{% raw %} Title: Create a Markdown Blog Post Integrating Research Details and a Featured Paper ==================================================================================== This task involves generating a Markdown file (ready for a GitHub-served Jekyll site) that integrates our research details with a featured research paper. The output must follow the exact format and conventions described below. ==================================================================================== Output Format (Markdown): ------------------------------------------------------------------------------------ --- layout: post title: "On the spatial distribution of the Large-Scale structure: An Unsupervised search for Parity Violation" date: 2024-10-21 categories: papers --- ![AI generated image](/assets/images/posts/2024-10-21-2410.16030.png) Samuel HewsonWill HandleyChris Lester Content generated by [gemini-1.5-pro](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/2024-10-21-2410.16030.txt). Image generated by [imagen-3.0-generate-002](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/2024-10-21-2410.16030.txt). ------------------------------------------------------------------------------------ ==================================================================================== Please adhere strictly to the following instructions: ==================================================================================== Section 1: Content Creation Instructions ==================================================================================== 1. **Generate the Page Body:** - Write a well-composed, engaging narrative that is suitable for a scholarly audience interested in advanced AI and astrophysics. - Ensure the narrative is original and reflective of the tone and style and content in the "Homepage Content" block (provided below), but do not reuse its content. - Use bullet points, subheadings, or other formatting to enhance readability. 2. **Highlight Key Research Details:** - Emphasize the contributions and impact of the paper, focusing on its methodology, significance, and context within current research. - Specifically highlight the lead author ({'name': 'Samuel Hewson'}). When referencing any author, use Markdown links from the Author Information block (choose academic or GitHub links over social media). 3. **Integrate Data from Multiple Sources:** - Seamlessly weave information from the following: - **Paper Metadata (YAML):** Essential details including the title and authors. - **Paper Source (TeX):** Technical content from the paper. - **Bibliographic Information (bbl):** Extract bibliographic references. - **Author Information (YAML):** Profile details for constructing Markdown links. - Merge insights from the Paper Metadata, TeX source, Bibliographic Information, and Author Information blocks into a coherent narrative—do not treat these as separate or isolated pieces. - Insert the generated narrative between the HTML comments: and 4. **Generate Bibliographic References:** - Review the Bibliographic Information block carefully. - For each reference that includes a DOI or arXiv identifier: - For DOIs, generate a link formatted as: [10.1234/xyz](https://doi.org/10.1234/xyz) - For arXiv entries, generate a link formatted as: [2103.12345](https://arxiv.org/abs/2103.12345) - **Important:** Do not use any LaTeX citation commands (e.g., `\cite{...}`). Every reference must be rendered directly as a Markdown link. For example, instead of `\cite{mycitation}`, output `[mycitation](https://doi.org/mycitation)` - **Incorrect:** `\cite{10.1234/xyz}` - **Correct:** `[10.1234/xyz](https://doi.org/10.1234/xyz)` - Ensure that at least three (3) of the most relevant references are naturally integrated into the narrative. - Ensure that the link to the Featured paper [2410.16030](https://arxiv.org/abs/2410.16030) is included in the first sentence. 5. **Final Formatting Requirements:** - The output must be plain Markdown; do not wrap it in Markdown code fences. - Preserve the YAML front matter exactly as provided. ==================================================================================== Section 2: Provided Data for Integration ==================================================================================== 1. **Homepage Content (Tone and Style Reference):** ```markdown --- layout: home --- ![AI generated image](/assets/images/index.png) The Handley Research Group is dedicated to advancing our understanding of the Universe through the development and application of cutting-edge artificial intelligence and Bayesian statistical inference methods. Our research spans a wide range of cosmological topics, from the very first moments of the Universe to the nature of dark matter and dark energy, with a particular focus on analyzing complex datasets from next-generation surveys. ## Research Focus Our core research revolves around developing innovative methodologies for analyzing large-scale cosmological datasets. We specialize in Simulation-Based Inference (SBI), a powerful technique that leverages our ability to simulate realistic universes to perform robust parameter inference and model comparison, even when likelihood functions are intractable ([LSBI framework](https://arxiv.org/abs/2501.03921)). This focus allows us to tackle complex astrophysical and instrumental systematics that are challenging to model analytically ([Foreground map errors](https://arxiv.org/abs/2211.10448)). A key aspect of our work is the development of next-generation SBI tools ([Gradient-guided Nested Sampling](https://arxiv.org/abs/2312.03911)), particularly those based on neural ratio estimation. These methods offer significant advantages in efficiency and scalability for high-dimensional inference problems ([NRE-based SBI](https://arxiv.org/abs/2207.11457)). We are also pioneering the application of these methods to the analysis of Cosmic Microwave Background ([CMB](https://arxiv.org/abs/1908.00906)) data, Baryon Acoustic Oscillations ([BAO](https://arxiv.org/abs/1701.08165)) from surveys like DESI and 4MOST, and gravitational wave observations. Our AI initiatives extend beyond standard density estimation to encompass a broader range of machine learning techniques, such as: * **Emulator Development:** We develop fast and accurate emulators of complex astrophysical signals ([globalemu](https://arxiv.org/abs/2104.04336)) for efficient parameter exploration and model comparison ([Neural network emulators](https://arxiv.org/abs/2503.13263)). * **Bayesian Neural Networks:** We explore the full posterior distribution of Bayesian neural networks for improved generalization and interpretability ([BNN marginalisation](https://arxiv.org/abs/2205.11151)). * **Automated Model Building:** We are developing novel techniques to automate the process of building and testing theoretical cosmological models using a combination of symbolic computation and machine learning ([Automated model building](https://arxiv.org/abs/2006.03581)). Additionally, we are active in the development and application of advanced sampling methods like nested sampling ([Nested sampling review](https://arxiv.org/abs/2205.15570)), including dynamic nested sampling ([Dynamic nested sampling](https://arxiv.org/abs/1704.03459)) and its acceleration through techniques like posterior repartitioning ([Accelerated nested sampling](https://arxiv.org/abs/2411.17663)). ## Highlight Achievements Our group has a strong publication record in high-impact journals and on the arXiv preprint server. Some key highlights include: * Development of novel AI-driven methods for analyzing the 21-cm signal from the Cosmic Dawn ([21-cm analysis](https://arxiv.org/abs/2201.11531)). * Contributing to the Planck Collaboration's analysis of CMB data ([Planck 2018](https://arxiv.org/abs/1807.06205)). * Development of the PolyChord nested sampling software ([PolyChord](https://arxiv.org/abs/1506.00171)), which is now widely used in cosmological analyses. * Contributions to the GAMBIT global fitting framework ([GAMBIT CosmoBit](https://arxiv.org/abs/2009.03286)). * Applying SBI to constrain dark matter models ([Dirac Dark Matter EFTs](https://arxiv.org/abs/2106.02056)). ## Future Directions We are committed to pushing the boundaries of cosmological analysis through our ongoing and future projects, including: * Applying SBI to test extensions of General Relativity ([Modified Gravity](https://arxiv.org/abs/2006.03581)). * Developing AI-driven tools for efficient and robust calibration of cosmological experiments ([Calibration for astrophysical experimentation](https://arxiv.org/abs/2307.00099)). * Exploring the use of transformers and large language models for automating the process of cosmological model building. * Applying our expertise to the analysis of data from next-generation surveys like Euclid, the Vera Rubin Observatory, and the Square Kilometre Array. This will allow us to probe the nature of dark energy with increased precision ([Dynamical Dark Energy](https://arxiv.org/abs/2503.08658)), search for parity violation in the large-scale structure ([Parity Violation](https://arxiv.org/abs/2410.16030)), and explore a variety of other fundamental questions. Content generated by [gemini-1.5-pro](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/index.txt). Image generated by [imagen-3.0-generate-002](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/index.txt). ``` 2. **Paper Metadata:** ```yaml !!python/object/new:feedparser.util.FeedParserDict dictitems: id: http://arxiv.org/abs/2410.16030v2 guidislink: true link: http://arxiv.org/abs/2410.16030v2 updated: '2024-11-08T00:01:03Z' updated_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2024 - 11 - 8 - 0 - 1 - 3 - 4 - 313 - 0 - tm_zone: null tm_gmtoff: null published: '2024-10-21T14:05:16Z' published_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2024 - 10 - 21 - 14 - 5 - 16 - 0 - 295 - 0 - tm_zone: null tm_gmtoff: null title: "On the spatial distribution of the Large-Scale structure: An\n Unsupervised\ \ search for Parity Violation" title_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: "On the spatial distribution of the Large-Scale structure: An\n Unsupervised\ \ search for Parity Violation" summary: 'We use machine learning methods to search for parity violations in the Large-Scale Structure (LSS) of the Universe, motivated by recent claims of chirality detection using the 4-Point Correlation Function (4PCF), which would suggest new physics during the epoch of inflation. This work seeks to reproduce these claims using methods originating from high energy collider analyses. Our machine learning methods optimise some underlying parity odd function of the data, and use it to evaluate the parity odd fraction. We demonstrate the effectiveness and suitability of these methods and then apply them to the Baryon Oscillation Spectroscopic Survey (BOSS) catalogue. No parity violation is detected at any significance.' summary_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: 'We use machine learning methods to search for parity violations in the Large-Scale Structure (LSS) of the Universe, motivated by recent claims of chirality detection using the 4-Point Correlation Function (4PCF), which would suggest new physics during the epoch of inflation. This work seeks to reproduce these claims using methods originating from high energy collider analyses. Our machine learning methods optimise some underlying parity odd function of the data, and use it to evaluate the parity odd fraction. We demonstrate the effectiveness and suitability of these methods and then apply them to the Baryon Oscillation Spectroscopic Survey (BOSS) catalogue. No parity violation is detected at any significance.' authors: - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Samuel Hewson - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Will J. Handley - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Christopher G. Lester author_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Christopher G. Lester author: Christopher G. Lester arxiv_comment: 21 pages, 15 figures; updated references links: - !!python/object/new:feedparser.util.FeedParserDict dictitems: href: http://arxiv.org/abs/2410.16030v2 rel: alternate type: text/html - !!python/object/new:feedparser.util.FeedParserDict dictitems: title: pdf href: http://arxiv.org/pdf/2410.16030v2 rel: related type: application/pdf arxiv_primary_category: term: astro-ph.CO scheme: http://arxiv.org/schemas/atom tags: - !!python/object/new:feedparser.util.FeedParserDict dictitems: term: astro-ph.CO scheme: http://arxiv.org/schemas/atom label: null - !!python/object/new:feedparser.util.FeedParserDict dictitems: term: astro-ph.IM scheme: http://arxiv.org/schemas/atom label: null ``` 3. **Paper Source (TeX):** ```tex % ****** Start of file apssamp.tex ****** % % This file is part of the APS files in the REVTeX 4.2 distribution. % Version 4.2a of REVTeX, December 2014 % % Copyright (c) 2014 The American Physical Society. % % See the REVTeX 4 README file for restrictions and more information. % % TeX'ing this file requires that you have AMS-LaTeX 2.0 installed % as well as the rest of the prerequisites for REVTeX 4.2 % % See the REVTeX 4 README file % It also requires running BibTeX. The commands are as follows: % % 1) latex apssamp.tex % 2) bibtex apssamp % 3) latex apssamp.tex % 4) latex apssamp.tex % \documentclass[ reprint, superscriptaddress, %[numbers]usenatbib, %groupedaddress, %unsortedaddress, %runinaddress, %frontmatterverbose, %preprint, %preprintnumbers, nofootinbib, %nobibnotes, %bibnotes, amsmath,amssymb, aps, prd, %rmp, %prstab, %prstper, floatfix, author-numerical, %author-year, ]{revtex4-2} \usepackage{graphicx}% Include