{FLRW})(X(t),V(t))\right\rvert_{{\gamma}}\lesssim\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}\,.$

Then, one can bound (4.18) by $\sqrt{\varepsilon}a^{-1-c\sqrt{\varepsilon}}$
instead of applying (3.12b), improving the error term bound (4.19) as before
and thus yielding the same bound as above for $\tilde{r}=r$. In total, we
obtain

$\sum_{r=0}^{\ell}(V^{0}_{\gamma}(t))^{\ell-r}\lvert\mathbf{B}^{\ell-r}\mathbf{A}^{r}(f-f_{FLRW})(X(t),V(t))\rvert_{\gamma}\lesssim\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}\,.$

Finally, note that (4.3e) and (4.3c) then imply

$\lvert\underline{\nabla}_{vert}^{\ell-r}\underline{\nabla}_{hor}^{r}\zeta\rvert_{\underline{G}_{0}}\lesssim\lvert\mathbf{B}^{\ell-r}\mathbf{A}^{r}\zeta\rvert_{\underline{G}_{0}}+a^{-c\sqrt{\varepsilon}}\lvert\underline{\nabla}^{\ell-1}\zeta\rvert_{\underline{G}_{0}}\lesssim\sqrt{\varepsilon}a^{-c\sqrt{\varepsilon}},$

and subsequently (4.15) at order $\ell$. This closes the induction over
$\ell\leq 11$, proving (4.15) ∎

Since the reference distribution function does not occur in Vlasov energies
that contain horizontal derivatives (see (3.3g)), we will usually use the
following low order bound:

######  Corollary 4.10 (Strong $C^{l}_{1,\underline{G}_{0}}$-norm estimates
for certain derivatives of the Vlasov distribution function).

Let $\mathcal{D}_{L,M}$ be a combination of $L-M$ vertical and $M$ horizontal
derivatives, and let $0<M\leq L\leq 11$. Then, one has

(4.20)
$\|\mathcal{D}_{L,M}f\|_{C^{0}_{1,\underline{G}_{0}}(M_{t})}\lesssim\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}$

###### Proof.

Let $\hat{\mathcal{D}}_{L,M}$ denote the analogue to $\mathcal{D}_{L,M}$ with
respect to the connection $\hat{\underline{\nabla}}$ instead of
$\underline{\nabla}$. Since one also has $\hat{\mathcal{D}}_{L,M}f_{FLRW}=0$,
we then write

$\mathcal{D}_{L,M}f=\mathcal{D}_{L,M}(f-f_{FLRW})+(\mathcal{D}_{L,M}-\hat{\mathcal{D}}_{L,M})f_{FLRW}\,.$

The first term can be bounded using (4.15), while the second term, using
(4.6), is bounded by

$a^{-c\sqrt{\varepsilon}}\left[\|\Gamma-\hat{\Gamma}\|_{C^{L-1}_{G}}+\|\text{{Ric}}[G]-\text{{Ric}}[\gamma]\|_{C^{L-2}_{G}}\right]\lesssim
a^{-c\sqrt{\varepsilon}}\|G-\gamma\|_{C^{L}_{G}}\,.$

Thus, the statement follows with (4.3c) and (4.3e). ∎

Additionally, the bounds on the Vlasov distribution function admit bounds on
its energy-momentum tensor:

######  Corollary 4.11 (Strong low order bounds on Vlasov matter quantities).

The following bounds hold:

(4.21)
$\|\rho^{Vl}-\rho^{Vl}_{FLRW}\|_{C^{11}_{G}(M_{t})}+\|\mathfrak{p}^{Vl}-\mathfrak{p}^{Vl}_{FLRW}\|_{C^{11}_{G}(M_{t})}+\|\jmath\|_{C^{11}_{G}(M_{t})}+\|S^{Vl,\parallel}\|_{C^{11}_{G}(M_{t})}\lesssim\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}$

###### Proof.

Starting with $\rho^{Vl}-\rho^{Vl}_{FLRW}$, we compute the following:

$\displaystyle\left(\rho^{Vl}-{\rho}_{FLRW}^{Vl}\right)(t,x)=$
$\displaystyle\,\int_{\mathbb{R}^{3}}\left[f(t,x,v)-\frac{v_{\gamma}^{0}}{v^{0}}\frac{\mu_{\gamma}}{\mu_{G}}f_{FLRW}(t,x,v)\right]v^{0}\mu_{G}dv$

Note that one has

$\left\lvert
v^{0}-v^{0}_{\gamma}\right\rvert=\frac{(v^{0})^{2}-(v^{0}_{\gamma})^{2}}{v^{0}+v^{0}_{\gamma}}\leq\frac{\left\lvert(G-\gamma)_{ij}v^{i}v^{j}\right\rvert}{v^{0}+v^{0}_{\gamma}}\lesssim\lvert
G-\gamma\rvert_{G}\cdot v^{0}.$

Furthermore, by expanding the determinant and using (4.4) and
$\lvert\mu_{\gamma}\rvert\lesssim 1$, one obtains with (4.3c) that

$\lvert\mu_{G}-\mu_{\gamma}\rvert\lesssim\lvert\det G-\det\gamma\rvert\lesssim
a^{-c\sqrt{\varepsilon}}\lvert G-\gamma\rvert_{G}\,.$

Since $f_{FLRW}$ is also uniformly bounded, one now obtains:

(4.22)
$\displaystyle\left\lvert\rho^{Vl}-{\rho}_{FLRW}^{Vl}\right\rvert\lesssim$
$\displaystyle\,\int_{\mathrm{supp}(f-f_{FLRW})}\left[\lvert
f-f_{FLRW}(\cdot,\cdot,v)\rvert+a^{-c\sqrt{\varepsilon}}\lvert
G-\gamma\rvert_{G}\right]v^{0}\,\mu_{G}dv$

By (4.6), the integral over any polynomial of $v^{0}$ is bounded by
$a^{-c\sqrt{\varepsilon}}$, and thus the bound follows at order $0$ by
applying the uniform estimates (4.13) and (4.3c). For higher orders, note that
$\nabla{\rho}_{FLRW}^{Vl}=0$. Thus, for any of the matter quantities, we pull
the covariant derivatives $\nabla^{L}$ past the integral according to (2.25)
and can directly bound the integrand uniformly using (4.20), proving (4.21)
for the energy density. The bounds for the remaining quantities are proven
identically. ∎

Finally, we collect the following lemma that will be needed to bound the
evolution of weights in our energy approach:

######  Lemma 4.12 (Evolution bound on momentum variables).

(4.23) $-\frac{\partial_{t}v^{0}}{v^{0}}\lesssim\varepsilon
a^{-3}\,,\quad-\frac{\partial_{t}\langle v\rangle_{G}}{\langle
v\rangle_{G}}\lesssim\varepsilon a^{-3}$

###### Proof.

Recalling $v^{i}=a^{2}p^{i}$, one computes the following:

$\partial_{t}v^{0}=\frac{1}{2v^{0}}\left[2m^{2}a\dot{a}+(\partial_{t}G)_{ij}v^{i}v^{j}+4\frac{\dot{a}}{a}\lvert
v\rvert_{G}^{2}\right]=-(N+1)a^{-3}\Sigma_{ij}\frac{v^{i}v^{j}}{v^{0}}+N\frac{\dot{a}}{a}\frac{\lvert
v\rvert_{G}^{2}}{v^{0}}+\frac{\dot{a}}{a}\cdot\frac{m^{2}a^{2}+2\lvert
v\rvert_{G}^{2}}{v^{0}}$

The statement then follows from (4.1b) and (3.12a), where the final summand
has favourable sign and is thus dropped. The estimate for $\langle
v\rangle_{G}$ is obtained identically. ∎

###  4.3. Semi-norm equivalences and commutators for the Vlasov distribution

In this section, we collect estimates on covariant commutator errors for
functions on the mass shell, that in particular will imply that the Vlasov
energies (3.3f)-(3.3g) are coercive up to error terms dependent on lower order
Vlasov energies as well as metric norms and energies. This will be sufficient
to deduce control of the Vlasov matter quantities in $\mathcal{H}$ and
$\mathcal{C}$ from improved energy bounds in Section 7.

######  Lemma 4.13 (Commutator formulas for the Sasaki metric).

Let $\xi$ be a sufficiently regular function on
$\displaystyle\cup_{t\in(t_{Boot},t_{0}],x\in M}P_{(t,x)}$ and let
$M_{1},M_{2}\in\mathbb{N}$. Then, understanding that terms do not occur if
they would involve negative orders of derivatives, the following schematic
expressions hold:

(4.24a)
$\displaystyle[\underline{\nabla}_{vert}^{M_{1}}\underline{\nabla}_{hor}^{M_{2}},\underline{\nabla}_{hor}]\xi=$
$\displaystyle\,v\ast\text{{Ric}}[G]\ast\underline{\nabla}_{vert}^{M_{1}-1}\underline{\nabla}_{hor}^{M_{2}}\xi+v\ast\text{{Ric}}[G]\ast\underline{\nabla}_{vert}^{M_{1}+1}\underline{\nabla}_{hor}^{M_{2}-1}\xi$
$\displaystyle\,+\underbrace{v\ast\nabla^{M_{2}}(\text{{Ric}}[G]-2\kappa
G)\ast\underline{\nabla}_{vert}^{M_{1}}\xi}_{\text{if
}M_{2}>0}+\Xi_{M_{1},M_{2},hor}\xi$ (4.24b)
$\displaystyle[\partial_{t},\underline{\nabla}_{vert}^{M_{1}}\underline{\nabla}_{hor}^{M_{2}}]\xi=$
$\displaystyle\,v\ast\nabla^{M_{2}-2}\partial_{t}\text{{Ric}}[G]\ast\underline{\nabla}_{vert}^{M_{1}+1}\xi+v\ast\partial_{t}\Gamma[G]\ast\underline{\nabla}_{vert}^{M_{1}+1}\underline{\nabla}_{hor}^{M_{2}-1}\xi$
$\displaystyle\,+\Xi_{M_{1},M_{2},t}\xi$

If, on the momentum support of $\xi(t,\cdot,\cdot)$, $v^{0}$ is uniformly
bounded by $a(t)^{-c\sqrt{\varepsilon}}$ up to constant, the lower order error
terms can be bounded as follows:

(4.25a) $\displaystyle\|\langle v\rangle_{G}$
$\displaystyle(v^{0})^{M_{1}}\Xi_{M_{1},M_{2},hor}\xi\|_{L^{2}_{1,\underline{G}}(TM_{t})}\lesssim
a(t)^{-c\sqrt{\varepsilon}}\|\xi\|_{H_{1,\underline{G}_{0}}^{M_{1}+M_{2}-1}(TM_{t})}$
$\displaystyle\,+a(t)^{-c\sqrt{\varepsilon}}\|\text{{Ric}}[G]-2\kappa
G\|_{H^{M_{2}-1}_{G}(M_{t})}\|\xi\|_{C^{M_{1}+1}_{1,\underline{G}_{0}}(TM_{t})}$
(4.25b) $\displaystyle\|\langle v\rangle_{G}$
$\displaystyle(v^{0})^{M_{1}}\Xi_{M_{1},M_{2},t}\xi\|_{L^{2}_{1,\underline{G}}(TM_{t})}\lesssim\varepsilon
a(t)^{-3-c\sqrt{\varepsilon}}\|\xi\|_{H^{M_{1}+M_{2}-1}_{1,\underline{G}_{0}}(TM_{t})}$
$\displaystyle\,+a(t)^{-c\sqrt{\varepsilon}}\left(\|\partial_{t}\text{{Ric}}[G]\|_{H^{M_{2}-2}_{G}(M_{t})}+\|\partial_{t}\Gamma[G]\|_{H^{M_{2}-2}_{G}(M_{t})}\right)\|\xi\|_{C^{M_{1}+1}_{1,\underline{G}_{0}}(TM_{t})}$

By Lemma 4.6, (4.25) applies for $\xi=f$.

###### Proof.

These formulas follow from straightforward computations using the commutator
formulas in Lemma 2.8 and the connection coefficient expressions from (2.19).
For the error bounds, we recall the bounds (4.6) on the momentum support, and
estimate lower order curvature and metric errors with (4.3e) and (4.3c) to
obtain (4.25a). For (4.24b), recall (2.22e) and (2.22d) and use (4.3b) as well
as the bootstrap assumption to bound lower order terms in the rescaled shear
and lapse respectively. ∎

######  Lemma 4.14 (Rearrangement of derivatives in intermediate Vlasov
energies).

Let $0<K<L$ and let $\mathcal{D}_{L,K}$ denote a combination of $L-K$ vertical
and $K$ horizontal covariant derivatives with respect to the Sasaki metric
$\underline{G}$. Then, the following holds: (4.26a)
$\displaystyle\left\lvert\|\mathcal{D}_{L,K}(f-f_{FLRW})\|_{L^{2}_{1,\underline{G}_{0}}(TM_{t})}^{2}-\|\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}(f-f_{FLRW})\|_{L^{2}_{1,\underline{G}_{0}}(TM_{t})}^{2}\right\rvert$
$\displaystyle\lesssim$
$\displaystyle\,a(t)^{-c\sqrt{\varepsilon}}\left(\mathfrak{E}^{(\leq
L-1)}_{1,\leq K}(f,t)+\varepsilon{\mathfrak{E}^{(\leq
K-1)}(\text{{Ric}},t)}+\|\Gamma-\hat{\Gamma}\|_{H^{K-1}_{G}(M_{t})}^{2}+\|G-\gamma\|_{H^{K-1}_{G}(M_{t})}^{2}\right)$
Similarly, one has for $1\leq K<L$, (4.26b)
$\displaystyle\left\lvert\|\mathcal{D}_{L,K}f\|_{L^{2}_{1,\underline{G}_{0}}(TM_{t})}^{2}-\mathfrak{E}^{(L)}_{L,K}(f,t)\right\rvert\lesssim$
$\displaystyle\,a(t)^{-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq L-1)}_{1,\leq
K}(f,t)+\varepsilon a(t)^{-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
K-1)}(\text{{Ric}},t)$
$\displaystyle\,+a(t)^{-c\sqrt{\varepsilon}}\left(\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}(M_{t})}^{2}+\|G-\gamma\|_{H^{1}_{G}(M_{t})}^{2}\right)\,.$
If one additionally $K\leq 10$ in any of the above bounds, all terms but the
Vlasov energy quantities can be dropped.

###### Proof.

Reviewing the commutator formulas from Lemma 2.8, one schematically has for
$K>0$:

$\displaystyle\mathcal{D}_{L,K}\xi-\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}\xi=$
$\displaystyle\,\sum_{I=0}^{K-1}\left[v\ast\nabla^{I}\text{{Riem}}[G]\ast\underline{\nabla}^{L-I-1}\xi+\nabla^{I}\text{{Riem}}[G]\ast\underline{\nabla}^{L-I}\xi\right]$
$\displaystyle\,+\langle\text{lower order nonlinear terms containing
}v,\text{{Riem}}[G],\xi\rangle\,$

where we recall $\text{{Riem}}[G]=\text{{Ric}}[G]\ast G$. Thus, the product
rule and the a priori estimates (4.3e), (4.6) and (4.15) yield the following:

$\displaystyle\,\left\lvert\|\mathcal{D}_{L,K}(f-f_{FLRW})\|_{L^{2}_{1,\underline{G}_{0}}(TM)}^{2}-\|\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}(f-f_{FLRW})\|_{L^{2}_{1,\underline{G}_{0}}(TM)}^{2}\right\rvert$
$\displaystyle\lesssim$
$\displaystyle\,a^{-c\sqrt{\varepsilon}}\|f-f_{FLRW}\|_{H^{L-1}_{1,\underline{G}_{0}}(TM)}^{2}+\varepsilon
a^{-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq K-1)}(\text{{Ric}},\cdot)$

Furthermore, after applying the triangle inequality to
$\|\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f-f_{FLRW}\|_{L^{2}_{1,\underline{G}_{0}}(TM)}$,
we can estimate the reference terms with (4.10a). (4.26a) then follows by
iterating over iteratively replacing mixed terms on the right hand side until
only Vlasov energy terms remain.
For (4.26b), we largely proceed as above replacing $f-f_{FLRW}$ with $f$.
However, whenever we encounter a term with only one horizontal derivative by
commuting, we replace $f$ with $(f-f_{FLRW})+f_{FLRW}$. For the former, we
simply use (4.26a). Regarding the latter, the analogous bounds hold as for
$\mathcal{D}_{L,1}f_{FLRW}$ as for
$\underline{\nabla}_{vert}^{L-1}\underline{\nabla}_{hor}f_{FLRW}$ in (4.10a),
except for incurring metric terms up to second order since taking vertical
derivatives causes metric terms to occur at order zero that need to be
differentiated: For example,

$\underline{\nabla}_{hor}\underline{\nabla}_{vert}f_{FLRW}=\underline{\nabla}_{hor}(\mathcal{F}^{\prime}(\lvert
v\rvert_{\gamma}^{2})\ast
v\ast\gamma)=v\ast\left(\underline{\nabla}_{hor}\mathcal{F}^{\prime}(\lvert
v\rvert_{\gamma}^{2})\ast\gamma+\mathcal{F}^{\prime}(\lvert
v\rvert_{\gamma}^{2})\ast\nabla(G-\gamma)\right)\,.$

Note that, since we only need to apply (4.10a) when only one horizontal
derivative occurs, we do not incur higher order metric errors in (4.26b). ∎

Since Lemma 4.14 allows us to exchange arrange covariant derivatives in
arbitrary order up to curvature errors, this will imply that Vlasov energies
are coercive up to lower order metric errors, along with bounds on the
reference distribution function to control the purely vertical case. We will
use this in Corollary 7.9 to improve bounds on Sobolev norms containing
$f-f_{FLRW}$. To then improve the bootstrap assumptions, we need to show that
these sufficiently control the energy momentum tensor, which is ensured by the
following lemma:

######  Lemma 4.15 (Energy bounds for Vlasov energy-momentum tensor
components).

For $\,L\in\mathbb{Z}_{+}$ and $t\in(t_{Boot},t_{0}]$, the following bounds
hold:

(4.27a)
$\displaystyle\|\rho^{Vl}-{\rho}_{FLRW}^{Vl}\|_{L^{2}_{G}(M_{t})}+\|\mathfrak{p}^{Vl}-{\mathfrak{p}}_{FLRW}^{Vl}\|_{L^{2}_{G}(M_{t})}\lesssim$
$\displaystyle\,a(t)^{-c\sqrt{\varepsilon}}\left(\sqrt{\mathfrak{E}^{(0)}_{1,0}(f,t)}+\|G-\gamma\|_{L^{2}_{G}(M_{t})}\right)$
(4.27b)
$\displaystyle\|\rho^{Vl}-{\rho}_{FLRW}^{Vl}\|_{\dot{H}^{L}_{G}(M_{t})}+\|\mathfrak{p}^{Vl}-{\mathfrak{p}}_{FLRW}^{Vl}\|_{\dot{H}^{L}_{G}(M_{t})}\lesssim$
$\displaystyle\,a(t)^{-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(L)}_{1,L}(f,t)}$
(4.27c) $\displaystyle\|\jmath^{Vl}\|_{\dot{H}^{L}_{G}(M_{t})}\lesssim$
$\displaystyle\,a(t)^{-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(L)}_{1,L}(f,t)}$
(4.27d)
$\displaystyle\|(S^{Vl})^{\parallel}\|_{\dot{H}^{L}_{G}(M_{t})}\lesssim$
$\displaystyle\,a(t)^{-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(L)}_{1,L}(f,t)}$

###### Proof.

Note that we can apply Jensen’s inequality on continuous functions $\xi$ on
$P$ that have compact support after renormalising, i.e., for
$\lvert\mathrm{supp}\xi\rvert_{G}=\sup_{v\in\mathrm{supp}\xi(t,x,\cdot)}\lvert
v\rvert_{G}$,

