# Cosmic topology. Part II. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

Johannes R Eskilt, Yashar Akrami, Stefano Anselmi, Craig J Copi, Andrew H Jaffe, Arthur Kosowsky, Deyan P Mihaylov, Glenn D Starkman, Andrius Tamosiunas, James B Mertens, et al.

► To cite this version:

Johannes R Eskilt, Yashar Akrami, Stefano Anselmi, Craig J Copi, Andrew H Jaffe, et al. Cos-
mic topology. Part II. Eigenmodes, correlation matrices, and detectability of orientable Euclidean
manifolds. Journal of Cosmology and Astroparticle Physics, 2024, 03, pp.036. 10.1088/1475-
7516/2024/03/036 . hal-04160052

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# Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

To cite this article: Johannes R. Eskilt et al JCAP03(2024)036

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# Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

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## The COMPACT collaboration

Johannes R. Eskilt,ᵃ,ᵇ Yashar Akrami ID,ᶜ,>d,ᵇ Stefano Anselmi,ᶜ,>f,g Craig J. Copi,>d

Andrew H. Jaffe,ᵇ Arthur Kosowsky,ʰ Deyan P. Mihaylov,ᵈ Glenn D. Starkman,>d,ᵇ

Andrius Tamosiunas,ᵈ James B. Mertens,ᵈ Pip Petersen,ᵈ Samanta Saha,ᵈ Quinn Taylorᵈ and Özenç Güngörᵈ

E-mail: j.r.eskilt@astro.uio.no, yashar.akrami@csic.es,
stefano.anselmi@pd.infn.it, craig.copi@case.edu, a.jaffe@imperial.ac.uk,
kosowsky@pitt.edu, deyan.mihaylov@case.edu, glenn.starkman@case.edu,
andrius.tamosiunas@case.edu, james.mertens@case.edu, sxp1171@case.edu,
samanta.saha@case.edu, qxt42@case.edu, ozenc.gungor@case.edu

<figure><img src="image_4.png" /></figure>

<footnote>ᵃInstitute of Theoretical Astrophysics, University of Oslo,
P.O. Box 1029 Blindern, Oslo N-0315, Norway</footnote>

<footnote>ᵇAstrophysics Group & Imperial Centre for Inference and Cosmology, Department of Physics,
Imperial College London, Blackett Laboratory,
Prince Consort Road, London SW7 2AZ, United Kingdom</footnote>

<footnote>ᶜInstituto de Física Teórica (IFT) UAM-CSIC,
C/ Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, Madrid 28049, Spain</footnote>

<footnote>ᵈCERCA/ISO, Department of Physics, Case Western Reserve University,
10900 Euclid Avenue, Cleveland, OH 44106, U.S.A.</footnote>

<footnote>ᵉDipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova,
via Marzolo 8, Padova I-35131, Italy</footnote>

<footnote>fINFN, Sezione di Padova,
via Marzolo 8, Padova I-35131, Italy</footnote>

<footnote>gLUTH, UMR 8102 CNRS, Observatoire de Paris, PSL Research University,
Université Paris Diderot,
Meudon 92190, France</footnote>

<footnote>ʰDepartment of Physics and Astronomy, University of Pittsburgh,
Pittsburgh, PA 15260, U.S.A.</footnote>
**ABSTRACT:** If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologies $E_1-E_6$, $E_{11}$, $E_{12}$, $E_{16}$, and $E_{18}$ that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys.

**KEYWORDS:** CMBR theory, cosmological parameters from CMBR, cosmology of theories beyond the SM, physics of the early universe

ARXIV ePRINT: 2306.17112
# Contents

1 Introduction 2

2 Topologies and manifolds of $E^3$: general considerations 6

3 Properties of orientable Euclidean topologies 12

3.1 $E_1$: 3-torus 14

3.2 $E_2$: half-turn space 16

3.3 $E_3$: quarter-turn space 19

3.4 $E_4$: third-turn space 20

3.5 $E_5$: sixth-turn space 22

3.6 $E_6$: Hantzsche-Wendt space 24

3.7 $E_{11}$: Chimney space 26

3.8 $E_{12}$: Chimney space with half turn 27

3.9 $E_{16}$: slab space including rotation 29

3.10 $E_{18}$: the covering space of $E^3$ 31

4 Eigenmodes of the scalar Laplacian and correlation matrices 31

4.1 General considerations for eigenmodes 34

4.2 $E_1$: 3-torus 35

4.3 $E_2$: half-turn space 37

4.4 $E_3$: quarter-turn space 38

4.5 $E_4$: third-turn space 40

4.6 $E_5$: sixth-turn space 41

4.7 $E_6$: Hantzsche-Wendt space 42

4.8 $E_{11}$: Chimney space 44

4.9 $E_{12}$: Chimney space with half turn 45

4.10 $E_{16}$: slab space including rotation 46

4.11 $E_{18}$: the covering space of $E^3$ 48

5 Numerical analysis 49

5.1 Evaluation of CMB temperature covariance matrices 49

5.2 KL divergence 53

5.3 Cosmic variance signal-to-noise ratio 53

5.4 KL divergence and signal-to-noise ratio for compact, orientable, Euclidean manifolds 55

5.5 Expectations for three-dimensional versus two-dimensional correlations 60

6 Conclusion 60
<table><tr><td><b>A Construction of general generators</b></td><td><b>63</b></td></tr><tr><td>A.1 E<sub>1</sub>: 3-torus</td><td>64</td></tr><tr><td>A.2 E<sub>2</sub>: half-turn space</td><td>64</td></tr><tr><td>A.3 E<sub>3</sub>: quarter-turn space</td><td>66</td></tr><tr><td>A.4 E<sub>4</sub>: third-turn space</td><td>69</td></tr><tr><td>A.5 E<sub>5</sub>: sixth-turn space</td><td>71</td></tr><tr><td>A.6 E<sub>6</sub>: Hantzsche-Wendt space</td><td>72</td></tr><tr><td>A.7 E<sub>11</sub>: Chimney space</td><td>75</td></tr><tr><td>A.8 E<sub>12</sub>: Chimney space with half turn</td><td>75</td></tr><tr><td>A.9 E<sub>16</sub>: slab space including rotations</td><td>76</td></tr></table>

# 1 Introduction

In the century since the proposal of general relativity (GR) [1] as the dynamical theory of spacetime, and therefore of cosmology [2], we have widely come to view space as a three-dimensional (Riemannian) manifold with a geometry that is inhomogeneous on small scales but homogeneous and isotropic on large scales [3–6]. This geometry evolves according to the Einstein field equations, which are local second-order differential equations in which the evolution of the geometry is sourced by the stress-energy content of space. Meanwhile, the evolution of that stress-energy content is governed by Euler-Lagrange equations that incorporate the influence of the geometry on the stress-energy.

It is useful to do a background/perturbation decomposition of the metric. The largest-scale geometry is, per this view, given by the Friedman-Lemaître-Robertson-Walker (FLRW) metric,

$$ds^2 = -dt^2 + a^2(t) \left[ \frac{dr^2}{1 - k(r/r_c)^2} + r^2(d\theta^2 + \sin^2\theta d\phi^2) \right] \quad (1.1)$$

(in a convenient choice of coordinates, with $r_c$ the comoving curvature scale). Deviations from this geometry on large scales are usually treated perturbatively, starting with a scalar-vector-tensor decomposition of the perturbations in which they are evolved analytically [3, 7] or numerically [8, 9], or in a Newtonian approximation with N-body simulations [10, 11]. On smaller scales, e.g., inside a galaxy, the deviations from eq. (1.1) may be highly non-linear, and eq. (1.1) may not even be a relevant approximation [12].

Equation (1.1) is the metric (also known as the local geometry) of a homogeneous isotropic space, whose sole dynamical variable is the scale factor $a(t)$. There are only three such metrics, characterized by the curvature constant $k \in \{-1, 0, +1\}$, representing, respectively, homogeneous isotropic negatively curved (“hyperbolic”) space ($k = -1, H^3$), homogeneous isotropic flat (“Euclidean”) space ($k = 0, E^3$), and homogeneous isotropic positively curved (“spherical”) space ($k = +1, S^3$).

The specific justification for this assumption of a homogeneous and isotropic background metric is a matter for some reconsideration, despite the near large-scale homogeneity and isotropy of our observed universe. An inflationary perspective may prefer a nearly homogeneous and isotropic geometry, but the considerable evidence for violations of statistical
isotropy in the cosmic microwave background (CMB) temperature data (see, e.g., refs. [13–17] for reviews, and ref. [18]) calls into question a strict enforcement of this received wisdom. Not all spatial three-manifolds admit a single homogeneous metric; those that do, may admit one of the above three, or they may admit one of five others that are not isotropic [19]. Nevertheless, we leave this expansion of the suite of possible cosmological background metrics to future consideration.

There is also widespread misconception that the three FLRW metrics (1.1) fully specify the three possible smooth three-spaces $H^3$, $E^3$, and $S^3$. This is not so. Those are the covering spaces associated with these three metrics, i.e., the manifolds that admit those geometries and in which all closed loops can be deformed continuously to a point. For each such geometry, there are many distinct manifolds that admit that geometry. The richness of possible three-manifolds and their connections with possible spatial geometries has received much attention from mathematicians in recent decades (perhaps most notably the Thurston conjecture [19], proven by Perelman [20, 21]), but exploring that richness is far beyond the scope of this paper (for an overview see refs. [22, 23]). For our purposes, it will suffice to note that there are exactly 18 topologically distinct possibilities for flat space: $E_1–E_{18}$ [22, 24–27]. For ten of these 18 topologies, $E_1–E_{10}$, the manifolds are compact — they have finite three-volume when calculated with the FLRW metric ($k=0$); $E_{11}–E_{15}$ have finite area two-dimensional slices; $E_{16}$ and $E_{17}$ are compact in only one dimension; $E_{18}$ is the covering space and is infinite in every direction. These are each described in detail below. For $H^3$ and $S^3$ the topological possibilities are far richer — there are a small number of classes of spherical three-manifolds each with either a finite number or a countable infinity of members; there is no known enumeration of the hyperbolic three-manifolds, even of the compact ones [19]. This richness from the topologists’ perspective should not be misunderstood by cosmologists as a statement that there are “more” curved three-manifolds than Euclidean ones — the 18 flat topologies each require up to 6 real parameters to characterize a manifold, whereas the compact hyperbolic topologies have only one real parameter.

At any given time in the last century, a small group of cosmologists have been interested in the possibility that space is not the covering space of one of the three FLRW geometries, but rather is one of the many other possibilities. This interest goes back at least to de Sitter [28], who remarked that Einstein’s original $S^3$ cosmology would have been improved if situated in $S^3/Z_2$ — the three-sphere with opposite points identified. Interest has continued ever since, albeit tempered by the success of the inflationary paradigm for the early Universe (for an overview of recent developments see ref. [29]). Inflation addresses questions about the initial conditions of the Universe by invoking a period of accelerated expansion during which information about those initial conditions is stretched beyond the apparent horizon [30–33]. If the Universe’s non-trivial topology is an initial condition, an extended conventional inflationary period would make topology hard or impossible to detect, however, as discussed in ref. [34], “in many inflationary models based on string theory there is no exponential suppression of creation of topologically nontrivial compact flat or open inflationary universes”, “suggest[ing] … that compact flat or open universes with nontrivial topology should be considered a rule rather than an exception”.

In the late 1990s and 2000s, attention turned to the possibility of gathering convincing evidence (or imposing strict constraints) on cosmic topology from impending observations,
especially the then-upcoming full-sky high-resolution survey of the CMB temperature fluctuations by the Wilkinson Microwave Anisotropy Probe (WMAP) [35, 36]. Two principal observational approaches were proposed, studied, and eventually implemented, for both WMAP and Planck [26, 37–52].

The “circles-in-the-sky” method builds on the observation that in a topologically non-trivial universe with any closed spatial loop shorter than the diameter of the last scattering surface ($L_{LSS}$) of the CMB, the LSS self-intersects [37–40]. Since the LSS is a thin spherical annulus centered on the observer, that self-intersection is a circular locus of points visible to the observer in two distinct directions on the sky. One can check all possible pairs of equal-radius circles and determine if the temperature patterns around any of them are more similar than would be likely in the covering space.

The “Bayesian” method relies on comparing the pixel-pixel correlations in the observed CMB temperature map to those expected to be induced in manifolds with non-trivial topology (e.g., the paired circles), and to those expected in the covering space to determine which is the most likely underlying manifold [49, 50, 53].

Both of these methods rely, for calculations of both the expected signal and its cosmic variance, on a statistical analysis of simulated realizations of the expected CMB sky in these topologically non-trivial manifolds. Such simulations are produced by summing eigenmodes of the Laplacian with coefficients whose statistics are predicted by a theory for the generation of metric fluctuations in the early Universe and of their evolution. This theory is generally taken to be inflation, and under certain conditions, which are usually taken to be applicable, results in the scale of any topology being inflated far beyond the current Hubble scale (so that functionally we can take our manifold to be the covering space of $E^3$), effects of the initial conditions being “inflated away”, and the coefficients of the Laplacian eigenmodes of scalar fluctuations being statistically independent Gaussian random variables of zero mean with variance that depends only on the magnitude of the eigenvalue of the Laplacian and a power spectrum that is nearly scale-free. We shall retain as an assumption this statistical characterization of the coefficients of the Laplacian eigenmodes, without specifically ascribing the fluctuations to inflation, and, clearly, without asserting that the topology scale is far beyond the Hubble scale. The success of the usual Bayesian fits of covering-space inflationary $\Lambda$CDM model parameters to CMB temperature and $E$-mode polarization data, especially at $\ell \gtrsim 50$, and the failure so far to detect any “primordial non-Gaussianity” [54–58] suggest that this Gaussian hypothesis is at least a reasonable approximation for those $\ell$. Nevertheless, we should certainly be aware of this limitation, and the associated assumption of an inflation-inspired nearly-scale-free power spectrum.

The program of searching for topology therefore relies explicitly on developing analytic expressions for Laplacian eigenmodes in the manifolds of interest — or at least on identifying rapid algorithms for numerical calculation of the eigenmodes. So far, this is not known to be possible in most $H^3$ or $S^3$ manifolds, but is straightforward, at least in principle, in all $E^3$ manifolds and select $S^3$ manifolds [59–61]. Not surprisingly, the program to search for topology therefore included the derivation of such eigenmodes for the $E^3$ manifolds [25]. In this case, the eigenmodes are calculable finite linear combinations of certain covering-space eigenmodes — Fourier modes. In addition to calculating the eigenmodes themselves, one has need for many purposes of their expansion in a spherical coordinate basis, giving
their contributions to individual spherical harmonics, and to the mode-mode or pixel-pixel correlation matrices on the sky for hypothetical observers.

Unfortunately, this program of calculating Laplacian eigenmodes on Euclidean three-manifolds was incompletely implemented. While eigenmodes were calculated for representative examples of all Euclidean manifolds, what is needed is a comprehensive exhaustive description of all eigenmodes for all possible manifolds. For each of the 17 non-trivial $E^3$ topologies, we must specify at least one real parameter to fully fix the manifold, specify the Laplacian eigenmodes, and enable statistical predictions for cosmological observables. Moreover, because the topological boundary conditions are not translation invariant, observers in different locations in the same manifold can have different statistical expectations for observables in most topologies. Because the boundary conditions are not invariant under rotations, the expectations for observables are not statistically isotropic in any manifold of non-trivial topology. Statistical anisotropy, and potentially statistical inhomogeneity, are key observational features of a topologically non-trivial universe.

The circles-in-the-sky searches for topology performed on WMAP [26, 41, 43-48] and Planck data [50, 62] (as well as a general unpublished search analogous to ref. [47]) constrain the shortest non-trivial closed loop through the Earth to have a length greater than 98.5% of the diameter of the LSS. The translation of this limit to limits on the parameters characterizing the manifolds $E_i$ is underway; the results for the orientable manifolds $E_1-E_6$, $E_{11}$, $E_{12}$, and $E_{16}$ are in ref. [63], and the non-orientable manifolds will be presented in an upcoming paper [64].

The Bayesian limits presented in refs. [49, 50] are, typically, mildly more constraining than the circles-in-the-sky limit where both apply. However, the Bayesian limits rely on the mode-mode correlation matrix, which depends on the geometric parameters of the manifold, as well as (for $E_2-E_{17}$) on the location of the observer. The limits therefore only apply to the special values of the topological parameters and specific origin choices that were considered. How they extend even to the neighborhoods of those special values is unclear without further study.

In this paper, we remedy the previously incomplete characterization of the orientable Euclidean manifolds ($E_1-E_6$, $E_{11}$, $E_{12}$, $E_{16}$, and $E_{18}$), providing general expressions for their eigenmodes, allowing for general parametrizations of the geometry of the space and for arbitrary observer location. We also present formulae for the mode-mode correlation matrices on the sphere and other useful quantities. In so doing, we will find that we are implementing known cases that were however omitted from previous attempts to model all possible manifolds.

These mathematical results will be central to future work modeling the consequences of non-trivial topology, and searching exhaustively for the topology of the Universe. In the meantime, they enable us to forecast the detectability of topology in the CMB and eventually in other cosmological observables. Previous works [49, 50, 65] have employed the Kullback-Leibler (KL) divergence [66, 67] to quantify the information available in the CMB to distinguish between a topologically non-trivial universe and a topologically trivial one. We show here that the inclusion of all necessary topological parameters does not qualitatively change the conclusion that non-trivial topology should be detectable in the CMB if the length $L_{\text{topology}}$ of the shortest closed geodesic through us is less than the diameter of the LSS, $d_{\text{LSS}}$, but probably becomes undetectable if $L_{\text{topology}} \gtrsim 1.2d_{\text{LSS}}$. We argue that this
should be expected to change once high-quality three-dimensional data (i.e., tomographic intensity mapping or deep-enough galaxy surveys) become available. On the other hand, if the large-angle anomalies in the CMB are caused by cosmic topology (one of the very few promising sources of large-angle violation of statistical isotropy), then this result reassures us that $L_{\text{topology}} \lesssim 1.2d_{\text{LSS}}$ and there is a reasonable prospect of discovering cosmic topology, and possibly identifying the specific spatial manifold, from the CMB alone.

In section 2 we discuss the notation and review some general properties of the topologies and manifolds of $E^3$. Section 3 discusses the key properties of the orientable Euclidean manifolds. In section 4, we present the eigenmodes of the scalar Laplacian along with the Fourier and spherical-harmonic correlation matrices. Section 5 contains our numerical results and a number of representative examples of spherical-harmonic covariance matrices for the compact orientable manifolds with selected values of their parameters, as well as the corresponding KL divergence and a useful new off-diagonal signal-to-noise statistic.

The GitHub repository associated with this study is publicly available at https://github.com/CompactCollaboration. Codes will be deposited there as publicly usable versions become available.

## 2 Topologies and manifolds of $E^3$: general considerations

The isometry group of Euclidean three-space $E^3$, denoted by $E(3)$, includes arbitrary rotations and reflections (i.e., elements of the orthogonal group in three dimensions $O(3)$), arbitrary translations, and all products of these. A group element of $E(3)$ is freely acting on $E^3$ if it takes no point of $E^3$ to itself. These are comprised of translations, rotations about arbitrary axes followed by translations with a component parallel to that axis (“corkscrew motions”), reflections across planes followed by translations with components parallel to the plane (“glide reflections”), and certain products of these. The non-trivial $E^3$ topologies $E_i$ are formed by modding out $E(3)$ by a discrete subgroup $\Gamma^{E_i}$ of freely acting elements, i.e., $E(3) \to E(3)/\Gamma^{E_i}$ (see, e.g., refs. [68, 69]).

There are 18 distinct possibilities for $\Gamma^{E_i}$ (including the trivial group), leading to the 18 distinct topologies $E_i$ for $i \in \{1, \dots, 18\}$ [22, 24, 25, 27]. These $\Gamma^{E_i}$ are characterized in whole or in part by the specific selection of $O(3)$ elements that each involves, but in all cases, when constructing actions of $\Gamma^{E_i}$, there are translations that must be characterized by up to 6 real parameters. If space is a Euclidean manifold, then these degrees of freedom are physical, and a general description of the manifold must include them.<fnref n="1" />

The situation can be illustrated in the most familiar case — a simple three-torus, which is referred to as $E_1$. Its symmetry group $\Gamma^{E_1}$ is generated by three pure translations (i.e., the only element of $O(3)$ involved is the identity), which we will represent as $g_i^{E_1}$, a translation by $T_i^{E_1}$, for $i \in \{1, 2, 3\}$:

$$g_i^{E_1} : x \to x + T_i^{E_1}. \qquad (2.1)$$

Figures 1 and 2 illustrate the actions of the generators for $E_1$ and some of the other Euclidean topologies. The simplest, special case is the cubic three-torus where the translations

<footnote n="1">¹In addition, to fully describe observational properties, one must typically specify the position and orientation of the observer, which introduces up to 6 more real parameters.</footnote>
<figure><img src="image_5.png" /><figcaption>**Figure 1.** Diagrams showing the actions of the generators for the topologies $E_1-E_6$. In each subdiagram, an observer at the origin is represented by an orthogonal triad $\hat{x}, \hat{y}, \hat{z}$ shown as short red, green, and blue arrows, respectively. Translation vectors for each of the three generators are rooted at the origin, and may be labelled $T_{A_i}$, $T_A$, $T_B$, and $T_C$, depending on the details of the topology. The angles $\alpha, \beta$, and $\gamma$ mark their orientation. At the head of each of those translation vectors is another red-green-blue triad representing one of the observer's topological clones, with the amount indicated by which they have been rotated compared to the observer at the origin. The colored coordinate planes are provided only as visual aids.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 2. Diagrams as in figure 1, but showing the actions of the generators for the topologies $E_{11}$, $E_{12}$, $E_{16}$, and $E_{18}$.</figcaption></figure>

are orthogonal and of equal length, e.g., $T_1^{E_1} = L\hat{e}_1$, $T_2^{E_1} = L\hat{e}_2$, $T_3^{E_1} = L\hat{e}_3$, with $\{\hat{e}_1, \hat{e}_2, \hat{e}_3\}$ a set of three orthogonal unit vectors. A general element of $\Gamma^{E_1}$ is a product of integer powers of these $g_i^{E_1}$, i.e., it is a translation by an integer linear combination of these three translations,

$$T_n^{E_1} = n_1 T_1^{E_1} + n_2 T_2^{E_1} + n_3 T_3^{E_1}. \qquad (2.2)$$

An observer in $E_1$ will perceive themselves to have a lattice of “clones” displaced from themselves by these vectors $T_n^{E_1}$, for all sets of integers $\{n_1, n_2, n_3\}$. They would also perceive any object they see around them to also have clones displaced from its closest instance by these same vectors. This lattice of clones is a real physical phenomenon and is observable if $L$ is small enough (see, e.g., refs. [70–77]). The observer might choose to interpret such a situation as living in a finite volume, for example, the cube with corners at

$$\{0, T_1^{E_1}, T_2^{E_1}, T_1^{E_1} + T_2^{E_1}, T_3^{E_1}, T_1^{E_1} + T_3^{E_1}, T_2^{E_1} + T_3^{E_1}, T_1^{E_1} + T_2^{E_1} + T_3^{E_1}\} \qquad (2.3)$$

and with all opposite faces identified. Alternately they might consider themselves to be living in the covering space of $E^3$ (i.e., $E_{18}$) with any pair of points differing by a $g_n$ identified with one another, i.e., space is simply tiled by this cube. Further, they might call this cube their
“fundamental domain” (FD) or unit cell [27]. These are equivalent descriptions of the same reality, and we often use them interchangeably in describing topologically non-trivial universes.

Although in this simple case, we associate the cube with this $E_1$, the shape of the FD is not itself an observable, and is certainly not a physical property of the manifold. In two dimensions this was perhaps most famously made evident by the many interesting FD shapes represented by the Dutch artist M.C. Escher — birds, fish, etc. [78]. What is physical is the set of group elements in $\Gamma^{E_1}$, or, equivalently, the relative locations (and orientations) of the “clones” of a given point in space — i.e., its images under elements of $\Gamma^{E_1}$. Each location in a manifold will therefore have a Dirichlet domain: the set of all points in the manifold that are closer to them than to any of their clones.

For example, in the specific case given above with $T_i^{E_1} = L\hat{e}_i$, an observer located at $\frac{1}{2}(T_1^{E_1} + T_2^{E_1} + T_3^{E_1})$ will have the cube with corners given by (2.3) as their Dirichlet domain. Different observers will have different Dirichlet domains centered on themselves. Generically, the shape of the Dirichlet domain will also change with the observer location.<fnref n="2" />

Connected with this ambiguity of the shape of the FD, we note that there are many distinct choices of $T_i^{E_1}$ that lead to the exact same lattice of clones. For example, we could replace $T_3^{E_1}$ by $T_3^{E_1'} \equiv T_3^{E_1} + T_1^{E_1}$. This results in the precisely identical set of clones, even though $T_1^{E_1}$, $T_2^{E_1}$, and $T_3^{E_1'}$ are not the same length and not orthogonal. The FD defined as in (2.3) is not a cube, nor even a rectangular prism, but a parallelepiped; it is not the Dirichlet domain of any observer.

Indeed, $T_i^{E_1}$ (for $i \in \{1, 2, 3\}$) need not be orthogonal, nor of equal length — they can be any three linearly independent vectors, with (2.3) forming a general parallelepiped,

$$T_1^{E_1} = \begin{pmatrix} L_{1x} \\ L_{1y} \\ L_{1z} \end{pmatrix}, \quad T_2^{E_1} = \begin{pmatrix} L_{2x} \\ L_{2y} \\ L_{2z} \end{pmatrix}, \quad T_3^{E_1} = \begin{pmatrix} L_{3x} \\ L_{3y} \\ L_{3z} \end{pmatrix}. \qquad (2.4)$$

However, we would prefer to choose $T_i^{E_1}$ so that (2.3) is the Dirichlet domain of some observer. This will constrain the relative values of $L_{iw}$ with $w \in \{x, y, z\}$ in ways that we discuss in detail below for each of the orientable topologies in section 3.

Three of these nine degrees of freedom $L_{iw}$ are degenerate with the three Euler angles describing the orientation of the coordinate system. While a search for topology will need to account for the orientation of the observer's coordinates, it is convenient, when cataloging manifolds, and when simulating the possible signals, to remove these degrees of freedom. Generically, we can order the three vectors by their length (from shortest to longest), and then choose the shortest of the three $T_1^{E_1}$ to point in the x-direction, and $T_2^{E_1}$ to lie in the xy-plane, so that $L_{1y} = L_{1z} = L_{2z} = 0$, i.e.,

$$T_1^{E_1} = \begin{pmatrix} L_{1x} \\ 0 \\ 0 \end{pmatrix}, \quad T_2^{E_1} = \begin{pmatrix} L_{2x} \\ L_{2y} \\ 0 \end{pmatrix}, \quad T_3^{E_1} = \begin{pmatrix} L_{3x} \\ L_{3y} \\ L_{3z} \end{pmatrix}. \qquad (2.5)$$

<footnote n="2">²$E_1$ and the other homogeneous topologies, $E_{11}$, $E_{16}^{(h)}$, and $E_{18}$, are special cases since an observer located at any point in the manifold has the same lattice of clones, and therefore all observers have the same shape Dirichlet domain.</footnote>
We may alternatively choose to write them as

$$ T_1^{E_1} = L_1 \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad T_2^{E_1} = L_2 \begin{pmatrix} \cos \alpha \\ \sin \alpha \\ 0 \end{pmatrix}, \quad T_3^{E_1} = L_3 \begin{pmatrix} \cos \beta \cos \gamma \\ \cos \beta \sin \gamma \\ \sin \beta \end{pmatrix}. \qquad (2.6) $$

Both of these parametrizations can be useful.

Thus the group $\Gamma^{E_1}$ associated with $E_1$ has 6 real parameters and all allowed choices of these parameters result in the same $E_1$ topology, but generically they result in different lattices of clones and so are physically distinct (and distinguishable) manifolds. However, there are equivalence classes, each with a countably infinite number of members, as we can replace these three vectors by any three linearly independent integer linear combinations of these three vectors without changing the lattice of clones. Thus if we are trying to characterize the allowed possibilities for $\Gamma$ without double counting we must take care in choosing the ranges of the parameters. In this case of $E_1$ with the alternative parametrization (2.6), we could require that $0 < L_1 \le L_2 \le L_3$. However, that is not sufficient to prevent double-counting the physical parameters of the manifold. We need to insist that $T_i^{E_1}$ cannot be shortened by adding integer linear combinations of $T_j^{E_1}$ and $T_k^{E_1}$ (with $i, j, k$ distinct elements of $\{1, 2, 3\}$). Details are provided below in section 3.1.

The lattice of clones of any given point is not rotationally invariant — rotational invariance, and thus statistical isotropy, is not an expected property of the Universe for an observer in $E_1$ [79]. The orientation of the observer relative to the lattice of clones is an observable. This is important to keep in mind when making use of the results of this paper — we will choose a particular orientation of the Cartesian coordinate axes (e.g., as reflected in (2.6)), but there is no reason for that to coincide with the orientation of the observer’s coordinate system. The Euler angles of the rotation between the manifold’s coordinate system and the observer’s coordinate system must be varied.

In the case of $E_1$, the lattice of an observer’s clones does not depend on the location of the observer. This is not generally true. Generically for all $E_i$ and with respect to an arbitrary location $x_0^{E_i}$, each generator $g_{aj}$ of $\Gamma^{E_i}$ acts on a point $x$ in the manifold as

$$ g_{aj}^{E_i} : x \to M_a^{E_i}(x - x_0^{E_i}) + T_{aj}^{E_i} + x_0^{E_i}, \qquad (2.7) $$

where $M_a^{E_i} \in O(3)$ and $T_{aj}^{E_i}$ is a translation vector appropriate for the given topology $E_i$. We use the index $a$ to distinguish among the up to three distinct $M_a^{E_i} \in O(3)$ and the index $j$ to label the distinct vectors $T_{aj}^{E_i}$ for a given $M_a^{E_i}$. The origin $x_0^{E_i}$ is the position (i.e., relative to some arbitrary coordinate origin) of a point on the axis about which $M_a^{E_i}$ rotates or on the plane across which it reflects. Since the $M_a^{E_i}$ are such that the axes about which they rotate or the normals to the planes across which they reflect are orthogonal [19], we can choose a single $x_0^{E_i}$ for all the generators. (This is why $x_0^{E_i}$ needs neither $a$ nor $j$ labels.)