figure files \usepackage{dcolumn}% Align table columns on decimal point \usepackage{bm}% bold math \usepackage[hidelinks]{hyperref}% add hypertext capabilities %\usepackage[mathlines]{lineno}% Enable numbering of text and display math %\linenumbers\relax % Commence numbering lines \usepackage[capitalise]{cleveref} \usepackage{gensymb} \usepackage{tabularx} \usepackage{booktabs} \usepackage{verbatim} \usepackage{bbold} \usepackage{braket} \usepackage{aas_macros} \usepackage{xcolor} \usepackage{comment} \newcommand*{\Comb}[2]{{}^{#1}C_{#2}}% \newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}} \newcommand\comCGL[1]{{\color{red} CGL[ #1 ]}} %\usepackage[showframe,%Uncomment any one of the following lines to test %%scale=0.7, marginratio={1:1, 2:3}, ignoreall,% default settings %%text={7in,10in},centering, %%margin=1.5in, %%total={6.5in,8.75in}, top=1.2in, left=0.9in, includefoot, %%height=10in,a5paper,hmargin={3cm,0.8in}, %]{geometry} \begin{document} \preprint{APS/123-QED} \title{On the spatial distribution of the Large-Scale structure:\\ An Unsupervised search for Parity Violation}% Force line breaks with \\ %\thanks{A footnote to the article title}% \author{Samuel Hewson} \affiliation{Astrophysics Group, Cavendish Laboratory, University of Cambridge} \affiliation{St John's College, St John's Street, Cambridge, CB2 1TP, United Kingdom} \author{Will Handley} %\email{w.handley@cam.ac.uk} \affiliation{Astrophysics Group, Cavendish Laboratory, University of Cambridge} \affiliation{Kavli Institute for Cosmology, University of Cambridge} \affiliation{Gonville \& Caius College, Trinity Street, Cambridge, CB2 1TA, United Kingdom} \author{Christopher G. Lester} \affiliation{High Energy Physics Group, Cavendish Laboratory, University of Cambridge} %\collaboration{MUSO Collaboration}%\noaffiliation %\collaboration{CLEO Collaboration}%\noaffiliation \date{\today}% It is always \today, today, % but any date may be explicitly specified \begin{abstract} We use machine learning methods to search for parity violations in the Large-Scale Structure (LSS) of the Universe, motivated by recent claims of chirality detection using the 4-Point Correlation Function (4PCF), which would suggest new physics during the epoch of inflation. This work seeks to reproduce these claims using methods originating from high energy collider analyses. Our machine learning methods optimise some underlying parity odd function of the data, and use it to evaluate the parity odd fraction. We demonstrate the effectiveness and suitability of these methods and then apply them to the Baryon Oscillation Spectroscopic Survey (BOSS) catalogue. No parity violation is detected at any significance. \end{abstract} %\keywords{Suggested keywords}%Use showkeys class option if keyword %display desired \maketitle %\tableofcontents \section{\label{sec:level1}Introduction} A parity transformation acts to spatially invert a system and parity has, for a long time, been a defining symmetry of the Standard Model of particle physics. Over the past 30 years, progress has been made in investigating whether parity has a role to play in cosmology. Until very recently most of that work was focused on Cosmic Microwave Background (CMB) Polarisation~\citep{PhysRevLett.83.1506.lue, Kamionkowski_2011, Shiraishi_2016, PhysRevLett.125.221301.komatsu, philcox2023cmb, philcox2024testing}, or on Gravitational Waves~\citep{Saito_2007, Yunes_2010, Wang_2013, Orlando_2021, jenks2023parameterized}. %Possible Chern-Simons discussion - also Hou et al In recent years, following the proposal of \citet{cahn2021test} the Large-Scale Structure (LSS) of the Universe has been probed for parity violations using the 4-Point Correlation Function (4PCF). So far these studies have used the Baryon Oscillation Spectroscopic Survey (BOSS)~\citep{10.1093/mnras/stx721} of the Sloan Digital Sky Survey (SDSS)-III~\citep{2011AJ....142...72E, Dawson_2013} and have reported findings of parity violation at the $8.1\sigma$~\citep{philcox2021detection}, $7.1\sigma$~\citep{Hou_2023} and $2.9\sigma$-levels~\citep{Philcox_2022}. In three dimensions the lowest order polyhedron subject to parity asymmetry is the tetrahedron, as shown in \cref{fig:tetrahedron_mirror}. The 4PCF is, therefore, the lowest order $N$-point correlation function which can be probed for parity violations~\citep{cahn2021test}. Tetrahedra are constructed with galaxies at each of the four vertices, and the relative positions are used to calculate the 4PCF. This method facilitates searches across groups of four galaxies but is, by design, constrained to only search for parity violations at the four-point level. This limits the type and order of parity violations that could be detected. It is possible to consider higher $N$-point correlation functions, but these quickly become intractable. On top of this there could be parity violations which manifest through multiple orders of correlation, which would be undetectable. A second drawback of the method is the apparent difficulty in quantifying the significance of the detection. \citet{Philcox_2022} performed both a rank test and a $\chi^2$ test, and compared the results to 2048 MultiDarkPATCHY simulations. These simulations are calibrated to match the BOSS two- and three-point clustering statistics~\citep{Rodr_guez_Torres_2016, Kitaura_2016}, not the 4-point, which could lead to an underestimate of the uncertainty in the 4PCF calculations. \citet{Hou_2023} quantify the significance by calculating the uncertainty in the covariance matrix, this could be influenced by observational effects and instrumental systematics ~\citep{Hou_2023}, and assumes that the 4PCF follows a Gaussian distribution~\citep{2022PhRvD.106d3515H}. In fact, recent work by \citet{krolewski2024evidenceparityviolationboss} shows that correctly accounting for 8-point contributions reduces the significance of detection down to between $0\ \mathrm{and}\ 2.5\sigma$. \begin{figure} \centering \includegraphics{Images_pdf/tetrahedron_mirror.pdf} \includegraphics{Images_pdf/triangle_mirror.pdf} \caption{\textbf{Top:} A tetrahedron is the lowest order polygon subject to parity violation in three dimensions, therefore the 4PCF is the lowest order $N$-point correlation function which can be probed for parity violations. \textbf{Bottom:} three points can be parity violating in two dimensions, but in three dimensions the 2d parity operation becomes equivalent to a 180\degree\ rotation. Consider rotating into the page about the reflection axis. Figure adapted from \citet{taylor2023}.} \label{fig:tetrahedron_mirror} \end{figure} The aim of this paper is to demonstrate that more general methods can be used to search for parity violations in the LSS. Such methods do not have to be specifically 4-point, or even $N$-point, but rather consider the data in spatially localised regions and search for underlying parity violations across all $N$-point correlations. A more specialised method will than be implemented for the 4-point case, constructed with no knowledge of a 4PCF, and implemented to see if the results of \citet{Hou_2023} and \citet{Philcox_2022} can be reproduced. Similar methods to the ones we applied have also been proposed and tested with CNNs and Neural Field Scattering Transforms (NSFT) \citep{taylor2023, craigie2024unsupervisedsearchescosmologicalparity}. With the recent data release from the Dark Energy Spectroscopic Instrument (DESI)~\citep{2016arXiv161100036D, https://doi.org/10.5281/zenodo.7964161} and the upcoming releases from EUCLID~\citep{2011_EUCLID_def, 2020_Euclid} and the Nancy Grace Roman Space Telescope~\citep{2015Roman, 2019Roman}, the methods proposed her will be useful in future exploration of the LSS. \vspace{-0.4em} \section{Theory and Methods} \subsection{Theory of Parity} In Quantum Mechanics the Parity Operator, $\hat{P}$ is defined to spatially invert the wavefunction of a particle, such that a wavefunction $\Psi(x)$ is Parity-even if $\Psi(\hat{P}x)\ =\ \Psi(x)$ and Parity-odd if $\Psi(\hat{P}x)\ =\ -\Psi(x)$. The importance of parity pervades throughout physics, most notably in the Standard Model of particle physics, where parity is a symmetry of both Quantum Electrodynamic (QED) and Quantum Chromodynamic (QCD) interactions, but is violated by Weak interactions~\citep{PhysRev.104.254-weaktheory, PhysRev.105.1413-weakexp}. The creation of the present-day matter-antimatter asymmetry would require baryogenesis processes that violate charge and parity conservation~\citep{Sakharov:1967dj, Carroll_1998}, but since gravity is parity symmetric~\citep{Hilbert1915}, all cosmological correlators should be too. This means that parity asymmetry in the LSS would require new physics in the epoch of inflation. \subsection{Method of Lester and Tombs} Current theoretical models predict parity violating mechanisms which are not observable at the Large Hadron Collider (LHC)~\citep{Lester_2019}, and accordingly the LHC has not been used to search for these. Parity may however be violated by unknown means, which could be detectable in LHC deposit data. Motivated by this possibility, \citet{lester2022using} demonstrated that machine learning can be used to detect parity violating functions within datasets without any required knowledge of what is causing the violation. By training an image classification Convolutional Neural Network (CNN) and rewarding it for detecting asymmetry, a measure of the parity of the image dataset can be determined. To demonstrate this method, \citet{lester2022using} applied their algorithm to the MNIST data as well as more general symmetries~\citep{Tombs_2022}. The success of the algorithm on the MNIST data also displays the generality of the method, and its applicability outside particle physics. To further demonstrate the general success of the \citet{lester2022using} method, a Master's project was run in 2022/3 aiming to use the method to test for parity violations in snails~\citep{lester_gibbon2022}. The population was known to be parity violating as most snail species have a strong preference for the handedness of their shells' spirals. It was successfully shown that the unsupervised learning method could detect parity violations within the images. \subsubsection{Algorithm} To quantify any parity violations existing in a sample, consider some function $g(x)$, where $g$ represents the outputs of the CNN with inputs $x$. Then construct $f(x)$ to be the difference between the outputs from $x$, and the parity transform of $x$: \begin{equation}\label{f} f(x) = g(x) - g(\hat{P}x) \end{equation} from which it is immediately clear that $f(\hat{P}x) = -f(x)$. In fact further symmetries of the system can be specified: let ${S = \{S_1, S_2, ..., S_i\}}$ be the set of all the symmetries of the system. Then $f(x)$ can be defined as the sum over $S_i$ of the difference between the outputs from the original and parity transformed inputs as demonstrated in \cref{general_parity_f}: \begin{equation}\label{general_parity_f} f(x) = \sum_{i}{g(\hat{S_i}x) - g(\hat{S_i}\hat{P}x)} \end{equation} \cref{f180_rot} is an example of this where $f(x)$ has been constructed to obey invariance under 180\degree\ rotations, represented by $\hat{R}_{180}$. \begin{equation}\label{f180_rot} f(x) = g(x) + g(\hat{R}_{180}x) - g(\hat{P}x) + g(\hat{R}_{180}\hat{P}x) \end{equation} This creates a function for which $f(x) = f(\hat{R}_{180}x)$, enforcing 180\degree\ rotational symmetry and removing any idea of a `top' from the images. Accounting for symmetries in $f$ means they do not need to be accounted for in the subnetwork $g$, reducing the processing required on the inputs. Further symmetries are enforced by max-pooling layers within the network. \cref{f} will output zero for a parity even function $g(x)$, and the network can be constructed to return outputs $g$ such that the difference of $f$ from zero is maximised. For each batch $B = \{x_1, x_2, ..., x_{\lvert B \rvert} \}$, the mean value of $f$ is calculated. \begin{equation}\label{mean_f} \mu_B = \frac{1}{\lvert B \rvert}\sum_{x \in B} f(x) \end{equation} The loss function is defined \begin{equation}\label{loss_func} \mathcal{L}_B = -\mu_B /\sigma_B, \end{equation} where \begin{equation} \sigma_\mathrm{B}^2 = \frac{1}{\lvert B \rvert}\left(\sum_{x\in B}[f(x)]^2\right) - \mu_B^2 \end{equation} is the variance of $f$ over batches, which ensures that the network will not trivially increase the loss under $f(x) \mapsto \lambda f(x)$ for $\lambda > 1$. After training the neural net, the function $f(x)$ can be evaluated on the testing data. If the mean value of $f$ on that dataset differs from zero by a statistically significant amount, some parity-violating feature in the data has been detected. We compute the fraction of the outputs for which $f(x) > 0$. This `positive fraction' quantifies the proportion of the inputs that exist in a parity violating way, which means achiral datasets are expected to output 50\%, and deviations from this imply a parity violating dataset. It is important to note here that when searching for a violation of some symmetry then a function $f$ can provide evidence if \textbf{both}: \begin{enumerate} \item $f$ has the correct symmetry properties to permit the discovery, and \item $f$ yields statistically significant outputs when evaluated on independent datasets that it has not seen before and that were not used in its construction \end{enumerate} Here we have constructed $f$ by hand to be parity-odd, which satisfies the first requirement. The second