# The field of moduli of a divisor on a rational curve

Giulio Bresciani  Scuola Normale Superiore
Piazza dei Cavalieri 7
56126 Pisa
Italy<EMAIL_ADDRESS>
###### Abstract.

Let $k$ be a field with algebraic closure $\bar{k}$ and
$D\subset\mathbb{P}^{1}_{\bar{k}}$ a reduced, effective divisor of degree
$n\geq 3$, write $k(D)$ for the field of moduli of $D$. A. Marinatto proved
that when $n$ is odd, or $n=4$, $D$ descends to a divisor on
$\mathbb{P}^{1}_{k(D)}$.

We analyze completely the problem of when $D$ descends to a divisor on a
smooth, projective curve of genus $0$ on $k(D)$, possibly with no rational
points. In particular, we study the remaining cases $n\geq 6$ even, and we
obtain conceptual proofs of Marinatto’s results and of a theorem by B. Huggins
about the field of moduli of hyperelliptic curves.

We work over a field $k$ with algebraic closure $\bar{k}$. A rational curve is
a smooth, projective, geometrically connected curve of genus $0$.

### The field of moduli

Let $D\subset\mathbb{P}^{1}_{\bar{k}}$ be an effective reduced divisor. If $k$
is perfect, consider the subgroup $H\subset\operatorname{Gal}(\bar{k}/k)$ of
elements $\sigma$ such that there exists an automorphism of
$\mathbb{P}^{1}_{\bar{k}}/\bar{k}$ mapping $D$ to $\sigma(D)$. The _field of
moduli_ $k(D)$ of $D$ is the subfield of $\bar{k}$ of elements fixed by $H$.
In our joint paper with A. Vistoli [BVb] we have generalized the definition of
the field of moduli to arbitrary $k$; we work in this generality.

If $\bar{k}/k^{\prime}/k$ is a sub-extension and there exists a rational curve
$P$ over $k^{\prime}$ with a divisor $D_{0}\subset P$ such that
$(P_{\bar{k}},D_{0,\bar{k}})\simeq(\mathbb{P}^{1}_{\bar{k}},D)$, we say that
$k^{\prime}$ is a field of definition for $(\mathbb{P}^{1}_{\bar{k}},D)$. The
field of moduli is contained in every field of definition.

### Marinatto’s results

Let $n$ be the degree of $D$, assume $n\geq 3$. Assuming $k$ perfect of
characteristic $\neq 2$, A. Marinatto [Mar13] showed that, if $n$ is odd or
$n=4$, then $D$ descends to a divisor of $\mathbb{P}^{1}_{k(D)}$. Hence, if
$n$ is odd or $n=4$, $k(D)$ is a field of definition for
$(\mathbb{P}^{1}_{\bar{k}},D)$.

For every even integer $n\geq 6$, Marinatto constructed examples of divisors
$D\subset\mathbb{P}^{1}_{\bar{k}}$ of degree $n$ which do not descend to
$\mathbb{P}^{1}_{k(D)}$. His examples descend to a non-trivial rational curve
over $k(D)$, though, hence $k(D)$ is a field of definition for
$(\mathbb{P}^{1}_{\bar{k}},D)$ in these cases.

### Our results

Because of this, the problem of fields of definition versus fields of moduli
for $n$ even and $\geq 6$ is still open. We solve it, and generalize
Marinatto’s results for $n$ odd and $n=4$ to an arbitrary base field. The
following theorem sums up the various results we obtain.

######  Theorem 1.

Let $k$ be any field, $n\geq 3$ an integer and
$D\subset\mathbb{P}^{1}_{\bar{k}}$ a reduced, effective divisor of degree $n$.
If at least one of the following conditions holds, then
$(\mathbb{P}^{1}_{\bar{k}},D)$ is defined over its field of moduli.

  * •

$\operatorname{Aut}(\mathbb{P}^{1},D)$ is not cyclic of degree even and prime
with $\operatorname{char}k$.

  * •

$n=4$.

  * •

$n=6$.

  * •

$n$ is odd.

In particular, the first condition implies that $(\mathbb{P}^{1}_{\bar{k}},D)$
is always defined over its field of moduli if $\operatorname{char}k=2$.

If $n=4$ or $n$ is odd, we can say more precisely that $D$ descends to a
divisor of $\mathbb{P}^{1}$ over the field of moduli.

On the other hand, if $\operatorname{char}k\neq 2$, $n\geq 8$ is even and the
$2$-torsion $\operatorname{Br}(k)[2]$ of the Brauer group of $k$ is non-
trivial, then we may choose $D$ such that the field of moduli is $k$ and
$(\mathbb{P}^{1}_{\bar{k}},D)$ is not defined over $k$.

For $n=4$ or $n$ odd Theorem 1 is a generalization of Marinatto’s theorem to
arbitrary fields. For $n$ even and $\geq 6$, it is new. In order to prove all
the various sub-statements of Theorem 1, we first characterize the divisors
$D\subset\mathbb{P}^{1}_{\bar{k}}$ such that the field of moduli is a field of
definition for $(\mathbb{P}^{1}_{\bar{k}},D)$, see Theorem 5.

### Discussion of proofs

Marinatto’s proof for $n$ odd is based on a case-by-case analysis of the
finite subgroups of $\mathrm{PGL}_{2}$. Our approach is more conceptual: for
$n$ odd, we only use general techniques for problems about fields of moduli
(not specific to divisors or rational curves) plus a parity counting. For $n$
even and $\geq 8$, we also need to make some elementary geometric arguments.
If $\operatorname{Aut}(\mathbb{P}^{1},D)$ is not cyclic of degree even and
prime with $\operatorname{char}k$, we have a conceptual argument based on the
Riemann-Hurwitz formula; from this case we get the cases $n=4,6$, too.

Marinatto also uses a theorem of Huggins [Hug07, Theorem 5.3] about fields of
moduli of hyperelliptic curves. Huggins’ proof is based on a case-by-case
analysis of the finite subgroups of $\mathrm{PGL}_{2}$, too. Our argument with
Riemann-Hurwitz directly yields a short, conceptual proof of Huggins’ result;
we include it at the end of the paper. Our version is slightly more general
than the original one since we do not assume the base field to be perfect.

While we use the general framework constructed in [BVb], we mention the fact
that, in dimension $1$ (which is our case), the main ideas are due to P. Dèbes
and M. Emsalem [DE99].

### Acknowledgements

This paper was born as part of my recent joint articles with A. Vistoli [BVa]
and [BVb]. I am grateful to him for many useful discussions, as well as for
pointing out Marinatto’s article to me.

##  1\. The residual gerbe and the compression

For the convenience of the reader, we recall some objects and constructions
from our joint paper with A. Vistoli [BVb].

Consider the functor $\mathcal{G}:\operatorname{Sch}/k\to\operatorname{Sets}$
sending a scheme $S$ over $k$ to the set of projective bundles $Q\to S$ of
relative dimension $1$ with an effective divisor $E\subset Q$ finite étale
over $S$ such that each geometric fiber is isomorphic to
$(\mathbb{P}^{1}_{\bar{k}},D)$, possibly after enlarging the base field. It
can be checked that $\mathcal{G}$ is an algebraic stack of finite type over
$k$ with finite, étale inertia.

The algebraic stack $\mathcal{G}$ is a gerbe over a field $k(D)$ finite over
$k$ which is, by definition, the field of moduli of
$(\mathbb{P}^{1}_{\bar{k}},D)$, and the base change
$\mathcal{G}\times_{k(D)}\bar{k}$ is isomorphic to the classifying stack
$\mathcal{B}_{\bar{k}}\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$ [BVb,
Proposition 3.10]. It is called the _residual gerbe_ of
$(\mathbb{P}^{1}_{\bar{k}},D)$. By definition, $(\mathbb{P}^{1}_{\bar{k}},D)$
is defined over $k(D)$ if and only if the residual gerbe $\mathcal{G}$ has a
$k(D)$-rational point. If $k$ is finite then this is always true [DPTZ20,
Theorem 8.1].

There is a universal projective bundle $\mathcal{P}\to\mathcal{G}$ over
$\mathcal{G}$ with a universal divisor $\mathcal{D}\subset\mathcal{P}$ finite
étale over $\mathcal{G}$. The coarse moduli space $P$ of $\mathcal{P}$ is
called the _compression_ of $(\mathbb{P}^{1}_{\bar{k}},D)$, it is a rational
curve over $k(D)$ whose pullback to $\bar{k}$ identifies naturally with
$\mathbb{P}^{1}_{\bar{k}}/\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$
[BVb, §5]. Let us call _compressed divisor_ the coarse moduli space
$\mathbf{D}\subset\mathbf{P}$ of $\mathcal{D}$, since $\mathcal{D}$ is finite
étale over $\mathcal{G}$ then $\mathbf{D}$ is finite étale over
$\operatorname{Spec}k$.

If $k$ is perfect, we can give a more down-to-earth construction of the
compression: $\operatorname{Gal}(\bar{k}/k(D))$ acts naturally on
$\mathbb{P}^{1}_{\bar{k}}/\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$
which thus descends to a rational curve $\mathbf{P}$ over $k(D)$, and
similarly $D/\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$ descends to a
divisor $\mathbf{D}\subset\mathbf{P}$.

