# Cosmic Topology I: Limits on Orientable Euclidean Manifolds from Circle
Searches

Pip Petersen  Yashar Akrami  Craig J. Copi  Andrew H. Jaffe  Arthur Kosowsky
Glenn D. Starkman  Andrius Tamosiunas  Johannes R. Eskilt  Özenç Güngör
Samanta Saha  and Quinn Taylor

###### Abstract

The Einstein field equations of general relativity constrain the local
curvature at every point in spacetime, but say nothing about the global
topology of the Universe. Cosmic microwave background anisotropies have proven
to be the most powerful probe of non-trivial topology since, within
$\Lambda$CDM, these anisotropies have well-characterized statistical
properties, the signal is principally from a thin spherical shell centered on
the observer (the last scattering surface), and space-based observations
nearly cover the full sky. The most generic signature of cosmic topology in
the microwave background is pairs of circles with matching temperature and
polarization patterns. No such circle pairs have been seen above noise in the
WMAP or Planck temperature data, implying that the shortest non-contractible
loop around the Universe through our location is longer than 98.5% of the
comoving diameter of the last scattering surface. We translate this generic
constraint into limits on the parameters that characterize manifolds with each
of the nine possible non-trivial orientable Euclidean topologies, and provide
a code which computes these constraints. In all but the simplest cases, the
shortest non-contractible loop in the space can avoid us, and be shorter than
the diameter of the last scattering surface by a factor ranging from 2 to at
least 6. This result implies that a broader range of manifolds is
observationally allowed than widely appreciated. Probing these manifolds will
require more subtle statistical signatures than matched circles, such as off-
diagonal correlations of harmonic coefficients.

##  1 Introduction

The basic assumption of most cosmological models is that spacetime is
isotropic and homogeneous as encoded in the Friedmann-Lemaître-Robertson-
Walker metric. However, this metric is only an expression of the local
spacetime geometry, and says nothing about the global topology. Cosmologists
have long speculated [1] about the possibility of a non-trivial spatial
topology, which would contain non-contractible space-like loops: travelling
far enough in a particular direction returns back to the original position.

Perhaps the most compelling case for our spatial manifold having a non-trivial
topology is ‘why not?’ We expect that a theory of quantum gravity may contain
topology-changing processes, e.g., quantum foam [2, 3]. If so, there would be
no particular reason that, when the Universe emerges from the very early
quantum-gravity era, the space would be the infinite-volume covering space of
one of the eight possible homogeneous 3-geometries.111 The covering space of a
given geometry, such as $E^{3}$ (zero curvature), $S^{3}$ (positive
curvature), or $H^{3}$ (negative curvature) is the manifold with no non-
contractible closed loops, i.e., it is the manifold with trivial topology. It
is also the manifold with the full isometry group of the geometry. Subsequent
evolution of the Universe could hide the topology by inflating all topological
scales far beyond the current Hubble scale; but, absent evidence for a
specific model of inflation, it is premature to discard the possibility of
detecting a non-trivial topology of the Universe through an inflationary
argument.

The case for considering cosmic topology is strengthened by the observation of
several so-called large-angle anomalies in the cosmic microwave background
(CMB) temperature data (see, e.g., Refs. [4, 5, 6, 7, 8] for reviews). These
anomalies seem to indicate a preferred length scale in the Universe that is
comparable to the current Hubble scale. If these anomalies are physical, i.e.,
not statistical flukes, then the anomaly scale has not been inflated beyond
detectability, and could be the scale of cosmic topology.

A topologically non-trivial manifold can be represented by some fundamental
domain with a finite extent in at least one spatial dimension, and a
prescription for how a geodesic intersecting a boundary surface of this domain
resumes at a different boundary surface. These loops exhibit one or more
characteristic length scales; if these length scales are sufficiently large
compared to the Hubble scale, any observational consequences of the topology
will fall below fundamental detection limits.

The most obvious observational consequence of such a topological structure
would be the appearance of multiple images of distant cosmic objects: light
coming from a far-away galaxy would arrive at our position via multiple
different paths. Each manifold would exhibit a predictable pattern of images
of each astrophysical source to each observer (that can depend on the position
of each); hence the name cosmic crystallography [9, 10, 11, 12, 13, 14, 15,
16] for the search for such patterns. Detection of topology through such
multiple images is complicated by the fact that different light paths lead to
views of sources (such as galaxies) from different vantage points and with
different look-back times, so both the morphology and time evolution of a
source must be well understood to identify its multiple images. Since deep
imaging over large sky areas became available, no objects have been discovered
with clones in well-separated locations on the sky [16]. However, the
observational difficulties and systematic uncertainties make precise
quantitative constraints on topology from multiple-image limits difficult.

In 1996, Cornish, Spergel, and Starkman [17, 18] made a large conceptual step
forward with their realization that topology would imprint a very specific
signature in the CMB radiation. Most CMB photons travel to an observer
unimpeded from near the last scattering surface, a spherical slice through the
primordial photon distribution centered on the observer at a comoving distance
of nearly the Hubble length. The insight was that, within a fundamental
domain, the last scattering surface, comprised of all points that are at a
fixed comoving distance from the observer along a geodesic of the spatial
geometry, may no longer be a complete sphere, but that it will always be
composed of spherical sections that intersect.

Any such intersection of spherical sections will be a circle if the metric is
homogeneous and isotropic; at points along this circle, photons can travel
from the last scattering surface in two different directions to arrive at one
spacetime location. In a homogeneous universe, every place would have the same
local density and temperature, and the topology would have no observable
consequence. But in a universe with small random spatial variations in
physical conditions at the time of last scattering, a circle of spatial points
encodes a specific pattern of fluctuations around a mean temperature. To the
extent that CMB photons reflect the local conditions of the spacetime region
from which they originated, an observer will see two separated circles in the
microwave temperature sky that have the same pattern of hot and cold spots.
The sizes, locations, and numbers of matched circle pairs depend on the
specific manifold with its characteristic length scales, and for some
topologies on the location of the observer within the manifold, but the
existence of matched temperature circle pairs (for some observers) is a
completely generic signature of any manifold having a length scale smaller
than twice the comoving distance to the last scattering surface, so long as
the geometry is sufficiently close to homogeneous and isotropic [18, 19, 20].
But an observer will not necessarily see such circles — only if there is a
non-contractible closed loop through them that is shorter than the last
scattering surface diameter $L_{\mathrm{LSS}}$. Similar circles should then be
found in polarization [21] (although this case is subtler because polarization
arises primarily from primordial velocities, and the direction of the velocity
vector relative to the photon direction depends on the photon path).

A generic search of the microwave sky for matched circles is computationally
challenging due to the large number of possible circle-pair centers, circle
radii, and circle orientations. In addition, even perfect CMB sky maps, with
no detector noise or foreground emission, have significant noise for this
signal, arising from contributions to CMB anisotropies other than variations
in the local gas temperature at last scattering. These include the local gas
velocity (“Doppler signal”) and the integrated Sachs-Wolfe (ISW) effect from
time-varying gravitational potentials along the photon paths. Once full-sky
microwave maps were created by the Wilkinson Microwave Anisotropy Probe
(WMAP), full-sky matched-circle searches were completed, finding no matched
circles in excess of those expected from noise [22, 23, 24, 25, 26, 27, 28,
29]. This result was later confirmed using higher-resolution maps from the
Planck satellite [30, 31, 32]. The null results imply that our Universe
contains no closed space-like geodesics passing through our location with
comoving proper length less than 98.5% of $L_{\mathrm{LSS}}$.