requirement is demonstrated in \cref{sec:Valid}. From these principles it follows that the most important step here is the construction of $f$. This method could, in principle, work without any training. The training step just acts to increase the likelihood of a discovery by optimising, but not changing the form, of $f$. This also means that the net has the benefit of never creating a false positive from overtraining. If there is a positive detection, the data must inherently contain the parity violation specified by the function $f$, regardless of whether that $f$ was obtained by overtraining. \section{Dataset} \label{sec:Data} Following from \citet{Hou_2023} and \citet{Philcox_2022}, we use data release 12 (DR12) of the BOSS catalogue, which contains LOWZ and CMASS samples, each being split into the North Galactic Cap (NGC) and South Galactic Cap (SGC). The target selection algorithm results in samples of primarily Luminous Red Galaxies (LRGs), with CMASS further specified to select objects of uniform redshift~\citep{2016_REID_BOSS}. This leads to a roughly mass-limited sample down to a stellar mass $M \sim 10^{11.3}h^{-1}M_\odot$~\citep{marston_mNRAS_2013, 10.1093/mnras/stw1080, 10.1093/mnras/stw117, Bundy_2017}. A redshift cutoff of $0.43 < z < 0.7$ is applied, which raises the purity of the CMASS sample by adding more LRGs~\citep{Hou_2023}. For this investigation, the CMASS NGC was used, and the catalogue of galaxies was treated as a field of points in three-dimensional space, a 2d projection of this is shown in \cref{fig:cmass_NGC}. The BOSS dataset contains data in the form of right ascension, declination and redshift. This was converted into spatial coordinates using the fiducial cosmology specified by the BOSS collaboration~\citep{Alam_2017}. Namely, a flat $\Lambda$CDM model, with matter density $\Omega_\mathrm{m} = 0.31$, Hubble constant $h \equiv H_0/(100\text{km\ s}^{-1}\ \text{Mpc}^{-1}) = 0.676$, baryon density $\Omega_\mathrm{b}h^2 = 0.022$, fluctuation amplitude $\sigma_8 = 0.8$ and spectral tilt $n_\mathrm{s} = 0.97$. From this, the question can be asked: Is the spatial distribution of the points in the field parity violating? More specifically, is the spatial distribution of the points in the field parity violating on a given scale? The scale of the parity violation is specified to match the work of \citet{Hou_2023}. This means for the geometric analyses, groups are constructed such that points within a group are separated by a distance, ranging from $20h^{-1}$Mpc to $160h^{-1}$Mpc. To each galaxy a weight was applied: \begin{equation} w = w_\mathrm{fkp}w_\mathrm{sys}(w_\mathrm{rf} + w_\mathrm{cp} - 1) \end{equation} where $w_\mathrm{rf}$ is the redshift failure weight and $w_\mathrm{cp}$ the fibre collision weight; these are additive weights with default value of unity. $w_\mathrm{sys}$ is the systematic weight, which accounts for stellar density and seeing~\citep{Anderson_2012_weights}, and the FKP weight is defined as ${w_\mathrm{fkp} = [1+n(z)P_0]^{-1}}$~\citep{Feldman_1994_fkp}, where $n(z)$ is the weighted number density as a function of redshift, and $P_0 = 10^4[h^{-1}\text{Mpc}]^{-3}$. \begin{figure} \centering \includegraphics[width=\linewidth]{Images_pdf/CMASS_global_pic.pdf} \caption{The North Galactic Cap of the CMASS catalogue.} \label{fig:cmass_NGC} \end{figure} \section{Validating the Methods}\label{sec:Valid} To begin, in \cref{sec:2d_conv} we use the CNN architecture of \citet{lester2022using} to examine a two-dimensional projection of the data. This is followed \cref{sec:3d_CNN} by the addition of colour channels to the images, with the aim of representing three-dimensional information via colour. Finally, we investigate more geometric approaches in \cref{sec:angle}, which have the drawback of being constrained to some order of polyhedron, much like $N$-point correlation functions, but consequently allow direct comparison with the 4PCF. \subsection{2D Convolution}\label{sec:2d_conv} We begin by using the same architecture as \citet{lester2022using}: 2d convolutional layers, with image inputs. For our investigations, the data can be treated as a field of points, from which we constructed images to represent localised regions of the field. To ensure that this method is suitable for detecting parity violations in image data, we ran a number of tests using simulated data. These not only verify that the network is able to detect parity asymmetric data, but also help to determine a level of significance i.e.\ what net fraction of the data must contain a parity violating object to result in a non-zero detection.\ The choice of how to represent galaxies in the images is somewhat arbitrary, and two possible examples are shown in \cref{fig:Demo_images}. After testing multiple methods (\cref{appendix:image_sizes}), we decided to use PNG images generated from scatter plots of data points. An outline of this method follows: \begin{figure} \centering \includegraphics[width=\linewidth]{Images_pdf/pixel_v_graph.pdf} \caption{Two demonstration images generated from random data. \textbf{Left:} An image where each `galaxy' has its localised pixels colourised. \textbf{Right:} An image where each `galaxy' is represented by a scatter point.} \label{fig:Demo_images} \end{figure} To begin, a field of random data points was generated, and image samples created in the following manner: \begin{enumerate} \item Sample a random point from the dataset, $O$ \item Generate a square of side length $l$, centred on $O$ \item Rotate the square by a random angle, $\theta \in [0, 2\pi]$ \item Get all the data points lying within the rotated square \item Create an image of the square of data points \end{enumerate} This ensures that all the images are the same size, 64x64 pixels, following the work of \citet{lester_gibbon2022}. Testing images are generated first, and then training ones with the added condition that these images may not overlap with the testing set, to avoid overfitting. An exaggerated sample is displayed in \cref{fig:demo_random_imgsample} to clearly show the method of sampling squares. \subsubsection{Testing method with toy data} The toy data tests were constructed on the same scale as the real images would be. As outlined in \cref{sec:results} images covering 0.5\degree x 0.5\degree\ were chosen, giving an average of 22.3 galaxies per image. For the toy models we generated 12,000 images, split into 9,600 training and 2,400 testing, with each image averaging 22.3 data points. \begin{comment} \comCGL{Thank you sam for making the footnote appear as a footnote. Alas, it somehow seems partially clipped. The end is missing? } \end{comment} A number of toy datasets were curated in order to investigate the effectiveness of the method under different scenarios. The critical datasets are outlined with a description and purpose in \cref{table:CNN_datacodes}, with example images displayed in \cref{fig:Demo_samples}. Each dataset has an associated control dataset, which ensures the function of the net. A control group is generated by iterating through the real group, and flipping each image with a 50\% probability. For any large parity odd dataset this symmetrisation process always creates an achiral dataset, which can act as a control.\footnote{A right-handed figure drawn on a flat piece of tracing paper becomes left-handed if you are allowed to turn the tracing paper over and view it from the back. The handedness of two-dimensional objects are therefore only well-defined if the surfaces on which they live are orientable or are oriented in some special way(s). The toy datasets in \cref{table:CNN_datacodes} are all intrinsically two-dimensional and so, for the reason just given, any of these datasets which violate parity only do so on account of the embedded objects being implicitly pasted the `inside' of the celestial sphere, centered on earth, from where they are viewed. Such geocentric parity violation is, of course, entirely unlike the sorts of large scale parity violation one might want to see in the real universe. Any large scale parity violation will (presumably) not rely on the earth being at a special place in the universe. The geocentric nature of the parity violation in these toy datasets does not prevent them being used for the purposes they were created, however. Furthermore, this geocentricity is completely removed in the parity violating datasets introduced later in \cref{table:3d_cnn_datacodes}. These later toy datasets contain projections of intrinsically three-dimensional (parity violating) structures without any preferred orientations, and so are representative of a sort of parity violation that one might hope to see in reality.} Each data type was created from the same randomly generated background field, repeated over 10 different background fields to create 10 variations of each data type. The average `positive fraction' over the 10 repeats is listed in \cref{table:CNN_datacodes}. \begin{figure} \centering \includegraphics[width=0.8\linewidth]{Images_pdf/test_demo_2.pdf} \caption{A demonstration of the image sampling technique, on a background random field of 500,000 data points, with 12 testing samples shown in black and 48 training samples shown in red.} \label{fig:demo_random_imgsample} \end{figure} \begin{figure} \centering \includegraphics[width=\linewidth]{Images_pdf/2d_demo_set.pdf} \caption{A sample of toy images labelled by their data codes from \cref{table:CNN_datacodes}. For the purpose of demonstration, smaller spirals that have been overlaid have been coloured green. During testing, all images where in black and white. } \label{fig:Demo_samples} \end{figure} \begin{table*} \begin{tabular}{ p{2.35cm} p{5.5cm} p{4.9cm} P{2.0cm} P{2.0cm}} \toprule \textbf{Data code} & \textbf{Description} & \textbf{Purpose} & \textbf{Detection} & \textbf{Control}\\\midrule RI & Random Generated Image Data & Check sampling method isn't introducing parity violation & $49.9\pm0.3\%$ & $49.8\pm0.3\%$ \\\hline MNIST-RI & RI with random MNIST image overlaid in random position & Check detection of complex parity violation & $87.5\pm0.3\%$ & $50.3\pm0.3\%$ \\\hline F-MNIST-RI & MNIST-RI, but each MNIST image has a 50\% chance of being flipped vertically before inserting & Check method of insertion is not causing spurious detection & $50.0\pm0.3\%$ & $50.4\pm0.3\%$ \\\hline SPR-RI-100 &RI with 100 point spiral inserted in random position and random radius & Is parity violation in the same form as the data detected & $98.2\pm0.1\%$ & $49.5\pm0.3\%$ \\\hline F-SPR-RI-100 &SPR-RI-100, but each spiral has a 50\% chance of being flipped vertically before inserting &Check method of insertion is not causing spurious detection & $49.6\pm0.3\%$ & $49.5\pm0.3\%$ \\\hline SPR-RI-10 &RI with 10 point spiral inserted in random position and random radius & Is parity violation detectable on a more realistic scale & $97.9\pm0.1\%$ & $50.1\pm0.3\%$ \\\hline F-SPR-RI-10 &SPR-RI-10, but each spiral has a 50\% chance of being flipped vertically before inserting &Check method of insertion is not causing spurious detection & $49.8\pm0.1\%$ & $51.6\pm0.3\%$ \\\hline SPR-RI-3 &RI with 3 point spirals (triangles) inserted in random position and of random radius & Is parity violation detectable on the smallest scales & $93.5\pm0.2\%$ & $51.1\pm0.3\%$ \\\hline F-SPR-RI-3 &SPR-RI-3, but each spiral has a 50\% chance of being flipped vertically before inserting &Check method of insertion is not causing spurious detection & $51.9\pm0.3\%$ & $49.2\pm0.3\%$ \\\hline LH-RI-0.1 & RI where every point has 2 extra points generated close to it forcing a parity odd `left-handed' triangle on the scale of 1 tenth of the average inter-point distance & Can other types of parity violation be detected & $74.6\pm0.3\%$ & $49.6\pm0.3\%$ \\\hline LH-RI-1 & LH-RI-0.1 with extra points on the scale of the average inter-point distance & Can more realistic types of LH-RI-0.1 parity violation be detected & $67.6\pm0.3\%$ & $49.2\pm0.3\%$ \\\hline LH-RI-10 & LH-RI-0.1 with extra points on the scale of ten times the average inter-point distance & Can more realistic types of LH-RI-0.1 parity violation be detected & $52.3\pm0.3\%$ & $49.13\pm0.3\%$ \\\hline SPR-RI-5-1\% &RI with 5 point spiral inserted in random position and random radius in 1\% of images & How sensitive is the detection & $57.6\pm0.3\%$ & $49.8\pm0.3\%$ \\\hline SPR-RI-5-0.1\% &RI with 5 point spiral inserted in random position and random radius in 0.1\% of images & How sensitive is the detection & $47.6\pm0.3\%$ & $49.5\pm0.3\%$ \\\hline SPR-RI-5-0.01\% &RI with 5 point spiral inserted in random position and random radius in 0.01\% of images & How sensitive is the detection & $48.0\pm0.3\%$ & $48.8\pm0.3\%$ \\\bottomrule \end{tabular} \caption{A summary of the crucial image based toy datasets. Each dataset is given a code and a description and has its testing purpose outlined. The detection column contains the positive fraction detection, and the control column shows the positive fraction for the control group for each test.