There is a rational map $P\dashrightarrow\mathcal{G}$ whose pullback
$P_{\bar{k}}=\mathbb{P}^{1}_{\bar{k}}/\operatorname{Aut}(\mathbb{P}^{1},D)\dashrightarrow\mathcal{G}_{\bar{k}}=\mathcal{B}_{\bar{k}}\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$
to $\bar{k}$ is associated with the projection
$\mathbb{P}^{1}_{\bar{k}}\to\mathbb{P}^{1}_{\bar{k}}/\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$,
see the proof of [BVb, Theorem 5.4]. In particular, if
$P\simeq\mathbb{P}^{1}_{k(D)}$ then $\mathcal{G}$ has a $k(D)$-rational point.

##  2\. Rational maps from rational curves to gerbes

As we have seen above, rational maps $P\dashrightarrow\mathcal{G}$ where $P$
is a rational curve, $\mathcal{G}$ is a finite étale gerbe and such that the
geometric fibers are connected of genus $0$ play a crucial role in our
problem. In this section, we study such maps.

######  Proposition 2.

Let $k$ be a field, $P$ a rational curve with $P(k)=\emptyset$, $\Phi$ a
finite, étale gerbe with a rational map $P\dashrightarrow\Phi$. Assume that
the geometric fibers of $P\dashrightarrow\Phi$ are irreducible curves of genus
$0$.

The gerbe $\Phi$ is abelian with cyclic band of order prime to
$\operatorname{char}k$, and there exists a separable extension $k^{\prime}/k$
of degree $2$ and a point $p\in P(k^{\prime})$ such that
$P\dashrightarrow\Phi$ restricts to a morphism
$P\smallsetminus\\{p\\}\to\Phi$.

Let us first sketch the proof in the particular case in which $k$ is perfect.
Write $\Phi_{\bar{k}}=\mathcal{B}_{\bar{k}}G$ for some finite group $G$, the
base change of $P\dashrightarrow\Phi$ to $\bar{k}$ corresponds to a
$G$-covering $\mathbb{P}^{1}_{\bar{k}}\to\mathbb{P}^{1}_{\bar{k}}$. The branch
locus of the $G$-covering descends to $P$: if $P(k)=\emptyset$, this forces
every branch point $p$ of $\mathbb{P}^{1}_{\bar{k}}$ to have at least one
Galois conjugate $\bar{p}$ such that the ramification data over $p$ and
$\bar{p}$ are equal. Using the Riemann-Hurwitz formula, it can be easily
checked that this only happens for $G$ cyclic.

###### Proof.

Let $\bar{k}$ be an algebraic closure and $\bar{k}\supseteq k^{s}\supseteq k$
be the separable closure, we have $P_{k^{s}}\simeq\mathbb{P}^{1}_{k^{s}}$, in
particular $\Phi_{k^{s}}$ is neutral since there is a rational map
$P\dashrightarrow\Phi$. Choose a section $\operatorname{Spec}k^{s}\to\Phi$ and
let $G$ be its automorphism group, it is a finite constant group and there is
an induced branched $G$-covering $f:\mathbb{P}^{1}_{k^{s}}\to
P_{k^{s}}\simeq\mathbb{P}^{1}_{k^{s}}$.

Let $R$ be the ramification divisor of $f$ as in [Sta21, Tag 0C1B], we think
of $R$ as a non-reduced, finite scheme over $k^{s}$. There is an action of $G$
on $R$, in particular on the underlying set. Given a point $r\in R$ denote by
$d_{r}$ the length of $R$ at $r$, $e_{r}$ the ramification index, $o_{r}$ the
cardinality of the set-theoretic $G$-orbit of $r$. Since $f$ has degree $n$,
then $n=o_{r}e_{r}[k^{s}(r):k^{s}(f(r))]$. By Riemann-Hurwitz [Sta21, Lemma
0C1F] we have

$2n-2=\sum_{r\in R}d_{r}[k^{s}(r):k^{s}]$

with $d_{r}=e_{r}-1$ if $f$ is tamely ramified at $r$ and $d_{r}\geq e_{r}$
otherwise.

There exists a largest open subset $U\subset P$ such that $U\to\Phi$ is
defined [Bre21, Corollary A.2]. Write $Z=P\smallsetminus U$, then $Z_{k^{s}}$
is the branch locus of $f$ [Bre21, Lemma A.1, Corollary A.2.iv], in particular
$f(R)=Z_{k^{s}}$ set-theoretically. Since $P$ is a non-trivial Brauer-Severi
variety, $Z$ has no rational points. This implies that, if $r\in R$ is a point
such that $k^{s}(f(r))=k^{s}$, there exists a point $r^{\prime}\in R$ such
that $f(r^{\prime})\neq f(r)$ is a Galois conjugate of $f(r)$, and hence
$o_{r}=o_{r^{\prime}}$, $d_{r}=d_{r^{\prime}}$, $e_{r}=e_{r^{\prime}}$,
$k^{s}(r)\simeq k^{s}(r^{\prime})$.

Let us show first that $f$ is tamely ramified at every point. Assume that $f$
has wild ramification at some point $r$, in particular $d_{r}\geq e_{r}$. If
$k^{s}(f(r))=k^{s}$, by what we have said above we have

$2n-2\geq 2o_{r}e_{r}[k^{s}(r):k^{s}]=2n,$

which is absurd. If $k^{s}(f(r))\neq k^{s}$, then

$2n-2\geq o_{r}e_{r}[k^{s}(r):k^{s}]\geq
2o_{r}e_{r}[k^{s}(r):k^{s}(f(r))]=2n,$

hence we get a contradiction in this case, too.

We may thus assume that $f$ has tame ramification: in particular,
$k^{s}(r)=k^{s}(f(r))$ for every $r$, $n=o_{r}e_{r}$ and $d_{r}=e_{r}-1$ for
every $r\in R$. Since $f$ is ramified at every $r\in R$, then $e_{r}\geq 2$
and $o_{r}\leq n/2$. For every $z\in Z_{k^{s}}$, write $e_{z}$, $o_{z}$ for
$e_{r}$, $o_{r}$, where $r\in R$ is some point with $f(r)=z$. We have

$2n-2=\sum_{r\in R}(e_{r}-1)[k^{s}(r):k^{s}]=\sum_{z\in
Z_{k^{s}}}o_{z}(e_{z}-1)[k^{s}(z):k^{s}]=$ $=\sum_{z\in
Z_{k^{s}}}(n-o_{z})[k^{s}(z):k^{s}]\geq\deg Z\cdot n/2$

which implies $\deg Z\leq 3$. Since $P$ is a non-trivial Brauer-Severi variety
of dimension $1$, then $\deg Z$ is even, hence we may assume $\deg Z=2$ (if
$\deg Z=0$, then $G$ is trivial).

Assume first $Z_{k^{s}}$ contains only one point $z$ with
$[k^{s}(z):k^{s}]=2$, in particular $\operatorname{char}k=2$. Then

$2n-2=o_{z}(e_{z}-1)[k^{s}(z):k^{s}]=2n-2o_{z},$

hence $o_{z}=1$ and $n=e_{z}$ is prime with $\operatorname{char}k=2$, i.e. it
is odd. Since $n$ is odd, the base change $f_{\bar{k}}$ of $f$ to $\bar{k}$ is
tamely ramified and since $f$ has only one point then the same is true for
$f_{\bar{k}}$, and the ramification index must be $n$. By Riemann-Hurwitz
applied to $f_{\bar{k}}$, we have $2n-2=n-1$, i.e. $n=1$ which is in
contradiction with $\deg Z=2$.

If $Z_{k^{s}}$ contains two $k^{s}$-rational points, the Galois action swaps
them because $Z$ has no rational points, hence

$2n-2=2o_{z}(e_{z}-1)=2n-2o_{z},$

which implies $o_{z}=1$ and $n=e_{z}$ is prime to $\operatorname{char}k$. If
$r\in R$ is one of the ramification points, since $|G|=n=e_{z}$ then $r$ is
fixed by $G$. Since $|G|=n$ is prime with $\operatorname{char}k$, then $G$ is
linearly reductive: this implies that $G$ acts faithfully on the tangent space
of $r$ and is thus cyclic. ∎

######  Corollary 3.

Let $k$ be a field and $D\subset\mathbb{P}^{1}_{\bar{k}}$ an effective,
reduced divisor of degree $n\geq 3$ defined over $\bar{k}$. If the compression
of $(\mathbb{P}^{1}_{\bar{k}},D)$ is not isomorphic to $\mathbb{P}^{1}$ over
the field of moduli, then $\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$ is
cyclic of order prime to the characteristic.

######  Lemma 4.

Let $P\dashrightarrow\Phi$ be as in Proposition 2. Then $\Phi$ is neutral if
and only if it has odd degree.

###### Proof.

By Proposition 2, $\Phi$ is abelian with cyclic band $A$ of order prime to
$\operatorname{char}k$. We have that $\Phi$ corresponds to a class
$\phi\in\operatorname{H}^{2}(k,A)$ and $\Phi$ is neutral if and only if $\phi$
is trivial. Let $n$ be the degree of $\Phi$, which corresponds to the degree
of $A$. The group $\operatorname{H}^{2}(k,A)$ is abelian and $n$-torsion.

Let $k^{\prime}$ be the residue field of the point $p$ given by Proposition 2,
then $k^{\prime}/k$ is a separable extension of degree $2$ and
$\phi_{k^{\prime}}=0$ since $k^{\prime}$ splits $P$. We have
$2\phi=\operatorname{cor}_{k^{\prime}/k}(\phi_{k^{\prime}})=0$: if $n$ is odd,
then $\phi=\frac{n+1}{2}2\phi=0$.