This result has widely been interpreted to mean that any topologically non-
trivial manifold with a characteristic length scale smaller than the observed
limit is observationally ruled out. However, this is not the case. Even among
the restricted class of spatially flat manifolds, there can be length scales
significantly shorter than the diameter of the last scattering surface and
still not violate the observational bounds. This is particularly true for
topologies for which identification of boundary surfaces involves a rotation
or reflection.222 In more mathematically precise language, in Euclidean
manifolds where the covering space has been modded-out by a discrete subgroup
of the $E^{3}$ isometry group that includes corkscrew motions or glide
reflections — i.e., all Euclidean manifolds except the simple 3-torus, the
chimney space, the unrotated slab space, and the covering space — the
statement that the shortest closed spatial geodesic through us must be longer
than $98.5\%$ of the diameter of the LSS does not imply that the shortest
closed spatial geodesic through every point in the manifold must be longer
than this. This is possible because the topological boundary conditions break
homogeneity: the length of the shortest closed geodesic through a particular
point can vary substantially within the volume of the manifold.  In Ref. [33]
we review the status of cosmic topology, arguing that past searches have been
less thorough and their interpretation overly restrictive than typically
thought. In particular, as we show here, the lower limit on the topology scale
of an orientable Euclidean manifold can be a factor of 2 to 6 shorter than
$L_{\mathrm{LSS}}$ (with even weaker limits possible in the rotated slab
space).

In this paper, we systematically display constraints on the length scale of
all spatially flat (Euclidean), orientable topologies from the condition that
they do not violate the observed lack of circle pairs in the CMB temperature
sky. In particular, we directly compute which observer locations in the
fundamental domain result in a violation of the observed constraint, for a
range of length scales in each topology. We do this by finding the comoving
distance to the nearest observer clone in the covering space for observer
locations throughout the manifold. A particular topology with a given length
scale will have a fixed percentage of the fundamental domain volume allowed;
the topology is disfavored if this percentage is too small, or strictly ruled
out when it is zero. Our results highlight a number of cases where the minimum
comoving length scale of allowed manifolds is significantly smaller than the
comoving diameter of the last scattering surface. The existence of many such
topologies motivates further observational searches using different CMB
signals, in particular the correlations between spherical harmonic components
characteristic of a given topology [34].

Observations of the CMB temperature and polarization imply that the mean
spatial curvature is quite small, i.e., $|\Omega_{K}|\ll 1$, though perhaps
$\Omega_{K}<0$ [35]. Early-universe inflation, which can produce the basic
properties of the observed Universe, is often taken to predict that spatial
curvature should be near zero, but models that result in measurable curvature
have also been considered. Furthermore, inflationary arguments would have to
be reconsidered if cosmic topology were discovered, since the duration of
inflation would then need to be short enough that the topology scale is not
stretched to far beyond the current horizon scale. Despite these
qualifications, for simplicity we take the background geometry of the Universe
to be Euclidean (spatially flat) and orientable. We will consider non-
orientable Euclidean manifolds333 Note that physical arguments about the
consistency of quantum field theory may rule out non-orientable spaces [36],
though we advocate caution in accepting the applicability of such no-go
theorems, which, for example, assume that there are no topological features on
small scales that would change the orientability of the manifold as encoded in
the large-scale topology.  elsewhere, as we have identified new
representations of their symmetries (loosely, new parameterizations of their
possible fundamental domains) that merit detailed presentation. Non-flat
geometries admit a countable infinity of topologies; in future work we will
present limits on representative non-Euclidean manifolds from circle searches.

We make freely available a code that identifies the location of (and distance
to) the nearest clone of any observer within a described fundamental domain.
We also make available a second code that allows the user to establish whether
specific parameters of an orientable Euclidean manifold are allowed or
forbidden, depending on a chosen threshold of the fraction of observer
locations producing matching circles. These codes444Available at
https://github.com/CompactCollaboration/EOT_FDSearch. will eventually be
extended to include a wider set of manifolds.

Section 2 of this paper reviews relevant results about characterizing
topology, fundamental domains, and covering spaces, and establishes our
mathematical conventions. Section 3 discusses how to compute distances to the
nearest clones. Section 4 displays results showing the portions of fundamental
domains that are allowed for each topology considered as a function of the
characteristic length scale of the fundamental domain given the observational
constraint on matched circle pairs in the CMB temperature maps. In Section 5,
we discuss how these results motivate further searches for topologies that are
not ruled out by current observations, but which may still display detectable
signatures more subtle than circle pairs. In the appendix we enumerate the
specific set of topologies considered in this work.

##  2 Properties of Orientable Euclidean Manifolds

Each Euclidean manifold is characterized by a representation of its associated
discrete, freely acting subgroup of the isometry group of $E^{3}$— the
rotations, reflections, and translations in three dimensions. In general, a
representation is determined by a set of $d$ generators $g$ (where $d$ is the
number of compact dimensions) that each acts on points $\bm{x}$ in the
manifold as

$g:\bm{x}\to\bm{\mathsf{M}}(\bm{x}-\bm{x}_{0})+\bm{T}+\bm{x}_{0}\,.$ (2.1)

Here $\bm{\mathsf{M}}$ is an element of $O(3)$, i.e., a proper rotation
(possibly the identity), a reflection, or some combination of both; $\bm{T}$
is a translation vector; and $\bm{x}_{0}$ encodes the choice of origin555 For
a proper rotation, it is the position of an arbitrary point on the rotation
axis with respect to the chosen origin; for a reflection, it is the position
of an arbitrary point on the plane(s) of reflection; for the identity it is
irrelevant. . In this work we only consider orientable manifolds; thus
$\bm{\mathsf{M}}$ will be an element of $SO(3)$, i.e., either the identity or
a proper rotation around an axis $\bm{\hat{n}}$ by an angle $\alpha$,
$\bm{\mathsf{R}}_{\bm{\hat{n}}}^{\alpha}$. The full complement of group
elements is obtained by repeated application of the generators and their
inverses.

From the point of view of the covering space, each group element maps a given
point to another distinct point in the manifold. These points are referred to
as “clones.” It is often convenient to think of the topologically non-trivial
manifolds in terms of tilings of the covering space. Each tile contains
exactly one copy of each point — i.e., no tile contains a point and any of its
clones. All tiles are exact copies of one another — one can map one tile into
any other by the application of some group element. The covering space is
fully covered by the tiles — i.e., no point in the covering space is omitted.
One of these tiles is often referred to as the fundamental domain. However, it
is worth recognizing that these tiles have no absolute physical meaning as one
can group the points in space into tiles in many different ways — consider,
e.g., M.C. Escher’s Flying Fish, a tiling of Euclidean 2-space by tiles with a
very complex shape like a flying fish [37]. Thus the fundamental domain should
not be misinterpreted as the shape of the Universe. A convenient, and
frequent, choice of fundamental domain is the Dirichlet domain of some
particular point — the set of all points in the manifold that are closer to
that base point than to any of its clones. However, the shape of the Dirichlet
domain typically depends on the choice of base point; thus it too should not
be misinterpreted as the shape of the Universe. A given observer will have a
Dirichlet domain shape which is not typically shared by other observers. The
physical characteristics of a topology are fully contained in the isometries
of the manifold — i.e., the generators and the group elements they generate.