} \label{table:CNN_datacodes} \end{table*} \subsubsection{Summary of tests} The results of the toy data samples demonstrate that the net does not find parity violations where there are none and that it does detect parity violations where they exist on large enough scales. Crucially, it also shows that the method of sampling is not creating spurious detection. The ability to detect varying types of complex parity violation suggests that this method should be successful in finding any such violations in the real data. These tests also give insight into the sensitivity of the method: having just 0.1\% of the images contain parity violation is enough for it to be successfully detected. Perhaps the most powerful tests are \textbf{SPR-RI-3} and \textbf{LH-RI-10}, which show that images without any visible asymmetry, can be detected by the net. \subsection{2D Convolution with colour}\label{sec:3d_CNN} The 2d CNN is constrained by its loss of 3 dimensional information, meaning not all forms of 3d parity violation are detectable. A first attempt to resolve this is made by introducing colour to the images. By normalising the axis into the page and translating it to pixel colour, it is possible to add extra information to the two-dimensional images. With this modification each image represents a voxel of the space displayed as a 2d projection, with the third dimensional information encoded in the colour of a given point, this creates a colour CNN (cCNN). As in the previous case, we generate a random field of points, now in 3d space. Images are constructed following the same process outlined in \cref{sec:2d_conv}, with the additional step of using z values and weights to colour the pixels. Under this construction, each pixel has two degrees of freedom we need to express. The first is the distance of the galaxy and the second is its weight. To represent these degrees of freedom we assign RGB values to the pixels in the following manner: \begin{equation} (r, g, b)_{jk} = \sum_{i \in jk}{w_i\cdot\bm{c}(z_i)} \end{equation} \begin{equation} \bm{c}(z) = z \cdot \bm{R} + (1 - z) \cdot \bm{B} \end{equation} Where $w_i$ is the weight of the $i$th galaxy in pixel $jk$ and $\bm{R} = (1, 0, 0)$ and $\bm{B} = (0, 0, 1)$. In this way the distance correlates to the ratio between the red and blue channels, and the weight is represented by the intensity of the pixel. The background is set to $(0, 0, 0)$, so the net is looking at $\mathcal{O}(1)$ deviations from 0. A demonstration of this is visible in \cref{fig:demo_helices}. To account for the slight issues that arise from the fact that some the weights are larger than 1.0 and the fact that the clipping of values to fit in the integer range (0, 255) will cause some rounding issues, we also create an array dataset. This dataset contains the same 3 colour channel arrays of 64x64 pixels as would be extracted from the images, but contains slightly improved accuracy values. This array can then easily be manipulated in the same way as the image data. Under this construction we use the following identity to simplify the parity operator, to just consider an $x$ flip: \begin{equation} \hat{P} \equiv \hat{P}_x\hat{P}_y\hat{P}_z = \hat{P}_{i\in \{x,y,z\}}\hat{R} \end{equation} We then enforce rotational invariance in the 2d plane, so for inputs $\alpha$ construct $f(\alpha)$ in the following way: \begin{equation} f(\alpha) =\sum_{\hat{R}_i}\ \left[g(\hat{R}_i\alpha) - g(\hat{R}_i\hat{P}_x\alpha) \right], \label{eq:comb_f} \end{equation} where $\hat{R}_i \in [\hat{R}_{0}, \hat{R}_{90}, \hat{R}_{180}, \hat{R}_{270}]$. For this setup then, we can use the same net as we used for the black and white images. \subsubsection{Testing method with Toy Data} \begin{table*} \begin{tabular}{ p{2.0cm} p{5.2cm} p{4.0cm} P{1.9cm} P{1.95cm} P{1.95cm}} \toprule \textbf{Data code} & \textbf{Description} & \textbf{Purpose} & \textbf{Detection} & \textbf{Control} & \textbf{Uncoloured} \\\midrule RCI & Random Generated Image Data & Check sampling method isn't introducing parity violation & $49.8\pm0.3\%$ & $50.3\pm0.3\%$ & $49.9\pm0.3\%$ \\\hline HE-RCI & RCI with 1 left-handed helix inserted into the dataset for every 100 random points & Check detection of complex parity violation & $57.5\pm0.3\%$ & $50.2\pm0.3\%$ & $49.8\pm0.3\%$ \\\hline F-HE-RCI & HE-RCI, but each helix has a 50\% chance of being left- or right-handed & Check method of insertion is not causing spurious detection & $50.5\pm0.3\%$ & $50.4\pm0.3\%$ & $50.2\pm0.3\%$ \\\hline PV-RCI & Parity violating groups of 4 points randomly oriented and distributed. Local separation $\ll$ average interpoint distance. & Can parity violations be detected where the 2d dataset does not detect & $87.2\pm0.1\%$ & $49.6\pm0.3\%$ & $50.8\pm0.3\%$ \\\bottomrule \end{tabular} \caption{A summary of the crucial colour image based toy datasets. Each dataset is given a code and a description and has its testing purpose outlined. The detection column contains the positive fraction detection; the control column shows the positive fraction for the control group and the uncoloured column shows the detection for the same images coloured monochrome.} \label{table:3d_cnn_datacodes} \end{table*} To effectively test the functionality of this method, various datasets were constructed in three-dimensional space. The key results follow in \cref{table:3d_cnn_datacodes} and a full list of the important test can be found in \cref{appendix: test_results.}. Because we are using the same net as for the 2d convolutions, we can directly compare the colouration scheme outlined in \cref{sec:3d_CNN} with inputs where all galaxies are monochrome points, as show in \cref{fig:helix_demonstration}. This shows significant improvement in detection, and results in detections that would not otherwise be made. These tests illustrate the success of the introduction of colour channels as a method for including the third dimensional information. % \begin{figure} % \centering % \includegraphics[width=0.9\linewidth]{Illustration images/image_without_background.png} % \caption{A demonstration of creating pixelised colour images. In each row the distance coordinate increases from 0 to 1.0; this changes the colour from blue to red. The top three rows have weights 1.0, 0.5 and 0.25 respectively; this changes the intensity. In the fourth row each pixel contains two `galaxies', each with weight 0.25. The fifth row contains the same points as the third row, but each occupied pixel also contains a second `galaxy' further away than the first; this creates pixels with a redder tinge.} % \label{fig:pixel_generation} % \end{figure} \begin{figure} \centering \includegraphics[width=\linewidth]{Images_pdf/Helix_demo_1_large_pdf.pdf} \caption{Sample images containing helices. The colouration creates a parity violating dataset. \textbf{Left:} all galaxies have weight 1.0, and therefore are all the same intensity. \textbf{Right:} Galaxies have a randomly assigned weight from (0.25, 0.5, 0.75), this affects intensity.} \label{fig:demo_helices} \end{figure} \begin{figure} \centering \includegraphics[width=0.9\linewidth]{Images_pdf/3d_colour_demo.pdf} \caption{Sample images from a 3d dataset containing left-handed helices and a random field of points. \textbf{Top:} Images where points are coloured by their distance into the page. The 3d information creates a parity violating dataset. \textbf{Bottom:} All points are the same colour, the loss of information creates a dataset which is not parity violating.} \label{fig:helix_demonstration} \end{figure} \subsection{2D Angle based Method}\label{sec:angle} To further the investigation into three dimensions, and facilitate testing a corollary to the 4PCF a new neural net was constructed with the same guiding principles as the CNN. Since the inputs were numeric, two options were considered: 1D convolutions or linear layers. We performed the tests on the toy data using both separately. We find that both perform equally well, but the linear layers are marginally faster, so these were chosen to be used on the real data.\\ We begin by considering the two-dimensional case. In 2d the lowest order polyhedron subject to parity violation is the triangle. A triangle in 2d space, is defined by six degrees of freedom (d.o.f.) - three two-dimensional coordinates - to which the constraints of rotational and scale invariance are added, these reduce the six d.o.f.\ down to two. Two element inputs are then sufficient, and a convenient choice is the internal angles of the triangle. To make the angles subject to parity violation, they require ordering in a way that would change under the action of a parity operator. The method used is outlined as follows for groups of 3 points: \begin{enumerate} \item Find the mean position of the group and shift it to be the origin \item Compute the distances of the points to the origin use this to order the points with the closest as the first \item Reorder the second and third in an anti-clockwise direction from the first \item Compute the angles between the points in an anti-clockwise fashion \item Define $\hat{P}$ to swap the first and third angles \item Use the first two angles in every group of 3 as the net inputs \end{enumerate} The principles of this method are displayed in \cref{fig:angles_method}, which illustrates why this is the appropriate parity operator. In fact this method is generally extensible to any $n$-gon in 2d, with the first $n-1$ angles being the input and the parity operator inverting the order of the angles as shown in \cref{eq:2d_n_angle}. Conveniently for the function of the neural net, the nth angle can be calculated from the others, as given in \cref{eq:nth_angle}. \begin{equation} \hat{P}[\theta_1, \theta_2, ..., \theta_{n-1}] = [\theta_n, \theta_{n-1}, ..., \theta_2] \label{eq:2d_n_angle} \end{equation} \begin{equation} \theta_n = 2\pi - \sum_{i=1}^{n-1}\theta_i \label{eq:nth_angle} \end{equation} \begin{figure} \centering \includegraphics[width=\linewidth]{Images_pdf/triangle_mirror_with_circles.pdf} \caption{A demonstration of the angle method in 2d. The line connecting the closest vertex to the centroid is shown in green. Working anticlockwise from this line, the initial angles are ordered [1,2,3], under a parity transformation this changes to [3,2,1].} \label{fig:angles_method} \end{figure} \subsubsection{Testing method with toy data} \begin{table*} \begin{tabular}{ p{2.35cm} p{5.5cm} p{4.9cm} P{2.0cm} P{2.0cm}} \toprule \textbf{Data code} & \textbf{Description} & \textbf{Purpose} & \textbf{Detection} & \textbf{Control}\\\midrule R2d & Random Generated 2d point Data & Check parity violation is not detected where there is none & $50.41\pm0.10\%$ & $50.10\pm0.09\%$ \\\hline PV-0.1-R2d & R2d with two extra points generated near each R2d point, these are generated in a way that all groups of three are of the same chirality. Groups of three are separated on the scale of one tenth the average inter-point distance & Check detection of parity violating inputs & $90.77\pm0.06\%$ & $49.74\pm0.10\%$ \\\hline S-PV-0.1-R2d &PV-0.1-R2d, but after generating parity violating groups all points shuffled together and new groups drawn at random &Can parity violations be detected when they are visibly present, and groups sampled randomly from the dataset & $49.70\pm0.10\%$ & $49.68\pm0.10\%$ \\\hline C-PV-0.1-R2d &S-PV-0.1-R2d, with a distance constraint when generating groups from the shuffled data & Can parity violations become detectable by constraining the scale & $65.10\pm0.04\%$ & $49.85\pm0.04\%$ \\\hline \end{tabular} \caption{2D toy datasets. Each dataset is given a code and a description and has its testing purpose outlined. The detection column contains the positive fraction detection, and the control column shows the positive fraction for the control group for each test.} \label{table:2d_ang_datacodes_summary} \end{table*} As with the CNN, our first step was to test the efficacy and sensitivity of this method. This began with testing a random field and then working through a number of toy data scenarios, the critical ones are outlined in \cref{table:2d_ang_datacodes_summary}, and the full list can be found in \cref{appendix: test_results.}. These results crucially demonstrate that this method does not detect parity violation if not present, and can detect parity violation if it is present in at least 1\% of the input data. The main drawback of this method is that for it to be successful, the input data must be sampled such that it faithfully represents the parity of the system. Conceptualise this by considering a random uniform field of two-dimensional data points. For every point in the field generate two more points, positioned relative to the first so that they form a chiral triangle, and enforce the average separation of the points in each group of three to be one tenth of the average inter-point distance. This creates a visually parity-violating dataset, as displayed in \cref{fig:2d_angle_failure}. If the inputs to the net are calculated from these groups of three, there is a strong parity detection, as expected. However, the real data will not have the benefit of being pre-selected into parity violating groups, and therefore to produce a fair test it is necessary to generate the field as before, and then sample groups of three randomly from the total field. In this scenario, it becomes almost impossible to find the parity violation, because each point in a group is being sampled from a 500,000-point field, so the chances of selecting a group that represents the chirality is negligible. This is shown in \cref{table:2d_ang_datacodes_summary} with datasets \textbf{PV-0.1-R2d} and \textbf{S-PV-0.1-R2d}. This issue can be mitigated by requiring samples to be drawn on a given scale. With this additional requirement, detection of parity violation becomes significant, as long as the scale constraint matches the scale on which the parity violation exists. This idea is demonstrated in \cref{fig:2d_angle_constraint} and shown in \cref{table:2d_ang_datacodes_summary} with dataset \textbf{C-PV-0.1-R2d}. Testing over different constraint scales, the data can be investigated in an analogous way to power spectrum analyses. \begin{figure} \centering \includegraphics[width=\linewidth]{Images_pdf/scale_issue.pdf} \caption{An illustration of the failure of the 2 dimensional angles method to find parity violations across arbitrary scales. \textbf{Left:} A localised region of the field demonstrating the observable parity violation in the data. \textbf{Right:} The full field of points, with a randomly sampled group of 3 three shown in red. This demonstrates the impossibility of finding the parity violation when sampling across any scale: the chance of picking three points that exist in a parity violating way is negligible.