$\displaystyle\int_{M}\left(\int_{\mathbb{R}^{3}}v^{0}\xi\mu_{G}dv\right)^{2}{\text{{vol}}}_{G}=$
$\displaystyle\int_{M}\mu_{G}^{2}\lvert\mathrm{supp}\xi\rvert_{G}^{2}\left(\lvert\mathrm{supp}\xi\rvert_{G}^{-1}\int_{\mathrm{supp}\xi}v^{0}\xi\,dv\right)^{2}\,{\text{{vol}}}_{G}$
$\displaystyle\,\leq\int_{TM}\mu_{G}\lvert\mathrm{supp}\xi\rvert_{G}(v^{0})^{2}\xi^{2}\,{\text{{vol}}}_{\underline{G}}\,,$

and that

$\mu_{G}\lvert\mathrm{supp}(f-f_{FLRW})\rvert_{G}\lesssim(1+\mathcal{P}+\|G-\gamma\|_{C^{0}_{G}(M)})^{3}\lesssim
a^{-c\sqrt{\varepsilon}}$

since $\mu_{G}$ is uniformly bounded due to (4.4) and using (4.6) and (4.3c).
Thus, recalling (4.22), we compute for the energy density:

$\displaystyle\|\rho^{Vl}-{\rho}_{FLRW}^{Vl}\|^{2}_{L^{2}_{G}(M_{t})}\lesssim$
$\displaystyle\,\int_{M_{t}}\left(\int_{\mathrm{supp}(f-f_{FLRW})(t,x,\cdot)}\left[\lvert
f-f_{FLRW}\rvert+\lvert
G-\gamma\rvert_{G}\right]v^{0}\,\mu_{G}dv\right)^{2}{\text{{vol}}}_{G}$
$\displaystyle\lesssim$
$\displaystyle\,a^{-c\sqrt{\varepsilon}}\int_{TM_{t}}(v^{0})^{2}\left[\lvert
f-f_{FLRW}\rvert^{2}+\lvert
G-\gamma\rvert_{G}^{2}\right]{\text{{vol}}}_{\underline{G}}$
$\displaystyle\lesssim$
$\displaystyle\,a^{-c\sqrt{\varepsilon}}\left(\mathfrak{E}^{(0)}_{1,0}(f,\cdot)+a^{-c\sqrt{\varepsilon}}\|G-\gamma\|_{L^{2}_{G}(M)}^{2}\right)$

The computations for the pressure are fully analogous, proving (4.27a) after
updating constants. By (2.15) and (2.16), we can express $\jmath^{Vl}$ and
$(S^{Vl,\parallel})^{i}_{j}$ as weighted integrals over $f-f_{FLRW}$, avoiding
metric error terms. Thus, (4.27c) and (4.27d) at $L=0$ follow directly by
applying the Jensen inequality as above. Higher orders are proven fully
analogously, observing as in the proof of Lemma 4.11 that
$\nabla\rho^{Vl}_{FLRW}=\nabla\mathfrak{p}^{Vl}_{FLRW}=0$, as well as the
Vlasov energy definition (3.3g) – this allows us directly apply the Jensen
inequality to all matter variables without having to expand the integrand. ∎

##  5\. Estimates for spacetime and scalar field variables

###  5.1. Lapse energy estimates

In this section, we collect estimates for the lapse that follow from
(2.22f)-(2.22g). In particular, in contrast to the estimates in [FU23, Section
5], we will also prove estimates for odd order lapse energies, since these
will be necessary to control high order lapse terms.

######  Definition 5.1 (Elliptic operators).

(5.1a) $\displaystyle\mathcal{L}\zeta=$
$\displaystyle\,a^{4}\Delta\zeta-h\cdot\zeta,$ $\displaystyle h=$
$\displaystyle-3\kappa a^{4}+12\pi C^{2}+12\pi
a^{2}\left({\rho}_{FLRW}^{Vl}+{\mathfrak{p}}_{FLRW}^{Vl}\right)+H$
$\displaystyle H=$
$\displaystyle\,\langle\Sigma,\Sigma\rangle_{G}+8\pi\Psi^{2}+16\pi C\Psi+4\pi
a^{2}\left(\rho^{Vl}-{\rho}_{FLRW}^{Vl}+3\mathfrak{p}^{Vl}-3{\mathfrak{p}}_{FLRW}^{Vl}\right)$
(5.1b) $\displaystyle\tilde{\mathcal{L}}\zeta=$
$\displaystyle\,a^{4}\Delta\zeta-\tilde{h}\cdot\zeta,$
$\displaystyle\quad\tilde{h}=$ $\displaystyle-3\kappa a^{4}+12\pi C^{2}+12\pi
a^{2}({\mathfrak{p}}_{FLRW}^{Vl}+{\rho}_{FLRW}^{Vl})+\tilde{H}$
$\displaystyle\tilde{H}=$
$\displaystyle\,a^{4}\left[R[G]+\frac{2}{3}-8\pi\lvert\nabla\phi\rvert^{2}_{G}\right]+12\pi
a^{2}\left[(\mathfrak{p}^{Vl}-{\mathfrak{p}}_{FLRW}^{Vl})-(\rho^{Vl}-{\rho}_{FLRW}^{Vl})\right]$

With these operators, the rescaled lapse equations (2.22f)-(2.22g) then read

(5.2a) $\displaystyle\mathcal{L}N=$ $\displaystyle\,H\text{ and }$ (5.2b)
$\displaystyle\tilde{\mathcal{L}}N=$ $\displaystyle\,\tilde{H}\,,$

and admit the following series of estimates:

######  Lemma 5.2 (Maximum principle for the lapse).

If $\varepsilon>0$ is chosen small enough, one has

(5.3)
$\|N\|_{C^{0}(M)}\lesssim\min\\{\|{H}\|_{C^{0}(M)},\|\tilde{H}\|_{C^{0}(M)}\\}\,.$

In particular, one has $N+1>0$ if $\varepsilon>0$ is suitably small.

###### Proof.

The proof is identical to that of [Spe18, Lemma 13.1], where we also observe
that, for $\varepsilon>0$ small enough, $h$ and $\tilde{h}$ are uniformly
bounded and strictly positive due to (3.12a). ∎

######  Lemma 5.3 (Elliptic estimates for $\mathcal{L}$ and
$\tilde{\mathcal{L}}$).

For scalar functions $\zeta,Z$ on $M_{t}$ with $\mathcal{L}\zeta=Z$ or
$\tilde{\mathcal{L}}\zeta=Z$, one has

(5.4)
$a^{4}\|\Delta\zeta\|_{L^{2}_{G}(M)}+a^{2}\|\nabla\zeta\|_{L^{2}_{G}(M)}+\|\zeta\|_{L^{2}_{G}(M)}\lesssim\|Z\|_{L^{2}_{G}(M)}$

###### Proof.

The only difference in the proof in the pure scalar-field case is that Vlasov
terms occur in $h$ and $\tilde{h}$. However, we observe that
${\rho}_{FLRW}^{Vl}+{\mathfrak{p}}_{FLRW}^{Vl}$ is nonnegative and only
depends on $t$, while one has $\lvert H\rvert\lesssim\varepsilon$ by the
(4.1b), (4.1a) and (3.12a), as well as
$\lvert\tilde{H}\rvert\lesssim\varepsilon a^{2-c\sigma}$ by (3.12a). Hence,
one has

(5.5) $h,\tilde{h}\geq 6\pi C^{2}\text{ and }\lvert\nabla
h\rvert_{G},\lvert\nabla\tilde{h}\rvert_{G}\lesssim\varepsilon$

for small enough $\varepsilon>0$, which were the only properties needed to
prove the above lemma in the scalar field case. ∎

######  Lemma 5.4 (Odd order elliptic estimates for $\mathcal{L}$ and
$\tilde{\mathcal{L}}$).

Consider scalar functions $\zeta,Z$ with $\mathcal{L}\zeta=Z$ or
$\tilde{\mathcal{L}}\zeta=Z$. Then, one has

(5.6)
$a^{4}\|\nabla\Delta\zeta\|_{L^{2}_{G}(M)}+a^{2}\|\Delta\zeta\|_{L^{2}_{G}(M)}+\|\nabla\zeta\|_{L^{2}_{G}(M)}^{2}\lesssim\|Z\|_{\dot{H}^{1}_{G}(M)}+\sqrt{\varepsilon}a^{-c\sqrt{\varepsilon}}\|Z\|_{L^{2}_{G}(M)}$

###### Proof.

We only prove the statement for $\mathcal{L}$, the proof is identical for
$\tilde{\mathcal{L}}$. First, note that

$\nabla
Z=\nabla(\mathcal{L}\zeta)=a^{4}\nabla\Delta\zeta-h\nabla\zeta+\zeta\nabla h.$

Contracting $\nabla_{i}\mathcal{L}\zeta$ with $-\nabla^{\sharp i}\zeta$ and
integrating, we thus obtain

$\displaystyle\int_{M}a^{4}\lvert\Delta\zeta\rvert_{G}^{2}+h\lvert\nabla\zeta\rvert^{2}-\zeta\langle\nabla
h,\nabla\zeta\rangle_{G}\,{\text{{vol}}}_{G}=-\int_{M}\langle\nabla
Z,\nabla\zeta\rangle_{G}\,{\text{{vol}}}_{G}\,.$

Using integration by parts to rewrite the third term on the left, applying
(5.5) and rearranging then yields

$\displaystyle\int_{M}a^{4}\lvert\nabla^{2}\zeta\rvert_{G}^{2}+\lvert\nabla\zeta\rvert_{G}^{2}\,{\text{{vol}}}_{G}\lesssim$
$\displaystyle\,\int_{M}\lvert\nabla Z\rvert_{G}^{2}+\|\Delta
h\|_{L^{\infty}_{G}}\lvert\zeta\rvert_{G}^{2}\,{\text{{vol}}}_{G}\,.$

Using the a priori estimates (4.3b) and (4.3a) as well as (3.12a) for the
Vlasov terms, along with Lemma 5.3, this then becomes

$\displaystyle\int_{M}a^{4}\lvert\nabla^{2}\zeta\rvert_{G}+\lvert\nabla\zeta\rvert_{G}^{2}\,{\text{{vol}}}_{G}\lesssim$
$\displaystyle\,\|Z\|_{\dot{H}^{1}}^{2}+\sqrt{\varepsilon}a^{-c\sqrt{\varepsilon}}\|Z\|_{L^{2}_{G}}^{2}\,.$

Now contracting $\nabla_{i}\mathcal{L}\zeta$ with $a^{4}\nabla^{\sharp
i}\Delta\zeta$ and integrating and using integration by parts, one has

$\displaystyle\,\int_{M}a^{8}\lvert\nabla\Delta\zeta\rvert_{G}^{2}+ha^{4}\lvert\Delta\zeta\rvert^{2}+a^{4}\langle\nabla
h,\nabla\zeta\rangle\Delta\zeta+a^{4}\zeta\langle\nabla
h,\nabla\Delta\zeta\rangle_{G}\,{\text{{vol}}}_{G}$ $\displaystyle=$
$\displaystyle\,\int_{M}a^{8}\lvert\nabla\Delta\zeta\rvert_{G}^{2}+ha^{4}\lvert\Delta\zeta\rvert^{2}-a^{4}\Delta
h\cdot\zeta\Delta\zeta\,{\text{{vol}}}_{G}$ $\displaystyle=$
$\displaystyle\,\int_{M}a^{4}\langle\nabla
Z,\nabla\Delta\zeta\rangle_{G}\,{\text{{vol}}}_{G}\,.$

Thus, by (5.5), the following holds for some $K>0$:

$\displaystyle\,\int_{M}a^{8}\lvert\nabla\Delta\zeta\rvert_{G}^{2}+6\pi
C^{2}a^{4}\lvert\Delta\zeta\rvert^{2}\,{\text{{vol}}}_{G}$ $\displaystyle\leq$
$\displaystyle\,\frac{1}{2}\|Z\|_{\dot{H}^{1}_{G}(M)}^{2}+\int_{M}\frac{1}{2}a^{8}\lvert\nabla\Delta\zeta\rvert_{G}^{2}+K\varepsilon\left(\lvert\zeta\rvert^{2}+a^{4}\lvert\Delta\zeta\rvert^{2}\right)\,{\text{{vol}}}_{G}$

The statement then follows for small enough $\varepsilon$ by rearranging and
combining both bounds. ∎

######  Lemma 5.5 (Lapse energy estimate).

The following estimate holds:

(5.7a)
$\displaystyle\,a^{8}\mathfrak{E}^{(2)}(N,\cdot)+a^{4}\mathfrak{E}^{(1)}(N,\cdot)+\mathfrak{E}^{(0)}(N,\cdot)$
$\displaystyle\lesssim$
$\displaystyle\,\varepsilon^{2}\mathfrak{E}^{(0)}(\Sigma,\cdot)+\mathfrak{E}^{(0)}(\phi,\cdot)+a^{4}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)+a^{4-c\sqrt{\varepsilon}}\|G-\gamma\|_{L^{2}_{G}(M)}^{2}$
and (5.7b)
$\displaystyle\,a^{8}\mathfrak{E}^{(2)}(N,\cdot)+a^{4}\mathfrak{E}^{(1)}(N,\cdot)+\mathfrak{E}^{(0)}(N,\cdot)$
$\displaystyle\lesssim$
$\displaystyle\,a^{8}\mathfrak{E}^{(0)}(\text{{Ric}},\cdot)+{\varepsilon}a^{8-c\sqrt{\varepsilon}}\|\nabla\phi\|_{L^{2}_{G}(M)}^{2}+a^{4-c\sqrt{\varepsilon}}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)+a^{4-c\sigma}\|G-\gamma\|_{L^{2}_{G}(M)}^{2}$

For any $L\in\\{1,\dots,18\\}$, one additionally has

(5.7c)
$\displaystyle\,a^{8}\mathfrak{E}^{(L+2)}(N,\cdot)+a^{4}\mathfrak{E}^{(L+1)}(N,\cdot)+\mathfrak{E}^{(L)}(N,\cdot)$
$\displaystyle\lesssim$
$\displaystyle\,\varepsilon^{2}\mathfrak{E}^{(L)}(\Sigma,\cdot)+\mathfrak{E}^{(L)}(\phi,\cdot)+\varepsilon^{2}a^{-c\sqrt{\varepsilon}}\left[\mathfrak{E}^{(\leq\max\\{0,L-2\\})}(\Sigma,\cdot)+\mathfrak{E}^{(\leq\max\\{0,L-2\\})}(\phi,\cdot)\right]$
$\displaystyle\,+\underbrace{\left(\varepsilon^{4}a^{-c\sqrt{\varepsilon}}+\varepsilon^{2}a^{2-c\sigma}\right)\mathfrak{E}^{(\leq
L-3)}(\text{{Ric}},\cdot)}_{\text{not present for }L\leq
2}+a^{4-c\sqrt{\varepsilon}}\mathfrak{E}^{(L)}_{1,L}(f,\cdot)$

and

(5.7d)
$\displaystyle\,a^{8}\mathfrak{E}^{(L+2)}(N,\cdot)+a^{4}\mathfrak{E}^{(L+1)}(N,\cdot)+\mathfrak{E}^{(L)}(N,\cdot)$
$\displaystyle\lesssim$ $\displaystyle\,a^{8}\mathfrak{E}^{(\leq
L)}(\text{{Ric}},\cdot)+{\varepsilon}a^{8-c\sqrt{\varepsilon}}\|\nabla\phi\|_{H^{L}_{G}(M)}^{2}+a^{4-c\sqrt{\varepsilon}}\mathfrak{E}^{(L)}_{1,L}(f,\cdot)$

###### Proof.

This proof now proceeds as in the pure scalar field setting (see [FU23,
Corollary 5.5, 5.8]): We commute (5.2) with $\Delta^{l}$ and
$\nabla\Delta^{l}$ for $l=\lfloor\frac{L}{2}\rfloor=0,\dots,9$ and apply the
elliptic estimates from Lemma 5.3 and 5.4 respectively. For the estimates with
respect to $\mathcal{L}$, the right hand sides at order $L$ are of the form

$\sum_{I=1}^{L-1}\nabla^{I}H\ast\nabla^{L-I}N+(N+1)\begin{cases}\Delta^{\frac{L}{2}}H&L\text{
even}\\\ \nabla\Delta^{\lfloor\frac{L}{2}\rfloor}H&L\text{ odd}\end{cases}\,.$

Again, the only changes occur when considering Vlasov terms: At high order, we
apply Lemma 4.15 to estimate the density and pressure terms by Vlasov energies
up to curvature errors. Conversely, low order Vlasov terms are bounded using
(4.21). Regarding the remaining quantities, we note that [FU23, Lemma 4.5]
fully extends to our setting, allowing us to move between Sobolev norms and
energies up to curvature errors at high orders. ∎

###  5.2. Energy and norm estimates for the scalar field and remaining
spacetime variables

In this section, we collect individual energy and Sobolev norm estimates for
the shear as well as scalar field and metric variables. Compared to the
estimates in [FU23, Section 6] and their respective proofs, there are two
differences we need to account for: Firstly, we now consider arbitrary
(constant) spatial sectional curvature instead of $\kappa=-\frac{1}{9}$, which
leads to a slightly more refined argument being necessary to prove Lemma 5.6.
Secondly, Vlasov matter now enters the respective equations as additional
linear terms. This is most relevant when applying the momentum constraint
(2.22i), where the respective matter terms exhibit the same asymptotic
behaviour as those associated with the scalar field, as well as whenever we
apply the now weaker bootstrap assumption (3.13) on the lapse. Similarly,
lapse energy estimates will need to be adapted to include the linear Vlasov
terms from Lemma 5.5. However, the proofs otherwise will follow by the same
arguments as in the pure scalar field case, so we will often only focus on how
these changes are accounted for in the respective proofs.