The choice of topology fixes the matrices $\bm{\mathsf{M}}$, leaving freedom
to choose the translation vectors $\bm{T}$ in the generators. If
$\bm{\mathsf{M}}$ is a proper rotation, i.e., not the identity, then a
redefinition of the origin $\bm{x}_{0}$ can be used to adjust the components
of $\bm{T}$ that are perpendicular to the axis of rotation. In each topology
where this applies, we use this freedom to choose $\bm{T}$ parallel to the
axis of rotation and relabel the coordinates so that $\bm{x}_{0}=\bm{0}$.
Thus, for $\bm{x}=\bm{0}$, $g$ is always a pure translation, while for
$\bm{x}\neq\bm{0}$ it is a so-called “corkscrew” motion.

There are six fully compact orientable Euclidean topologies: “simple 3-torus,”
“half-turn space,” “quarter-turn space,” “third-turn space,” “sixth-turn
space,” and “Hantzsche-Wendt space” ($E_{1}$–$E_{6}$). Two other orientable
“chimney spaces” $E_{11}$ and $E_{12}$ are compact in two dimensions but not
the third. Finally the “slab space” $E_{16}$ is a manifold compact in just one
dimension. The matrices $\bm{\mathsf{M}}$ and the associated translations
$\bm{T}$ for all of these orientable Euclidean topologies are given in the
appendix. Further for all of these topologies (except for the special case of
$E_{16}$, which we discuss in the appendix), there is, for each of the
$\bm{\mathsf{M}}$’s included in the set of group elements, a smallest positive
integer $N$ such that

$\bm{\mathsf{M}}^{N}=\mathds{1}\,,$ (2.2)

where $\mathds{1}$ is the identity. This means that in $E_{2}$–$E_{6}$ it is
always possible to build three linearly independent pure translations out of
products of the generators. For example, $g^{N}$ (the application of the
generator $g$ a total of $N$ times) is always a pure translation. These three
linearly independent translations serve as the generators of an “associated
$E_{1}$.” Properly chosen, these translations generate the subgroup of the
isometry group of the manifold that are all the pure translations. Similarly,
for $E_{12}$, we can construct an associated $E_{11}$. For each manifold,
$E_{1}$–$E_{6}$, $E_{11}$, and $E_{12}$, the associated $E_{1}$ or $E_{11}$ is
provided in the appendix.

Figure 1: A representation of a section of the $E_{4}$ covering space with
parameter values $L_{A}=2$ and $L_{B}=0.95$ in units of $L_{\mathrm{LSS}}$,
the comoving diameter of the last scattering surface. Observers who see
matching circles are in red, and observers who do not see circles are in blue.
The solid black lines represents two potential “associated $E_{1}$’s”,
depending on the choice of origin. The fraction of points in any associated
$E_{1}$ is equal, and in this case both $E_{1}$ representations have a
fraction of points allowed at 0.91.

These associated representations are of computational utility, as the natural
fundamental domains constructed from those pure translations have simple
shapes, allowing for the easy random selection of observers with uniform
probability inside the associated $E_{1}$ or $E_{11}$. Uniform sampling of the
associated $E_{1}$ or $E_{11}$ guarantees uniform sampling of the manifold. A
finite portion of an example covering space of $E_{4}$ is shown in Fig. 1,
where every point represents an observer location and each black parallelogram
encloses a choice of associated $E_{1}$. The red and blue points represent,
respectively, whether or not an observer at that location would see matching
circles in their sky. We do not provide a fundamental domain for each topology
because it has no objective physical significance, nor do we provide a
Dirichlet domain because it varies from observer to observer except in
$E_{1}$, $E_{11}$, and an unrotated $E_{16}$.

##  3 Clone Search

The main goal of this work is to translate the non-detection of matched circle
pairs in the CMB to constraints on the length scales of each compact,
orientable, Euclidean topology. For each topology and any set of topological
parameter values characterizing a manifold with that topology, we determine
the fraction of randomly distributed observers in that manifold who would have
seen matched circle pairs. If every observer in the manifold would have seen
circles, then the circle searches based on the WMAP and Planck data would
completely rule out that manifold. In general, if a fraction $p$ of observers
would see circles, then the manifold can be ruled out at approximately this
$p$-value.

To achieve this goal, we randomly populate the manifold with observers, and
determine the distance to each observer’s nearest clone. For a given topology,
we first choose a manifold based on the allowed range of parameters (see the
appendix) and uniformly populate that manifold’s associated $E_{1}$ or
$E_{11}$ with potential observers. Next, the eligible nearest clones of each
observer are found by repeated application of generators. This list is
composed of those clones lying in the region of the covering space adjoining
the observer’s associated $E_{1}$ or $E_{11}$. The distance to each of these
clones is computed and the shortest distance, corresponding to the nearest
clone, is noted. If this shortest distance is less than $L_{\mathrm{LSS}}$
then the observer point is identified as an “excluded point,” otherwise it is
an “allowed point.” This identification is made since an observer that has a
clone closer than $L_{\mathrm{LSS}}$ will see matched circle pairs, violating
the null result of the previous searches.666More accurately, the searches did
not find a value of the circle radii and center separations for which there
was an excess of circle pairs exhibiting a value of the matching statistic
that exceeded the detection threshold for a 99.7% C.L. detection, nor for a
95% C.L. detection, for clone separations from the observer up to 98.5% of
$L_{\mathrm{LSS}}$. However, for slightly smaller distances, the chances of a
false negative fall precipitously.  Once the associated $E_{1}$ or $E_{11}$ is
sampled, the fraction $p$ of excluded points is determined and the manifold is
classified based on the $p$-value: allowed ($p=0$), ruled out ($p=1$), or
somewhere in between (all other cases are quantified by $p$).

The procedure for determining the closest clone to a point can be divided into
two steps: applying non-trivial generators and applying pure translations. The
first step involves applying the non-trivial generators to the sampled point a
number of times, up to the order of that generator in the associated $E_{1}$,
described by $N$ in Eq. 2.2. This step produces a set of clones, each with a
unique configuration in the associated $E_{1}$. For example, in $E_{2}$, we
have two points in the set: the fiducial point and the point determined by a
single application of $g^{E_{2}}_{B}$ (see Section A.2 for this case and the
appendix for the required generators of the other topologies). The second step
involves applying the group of pure translations to each point in this new set
of clones. With the correct choice of translations, these two steps will give
clones only within the $3\times 3\times 3$ cube of associated $E_{1}$, with
the sampled associated $E_{1}$ in the center. The closest clone will always be
in this small set of new points, and so drastically reduces the number of
clones to be evaluated. The minimum Euclidean distance of these clones is then
found and compared to $L_{\mathrm{LSS}}$ to determine if the observer location
is allowed or excluded.

##  4 Topological Constraints from Observations

To examine how the distribution of allowed (or excluded) points in a manifold
depends on their generators, we first fix the trivial generators — those that
are pure translations — to a characteristic length scale greater than
$L_{\mathrm{LSS}}$. This is required since when the length scales of those
trivial generators are smaller than the diameter of the last scattering
surface every point in the manifold would have a clone close enough that
matched circles would be observable. By adjusting the remaining length scales,
we are able to generate the regions of disallowed points for each topology and
examine how those regions change as a function of the chosen length scales.