If $\Phi$ has even degree, by Proposition 2 $\operatorname{char}k$ is prime
with $n$ and hence $\operatorname{char}k\neq 2$. In particular, the $2$-adic
étale fundamental group of $\mathbb{G}_{m}$ over $k^{s}$ is $\mathbb{Z}_{2}$.
Write $U=P\smallsetminus\\{p\\}$, to prove that $\Phi$ is not neutral it is
then enough to show that there exists a morphism $U\to\Gamma$ where $\Gamma$
is a non-neutral gerbe banded by $\mu_{2}$, since the fact the the degree of
$\Phi$ is even gives us a factorization $U\to\Phi\to\Gamma$. Such a morphism
$U\to\Gamma$ is constructed in the proof of [BV15, Proposition 13.2] using
parameters $p=r=2$ and $X=U$. ∎

##  3\. Which divisors descend to a rational curve

We characterize which pairs $(\mathbb{P}^{1}_{\bar{k}},D)$ descend to the
field of moduli. We’ll then prove Theorem 1 using this characterization.

######  Theorem 5.

Let $k$ be a field and $D\subset\mathbb{P}^{1}_{\bar{k}}$ an effective,
reduced divisor of degree $n\geq 3$ on $\mathbb{P}^{1}_{\bar{k}}$ with $n\geq
3$. The following are equivalent.

  * •

$(\mathbb{P}^{1}_{\bar{k}},D)$ is not defined over its field of moduli.

  * •

The compression has no rational points and
$\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$ has even degree.

  * •

The compression has no rational points and
$\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$ is cyclic of degree even and
prime to $\operatorname{char}k$.

###### Proof.

Thanks to [BVb, Proposition 4.1], we may assume that $k$ is infinite.
Furthermore, by base change we may assume that $k$ is the field of moduli. Let
$\mathcal{G},\mathbf{P}$ be the residual gerbe and the compression
respectively, there is a rational map $\mathbf{P}\dashrightarrow\mathcal{G}$
whose geometric fibers are irreducible of genus $0$.

If $(\mathbb{P}^{1}_{\bar{k}},D)$ is not defined over $k$, i.e. $\mathcal{G}$
is not neutral, then $\mathbf{P}$ is not isomorphic to $\mathbb{P}^{1}$ and
hence $\mathcal{G}$ is abelian with cyclic band of degree even and prime with
$\operatorname{char}k$ by Proposition 2 and Lemma 4. Since
$\mathcal{G}_{\bar{k}}=\mathcal{B}_{\bar{k}}\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$,
we obtain that the first condition implies the third, which in turn clearly
implies the second. If $\mathbf{P}(k)=\emptyset$ and
$\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$ has even degree, then by
Lemma 4 the residual gerbe $\mathcal{G}$ is not neutral, hence we conclude. ∎

######  Corollary 6.

Let $k$ be a field of characteristic $2$ and
$D\subset\mathbb{P}^{1}_{\bar{k}}$ an effective, reduced divisor with $n\geq
3$. Then $(\mathbb{P}^{1}_{\bar{k}},D)$ is defined over its field of moduli.

##  4\. Divisors of degree $6$

######  Proposition 7.

Let $k$ be a field and $D\subset\mathbb{P}^{1}_{\bar{k}}$ an effective,
reduced divisor of degree $6$. Then $(\mathbb{P}^{1}_{\bar{k}},D)$ is defined
over its field of moduli.

###### Proof.

Thanks to Corollary 6, we may assume $\operatorname{char}k\neq 2$. Up to base
change, we may assume that $k$ is the field of moduli, let $\mathbf{P}$ be the
compression and $\mathbf{D}\subset\mathbf{P}$ the compressed divisor, we have
that $\mathbf{D}$ is finite étale over $k$. Assume by contradiction that
$(\mathbb{P}^{1}_{\bar{k}},D)$ is not defined over $k$, then by Theorem 5
$\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$ is cyclic of degree even and
prime to $\operatorname{char}k$, and $\mathbf{P}$ is a non-trivial rational
curve.

Since $\mathbf{P}$ is non-trivial, then $\mathbf{D}$ has even degree. Since
$\operatorname{char}k\neq 2$, then
$\mathbf{D}_{\bar{k}}=D/\operatorname{Aut}(\mathbb{P}^{1},D)$ has either $2,4$
or $6$ points. Since $\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$ is
cyclic of degree even and prime with $\operatorname{char}k$, then its action
on $\mathbb{P}^{1}_{\bar{k}}$ has two fixed points, while all the other orbits
have even cardinality equal to
$|\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)|$. It then follows that $D$
contains the $2$ fixed points, $\mathbf{D}_{\bar{k}}$ has $4$ points and
$\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$ has order $2$.

Choose coordinates on $\mathbb{P}^{1}_{\bar{k}}$ such that $0,\infty\in D$ are
the two fixed points, and $1$ is another point of $D$. With respect to these
coordinates, we have that the only non-trivial element of
$\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$ is the map $x\mapsto-x$, we
may thus write $D=\\{0,\infty,1,-1,\lambda,-\lambda\\}$. It is immediate to
check that $x\mapsto\lambda/x$ defines another non-trivial automorphism of
$(\mathbb{P}^{1},D)$, hence we get a contradiction. ∎

##  5\. Divisors of even degree $n\geq 8$

In order to construct divisors of even degree $n\geq 8$ not defined over their
field of moduli, we first construct a divisor over a non-trivial rational
curve with certain special properties.

######  Lemma 8.

Let $k$ be an infinite field of characteristic $\neq 2$, $P$ a non-trivial
rational curve, $n\geq 8$ an even integer.

There exists a reduced, effective divisor $E\subset P$, a quadratic extension
$k^{\prime}/k$, a point $p\in P(k^{\prime})$ with Galois conjugate $\bar{p}\in
P(k^{\prime})$ and a cyclic cover $f:\mathbb{P}^{1}_{\bar{k}}\to P_{\bar{k}}$
of degree $2$ ramified over $p,\bar{p}$ such that the divisor
$D=f^{-1}(E_{\bar{k}})$ with the reduced structure has degree $n$ and
$\operatorname{Aut}(f)$ is its own centralizer in
$\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$.

###### Proof.

Let $\omega$ be the canonical bundle of $P$. Write either $n=4m$ or $n=4m-2$
with $m\geq 2$, depending on the class of $n$ modulo $4$. Consider the natural
map
$\mathbb{P}(\operatorname{H}^{0}(\omega^{-1}))\times\mathbb{P}(\operatorname{H}^{0}(\omega^{-m+1}))\to\mathbb{P}(\operatorname{H}^{0}(\omega^{-m}))$
corresponding to the sum of divisors, it is dominant.

If $n\geq 10$, i.e. $m\geq 3$, a generic rational point of
$\mathbb{P}(\operatorname{H}^{0}(\omega^{-m}))$ corresponds to a divisor étale
over $k$ with trivial automorphism group scheme. Moreover, any rational point
of $\mathbb{P}(\operatorname{H}^{0}(\omega^{-1}))$ corresponds to a point of
$P$ whose residue field is a quadratic extension of $k$. Taking a generic
rational point of
$\mathbb{P}(\operatorname{H}^{0}(\omega^{-1}))\times\mathbb{P}(\operatorname{H}^{0}(\omega^{-m+1}))$,
we can thus find a divisor $E\subset P$ of degree $2m$ étale over $k$ with
trivial automorphism group scheme and a quadratic extension $k^{\prime}/k$
such that $E(k^{\prime})\neq\emptyset$. If $n=8$, with an analogous argument
we get a reduced divisor $E\subset P$ of degree $4$ such that
$\operatorname{Aut}(\mathbb{P}^{1},E_{\bar{k}})$ is the Klein group.

If $p\in P(k^{\prime})$ is a $k^{\prime}$-rational point, denote by $\bar{p}$
its Galois conjugate. If $n=4m-2$, choose $p\in E(k^{\prime})$, otherwise
$p\in P\smallsetminus E(k^{\prime})$. If $n=8$, for every non-trivial
$g\in\operatorname{Aut}(P_{\bar{k}},E_{\bar{k}})$ the set of points $p\in
E(k^{\prime})$ such that either $g(p)=p$ or $g(p)=\bar{p}$ is not Zariski-
dense, since $k$ is infinite we may choose $p\in P\smallsetminus
E(k^{\prime})$ so that $g(p)\neq p$, $g(p)\neq\bar{p}$ for every non-trivial
$g\in\operatorname{Aut}(P_{\bar{k}},E_{\bar{k}})$.

Consider a cyclic cover $f:\mathbb{P}^{1}_{\bar{k}}\to P_{\bar{k}}$ of degree
$2$ which ramifies at $p$ and $\bar{p}$ and let
$D\subset\mathbb{P}^{1}_{\bar{k}}$ be the inverse image of $E_{\bar{k}}$ with
the reduced structure, it is an effective divisor of degree $n$. The group of
automorphisms $\operatorname{Aut}(\mathbb{P}^{1},D)$ has a cyclic subgroup
$\operatorname{Aut}(f)$ of order $2$. Let
$C\subset\operatorname{Aut}(\mathbb{P}^{1},D)$ be the centralizer of
$\operatorname{Aut}(f)$, then $C/\operatorname{Aut}(f)$ acts faithfully on
$(P_{\bar{k}},E_{\bar{k}})$. If $n\geq 10$,
$\operatorname{Aut}(P_{\bar{k}},E_{\bar{k}})$ is trivial and hence
$C=\operatorname{Aut}(f)$. If $n=8$ and $g\in
C/\operatorname{Aut}(f)\subset\operatorname{Aut}(P_{\bar{k}},E_{\bar{k}})$,
then $g(p)$ is a branch point for $f$ and hence either $g(p)=p$ or
$g(p)=\bar{p}$. By our choice of $p$, we get that $g$ is trivial, hence
$C=\operatorname{Aut}(f)$ in this case, too. ∎

######  Lemma 9.

Let $G$ be a finite group and $g\in G$ a non trivial element with
$g^{2}=\mathrm{id}$ and such that the centralizer of $g$ is $<g>$. Then
$|G|/2$ is odd.

