will simplify some arguments in the derivation of Theorem 2. We are going to
use the following result [9, Theorem 1.4]

$\displaystyle\Theta_{0}\big{(}\exp(-\lambda|B^{\bullet}_{r}|)\>|\>Z_{r}=l\big{)}=$
$\displaystyle
r^{3}(2\lambda)^{\frac{3}{4}}\frac{\cosh((2\lambda)^{\frac{1}{4}}r)}{\sinh^{3}((2\lambda)^{\frac{1}{4}}r)}$
$\displaystyle\cdot\exp\Big{(}-l\big{(}\sqrt{\frac{\lambda}{2}}(3\coth^{2}((2\lambda)^{\frac{1}{4}}r)-2)-\frac{3}{2r^{2}}\big{)}\Big{)}$
(35)

for every $\lambda>0$. In particular, using (13), we obtain that for every
$\lambda\geqslant 0$:

$\Theta_{0}\big{(}\exp(-\lambda|B^{\bullet}_{r}|)\big{)}=3^{\frac{3}{2}}\cosh((2\lambda)^{\frac{1}{4}}r)\>(\cosh^{2}((2\lambda)^{\frac{1}{4}}r)+2)^{-\frac{3}{2}}$
(36)

(this formula also appears in [9]).

######  Corollary 3.

There exists a constant $C>0$ such that for every $z>0$ and $r>0$:

$\Theta_{z}\big{(}|B_{r}^{\bullet}|\big{)}\leqslant C(r+\sqrt{z})^{4}.$

###### Proof.

Fix $z>0$ and $r>0$. First remark that, under
$\Theta_{0}(\cdot\,|\,T_{z}\leqslant\sqrt{z})$, the hull
$B_{T_{z}+r}^{\bullet}$ is contained in $B_{\sqrt{z}+r}^{\bullet}$. So an
application of Corollary 1 gives:

$\displaystyle\Theta_{0}\big{(}|B_{r+\sqrt{z}}^{\bullet}|\big{)}$
$\displaystyle\geqslant\Theta_{0}\big{(}|B_{r+\sqrt{z}}^{\bullet}|\mathbbm{1}_{T_{z}<\sqrt{z}}\big{)}$
$\displaystyle\geqslant\Theta_{0}\big{(}(|B_{T_{z}+r}^{\bullet}|-|B_{T_{z}}^{\bullet}|)\mathbbm{1}_{T_{z}<\sqrt{z}}\big{)}$
$\displaystyle=\Theta_{z}\big{(}|B_{r}^{\bullet}|\big{)}\>\Theta_{0}(T_{z}<\sqrt{z}).$

On the other hand, we have
$\\{Z_{\sqrt{z}}>z\\}\subset\\{T_{z}\leqslant\sqrt{z}\\}$. By scaling we also
have $\Theta_{0}(Z_{\sqrt{z}}>z)=\Theta_{0}(Z_{1}>1)>0$ and
$\Theta_{0}\big{(}|B_{r+\sqrt{z}}^{\bullet}|\big{)}=(r+\sqrt{z})^{4}\Theta_{0}\big{(}|B_{1}^{\bullet}|\big{)}$.
Finally, it is easy to deduce from (36) that
$\Theta_{0}\big{(}|B_{1}^{\bullet}|\big{)}$ is finite. This gives the
statement of the corollary with
$C=\Theta_{0}(|B_{1}^{\bullet}|)/\Theta_{0}(Z_{1}>1)$. ∎

We now give two lemmas that will be useful to control the fluctuations of the
volume of hulls in the Brownian plane.

######  Lemma 9.

For every $\beta\in\mathbb{R}$, we have
$\Theta_{0}(|B^{\bullet}_{1}|^{\beta})<\infty$ if and only if
$\beta<\frac{3}{2}$.

###### Proof.

To simplify notation write
$F(\lambda):=\Theta_{0}(\exp(-\lambda|B^{\bullet}_{1}|))$ for every
$\lambda\geqslant 0$. Remark that for every $\lambda>0$, we have
$F^{\prime\prime}(\lambda)=\Theta_{0}(|B^{\bullet}_{1}|^{2}\exp(-\lambda|B^{\bullet}_{1}|))$.
If $\alpha>0$, we have:

$\displaystyle\int_{\mathbb{R}_{+}}\lambda^{\alpha-1}F^{\prime\prime}(\lambda)\>d\lambda$
$\displaystyle=\Theta_{0}\Big{(}|B^{\bullet}_{1}|^{2}\int_{\mathbb{R}_{+}}\lambda^{\alpha-1}\exp(-\lambda|B^{\bullet}_{1}|)\>d\lambda\Big{)}=\Gamma(\alpha)\,\Theta_{0}\big{(}|B^{\bullet}_{1}|^{2-\alpha}\big{)}.$

From the explicit expression of $F$ given in (36) we get:

$F^{\prime\prime}(\lambda)=\frac{1}{9\sqrt{2}}\lambda^{-\frac{1}{2}}+O(1)$

as $\lambda\downarrow 0$, and

$F^{\prime\prime}(\lambda)=O\big{(}\exp(-2(2\lambda)^{\frac{1}{4}})\big{)}$

as $\lambda\uparrow\infty$. So for $\alpha=\frac{1}{2}$,
$\Theta_{0}(|B^{\bullet}_{1}|^{\frac{3}{2}})=\int_{\mathbb{R}_{+}}\lambda^{-\frac{1}{2}}F^{\prime\prime}(\lambda)\>d\lambda=\infty$
and for every $\alpha>\frac{1}{2}$,
$\Theta_{0}(|B^{\bullet}_{1}|^{2-\alpha})<\infty$. ∎

We conclude this section with the following consequence of Lemma 9:

######  Lemma 10.

For every $\beta_{1}>0$ and $\beta_{2}>2/3$ we have $\Theta_{0}$-a.s.

$\inf_{r>0}\frac{|B^{\bullet}_{r}|}{r^{4}(1+|\log(r)|)^{-\beta_{1}}}>0$

and

$\sup_{r>0}\frac{|B^{\bullet}_{r}|}{r^{4}(1+|\log(r)|)^{\beta_{2}}}<\infty.$

###### Proof.

Fix $\beta_{1}$ and $\beta_{2}$ as in the statement. By Lemma 9, the
quantities $\Theta_{0}(|B^{\bullet}_{1}|^{-1/\beta_{1}})$ and
$\Theta_{0}(|B^{\bullet}_{1}|^{1/\beta_{2}})$ are finite. This implies by the
scaling invariance of $\mathcal{M}_{\infty}$:

$\displaystyle\sum\limits_{m\in\mathbb{Z}}\Theta_{0}(|B^{\bullet}_{2^{m}}|^{-1}>|m|^{\beta_{1}}2^{-4m})$
$\displaystyle\leqslant
2\sum\limits_{m=0}^{\infty}\Theta_{0}(|B^{\bullet}_{1}|^{-\frac{1}{\beta_{1}}}>m)<\infty$

and

$\displaystyle\sum\limits_{m\in\mathbb{Z}}\Theta_{0}(|B^{\bullet}_{2^{m}}|>|m|^{\beta_{2}}2^{4m})$
$\displaystyle\leqslant
2\sum\limits_{m=0}^{\infty}\Theta_{0}(|B^{\bullet}_{1}|^{\frac{1}{\beta_{2}}}>m)<\infty\>.$

The result then follows by the Borel-Cantelli lemma. ∎

###  4.2 Proof of Part $\rm(i)$ of Theorem 2

The goal of this section is to prove the following slightly more precise form
of Theorem 2 (ii).

######  Proposition 8.

Let $f:\mathbb{R}_{+}\to\mathbb{R}_{+}^{*}$ be a positive nondecreasing
function such that $\sum\limits_{m\in\mathbb{N}}f(m)^{-2}=\infty\>.$ Then,

$\mathop{\inf_{A\in\mathcal{K}}}_{A\subset
B_{1}^{\bullet}}\frac{\Delta(\partial
A)}{|A|^{\frac{1}{4}}}f(|\log(|A|)|)=0,\>\>\Theta_{0}\text{-a.s.}$ (37)

and

$\mathop{\inf_{A\in\mathcal{K}}}_{B_{1}^{\bullet}\subset
A}\frac{\Delta(\partial
A)}{|A|^{\frac{1}{4}}}f(|\log(|A|)|)=0,\>\>\Theta_{0}\text{-a.s.}$ (38)

###### Proof.

Fix $r>2\tilde{c}_{2}/\tilde{c}_{1}\geqslant 2$, where $\tilde{c}_{1}$ and
$\tilde{c}_{2}$ are as in Proposition 6, and let
$f:\mathbb{R}_{+}\to\mathbb{R}_{+}^{*}$ be a positive nondecreasing function
such that $\sum\limits_{m\in\mathbb{N}}f(m)^{-2}=\infty\>.$ We give a detailed
proof of the (38) since (37) can be obtained, mutatis mutandis, by the same
method. Recall the notation $T_{r}:=\inf\\{t\geqslant 0:\>Z_{t}\geqslant
r\\}$. Since $Z$ does not have positive jumps, we have $Z_{T_{r}}=r$. For
every $n\geqslant 1$, a separating cycle taking values in
$B_{T_{r^{n}},T_{r^{n+1}}}^{\bullet}$ bounds a Jordan domain $A\in\mathcal{K}$
such that $B_{T_{r^{n}}}^{\bullet}\subset A$ and for $n$ large enough we also
have $B_{1}^{\bullet}\subset A$. Hence,

$\mathop{\inf_{A\in\mathcal{K}}}_{B_{1}^{\bullet}\subset
A}\frac{\Delta(\partial
A)}{|A|^{\frac{1}{4}}}f(|\log(|A|)|)\leqslant\liminf\limits_{n\to\infty}\frac{L_{T_{r^{2n+1}},T_{r^{2n+2}}}}{|B_{T_{r^{2n+1}}}^{\bullet}|^{\frac{1}{4}}}f(\log(|B_{T_{r^{2n+2}}}^{\bullet}|)),\>\>\Theta_{0}\text{-a.s.}$

So to obtain (38) it is enough to show that:

$\liminf\limits_{n\to\infty}\frac{L_{T_{r^{2n+1}},T_{r^{2n+2}}}}{|B_{T_{r^{2n+1}}}^{\bullet}|^{\frac{1}{4}}}f(\log(|B_{T_{r^{2n+2}}}^{\bullet}|))=0,\>\>\Theta_{0}\text{-a.s.}$
(39)

Let us study the growth of the sequence $(T_{r^{n}})_{n\in\mathbb{N}}$. First
note that:

$\Theta_{0}(T_{1}\geqslant u)\leqslant\Theta_{0}(Z_{u}\leqslant
1)=\Theta_{0}(Z_{1}\leqslant
u^{-2})=\frac{3^{\frac{3}{2}}}{\sqrt{2\pi}}\int_{0}^{u^{-2}}\sqrt{x}\exp(-\frac{3}{2}x)\>dx\leqslant\sqrt{\frac{6}{\pi}}u^{-3}$

where in the first equality we apply the scaling invariance of the Brownian
plane and in the second one we use the density of $Z_{1}$ (see (13)). In
particular, we have $\Theta_{0}(T_{1})<\infty$. So by scaling invariance we
obtain:

$\sum\limits_{n\in\mathbb{N}^{*}}\Theta_{0}(T_{r^{n}}>nr^{\frac{n}{2}})=\sum\limits_{n\in\mathbb{N}^{*}}\Theta_{0}(T_{1}>n)<\infty.$

The Borel-Cantelli lemma then implies that
$\limsup\limits_{n\to\infty}(nr^{n/2})^{-1}T_{r^{n}}\leqslant 1$,
$\Theta_{0}$-a.s. Since $\lim\limits_{s\to\infty}T_{s}=\infty$,
$\Theta_{0}$-a.s. , Lemma 10 gives:

$\limsup\limits_{n\to\infty}\frac{\log(|B_{T_{r^{2n+2}}}^{\bullet}|)}{4\log(T_{r^{2n+2}})}\leqslant
1,\>\>\Theta_{0}\text{-a.s.}$

and then we deduce:

$\limsup\limits_{n\to\infty}\frac{\log(|B_{T_{r^{2n+2}}}^{\bullet}|)}{2n\log(r)}\leqslant
1,\>\>\Theta_{0}\text{-a.s.}$

Fix $h>2\log(r)$. As $f$ is nondecreasing we have:

$\liminf\limits_{n\to\infty}\frac{L_{T_{r^{2n+1}},T_{r^{2n+2}}}}{|B_{T_{r^{2n+1}}}^{\bullet}|^{\frac{1}{4}}}f(\log(|B_{T_{r^{2n+2}}}^{\bullet}|))\leqslant\liminf\limits_{n\to\infty}\frac{L_{T_{r^{2n+1}},T_{r^{2n+2}}}}{|B_{T_{r^{2n}},T_{r^{2n+1}}}^{\bullet}|^{\frac{1}{4}}}f(hn).$
(40)

We will use the Borel-Cantelli lemma to conclude. By Theorem 4, under
$\Theta_{0}$ :

$\Big{(}\>\frac{1}{r^{4n}}|B^{\bullet}_{T_{r^{2n}},T_{r^{2n+1}}}|,\frac{1}{r^{n+\frac{1}{2}}}L_{T_{r^{2n+1}},T_{r^{2n+2}}}\>\Big{)}_{n\in\mathbb{N}}\>.$

is an i.i.d. sequence of random variables and for every $n\geqslant 0$ the
variables:

$\frac{1}{r^{4n}}|B^{\bullet}_{T_{r^{2n}},T_{r^{2n+1}}}|,\>\>\text{and}\>\>\frac{1}{r^{n+\frac{1}{2}}}L_{T_{r^{2n+1}},T_{r^{2n+2}}}$

are independent. Moreover
$\frac{1}{r^{4n}}|B^{\bullet}_{T_{r^{2n}},T_{r^{2n+1}}}|$ (resp.
$\frac{1}{r^{n+\frac{1}{2}}}L_{T_{r^{2n+1}},T_{r^{2n+2}}}$) is distributed
under $\Theta_{0}$ as $B_{T_{r}}^{\bullet}$ (resp. $L_{0,T_{r}}$) under
$\Theta_{1}$. Fix $\delta>0$, such that
$\Theta_{1}(B_{T_{r}}^{\bullet}>\delta)>0$ and let $\varepsilon>0$. By the
previous remark we have:

$\displaystyle\sum\limits_{n=0}^{\infty}\Theta_{0}\Big{(}|B^{\bullet}_{T_{r^{2n}},T_{r^{2n+1}}}|>\delta
r^{4n},\>$ $\displaystyle
L_{T_{r^{2n+1}},T_{r^{2n+2}}}<\frac{\varepsilon}{f(hn)}r^{n+\frac{1}{2}}\Big{)}$
$\displaystyle=\sum\limits_{n=0}^{\infty}\Theta_{1}(|B^{\bullet}_{T_{r}}|>\delta)\>\Theta_{1}\big{(}L_{0,T_{r}}<\frac{\varepsilon}{f(hn)}\big{)}.$

By Proposition 7 the right-hand side of the last display is greater than

$\tilde{c}_{3}\>\Theta_{1}(|B^{\bullet}_{T_{r}}|>\delta)\sum\limits_{n=0}^{\infty}\left(\frac{\varepsilon^{2}}{f(hn)^{2}}\wedge
1\right)$

which is infinite since $\sum\limits_{m\in\mathbb{N}}f(m)^{-2}=\infty$. The
Borel-Cantelli lemma then implies that:

$\liminf\limits_{n\to\infty}\frac{L_{T_{r^{2n+1}},T_{r^{2n+2}}}}{|B_{T_{r^{2n}},T_{r^{2n+1}}}^{\bullet}|^{\frac{1}{4}}}f(hn)\leqslant\varepsilon\delta^{-\frac{1}{4}}\sqrt{r},\>\>\Theta_{0}\text{-a.s.}$

This holds for every $\varepsilon>0$, which together with (40) gives (39). ∎

###  4.3 Proof of Part $\rm(ii)$ of Theorem 2

We need to show that for any positive nondecreasing function $f$, the
condition $\sum\limits_{m\in\mathbb{N}}f(m)^{-2}<\infty$ implies
$\inf\limits_{A\in\mathcal{K}}\frac{\Delta(\partial
A)}{|A|^{\frac{1}{4}}}f(|\log(|A|)|)>0$, $\Theta_{0}$-a.s.

We begin with a technical lemma.

######  Lemma 11.

Let $\beta\in[0,1)$. There exists a constant $C_{\beta}>0$, which only depends
on $\beta$, such that for every $r>0$ and $\varepsilon>0$:

$\Theta_{0}\big{(}\mathbbm{1}_{L_{1}<\varepsilon}|B_{r}^{\bullet}|^{\beta}\big{)}\leqslant
C_{\beta}\>r^{4\beta}\varepsilon^{2}.$

The reason for taking $\beta<1$ is just technical and one can extend the
result to $\beta<\frac{3}{2}$ but the proof will be more tedious. At an
intuitive level, Lemma 11 states that if we know that there exists a small
cycle separating the hull of radius $1$ then the expected volume of the hull
of radius $r$ stays at most of order $r^{4}$ (with a uniform control).

###### Proof.

Fix $\beta\in(0,1)$ and let $r>0$ and $\varepsilon>0$.

To simplify notation, set $m:=\frac{1}{\varepsilon}$, $q:=\frac{1}{\beta}$ and
$p:=\frac{q}{q-1}=\frac{1}{1-\beta}$. By the scaling property of
$\mathcal{M}_{\infty}$:

$\Theta_{0}\big{(}\mathbbm{1}_{L_{1}<\varepsilon}|B_{r}^{\bullet}|^{\beta}\big{)}=\frac{1}{m^{4\beta}}\Theta_{0}\big{(}\mathbbm{1}_{L_{m}<1}|B_{mr}^{\bullet}|^{\beta}\big{)}.$

If $L_{m}<1$, there is a separating cycle of length smaller than $1$ that is
contained in $\check{B}_{m}^{\circ}$, and necessarily this separating cycle is
contained in $B_{m+k,m+k+2}^{\bullet}$ for some integer $k\geqslant 0$. Hence,

$\Theta_{0}\big{(}\mathbbm{1}_{L_{m}<1}|B_{mr}^{\bullet}|^{\beta}\big{)}\leqslant\sum\limits_{k=0}^{\infty}\Theta_{0}\big{(}\mathbbm{1}_{L_{m+k,m+k+2}<1}|B_{mr}^{\bullet}|^{\beta}\big{)}.$

Applying the conditional version of the Hölder inequality with respect to
$Z_{m+k+4}$ we obtain:

$\displaystyle\Theta_{0}\big{(}\mathbbm{1}_{L_{m+k,m+k+2}<1}|B_{mr}^{\bullet}|^{\beta}\big{)}\leqslant\Theta_{0}\Big{(}\>\Theta_{0}\big{(}L_{m+k,m+k+2}<1\>\big{|}\>Z_{m+k+4}\big{)}^{\frac{1}{p}}\>\Theta_{0}\big{(}|B_{mr}^{\bullet}|\>\big{|}\>Z_{m+k+4}\big{)}^{\frac{1}{q}}\>\Big{)}.$

By Proposition 5, there exists $\alpha_{1}>0$ such that

$\Theta_{0}\big{(}L_{m+k,m+k+2}<1\>\big{|}\>Z_{m+k+4}\big{)}\leqslant
e\cdot\exp(-\alpha_{1}Z_{m+k+4})$

for every $k\geqslant 0$ and thus we get :

$\Theta_{0}\big{(}\mathbbm{1}_{L_{m}<1}|B_{mr}^{\bullet}|^{\beta}\big{)}\leqslant
e\>\sum\limits_{k=0}^{\infty}\>\Theta_{0}\Big{(}\exp(-\alpha
Z_{m+k+4})\>\Theta_{0}\big{(}|B_{mr}^{\bullet}|\>\big{|}\>Z_{m+k+4}\big{)}^{\frac{1}{q}}\>\Big{)}$
(41)

where $\alpha:=\alpha_{1}/p$. Then again by the Hölder inequality:

$\displaystyle\Theta_{0}\Big{(}\exp(-\alpha Z_{m+k+4})\leavevmode\nobreak\ $
$\displaystyle\Theta_{0}\big{(}|B_{mr}^{\bullet}|\>\big{|}\>Z_{m+k+4}\big{)}^{\frac{1}{q}}\>\Big{)}$
$\displaystyle=\Theta_{0}\Big{(}\exp(-\frac{\alpha}{p}Z_{m+k+4})\exp(-\frac{\alpha}{q}Z_{m+k+4})\Theta_{0}\big{(}|B_{mr}^{\bullet}|\>\big{|}\>Z_{m+k+4}\big{)}^{\frac{1}{q}}\>\Big{)}$
$\displaystyle\leqslant\Theta_{0}\big{(}\exp(-\alpha
Z_{m+k+4})\big{)}^{\frac{1}{p}}\>\Theta_{0}\big{(}\exp(-\alpha
Z_{m+k+4})|B_{mr}^{\bullet}|\big{)}^{\frac{1}{q}}.$ (42)

By (13), $Z_{m+k+4}$ follows the Gamma distribution with parameter
$\frac{3}{2}$ and mean $(m+k+4)^{2}$. So we can find $d_{1}(\alpha)>0$
independent of $m$ and $k$ such that:

$\Theta_{0}(\exp(-\alpha
Z_{m+k+4}))\leqslant\frac{d_{1}(\alpha)}{(m+k+4)^{3}}\,,\quad\Theta_{0}(Z_{m+k+4}\exp(-\alpha
Z_{m+k+4}))\leqslant\frac{d_{1}(\alpha)}{(m+k+4)^{3}}.$ (43)

Moreover, noting that $|\partial B_{m+k+4}^{\bullet}|=0$, $\Theta_{0}$-a.s. ,
we obtain:

$\displaystyle\Theta_{0}\big{(}\exp(-\alpha
Z_{m+k+4})|B_{mr}^{\bullet}|\big{)}$
$\displaystyle=\Theta_{0}\big{(}\exp(-\alpha
Z_{m+k+4})|B_{(m+k+4)\>\wedge\>mr}^{\bullet}|\big{)}$
$\displaystyle+\Theta_{0}\big{(}\exp(-\alpha
Z_{m+k+4})|B_{m+k+4,mr}^{\bullet}|\big{)}$

where by convention $|B_{s_{1},s_{2}}^{\bullet}|=0$ if $s_{2}\leqslant s_{1}$.

Let $0<s_{1}\leqslant s_{2}$. We observe that $|B^{\bullet}_{s_{1}}|$ is
independent of $Z_{s_{2}}$ conditionally on $Z_{s_{1}}$. This follows from the
special Markov property and the spine independence property, using the fact
that $|B^{\bullet}_{s_{1}}|$ is determined by the excursions of $\omega$ below
$s_{1}$ for all atoms of $\mathfrak{L}$ and $\mathfrak{R}$ such that
$t\geqslant\tau_{s_{1}}$, and by the atoms $(t,\omega)$ such that
$t\leqslant\tau_{s_{1}}$. Thanks to this conditional independence property, we
have

$\Theta_{0}\Big{(}|B_{s_{1}}^{\bullet}|\>\big{|}\>Z_{s_{2}}\Big{)}=\Theta_{0}\Big{(}\Theta_{0}\big{(}|B_{s_{1}}^{\bullet}|\>\big{|}\>Z_{s_{1}}\big{)}\>\big{|}\>Z_{s_{2}}\Big{)}\leavevmode\nobreak\
.$ (44)

By differentiating the right-hand side of $\eqref{eq_B_sachant_Z}$ at
$\lambda=0$ we get

$\Theta_{0}\left(|B_{s_{1}}^{\bullet}|\>\big{|}\>Z_{s_{1}}\right)=\frac{2}{15}s_{1}^{4}+\frac{1}{5}s_{1}^{2}Z_{s_{1}}$

similarly we have from (14):

$\Theta_{0}(Z_{s_{1}}\>\big{|}\>Z_{s_{2}})=\frac{s_{1}^{3}}{s_{2}^{3}}Z_{s_{2}}+\frac{s_{2}-s_{1}}{s_{2}}s_{1}^{2}.$

So by (44), for every $0<s_{1}\leqslant s_{2}$

$\Theta_{0}\Big{(}|B_{s_{1}}^{\bullet}|\>\big{|}\>Z_{s_{2}}\Big{)}=\frac{1}{3}s_{1}^{4}-\frac{s_{1}^{5}}{5s_{2}}+\frac{s_{1}^{5}}{5s_{2}^{3}}Z_{s_{2}}.$

Taking $s_{1}=(m+k+4)\wedge(mr)$ and $s_{2}=m+k+4$, we deduce from the last
two formulas and (43) that there exists $d_{2}(\alpha)>0$ independent of $m$
and $k$ such that:

$\Theta_{0}\big{(}\exp(-\alpha
Z_{m+k+4})|B_{(m+k+4)\>\wedge\>mr}^{\bullet}|\big{)}\leqslant\frac{d_{2}(\alpha)}{(m+k+4)^{3}}r^{4}m^{4}.$

Suppose that $m+k+4<mr$. Then by the spatial Markov property and Corollary 3,

$\displaystyle\Theta_{0}(\>\exp(-\alpha Z_{m+k+4})|B_{m+k+4,mr}^{\bullet}|\>)$
$\displaystyle=\Theta_{0}\Big{(}\exp(-\alpha
Z_{m+k+4})\>\Theta_{Z_{m+k+4}}\big{(}|B^{\bullet}_{mr-m-k-4}|\big{)}\>\Big{)}$
$\displaystyle\leqslant C^{\prime}\>\Theta_{0}(\exp(-\alpha
Z_{m+k+4})(m^{4}r^{4}+Z_{m+k+4}^{2})).$

where $C^{\prime}=16C$, if $C$ is the constant appearing in Corollary 3. Using
again the distribution of $Z_{m+k+4}$ , we get that there exists a constant
$d_{3}(\alpha)>0$ independent of $m$ and $k$ such that, if $m+k+4<mr$,

$\Theta_{0}(\exp(-\alpha
Z_{m+k+4})\>|B_{m+k+4,mr}^{\bullet}|\>)\leqslant\frac{d_{3}(\alpha)}{(m+k+4)^{3}}\>r^{4}m^{4}.$

Summarizing, we get in both cases $m+k+4<mr$ and $m+k+4\geqslant mr$,

$\Theta_{0}\big{(}\exp(-\alpha
Z_{m+k+4})|B_{mr}^{\bullet}|\big{)}\leqslant\frac{d_{2}(\alpha)+d_{3}(\alpha)}{(m+k+4)^{3}}\>r^{4}m^{4}.$

Coming back to (41) and (4.3), using (43) once again, and recalling that
$q=1/\beta$ and $m=1/\varepsilon$,

we can find a constant $d(\alpha)>0$ such that:

$\displaystyle\Theta_{0}\big{(}\mathbbm{1}_{L_{1}<\varepsilon}|B_{r}^{\bullet}|^{\beta}\big{)}$
$\displaystyle=\frac{1}{m^{4\beta}}\Theta_{0}(\mathbbm{1}_{L_{m}<1}|B_{mr}^{\bullet}|^{\beta})$
$\displaystyle\leqslant\frac{c}{m^{4\beta}}\sum\limits_{k=0}^{\infty}\Theta_{0}\big{(}\exp(-\alpha
Z_{m+k+4})\big{)}^{\frac{1}{p}}\>\Theta_{0}\big{(}\exp(-\alpha
Z_{m+k+4})|B_{mr}^{\bullet}|\big{)}^{\frac{1}{q}}$
$\displaystyle\leqslant\sum\limits_{k=0}^{\infty}\frac{d(\alpha)}{(m+k+4)^{3}}r^{4\beta}$

and the lemma follows since $m=\varepsilon^{-1}$. ∎

We now use Lemma 11 to prove that for any nondecreasing positive function $f$:

$\sum\limits_{m\in\mathbb{N}}\frac{1}{f(m)^{2}}<\infty\implies\inf\limits_{A\in\mathcal{K}}\frac{\Delta(\partial
A)}{|A|^{\frac{1}{4}}}f(|\log(|A|)|)>0\>,\>\Theta_{0}\text{-a.s.}$ (45)

###### Proof.

Fix a nondecreasing function $f:\mathbb{R}_{+}\to\mathbb{R}_{+}^{*}$ such that
$\sum\limits_{m\in\mathbb{N}}f(m)^{-2}<\infty.$ We begin by showing that for
every $A\in\mathcal{K}$ we have:

$\frac{\Delta(\partial
A)}{|A|^{\frac{1}{4}}}f(|\log(|A|)|)\geqslant\Big{(}\inf\limits_{m\in\mathbb{Z}}\frac{2^{m-1}}{|B_{2^{m+1}}^{\bullet}|^{\frac{1}{4}}}f\big{(}|\log(|B_{2^{m}}^{\bullet}|)|\big{)}\Big{)}\wedge\Big{(}\inf\limits_{m\in\mathbb{Z}}\frac{L_{2^{m-1}}}{|B_{2^{m+1}}^{\bullet}|^{\frac{1}{4}}}f\big{(}|\log(|B_{2^{m}}^{\bullet}|)|\big{)}\Big{)}.$
(46)

Let $A\in\mathcal{K}$ and let $m$ be the unique element of $\mathbb{Z}$ such
that:

$|B_{2^{m}}^{\bullet}|<|A|\leqslant|B_{2^{m+1}}^{\bullet}|.$

We divide the proof of (46) in two cases:

$\bullet$ Case 1: Assume that $\partial A$ intersects $\partial
B^{\bullet}_{2^{m-1}}$. As $|B^{\bullet}_{2^{m}}|<|A|$, there exists $x\in
A\setminus B^{\bullet}_{2^{m}}$. Consider a path $p_{\infty}$ connecting $x$
to $\infty$ that does not hit $B^{\bullet}_{2^{m}}$ and let $y$ be the last
point of $p_{\infty}$ that belongs to $A$. By construction we have
$y\in\partial A$ and $y\notin B^{\bullet}_{2^{m}}$. Finally let $z\in\partial
A\cap\partial B^{\bullet}_{2^{m-1}}$. Since $\partial A$ connects $y$ and $z$
we have

$\Delta(\partial A)\geqslant\Delta(y,z)\geqslant 2^{m-1}.$

This implies that:

$\frac{\Delta(\partial
A)}{|A|^{\frac{1}{4}}}f(|\log(|A|)|)\geqslant\frac{2^{m-1}}{|B^{\bullet}_{2^{m+1}}|^{\frac{1}{4}}}f(|\log(|B^{\bullet}_{2^{m}}|)|).$

$\bullet$ Case 2: Assume $\partial A$ does not intersect $\partial
B^{\bullet}_{2^{m-1}}$. Since $|A|>|B_{2^{m-1}}^{\bullet}|$, the set $A$ is
not contained in $B_{2^{m-1}}^{\bullet}$. This implies that $\partial A$
separates $B^{\bullet}_{2^{m-1}}$ from infinity, and consequently
$\Delta(\partial A)\geqslant L_{2^{m-1}}$. It follows that:

$\frac{\Delta(\partial
A)}{|A|^{\frac{1}{4}}}f(|\log(|A|)|)\geqslant\frac{L_{2^{m-1}}}{|B^{\bullet}_{2^{m+1}}|^{\frac{1}{4}}}f(|\log(|B^{\bullet}_{2^{m}}|)|)$

and this completes the proof of (46).

Figure 4: Illustration of (46). In red we represent the boundary of $A$. On
the left we are in case 1 and we have $\Delta(\partial A)\geqslant 2^{m-1}$.
On the right we are in case 2 and we have $\Delta(\partial A)\geqslant
L_{2^{m-1}}$.

Thanks to (46), the proof of (45) will be complete if we can verify that:

$\inf\limits_{m\in\mathbb{Z}}\frac{2^{m-1}}{|B_{2^{m+1}}^{\bullet}|^{\frac{1}{4}}}f\big{(}|\log(|B_{2^{m}}^{\bullet}|)|\big{)}>0,\>\>\Theta_{0}\text{-a.s.}$
(47)

and

$\inf\limits_{m\in\mathbb{Z}}\frac{L_{2^{m-1}}}{|B_{2^{m+1}}^{\bullet}|^{\frac{1}{4}}}f\big{(}|\log(|B_{2^{m}}^{\bullet}|)|\big{)}>0,\>\>\Theta_{0}\text{-a.s.}$
(48)

Let us start by proving (47). By Lemma 10, $\Theta_{0}\text{-a.s.}$, there is
a positive integer $M$ such that for every $m\in\mathbb{Z}$ with $|m|\geqslant
M$:

$\frac{1}{|m|}2^{4m}\leqslant|B_{2^{m}}^{\bullet}|\leqslant|m|2^{4m}$ (49)

In particular, we have:

$\mathop{\inf_{m\in\mathbb{Z}}}_{|m|>M}\frac{2^{m-1}}{|B_{2^{m+1}}^{\bullet}|^{\frac{1}{4}}}f(|\log(|B_{2^{m}}^{\bullet}|)|)\geqslant\frac{1}{4}\mathop{\inf_{m\in\mathbb{Z}}}_{|m|>M}\frac{f\big{(}|4\log(2)m-\log(|m|)|\big{)}}{(|m|+1)^{\frac{1}{4}}}\>\>\Theta_{0}\text{-a.s.}$

On the other hand, by the Cauchy-Schwarz inequality:

$\frac{1}{f(n)}\leqslant\frac{1}{n}\sum\limits_{k=1}^{n}\frac{1}{f(k)}\leqslant\frac{1}{n^{\frac{1}{2}}}\big{(}\sum\limits_{k=1}^{n}\frac{1}{f(k)^{2}}\big{)}^{\frac{1}{2}}\leqslant\frac{1}{n^{\frac{1}{2}}}\big{(}\sum\limits_{k=1}^{\infty}\frac{1}{f(k)^{2}}\big{)}^{\frac{1}{2}}.$

Consequently $\inf\limits_{n\in\mathbb{N}}n^{-\frac{1}{2}}f(n)>0$ and we
obtain (47). Actually here it will be enough to have
$\inf\limits_{n\in\mathbb{N}}n^{-\frac{1}{4}}f(n)>0$.

Let us prove (48). By (49), if we can verify that

$\sum\limits_{m\in\mathbb{Z}}\Theta_{0}\big{(}\frac{L_{2^{m-1}}}{|B^{\bullet}_{2^{m+1}}|^{\frac{1}{4}}}<\frac{1}{f(|m|)}\big{)}<\infty$

we will conclude by an application of the Borel-Cantelli lemma. Fix
$\beta\in(\frac{3}{4},1)$. For every $m\in\mathbb{Z}$, by scaling we have:

$\displaystyle\Theta_{0}\big{(}\frac{L_{2^{m-1}}}{|B^{\bullet}_{2^{m+1}}|^{\frac{1}{4}}}<\frac{1}{f(|m|)}\big{)}$
$\displaystyle=\Theta_{0}\big{(}\frac{L_{1}}{|B^{\bullet}_{4}|^{\frac{1}{4}}}<\frac{1}{f(|m|)}\big{)}.$

Consequently, we get :

$\displaystyle\Theta_{0}\big{(}\frac{L_{2^{m-1}}}{|B^{\bullet}_{2^{m+1}}|^{\frac{1}{4}}}<\frac{1}{f(|m|)}\big{)}$
$\displaystyle\leqslant\Theta_{0}\big{(}L_{1}<\frac{1}{f(|m|)}\>,\>|B_{4}^{\bullet}|\leqslant
1\big{)}$
$\displaystyle+\sum\limits_{n=1}^{\infty}\Theta_{0}\big{(}|B^{\bullet}_{4}|^{\frac{1}{4}}\in[n,n+1]\>,\>L_{1}<\frac{n+1}{f(|m|)}\big{)}.$

Now remark that the right term of the above display is bounded above by

$\displaystyle\Theta_{0}\big{(}L_{1}<\frac{1}{f(|m|)}\big{)}+\sum\limits_{n=1}^{\infty}\Theta_{0}\big{(}|B^{\bullet}_{4}|^{\beta}\geqslant
n^{4\beta},\>L_{1}<\frac{n+1}{f(|m|)}\big{)}.$

We deduce by an application of Markov inequality that

$\Theta_{0}\big{(}\frac{L_{2^{m-1}}}{|B^{\bullet}_{2^{m+1}}|^{\frac{1}{4}}}<\frac{1}{f(|m|)}\big{)}\leqslant\Theta_{0}(L_{1}<\frac{1}{f(|m|)})+\sum\limits_{n=1}^{\infty}\frac{1}{n^{4\beta}}\Theta_{0}(|B_{4}^{\bullet}|^{\beta}\mathbbm{1}_{L_{1}<\frac{n+1}{f(|m|)}})$

Lemma 11 and Theorem 1 imply that there exist two constants
$c_{2}\in(0,\infty)$ and $C\in(0,\infty)$ such that:

$\sum\limits_{m\in\mathbb{Z}}\Theta_{0}(\frac{L_{2^{m-1}}}{|B^{\bullet}_{2^{m+1}}|^{\frac{1}{4}}}<\frac{1}{f(|m|)})\leqslant
2c_{2}\sum\limits_{m\in\mathbb{N}}^{\infty}\frac{1}{f(m)^{2}}+C\sum\limits_{n=1}^{\infty}\frac{(n+1)^{2}}{n^{4\beta}}\sum\limits_{m\in\mathbb{N}}\frac{1}{f(m)^{2}}<\infty.$

This completes the proof. ∎

We observe that the same proof will work mutatis mutandis if we replace $|A|$
by $\Delta(\partial A)$ inside the logarithm in theorem 2.

Recall that $\mathcal{M}_{\infty}^{(z)}$ stands for an infinite volume
Brownian disk with perimeter $z$. With a slight abuse of terminology, we call
Jordan domain of $\mathcal{M}_{\infty}^{(z)}$ the closure of the bounded
component of the complement of an injective cycle of
$\mathcal{M}_{\infty}^{(z)}$. As a direct consequence of Corollary 1,
Proposition 8 $\rm(i)$ and Theorem 2 we obtain:

######  Corollary 4.

Fix $z>0$. Let $\mathcal{M}_{\infty}^{(z)}$ be the infinite volume Brownian
disk with perimeter $z$ defined under the probability measure $\Theta_{z}$.
Consider the collection $\mathcal{K}^{(z)}$ of all Jordan domains of
$\mathcal{M}_{\infty}^{(z)}$ whose interior contains the boundary of
$\mathcal{M}_{\infty}^{(z)}$. For any nondecreasing function
$f:\mathbb{R}_{+}\to\mathbb{R}_{+}^{*}$

$\rm(i)$ We have

$\inf\limits_{A\in\mathcal{K}^{(z)}}\frac{\Delta\big{(}\partial
A\big{)}}{|A|^{\frac{1}{4}}}f(|\log(|A|)|)=0\>,\>\Theta_{z}\text{-a.s.}\>,\>\text{if}\>\>\sum\limits_{m\in\mathbb{N}}\frac{1}{f(m)^{2}}=\infty.$

$\rm(ii)$ We have

$\inf\limits_{A\in\mathcal{K}^{(z)}}\frac{\Delta\big{(}\partial
A\big{)}}{|A|^{\frac{1}{4}}}f(|\log(|A|)|)>0\>,\>\Theta_{z}\text{-a.s.}\>,\>\text{if}\>\>\sum\limits_{m\in\mathbb{N}}\frac{1}{f(m)^{2}}<\infty.$

### Appendix: Proof of Lemma 1

This appendix is devoted to the proof of Lemma 1, which relies on [18,
Proposition 8]. We use the notation of Subsection 2.4.1.

Proof of Lemma 1.

First fix $0<r_{1}<r_{2}<\infty$. The lemma will follow if we prove that,
$\Theta_{0}$-a.s. , for every $r_{1}\leqslant r\leqslant r_{2}$, we have:

$Z_{r}=\lim\limits_{\varepsilon\downarrow
0}\frac{1}{\varepsilon^{2}}|\check{B}^{\circ}_{r}\cap B_{r+\varepsilon}|.$

In order to prove this, we introduce the event
$A(s):=\\{Z_{r_{2}}^{s,\infty}=0\\}$, for every $s>r_{2}$. In particular,
under the event $A(s)$, we have $Z_{r}=Z_{r}^{r,s}$ for every $r\leqslant
r_{2}$.