Fig. 2 shows two-dimensional projections of manifolds with the topologies
$E_{2}$, $E_{3}$, $E_{4}$, $E_{5}$, and $E_{12}$ (along the single corkscrew
direction), and a three-dimensional plot of an $E_{6}$ manifold, which has
three corkscrew motions. For the specific choices of the topology parameters
of those manifolds indicated, observers in the blue locations would not see
matched circle pairs, while observers in the red locations (which comprise
cylinders in the manifold, of which Fig. 2 shows cross-sections) would see
circle pairs. For $E_{2}$, $E_{3}$, $E_{4}$, $E_{5}$, and $E_{12}$, the
translations orthogonal to the corkscrew motion are chosen to be of length
$L_{\mathrm{LSS}}$ — large enough that the clones in those directions are too
far to result in matched circle pairs. The translations associated with the
corkscrews are chosen such that just under 10% of observers see circle pairs.
$E_{1}$ and $E_{11}$ are not shown in the figure because they are homogeneous
— if any topology scale is less than $L_{\mathrm{LSS}}$, then all observers
can see a matched circle pair, otherwise none can. We discuss $E_{16}$ below.

While the shape of the associated $E_{1}$ and the pattern of excluded points
are particular to each manifold, they do share common features. Most
interestingly, for appropriate parameters there are excluded cylindrical
“columns” centered on each corkscrew axis (and its clones). This property is
most clear in the three-dimensional plot of the circle-seeing (red) and non-
circle-seeing (blue) regions of an $E_{6}$ manifold in the bottom-left panel
of Fig. 2. In $E_{6}$, each non-trivial generator ($g_{A}$, $g_{B}$, and
$g_{C}$ as discussed in the appendix) has a corkscrew motion, and we see the
intersecting orthogonal cylinders in the figure.

Figure 2: Classification of observer locations into excluded points
represented in red (those who would see matched circle pairs), and allowed
points represented in blue (those who would not) in six non-trivial,
orientable topologies that have symmetry groups including corkscrew motions,
i.e., $E_{2}$, $E_{3}$, $E_{4}$, $E_{5}$, $E_{6}$, and $E_{12}$. ($E_{1}$ and
$E_{11}$ are homogeneous — if any topology scale is less than
$L_{\mathrm{LSS}}$, then all observers can see a matched circle pair,
otherwise none can. $E_{16}$ is discussed separately in the text.) The black
circles indicate the analytical results for each of these topologies — making
use of the method described in Eq. 4.2 — to compare to the numerical results.
The coordinates $x$, $y$, $z$ denote the locations of the sampled observers
relative to an origin on the axis of the corkscrew motion(s) of each manifold.
The length scales associated with the fundamental domains are in units of
$L_{\mathrm{LSS}}$. (See the appendix for details regarding the meaning of
each length scale in each topology.) Parameters defining the manifolds are
chosen such that more than $90\%$ of the observers do not see matching circles
in the sky. Note: $E_{6}$ requires a choice of
$L_{A}=L_{B}=L_{C}=0.98L_{\mathrm{LSS}}$ to reveal its symmetries.

For the simplest orientable topologies, the locations of these cylinders of
excluded observers can be readily calculated analytically. For example, in the
half-turn space, $E_{2}$ (with corkscrew axis $\hat{z}$), for excluded points
the shortest distance between clones satisfies

$\left\lVert\begin{pmatrix}x\\\ y\\\ z\end{pmatrix}-\begin{pmatrix}-x\\\ -y\\\
z+L_{B}\end{pmatrix}\right\rVert\leq L_{\mathrm{LSS}}\,,$ (4.1)

which reduces to the region

$x^{2}+y^{2}\leq\frac{L_{\mathrm{LSS}}^{2}-L_{B}^{2}}{4}\,.$ (4.2)

This is a cylinder with axis centered on the origin in the $xy$-plane, with
radius $\frac{1}{2}\sqrt{L_{\mathrm{LSS}}^{2}-L_{B}^{2}}$. Similar
calculations can be performed for other topologies, which produce the
boundaries shown by the black circles in Fig. 2. As expected, the radii of the
excluded cylinders depend only on the length of the translation vector along
the corkscrew axis.

Figure 3: Locations $(x,y)$ of observers in planes of constant $L_{B}$ having
clones closer than $L_{\mathrm{LSS}}$, the comoving diameter of the last
scattering surface, as a function of $L_{B}$ for $E_{3}$, $E_{4}$, and $E_{5}$
with $L_{A}=2$ in units of $L_{\mathrm{LSS}}$. As the non-trivial length scale
($L_{B}$) decreases, the allowed region (in white) decreases. At
$L_{B}\lesssim L_{\mathrm{LSS}}$, only a small fraction of observers see
circle pairs.

In Fig. 3, we make use of these analytical expressions and numerical results
to show how the pattern of allowed (or excluded) points changes as the length
scale $L_{B}$ (the length of the corkscrew motion about the $\hat{z}$ axis) is
varied in $E_{3}$, $E_{4}$, and $E_{5}$. $L_{A}$, the length of the two pure-
translation generators in the $xy$-plane, is chosen to be $2L_{\mathrm{LSS}}$,
so clones closer than $L_{\mathrm{LSS}}$ must be due, at least in part, to the
corkscrew motion. For all three topologies, increasing $L_{B}$ gradually
reduces the fraction of observers with a clone closer than $L_{\mathrm{LSS}}$.
We have restricted these contour plots to the topologies with two free
parameters; however, similar figures can be produced for the other orientable
topologies in a higher dimension parameter space.

We see in Fig. 3 that as soon as $L_{B}$ is slightly less than
$L_{\mathrm{LSS}}$ ($L_{B}=0.98L_{\mathrm{LSS}}$ shown in dark grey), points
in a disk centered at $(x,y)=(0,0)$ (and with any value of $z$) should see
matched circle pairs and so are excluded. The radius of the disk depends on
$L_{B}$, and grows with $|L_{\mathrm{LSS}}-L_{B}|$. These are seen in all
panels as partial disks centered on $(x,y)=(0,0)$, but also at the images of
$(0,0)$ under the pure translations in the $xy$-plane — i.e., at the corners
of the parallelograms outlined in black. However, another set of points is
also excluded in $E_{3}$ and $E_{4}$ — disks around points in the $xy$-plane
whose images under the corkscrew motions have $x$ and $y$ coordinates that are
clones of the original point. For example, for $E_{3}$, the $\pi$-corkscrew
takes $(L_{\mathrm{LSS}},L_{\mathrm{LSS}},z)$ to
$(-L_{\mathrm{LSS}},L_{\mathrm{LSS}},z+L_{B})$; but the application of one of
the pure translations (by $(2L_{\mathrm{LSS}},0,0)$) takes this to
$(L_{\mathrm{LSS}},L_{\mathrm{LSS}},z+L_{B})$, a distance $L_{B}$ directly
above the original point. The same happens in $E_{4}$ for observers at
$(0,2L_{\mathrm{LSS}}/\sqrt{3},z)$ and
$(L_{\mathrm{LSS}},L_{\mathrm{LSS}}/\sqrt{3},z)$; it does not happen in
$E_{5}$.

In $E_{3}$ and $E_{5}$, new excluded disks emerge when
$L_{B}<L_{\mathrm{LSS}}/2$, centered at points where two applications of the
corkscrew motion carry the disks to points that are clones of the original
point. For $E_{3}$, for example, $(0,L_{\mathrm{LSS}},z)$ is taken to
$(0,-L_{\mathrm{LSS}},z+2L_{B})$, which can be taken back to
$(0,L_{\mathrm{LSS}},z+2L_{B})$ with a translation by
$(0,2L_{\mathrm{LSS}},0)$. For $E_{5}$ it is no surprise that the disks appear
at the same locations where there are “interior disks" in $E_{4}$, since the
corkscrew motion for $E_{5}$ is a rotation by half the angle as in $E_{4}$.