###### Proof.

Let $P\subset G$ be a $2$-Sylow subgroup containing $g$, the center $Z$ of $P$
is non-trivial. Since the centralizer of $g$ is $<g>$, we get that
$Z\subset<g>$ and hence $Z=<g>$. It follows that $P$ centralizes $g$, hence
$P=Z=<g>$ and $|G|/2=[G:P]$ is odd. ∎

######  Proposition 10.

Let $k$ be a field of characteristic $\neq 2$, assume that the $2$-torsion
$\operatorname{Br}(k)[2]$ of the Brauer group is non-trivial. Let $n\geq 8$ be
an even integer. There exists an effective, reduced divisor
$D\subset\mathbb{P}^{1}_{\bar{k}}$ of degree $n$ with field of moduli equal to
$k$ such that $(\mathbb{P}^{1}_{\bar{k}},D)$ is not defined over $k$.

###### Proof.

Since the $2$-torsion of the Brauer group is non trivial, in particular $k$ is
infinite, and there exists a non-trivial rational curve $P$. Let $E\subset P$,
$p\in E(k^{\prime})$, $f:\mathbb{P}^{1}_{\bar{k}}\to P_{\bar{k}}$,
$D\subset\mathbb{P}^{1}_{\bar{k}}$ as in Lemma 8.

Let $k(D)/k$ be the field of moduli of $(\mathbb{P}^{1}_{\bar{k}},D)$ and
$\mathcal{G}$ the residue gerbe. Consider $p$ as a divisor of degree $2$, let
$\sqrt[2]{P,p}$ be $2$nd root stack and $\sqrt[2]{P,p}\to\Pi$ its étale
fundamental gerbe. We have that $\Pi$ is an abelian gerbe banded by $\mu_{2}$
and $\sqrt[2]{P,p}\to\Pi$ is representable, it can be checked after base
changing to $\bar{k}$.

Let $E^{\prime}\subset\sqrt[2]{P,p}$ be the inverse image of $E\subset P$ with
the reduced structure, since $\sqrt[2]{P,p}\to\Pi$ is representable then
$E^{\prime}$ finite étale over $\Pi$ and the geometric fibers of
$(\sqrt[2]{P,p},E^{\prime})\to\Pi$ are isomorphic to
$(\mathbb{P}^{1}_{\bar{k}},D)$. Let $\mathcal{G}$, $\mathcal{P}\to\mathcal{G}$
be the residual gerbe and the universal family of
$(\mathbb{P}^{1}_{\bar{k}},D)$ respectively. By definition of residual gerbe,
we have a $2$-cartesian diagram

${\sqrt[2]{P,p}}$${\mathcal{P}}$${\Pi}$${\mathcal{G}.}$

Since $\Pi$ is a gerbe over $k$ and there is a factorization
$\Pi\to\mathcal{G}\to\operatorname{Spec}k(D)\to\operatorname{Spec}k$, we get
that $k(D)=k$.

By Lemma 9, $\operatorname{Aut}(f)$ has odd index in
$\operatorname{Aut}(\mathbb{P}^{1},D)$. Let $2a$ with $a$ odd be the order of
$\operatorname{Aut}(\mathbb{P}^{1},D)$. Let $\mathbf{P}$ be the compression of
$(\mathbb{P}^{1}_{\bar{k}},D)$, by the diagram above there is a natural
induced map $P\to\mathbf{P}$ of degree $a$. Since $a$ is odd and $P$ is a non-
trivial rational curve, it follows that $\mathbf{P}$ is a non-trivial rational
curve, since otherwise $P$ would have a divisor of odd degree. By Corollary 3,
we get that $\operatorname{Aut}(\mathbb{P}^{1},D)$ is cyclic of even order and
hence $(\mathbb{P}^{1}_{\bar{k}},D)$ is not defined over $k$ by Theorem 5. ∎

##  6\. Generalizations of results of A. Marinatto and B. Huggins

The following lemma connects our results to those of Marinatto.

######  Lemma 11.

Let $k$ be a field and $D\subset\mathbb{P}^{1}_{\bar{k}}$ an effective,
reduced divisor of degree $n\geq 3$ on $\mathbb{P}^{1}_{\bar{k}}$ with $n\geq
3$. The following are equivalent.

  * •

$D$ descends to a divisor of $\mathbb{P}^{1}$ over the field of moduli.

  * •

The universal family $\mathcal{P}\to\mathcal{G}$ has a rational point.

  * •

The compression $\mathbf{P}$ has a rational point.

###### Proof.

By base change, we may assume that $k$ is the field of moduli. If $D$ descends
to a divisor of $\mathbb{P}^{1}$ over $k$, then by definition there is a
morphism $\mathbb{P}^{1}_{k}\to\mathcal{P}$, hence the universal family
$\mathcal{P}$ has a rational point, which in turn implies that the compression
$\mathbf{P}$ has a rational point. Now assume that $\mathbf{P}(k)$ is non-
empty, we want to show that $D$ descends to a divisor of $\mathbb{P}^{1}$ over
$k$.

Assume first that $k$ is finite. By [DPTZ20, Theorem 8.1], $\mathcal{G}(k)$ is
non-empty, hence there exists a model $(P,D_{0})$ of
$(\mathbb{P}^{1}_{\bar{k}},D)$ over $k$. Since $k$ is finite, then
$P=\mathbb{P}^{1}_{k}$.

Assume now that $k$ is infinite. As shown in the proof of [BVb, Theorem 5.4],
there exists an open subset $U\subset\mathcal{P}$ which is a scheme. Since $k$
is infinite and $U(k)\subset\mathbf{P}(k)$ is open, then $U(k)$ is non-empty,
hence there exists a rational point $p\in\mathcal{P}(k)$. The image of $p$ in
$\mathcal{G}(k)$ gives, by definition of $\mathcal{G}$ and $\mathcal{P}$, a
model $(P,D_{0})$ of $(\mathbb{P}^{1}_{\bar{k}},D)$ over $k$ such that
$P(k)\neq\emptyset$. ∎

######  Theorem 12 ([Mar13, Theorem 1]).

Let $k$ be a field and $D\subset\mathbb{P}_{\bar{k}}^{1}$ a reduced, effective
divisor of $\mathbb{P}^{1}_{\bar{k}}$ of degree $n\geq 3$. If $n$ is odd, then
$D$ descends to a divisor of $\mathbb{P}^{1}$ over the field of moduli.

###### Proof.

Thanks to [BVb, Proposition 4.1] and Corollary 6, we may assume that $k$ is
infinite with $\operatorname{char}k\neq 2$. By base change, we can also assume
that $k$ is the field of moduli of $(\mathbb{P}_{\bar{k}}^{1},D)$. Let
$\mathbf{P}$ be the compression and $\mathbf{D}\subset\mathbf{P}$ the
compressed divisor, we have that $\mathbf{D}$ is a finite étale scheme over
$k$.

By Lemma 11 it is enough to show that, if $\mathbf{P}(k)=\emptyset$, then $n$
is even. If $p,q\in\mathbf{D}(k^{s})=\mathbf{D}(\bar{k})$ are in the same
Galois orbit, then the fibers of
$D\to\mathbf{D}_{\bar{k}}=D/\operatorname{Aut}(\mathbb{P}^{1},D)$ over $p$ and
$q$ have the same cardinality. Since $\mathbf{P}$ is non-trivial and
$\mathbf{D}$ is étale over $k$, the Galois-orbits of $\mathbf{D}(k^{s})$ have
even cardinality. It follows that $D$ has even cardinality, too. ∎

######  Proposition 13 ([Mar13, Proposition 2.12]).

Let $k$ be a field and $D\subset\mathbb{P}^{1}_{\bar{k}}$ an effective,
reduced divisor of degree $4$. Then $D$ descends to a divisor of
$\mathbb{P}^{1}$ over the field of moduli.

###### Proof.

For every such divisor, there is a copy of $\mathbb{Z}/2\times\mathbb{Z}/2$
inside $\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$, hence the compression
has a rational point by Proposition 2. We conclude by Lemma 11. ∎

######  Theorem 14 ([Hug07, Theorem 5.3]).

Let $k$ be a field of characteristic $\neq 2$ and $H$ a hyperelliptic curve
over $\bar{k}$ with hyperelliptic involution $\iota$. If $H$ is not defined
over its field of moduli, then
$\mathop{\underline{\mathrm{Aut}}}\nolimits(H)/<\iota>$ is cyclic of order
prime to $\operatorname{char}k$.