######  Lemma 5.6 (Energy estimates for the scalar field).

For $L\in 2\mathbb{Z}_{+},\,L\leq 18$, the following estimate holds:

(5.8a)
$\displaystyle\,\mathfrak{E}^{(L)}(\phi,t)+\int_{t}^{t_{0}}\dot{a}(s)a(s)^{3}\mathfrak{E}^{(L+1)}(N,s)+\frac{\dot{a}(s)}{a(s)}\mathfrak{E}^{(L)}(N,s)\,ds$
$\displaystyle\lesssim$
$\displaystyle\,\mathfrak{E}^{(L)}(\phi,t_{0})+\int_{t}^{t_{0}}\left({\varepsilon}a(s)^{-3}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(L)}(\phi,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\left\\{\varepsilon
a(s)^{-3}\mathfrak{E}^{(L)}(\Sigma,s)+\varepsilon^{\frac{3}{2}}a(s)^{-3}\mathfrak{E}^{(L-2)}(\text{{Ric}},s)\right.$
$\displaystyle\,\phantom{+\int_{t}^{t_{0}}}+a(s)^{-2-c\sqrt{\varepsilon}-(L+1)\omega}\cdot
a(s)^{(L+1)\omega}\mathfrak{E}^{(\leq L)}_{1,\leq
L}(f,s)+\sqrt{\varepsilon}a(s)^{-1-c\sqrt{\varepsilon}}\|G-\gamma\|_{L^{2}_{G}(M_{s})}^{2}$
$\displaystyle\,\phantom{+\int_{t}^{t_{0}}}\left.+\sqrt{\varepsilon}a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-2)}(\phi,s)+\varepsilon a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-2)}(\Sigma,s)\right\\}\,ds$
$\displaystyle\underbrace{+\int_{t}^{t_{0}}\varepsilon^{\frac{3}{2}}a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-4)}(\text{{Ric}},s)\,ds}_{\text{if }L\geq 4}$ Similarly, one has (5.8b)
$\displaystyle\,\mathfrak{E}^{(0)}(\phi,t)+\int_{t}^{t_{0}}\dot{a}(s)a(s)^{3}\mathfrak{E}^{(1)}(N,s)+\frac{\dot{a}(s)}{a(s)}\mathfrak{E}^{(0)}(N,s)\,ds$
$\displaystyle\lesssim$
$\displaystyle\,\mathfrak{E}^{(0)}(\phi,t_{0})+\int_{t}^{t_{0}}\varepsilon
a(s)^{-3}\mathfrak{E}^{(0)}(\phi,s)+\varepsilon
a^{-3}\mathfrak{E}^{(0)}(\Sigma,s)\,ds\,$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-2-c\sqrt{\varepsilon}-\omega}\left(a(s)^{\omega}\mathfrak{E}^{(0)}_{1,0}(f,s)+a(s)^{\omega}\|G-\gamma\|_{L^{2}_{G}(M_{s})}^{2}\right)\,ds$
as well as (5.8c)
$\displaystyle\,a(t)^{4}\mathfrak{E}^{(L+1)}(\phi,t)+\int_{t}^{t_{0}}\left\\{\dot{a}(s)a(s)^{7}\mathfrak{E}^{(L+2)}(N,s)+\dot{a}(s)a(s)^{3}\mathfrak{E}^{(L+1)}(N,s)\right\\}\,ds$
$\displaystyle\lesssim$
$\displaystyle\,a(t_{0})^{4}\mathfrak{E}^{(L+1)}(\phi,t_{0})+\int_{t}^{t_{0}}\left(\varepsilon
a(s)^{-3}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\cdot
a(s)^{4}\mathfrak{E}^{(L+1)}(\phi,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\left\\{\varepsilon a(s)^{-3}\cdot
a(s)^{4}\mathfrak{E}^{(L+1)}(\Sigma,s)+\left(\varepsilon
a(s)^{-3}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(L)}(\phi,s)+\varepsilon
a(s)^{-3}\mathfrak{E}^{(L)}(\Sigma,s)\right.$
$\displaystyle\,\phantom{+\int_{t}^{t_{0}}}+\varepsilon
a(s)^{-1-c\sqrt{\varepsilon}}\cdot
a(s)^{4}\mathfrak{E}^{(L-1)}(\text{{Ric}},s)+\left(\varepsilon^{3}a^{-3}+\varepsilon
a^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(L-2)}(\text{{Ric}},s)$
$\displaystyle\,\phantom{+\int_{t}^{t_{0}}}+a(s)^{-2-c\sqrt{\varepsilon}-(L+2)\omega}\cdot
a(s)^{4+(L+2)\omega}\mathfrak{E}^{(L+1)}_{1,L+1}(f,s)+a^{2-c\sqrt{\varepsilon}-(L+1)\omega}\cdot
a(s)^{(L+1)\omega}\mathfrak{E}^{(\leq L)}_{1,\leq L}(f,s)\,$
$\displaystyle\,\phantom{+\int_{t}^{t_{0}}}\left.+\sqrt{\varepsilon}a^{3-c\sqrt{\varepsilon}}\|G-\gamma\|_{L^{2}_{G}(M_{s})}^{2}\right\\}\,ds$
and (5.8d)
$\displaystyle\,a(t)^{4}\mathfrak{E}^{(1)}(\phi,t)+\int_{t}^{t_{0}}\left\\{\dot{a}(s)a(s)^{7}\mathfrak{E}^{(2)}(N,s)+\dot{a}(s)a(s)^{3}\mathfrak{E}^{(1)}(N,s)\right\\}\,ds$
$\displaystyle\lesssim$
$\displaystyle\,a(t_{0})^{4}\mathfrak{E}^{(1)}(\phi,t_{0})+\int_{t}^{t_{0}}\left(\varepsilon
a(s)^{-3}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\cdot
a(s)^{4}\mathfrak{E}^{(1)}(\phi,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\left\\{\varepsilon a(s)^{-3}\cdot
a(s)^{4}\mathfrak{E}^{(1)}(\Sigma,s)+\left(\varepsilon
a(s)^{-3}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(0)}(\phi,s)+\varepsilon
a(s)^{-3}\mathfrak{E}^{(0)}(\Sigma,s)\right.$
$\displaystyle\,\phantom{+\int_{t}^{t_{0}}}+a(s)^{-2-c\sqrt{\varepsilon}-(L+2)\omega}\cdot
a(s)^{4+2\omega}\mathfrak{E}^{(1)}_{1,1}(f,s)$
$\displaystyle\,\phantom{+\int_{t}^{t_{0}}}\left.+a(s)^{2-c\sqrt{\varepsilon}-\omega}\cdot
a(s)^{\omega}\left(\mathfrak{E}^{(0)}_{1,0}(f,s)+\|G-\gamma\|_{L^{2}_{G}(M_{s})}^{2}\right)\right\\}\,ds$

###### Proof.

We only sketch how to adapt the proof of (5.8a) from that of [FU23, Lemma 6.2]
– analogous adaptations can then be applied to the proof of [FU23, Lemma 6.4]
to obtain (5.8c), as well as the low order bounds (5.8b) and (5.8d).
After taking the time derivative, inserting (2.22p) and integrating by parts,
the following schematic equality holds:

(5.9a) $\displaystyle-\partial_{t}\mathfrak{E}^{(L)}(\phi,\cdot)\lesssim$
$\displaystyle\,\int_{M}\left\\{6C\frac{\dot{a}}{a}\Delta^{\frac{L}{2}}N\cdot\Delta^{\frac{L}{2}}\Psi+3a\langle\nabla\Delta^{\frac{L}{2}}N,\nabla\phi\rangle_{G}\cdot\Delta^{\frac{L}{2}}\Psi\right.$
(5.9b)
$\displaystyle\,\phantom{\int_{M}}\left.-2Ca\langle\nabla\Delta^{\frac{L}{2}}N,\nabla\Delta^{\frac{L}{2}}\phi\rangle_{G}\right\\}-4\dot{a}a^{3}\lvert\nabla\Delta^{\frac{L}{2}}\phi\rvert_{G}^{2}\,{\text{{vol}}}_{G}+\langle\text{error
terms}\rangle$

Regarding the first term in (5.9a), we insert the following zero using (2.22f)
and (2.3a):

$\displaystyle 0=$
$\displaystyle\,\int_{M}\text{{div}}_{G}\left(\nabla\Delta^{\frac{L}{2}}N\cdot\Delta^{\frac{L}{2}}N\right)\,{\text{{vol}}}_{G}$
$\displaystyle=$
$\displaystyle\,\int_{M}-\frac{3}{8\pi}\dot{a}a^{3}\lvert\nabla\Delta^{\frac{L}{2}}N\rvert_{G}^{2}-\frac{3}{8\pi}\left(-3\kappa\dot{a}a^{3}+12\pi
C^{2}\frac{\dot{a}}{a}+12\pi
a^{-2}\left({\rho}_{FLRW}^{Vl}-{\mathfrak{p}}_{FLRW}^{Vl}\right)\right)\lvert\Delta^{\frac{L}{2}}N\rvert^{2}$
$\displaystyle\quad-6C\frac{\dot{a}}{a}\Delta^{\frac{L}{2}}N\cdot\Delta^{\frac{L}{2}}\Psi+\frac{9}{2}\pi
a^{-2}\Delta^{\frac{L}{2}}\left[\left(\rho^{Vl}-{\rho}_{FLRW}^{Vl}\right)-\left(\mathfrak{p}^{Vl}-{\mathfrak{p}}^{Vl}_{FLRW}\right)\right]$
$\displaystyle\quad+\langle\text{borderline and junk
terms}\rangle\,{\text{{vol}}}_{G}$

Regarding the first line, the first and third term can be pulled to the left
hand side, while the second and fourth can be bounded by $\lesssim
a^{-2}\mathfrak{E}^{(L)}(N,\cdot)$. The first term in the second line
precisely cancels the first term in (5.9a), and the Vlasov term can be bounded
by

(5.10) $a^{-2-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq L)}_{1,\leq
L}(f,\cdot)\,$

by Lemma 4.15. Note that, for the same argument at $L=0$, this also incurs a
metric term.
The second term in (5.9a) and the explicit term in (5.9b) can be treated
similarly, so we focus on the latter, denoting it by $(\ast)$: Applying (2.5),
we obtain the following:

$\displaystyle\lvert(\ast)\rvert\leq$
$\displaystyle\,\int_{M}2\sqrt{\frac{3}{4\pi}}\left(\dot{a}a^{3}+\kappa
a^{3}\right)\lvert\nabla\Delta^{\frac{L}{2}}N\rvert_{G}\lvert\nabla\Delta^{\frac{L}{2}}\phi\rvert_{G}\,{\text{{vol}}}_{G}$
$\displaystyle\leq$
$\displaystyle\,\int_{M}4\dot{a}a^{3}\lvert\nabla\Delta^{\frac{L}{2}}\phi\rvert_{G}^{2}+\frac{3}{16\pi}\dot{a}a^{3}\lvert\nabla\Delta^{\frac{L}{2}}N\rvert_{G}^{2}\,{\text{{vol}}}_{G}+\lvert\kappa\rvert
a^{-1}\left(\mathfrak{E}^{(L)}(\phi,\cdot)+a^{4}\mathfrak{E}^{(L+1)}(N,\cdot)\right)$

The first term cancels with the second term in (5.9b), the second can be
pulled to the left and absorbed into already present lapse energy terms after
adapting constants.
Combining these observations and dealing with borderline and junk terms as in
the pure scalar field setting, one obtains the following differential
estimate:

$\displaystyle-\partial_{t}\mathfrak{E}^{(L)}(\phi,\cdot)+\dot{a}a^{3}\mathfrak{E}^{(L+1)}(N,\cdot)+\frac{\dot{a}}{a}\mathfrak{E}^{(L)}(N,\cdot)$
(5.11a) $\displaystyle\lesssim$ $\displaystyle\,\left(\varepsilon
a^{-3}+a^{-1-c{\sigma}}\right)\mathfrak{E}^{(L)}(\phi,\cdot)+\left(\varepsilon
a^{-3}+a^{-1-c\sqrt{\varepsilon}}\right)\left(a^{4}\mathfrak{E}^{(L+1)}(N,\cdot)+\mathfrak{E}^{(L)}(N,\cdot)\right)$
(5.11b) $\displaystyle\,+\varepsilon
a^{-3}\mathfrak{E}^{(L)}(\Sigma,\cdot)+\varepsilon^{\frac{3}{2}}a^{-3}\mathfrak{E}^{(L-2)}(\text{{Ric}},\cdot)+{\varepsilon}a^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-2)}(\phi,\cdot)$ (5.11c) $\displaystyle\,+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-2)}(\Sigma,\cdot)+\left[\varepsilon
a^{-3-c\sqrt{\varepsilon}}+\sqrt{\varepsilon}a^{-1-c\sqrt{\varepsilon}}\right]\mathfrak{E}^{(\leq
L-2)}(N,\cdot)$ (5.11d)
$\displaystyle\,+a^{-2-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq L)}_{1,\leq
L}(f,\cdot)+\underbrace{\varepsilon^{\frac{3}{2}}a^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-4)}(\text{{Ric}},\cdot)}_{\text{not present for }L=2}$

The statement then follows by applying Lemma 5.5 to the lapse energies in
(5.11a) and (5.11c) and integrating. We note that, for $L=0$, one not only
incurs metric norms by applying elliptic lapse estimates at low orders, but
also when estimating (5.10). ∎

We will need an additional norm bound to obtain optimal control for the scalar
field:

######  Lemma 5.7 (Norm bound for $\nabla\phi$).

$\|\nabla\phi\|_{H^{l}_{G}(M_{t})}\lesssim(1+\varepsilon
a^{-c\sqrt{\varepsilon}})\|\Sigma\|_{H^{l+1}_{G}(M_{t})}+\sqrt{\varepsilon}a(t)^{-c\sqrt{\varepsilon}}\|\Psi\|_{H^{l}_{G}(M_{t})}+(1+\varepsilon
a^{-c\sqrt{\varepsilon}})\|\jmath^{Vl}\|_{H^{l}_{G}(M_{t})}$

###### Proof.

The momentum constraint (2.22i) leads to the following expression after
rearranging and using (4.1a) to ensure $\Psi+C>\frac{C}{2}$:

$\displaystyle\lvert\nabla^{l}\nabla\phi\rvert_{G}=$
$\displaystyle\,\left\lvert\nabla^{l}\left(\frac{\frac{1}{8\pi}\text{{div}}_{G}\Sigma-\jmath^{Vl}}{\Psi+C}\right)\right\rvert_{G}$
$\displaystyle\lesssim$
$\displaystyle\,\lvert\nabla^{l+1}\Sigma\rvert_{G}+\lvert\nabla^{l}\jmath^{Vl}\rvert_{G}+\sum_{I_{1}+I_{2}=l-1}\left(\lvert\nabla^{I_{1}+1}\Sigma\rvert_{G}+\lvert\nabla^{I_{1}}\jmath^{Vl}\rvert_{G}\right)\lvert\nabla^{I_{2}+1}\Psi\rvert_{G}$

The statement then follows by applying the Young inequality with low order
bounds (4.3b), (4.21) and (4.1a). ∎

Before moving on the energy estimates for Bel-Robinson variables, we prepare
the necessary bounds on the Vlasov matter components that occur in the Bel-
Robinson evolution equations (2.22l) and (2.22m):

######  Lemma 5.8 (Bounds for Bel-Robinson Vlasov matter components).

Recalling (2.22n) and (2.22o), writing

$(J^{Vl})^{(\ast)}_{i0j,symm}=\frac{1}{2}\left[(J^{Vl})^{(\ast)}_{i0j}+(J^{Vl})^{(\ast)}_{j0i}\right]$

and using $J^{Vl,\parallel}_{(\cdot)0(\cdot),symm}$ to denote the same
expression as in (2.22n) where pure trace terms are dropped, one has

$\displaystyle\|a^{4}(N+1)J^{Vl,\parallel}_{(\cdot)0(\cdot),symm}\|_{L^{2}_{G}(M)}\lesssim$
$\displaystyle\,a^{-1-c\sqrt{\varepsilon}-\frac{\omega}{2}}\sqrt{a^{\omega}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)}+a^{-1-c\sqrt{\varepsilon}-\omega}\sqrt{a^{4+2\omega}\mathfrak{E}^{(1)}_{1,\leq
1}(f,\cdot)}$
$\displaystyle\,+a^{-1-c\sqrt{\varepsilon}}\sqrt{a^{4}\mathfrak{E}^{(1)}(N,\cdot)}+a^{1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(0)}(N,\cdot)}+a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(0)}(\Sigma,\cdot)}$

as well as, for $L\in 2\mathbb{Z}_{+},L\leq 18$,

$\displaystyle\,\|a^{4}\Delta^{\frac{L}{2}}\left((N+1)J^{Vl,\parallel}_{(\cdot)0(\cdot),symm}\right)\|_{L^{2}_{G}(M)}$
$\displaystyle\lesssim$
$\displaystyle\,a^{-1-c\sqrt{\varepsilon}-\frac{L+2}{2}\omega}\sqrt{a^{4+(L+2)\omega}\mathfrak{E}^{(L+1)}_{1,L+1}(f,\cdot)}+a^{-1-c\sqrt{\varepsilon}-\frac{L+1}{2}\omega}\sqrt{a^{(L+1)\omega}\mathfrak{E}^{(\leq
L)}_{1,\leq L}(f,\cdot)}$
$\displaystyle\,+a^{-1-c\sqrt{\varepsilon}}\sqrt{a^{4}\mathfrak{E}^{(L+1)}(N,\cdot)}+a^{1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(L)}(N,\cdot)}$
$\displaystyle\,+\sqrt{\varepsilon}a^{1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(\leq
L-2)}(N,\cdot)}+a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(\leq
L)}(\Sigma,\cdot)}$
$\displaystyle\,+\sqrt{\varepsilon}a^{-1-c\sigma}\sqrt{a^{4}\mathfrak{E}^{(L-1)}(\text{{Ric}},\cdot)}+\sqrt{\varepsilon}a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}$

and, for $L\in 2\mathbb{N},L\leq 18$,

$\displaystyle\|a^{4}\Delta^{\frac{L}{2}}\left((N+1)J^{Vl,\ast}_{(\cdot)0(\cdot),symm}\right)\|_{L^{2}_{G}}\lesssim$
$\displaystyle\,a^{-2-c\sqrt{\varepsilon}}\sqrt{a^{4}\mathfrak{E}^{(L+1)}_{1,L+1}(f,\cdot)}+\varepsilon
a^{-1-c\sqrt{\varepsilon}-\frac{L+1}{2}\omega}\sqrt{a^{(L+1)\omega}\mathfrak{E}^{(\leq
L)}_{1,L}(f,\cdot)}$
$\displaystyle\,+a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(\leq
L)}(\Sigma,\cdot)}+\underbrace{\varepsilon
a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}}_{\text{if }L\neq 0}$

###### Proof.

We can directly use Lemma 4.15 to control $(S^{Vl})^{\parallel}$ and
$\nabla\jmath^{Vl}$. For the remaining terms, recall that we can apply (2.25)
to pull the covariant derivative past the respective momentum integrals. By
Lemma 2.7, any of the prefactors depending only on the momentum are uniformly
bounded by $v^{0}$ up to constant and vanish when hit by a horizontal
derivative. Keeping this in mind, we apply the product rule: When estimating
high order Vlasov terms, we apply the Jensen inequality as in the proof of
Lemma 4.15 to connect the respectice quantities to our Vlasov energies, and
apply the low order bounds (4.3b) and (3.12a) for shear and lapse
respectively. On the other hand, if most derivatives hit lapse or shear terms,
we estimate the respective distribution function terms with (4.15) by
$a^{-c\sqrt{\varepsilon}}$, and simply apply (4.6) to deal with the remaining
integrals over functions of $v$. Finally, recall [FU23, Lemma 4.5] to bound
Sobolev norms in $\Sigma$ and $N$ to their respective energy terms up to
curvature errors. ∎

######  Corollary 5.9 (Energy estimates for Bel-Robinson variables).

The following holds:

(5.12)
$\displaystyle\,\mathfrak{E}^{(0)}(W,t)+\int_{t}^{t_{0}}\int_{M}\left[8\pi
C^{2}a(s)^{-3}(N+1)\langle\Sigma,\mathcal{E}\rangle_{G}+6\frac{\dot{a}(s)}{a(s)}(N+1)\lvert\mathcal{E}\rvert_{G}^{2}\right]\,{\text{{vol}}}_{G}\,ds$
$\displaystyle\lesssim$
$\displaystyle\,\mathfrak{E}^{(0)}(W,t_{0})+\int_{t}^{t_{0}}\left({\varepsilon}a(s)^{-3}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(0)}(W,s)+\varepsilon^{-\frac{1}{8}}a(s)^{-3}\cdot
a(s)^{4}\mathfrak{E}^{(1)}(\phi,s)\,ds$
$\displaystyle\,\phantom{\mathfrak{E}^{(0)}(W,t_{0})}+\int_{t}^{t_{0}}\varepsilon
a(s)^{-1-c\sqrt{\varepsilon}}\mathfrak{E}^{(0)}(\phi,s)+\varepsilon
a(s)^{-3}\mathfrak{E}^{(0)}(\Sigma,s)\,ds$
$\displaystyle\,\phantom{\mathfrak{E}^{(0)}(W,t_{0})}+\int_{t}^{t_{0}}a(s)^{-2-c\sqrt{\varepsilon}-2\omega}\cdot
a(s)^{4+2\omega}\mathfrak{E}^{(1)}_{1,\leq
1}(f,s)+a(s)^{1-c\sqrt{\varepsilon}-\omega}\cdot
a(s)^{4}\|G-\gamma\|_{L^{2}_{G}(M_{s})}^{2}\,ds$

as well as, for $L\in 2\mathbb{N},\,2\leq L\leq 18$,

(5.13)
$\displaystyle\,\mathfrak{E}^{(L)}(W,t)+\int_{t}^{t_{0}}\int_{M}\left[8\pi
C^{2}a(s)^{-3}(N+1)\langle\Delta^{\frac{L}{2}}\Sigma,\Delta^{\frac{L}{2}}\mathcal{E}\rangle_{G}+6\frac{\dot{a}(s)}{a(s)}\lvert\Delta^{\frac{L}{2}}\mathcal{E}\rvert_{G}^{2}\right]\,{\text{{vol}}}_{G}\,ds$
$\displaystyle\,\lesssim\mathfrak{E}^{(L)}(W,t_{0})+\int_{t}^{t_{0}}\left(\varepsilon^{\frac{1}{8}}a(s)^{-3}+a(s)^{-2-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(L)}(W,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\left\\{\varepsilon^{-\frac{1}{8}}a(s)^{-3}\cdot
a(s)^{4}\mathfrak{E}^{(L+1)}(\phi,s)+\left(\varepsilon^{\frac{1}{8}}a(s)^{-3}+a(s)^{-1}\right)\mathfrak{E}^{(L)}(\phi,s)\right.$
$\displaystyle\,\phantom{\int_{t}^{t_{0}}}+\left(\varepsilon
a(s)^{-3}+a^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(L)}(\Sigma,s)$
$\displaystyle\,\phantom{\int_{t}^{t_{0}}}+\varepsilon^{\frac{7}{8}}a(s)^{-3}\cdot
a(s)^{4}\mathfrak{E}^{(L-1)}(\text{{Ric}},s)+\left(\varepsilon^{\frac{31}{8}}a(s)^{-3}+\varepsilon
a^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(\leq L-2)}(\text{{Ric}},s)$
$\displaystyle\,\phantom{\int_{t}^{t_{0}}}+a(s)^{-2-c\sqrt{\varepsilon}-(L+2)\omega}\cdot
a(s)^{4+(L+2)\omega}\mathfrak{E}^{(L+1)}_{1,L+1}(f,s)+a(s)^{-1-c\sqrt{\varepsilon}-(L+1)\omega}\cdot
a(s)^{(L+1)\omega}\mathfrak{E}^{(\leq L)}_{1,\leq L}(f,s)$
$\displaystyle\,\phantom{\int_{t}^{t_{0}}}+\left(\varepsilon^{\frac{15}{8}}a(s)^{-3-c\sqrt{\varepsilon}}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(\leq
L-2)}(\phi,s)$ $\displaystyle\,\left.\phantom{\int_{t}^{t_{0}}}+\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}}\left(\mathfrak{E}^{(\leq
L-2)}(\Sigma,s)+\mathfrak{E}^{(\leq L-2)}(W,s)\right)+\underbrace{\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-4)}(\text{{Ric}},s)\,ds}_{\text{not present for }L=2}\right\\}\,ds\,.$