Figure 4: Excluded regions of the parameter space of the $E_{3}$, $E_{4}$, and
$E_{5}$ topologies, with the length scales measured in units of
$L_{\mathrm{LSS}}$, the comoving diameter of the last scattering surface. Each
region is limited by the percentage of observers having a close enough clone
(closer than the $L_{\mathrm{LSS}}$) to produce a matched circle pair in the
sky. For example, for any choice of parameters within the lightest gray
region, 0 to $50\%$ of observers in the fundamental domain will have a clone
close enough to produce a matched circle pair. $L_{A}$ and $L_{B}$ are given
in units of $L_{\mathrm{LSS}}$.  Figure 5:  Regions of the parameter space of
the $E_{1}$–$E_{5}$ topologies where an observer would or would not have a
clone close enough to generate a matched circle pair in the sky. In the shaded
region for a topology (and the topologies above it in the legend) $\geq 95\%$
of observers have a clone closer than $L_{\mathrm{LSS}}$, the comoving
diameter of the last scattering surface. In the exterior (white) region, no
topology will produce circles in the sky. For $E_{2}$ we choose the right
rectangular prism, $L_{A_{1}}=L_{A_{2}}\equiv L_{A}$. $E_{2}$–$E_{5}$ are
arranged in increasing order of their associated rotation. $L_{A}$ and $L_{B}$
are given in units of $L_{\mathrm{LSS}}$.

To this point, we have explicitly chosen parameters such that only a small
fraction ($<10\%$) of points in the manifold would have seen clones. We also
aim to identify when a specific topology becomes either disfavored or strictly
ruled out. This can be done by examining the unique parameter space for each
topology. The dimensionality of this parameter space depends on the topology,
and for the orientable Euclidean manifolds ranges from 2 to 6. In some
topologies, we can straightforwardly calculate analytically the shortest
distance to each observer’s nearest clone, and so determine how the constraint
on the shortest distance around the Universe through us translates into a
constraint on the parameters of that topology. In other topologies, the
analytic approach is more fraught, or perhaps impossible. Here, we choose to
uniformly sample manifolds of each topology in order to establish the
boundaries in parameter space between allowed regions (sufficiently low
fraction of observers would see circles) and excluded regions (too high a
fraction of observers would see circles). That boundary depends on the
fraction of circle-seeing observers one permits on the topology and on the
topology parameters that specify the manifold.

The results of this sampling of the topology parameter spaces are shown in
Fig. 4 for $E_{3}$, $E_{4}$, and $E_{5}$ — the three manifolds with just two
topology parameters. In the most strict case — no observer sees circles — we
have sharp cutoffs defined by the condition
$L_{\mathrm{max}}=L_{\mathrm{LSS}}$, where $L_{\mathrm{max}}$ is the length of
the longest translation vector of the three generators.777 There is an
important subtlety here: one must have chosen the three generators with the
shortest possible translation vectors. The parameterizations presented in the
appendix make that choice.  This makes intuitive sense — for pure
translations, the distance to the nearest clone is the length of the
translation. For corkscrew motions, the points along the axis of rotation
experience a pure translation; they are the closest to their clones, and only
become excluded when the pure translation does not produce matching circles.
If we allow a growing percentage of observers to see circles, we begin to
expand the parameter space, until we reach the other limit: manifolds in which
no observer sees circles. These results are summarized in Fig. 5 for the case
where at least 95% of observers in the manifold see matched circles. We
include $E_{2}$ (which has four parameters) with $L_{A_{1}}=L_{A_{2}}\equiv
L_{A}$ and $\alpha=\pi/2$ (see the appendix for details of the $E_{2}$
parameters).

Figure 6:  Constraints on parameter space of the $E_{2}$ topology with length
scales $L_{A_{1}}$, $L_{A_{2}}$, and $L_{B}$ measured in units of
$L_{\mathrm{LSS}}$, the comoving diameter of the last scattering surface.
Every point within a graph describes a unique representation of an $E_{2}$
fundamental domain. Left to right: the ‘fraction threshold’ describing how
many points within the fundamental domains must not see circles in order for
the parameter values to be considered excluded. Top to bottom: the angle
$\alpha$ of the fundamental domains. Figure 7: As in Fig. 6, but for the
$E_{6}$ topology with length scales $L_{A}$, $L_{B}$, and $L_{C}$.

Topologies with more than two free parameters, i.e., $E_{2}$ and $E_{6}$, are
more difficult to produce analytic results for, and so we rely on numerical
visualizations of the three-dimensional parameter space. Fig. 6 and Fig. 7
show constraints on the parameter spaces for these two topologies. More
precise results for values near the boundary can be identified using the
provided associated $E_{1}$ sampling code on GitHub.

$E_{16}$ is a special case. If the rotation angle $\zeta$ associated with the
corkscrew is zero (see section A.9 of the appendix), then it is like $E_{1}$
and $E_{11}$. If $\zeta\neq 0$ and $L_{\mathrm{LSS}}<L$, then no observer sees
a circle. If $\zeta\neq 0$ and $L_{\mathrm{LSS}}/2<L<L_{\mathrm{LSS}}$, then
only observers in a cylinder around the rotation axis would see circles; the
larger $|\zeta|$ is, the larger the radius of the cylinder. If
$L_{\mathrm{LSS}}/3<L<L_{\mathrm{LSS}}/2$, then the cylinder of observers who
see circles grows — the closer $\zeta$ is to $\pm\pi$, the larger the radius;
if $\zeta=\pm\pi$, then all observers would see circles. A similar transition
happens for $L_{\mathrm{LSS}}/4<L<L_{\mathrm{LSS}}/3$ — a larger cylinder of
circle-observing observers, with the radius determined by how close $\zeta$ is
to $2p\pi/3$, for $p=\pm 1$ or $\pm 2$. And so on, as $L$ falls below
$L_{\mathrm{LSS}}/q$ for each successive positive integer $q$.

##  5 Conclusions

We have shown herein the limits on the parameters of the orientable Euclidean
topologies from non-observation of matching circle pairs in the CMB sky are
much less restrictive than previously assumed. For the $E_{1}$ and $E_{11}$
topologies, the shortest closed spatial geodesic around the Universe
$L_{\mathrm{min}}$, which is also the length of the shortest of the
translations associated with the topology, must be longer than approximately
$L_{\mathrm{LSS}}$. However, in $E_{2}$–$E_{6}$, $E_{12}$, and $E_{16}$ the
constraint on $L_{\mathrm{min}}$ is weaker by a topology-dependent factor that
ranges from 2 in $E_{2}$ and $E_{12}$ to 6 in $E_{5}$, and an arbitrarily
large factor in $E_{16}$. Sampling of observer locations in each manifold has
been used to identify the allowed and excluded regions within each manifold
and to place constraints on the parameter space of each topology.

We have described a method for identifying the location of the nearest clone
in the covering space of $E^{3}$, depending on the chosen parameters of a
manifold with a given topology. We demonstrated in Fig. 3 how the regions of a
manifold in which observers see matched circle pairs in the CMB sky depend on
the parameters of the topology. We have also provided instructive
visualisations of the allowed and excluded parameters of each topology. Tools
will be made publicly available on GitHub for finding the location and
distance to the nearest clone; calculating the fraction of observer locations
in a given manifold with matched circle pairs; and searching parameter space
of a topology at a specified exclusion threshold.