} \label{fig:2d_angle_failure} \end{figure} \begin{figure} \centering \includegraphics[width=\linewidth]{Images_pdf/constraint.pdf} \caption{By constraining the scale on which to construct a group of points, the net has a significantly improved likelihood of detection. \textbf{Left:} A localised region of the field of points. The field is chiral by construction, and each group of three has been coloured to show the asymmetry. \textbf{Right:} Adding a constraint on the scale over which groups can be constructed significantly increases the chances of detection.} \label{fig:2d_angle_constraint} \end{figure} \subsubsection{Summary of tests} After adding the constraint on the scale of the groups, a detection is made with a high level of significance if the constraint appropriately selects groups of data with at least $\sim1.0\%$ containing parity violation. These tests were also extended to groups of 4 and 5 points, the full results of which can be found in \cref{appendix:2d_toy_test} \subsection{3D Angle based Method} Now that the success of an angle based method in 2d has been demonstrated, this motivates exploration of a 3d equivalent. To detect parity violation, groups of four points are the smallest scale on which this method will work, and so this can act as a direct comparison to the 4PCF. As in the two-dimensional case, the critical part of this method is defining an ordering of the angles and understanding how this ordering changes under a parity operator. There are a number of ways to define an ordering, the process used for groups of four points is outlined as follows, and displayed in \cref{fig:3D_angle_demo}: \begin{enumerate} \item Compute the mean position of the group and shift it to be the origin \item Calculate the distances of the points to the origin, and order the points by distance with the closest point as the first \item Use the first position as an axis, and starting from the second-closest point, work around the axis in a right-handed manner to determine the order of points 3 and 4 \item Compute the internal angles and order them as $[\theta_{12}, \theta_{13}, \theta_{14}, \theta_{23}, \theta_{34}, \theta_{24}]$ \end{enumerate} This method fixes the positions of the first two points because that only depends on their distance to the mean. The ordering of the third and fourth points is dependent on the chirality of the group of points. This suggests an appropriate parity operator would swap positions three and four, which would reorder the angles as shown in \cref{eq:3d_angle_P}. \begin{equation}\label{eq:3d_angle_P} \hat{P}[\theta_{12}, \theta_{13}, \theta_{14}, \theta_{23}, \theta_{34}, \theta_{24}] = [\theta_{12}, \theta_{14}, \theta_{13}, \theta_{24}, \theta_{34}, \theta_{23}] \end{equation} To ensure this is always the appropriate parity operator, we generated 100,000 tetrahedra, and determined their orders by the process above, giving [1, 2, 3, 4]. Parity operators in three-dimensional space were defined: $\hat{P}_x, \hat{P}_y, \hat{P}_z$, flipping in the $yz,\ zx\ \text{and}\ xy$ planes respectively. Combinations of these parity operators were applied to each tetrahedral group. The four combinations that do not change the parity are $[\mathbb{1},\ \hat{P}_x\hat{P}_y,\ \hat{P}_y\hat{P}_z,\ \hat{P}_z\hat{P}_x]$ and these all leave the order as [1, 2, 3, 4]. The four parity flipping combinations are $[\hat{P}_x,\ \hat{P}_y,\ \hat{P}_z,\ \hat{P}_x\hat{P}_y\hat{P}_z]$ and all were found to change the order of the points to [1, 2, 4, 3]. This confirmed that the parity operator given in \cref{eq:3d_angle_P} is the correct operator in this angular description.\\ A group of 4 points in three-dimensional space has 12 degrees of freedom, - 4, 3d coordinates - and after accounting for rotational and scaling invariance, this reduces to 5 d.o.f. Since a tetrahedron has 6 internal angles ($\Comb{6}{2}$), the inputs are chosen to be the first 5: $[\theta_{12}, \theta_{13}, \theta_{14}, \theta_{23}, \theta_{34}]$. In the 2d case the final angle, which is needed for the parity transformation, can be calculated from the others, as shown in \cref{eq:nth_angle}. In the 3d case it is not so simple. Trying to use the first five angles gives four choices for the sixth. To overcome this issue, we created a custom data class, so that every data entry has all six internal angles, the first five of which provide the basic inputs to the net, and the sixth, which can be accessed to generate the parity transformed inputs. \begin{figure} \centering \includegraphics[width=\linewidth]{Images_pdf/3D_angles_demo_1.pdf} \caption{An illustration of the ordering of angles in 3 dimensions. The images show a group of four points with their mean position in red. \textbf{Left:} The points are initially ordered by distance from the mean. \textbf{Right:} The ordering is finalised by starting from the second point and working in a right-handed manner about the vector describing the position of the first point. In this case points three and four are reordered. } \label{fig:3D_angle_demo} \end{figure} \subsubsection{Testing method with toy data}\label{sec:test_3dang} Once again, the significant work in creating the net was establishing its accuracy and sensitivity. Numerous tests were carried out, with the results listed in \cref{table:3d_ang_datacodes_summary}. \begin{table*} \begin{tabular}{ p{2.35cm} p{5.5cm} p{4.9cm} P{2.0cm} P{2.0cm}} \toprule \textbf{Data code} & \textbf{Description} & \textbf{Purpose} & \textbf{Detection} & \textbf{Control}\\\midrule R3 & Random Generated 3d point Data & Check parity violation is not detected where there is none & $49.90\pm0.11\%$ & $50.11\pm0.10\%$ \\\hline PV-0.1-R3 & R3 with three extra points generated near each R3 point, these are generated in a way that all groups of four are of the same chirality. Groups of four are separated on the scale of one tenth the average inter point distance & Check detection of parity violating inputs & $87.59\pm0.11\%$ & $49.69\pm0.07\%$ \\\hline S-PV-0.1-R3 &PV-0.1-R3, but after generating parity violating groups all points shuffled together and new groups drawn at random &Can parity violations be detected when they are visibly present, and groups sampled randomly from the dataset & $50.37\pm0.08\%$ & $50.15\pm0.08\%$ \\\hline C-PV-0.1-R3 &S-PV-0.1-R3, with a distance constraint when generating groups from the shuffled date & Can parity violations become detectable by constraining the scale & $58.73\pm0.08\%$ & $49.87\pm0.08\%$ \\\hline C-PV-1-R3 &PV-0.1-R3, but scale of parity violations now same scale as average inter-point distance. No longer visibly parity violating. Groups drawn with scale constraint & Check parity violations detectable on more realistic scales with scale constraint & $54.37\pm0.05\%$ & $49.87\pm0.06\%$ \\\bottomrule \end{tabular} \caption{3D toy datasets. Each dataset is given a code and a description and has its testing purpose outlined. The detection column contains the positive fraction detection, and the control column shows the positive fraction for the control group for each test.} \label{table:3d_ang_datacodes_summary} \end{table*} The main conclusions of the tests were that the method does detect significant parity violations, as long as at least 1.2\% of the inputs are net parity violating. The main drawback of the method is the difficulty of ensuring the groups of four points are sampled such that if there is parity violation, it is appropriately represented. The dataset \textbf{S-PV-0.1-R3} shows that some constraints are needed, and as in the 2d case, it was shown (\textbf{C-PV-0.1-R3}) that constraining the scale on which groups are generated is sufficient to capture the parity violation, as long as the scales are aligned. Group \textbf{C-PV-1-R3} shows that parity violations are detected where the human eye would be unable to, a powerful demonstration of this method. \subsection{Vector input method} To finalise the search for parity violations at the 4-point level, we construct a vector method. For this net the inputs are components of the vectors describing a group of points in their mean at origin frame. To account for the scaling invariance, all inputs are normalised to make the longest vector in the group have a length of unity. The rotational invariance of the setup is accounted for in the construction of $f(x)$: \begin{multline} f(x) = \sum_{\hat{R}_i}\ \left[g(\hat{R}_ix)\ -\ \sum_{i}g(\hat{R}_i\hat{P}_ix)\ + \right. \\ \ \left. \frac{1}{2}\sum_{i \neq j}g(\hat{R}_i\hat{P}_i\hat{P}_jx)\ -\ g(\hat{R}_i\hat{P}_x\hat{P}_y\hat{P}_zx)\right] \label{eq:strange} \end{multline} where $\hat{R}_i \in [\hat{R}_{0}, \hat{R}_{90}, \hat{R}_{180}, \hat{R}_{270}]$. This ensures that every orientation of each handedness is equally represented. \begin{comment} \comCGL{I can prove to myself that $f(x)$ as given in \eqref{eq:strange} is odd with respect to any $\hat P_i$ individually. I.e. ~I can show that $f(x)=-f(\hat P_i x)$ for all $i$. From this it follows trivially that the given form of $f(x)$ is parity odd, i.e.~that $f(x)=-f(\hat P x)$ where $\hat P=\hat P_x \hat P_y \hat P_z$. Parity oddness should be the minimum requirement for using any $f(x)$, so that's good so far. However, there are simpler $f(x)$ which are also parity odd, namely $f(x)=g(x)-g(\hat P x)$, and Sam has chosen not to use those simpler choices, so I have tried to understand/guess why Sam has chosen to use the specific form of $f(x)$ given in \eqref{eq:strange}. Presumably the reason is that the chosen form has more symmetries -- so let us find out what symmetries it has! Since $f(x)$ is odd under any $\hat P _i$ (already discussed) it will be even under the application of any two $\hat P_i$ \ldots and the product of any two dissimilar $P_i$ is a 180 degree rotation about some axis (e.g.~$\hat P_x \hat P_y=R_z^{180}$). Hence it follows that the given form of $f(x)$ is invariant under 180 degree rotation about any of the three principle axes. However, so far as I have been able to show, that is where it ends. I.e.~I have not managed to demonstrate that the supplied $f(x)$ is (say) invariant under 90 degree rotations about any axis.\footnote{That I have not managed to show that invariance does not imply that the invariance is not present.} Thus I am a little confused as to what the reason was for choosing this particular $f(x)$. If it was to provide rotation invariance, why only 180 degree rotation invariance but not 90 degree? Was there a particular reason? Or was this accidental? The text says that $f(x)$ ``The rotational invariance'' is there as it ``reduces work needed on the inputs''. Perhaps that text should be changed to say more specifically that the $f(x)$ has been made invariant w.r.t.~180 degree rotations about each coordinate axis (if that's all it is) and a few words added to explain why symmetrisation stopped there rather than adding in full rotational invariance. Was it practicality? Complexity? Resources?} \end{comment} \subsubsection{Testing method with toy data} Because this also considers 3d distributions, we used the same tests as those in \cref{sec:test_3dang} on the 3d angle method. The results are listed in \cref{table:3d_vec_datacodes_summary} and show the efficacy and suitability of this method. As with the previous geometric methods, the only way to detect parity violations is to issue a constraint on the scale of the testing groups. It is important to note that the deviations from 50\% are much smaller in this case, because of the sampling over 3d, thus significance is shown using $\sigma$-levels in \cref{sec:results}. \begin{table} \centering \begin{tabular}{P{2.5cm} P{2.2cm} P{2.2cm}} \toprule \textbf{Data code} & \textbf{Detection} & \textbf{Control}\\\midrule R3 & $50.11\pm0.07\%$ & $50.21\pm0.06\%$ \\\hline PV-0.1-R3 & $99.83\pm0.02\%$ & $50.3\pm0.31\%$ \\\hline S-PV-0.1-R3 & $50.16\pm0.07\%$ & $49.84\pm0.07\%$ \\\hline C-PV-0.1-R3 & $52.44\pm0.10\%$ & $50.11\pm0.11\%$ \\\hline C-PV-1-R3 & $52.68\pm0.07\%$ & $49.91\pm0.07\%$ \\\bottomrule \end{tabular} \caption{Results for the vector input method on the 3d toy datasets, as defined in \cref{table:3d_ang_datacodes_full} The detection column contains the positive fraction detection, and the control column shows the positive fraction for the control group for each test. } \label{table:3d_vec_datacodes_summary} \end{table} \subsection{Summary of Sensitivities} After extensively testing each method, the net fraction of the inputs required to contain parity violating objects in order to make a detection significant from non-zero was determined. The level for each method is outlined in \cref{tab:frac_detect}. For a null detection, these are the levels above which we can conclude that parity violations are not present. \begin{table} \centering \begin{tabular}{c c} \toprule \textbf{Method} & \textbf{Fraction to detect}\\\midrule 2D Images & $<0.1\%$ \\\hline 3D Images & $\sim0.5\%$ \\\hline 2D Angles & $\sim1.0\%$\\\hline 3D Angles & $\sim1.0\%$ \\\hline 3D Vectors & $\sim2.0\%$\\\hline \end{tabular} \caption{Summary of the required net fraction of the inputs containing a ``parity violating'' object for the network to return a fraction significantly different from random input data.