All elements of $\Gamma^{E_i}$ can be obtained by successive actions of these generators and their inverses. For $E_1-E_{10}$ three generators are required (one can choose to include extra redundant generators, though we mostly refrain from doing so in this work). These are the compact Euclidean topologies. For $E_{11}-E_{15}$ two generators are required; for $E_{16}$ and
$E_{17}$ one generator is required; $E_{18}$ is the covering space of the Euclidean geometry $E^3$, for which no generators are required.

In the case of $E_1$ described above, $M_a^{E_i}$ was the identity for all three generators. More generally the generators can be chosen so that each $M_a^{E_i}$ is one of: the identity, a rotation about a coordinate axis, or the reflection of a single coordinate. $T_{aj}^{E_i}$ can never be $0$, if it were then $g_{aj}^{E_i}$ would not be freely acting. As described above, the three types of generators are referred to as translations ($M_a^{E_i} = 1$), corkscrew motions ($M_a^{E_i}$ a rotation), and glide reflections ($M_a^{E_i}$ a reflection).

In the case where one or more of the $M_a^{E_i}$ is not the identity, the manifold is not homogeneous, i.e., the lattice of clones of an observer depends on the location of the invariant axis/plane of $M_a^{E_i}$ relative to the observer, as encoded in $x_0^{E_i}$. One way to understand this inhomogeneity is that a change of choice of origin changes $x_0^{E_i}$, which can change $T_{aj}^{E_i}$.

For example, consider a shift of origin by $-v$, which takes $x_0^{E_i} \rightarrow x_0'^{E_i} \equiv x_0^{E_i} + v$. In this case, we would rewrite the generator (2.7) as

$$g_{aj}^{E_i} : x \rightarrow M_a^{E_i}(x - x_0'^{E_i}) + (T_{aj}^{E_i} - (1 - M_a^{E_i})v) + x_0'^{E_i}, \qquad (2.8)$$

so that

$$T_{aj}^{E_i} \rightarrow T_{aj}^{E_i} - (1 - M_a^{E_i})v. \qquad (2.9)$$

This is precisely the statement that observers at different locations have different clone lattices, or equivalently that the shape of the Dirichlet domain depends on the position of the observer (see, e.g., ref. [63] for a wider discussion).

For certain purposes, it might be useful to use the shift in origin to “simplify” the set of $T_{aj}^{E_i}$, for example, to set certain components to zero, or to equate certain components to one another. However, $1 - M_a^{E_i}$ always has some eigenvectors with eigenvalue 0, and these components of $T_{aj}^{E_i}$ are unaffected by such shifts:

* If $M_a^{E_i}$ is a proper rotation about an axis, then the component of $T_{aj}^{E_i}$ normal to the plane of rotation cannot be altered by shifting $x_0^{E_i}$, but the components in the plane of rotation can be adjusted. For example, if $M_a^{E_i} = R_z(\theta)$, then we can shift the origin such that $T_{aj}^{E_i} \parallel \hat{z}$, absorbing the other components of $T_{aj}^{E_i}$. In other words, we can shift the origin of the coordinate system so that it lies on the axis of rotation.

* If $M_a^{E_i}$ is a reflection across a plane, then the components of $T_{aj}^{E_i}$ in the plane of reflection cannot be altered by shifting $x_0^{E_i}$, but the components normal to the plane of reflection can be adjusted. For example, if $M_a^{E_i} = \text{diag}(1, -1, 1)$, then $1 - M_a^{E_i} = \text{diag}(0, 2, 0)$ and we can absorb only the y-component of $T_{aj}^{E_i}$, leaving $T_{aj}^{E_i}$ to be a general vector in the $xz$-plane. In other words, we can shift the origin of the coordinate system to lie in the plane of reflection.

* If $M_a^{E_i} = 1$, then $1 - M_a^{E_i} = 0$ and none of the components of $T_{aj}^{E_i}$ can be absorbed: $T_{aj}^{E_i}$ remains an arbitrary vector.

If more than one of the $M_a^{E_i}$ is not the identity, then their axes/planes must be orthogonal to one another (see, for example, refs. [19]); since there are never more than three distinct $M_a^{E_i}$, the associated axes/planes can always be taken to be parallel to coordinate axes/planes.
This orthogonality will also simplify the choices of which $\mathcal{T}_{a_j}^{E_i}$ to modify using the freedom to choose the coordinate origin. A shift in origin, with the resulting shift in the positions of the axes of rotation and planes of reflection associated with the generators $g_{a_j}^{E_i}$, changes the $\mathcal{T}_{a_j}^{E_i}$, and results in a different lattice of clones for an observer located at the new origin versus the old one. It also results in a different Dirichlet domain. We might have been tempted to interpret the unit cell of the clone lattice or the observer's Dirichlet domain as "the shape of the Universe". This would then lead to the conclusion that the shape of the Universe depends on the choice of origin. However "the shape of the Universe" is ambiguous. What is physical is the lattice of clones (and their orientations) seen by an observer for themselves (and for any other objects in the manifold); except for $E_1, E_{11}, E_{16}^{(h)}$, and $E_{18}$, this lattice depends on where the observer is located relative to the invariant axes/planes of the $\mathcal{M}_a^{E_i}$.

From a mathematical point of view, we could use our freedom to choose the origin to eliminate or relate as many as three of the components of $\mathcal{T}_{a_j}^{E_i}$. While this ability to simplify the $\mathcal{T}_{a_j}^{E_i}$ may prove useful for enumerating manifolds or for simulating cosmological observables, for an observer, the most sensible choice of origin is likely to be their own position, which may be very far from the point one would choose to yield a simplified set of generators. We therefore preserve both $x_0^{E_i}$ and $\mathcal{T}_{a_j}^{E_i}$ in our expressions for eigenmodes, and comment appropriately.

One important property of manifolds is their orientability or non-orientability. Loosely, a manifold is orientable if a right-handed triad remains right-handed when carried around all possible closed loops; it is non-orientable if there are closed loops for which it becomes left-handed when carried around them. The properties of Euclidean manifolds of various topologies are discussed below and summarized in table 1.

### 3 Properties of orientable Euclidean topologies

The 18 Euclidean topologies can be classified by their number of compact dimensions and whether they are orientable, homogeneous, and/or isotropic. The topologies, with their names, symbols, and properties, are listed in table 1. The balance of this paper concerns only topologies with orientable manifolds: the fully compact ($E_1-E_6$, illustrated in figure 1), those with compact-cross-sectional area ($E_{11}-E_{12}$, see figure 2), those that are compact in one dimension and orientable ($E_{16}$, see figure 2), and the covering space $E_{18}$, as highlighted in table 1. The other topologies will be addressed in upcoming papers.

In this section, we summarize the important features of each of these topologies organized as follows. First, we list its important properties as summarized in table 1. Next we provide an action of the generators $g_{a_j}^{E_i}$ of its associated discrete subgroup $\Gamma^{E_i}$ of $E(3)$. In other words, we specify the matrices $\mathcal{M}_a^{E_i}$ and the associated non-zero translation vectors $\mathcal{T}_{a_j}^{E_i}$ that characterize a manifold of each topology. For orientable manifolds, we can always choose the generators such that all the matrices $\mathcal{M}_a^{E_i}$ are either the identity, so that $g_{a_j}^{E_i}$ is a pure translation, or a rotation about an axis parallel to a coordinate axis through some origin $x_0^{E_i}$, so that $g_{a_j}^{E_i}$ is a “corkscrew motion” (e.g., see ref. [19]). With this restriction to orientable manifolds, none of the generators is a glide reflection, so $\mathcal{M}_a^{E_i} \in \text{SO}(3)$.
<table><caption>**Table 1.** Properties of the 18 three-dimensional Euclidean topologies. The orientable topologies, the focus of this work, are highlighted. Manifolds are labelled as homogeneous if the statistics of observables are independent of observer location, and isotropic if they are independent of observer orientation.</caption><thead><tr><th>Symbol</th><th>Name</th><th>Compact Dimensions</th><th>Orientable</th><th>Homogeneous</th><th>Isotropic</th></tr></thead><tbody><tr><td>E<sub>1</sub></td><td>3-torus</td><td>3</td><td>Yes</td><td>Yes</td><td>No</td></tr><tr><td>E<sub>2</sub></td><td>Half-turn</td><td>3</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>E<sub>3</sub></td><td>Quarter-turn</td><td>3</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>E<sub>4</sub></td><td>Third-turn</td><td>3</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>E<sub>5</sub></td><td>Sixth-turn</td><td>3</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>E<sub>6</sub></td><td>Hantz sche-Wendt</td><td>3</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>E<sub>7</sub></td><td>Klein space</td><td>3</td><td>No</td><td>No</td><td>No</td></tr><tr><td>E<sub>8</sub></td><td>— (horizontal flip)</td><td>3</td><td>No</td><td>No</td><td>No</td></tr><tr><td>E<sub>9</sub></td><td>— (vertical flip)</td><td>3</td><td>No</td><td>No</td><td>No</td></tr><tr><td>E<sub>10</sub></td><td>— (half-turn)</td><td>3</td><td>No</td><td>No</td><td>No</td></tr><tr><td>E<sub>11</sub></td><td>Chimney space</td><td>2</td><td>Yes</td><td>Yes</td><td>No</td></tr><tr><td>E<sub>12</sub></td><td>— (half-turn)</td><td>2</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>E<sub>13</sub></td><td>— (vertical flip)</td><td>2</td><td>No</td><td>No</td><td>No</td></tr><tr><td>E<sub>14</sub></td><td>— (horizontal flip)</td><td>2</td><td>No</td><td>No</td><td>No</td></tr><tr><td>E<sub>15</sub></td><td>— (half-turn + flip)</td><td>2</td><td>No</td><td>No</td><td>No</td></tr><tr><td>E<sub>16</sub><sup>(h)</sup></td><td>Slab (unrotated)</td><td>1</td><td>Yes</td><td>Yes</td><td>No</td></tr><tr><td>E<sub>16</sub><sup>(i)</sup></td><td>Slab (rotated)</td><td>1</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>E<sub>17</sub></td><td>Slab (flip)</td><td>1</td><td>No</td><td>No</td><td>No</td></tr><tr><td>E<sub>18</sub></td><td>Covering space</td><td>0</td><td>Yes</td><td>Yes</td><td>Yes</td></tr></tbody></table>

Within each set of topologies with the same number of compact dimensions, there is exactly one for which all of its generators, and thus all the elements of $\Gamma^{E_i}$, are pure translations. These are the 3-torus, $E_1$, with three compact dimensions; the chimney space, $E_{11}$, with two compact dimensions; the unrotated slab space, $E_{16}^{(h)}$, with one compact dimension; and, trivially, the covering space (i.e., the full Euclidean space), $E_{18}$, with no compact dimensions. The other topologies in each set can be viewed as “roots” of $E_1$, $E_{11}$, or $E_{16}^{(h)}$.

As noted above, for each group $\Gamma^{E_i}$, either each SO(3) matrix $M_a^{E_i}$ of its group elements, and in particular of its generators, is itself the identity, or there is a positive integer $N$ such that $(M_a^{E_i})^N = 1$. For $E_{16}^{(i)}$ this is only true if the associated rotation angle is a rational multiple of $2\pi$. Thus the generator applied $N$ times is a pure translation, and we are always able to construct a subgroup of $\Gamma^{E_i}$ of the same rank composed of such pure translations. For $E_1-E_6$ this subgroup is rank 3, for $E_{11}$ and $E_{12}$ it is rank 2, and for $E_{16}^{(i)}$ it is rank 1. In other words, those integer-powers of generators generate an associated homogeneous manifold: for $E_2-E_6$ we call this $E_1$ the “associated $E_1$” of this manifold; for $E_{12}$ it is called the associated $E_{11}$; $E_{16}^{(i)}$ has an associated $E_{16}^{(h)}$ only for rotation angles that are rational multiples of $2\pi$.

We provide the associated $E_1$, $E_{11}$, or $E_{16}^{(h)}$ of each manifold. It has the important property that it is homogeneous, i.e., every observer agrees on it. This can prove useful. For
example, as detailed in ref. [63] for the compact manifolds, we can construct a unit cell from these translations. In this case, when we center that cell on the origin and construct the 3 × 3 × 3 block of neighboring unit cells, then the nearest clone to any point will always be located within that neighborhood. The associated $E_1$ is also used to calculate the volume of the compact manifolds, which are provided next.

As remarked above, the action of the generators is affected by the choice of orientation and origin of the coordinate system in conjunction with the orientation and origin of an observer's coordinate system. The choices made are contained in the description of each manifold and fall into two broad categories:

* The orientation of the coordinate system used in the action of the generators allows for the simplification of the translation vectors $T_{aj}^{E_i}$ and/or to fix the ratios of some of their parameters. In particular, we will first use the rotational freedom to fix the axis associated with any corkscrew motions to be along a coordinate axis. Next, when additional rotational freedom remains, we use it to fix one of the components of a translation vector.

* Shifting the origin of the coordinate system, $x_0$, allows us to freely adjust the two components of $T_{aj}^{E_i}$ perpendicular to it.

We describe how these are implemented and components could be adjusted by the freedom to shift the origin.

Care must be taken when varying the parameters in generators to ensure that choices are not redundant, i.e., that choices of parameters that appear different actually generate the same lattice of clones. A list of conditions is provided to allow one to vary the parameters over all allowed values without “double-counting”.

As noted above, the fundamental domain is commonly used as a tool to describe homogeneous spaces, but it is observer-dependent in inhomogeneous ones. Due to this, convenient representations of the fundamental domain are given for the homogeneous manifolds, $E_1$, $E_{11}$, and $E_{16}^{(h)}$, but not for any of the others.

## 3.1 $E_1$: 3-torus

*Properties:* as listed in table 1, manifolds of this topology are compact, orientable, homogeneous, and anisotropic. Further, all the compact topologies are roots of $E_1$.

*Generators:* the generators are given by

$M_A^{E_1} = 1$, with

$$ T_{A_1}^{E_1} = \begin{pmatrix} L_{A_1 x} \\ 0 \\ 0 \end{pmatrix} \equiv T_1^{E_1}, \quad T_{A_2}^{E_1} = \begin{pmatrix} L_{A_2 x} \\ L_{A_2 y} \\ 0 \end{pmatrix} \equiv T_2^{E_1}, \quad T_{A_3}^{E_1} = \begin{pmatrix} L_{A_3 x} \\ L_{A_3 y} \\ L_{A_3 z} \end{pmatrix} \equiv T_3^{E_1}. \quad (3.1) $$

We may also write the translation vectors as

$$ T_{A_1}^{E_1} = L_{A_1} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad T_{A_2}^{E_1} = L_{A_2} \begin{pmatrix} \cos \alpha \\ \sin \alpha, \\ 0 \end{pmatrix}, \quad T_{A_3}^{E_1} = L_{A_3} \begin{pmatrix} \cos \beta \cos \gamma \\ \cos \beta \sin \gamma \\ \sin \beta \end{pmatrix}. \quad (3.2) $$
Here and throughout, when written in the alternative form we will always choose the lengths to be positive (here meaning $0 < L_{A_i}$) with the orientation of the vector determined by the angles (here $\alpha, \beta$, and $\gamma$). The alternative names $T_i^{E_1}$ introduce a notation that will be useful for the rest of the non-trivial topologies, $E_2-E_{17}$.

Since $E_1$ is homogeneous, all the generators are pure translations associated with the same matrix $M_A^{E_1} = 1$; the “A” label is extraneous and can become cumbersome. For compactness of expressions we will often drop the A in the subscript of $L_{Aiw}$ and $L_{A_i}$ so that

$$ 
\begin{aligned}
L_{iw} &\equiv L_{Aiw}, \quad \text{for } i \in \{1, 2, 3\}, w \in \{x, y, z\}; \\
L_i &\equiv L_{A_i}, \quad \text{for } i \in \{1, 2, 3\}.
\end{aligned} 
\qquad (3.3) 
$$

Volume:

$$ 
\begin{aligned}
V_{E_1} &= |(T_1^{E_1} \times T_2^{E_1}) \cdot T_3^{E_1}| \\
&= |L_{1x} L_{2y} L_{3z}| = L_1 L_2 L_3 |\sin \alpha \sin \beta|.
\end{aligned} 
\qquad (3.4) 
$$

Origin: since $M_A^{E_1} = 1$ the manifold is homogeneous and the clone lattice of an observer is independent of the location of the observer.

Real parameters (6 independent): there are 6 independent parameters required to fully define $E_1$. Since $E_1$ is homogeneous they are all required, none can be traded for shifts of the origin.

Parameter ranges: we want to ensure that we do not double-count parameter choices that appear different but actually generate the same lattice of clones. To this end, we choose our coordinate system such that the shortest translation vector is $T_{A_1}^{E_1}$ and is oriented along the $+\hat{x}$ direction, while the two shortest translation vectors, $T_{A_1}^{E_1}$ and $T_{A_2}^{E_1}$, define the $xy$-plane, with the $y$-component of $T_{A_2}^{E_1}$ positive. $T_{A_3}^{E_1}$ is then an arbitrary vector subject to the following conditions. This serves as the base set of conditions that will be applied to all the compact orientable spaces, $E_1-E_6$.

1. $0 < L_{A_1x} = L_{A_1}, 0 < L_{A_2y} = L_{2y}$, and $0 \neq L_{A_3z} = L_{3z}$, i.e., choice of orientation;

2. $0 < |T_{A_1}^{E_1}|/|T_{A_3}^{E_1}| \le |T_{A_1}^{E_1}|/|T_{A_2}^{E_1}| \le 1$, i.e., choice of ordering;

3. $|T_{A_2}^{E_1} \cdot T_{A_1}^{E_1}| \le \frac{1}{2}|T_{A_1}^{E_1}|^2$, i.e., $T_{A_2}^{E_1}$ cannot be shortened by adding or subtracting $T_{A_1}^{E_1}$;

4. $|T_{A_3}^{E_1} \cdot T_{A_1}^{E_1}| \le \frac{1}{2}|T_{A_1}^{E_1}|^2$, i.e., $T_{A_3}^{E_1}$ cannot be shortened by adding or subtracting $T_{A_1}^{E_1}$;

5. $|T_{A_3}^{E_1} \cdot T_{A_2}^{E_1}| \le \frac{1}{2}|T_{A_2}^{E_1}|^2$, i.e., $T_{A_3}^{E_1}$ cannot be shortened by adding or subtracting $T_{A_2}^{E_1}$;

6. $|T_{A_3}^{E_1} \cdot (T_{A_1}^{E_1} \pm T_{A_2}^{E_1})| \le \frac{1}{2}|T_{A_1}^{E_1} \pm T_{A_2}^{E_1}|^2$, i.e., $T_{A_3}^{E_1}$ cannot be shortened by adding or subtracting $T_{A_1}^{E_1} \pm T_{A_2}^{E_1}$;

7. $|T_{A_1}^{E_1} \cdot (T_{A_2}^{E_1} \pm T_{A_3}^{E_1})| \le \frac{1}{2}|T_{A_2}^{E_1} \pm T_{A_3}^{E_1}|^2$, i.e., $T_{A_1}^{E_1}$ cannot be shortened by adding or subtracting $T_{A_2}^{E_1} \pm T_{A_3}^{E_1}$;

8. $|T_{A_2}^{E_1} \cdot (T_{A_3}^{E_1} \pm T_{A_1}^{E_1})| \le \frac{1}{2}|T_{A_3}^{E_1} \pm T_{A_1}^{E_1}|^2$, i.e., $T_{A_2}^{E_1}$ cannot be shortened by adding or subtracting $T_{A_3}^{E_1} \pm T_{A_1}^{E_1}$.

<footnote n="3">³Strictly speaking, the constraint is $-\frac{1}{2}|T_{A_1}^{E_1}|^2 < T_{A_2}^{E_1} \cdot T_{A_1}^{E_1} \le \frac{1}{2}|T_{A_1}^{E_1}|^2$. To keep these conditions terse we will continue to use the stated shorter form. A similar simplification holds for the conditions when written in terms of the parameters.</footnote>
In terms of the parameters, these conditions become:

1. $0 < \alpha < \pi$ so that $0 < \sin \alpha \le 1$ (recall that by definition $0 < L_i$ for $i \in \{1, 2, 3\}$);

2. $0 < L_1/L_3 \le L_1/L_2 \le 1$;

3. $|\cos \alpha| \le \frac{1}{2} L_1/L_2$;

4. $|\cos \beta \cos \gamma| \le \frac{1}{2} L_1/L_3$;

5. $|\cos \beta \cos(\alpha - \gamma)| \le \frac{1}{2} L_2/L_3$;

6. $L_3 |\cos \beta| L_2 \cos(\alpha - \gamma) \pm L_1 \cos \gamma| \le \frac{1}{2} [L_1^2 + L_2^2 \pm 2L_1 L_2 \cos \alpha]$;

7. $L_1|L_2 \cos \alpha \pm L_3 \cos \beta \cos \gamma| \le \frac{1}{2} [L_2^2 + L_3^2 \pm 2L_2 L_3 \cos \beta \cos(\alpha - \gamma)]$;

8. $L_2|L_3 \cos \beta \cos(\alpha - \gamma) \pm L_1 \cos \alpha| \le \frac{1}{2} [L_1^2 + L_3^2 \pm 2L_1 L_3 \cos \beta \cos \gamma]$.

*Convenient fundamental domain:* a convenient choice of FD is a parallelepiped, the vertices of which are any base point and seven clones.

There are two convenient choices of base point:

(A) Origin-centered FD.

• Base point: $x_A \equiv \frac{1}{2}(-T_{A_1}^{E_1} - T_{A_2}^{E_1} - T_{A_3}^{E_1})$;

• 3 other corners of the bottom face:

$$-x_B = g_{A_1}^{E_1} x_A = \frac{1}{2}(T_{A_1}^{E_1} - T_{A_2}^{E_1} - T_{A_3}^{E_1});$$

$$-x_C = g_{A_2}^{E_1} g_{A_1}^{E_1} x_A = \frac{1}{2}(T_{A_1}^{E_1} + T_{A_2}^{E_1} - T_{A_3}^{E_1});$$

$$-x_D = g_{A_2}^{E_1} x_A = \frac{1}{2}(-T_{A_1}^{E_1} + T_{A_2}^{E_1} - T_{A_3}^{E_1});$$

• Top face:

$$-x_E = g_{A_3}^{E_1} x_A = x_A + T_{A_3}^{E_1};$$

$$-x_F = g_{A_3}^{E_1} x_B = x_B + T_{A_3}^{E_1};$$

$$-x_G = g_{A_3}^{E_1} x_C = x_C + T_{A_3}^{E_1};$$

$$-x_H = g_{A_3}^{E_1} x_D = x_D + T_{A_3}^{E_1}.$$

(B) Origin-rooted FD.

• Base point: $x_A \equiv 0$;

• 3 other corners of $z=0$ face:

$$-x_B = g_{A_1}^{E_1} x_A = T_{A_1}^{E_1};$$

$$-x_C = g_{A_2}^{E_1} g_{A_1}^{E_1} x_A = T_{A_1}^{E_1} + T_{A_2}^{E_1};$$

$$-x_D = g_{A_2}^{E_1} x_A = T_{A_2}^{E_1};$$

• $z \ne 0$ face: same as “top face” from case (A).

## 3.2 $E_2$: half-turn space

*Properties:* as listed in table 1, this manifold is compact, orientable, inhomogeneous, and anisotropic.
Generators: in general (see appendix A.2) the generators of $E_2$ can be written as<fnref n="4" />

$$ M_A^{E_2} = \mathbf{1}, \quad M_B^{E_2} = \mathbf{R}_{\hat{z}}(\pi) = \mathrm{diag}(-1, -1, 1), \quad \text{with} \\ \mathbf{T}_{A_1}^{E_2} = \begin{pmatrix} L_{A_1 x} \\ 0 \\ 0 \end{pmatrix} \equiv \mathbf{T}_1^{E_2}, \quad \mathbf{T}_{A_2}^{E_2} = \begin{pmatrix} L_{A_2 x} \\ L_{A_2 y} \\ 0 \end{pmatrix} \equiv \mathbf{T}_2^{E_2}, \quad \mathbf{T}_B^{E_2} = \begin{pmatrix} L_{B x} \\ L_{B y} \\ L_{B z} \end{pmatrix}. \quad (3.5) $$

Alternatively, we can write the translation vectors as

$$ \mathbf{T}_{A_1}^{E_2} = L_{A_1} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{T}_{A_2}^{E_2} = L_{A_2} \begin{pmatrix} \cos \alpha \\ \sin \alpha \\ 0 \end{pmatrix}, \quad \mathbf{T}_B^{E_2} = L_B \begin{pmatrix} \cos \beta \cos \gamma \\ \cos \beta \sin \gamma \\ \sin \beta \end{pmatrix}. \quad (3.6) $$

Similar to $E_1$, we often simplify the notation when working with parameters of $E_2$ by dropping the A label and instead using

$$ \begin{aligned} L_{iw} &\equiv L_{A_i w}, & \text{for } i \in \{1, 2\}, w \in \{x, y, z\}; \\ L_i &\equiv L_{A_i}, & \text{for } i \in \{1, 2\}. \end{aligned} \quad (3.7) $$

Associated $E_1$: in addition to $\mathbf{T}_1^{E_2}$ and $\mathbf{T}_2^{E_2}$ defined above, a third independent translation is

$$ g_3^{E_2} \equiv (g_B^{E_2})^2 : \boldsymbol{x} \rightarrow \boldsymbol{x} + \mathbf{T}_3^{E_2}, \quad (3.8) $$

for

$$ \mathbf{T}_3^{E_2} \equiv \begin{pmatrix} 0 \\ 0 \\ 2L_{Bz} \end{pmatrix} = 2L_B \begin{pmatrix} 0 \\ 0 \\ \sin \beta \end{pmatrix}. \quad (3.9) $$

The three vectors $\mathbf{T}_1^{E_2}$, $\mathbf{T}_2^{E_2}$, and $\mathbf{T}_3^{E_2}$ define the associated $E_1$.

Volume:

$$ V_{E_2} = \frac{1}{2} |(\mathbf{T}_1^{E_2} \times \mathbf{T}_2^{E_2}) \cdot \mathbf{T}_3^{E_2}| = |L_{1x}L_{2y}L_{Bz}| = L_1 L_2 L_B |\sin \alpha \sin \beta|. \quad (3.10) $$

Tilts versus origin position: when shifting the origin, $x_{0x}^{E_2}$ and $x_{0y}^{E_2}$ change $L_{Bx}$ and $L_{By}$, but $L_{Bz}$ is unaffected. Equivalently, when shifting $L_B$, $\beta$, and $\gamma$ are changed while holding $L_B \sin \beta$ fixed. This shows that shifting the origin is equivalent to tilting the translation vector associated with the rotation, $\mathbf{T}_B^{E_2}$.

Explicitly, since $M_A^{E_2} = \mathbf{1}$, neither $\mathbf{T}_{A_1}^{E_2}$ nor $\mathbf{T}_{A_2}^{E_2}$ is affected by shifts in $x_0^{E_2}$. In contrast, tilting $\mathbf{T}_B^{E_2}$ out of the $\hat{z}$ direction is equivalent to a shift of origin; i.e., $\beta$ and $\gamma$ can be adjusted (for example to $\pi/2$) by the choice of $x_{0x}^{E_2}$ and $x_{0y}^{E_2}$. In particular, one can set $\beta = \pi/2$, so that $\mathbf{T}_B^{E_3} = L_B \hat{e}_z$. Of course, the observer will then not sit at the origin of coordinates and may be up to $|\mathbf{T}_{A_1}^{E_2} + \mathbf{T}_{A_2}^{E_2}|$ away in the $xy$-plane. (Their $z$ position is immaterial.)

<footnote n="4">⁴Here and throughout all rotations will be treated as active.</footnote>
Real parameters (6 independent): there are 6 independent parameters required to fully define $E_2$. As noted above, some are redundant with shifting the origin, or equivalently, a tilt. Thus we have:

• $L_{1x}$, $L_{2x}$, $L_{2y}$, and $L_{Bz}$ are intrinsic parameters of the manifold;

• $L_{Bx}$ and $L_{By}$ can be traded for $x_{0x}^{E_2}$ and $x_{0y}^{E_2}$;

• the standard (special origin, i.e., “untilted”) form is $0 = L_{2x} = L_{Bx} = L_{By}$.