Moreover, by Proposition 1 (taking the limit when $t\to\infty$) we also have:

$\Theta(A(s))=\big{(}\frac{s-r}{s}\big{)}^{3},$

which converges to $1$ when $s\to\infty$. Consequently, to obtain the desired
result it is sufficient to show that, for every $s>r_{2}$, under $A(s)$ we
have:

$Z_{r}^{r,s}=\lim\limits_{\varepsilon\downarrow
0}\frac{1}{\varepsilon^{2}}|\check{B}^{\circ}_{r}\cap B_{r+\varepsilon}|,$

for every $r\in[r_{1},r_{2}]$. Let us now introduce, for every
$r\in\mathbb{R}$ and $\omega\in\mathcal{S}$ with $\omega_{0}>r$, the quantity:

$\mathcal{Z}_{r}^{\varepsilon}(\omega):=\frac{1}{\varepsilon^{2}}\int_{0}^{\sigma}ds\mathbbm{1}_{\text{hit}_{r}(\omega_{s})=\infty,\leavevmode\nobreak\
\widehat{\omega}_{s}<r+\varepsilon}.$

In particular, note that $\mathcal{Z}_{r}(\omega)=\liminf_{\varepsilon\to
0}\mathcal{Z}_{r}^{\varepsilon}(\omega)$. We set:

$Z_{r}^{r,s}(\varepsilon):=\int\mathcal{Z}_{r}^{\varepsilon}(\omega)\mathfrak{R}^{r,s}(d\ell
d\omega)+\int\mathcal{Z}_{r}^{\varepsilon}(\omega)\mathfrak{L}^{r,s}(d\ell
d\omega).$

Under $A(s)$, all the labels appearing after the point $\tau_{s}$ of the spine
are greater than $r_{2}$. This implies that, under $A(s)$, the quantity
$\varepsilon^{2}Z_{r}^{r,s}(\varepsilon)$ is exactly
$|\check{B}^{\circ}_{r}\cap B_{r+\varepsilon}|$ (since $|\cdot|$ is the
pushforward of Lebesgue measure under $\Pi\circ\mathcal{E}$). To conclude, we
are going to use [18, Proposition 8], which states that, for every $s>0$ and
$\beta>0$, we have

$\sup_{r\in(-\infty,s-\beta]}|\mathcal{Z}_{r}^{\varepsilon}-\mathcal{Z}_{r}|\leavevmode\nobreak\
\underset{\varepsilon\to 0}{\longrightarrow}\leavevmode\nobreak\
0\;,\qquad\mathbb{N}_{s}\hbox{-a.e.}$ (50)

To translate (50) in terms of $Z_{r}^{r,s}$ and $Z_{r}^{r,s}(\varepsilon)$, we
recall that $(X_{(\tau_{s}-\ell)\vee 0})_{\ell\geqslant 0}$ is a Bessel
process of dimension $-5$ started from $s$, and that, conditionally on
$(X_{(\tau_{s}-\ell)\vee 0})_{\ell\geqslant 0}$, the measures
$\mathfrak{R}^{0,s}$ and $\mathfrak{L}^{0,s}$ are two independent Poisson
point measures on $\mathbb{R}_{+}\times\mathcal{S}$ with intensity:

$2\mathbbm{1}_{[0,\tau_{s}]}(\ell)\leavevmode\nobreak\
d\ell\leavevmode\nobreak\
\mathbb{N}_{X_{\ell}}(d\omega\cap\\{\omega_{*}>0\\}).$

In particular, the distribution $(X_{(\tau_{s}-\ell)\vee
0})_{0\leqslant\ell\leqslant\tau_{s}-\tau_{r_{1}}}$ is absolute continuous
with respect to the distribution of a Brownian motion started from $s$ and
stopped when it hits $r_{1}$. We can now apply [17, Proposition 2] to deduce
that the distribution of
$(Z_{r}^{r,s},Z_{r}^{r,s}(\varepsilon))_{r\in[r_{1},r_{2}]}$ is absolute
continuous with respect to the distribution
$(\mathcal{Z}_{r-r_{1}},\mathcal{Z}^{\varepsilon}_{r-r_{1}})_{r\in[r_{1},r_{2}]}$
under $\mathbb{N}_{s-r_{1}}$. Consequently (50) gives that:

$\sup_{r\in[r_{1},r_{2}]}|Z_{r}^{r,s}(\varepsilon)-Z_{r}^{r,s}|\leavevmode\nobreak\
\underset{\varepsilon\to 0}{\longrightarrow}\leavevmode\nobreak\
0\;,\qquad\Theta_{0}\hbox{-a.s.}$

which completes the proof. $\square$

## Acknowledgements

I warmly thank Jean-François Le Gall for all his suggestions and for carefully
reading earlier versions of this manuscript. The present work was supported by
the ERC Advanced Grant 740943 GEOBROWN.

## References

  * [1] C. Abraham, J.-F. Le Gall, Excursion theory for Brownian motion indexed by the Brownian tree. J. Eur. Math. Soc. (JEMS) 20, 2951–3016 (2018)
  * [2] E. Baur, G. Miermont, G. Ray, Classification of scaling limits of uniform quadrangulations with a boundary. Ann. Probab. 47, 3397–3477 (2019)
  * [3] J. Bertoin, Lévy processes. Cambridge Tracts in Mathematics (1998)
  * [4] J. Bertoin, T. Budd, N. Curien, I. Kortchemski, Martingales in self-similar growth-fragmentations and their connections with random planar maps. Probab. Theory Related Fields 172, 663–724 (2018)
  * [5] J. Bouttier, E. Guitter, G. Miermont, Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants. (2021), arXiv:2104.10084
  * [6] T. Budzinski, The hyperbolic Brownian plane. Probab. Theory Related Fields, 171, 503–541, (2018)
  * [7] P. Chassaing, B. Durhuus, Local limit of labeled trees and expected volume growth in a random quadrangulation. Ann. Probab., 34:879–917, (2006)
  * [8] N. Curien, J.-F. Le Gall, The Brownian plane. J. Theoret. Probab. 27, 1240–1291 (2014)
  * [9] N. Curien, J.-F. Le Gall, The hull process of the Brownian plane. Probab. Theory Related Fields 166, 187–231 (2016)
  * [10] T. Duquesne, J.-F. Le Gall, Random trees, Lévy processes and Spatial branching processes. Astérisque 281, (2002)
  * [11] J.-F. Le Gall, The Brownian snake and solutions of $\Delta u=u^{2}$ in a domain. Probab. Th. Rel. Fields 102, (1995)
  * [12] J.-F. Le Gall, Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Boston, (1999)
  * [13] J.-F. Le Gall, Subordination of trees and the Brownian map.Probab. Theory Related Fields 171, 819–864 (2018)
  * [14] J.-F. Le Gall, Brownian disks and the Brownian snake. Ann. Inst. H. Poincaré Probab. Stat. 55, 237–313 (2019)
  * [15] J.-F. Le Gall, T. Lehéricy, Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation. Ann. Probab. 47 (3) 1498 - 1540, (2019)
  * [16] J-F. Le Gall, F. Paulin, Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geometric and Functional Analysis 18 (2008)
  * [17] J.-F. Le Gall, A. Riera, Spine representations for non-compact models of random geometry. Probab. Theory Related Fields, to appear
  * [18] J.-F. Le Gall, A. Riera, Growth-fragmentation processes in Brownian motion indexed by the Brownian tree. Ann. Probab. 48 (4) 1742 - 1784, (2020)
  * [19] K. Krikun, Local structure of random quadrangulations. arXiv:math/0512304, (2014)
  * [20] J. Miller, S. Sheffield, Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding. Preprint, available at arXiv:1605.03563
  * [21] D. Williams, Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. London Math. Soc., Ser. 3, 28, 738–768 (1974)


aainstitutetext: Rudolf Peierls Centre for Theoretical Physics
Beecroft Building, Clarendon Laboratory, Parks Road, University of Oxford, OX1
3PU, UKbbinstitutetext: Institute for Theoretical Physics
Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands

# Kination, Meet Kasner: On The Asymptotic Cosmology of String
Compactifications

Fien Apers a Joseph P. Conlon a Martin Mosny b and Filippo Revello
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study runaway, kination-dominated epochs in string cosmology. We show how
the apparent classical decompactification runaway of the volume modulus,
described by a kination epoch in the 4-dimensional EFT, can be uplifted to a
classical Kasner solution in 10d in which the non-compact dimensions collapse
towards a Big Crunch. This can also be generalised for arbitrary spacetime and
compactification dimensions. We conclude with some comments on how this
picture is modified by quantum effects, and the need for both dynamical and
kinematical Swampland constraints.

##  1 Introduction

Cosmological solutions of string theory are important for several reasons. If
a string cosmology ever described the actual early universe, its dynamical
evolution is important for understanding the structure of this universe, our
home. On a more general note, cosmological solutions are also interesting for
the slightly more formal question of understanding the behaviours which are
allowed, even in principle, within string theory.

One common feature of string theory vacua is the presence of runaway
potentials towards an asymptotic limit Dine:1985he ; BrusteinSteinhardt . Such
runaway potentials can arise from gaugino condensation in heterotic string
theory, from non-perturbative superpotentials in type IIB compactifications,
and also from perturbative corrections in type IIA, type IIB and other models.

The simplest asymptotic limits of moduli space are the large volume / weak
coupling limits. In most scenarios, the moduli that control the approach to
these asymptotic limits are the Kähler (volume) and dilaton (string coupling)
moduli. The Kähler potential tends to be logarithmic in the asymptotic moduli,
for example

$K=-3\ln\left(T+\bar{T}\right)-\ln\left(S+\bar{S}\right)$ (1)

for a 1-modulus Calabi-Yau in the context of type IIB D3/D7 compactifications.
The canonically normalised fields are also logarithmic in these moduli,
$\Phi=\sqrt{\frac{3}{2}}\ln\tau$ and $\Psi=\sqrt{\frac{1}{2}}\ln s$, where
$\tau={\rm Re}(T)$ and $s={\rm Re}(S)=\frac{1}{g_{s}}$. It follows that
potentials with power-law expansions in $\tau^{-1}$ and $g_{s}$, as one would
expect for physics originating in the perturbative $\alpha^{{}^{\prime}}$ and
$g_{s}$ expansions in string theory, are exponential when written in the
canonical fields $\Phi$ and $\Psi$.

The development of precise scenarios of moduli stabilisation allows such
dynamics to be studied in a robust setting rather than through _ad hoc_
assumptions. In our analysis, we first restrict to dynamics where only a
single volume modulus controls the approach to the asymptotic limit, and
regard the other fields (such as the dilaton) as rigid. This assumption is
well motivated in the context of LVS Balasubramanian:2005zx , where the moduli
mass spectrum and generalised no-scale structure Conlon:2005ki  sees the
volume modulus singled out as the unique light field, while the dilaton is
flux-stabilised together with the other complex structure moduli at
parametrically massive values Giddings:2001yu ; Dasgupta:1999ss .

In this case, the simplest asymptotic potential (corresponding to a power-law
expansion in $\tau^{-1}$, i.e. in inverse volume) is

$V(\Phi)=V_{0}\exp\left(-\lambda\Phi\right),$ (2)

with $\lambda$ is a constant (although we can also consider double exponential
potentials, $V(\Phi)\sim\exp\left(-\exp\left(\lambda\Phi\right)\right)$,
arising from a purely non-perturbative superpotential $W=e^{-\alpha T}$). This
familiar behaviour has also recently been reinterpreted in terms of the
swampland de Sitter conjecture, at least in the asymptotic limit (see
Obied:2018sgi ).

As long as $\lambda>\sqrt{2}$, it has long been understood that the scalar
field evolution will result in kinetic energy domination and the scalar field
rolling away to the asymptotic $\Phi\to\infty$ limit Wetterich:1987fm ;
Ferreira:1997hj ; Copeland:1997et  (often referred to as a decompactification
limit). Values of $\lambda<\sqrt{2}$ can give accelerated expansion but are
hard to realise in string theory (see Garg:2018zdg ; Hebecker:2019csg ;
ValeixoBento:2020ujr ; Andriot:2020vlg ; Cicoli:2021fsd ; Cicoli:2021skd ;
Calderon-Infante:2022nxb ; Andriot:2022brg  for recent analyses). As well as
asymptotic limits, similar periods of kinetic energy domination can arise in
string cosmology as transitory phases, as an intermediary epoch between the
end of inflation and fields settling down into the present-day vacuum (see
Conlon:2022pnx  for a recent study).

In the asymptotic limit, such kinetic-dominated runaways are normally viewed
as phenomenologically undesirable, but not dangerous. The main point of this
paper is to emphasise that this runaway behaviour conceals a more pathological
aspect: its correct interpretation is as a 10-dimensional Kasner solution in
the string frame, with the three external dimensions contracting and the six
internal dimensions expanding. In an asymptotic limit, such a runaway solution
is headed not towards decompactification: but instead towards a Big Crunch
singularity in which the ‘non-compact’ dimensions all collapse.

The basic intuition for this is that, in a kinetic-dominated runaway, the
potential becomes negligible for the dynamics. As the potential tends to be
sourced by fluxes and branes within the compactified theory, in a limit of
zero potential the dynamics of compactified effective field theories, at least
in terms of the runaway field, should map onto sourceless cosmological
solutions in the higher-dimensional theory. However, sourceless cosmological
solutions do not allow for uniform expansion, but rather require the
inhomogeneous behaviour of Kasner cosmologies.

The study of time-dependent cosmological backgrounds in string theory has a
long history, including the study of the runaway epoch and generalisations of
Kasner cosmologies (for example, see Antoniadis:1988vi ; Mueller:1989in ;
Arkani-Hamed:1999fet ; Brandle:2000qp ; Wands:2002ka ; Giddings:2003zw ;
Giddings:2004vr  and for more recent work see Cicoli:2021fsd ; Cicoli:2021skd
; Rudelius:2022gbz ). However, this link of runaway stringy decompactification
potentials to higher-dimensional Kasner cosmologies, leading to a subsequent
Big Crunch in string frame, seems either not to have been made or to have so
dropped out of general awareness that it needs restating.

The structure of this paper is as follows. In section 2 we demonstrate the
relationship of Kasner and kination solution in 4d compactifications, first
for one-modulus models and then in more general scenarios. In section 3 we
consider the same question in compactifications to dimensions other than 4,
finding that the same picture can be generalised for an arbitrary number of
dimensions. In section 4 we consider the relationship to the swampland and the
question of which low-energy behaviours are dynamically possible within
quantum gravity, before concluding.