} \label{tab:frac_detect} \end{table} \section{Generating inputs and Results}\label{sec:results} \subsection{Curation for 2d CNN} To generate images from the BOSS catalogue the method outlined in \cref{sec:2d_conv} was used. 12,000 images were sampled, split into 2,400 testing and 9,600 training images, each covering a 0.5\degree x 0.5\degree\ patch of the sky. This size was chosen to create a large enough image dataset that suitably covered the full field of galaxy data. Smaller images also offset issues with approximating a spherical surface onto a flat projection and reduce overlap of images onto survey artifact regions. A full list of tests on different sizes can be found in \cref{appendix:image_sizes}. \begin{figure} \centering \includegraphics[width = \linewidth]{Images_pdf/Cmass_tot_2.pdf} \caption{A sample of images taken from CMASS. The black squares show testing images, and the red squares show training images.} \label{fig:CNN_demo_images} \end{figure} \subsection{Curation for 2d CNN with colour} To generate the 3d images, the method outlined in \cref{sec:3d_CNN} was used. Each voxel of the space had a side length of 0.5\degree and a redshift depth of 0.33. Since there is no approximation of the 3d distribution, there is no constraint on the upper size of the images, and so varying image sizes were tested. \subsection{Curation for geometric nets} For the 2d angle method, groups were constructed from the 2d projection of the data, with points in a group constrained to be within 1\degree\ of each other. Groups were constructed on the 3, 4 and 5 point levels. In the 3d case, for both the angle and vector input methods, groups were constructed to contain 4 points, allowing for direct comparison to the 4PCF method. These groups were drawn from the CMASS catalogue, with the constraint that the points must be separated by a distance in the range of $\ 20h^{-1}$Mpc to $160h^{-1}$Mpc. In all cases datasets of 500,000 data groups were constructed; 400,000 for training and 100,000 for testing. \subsection{Results} The average level of detection from bootstrapping over the testing sets and their controls for each method, as well as the random toy tests for comparison are given in \cref{tab:method_results}. For every method, no significant parity violation was detected. This is confirmed in the $\sigma$ levels displayed in \cref{tab:t_test_results}. These show no significant deviation from the control groups or from random generated data. \begin{table} \centering \begin{tabular}{P{1.6cm} P{1.85cm} P{1.85cm} P{1.85cm}} \toprule \textbf{Method} & \textbf{Detection} & \textbf{Control} & \textbf{Random}\\\midrule 2D Images & $50.4\pm0.3\%$ & $50.1\pm0.3\%$ & $49.9\pm0.3\%$\\\hline Colour Images & $49.7\pm0.3\%$ & $49.4\pm0.3$ & $50.2\pm0.3\%$ \\\hline 2D Angles, n=3 & $50.15\pm0.09\%$ & $50.02\pm0.09\%$ & $50.41\pm0.10\%$\\\hline 2D Angles, n=4 & $50.05\pm0.06\%$ & $49.85\pm0.06\%$ & $50.16\pm0.05\%$\\\hline 2D Angles, n=5 & $50.30\pm0.06\%$ & $50.27\pm0.06\%$ & $49.60\pm0.05\%$\\\hline 3D Angles & $49.91\pm0.06\%$ & $50.09\pm0.05\%$ & $49.92\pm0.05\%$\\\hline 3D Vectors & $50.08\pm0.03\%$ & $50.09\pm0.03\%$ & $50.03\pm0.03\%$\\\hline \end{tabular} \caption{The average deviation from parity even of the CMASS NGC catalogue for the different methods.} \label{tab:method_results} \end{table} \begin{table} \centering \begin{tabular}{c c c} \toprule \textbf{Method} & \textbf{CMASS-Control} & \textbf{CMASS-Random}\\\midrule 2D Images & 0.662 &0.662 \\\hline Colour Images &1.00 &0.333 \\\hline 2D Angles, n=3 & 1.02 & 1.93\\\hline 2D Angles, n=4 & 1.17 & 1.41\\\hline 2D Angles, n=5 & 0.353 & 1.28 \\\hline 3D Angles & 0.00 & 0.128\\\hline 3D Vectors & 0.235 & 1.18\\\hline \end{tabular} \caption{The $\sigma$-level deviation, assuming Gaussian error, calculated from differences from 50\%. This shows no significant parity violations.} \label{tab:t_test_results} \end{table} \section{Discussion} The results suggest there is no parity violation to be found in the LSS represented in the NGC of the CMASS catalogue. For the 2d projections, parity violation is not expected, because compression of a three-dimensional dataset down to two dimensions removes information about the three-dimensional distribution. For the three-dimensional cases, the results obtained are at odds with the work of \citet{Philcox_2022} and \citet{Hou_2023}. Recent analyses which account for the 8PCF bias term, have also disputed the detection of parity violation~\citep{krolewski2024evidenceparityviolationboss}. During a recent talk at the Royal Society~\citep{RS_A}, Philcox announced results that cast doubt on the parity violation detection~\citep{philcox2021detection, Philcox_2022}, which will be released in an upcoming issue of \citet{PT_RSA}. The final important consideration is the `artifacts' of the dataset. The 2d projection in \cref{fig:cmass_NGC} displays notable sample artifacts, which could interfere with the analyses. In all methods, the weights applied \cref{sec:Data} act to account for sampling issues, reducing the impact. In the case of the CNN, images were constructed to further offset the impact: sampling a large enough set of small enough patches minimised overlap with artifact regions, displayed in \cref{fig:Artifact_distribution}. That the overall impact is negligible is perhaps best displayed by the null results in all cases. Given that no parity violation is detected, if the artifacts were interfering, it would have to be in a way that perfectly cancels with some underlying parity violation of the dataset. Since no detection is made in either 2d or 3d, the probability of this being the case is negligible. To make one final confirmation of this belief, the sky was split into 8 bins by position, displayed in \cref{fig:Split_cmass}. Treating each of these regions separately, no significant difference was found between regions (\cref{appendix:binned_sky}). Since every region is differently affected by artifacts, this suggests they have no noticeable impact on results. Given the sensitivity of the methods, and the issues with searching on different scales, the CNN seems to be the most suitable method for these analyses. Its main advantage is that each image contains the full distribution for a region of space, making it the most general search method for parity violations across scales. In this work we proposed and demonstrated the effectivity of conveying three-dimensional information through colour creating a cCNN. We do not claim that this is the most suitable method, but the success of our test cases provides evidence to show the power of this method. To further this pursuit, convolutions could be extended into three dimensions; extensive further work would be required to ensure the model would correctly detect chirality. A possible implementation of this three-dimensional structure could be layers of two-dimensional images, stacked into three-dimensional voxel inputs, similar to the application in classification of medical images~\citep{med_3dCNN1, singh20203d_CNN}. \begin{figure}[t!] \centering \includegraphics[width=\linewidth]{Images_pdf/mini_dist_5.pdf} \caption{The set of images generated from points near the large artifact between (210, 35) and (240, 50). Very few images are affected by the presence of the artifact, and those which are affected have small overlaps into the affected region.} \label{fig:Artifact_distribution} \end{figure} \begin{figure}[t!] \centering \includegraphics[width=\linewidth]{Images_pdf/CMASS_split_Regions.pdf} \caption{The CMASS catalogue split into 8 regions, each of which was investigated separately. Each region showed the same level of parity violation, as shown in \cref{appendix:binned_sky}. This further suggests that the artifacts are not skewing the results.} \label{fig:Split_cmass} \end{figure} \section{Conclusion} We used machine learning methods to search for parity violations in the Large-Scale Structure of the Universe. Five variations of the method were employed, two investigating the two-dimensional projection of the data, and three investigating the three-dimensional distribution. It was shown that all of these methods can detect chiral datasets down to significance fractions outlined in \cref{tab:frac_detect}. Over various samples from the BOSS CMASS catalogue, no parity violation was detected in any of these methods. As such, we cannot provide further support for the recent claims by \citet{philcox2021detection} and \citet{Hou_2023}. To consolidate the work outlined in this paper, the CMASS SGC and LOWZ catalogues should be explored. Uncertainties in the fiducial cosmology used (\cref{sec:Data}) would alter the spatial distribution in three dimensions, and so the implications of these uncertainties should be investigated. Additionally, it would be useful to directly compare these methods to the 4PCF, and see how each performs against mock parity violating datasets. Further work should also be made in creating a three-dimensional CNN architecture to better explore local regions for parity violations across all orders in three dimensions. All of these methods should be applied to upcoming DESI, EUCLID and Roman data releases. \begin{acknowledgments} SH thanks the Cavendish Laboratory for the Part III Project opportunity. SH also thanks St John's College and the University of Cambridge for funding throughout the year. WH was supported by a Royal Society University Research Fellowship. \end{acknowledgments} % The \nocite command causes all entries in a bibliography to be printed out % whether or not they are actually referenced in the text. This is appropriate % for the sample file to show the different styles of references, but authors % most likely will not want to use it. %\nocite{*} \newpage \bibliographystyle{unsrtnat} \bibliography{apssamp}% Produces the bibliography via BibTeX. \newpage \appendix \section{Image size and type tests}\label{appendix:image_sizes} The work carried out by \citet{lester_gibbon2022} demonstrated that using different image types has no effect on the function of the net, JPEG and PNG images could both be used equally. We chose to use PNG images because space-saving was not an issue, and the higher quality would make the pngs easier to work with. When investigating the size, the primary considerations were: \begin{enumerate} \item Does each image contain enough points to be able to detect parity violation on real scales. We required $3 < \braket{N}$, since we need at least 3 points to create a chiral object in 2d. \item Are images small enough to get a large sample of images, where training and testing images do not overlap, and each set suitably represents the data. \item Are images small enough to minimise effects of approximating 3d distribution to 2d plane. \end{enumerate} These considerations led to square side lengths in the range ${0.3 < l < 1.0}$. A few sizes were tested on real and random data, the results of which are given in \cref{tab:image_size_results}. No method had a specific advantage in testing, so we chose to use 0.5\degree\ square images because whilst larger point clouds could be used, 22.3 points is a reasonable level to look for parity violations whilst minimising any effects of projecting onto 2d space. \begin{table} \centering \begin{tabular}{c c c c} \toprule \textbf{Side length} & \textbf{Avg. points} & \textbf{Detection} & \textbf{Rand. Detection}\\\midrule $l = 0.3$ & 8.0 & 50.8 & 51.2\\\hline $l = 0.5$ & 22.3 & 50.1 & 50.6\\\hline $l = 0.7$ & 43.7 & 51.1 &50.5\\\hline $l = 1.0$ &89.2 & 50.1 & 49.3\\\hline \end{tabular} \caption{Test results on different size image patches. On each size, tests were run on the real CMASS data and randomly generated data.} \label{tab:image_size_results} \end{table} \section{Important test results}\label{appendix: test_results.} The following tables list more fully the important results from the validation of the methods. \cref{table:3d_cnn_datacodes_full} gives the results from the cCNN; \cref{table:2d_ang_datacodes_full} the results from the 2d angle method; \cref{table:3d_ang_datacodes_full} results from the 3d angle method; and \cref{table:3d_vec_datacodes_full} the results from the 3d vector method. \begin{table*} \begin{tabular}{ p{2.0cm} p{5.1cm} p{4.2cm} P{1.9cm} P{1.94cm} P{1.94cm}} \toprule \textbf{Data code} & \textbf{Description} & \textbf{Purpose} & \textbf{Detection} & \textbf{Control} & \textbf{Uncoloured} \\\midrule RCI & Random Generated Image Data & Check sampling method isn't introducing parity violation & $49.8\pm0.3\%$ & $50.3\pm0.3\%$ & $49.9\pm0.3\%$ \\\hline HE-CI & Field of left-handed helices & Check detection of complex parity violation & $97.6\pm0.3\%$ & $50.5\pm0.3\%$ & $62.8\pm0.3\%$ \\\hline F-HE-CI & HE-CI, but each helix 50\% chance of being left- or right-handed & Check method does not detect where there is none & $51.0\pm0.3\%$ & $50.6\pm0.3\%$ & $50.5\pm0.3\%$ \\\hline HE-RCI & RCI with 1 left-handed helix inserted into the dataset for every 100 random points & Check detection of complex parity violation & $57.5\pm0.3\%$ & $50.2\pm0.3\%$ & $49.8\pm0.3\%$ \\\hline F-HE-RCI & HE-RCI, but each helix has a 50\% chance of being left- or right-handed & Check method of insertion is not causing spurious detection & $50.5\pm0.3\%$ & $50.4\pm0.3\%$ & $50.2\pm0.3\%$ \\\hline PV-RCI & Parity violating groups of 4 points randomly oriented and distributed. Local separation $\ll$ average interpoint distance. & Can parity violations be detected where the 2d dataset does not detect. & $87.2\pm0.1\%$ & $49.6\pm0.3\%$ & $50.8\pm0.3\%$ \\\hline PV-RCI-W &PV-RCI with wider field of view, each image contains $\sim5$ groups of 4. &Check detection still possible when parity violating groups have very similar pixel colours & $67.0\pm0.1\%$ & $50.2\pm0.3\%$ & $50.1\pm0.3\%$ \\\hline CHE-RCI & HE-RCI, but now each helix tapers in, giving it an orientation as well as a handedness & Check detection of more complex parity violation & $58.9\pm0.3\%$ & $49.8\pm0.3\%$ & $49.6\pm0.3\%$ \\\hline F-CHE-RCI & CHE-RCI, but each helix has a 50\% chance of being left- or right-handed & Check method of insertion is not causing spurious detection & $50.5\pm0.3\%$ & $50.4\pm0.3\%$ & $50.2\pm0.3\%$ \\\hline HE-RCI-0.1 & HE-RCI with 10 times fewer helices & Check detection of complex parity violation & $54.3\pm0.3\%$ & $50.6\pm0.3\%$ & $49.9\pm0.3\%$ \\\hline HE-RCI-0.01 & HE-RCI with 100 times fewer helices & Check detection of complex parity violation & $52.2\pm0.3\%$ & $50.1\pm0.3\%$ & $49.8\pm0.3\%$ \\\bottomrule \end{tabular} \caption{A summary of the important colour image based toy datasets. Each dataset is given a code and a description and has its testing purpose outlined. The detection column contains the positive fraction detection; the control column shows the positive fraction for the control group and the uncoloured column shows the detection for the same images coloured monochrome.