In terms of the alternative parameter form we have:

• $L_1$, $L_1/L_2$, $L_1/L_B$, and $\alpha$ are intrinsic parameters of the manifold;

• $\beta$ and $\gamma$ can be traded for $x_{0x}^{E_2}$ and $x_{0y}^{E_2}$;

• the standard (special origin, i.e., “untilted”) form is $\alpha = \pi/2$, $\beta = \pi/2$;

Parameter ranges: we want to ensure that we do not double-count parameter choices that appear different but actually generate the same lattice of clones. Similar to $E_1$, we therefore require:

1. $0 < L_{A_1 x} = L_{A_1}, 0 < L_{A_2 y} \equiv L_{2y}$, and $0 \neq L_{Bz}$, i.e., choice of orientation;

2. $0 < |\mathbf{T}_{A_1}^{E_2}|/|\mathbf{T}_{A_2}^{E_2}| \le 1$, i.e., choice of ordering (note, in contrast to $E_1$, we cannot constrain $|\mathbf{T}_B^{E_2}|$ since it is identified not as the longest vector but as the one associated with $\mathbf{M}_B^{E_2} \neq \mathbf{1}$);

3. $|\mathbf{T}_{A_2}^{E_2} \cdot \mathbf{T}_{A_1}^{E_2}| \le \frac{1}{2}|\mathbf{T}_{A_1}^{E_2}|^2$, i.e., $\mathbf{T}_{A_2}^{E_2}$ cannot be shortened by adding or subtracting $\mathbf{T}_{A_1}^{E_2}$;

4. $|\mathbf{T}_B^{E_2} \cdot \mathbf{T}_{A_1}^{E_2}| \le \frac{1}{2}|\mathbf{T}_{A_1}^{E_2}|^2$, i.e., $\mathbf{T}_B^{E_2}$ cannot be shortened by adding or subtracting $\mathbf{T}_{A_1}^{E_2}$;

5. $|\mathbf{T}_B^{E_2} \cdot \mathbf{T}_{A_2}^{E_2}| \le \frac{1}{2}|\mathbf{T}_{A_2}^{E_2}|^2$, i.e., $\mathbf{T}_B^{E_2}$ cannot be shortened by adding or subtracting $\mathbf{T}_{A_2}^{E_2}$;

6. $|\mathbf{T}_B^{E_2} \cdot (\mathbf{T}_{A_1}^{E_2} \pm \mathbf{T}_{A_2}^{E_2})| \le \frac{1}{2}|\mathbf{T}_{A_1}^{E_2} \pm \mathbf{T}_{A_2}^{E_2}|^2$, i.e., $\mathbf{T}_B^{E_2}$ cannot be shortened by adding or subtracting $\mathbf{T}_{A_1}^{E_2} \pm \mathbf{T}_{A_2}^{E_2}$;

7. $|\mathbf{T}_{A_1}^{E_2} \cdot (\mathbf{T}_{A_2}^{E_2} \pm \mathbf{T}_3^{E_1})| \le \frac{1}{2}|\mathbf{T}_{A_2}^{E_2} \pm \mathbf{T}_3^{E_2}|^2$, automatically satisfied by condition (3) and since $\mathbf{T}_{A_1}^{E_2} \cdot \mathbf{T}_3^{E_2} = 0$ (note that it is $\mathbf{T}_3^{E_2}$ that appears here, not $\mathbf{T}_B^{E_2}$);

8. $|\mathbf{T}_{A_2}^{E_2} \cdot (\mathbf{T}_3^{E_2} \pm \mathbf{T}_{A_1}^{E_2})| \le \frac{1}{2}|\mathbf{T}_3^{E_2} \pm \mathbf{T}_{A_1}^{E_2}|^2$, automatically satisfied by condition (3) and since $\mathbf{T}_{A_2}^{E_2} \cdot \mathbf{T}_3^{E_2} = 0$ (note that it is $\mathbf{T}_3^{E_2}$ that appears here, not $\mathbf{T}_B^{E_2}$).

In terms of the parameters, the necessary conditions become:

1. $0 < \alpha < \pi$ so that $0 < \sin \alpha \le 1$, $0 < |\beta| \le \pi/2$ so that $0 \neq \sin \beta$, and $0 \le \gamma < 2\pi$;

2. $0 < L_1/L_2 \le 1$, $0 < L_1/L_B < \infty$;

3. $|\cos \alpha| \le \frac{1}{2}L_1/L_2$;

4. $|\cos \beta \cos \gamma| \le \frac{1}{2}L_1/L_B$;

5. $|\cos \beta \cos(\alpha - \gamma)| \le \frac{1}{2}L_2/L_B$;

6. $L_B|\cos \beta||L_2 \cos(\alpha - \gamma) \pm L_1 \cos \gamma| \le \frac{1}{2}[L_1^2 + L_2^2 \pm 2L_1L_2 \cos \alpha]$.
### 3.3 $E_3$: quarter-turn space

**Properties:** as listed in table 1, this manifold is compact, orientable, inhomogeneous, and anisotropic.

**Generators:** in general (see appendix A.3) the generators of $E_3$ can be written as

$$ \mathbf{M}_A^{E_3} = \mathbf{1}, \quad \mathbf{M}_B^{E_3} = \mathbf{R}_{\hat{z}}(\pi/2) = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad \text{with} \\ \mathbf{T}_{A_1}^{E_3} = \begin{pmatrix} L_A \\ 0 \\ 0 \end{pmatrix} \equiv \mathbf{T}_1^{E_3}, \quad \mathbf{T}_{A_2}^{E_3} = \begin{pmatrix} 0 \\ L_A \\ 0 \end{pmatrix} \equiv \mathbf{T}_2^{E_3}, \quad \mathbf{T}_B^{E_3} = \begin{pmatrix} L_{Bx} \\ L_{By} \\ L_{Bz} \end{pmatrix}. \qquad (3.11) $$

Alternatively, we can write the translation vectors as

$$ \mathbf{T}_{A_1}^{E_3} = L_A \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{T}_{A_2}^{E_3} = L_A \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \mathbf{T}_B^{E_3} = L_B \begin{pmatrix} \cos \beta \cos \gamma \\ \cos \beta \sin \gamma \\ \sin \beta \end{pmatrix}. \qquad (3.12) $$

Note that the $\mathbf{T}_{A_1}^{E_3}$ and $\mathbf{T}_{A_2}^{E_3}$ are the same length and orthogonal to one another.

**Associated $E_1$:** in addition to $\mathbf{T}_1^{E_3}$ and $\mathbf{T}_2^{E_3}$ defined above, a third independent translation follows from $(\mathbf{M}_B^{E_3})^4 = \mathbf{1}$:

$$ g_3^{E_3} \equiv (g_B^{E_3})^4 : \mathbf{x} \to \mathbf{x} + \mathbf{T}_3^{E_3}, \qquad (3.13) $$

for

$$ \mathbf{T}_3^{E_3} \equiv \begin{pmatrix} 0 \\ 0 \\ 4L_{Bz} \end{pmatrix} = 4L_B \begin{pmatrix} 0 \\ 0 \\ \sin \beta \end{pmatrix}. \qquad (3.14) $$

**Volume:**

$$ V_{E_3} = \frac{1}{4} |(\mathbf{T}_1^{E_3} \times \mathbf{T}_2^{E_3}) \cdot \mathbf{T}_3^{E_3}| = L_A^2 L_B |\sin \beta|. \qquad (3.15) $$

**Tilts versus origin position:** as in $E_2$, a shift of origin will change $L_{Bx}$ and $L_{By}$ but will not affect $L_{Bz}$. This again leads to a tilt being equivalent to a shift of origin, which allows us to choose $\mathbf{T}_B^{E_3} = L_B \hat{\mathbf{e}}_z$ at the expense of the observer no longer being at the origin.

**Real parameters (4 independent):** there are 4 independent parameters required to fully define $E_3$. As in $E_2$ some are redundant with shifting the origin. Thus we have:

* $L_A$ and $L_A/L_B$ are intrinsic parameters of the manifold;

* $L_{Bx}$ and $L_{By}$, or equivalently $\beta$ and $\gamma$, can be traded for $x_{0x}^{E_3}$ and $x_{0y}^{E_3}$;

* the standard (special origin, i.e., “untilted”) form is $0 = L_{Bx} = L_{By}$, or equivalently $\beta = \pi/2$ (with $\gamma$ irrelevant).
*Parameter ranges:* we want to ensure that we do not double-count parameter choices that appear different but actually generate the same lattice of clones. Similar to $E_1$, we therefore require:

1. $0 < L_A$ by definition as in $E_1$ and $0 \neq L_{Bz}$, i.e., choice of orientation;

2. $|\mathbf{T}_{A_1}^{E_3}|/|\mathbf{T}_{A_2}^{E_3}| = 1$, automatically enforced by parametrization (note that we cannot constrain $|\mathbf{T}_B^{E_3}|$ since it is identified not as the longest vector but as the one associated with $\mathbf{M}_B^{E_3} \neq \mathbf{1}$);

3. $|\mathbf{T}_{A_2}^{E_3} \cdot \mathbf{T}_{A_1}^{E_3}| = 0$; automatically enforced by parametrization;

4. $|\mathbf{T}_B^{E_3} \cdot \mathbf{T}_{A_1}^{E_3}| \leq \frac{1}{2}|\mathbf{T}_{A_1}^{E_3}|^2$, i.e., $\mathbf{T}_B^{E_3}$ cannot be shortened by adding or subtracting $\mathbf{T}_{A_1}^{E_3}$;

5. $|\mathbf{T}_B^{E_3} \cdot \mathbf{T}_{A_2}^{E_3}| \leq \frac{1}{2}|\mathbf{T}_{A_2}^{E_3}|^2$, i.e., $\mathbf{T}_B^{E_3}$ cannot be shortened by adding or subtracting $\mathbf{T}_{A_2}^{E_3}$;

6. $|\mathbf{T}_B^{E_3} \cdot (\mathbf{T}_{A_1}^{E_3} \pm \mathbf{T}_{A_2}^{E_3})| \leq \frac{1}{2}|\mathbf{T}_{A_1}^{E_3} \pm \mathbf{T}_{A_2}^{E_3}|^2$, automatically enforced given conditions (4) and (5);

7. $|\mathbf{T}_{A_1}^{E_3} \cdot (\mathbf{T}_{A_2}^{E_3} \pm \mathbf{T}_3^{E_3})| = 0$, automatically enforced (note that it is $\mathbf{T}_3^{E_3}$ that appears here, not $\mathbf{T}_B^{E_3}$);

8. $|\mathbf{T}_{A_2}^{E_3} \cdot (\mathbf{T}_3^{E_3} \pm \mathbf{T}_{A_1}^{E_3})| = 0$, automatically enforced (note that it is $\mathbf{T}_3^{E_3}$ that appears here, not $\mathbf{T}_B^{E_3}$).

In terms of the parameters, the necessary conditions become:

1. $0 < L_A$, $0 < |\beta| \leq \pi/2$ so that $0 \neq \sin \beta$, and $0 \leq \gamma < 2\pi$;

2. $0 < L_A/L_B < \infty$;

4. $|L_{Bx}| \leq \frac{1}{2}L_A$, or equivalently $\cos \beta|\cos \gamma| \leq \frac{1}{2}L_A/L_B$;

5. $|L_{By}| \leq \frac{1}{2}L_A$, or equivalently $\cos \beta|\sin \gamma| \leq \frac{1}{2}L_A/L_B$.

### 3.4 $E_4$: third-turn space

*Properties:* as listed in table 1, this manifold is compact, orientable, inhomogeneous, and anisotropic.

*Generators:* in general (see appendix A.4) the generators of $E_4$ can be written as

$$ \mathbf{M}_A^{E_4} = \mathbf{1}, \quad \mathbf{M}_B^{E_4} = \mathbf{R}_z(2\pi/3) = \begin{pmatrix} -1/2 & -\sqrt{3}/2 & 0 \\ \sqrt{3}/2 & -1/2 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad \text{with} \\ \mathbf{T}_{A_1}^{E_4} = \begin{pmatrix} L_A \\ 0 \\ 0 \end{pmatrix} \equiv \mathbf{T}_1^{E_4}, \quad \mathbf{T}_{A_2}^{E_4} = \begin{pmatrix} -L_A/2 \\ \sqrt{3}L_A/2 \\ 0 \end{pmatrix} \equiv \mathbf{T}_2^{E_4}, \quad \mathbf{T}_B^{E_4} = \begin{pmatrix} L_{Bx} \\ L_{By} \\ L_{Bz} \end{pmatrix}. \quad (3.16) $$

Notice that $\mathbf{T}_{A_2}^{E_4} = \mathbf{R}_z(2\pi/3)\mathbf{T}_{A_1}^{E_4}$. It can also be useful to define and use

$$ \mathbf{T}_{A_3}^{E_4} \equiv \mathbf{R}_z(-2\pi/3)\mathbf{T}_{A_1}^{E_4} = \begin{pmatrix} -L_A/2 \\ -\sqrt{3}L_A/2 \\ 0 \end{pmatrix} = -\mathbf{T}_{A_1}^{E_4} - \mathbf{T}_{A_2}^{E_4}. \quad (3.17) $$
Alternatively, we can write the translation vectors as

$$ T_{A_1}^{E_4} = L_A \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad T_{A_2}^{E_4} = L_A \begin{pmatrix} -1/2 \\ \sqrt{3}/2 \\ 0 \end{pmatrix}, \quad T_B^{E_4} = L_B \begin{pmatrix} \cos \beta \cos \gamma \\ \cos \beta \sin \gamma \\ \sin \beta \end{pmatrix}. \quad (3.18) $$

Associated $E_1$: in addition to $T_1^{E_4}$ and $T_2^{E_4}$ defined above, a third independent translation follows from $(M_B^{E_4})^3 = 1$:

$$ g_3^{E_4} = (g_B^{E_4})^3 : \boldsymbol{x} \to \boldsymbol{x} + \boldsymbol{T}_3^{E_4}, \quad (3.19) $$

for

$$ T_3^{E_4} = \begin{pmatrix} 0 \\ 0 \\ 3L_{Bz} \end{pmatrix} = 3L_B \begin{pmatrix} 0 \\ 0 \\ \sin \beta \end{pmatrix}. \quad (3.20) $$

Volume:

$$ V_{E_4} = \frac{1}{3} |(T_1^{E_4} \times T_2^{E_4}) \cdot T_3^{E_4}| = \frac{\sqrt{3}}{2} L_A^2 L_B |\sin \beta|. \quad (3.21) $$

Tilts versus origin position: as in $E_2$, a shift of origin will change $L_{Bx}$ and $L_{By}$ but will not affect $L_{Bz}$. This again leads to a tilt being equivalent to a shift of origin, which allows us to choose $T_B^{E_4} = L_B \hat{e}_z$ at the expense of the observer no longer being at the origin.

Real parameters (4 independent): there are 4 independent parameters required to fully define $E_4$. As in $E_2$, some are redundant with shifting the origin. Thus we have:

* $L_A$ and $L_A/L_B$ are intrinsic parameters of the manifold;

* $L_{Bx}$ and $L_{By}$, or equivalently $\beta$ and $\gamma$, can be traded for $x_{0x}^{E_4}$ and $x_{0y}^{E_4}$;

* the standard (special origin, i.e., “untilted”) form is $0 = L_{Bx} = L_{By}$, or equivalently $\beta = \pi/2$ (with $\gamma$ irrelevant).

Parameter ranges: we want to ensure that we do not double-count parameter choices that appear different but actually generate the same lattice of clones. Similar to $E_1$, we therefore require:

1. $0 < L_A$ and $0 \neq L_{Bz}$, i.e., choice of orientation;

2. $|T_{A_1}^{E_4}|/|T_{A_2}^{E_4}| = 1$, automatically enforced by parametrization (note that we cannot constrain $|T_B^{E_4}|$ since it is identified not as the longest vector but as the one associated with $M_B^{E_4} \neq 1$);

3. $|T_{A_2}^{E_4} \cdot T_{A_1}^{E_4}| = \frac{1}{2}|T_{A_1}^{E_4}|^2$, automatically enforced by parametrization;

4. $|T_B^{E_4} \cdot T_{A_1}^{E_4}| \le \frac{1}{2}|T_{A_1}^{E_4}|^2$, i.e., $T_B^{E_4}$ cannot be shortened by adding or subtracting $T_{A_1}^{E_4}$;

5. $|T_B^{E_4} \cdot T_{A_2}^{E_4}| \le \frac{1}{2}|T_{A_2}^{E_4}|^2$, i.e., $T_B^{E_4}$ cannot be shortened by adding or subtracting $T_{A_2}^{E_4}$;

6. $|T_B^{E_4} \cdot (T_{A_1}^{E_4} \pm T_2^{E_4})| \le \frac{1}{2}|T_{A_1}^{E_4} \pm T_{A_2}^{E_4}|^2$, i.e., $T_B^{E_4}$ cannot be shortened by adding or subtracting $T_{A_1}^{E_4} \pm T_{A_2}^{E_4}$;

7. $|T_{A_1}^{E_4} \cdot (T_{A_2}^{E_4} \pm T_3^{E_4})| \le \frac{1}{2}|T_{A_2}^{E_4} \pm T_3^{E_4}|^2$, automatically enforced given that $T_{A_1}^{E_4} \cdot T_3^{E_4} = 0$ and $T_{A_2}^{E_4} \cdot T_3^{E_4} = 0$ (note that it is $T_3^{E_4}$ that appears here, not $T_B^{E_4}$);
8. $|T_{A_2}^{E_4} \cdot (T_3^{E_4} \pm T_{A_1}^{E_4})| \le \frac{1}{2}|T_3^{E_4} \pm T_{A_1}^{E_4}|^2$, automatically enforced given that $T_{A_1}^{E_4} \cdot T_3^{E_4} = 0$ and $T_{A_2}^{E_4} \cdot T_3^{E_4} = 0$ (note that it is $T_3^{E_4}$ that appears here, not $T_B^{E_4}$).

In terms of the parameters the necessary conditions become:

1. $0 < L_A, 0 < |\beta| \le \pi/2$ so that $0 \ne \sin \beta$, and $0 \le \gamma < 2\pi$;

2. $0 < L_A/L_B < \infty$;

4. $\cos \beta |\cos \gamma| \le \frac{1}{2} L_A / L_B$;

5. $\cos \beta |\cos(\gamma + \pi/3)| \le \frac{1}{2} L_A / L_B$;

6. $\cos \beta |\cos(\gamma - \pi/3)| \le \frac{1}{2} L_A / L_B$ and $\cos \beta |\cos(\gamma + \pi/6)| \le \frac{\sqrt{3}}{2} L_A / L_B$.

These conditions can be written compactly as $0 < L_A, 0 \ne L_{Bz}$, and

1. $-\frac{1}{2} L_A < L_{Bx} \le 0$ and $-L_A + L_{Bx} < \sqrt{3} L_{By} \le L_A + 3 L_{Bx}$, or

2. $0 < L_{Bx} \le \frac{1}{2} L_A$ and $-L_A + 3 L_{Bx} < \sqrt{3} L_{By} \le L_A + L_{Bx}$.

### 3.5 $E_5$: sixth-turn space

*Properties:* as listed in table 1, this manifold is compact, orientable, inhomogeneous, and anisotropic.

*Generators:* in general (see appendix A.5) the generators of $E_5$ can be written as

$$ \mathbf{M}_A^{E_5} = 1, \quad \mathbf{M}_B^{E_5} = \mathbf{R}_z(\pi/3) = \begin{pmatrix} 1/2 & -\sqrt{3}/2 & 0 \\ \sqrt{3}/2 & 1/2 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad \text{with} \\ \mathbf{T}_{A_1}^{E_5} = \begin{pmatrix} L_A \\ 0 \\ 0 \end{pmatrix} \equiv \mathbf{T}_1^{E_5}, \quad \mathbf{T}_{A_2}^{E_5} = \begin{pmatrix} -L_A/2 \\ \sqrt{3}L_A/2 \\ 0 \end{pmatrix} \equiv \mathbf{T}_2^{E_5}, \quad \mathbf{T}_B^{E_5} = \begin{pmatrix} L_{Bx} \\ L_{By} \\ L_{Bz} \end{pmatrix}. \quad (3.22) $$

Notice that $\mathbf{T}_{A_2}^{E_5} = -\mathbf{R}_z(-\pi/3)\mathbf{T}_{A_1}^{E_5}$. It can also be useful to define and use

$$ \mathbf{T}_{A_3}^{E_5} \equiv -\mathbf{R}_z(\pi/3)\mathbf{T}_{A_1}^{E_5} = \begin{pmatrix} -L_A/2 \\ -\sqrt{3}L_A/2 \\ 0 \end{pmatrix} = -\mathbf{T}_{A_1}^{E_5} - \mathbf{T}_{A_2}^{E_5}. \quad (3.23) $$

Alternatively, we can write the translation vectors as

$$ \mathbf{T}_{A_1}^{E_5} = L_A \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{T}_{A_2}^{E_5} = L_A \begin{pmatrix} -1/2 \\ \sqrt{3}/2 \\ 0 \end{pmatrix}, \quad \mathbf{T}_B^{E_5} = L_B \begin{pmatrix} \cos \beta \cos \gamma \\ \cos \beta \sin \gamma \\ \sin \beta \end{pmatrix}. \quad (3.24) $$

*Associated $E_1$:* in addition to $\mathbf{T}_1^{E_5}$ and $\mathbf{T}_2^{E_5}$ defined above, a third independent translation follows from $(\mathbf{M}_B^{E_5})^6 = 1$:

$$ g_3^{E_5} \equiv (g_B^{E_5})^6 : \boldsymbol{x} \to \boldsymbol{x} + \mathbf{T}_3^{E_5}, \quad (3.25) $$

for

$$ \mathbf{T}_3^{E_5} \equiv \begin{pmatrix} 0 \\ 0 \\ 6L_{Bz} \end{pmatrix} = 6L_B \begin{pmatrix} 0 \\ 0 \\ \sin \beta \end{pmatrix}. \quad (3.26) $$
Volume:

$$V_{E_5} = \frac{1}{6} |(\mathbf{T}_1^{E_5} \times \mathbf{T}_2^{E_5}) \cdot \mathbf{T}_3^{E_5}| = \frac{\sqrt{3}}{2} L_A^2 L_B |\sin \beta|. \quad (3.27)$$

**Tilts versus origin position:** as in $E_2$, a shift of origin will change $L_{Bx}$ and $L_{By}$ but will not affect $L_{Bz}$. This again leads to a tilt being equivalent to a shift of origin, which allows us to choose $\mathbf{T}_B^{E_5} = L_B \hat{e}_z$ at the expense of the observer no longer being at the origin.

**Real parameters (4 independent):** there are 4 independent parameters required to fully define $E_5$. As in $E_2$, some are redundant with shifting the origin. Thus we have:

* $L_A$ and $L_A/L_B$ are intrinsic parameters of the manifold;

* $L_{Bx}$ and $L_{By}$, or equivalently $\beta$ and $\gamma$, can be traded for $x_{0x}^{E_5}$ and $x_{0y}^{E_5}$;

* the standard (special origin, i.e., “tilted”) form is $0 = L_{Bx} = L_{By}$, or equivalently $\beta = \pi/2$ (with $\gamma$ irrelevant).

**Parameter ranges:** we want to ensure that we do not double-count parameter choices that appear different but actually generate the same lattice of clones. Similar to $E_1$, we therefore require:

1. $0 < L_A$ and $0 \neq L_{Bz}$, i.e., choice of orientation;

2. $|\mathbf{T}_{A_1}^{E_5}|/|\mathbf{T}_{A_2}^{E_5}| = 1$, automatically enforced by parametrization (note that we cannot constrain $|\mathbf{T}_B^{E_5}|$ since it is identified not as the longest vector but as the one associated with $\mathbf{M}_B^{E_5} \neq 1$);

3. $|\mathbf{T}_{A_2}^{E_5} \cdot \mathbf{T}_{A_1}^{E_5}| = \frac{1}{2}|\mathbf{T}_{A_1}^{E_5}|^2$, automatically enforced by parametrization;

4. $|\mathbf{T}_B^{E_5} \cdot \mathbf{T}_{A_1}^{E_5}| \leq \frac{1}{2}|\mathbf{T}_{A_1}^{E_5}|^2$, i.e., $\mathbf{T}_B^{E_5}$ cannot be shortened by adding or subtracting $\mathbf{T}_{A_1}^{E_5}$;

5. $|\mathbf{T}_B^{E_5} \cdot \mathbf{T}_{A_2}^{E_5}| \leq \frac{1}{2}|\mathbf{T}_{A_2}^{E_5}|^2$, i.e., $\mathbf{T}_B^{E_5}$ cannot be shortened by adding or subtracting $\mathbf{T}_{A_2}^{E_5}$;

6. $|\mathbf{T}_B^{E_5} \cdot (\mathbf{T}_{A_1}^{E_5} \pm \mathbf{T}_{A_2}^{E_5})| \leq \frac{1}{2}|\mathbf{T}_{A_1}^{E_5} \pm \mathbf{T}_{A_2}^{E_5}|^2$, i.e., $\mathbf{T}_B^{E_5}$ cannot be shortened by adding or subtracting $\mathbf{T}_{A_1}^{E_5} \pm \mathbf{T}_{A_2}^{E_5}$;

7. $|\mathbf{T}_{A_1}^{E_5} \cdot (\mathbf{T}_{A_2}^{E_5} \pm \mathbf{T}_3^{E_5})| \leq \frac{1}{2}|\mathbf{T}_{A_2}^{E_5} \pm \mathbf{T}_3^{E_5}|^2$, automatically enforced given that $\mathbf{T}_{A_1}^{E_5} \cdot \mathbf{T}_3^{E_5} = 0$ and $\mathbf{T}_{A_2}^{E_5} \cdot \mathbf{T}_3^{E_5} = 0$ (note that it is $\mathbf{T}_3^{E_5}$ that appears here, not $\mathbf{T}_B^{E_5}$);

8. $|\mathbf{T}_{A_2}^{E_5} \cdot (\mathbf{T}_3^{E_5} \pm \mathbf{T}_{A_1}^{E_5})| \leq \frac{1}{2}|\mathbf{T}_3^{E_5} \pm \mathbf{T}_{A_1}^{E_5}|^2$, automatically enforced given that $\mathbf{T}_{A_1}^{E_5} \cdot \mathbf{T}_3^{E_5} = 0$ and $\mathbf{T}_{A_2}^{E_5} \cdot \mathbf{T}_3^{E_5} = 0$ (note that it is $\mathbf{T}_3^{E_5}$ that appears here, not $\mathbf{T}_B^{E_5}$).

In terms of the parameters these conditions are the same as those for $E_4$, so the necessary conditions become:

1. $0 < L_A$, $0 < |\beta| \leq \pi/2$ so that $0 \neq \sin \beta$, and $0 \leq \gamma < 2\pi$;

2. $0 < L_A/L_B < \infty$;

4. $\cos \beta |\cos \gamma| \leq \frac{1}{2} L_A/L_B$;

5. $\cos \beta |\cos(\gamma + \pi/3)| \leq \frac{1}{2} L_A/L_B$;

6. $\cos \beta |\cos(\gamma - \pi/3)| \leq \frac{1}{2} L_A/L_B$ and $\cos \beta |\cos(\gamma + \pi/6)| \leq \frac{\sqrt{3}}{2} L_A/L_B$.

These conditions can be written compactly as $0 < L_A$, $0 \neq L_{Bz}$, and

1. $-\frac{1}{2} L_A < L_{Bx} \leq 0$ and $-L_A + L_{Bx} < \sqrt{3} L_{By} \leq L_A + 3 L_{Bx}$, or

2. $0 < L_{Bx} \leq \frac{1}{2} L_A$ and $-L_A + 3 L_{Bx} < \sqrt{3} L_{By} \leq L_A + L_{Bx}$.
## 3.6 $E_6$: Hantzsche-Wendt space

*Properties:* as listed in table 1, this manifold is compact, orientable, inhomogeneous, and anisotropic (for more information see ref. [80]).

*Generators:* in general (see appendix A.6) the generators of $E_6$ can be written as

$$ \mathbf{M}_A^{E_6} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \quad \mathbf{M}_B^{E_6} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \quad \mathbf{M}_C^{E_6} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad \text{with} $$

$$ \mathbf{T}_A^{E_6} = \begin{pmatrix} L_{Ax} \\ L_{By} + L_{Cy} \\ L_{Az} \end{pmatrix}, \quad \mathbf{T}_B^{E_6} = \begin{pmatrix} L_{Bx} \\ L_{By} \\ L_{Cz} + L_{Az} \end{pmatrix}, \quad \mathbf{T}_C^{E_6} = \begin{pmatrix} L_{Ax} + L_{Bx} \\ L_{Cy} \\ L_{Cz} \end{pmatrix}. \quad (3.28) $$

Alternatively, we can reparametrize some of the lengths via

$$ L_{Bx} = \left(r_x - \frac{1}{2}\right) L_{Ax}, \quad L_{Cy} = \left(r_y - \frac{1}{2}\right) L_{By}, \quad L_{Az} = \left(r_z - \frac{1}{2}\right) L_{Cz}, \quad (3.29) $$

so that the translation vectors become

$$ \mathbf{T}_A^{E_6} = \begin{pmatrix} L_{Ax} \\ \left(r_y + \frac{1}{2}\right) L_{By} \\ \left(r_z - \frac{1}{2}\right) L_{Cz} \end{pmatrix}, \quad \mathbf{T}_B^{E_6} = \begin{pmatrix} \left(r_x - \frac{1}{2}\right) L_{Ax} \\ L_{By} \\ \left(r_z + \frac{1}{2}\right) L_{Cz} \end{pmatrix}, \quad \mathbf{T}_C^{E_6} = \begin{pmatrix} \left(r_x + \frac{1}{2}\right) L_{Ax} \\ \left(r_y - \frac{1}{2}\right) L_{By} \\ L_{Cz} \end{pmatrix}. \quad (3.30) $$

*Associated $E_1$:* since $(\mathbf{M}_A^{E_6})^2 = (\mathbf{M}_B^{E_6})^2 = (\mathbf{M}_C^{E_6})^2 = 1$ we define three independent translations as

$$ \begin{aligned} g_1^{E_6} &\equiv (g_A^{E_6})^2 : \boldsymbol{x} \to \boldsymbol{x} + \mathbf{T}_1^{E_6}, \\ g_2^{E_6} &\equiv (g_B^{E_6})^2 : \boldsymbol{x} \to \boldsymbol{x} + \mathbf{T}_2^{E_6}, \\ g_3^{E_6} &\equiv (g_C^{E_6})^2 : \boldsymbol{x} \to \boldsymbol{x} + \mathbf{T}_3^{E_6}, \end{aligned} \quad (3.31) $$

for

$$ \mathbf{T}_1^{E_6} = \begin{pmatrix} 2L_{Ax} \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{T}_2^{E_6} = \begin{pmatrix} 0 \\ 2L_{By} \\ 0 \end{pmatrix}, \quad \mathbf{T}_3^{E_6} = \begin{pmatrix} 0 \\ 0 \\ 2L_{Cz} \end{pmatrix}. \quad (3.32) $$

*Volume:*

$$ V_{E_6} = \frac{1}{4} |(\mathbf{T}_1^{E_6} \times \mathbf{T}_2^{E_6}) \cdot \mathbf{T}_3^{E_6}| = 2|L_{Ax}L_{By}L_{Cz}|. \quad (3.33) $$

*Tilts versus origin position:* similar to $E_2$, a shift of the origin affects the tilts of the translation vectors. Here since there are rotations around all three axes, a shift of origin will affect all three of the translation vectors. In this case, $x_0^{E_6}$ can replace $L_{Bx}$, $L_{Cy}$, and $L_{Az}$, or equivalently $r_x$, $r_y$, and $r_z$. This leads to the translation vectors with respect to the special origin given by

$$ \mathbf{T}_A^{E_6} = \begin{pmatrix} L_{Ax} \\ L_{By} \\ 0 \end{pmatrix}, \quad \mathbf{T}_B^{E_6} = \begin{pmatrix} 0 \\ L_{By} \\ L_{Cz} \end{pmatrix}, \quad \mathbf{T}_C^{E_6} = \begin{pmatrix} L_{Ax} \\ 0 \\ L_{Cz} \end{pmatrix}, \quad \text{[special origin]} \quad (3.34) $$

at the expense of the observer no longer being at the origin.
Real parameters (6 independent): there are 6 independent parameters required to fully define $E_6$. As in $E_2$, some are redundant with shifting the origin. Thus we have:

• $L_{Ax}$, $L_{By}$, and $L_{Cz}$ are intrinsic parameters of the manifold;

• $L_{Bx}$, $L_{Cy}$, and $L_{Az}$, or equivalently $r_x$, $r_y$, and $r_z$, can be traded for $x_0^{E_6}$;

• the standard (special origin, i.e., “untilted”) form is $0 = L_{Bx} = L_{Cy} = L_{Az}$, or equivalently $1/2 = r_x = r_y = r_z$.