###### Proof.

The proof proceeds exactly as in the pure scalar field case (see [FU23, Lemma
6.6]), where we use Lemma 5.8 to bound the Vlasov matter terms, and apply
Lemma 5.5 to bound lapse energies. ∎

######  Lemma 5.10 (Integral energy estimates for the second fundamental
form).

One has:

(5.14)
$\displaystyle\,\mathfrak{E}^{(0)}(\Sigma,t)+2\int_{t}^{t_{0}}\int_{M}\left[a(s)^{-3}(N+1)\langle\mathcal{E},\Sigma\rangle_{G}+\frac{\dot{a}(s)}{a(s)}(N+1)\lvert\Delta^{\frac{L}{2}}\Sigma\rvert_{G}^{2}\right]\,{\text{{vol}}}_{G}\,ds$
$\displaystyle\lesssim$
$\displaystyle\,\mathfrak{E}^{(0)}(\Sigma,t_{0})+\int_{t}^{t_{0}}\left(\varepsilon^{\frac{1}{8}}a(s)^{-3}+a^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(0)}(\Sigma,s)\,ds+\int_{t}^{t_{0}}\varepsilon^{-\frac{1}{8}}a(s)^{-3}\mathfrak{E}^{(0)}(\phi,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-1-c\sqrt{\varepsilon}-\omega}\left(a(s)^{\omega}\mathfrak{E}^{(0)}_{1,0}(f,s)+\|G-\gamma\|_{L^{2}_{G}(M_{s})}^{2}\right)\,ds$

For $L\in 2\mathbb{Z}_{+},L\leq 18$, the following holds:

(5.15)
$\displaystyle\,\mathfrak{E}^{(L)}(\Sigma,t)+2\int_{t}^{t_{0}}\int_{M}\left[a(s)^{-3}(N+1)\langle\Delta^{\frac{L}{2}}\mathcal{E},\Delta^{\frac{L}{2}}\Sigma\rangle_{G}\,{\text{{vol}}}_{G}\,ds+\frac{\dot{a}(s)}{a(s)}(N+1)\lvert\Delta^{\frac{L}{2}}\Sigma\rvert_{G}^{2}\right]\,{\text{{vol}}}_{G}\,ds$
$\displaystyle\,\lesssim\mathfrak{E}^{(L)}(\Sigma,t_{0})+\int_{t}^{t_{0}}\left(\varepsilon^{\frac{1}{8}}a(s)^{-3}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(L)}(\Sigma,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\Bigr{\\{}\varepsilon^{-\frac{1}{8}}a(s)^{-3}\mathfrak{E}^{(L)}(\phi,s)+\varepsilon^{\frac{15}{8}}a(s)^{3-c\sigma}\mathfrak{E}^{(L-1)}(\text{{Ric}},s)+\varepsilon^{2}a(s)^{-3}\mathfrak{E}^{(L-2)}(\text{{Ric}},s)$
$\displaystyle\,\phantom{\int_{t}^{t_{0}}}+\varepsilon^{2}a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-2)}(\Sigma,s)+\left(\varepsilon^{\frac{15}{8}}a(s)^{-3-c\sqrt{\varepsilon}}+\varepsilon
a(s)^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(\leq L-2)}(\phi,s)$
$\displaystyle\,\phantom{\int_{t}^{t_{0}}}+a(s)^{-1-c\sqrt{\varepsilon}-(L+1)\omega}\cdot
a(s)^{(L+1)\omega}\mathfrak{E}^{(\leq L)}_{1,\leq
L}(f,s)+\underbrace{\varepsilon^{2}a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-4)}(\text{{Ric}},s)}_{\text{not present for }L=2}\Bigr{\\}}\,ds$

###### Proof.

Note that the only term that is not pure trace that the presence of Vlasov
matter adds to the evolution equation is $S^{Vl,\parallel}$. Hence, commuting
(2.22c) with $\Delta^{\frac{L}{2}}$ and carrying over error term notation from
[FU23, (11.14)], we have the following:

(5.16a) $\displaystyle-\partial_{t}\mathfrak{E}^{(L)}(\Sigma,\cdot)=$
$\displaystyle\,\int_{M}\left\\{2a\langle\nabla^{2}\Delta^{\frac{L}{2}}N,\Delta^{\frac{L}{2}}\Sigma\rangle_{G}-2a(N+1)\langle\Delta^{\frac{L}{2}}\text{{Ric}}[G],\Delta^{\frac{L}{2}}\Sigma\rangle_{G}\right.$
(5.16b) $\displaystyle\,\qquad+16\pi
a^{-1}\langle\Delta^{\frac{L}{2}}S^{Vl,\parallel},\Delta^{\frac{L}{2}}\Sigma\rangle_{G}$
(5.16c) $\displaystyle\,\qquad+(\partial_{t}G^{-1})\ast
G^{-1}\ast\Delta^{\frac{L}{2}}\Sigma\ast\Delta^{\frac{L}{2}}\Sigma-3N\frac{\dot{a}}{a}\lvert\Delta^{\frac{L}{2}}\Sigma\rvert_{G}^{2}$
(5.16d)
$\displaystyle\,\qquad\left.-2\langle\mathfrak{S}_{L,Border},\Delta^{\frac{L}{2}}\Sigma\rangle_{G}-2\langle\mathfrak{S}_{L,Junk}^{\parallel},\Delta^{\frac{L}{2}}\Sigma\rangle_{G}\right\\}\,{\text{{vol}}}_{G}$

Regarding (5.16a), we apply the lapse energy estimate (5.7c) (resp., for
$L=0$, (5.7a)) for the first term and have the following:

$\displaystyle\,a\|\nabla^{2}\Delta^{\frac{L}{2}}N\|_{L^{2}_{G}}\sqrt{\mathfrak{E}^{(L)}(\Sigma,\cdot)}\lesssim
a^{-3}\sqrt{a^{8}\mathfrak{E}^{(L+2)}(N,\cdot)+\mathfrak{E}^{(L)}(N,\cdot)}\sqrt{\mathfrak{E}^{(L)}(\Sigma,\cdot)}$
$\displaystyle\lesssim$
$\displaystyle\,\left(\varepsilon^{\frac{1}{8}}a^{-3}+a^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(L)}(\Sigma,\cdot)+\varepsilon^{-\frac{1}{8}}a^{-3}\mathfrak{E}^{(L)}(\phi,\cdot)+a^{-1-c\sqrt{\varepsilon}-(L+1)\omega}+a^{(L+1)\omega}\mathfrak{E}^{(L)}_{1,L}(f,\cdot)$
$\displaystyle\,+\begin{cases}a^{-1-c\sqrt{\varepsilon}}\cdot
a^{\omega}\|G-\gamma\|_{L^{2}_{G}(M)}^{2}&L=0\\\
\varepsilon^{\frac{15}{8}}a^{-3-c\sqrt{\varepsilon}}\left(\mathfrak{E}^{(\leq
L-2)}(\phi,\cdot)+\mathfrak{E}^{(\leq
L-2)}(\Sigma,\cdot)\right)+\varepsilon^{\frac{15}{8}}a^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-3)}(\text{{Ric}},\cdot)&L\geq 2\end{cases}$

For the second term in (5.16a), we insert the Bel-Robinson constraint (2.22j).
High order lapse terms are treated similarly as above, and the only Vlasov
term that is tracefree and thus isn’t cancelled by the inner product is

$-2(N+1)a^{-3}\cdot 4\pi
a^{2}\langle\Delta^{\frac{L}{2}}S^{Vl,\parallel},\Delta^{\frac{L}{2}}\Sigma\rangle_{G}\,.$

This term is bounded in absolute value by
$a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(L)}_{1,L}(f,\cdot)}\sqrt{\mathfrak{E}^{(L)}(\Sigma,\cdot)}$
up to constant with Lemma 4.15, as is the term in (5.16b). (5.16c)-(5.16d)
only contribute error terms that can be dealt with as in the scalar field
setting, accounting for (3.13). ∎

######  Lemma 5.11 (Elliptic energy estimate for the second fundamental form).

For any $L\in 2\mathbb{Z}_{+}$, $2\leq L\leq 18$, we have

(5.17a) $\displaystyle a^{4}\mathfrak{E}^{(L+1)}(\Sigma,\cdot)$
$\displaystyle\,\lesssim\left(a^{4-c\sqrt{\varepsilon}}+\varepsilon
a^{2-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(L)}(\Sigma,\cdot)+\mathfrak{E}^{(L)}(\phi,\cdot)+\mathfrak{E}^{(L)}(W,\cdot)$
$\displaystyle\,+\varepsilon^{2}a^{4}\mathfrak{E}^{(L-1)}(\text{{Ric}},\cdot)+a^{4-c\sqrt{\varepsilon}}\mathfrak{E}^{(L)}_{1,L}(f,\cdot)$
$\displaystyle\,+\varepsilon a^{2-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-2)}(\phi,\cdot)+a^{2-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-2)}(\Sigma,\cdot)+\varepsilon a^{2-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)\,.$ For $L=0$, one analogously has (5.17b)
$a^{4}\mathfrak{E}^{(1)}(\Sigma,\cdot)\lesssim\,\left(a^{4-c\sqrt{\varepsilon}}+\varepsilon
a^{2-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(0)}(\Sigma,\cdot)+\mathfrak{E}^{(0)}(\phi,\cdot)+\mathfrak{E}^{(0)}(W,\cdot)+a^{4}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)\,.$

###### Proof.

Recall from the proof of [FU23, Lemma 6.10] that, since
$\Delta^{\frac{L}{2}}\Sigma$ is tracefree, one has

$\mathfrak{E}^{(L+1)}(\Sigma,\cdot)\lesssim
a^{-c\sqrt{\varepsilon}}\mathfrak{E}^{(L)}(\Sigma,\cdot)+\int_{M}\lvert\text{{div}}_{G}\Delta^{\frac{L}{2}}\Sigma\rvert_{G}^{2}+\lvert\text{{curl}}\Delta^{\frac{L}{2}}\Sigma\rvert_{G}^{2}\,{\text{{vol}}}_{G}$

Inserting commuted analogues of (2.22i) and (2.22k), we can argue as in the
scalar field setting, where the commuted analogue of (2.22i) now additionally
contains the term

$8\pi\Delta^{\frac{L}{2}}\jmath^{Vl},$

where we again apply Lemma 4.15, leading to the respective Vlasov energy
terms. ∎

We close out this section by collecting all necessary bounds for metric
variables:

######  Lemma 5.12 (Sobolev norm estimates for metric variables).

For any $l\in\mathbb{N},\,l\leq 18$, we have the following:

(5.18a) $\displaystyle
a(t)^{\omega}\|G-\gamma\|_{H^{l}_{G}(M_{t})}^{2}\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-3+\frac{\omega}{2}-c\sqrt{\varepsilon}}+a(s)^{-1-c\sigma}\right)\cdot
a(s)^{\omega}\|G-\gamma\|_{H^{l}_{G}(M_{s})}^{2}\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-3+\frac{\omega}{2}}\left(\mathfrak{E}^{(\leq
l)}(\Sigma,s)+\mathfrak{E}^{(\leq l)}(\phi,s)+\underbrace{\mathfrak{E}^{(\leq
l-2)}(\text{{Ric}},s)}_{\text{if }l\geq 3}\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-1-c\sqrt{\varepsilon}-(l+1)\omega}\cdot
a(s)^{(l+1)\omega}\mathfrak{E}^{(\leq l)}_{1,\leq l}(f,s)\,ds\,,$ (5.18b)
$\|G-\gamma\|_{H^{l}_{G}(M_{t})}^{2}\lesssim
a^{-c\varepsilon^{\frac{1}{8}}}\left(\varepsilon^{4}+\varepsilon^{-\frac{1}{4}}\sup_{s\in(t,t_{0})}\left(\|N\|_{H^{l}_{G}(M_{s})}^{2}+\|\Sigma\|_{H^{l}_{G}(M_{s})}^{2}\right)\right)$

For $l>0$, one additionally has

(5.19a) $\displaystyle
a(t)^{\beta+\omega}\|\Gamma-\hat{\Gamma}\|_{H^{l-1}_{G}(M_{t})}^{2}\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-3+\frac{\omega}{2}-c\sqrt{\varepsilon}}+a(s)^{-1-c\sigma}\right)\cdot
a(s)^{\beta+\omega}\|\Gamma-\hat{\Gamma}\|_{H^{l-1}_{G}(M_{s})}^{2}\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-3+\frac{\omega}{2}}\cdot
a(s)^{\beta+\omega}\left(\mathfrak{E}^{(\leq l)}(\Sigma,s)+\mathfrak{E}^{(\leq
l)}(\phi,s)+\underbrace{\mathfrak{E}^{(\leq l-2)}(\text{{Ric}},s)}_{\text{if
}l\geq 3}\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-1-c\sqrt{\varepsilon}-l\omega}\cdot
a(s)^{\beta+(l+1)\omega}\mathfrak{E}^{(\leq l)}_{1,\leq l}(f,s)\,ds\,\text{
and}$ (5.19b)
$\displaystyle\|\Gamma-\hat{\Gamma}\|_{H^{l-1}_{G}(M_{t})}^{2}\lesssim$
$\displaystyle\,a(t)^{-c\varepsilon^{\frac{1}{8}}}\left(\varepsilon^{4}+\varepsilon^{-\frac{1}{4}}\sup_{s\in(t,t_{0})}\left(\|N\|_{H^{l}_{G}(M_{s})}^{2}+\|\Sigma\|_{H^{l}_{G}(M_{s})}^{2}\right)\right)$

as well as the following:

(5.20a) $\displaystyle\mathfrak{E}^{(L-2)}(\text{{Ric}},t)\lesssim$
$\displaystyle\,\mathfrak{E}^{(L-2)}(\text{{Ric}},t_{0})+\int_{t}^{t_{0}}\left(\varepsilon^{\frac{1}{8}}a(s)^{-3}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(L-2)}(\text{{Ric}},s)\,ds$
$\displaystyle+\int_{t}^{t_{0}}\Bigr{\\{}$
$\displaystyle\,\varepsilon^{-\frac{1}{8}}a(s)^{-3}\left(\mathfrak{E}^{(L)}(\phi,s)+\mathfrak{E}^{(L)}(\Sigma,s)\right)+a(s)^{-1-c\sqrt{\varepsilon}-(L+1)\omega}\cdot
a(s)^{(L+1)\omega}\mathfrak{E}^{(\leq L)}_{1,\leq L}(f,s)$
$\displaystyle\,+\varepsilon^{-\frac{1}{8}}a(s)^{-3-c\sqrt{\varepsilon}}\left(\mathfrak{E}^{(\leq
L-2)}(\phi,s)+\mathfrak{E}^{(\leq L-2)}(\Sigma,s)\right)$
$\displaystyle\,+\varepsilon^{\frac{7}{8}}a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-4)}(\text{{Ric}},s)\Bigr{\\}}\,ds\,,$ (5.20b) $\displaystyle
a(t)^{4}\mathfrak{E}^{(L-1)}(\text{{Ric}},t)\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-1-c\sqrt{\varepsilon}}+\varepsilon^{\frac{1}{8}}a(s)^{-3}\right)\cdot
a(s)^{4}\mathfrak{E}^{(L-1)}(\text{{Ric}},s)\,ds$ $\displaystyle+$
$\displaystyle\,\int_{t}^{t_{0}}\left\\{\varepsilon^{-\frac{1}{8}}a(s)^{-3}\cdot
a(s)^{4}\mathfrak{E}^{(L+1)}(\Sigma,s)+\left(a(s)^{-1-c\sqrt{\varepsilon}}+\varepsilon^{\frac{15}{8}}a(s)^{-3}\right)\mathfrak{E}^{(L)}(\Sigma,s)\right.$
$\displaystyle\,\qquad+\left(a(s)^{-1-c\sqrt{\varepsilon}}+\varepsilon^{-\frac{1}{8}}a(s)^{-3}\right)\mathfrak{E}^{(L)}(\phi,s)$
$\displaystyle\,\qquad+\left(a(s)^{-1-c\sqrt{\varepsilon}}+\varepsilon^{\frac{15}{8}}a(s)^{-3-c\sqrt{\varepsilon}}\right)\left(\mathfrak{E}^{(\leq
L-2)}(\phi,s)+\mathfrak{E}^{(\leq L-2)}(\Sigma,s)\right)$
$\displaystyle\,\qquad+a(s)^{-1-c\sigma-(L+1)\omega}\cdot
a(s)^{(L+1)\omega}\mathfrak{E}^{(\leq L)}_{1,\leq L}(f,s)$
$\displaystyle\,\qquad\left.+\varepsilon
a(s)^{-3}\mathfrak{E}^{(L-2)}(\text{{Ric}},s)+\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-4)}(\text{{Ric}},s)\right\\}\,ds$ Additionally, the following holds: (5.20c)
$\displaystyle\mathfrak{E}^{(0)}$
$\displaystyle\,(\text{{Ric}},t)\lesssim\mathfrak{E}^{(0)}(\text{{Ric}},t_{0})+\int_{t}^{t_{0}}\left(\varepsilon^{\frac{1}{8}}a(s)^{-3}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(0)}(\text{{Ric}},s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\varepsilon^{-\frac{1}{8}}a(s)^{-3}\left(\mathfrak{E}^{(0)}(\phi,s)+\mathfrak{E}^{(0)}(\Sigma,s)\right)+a(s)^{-1-c\sqrt{\varepsilon}}\left(\mathfrak{E}^{(0)}_{1,0}(f,s)+\|G-\gamma\|_{L^{2}_{G}(M_{s})}^{2}\right)\,ds$
Finally, at highest order, one has the following bound: (5.20d) $\displaystyle
a^{8}\mathfrak{E}^{(l)}(\text{{Ric}},\cdot)\lesssim$
$\displaystyle\,\mathfrak{E}^{(l)}(W,\cdot)+\mathfrak{E}^{(l)}(\Sigma,\cdot)+\varepsilon
a^{-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
l-2)}(\Sigma,\cdot)+\mathfrak{E}^{(\leq l)}(\phi,\cdot)$
$\displaystyle\,+a^{4-c\sqrt{\varepsilon}}\left(\mathfrak{E}^{(\leq
l)}_{1,\leq l}(f,\cdot)\underbrace{+\|G-\gamma\|_{L^{2}_{G}}^{2}}_{\text{if
}l=0}\right)$

###### Proof.

Since (2.22a), (2.22d) and (2.22e) do not contain Vlasov matter variables, the
proofs for (5.18b), (5.19b) and (5.20a)-(5.20c) are largely unchanged to those
of [FU23, Lemma 6.11 - 6.14], where we apply Lemma 5.5 to estimate lapse
energy terms. Regarding the scaled estimates, we consider (5.18a) for even
$l\geq 2$; the other cases as well as (5.19a) are proven similarly: Commuting
(2.22a) with $\Delta^{\frac{l}{2}}$, we have the following:

$\displaystyle-\partial_{t}\left(a(t)^{\omega}\|G-\gamma\|_{H^{l}_{G}}^{2}\right)\lesssim$
$\displaystyle\,a^{-3-c\sqrt{\varepsilon}+\omega}\left(\sqrt{\mathfrak{E}^{(\leq
l)}(\Sigma,\cdot)}+\sqrt{\mathfrak{E}^{(\leq
l)}(N,\cdot)}+\sqrt{\mathfrak{E}^{(\leq
l-2)}(\text{{Ric}},\cdot)}\right)\|G-\gamma\|_{H^{l}_{G}}$

We apply the Young inequality to, for example, the shear term as follows:

$a^{-3-c\sqrt{\varepsilon}+\omega}\sqrt{\mathfrak{E}^{(\leq
l)}(\Sigma,\cdot)}\|G-\gamma\|_{H^{l}_{G}}\lesssim
a^{-3+\frac{\omega}{2}-c\sqrt{\varepsilon}}\cdot
a^{\omega}\|G-\gamma\|_{H^{l}_{G}}^{2}+a^{-3+\frac{\omega}{2}}\mathfrak{E}^{(\leq
l)}(\Sigma,\cdot)$

The statement then follows by inserting Lemma 5.5, applying the Young
inequality similarly to the other terms and integrating.
Finally, (5.20d) is a direct consequence of applying $\Delta^{\frac{l}{2}}$
(resp. $\nabla\Delta^{\frac{l-1}{2}}$) to the constraint equation (2.22j),
rearranging and applying the a priori estimates (4.1b), (4.3b), (4.3d), (4.1a)
and (4.3a). In particular, regarding the Vlasov terms, one uses (4.27d) and
(4.27a) to connect their norms to our energy quantities. ∎

##  6\. Vlasov energy estimates

In this section, we collect the necessary energy estimates to obtain the
bootstrap improvements in the following section. In particular, for each
derivative order and each number of horizontal derivatives in the energy, we
prove both a time-scaled estimate, which are all combined in Lemma 7.3 to
obtain bootstrap improvements for the spacetime and scalar field variables, as
well as non-scaled estimates that lose smallness, which we use to improve the
bootstrap assumptions on Vlasov matter. Regarding the time-scaled estimates,
we scale by $a^{\omega}$ and powers there of to indicate scaling necessary
within the hierarchy of vertical and horizontal derivatives at a given order
(foreshadowing (7.2)), and by $a^{\beta}$ to indicate additional scaling
necessary for the total energy at odd derivative orders (foreshadowing the
final line of (7.5a)).