###### Proof.

Let $D\subset\mathbb{P}^{1}_{\bar{k}}=H/\iota$ be the branch divisor, we have
$\operatorname{Aut}(X)/<\iota>=\operatorname{Aut}(\mathbb{P}^{1}_{\bar{k}},D)$.
Let $\mathcal{G}_{H},\mathcal{P}_{H},\mathcal{G}_{D},\mathcal{P}_{D}$ be the
residue gerbes and the universal families of $H,D$ respectively cf. [BVb, §3,
§5], there is a cartesian diagram

${\mathcal{P}_{H}}$${\mathcal{P}_{D}}$${\mathcal{G}_{H}}$${\mathcal{G}_{D}}$

where the morphism $\mathcal{G}_{H}\to\mathcal{G}_{D}$ is an abelian gerbe
banded by $\mu_{2}$. It follows that $(\mathbb{P}^{1}_{\bar{k}},D)$ and $H$
have equal fields of moduli and equal compression $\mathbf{P}$. Thanks to
[BVb, Proposition 4.1], we may assume that $k$ is infinite. Since $H$ is not
defined over the field of moduli, i.e. $\mathcal{G}_{H}(k)=\emptyset$, and
there is a rational map $\mathbf{P}\dashrightarrow\mathcal{G}_{H}$ [BVb, Proof
of Theorem 5.4], we get that $\mathbf{P}(k)=\emptyset$. We conclude by
applying Corollary 3 to $\mathbf{P}\dashrightarrow\mathcal{G}_{D}$. ∎

## References

  * [Bre21] Giulio Bresciani, _Essential dimension and pro-finite group schemes_ , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), no. 4, 1899–1936.
  * [BVa] Giulio Bresciani and Angelo Vistoli, _An arithmetic valuative criterion for proper maps of tame algebraic stacks_ , arxiv:2210.03406.
  * [BVb] by same author, _Fields of moduli and the arithmetic of tame quotient singularities_ , arxiv:2210.04789.
  * [BV15] Niels Borne and Angelo Vistoli, _The Nori fundamental gerbe of a fibered category_ , J. Algebraic Geom. 24 (2015), no. 2, 311–353.
  * [DE99] Pierre Dèbes and Michel Emsalem, _On fields of moduli of curves_ , J. Algebra 211 (1999), no. 1, 42–56.
  * [DPTZ20] Valentina Di Proietto, Fabio Tonini, and Lei Zhang, _Frobenius fixed objects of moduli_ , arXiv:2012.14075, 2020.
  * [Hug07] Bonnie Huggins, _Fields of moduli of hyperelliptic curves_ , Math. Res. Lett. 14 (2007), no. 2, 249–262.
  * [Mar13] Andrea Marinatto, _The field of definition of point sets in $\mathbb{P}^{1}$_, J. Algebra 381 (2013), 176–199.
  * [Sta21] The Stacks Project Authors, _Stacks project_ , http://stacks.math.columbia.edu, 2021.


Periodic boundary conditions and $G_{2}$ cosmology

A A Coley,

Department of Mathematics & Statistics, Dalhousie University,

Halifax, Nova Scotia, Canada B3H 3J5

Email<EMAIL_ADDRESS>
W C Lim,

Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton
3240, New Zealand

Email<EMAIL_ADDRESS>
###### Abstract

In the standard concordance cosmology the spatial curvature is assumed to be
constant and zero (or at least very small). In particular, in numerical
computations of the structure of the universe using N-body simulations, exact
periodic boundary conditions are assumed which constrains the spatial
curvature. In order to confirm this qualitatively, we numerically evolve a
special class of spatially inhomogeneous $G_{2}$ models with both periodic
initial data and non periodic initial data using zooming techniques. We
consequently demonstrate that in these models periodic initial conditions do
indeed suppress the growth of the spatial curvature as the models evolve away
from their initial isotropic and spatially homogeneous state, thereby
verifying that the spatial curvature is necessarily very small in standard
cosmology.

##  1 Introduction

Cosmology is concerned with the large scale behaviour of the Universe within
General Relativity (GR) [1]. The cosmological principle then implies that on
the largest scales the Universe can be accurately modeled by an exact solution
to Einstein’s field equations (EFE) which is both spatially homogeneous and
isotropic leading to a (background) Friedmann-Lemaıtre-Robertson-Walker (FLRW)
model (with constant spatial curvature) with the cosmological constant,
$\Lambda$, representing dark energy (or the so-called $\Lambda$CDM standard
concordance cosmology, where CDM denotes cold dark matter). Inflation in the
early universe is often regarded as a part of the standard model, in which the
background spatial curvature, characterized by the normalized curvature
parameter, is predicted to be negligible [2]. Regardless of whether inflation
is included as part of the standard model or not, spatial curvature is
implicitly assumed to be zero.

An important goal in cosmology is to understand the origin of the structure of
the universe. Assuming that cosmic structure grew out of small initial
fluctuations, we can investigate their evolution on sufficiently large scales
using linear perturbation theory (LPT), where the spatially inhomogeneous
perturbations live on the flat FLRW background spacetime. Inflation then
provides a causal mechanism for generating primordial cosmological
perturbations via quantum fluctuations in the inflaton field, which
subsequently act as seeds for the anisotropies in the cosmic microwave
background (CMB) [3] and give rise to the large scale structure of our
universe. At late times and scales much smaller than the Hubble scale
fluctuations of the cosmic density are not necessarily small, and LPT is not
sufficient and clustering needs to be treated non-linearly. Usually this is
studied with (non-relativistic) N-body simulations. Recently non-linear
perturbations have been investigated at second-order and non perturbative
relativistic effects have been studied computationally [1].

Although standard cosmology has been very successful in describing current
observations, there are a number of possible anomalies [4], including the
tension between the recent determination of the local value of the Hubble
constant based on direct measurements of supernovae [5] and the value derived
from the most recent CMB data [3], as well as various self-inconsistencies
[6]. In addition, the values of cosmological parameters inferred from the CMB
perhaps suggest a number of directional dependences (e.g., possible greater
than anticipated bulk flows on large scales and dark flow [7]).

In principle, since the Universe is not isotropic or spatially homogeneous on
small local scales, the effective smoothed out macroscopic gravitational FE on
large scales should be obtained by averaging the EFE of GR. The averaging of
the EFE for local inhomogeneities can lead to important backreaction effects
on the average evolution of the Universe [8], and can significantly affect
precision cosmology at the level of 1 % [1].

###  1.1 Spatial curvature

As noted above, in standard cosmology the spatial curvature is assumed to be
constant and zero (or at least perturbatively small). 111By global constant
spatial curvature we essentially mean the value of $k$ in the background FLRW
spacetime (and hence the initial conditions). By the local value of the
spatial curvature we primarily mean the local value obtained later in the
paper by the numerical integration across the region of simulation for
different specific values of $x_{zoom}$. In the later numerical simulations we
also discuss an averaged spatial curvature at a given time. However, there is
no completely independent constraint that gaurentees a value for the magnitude
of the effective normalized spatial curvature $\Omega_{k}$ of less than about
1 %. Moreover, any possible such small non-zero measurement of $\Omega_{k}$
can perhaps be taken to indicate that the assumptions in the standard model
are not valid. We also note that spatial curvature is, in general, evolving in
relativistic cosmological models [8].

In order to develop models to be utilized for cosmological predictions and
comparison with observational data, it is necessary to make assumptions.
However, it is crucial to check how the assumptions affect the results. More
importantly, we can only validate the consistency of assumptions and we cannot
rule out alternative explanations. The assumption of a FLRW background on
cosmological scales presents a number of problems [1]. For example, the
assumption that there exists a 1+3 spacetime split and a global time and a
background inertial (Gaussian normal) coordinate system over a complete Hubble
scale background patch in the standard model leads to the constraint that the
spatial curvature (and the vorticity) must be sufficiently small. In addition,
any assumptions of exact periodic boundary conditions (PBC) (appropriate on
scales comparable to the homogeneity scale) necessarily imply that the global
spatial curvature is exactly zero [1] (since in the actual standard model the
Universe is taken to be simply connected), and thus any suitable approximation
will amount to $\Omega_{k}$ being less than the perturbation (e.g., LPT)
scale. We shall return to this point below.

There are also assumptions related to the weak field approach, the actual
applicability of perturbation theory and Gaussian initial conditions, that
imply spatial curvature is neglected. It is also asserted that cosmological
backreaction can be neglected, but in LPT the initial fluctuations are usually
taken to be Gaussian, which implies that at the linear level all averages are
zero by construction. In addition, we should be aware that any intuition based
on Newtonian theory may be misplaced; e.g., the average spatial curvature of
voids and clustered matter (with negative and positive curvature,
respectively) is not necessarily zero within GR. Thus, within standard
cosmology the spatial curvature is assumed to be zero, or at least very small
and at most first order in terms of the perturbation approximation, in order
for any subsequent analysis to be valid. Any prediction larger than this
indicates an inconsistency in the approach. In this sense the standard model
cannot be used to predict a small spatial curvature.

It has recently been re-emphasized that, in an era of precision cosmology, any
physical approximations (including, for example, the use of LPT), in standard
cosmological models should be analysed to check if they are sufficiently
accurate for comparison with observations used to draw conclusions about the
actual properties of our Universe [9].

###  1.2 Periodic boundary conditions

In the general inhomogeneous cosmological case, the formulation of the
evolution eqns. consists of utilizing global Cartesian coordinates and metric
variables, where initial conditions on the metric are specified. In the actual
N-body simulations a cell of length L (a fraction of the Hubble scale) is
taken and the system is integrated where the PBC at L is applied at all times
(and is used at every step to compute spatial derivatives near the edges of
the cell). Hence spatial curvature is held zero on the boundary, and in this
sense the growth of spatial curvature is suppressed, relatively speaking. It
is of importance to verify numerically that initial PBC constrain the spatial
curvature to be zero at the boundary for all time, which is maintained at zero
even below numerical errors that begin to grow.