##  2 Kasner to Kination And Back Again

The Kasner solution Kasner:1921zz  is one of the oldest cosmological solutions
of vacuum general relativity. In 10-dimensional spacetime, and splitting
coordinates into $x_{i}$ (external) and $y_{j}$ (internal), the Kasner metric
becomes

$ds^{2}=-dt^{2}+\sum_{i=1}^{3}t^{2p_{i}}dx_{i}^{2}+\sum_{j=1}^{6}t^{2q_{j}}dy_{j}^{2},$
(3)

where

$\sum_{i}p_{i}+\sum_{j}q_{j}=1,\qquad\sum_{i}p_{i}^{2}+\sum_{j}q_{j}^{2}=1.$
(4)

We now relate various forms of this solution to the dynamics of 4-dimensional
effective field theories.

###  2.1 EFTs with a single modulus

In terms of 4-dimensional kination dynamics, we first consider the case where
only a single volume modulus appears in the 4-dimensional EFT dynamics. We can
then restrict the Kasner solution to homogeneous behaviour in the extra
dimensions. In this case, two possible Kasner solutions exist. The first has
$p_{i}=-1/3$ and $q_{j}=1/3$ (describing contraction in the 4-dimensional
space and expansion in the compact space), while the latter has $p_{i}=5/9$
and $q_{j}=-1/9$ (with the 4-dimensional space expanding and the compact space
contracting).

We first show that the former solution,

$ds^{2}=-dt^{2}+\sum_{i=1}^{3}t^{-2/3}dx_{i}^{2}+\sum_{j=1}^{6}t^{2/3}dy_{j}^{2},$

dimensionally reduces to the 4-dimensional runaway kination solution, with a
scalar field rolling asymptotically towards infinity.

Treating the extra dimensions as compact, it follows from the Kasner metric
that the extra-dimensional volume behaves as

$\left(\frac{{\rm Vol}(CY,t)}{{\rm
Vol}(CY,t_{0})}\right)\sim\left(\frac{t}{t_{0}}\right)^{2}.$ (5)

As long as this volume is small, we must be able to view the physics from the
perspective of 4-dimensional EFT, according to the standard procedures of
dimensional reduction Polchinski:1998rr . As applicable for LVS, we are
regarding both the string coupling and other moduli as rigid and fixed with a
high mass, allowing us to treat the only relevant rolling field as the volume
(note that developments in moduli stabilisation are crucial for this, as
otherwise it would be inconsistent to restrict the dynamics to only the volume
mode and neglect all the other moduli). Dropping the non-dynamical and
constant $g_{s}$ factors, the 4d Einstein frame metric relates to the 10d
string frame metric as

$g_{4d,Einstein}=\mathcal{V}g_{4d,s},$ (6)

where $\mathcal{V}$ is the internal volume in units of the string length
$l_{s}$ and $g_{4d,s}$ is the restriction of the 10d string frame metric to
the four large dimensions. We therefore have

$ds^{2}_{4d,E}=-t^{2}dt^{2}+t^{4/3}\left(dx_{1}^{2}+dx^{2}+dx_{3}^{2}\right).$
(7)

To put this in a more canonical form, we write $t_{E}=\frac{t^{2}}{2}$, to
give

$ds^{2}_{4d,E}=-dt_{E}^{2}+(2t_{E})^{2/3}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right),$
(8)

with the growing scale factor $a(t_{E})\sim t_{E}^{1/3}$ appropriate for a
kinetic energy-dominated universe.

In a 4-dimensional kinating phase, the evolution of the rolling scalar is

$\Phi(t_{E})=\Phi(t_{E,0})+\sqrt{\frac{2}{3}}M_{P}\ln\left(\frac{t_{E}}{t_{E,0}}\right).$
(9)

Using the standard Kähler potential $K=-3\ln\left(T+\bar{T}\right)$, it
follows that the canonical field $\Phi=\sqrt{\frac{3}{2}}\ln{\rm
Re}(T)=\sqrt{\frac{2}{3}}\ln\mathcal{V}$. In the kination effective field
theory, the compact volume therefore grows as $\mathcal{V}\sim t_{E}\sim
t^{2}$ – entirely consistent with the behaviour of the 10-dimensional Kasner
solution.

We can also run this argument upwards, starting with the kination dynamics of
a scalar field in 4-dimensions, with

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\Phi_{0}+\sqrt{\frac{2}{3}}M_{P}\ln\left(\frac{t_{E}}{t_{E,0}}\right),$
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
dt_{E}^{2}+t_{E}^{2/3}\left(dx_{1}^{2}+dx^{2}+dx_{3}^{2}\right).$ (10)

As above, our knowledge of the physical interpretation of moduli implies that
the Calabi-Yau volume grows as ${\rm Vol(CY)}\sim t_{E}/t_{E,0}$. Together
with our knowledge of the rules of dimensional reduction, this implies that
the corresponding 10-dimensional string frame metric, restricted to 4d,
behaves as

$ds_{string}^{2}=-\frac{dt_{E}^{2}}{t_{E}}+t_{E}^{-1/3}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right).$
(11)

Performing a coordinate redefinition $t=t_{E}^{1/2}$, we obtain

$ds_{string}^{2}=-dt^{2}+t^{-2/3}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right),$
(12)

which is the expected restriction of the Kasner solution to 4d.

This argument shows that pure runaway kination (i.e. a classical scalar field
rolling to the infinite decompactification limit in an otherwise empty
universe) should be interpreted as representing the higher dimensional Kasner
solution. The Kasner solution reveals a hidden problem with the asymptotics:
the 3 ordinary dimensions are actually shrinking in string frame, and so the
asymptotic limit is not 10-dimensional flat space but instead a singular
crunch in the ‘non-compact’ dimensions.

Note that this point was already lurking and implicitly present in the stringy
interpretation of the 4d kination solution. Although this solution may appear
unproblematic as the scale factor is growing as $a(t_{E})\sim t_{E}^{1/3}$,
this is within a frame where the 4-dimensional Planck scale, $M_{P}=2.4\times
10^{18}{\rm GeV}$, is fixed. As ${\rm Vol(CY)}\sim t_{E}/t_{E,0}$, we see that
the string scale actually behaves as
$m_{s}\sim\frac{M_{P}}{\sqrt{\mathcal{V}}}\sim\frac{M_{P}}{\sqrt{t_{E}}}$ and
so the string length behaves as $l_{s}\sim t_{E}^{1/2}$. So, although this
kinating universe does have a growing scale factor, its fundamental scale
$l_{s}$ is growing faster than the scale factor: given time, the fundamental
scale much catch up with the expanding universe, with catastrophic
consequences.

A similar analysis holds for the inwards kination solution, describing a roll
in from large compact volumes. Here, the appropriate 10d Kasner metric is

$ds^{2}=-dt^{2}+\sum_{i=1}^{3}t^{10/9}dx_{i}^{2}+\sum_{j=1}^{6}t^{-2/9}dy_{j}^{2},$

and so $\mathcal{V}_{CY}\sim t^{-2/3}$. The conversion to 4d Einstein frame
gives

$ds_{4d,E}^{2}=-t^{-2/3}dt^{2}+t^{4/9}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right).$

Defining $t_{E}=t^{2/3}$, this metric becomes (up to numerical factors)

$ds_{4d,E}^{2}=-dt_{E}^{2}+t_{E}^{2/3}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right),$

which is the kination metric. In the 4-dimensional stringy effective field
theory, this corresponds to cosmological evolution of the rolling volume field
with

$\Phi(t)=\Phi(t_{0})-\sqrt{\frac{2}{3}}M_{P}\ln\left(\frac{t_{E}}{t_{E,0}}\right).$
(13)

It is easy to see that this does indeed give $\mathcal{V}\sim t_{E}^{-1}\sim
t^{-2/3}$, as expected for consistency with the 10d Kasner solution.

###  2.2 Multiple Moduli and Toroidal Compactifications

It is instructive to extend this analysis to the case where multiple moduli
control the extra-dimensional volume. A useful example is when the Calabi-Yau
volume can be split as $T^{2}\times T^{2}\times T^{2}$, with all tori
dynamical. Cosmologically, this also describes fibred versions of LVS in which
the fibre (torus) moduli are lighter than the volume modulus and so remain
dynamical in the cosmological evolution Cicoli:2008gp , even though the blow-
up Kähler moduli are fixed along with the dilaton and complex structure
moduli.

We therefore consider a toroidal configuration, with an internal manifold of
$T^{2}\times T^{2}\times T^{2}$. In IIB compactifications, the chiral
multiplets are defined in terms of 4-cycle volumes $\tau_{i}$, so we write

$\mathcal{V}=\sqrt{\tau_{1}\tau_{2}\tau_{3}},$ (14)

(any numerical prefactor in Eq (14) is irrelevant). The corresponding Kähler
potential is

$K=-\ln\left(T_{1}+\bar{T}_{1}\right)-\ln\left(T_{2}+\bar{T}_{2}\right)-\ln\left(T_{3}+\bar{T}_{3}\right),$
(15)

with $T_{i}=\tau_{i}+ic_{i}$, where $c_{i}$ is the RR axion (as these RR
axions are not important for our subsequent dynamics, we shall not discuss
them further). The kinetic terms are rendered canonical in terms of new fields
$\phi_{i}=\sqrt{\frac{1}{2}}\ln\tau_{i}$. Within a kination-dominated runaway,
the effective equations of motion for the $\phi_{i}$ fields are

$\ddot{\phi}_{i}+3H\dot{\phi}_{i}=0,$ (16)

and are solved by

$\phi_{i}(t)=\phi_{i}(t_{E,0})+\alpha_{i}M_{P}\ln\left(\frac{t_{E}}{t_{E,0}}\right).$
(17)

The Hubble scale is given by

$H=\frac{1}{M_{P}}\sqrt{\frac{1}{6}\left(\dot{\phi}_{1}^{2}+\dot{\phi}_{2}^{2}+\dot{\phi}_{3}^{2}\right)}=\frac{1}{t_{E}}\sqrt{\frac{\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}}{6}}.$
(18)

and so solutions of the equations of motion satisfy

$\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}=\frac{2}{3}.$ (19)

The physical volume of each 4-cycle behaves as

$\left(\frac{\tau_{i}}{\tau_{i,0}}\right)=\left(\frac{t_{E}}{t_{E,0}}\right)^{\alpha_{i}\sqrt{2}},$
(20)

and so the overall volume behaves as

$\frac{\mathcal{V}}{\mathcal{V}_{0}}\sim\sqrt{\tau_{1}\tau_{2}\tau_{3}}=\left(\frac{t_{E}}{t_{E,0}}\right)^{\left(\alpha_{1}+\alpha_{2}+\alpha_{3}\right)/\sqrt{2}}.$
(21)

To connect this to the Kasner solution, note that the volume of 4-cycle moduli
$\tau_{i}$ relates to the volume of the 2-tori $t_{i}$ as
$\tau_{i}\sim\epsilon_{ijk}t_{j}t_{k}$, and so $\tau_{1}\sim t_{2}t_{3}$. It
follows from Eq. (20) that the time-dependences of the 2-cycle volumes $t_{i}$
are

$t_{1}\sim t_{E}^{\lambda_{1}},\qquad t_{2}\sim t_{E}^{\lambda_{2}},\qquad
t_{3}\sim t_{E}^{\lambda_{3}},$

where

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{2}+\alpha_{3}-\alpha_{1}}{\sqrt{2}},$ (22)
$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}+\alpha_{3}-\alpha_{2}}{\sqrt{2}},$ (23)
$\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}+\alpha_{2}-\alpha_{3}}{\sqrt{2}}.$ (24)

As the exponents $\lambda_{i}$ directly tell us about the growth of the three
2-tori within the 6-dimensional space, they allow us to re-interpret the
4-dimensional kination solution as a Kasner solution within the 10-dimensional
theory. Using Eq. (21) to Eq. (24) and rescaling the 4d metric from 4d
Einstein frame to 10d string frame, the required Kasner metric is

$\displaystyle ds_{10d}^{2}$ $\displaystyle=$
$\displaystyle\frac{-dt_{E}^{2}}{t_{E}^{\frac{\left(\alpha_{1}+\alpha_{2}+\alpha_{3}\right)}{\sqrt{2}}}}+t_{E}^{\frac{2}{3}-\frac{\left(\alpha_{1}+\alpha_{2}+\alpha_{3}\right)}{\sqrt{2}}}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right)$
$\displaystyle+t_{E}^{\frac{\alpha_{2}+\alpha_{3}-\alpha_{1}}{\sqrt{2}}}\left(dy_{1}^{2}+dy_{2}^{2}\right)+t_{E}^{\frac{\alpha_{1}+\alpha_{3}-\alpha_{2}}{\sqrt{2}}}\left(dy_{3}^{2}+dy_{4}^{2}\right)+t_{E}^{\frac{\alpha_{2}+\alpha_{3}-\alpha_{1}}{\sqrt{2}}}\left(dy_{5}^{2}+dy_{6}^{2}\right).$

To turn this into canonical Kasner form, we write
$t_{E}=t^{\frac{1}{1-\frac{\left(\alpha_{1}+\alpha_{2}+\alpha_{3}\right)}{2\sqrt{2}}}}$.
The metric then becomes

$ds_{10d}^{2}=-dt^{2}+t^{2p}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right)+t^{2q_{1}}\left(dy_{1}^{2}+dy_{2}^{2}\right)+t^{2q_{2}}\left(dy_{3}^{2}+dy_{4}^{2}\right)+t^{2q_{3}}\left(dy_{5}^{2}+dy_{6}^{2}\right),$
(26)

where the Kasner exponents are given by

$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{2}-3\left(\alpha_{1}+\alpha_{2}+\alpha_{3}\right)}{6\sqrt{2}-3\left(\alpha_{1}+\alpha_{2}+\alpha_{3}\right)},$
(27) $\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle\frac{\left(\alpha_{2}+\alpha_{3}-\alpha_{1}\right)}{2\sqrt{2}-\left(\alpha_{1}+\alpha_{2}+\alpha_{3}\right)},$
(28) $\displaystyle q_{2}$ $\displaystyle=$
$\displaystyle\frac{\left(\alpha_{1}+\alpha_{3}-\alpha_{2}\right)}{2\sqrt{2}-\left(\alpha_{1}+\alpha_{2}+\alpha_{3}\right)},$
(29) $\displaystyle q_{3}$ $\displaystyle=$
$\displaystyle\frac{\left(\alpha_{1}+\alpha_{2}-\alpha_{3}\right)}{2\sqrt{2}-\left(\alpha_{1}+\alpha_{2}+\alpha_{3}\right)}.$
(30)

It can be verified that these exponents indeed satisfy the Kasner conditions,

$\displaystyle 3p+2\left(q_{1}+q_{2}+q_{3}\right)$ $\displaystyle=$
$\displaystyle 1,$ (31) $\displaystyle
3p^{2}+2\left(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}\right)$ $\displaystyle=$
$\displaystyle 1,$ (32)

provided that the condition of Eq. (19) is satisfied, namely
$\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}=\frac{2}{3}$ – thus showing that
the 4-dimensional kination solution can indeed be interpreted as a higher-
dimensional Kasner cosmology in string frame.