Parameter ranges: we want to ensure that we do not double-count parameter choices that appear different but actually generate the same lattice of clones. Although most of the rotational freedom was used to set the three orthogonal rotation axes as the coordinate axes, there remains the freedom to perform a half turn (rotation by $\pi$) about any two of the axes. With this freedom, we can use a rotation by $\pi$ around the $y$-axis to always have $0 < L_{Ax}$ and a rotation by $\pi$ around the $x$-axis to always have $0 < L_{By}$. Further, we have the freedom to order the axes by the lengths of the associated $E_1$ vectors. Similar to $E_1$, we therefore require:<fnref n="5" />

1. $0 < L_{Ax}$, $0 < L_{By}$, and $0 \neq L_{Cz}$, i.e., choice of orientation;

2. $0 < |\mathbf{T}_1^{E_6}|/|\mathbf{T}_3^{E_6}| \leq |\mathbf{T}_1^{E_6}|/|\mathbf{T}_2^{E_6}| \leq 1$, i.e., choice of ordering;

3. $|\mathbf{T}_A^{E_6} \cdot \mathbf{T}_2^{E_6}| \leq \frac{1}{2}|\mathbf{T}_2^{E_6}|^2$, i.e., $\mathbf{T}_A^{E_6}$ cannot be shortened by adding or subtracting $\mathbf{T}_2^{E_6}$;

4. $|\mathbf{T}_A^{E_6} \cdot \mathbf{T}_3^{E_6}| \leq \frac{1}{2}|\mathbf{T}_3^{E_6}|^2$, i.e., $\mathbf{T}_A^{E_6}$ cannot be shortened by adding or subtracting $\mathbf{T}_3^{E_6}$;

5. $|\mathbf{T}_B^{E_6} \cdot \mathbf{T}_1^{E_6}| \leq \frac{1}{2}|\mathbf{T}_1^{E_6}|^2$, i.e., $\mathbf{T}_B^{E_6}$ cannot be shortened by adding or subtracting $\mathbf{T}_1^{E_6}$;

6. $|\mathbf{T}_B^{E_6} \cdot \mathbf{T}_3^{E_6}| \leq \frac{1}{2}|\mathbf{T}_3^{E_6}|^2$, i.e., $\mathbf{T}_B^{E_6}$ cannot be shortened by adding or subtracting $\mathbf{T}_3^{E_6}$;

7. $|\mathbf{T}_C^{E_6} \cdot \mathbf{T}_1^{E_6}| \leq \frac{1}{2}|\mathbf{T}_1^{E_6}|^2$, i.e., $\mathbf{T}_C^{E_6}$ cannot be shortened by adding or subtracting $\mathbf{T}_1^{E_6}$;

8. $|\mathbf{T}_C^{E_6} \cdot \mathbf{T}_2^{E_6}| \leq \frac{1}{2}|\mathbf{T}_2^{E_6}|^2$, i.e., $\mathbf{T}_C^{E_6}$ cannot be shortened by adding or subtracting $\mathbf{T}_2^{E_6}$.

In terms of the parameters, a direct application of these conditions gives:

1. $0 < L_{Ax} \leq L_{By}$ and $0 \neq L_{Cz}$ (including the next condition);

2. $0 < L_{Ax}/|L_{Cz}| \leq L_{Ax}/L_{By} \leq 1$;

3. $-2L_{By} < L_{Cy} \leq 0$;

4. $|L_{Az}| \leq |L_{Cz}|$;

5. $|L_{Bx}| \leq L_{Ax}$;

6. $-2|L_{Cz}| < L_{Az} < 2|L_{Cz}|$;

7. $-2L_{Ax} < L_{Bx} \leq 0$;

8. $|L_{Cy}| \leq L_{By}$.

These conditions are not all independent and can be written more compactly as:

1. $0 < L_{Ax} \leq L_{By} \leq |L_{Cz}|$;

2. $-L_{Ax} < L_{Bx} \leq 0$;

<footnote n="5">⁵Note that because $\mathbf{T}_1^{E_6}$, $\mathbf{T}_2^{E_6}$, and $\mathbf{T}_3^{E_6}$ are orthogonal to one another, we do not need to require that $\mathbf{T}_a^{E_6}$ ($a = A, B, C$) cannot be shortened by adding/subtracting linear combinations of $\mathbf{T}_i^{E_6}$ ($i = 1, 2, 3$).</footnote>
3. $-L_{By} < L_{Cy} \le 0$;

4. $|L_{Az}| \le |L_{Cz}|$.

Finally, in terms of the alternative form parameters these conditions (2)–(4) are equivalent to

$$ \{r_x, r_y, r_z\} \in (-1/2, 1/2]. \qquad (3.35) $$

## 3.7 $E_{11}$: Chimney space

The chimney space $E_{11}$ is the basis for all the Euclidean manifolds with two compact dimensions (i.e., compact cross-sections), in much the same way as $E_1$ is for all the compact manifolds. It can be thought of $E_1$ with one non-compact dimension. All of $E_{12}-E_{15}$ are roots of $E_{11}$.

**Properties:** as listed in table 1, this manifold has compact cross-sections and is orientable, homogeneous, and anisotropic.

**Generators:** since $E_{11}$ only has two compact dimensions, it is described by two generators. In general (see appendix A.7) the $z$ direction is chosen to be non-compact and the generators of $E_{11}$ are given by

$$ \mathbf{M}_A^{E_{11}} = 1, \quad \text{with} \\ \mathbf{T}_{A_1}^{E_{11}} = \begin{pmatrix} L_{A_1 x} \\ 0 \\ 0 \end{pmatrix} \equiv \mathbf{T}_1^{E_{11}}, \quad \mathbf{T}_{A_2}^{E_{11}} = \begin{pmatrix} L_{A_2 x} \\ L_{A_2 y} \\ 0 \end{pmatrix} \equiv \mathbf{T}_2^{E_{11}}, \qquad (3.36) $$

or alternately

$$ \mathbf{T}_{A_1}^{E_{11}} = L_{A_1} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{T}_{A_2}^{E_{11}} = L_{A_2} \begin{pmatrix} \cos \alpha \\ \sin \alpha \\ 0 \end{pmatrix}. $$

Similar to $E_1$, it is often convenient to simplify notation by dropping the $A$ label and instead use

$$ \begin{aligned} L_{iw} &\equiv L_{A_i w}, && \text{for } i \in \{1, 2\}, w \in \{x, y, z\}; \\ L_i &\equiv L_{A_i}, && \text{for } i \in \{1, 2\}. \end{aligned} \qquad (3.37) $$

**Cross-sectional area:** since the chimney spaces have two compact dimensions their volumes are infinite, but their cross-sections perpendicular to the non-compact direction are finite:

$$ A_{E_{11}} = |\mathbf{T}_1^{E_{11}} \times \mathbf{T}_2^{E_{11}}| = L_1 L_2 |\sin \alpha|. \qquad (3.38) $$

**Origin:** since $\mathbf{M}_A^{E_{11}} = 1$ the manifold is homogeneous and the lattice of an observer is independent of the location of the observer.

**Real parameters (3 independent):** there are 3 independent parameters to fully define $E_{11}$. Since $E_{11}$ is homogeneous they are all required, none can be traded for shifts of the origin.
*Parameter ranges:* we want to ensure that we do not double-count parameter choices that appear different but actually generate the same lattice of clones. The constraints on the parameter ranges are similar to those in $E_1$, though simplified due to $E_{11}$ having one less compact dimension. The two translation vectors $\mathcal{T}_{A_j}^{E_{11}}$ can be used to define the *xy*-plane. They can be ordered such that $|\mathcal{T}_{A_1}^{E_{11}}| \le |\mathcal{T}_{A_2}^{E_{11}}|$. Finally, we can rotate around the *x*-axis to always choose $0 < L_{A_{2y}}$. With this we have:

1. $0 < L_{A_1 x} = L_{A_1}$ and $0 < L_{A_{2y}}$, i.e., choice of orientation;

2. $0 \le |\mathcal{T}_{A_1}^{E_{11}}|/|\mathcal{T}_{A_2}^{E_{11}}| \le 1$, i.e., choice of ordering;

3. $|\mathcal{T}_{A_2}^{E_{11}} \cdot \mathcal{T}_{A_1}^{E_{11}}| \le \frac{1}{2}|\mathcal{T}_{A_1}^{E_{11}}|^2$, i.e., $\mathcal{T}_{A_2}^{E_{11}}$ cannot be shortened by adding or subtracting $\mathcal{T}_{A_1}^{E_{11}}$.

In terms of the parameters the conditions become:

1. $0 < \alpha \le \pi/2$ so that $0 < \sin \alpha \le 1$;

2. $0 < L_1/L_2 \le 1$;

3. $|\cos \alpha| \le \frac{1}{2}L_1/L_2$.

*Convenient fundamental domain:* a convenient choice of FD in the *xy*-plane is an infinite “cylinder” with a parallelogram cross-section in any constant-*z* plane, the vertices of which are any base point and three clones.

There are two convenient choices of base point:

(A) Origin-centered FD.

• Base point: $\boldsymbol{x}_A \equiv \frac{1}{2}(-\mathcal{T}_{A_1}^{E_{11}} - \mathcal{T}_{A_2}^{E_{11}})$;

• Three other corners of the face:

$$- \boldsymbol{x}_B = g_{A_1}^{E_{11}} \boldsymbol{x}_A = \frac{1}{2} (\mathcal{T}_{A_1}^{E_{11}} - \mathcal{T}_{A_2}^{E_{11}});$$

$$- \boldsymbol{x}_C = g_{A_2}^{E_{11}} g_{A_1}^{E_{11}} \boldsymbol{x}_A = \frac{1}{2} (\mathcal{T}_{A_1}^{E_{11}} + \mathcal{T}_{A_2}^{E_{11}});$$

$$- \boldsymbol{x}_D = g_{A_2}^{E_{11}} \boldsymbol{x}_A = \frac{1}{2} (-\mathcal{T}_{A_1}^{E_{11}} + \mathcal{T}_{A_2}^{E_{11}}).$$

(B) Origin-rooted FD.

• Base point: $\boldsymbol{x}_A \equiv \mathbf{0}$;

• Three other corners of the face:

$$- \boldsymbol{x}_B = g_{A_1}^{E_{11}} \boldsymbol{x}_A = \mathcal{T}_{A_1}^{E_{11}};$$

$$- \boldsymbol{x}_C = g_{A_2}^{E_{11}} g_{A_1}^{E_{11}} \boldsymbol{x}_A = \mathcal{T}_{A_1}^{E_{11}} + \mathcal{T}_{A_2}^{E_{11}};$$

$$- \boldsymbol{x}_D = g_{A_2}^{E_{11}} \boldsymbol{x}_A = \mathcal{T}_{A_2}^{E_{11}}.$$

## 3.8 $E_{12}$: Chimney space with half turn

The chimney space with half turn is a root of $E_{11}$ and can be thought of as $E_2$ with one non-compact dimension.

*Properties:* as listed in table 1, this manifold has compact cross-sections and is orientable, inhomogeneous, and anisotropic.
**Generators:** similar to $E_{11}$, there are two generators, and similar to $E_2$, one of the matrices is a rotation by $\pi$. Conventionally this rotation is chosen to be around the y-axis (cf. ref. [25]). Here we instead choose the rotation to be around the z-axis, as is done in $E_2$. This makes it clear that $E_{12}$ is the limit of $E_2$ with $|T_{A_2}^{E_2}| \to \infty$. In general (see appendix A.8), the generators of $E_{12}$ can be written as

$$ \mathbf{M}_A^{E_{12}} = \mathbf{1}, \quad \mathbf{M}_B^{E_{12}} = \mathbf{R}_z(\pi) = \operatorname{diag}(-1, -1, 1), \quad \text{with} \\ \mathbf{T}_A^{E_{12}} = \begin{pmatrix} L_{Ax} \\ 0 \\ 0 \end{pmatrix} \equiv \mathbf{T}_1^{E_{12}}, \quad \mathbf{T}_B^{E_{12}} = \begin{pmatrix} L_{Bx} \\ L_{By} \\ L_{Bz} \end{pmatrix}, \qquad (3.39) $$

or alternately,

$$ \mathbf{T}_A^{E_{12}} = L_A \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \mathbf{T}_B^{E_{12}} = L_B \begin{pmatrix} \cos \beta \cos \gamma \\ \cos \beta \sin \gamma \\ \sin \beta \end{pmatrix}. \qquad (3.40) $$

**Associated $E_{11}$:** in addition to $T_1^{E_{12}}$ defined above, a second independent translation is

$$ (g_B^{E_{12}})^2 : \boldsymbol{x} \to \boldsymbol{x} + \mathbf{T}_2^{E_{12}}, \qquad (3.41) $$

for

$$ \mathbf{T}_2^{E_{12}} \equiv \begin{pmatrix} 0 \\ 0 \\ 2L_{Bz} \end{pmatrix} = 2L_B \begin{pmatrix} 0 \\ 0 \\ \sin \beta \end{pmatrix}. \qquad (3.42) $$

**Cross-sectional area:**

$$ A_{E_{12}} = \frac{1}{2} |\mathbf{T}_1^{E_{12}} \times \mathbf{T}_2^{E_{12}}| = L_A L_B |\sin \beta|. \qquad (3.43) $$

**Tilts versus origin position:** a shift of origin will change $L_{Bx}$ and $L_{By}$ (the two components of $T_B^{E_{12}}$ perpendicular to the axis of rotation) but will not affect $L_{Bz}$. By special choice of origin on the axis of rotation, the tilt can be traded for a shift and we can choose $T_B^{E_{12}} = L_B \hat{e}_z$ at the expense of the observer no longer being at the origin.

**Real parameters (4 independent):** there are 4 independent parameters required to fully define $E_{12}$ with some being redundant with shifting the origin. Thus we have:

* $L_A$ and $L_A/L_B$ are intrinsic parameters of the manifold;

* $L_{Bx}$ and $L_{By}$, or equivalently $\beta$ and $\gamma$, can be traded for $x_{0x}^{E_{12}}$ and $x_{0y}^{E_{12}}$;

* the standard (special origin, i.e., “untilted”) form is $0 = L_{Bx} = L_{By}$, or equivalently, $\beta = \pi/2$ (with $\gamma$ irrelevant).

**Parameter ranges:** we want to ensure that we do not double-count parameter choices that appear different but actually generate the same lattice of clones. Similar to $E_{11}$, we therefore require:

1. $0 < L_A$ and $0 < L_{Bz}$, i.e., choice of orientation;

2. the lengths of $T_A^{E_{12}}$ and $T_B^{E_{12}}$ are unconstrained;

3. $|T_B^{E_{12}} \cdot T_A^{E_{12}}| \le \frac{1}{2}(T_A^{E_{12}})^2$, i.e., $T_B^{E_{12}}$ cannot be shortened by adding or subtracting $T_A^{E_{12}}$.
In terms of the parameters, the necessary conditions become:

1. $0 < \beta \le \pi/2$ so that $0 < \cos \beta \le 1$;

3. $|L_{Bx}| \le \frac{1}{2}L_{Ax}$, or equivalently, $\cos \beta |\cos \gamma| \le \frac{1}{2}L_A/L_B$.

## 3.9 $E_{16}$: slab space including rotation

The slab space $E_{16}$ is the basis for all Euclidean three-manifolds with one compact dimension (i.e., compact lengths), in much the same way as $E_1$ and $E_{11}$ are for all compact and two compact dimensions, respectively. The possibility of having a corkscrew (as opposed to a pure translation) in the slab space appears to be new, at least in the cosmology literature. While topologically the corkscrew is continuously deformable to the unrotated slab space, physically the corkscrew leads to a distinguishable pattern of clones. Due to this, we split the description of $E_{16}$ into two cases: $E_{16}^{(h)}$ and $E_{16}^{(i)}$.

### 3.9.1 $E_{16}^{(h)}$: conventional unrotated slab space

The conventional definition of $E_{16}$ only includes a translation. Here we call this choice $E_{16}^{(h)}$, because the space is homogeneous.

**Properties:** as listed in table 1, this manifold has a compact length and is orientable, homogeneous, and anisotropic.

**Generators:** in general, since $E_{16}$ has one compact dimension it is described by one generator, which we may take to be a translation in the $z$ direction (see appendix A.9.1), so the generator of $E_{16}^{(h)}$ is

$$ M_A^{E_{16}^{(h)}} = 1, \quad \text{with} \quad T_A^{E_{16}^{(h)}} = L \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}. \qquad (3.44) $$

Even though there is only one generator, since this generator is a pure translation, we follow the convention of using $A$ to label it.

**Length:** since the slab spaces have only one compact dimension their volumes and cross-sectional areas are infinite. The shortest path length around the manifold at any point is $L$.

**Origin:** since $M_A^{E_{16}^{(h)}} = 1$ the manifold is homogeneous and the lattice of an observer is independent of the location of the observer.

**Real parameters (1 independent):** there is 1 independent parameter required to fully define $E_{16}^{(h)}$: the length of the compact dimension.

**Parameter ranges:** we want to ensure that we do not double-count parameter choices that appear different but actually generate the same lattice of clones. In this case, we can always choose $0 < L$ through the orientation of the coordinate axes.
### 3.9.2 $E_{16}^{(i)}$: general rotated slab space

The orientable slab space also allows for a corkscrew motion. Physically this corkscrew is distinguishable and must be treated as a separate case. As discussed in appendix A.9.2, the rotation angle must be a rational multiple of $2\pi$ in order for the eigenmodes of the Laplacian not to have azimuthal symmetry around the corkscrew axis. Such a symmetry would exclude them as a basis for general smooth functions on the manifold.

**Properties:** due to the corkscrew motion this differs from $E_{16}^{(h)}$ in that it is inhomogeneous. As listed in table 1, this manifold has a compact length and is orientable, inhomogeneous, and anisotropic.

**Generators:** similar to $E_{16}^{(h)}$, there is one generator. In general the generator of $E_{16}^{(i)}$ (see appendix A.9.2) can be written as

$$ \mathbf{M}_{B}^{E_{16}^{(i)}} = \mathbf{R}_{\hat{z}}(2\pi p/q) = \begin{pmatrix} \cos(2\pi p/q) & -\sin(2\pi p/q) & 0 \\ \sin(2\pi p/q) & \cos(2\pi p/q) & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad \text{with} \\ T_{B}^{E_{16}^{(i)}} = \begin{pmatrix} L_{x} \\ 0 \\ L_{z} \end{pmatrix} = L \begin{pmatrix} \cos \beta \\ 0 \\ \sin \beta \end{pmatrix}, \qquad (3.45) $$

and $p \in \mathbb{Z}^{\neq 0}$, $q \in \mathbb{Z}^{> 0}$, and $|p|$ and $q$ relatively prime. As in $E_{16}^{(h)}$, here since the generator is a rotation we use $B$ to label it.

**Associated $E_{16}^{(h)}$:** a pure translation can be defined for $E_{16}^{(i)}$ as

$$ g_{1}^{E_{16}^{(i)}} \equiv (g_{B}^{E_{16}^{(i)}})^{q} : \boldsymbol{x} \rightarrow \boldsymbol{x} + \boldsymbol{T}_{1}^{E_{16}^{(i)}}, \quad \text{for} \quad \boldsymbol{T}_{1}^{E_{16}^{(i)}} \equiv \begin{pmatrix} 0 \\ 0 \\ qL_{z} \end{pmatrix} = qL \sin \beta \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}. \qquad (3.46) $$

**Length:** the length of the associated $E_{16}^{(h)}$ is

$$ L_{E_{16}^{(i)}} = qL|\sin \beta|. \qquad (3.47) $$

**Tilts versus origin position:** a shift of origin will change $L_x$ and $L_y$. Rotational freedom can always be used to restore $L_y = 0$. By special choice of origin on the axis of rotation, the tilt can be traded for a shift and we can choose $T_B^{E_{16}^{(i)}} = L\hat{e}_z$ at the expense of the observer no longer being at the origin.

**Real parameters (2 independent):** there are 2 independent parameters required to fully define $E_{16}^{(i)}$ with 1 parameter interchangeable with a shift of origin. Thus we have:

* $L$ is an intrinsic parameter of the manifold;

* $L_x$, or equivalently $\beta$, can be traded for $x_{0x}^{E_{16}^{(i)}}$;
* $L_y = 0$ is not a parameter, even though it can be traded for $x_{0y}^{E_{16}^{(i)}}$; it is redundant with the orientation of the coordinate system, and thus the rotation of an observer's coordinate system, about the topology rotation axis ($\hat{e}_z$);

* the standard (special origin, i.e., “untilted”) form is $0 = L_x$, or equivalently $\beta = \pi/2$.

*Parameter ranges:* we want to ensure that we do not double-count parameter choices that appear different but actually generate the same lattice of clones. Similar to $E_{16}^{(h)}$, we can always require $0 < L_x$ and $0 < L_z$, or equivalently $0 < \beta \le \pi/2$, through orientation of the coordinate system.

### 3.10 $E_{18}$: the covering space of $E^3$

Three-dimensional Euclidean space $E_{18}$ is the covering space of the $E^3$ geometry. It is infinite in all directions and has no generators. As listed in table 1, it has no compact dimensions and is orientable, homogeneous, and the only Euclidean topology that is isotropic.

## 4 Eigenmodes of the scalar Laplacian and correlation matrices

A key ingredient of cosmological perturbation theory is the set of the scalar (and tensor) eigenmodes of the Laplacian. Characteristically, it is the amplitudes of these modes for which theories give statistical predictions [81, 82].

In this section, we present the scalar eigenmodes for the orientable Euclidean manifolds in their full generality. While such eigenmodes have been presented before [25, 59–61, 81], it has been in a context where the full topology parameter space has not been included, even when its existence has been hinted at. Note that we are not faithful to the notational conventions of those works, so any comparisons should be made carefully.

In the covering space, $E_{18}$, the eigenmodes of the Laplacian are<fnref n="6" />

$$ \Upsilon_k^{E_{18}}(\boldsymbol{x}) = e^{i\boldsymbol{k}\cdot(\boldsymbol{x}-\boldsymbol{x}_0)}. \qquad (4.1) $$

Here $\boldsymbol{x}_0$ is the position of an arbitrary origin relative to the observer's coordinate system<fnref n="7" /> and $\boldsymbol{k} = (k_x, k_y, k_z)^T$, referred to as the wavevector, is any triplet of real numbers with its magnitude $k$ referred to as the wavenumber. Since

$$ \nabla^2 \Upsilon_k^{E_{18}}(\boldsymbol{x}) = -|\boldsymbol{k}|^2 \Upsilon_k^{E_{18}}(\boldsymbol{x}), \qquad (4.2) $$

the eigenvalue associated with $\Upsilon_k^{E_{18}}$ is $-|\boldsymbol{k}|^2 \equiv -k^2$. It can assume any non-positive real value.

In standard inflationary cosmological theory, the adiabatic curvature perturbation field $\delta^R$ is the sum of the eigenmodes $\Upsilon_k^{E_{18}}(\boldsymbol{x})$ with amplitudes $\delta^R(\boldsymbol{k})$ that are described by Gaussian random variables of zero mean and dimensionless power spectrum $P^R(k)$.<fnref n="8" /> We

<footnote n="6">⁶There are many conventions for normalizing the eigenmodes. Here we choose not to include any additional factors and will discuss the implications of this for each manifold below.</footnote>

<footnote n="7">⁷The inclusion of $\boldsymbol{x}_0$ has no particular role for the covering space, for $E_1$, for $E_{11}$, or for $E_{16}^{(h)}$, but is crucial for $E_2-E_6$, $E_{12}$, and $E_{16}^{(i)}$.</footnote>

<footnote n="8">⁸Some small amount of non-Gaussianity is often predicted, but we reserve such considerations in a topological context for future work.</footnote>
can write the resulting three-dimensional scalar field as

$$ \delta^R(\boldsymbol{x}) = \int \frac{d^3k}{(2\pi)^3} \delta^R(\boldsymbol{k}) \Upsilon_k^{E_{1s}}(\boldsymbol{x}). \qquad (4.3) $$

However, we will be interested in other scalar fields $\delta^X$ that are linearly related to $\delta^R$ by a transfer function that we should write as $\Delta^{RX}$, but we will drop the $\mathcal{R}$. The expectation value of any pair of $\delta^X(\boldsymbol{k})$ is

$$ C_{kk'}^{E_{1s};XY} = \langle \delta^X(\boldsymbol{k}) \delta^{Y*}(\boldsymbol{k}') \rangle = (2\pi)^3 \frac{2\pi^2}{k^3} \mathcal{P}^R(k) \Delta^X(k) \Delta^{Y*}(k') \delta^{(D)}(\boldsymbol{k} - \boldsymbol{k}'), \quad (4.4) $$

where $\delta^{(D)}(\boldsymbol{k} - \boldsymbol{k}')$ is the three-dimensional Dirac delta function and we have assumed that the transfer function depends only on the magnitude of $\boldsymbol{k}$.<fnref n="9" /> For the adiabatic curvature $\delta^R$, the primordial power spectrum is often written as<fnref n="10" />

$$ \mathcal{P}^R(k) = A_s \left(\frac{k}{k_*}\right)^{n_s-1}, \qquad (4.5) $$

with the scalar amplitude $A_s$ defined at the fiducial wavenumber $k_*$, and the scalar spectral tilt $n_s$. We assume throughout that $\mathcal{P}^R$ is the same function for $E_1-E_{18}$, as might be expected to result, for example, from an epoch of inflation, and so do not add a topology label to $\mathcal{P}^R$.

Since Euclidean geometry has both translational and rotational isometry, there are other natural bases of the eigenmodes of the Laplacian. Given the nature of cosmological observations, in particular those of the cosmic microwave background, it is more convenient to work in spherical coordinates $(r, \theta, \phi)$ with the plane waves expanded in terms of spherical harmonics as

$$ e^{ik \cdot (\boldsymbol{x} - \boldsymbol{x}_0)} = 4\pi e^{-ik \cdot \boldsymbol{x}_0} \sum_{\ell m} i^\ell j_\ell(kr) Y_{\ell m}^*(\hat{\boldsymbol{k}}) Y_{\ell m}(\theta, \phi), \qquad (4.6) $$

where $j_\ell$ are the spherical Bessel functions and $Y_{\ell m}$ are the (scalar) spherical harmonics. This allows us to always expand the eigenmode as

$$ \Upsilon_k^{E_i}(\boldsymbol{x}) = 4\pi \sum_{\ell m} j_\ell(kr) \xi_{k\ell m}^{E_i;\hat{k}} Y_{\ell m}(\theta, \phi). \qquad (4.7) $$

For $E_{18}$

$$ \xi_{k\ell m}^{E_{18};\hat{k}} = e^{-ik \cdot x_0} i^\ell Y_{\ell m}^*(\hat{\boldsymbol{k}}). \qquad (4.8) $$

In certain cases, e.g., CMB fluctuations, observations project the scalar field $\delta^X$ onto the sphere of the sky, integrating along the line of sight with an appropriate transfer function, so

$$ \delta^X(\theta, \phi) = \sum_{\ell m} a_{\ell m}^{E_{18}; X} Y_{\ell m}(\theta, \phi), \qquad (4.9) $$

<footnote n="9">⁹For three-dimensional scalar quantities X and Y the transfer function will typically depend only on the magnitude of $\boldsymbol{k}$. However, it may be useful to consider a more general dependence on $\boldsymbol{k}$, for example when deriving a transfer function of CMB temperature and polarization on the sky. We will continue to write $\Delta^X(k)$.</footnote>

<footnote n="10">¹⁰The normalization by $2\pi^2/k^3$ in (4.4) is a common convention, but not universal. In this convention, $\mathcal{P}^X(k)$ is the contribution to the variance per logarithmic interval of wavenumber: the total variance of $\delta^X$ is $\int d(\ln k) \mathcal{P}^X(k)$. In the large-scale structure literature, the matter power spectrum is usually denoted by the quantity $P(k) = 2\pi^2\mathcal{P}(k)/k^3$.</footnote>
where

$$a_{\ell m}^{E_{18};X} = \frac{4\pi}{(2\pi)^3} \int d^3k \, \delta^R(\mathbf{k}) \xi_{k\ell m}^{E_{18};\hat{k}} \Delta_{\ell}^{X}(k). \quad (4.10)$$

Here $\Delta_{\ell}^{X}(k)$ is the spherical-harmonic transfer function from $\mathcal{R}$ to $X$, and which, relative to $\Delta^{X}(\mathbf{k})$ absorbs the $j_{\ell}(kr)$ that contributed to the integrand of the radial integral.<fnref n="11" />

If, as usual, $X$ represents a real scalar quantity, then the spherical-harmonic coefficients satisfy $a_{\ell m}^* = (-1)^m a_{-\ell m}$ and we only obtain unique physical information about the $a_{\ell m}$ from $m \ge 0$. In this paper, we will be particularly interested in the properties of the CMB temperature fluctuations, i.e., in $X = T$.