All of the estimates in this section hold for $t\in(t_{Boot},t_{0}]$.
Furthermore, we let $\omega\gg\sigma>0$ be a constant that is suitably small:
For example, if $\sigma$ is chosen small enough that $c\sigma<\frac{1}{2}$
holds throughout the proof, $\omega<\frac{1}{40}$ is an admissible choice if
$\sigma$ is chosen small enough, and in general, $\omega$ only needs to be
small compared to the order of derivatives used in our estimates once $\sigma$
is chosen to be small enough compared to $c>0$.

###  6.1. Preliminaries

First, we collect some tools that will be used throughout:

######  Remark 6.1 (Integrating $M_{t}$-tangent tensors over the tangent
space).

We recall that by Lemma 4.6, positive powers of $\lvert v\rvert_{G}$ can be
bounded by $a^{-c\sqrt{\varepsilon}}$, and thus integrals over powers of
$\lvert v\rvert_{G}$ and $v^{0}$ on the support of $f$, $f_{FLRW}$ or
$f-f_{FLRW}$ are bounded by $a^{-c{\varepsilon}}$. In particular, for any
$M_{t}$-tangent tensor $\mathfrak{T}$ and any $\alpha>-3$, this implies

$\int_{TM_{t}}\lvert\mathfrak{T}\rvert_{G}^{2}\cdot\langle
v\rangle_{G}^{2\alpha}\,{\text{{vol}}}_{\underline{G}}=\int_{M_{t}}\lvert\mathfrak{T}\rvert_{G}^{2}\int_{T_{x}M_{t}}\left(\langle
v\rangle_{G}^{2\alpha}\mu_{G}dv\right){\text{{vol}}}_{G}\lesssim\|\mathfrak{T}\|_{L^{2}_{G}(M_{t})}^{2}\cdot
a^{-c\sqrt{\varepsilon}}\,.$

We use this in the sequel without further comment, usually when needing to
estimate high order terms of $G-\gamma,\,\Sigma,\,N$ or scalar field
quantities.

Next, we collect the core estumate that ensures the transport equation almost
conserves Vlasov energies:

######  Lemma 6.2 (Energy mechanism for the Vlasov equation).

Let $\mu\geq 0$ and $0\leq L\leq 19$. Then, the following holds:

$\displaystyle\int_{TM_{t}}\langle
v\rangle^{2\mu}\langle\bm{X}\underline{\nabla}_{vert}^{L}(f-f_{FLRW}),\underline{\nabla}_{vert}^{L}(f-f_{FLRW})\rangle_{\underline{G}_{0}}{\text{{vol}}}_{\underline{G}}\lesssim$
$\displaystyle\,\varepsilon a^{-3}\mathfrak{E}^{(L)}_{\mu,0}(f,\cdot)$

Similarly, if one additionally has $1\leq K\leq L$, we have

$\displaystyle\int_{TM_{t}}\langle
v\rangle^{2\mu}\langle\bm{X}\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f,\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\rangle_{\underline{G}_{0}}{\text{{vol}}}_{\underline{G}}\lesssim$
$\displaystyle\,\varepsilon a^{-3}\cdot\mathfrak{E}^{(L)}_{\mu,K}(f,\cdot)\,.$

###### Proof.

Both estimates are proved identically, so we only write out the proof for the
second. Writing out $\bm{X}$ (see (2.21b)), the left hand side takes the
following form:

$\displaystyle\int_{TM_{t}}$
$\displaystyle\left\langle\left[-a^{-1}\frac{v^{j}}{v^{0}}\underline{\nabla}_{j}+a^{-1}v^{0}\nabla^{\sharp
j}N\mathbf{B}_{j}+2N\frac{\dot{a}}{a}v^{j}\mathbf{B}_{j}-2(N+1)a^{-3}\Sigma^{\sharp
j}_{i}v^{i}\mathbf{B}_{j}\right]\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f,\right.$
$\displaystyle\,\qquad\left.\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\right\rangle_{\underline{G}}\cdot\langle
v\rangle^{2\mu}(v^{0})^{2(L-K)}\,{\text{{vol}}}_{\underline{G}}$

The first term vanishes after integrating by parts in $\underline{\nabla}$
over $TM_{t}$, since $\underline{\nabla}_{j}\frac{v^{j}}{v^{0}}=0$ holds by
(2.26a). For the remaining three terms, we use Fubini’s theorem to replace the
integral by one over $\int_{M}\int_{T_{x}M_{t}}$ and then integrate by parts
in $\mathbf{B}$. This leads to the following after rearranging and applying
(2.26c):

$\displaystyle\int_{TM_{t}}$ $\displaystyle\,\langle
v\rangle^{2\mu}\langle\bm{X}\underline{\nabla}_{vert}^{L}(f-f_{FLRW}),\underline{\nabla}_{vert}^{L}(f-f_{FLRW})\rangle_{\underline{G}_{0}}{\text{{vol}}}_{\underline{G}}$
$\displaystyle=-\frac{1}{2}\,\int_{TM_{t}}$
$\displaystyle\,\left\\{a^{-1}\nabla^{\sharp
j}N\frac{v^{\flat}_{j}}{v^{0}}+6N\frac{\dot{a}}{a}-2(N+1)a^{-3}\Sigma^{\sharp
j}_{j}\right\\}\langle
v\rangle^{2\mu}_{G}\lvert\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\rvert_{\underline{G}_{0}}^{2}\,{\text{{vol}}}_{\underline{G}}$
$\displaystyle+\,\int_{TM_{t}}$
$\displaystyle\,\left\\{a^{-1}v^{j}v^{0}\nabla_{j}N+2\lvert
v\rvert_{G}^{2}N\frac{\dot{a}}{a}-2(N+1)a^{-3}v^{i}v^{j}\Sigma_{ij}\right\\}$
$\displaystyle\,\cdot\left(2\mu\langle
v\rangle^{-2}+2(L-K)(v^{0})^{-2}\right)\langle
v\rangle^{2\mu}_{G}\lvert\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\rvert_{\underline{G}_{0}}^{2}\,{\text{{vol}}}_{\underline{G}}$

Note that the last term in the second line vanishes since one has
$\Sigma^{\sharp j}_{j}=0$. The remaining terms can be estimated by
$\varepsilon a^{-3}\mathfrak{E}^{(L)}_{\mu,K}(f,\cdot)$ using (4.1b) and
(3.13). ∎

To be able to apply the above identity, we need to commute $\partial_{t}$ past
the respective covariant Sasaki derivatives, apply (2.21a), and then commute
$\bm{X}$ back past them. We have already collected error term formulas for the
former in Lemma 4.13, and now collect schematic formulas for the latter:

######  Lemma 6.3 (The commuted rescaled Vlasov equation).

The following commuted analogues of (2.21a) hold: For $1\leq L\leq 19$, one
has

(6.1a) $\displaystyle\underline{\nabla}_{vert}^{L}\partial_{t}f=$
$\displaystyle\,\bm{X}\underline{\nabla}_{vert}^{L}(f-f_{FLRW})+a^{-1}\frac{(v^{0})^{2}+v\ast
v}{(v^{0})^{3}}\ast\underline{\nabla}_{vert}^{L-1}\underline{\nabla}_{hor}(f-f_{FLRW})$
$\displaystyle\,+a^{-1}\frac{v\ast
v}{v^{0}}\ast\text{{Ric}}[G]\ast\underline{\nabla}_{vert}^{L-1}\underline{\nabla}_{hor}(f-f_{FLRW})$
$\displaystyle\,+a^{-1}\frac{v}{v^{0}}\ast\nabla
N\ast\underline{\nabla}_{vert}^{L}(f-f_{FLRW})-2N\frac{\dot{a}}{a}\underline{\nabla}^{L}_{vert}(f-f_{FLRW})$
$\displaystyle\,+a^{-3}(N+1)\Sigma\ast\underline{\nabla}_{vert}^{L}(f-f_{FLRW})$
$\displaystyle\,+\mathfrak{F}_{L,0}+\underline{\nabla}_{vert}^{L}\bm{X}f_{FLRW}$
where one has, for $L\geq 2$, (6.1b)
$\|(v^{0})^{L}\mathfrak{F}_{L,0}\|_{L^{2}_{1,\underline{G}}(TM)}\lesssim
a^{-1-c\sigma}\sqrt{\mathfrak{E}^{(\leq L-1)}_{1,\leq
1}(f,\cdot)}+\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}(M)}+\|G-\gamma\|_{H^{1}_{G}(M)}$

and said error term vanishes for $L=1$, as well as

(6.2) $\displaystyle\|(v^{0})^{L}\underline{\nabla}_{vert}^{L}$
$\displaystyle\bm{X}f_{FLRW}\|_{L^{2}_{1,\underline{G}}(TM)}\lesssim
a^{-3-c\sqrt{\varepsilon}}\left(\sqrt{\mathfrak{E}^{(0)}(\Sigma,\cdot)}+\sqrt{a^{4}\mathfrak{E}^{(1)}(N,\cdot)}+\sqrt{\mathfrak{E}^{(0)}(N,\cdot)}\right)$
$\displaystyle\,+a^{-1-c\sqrt{\varepsilon}}\left(\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}(M)}+\|G-\gamma\|_{L^{2}_{G}(M)}\right)\,.$

The latter bound also extends to $L=0$.
Furthermore, for $0<K<L$, we have

(6.3a)
$\displaystyle\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}\partial_{t}f=$
$\displaystyle\,\bm{X}\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f+a^{-1}\frac{v\ast
v}{v^{0}}\ast\nabla^{K-1}\text{{Ric}}[G]\ast\underline{\nabla}_{vert}^{L-K+1}f$
$\displaystyle\,+a^{-1}\left(\frac{1}{v^{0}}+\frac{v\ast
v}{(v^{0})^{3}}\right)\ast\underline{\nabla}_{vert}^{L-K-1}\underline{\nabla}_{hor}^{K+1}f$
$\displaystyle+a^{-1}\frac{v}{v^{0}}\ast\nabla^{\sharp}N\ast\mathbf{B}\underline{\nabla}_{vert}^{L-K-1}\underline{\nabla}_{hor}^{K}f-2N\frac{\dot{a}}{a}\mathbf{B}\underline{\nabla}_{vert}^{L-K-1}\underline{\nabla}_{hor}^{K}f$
$\displaystyle\,+a^{-3}(N+1)\Sigma^{\sharp}\ast\mathbf{B}\underline{\nabla}_{vert}^{L-K-1}\underline{\nabla}_{hor}^{K}f$
$\displaystyle\,+a^{-3}(N+1)\nabla\Sigma\ast
v\ast\mathbf{B}\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K-1}f$
$\displaystyle\,+\mathfrak{F}_{L,K}$ with (6.3b)
$\displaystyle\|(v^{0})^{L-K}$
$\displaystyle\mathfrak{F}_{L,K}\|_{L^{2}_{1,\underline{G}}(TM)}\lesssim\left(a^{-1-c\sigma}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\right)\sqrt{\mathfrak{E}^{(\leq L-1)}_{1,\leq
K}(f,\cdot)}$
$\displaystyle\,+a^{-3-c\sqrt{\varepsilon}}\left(\sqrt{\mathfrak{E}^{(\leq
K)}(\Sigma,\cdot)}+\sqrt{\mathfrak{E}^{(\leq
K)}(N,\cdot)}\right)+a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(K+1)}(N,\cdot)}$
$\displaystyle\,\underbrace{\left(\varepsilon
a^{-3-c\sqrt{\varepsilon}}+\sqrt{\varepsilon}a^{-1-c\sqrt{\varepsilon}}\right)\left(\sqrt{\mathfrak{E}^{(\leq
K-1)}(\text{{Ric}},\cdot)}+\|\Gamma-\hat{\Gamma}\|_{H^{1}_{G}(M)}+\|G-\gamma\|_{H^{1}_{G}(M)}\right)}_{\text{if
}K\geq 2}$

as well as

(6.4a) $\displaystyle\underline{\nabla}_{hor}^{L}\partial_{t}f=$
$\displaystyle\,\bm{X}\underline{\nabla}_{hor}^{L}f+a^{-1}\frac{v\ast
v}{v^{0}}\ast\nabla^{L-1}\text{{Ric}}[G]\ast\underline{\nabla}_{vert}f$
$\displaystyle\,+a^{-3}(N+1)\nabla\Sigma\ast
v\ast\mathbf{B}\underline{\nabla}_{hor}^{L-1}f+\mathfrak{F}_{L,L}$ with (6.4b)
$\displaystyle\|$
$\displaystyle\mathfrak{F}_{L,L}\|_{L^{2}_{1,\underline{G}}(TM)}\lesssim\varepsilon
a^{-1-c\sigma}\sqrt{\mathfrak{E}^{(L)}_{1,\leq
L-1}(f,\cdot)}+\left(a^{-1-c\sigma}+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\right)\sqrt{\mathfrak{E}^{(\leq L-1)}_{1,\leq
L-1}(f,\cdot)}$
$\displaystyle\,+a^{-3-c\sqrt{\varepsilon}}\left(\sqrt{\mathfrak{E}^{(\leq
L)}(\Sigma,\cdot)}+\sqrt{\mathfrak{E}^{(\leq
L)}(N,\cdot)}\right)+a^{-1-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(L+1)}(N,\cdot)}$
$\displaystyle\,+\underbrace{\left(\varepsilon
a^{-3-c\sqrt{\varepsilon}}+\sqrt{\varepsilon}a^{-1-c\sqrt{\varepsilon}}\right)\left(\sqrt{\mathfrak{E}^{(\leq
L-2)}(\text{{Ric}},\cdot)}+\|\Gamma-\hat{\Gamma}\|_{H^{1}_{G}(M)}+\|G-\gamma\|_{H^{1}_{G}(M)}\right)}_{\text{
if }L\geq 2}.$

###### Proof.

We first consider the purely vertical case (6.1a), then the purely horizontal
case (6.4a), and then briefly (6.3a) as a combination of both cases before
discussing the respective error bounds.

For (6.1a), we first replace $f$ with $(f-f_{FLRW})+f_{FLRW}$, which allows us
to collect all reference error terms in
$\underline{\nabla}_{vert}^{L}\bm{X}f_{FLRW}$. Since $\mathbf{B}_{i}$ and
$\underline{\nabla}_{i+3}$ coincide when acting on purely vertical tensors
(see (2.19)) and vertical frame derivatives commute, the only error terms
generated in computing (6.1a) are down to applying the product rule to
momentum prefactors, except for the horizontal term in $\bm{X}$. Here, we
additionally need to apply (4.24a) for $M_{2}=0$, which yields the term in the
second line of (6.1a) as the leading error term, while all others are absorbed
into the error terms. In particular, we note that this commutation only
generates terms of the type in the second line of (6.1a), where we can always
pull the horizontal derivative back to the front up to error terms as in Lemma
4.14 that are absorbed into $\mathfrak{F}_{L,0}$. We also note that, whenever
vertical derivatives hit $v$-prefactors, both a power of $v$ and a derivative
are lost, and thus the momentum scaling of lower order error terms always
precisely matches that of the respective $\underline{G}_{0}$-norm, up to
estimating $v$ by $v^{0}$ in $\lvert\cdot\rvert_{G}$.