It is again straightforward to reverse this argument, so that it now runs from
the 10-dimensional Kasner solution into a 4-dimensional kination solution. We
start with a 10-dimensional Kasner solution,

$ds_{10d}^{2}=-dt^{2}+t^{2p}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right)+t^{2q_{1}}\left(dy_{1}^{2}+dy_{2}^{2}\right)+t_{s}^{2q_{2}}\left(dy_{3}^{2}+dy_{4}^{2}\right)+t^{2q_{3}}\left(dy_{5}^{2}+dy_{6}^{2}\right).$
(33)

As the volume of the internal dimensions is $\mathcal{V}_{CY}\sim
t_{s}^{2\left(q_{1}+q_{2}+q_{3}\right)}$, the 4d Einstein frame metric is

$ds_{4d,E}^{2}=-t_{s}^{2\left(q_{1}+q_{2}+q_{3}\right)}dt_{s}^{2}+t_{s}^{2p+2\left(q_{1}+q_{2}+q_{3}\right)}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right).$
(34)

Using the coordinate redefinition
$t_{E}=t_{s}^{1+\left(q_{1}+q_{2}+q_{3}\right)}$, we obtain

$ds_{4d,E}^{2}=-dt_{E}^{2}+t_{E}^{\frac{2p+2\left(q_{1}+q_{2}+q_{3}\right)}{1+\left(q_{1}+q_{2}+q_{3}\right)}}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right),$
(35)

which indeed reduces to the kination metric

$ds_{4d,E}^{2}=-dt_{E}^{2}+t_{E}^{2/3}\left(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\right)$
(36)

provided the exponents satisfy the Kasner condition
$3p+2\left(q_{1}+q_{2}+q_{3}\right)=1$.

###  2.3 Perturbations of Kasner

If a 4-dimensional runway kination solution is equivalent to a higher-
dimensional Kasner solution, then it should also be the case that
perturbations of this Kasner solution should have an interpretation as
perturbations within the 4-d kination solution. In the case of the kination
solution, it is long understood that a radiation perturbation of the kination
solution will grow relative to the background, ending up in a tracker solution
Wetterich:1987fm ; Copeland:1997et ; Ferreira:1997hj . A growing perturbation
of the Kasner solution, which places energy in a mode which corresponds, under
dimensional reduction, to radiation degrees of freedom in the 4-dimensional
theory, should then map under dimensional reduction to the growth of radiation
in a kination background.

While we leave a full analysis including wave-like behaviour for future work,
we discuss here some simple perturbations of the Kasner solution. Hence, we
consider the perturbed Kasner metric

$\begin{split}ds^{2}=ds_{\text{kasner}}^{2}&+\frac{2\epsilon
t^{2/3}}{\sqrt{6}}\sum_{\begin{subarray}{c}i=1,2,3\\\
j={4,\dots,9}\end{subarray}}h_{ij}(t)dx_{i}dy_{j-3}\\\
&+2\epsilon^{2}t^{2/3}\sum_{\begin{subarray}{c}i,j=1,2,3\\\
i<j\end{subarray}}h_{ij}^{2}(t)dx_{i}dx_{j}+\epsilon^{2}t^{2/3}\sum_{i=1,2,3}h_{ii}^{2}(t)dx_{i}^{2}.\end{split}$
(37)

This involves turning on components of the metric (the mixed $g_{xy}$
components) which correspond on toroidal dimensional reduction to vector
fields, $A_{\mu}$.111Although we note that on a full Calabi-Yau the absence of
1-cycles implies that no $U(1)$ gauge field would arise in this manner. We can
solve the vacuum Einstein equations to first order in $\epsilon$ if

$h_{ij}^{\prime}(t)=0,$ (38)

for all $i$ and $j$, whereas if we further restrict to

$h_{ij}(t)=1$ (39)

the solution is then exact to all orders.

Note that locally this perturbation is equivalent to the Kasner metric, as
they are related through the coordinate transformation

$x_{i}\rightarrow x_{i},\ \ \ \ \ \ \ y_{j}\rightarrow
y_{j}+\tfrac{\epsilon}{\sqrt{6}}(x_{1}+x_{2}+x_{3}).$ (40)

However, such a coordinate transformation violates the global structure of the
metric. In particular, it mixes the compact and noncompact coordinates and
destroys the coordinate periodicity of the compactification structure

$x_{i}\sim x_{i}+2\pi R_{i},\ \ \ \ \ \ \ y_{j}\sim y_{j}+2\pi R_{j}.$ (41)

In contrast, the perturbed metric is defined within the compactification
coordinates which satisfy the periodicity conditions, allowing for a simple
and physical interpretation of compactification on reduction to the
4-dimensional theory.

Reducing to 4d as previously, we now get a string frame metric

$ds_{\text{string}}^{2}=-dt^{2}+t^{-2/3}(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2})+\epsilon^{2}t^{2/3}(dx_{1}+dx_{2}+dx_{3})^{2}.$
(42)

In Einstein frame, we have

$ds_{4,E}^{2}=t^{2}ds^{2}_{\text{string}}=-d\tau^{2}+(2\tau)^{2/3}(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2})+\epsilon^{2}(2\tau)^{4/3}(dx_{1}+dx_{2}+dx_{3})^{2},$
(43)

with $\tau=t^{2}/2$. To first order in $\epsilon$ the scale factor is
$a(\tau)=\tau^{1/3}$ while the energy density associated to (43) is

$\rho=\tfrac{1}{24\pi}\tau^{-2}+\tfrac{2^{5/3}}{24\pi}\epsilon^{2}\tau^{-4/3}\sim
a^{-6}+\epsilon^{2}a^{-4},$ (44)

which is the energy density applicable to kination with a small but growing
radiation fraction, consistent with the expected behaviour in 4d EFT. Although
we leave a full analysis of perturbations to future work, this provides
evidence that perturbed Kasner solutions indeed capture the growth of
radiation in a kination background, viewed from a 4d perspective.

##  3 Kasner and Kination in Higher Dimensions

We next investigate whether similar behaviours hold for compactifications from
$D$ to $(d+1)$ spacetime dimensions, where $D$ and $d$ can take any values
(provided that $D>d+1>1$).

###  3.1 Kasner to Kination

In a $(d+1)$-dimensional FLRW spacetime with vanishing potential, the
Friedmann equation is (See Appendix B for details)

$\left(\frac{\dot{a}}{a}\right)^{2}=H^{2}=\frac{2}{d(d-1)M_{P,d+1}^{2}}\rho=\frac{\dot{\phi}^{2}}{M_{P,d+1}^{2}d(d-1)},$
(45)

where $M_{P,d+1}$ is the $(d+1)$-dimensional Planck mass. The scalar equation
of motion is

$\ddot{\phi}+dH\dot{\phi}=0.$ (46)

Together with Eq. (45), this implies

$\phi(t_{E})=\phi_{0}(t_{E})+\sqrt{\frac{d-1}{d}}M_{P,d+1}\log\left(\frac{t_{E}}{t_{E,0}}\right),\quad\quad\quad\rho(t_{E})=\frac{d-1}{2d}\frac{M_{P,d+1}^{2}}{t_{E}^{2}},$
(47)

with the scale factor and Hubble scale behaving as

$\displaystyle H(t_{E})$ $\displaystyle=$ $\displaystyle\frac{1}{d\,t_{E}},$
(48) $\displaystyle a(t_{E})$ $\displaystyle=$
$\displaystyle\left(\frac{t_{E}}{t_{E,0}}\right)^{1/d},$ (49)

corresponding to a metric

$ds_{E,d}^{2}=dt_{E}^{2}-t_{E}^{\frac{2}{d}}\sum_{i=1}^{d}dx_{i}^{2}.$ (50)

We ask whether Eq. (50) can be obtained through compactification of a
$D$-dimensional Kasner solution. For a generic $D-$dimensional spacetime, a
Kasner solution compatible with an isotropic $d$-dimensional space has a
metric

$ds^{2}=dt^{2}-t^{2p}\sum_{i=1}^{d}dx_{i}^{2}-\sum_{j=1}^{D-d-1}t^{2q_{i}}dy_{j}^{2},$
(51)

where the exponents must satisfy

$\begin{cases}pd+\sum_{j=1}^{D-d-1}q_{j}=1,\\\
p^{2}d+\sum_{j=1}^{D-d-1}q_{j}^{2}=1.\\\ \end{cases}$ (52)

For homogeneous internal dimensions, $q_{i}=q$, with the two solutions

$\bar{p}_{\pm}=\frac{1\pm\sqrt{\frac{(D-2)(D-d-1)}{d}}}{D-1}\quad\quad\bar{q}_{\pm}=\frac{1\mp\sqrt{\frac{(D-2)d}{D-d-1}}}{D-1}.$
(53)

To see whether (51) can reproduce $(d+1)$-dimensional kination, as in section
2 we move to Einstein frame via
$g_{\mu\nu,E}=\left(\mathcal{V}_{D-d-1}\right)^{2/(d-1)}\,g_{\mu\nu,s}$ (See
Appendix A). As the higher dimensional volume scales as

$\mathcal{V}_{D-d-1}\sim t^{\sum_{j=1}^{D-d-1}q_{j}},$ (54)

the $(d+1)$-dimensional part of the metric becomes

$ds_{4d,E}^{2}=t^{\frac{2(1-pd)}{d-1}}dt^{2}-t^{\frac{2(1-p)}{d-1}}\sum_{i=1}^{d}dx_{i}^{2},$
(55)

where we have used (52). With the substitution

$t=\left(\frac{d-pd}{d-1}t_{E}\right)^{\frac{d-1}{d-pd}},$ (56)

we obtain

$ds_{4d,E}^{2}=dt_{E}^{2}-t_{E}^{\frac{2}{d}}\sum_{i=1}^{d}dx_{i}^{2},$ (57)

matching with (50).

Furthermore, as happens in $d=3$, the kinating solution is unstable to
perturbations (such as the presence of initial radiation), which drive it
towards a tracker solution in any dimension $d>1$. Indeed, radiation and
kination energy densities redshift as222Matter, which redshifts as
$a(t_{E})^{-d}$, can also drive kination towards a tracker solution, at a rate
faster than radiation. In practice though, it is hard to have states behaving
as matter when the Hubble scale is very high, for example after inflation.

$\rho_{{\text{rad}}}(t_{E})=\frac{\rho_{{\text{rad}},0}}{a(t_{E})^{d+1}}\quad\quad{\text{and}}\quad\quad\rho_{{\text{kin}}}(t_{E})=\frac{\rho_{{\text{kin}},0}}{a(t_{E})^{2d}}$
(58)

respectively, so that that the latter will eventually catch up with the
former. More precisely, one can repeat the analysis of Copeland:1997et , and
find that kination along a sufficiently steep exponential potential, in the
presence of radiation, always evolves to an attractor solution where the
energy densities have fixed ratios. The details are presented in Appendix B.

###  3.2 Kination to Kasner

We remark that, so far, we haven’t imposed the requirement that the scalar
field giving rise to kination be identified with the extra dimensional
volume(s). This aspect is less developed in $d\neq 3$ spatial dimensions, as
the nature of light scalar fields depends on the moduli stabilisation
scenario, and these have been most studied for compactifications to 3 spatial
dimensions. Since the volume has a fixed time dependence during kination, this
leads to an additional condition that must be satisfied in order for the
uplift to work.333It would also be interesting to consider the cases where
fields other than the volume, e.g. the dilaton, are driving kination.

For simplicity, let us firstly treat the case where a single scalar is
controlling the size of the extra dimensions, and all the exponents $q_{j}=q$
are equal (analogous to the LVS case considered in section 2). In the string
frame, the volume scales as

$\mathcal{V}_{S}\sim t^{\sum_{j=1}^{D-d+1}q_{j}}=t^{1-pd}\sim
t_{E}^{\frac{(1-pd)(d-1)}{d-pd}}$ (59)

If there is a single modulus controlling the size of the volume, the
canonically normalised scalar is (See Appendix A)

$\Phi=\sqrt{\frac{D-2}{(D-d-1)(d-1)}}M_{P,d+1}\log\mathcal{V}_{S},\quad\mathcal{V}_{S}\sim
t_{E}^{(d-1)\sqrt{\frac{D-d-1}{d(D-2)}}}.$ (60)

If $\Phi$ is the field responsible for kination, this should agree with (59).
Indeed, the two exponents are equal if

$p^{2}d(D-1)-2pd+2+d-D=0,$ (61)

which is implied by (52). Therefore, a kinating volume modulus can be uplifted
to a Kasner solution for any total spacetime dimension $D$ and
compactification dimension $d+1$.