It is surprising, but easily proved, that the isotropy of $E_{18}$ means that if the $\delta^X(\mathbf{k})$ are independent Gaussian random variables of zero mean, with variance only dependent upon the magnitude $k$, then the coefficients $a_{\ell m}^X$ are independent with variance only dependent upon $\ell$. This leads to the customary statement of statistical isotropy,

$$C_{\ell m \ell' m'}^{E_{18};XY} \equiv \langle a_{\ell m}^X a_{\ell' m'}^{Y*} \rangle = C_{\ell}^{XY} \delta_{\ell \ell'}^{(K)} \delta_{mm'}^{(K)}, \quad (4.11)$$

where $\delta_{ij}^{(K)}$ is the Kronecker delta.

Non-trivial topological boundary conditions have two important effects on the eigenmodes of the Laplacian:

1. Only certain wavevectors $\mathbf{k}$ are “allowed” by the boundary conditions. For the fully compact topologies $E_1-E_{10}$, the allowed wavevectors form a discrete lattice. We write $\delta_{\mathbf{k}}^X$, not $\delta^X(\mathbf{k})$. Thus the correlator $\langle \delta_{\mathbf{k}}^X \delta_{\mathbf{k}'}^{Y*} \rangle$ contains terms involving $\mathcal{P}^X(k)\delta_{\mathbf{k}\mathbf{k}'}^{(K)}$ (i.e., a Kronecker, rather than Dirac, delta, although there can be a mix of the two for the chimney and slab spaces with a mix of finite and infinite directions).

2. Except for $E_1, E_{11}$, and $E_{16}^{(h)}$, the eigenmodes are not single covering-space eigenmodes but instead linear combinations thereof, with different $\mathbf{k}$ of the same magnitude. This induces extra terms in the correlator coupling $\mathbf{k}$ to the generator's rotations of $\mathbf{k}$ with Kronecker or Dirac deltas.

Each of these effects encodes the violation of statistical isotropy, and each of them breaks the surprising connection presented above between the statistics of $\delta^X(\mathbf{k})$ and the statistics of $a_{\ell m}^X$. Equations (4.4) and (4.11) no longer hold. Instead of $C_{\mathbf{k}\mathbf{k}'}^{E_{18};XY} \equiv \langle \delta^X(\mathbf{k}) \delta^{Y*}(\mathbf{k}') \rangle$ being proportional to a Dirac delta function of $\mathbf{k}$ and $\mathbf{k}'$, it vanishes except for certain allowed $\mathbf{k}$ and generically connects all pairs of allowed $\mathbf{k}$ with correlations of equal magnitude and location-dependent phase. Meanwhile, $\langle a_{\ell m}^X a_{\ell' m'}^{Y*} \rangle$ is also not diagonal:

$$\langle a_{\ell m}^X a_{\ell' m'}^{Y*} \rangle = C_{\ell m \ell' m'}^{XY}. \quad (4.12)$$

Despite the reality condition on the spherical-harmonic coefficients themselves, the quantity $C_{\ell-m\ell'm'}^{XY} = \langle a_{\ell-m}^X a_{\ell'm'}^{Y*} \rangle = (-1)^m \langle a_{\ell m}^{X*} a_{\ell'm'}^{Y*} \rangle$ does contain independent information. Rather, the $C_{\ell m \ell' m'}^{XY}$ matrix is Hermitian in the $(X, \ell m)$, $(Y, \ell' m')$ index sets.

<footnote n="11">¹¹As with $\Delta^X(k)$, $\Delta_{\ell}^{X}(k)$ is typically only a function of the wavevector magnitude $k$, but in more generality we can have $\Delta_{\ell}^{X}(\mathbf{k})$ dependent on the full three-dimensional wavevector.</footnote>
In the subsections below, we present the eigenmodes and eigenspectra of the orientable Euclidean manifolds as functions of their topological parameters in their full generality. Assuming that it is the amplitudes of these eigenmodes that are Gaussian random variables of zero mean and dimensionless power spectrum $P^R(k)$, we present the correlation matrices for Fourier-mode amplitudes $C_{kk'}^{E_i;XY}$ and spherical-harmonic amplitudes $C_{lm'l'm'}^{E_i;XY}$. The important results for each topology are boxed. The generality of the results employs the orientation and other choices described in section 3, but also includes both an arbitrary origin for the definition of the manifold parameters and an arbitrary location for the observer. As discussed in section 3, there are redundancies in these choices. Any comprehensive search over parameters must take care to avoid overweighting some parts of parameter space. In practice it is convenient to make one of two choices when employing the results below, either

1. choose the observer to be at the origin, $\mathbf{x}_0 = \mathbf{0}$, and use the “tilted” parameters of the manifold, or

2. choose the “special” origin for the coordinate system, in which case the manifold parameters are simplified, but some of the components of observer location, $\mathbf{x}_0$, become significant.

## 4.1 General considerations for eigenmodes

In each of the manifolds, the eigenmodes of the scalar Laplacian $\Upsilon_k^{E_i}(\mathbf{x})$ must be invariant under every possible group transformation $G_\alpha \in \Gamma^{E_i}$:

$$ \Upsilon_k^{E_i}(G_\alpha \mathbf{x}) = \Upsilon_k^{E_i}(\mathbf{x}). \qquad (4.13) $$

Formally, the solution is that $\Upsilon_k^{E_i}(\mathbf{x})$ is a simple linear combination of all covering-space eigenmodes related by the group transformations

$$ \Upsilon_k^{E_i}(\mathbf{x}) \propto \sum_{G_\alpha \in \Gamma^{E_i}} e^{i\mathbf{k} \cdot G_\alpha \mathbf{x}}. \qquad (4.14) $$

More practically, we can limit the sum to a small, finite set of group elements $G_\alpha \in \mathcal{G}^{E_i}$,

$$ \Upsilon_k^{E_i}(\mathbf{x}) = \frac{1}{\sqrt{N(\mathcal{G}^{E_i})}} \sum_{G_\alpha \in \mathcal{G}^{E_i}} e^{i\mathbf{k} \cdot G_\alpha \mathbf{x}}, \qquad (4.15) $$

where $N(\mathcal{G}^{E_i})$ is the number of elements in $\mathcal{G}^{E_i}$. $\mathcal{G}^{E_i}$ includes one group element for each of the SO(3) matrices $\mathbf{M}^{(G_\alpha)}$ that appears when we explicitly write the action of the group elements,

$$ G_\alpha : (\mathbf{x} - \mathbf{x}_0) \rightarrow \mathbf{M}^{(G_\alpha)}(\mathbf{x} - \mathbf{x}_0) + \mathbf{v}^{(G_\alpha)}. \qquad (4.16) $$

These $\mathbf{M}^{(G_\alpha)}$ are then just the matrices $\mathbf{M}_a^{E_i}$ that appear in the generators, as described in section 2, plus all non-identical SO(3) matrices that can be built from arbitrary products of those $\mathbf{M}_a^{E_i}$. Below, we will present the $\mathcal{G}^{E_i}$ for each $E_i$.

Equation (4.13) must still be satisfied for every group element $G_\alpha \in \Gamma^{E_i}$. Among those group elements are a subgroup of pure translations, which are all the integer linear combinations of the $\mathbf{T}_j^{E_i}$, i.e., the translations of the associated homogeneous space ($E_1, E_{11}$,
or $E_{16}^{(h)}$) of that manifold. Considering the invariance of $\Upsilon_k^{E_i}(x)$ under the translation by $T_j^{E_i}$, and recognizing that $G^{E_i}$ always includes the identity matrix, we learn that one must have

$$e^{ik \cdot [(x-x_0)+T_j^{E_i}]} = e^{ik \cdot (x-x_0)} \qquad (4.17)$$

or more compactly

$$k \cdot T_j^{E_i} = 2\pi n_j, \quad \text{for } n_j \in \mathbb{Z}. \qquad (4.18)$$

This is exactly the discretization condition that we get with an $E_1$, $E_{11}$, or $E_{16}^{(h)}$. In other words, the eigenmodes of the Laplacian on an $E_i$ manifold are linear combinations of the Fourier modes that are eigenmodes of the associated homogeneous space. For each $E_i$ below, we present those discretization conditions.

Equation (4.15) satisfies the invariance condition (4.13) for all $k$ allowed by (4.18), however in some cases the sum over $G_\alpha \in G^{E_i}$ yields more than one identical term. This occurs when $M^{(G_\alpha)}k = k$ for certain $k$ allowed by (4.18). More specifically, for the manifolds in question, where $M^{(G_\alpha)}$ are matrices $M_a^{E_i}$ representing rotations by $2\pi p_a/q_a$ ($p_a \in \mathbb{Z}^\neq 0$, $q_a \in \mathbb{Z}^> 0$, $|p|$ and $q$ relatively prime) about one of the three coordinate axes, this occurs when $(M_a^{E_i})^{N_a}k = k$ has a solution for $N_a < q_a$. We will consider those cases explicitly for each $E_i$.

## 4.2 $E_1$: 3-torus

The 3-torus is the simplest of the compact Euclidean topologies and will serve as a model for determining the eigenspectrum and eigenmodes of all the Euclidean three-manifolds. In this subsection we determine which of the eigenvalues and eigenmodes of the scalar Laplacian acting on the covering space $E_{18}$ are preserved by the isometries of the topology. We then use that information to present the Fourier space and spherical-harmonic space correlation matrices of any fluctuations that are linearly related to independent Gaussian random fluctuations of the amplitudes of those eigenmodes.

We begin with the covering-space ($E_{18}$) eigenmodes (4.1). Though in general the eigenmodes of $E_i$ can be linear combinations of the $E_{18}$ eigenmodes, the $E_1$ eigenmodes are the subset of the $E_{18}$ eigenmodes that respect the $E_1$ symmetries,

$$\Upsilon_k^{E_1}(g_{A_j}^{E_1}x) = \Upsilon_k^{E_1}(x). \qquad (4.19)$$

This is because all the group elements of $\Gamma^{E_1}$ are pure translations, i.e., $M^{(G_\alpha)} = 1$ for all $G_\alpha \in G^{E_1}$, so $M^{(G_\alpha)}k = k$ trivially.

As discussed above in general (cf. (4.18)), the symmetry condition (4.19) leads to the discretization of the allowed $k$ in $E_1$:

$$\begin{align}
2\pi n_1 &= (k_n)_x L_1, \notag \\
2\pi n_2 &= (k_n)_x L_2 \cos \alpha + (k_n)_y L_2 \sin \alpha, \tag{4.20} \\
2\pi n_3 &= (k_n)_x L_3 \cos \beta \cos \gamma + (k_n)_y L_3 \cos \beta \sin \gamma + (k_n)_z L_3 \sin \beta. \notag
\end{align}$$

Since the wavenumbers are now discretized, they are labeled by integers $n_i \in \mathbb{Z}$ and we denote this explicitly by writing the wavevector as $k_n$ for $n = (n_1, n_2, n_3)$. Here and below
we will use either the $n_i$ or $(\mathbf{k}_n)_i$ labels as convenient for the situation. Inverting these requirements, the components of the wavevectors are

$$ 
\begin{align} 
(\mathbf{k}_n)_x &= \frac{2\pi n_1}{L_1}, \nonumber \\ 
(\mathbf{k}_n)_y &= \frac{2\pi n_2}{L_2 \sin \alpha} - \frac{2\pi n_1 \cos \alpha}{L_1 \sin \alpha}, \tag{4.21} \\ 
(\mathbf{k}_n)_z &= \frac{2\pi n_3}{L_3 \sin \beta} - \frac{2\pi n_2 \cos \beta \sin \gamma}{L_2 \sin \alpha \sin \beta} - \frac{2\pi n_1 \cos \beta(\sin \alpha \cos \gamma - \cos \alpha \sin \gamma)}{L_1 \sin \alpha \sin \beta}. \nonumber 
\end{align} 
$$

Clearly the eigenvalues $k_n = |\mathbf{k}_n|$ are complicated functions of $\mathbf{n}$.

Thus

$$ \Upsilon_{k_n}^{E_1}(\mathbf{x}) = e^{ik_n \cdot (\mathbf{x} - \mathbf{x}_0)}, \quad \text{for } \mathbf{n} \in \mathcal{N}^{E_1}, \tag{4.22} $$

where

$$ \mathcal{N}^{E_1} = \{(n_1, n_2, n_3) | n_i \in \mathbb{Z} \} \setminus (0, 0, 0). \tag{4.23} $$

Following (4.4) the Fourier-mode correlation matrix for $E_1$ is

$$ C_{k_n k_{n'}}^{E_1;XY} = V_{E_1} \frac{2\pi^2}{k_n^3} \mathcal{P}^{\mathcal{R}}(k_n) \Delta^X(k_n) \Delta^{Y*}(k_n) \delta_{k_n k_{n'}}^{(K)}. \tag{4.24} $$

In transitioning from the covering space $E_{18}$ we have replaced $(2\pi)^3 \delta^{(D)}(\mathbf{k}-\mathbf{k}')$ with $V_{E_1} \delta_{k_n k_{n'}}^{(K)}$, where the volume factor $V_{E_1}$ is given by (3.4).

As for $E_{18}$ above, we can project the field $\delta^X$ onto the sky by performing a radial integral with suitable weight function and transfer function, giving

$$ a_{\ell m}^{E_1;X} = \frac{4\pi}{V_{E_1}} \sum_{n \in \mathcal{N}^{E_1}} \delta_{k_n}^{\mathcal{R}} \xi_{k_n \ell m}^{E_1; \hat{k}_n} \Delta_{\ell}^{X}(k_n), \tag{4.25} $$

with

$$ \xi_{k_n \ell m}^{E_1; \hat{k}_n} \equiv e^{-ik_n \cdot x_0} i^\ell Y_{\ell m}^*(\hat{\mathbf{k}}_n). \tag{4.26} $$

Because $\mathcal{N}^{E_1}$ labels only a discrete set of $\mathbf{k}_n$, the integral over $d^3k$ in eq. (4.10) is replaced by a sum over $\mathbf{n} \in \mathcal{N}^{E_1}$.

For the compact topologies $E_i$ with $i \in \{1, \dots, 6\}$, the spherical-harmonic space covariance matrix has the general form<fnref n="12" />

$$ C_{\ell m \ell' m'}^{E_i;XY} = \frac{(4\pi)^2}{V_{E_i}} \sum_{n \in \mathcal{N}^{E_i}} \Delta_{\ell}^{X}(k_n) \Delta_{\ell'}^{Y*}(k_n) \frac{2\pi^2 \mathcal{P}^{\mathcal{R}}(k_n)}{k_n^3} \xi_{k_n \ell m}^{E_i; \hat{k}_n} \xi_{k_n \ell' m'}^{E_i; \hat{k}_{n*}}. \tag{4.27} $$

<footnote n="12">¹²Note that, while $\Delta^Y(\mathbf{k})$ is complex, $\Delta_\ell^Y$ is real for the usual cases of CMB temperature and polarization; nevertheless, we retain the complex conjugate for generic $Y$.</footnote>
## 4.3 $E_2$: half-turn space

The eigenspectrum and eigenmodes of the half-turn space can be determined in a manner analogous to that of the 3-torus. We could begin from the covering space, but it is more expedient to recognize that $E_2$ is $E_1$ with extra symmetries imposed. With this, the eigenspectrum of $E_2$ will be discretized with wavevectors $k_n$ and the eigenfunctions $\Upsilon_{k_n}^{E_2}(x)$ will be linear combinations of $\Upsilon_{k_n}^{E_1}(x)$. For $E_2$, the discretization condition (4.18) from the translation vectors $T_j^{E_2}$ leads to the components of the allowed wavevectors,

$$ 
\begin{align} 
(k_n)_x &= \frac{2\pi n_1}{L_1}, \\ 
(k_n)_y &= \frac{2\pi n_2}{L_2 \sin \alpha} - \frac{2\pi n_1 \cos \alpha}{L_1 \sin \alpha}, \tag{4.28} \\ 
(k_n)_z &= \frac{2\pi n_3}{2L_B \sin \beta}. 
\end{align} 
$$

Unlike in $E_1$, the eigenmodes of $E_2$ can include a linear combination of two $E_1$ eigenmodes. This follows in the application of eq. (4.15) since the condition $(M_B^{E_2})^N k_n = k_n$ has more than one solution for the minimum positive $N$, depending on $k_n$, namely, $N = 1$ for $(k_n)_x = (k_n)_y = 0$ and $N = 2$ otherwise. Written explicitly,

N = 1 eigenmodes: $k_n = (0, 0, (k_n)_z)^T$, i.e., $n = (0, 0, n_3)$, $n_3 \in 2\mathbb{Z}^\neq^{13}$ with

$$ \Upsilon_{k_n}^{E_2}(x) = e^{i k_n \cdot (x - x_0)} = e^{i(k_n)_z(z - z_0)}, \tag{4.29} $$

N = 2 eigenmodes: $((k_n)_x, (k_n)_y) \neq (0, 0)$, i.e., $(n_1, n_2) \neq (0, 0)$, and per eq. (4.15),

$$ \Upsilon_{k_n}^{E_2}(x) = \frac{1}{\sqrt{2}} \left( e^{i k_n \cdot (x - x_0)} + e^{i k_n \cdot (M_B^{E_2}(x - x_0) + T_B^{E_2})} \right). \tag{4.30} $$

The linear combination in the $N = 2$ modes requires some care. Notice that $k_n^T M_B^{E_2} = -(k_n)_x, -(k_n)_y, (k_n)_z$), i.e., $M_B^{E_2}$ maps $(n_1, n_2, n_3) \to (-n_1, -n_2, n_3)$. One implication of this is that summing over $(n_1, n_2, n_3)$ would double-count eigenmodes if all $n_1 \in \mathbb{Z}$ and all $n_2 \in \mathbb{Z}$ were included. Hence, we define two sets of allowed modes, one for $N = 1$ and another for $N = 2$,

$$ 
\begin{align} 
\mathcal{N}_1^{E_2} &= \{(0, 0, n_3) | n_3 \in 2\mathbb{Z}^\neq^0\}, \\ 
\mathcal{N}_2^{E_2} &= \{(n_1, n_2, n_3) | n_1 \in \mathbb{Z}^{>0}, n_2 \in \mathbb{Z}, n_3 \in \mathbb{Z}\} \cup \{(0, n_2, n_3) | n_2 \in \mathbb{Z}^{>0}, n_3 \in \mathbb{Z}\}, \tag{4.31} \\ 
\mathcal{N}^{E_2} &= \mathcal{N}_1^{E_2} \cup \mathcal{N}_2^{E_2}. 
\end{align} 
$$

<footnote n="13">¹³Odd $n_3$ is excluded because when $n_3$ is odd $e^{ik_n \cdot T_B^{E_2}} = -1$, and so $\Upsilon_{(0,0,n_3)}^{E_2}(x)$ does not then satisfy the boundary conditions. Similar conditions apply to the $N = 1$ eigenmodes of other manifolds for similar reasons.</footnote>
With these the Fourier-mode correlation matrix can now be expressed as

$$C_{knkn'}^{E2;XY} = V_{E2} \frac{2\pi^2}{k_n^3} \mathcal{P}^R(k_n) \Delta^X(k_n) \Delta^{Y*}(k_n) e^{i(k_{n'}-k_n)\cdot x_0} \left[ \sum_{\tilde{n} \in \mathcal{N}_1^{E2}} \delta_{kn\tilde{n}}^{(K)} \delta_{k_{n'}\tilde{n}}^{(K)} \right. \\ \left. + \frac{1}{2} \sum_{\tilde{n} \in \mathcal{N}_2^{E2}} \sum_{a=0}^{1} \sum_{b=0}^{1} e^{ik_{\tilde{n}}\cdot(T^{(a)}-T^{(b)})} \delta_{kn}^{(K)} (\mathbf{M}_B^{E2})^T {}^a\mathbf{k}_{\tilde{n}} \delta_{k_{n'}}^{(K)} (\mathbf{M}_B^{E2})^T {}^b\mathbf{k}_{\tilde{n}} \right], \quad (4.32)$$

where $V_{E2}$ is given in (3.10), $T^{(0)} \equiv 0$, and $T^{(1)} \equiv T_B^{E2}$.<fnref n="14" /> Note that $C_{knkn'}^{E2;XY} = 0$ for $|k_{n'}| \neq |k_n|$, so $\mathcal{P}^R(k_n)$, $\Delta^X(k_n)$, and $\Delta^{Y*}(k_n)$ are each a function only of $k_n$.<fnref n="15" />

Another implication of the rotation in $\mathbf{M}_B^{E2}$ comes when representing the eigenmodes in the harmonic basis. Here we will combine modes with the same eigenvalue $k_n$ and orientations $\hat{\mathbf{k}}_n$ and $(\mathbf{M}_B^{E2})^T \hat{\mathbf{k}}_n$. Since the half turn is a rotation around the z-axis by $\pi$ we can use the rotation properties of the spherical harmonics to simplify our expressions. In particular

$$Y_{\ell m}^* ((\mathbf{M}_B^{E2})^T \hat{\mathbf{k}}_n) = e^{im\pi} Y_{\ell m}^* (\hat{\mathbf{k}}_n) = (-1)^m Y_{\ell m}^* (\hat{\mathbf{k}}_n). \quad (4.33)$$

This gives for the eigenmodes in the harmonic basis

$$\begin{align}
\xi_{k_n \ell m}^{E2; \hat{\mathbf{k}}_n} &= e^{-ik_n \cdot x_0} i^\ell Y_{\ell m}^*(\hat{\mathbf{k}}_n), \quad & \text{for } \mathbf{n} \in \mathcal{N}_1^{E2}, \nonumber \\
\xi_{k_n \ell m}^{E2; \hat{\mathbf{k}}_n} &= \frac{1}{\sqrt{2}} i^\ell Y_{\ell m}^*(\hat{\mathbf{k}}_n) \sum_{j=0}^{1} (-1)^{jm} e^{-ik_n \cdot (\mathbf{M}_B^{E2})^j x_0} e^{ik_n \cdot \mathbf{M}_{0j}^{E2} \mathbf{T}_B^{E2}}, \quad & \text{for } \mathbf{n} \in \mathcal{N}_2^{E2}, \tag{4.34}
\end{align}$$

where it will prove useful for many of the topologies to define

$$\mathbf{M}_{00}^{Ei} \equiv 0, \quad \mathbf{M}_{01}^{Ei} \equiv 1, \quad \text{and} \quad \mathbf{M}_{0j}^{Ei} \equiv \sum_{r=0}^{j-1} (\mathbf{M}_B^{Ei})^r \quad \text{for } j > 1. \quad (4.35)$$

Finally, the spherical-harmonic space covariance matrix has the form (4.27).

## 4.4 E3: quarter-turn space

The eigenspectrum and eigenmodes of the quarter-turn space can be determined in a manner analogous to that for $E_2$. For $E_3$, the discretization condition (4.18) from the translation vectors $T_j^{E3}$ leads to the components of the allowed wavevectors,

$$(\mathbf{k}_n)_x = \frac{2\pi n_1}{L_A}, \quad (\mathbf{k}_n)_y = \frac{2\pi n_2}{L_A}, \quad (\mathbf{k}_n)_z = \frac{2\pi n_3}{4L_B \sin \beta}. \quad (4.36)$$

<footnote n="14">¹⁴In eq. (4.32), $\mathbf{k}_n$ and $\mathbf{k}_{n'}$ are wavevectors of the associated $E_1$, as specified above in (4.28). $C_{knkn'}^{E2;XY}$ describes correlations between amplitudes of the plane waves that comprise the eigenmodes of a specific manifold — i.e., of a specific topology, with specific values of its parameters. It is this object that would be used, for example, in creating realizations of initial conditions for large-scale structure simulations. If one was, instead, constructing a likelihood function to compare data with expectations from $E_2$ manifolds, one would need to convolve $C_{knkn'}^{E2;XY}$ with a kernel characterizing the Fourier structure of the survey of interest.</footnote>

<footnote n="15">¹⁵The terms in the square brackets of eq. (4.32) encode the correlations between Fourier modes with wavevectors related by application of $\mathbf{M}_B^{E2}$. Similar terms will appear in the expressions for the correlation matrices of other topologies.</footnote>
As in $E_2$, the eigenmodes of $E_3$ can include linear combinations of $E_1$ eigenmodes since $(\mathbf{M}_B^{E_3})^N \mathbf{k}_n = \mathbf{k}_n$ has more than one solution for the minimum positive $N$, depending on $\mathbf{k}_n$. Here we have

$N = 1$ eigenmodes: $\mathbf{k}_n = (0, 0, (\mathbf{k}_n)_z)^T$, i.e., $\mathbf{n} = (0, 0, n_3)$, $n_3 \in 4\mathbb{Z}^\pm$, with

$$ \Upsilon_{\mathbf{k}_n}^{E_3}(\mathbf{x}) = e^{i\mathbf{k}_n \cdot (\mathbf{x} - \mathbf{x}_0)}, \qquad (4.37) $$

$N = 4$ eigenmodes: $((\mathbf{k}_n)_x, (\mathbf{k}_n)_y) \neq (0, 0)$, i.e., $(n_1, n_2) \neq (0, 0)$, with

$$ \Upsilon_{\mathbf{k}_n}^{E_3}(\mathbf{x}) = \frac{1}{\sqrt{4}} \sum_{j=0}^{3} e^{i\mathbf{k}_n \cdot ((\mathbf{M}_B^{E_3})^j(\mathbf{x} - \mathbf{x}_0) + \mathbf{M}_{0j}^{E_3} \mathbf{T}_B^{E_3})}, \qquad (4.38) $$

and $\mathbf{M}_{0j}^{E_3}$ defined in (4.35).

As in $E_2$, the cyclic properties of $\mathbf{M}_B^{E_3}$ would lead to repeated counting of eigenmodes if all $n_1 \in \mathbb{Z}$ and $n_2 \in \mathbb{Z}$ were included. In this case, under the repeated action of $\mathbf{M}_B^{E_3}$ we have the mappings $(n_1, n_2) \to (n_2, -n_1) \to (-n_1, -n_2) \to (-n_2, n_1)$. To avoid this, we define two sets of allowed modes, now for $N = 1$ and $N = 4$,

$$ \begin{aligned} \mathcal{N}_1^{E_3} &= \{(0, 0, n_3) | n_3 \in 4\mathbb{Z}^\pm\}, \\ \mathcal{N}_4^{E_3} &= \{(n_1, n_2, n_3) | n_1 \in \mathbb{Z}^{\ge 0}, n_2 \in \mathbb{Z}^{> 0}, n_3 \in \mathbb{Z}\}, \\ \mathcal{N}^{E_3} &= \mathcal{N}_1^{E_3} \cup \mathcal{N}_4^{E_3}. \end{aligned} \qquad (4.39) $$

With these the Fourier-mode correlation matrix can now be expressed as

$$ C_{\mathbf{k}_n \mathbf{k}_{n'}}^{E_3;XY} = V_{E_3} \frac{2\pi^2}{k_n^3} \mathcal{P}^R(k_n) \Delta^X(k_n) \Delta^{Y*}(k_n) e^{i(\mathbf{k}_{n'} - \mathbf{k}_n) \cdot \mathbf{x}_0} \left[ \sum_{\tilde{n} \in \mathcal{N}_1^{E_3}} \delta_{\mathbf{k}_n \mathbf{k}_{\tilde{n}}}^{(K)} \delta_{\mathbf{k}_{n'} \mathbf{k}_{\tilde{n}}}^{(K)} \right. \\ \left. + \frac{1}{4} \sum_{\tilde{n} \in \mathcal{N}_4^{E_3}} \sum_{a=0}^3 \sum_{b=0}^3 e^{i\mathbf{k}_{\tilde{n}} \cdot (\mathbf{T}^{(a)} - \mathbf{T}^{(b)})} \delta_{\mathbf{k}_n ([\mathbf{M}_B^{E_3}]^T)^a \mathbf{k}_{\tilde{n}}}^{(K)} \delta_{\mathbf{k}_{n'} ([\mathbf{M}_B^{E_3}]^T)^b \mathbf{k}_{\tilde{n}}}^{(K)} \right], \qquad (4.40) $$

where $V_{E_3}$ is given in (3.15), $\mathbf{T}^{(0)} = \mathbf{0}$, and $\mathbf{T}^{(a)} = \mathbf{M}_{0a}^{E_3} \mathbf{T}_B^{E_3}$ for $a \in \{1, 2, 3\}$.