} \label{table:3d_cnn_datacodes_full} \end{table*} \begin{table*} \begin{tabular}{ p{2.35cm} p{5.5cm} p{4.9cm} P{2.0cm} P{2.0cm}} \toprule \textbf{Data code} & \textbf{Description} & \textbf{Purpose} & \textbf{Detection} & \textbf{Control}\\\midrule R2d & Random Generated 2d point Data & Check parity violation is not detected where there is none & $50.41\pm0.10\%$ & $50.10\pm0.09\%$ \\\hline PV-0.1-R2d & R2d with two extra points generated near each R2d point, these are generated in a way that all groups of three are of the same chirality. Groups of three are separated on the scale of one tenth the average inter-point distance & Check detection of parity violating inputs & $90.77\pm0.06\%$ & $49.74\pm0.10\%$ \\\hline F$x$-PV-0.1-R2d & PV-0.1-R2d with 50\% of the extra points inserted with their relative $x$ coordinate to the original point inverted & Check method of generating points is not injecting parity violation & $49.66\pm0.10\%$ & $49.98\pm0.10\%$ \\\hline F$y$-PV-0.1-R2d & F$x$-PV-0.1-R2d flipped in $y$ instead of $x$ & Check method of generating points is not injecting parity violation & $49.98\pm0.10\%$ & $49.57\pm0.10\%$ \\\hline PV-0.1-R2d-5\% & PV-0.1-R2d, but only 5\% of inputs generated in this way. The rest are random & Check fraction of inputs needed to detect & $52.46\pm0.10\%$ & $50.0\pm0.10\%$ \\\hline PV-0.1-R2d-1\% &PV-0.1-R2d, but only 1\% of inputs generated in this way. The rest are random & Check fraction of inputs needed to detect & $50.96\pm0.10\%$ & $50.0\pm0.10\%$ \\\hline S-PV-0.1-R2d &PV-0.1-R2d, but after generating parity violating groups all points shuffled together and new groups drawn at random &Can parity violations be detected when they are visibly present, and groups sampled randomly from the dataset & $49.70\pm0.10\%$ & $49.68\pm0.10\%$ \\\hline C-PV-0.1-R2d &S-PV-0.1-R2d, with a distance constraint when generating groups from the shuffled data & Can parity violations become detectable by constraining the scale & $65.10\pm0.04\%$ & $49.85\pm0.04\%$ \\\hline C-R2d &R2d with groups constructed after constraining the scale &Check action of constraining scale is not causing spurious detection & $50.34\pm0.09\%$ & $50.12\pm0.10\%$ \\\hline C-PV-1-R2d &PV-0.1-R2d, but scale of parity violations now same scale as average inter-point distance. No longer visibly parity violating. Groups drawn with scale constraint & Check parity violations detectable on more realistic scales with scale constraint & $55.12\pm0.10\%$ & $49.63\pm0.10\%$ \\\hline C-R2d-LH5 &Generate groups of three from R2d with distance in range 1-5 x average inter-point distance. Remove groups of three where $\theta_{12} < \theta_{13}$. Reshuffle and drawn new groups on same scale &Can different methods of generating parity violations be detected & $52.46\pm0.10\%$ & $50.07\pm0.10\%$ \\\hline \end{tabular} \caption{2D toy datasets. Each dataset is given a code and a description and has its testing purpose outlined. The detection column contains the positive fraction detection, and the control column shows the positive fraction for the control group for each test.} \label{table:2d_ang_datacodes_full} \end{table*} \begin{table*} \begin{tabular}{ p{2.35cm} p{5.5cm} p{4.9cm} P{2.0cm} P{2.0cm}} \toprule \textbf{Data code} & \textbf{Description} & \textbf{Purpose} & \textbf{Detection} & \textbf{Control}\\\midrule R3 & Random Generated 3d point Data & Check parity violation is not detected where there is none & $49.90\pm0.11\%$ & $50.11\pm0.10\%$ \\\hline PV-0.1-R3 & R3 with three extra points generated near each R3 point, these are generated in a way that all groups of four are of the same chirality. Groups of four are separated on the scale of one tenth the average inter point distance & Check detection of parity violating inputs & $87.59\pm0.11\%$ & $49.69\pm0.07\%$ \\\hline F-PV-0.1-R3d & PV-0.1-R3d with 50\% of the extra points inserted with their relative x coordinate to the original point inverted & Check method of generating points is not injecting parity violation & $50.17\pm0.11\%$ & $50.21\pm0.10\%$ \\\hline PV-0.1-R3-5\% & PV-0.1-R3, but only 5\% of inputs generated in this way. The rest are random & Check fraction of inputs needed to detect & $53.67\pm0.07\%$ & $50.01\pm0.08\%$ \\\hline PV-0.1-R3-1\% &PV-0.1-R3, but only 1\% of inputs generated in this way. The rest are random & Check fraction of inputs needed to detect & $49.37\pm0.08\%$ & $50.05\pm0.07\%$ \\\hline S-PV-0.1-R3 &PV-0.1-R3, but after generating parity violating groups all points shuffled together and new groups drawn at random &Can parity violations be detected when they are visibly present, and groups sampled randomly from the dataset & $50.37\pm0.08\%$ & $50.15\pm0.08\%$ \\\hline C-PV-0.1-R3 &S-PV-0.1-R3, with a distance constraint when generating groups from the shuffled date & Can parity violations become detectable by constraining the scale & $58.73\pm0.08\%$ & $49.87\pm0.08\%$ \\\hline C-R3 &R3 with groups constructed after constraining the scale &Check action of constraining scale is not causing spurious detection & $49.87\pm0.04\%$ & $50.32\pm0.07\%$ \\\hline C-PV-1-R3 &PV-0.1-R3, but scale of parity violations now same scale as average inter-point distance. No longer visibly parity violating. Groups drawn with scale constraint & Check parity violations detectable on more realistic scales with scale constraint & $54.37\pm0.05\%$ & $49.87\pm0.06\%$ \\\hline C-R3-LH5 &Generate groups of 4 from R3 with distance in range 1-5x average inter-point distance. Remove groups of 4 where $\theta_{13} < \theta_{14}$. Reshuffle and drawn new groups on same scale &Can different methods of generating parity violations be detected & $53.11\pm0.07\%$ & $50.64\pm0.06\%$ \\\bottomrule \end{tabular} \caption{3D toy datasets. Each dataset is given a code and a description and has its testing purpose outlined. The detection column contains the positive fraction detection, and the control column shows the positive fraction for the control group for each test.} \label{table:3d_ang_datacodes_full} \end{table*} \begin{table} \centering \begin{tabular}{P{2.5cm} P{2.2cm} P{2.2cm}} \toprule \textbf{Data code} & \textbf{Detection} & \textbf{Control}\\\midrule R3 & $50.11\pm0.07\%$ & $50.21\pm0.06\%$ \\\hline PV-0.1-R3 & $99.83\pm0.02\%$ & $50.3\pm0.31\%$ \\\hline F-PV-0.1-R3d & $50.35\pm0.07\%$ & $49.87\pm0.07\%$ \\\hline PV-0.1-R3-5\% & $53.18\pm0.07\%$ & $49.78\pm0.07\%$ \\\hline PV-0.1-R3-1\% & $50.43\pm0.07\%$ & $49.91\pm0.07\%$ \\\hline S-PV-0.1-R3 & $50.16\pm0.07\%$ & $49.84\pm0.07\%$ \\\hline C-PV-0.1-R3 & $47.56\pm0.10\%$ & $50.11\pm0.11\%$ \\\hline C-R3 & $49.98\pm0.11\%$ & $50.09\pm0.10\%$ \\\hline C-PV-1-R3 & $47.36\pm0.07\%$ & $49.91\pm0.07\%$ \\\hline C-R3-LH5 & $48.12\pm0.07\%$ & $50.04\pm0.07\%$ \\\bottomrule \end{tabular} \caption{Results for the vector input method on the 3d toy datasets, as defined in \cref{table:3d_ang_datacodes_full} The detection column contains the positive fraction detection, and the control column shows the positive fraction for the control group for each test. } \label{table:3d_vec_datacodes_full} \end{table} \section{Additional tests on networks}\label{appendix:2d_toy_test} \subsection{2D angles: n=4,5} As with the method using angles from groups of 3, the methods using 4 and 5 points per group were also tested. These were tested using the same tests as the n=3 case. The results are given in \cref{table:2d_4_datacodes} and \cref{table:2d_5_datacodes}. The tests demonstrate that these methods are equally suitable. \begin{table}[!htbp] \centering \begin{tabular}{ c c c} \toprule \textbf{Data code} & \textbf{Detection} & \textbf{Control}\\\midrule R2d & $50.22\pm1.0\%$ & $49.87\pm1.0\%$ \\\hline PV-0.1-R2d & $98.32\pm0.9\%$ & $50.91\pm1.0\%$ \\\hline PV-0.1-R2d-1\% & $52.62\pm1.0\%$ & $50.43\pm1.0\%$ \\\hline PV-0.1-R2d-0.5\% & $51.56\pm0.3\%$ & $50.52\pm1.0\%$ \\\hline S-PV-0.1-R2d & $50.36\pm0.3\%$ & $50.22\pm1.0\%$ \\\hline C-PV-0.1-R2d & $55.57\pm0.3\%$ & $50.22\pm1.0\%$ \\\hline C-R2d & $50.61\pm0.3\%$ & $50.21\pm1.0\%$ \\\hline C-PV-1-R2d & $53.69\pm0.3\%$ & $50.32\pm1.0\%$ \\\hline C-R2d-LH5 & $51.26\pm0.3\%$ & $50.62\pm1.0\%$\\\bottomrule \end{tabular} \caption{2d toy datasets for n=4. The datacodes are defined in \cref{table:2d_ang_datacodes_full}} \label{table:2d_4_datacodes} \end{table} \begin{table} \centering \begin{tabular}{ c c c} \toprule \textbf{Data code} & \textbf{Detection} & \textbf{Control}\\\midrule R2d & $50.24\pm1.0\%$ & $50.17\pm1.0\%$ \\\hline PV-0.1-R2d & $99.49\pm0.9\%$ & $50.31\pm1.0\%$ \\\hline PV-0.1-R2d-1\% & $53.32\pm1.0\%$ & $50.38\pm1.0\%$ \\\hline PV-0.1-R2d-0.5\% & $51.76\pm0.3\%$ & $50.21\pm1.0\%$ \\\hline S-PV-0.1-R2d & $50.29\pm0.3\%$ & $50.27\pm1.0\%$ \\\hline C-PV-0.1-R2d & $54.77\pm0.3\%$ & $50.18\pm1.0\%$ \\\hline C-R2d & $50.51\pm0.3\%$ & $50.43\pm1.0\%$ \\\hline C-PV-1-R2d & $53.72\pm0.3\%$ & $50.23\pm1.0\%$ \\\hline C-R2d-LH5 & $51.07\pm0.3\%$ & $50.52\pm1.0\%$\\\bottomrule \end{tabular} \caption{2d toy datasets for n=5. The datacodes are defined in \cref{table:2d_ang_datacodes_full}} \label{table:2d_5_datacodes} \end{table} \section{Patches of the sky}\label{appendix:binned_sky} In order to confirm the arguments which propose that the artifacts are having no significant impact, the CMASS NGC was split into eight regions. And image datasets compiled from each separately, the full results follow in \cref{table:regions_results}, where we see that no significant difference occurs between the regions. Suggesting that the artifacts are indeed having no impact. For these tests, the same size patches of the sky were used, and each region had 3,000 images generated from it. The image distribution for the first region is shown in \cref{fig:Region_1_demo} \begin{figure} \centering \includegraphics[width=0.9\linewidth]{Images_pdf/bottom_right_4.pdf} \caption{The image samples drawn from the first region for localised ananlysis.} \label{fig:Region_1_demo} \end{figure} \begin{table} \centering \begin{tabular}{ c c c} \toprule \textbf{Data code} & \textbf{Detection} & \textbf{Control}\\\midrule R1 & $49.96\pm2.0\%$ & $53.43\pm2.0\%$ \\\hline R2 & $48.95\pm2.0\%$ & $51.33\pm2.0\%$ \\\hline R3 & $50.91\pm2.0\%$ & $50.4\pm2.0\%$ \\\hline R4 & $51.31\pm2.0\%$ & $50.92\pm2.0\%$ \\\hline R5 & $49.65\pm2.0\%$ & $52.43\pm2.0\%$ \\\hline R6 & $52.46\pm2.0\%$ & $51.04\pm2.0\%$ \\\hline R7 & $48.76\pm2.0\%$ & $52.93\pm2.0\%$ \\\hline R8 & $49.96\pm2.0\%$ & $53.43\pm2.0\%$ \\\bottomrule \end{tabular} \caption{Levels of detection in the separate regions, used to investigate the impact of the artifacts.} \label{table:regions_results} \end{table} \end{document} % % ****** End of file apssamp.tex ****** ``` 4. **Bibliographic Information:** ```bbl \begin{thebibliography}{51} \providecommand{\natexlab}[1]{#1} \providecommand{\url}[1]{\texttt{#1}} \expandafter\ifx\csname urlstyle\endcsname\relax \providecommand{\doi}[1]{doi: #1}\else \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi \bibitem[Lue et~al.(1999)Lue, Wang, and Kamionkowski]{PhysRevLett.83.1506.lue} Arthur Lue, Limin Wang, and Marc Kamionkowski. \newblock Cosmological signature of new parity-violating interactions. \newblock \emph{Phys. Rev. Lett.}, 83:\penalty0 1506--1509, Aug 1999. \newblock \doi{10.1103/PhysRevLett.83.1506}. \newblock URL \url{https://link.aps.org/doi/10.1103/PhysRevLett.83.1506}. \bibitem[Kamionkowski and Souradeep(2011)]{Kamionkowski_2011} Marc Kamionkowski and Tarun Souradeep. \newblock Odd-parity cosmic microwave background bispectrum. \newblock \emph{Physical Review D}, 83\penalty0 (2), January 2011. \newblock ISSN 1550-2368. \newblock \doi{10.1103/physrevd.83.027301}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevD.83.027301}. \bibitem[Shiraishi(2016)]{Shiraishi_2016} Maresuke Shiraishi. \newblock Parity violation in the cmb trispectrum from the scalar sector. \newblock \emph{Physical Review D}, 94\penalty0 (8), October 2016. \newblock ISSN 2470-0029. \newblock \doi{10.1103/physrevd.94.083503}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevD.94.083503}. \bibitem[Minami and Komatsu(2020)]{PhysRevLett.125.221301.komatsu} Yuto Minami and Eiichiro Komatsu. \newblock New extraction of the cosmic birefringence from the planck 2018 polarization data. \newblock \emph{Phys. Rev. Lett.