In addition, if in N-body simulations the cell is restricted to a finite size
L, then periodicity implies that longer wavelength inhomogeneities than L are
effectively neglected. This will have an effect on the numerical simulations,
which is a general concern, especially when the large-scale cutoff is indeed
below the Hubble scale as will be necessary if small ($\sim Mpc$) scales are
to be resolved. (There will be a cutoff in practice in the power spectrum of
the inhomogeneities immediately from the initial conditions.)

Finally, it is not absolutely clear whether periodic initial conditions
guarantee periodic evolution for all times. There is no general mathematical
theorem on the preservation of discrete symmetries (that periodicity implies;
however, see [10]). But note that if it were true that periodic initial
conditions are propagated, then it follows that the spatial curvature is
indeed constrained at L for all times.

In the few current relativistic cosmological simulations, the spatial
curvature can have large local fluctuations but remains small when averaged
over large scales. It becomes extremely small, $\Omega_{R}\sim 10^{-8}$, when
averaged over the whole periodic box of the simulation [11]. Work is in
progress for a more in-depth investigation of average spatial curvature from
numerical relativity, along with the constraints due to the PBC, in a more
frame-independent formalism [12]. However, the preliminary evidence is that
these numerical solutions have an exceptionally and unnaturally small spatial
curvature in a neighbourhood of the periodic boundary, which affects the
average spatial curvature in the whole cell, and can perhaps be ascribed to
the use of PBC.

In different formulations of the EFE (e.g., the $G_{2}$ case below), it is
difficult to compare with the standard numerical results of the full 3D code
and the usual N-body simulations. Indeed, since the EFE are hyperbolic, and
depending on the particular formulation of the evolution equations, once the
initial conditions are fully specified it is not clear whether the application
of spatial PBC are even required (beyond the specification of the initial data
that is).

##  2 Spatially inhomogeneous models

The FLRW cosmology (or Friedmann–Lemaıtre (FL) model as we will refer to as
here when regarding the solution as a dynamical equilibrium state) is an
expanding exactly spatially homogeneous and spatially isotropic universe,
which is believed to describe the Universe on a sufficiently large spatial
scale, at least since the time of last scattering. However, the observable
part of the Universe is not exactly spatially homogeneous and isotropic on any
spatial scale. From a practical point of view, we are interested in studying
models that are close to FL in some appropriate dynamical sense, perhaps using
deviations from an FL model by applying LPT. But it is not known how reliable
the linear theory is and, moreover, in using it we are a priori excluding the
possibility of uncovering important non-linear effects.

Therefore, it is of interest to consider (more) general spatially
inhomogeneous models (than FL) in order to investigate the constraints that
observations impose [13]. Indeed, it could be argued that we should not study
exotic ideas until conventional GR (with inhomogeneities) has been fully
explored. In the first instance, it is perhaps advisable to investigate a
special inhomogeneous model where some analytical and qualitative insights are
possible. In this paper we wish to study the effect of spatial curvature and
PBC numerically in the special case of $G_{2}$ models.

###  2.1 $G_{2}$ models

The simplest spatially inhomogeneous cosmological models have two commuting
Killing vector fields (i.e., models admitting a 2-parameter Abelian isometry
group acting transitively on spacelike 2-surfaces), which consequently have a
single degree of spatial inhomogeneity. This class of models, which are
referred to briefly as $G_{2}$ cosmologies, are governed by the EFE which
constitute partial differential equations (PDE) in two independent variables
[14]. A lot of the research to date has focused on the so-called Gowdy vacuum
spacetimes [15], because such models are much more tractable than non-vacuum
ones and the existence of compact space sections facilitates numerical
simulations since the problem of boundary conditions at spatial infinity is
avoided. Recently, a plane-parallel model (or wall universe) [16], in which
light rays pass through a series of plane-symmetric perturbations around a
FLRW background, was used to study the back-reaction from the small-scale
inhomogeneities [17, 18].

Belinskii, Khalatnikov and Lifshitz (BKL) [19] have conjectured that within GR
the approach to the generic spacelike singularity to the past is vacuum
dominated (assuming $p<\rho$), local, and oscillatory (labeled ‘Mixmaster’).
Localilty implies that the contribution of terms in the evolution eqns. with
spatial derivatives can be neglected. Numerical studies of the asymptotics of
the simplest inhomogeneous vacuum Gowdy models demonstrate that on approach to
the singularity, spiky structures form [20]. These spikes become narrower as
the singularity is approached. Studies of $G_{2}$ and more general models have
produced numerical evidence that the BKL conjecture generally holds except
possibly at isolated points where spiky structures form [21, 22].

We are particularly interested in late time evolution here. We shall
concentrate on $G_{2}$ cosmologies with a perfect fluid matter content, and
especially dust [23]. At any instant of time, the state of a $G_{2}$ cosmology
is described by a finite-dimensional dynamical state vector of functions of
the spatial coordinate. That is, the dynamical state space of $G_{2}$
cosmologies is a function space and is thus infinite-dimensional. The
evolution of a $G_{2}$ cosmology is consequently described by an orbit in this
infinite-dimensional dynamical state space. In a $G_{2}$ cosmology, at each
point there exists a preferred timelike 2-space that is orthogonal to the
$G_{2}$–orbits. Therefore, there is an infinite family of geometrically
preferred timelike congruences and gauge freedom in the choice of the
orthonormal frame (and associated with the choice of the local coordinates).
Following [24, 14], we shall utilize the orthonormal frame formalism adapted
to the $G_{2}$ orbits. Such a dynamical formulation has proved effective in
studying $G_{2}$ cosmologies. Here we define scale-invariant dependent
variables by normalisation with the area expansion rate of the $G_{2}$–orbits,
in order to obtain the evolution eqns. as a system of PDE in first-order
symmetric hyperbolic (FOSH) format. This FOSH format also provides a natural
framework for numerical studies.

###  2.2 $G_{2}$ evolution equations in timelike area gauge

The geometry of the general $G_{2}$ class of spacetimes is given in [22],
where all of the metric functions depend only on the time coordinate $t$ and
the spatial coordinate $x$. We will display the $G_{2}$ evolution system in
terms of the timelike area gauge in the Gowdy subcase $\Sigma_{2}=0$ (with
unit lapse, thereby defining $t$ time). We utilize $\beta$-normalized
variables [14, 23] in the orthonormal frame formalism [24], where the $t$ and
$x$ subscripts denote partial differentiation. We also assume dust
($\gamma=1$) and a single non-zero tilt component, $v$. We are particularly
interested in future dynamics close to ’FL’. We also assume that the
cosmological constant is zero, so that we have an unconstrained evolution
system in FOSH format [14]:

$\displaystyle\partial_{t}E_{1}{}^{1}$ $\displaystyle=$
$\displaystyle(q+3\Sigma_{+})\,E_{1}{}^{1}$ (1)
$\displaystyle\partial_{t}\Sigma_{-}+E_{1}{}^{1}\,\partial_{x}N_{\times}$
$\displaystyle=$
$\displaystyle(q+3\Sigma_{+}-2)\,\Sigma_{-}+2\sqrt{3}\,\Sigma_{\times}^{2}-2\sqrt{3}\,N_{-}^{2}$
(2) $\displaystyle\partial_{t}N_{\times}+E_{1}{}^{1}\,\partial_{x}\Sigma_{-}$
$\displaystyle=$ $\displaystyle(q+3\Sigma_{+})\,N_{\times}$ (3)
$\displaystyle\partial_{t}\Sigma_{\times}-E_{1}{}^{1}\,\partial_{x}N_{-}$
$\displaystyle=$
$\displaystyle(q+3\Sigma_{+}-2-2\sqrt{3}\Sigma_{-})\,\Sigma_{\times}-2\sqrt{3}\,N_{\times}\,N_{-}$
(4) $\displaystyle\partial_{t}N_{-}-E_{1}{}^{1}\,\partial_{x}\Sigma_{\times}$
$\displaystyle=$
$\displaystyle(q+3\Sigma_{+}+2\sqrt{3}\Sigma_{-})\,N_{-}+2\sqrt{3}\,\Sigma_{\times}\,N_{\times}$
(5) $(\partial_{t}+v\,E_{1}{}^{1}\,\partial_{x})\,\Omega+\Omega
E_{1}{}^{1}\,\partial_{x}v=2\,\Omega\ [(q+1)-{\textstyle{1\over
2}}\,(1-3\Sigma_{+})\,(1+v^{2})-1]$ (6)
$(\partial_{t}+v\,E_{1}{}^{1}\,\partial_{x})\,v=(1-v^{2})\,[\
3(N_{\times}\Sigma_{-}-N_{-}\Sigma_{\times})+{\textstyle{3\over 2}}\Omega
v-(1-3\Sigma_{+})\,v]$ (7)

where

$(q+3\Sigma_{+})\equiv 2-{\textstyle{3\over 2}}\,(1-v^{2})\,\Omega\ $ (8)

and

$\Sigma_{+}\equiv{\textstyle{1\over
2}}\,(1-\Sigma_{-}^{2}-N_{\times}^{2}-\Sigma_{\times}^{2}-N_{-}^{2}-\Omega).\\\
$ (9)