Also as in $E_2$, we can use the rotation properties of the spherical harmonics along with the fact that $\mathbf{M}_B^{E_3}$ is a rotation around the z-axis by $\pi/2$ to note that

$$ Y_{\ell m}^* ([ (\mathbf{M}_B^{E_3})^T ]^j \hat{\mathbf{k}}_n) = e^{imj\pi/2} Y_{\ell m}^* (\hat{\mathbf{k}}_n) = i^{mj} Y_{\ell m}^* (\hat{\mathbf{k}}_n). \qquad (4.41) $$

This gives for the eigenmodes in the harmonic basis

$$ \begin{aligned} \xi_{k_n \ell m}^{E_3; \hat{\mathbf{k}}_n} &= e^{-i \mathbf{k}_n \cdot \mathbf{x}_0} i^\ell Y_{\ell m}^*(\hat{\mathbf{k}}_n), & \text{for } \mathbf{n} \in \mathcal{N}_1^{E_3}, \\ \xi_{k_n \ell m}^{E_3; \hat{\mathbf{k}}_n} &= \frac{1}{\sqrt{4}} i^\ell Y_{\ell m}^*(\hat{\mathbf{k}}_n) \sum_{j=0}^3 i^{mj} e^{-i \mathbf{k}_n \cdot (\mathbf{M}_B^{E_3})^j \mathbf{x}_0} e^{i \mathbf{k}_n \cdot \mathbf{M}_{0j}^{E_3} \mathbf{T}_B^{E_3}}, & \text{for } \mathbf{n} \in \mathcal{N}_4^{E_3}, \end{aligned} \qquad (4.42) $$

and the spherical-harmonic space covariance matrix has the form (4.27).
## 4.5 $E_4$: third-turn space

The eigenspectrum and eigenmodes of the third-turn space can be determined in a manner analogous to those from above. For $E_4$, the discretization condition (4.18) from the translation vectors $T_j^{E_4}$ leads to the components of the allowed wavevectors,

$$ (\mathbf{k}_n)_x = \frac{2\pi n_1}{L_A}, \quad (\mathbf{k}_n)_y = \frac{2\pi}{\sqrt{3}L_A}(n_1 + 2n_2), \quad (\mathbf{k}_n)_z = \frac{2\pi n_3}{3L_B \sin \beta}. \quad (4.43) $$

As in $E_2$, the eigenmodes of $E_4$ can include linear combinations of $E_1$ eigenmodes since $(\mathbf{M}_B^{E_4})^N \mathbf{k}_n = \mathbf{k}_n$ has more than one solution for the minimum positive $N$, depending on $\mathbf{k}_n$. Here we have

N = 1 eigenmodes: $\mathbf{k}_n = (0, 0, (\mathbf{k}_n)_z)^T$, i.e., $\mathbf{n} = (0, 0, n_3)$, $n_3 \in 3\mathbb{Z}^\pm$, with

$$ \Upsilon_{\mathbf{k}_n}^{E_4}(\mathbf{x}) = e^{i\mathbf{k}_n \cdot (\mathbf{x} - \mathbf{x}_0)}, \quad (4.44) $$

N = 3 eigenmodes: $((\mathbf{k}_n)_x, (\mathbf{k}_n)_y) \neq (0, 0)$, i.e., $(n_1, n_2) \neq (0, 0)$, with

$$ \Upsilon_{\mathbf{k}_n}^{E_4}(\mathbf{x}) = \frac{1}{\sqrt{3}} \sum_{j=0}^{2} e^{i\mathbf{k}_n \cdot ((\mathbf{M}_B^{E_4})^j(\mathbf{x} - \mathbf{x}_0) + \mathbf{M}_{0j}^{E_4}\mathbf{T}_B^{E_4})}, \quad (4.45) $$

and $\mathbf{M}_{0j}^{E_4}$ defined in (4.35).

As in $E_2$, the cyclic properties of $\mathbf{M}_B^{E_4}$ would lead to repeated counting of eigenmodes if all $n_1 \in \mathbb{Z}$ and $n_2 \in \mathbb{Z}$ were included. To avoid this, we define two sets of allowed modes, now for $N=1$ and $N=3$,

$$ \begin{aligned} \mathcal{N}_1^{E_4} &= \{(0, 0, n_3) | n_3 \in 3\mathbb{Z}^\pm\}, \\ \mathcal{N}_3^{E_4} &= \{(n_1, n_2, n_3) | n_1 \in \mathbb{Z}^\pm, n_2 \in \mathbb{Z}, n_1 n_2 \ge 0, n_3 \in \mathbb{Z}\}, \\ \mathcal{N}^{E_4} &= \mathcal{N}_1^{E_4} \cup \mathcal{N}_3^{E_4}. \end{aligned} \quad (4.46) $$

With these the Fourier-mode correlation matrix can now be expressed as

$$ C_{\mathbf{k}_n \mathbf{k}_{n'}}^{E_4;XY} = V_{E_4} \frac{2\pi^2}{k_n^3} P^R(\mathbf{k}_n) \Delta^X(\mathbf{k}_n) \Delta^{Y*}(\mathbf{k}_n) e^{i(\mathbf{k}_{n'} - \mathbf{k}_n) \cdot \mathbf{x}_0} \left[ \sum_{\tilde{n} \in \mathcal{N}_1^{E_4}} \delta_{\mathbf{k}_n \mathbf{k}_{\tilde{n}}}^{(K)} \delta_{\mathbf{k}_{n'} \mathbf{k}_{\tilde{n}}}^{(K)} \right. \\ \left. + \frac{1}{3} \sum_{\tilde{n} \in \mathcal{N}_3^{E_4}} \sum_{a=0}^2 \sum_{b=0}^2 e^{i\mathbf{k}_{\tilde{n}} \cdot (\mathbf{T}^{(a)} - \mathbf{T}^{(b)})} \delta_{\mathbf{k}_n ([([\mathbf{M}_B^{E_4}]^T)^a \mathbf{k}_{\tilde{n}})]}^{(K)} \delta_{\mathbf{k}_{n'} ([([\mathbf{M}_B^{E_4}]^T)^b \mathbf{k}_{\tilde{n}})]}^{(K)} \right], \quad (4.47) $$

where $V_{E_4}$ is given in (3.21), $\mathbf{T}^{(0)} = \mathbf{0}$, and $\mathbf{T}^{(a)} = \mathbf{M}_{0a}^{E_4} \mathbf{T}_B^{E_4}$ for $a \in \{1, 2\}$.

Also as in $E_2$, we can use the rotation properties of the spherical harmonics along with the fact that $\mathbf{M}_B^{E_4}$ is a rotation around the z-axis by $2\pi/3$ to note that

$$ Y_{\ell m}^* ([([\mathbf{M}_B^{E_4}]^T)^j \hat{\mathbf{k}}_n) = e^{imj2\pi/3} Y_{\ell m}^* (\hat{\mathbf{k}}_n). \quad (4.48) $$
This gives for the eigenmodes in the harmonic basis

$$
\begin{align}
\xi_{kn\ell m}^{E_4; \hat{k}_n} &= e^{-ik_n \cdot x_0} i^\ell Y_{\ell m}^*(\hat{k}_n), && \text{for } \mathbf{n} \in \mathcal{N}_1^{E_4}, \nonumber \\
\xi_{kn\ell m}^{E_4; \hat{k}_n} &= \frac{1}{\sqrt{3}} i^\ell Y_{\ell m}^*(\hat{k}_n) \sum_{j=0}^2 e^{imj2\pi/3} e^{-ik_n \cdot (\mathbf{M}_B^{E_4})^j x_0} e^{ik_n \cdot \mathbf{M}_{0j}^{E_4} \mathbf{T}_B^{E_4}}, && \text{for } \mathbf{n} \in \mathcal{N}_3^{E_4}, \tag{4.49}
\end{align}
$$

and the spherical-harmonic space covariance matrix has the form (4.27).

4.6 E5: sixth-turn space

The eigenspectrum and eigenmodes of the sixth-turn space can be determined in a manner
analogous to those from above, in particular it is very similar to E₄. For E₅, the discretization
condition (4.18) from the translation vectors Tⱼ:E₅ leads to the components of the allowed
wavevectors,

$$
(\mathbf{k}_n)_x = \frac{2\pi n_1}{L_A}, \quad (\mathbf{k}_n)_y = \frac{2\pi}{\sqrt{3}L_A}(n_1 + 2n_2), \quad (\mathbf{k}_n)_z = \frac{2\pi n_3}{6L_B \sin \beta}. \quad (4.50)
$$

As in $E_2$, the eigenmodes of $E_5$ can include linear combinations of $E_1$ eigenmodes since $(\mathbf{M}_B^{E_5})^N \mathbf{k}_n = \mathbf{k}_n$ has more than one solution for the minimum positive $N$, depending on $\mathbf{k}_n$. Here we have

$N = 1$ eigenmodes: $\mathbf{k}_n = (0, 0, (\mathbf{k}_n)_z)^T$, i.e., $\mathbf{n} = (0, 0, n_3)$, $n_3 \in 6\mathbb{Z}^{\neq 0}$, with

$$
\Upsilon_{k_n}^{E_5}(\mathbf{x}) = e^{ik_n \cdot (\mathbf{x} - \mathbf{x}_0)}, \quad (4.51)
$$

$N = 6$ eigenmodes: $((\mathbf{k}_n)_x, (\mathbf{k}_n)_y) \neq (0, 0)$, i.e., $(n_1, n_2) \neq (0, 0)$, with

$$
\Upsilon_{k_n}^{E_5}(\mathbf{x}) = \frac{1}{\sqrt{6}} \sum_{j=0}^{5} e^{i \mathbf{k}_n \cdot ((\mathbf{M}_B^{E_5})^j (\mathbf{x} - \mathbf{x}_0) + \mathbf{M}_{0j}^{E_5} \mathbf{T}_B^{E_5})}, \quad (4.52)
$$

and $\mathbf{M}_{0j}^{E_5}$ defined in (4.35).

As in $E_2$, the cyclic properties of $\mathbf{M}_B^{E_5}$ would lead to repeated counting of eigenmodes if all $n_1 \in \mathbb{Z}$ and $n_2 \in \mathbb{Z}$ were included. To avoid this, we define two sets of allowed modes, now for $N=1$ and $N=6$,

$$
\begin{align}
\mathcal{N}_1^{E_5} &= \{(0, 0, n_3) | n_3 \in 6\mathbb{Z}^{\neq 0}\}, \notag \\
\mathcal{N}_6^{E_5} &= \{(n_1, n_2, n_3) | n_1 \in \mathbb{Z}^{>0}, n_2 \in \mathbb{Z}^{\ge 0}, n_3 \in \mathbb{Z}\}, \tag{4.53} \\
\mathcal{N}^{E_5} &= \mathcal{N}_1^{E_5} \cup \mathcal{N}_6^{E_5}. \notag
\end{align}
$$

With these the Fourier-mode correlation matrix can now be expressed as

$$
C_{knk'n}^{E5;XY} = V_{E5} \frac{2\pi^2}{k_n^3} \mathcal{P}^R(k_n) \Delta^X(k_n) \Delta^{Y*}(k_n) e^{i(k_n'-k_n)\cdot x_0} \left[ \sum_{\tilde{n} \in \mathcal{N}_1^{E5}} \delta_{kn\tilde{n}}^{(K)} \delta_{k'n\tilde{n}}^{(K)} \right. \\
\left. + \frac{1}{6} \sum_{\tilde{n} \in \mathcal{N}_6^{E5}} \sum_{a=0}^{5} \sum_{b=0}^{5} e^{ik_{\tilde{n}} \cdot (T^{(a)} - T^{(b)})} \delta_{kn([([M_B^{E5})^T]^a k_{\tilde{n}})}^{(K)} \delta_{k'n([([M_B^{E5})^T]^b k_{\tilde{n}})}^{(K)} \right], \quad (4.54)
$$

where $V_{E_5}$ is given in (3.27), $\mathbf{T}^{(0)} \equiv \mathbf{0}$, and $\mathbf{T}^{(a)} \equiv \mathbf{M}_{0a}^{E_5} \mathbf{T}_B^{E_5}$ for $a \in \{1, \dots, 5\}$.
Also as in $E_2$, we can use the rotation properties of the spherical harmonics along with the fact that $\mathbf{M}_B^{E_5}$ is a rotation around the z-axis by $\pi/3$ to note that

$$Y_{lm}^* ((\mathbf{M}_B^{E_5})^T)^j \hat{\mathbf{k}}_n) = e^{imj\pi/3} Y_{lm}^*(\hat{\mathbf{k}}_n). \quad (4.55)$$

This gives for the eigenmodes in the harmonic basis

$$ \begin{aligned} \xi_{k_n l m}^{E_5; \hat{\mathbf{k}}_n} &= e^{-ik_n \cdot x_0} i^l Y_{lm}^*(\hat{\mathbf{k}}_n), && \text{for } \mathbf{n} \in \mathcal{N}_1^{E_5}, \\ \xi_{k_n l m}^{E_5; \hat{\mathbf{k}}_n} &= \frac{1}{\sqrt{6}} i^l Y_{lm}^*(\hat{\mathbf{k}}_n) \sum_{j=0}^5 e^{imj\pi/3} e^{-ik_n \cdot (\mathbf{M}_B^{E_5})^j x_0} e^{ik_n \cdot \mathbf{M}_{0j}^{E_5} \mathbf{T}_B^{E_5}}, && \text{for } \mathbf{n} \in \mathcal{N}_6^{E_5}, \end{aligned} \quad (4.56) $$

and the spherical-harmonic space covariance matrix has the form (4.27).

## 4.7 $E_6$: Hantzsche-Wendt space

The eigenspectrum and eigenmodes of the Hantzsche-Wendt space can be determined in a manner analogous to those from above. Complications arise from the fact that $E_6$ contains rotations around multiple axes so a more careful discussion is warranted. The discretization condition (4.18) from the translation vectors $\mathbf{T}_j^{E_6}$ is still straightforward and leads to the components of the allowed wavevectors,

$$(\mathbf{k}_n)_x = \frac{2\pi n_1}{2L_{Ax}}, \quad (\mathbf{k}_n)_y = \frac{2\pi n_2}{2L_{By}}, \quad (\mathbf{k}_n)_z = \frac{2\pi n_3}{2L_{Cz}}. \quad (4.57)$$

As in $E_2$, the eigenmodes of $E_6$ can include linear combinations of $E_1$ eigenmodes since $(\mathbf{M}_a^{E_6})^{N_a} \mathbf{k}_n = \mathbf{k}_n$ has more than one solution for the minimum positive $N_a$, depending on $\mathbf{k}_n$, for $a \in \{A, B, C\}$. Since all the rotations are half turns, i.e., all $(\mathbf{M}_a^{E_6})^2 = 1$, there are linear combinations with the $N_a = 1$ and $N_a = 2$. At first glance eq. (4.15) seems to suggest that the eigenmodes with $N_a = 1$ will be a linear combination of four eigenmodes of $E_1$. However, since $\mathbf{M}_A^{E_6} \mathbf{M}_B^{E_6} = \mathbf{M}_C^{E_6}$ (and all permutations of $\{A, B, C\}$) along with the invariance of the eigenmodes under the group action (4.13), a linear combination of only two eigenmodes of $E_1$ is required in this case. Based on this we can choose

$N_A = 1$ eigenmodes: $\mathbf{k}_n = ((\mathbf{k}_n)_x, 0, 0)^T$, i.e., $\mathbf{n} = (n_1, 0, 0)$, $n_1 \in 2\mathbb{Z}^\neq 0$, with

$$ \Upsilon_{k_n}^{E_6}(\mathbf{x}) = \frac{1}{\sqrt{2}} \left[ e^{ik_n \cdot (\mathbf{x} - \mathbf{x}_0)} + e^{-ik_n \cdot (\mathbf{x} - \mathbf{x}_0)} e^{ik_n \cdot \mathbf{T}_B^{E_6}} \right], \quad (4.58) $$

$N_B = 1$ eigenmodes: $\mathbf{k}_n = (0, (\mathbf{k}_n)_y, 0)^T$, i.e., $\mathbf{n} = (0, n_2, 0)$, $n_2 \in 2\mathbb{Z}^\neq 0$, with

$$ \Upsilon_{k_n}^{E_6}(\mathbf{x}) = \frac{1}{\sqrt{2}} \left[ e^{ik_n \cdot (\mathbf{x} - \mathbf{x}_0)} + e^{-ik_n \cdot (\mathbf{x} - \mathbf{x}_0)} e^{ik_n \cdot \mathbf{T}_C^{E_6}} \right], \quad (4.59) $$

$N_C = 1$ eigenmodes: $\mathbf{k}_n = (0, 0, (\mathbf{k}_n)_z)^T$, i.e., $\mathbf{n} = (0, 0, n_3)$, $n_3 \in 2\mathbb{Z}^\neq 0$, with

$$ \Upsilon_{k_n}^{E_6}(\mathbf{x}) = \frac{1}{\sqrt{2}} \left[ e^{ik_n \cdot (\mathbf{x} - \mathbf{x}_0)} + e^{-ik_n \cdot (\mathbf{x} - \mathbf{x}_0)} e^{ik_n \cdot \mathbf{T}_A^{E_6}} \right], \quad (4.60) $$
**N<sub>a</sub> = 2 eigenmodes:** at most one component of **k**<sub>n</sub> zero, i.e., at most one of the n<sub>i</sub> equal to zero, with

$$
\Upsilon_{\mathbf{k}_n}^{E_6}(\mathbf{x}) = \frac{1}{\sqrt{4}} \left[ e^{i\mathbf{k}_n \cdot (\mathbf{x} - \mathbf{x}_0)} + \sum_{a \in \{A, B, C\}} e^{i\mathbf{k}_n \cdot \mathbf{M}_a^{E_6}(\mathbf{x} - \mathbf{x}_0)} e^{i\mathbf{k}_n \cdot \mathbf{T}_a^{E_6}} \right]. \quad (4.61)
$$

The cyclic properties of the $\mathbf{M}_a^{E_6}$ would lead to repeated counting of eigenmodes. To avoid this, we define the sets of allowed modes as

$$
\begin{align}
\mathcal{N}_{1A}^{E_6} &= \{(n_1, 0, 0) | n_1 \in 2\mathbb{Z}^+\}, \nonumber \\
\mathcal{N}_{1B}^{E_6} &= \{(0, n_2, 0) | n_2 \in 2\mathbb{Z}^+\}, \nonumber \\
\mathcal{N}_{1C}^{E_6} &= \{(0, 0, n_3) | n_3 \in 2\mathbb{Z}^+\}, \nonumber \\
\mathcal{N}_2^{E_6} &= \{(n_1, n_2, n_3) | n_1 \in \mathbb{Z}^+, n_2 \in \mathbb{Z}^+, n_3 \in \mathbb{Z}\} \tag{4.62} \\
&\quad \cup \{(0, n_2, n_3) | n_2 \in \mathbb{Z}^+, n_3 \in \mathbb{Z}^+\} \nonumber \\
&\quad \cup \{(n_1, 0, n_3) | n_1 \in \mathbb{Z}^+, n_3 \in \mathbb{Z}^+\}, \nonumber \\
\mathcal{N}^{E_6} &= \mathcal{N}_{1A}^{E_6} \cup \mathcal{N}_{1B}^{E_6} \cup \mathcal{N}_{1C}^{E_6} \cup \mathcal{N}_2^{E_6}. \nonumber
\end{align}
$$

With these the Fourier-mode correlation matrix can now be expressed as

$$
C_{\mathbf{k}_n \mathbf{k}_{n'}}^{E_6;XY} = V_{E_6} \frac{2\pi^2}{k_n^3} \mathcal{P}^{\mathcal{R}}(\mathbf{k}_n) \Delta^X(\mathbf{k}_n) \Delta^{Y*}(\mathbf{k}_n) e^{i(\mathbf{k}_{n'} - \mathbf{k}_n) \cdot \mathbf{x}_0} \\
\times \left[ \frac{1}{2} \sum_{\tilde{n} \in \mathcal{N}_{1A}^{E_6}} \sum_{a,b \in \{0,B\}} e^{i\mathbf{k}_n \cdot (\mathbf{T}^{(a)} - \mathbf{T}^{(b)})} \delta^{(K)}_{\mathbf{k}_n((\mathbf{M}_a^{E_6})^T \mathbf{k}_{\tilde{n}})} \delta^{(K)}_{\mathbf{k}_{n'}((\mathbf{M}_b^{E_6})^T \mathbf{k}_{\tilde{n}})} \right. \\
+ \frac{1}{2} \sum_{\tilde{n} \in \mathcal{N}_{1B}^{E_6}} \sum_{a,b \in \{0,C\}} e^{i\mathbf{k}_n \cdot (\mathbf{T}^{(a)} - \mathbf{T}^{(b)})} \delta^{(K)}_{\mathbf{k}_n((\mathbf{M}_a^{E_6})^T \mathbf{k}_{\tilde{n}})} \delta^{(K)}_{\mathbf{k}_{n'}((\mathbf{M}_b^{E_6})^T \mathbf{k}_{\tilde{n}})} \\
+ \frac{1}{2} \sum_{\tilde{n} \in \mathcal{N}_{1C}^{E_6}} \sum_{a,b \in \{0,A\}} e^{i\mathbf{k}_n \cdot (\mathbf{T}^{(a)} - \mathbf{T}^{(b)})} \delta^{(K)}_{\mathbf{k}_n((\mathbf{M}_a^{E_6})^T \mathbf{k}_{\tilde{n}})} \delta^{(K)}_{\mathbf{k}_{n'}((\mathbf{M}_b^{E_6})^T \mathbf{k}_{\tilde{n}})} \\
\left. + \frac{1}{4} \sum_{\tilde{n} \in \mathcal{N}_2^{E_6}} \sum_{a,b \in \{0,A,B,C\}} e^{i\mathbf{k}_n \cdot (\mathbf{T}^{(a)} - \mathbf{T}^{(b)})} \delta^{(K)}_{\mathbf{k}_n((\mathbf{M}_a^{E_6})^T \mathbf{k}_{\tilde{n}})} \delta^{(K)}_{\mathbf{k}_{n'}((\mathbf{M}_b^{E_6})^T \mathbf{k}_{\tilde{n}})} \right], \quad (4.63)
$$

where $V_{E_6}$ is given in (3.33), $\mathbf{T}^{(0)} = \mathbf{0}$, and $\mathbf{T}^{(a)} = \mathbf{T}_a^{E_6}$ for $a \in \{A, B, C\}$.

The rotation properties of the spherical harmonics can again be used to simplify the eigenmodes of $E_6$ in the harmonic basis. Though there are multiple axes of rotation, the fact that they are half turns allows by direct computation to show that

$$
\begin{align}
Y_{\ell m}^* ((\mathbf{M}_A^{E_6})^T \hat{\mathbf{k}}_n) &= (-1)^{\ell} Y_{-\ell}^* (\hat{\mathbf{k}}_n), \nonumber \\
Y_{\ell m}^* ((\mathbf{M}_B^{E_6})^T \hat{\mathbf{k}}_n) &= (-1)^{\ell+m} Y_{-\ell-m}^* (\hat{\mathbf{k}}_n), \tag{4.64} \\
Y_{\ell m}^* ((\mathbf{M}_C^{E_6})^T \hat{\mathbf{k}}_n) &= (-1)^m Y_{\ell m}^* (\hat{\mathbf{k}}_n). \nonumber
\end{align}
$$
With these the eigenmodes in the harmonic basis can be written in a number of useful forms. We have general expressions patterned after the expressions given for $E_1-E_5$,

$$
\begin{align}
\xi_{k_n l m}^{E_6; \hat{k}_n} &= \frac{1}{\sqrt{2}} i^l Y_{l m}^*(\hat{k}_n) \left[ e^{-i k_n \cdot x_0} + (-1)^l e^{i k_n \cdot x_0} e^{i k_n \cdot T_B^{E_6}} \right], && \text{for } \mathbf{n} \in \mathcal{N}_{1A}^{E_6}, \nonumber \\
\xi_{k_n l m}^{E_6; \hat{k}_n} &= \frac{1}{\sqrt{2}} i^l Y_{l m}^*(\hat{k}_n) \left[ e^{-i k_n \cdot x_0} + (-1)^l e^{i k_n \cdot x_0} e^{i k_n \cdot T_C^{E_6}} \right], && \text{for } \mathbf{n} \in \mathcal{N}_{1B}^{E_6}, \nonumber \\
\xi_{k_n l m}^{E_6; \hat{k}_n} &= \frac{1}{\sqrt{2}} i^l Y_{l m}^*(\hat{k}_n) \left[ e^{-i k_n \cdot x_0} + (-1)^l e^{i k_n \cdot x_0} e^{i k_n \cdot T_A^{E_6}} \right], && \text{for } \mathbf{n} \in \mathcal{N}_{1C}^{E_6}, \nonumber \\
\xi_{k_n l m}^{E_6; \hat{k}_n} &= \frac{1}{\sqrt{4}} i^l \left[ Y_{l m}^*(\hat{k}_n) \left( 1 + (-1)^m e^{-i k_n \cdot \mathbf{M}_C^{E_6} x_0} e^{i k_n \cdot \mathbf{T}_C^{E_6}} \right) + (-1)^l Y_{l-m}^*(\hat{k}_n) \right. \nonumber \\
&\qquad \left. \times \left( e^{-i k_n \cdot \mathbf{M}_A^{E_6} x_0} e^{i k_n \cdot \mathbf{T}_A^{E_6}} + (-1)^m e^{-i k_n \cdot \mathbf{M}_B^{E_6} x_0} e^{i k_n \cdot \mathbf{T}_B^{E_6}} \right) \right], && \text{for } \mathbf{n} \in \mathcal{N}_2^{E_6}, \tag{4.65}
\end{align}
$$

and the spherical-harmonic space covariance matrix has the form (4.27).

## 4.8 $E_{11}$: Chimney space

The chimney space is similar to the 3-torus though with only two compact dimensions and thus a finite “cross-sectional area” (3.38). As in $E_1$, we begin with the covering-space ($E_{18}$) eigenmodes (4.1) and restrict ourselves to the set that respect the $E_{11}$ symmetries,

$$ \Upsilon_k^{E_{11}}(g_{A_j}^{E_{11}}\mathbf{x}) = \Upsilon_k^{E_{11}}(\mathbf{x}). \qquad (4.66) $$

Also as in $E_1$, since all the group elements of $\Gamma^{E_{11}}$ are pure translations, only one $E_{18}$ eigenmode will contribute for each allowed $\mathbf{k}$. The two generators of $E_{11}$ (3.36) lead to two (rather than three as in $E_1$) discretization conditions following (4.18),

$$
\begin{align}
2\pi n_1 &= (\mathbf{k}_n)_x L_1, \tag{4.67} \\
2\pi n_2 &= (\mathbf{k}_n)_x L_2 \cos \alpha + (\mathbf{k}_n)_y L_2 \sin \alpha. \nonumber
\end{align}
$$

These can be inverted to determine the components of the allowed wavevectors,

$$
\begin{align}
(\mathbf{k}_n)_x &= \frac{2\pi n_1}{L_1}, \tag{4.68} \\
(\mathbf{k}_n)_y &= \frac{2\pi n_2}{L_2 \sin \alpha} - \frac{2\pi n_1 \cos \alpha}{L_1 \sin \alpha}, \nonumber
\end{align}
$$

while $(\mathbf{k}_n)_z \equiv k_z$ is unconstrained. We will write $\mathbf{k}_n$ as a shorthand for the wavevector parametrized by $(n_1, n_2; k_z)$, i.e., by the integers $n_1$ and $n_2$ and the real variable $k_z$. The usual (untilted) results are recovered for $\alpha = \pi/2$.

Thus

$$ \Upsilon_{k_n}^{E_{11}}(\mathbf{x}) = e^{i k_n \cdot (\mathbf{x} - \mathbf{x}_0)}, \quad \text{for } \mathbf{n} \in \mathcal{N}^{E_{11}}, \qquad (4.69) $$

where

$$ \mathcal{N}^{E_{11}} = \{(n_1, n_2) | n_i \in \mathbb{Z}\}. \qquad (4.70) $$
Following (4.4) the Fourier-mode correlation matrix for $E_{11}$ is

$$C_{knkn'}^{E_{11};XY} = 2\pi A_{E_{11}} \frac{2\pi^2}{k_n^3} \mathcal{P}^{\mathcal{R}}(k_n) \Delta^X(k_n) \Delta^{Y*}(k_n) \delta_{(kn)_x(kn')_x}^{(K)} \delta_{(kn)_y(kn')_y}^{(K)} \delta^{(D)}(k_z - k_z'). \quad (4.71)$$

In transitioning from $E_1$ (compare eq. (4.71) to eq. (4.24)) we have replaced $V_{E_1}$ with $2\pi A_{E_{11}}$ (the cross-sectional area given by (3.38)) and $\delta_{knkn'}^{(K)}$ by a Kronecker delta for the $x$ and $y$ components of $k_n$ and a Dirac delta function for the $z$ component.

We can project the field $\delta^X$ onto the sky by performing a radial integral with a suitable weight function and transfer function, giving

$$a_{\ell m}^{E_{11};X} = \frac{4\pi}{2\pi A_{E_{11}}} \sum_{(n_1, n_2) \in \mathcal{N}^{E_{11}}} \int_{-\infty}^{\infty} dk_z \delta_{k_n}^{\mathcal{R}} \xi_{k_n \ell m}^{E_{11}; \hat{k}_n} \Delta_{\ell}^{X}(k_n), \quad (4.72)$$

with

$$\xi_{k_n \ell m}^{E_{11}; \hat{k}_n} \equiv e^{-ik_n \cdot x_0} i^\ell Y_{\ell m}^*(\hat{k}_n), \quad (4.73)$$

where a similar transition to that above was performed in starting from eq. (4.25) and replacing the sum over the $(k_n)_z$ with the integral over $k_z$.