We have considered only orientable Euclidean topologies, ignoring the non-flat
and non-orientable cases. In future studies we will constrain the parameter
spaces for those topologies, and provide additional descriptions of their
properties. An upcoming paper [34] will provide more details about the
properties of the Euclidean manifolds, both orientable and non-orientable,
including detailed descriptions of the most general parameterizations of the
manifolds of each topology, and present the resulting eigenmodes of the scalar
Laplacian. These are basic building blocks necessary to further the generic
search for non-trivial cosmic topology.

Certainly the available parameter space in which to search for cosmic topology
is considerably more open than previously believed. In Ref. [34], we will show
that the accessible microwave background information available due to any non-
trivial topology can also be larger than previously understood.

## Acknowledgments

We thank Jeffrey Weeks and David Singer for valuable conversations. Y.A.
acknowledges support by the Richard S. Morrison Fellowship, from research
projects PGC2018-094773-B-C32 and PID2021-123012NB-C43, by the Spanish
Research Agency (Agencia Estatal de Investigación)’s grant
RYC2020-030193-I/AEI/10.13039/501100011033 and the European Social Fund (Fondo
Social Europeo) through the Ramón y Cajal program within the State Plan for
Scientific and Technical Research and Innovation (Plan Estatal de
Investigación Científica y Técnica y de Innovación) 2017-2020, and by the
Spanish Research Agency through the grant IFT Centro de Excelencia Severo
Ochoa No CEX2020-001007-S funded by MCIN/AEI/10.13039/501100011033. C.J.C.,
A.K. and G.D.S. acknowledge partial support from NASA ATP grant RES240737;
G.D.S. from DOE grant DESC0009946; P.P., Y.A., G.D.S., O.G. and S.S. from the
Simons Foundation; Y.A., A.H.J. and G.D.S. from the Royal Society (UK); and
A.H.J. from STFC in the UK. A.T. is supported by the Richard S. Morrison
Fellowship. J.R.E. acknowledges support from the European Research Council
under the Horizon 2020 Research and Innovation Programme (Grant agreement No.
819478).

##  Appendix A Generators of Orientable Euclidean Topologies

This appendix collects expressions for the generators of the topologies
considered in this paper, most of which have appeared previously in the
literature though we do provide explicit expressions for the most general form
of the $E_{2}$ generators which had previously only been suggested (see, e.g.,
Ref. [38]). Also the possibility of a corkscrew motion rather than a pure
translation appears to have been overlooked in $E_{16}$. In Ref. [34], we will
find that for non-orientable Euclidean topologies, the possible manifolds are
considerably more diverse than previously expected.

These generators depend on one or more free parameters. Our parameter sets are
chosen to represent all possible cases while simplifying the form of the
generators. We provide restrictions on the parameter values to avoid duplicate
representations of the same physical space, defined by identical patterns of
clones. The rationale behind these conditions on the parameters is given in
Ref. [34].

###  A.1 $E_{1}$: Simple 3-torus

The simple 3-torus is the most basic compact Euclidean manifold, only
containing translations in its symmetry group. We use the freedom to orient
the observer to align the shortest translation along the $x$-direction, and
then to set the $z$-component of the second shortest translation to zero.
Equivalently, this means that two translation vectors $\bm{T}^{E_{1}}_{A_{1}}$
and $\bm{T}^{E_{1}}_{A_{2}}$ lie in the $xy$-plane and the third
$\bm{T}^{E_{1}}_{A_{3}}$ has a non-zero $z$-component. The generators of the
simple 3-torus are then given in the usual orthogonal Cartesian basis by

$\displaystyle\bm{\mathsf{M}}_{A}^{E_{1}}$ $\displaystyle=\mathds{1}\,,$
$\displaystyle\bm{T}^{E_{1}}_{A_{1}}$
$\displaystyle=L_{A_{1}}(1,\,0,\,0)\equiv\bm{T}_{1}^{E_{1}}\,,$ (A.1)
$\displaystyle\bm{T}^{E_{1}}_{A_{2}}$
$\displaystyle=L_{A_{2}}(\cos\alpha,\,\sin\alpha,\,0)\equiv\bm{T}_{2}^{E_{1}}\,,$
$\displaystyle\bm{T}^{E_{1}}_{A_{3}}$
$\displaystyle=L_{A_{3}}(\cos\beta\cos\gamma,\,\cos\beta\sin\gamma,\,\sin\beta)\equiv\bm{T}_{3}^{E_{1}}\,.$

Here, $\alpha$ is the angle between the two vectors in the $xy$-plane, and
$\beta$ and $\gamma$ are the angles describing the orientation of the third
vector. We take $L_{A_{1}}$, $L_{A_{2}}$, and $L_{A_{3}}$ as the “length
scales” of each of these translation vectors, which in this simple case
correspond to side lengths of the fundamental domain. This choice simply
defines a convenient coordinate system for any set of three linearly
independent vectors in three dimensions.

Note that we have introduced indices on the matrices $\bm{\mathsf{M}}$, the
translations $\bm{T}$, and the lengths associated with each generator. We
label each distinct matrix within a given topology as
$\bm{\mathsf{M}}^{E_{i}}_{a}$, $a\in\\{A,B,C\\}$. For $E_{1}$ all the matrices
are the same, $\bm{\mathsf{M}}^{E_{1}}_{A}$. For each fixed $a$, we label the
translations as $\bm{T}^{E_{i}}_{a_{j}}$ and the lengths as $L_{a_{j}}$, where
the additional index $j\in\\{1,2,3\\}$ is only included when required. For
$E_{1}$, the three vectors associated with $\bm{\mathsf{M}}^{E_{1}}_{A}$ are
thus $\bm{T}^{E_{i}}_{A_{1}}$, $\bm{T}^{E_{i}}_{A_{2}}$, and
$\bm{T}^{E_{i}}_{A_{3}}$.

###  A.2 $E_{2}$: Half-turn space

The half-turn space replaces one of the translations in $E_{1}$ with a
corkscrew motion by $\pi$. We have used the freedom to rotate the coordinate
system to set the axis of the associated rotation
$\bm{\mathsf{M}}_{B}^{E_{2}}$ parallel to $\bm{\hat{z}}$, and then to set the
$y$-component of the shorter of the two pure translations to zero, i.e.,
$\bm{T}^{E_{2}}_{A_{1}y}=0$. (The orthogonality of the pure translations to
the axis of rotation is not a choice, but a necessity [34].) The generators of
the half-turn space are then given by

$\displaystyle\bm{\mathsf{M}}^{E_{2}}_{A}$ $\displaystyle=\mathds{1}\,,$
$\displaystyle\bm{T}^{E_{2}}_{A_{1}}$
$\displaystyle=L_{A_{1}}(1,\,0,\,0)\equiv\bm{T}^{E_{2}}_{1}\,,$
$\displaystyle\bm{T}^{E_{2}}_{A_{2}}$
$\displaystyle=L_{A_{2}}(\cos\alpha,\,\sin\alpha,\,0)\equiv\bm{T}^{E_{2}}_{2}\,;$
(A.2) $\displaystyle\bm{\mathsf{M}}^{E_{2}}_{B}$
$\displaystyle=\bm{\mathsf{R}}_{\bm{\hat{z}}}^{-\pi}=\operatorname{diag}(-1,-1,1)\,,$
$\displaystyle\bm{T}^{E_{2}}_{B}$ $\displaystyle=L_{B}(0,\,0,\,1)\,.$

The associated $E_{1}$ is given by the translation vectors
$\bm{T}^{E_{2}}_{1}$ and $\bm{T}^{E_{2}}_{2}$, and a third translation vector
$\bm{T}^{E_{2}}_{3}$ obtained from $(g_{B}^{E_{2}})^{2}$. Explicitly,

$(g_{B}^{E_{2}})^{2}:\bm{x}\to\bm{x}+(\mathds{1}+\bm{\mathsf{M}}^{E_{2}}_{B})\bm{T}^{E_{2}}_{B}\,,$
(A.3)

from which we define

$\bm{T}^{E_{2}}_{3}\equiv(\mathds{1}+\bm{\mathsf{M}}^{E_{2}}_{B})\bm{T}^{E_{2}}_{B}=2L_{B}(0,\,0,\,1)\,.$
(A.4)

The following restrictions on the $E_{2}$ parameters ensure we do not have
duplicate representations of the same physical space:

  1. 1.