Note that in the present case we have from Eqs. (8) and (9) that

$q=\frac{1}{2}+\frac{3}{2}(\Sigma_{-}^{2}+N_{\times}^{2}+\Sigma_{\times}^{2}+N_{-}^{2})+\frac{3}{2}v^{2}\Omega\geq\frac{1}{2},$
(10)

which, together with

$\partial_{t}\beta=-(q+1)\beta,$ (11)

guarantees that $\beta$ is single signed and is monotone (for small $\epsilon$
– see below). The (negative) spatial curvature ($-\Omega_{k}$) is defined via

$\Omega_{k}\equiv N_{\times}^{2}+N_{-}^{2}.$ (12)

The relation of the variables above with metric components is given by (see
(6.39) and (6.40) of [23], or Appendix A.3 of [14]):

$ds^{2}=\beta^{-2}[-dt^{2}+(E_{1}{}^{1})^{-2}dx^{2}]+e^{2t}[e^{P(t,x)}(dy+Q(t,x)dz)^{2}+e^{-P(t,x)}dz^{2}]$
(13) $\displaystyle(\Sigma_{-},\ N_{\times})$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{3}}(\partial_{t},\ -E_{1}{}^{1}\partial_{x})P$
(14) $\displaystyle(\Sigma_{\times},\ N_{-})$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{3}}e^{P}(\partial_{t},E_{1}{}^{1}\partial_{x})Q.$
(15)

####  2.2.1 Initial data

Since we have an unconstrained FOSH system no constraints need be satisfied
and we freely specify
$\\{E_{1}{}^{1},~{}\Sigma_{-},~{}N_{\times},~{}\Sigma_{\times},~{}N_{-},~{}\Omega,v\\}$
at $t=0$ on $\\{-L\leq x\leq L\\}$.

We shall consider initial data close to a (flat) FL model of the form (for
small $\epsilon\sim 10^{-4}$):

$\\{E_{1}{}^{1}=1+\epsilon^{2}{\tilde{E}}$ },

$\\{\Sigma_{-}=\epsilon{\tilde{\Sigma}}_{-},N_{\times}=\epsilon{\tilde{N}}_{\times},\Sigma_{\times}=\epsilon{\tilde{\Sigma}}_{\times},N_{-}=\epsilon{\tilde{N}}_{-}\\}$,

$\\{\Omega=1-\epsilon^{2}{\tilde{\Omega}},v=\epsilon^{2}{\tilde{v}}\\}$.

Note that it follows that

$\\{q=\frac{1}{2}+\epsilon^{2}{\tilde{q}},\Sigma_{+}=\epsilon^{2}{\tilde{\Sigma}}_{+},\Omega_{k}=\epsilon^{2}{\tilde{\Omega}}_{k}\\}$.

The initial data is given by (for example) $\epsilon{\tilde{\Sigma}}_{-}$ at
$t=0$ (etc.). Since shell crossings can occur in dust models, we either cease
integration once any singular behaviour develops or we can include a small
fiducial pressure in the models.

####  2.2.2 Linear regime

For small $\epsilon$, we integrate eqn. (1) to obtain:

${\tilde{E}}=e^{\frac{t}{2}}+o(\epsilon^{2}),$ (16)

where we have normalized ${\tilde{E}}$ to be unity at $t=0$. In the linear
regime we have that $e^{\frac{t}{2}}$ is small (which for $\epsilon\sim
10^{-4}$ corresponds to $t<10$). Eqns. (6,7) also constitute o($\epsilon^{2}$)
corrections to zeroth order evolution equations. In the linear regime,
$\Omega_{k}\sim o(\epsilon^{2})$.

####  2.2.3 Shear and curvature

Eqns. (2) – (5) represent the first order evolution eqns. for the normalized
shear and curvature variables with second order corrections. We immediately
see the growing modes for the normalized curvature variables $\sim
e^{\frac{t}{2}}$ and the decaying modes for the normalized shear variables
$\sim e^{-\frac{3t}{2}}$, corresponding to the familiar eigenvalues
$\\{\frac{1}{2},-\frac{3}{2}\\}$ for the flat FL ”saddle point” solution [so
that the growth of the shear is suppressed relative to that of the spatial
curvature in the initial linear regime].

Solving eqns. (2) - (5) we obtain:

$\displaystyle{\tilde{\Sigma}}_{-}$ $\displaystyle=$ $\displaystyle
e^{-\frac{3t}{2}}[\sigma_{-}+\epsilon{\bar{\Sigma}}_{-}],$ (17)
$\displaystyle{\tilde{N}}_{\times}$ $\displaystyle=$ $\displaystyle
e^{\frac{t}{2}}[\nu_{\times}+\epsilon{\bar{N}}_{\times}],$ (18)
$\displaystyle{\tilde{\Sigma}}_{\times}$ $\displaystyle=$ $\displaystyle
e^{-\frac{3t}{2}}[\sigma_{\times}+\epsilon{\bar{\Sigma}}_{\times}],$ (19)
$\displaystyle{\tilde{N}}_{-}$ $\displaystyle=$ $\displaystyle
e^{\frac{t}{2}}[\nu_{-}+\epsilon{\bar{N}}_{-}],$ (20)

where $\\{\sigma_{-},\sigma_{\times},\nu_{-},\nu_{\times}\\}$ are slowly
varying (and, for example, $\partial_{x}\Sigma_{-}\sim(\epsilon
e^{\frac{t}{2}}){\bar{\Sigma}}_{-}{}^{\prime}$,
$\partial_{x}\Sigma_{\times}\sim(\epsilon
e^{\frac{t}{2}}){\bar{\Sigma}}_{\times}{}^{\prime}$, where a prime denotes
$\partial_{x}$). In the regime in which the shear is sub-dominant, these
quantities are approximately constant.

####  2.2.4 Spatial curvature

From eqns. (3) and (5), to $o(\epsilon)$ eqn. (12) becomes:

$\partial_{t}\Omega_{k}=\Omega_{k},$ (21)

so that $\Omega_{k}$ remains small (and of second order initially). More
precisely, to $o(\epsilon^{4})$:

$\partial_{t}\Omega_{k}=(\epsilon
e^{\frac{t}{2}})^{2}[(\nu_{-}^{2}+\nu_{\times}^{2})+\epsilon(\nu_{\times}{\bar{N}_{\times}}+\nu_{-}{\bar{N}_{-}})+4\sqrt{3}(\epsilon
e^{\frac{t}{2}})e^{-2t}(\sigma_{-}\nu_{-}+\sigma_{\times}\nu_{\times})+(\epsilon
e^{\frac{t}{2}})e^{-2t}({\nu_{-}}{\bar{\Sigma}}_{\times}{}^{\prime}+\nu_{\times}{\bar{\Sigma}}_{-}{}^{\prime}).$
(22)

Clearly to second order we duplicate eqn. (21); the leading order correction
to this eqn. comes from the term
$\epsilon(\nu_{-}{\bar{N}_{\times}}+\nu_{\times}{\bar{N}_{-}})$, which
corresponds to eqn. (22) to next order. The remaining terms are of order
$o(\epsilon e^{-2t})$ and $o(\epsilon^{2})$. For early times
$\partial_{t}\Omega_{k}>0$, and so the magnitude of the spatial curvature
grows. The sign of the next order terms is not necessarily positive.

####  2.2.5 Boundary conditions

We can consider PBC at $X=\pm L$. Any periodic initial data for any variable
$X$ on $[-L,L]$, for $L$ sufficiently large, can be Fourier decomposed. Note
that at $t=0$ the average values are $\langle X\rangle=c_{0}$, which is
usually taken to be zero, and $\langle X^{\prime}\rangle=0$. We are also
interested in non periodic initial data, since in the formulation of $G_{2}$
models here PBC are not necessary

####  2.2.6 Numerical methods: Zooming

One numerical method for resolving small scale structure is adaptive mesh
refinement (AMR). However, if we know beforehand the location of the
structure, we need not use AMR. Because we wish to numerically study non-PBC,
we shall consequently use zooming techniques, in which we use a coordinate
system adapted to the structure that is under investigation. In particular, we
can choose a coordinate system that shrinks exponentially with time, in which
the new coordinates zoom in on the worldline, and where the constant rate of
focus $A$ is controlled ($A=1$ is the natural choice, which ensures a good
excision boundary throughout the simulation) [22]. The numerical grid ends at
a fixed coordinate value $X=L$. Ordinarily, that would call for a boundary
condition at $L$, but we can use the method of excision, which can be applied
to any hyperbolic equations where the outer boundary is chosen so that all
causal influences are outgoing. In that case the equations of motion are
simply implemented at the outer boundary; no boundary condition is needed (or
even allowed).

##  3 Results

Figure 1: In Figure 1–3 we consider three numerical runs for the initial
conditions (a) $\Sigma_{-}=0.01\sin\pi x$ (b) $\Sigma_{-}=0.01\sin\pi x+0.01$
(c) $\Sigma_{-}=0.01\sin\pi x+0.01x$ (each zooming in at
$x_{\text{zoom}}=0,0.25,0.5$, plotted in blue, green, red, respectively) in
the diagonal $G_{2}$ models. The domain is $x-x_{\text{zoom}}\in[-10,10]$,
with $t$ running from $t=0$ to $t=3$. The initial conditions can be
interpreted as a perturbed open FL model (small $\Sigma_{-}$) that is also
close to a flat FL model (small $v$). For all initial conditions $v$ grows
from 0.1 to 0.4, taking the solution further away from the flat FL model. The
growth of the spatial curvature is displayed in the final frame in each figure
for each initial condition. Figure 2: (See Figure 1 caption.) Figure 3: (See
Figure 1 caption.)