For the chimney spaces $E_i$ with $i = 11$ or $i = 12$, the spherical-harmonic space covariance matrix now has the form

$$C_{\ell m \ell' m'}^{E_i;XY} = \frac{(4\pi)^2}{2\pi A_{E_i}} \sum_{(n_1, n_2) \in \mathcal{N}^{E_i}} \int_{-\infty}^{\infty} dk_z \Delta_{\ell}^{X}(k_n) \Delta_{\ell'}^{Y*}(k_n) \frac{2\pi^2 \mathcal{P}^{\mathcal{R}}(k_n)}{k_n^3} \xi_{k_n \ell m}^{E_i; \hat{k}_n} \xi_{k_n \ell' m'}^{E_i; \hat{k}_{n'}}. \quad (4.74)$$

## 4.9 E$_{12}$: Chimney space with half turn

The eigenspectrum and eigenmodes of the chimney space with half turn can be determined in a manner analogous to that of the chimney space in much the same way that these quantities for $E_2$ were determined from $E_1$. For $E_{12}$, the discretization condition (4.18) leads to the components of the allowed wavevectors,

$$(\mathbf{k}_n)_x = \frac{2\pi n_1}{L_{Ax}}, \quad (\mathbf{k}_n)_z = \frac{\pi n_2}{L_{Bz}}, \quad (4.75)$$

with $k_y$ again unconstrained. As in $E_{11}$, we will write $\mathbf{k}_n$ as a shorthand for the wavevector parametrized by the integer array $\mathbf{n} = (n_1, n_2)$ and the real variable $k_y$.

Unlike in $E_{11}$, the eigenmodes of $E_{12}$ can include a linear combination of two $E_{11}$ eigenmodes. Here $(\mathbf{M}_B^{E_{12}})^N \mathbf{k}_n = \mathbf{k}_n$ has two solutions, $N = 1$ and $N = 2$. Written explicitly,

$N = 1$ eigenmodes: $\mathbf{k}_n = (0, 0, (\mathbf{k}_n)_z)^T$, i.e., $\mathbf{n} = (0, n_2)$, $n_2 \in 2\mathbb{Z}^\neq 0$, $k_y = 0$, with

$$\Upsilon_{k_n}^{E_{12}}(\mathbf{x}) = e^{ik_n \cdot (\mathbf{x}-\mathbf{x}_0)} = e^{i\pi n_2(z-z_0)/L_{Bz}}, \quad (4.76)$$

$N = 2$ eigenmodes: $((\mathbf{k}_n)_x, (\mathbf{k}_n)_z) \neq (0, 0)$, i.e., $(n_1, n_2) \neq (0, 0)$, with

$$\Upsilon_{k_n}^{E_{12}}(\mathbf{x}) = \frac{1}{\sqrt{2}} \left( e^{ik_n \cdot (\mathbf{x}-\mathbf{x}_0)} + e^{ik_n \cdot (\mathbf{M}_B^{E_{12}}(\mathbf{x}-\mathbf{x}_0)+\mathbf{T}_B^{E_{12}})} \right). \quad (4.77)$$
Here the two sets of allowed modes are defined by

$$
\begin{align}
\mathcal{N}_1^{E_{12}} &\equiv \{(0, n_2)|n_2 \in 2\mathbb{Z}^\neq 0\}, \nonumber \\
\mathcal{N}_2^{E_{12}} &\equiv \{(n_1, n_2)|n_1 \in \mathbb{Z}^> 0, n_2 \in \mathbb{Z}\}, \nonumber \\
\mathcal{N}^{E_{12}} &\equiv \mathcal{N}_1^{E_{12}} \cup \mathcal{N}_2^{E_{12}}. \tag{4.78}
\end{align}
$$

With these the Fourier-mode correlation matrix can now be expressed as

$$
\begin{equation}
\begin{split}
C_{knk'n}^{E_{12};XY} = {}& 2\pi A_{E_{12}} \frac{2\pi^2}{k_n^3} \mathcal{P}^{\mathcal{R}}(k_n) \Delta^X(k_n) \Delta^{Y*}(k_n) e^{i(k_{n'}-k_n)\cdot x_0} \\
& \times \frac{1}{2} \sum_{(\tilde{n}_1, \tilde{n}_2) \in \mathcal{N}_2^{E_{12}}} \int_{-\infty}^{\infty} d\tilde{k}_z \sum_{a=0}^{1} \sum_{b=0}^{1} e^{ik_{\tilde{n}}(T^{(a)} - T^{(b)})} \delta^{(\mathrm{K})}(k_n)_x \delta^{(\mathrm{K})}(k_{n'})_x \delta^{(\mathrm{K})}(k_{n'})_x \delta^{(\mathrm{K})}(k_{n'})_z \delta^{(\mathrm{D})}(k_y - \tilde{k}_y^{(a)}) \delta^{(\mathrm{D})}(k_y' - \tilde{k}_y^{(b)}),
\end{split}
\tag{4.79}
\end{equation}
$$

where the terms with $n \in \mathcal{N}_1^{E_{12}}$ are of measure zero and have been dropped, $A_{E_{12}}$ is given in (3.43), $T^{(0)} \equiv 0$, $T^{(1)} \equiv T_B^{E_{12}}$, $k_{\tilde{n}}^{(a)} \equiv [(M_B^{E_{12}})^T]^a k_{\tilde{n}}$, and $\tilde{k}_y^{(a)} \equiv (k_{\tilde{n}}^{(a)})_y$.

In the harmonic basis we have

$$
\begin{align}
\xi_{k_n \ell m}^{E_{12}; \hat{k}_n} &= e^{-ik_n \cdot x_0} i^\ell Y_{\ell m}^*(\hat{k}_n), && \text{for } n \in \mathcal{N}_1^{E_{12}}, \nonumber \\
\xi_{k_n \ell m}^{E_{12}; \hat{k}_n} &= \frac{1}{\sqrt{2}} i^\ell Y_{\ell m}^*(\hat{k}_n) \sum_{j=0}^{1} i^{mj} e^{-ik_n \cdot (M_B^{E_{12}})^j x_0} e^{ik_n \cdot M_{0j}^{E_{12}} T_B^{E_{12}}}, && \text{for } n \in \mathcal{N}_2^{E_{12}}, \tag{4.80}
\end{align}
$$

and the spherical-harmonic space covariance matrix has the form (4.74).

4.10 E₁₆: slab space including rotation

As discussed above (section 3.9), there are two physically distinct cases for E₁₆, which will
be discussed separately.

4.10.1 $E_{16}^{(h)}$: conventional unrotated slab space

Similar to $E_1$ and $E_{11}$, $E_{16}^{(h)}$ is homogeneous. It is compact in only one dimension so
there is only one generator of the topology. The one discretization condition following
from (4.18) leads to

$$
(\mathbf{k}_n)_z = \frac{2\pi n}{L}, \tag{4.81}
$$

while $(\mathbf{k}_n)_x \equiv k_x$ and $(\mathbf{k}_n)_y \equiv k_y$ are unconstrained. As in the chimney spaces, we will
again write $\mathbf{k}_n$ as a shorthand for the wavevector characterized by the integer $n$ and the
real variables $k_x$ and $k_y$.

Thus

$$
\Upsilon_{k_n}^{E_{16}^{(h)}}(\boldsymbol{x}) = e^{ik_n \cdot (\boldsymbol{x} - \boldsymbol{x}_0)}, \quad \text{for } n \in \mathcal{N}_{16}^{E_{16}} \tag{4.82}
$$
where

$$ \mathcal{N}^{E_{16}^{(h)}} = \{n \in \mathbb{Z}\}. \qquad (4.83) $$

Following (4.4) the Fourier-mode correlation matrix for $E_{16}^{(h)}$ is

$$ C_{\mathbf{k}_n \mathbf{k}_{n'}}^{E_{16}^{(h)};XY} = (2\pi)^2 L \frac{2\pi^2}{k^3} \mathcal{P}^R(k) \Delta^X(k_n) \Delta^{Y*}(k_n) \times \delta^{(D)}(k_x - k_x') \delta^{(D)}(k_y - k_y') \delta^{(K)}_{(\mathbf{k}_n)_z (\mathbf{k}_{n'})_z}. \qquad (4.84) $$

In transitioning from $E_1$ (compare eq. (4.84) to eq. (4.24)) we have replaced $V_{E_1}$ with $(2\pi)^2 L$ and $\delta_{\mathbf{k}_n \mathbf{k}_{n'}}^{(K)}$ by a Kronecker delta for the z component of $\mathbf{k}_n$ and Dirac delta functions for the x and y components.

We can project the field $\delta^X$ onto the sky by performing a radial integral with a suitable weight function and transfer function, giving

$$ a_{\ell m}^{E_{16}^{(h)};X} = \frac{4\pi}{(2\pi)^2 L} \sum_{n \in \mathcal{N}^{E_{16}^{(h)}}} \int_{-\infty}^{\infty} dk_x \int_{-\infty}^{\infty} dk_y \delta_{\mathbf{k}_n}^{\mathcal{R}} \xi_{\mathbf{k}_n \ell m}^{E_{16}^{(h)}; \hat{\mathbf{k}}_n} \Delta_{\ell}^{X}(k_n), \qquad (4.85) $$

with

$$ \xi_{\mathbf{k}_n \ell m}^{E_{16}^{(h)}; \hat{\mathbf{k}}_n} \equiv e^{-i \mathbf{k}_n \cdot \mathbf{x}_0} i^{\ell} Y_{\ell m}^{*}(\hat{\mathbf{k}}_n), \quad n \in \mathcal{N}^{E_{16}^{(h)}}, \qquad (4.86) $$

where a similar transition to that above was performed in starting from eq. (4.25).

For the slab spaces the spherical-harmonic space covariance matrix now has the form

$$ C_{\ell m \ell' m'}^{E_{16}^{(a)};XY} = \frac{(4\pi)^2}{(2\pi)^2 L} \sum_{n \in \mathcal{N}^{E_{16}^{(a)}}} \int_{-\infty}^{\infty} dk_x \int_{-\infty}^{\infty} dk_y \Delta_{\ell}^{X}(k_n) \Delta_{\ell'}^{Y*}(k_n) \frac{2\pi^2 \mathcal{P}^{\mathcal{R}}(k)}{k_n^3} \xi_{\mathbf{k}_n \ell m}^{E_{16}^{(a)}; \hat{\mathbf{k}}_n} \xi_{\mathbf{k}_n \ell' m'}^{E_{16}^{(a)}; \hat{\mathbf{k}}_{n*}}, \qquad (4.87) $$

where $a \in \{\text{h}, \text{i}\}$ labels the homogeneous ($E_{16}^{(h)}$) and inhomogeneous ($E_{16}^{(i)}$) slab spaces.

### 4.10.2 $E_{16}^{(i)}$: general rotated slab space

The eigenspectrum and eigenmodes of $E_{16}^{(i)}$ can be determined from $E_{16}^{(h)}$ in a manner similar to that for $E_{12}$ from $E_{11}$ since $M_B^{E_{16}^{(i)}}$ is a rotation around the z-axis. For $E_{16}^{(i)}$, the discretization condition (4.18) leads to the component of the allowed wavevectors,

$$ (\mathbf{k}_n)_z = \frac{2\pi n}{qL \sin \beta}, \qquad (4.88) $$

while $(\mathbf{k}_n)_x \equiv k_x$ and $(\mathbf{k}_n)_y \equiv k_y$ are again unconstrained. As in $E_{16}^{(h)}$, we will again write $\mathbf{k}_n$ as a shorthand for the wavevector characterized by the integer $n$ and the real variables $k_x$ and $k_y$.

Unlike in $E_{16}^{(h)}$, the eigenmodes of $E_{16}^{(i)}$ can include linear combinations of $E_{16}^{(h)}$ eigenmodes.

Here $(M_B^{E_{16}^{(i)}})^N \mathbf{k}_n = \mathbf{k}_n$ has two solutions, $N=1$ and $N=q$. Written explicitly,

$N=1$ eigenmodes: $\mathbf{k}_n = (0, 0, (\mathbf{k}_n)_z)^T$, i.e., $n \in q\mathbb{Z}^\mathcal{I}$, $k_x = k_y = 0$, with

$$ \Upsilon_{\mathbf{k}_n}^{E_{16}^{(i)}}(\mathbf{x}) = e^{i\mathbf{k}_n \cdot (\mathbf{x} - \mathbf{x}_0)}, \qquad (4.89) $$
N = q eigenmodes: $(k_x, k_y) \neq (0, 0)$, $n \in \mathbb{Z}$, with

$$ \Upsilon_{k_n}^{E_{16}^{(i)}}(\mathbf{x}) = \frac{1}{\sqrt{q}} \sum_{j=0}^{q-1} e^{ik_n \cdot (\mathbf{M}_B^{E_{16}^{(i)}})^j (\mathbf{x}-\mathbf{x}_0)} e^{ik_n \cdot \mathbf{M}_{0j}^{E_{16}^{(i)}} \mathbf{T}_B^{E_{16}^{(i)}}}, \quad (4.90) $$

and $\mathbf{M}_{00}^{E_{16}^{(i)}} \equiv 0$ and $\mathbf{M}_{0j}^{E_{16}^{(i)}}$ defined in (4.35). Here the two sets of allowed modes are defined by

$$ \begin{aligned} \mathcal{N}_1^{E_{16}^{(i)}} &= \{n \in q\mathbb{Z} \neq 0\}, \\ \mathcal{N}_q^{E_{16}^{(i)}} &= \{n \in \mathbb{Z}\}, \\ \mathcal{N}^{E_{16}^{(i)}} &= \mathcal{N}_1^{E_{16}^{(i)}} \cup \mathcal{N}_q^{E_{16}^{(i)}}. \end{aligned} \quad (4.91) $$

With these the Fourier-mode correlation matrix can now be expressed as

$$ \begin{aligned} C_{k_n k_{n'}}^{E_{16}^{(i)};XY} ={}& (2\pi)^2 L_{E_{16}^{(i)}} \frac{2\pi^2}{k_n^3} \mathcal{P}^R(k_n) \Delta^X(k_n) \Delta^{Y^*}(k_n) e^{i(k_{n'} - k_n) \cdot x_0} \\ & \times \frac{1}{q} \sum_{\tilde{n} \in \mathcal{N}_q^{E_{16}^{(i)}}} \int_{-\infty}^{\infty} dk_x \int_{-\infty}^{\infty} dk_y \sum_{a=0}^{q-1} \sum_{b=0}^{q-1} e^{ik_{\tilde{n}} \cdot (T^{(a)} - T^{(b)})} \delta^{(K)}(k_n)_z (k_{\tilde{n}}^{(a)})_z \delta^{(K)}(k_{n'})_z (k_{\tilde{n}}^{(b)})_z \\ & \times \delta^{(D)}(k_x - \tilde{k}_x^{(a)}) \delta^{(D)}(k_{x'} - \tilde{k}_{x'}^{(b)}) \delta^{(D)}(k_y - \tilde{k}_y^{(a)}) \delta^{(D)}(k_{y'} - \tilde{k}_{y'}^{(b)}), \end{aligned} \quad (4.92) $$

where the terms with $n \in \mathcal{N}_1^{E_{16}^{(i)}}$ are of measure zero and have been dropped, $L_{E_{16}^{(i)}}$ is given in (3.47), $T^{(0)} \equiv 0$, $T^{(a)} \equiv \mathbf{M}_{0a}^{E_{16}^{(i)}} \mathbf{T}_B^{E_{16}^{(i)}}$ for $a \in \{1, \dots, q-1\}$, $\mathbf{k}_{\tilde{n}}^{(a)} \equiv [(\mathbf{M}_B^{E_{16}^{(i)}})^T]^a \mathbf{k}_{\tilde{n}}$, and $\tilde{k}_w^{(a)} \equiv (\mathbf{k}_{\tilde{n}}^{(a)})_w$ for $w \in \{x, y\}$.

In the harmonic basis we have

$$ \begin{aligned} \xi_{k_n \ell m}^{E_{16}^{(i)}; \hat{k}_n} &= e^{-ik_n \cdot x_0} i^\ell Y_{\ell m}^*(\hat{k}_n), & \text{for } n \in \mathcal{N}_1^{E_{16}^{(i)}}, \\ \xi_{k_n \ell m}^{E_{16}^{(i)}; \hat{k}_n} &= \frac{1}{\sqrt{q}} i^\ell Y_{\ell m}^*(\hat{k}_n) \sum_{j=0}^{q-1} e^{i\pi mjp/q} e^{-ik_n \cdot (\mathbf{M}_B^{E_{16}^{(i)}})^j x_0} e^{ik_n \cdot \mathbf{M}_{0j}^{E_{16}^{(i)}} \mathbf{T}_B^{E_{16}^{(i)}}}, & \text{for } n \in \mathcal{N}_q^{E_{16}^{(i)}}, \end{aligned} \quad (4.93) $$

and the spherical-harmonic space covariance matrix has the form (4.87).

### 4.11 E$_{18}$: the covering space of E$^3$

The covering space itself has all of the usual Fourier and spherical-harmonic mode functions, along with a simple isotropic correlation structure. The eigenmodes are

$$ \Upsilon_k^{E_{18}}(\mathbf{x}) = e^{ik \cdot (\mathbf{x} - \mathbf{x}_0)}, \quad (4.94) $$

the Fourier-mode correlation matrix is

$$ C_{kk'}^{E_{18};XY} = \frac{2\pi^2}{k^3} \mathcal{P}^R(k) \Delta^X(k) \Delta^{Y^*}(k) \delta^{(D)}(\mathbf{k} - \mathbf{k}'), \quad (4.95) $$
and the spherical-harmonic space covariance matrix has the form

$$C_{\ell m \ell' m'}^{E_{18};XY} = \frac{2}{\pi} \int d^3 \mathbf{k} \Delta_{\ell}^{X}(\mathbf{k}) \Delta_{\ell'}^{Y*}(\mathbf{k}) \frac{2\pi^2 \mathcal{P}^{\mathcal{R}}(k)}{k^3} \xi_{\ell m}^{E_{18};\mathbf{k}} \xi_{\ell' m'}^{E_{18};\mathbf{k}*}, \quad (4.96)$$

with

$$\xi_{\ell m}^{E_{18};\mathbf{k}} = e^{-i\mathbf{k} \cdot \mathbf{x}_0} i^{\ell} Y_{\ell m}^{*}(\hat{\mathbf{k}}). \quad (4.97)$$

In the typical case (e.g., CMB temperature and polarization) where $\Delta_{\ell}^{X}(\mathbf{k}) = \Delta_{\ell}^{X}(k)$, eq. (4.96) gives the isotropic

$$C_{\ell m \ell' m'}^{E_{18};XY} = C_{\ell}^{E_{18};XY} \delta_{\ell \ell'}^{(K)} \delta_{mm'}^{(K)}, \quad (4.98)$$

with the angular power spectrum given by

$$C_{\ell}^{E_{18};XY} = 4\pi \int \frac{dk}{k} \mathcal{P}^{\mathcal{R}}(k) \Delta_{\ell}^{X}(k) \Delta_{\ell}^{Y*}(k). \quad (4.99)$$

# 5 Numerical analysis

As described above, to the extent that the CMB is Gaussian, all the information about the temperature anisotropies is given by the 2-point correlation matrix of the spherical-harmonic coefficients, namely the covariance matrix $C_{\ell m \ell' m'}^{E_{i};XY} = \langle a_{\ell m}^{E_{i};X} a_{\ell' m'}^{E_{i};Y*} \rangle$ [83]. In the familiar case of the isotropic covering space, we have the usual formula $\langle a_{\ell m}^{E_{18};X} a_{\ell' m'}^{E_{18};Y*} \rangle = C_{\ell}^{E_{18};XY} \delta_{\ell \ell'}^{(K)} \delta_{mm'}^{(K)}$, where only the diagonal elements are non-zero and they are independent of $m$. But a non-trivial topology breaks the assumption of isotropy (see, e.g., ref. [25]), inducing non-zero values for off-diagonal components of the correlation matrix. In this section, we numerically compute the correlation matrix elements for sample manifolds of each of the compact, orientable, Euclidean topologies $E_1-E_6$.<fnref n="16" /> We then estimate the detectability of this off-diagonal signal of topology given the cosmic variance associated with the primary CMB temperature anisotropies.

## 5.1 Evaluation of CMB temperature covariance matrices

The covariance matrices for the fully compact, orientable, Euclidean topologies can generally be written like eq. (4.27), which we rewrite here for auto-correlation of the CMB temperature (T) fluctuations:

$$C_{\ell m \ell' m'}^{E_i;TT} = \frac{(4\pi)^2}{V_{E_i}} \sum_{n \in \mathcal{N}^{E_i}} \Delta_{\ell}^{T}(k_n) \Delta_{\ell'}^{T}(k_n) \frac{2\pi^2 \mathcal{P}^{\mathcal{R}}(k_n)}{k_n^3} \xi_{k_n \ell m}^{E_i;\hat{k}_n} \xi_{k_n \ell' m'}^{E_i;\hat{k}_{n*}}. \quad (5.1)$$

For brevity, we will drop the T and TT labels in this section. The covering space, $E_{18}$, reduces this three-dimensional summation to a one-dimensional integral over the magnitude of the wavevector, $|\mathbf{k}|$ (or $k$), which is computationally fast to evaluate numerically. However, for non-trivial topologies, we must perform computationally costly summations. In principle, we need to perform the sum over an infinite set of wavevectors, but in practice, we must choose a maximum value of $|\mathbf{k}|$, $|\mathbf{k}_{\text{max}}(\ell)|$, at which we stop the summation. As we are interested

<footnote n="16">¹⁶The non-compact cases $E_{11}$, $E_{12}$, and $E_{16}^{(h)}$ can be regarded as limiting cases of $E_1$ and $E_2$. We reserve detailed investigation of $E_{16}^{(h)}$ for future specific work.</footnote>
in probing Dirichlet domains that fully contain the last scattering surface, we are forced to prioritize large topology scales (heuristically, the various $L_i$) over the computational accuracy that might be achieved by calculating the summation up to a large $|k|$.

As an example, consider the cubic $E_1$, the 3-torus with equal lengths $L_{A_1} = L_{A_2} = L_{A_3} = L$. Here, the wavevector is

$$k = \frac{2\pi}{L} n. \qquad (5.2)$$

In making $L$ larger, we are forced to go to larger values of $|n|$ to achieve a given accuracy for a given element of $C_{\ell m \ell' m'}^{E_1}$, meaning we must execute a larger number of summations. Since we live in three spatial dimensions, the number of operations we must perform for a given $\{\ell, m, \ell', m'\}$ scales as $O(L^3)$. Therefore, we need to make a clear definition for our multipole-dependent cutoff $|k_{\max}(\ell)|$. We define the ratio function

$$R_{\ell}(|k|) = \frac{C_{\ell}^{|k|}}{C_{\ell}^{\Lambda \text{CDM}}}, \qquad (5.3)$$

where

$$C_{\ell}^{|k|} = 4\pi \int_{0}^{|k|} dk' \frac{\mathcal{P}^{\mathcal{R}}(k')}{k'} \Delta_{\ell}(k')^2, \qquad (5.4)$$

and $C_{\ell}^{\Lambda \text{CDM}}$ is the standard $\Lambda$CDM angular power spectrum produced by CAMB [84, 85] for $E_{18}$. We then find $|k_{\max}(\ell)|$ such that wavevectors with $|k| \le |k_{\max}(\ell)|$ would contribute at least 99% of $C_{\ell}^{\Lambda \text{CDM}}$. In the limit of $L \to \infty$, we would find a power spectrum that is 1% smaller than the actual power spectrum. For computing the off-diagonal elements $\ell \ne \ell'$, we perform the sum in eq. (5.1) up to $|k_{\max}(\max(\ell, \ell'))|$. Increasing the precision to a higher number than 99% has a negligible impact on the KL divergence we define later.

We solve these equations by developing a special-purpose Python code. To obtain the transfer function $\Delta_{\ell}(k)$ and the primordial power spectrum $\mathcal{P}^{\mathcal{R}}(k)$, we use CAMB with the Planck 2018 best-fit $\Lambda$CDM cosmological parameters [86]. Before we present the covariance matrices for all the compact, orientable topologies, we note that the choice of coordinate system will affect the appearance and symmetries of the covariance matrix. We will, in this paper, plot the rescaled covariance matrix, which we define as

$$\Xi_{\ell m \ell' m'}^{E_i} = \frac{C_{\ell m \ell' m'}^{E_i}}{\sqrt{C_{\ell}^{\Lambda \text{CDM}} C_{\ell'}^{\Lambda \text{CDM}}}}, \qquad (5.5)$$

and will be of importance later when we discuss the KL divergence.

In figure 3, we show absolute values of the rescaled covariance matrix for the unshifted and untilted $E_2$ in the coordinate system used for this work. In the presented plots, we order the $\{\ell, m\}$ elements as $s = \ell(\ell+1)+m$, with $m$ in the range $-\ell \le m \le \ell$. In the right panel of the figure, we randomly rotate the coordinate system by applying the Wigner D-matrices $D_{m\bar{m}}^{\ell}(\theta_0, \phi_0)$ (see, e.g., ref. [87]). Here, the new coordinates are $\theta' = \theta + \theta_0$ and $\phi' = \phi + \phi_0$ so that the covariance matrix in the new coordinates becomes

$$C_{\ell m \ell' m'}^{E_i} = \sum_{\bar{m}=-\ell}^{\ell} \sum_{\bar{m}'=-\ell'}^{\ell'} D_{m\bar{m}}^{\ell}(\theta_0, \phi_0) D_{m'\bar{m}'}^{*\ell'}(\theta_0, \phi_0) C_{\ell\bar{m}\ell'\bar{m}'}^{E_i}. \qquad (5.6)$$
<figure><img src="image_7.png" /><figcaption>**Figure 3.** Effect of rotating the coordinate system on the CMB temperature covariance matrix. Here, the topology is the untilted $E_2$ with $L_{A_1} = L_{A_2} = 1.4L_{LSS}$ and $L_B = L_{LSS}$, where $L_{LSS}$ is the diameter of the last scattering surface, and with an on-axis observer. The left panel shows the absolute value of the rescaled covariance matrix $\Xi_{\ell m \ell' m'}^{E_2; \text{TT}}$ (at low multipoles $\ell$) in the coordinate system used for this work as opposed to a random rotation of the coordinate system in the right panel. For each $(\ell, \ell')$ block the matrix elements show the corresponding $(m, m')$ in increasing order, i.e., $-\ell \le m \le \ell$. Although the right panel seems brighter, both plots contain an equal amount of information about the topology of the Universe, as confirmed by both resulting in the same KL divergence.</figcaption></figure>

The rotation operation only mixes the $m$ and $m'$ elements in each $(\ell, \ell')$ block, meaning that all the dark, uncorrelated elements in the $\ell - \ell' \equiv 1 \pmod 2$ of the left panel of figure 3 stay uncorrelated after the rotation. Unlike this configuration for $E_2$, some topologies will induce correlations in all $(\ell, \ell')$ blocks, meaning that a random rotation of the coordinate system can induce correlations between all combinations of $\{\ell, m\}$ and $\{\ell', m'\}$.

The absolute values of the rescaled covariance matrices for the untilted and unshifted (or on-axis) topologies $E_1-E_6$ with topological length scales equal to $L_{LSS}$ are shown in the left panel of figure 4. All these topologies have non-zero correlations for the $\ell - \ell' \equiv 0 \pmod 2$ blocks in their on-axis and untilted configurations, but we also find correlations in the $\ell - \ell' = \text{odd blocks of } E_3, E_4$, and $E_5$. This can be explained by the lack of symmetry in the $z$ direction. For $E_2$, the $z \to z \pm L_B$ transformations both give a $180^\circ$ corkscrew rotation, preserving the $z$-symmetry when the observer is on-axis. But $E_3, E_4$, and $E_5$ do not have this symmetry as the transformation $z \to z + L_B$ gives the positive corkscrew rotations of $90^\circ, 120^\circ$, and $60^\circ$, respectively, while the transformation in the opposite direction $z \to z - L_B$ gives a negative rotation, breaking the $z$-direction symmetry. Both $E_2$ and $E_6$ lose this symmetry when they are off-axis, creating $\ell - \ell' = \text{odd correlations.}$ $E_1$, on the other hand, will never have $\ell - \ell' = \text{odd correlations.}$
<figure><img src="image_8.png" /><figcaption>**Figure 4.** Left panel: absolute values of the rescaled CMB temperature covariance matrix $\Xi^{E_i;TT}_{\ell m \ell' m'}$ (at low multipoles $\ell$) for the cubic, untilted $E_1-E_6$ with $L=L_{LSS}$, where $L_{LSS}$ is the diameter of the last scattering surface, and for an on-axis observer. Right panel: the two KL divergences $D_{KL}(p \parallel q)$ and $D_{KL}(q \parallel p)$, and the off-diagonal signal-to-noise ratio $\sigma_{\bar{R}}^{-2}$ for the cubic, untilted $E_1-E_6$ with an on-axis observer. We find that the curves for $E_1-E_3$ are indistinguishable, meaning that the $E_1, E_2$, and $E_3$ cases carry the same information. The same is true for $E_4$ and $E_5$. We have varied all lengths $L$ and have set $\ell_{max} = 30$.</figcaption></figure>

We can learn something about the symmetries between $(m, m')$ elements in the covariance plots. It is straightforward to see that an on-axis observer of untilted $E_2, E_3, E_4$, and $E_5$ has

$$ \xi_{k_n \ell m}^{E_i; \hat{k}_n} \propto \sqrt{\lambda^{E_i} \delta_{mn3}^{(K) \text{mod } \lambda^{E_i}}}, \qquad (5.7) $$

where $(\text{mod } \lambda^{E_i})$ indicates that the Kronecker delta is 1 if $m \equiv n_3 (\text{mod } \lambda^{E_i})$. Here the integer $\lambda^{E_i}$ is the number of times one has to translate by $z \to z + L_B$ to get a corkscrew rotation of $360^\circ$, namely, $\lambda^{E_2} = 2$, $\lambda^{E_3} = 4$, $\lambda^{E_4} = 3$, and $\lambda^{E_6} = 6$. By combining the two eigenmodes that enter the sum in the covariance matrix, we find

$$ \xi_{k_n \ell m}^{E_i; \hat{k}_n} \xi_{k_n \ell' m'}^{E_i; \hat{k}_{n'}} \propto \lambda^{E_i} \delta_{mn3}^{(K)} \delta_{m'n3}^{(K)} (\text{mod } \lambda^{E_i}) = \lambda^{E_i} \delta_{mm'}^{(K)} \delta_{n3m}^{(K)} (\text{mod } \lambda^{E_i}). \qquad (5.8) $$

This then results in a clear pattern in the covariance matrix, $C^{E_i}_{\ell m \ell' m'} \propto \delta^{(K)}_{mm'} (\text{mod } \lambda^{E_i})$. As mentioned, this pattern generically disappears once we arbitrarily rotate the coordinate system.