##  4 Applications to the Swampland

The physics we have described involves evolution that, from a 4-dimensional
perspective, appears as an ordinary kinating scalar field. However, the
trajectory of the field leads towards a Big Crunch and so cannot be continued
indefinitely within the full theory. This is reminiscent of swampland
restrictions on allowed behaviours within 4d effective field theories within
quantum gravity UV completions.

###  4.1 What goes wrong in the effective field theory?

In a kination phase in 4d EFT, the field moves through roughly one Planckian
distance every Hubble time

$\Delta\Phi=\sqrt{\frac{2}{3}}M_{P}\ln\left(\frac{t_{E}}{t_{E,0}}\right).$
(62)

We have seen in section 2 that, even classically, this field profile cannot be
sustained indefinitely due to the presence of a Crunch singularity. But what
goes wrong? How can we see this from a 4-dimensional perspective and how does
the 4-dimensional effective field theorist know about his or her imminent
demise? When measuring scales, the effective field theorist sees
$\frac{H}{m_{KK}}\to 0$ and $\frac{H}{m_{\Psi}}\to 0$ along the kination roll,
an apparent indicator that they are being rapidly transported to the Isles of
the Blessed in which all couplings are weak and all volumes are large. But, in
fact, they are doing Turkey Effective Field Theory: and Christmas (or
Thanksgiving) is coming in the form of the Crunch.

In a theory such as LVS, the light moduli are the volume and the axion; all
other moduli obtain large masses and so can consistently be integrated out of
the low-energy effective field theory. Within the cosmological kination
solution, the masses of these other moduli satisfy $m_{\Psi}>H$ and so will
not be subject to gravitational particle production. Although it is true that
a tower of particles becomes progressively lighter as the kinating field
evolves to the boundary of moduli space, consistent with the Swampland
Distance Conjecture Klaewer:2016kiy , it is always the case that
$m_{\text{tower}}>H$, and so the single-modulus description remains valid
within low-energy effective field theory. The Transplanckian Censorship
Conjecture (TCC) Bedroya:2019snp  is also satisfied,444As is a refined version
proposed in Andriot:2022brg . as

$\frac{a(t_{E})}{a(t_{E,0})}\frac{H(t_{E})}{M_{P}}=\left(\frac{t_{E}}{t_{E,0}}\right)^{1/3}\frac{1}{3M_{P}t}\ll
1$ (63)

for any times $t_{E}>t_{E,0}$. Indeed, what is interesting is that the
singularity arising en route to decompactification does not appear to trace
back to a violation of the refined distance conjecture Ooguri:2006in ;
Klaewer:2016kiy ; Baume:2016psm ; Ooguri:2018wrx , the EFT paradigm555We refer
to Section 2.1 of Conlon:2022pnx  for a more detailed discussion on the
validity of the EFT approach. or any other swampland constraint. The behaviour
involved instead appears harmless: a runway towards the weak-coupling, large-
volume decompactification limit.

One 4-dimensional perspective on the underlying problem is that, although the
scale factor $a(t_{E})\sim t_{E}^{1/3}$ is growing in Einstein frame, the
fundamental string length $l_{s}$ is growing faster ($l_{s}\sim t_{E}^{1/2}$)
and the Crunch corresponds to the latter catching up with the former. However,
to foresee this within 4-dimensional effective field theory, we need to know
enough of the fundamental theory to know that a string length exists and what
size it has.

One further 4-dimensional aspect to the question can be identified. Our
analysis has focused on classical evolution and a classical runaway, starting
with an empty universe with just a rolling modulus. In a quantum universe,
even if initially empty, quantum particle production will occur and generate a
small fraction of radiation (for example, for either the graviton or the
massless axion always present in LVS). Along the kination roll, any initial
amount of radiation, however small, will grow in relative importance (as
$\rho_{\gamma}\sim a^{-4}$ while $\rho_{kination}\sim a^{-6}$) and the
solution will evolve towards a tracker solution, moving away from the kination
(and uplifted Kasner) dynamics. So, it is clear that quantum effects will
disrupt this analysis and allow the kination phase only to last for a finite
amount of time.

###  4.2 Dynamics and Kinematics in the Swampland

There is a large literature on swampland conjectures: bounds on the possible
allowed forms of low-energy field theories consistent with quantum gravity.
The vast majority of these, however, are _kinematical_ in nature: they are
statements about the masses of towers of particles at different points in
moduli space (for example, as in Grimm:2018ohb ; Blumenhagen:2018nts ;
Grimm:2018cpv ; Corvilain:2018lgw ) or the charge-to-mass ratio of particles
at points in moduli space (as in the weak gravity conjecture Lee:2018urn ;
Lee:2018spm ; Gendler:2020dfp ; Bastian:2020egp ).

Such conjectures do not address _dynamical_ evolution, except for a limited
number of examples such as the TCC Bedroya:2019snp  or the Festina Lente
Montero:2019ekk ; Montero:2021otb  bound. The results of section 2 show that
the ‘obvious’ route to decompactification is anything but: rather than leading
to decompactification into flat and empty 10-dimensional space, it instead
produces a Big Crunch singularity within our three spatial dimensions. This
motivates the question of whether there are restrictions on the nature of
dynamical trajectories towards the asymptotic regions of moduli space, such as
the large-volume and weak-coupling limits? More in general, it would be
interesting to come up with constraints on the allowed trajectories in field
space, in a similar spirit to Calderon-Infante:2022nxb ; Calderon-
Infante:2020dhm . Although our focus has been on evolution towards
decompactified large-volume weak-coupling limits of moduli space, it would
also be interesting to study an evolution towards other singular loci in
moduli space, such as the ones analysed in Grimm:2019ixq .

Another perspective on this is that in a kination phase, the fields moves
through roughly a Planckian distance every Hubble time

$\Delta\Phi=\sqrt{\frac{2}{3}}M_{P}\ln\left(\frac{t_{E}}{t_{E,0}}\right)$ (64)

and so kination defines a field profile in spacetime in which the field vev
alters by a Planckian distance every Hubble time. The essence of section 2 was
that there are significant problems with such a profile. This suggests that
fundamental constraints may exist on field profiles with parametrically trans-
Planckian temporal field excursions. We leave, however, a precise formulation
of such constraints for future work.

##  5 Conclusions

We have seen in this paper how the runaway, kination-dominated behaviour of
scalar fields in string theory conceals an interesting higher-dimensional
interpretation in terms of a 10-dimensional Kasner solution. This behaviour is
interesting for both phenomenological and more formal reasons. From a
phenomenological side, in various scenarios in string cosmology the universe
goes through a kination phase between inflation and the final minimum (e.g.
see Conlon:2022pnx  for a recent study). The higher-dimensional solution may
give a novel perspective on the physics of this epoch.

From a more formal perspective, the kination runaway represents behaviour
which appears satisfactory within 4-dimensional EFT, but conceals a pathology
within the full theory. This is reminiscent of the swampland, and motivates
the development of swampland constraints which are _dynamical_ in nature,
corresponding to the time evolution of systems, rather than merely _kinematic_
statements about theories at particular points in moduli space.

Kination epochs in string cosmology are common, result in trans-Planckian
field excursions, but are relatively understudied in the string cosmology
literature. We regard the physics described in this paper as an example of why
they are interesting and merit more detailed study.

### Acknoweldgements

We thank Prateek Agrawal, Ed Copeland, Thomas Grimm and Sirui Ning for
discussions. FA is supported by the Clarendon Scholarship in partnership with
the Scatcherd European Scholarship, Saven European Scholarship, and the
Hertford College Peter Howard Scholarship. We acknowledge support from STFC
grant ST/T000864/1. The research of FR is partly supported by the Dutch
Research Council (NWO) via a Start-Up grant and a Vici grant. For the purpose
of Open Access, the authors have applied a CC BY public copyright licence to
any Author Accepted Manuscript version arising from this submission.

##  Appendix A Dimensional reduction

We briefly review here how one can obtain the relation between the canonically
normalised volume modulus in $d+1$ dimensions and the volume of the compact
space. Although this can easily be achieved within a supergravity description
(such as $4d$ $\mathcal{N}=1$ theories), direct dimensional reduction gives
the result for any dimension. We start from a $D$-dimensional Einstein Hilbert
action,

$S_{EH}=\frac{1}{2k_{D}^{2}}\int
d^{D}x\,\sqrt{-\tilde{g}_{D}}\mathcal{\tilde{R}}_{D},$ (65)

and decompose the higher dimensional metric as

$ds^{2}=\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2\omega(x)}\tilde{g}_{mn}dy^{m}dy^{n}\quad\mu,\nu=0,..,d\quad
m,n=d+1,..,D-1.$ (66)

Therefore, the higher dimensional volume (in the string frame) is given by

$\mathcal{V}=e^{(D-d-1)\omega(x)}.$ (67)

Let us first perform a Weyl rescaling on the $D$-dimensional metric, through
the field redefinition

$\tilde{g}_{ab}^{D}=e^{2\omega(x)}g_{ab}^{D}.$ (68)

By use of standard identities in GR, the Lagrangian density can be rewritten
as

$\sqrt{-\tilde{g}_{D}}\mathcal{\tilde{R}}_{D}=\sqrt{-g_{D}}e^{(D-2)\omega(x)}\left(\mathcal{R}_{D}-2(D-1)\nabla^{2}\omega-(D-1)(D-2)\partial_{a}\omega\partial^{a}\omega\right),$
(69)

where all the contractions and covariant derivatives on the right-hand side
are performed with respect to the new metric $g_{ab}^{D}$. Since the metric is
now a product metric of two spaces, the higher dimensional Ricci scalar
decomposes as the sum of the Ricci scalar in the two spaces, _i.e._

$\mathcal{R}_{D}=\mathcal{R}_{d+1}+\mathcal{R}_{D-d-1},$ (70)

with $g_{\mu\nu}^{d+1}$ the $(d+1)$-dimensional block of the full metric
$g_{ab}^{D}$. In the bulk we are assuming the source-less Einstein equations
to be satisfied, so $\mathcal{R}_{(D-d-1)}=0$. We can then compactify to $d+1$
dimensions. Integrating out the KK modes from the spectrum, the action becomes

$\displaystyle S_{EH}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{P,d+1}}{2}\int
d^{d+1}x\sqrt{-g_{d+1}}e^{(D-2)\omega(x)}$
$\displaystyle\times\left(\mathcal{R}_{d+1}-2(D-1)\nabla^{2}\omega-(D-1)(D-2)\partial_{\mu}\omega\partial^{\mu}\omega\right).$

We have defined the $d$-dimensional (reduced) Planck mass as

$\frac{M^{2}_{P,d+1}}{2}=\frac{\tilde{\mathcal{V}}}{2k_{D}^{2}},$ (72)

where ${\tilde{\mathcal{V}}}$ is the volume calculated with the metric
$\tilde{g}_{mn}$. We can now perform a second Weyl rescaling of the
$d+1$-dimensional metric only,

$g_{\mu\nu}^{d+1}=e^{-\frac{2(D-2)}{d-1}\omega(x)}h_{\mu\nu},$ (73)

to make the Ricci scalar in $d+1$ dimensions canonically normalised. The
latter will transform as dictated by (69) under a scale transformation (with
$D=d+1$), while the Laplacian transforms as

$\nabla_{g}^{2}\,\omega=e^{-\frac{2(D-2)}{d-1}\omega(x)}\left(\nabla^{2}_{h}\,\omega-(D-2)h^{\mu\nu}\partial_{\mu}\omega\partial_{\nu}\omega\right).$
(74)

Putting everything together, and turning any total derivatives into boundary
terms, the action becomes

$S_{EH}=\frac{M^{2}_{P,d+1}}{2}\int
d^{d+1}x\sqrt{-h}\left(\mathcal{R}_{h}-\frac{(D-2)(D-d-1)}{d-1}\partial_{\mu}\omega\partial^{\mu}\omega\right),$
(75)

so that the normalised volume scalar takes the form

$\Phi=\sqrt{\frac{D-2}{(D-d-1)(d-1)}}M_{P,d+1}\log\mathcal{V}.$ (76)

For $D=10$ and $d=3$, we recover
$\Phi=\sqrt{\frac{2}{3}}M_{P}\log\mathcal{V}$. Combining Eqs. (67), (68) and
(73), we also see that the Einstein frame metric $h_{\mu\nu}$ is related to
the string frame metric $\tilde{g}_{\mu\nu}$ by

$h_{\mu\nu}=\mathcal{V}^{\frac{2}{(d-1)}}\tilde{g}_{\mu\nu},$ (77)

which correctly reduces to $h_{\mu\nu}=\mathcal{V}\tilde{g}_{\mu\nu}$ for
$d=3$.