$0<L_{A_{1}}$ ;

  2. 2.

$0<L_{A_{1}}/L_{A_{2}}\leq 1$ , $0<L_{A_{1}}/L_{B}<\infty$ ;

  3. 3.

$|\cos\alpha|<L_{A_{1}}/(2L_{A_{2}})$ .

###  A.3 $E_{3}$: Quarter-turn space

The quarter-turn space replaces the $\pi$ corkscrew of $E_{2}$ with a
corkscrew motion by $\pi/2$. The same freedom to rotate the coordinate system
as in $E_{2}$ has been used to set the axis of rotation of
$\bm{\mathsf{M}}_{B}^{E_{3}}$ parallel to $\bm{\hat{z}}$, and then to set the
$y$-component of one of the two pure translations to zero, i.e.,
$\bm{T}^{E_{3}}_{A_{1}y}=0$. (The orthogonality of the pure translations to
the axis of rotation is not a choice, but a necessity [34], as is the equal
length of the two pure translations and their orthogonality.) The generators
of the quarter-turn space are then given by

$\displaystyle\bm{\mathsf{M}}^{E_{3}}_{A}$ $\displaystyle=\mathds{1}\,,$
$\displaystyle\bm{T}^{E_{3}}_{A_{1}}$
$\displaystyle=L_{A}(1,\,0,\,0)\equiv\bm{T}^{E_{3}}_{1}\,,$
$\displaystyle\bm{T}^{E_{3}}_{A_{2}}$
$\displaystyle=L_{A}(0,\,1,\,0)\equiv\bm{T}^{E_{3}}_{2}\,;$ (A.5)
$\displaystyle\bm{\mathsf{M}}_{B}^{E_{3}}$
$\displaystyle=R_{\bm{\hat{z}}}^{-\pi/2}=\begin{pmatrix}0&1&0\\\ -1&0&0\\\
0&0&1\end{pmatrix}\,,$ $\displaystyle\bm{T}^{E_{3}}_{B}$
$\displaystyle=L_{B}(0,\,0,\,1)\,.$

The associated $E_{1}$ is given by the translation vectors
$\bm{T}^{E_{3}}_{1}$, $\bm{T}^{E_{3}}_{2}$, and a third translation vector
$\bm{T}^{E_{3}}_{3}$ determined from $(g_{B}^{E_{3}})^{4}$ in a fashion
analogous to Eq. A.4,

$\bm{T}^{E_{3}}_{3}\equiv 4L_{B}(0,\,0,\,1)\,.$ (A.6)

The following restrictions on the $E_{3}$ parameters ensure we do not have
duplicate representations of the same physical space:

  1. 1.

$0<L_{A}$ ;

  2. 2.

$0<L_{A}/L_{B}<\infty$ .

###  A.4 $E_{4}$: Third-turn space

The third-turn space replaces the $\pi$ corkscrew of $E_{2}$ with a corkscrew
motion by $2\pi/3$. The same freedom to rotate the coordinate system as in
$E_{2}$ can be used to set the axis of rotation of
$\bm{\mathsf{M}}_{B}^{E_{4}}$ parallel to $\bm{\hat{z}}$, and then to set the
$y$-component of one of the two pure translations to zero, i.e.,
$\bm{T}^{E_{4}}_{A_{1}y}=0$. (The orthogonality of the pure translations to
the axis of rotation is not a choice, but a necessity [34], as is the equal
length of the two pure translations and the angle between them.) The
generators of the third-turn space are then given by

$\displaystyle\bm{\mathsf{M}}^{E_{4}}_{A}$ $\displaystyle=\mathds{1}\,,$
$\displaystyle\bm{T}^{E_{4}}_{A_{1}}$
$\displaystyle=L_{A}(1,\,0,\,0)\equiv\bm{T}^{E_{4}}_{1}\,,$ (A.7)
$\displaystyle\bm{T}^{E_{4}}_{A_{2}}$
$\displaystyle=R_{\bm{\hat{z}}}^{-2\pi/3}\bm{T}^{E_{4}}_{A_{1}}=L_{A}(-1/2,\,\sqrt{3}/2,\,0)\equiv\bm{T}^{E_{4}}_{2}\,;$
$\displaystyle\bm{\mathsf{M}}_{B}^{E_{4}}$
$\displaystyle=\bm{\mathsf{R}}_{\bm{\hat{z}}}^{-2\pi/3}=\begin{pmatrix}-1/2&\sqrt{3}/2&0\\\
-\sqrt{3}/2&-1/2&0\\\ 0&0&1\end{pmatrix}\,,$ $\displaystyle\bm{T}^{E_{4}}_{B}$
$\displaystyle=L_{B}(0,\,0,\,1)\,.$

The associated $E_{1}$ is given by the translation vectors
$\bm{T}^{E_{4}}_{1}$ and $\bm{T}^{E_{4}}_{2}$, and a third translation vector
$\bm{T}^{E_{4}}_{3}$ determined from $(g_{B}^{E_{4}})^{3}$, as in $E_{2}$ and
$E_{3}$,

$\bm{T}^{E_{4}}_{3}\equiv 3L_{B}(0,\,0,\,1)\,.$ (A.8)

The following restrictions on the $E_{4}$ parameters ensure we do not have
duplicate representations of the same physical space:

  1. 1.

$0<L_{A}$ ;

  2. 2.

$0<L_{A}/L_{B}<\infty$ .

###  A.5 $E_{5}$: Sixth-turn space

The sixth-turn space replaces the $\pi$ corkscrew of $E_{2}$ with a corkscrew
motion by $\pi/3$. The same freedom to rotate the coordinate system as in
$E_{2}$ has been used to set the axis of rotation of
$\bm{\mathsf{M}}_{B}^{E_{5}}$ parallel to $\bm{\hat{z}}$, and then to set the
$y$-component of one of the two pure translations to zero, i.e.,
$\bm{T}^{E_{5}}_{A_{1}y}=0$. (The orthogonality of the pure translations to
the axis of rotation is a necessity [34], as is the equal length of the two
pure translations and the angle between them.) The generators of the sixth-
turn space are then given by

$\displaystyle\bm{\mathsf{M}}^{E_{5}}_{A}$ $\displaystyle=\mathds{1}\,,$
$\displaystyle\bm{T}^{E_{5}}_{A_{1}}$ $\displaystyle=L_{A}(1,\,0,\,0)\,,$
$\displaystyle\bm{T}^{E_{5}}_{A_{2}}$
$\displaystyle=L_{A}(-1/2,\,\sqrt{3}/2,\,0)\,;$ (A.9)
$\displaystyle\bm{\mathsf{M}}_{B}^{E_{5}}$
$\displaystyle=\bm{\mathsf{R}}_{\bm{\hat{z}}}^{-\pi/3}=\begin{pmatrix}1/2&\sqrt{3}/2&0\\\
-\sqrt{3}/2&1/2&0\\\ 0&0&1\end{pmatrix}\,,$ $\displaystyle\bm{T}^{E_{5}}_{B}$
$\displaystyle=L_{B}(0,\,0,\,1)\,.$

The associated $E_{1}$ is given by the translation vectors
$\bm{T}^{E_{5}}_{1}$ and $\bm{T}^{E_{5}}_{2}$, and a third translation vector
$\bm{T}^{E_{5}}_{3}$ determined from $(g_{B}^{E_{5}})^{6}$,

$\bm{T}^{E_{5}}_{3}\equiv 6L_{B}(0,\,0,\,1)\,.$ (A.10)

The following restrictions on the $E_{5}$ parameters ensure we do not have
duplicate representations of the same physical space:

  1. 1.