In Figures 1–3 we present three numerical simulations for the $G_{2}$ dust
evolution system in terms of the timelike area gauge and using
$\beta$-normalized variables with the initial conditions
$\\{E_{1}{}^{1}=1,\Sigma_{-}={\tilde{\Sigma}_{-}},N_{\times}=0,\Sigma_{\times}=0,N_{-}=0,\Omega=1-{\tilde{\Sigma}_{-}}^{2},v=0.1\\}$
(which actually correspond to diagonal $G_{2}$ models, with
$\Sigma_{\times}=N_{-}=0$ at all times) and with (a)
${\tilde{\Sigma}_{-}}=0.01\sin\pi x$ (b) ${\tilde{\Sigma}_{-}}=0.01\sin\pi
x+0.01$ (c) ${\tilde{\Sigma}_{-}}=0.01\sin\pi x+0.01x$ (each zooming in at
$x_{\text{zoom}}=0,0.25,0.5$, plotted in blue, green, red, respectively). The
domain is $x-x_{\text{zoom}}$ in $[-10,10]$ with zooming, with $t$ running
from $t=0$ to $t=3$. The initial conditions can be interpreted as a perturbed
open FL model (small $\Sigma_{-}$) that is also close to a flat FL model
(small $v$). The numerical results demonstrate that for all initial conditions
$v$ grows from 0.1 to 0.4, taking the solution further (drifting) away from
flat FL (regarded as a saddle point in the dynamical state space), regardless
of the periodicity or aperiodicity of the initial conditions.

From the final frame in the Figs. it is clearly demonstrated that $\Omega_{k}$
is suppressed in Fig. 1(a) with periodic initial conditions, and the growth of
$\Omega_{k}$ is evident when periodicity is broken in Fig. 1(c), which can be
attributed to the non-periodic term $0.01x$ in the initial conditions for
$\Sigma_{-}$. Note that this non-periodic term has a very large wavelength.
However, the simulations Fig. 1(b) show that the presence of a constant term
0.01 in $\Sigma_{-}$ is found to have a negligible effect on the spatial
curvature $N_{\times}$ and $\Omega_{k}$, since the large background
homogeneous term in the shear in this model (i.e., $\Sigma_{-}=0.01\sin\pi
x+0.01$, which is still periodic) does not pass into $N_{\times}$. Additional
simulations with various initial conditions were not found to change the
qualitative behaviour of these results.

##  4 Discussion

We have shown that in this particular $G_{2}$ model periodic initial
conditions do indeed suppress the spatial curvature, and hence we have
verified that their use implies a small spatial curvature a priori. More
comprehensive numerical analysis is possible but, as we noted above, this may
not directly improve the qualitative conclusions regarding more general 3D
simulations.

Indeed, in the $G_{2}$ FOSH formulation of the evolution eqns., the variables
are not metric variables and consequently the initial conditions are not
specified on the metric functions. Hence the meaning of periodic initial data
has a different meaning to that in the usual standard sense. Indeed, in this
formulation once the initial conditions are fully specified the application of
spatial boundary conditions are not even required. It is difficult to compare
directly with the results obtained in the usual full 3D numerical simulations.
Hence it is important to repeat this type of analysis in general cosmological
simulations [12].

If the periodic initial conditions are maintained, the only arbitrarily large
wavelength modes allowed are the background homogeneous terms, and they are
only allowed in the shear components (which are time derivatives of the
metric) but not in the spatial curvature components (which are spatial
derivatives of the metric). That is, the spatially homogeneous growing mode is
turned off, and long-wavelength modes which behave like Bianchi dynamics,
leaving only short-wavelength modes which grow slower (if they grow at all).
Periodicity generally disallows arbitrarily large wavelengths in the spatial
curvature.

In the analysis there is a potential gauge issue which concerns the choice of
the timelike area gauge (which affects the normalization of the independent
variables such as $\Omega_{k}$). (The usual gauge issues when using LPT are
not relevent here in the numerical simulations). The emphasis in this short
paper is to qualitatively illustrate the relative growth of $\Omega_{k}$ in
the evolutions without PBC compared to those with PBC. Other numerical
investigations were undertaken, but none challenge this conclusion and none,
in our opinion, really led to any further insights. So although the gauge
affects the actual quantitative results, it does not affect the qualitative
results (at least according to our computations). In particular, the behaviour
of the growth of $\Omega_{k}$ (in the figures) is typical as found across the
region of simulation, and in other timelike area gauge numerical simulations,
indicating that the relative average spatial curvature is increasing in the
models without PBC. In addition, comparison of $\Omega$ and $\Omega_{k}$ in
the figures (where the influence of the normalization factor is then
minimised) illustrates that the qualitative growth of $\Omega_{k}$ in the
evolutions without PBC compared to those with PBC does not depend on the
choice of the timelike area gauge.

## Data availability statement

The data that support the findings of this study are openly available at the
following URL/DOI:

    www.math.waikato.ac.nz/~wclim/pbcpaper2023_source_files/.

## Acknowledgement

AAC would like to acknowledge NSERC for financial assistance. We would like to
thank Pierre Mourier and Hayley Macpherson for helpful comments and private
communications.

## References

  * [1] A. A. Coley and G. F. R. Ellis, Class. Quant. Grav. 37 013001 (2020) [arXiv:1909.05346].
  * [2] J. Martin [arXiv:1902.05286].
  * [3] Planck Collaboration: Planck 2018 results, Astronomy & Astrophysics 641 A6 & A7 (2020); Y. Akrami et al. [arxiv:1807.06205 & 06209 & 06211];
  * [4] T. Buchert et al., Int. J. Mod. Phys. D 25 1630007 (2016).
  * [5] A. G. Riess et al., A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team [arXiv:2112.04510]; A. G. Riess et al., Astrophys. J. 861 126 (2018) [arxiv:1804.10655].
  * [6] E. Di Valentino et al., Class. Quant. Grav. 38 153001 (2021); M. Asgari et al., Astronomy & Astrophysics 645 A104 (2021); T. M. Abbott et al., Phys. Rev. D 105 023520 (2022).
  * [7] A. Kashlinsky et al., Astrophysical Journal Letters 686 L49 (2008); C. Howlett et al., Monthly Notices of the Royal Astronomical Society 515 953 (2022).
  * [8] K. Bolejko, Phys. Rev. D 97 103529 (2018); D. L. Wiltshire, New J. Phys. 9 377 (2007) & Class. Quant. Grav. 28 164006 (2011); T. Buchert et al., Class. Quant. Grav. 32 215021 (2015); A. Coley, N. Pelavas and R. Zalaletdinov, Phys. Rev Letts. 95 151102 (2005).
  * [9] J. T. Giblin Jr, J. B. Mertens, G. D. Starkman and C. Tian, Phys. Rev. D 99 023527 (2019) [arXiv:1810.05203 [astro-ph.CO]]; P. K. Aluri et al., Is the observable universe consistent with the cosmological principle? [arXiv:2207.05765].
  * [10] H. Friedrich and A. Rendall. ”The Cauchy problem for the Einstein equations.” Einstein’s Field Equations and Their Physical Implications: Selected Essays in Honour of Jürgen Ehlers (Springer, Berlin Heidelberg, pp 127-223, 2000).
  * [11] H. Macpherson, J. Price and P. Lasky, Phys. Rev. D 99 3522 (2019).
  * [12] H. Macpherson and P. Mourier, in preparation.
  * [13] J. Wainwright and G. F. R. Ellis (eds) Dynamical Systems in Cosmology (Cambridge Cambridge University Press, 1997).
  * [14] H. van Elst, C. Uggla, and J. Wainwright, Class. Quant. Grav. 19 51 (2002) [arXiv:gr/qc/0107041].
  * [15] R. H. Gowdy, Phys. Rev. Lett. 27 826 (1971) & Ann. Phys. (N.Y.) 83 203 (1971).
  * [16] M. Grasso, E. Villa, M. Korzyński and S. Matarrese, Isolating non-linearities of light propagation in inhomogeneous cosmologies [arxiv/2105.04552].
  * [17] E. Villa, S. Matarrese, and D. Maino, JCAP 2011 024 (2011).
  * [18] J. Adamek, E. Di Dio, R. Durrer and M. Kunz, Phys. Rev. D 89 063543 (2014).
  * [19] V. A. Belinskiı, I. M. Khalatnikov, and E. M. Lifshitz, Adv. Phys., 12 185 (1963) & 19 525 (1970) & 31 639 (1982).
  * [20] B. K. Berger and V. Moncrief, Phys. Rev. D 48 4676 (1993); B. K. Berger and D. Garfinkle, Phys. Rev. D 57 4767 1998); B. K. Berger, J. Isenberg and M. Weaver, Phys. Rev. D 64 084006 (2001); D. Garfinkle, Phys. Rev. Lett. 93 161101 (2004).
  * [21] L. Andersson, H. van Elst, W. C. Lim and C. Uggla, Phys. Rev. Lett. 94 051101 (2005).
  * [22] W. C. Lim, L. Andersson, D. Garfinkle and F. Pretorius, Phys. Rev. D 79 123526 (2009) [arXiv: 0904.1546].
  * [23] W. C. Lim, Ph.D. thesis, University of Waterloo 2004) [gr-qc/0410126].
  * [24] H. van Elst and C. Uggla, Class. Quant. Grav. 14 2673 (1997).