There are other patterns, such as an untilted, cubic $E_1$ being invariant under a $\pi/4$ rotation, yielding only $m - m' \equiv 0 (\text{mod } 4)$ correlations, but we will not find and categorize all these symmetries in this work.
## 5.2 KL divergence

Searching for repeating circles in the temperature fluctuations of the CMB has been a widely used method to look for evidence for a non-trivial cosmic topology. However, when the Dirichlet domain does not fully contain the last scattering surface, no matched circles appear in the CMB and other methods must be applied. We can then ask the question “at what size of the Dirichlet domain will a non-trivial topology of the Universe be indistinguishable from the trivial topology, i.e., from the covering space?”

To answer this question we can apply the KL divergence method [66, 67] to a given set of “data” {$a_{\ell m}$} to compare the probability distribution of a non-trivial topology, $p(\{a_{\ell m}\})$, to that of the trivial one, $q(\{a_{\ell m}\})$. We imagine an ideal experiment with no noise, no foreground emission, and no mask to assess the maximum detectability of topology from the CMB. The KL divergence $D_{\text{KL}}(p \parallel q)$, which quantifies the loss of information if we assume the (“correct”) model $p$ is represented by the (“incorrect”) model $q$, is given as

$$D_{\text{KL}}(p \parallel q) = \int d\{a_{\ell m}\} p(\{a_{\ell m}\}) \ln \left[ \frac{p(\{a_{\ell m}\})}{q(\{a_{\ell m}\})} \right]. \qquad (5.9)$$

This can be written in terms of the eigenvalues $\lambda_j$ of the rescaled covariance matrix, $\Xi_{\ell m \ell' m'}^{E_i}$ [65]. The KL divergence then becomes

$$D_{\text{KL}}(p \parallel q) = \frac{1}{2} \sum_{j} \left( \ln |\lambda_j| + \lambda_j^{-1} - 1 \right). \qquad (5.10)$$

We could also turn the question around: “how much information is lost if we assume the model $q$ is represented by the model $p$?” As the eigenvalues of the inverse of the matrix $\Xi_{\ell m \ell' m'}^{E_i}$ are simply $1/\lambda_j$, we find

$$D_{\text{KL}}(q \parallel p) = \frac{1}{2} \sum_{i} (-\ln |\lambda_j| + \lambda_j - 1). \qquad (5.11)$$

## 5.3 Cosmic variance signal-to-noise ratio

We can also evaluate the signal-to-noise ratio for detecting the off-diagonal elements of a specific correlation matrix in the ideal case of a full-sky CMB map with a given angular resolution and zero instrumental noise. In this case, the noise is due to the cosmic variance [88] from a particular random sky realization, and thus is in principle the best we can do to detect topology from a CMB temperature map.

The spherical-harmonic coefficients are random complex numbers with zero mean, random phase, and amplitude variance given by

$$\sigma_{\ell m}^2 \equiv \langle |a_{\ell m}|^2 \rangle = C_{\ell m \ell m}, \qquad (5.12)$$

ignoring instrumental noise and beam effects as appropriate for a cosmic variance-limited observation. The $a_{\ell m}$ satisfy the reality condition $a_{\ell-m} = (-1)^m a_{\ell m}^*$. For $m \neq 0$, they are described by two independent scalar Gaussian random variables for their real and imaginary parts, each with variance $\sigma_{\ell m}^2/2$, and for $m = 0$ by a single scalar Gaussian random variable of variance $\sigma_{\ell m}^2$.
Products of two different spherical-harmonic coefficients are also complex random variables, which can be written as

$$a_{\ell m} a_{\ell' m'}^* = C_{\ell m \ell' m'} \pm S_{\ell m \ell' m'}, \qquad (5.13)$$

where the $C_{\ell m \ell' m'}$ is the expectation value of $a_{\ell m} a_{\ell' m'}^*$, and the standard error term $S_{\ell m \ell' m'}$ is a random complex number with zero mean for both real and imaginary components. To obtain the standard error, the product $a_{\ell m} a_{\ell' m'}^*$ can be expanded into its real and imaginary parts; each contains two terms which are products of two random normal variables (the real and/or imaginary parts of $a_{\ell m}$). We approximate the two as being independent; this is exact for a Gaussian random curvature field in the covering space because of statistical isotropy. It is a good approximation whenever the correlations between spherical-harmonic coefficients are small compared to their amplitudes, which is the case for all the $E_i$ topologies, except, possibly, for low $\ell$ values. It also appears to be the case for the observed sky when analyzed assuming statistical isotropy, again except, possibly, for low $\ell$ values. Since the variance of the product of two independent random normal variables is the product of the variances, we have

$$\begin{aligned}
\mathrm{Var}(\mathrm{Re} \, a_{\ell m} \, \mathrm{Re} \, a_{\ell' m'}^*) &= \frac{1}{2} \sigma_{\ell m}^2 (1 + \delta_{m0}^{(K)}) \frac{1}{2} \sigma_{\ell' m'}^2 (1 + \delta_{m'0}^{(K)}), \\
\mathrm{Var}(\mathrm{Re} \, a_{\ell m} \, \mathrm{Im} \, a_{\ell' m'}^*) &= \frac{1}{2} \sigma_{\ell m}^2 (1 + \delta_{m0}^{(K)}) \frac{1}{2} \sigma_{\ell' m'}^2 (1 - \delta_{m'0}^{(K)}),
\end{aligned} \qquad (5.14)$$

and the same result for the other possible combinations of real and imaginary parts on the left side. Combining these,

$$\begin{aligned}
\mathrm{Var} \, \mathrm{Re} \, (a_{\ell m} a_{\ell' m'}^*) &= \frac{1}{2} \sigma_{\ell m}^2 \sigma_{\ell' m'}^2 (1 + \delta_{m0}^{(K)} \delta_{m'0}^{(K)}), \\
\mathrm{Var} \, \mathrm{Im} \, (a_{\ell m} a_{\ell' m'}^*) &= \frac{1}{2} \sigma_{\ell m}^2 \sigma_{\ell' m'}^2 (1 - \delta_{m0}^{(K)} \delta_{m'0}^{(K)}).
\end{aligned} \qquad (5.15)$$

Note that the random variable $a_{\ell m} a_{\ell' m'}^*$ is not itself a Gaussian variable (its amplitude is distributed as the Bessel function $K_0$), but this is not important for our conclusions.

To estimate the possible signal-to-noise ratio with which a given topology can be detected, we write the off-diagonal signal of a particular topology as $RC_{\ell m \ell' m'}^{E_i}$ with $R = 1$ corresponding to the actual signal. We can then estimate $R$ from the off-diagonal elements, where each individual element can serve as a noisy estimator $\tilde{R}$ of $R$; for example, $\tilde{R}_{\ell m \ell' m'}^{\mathrm{Re}} = \mathrm{Re} (a_{\ell m} a_{\ell' m'}^*) / \mathrm{Re} C_{\ell m \ell' m'}^{E_i}$ or the same with the imaginary component (except for $m = m' = 0$ where the latter vanishes). We can also estimate $R$ using any weighted normalized sum of the individual estimates; a textbook result is that the minimum variance estimator uses the inverse variance weight for each term. In this case, the inverse variance of the estimate is the sum of the inverse variances of the terms, so

$$\begin{aligned}
\sigma_{\tilde{R}}^{-2} = \frac{1}{\mathrm{Var} \, \tilde{R}} = \frac{1}{2} \sum_{(\ell m) \neq (\ell' m')} \frac{1 + \delta_{m0}^{(K)} + \delta_{m'0}^{(K)} - \delta_{m0}^{(K)} \delta_{m'0}^{(K)}}{2} \\
\times \left[ \frac{2 \left( \mathrm{Re} \, C_{\ell m \ell' m'}^{E_i} \right)^2}{(1 + \delta_{m0}^{(K)} \delta_{m'0}^{(K)}) \sigma_{\ell m}^2 \sigma_{\ell' m'}^2} + \frac{2 (\mathrm{Im} \, C_{\ell m \ell' m'}^{E_i})^2}{\sigma_{\ell m}^2 \sigma_{\ell' m'}^2} \right]
\end{aligned}$$
$$ 
\begin{aligned}
&= \frac{1}{2} \sum_{(\ell m) \neq (\ell' m')} \frac{1 + \delta_{m0}^{(K)} + \delta_{m'0}^{(K)} - \delta_{m0}^{(K)} \delta_{m'0}^{(K)}}{1 + \delta_{m0}^{(K)} \delta_{m'0}^{(K)}} \frac{|C_{\ell m \ell' m'}^{E_i}|^2}{\sigma_{\ell m}^2 \sigma_{\ell' m'}^2} \\
&\approx \frac{1}{2} \sum_{(\ell m) \neq (\ell' m')} \frac{|C_{\ell m \ell' m'}^{E_i}|^2}{\sigma_{\ell m}^2 \sigma_{\ell' m'}^2}. \qquad (5.16)
\end{aligned}
$$

The sums in (5.16) are over all pairs of indices except along the diagonal where $\ell = \ell'$ and $m = m'$, between $\ell = 2$ and some $\ell_{max}$. The Hermitian redundancy of the correlation matrix<fnref n="17" /> means that the sum over-counts the off-diagonal terms by exactly a factor of 2. This is accounted for by taking 1/2 of the sum, canceling the factors of 2 in each term. The imaginary-part term vanishes when $m = m' = 0$. The terms then combine to give the squared amplitude. We account for the symmetry $C_{\ell-m\ell'-m'}^{E_i} = (-1)^{m+m'} C_{\ell'm'\ell m}^{E_i}$ when neither $m = 0$ nor $m' = 0$, with the factor $(1 + \delta_{m0}^{(K)} + \delta_{m'0}^{(K)} - \delta_{m0}^{(K)} \delta_{m'0}^{(K)})/2$ in the sum. Finally, in practice, we ignore the Kronecker delta terms in the final expression, which account for only a fraction $1/\ell_{max}^2$ of terms.

The maximum possible signal-to-noise ratio of the off-diagonal elements in the covariance matrix for measuring a given topology is simply $\sigma_R^{-2}$, and it is proportional to the mean-square correlation coefficient of the correlation matrix $C_{\ell m \ell' m'}^{E_i}$. Eq. (5.16) can be directly compared with the KL statistics.<fnref n="18" /> This maximum signal-to-noise ratio is obtained if we know the topology and know our position and orientation in the Dirichlet domain. In practice, we cannot know these things a priori, so we must search over a space of locations and orientations for any particular topology, with only the correct choice giving the maximum signal-to-noise ratio computed here.

## 5.4 KL divergence and signal-to-noise ratio for compact, orientable, Euclidean manifolds

In figure 5 we compare the three statistics $D_{KL}(p \parallel q)$, $D_{KL}(q \parallel p)$, and $\sigma_R^{-2}$ for the cubic $E_1$ as a function of $L_1 = L_2 = L_3 \equiv L$ in units of $L_{SS}$, and find that they convey similar information. In the right panel of figure 4, we compare the three statistics for the untilted $E_1-E_6$ with an on-axis observer. We find that they qualitatively tell the same story for all cases; once the size of the Dirichlet domain is larger than the diameter of the last scattering surface, the information content of the topology of the Universe in the CMB data drops by increasing the length scale and will eventually cross the no-detection line $D_{KL} = 1$. As these three statistics qualitatively behave similarly, from now on we will only quote $D_{KL}(p \parallel q)$. However, the off-diagonal-signal-to-noise statistic has the advantage over the KL divergence in that calculating it does not involve inverting a large matrix. The sums in (5.16) can

<footnote n="17">¹⁷For $X \neq Y$ in $C^{XY}$, we would drop the factor of 1/2 in (5.16).</footnote>

<footnote n="18">¹⁸$D_{KL} = 1$ can be taken as a theoretical detectability threshold for non-trivial topology in the absence of noise, foregrounds, and mask. $D_{KL}$ characterizes the relative odds of two models — here a non-trivial topology versus the covering space. According to Wilks’ theorem twice the likelihood ratio is asymptotically $\chi^2$ distributed with number of degrees of freedom equal to the difference between those of the two distributions. Thus, for example, $D_{KL} \gtrsim 3$ implies that topology can be strongly detected in cases where there are three orientation degrees of freedom and three more that characterize the manifold of the particular topology.</footnote>
<figure><img src="image_9.png" /><figcaption>**Figure 5.** Comparison of the $D_{\text{KL}}(p \parallel q)$, $D_{\text{KL}}(q \parallel p)$, and $\sigma_{\bar{R}}^{-2}$ statistics for cubic $E_1$ as a function of $L_1 = L_2 = L_3 \equiv L$ in units of $L_{\text{LSS}}$, where $L_{\text{LSS}}$ is the diameter of the last scattering surface.</figcaption></figure>

become unwieldy, but they can be approximated by using a randomly chosen smaller number of terms and scaling, in the style of Monte Carlo integration.

We show in figures 6 to 8 the absolute values of the rescaled covariance matrices $|\Xi_{\ell m \ell' m'}^{E_i}|$ of each topology for two different configurations: an on-axis observer $\mathbf{x}_0 = (0, 0, 0)^T$ and an off-axis observer $\mathbf{x}_0 \neq (0, 0, 0)^T$. As the tilt and off-axis position of the observer are degenerate, we set the tilt parameters to zero. The exception is for $E_1$ as this topology is homogeneous. For $E_1$ we have chosen an untilted ($\beta = 90^\circ$) configuration and a tilted ($\beta = 75^\circ$) configuration in figure 6.

For $E_2-E_5$, we set $L_A = 1.4L_{\text{LSS}}$ and vary $L_B$ in the KL divergence plots.<fnref n="19" /> The x-axis is shown as $L_B/L_{\text{circle}}$, where $L_{\text{circle}}$ is defined as the smallest $L_B$ for which no pairs of identical circles appear in the CMB. For an on-axis observer, this is simply $L_{\text{LSS}}$ as long as $L_A > L_{\text{LSS}}$.<fnref n="20" /> Hence, when $L_B/L_{\text{circle}} > 1$ no identical circles appear in the CMB and the information content of cosmic topology in the CMB drops as seen by the falling KL divergence. For the figures of the rescaled covariance matrices, we set $L_B = L_{\text{circle}}$, meaning that these covariance matrices can never be ruled out by CMB circle searches. For $E_1$, we do the same as above, but with $L_A \to L_1 = L_2$ and $L_B \to L_3$. For $E_2-E_6$, $L_{\text{circle}}$ depends on the parameters of the topology, while $E_1$ always has $L_3 = L_{\text{circle}} = L_{\text{LSS}}$ as its circle limit even with a tilted topology ($\beta \neq 90^\circ$).

The off-axis positions in figures 6 to 8 are not randomly chosen. We have chosen locations that are in the middle between two excluded regions for $L_B < L_{\text{LSS}}$ of figure 2 of ref. [63]. What this means is that the position is chosen so that the observer has the highest number of nearest clones separated by the same distance to the observer. We, therefore, hypothesize that these off-axis positions represent maximum possible information of non-trivial topology in the CMB.

<footnote n="19">¹⁹Note that for $E_2$ the length parameters are $L_{A1}$, $L_{A2}$, and $L_B$, instead of $L_A$ and $L_B$. Here, for simplicity, we refer to $L_{A1} = L_{A2}$ as $L_A$ for $E_2$.</footnote>

<footnote n="20">²⁰If $L_A < L_{\text{LSS}}$ there would always be identical circles in the CMB for all values of $L_B$.</footnote>
<figure><img src="image_10.png" /><figcaption>**Figure 6.** Absolute values of the rescaled CMB temperature covariance matrix $\Xi^{E_i; \text{TT}}_{\ell m \ell' m'}$ (at low multipoles $\ell$) and the KL divergence for $E_1$ and $E_2$, where we have set $L_1 = L_2 = 1.4 L_{\text{LSS}}$ for $E_1$ and $L_{A_1} = L_{A_2} = 1.4 L_{\text{LSS}}$ for $E_2$. The KL divergence is calculated with $\ell_{\text{max}} = 30$. In $E_1$, the circle limit for both tilt angles $\beta = 90^\circ$ and $\beta = 75^\circ$ is $L_3 = L_{\text{circle}} = L_{\text{LSS}}$, where $L_{\text{circle}}$ is defined as the smallest $L_3$ for which no pairs of identical circles appear in the CMB and $L_{\text{LSS}}$ is the diameter of the last scattering surface. For $E_2$, the limit for an on-axis observer is $L_B = L_{\text{circle}} = L_{\text{LSS}}$, where $L_{\text{circle}}$ is the smallest $L_B$ for which no pairs of identical circles appear in the CMB, while the observer at $x_0 = L_{\text{LSS}}(0.35, 0, 0)^T$ reaches the circle limit when $L_B = L_{\text{circle}} = \sqrt{1-0.7^2 L_{\text{LSS}}} \approx 0.71 L_{\text{LSS}}$. For the rescaled covariance matrix plots we have used $L_3 = L_{\text{circle}}$ for $E_1$ and $L_B = L_{\text{circle}}$ for $E_2$.</figcaption></figure>

For example, for $E_2$ with $L_A = 1.4 L_{\text{LSS}}$ and $x_0 = L_{\text{LSS}}(0.35, 0, 0)^T$ in figure 6 there are four nearest clones with the same distance to the observer. Namely at positions $\boldsymbol{x} = ((1.4 - 0.35)L_{\text{LSS}}, 0, L_B)^T$, $\boldsymbol{x} = (-0.35L_{\text{LSS}}, 0, L_B)^T$, $\boldsymbol{x} = ((1.4 - 0.35)L_{\text{LSS}}, 0, -L_B)^T$, and $\boldsymbol{x} = (-0.35L_{\text{LSS}}, 0, -L_B)^T$. The distance to these four clones is $\sqrt{(0.7L_{\text{LSS}})^2 + L_B^2}$. When the distance is equal to $L_{\text{LSS}}$, we get $L_B = L_{\text{circle}} = \sqrt{1 - 0.7^2 L_{\text{LSS}}}$. An on-axis observer $x_0 = (0, 0, 0)^T$ has only two nearest clones, $x = (0, 0, L_B)^T$ and $x = (0, 0, -L_B)^T$, both with a distance $L_B$ yielding $L_B = L_{\text{circle}} = L_{\text{LSS}}$. Fewer nearest clones with the same distance means fewer correlations in the covariance matrix and a lower KL divergence.

$E_6$ is more complex, and we found an off-axis position from numerical considerations. From figure 8 we already find that the on-axis position contains more information than the chosen off-axis one, meaning that there could be a more optimal off-axis position yielding higher information.

We have also looked into whether including higher multipoles $\ell_{\text{max}}$ increases the KL divergence for $E_1$ to $E_6$. Although going to higher multipoles increases the KL divergence when the distance to the nearest clone is less than $L_{\text{LSS}}$, we find that increasing $\ell_{\text{max}} = 30$ to $\ell_{\text{max}} = 50$ makes a negligible difference when there are no identical circles in the CMB. This agrees with the findings of ref. [29] (see the lower panel of figure 4 in that reference).
<figure><img src="image_11.png" /><figcaption>**Figure 7.** As in figure 6, but for $E_3$ and $E_4$, where we have set $L_A = 1.4L_{\text{LSS}}$. In both cases, the circle limit for the on-axis observers is $L_B = L_{\text{circle}} = L_{\text{LSS}}$, while the off-axis observers have a circle limit of $L_B = L_{\text{circle}} = \sqrt{1-0.7^2}L_{\text{LSS}} \approx 0.71L_{\text{LSS}}$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 8.** As in figures 6 and 7, but for $E_5$ and $E_6$. For the covariance matrix plots of $E_5$, we have set $L_A = 1.4L_{\text{LSS}}$ and $L_B = L_{\text{circle}}$. $L_{\text{circle}} = L_{\text{LSS}}$ and $L_{\text{circle}} = \sqrt{1-0.7^2}L_{\text{LSS}} \approx 0.71L_{\text{LSS}}$ for the observer locations $\mathbf{x}_0 = L_{\text{LSS}}(0,0,0)^T$ and $\mathbf{x}_0 = L_{\text{LSS}}(0.35,0.61,0)^T$, respectively. For the covariance matrix plots of $E_6$, we have set $L_{Ax} = L_{By} = L_{Cz} = L_{\text{circle}}$, where $L_{\text{circle}} = 1/\sqrt{2}L_{\text{LSS}} \approx 0.71$ for $\mathbf{x}_0 = L_{\text{LSS}}(0,0,0)^T$ and $L_{\text{circle}} \approx 0.90$ for $\mathbf{x}_0 = L_{\text{LSS}}(-0.5,0.25,0)^T$. In the KL divergence plot of $E_5$, we have set $L_A = 1.4L_{\text{LSS}}$ while varying $L_B$. For $E_6$, we have varied $L \equiv L_{Ax} = L_{By} = L_{Cz}$, and we have set $r_x = r_y = r_z = 1/2$.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Figure 9.** The quantity $C_{\ell}^{E_i;TT} \equiv \frac{1}{2\ell+1} \sum_m C_{\ell m \ell m}^{E_i;TT}$ for the CMB temperature sky for the configurations with off-axis observers in figures 6 to 8. The gray band shows the $1\sigma$ cosmic variance for $\Lambda$CDM, i.e., the trivial topology.</figcaption></figure>

Finally, we show the diagonal $C_{\ell}^{E_i;TT}$ in figure 9, defined as $C_{\ell}^{E_i;TT} \equiv \frac{1}{2\ell+1} \sum_m C_{\ell m \ell m}^{E_i;TT}$ for the non-trivial topologies. We have divided by $R_{\ell}(|\mathbf{k}|) = 0.99$ in eq. (5.3) to make up for the power lost by not summing over an infinite number of wavevectors. These topologies are chosen to have $L_3 = L_B = L = L_{\text{circle}}$ so that their nearest clones are exactly $L_{\text{LSS}}$ away, meaning that the CMB contains no identical circles.

We end this section by briefly discussing how to generate $a_{\ell m}^{E_i}$ realizations of the CMB sky signal assuming a non-trivial topology, even though we have not used such realizations in this paper. There are three different ways for doing that. As the first possibility, one can sum over all the Fourier modes that contribute to each spherical-harmonic coefficient by generating realizations of the primordial density fluctuations $\delta_{k_n}^R$ for each discrete wavevector $\mathbf{k}_n$. Namely, one would compute

$$a_{\ell m}^{E_i} = \frac{4\pi}{V_{E_i}} \sum_{n \in \mathcal{N}^{E_i}} \delta_{k_n}^{R} \xi_{k_n \ell m}^{E_i; \hat{k}_n} \Delta_{\ell}(k_n). \qquad (5.17)$$

As all the information is in the covariance matrix, the second possibility is to generate the $C_{\ell m \ell' m'}^{E_i}$ and create a realization by using Cholesky decomposition [89, 90]. However, calculating the full covariance matrix is computationally more expensive than performing the summation in eq. (5.17). A third option is to generate three-dimensional real-space density fluctuations for a given topology and its parameters and then take a spherical slice to create the temperature fluctuations of the CMB. We expect this to be computationally significantly faster than solving eq. (5.17), especially for topology length scales of the order of or larger than $L_{\text{LSS}}$.
## 5.5 Expectations for three-dimensional versus two-dimensional correlations

We have seen that the reach of the CMB for extracting topological information is likely limited to the case where the distance to our nearest clone is not much more than the diameter of the last scattering surface. We expect that three-dimensional data from the full interior of the last scattering surface — such as might eventually be obtained using some combination of large-scale structure surveys, intensity mapping, and other data types — would allow us to extend this reach, possibly substantially.

In section 4, we identified two topological effects on eigenmodes that could result in measurable effects on observables — the discretization of the set of allowed wavevectors $k_n$, and, in the non-homogeneous manifolds, the induced correlation between Fourier modes of different allowed wavevectors of the same magnitude. The observability of the discretization is limited by the observer's ability to resolve different wavevectors. The separation between allowed wavevectors is $\sim 2\pi/L_{top}$, where $L_{top}$ is some characteristic length scale associated with the topology. The observer's resolution is limited in principle to $\sim 2\pi/L_{SS}$, leading the detectability to fall as $L_{SS}/L_{top} \to \infty$. However, as the magnitude of the wavevector increases, and the number of modes with the same magnitude of wavevector grows as $k^2$, it may become more and more difficult to attain the target angular resolution in the wavevector space, further degrading the ability to detect topological information.

Meanwhile, if we are in an inhomogeneous space in which one (or more) of the group generators involves a rotation about some axis, then all pairs of Fourier modes with wavevectors related by that same rotation have perfectly correlated amplitudes. This is true no matter the magnitude of the wavevector. That sounds like a potentially very powerful signal of topology. However, the phase angle of the correlation grows as the magnitude of the wavevector increases. For large enough $k$, neighboring wavevectors have essentially random phases. This will degrade the utility of the correlation information if observations average over nearby wavevectors.

All this needs to be evaluated carefully and is the subject of an upcoming paper [91] targeted at this and related topics.

## 6 Conclusion

The local spatial geometry of the Universe may be Euclidean (i.e., flat) when averaged over volumes that are large compared to the largest structures but small compared to our Hubble volume or to the size of the last scattering sphere; however, that does not mean that the topology is that of the infinite-in-all-directions “covering space” of Euclidean geometry $E^3$. There are eighteen possible topologies (labeled $E_1-E_{18}$) for three-manifolds that have homogeneous local flat geometry, the covering space is just one of them — $E_{18}$. In this paper, we have considered the ten topologies that have orientable Euclidean manifolds; in an upcoming paper, we will consider the remaining eight topologies that have non-orientable Euclidean manifolds [92]. These ten topologies of orientable manifolds fall into four classes: six with compact manifolds, i.e., all three spatial dimensions are compact, ($E_1-E_6$); two that have two compact dimensions ($E_{11}$ and $E_{12}$); one that has one compact dimension ($E_{16}$); and the covering space $E_{18}$, which has no compact dimensions.

For each of these topologies (cf. section 3) we have provided a completely general parametrization of all the possible homogeneous manifolds. This builds on previous work (see
especially ref. [25]), which included all these topologies, but not, or not explicitly, with their
most general parametrization. In particular, we have allowed the full range of possibilities
for the translation vectors associated with the generators of $E_2-E_5$ (that was already the
case for $E_1$), the two generators of $E_{11}$ and $E_{12}$, and the generator of $E_{16}$. Most new,
however, is the recognition that group actions that lead to the so-called slab space $E_{16}$
are not generically pure translations, but corkscrew motions; the standard group action
based on pure translations is a very special case because it is the only $E_{16}$ manifold that is
homogeneous. We have also identified an "associated $E_1$" for each manifold of topologies
$E_2-E_6$, an associated $E_{11}$ for each $E_{12}$ manifold, and an associated unrotated, and therefore
homogeneous, $E_{16}$ (i.e., $E_{16}^{(h)}$) for each rotated $E_{16}$ (i.e., $E_{16}^{(i)}$) where that is possible, i.e.,
where the rotation is a rational multiple of $2\pi$.

In section 4, we have provided the general analytic expressions for the eigenmodes of the
(scalar) Laplacian for each of these orientable manifolds, which are linear combinations of
Fourier modes. Once again, we realize that $E_{16}^{(h)}$ is a very special case of $E_{16}$. In considering a
generic slab manifold, $E_{16}^{(i)}$, based on a corkscrew motion we find that the rotation associated
with that corkscrew must be a rational multiple of $2\pi$ in order for the eigenmodes (of the
scalar Laplacian) not to have azimuthal symmetry around the corkscrew axis, and thus
to be a basis set for general functions on the manifold. This is a situation unlike any of
the other $E^3$ topologies.

We have also presented the Fourier-mode correlation matrices $C_{knk'n'}^{E_i;XY}$ for the amplitudes
of the Fourier modes of the associated $E_1$, $E_{11}$, and $E_{16}^{(h)}$, under the standard assumption
that the primordial amplitudes of the eigenmodes of a particular scalar field, e.g., the gauge-
invariant curvature potential $\mathcal{R}$, are Gaussian random statistically independent variables of
zero mean with a variance that depends only on the eigenvalue of the Laplacian. The $X$ and
$Y$ label two scalar fields (potentially at some later epoch) that are linearly related to that
primordial field through transfer functions $\Delta^X(k)$ and $\Delta^Y(k)$. ($X$ and $Y$ are not necessarily
different.) Finally, we have presented, for similar scalar fields "projected" onto the sky, the
spherical-harmonic-spherical-harmonic correlation matrix $C_{\ell m \ell' m'}^{E_i;XY}$ for each topology as a
function of manifold parameters. Once again, this builds on previous work, especially ref. [25],
but allows for the most general parametrization of each manifold and of the observer's
position therein.

We have explicitly included the effect of the choice of origin $\mathbf{x}_0$ on the analytic expressions
for the eigenmodes, and the correlation matrices. This is an important addition because,
except for $E_1$, $E_{11}$, $E_{16}^{(h)}$, and $E_{18}$, the manifolds are inhomogeneous (i.e. expectation values
of observables are inhomogeneous), and the expectations for the distributions of observables
depend on the location of an observer in the manifold.

In section 5, we have studied the CMB temperature spherical-harmonic correlation
matrices $C_{\ell m \ell' m'}^{E_i;TT}$ for $E_1-E_6$. We have shown, in figure 3, how a preferred orientation of
the observer concentrates the correlations in a small fraction of the matrix elements, while
a random orientation diffuses it broadly across the matrix. We have suggested for future
consideration that this existence of a preferred orientation of the observer may constitute a
signature of topology that could be used to identify the presence of a non-trivial topology. In
figure 4, as well as figures 6 to 8, we have presented the spherical-harmonic correlation matrices
for $E_1-E_6$ for specific simple choices of the manifold parameters and the observer location.