For (6.4a), we need to ensure that, after taking horizontal derivatives of
$\bm{X}f$ and applying the product rule, we can still express all terms
covariantly despite Christoffel symbols occuring explicity in, for example,
the commutator
$[\mathbf{A}_{i},\mathbf{B}_{j}]f=\Gamma[G]^{l}_{ij}\mathbf{B}_{l}f$. To this
end, we treat these summands in $\bm{X}f$ as scalar functions, apply the
product rule with respect to the horizontal frame derivative, commute the
frame derivatives and rearrange to obtain a coordinate invariant expression at
first order. For example, for the summand $a^{-3}(N+1)\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f$ in $\bm{X}f$, this looks as follows at first
order, applying (2.27b):

$\displaystyle\underline{\nabla}_{i}\left[(N+1)\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f\right]=$
$\displaystyle\,\partial_{i}N\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f+(N+1)\partial_{i}(\Sigma^{\sharp
j}_{k})v^{k}\mathbf{B}_{j}f+(N+1)\Sigma^{\sharp
j}_{k}(\mathbf{A}_{i}v^{k})\mathbf{B}_{j}f$
$\displaystyle\,+(N+1)\Sigma^{\sharp
j}_{k}v^{k}\left([\mathbf{A}_{i},\mathbf{B}_{j}]\xi+\mathbf{B}_{j}\mathbf{A}_{i}f\right)$
$\displaystyle=$ $\displaystyle\,\nabla_{i}N\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}\xi+(N+1)\left[\left(\nabla_{i}\Sigma^{\sharp
j}_{k}-\Gamma[G]_{il}^{j}\Sigma^{\sharp
l}_{k}+\Gamma[G]^{l}_{ik}\Sigma^{\sharp
j}_{l}\right)v^{k}\mathbf{B}_{j}f\right.$
$\displaystyle\,+\left.\Sigma^{\sharp
j}_{k}\left(-v^{l}\Gamma[G]^{k}_{il}\mathbf{B}_{j}f+v^{k}\Gamma[G]^{l}_{ij}\mathbf{B}_{l}f\right)+\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}\mathbf{A}_{i}f\right]$ $\displaystyle=$
$\displaystyle\,(N+1)\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}\underline{\nabla}_{i}f+\nabla_{i}N\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f+(N+1)\nabla_{i}\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f$

This can be repeated at higher orders while curvature errors are incurred:
Considering, for example, the shear part of the same term at second order, we
apply the observations from the first order case consecutively while keeping
track of curvature terms that arise from Sasaki connection coefficients
containing horizontal and vertical indices (see (2.19)) and commuting the
vertical frame derivative past them:

$\displaystyle\,\underline{\nabla}_{i_{1}}\underline{\nabla}_{i_{2}}\left(\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f\right)$ $\displaystyle=$
$\displaystyle\,\mathbf{A}_{i_{1}}\mathbf{A}_{i_{2}}\left(\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f\right)-\Gamma[G]_{i_{1}i_{2}}^{l}\mathbf{A}_{l}\left(\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f\right)-\frac{1}{2}v^{l}{\text{{Riem}}[G]^{m}}_{li_{1}i_{2}}\mathbf{B}_{m}\left(\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f\right)$ $\displaystyle=$
$\displaystyle\,\mathbf{A}_{i_{1}}\left(\nabla_{i_{2}}\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f+\Sigma_{k}^{\sharp
j}\mathbf{B}_{j}\mathbf{A}_{i_{2}}f\right)-v^{k}\left(\Gamma[G]_{i_{1}i_{2}}^{l}\Sigma^{\sharp
j}_{k}\mathbf{B}_{j}\mathbf{A}_{l}f+\Gamma[G]_{i_{1}i_{2}}^{l}\nabla_{l}\Sigma^{\sharp
j}_{k}\mathbf{B}_{j}f\right)$
$\displaystyle\,-\frac{1}{2}v^{l}{\text{{Riem}}[G]^{m}}_{li_{1}i_{2}}\left(\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}\mathbf{B}_{m}f+\Sigma_{m}^{\sharp
j}\mathbf{B}_{j}f\right)$ $\displaystyle=$
$\displaystyle\,\left(\nabla_{i_{1}}\nabla_{i_{2}}\Sigma^{\sharp
j}_{k}+\Gamma[G]_{i_{1}i_{2}}^{l}\nabla_{l}\Sigma^{\sharp
j}_{k}\right)v^{k}\mathbf{B}_{j}f+\nabla_{i_{2}}\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}\mathbf{A}_{i_{1}}f+\nabla_{i_{1}}\Sigma^{\sharp
j}_{k}\mathbf{B}_{j}\mathbf{A}_{i_{2}}f$ $\displaystyle\,+\Sigma^{\sharp
j}_{k}\mathbf{B}_{j}\mathbf{A}_{i_{1}}\mathbf{A}_{i_{2}}f-\Gamma[G]_{i_{1}i_{2}}^{l}\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}\mathbf{A}_{l}f-\Gamma[G]_{i_{1}i_{2}}^{l}\nabla_{l}\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}f$ $\displaystyle\,-\frac{1}{2}\Sigma_{k}^{\sharp
j}v^{k}\mathbf{B}_{j}\left(v^{l}{\text{{Riem}}[G]^{m}}_{li_{1}i_{2}}\mathbf{B}_{m}f\right)+\frac{1}{2}v^{k}{\text{{Riem}}[G]^{m}}_{ji_{1}i_{2}}\Sigma_{k}^{\sharp
j}\mathbf{B}_{m}f-\frac{1}{2}v^{l}{\text{{Riem}}[G]^{k}}_{li_{1}i_{2}}\Sigma_{k}^{\sharp
j}\mathbf{B}_{j}f$ $\displaystyle=$
$\displaystyle\,\left(\nabla_{i_{1}}\nabla_{i_{2}}\Sigma^{\sharp
j}_{k}\right)v^{k}\mathbf{B}_{j}f+\nabla_{i_{2}}\Sigma_{k}^{\sharp
j}v^{k}\mathbf{B}_{j}\underline{\nabla}_{i_{1}}f+\nabla_{i_{1}}\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}\underline{\nabla}_{i_{2}}f+\Sigma_{k}^{\sharp
j}v^{k}\mathbf{B}_{j}\underline{\nabla}_{i_{1}}\underline{\nabla}_{i_{2}}f$
$\displaystyle\,+\frac{1}{2}v^{k}{\text{{Riem}}[G]^{m}}_{ji_{1}i_{2}}\Sigma_{k}^{\sharp
j}\mathbf{B}_{m}f$

Repeating this argument at higher orders, we obtain the following schematic
representation:

$\displaystyle\,[\underline{\nabla}_{hor}^{L},(N+1)\Sigma^{\sharp
j}_{k}v^{k}\mathbf{B}_{j}]f$ $\displaystyle=$
$\displaystyle\,\sum_{K_{N}+K_{\Sigma}+K_{f}=L,\,K_{f}\neq
L}v\ast\nabla^{K_{N}}(N+1)\ast\nabla^{K_{\Sigma}}\Sigma\ast\mathbf{B}\underline{\nabla}_{hor}^{K_{f}}f$
$\displaystyle\,+\sum_{K_{\text{{Ric}}}+K_{N}+K_{\Sigma}+K_{f}=L-2}v\ast\nabla^{K_{\text{{Ric}}}}\text{{Ric}}[G]\ast\nabla^{K_{N}}(N+1)\ast\nabla^{K_{\Sigma}}\Sigma\ast\mathbf{B}\underline{\nabla}_{hor}^{K_{f}}f$
$\displaystyle\,+\langle\text{low order nonlinear error terms with multiple
curvature terms}\rangle$

The other vertical terms in $\bm{X}f$ are treated similarly, where
$[\mathbf{A}_{i},v^{j}\mathbf{B}_{j}]f=0$ can be used to simplify some
arguments. Finally, regarding the horizontal term in $\bm{X}f$, we apply
(4.24a) with $M_{1}=0,M_{2}=L$, again noting that we can rearrange the
covariant derivatives up to error terms into the order presented, and can deal
with the error terms as in Lemma 4.14. Finally, (6.3a) follows from combining
the ideas from both of these proofs, commuting with horizontal derivative
terms as in the purely vertical case and noting that $\mathbf{B}$ and
$\underline{\nabla}_{vert}^{L-K}$ commute.

The error bounds (6.4b), (6.3b) and (6.1b) simply follow
$L^{2}-L^{\infty}-$estimating the remaining terms and using the a priori
estimates (4.1b) and (4.3b) for the shear, (4.15) for low order Vlasov terms,
(4.3e) for the curvature and (4.6) to control momentum terms as well as the
lapse bootstrap assumption from (3.12a). In particular, we note that
$\mathbf{B}f$ and higher order vertical derivatives of it are only bounded by
$a^{-c\sqrt{\varepsilon}}$, leading to the lapse and shear energies in the
second line of (6.4b). Furthermore, we apply (4.26b) to order the derivatives
in the Vlasov error terms as needed, and note that the leading curvature terms
arise from this procedure.

Finally, for (6.2), we recall that $v^{0}$, $\langle v\rangle_{G}$ and
$\mathbf{B}f_{FLRW}$ are bounded by $a^{-c\sqrt{\varepsilon}}$ on the support
of $f_{FLRW}$ since the latter has bounded support with respect to $\gamma$
and one has (4.3c). Using (4.11) for the horizontal term in $\bm{X}$ then
directly leads to the stated bound. ∎

###  6.2. Vlasov energy estimates

Recalling our conventions on $t$ and $\omega$ from the start of this section,
we now start by proving the Vlasov integral energy estimate at order $0$:

######  Lemma 6.4 (Zero order Vlasov energy estimates).

The following estimate holds:

(6.5a) $\displaystyle a(t)^{\omega}\mathfrak{E}^{(0)}_{1,0}(f,t)\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}a(s)^{-3+\frac{\omega}{2}}\cdot
a(s)^{\omega}\mathfrak{E}^{(0)}_{1,0}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-3-c\sqrt{\varepsilon}+\frac{\omega}{2}}\left(\mathfrak{E}^{(0)}(\Sigma,s)+\mathfrak{E}^{(0)}(\phi,s)+a(s)^{4}\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}(M_{s})}^{2}\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-1-c\sqrt{\varepsilon}+\frac{\omega}{2}}\|G-\gamma\|_{L_{G}^{2}(M_{s})}^{2}\,ds$
(6.5b) $\displaystyle\mathfrak{E}^{(0)}_{1,0}(f,t)\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-1-c\sigma}+\varepsilon^{\frac{1}{8}}a(s)^{-3}\right)\mathfrak{E}^{(0)}_{1,0}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\varepsilon^{-\frac{1}{8}}a(s)^{-3-c\sqrt{\varepsilon}}\left(\mathfrak{E}^{(0)}(\Sigma,s)+\mathfrak{E}^{(0)}(\phi,s)\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-1-c\sqrt{\varepsilon}}\left(\|\Gamma-\hat{\Gamma}\|_{L_{G}^{2}(M_{s})}^{2}+\|G-\gamma\|_{L_{G}^{2}(M_{s})}^{2}\right)\,ds$

###### Proof.

We first prove the unscaled estimate (6.5b) before sketching the necessary
adaptations to arrive at (6.5a):

(6.6) $\displaystyle-\partial_{t}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)=$
$\displaystyle\,\int_{TM}\langle
v\rangle^{2}_{G}\left(-2\frac{\partial_{t}\langle v\rangle_{G}}{\langle
v\rangle_{G}}-6N\frac{\dot{a}}{a}\right)\lvert
f-f_{FLRW}\rvert^{2}\,{\text{{vol}}}_{\underline{G}}$
$\displaystyle\,-2\int_{TM}\langle
v\rangle_{G}^{2}(f-f_{FLRW})\cdot\partial_{t}f\,{\text{{vol}}}_{\underline{G}}$

The first line can be bounded by $\lesssim\varepsilon
a^{-3}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)$ with (4.23) and (3.12a). We insert
the rescaled Vlasov equation (2.21a) in the second line and expand as follows:

$\displaystyle\,-2\int_{TM}\langle
v\rangle_{G}^{2}(f-f_{FLRW})\cdot\partial_{t}f\,{\text{{vol}}}_{\underline{G}}$
(6.7a) $\displaystyle=$ $\displaystyle\,-2\int_{TM}\langle
v\rangle_{G}^{2}(f-f_{FLRW})\cdot\bm{X}(f-f_{FLRW})\,{\text{{vol}}}_{\underline{G}}$
(6.7b) $\displaystyle\,-2\int_{TM}\langle
v\rangle_{G}^{2}(f-f_{FLRW})\bm{X}f_{FLRW}$

(6.7a) is bounded by $\varepsilon a^{-3}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)$ due
to core energy mechanism from Lemma 6.2. Regarding (6.7b), we apply Lemma 5.5
for $l=0$ to (6.2) for $L=0$ and collect:

$\displaystyle\|\bm{X}f_{FLRW}\|_{L^{2}_{1,\underline{G}}(TM)}\lesssim$
$\displaystyle\,a^{-1-c\sqrt{\varepsilon}}\left[\|G-\gamma\|_{L^{2}_{G}(M)}+\sqrt{\mathfrak{E}^{(0)}_{1,0}(f,\cdot)}\right]$
$\displaystyle\,+a^{-3-c\sqrt{\varepsilon}}\left[\sqrt{\mathfrak{E}^{(0)}(\Sigma,\cdot)}+\sqrt{\mathfrak{E}^{(0)}(\phi,\cdot)}+a^{4}\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}(M)}\right]$

Putting all of the above together and applying the Young inequality on each
summand, we obtain

$\displaystyle-\partial_{t}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)\lesssim$
$\displaystyle\,\left(a^{-1-c\sigma}+\varepsilon^{\frac{1}{8}}a^{-3}\right)\mathfrak{E}^{(0)}_{1,0}(f,\cdot)+\varepsilon^{-\frac{1}{8}}a^{-3-c\sqrt{\varepsilon}}\left(\mathfrak{E}^{(0)}(\Sigma,\cdot)+\mathfrak{E}^{(0)}(\phi,\cdot)+a^{4}\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}}^{2}\right)$
$\displaystyle\,+a^{-1-c\sqrt{\varepsilon}}\|G-\gamma\|_{L^{2}}^{2}$

and thus the result after applying the Gronwall lemma.

For (6.5a), we argue along the same lines, with the estimates for (6.6) and
(6.7a) treated identically and where
$(-\partial_{t}a^{\omega})\mathfrak{E}^{(0)}(f,\cdot)$ can be dropped since it
is nonpositive. Regarding the terms in (6.7b), we apply the same estimates,
but now distribute the additional Vlasov energy scaling to mitigate the
divergence of the other energies:

$\displaystyle\,a^{-3}\cdot\left(a^{\frac{\omega}{4}}\|\bm{X}f_{FLRW}\|_{L^{2}_{\underline{G}}}\right)\cdot\left(a^{\frac{3\omega}{4}}\sqrt{\mathfrak{E}_{1,0}^{(0)}(f,\cdot)}\right)\,$
$\displaystyle\lesssim$
$\displaystyle\,a^{-3+\frac{\omega}{2}}\left(a^{\omega}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)\right)+a^{-3-c\sqrt{\varepsilon}+\frac{\omega}{2}}\left(\mathfrak{E}^{(0)}(\Sigma,\cdot)+\mathfrak{E}^{(0)}(\phi,\cdot)+a^{4}\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}}\right)$
$\displaystyle\,+a^{-1-c\sqrt{\varepsilon}+\frac{\omega}{2}}\|G-\gamma\|_{L^{2}_{G}}^{2}\,.$

∎

######  Lemma 6.5 (Vertical Vlasov energy estimates).

Let $\omega^{\prime}\in\\{0,\omega\\}$ and $\beta\in\\{0,4\\}$. Then:

(6.8) $\displaystyle a(t)^{\beta+\omega^{\prime}}$
$\displaystyle\,\mathfrak{E}^{(L)}_{1,0}(f,t)\lesssim\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-1-c\sigma}+\varepsilon
a(s)^{-3}\right)a(s)^{\beta+{\omega^{\prime}}}\mathfrak{E}^{(L)}_{1,0}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-1-c\sigma-{\omega^{\prime}}}\cdot
a(s)^{\beta+2{\omega^{\prime}}}\mathfrak{E}_{1,\leq 1}^{(\leq L)}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-3-c\sqrt{\varepsilon}+\beta+{\omega^{\prime}}}\left(\mathfrak{E}^{(0)}(\phi,s)+\mathfrak{E}^{(0)}(\Sigma,s)\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-1-c\sqrt{\varepsilon}+\beta+{\omega^{\prime}}}\left(\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}(M_{s})}^{2}+\|G-\gamma\|_{L^{2}_{G}(M_{s})}^{2}\right)\,ds$

###### Proof.

We only prove the case of $\omega^{\prime}=\beta=0$ in full since the other
cases are proven similarly, keeping in mind that
$-\partial_{t}(a^{\beta+{\omega^{\prime}}})\mathfrak{E}^{(L)}_{1,0}\leq 0$.
Since $\partial_{t}$ and $\mathbf{B}_{i}$ commute and since
$\underline{\nabla}_{vert}^{L}\xi=\mathbf{B}^{L}\xi$ holds, we get the
following:

$\displaystyle-\partial_{t}\mathfrak{E}^{(L)}_{1,0}(f,\cdot)=$
$\displaystyle\,-\int_{TM}\langle
v\rangle_{G}^{2}\left(2\frac{\partial_{t}\langle v\rangle_{G}}{\langle
v\rangle_{G}}+2L\frac{\partial_{t}v^{0}}{v^{0}}+6N\frac{\dot{a}}{a}\right)\lvert\underline{\nabla}_{vert}^{L}(f-f_{FLRW})\rvert^{2}_{\underline{G}_{0}}{\text{{vol}}}_{\underline{G}}$
$\displaystyle\,-\int_{TM}\langle
v\rangle_{G}^{2}(v^{0})^{2L}\partial_{t}G\ast_{\underline{G}}\underline{\nabla}_{vert}^{L}(f-f_{FLRW})\ast_{\underline{G}}^{L}\underline{\nabla}_{vert}(f-f_{FLRW})\,{\text{{vol}}}_{\underline{G}}$
$\displaystyle\,-2\int_{TM}\langle
v\rangle_{G}^{2}\langle\underline{\nabla}_{vert}(f-f_{FLRW}),\underline{\nabla}_{vert}\partial_{t}f\rangle_{\underline{G}_{0}}\,{\text{{vol}}}_{\underline{G}}$

The first two lines are bounded by $\varepsilon
a^{-3}\mathfrak{E}^{(1)}_{1,0}(f,\cdot)$ up to constant, with the first line
due to (4.23) and (3.13), and with the second since
$\lvert\partial_{t}G\rvert_{G}\lesssim\varepsilon a^{-3}$ holds by (4.1b) and
(3.13) applied to (2.22a). Regarding the final line, we insert (6.1a):

$\displaystyle\,-2\int_{TM}\langle
v\rangle_{G}^{2}\langle\underline{\nabla}_{vert}^{L}(f-f_{FLRW}),\underline{\nabla}_{vert}^{L}\partial_{t}f\rangle_{\underline{G}_{0}}\,{\text{{vol}}}_{\underline{G}}$
(6.9a) $\displaystyle\lesssim$ $\displaystyle\,-2\int_{TM}\langle
v\rangle_{G}^{2}(v^{0})^{2L}\langle\underline{\nabla}_{vert}^{L}(f-f_{FLRW}),\underline{\nabla}_{vert}^{L}\bm{X}f_{FLRW})\rangle_{\underline{G}}{\text{{vol}}}_{\underline{G}}$
(6.9b) $\displaystyle+\int_{TM}\langle
v\rangle_{G}^{2}\left[a^{-1}\left(1+\lvert
v\rvert_{G}^{2}\lvert\text{{Ric}}[G]\rvert_{G}\right)\lvert\underline{\nabla}_{vert}^{L-1}\underline{\nabla}_{hor}(f-f_{FLRW})\rvert_{\underline{G}_{0}}(v^{0})\right.$
(6.9c) $\displaystyle\quad\left.+\left(a^{-1}\lvert\nabla
N\rvert_{G}+\frac{\dot{a}}{a}\lvert
N\rvert+a^{-3}\lvert\Sigma\rvert_{G}\right)\lvert\underline{\nabla}_{vert}^{L}(f-f_{FLRW})\rvert_{\underline{G}_{0}}\right]\lvert\underline{\nabla}_{vert}^{L}(f-f_{FLRW})\rvert_{\underline{G}_{0}}\,{\text{{vol}}}_{\underline{G}}$
(6.9d)
$\displaystyle\,+\left(\|\mathfrak{F}_{L,0}\|_{L^{2}_{1,\underline{G}}}+\|\underline{\nabla}_{vert}^{L}\bm{X}f_{FLRW}\|_{L^{2}_{1,\underline{G}}}\right)\sqrt{\mathfrak{E}^{(L)}_{1,0}(f,\cdot)}$

Going through the right hand side line by line, (6.9a) can be bounded by
$\varepsilon a^{-3}\mathfrak{E}^{(L)}_{1,0}(f,\cdot)$ by Lemma 6.2. For
(6.9b), we apply (4.3e) and (4.6) on the prefactor to estimate it as follows:

$\lvert\eqref{eq:vlasov-vert-en2}\rvert\lesssim
a^{-1-c\sqrt{\varepsilon}}\left(\sqrt{\mathfrak{E}^{(L)}_{1,1}(f,\cdot)}+\|\underline{\nabla}_{vert}^{L-1}\underline{\nabla}_{hor}f_{FLRW}\|_{L^{2}_{1,\underline{G}_{0},m}}\right)\sqrt{\mathfrak{E}^{(L)}_{1,0}(f,\cdot)}$

For the reference term, we apply (4.10a) for $K=1$, and obtain the bound

$\lvert\eqref{eq:vlasov-vert-en2}\rvert\lesssim
a^{-1-c\sqrt{\varepsilon}}\mathfrak{E}^{(L)}_{1,1}(f,\cdot)+a^{-1-c\sqrt{\varepsilon}}\left(\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}(M_{t})}^{2}+\|G-\gamma\|_{L^{2}_{G}(M_{t})}^{2}\right)+a^{-1-c\sqrt{\varepsilon}}\mathfrak{E}^{(L)}_{1,0}(f,\cdot)\,.$