$0<L_{A}$ ;

  2. 2.

$0<L_{A}/L_{B}<\infty$ .

###  A.6 $E_{6}$: Hantzsche-Wendt space

The Hantzsche-Wendt space replaces each of the translations of $E_{1}$ with a
corkscrew motion by angle $\pi$. We have used the freedom to rotate the
coordinate system to set the axes of rotation of these three corkscrews to be
parallel to the three coordinate axes. Their orthogonality to one another has
been proven [39]. The appropriate parameterization of the associated
translations is demonstrated in Ref. [34]. The generators of the Hantzsche-
Wendt space are then given by

$\displaystyle\bm{\mathsf{M}}^{E_{6}}_{A}$
$\displaystyle=\bm{\mathsf{R}}_{\bm{\hat{x}}}^{-\pi}=\operatorname{diag}(1,-1,-1)\,,$
$\displaystyle\bm{T}^{E_{6}}_{A}$ $\displaystyle=(L_{A},\,L_{B},\,0)\,;$
$\displaystyle\bm{\mathsf{M}}^{E_{6}}_{B}$
$\displaystyle=\bm{\mathsf{R}}_{\bm{\hat{y}}}^{-\pi}=\operatorname{diag}(-1,1,-1)\,,$
(A.11) $\displaystyle\bm{T}^{E_{6}}_{B}$
$\displaystyle=(0,\,L_{B},\,L_{C})\,;$
$\displaystyle\bm{\mathsf{M}}^{E_{6}}_{C}$
$\displaystyle=\bm{\mathsf{R}}_{\bm{\hat{z}}}^{-\pi}=\operatorname{diag}(-1,-1,1)\,,$
$\displaystyle\bm{T}^{E_{6}}_{C}$ $\displaystyle=(L_{A},\,0,\,L_{C})\,.$

The associated $E_{1}$ is given by the translation vectors generated from
$(g_{a}^{E_{6}})^{2}$ for $a\in\\{A,B,C\\}$,

$\displaystyle\bm{T}^{E_{6}}_{1}$ $\displaystyle\equiv(2L_{A},\,0,\,0)\,,$
$\displaystyle\bm{T}^{E_{6}}_{2}$ $\displaystyle\equiv(0,\,2L_{B},\,0)\,,$
(A.12) $\displaystyle\bm{T}^{E_{6}}_{3}$
$\displaystyle\equiv(0,\,0,\,2L_{C})\,.$

The following restrictions on the $E_{5}$ parameters ensure we do not have
duplicate representations of the same physical space:

  1. 1.

$0<|L_{A}|$ ,

  2. 2.

$1\geq|L_{A}/L_{B}|\geq|L_{A}/L_{C}|$ .

###  A.7 $E_{11}$: Chimney space

The chimney space has only two compact dimensions, and has only translations
in its group. We take the shorter of the two translations to be in the
$x$-direction, and then set the $z$-component of the other translation to
zero. The generators of the chimney space are then given by

$\displaystyle\bm{\mathsf{M}}_{A}$ $\displaystyle=\mathds{1}\,,$
$\displaystyle\bm{T}^{E_{11}}_{A_{1}}$
$\displaystyle=L_{A_{1}}(1,\,0,\,0)\equiv\bm{T}^{E_{11}}_{1}\,,$ (A.13)
$\displaystyle\bm{T}^{E_{11}}_{A_{2}}$
$\displaystyle=L_{A_{2}}(\cos\alpha,\,\sin\alpha,\,0)\equiv\bm{T}^{E_{11}}_{2}\,.$

Much like $E_{1}$, these two translations can be taken to define the
fundamental domain.

###  A.8 $E_{12}$: Chimney space with half turn

The chimney space with a half turn replaces one of the two translations of
$E_{11}$ with a corkscrew motion by $\pi$. We choose the axis of rotation of
$\bm{\mathsf{M}}_{B}^{E_{12}}$ parallel to $\bm{\hat{y}}$, and then set the
$z$-component of the pure translation to zero, i.e.,
$\bm{T}^{E_{2}}_{A_{1}z}=0$. The generators of the chimney space with a half
turn are then given by

$\displaystyle\bm{\mathsf{M}}^{E_{12}}_{A}$ $\displaystyle=\mathds{1}\,,$
$\displaystyle\bm{T}^{E_{12}}_{A}$
$\displaystyle=L_{A}(\cos\alpha,\,\sin\alpha,\,0)\equiv\bm{T}^{E_{12}}_{1}\,;$
(A.14) $\displaystyle\bm{\mathsf{M}}^{E_{12}}_{B}$
$\displaystyle=\bm{\mathsf{R}}_{\bm{\hat{y}}}^{-\pi}=\operatorname{diag}(-1,1,1)\,,$
$\displaystyle\bm{T}^{E_{12}}_{B}$
$\displaystyle=L_{B}(\cos\beta\cos\gamma,\,\cos\beta\sin\gamma,\,\sin\beta)\,.$

The associated $E_{11}$ is given by the translation vectors
$\bm{T}^{E_{12}}_{1}$ and the second determined from $(g_{B}^{E_{12}})^{2}$,

$\bm{T}^{E_{12}}_{2}\equiv(0,\,2L_{B}\cos\beta\sin\gamma,\,0)\,.$ (A.15)

###  A.9 $E_{16}$: Slab space with rotation

The slab space is compact in only one dimension. All group elements are
integer repetitions of the generator (or its inverse). We can always choose
the compact dimension to be in the $z$-direction. The standard representation
of $E_{16}$ is a pure translation. However, the pure translation is a special
case as it is homogeneous — all observers see the same pattern of clones. An
additional possibility is to also include a corkscrew motion by an arbitrary
angle $\zeta$. This possibility in the slab space appears to be new, at least
in the cosmology literature. The generator that includes a corkscrew motion is
given by

$\displaystyle\bm{\mathsf{M}}^{E_{16}}(\zeta)$
$\displaystyle=\bm{\mathsf{R}}_{\bm{\hat{z}}}^{\zeta}\,,\qquad
0\leq\zeta<2\pi\,,$ $\displaystyle\bm{T}^{E_{16}}$
$\displaystyle=L(0,\,0,\,1)\,,\qquad 0<L\,.$ (A.16)

For $\zeta\neq 0$ we have changed the origin to eliminate the $x$ and $y$
components of $\bm{T}^{E_{16}}$; for $\zeta=0$ we have set
$\bm{T}^{E_{16}}\|\hat{z}$ by rotating the coordinate system.

This is a strange case, see Section 4 (and Ref. [34] for full details).

## References

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