##  Appendix B Tracker solution in arbitrary dimensions

We here show the existence of tracker solutions in arbitrary dimensions, for a
kinating scalar field rolling along an exponential potential and in the
presence of radiation. This confirms the expectations of Eq (58). Our analysis
will closely parallel that of Copeland:1997et , generalising it to an
arbitrary number of dimensions. We start from a $(d+1)$ dimensional FLRW
metric,

$ds^{2}=-dt_{E}^{2}+a(t_{E})^{2}\sum_{i=1}^{d}dx_{i}^{2},$ (78)

and assume that (together with the kinating scalar) another cosmic fluid is
present. For simplicity, we only consider the case of radiation, whose
conservation law and equation of state are given by

$\dot{\rho}+dH\left(\rho_{\gamma}+P_{\gamma}\right)=0,\quad\quad\quad
P_{\gamma}=\frac{\rho_{\gamma}}{d}.$ (79)

The $(d+1)$-dimensional Friedmann equations can be written as (see for example
Nojiri:2002hz ; Chen:2014fqa )

$\dot{H}=-\frac{M^{2}_{P,d+1}}{d-1}\left(\rho_{\gamma}+P_{\gamma}+\frac{\dot{\Phi}^{2}}{2}\right),$
(80)
$H^{2}=\frac{2}{d(d-1)M_{P,d+1}^{2}}\left(\rho_{\gamma}+\frac{1}{2}\dot{\Phi}^{2}+V(\Phi)\right),$
(81)

and the equation of motion for the spatially homogeneous scalar is given by

$\ddot{\Phi}+dH\dot{\Phi}+\frac{\partial V}{\partial\Phi}=0.$ (82)

We define the variables

$x=\frac{1}{HM_{P,d}}\frac{\dot{\Phi}}{\sqrt{d(d-1)}},\quad\quad
y=\frac{1}{HM_{P,d}}\sqrt{\frac{2V(\Phi)}{d(d-1)}},$ (83)

and take the specific choice of an exponential potential
$V=V_{0}e^{-\frac{\lambda\Phi}{M_{P,d+1}}}$. The Friedmann equations can then
be recast as the dynamical system

$\begin{cases}x^{\prime}=-dx+\lambda\sqrt{\frac{d(d-1)}{4}}y^{2}+x\left(dx^{2}+\frac{d+1}{2}\left(1-x^{2}-y^{2}\right)\right),\\\
y^{\prime}=-\lambda\sqrt{\frac{d(d-1)}{4}}xy+y\left(dx^{2}+\frac{d+1}{2}\left(1-x^{2}-y^{2}\right)\right),\\\
\end{cases}$ (84)

with the constraint that

$0\leq\Omega_{\gamma}\leq
1,\quad\quad\Omega_{\gamma}\equiv\frac{2\rho_{\gamma}}{d(d-1)H^{2}}=1-x^{2}-y^{2}.$
(85)

The primed quantities in (84) represent derivatives with respect to $N=\log
a$. Without loss of generality, we can assume $y\geq 0$, since sending
$y\rightarrow-y$ is equivalent to inverting the sign of $t$. The system above
always has the trivial critical points

$(x,y)_{1}=(0,0),\quad\quad\quad(x,y)_{2}=(1,0),\quad\quad\quad(x,y)_{3}=(-1,0),$
(86)

which correspond to the cases of first an empty universe and then kination (in
either direction for $\Phi$) respectively. A linear stability analysis around
those points shows they always have at least one runaway direction, and are
thus an unstable node or a saddle point. There are also two non-trivial
critical points for $d>1$, given by

$(x,y)_{4}=\left(\frac{\lambda}{2}\sqrt{\frac{d-1}{d}},\frac{1}{2}\sqrt{4-\lambda^{2}\frac{d-1}{d}}\right),\quad(x,y)_{5}=\left(\frac{d+1}{\lambda\sqrt{d(d-1)}},\frac{1}{\lambda}\sqrt{\frac{d+1}{d}}\right).$
(87)

The first critical point is characterised by $\Omega_{\gamma,4}=0$, and is
also present in the absence of radiation Rudelius:2022gbz . It is a solution
where the scalar field is not kinating ($y\neq 0$, so there is energy density
in the potential), but there no radiation is present. It is an attractor for

$\lambda^{2}<\frac{2(d+1)}{(d-1)},$ (88)

and a saddle point otherwise (if it exists). For the second critical point,
the constraint (85) implies it only exists for

$\lambda^{2}\geq\frac{2(d+1)}{(d-1)}\quad\quad{\text{as}}\quad\quad\Omega_{\gamma,5}=1-\frac{2}{\lambda^{2}}\frac{d+1}{d-1}.$
(89)

Linearising (84) around $(x,y)_{5}$, one obtains that this last solution is a
stable node for

$\frac{2(d+1)}{(d-1)}<\lambda^{2}<\frac{16(d+1)^{2}}{(9+7d)(d-1)}$ (90)

and a stable spiral for

$\lambda^{2}>\frac{16(d+1)^{2}}{(9+7d)(d-1)}.$ (91)

This analysis is in agreement with Table I of Copeland:1997et  for $d=3$.

## References

  * [1] M. Dine and N. Seiberg, _Is the Superstring Weakly Coupled?_ , _Phys. Lett. B_ 162 (1985) 299.
  * [2] R. Brustein and P. J. Steinhardt, _Challenges for superstring cosmology_ , _Phys. Lett. B_ 302 (1993) 196 [hep-th/9212049].
  * [3] V. Balasubramanian, P. Berglund, J. P. Conlon and F. Quevedo, _Systematics of moduli stabilisation in Calabi-Yau flux compactifications_ , _JHEP_ 03 (2005) 007 [hep-th/0502058].
  * [4] J. P. Conlon, F. Quevedo and K. Suruliz, _Large-volume flux compactifications: Moduli spectrum and D3/D7 soft supersymmetry breaking_ , _JHEP_ 08 (2005) 007 [hep-th/0505076].
  * [5] S. B. Giddings, S. Kachru and J. Polchinski, _Hierarchies from fluxes in string compactifications_ , _Phys. Rev. D_ 66 (2002) 106006 [hep-th/0105097].
  * [6] K. Dasgupta, G. Rajesh and S. Sethi, _M theory, orientifolds and G - flux_ , _JHEP_ 08 (1999) 023 [hep-th/9908088].
  * [7] G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, _De Sitter Space and the Swampland_ , 1806.08362.
  * [8] C. Wetterich, _Cosmology and the Fate of Dilatation Symmetry_ , _Nucl. Phys. B_ 302 (1988) 668 [1711.03844].
  * [9] P. G. Ferreira and M. Joyce, _Cosmology with a primordial scaling field_ , _Phys. Rev. D_ 58 (1998) 023503 [astro-ph/9711102].
  * [10] E. J. Copeland, A. R. Liddle and D. Wands, _Exponential potentials and cosmological scaling solutions_ , _Phys. Rev. D_ 57 (1998) 4686 [gr-qc/9711068].
  * [11] S. K. Garg, C. Krishnan and M. Zaid Zaz, _Bounds on Slow Roll at the Boundary of the Landscape_ , _JHEP_ 03 (2019) 029 [1810.09406].
  * [12] A. Hebecker, T. Skrzypek and M. Wittner, _The $F$-term Problem and other Challenges of Stringy Quintessence_, _JHEP_ 11 (2019) 134 [1909.08625].
  * [13] B. Valeixo Bento, D. Chakraborty, S. L. Parameswaran and I. Zavala, _Dark Energy in String Theory_ , _PoS_ CORFU2019 (2020) 123 [2005.10168].
  * [14] D. Andriot, P. Marconnet and T. Wrase, _Intricacies of classical de Sitter string backgrounds_ , _Phys. Lett. B_ 812 (2021) 136015 [2006.01848].
  * [15] M. Cicoli, F. Cunillera, A. Padilla and F. G. Pedro, _Quintessence and the Swampland: The Parametrically Controlled Regime of Moduli Space_ , _Fortsch. Phys._ 70 (2022) 2200009 [2112.10779].
  * [16] M. Cicoli, F. Cunillera, A. Padilla and F. G. Pedro, _Quintessence and the Swampland: The Numerically Controlled Regime of Moduli Space_ , _Fortsch. Phys._ 70 (2022) 2200008 [2112.10783].
  * [17] J. Calderón-Infante, I. Ruiz and I. Valenzuela, _Asymptotic Accelerated Expansion in String Theory and the Swampland_ , 2209.11821.
  * [18] D. Andriot, L. Horer and G. Tringas, _Negative scalar potentials and the swampland: an Anti-Trans-Planckian Censorship Conjecture_ , 2212.04517.
  * [19] J. P. Conlon and F. Revello, _Catch-me-if-you-can: the overshoot problem and the weak/inflation hierarchy_ , _JHEP_ 11 (2022) 155 [2207.00567].
  * [20] I. Antoniadis, C. Bachas, J. R. Ellis and D. V. Nanopoulos, _An Expanding Universe in String Theory_ , _Nucl. Phys. B_ 328 (1989) 117.
  * [21] M. T. Mueller, _Rolling Radii and a Time Dependent Dilaton_ , _Nucl. Phys. B_ 337 (1990) 37.
  * [22] N. Arkani-Hamed, S. Dimopoulos, N. Kaloper and J. March-Russell, _Rapid asymmetric inflation and early cosmology in theories with submillimeter dimensions_ , _Nucl. Phys. B_ 567 (2000) 189 [hep-ph/9903224].
  * [23] M. Brandle, A. Lukas and B. A. Ovrut, _Heterotic M theory cosmology in four-dimensions and five-dimensions_ , _Phys. Rev. D_ 63 (2001) 026003 [hep-th/0003256].
  * [24] D. Wands, _String inspired cosmology_ , _Class. Quant. Grav._ 19 (2002) 3403 [hep-th/0203107].
  * [25] S. B. Giddings, _The Fate of four-dimensions_ , _Phys. Rev. D_ 68 (2003) 026006 [hep-th/0303031].
  * [26] S. B. Giddings and R. C. Myers, _Spontaneous decompactification_ , _Phys. Rev. D_ 70 (2004) 046005 [hep-th/0404220].
  * [27] T. Rudelius, _Asymptotic scalar field cosmology in string theory_ , _JHEP_ 10 (2022) 018 [2208.08989].
  * [28] E. Kasner, _Geometrical theorems on Einstein’s cosmological equations_ , _Am. J. Math._ 43 (1921) 217.
  * [29] J. Polchinski, _String theory. Vol. 2: Superstring theory and beyond_ , Cambridge Monographs on Mathematical Physics. Cambridge University Press, 12, 2007, 10.1017/CBO9780511618123.
  * [30] M. Cicoli, C. P. Burgess and F. Quevedo, _Fibre Inflation: Observable Gravity Waves from IIB String Compactifications_ , _JCAP_ 03 (2009) 013 [0808.0691].
  * [31] D. Klaewer and E. Palti, _Super-Planckian Spatial Field Variations and Quantum Gravity_ , _JHEP_ 01 (2017) 088 [1610.00010].
  * [32] A. Bedroya and C. Vafa, _Trans-Planckian Censorship and the Swampland_ , _JHEP_ 09 (2020) 123 [1909.11063].
  * [33] H. Ooguri and C. Vafa, _On the Geometry of the String Landscape and the Swampland_ , _Nucl. Phys. B_ 766 (2007) 21 [hep-th/0605264].
  * [34] F. Baume and E. Palti, _Backreacted Axion Field Ranges in String Theory_ , _JHEP_ 08 (2016) 043 [1602.06517].
  * [35] H. Ooguri, E. Palti, G. Shiu and C. Vafa, _Distance and de Sitter Conjectures on the Swampland_ , _Phys. Lett. B_ 788 (2019) 180 [1810.05506].
  * [36] T. W. Grimm, E. Palti and I. Valenzuela, _Infinite Distances in Field Space and Massless Towers of States_ , _JHEP_ 08 (2018) 143 [1802.08264].
  * [37] R. Blumenhagen, D. Kläwer, L. Schlechter and F. Wolf, _The Refined Swampland Distance Conjecture in Calabi-Yau Moduli Spaces_ , _JHEP_ 06 (2018) 052 [1803.04989].
  * [38] T. W. Grimm, C. Li and E. Palti, _Infinite Distance Networks in Field Space and Charge Orbits_ , _JHEP_ 03 (2019) 016 [1811.02571].
  * [39] P. Corvilain, T. W. Grimm and I. Valenzuela, _The Swampland Distance Conjecture for Kähler moduli_ , _JHEP_ 08 (2019) 075 [1812.07548].
  * [40] S.-J. Lee, W. Lerche and T. Weigand, _Tensionless Strings and the Weak Gravity Conjecture_ , _JHEP_ 10 (2018) 164 [1808.05958].
  * [41] S.-J. Lee, W. Lerche and T. Weigand, _A Stringy Test of the Scalar Weak Gravity Conjecture_ , _Nucl. Phys. B_ 938 (2019) 321 [1810.05169].
  * [42] N. Gendler and I. Valenzuela, _Merging the weak gravity and distance conjectures using BPS extremal black holes_ , _JHEP_ 01 (2021) 176 [2004.10768].
  * [43] B. Bastian, T. W. Grimm and D. van de Heisteeg, _Weak gravity bounds in asymptotic string compactifications_ , _JHEP_ 06 (2021) 162 [2011.08854].
  * [44] M. Montero, T. Van Riet and G. Venken, _Festina Lente: EFT Constraints from Charged Black Hole Evaporation in de Sitter_ , _JHEP_ 01 (2020) 039 [1910.01648].
  * [45] M. Montero, C. Vafa, T. Van Riet and G. Venken, _The FL bound and its phenomenological implications_ , _JHEP_ 10 (2021) 009 [2106.07650].
  * [46] J. Calderón-Infante, A. M. Uranga and I. Valenzuela, _The Convex Hull Swampland Distance Conjecture and Bounds on Non-geodesics_ , _JHEP_ 03 (2021) 299 [2012.00034].
  * [47] T. W. Grimm, C. Li and I. Valenzuela, _Asymptotic Flux Compactifications and the Swampland_ , _JHEP_ 06 (2020) 009 [1910.09549].
  * [48] S. Nojiri, S. D. Odintsov and S. Ogushi, _Friedmann-Robertson-Walker brane cosmological equations from the five-dimensional bulk (A)dS black hole_ , _Int. J. Mod. Phys. A_ 17 (2002) 4809 [hep-th/0205187].
  * [49] S. Chen, G. W. Gibbons, Y. Li and Y. Yang, _Friedmann’s Equations in All Dimensions and Chebyshev’s Theorem_ , _JCAP_ 12 (2014) 035 [1409.3352].