For (6.9c), we apply (4.6), (3.13) and (4.1b) to bound the terms by
$\varepsilon a^{-3}\mathfrak{E}^{(L)}_{1,0}$. Finally, we apply (6.1b) and
(6.2) to bound (6.9d), where we insert Lemma 5.5 at order $l=0$ for the
occuring lapse energy terms and then apply the Young inequality. The integral
estimate (6.8) then follows by integrating.
For $\omega^{\prime}=\omega$, we again use the additional scaling to prevent
otherwise strongly divergent energy terms to diverge more than $a^{-3}$ as in
the zero order case. For the Vlasov energy gaining a horizontal derivative, we
distribute the scaling as follows:

$a^{\omega}\sqrt{\mathfrak{E}^{(L)}_{1,0}(f,\cdot)}\sqrt{\mathfrak{E}^{(L)}_{1,1}(f,\cdot)}=\sqrt{a^{\omega}\mathfrak{E}^{(L)}_{1,0}(f,\cdot)}\cdot\sqrt{a^{-\omega}\cdot
a^{2\omega}\mathfrak{E}^{(L)}_{1,1}(f,\cdot)}$

The respective proofs with $\beta=4$ instead of $\beta=0$ are identical. ∎

######  Lemma 6.6 (Mixed Vlasov energy estimates).

Let $0<K<L$ and $\beta\in\\{0,4\\}$. Then, the following estimates hold:

(6.10) $\displaystyle a(t)^{\beta+(K+1){\omega}}$
$\displaystyle\,\mathfrak{E}^{(L)}_{1,K}(f,t)\lesssim\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-3+\frac{\omega}{2}}+\varepsilon^{\frac{1}{8}}a(s)^{-3}\right)a(s)^{\beta+(K+1){\omega}}\mathfrak{E}^{(L)}_{1,K}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-1-{\omega}}\cdot
a(s)^{\beta+(K+2){\omega}}\mathfrak{E}^{(L)}_{1,K+1}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\left(a(s)^{-1-c\sqrt{\varepsilon}+\frac{\omega}{2}}+\varepsilon^{2}a(s)^{-3-c\sqrt{\varepsilon}+\frac{\omega}{2}}\right)\cdot
a(s)^{\beta+((K-1)+1){\omega}}\mathfrak{E}^{(\leq L)}_{1,\leq K-1}(f,s)$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-3-c\sqrt{\varepsilon}+\frac{\omega}{2}}a(s)^{\beta}\left(\mathfrak{E}^{(\leq
K)}(\Sigma,s)+\mathfrak{E}^{(\leq K)}(\phi,s)\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\left(a(s)^{-1-c\sqrt{\varepsilon}+(K+1){\omega}}+\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}+(K+1){\omega}}\right)\cdot
a(s)^{\beta}\mathfrak{E}^{(\leq K-1)}(\text{{Ric}},s)\,ds$ (6.11)
$\displaystyle\mathfrak{E}^{(L)}_{1,K}(f,t)\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-1-c\sigma}+\varepsilon^{\frac{1}{8}}a(s)^{-3}\right)\mathfrak{E}^{(L)}_{1,K}(f,s)+a(s)^{-1}\mathfrak{E}^{(L)}_{1,K+1}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\left(\varepsilon^{\frac{7}{8}}a(s)^{-3-c\sqrt{\varepsilon}}+a(s)^{-1-c\sqrt{\varepsilon}}\right)\left(\mathfrak{E}^{(L)}_{1,K-1}(f,s)+\mathfrak{E}^{(\leq
L-1)}_{1,\leq K-1}(f,s)\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\varepsilon^{-\frac{1}{8}}a(s)^{-3-c\sqrt{\varepsilon}}\left(\mathfrak{E}^{(\leq
K)}(\Sigma,s)+\mathfrak{E}^{(\leq K)}(\phi,s)\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\left(a(s)^{-1-c\sqrt{\varepsilon}}+\varepsilon^{\frac{15}{8}}a(s)^{-3-c\sqrt{\varepsilon}}\right)\mathfrak{E}^{(\leq
K-1)}(\text{{Ric}},s)\,ds$

For $K<10$, the curvature energies can be dropped.

###### Proof.

We again focus on proving the unscaled energy inequality (6.11), and touch on
how the scale factors are distributed for (6.10) at the end. One computes:

$\displaystyle\,-\partial_{t}\int_{TM}\lvert
v\rvert_{G}^{2}\lvert\underline{\nabla}^{L-K}_{vert}\underline{\nabla}_{hor}^{K}f\rvert_{\underline{G}_{0}}^{2}\,{\text{{vol}}}_{\underline{G}}$
(6.12a) $\displaystyle=$ $\displaystyle\,-\int_{TM}\langle
v\rangle_{G}^{2}\left(2\frac{\partial_{t}\langle v\rangle_{G}}{\langle
v\rangle_{G}}+2(L-K)\frac{\partial_{t}v^{0}}{v^{0}}+6N\frac{\dot{a}}{a}\right)\lvert\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\rvert^{2}_{\underline{G}_{0}}{\text{{vol}}}_{\underline{G}}$
(6.12b) $\displaystyle\,-\int_{TM}\langle
v\rangle_{G}^{2}(v^{0})^{2(L-K)}(\partial_{t}\underline{G}^{-1})\ast_{\underline{G}}\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\ast_{\underline{G}}\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\,{\text{{vol}}}_{\underline{G}}$
(6.12c) $\displaystyle\,-2\int_{TM}\langle
v\rangle_{G}^{2}\langle[\partial_{t},\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}]f,\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\rangle_{\underline{G}_{0}}\,{\text{{vol}}}_{\underline{G}}$
(6.12d) $\displaystyle\,-2\int_{TM}\langle
v\rangle_{G}^{2}\langle\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}\partial_{t}f,\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\rangle_{\underline{G}_{0}}\,{\text{{vol}}}_{\underline{G}}$

(6.12a)-(6.12b) can be bounded by $\varepsilon
a^{-3}\mathfrak{E}^{(L)}_{1,K}(f,\cdot)$ up to constant as in the purely
vertical case. Regarding (6.12c), (4.24b) implies the following:

$\displaystyle\,\left\lvert\langle[\partial_{t},\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}]f,\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\rangle_{\underline{G}_{0}}\right\rvert$
$\displaystyle\lesssim$
$\displaystyle\,\lvert\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\rvert_{\underline{G}_{0}}\cdot\left[\frac{\lvert
v\rvert_{G}}{v^{0}}\lvert\underline{\nabla}_{vert}^{L-K+1}f\rvert_{\underline{G}_{0}}\lvert\nabla^{K-2}\partial_{t}\text{{Ric}}[G]\rvert_{G}\right.$
$\displaystyle\,\left.+v^{0}\lvert\nabla^{K-1}\partial_{t}\text{{Ric}}[G]\rvert_{G}\lvert\underline{\nabla}_{vert}^{L-K-1}\underline{\nabla}_{hor}f\rvert_{\underline{G}_{0}}+\frac{\lvert
v\rvert_{G}}{v^{0}}\lvert\partial_{t}\Gamma[G]\rvert_{G}\lvert\underline{\nabla}_{vert}^{L-K+1}\underline{\nabla}_{hor}^{K-1}f\rvert_{\underline{G}_{0}}\right.$
$\displaystyle\,\left.+(v^{0})^{L-K}\lvert\Xi_{L-K,K,t}f\rvert_{\underline{G}}\right]$

Recall that, by integration by parts and the Young inequality, one has

$a^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(l-1)}(\Sigma,\cdot)\lesssim
a^{-3}\mathfrak{E}^{(l)}(\Sigma,\cdot)+a^{-3-2c\sqrt{\varepsilon}}\mathfrak{E}^{(l-2)}(\Sigma,\cdot)$

and the analogous estimates for $\mathfrak{E}^{(l-1)}(N,\cdot)$ and
$\mathfrak{E}^{(l-1)}(\text{{Ric}},\cdot)$. Applying (4.3b) and (3.13) to
(2.22e) and (2.22d), as well as updating $c>0$, one then gets

$\displaystyle\,\|\partial_{t}\text{{Ric}}[G]\|_{H^{l-2}_{G}}+\|\partial_{t}\Gamma[G]\|_{H^{l-1}_{G}}$
$\displaystyle\lesssim$
$\displaystyle\,a^{-3}\left(\sqrt{\mathfrak{E}^{(l)}(\Sigma,\cdot)}+\sqrt{\mathfrak{E}^{(l)}(N,\cdot)}\right)+a^{-3-c\sqrt{\varepsilon}}\left(\sqrt{\mathfrak{E}^{(\leq
l-2)}(\Sigma,\cdot)}+\sqrt{\mathfrak{E}^{(\leq l-2)}(N,\cdot)}\right)$
$\displaystyle\,+\varepsilon
a^{-3-c\sqrt{\varepsilon}}\sqrt{\mathfrak{E}^{(\leq
l-2)}(\text{{Ric}},\cdot)}$

as well as

$\|\partial_{t}\text{{Ric}}[G]\|_{C^{10}_{G}(M)}+\|\partial_{t}\Gamma\|_{C^{11}_{G}(M)}\lesssim\varepsilon
a^{-3-c\sqrt{\varepsilon}}\,.$

Further, the supremum norm over $\xi=f$ in (4.25b) can be bounded by
$a^{-c\sqrt{\varepsilon}}$ due to (4.15) whenever it occurs. Altogether, this
implies

$\displaystyle\lvert\eqref{eq:vlasov-interim-3}\rvert\lesssim$
$\displaystyle\,\varepsilon
a^{-3-c\sqrt{\varepsilon}}\left(\sqrt{\mathfrak{E}^{(L)}_{1,K-1}(f,\cdot)}+\sqrt{\mathfrak{E}^{(\leq
L-1)}_{1,\leq K-1}(f,\cdot)}\right)$
$\displaystyle\,+a^{-3-c\sqrt{\varepsilon}}\left(\sqrt{\mathfrak{E}^{(\leq
K)}(\Sigma,\cdot)}+\sqrt{\mathfrak{E}^{(\leq
K)}(N,\cdot)}+\varepsilon\sqrt{\mathfrak{E}^{(\leq
K-2)}(\text{{Ric}},\cdot)}\right)\,.$

Considering (6.12d), we can apply (6.3a) and using $\lvert v\rvert_{G}\leq
v^{0}$ to obtain the following:

$\displaystyle\eqref{eq:vlasov-interim-4}+$ $\displaystyle\,2\int_{TM}\langle
v\rangle_{G}^{2}\langle\bm{X}\underline{\nabla}^{L-K}_{vert}\underline{\nabla}^{K}_{hor}f,\underline{\nabla}^{L-K}_{vert}\underline{\nabla}^{K}_{hor}f\rangle{\text{{vol}}}_{\underline{G}}$
(6.13a) $\displaystyle\lesssim$ $\displaystyle\,\int_{TM}\langle
v\rangle_{G}^{2}\left[a^{-1}\lvert
v\rvert_{G}\lvert\nabla^{K-1}\text{{Ric}}[G]\rvert_{G}\lvert\underline{\nabla}_{vert}^{L-K+1}f\rvert_{\underline{G}_{0}}+a^{-1}\lvert\underline{\nabla}_{vert}^{L-K-1}\underline{\nabla}_{hor}^{K+1}f\rvert_{\underline{G}_{0}}\right.$
(6.13b) $\displaystyle\,+a^{-1}(v^{0})^{L-K}\lvert\nabla
N\rvert_{G}\lvert\mathbf{B}\underline{\nabla}_{vert}^{L-K-1}\underline{\nabla}_{hor}^{K}f\rvert_{\underline{G}}$
(6.13c) $\displaystyle\,+a^{-3}(v^{0})^{L-K}\lvert
N\rvert\lvert\mathbf{B}\underline{\nabla}_{vert}^{L-K-1}\underline{\nabla}_{hor}^{K}f\rvert_{\underline{G}}+a^{-3}(v^{0})^{L-K}\lvert\Sigma\rvert_{G}\lvert\mathbf{B}\underline{\nabla}_{vert}^{L-K-1}\underline{\nabla}_{hor}^{K}f\rvert_{\underline{G}}$
(6.13d)
$\displaystyle\,+a^{-3}(v^{0})^{L-K+1}\lvert\nabla\Sigma\rvert_{G}\lvert\mathbf{B}\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K-1}f\rvert_{\underline{G}}+\left.(v^{0})^{L-K}\lvert\mathfrak{F}_{L,K}\rvert_{\underline{G}}\right]\cdot\lvert\underline{\nabla}^{L-K}_{vert}\underline{\nabla}^{K}_{hor}f\rvert_{\underline{G}_{0}}\,{\text{{vol}}}_{\underline{G}}$

The extra term on the left hand side is dealt with using Lemma 6.2. Regarding
(6.13b)-(6.13c), note that one schematically has

$\mathbf{B}\underline{\nabla}_{vert}^{L-K-1}\underline{\nabla}_{hor}^{K}f=\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f+\Gamma[G]\ast\underline{\nabla}^{L-1}f,$

where the second term always contains horizontal derivatives except for the
case $K=1$. Then, one expands the term as follows:

$\lvert(v^{0})^{L-1}\Gamma[G]\ast\underline{\nabla}_{vert}^{L-1}f\rvert_{G}\lesssim
a^{-c\sqrt{\varepsilon}}\left(\lvert\underline{\nabla}^{L-1}_{vert}(f-f_{FLRW})\rvert_{\underline{G}_{0}}+\lvert\underline{\nabla}_{vert}^{L-1}f_{FLRW}\rvert_{\underline{G}_{0}}\right)\,.$

The former can be related back to $\mathfrak{E}^{(L-1)}_{1,0}(f,\cdot)$ in
estimation and the latter is simply bounded uniformly by
$a^{-c\sqrt{\varepsilon}}$. Else, this lower order term can be bounded by
$\sqrt{\mathfrak{E}^{(L-1)}_{1,\leq K-1}(f,\cdot)}$ in estimation. We can
treat
$\mathbf{B}\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K-1}f$
identically, except that the case of only vertical derivatives appearing in
error terms now happens for $K\leq 2$. Thus and applying (4.3e), (4.1b) and
(4.15) as well as (3.12a) for the lapse, we obtain

$\displaystyle\eqref{eq:vlasov-interim-4}\lesssim$
$\displaystyle\,\left(a^{-1-c\sigma}+\varepsilon^{\frac{1}{8}}a^{-3}\right)\mathfrak{E}^{(L)}_{1,K}(f,\cdot)+a^{-1}\mathfrak{E}^{(L)}_{1,K+1}(f,\cdot)+\varepsilon^{\frac{15}{8}}a^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(L)}_{1,K-1}(f,\cdot)$
$\displaystyle\,+\varepsilon a^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-1)}_{1,\leq K-1}(f,\cdot)+\varepsilon
a^{-1-c\sqrt{\varepsilon}}\mathfrak{E}^{(K-1)}(\text{{Ric}},\cdot)$
$\displaystyle\,+a^{-3-c\sqrt{\varepsilon}}\left({\mathfrak{E}^{(\leq
1)}(\Sigma,\cdot)}+{a^{4}\mathfrak{E}^{(1)}(N,\cdot)}+\mathfrak{E}^{(0)}(N,\cdot)\right)$
$\displaystyle\,+\|(v^{0})^{(L-K)}\mathfrak{F}_{L,K}\|_{L^{2}_{1,\underline{G}}}\sqrt{\mathfrak{E}^{(L)}_{1,K}(f,\cdot)}$

Note that the penultimate line comes from the discussed error terms arising
from replacing $\mathbf{B}$ with $\underline{\nabla}_{vert}$ at low horizontal
orders.
The statement now follows by inserting (6.3b), combining the bounds for
(6.12a)-(6.12d), inserting Lemma 5.5 and integrating. However, the terms in
(6.3b) dominate the low order shear and lapse energies present above. (6.11)
then follows by inserting (6.3b), combining all the discussed bounds for
(6.12a)-(6.12d) (with the latter combining all bounds for (6.13a)-(6.13d)) and
integrating.

For the scaled version, the main point is that one needs to be careful to
ensure that $\mathfrak{E}^{(L)}_{1,K-1}(f,\cdot)$ terms do not enter at a
power worse than $a^{-3}$, which is the case in the unscaled version. To this
end, we distribute the scaling as follows:

$a^{-3-c\sqrt{\varepsilon}+(K+1){\omega}}\sqrt{\mathfrak{E}^{(L)}_{1,K}(f,\cdot)}\sqrt{\mathfrak{E}^{(L)}_{1,K-1}(f,\cdot)}=a^{-3+\frac{\omega}{2}}\sqrt{a^{(K+1)\omega}\mathfrak{E}^{(L)}_{1,K}(f,\cdot)}\sqrt{a^{K\omega-2c\sqrt{\varepsilon}}\mathfrak{E}^{(L)}_{1,K}(f,\cdot)}$

Then, we apply the Young inequality and update $c$. We note that one never
encounters more than $K$ horizontal derivatives in the error terms and thus
can never lose scaling. For the non-Vlasov energies, we distribute scaling as
in the previous arguments. Again, replacing $\beta=0$ with $\beta=4$ goes
through identically. ∎

######  Lemma 6.7 (Horizontal Vlasov energy estimates).

Let $\beta\in\\{0,4\\}$. Then:

(6.14) $\displaystyle a(t)^{\beta+(L+1)\omega}$
$\displaystyle\mathfrak{E}^{(L)}_{1,L}(f,t)\lesssim\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-3+\frac{\omega}{2}}+\varepsilon^{\frac{1}{8}}a(s)^{-3}\right)a(s)^{\beta+(L+1)\omega}\mathfrak{E}^{(L)}_{1,L}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}+\frac{\omega}{2}}\cdot
a(s)^{\beta+L{\omega}}\mathfrak{E}^{(\leq L)}_{1,\leq L-1}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-3-c\sqrt{\varepsilon}+\omega}\cdot
a(s)^{\beta}\left(\mathfrak{E}^{(\leq L)}(\Sigma,s)+\mathfrak{E}^{(\leq
L)}(\phi,s)\right)\,ds$ $\displaystyle\,+\int_{t}^{t_{0}}\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}+(L+1){\omega}}\left(a(s)^{4+\beta}\mathfrak{E}^{(L-1)}(\text{{Ric}},s)+a(s)^{\beta}\mathfrak{E}^{(\leq
L-2)}(\text{{Ric}},s)\right)\,ds$

(6.15) $\displaystyle\mathfrak{E}^{(L)}_{1,L}(f,t)\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-1-c\sigma}+\varepsilon^{\frac{1}{8}}a(s)^{-3}\right)\mathfrak{E}^{(L)}_{1,L}(f,s)+\varepsilon^{\frac{15}{8}}a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(L)}_{1,L-1}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\varepsilon^{-\frac{1}{8}}a(s)^{-3-c\varepsilon^{\frac{1}{8}}}\left(\mathfrak{E}^{(\leq
L)}(\Sigma,s)+\mathfrak{E}^{(\leq L)}(\phi,s)\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\varepsilon^{\frac{7}{8}}a(s)^{-3-c\sqrt{\varepsilon}}\cdot\left(a(s)^{4}\mathfrak{E}^{(L-1)}(\text{{Ric}},s)+\mathfrak{E}^{(\leq
L-2)}(\text{{Ric}},s)\right)\,ds$

For $L\leq 10$, the curvature energies can be dropped. For $L=1$, one in
particular has

(6.16) $\displaystyle a(t)^{4+2\omega}\mathfrak{E}^{(1)}_{1,1}(f,t)\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-3+\frac{\omega}{2}}+\varepsilon^{\frac{1}{8}}a(s)^{-3}\right)\cdot
a(s)^{4+2\omega}\mathfrak{E}^{(1)}_{1,1}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}+\frac{\omega}{2}}\cdot
a(s)^{4+\omega}\mathfrak{E}^{(1)}_{1,0}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-3-c\sqrt{\varepsilon}}\cdot
a(s)^{4}\left(\mathfrak{E}^{(1)}(\Sigma,s)+\mathfrak{E}^{(1)}(\phi,s)\right)\,ds$
(6.17) $\displaystyle\mathfrak{E}^{(1)}_{1,1}(f,t)\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-1-c\sigma}+\varepsilon^{\frac{1}{8}}a(s)^{-3}\right)\mathfrak{E}^{(1)}_{1,1}(f,s)+\varepsilon^{\frac{15}{8}}a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(1)}_{1,0}(f,s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\varepsilon^{-\frac{1}{8}}a(s)^{-3-c\varepsilon^{\frac{1}{8}}}\left(\mathfrak{E}^{(\leq
1)}(\Sigma,s)+\mathfrak{E}^{(\leq 1)}(\phi,s)\right)\,ds$

###### Proof.

The proof of these estimates is analogous to that of Lemma 6.6 since (6.4a) is
just a simpler version of (6.3a), and the commutator formula (4.24b) is also
used for $M_{1}=0$ in a similar fashion to the previous proof. As a difference
in estimation, we note that the high order curvature term present in (6.4a) is
now of order $L-1$, and thus needs to be thought of as being of order $L+1$ in
the metric, i.e. of higher order than the Vlasov energy itself. Consequently,
the resulting energy controlling this term needs to be additionally scaled to
allow the argument to close, which makes the resulting energy term borderline.
∎

##  7\. Improving the Boootstrap assumptions

In this section, we combine the energy estimates from Sections 5, and 6 to
improve the bootstrap assumption (3.12a) – recall that (3.12b) is already
improved due to (4.15). To this end, we first derive improved energy bounds
for the spacetime and scalar field variables and then use these to derive
bounds for Vlasov energies. We then translate these bounds to Sobolev norm
bounds, and finally to bounds on $\mathcal{C}$.
In the following, we again assume $t\in(t_{Boot},t_{0}]$ and that
$\omega\gg\sigma$ is a suitably small constant.

To deal with linear borderline terms in the energy estimates for the shear
(see Lemma 5.10) and Bel-Robinson variables (see Lemma 5.9), we need to extend
[FU23, Lemma 7.1] to ensure that the error terms essentially have favourable
sign even for flat and spherical spatial geometry:

######  Lemma 7.1.

Let $L\in 2\mathbb{N}$. Then, the following estimate is satisfied:

(7.1) $\displaystyle\,\int_{M}\left[16\pi
C^{2}a^{-3}(N+1)\langle\Delta^{\frac{L}{2}}\mathcal{E},\Delta^{\frac{L}{2}}\Sigma\rangle_{G}+8\pi
C^{2}\frac{\dot{a}}{a}(N+1)\lvert\Delta^{\frac{L}{2}}\Sigma\rvert_{G}^{2}+6\frac{\dot{a}}{a}(N+1)\lvert\Delta^{\frac{L}{2}}\mathcal{E}\rvert_{G}^{2}\right]{\text{{vol}}}_{G}$
$\displaystyle\begin{cases}\geq 0&\kappa\leq 0\\\ \gtrsim-a^{-1}\left[4\pi
C^{2}\mathfrak{E}^{(L)}(\Sigma,\cdot)+\mathfrak{E}^{(L)}(W,\cdot)\right]&\kappa>0\end{cases}$

###### Proof.

First, recall that Lemma 5.2 implies $N+1>0$. For $\kappa\leq 0$, we have
$a^{-3}\leq\frac{3}{4\pi C^{2}}\frac{\dot{a}}{a}$ by (2.3a) since the Vlasov
density is nonnegative. Then, the statement follows directly from this and the
Young inequality as in [FU23, Lemma 7.1]. Else, one has

$a^{-3}\leq\frac{3}{4\pi
C^{2}}\frac{\sqrt{\dot{a}^{2}+\kappa}}{a}\leq\frac{3}{4\pi
C^{2}}\left(\frac{\dot{a}}{a}+\sqrt{\kappa}a^{-1}\right)$

and thus the integrand on the left hand side in (7.1) is bounded from below by

$\gtrsim-a^{-1}\lvert\Delta^{\frac{L}{2}}\Sigma\rvert_{G}\lvert\Delta^{\frac{L}{2}}\mathcal{E}\rvert_{G}\,,$

implying the statement. ∎

To obtain bounds on the spacetime and scalar field energies, we need to
combine Vlasov energies at order $L$ with appropriate scaling as follows:

######  Definition 7.2 (Total scaled Vlasov energies).

For $0\leq L\leq 19$, we define

(7.2)
$\mathfrak{E}^{(L)}_{total,Vl}=\sum_{K=0}^{L}a^{(K+1)\omega}\mathfrak{E}^{(L)}_{1,K}(f,\cdot)\,.$

With this in hand, we can combine the estimates from Section 6 as follows:

######  Lemma 7.3 (Total scaled Vlasov energy estimate).

For $L=1$, one has

(7.3) $\displaystyle a(t)^{4}\mathfrak{E}^{(1)}_{total,Vl}$
$\displaystyle(t)\lesssim\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-1-{\omega}}+a(s)^{-3+\frac{\omega}{2}}+\varepsilon^{\frac{1}{8}}a(s)^{-3}\right)a(s)^{4}\mathfrak{E}^{(1)}_{total,Vl}(s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-3-c\sqrt{\varepsilon}+\omega}\cdot
a(s)^{4}\left(\mathfrak{E}^{(\leq 1)}(\phi,s)+\mathfrak{E}^{(\leq
1)}(\Sigma,s)\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-1-c\sqrt{\varepsilon}}\left(a(s)^{4+\omega}\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}(M_{s})}^{2}+a(s)^{4+\omega}\|G-\gamma\|_{L^{2}_{G}(M_{s})}^{2}\right)\,ds$

Let $L\geq 2$. Then, the following holds:

(7.4) $\displaystyle a(t)^{\beta}\mathfrak{E}^{(L)}_{total,Vl}(t)\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(a(s)^{-1-{\omega}}+a(s)^{-3+\frac{\omega}{2}}+\varepsilon^{\frac{1}{8}}a(s)^{-3}\right)a(s)^{\beta}\mathfrak{E}^{(L)}_{total,Vl}(s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}a(s)^{-3-c\sqrt{\varepsilon}+\omega}\cdot
a(s)^{\beta}\left(\mathfrak{E}^{(L)}(\phi,s)+\mathfrak{E}^{(L)}(\Sigma,s)\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}+(L+1)\omega}\cdot
a(s)^{4+\beta}\mathfrak{E}^{(L-1)}(\text{{Ric}},s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}+L\omega}\cdot
a(s)^{\beta}\mathfrak{E}^{(L-2)}(\text{{Ric}},s)\,ds$
$\displaystyle+\int_{t}^{t_{0}}a(s)^{-3-c\sqrt{\varepsilon}+\omega}\cdot
a(s)^{\beta}\left(\mathfrak{E}^{(\leq L-2)}(\phi,s)+\mathfrak{E}^{(\leq
L-2)}(\Sigma,s)\right)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\left(a(s)^{-1-\omega}+a(s)^{-3-c\sqrt{\varepsilon}+\frac{\omega}{2}}\right)a(s)^{\beta+(L-1)\omega}\mathfrak{E}^{(\leq
L-2)}_{1,\leq L-2}(f,s)\,ds$ $\displaystyle\,+\int_{t}^{t_{0}}\varepsilon
a(s)^{-3-c\sqrt{\varepsilon}+\omega}\mathfrak{E}^{(\leq
L-4)}(\text{{Ric}},s)\,ds$
$\displaystyle\,+\int_{t}^{t_{0}}\Big{(}a(s)^{-1-c\sqrt{\varepsilon}+\beta}+a(s)^{-3-c\sqrt{\varepsilon}+\beta+\frac{3}{2}{\omega}}\Big{)}\cdot$
$\displaystyle\,\qquad\cdot
a(s)^{\omega}\left(\|\Gamma-\hat{\Gamma}\|_{H^{1}_{G}(M_{s})}^{2}+\|G-\gamma\|_{H^{2}_{G}(M_{s})}^{2}\right)\,ds$

###### Proof.

This is a combination of Lemma 6.5, 6.6 and 6.7. Observe that the final line
arises from Lemma 6.5, while the remaining lines arise from Lemma 6.6 for
$K=L-1$ and Lemma 6.7. Furthermore, we immediately rewrote
$\mathfrak{E}^{(L-1)}_{1,\leq K}(f,\cdot)$ as follows: First, let
$\mathcal{K}_{L,K}>0$ be a suitable constant depending on $L$ and $K$ and
$\underline{\nabla}_{v}=v^{l}\underline{\nabla}_{l+3}$. For $L-1>K>0$, we
integrate by parts:

$\displaystyle a^{-c\sqrt{\varepsilon}}\mathfrak{E}^{(L-1)}_{1,K}(f,\cdot)=$
$\displaystyle\,-a^{-c\sqrt{\varepsilon}}\int_{TM}\langle
v\rangle_{G}^{2}(v^{0})^{2(L-K-1)}\langle\underline{\nabla}_{vert}^{L-K-2}\underline{\nabla}_{hor}^{K}f,\underline{\nabla}_{vert}^{L-K}\underline{\nabla}_{hor}^{K}f\rangle_{\underline{G}}\,{\text{{vol}}}_{\underline{G}}$
$\displaystyle\,-2a^{-c\sqrt{\varepsilon}}\int_{TM}\left((L-K-1)(v^{0})^{2(L-K-2)}\langle
v\rangle_{G}^{2}+(v^{0})^{2(L-K-1)}\right)\cdot$
$\displaystyle\,\qquad\qquad\qquad\cdot\langle\underline{\nabla}_{vert}^{L-K-2}\underline{\nabla}_{hor}^{K}f,\underline{\nabla}_{v}\underline{\nabla}_{vert}^{L-K-2}\underline{\nabla}_{hor}^{K}f\rangle\,{\text{{vol}}}_{\underline{G}}$

Integrating by parts again for the second term, one obtains

$\displaystyle
a^{-c\sqrt{\varepsilon}}\mathfrak{E}^{(L-1)}_{1,K}(f,\cdot)\leq$
$\displaystyle\,\frac{1}{2}\left(\mathfrak{E}^{(L)}_{1,K}(f,\cdot)+a^{-2c\sqrt{\varepsilon}}\mathfrak{E}^{(L-2)}_{1,K}(f,\cdot)\right)$
$\displaystyle\,+\mathcal{K}_{L,K}a^{-c\sqrt{\varepsilon}}\int_{TM}(v^{0})^{2(L-K-2)}\langle
v\rangle_{G}^{2}\lvert\underline{\nabla}_{vert}^{L-K-2}\underline{\nabla}_{hor}^{K}f\rvert_{\underline{G}}^{2}\,{\text{{vol}}}_{\underline{G}}$
$\displaystyle\lesssim$
$\displaystyle\,\mathfrak{E}^{(L)}_{1,K}(f,\cdot)+a^{-c\sqrt{\varepsilon}}\mathfrak{E}^{(L-2)}_{1,K}(f,\cdot)\,.$

For $K=L-1>1$, one can argue identically without an error term after the first
integration by parts since $\underline{\nabla}_{hor}v^{0}=0$, obtaining

$a^{-c\sqrt{\varepsilon}}\mathfrak{E}^{(L-1)}_{1,L-1}(f,\cdot)\lesssim\mathfrak{E}^{(L)}_{1,L}(f,\cdot)+a^{-c\sqrt{\varepsilon}}\mathfrak{E}^{(L-2)}_{1,L-2}(f,\cdot).$

For $K=0$, one can argue as above replacing $f$ with $f-f_{FLRW}$. Finally,
for $L-1=1=K$, we need to perform the following replacement after integrating
by parts:

$\displaystyle\,a^{-3-c\sqrt{\varepsilon}+\frac{5}{2}\omega+\beta}\mathfrak{E}^{(1)}_{1,1}(f,\cdot)$
$\displaystyle=$
$\displaystyle\,a^{-3-c\sqrt{\varepsilon}+\frac{3}{2}\omega+\beta}\left[-\int_{TM}\langle
v\rangle_{G}^{2}(f-f_{FLRW})\cdot({\underline{G}}^{-1})^{ij}\underline{\nabla}_{i}\underline{\nabla}_{j}f\,{\text{{vol}}}_{\underline{G}}\right.$
$\displaystyle\,-\left.\int_{TM}\langle
v\rangle_{G}^{2}(f-f_{FLRW})({\underline{G}}^{-1})^{ij}\underline{\nabla}_{i}\underline{\nabla}_{j}f_{FLRW}+\langle
v\rangle_{G}^{2}\lvert\underline{\nabla}_{hor}f_{FLRW}\rvert_{\underline{G}_{0}}^{2}\,{\text{{vol}}}_{G}\right]$
$\displaystyle\lesssim$
$\displaystyle\,a^{-3+\frac{\omega}{2}}\left(a^{\beta+3\omega}\mathfrak{E}^{(2)}_{1,2}(f,\cdot)+a^{-c\sqrt{\varepsilon}}\cdot
a^{\beta+\omega}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)\right)$
$\displaystyle\,+a^{-3-c\sqrt{\varepsilon}+\beta+\frac{5}{2}\omega}\|\underline{\nabla}_{hor}f_{FLRW}\|_{L^{2}_{1,\underline{G}}(TM)}^{2}+a^{3-c\sqrt{\varepsilon}+\beta+\frac{7}{2}\omega}\|\underline{\nabla}_{hor}^{2}f_{FLRW}\|_{L^{2}_{1,\underline{G}}(TM)}^{2}$
$\displaystyle\lesssim$
$\displaystyle\,a^{-3+\frac{\omega}{2}}\left(a^{\beta+3\omega}\mathfrak{E}^{(2)}_{1,2}(f,\cdot)+a^{-c\sqrt{\varepsilon}}\cdot
a^{\beta+\omega}\mathfrak{E}^{(0)}_{1,0}(f,\cdot)\right)$
$\displaystyle\,+a^{-3-c\sqrt{\varepsilon}+\beta+\frac{5}{2}\omega}\left(\|\Gamma-\hat{\Gamma}\|_{H^{1}_{G}(M)}^{2}+\|G-\gamma\|^{2}_{H^{2}_{G}(M)}\right)\,$

The final step follows by applying (4.10b). ∎

These combined estimates indicate that, along with scaled Vlasov energies, we
will also need do include scaled metric and Christoffel symbol errors at low
orders as well as scaled high order curvature terms at high order when
combining Vlasov estimates with the rest of the Einstein evolution. Recalling
the energy notation from Definition 3.4, this leads to the following total
energies:

######  Definition 7.4 (Total energies).

We define: $\displaystyle\mathfrak{E}^{(0)}_{total}=$
$\displaystyle\,\mathfrak{E}^{(0)}(\phi,\cdot)+\varepsilon^{\frac{1}{4}}\left(\mathfrak{E}^{(0)}(W,\cdot)+4\pi
C^{2}\mathfrak{E}^{(0)}(\Sigma,\cdot)\right)+a^{4}\mathfrak{E}^{(1)}(\phi,\cdot)+\varepsilon^{\frac{1}{2}}a^{4}\mathfrak{E}^{(1)}(\Sigma,\cdot)$
$\displaystyle\,+\varepsilon^{\frac{1}{4}}\mathfrak{E}^{(0)}_{total,Vl}+\varepsilon^{\frac{1}{2}}a^{4}\mathfrak{E}^{(1)}_{total,Vl}+a^{\omega}\|G-\gamma\|_{L^{2}_{G}(M)}^{2}+a^{4+\omega}\|\Gamma-\hat{\Gamma}\|_{L^{2}_{G}(M)}$
and, for $L\in 2\mathbb{N},2\leq L\leq 18$ (7.5a)
$\displaystyle\mathfrak{E}^{(L)}_{total}=$
$\displaystyle\,\mathfrak{E}^{(L)}(\phi,\cdot)+\varepsilon^{\frac{1}{4}}\left(\mathfrak{E}^{(L)}(W,\cdot)+4\pi
C^{2}\mathfrak{E}^{(L)}(\Sigma,\cdot)\right)+a^{4}\mathfrak{E}^{(L+1)}(\phi,\cdot)+\varepsilon^{\frac{1}{2}}a^{4}\mathfrak{E}^{(L+1)}(\Sigma,\cdot)$
$\displaystyle\,+\varepsilon^{\frac{1}{2}}\mathfrak{E}^{(L-2)}(\text{{Ric}},\cdot)+\varepsilon^{\frac{3}{4}}a^{4}\mathfrak{E}^{(L-1)}(\text{{Ric}},\cdot)$
$\displaystyle\,+\varepsilon^{\frac{1}{4}}\mathfrak{E}^{(L)}_{total,Vl}+\varepsilon^{\frac{1}{2}}a^{4}\mathfrak{E}^{(L+1)}_{total,Vl}+\varepsilon^{\frac{1}{2}}a^{8+\omega}\mathfrak{E}^{(L)}(\text{{Ric}},\cdot)+\underbrace{a^{\omega}\left(\|\Gamma-\hat{\Gamma}\|_{H^{1}_{G}(M)}^{2}+\|G-\gamma\|_{H^{2}_{G}(M)}^{2}\right)}_{\text{if
}L=2}$

######  Proposition 7.5 (Total energy bound).

For $L\in 2\mathbb{N},L\leq 18$, the following holds:

(7.6)
$\mathfrak{E}^{(L)}_{total}\lesssim\varepsilon^{4}a^{-c\varepsilon^{\frac{1}{8}}}$

###### Proof.

We start proving the case of $L=0$ by combining the following energy
estimates: The zero order energy estimates (5.8b), (5.12) and (5.14), the
scalar field estimate (5.8d) and the elliptic shear estimate (5.17b) as well
as the Vlasov estimates (6.5a), (7.3) and the metric estimate (5.18a). Notice
that, by (7.1), the indefinite terms on the left hand sides of (5.12) and
(5.14) are nonnegative for $\kappa\leq 0$ or are so up to a term that can be
pulled to right and estimated from above by
$\int_{t}^{t_{0}}a(s)^{-1}\mathfrak{E}^{(0)}_{total}(s)\,ds$. This leads to
the following estimate:

$\displaystyle\mathfrak{E}^{(0)}_{total}(t)\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left(\varepsilon^{\frac{1}{8}}a(s)^{-3}+a(s)^{-3+\frac{\omega}{2}}\right)\mathfrak{E}^{(0)}_{total}(s)\,ds+\varepsilon^{\frac{1}{4}}\mathfrak{E}^{(0)}_{total}(t)$

After rearranging and having chosen $\varepsilon>0$ to have been small enough,
the Gronwall lemma now yields

$\mathfrak{E}^{(0)}_{total}(t)\lesssim\varepsilon^{4}a(t)^{-c\varepsilon^{\frac{1}{8}}}\,.$

For $L\geq 2$, we proceed iteratively, assuming the bound to have been shown
up to ${\mathfrak{E}}_{total}^{\leq L-2}$. One now similarly combines the
following estimates, going through (7.5a):

  * •

For the first line: (5.8a), (5.13), (5.15), (5.8c), (5.17a)

  * •

For the second line: (5.20a) (respectively (5.20c) for $L=2$) and (5.20b)

  * •

For the Vlasov energies in the third line: (7.4) with $\beta=0$ at order $L$
and with $\beta=4$ at order $L+1$

  * •

For the curvature energy in the final line: (5.20d) with $l=L$

  * •

For the metric terms if $L=2$: (5.19a) with $l=2,\beta=0$ and (5.18a) with
$l=2$.

Note that, for $L=2$, the metric and Christoffel symbol norms become a part of
the total energy, and else they are absorbed into the lower order terms since
they have already been improved.
Altogether, this yields the following:

$\displaystyle\mathfrak{E}^{(L)}_{total}(t)\lesssim$
$\displaystyle\,\varepsilon^{4}+\int_{t}^{t_{0}}\left[\left(\varepsilon^{\frac{1}{8}}a(s)^{-3}+a(s)^{-3-c\sqrt{\varepsilon}+\frac{\omega}{2}}\right)\mathfrak{E}^{(L)}_{total}(s)+\varepsilon^{\frac{1}{8}}a(s)^{-3-c\sqrt{\varepsilon}}\mathfrak{E}^{(\leq
L-2)}_{total}(s)\right]\,ds$