}, 125:\penalty0 221301, Nov 2020. \newblock \doi{10.1103/PhysRevLett.125.221301}. \newblock URL \url{https://link.aps.org/doi/10.1103/PhysRevLett.125.221301}. \bibitem[Philcox(2023)]{philcox2023cmb} Oliver H.~E. Philcox. \newblock Do the cmb temperature fluctuations conserve parity?, 2023. \bibitem[Philcox and Shiraishi(2024)]{philcox2024testing} Oliver H.~E. Philcox and Maresuke Shiraishi. \newblock Testing parity symmetry with the polarized cosmic microwave background, 2024. \bibitem[Saito et~al.(2007)Saito, Ichiki, and Taruya]{Saito_2007} Shun Saito, Kiyotomo Ichiki, and Atsushi Taruya. \newblock Probing polarization states of primordial gravitational waves with cosmic microwave background anisotropies. \newblock \emph{Journal of Cosmology and Astroparticle Physics}, 2007\penalty0 (09):\penalty0 002–002, September 2007. \newblock ISSN 1475-7516. \newblock \doi{10.1088/1475-7516/2007/09/002}. \newblock URL \url{http://dx.doi.org/10.1088/1475-7516/2007/09/002}. \bibitem[Yunes et~al.(2010)Yunes, O’Shaughnessy, Owen, and Alexander]{Yunes_2010} Nicolás Yunes, Richard O’Shaughnessy, Benjamin~J. Owen, and Stephon Alexander. \newblock Testing gravitational parity violation with coincident gravitational waves and short gamma-ray bursts. \newblock \emph{Physical Review D}, 82\penalty0 (6), September 2010. \newblock ISSN 1550-2368. \newblock \doi{10.1103/physrevd.82.064017}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevD.82.064017}. \bibitem[Wang et~al.(2013)Wang, Wu, Zhao, and Zhu]{Wang_2013} Anzhong Wang, Qiang Wu, Wen Zhao, and Tao Zhu. \newblock Polarizing primordial gravitational waves by parity violation. \newblock \emph{Physical Review D}, 87\penalty0 (10), May 2013. \newblock ISSN 1550-2368. \newblock \doi{10.1103/physrevd.87.103512}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevD.87.103512}. \bibitem[Orlando et~al.(2021)Orlando, Pieroni, and Ricciardone]{Orlando_2021} Giorgio Orlando, Mauro Pieroni, and Angelo Ricciardone. \newblock Measuring parity violation in the stochastic gravitational wave background with the lisa-taiji network. \newblock \emph{Journal of Cosmology and Astroparticle Physics}, 2021\penalty0 (03):\penalty0 069, March 2021. \newblock ISSN 1475-7516. \newblock \doi{10.1088/1475-7516/2021/03/069}. \newblock URL \url{http://dx.doi.org/10.1088/1475-7516/2021/03/069}. \bibitem[Jenks et~al.(2023)Jenks, Choi, Lagos, and Yunes]{jenks2023parameterized} Leah Jenks, Lyla Choi, Macarena Lagos, and Nicolás Yunes. \newblock Parameterized parity violation in gravitational wave propagation, 2023. \bibitem[Cahn et~al.(2021)Cahn, Slepian, and Hou]{cahn2021test} Robert~N. Cahn, Zachary Slepian, and Jiamin Hou. \newblock A test for cosmological parity violation using the 3d distribution of galaxies, 2021. \bibitem[{BOSS collaboration}(2017{\natexlab{a}})]{10.1093/mnras/stx721} {BOSS collaboration}. \newblock {The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample}. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 470\penalty0 (3):\penalty0 2617--2652, 03 2017{\natexlab{a}}. \newblock ISSN 0035-8711. \newblock \doi{10.1093/mnras/stx721}. \newblock URL \url{https://doi.org/10.1093/mnras/stx721}. \bibitem[{BOSS collaboration}(2011)]{2011AJ....142...72E} {BOSS collaboration}. \newblock {SDSS-III: Massive Spectroscopic Surveys of the Distant Universe, the Milky Way, and Extra-Solar Planetary Systems}. \newblock \emph{\aj}, 142\penalty0 (3):\penalty0 72, September 2011. \newblock \doi{10.1088/0004-6256/142/3/72}. \bibitem[{BOSS collaboration}(2012{\natexlab{a}})]{Dawson_2013} {BOSS collaboration}. \newblock The baryon oscillation spectroscopic survey of sdss-iii. \newblock \emph{The Astronomical Journal}, 145\penalty0 (1):\penalty0 10, dec 2012{\natexlab{a}}. \newblock \doi{10.1088/0004-6256/145/1/10}. \newblock URL \url{https://dx.doi.org/10.1088/0004-6256/145/1/10}. \bibitem[Philcox et~al.(2021)Philcox, Hou, and Slepian]{philcox2021detection} Oliver H.~E. Philcox, Jiamin Hou, and Zachary Slepian. \newblock A first detection of the connected 4-point correlation function of galaxies using the boss cmass sample, 2021. \bibitem[Hou et~al.(2023)Hou, Slepian, and Cahn]{Hou_2023} Jiamin Hou, Zachary Slepian, and Robert~N Cahn. \newblock Measurement of parity-odd modes in the large-scale 4-point correlation function of sloan digital sky survey baryon oscillation spectroscopic survey twelfth data release cmass and lowz galaxies. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 522\penalty0 (4):\penalty0 5701–5739, May 2023. \newblock ISSN 1365-2966. \newblock \doi{10.1093/mnras/stad1062}. \newblock URL \url{http://dx.doi.org/10.1093/mnras/stad1062}. \bibitem[Philcox(2022)]{Philcox_2022} Oliver~H.E. Philcox. \newblock Probing parity violation with the four-point correlation function of boss galaxies. \newblock \emph{Physical Review D}, 106\penalty0 (6), September 2022. \newblock ISSN 2470-0029. \newblock \doi{10.1103/physrevd.106.063501}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevD.106.063501}. \bibitem[{BOSS collaboration}(2016{\natexlab{a}})]{Rodr_guez_Torres_2016} {BOSS collaboration}. \newblock The clustering of galaxies in the sdss-iii baryon oscillation spectroscopic survey: modelling the clustering and halo occupation distribution of boss cmass galaxies in the final data release. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 460\penalty0 (2):\penalty0 1173–1187, April 2016{\natexlab{a}}. \newblock ISSN 1365-2966. \newblock \doi{10.1093/mnras/stw1014}. \newblock URL \url{http://dx.doi.org/10.1093/mnras/stw1014}. \bibitem[Kitaura et~al.(2016)Kitaura, Rodríguez-Torres, Chuang, Zhao, Prada, Gil-Marín, Guo, Yepes, Klypin, Scóccola, Tinker, McBride, Reid, Sánchez, Salazar-Albornoz, Grieb, Vargas-Magana, Cuesta, Neyrinck, Beutler, Comparat, Percival, and Ross]{Kitaura_2016} Francisco-Shu Kitaura, Sergio Rodríguez-Torres, Chia-Hsun Chuang, Cheng Zhao, Francisco Prada, Héctor Gil-Marín, Hong Guo, Gustavo Yepes, Anatoly Klypin, Claudia~G. Scóccola, Jeremy Tinker, Cameron McBride, Beth Reid, Ariel~G. Sánchez, Salvador Salazar-Albornoz, Jan~Niklas Grieb, Mariana Vargas-Magana, Antonio~J. Cuesta, Mark Neyrinck, Florian Beutler, Johan Comparat, Will~J. Percival, and Ashley Ross. \newblock The clustering of galaxies in the sdss-iii baryon oscillation spectroscopic survey: mock galaxy catalogues for the boss final data release. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 456\penalty0 (4):\penalty0 4156–4173, January 2016. \newblock ISSN 1365-2966. \newblock \doi{10.1093/mnras/stv2826}. \newblock URL \url{http://dx.doi.org/10.1093/mnras/stv2826}. \bibitem[{Hou} et~al.(2022){Hou}, {Cahn}, {Philcox}, and {Slepian}]{2022PhRvD.106d3515H} Jiamin {Hou}, Robert~N. {Cahn}, Oliver H.~E. {Philcox}, and Zachary {Slepian}. \newblock {Analytic Gaussian covariance matrices for galaxy N -point correlation functions}. \newblock \emph{\prd}, 106\penalty0 (4):\penalty0 043515, August 2022. \newblock \doi{10.1103/PhysRevD.106.043515}. \bibitem[Krolewski et~al.(2024)Krolewski, May, Smith, and Hopkins]{krolewski2024evidenceparityviolationboss} Alex Krolewski, Simon May, Kendrick Smith, and Hans Hopkins. \newblock No evidence for parity violation in boss, 2024. \newblock URL \url{https://arxiv.org/abs/2407.03397}. \bibitem[Taylor et~al.(2023)Taylor, Craigie, and Ting]{taylor2023} Peter~L. Taylor, Matthew Craigie, and Yuan-Sen Ting. \newblock Unsupervised searches for cosmological parity-violation: An investigation with convolutional neural networks, 2023. \newblock URL \url{https://arxiv.org/abs/2312.09287}. \bibitem[Craigie et~al.(2024)Craigie, Taylor, Ting, Cuesta-Lazaro, Ruggeri, and Davis]{craigie2024unsupervisedsearchescosmologicalparity} Matthew Craigie, Peter~L. Taylor, Yuan-Sen Ting, Carolina Cuesta-Lazaro, Rossana Ruggeri, and Tamara~M. Davis. \newblock Unsupervised searches for cosmological parity violation: Improving detection power with the neural field scattering transform, 2024. \newblock URL \url{https://arxiv.org/abs/2405.13083}. \bibitem[{DESI collaboration}(2016)]{2016arXiv161100036D} {DESI collaboration}. \newblock {The DESI Experiment Part I: Science,Targeting, and Survey Design}. \newblock \emph{arXiv e-prints}, art. arXiv:1611.00036, October 2016. \newblock \doi{10.48550/arXiv.1611.00036}. \bibitem[{DESI Collaboration Et Al}(2023)]{https://doi.org/10.5281/zenodo.7964161} {DESI Collaboration Et Al}. \newblock The early data release of the dark energy spectroscopic instrument, 2023. \newblock URL \url{https://zenodo.org/record/7964161}. \bibitem[{Euclid Collaboration}(2011)]{2011_EUCLID_def} {Euclid Collaboration}. \newblock {Euclid Definition Study Report}. \newblock \emph{arXiv e-prints}, art. arXiv:1110.3193, October 2011. \newblock \doi{10.48550/arXiv.1110.3193}. \bibitem[{Euclid collaboration}(2020)]{2020_Euclid} {Euclid collaboration}. \newblock Euclid preparation: Vii. forecast validation for euclid cosmological probes. \newblock \emph{Astronomy \& Astrophysics}, 642:\penalty0 A191, October 2020. \newblock ISSN 1432-0746. \newblock \doi{10.1051/0004-6361/202038071}. \newblock URL \url{http://dx.doi.org/10.1051/0004-6361/202038071}. \bibitem[{Spergel} et~al.(2015){Spergel}, {Gehrels}, {Baltay}, {Bennett}, {Breckinridge}, {Donahue}, {Dressler}, {Gaudi}, {Greene}, {Guyon}, {Hirata}, {Kalirai}, {Kasdin}, {Macintosh}, {Moos}, {Perlmutter}, {Postman}, {Rauscher}, {Rhodes}, {Wang}, {Weinberg}, {Benford}, {Hudson}, {Jeong}, {Mellier}, {Traub}, {Yamada}, {Capak}, {Colbert}, {Masters}, {Penny}, {Savransky}, {Stern}, {Zimmerman}, {Barry}, {Bartusek}, {Carpenter}, {Cheng}, {Content}, {Dekens}, {Demers}, {Grady}, {Jackson}, {Kuan}, {Kruk}, {Melton}, {Nemati}, {Parvin}, {Poberezhskiy}, {Peddie}, {Ruffa}, {Wallace}, {Whipple}, {Wollack}, and {Zhao}]{2015Roman} D.~{Spergel}, N.~{Gehrels}, C.~{Baltay}, D.~{Bennett}, J.~{Breckinridge}, M.~{Donahue}, A.~{Dressler}, B.~S. {Gaudi}, T.~{Greene}, O.~{Guyon}, C.~{Hirata}, J.~{Kalirai}, N.~J. {Kasdin}, B.~{Macintosh}, W.~{Moos}, S.~{Perlmutter}, M.~{Postman}, B.~{Rauscher}, J.~{Rhodes}, Y.~{Wang}, D.~{Weinberg}, D.~{Benford}, M.~{Hudson}, W.~S. {Jeong}, Y.~{Mellier}, W.~{Traub}, T.~{Yamada}, P.~{Capak}, J.~{Colbert}, D.~{Masters}, M.~{Penny}, D.~{Savransky}, D.~{Stern}, N.~{Zimmerman}, R.~{Barry}, L.~{Bartusek}, K.~{Carpenter}, E.~{Cheng}, D.~{Content}, F.~{Dekens}, R.~{Demers}, K.~{Grady}, C.~{Jackson}, G.~{Kuan}, J.~{Kruk}, M.~{Melton}, B.~{Nemati}, B.~{Parvin}, I.~{Poberezhskiy}, C.~{Peddie}, J.~{Ruffa}, J.~K. {Wallace}, A.~{Whipple}, E.~{Wollack}, and F.~{Zhao}. \newblock {Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report}. \newblock \emph{arXiv e-prints}, art. arXiv:1503.03757, March 2015. \newblock \doi{10.48550/arXiv.1503.03757}. \bibitem[{WFIRST collaboration}(2019)]{2019Roman} {WFIRST collaboration}. \newblock {The Wide Field Infrared Survey Telescope: 100 Hubbles for the 2020s}. \newblock \emph{arXiv e-prints}, art. arXiv:1902.05569, February 2019. \newblock \doi{10.48550/arXiv.1902.05569}. \bibitem[Lee and Yang(1956)]{PhysRev.104.254-weaktheory} T.~D. Lee and C.~N. Yang. \newblock Question of parity conservation in weak interactions. \newblock \emph{Phys. Rev.}, 104:\penalty0 254--258, Oct 1956. \newblock \doi{10.1103/PhysRev.104.254}. \newblock URL \url{https://link.aps.org/doi/10.1103/PhysRev.104.254}. \bibitem[Wu et~al.(1957)Wu, Ambler, Hayward, Hoppes, and Hudson]{PhysRev.105.1413-weakexp} C.~S. Wu, E.~Ambler, R.~W. Hayward, D.~D. Hoppes, and R.~P. Hudson. \newblock Experimental test of parity conservation in beta decay. \newblock \emph{Phys. Rev.}, 105:\penalty0 1413--1415, Feb 1957. \newblock \doi{10.1103/PhysRev.105.1413}. \newblock URL \url{https://link.aps.org/doi/10.1103/PhysRev.105.1413}. \bibitem[Sakharov(1967)]{Sakharov:1967dj} A.~D. Sakharov. \newblock {Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe}. \newblock \emph{Pisma Zh. Eksp. Teor. Fiz.}, 5:\penalty0 32--35, 1967. \newblock \doi{10.1070/PU1991v034n05ABEH002497}. \bibitem[Carroll(1998)]{Carroll_1998} Sean~M. Carroll. \newblock Quintessence and the rest of the world: Suppressing long-range interactions. \newblock \emph{Physical Review Letters}, 81\penalty0 (15):\penalty0 3067–3070, October 1998. \newblock ISSN 1079-7114. \newblock \doi{10.1103/physrevlett.81.3067}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevLett.81.3067}. \bibitem[Hilbert(1915)]{Hilbert1915} D.~Hilbert. \newblock Die grundlagen der physik . (erste mitteilung.). \newblock \emph{Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse}, 1915:\penalty0 395--408, 1915. \newblock URL \url{http://eudml.org/doc/58946}. \bibitem[Lester and Schott(2019)]{Lester_2019} Christopher~G. Lester and Matthias Schott. \newblock Testing non-standard sources of parity violation in jets at the lhc, trialled with cms open data. \newblock \emph{Journal of High Energy Physics}, 2019\penalty0 (12), December 2019. \newblock ISSN 1029-8479. \newblock \doi{10.1007/jhep12(2019)120}. \newblock URL \url{http://dx.doi.org/10.1007/JHEP12(2019)120}. \bibitem[Lester and Tombs(2022)]{lester2022using} Christopher~G. Lester and Rupert Tombs. \newblock Using unsupervised learning to detect broken symmetries, with relevance to searches for parity violation in nature. (previously: ``{Stressed GANs} snag desserts''), 2022. \newblock ISSN 2835-8856. \newblock URL \url{https://openreview.net/forum?id=QFJ3gtbwHR}. \newblock Published in ``Transactions on Machine Learning Research''. \bibitem[Tombs and Lester(2022)]{Tombs_2022} Rupert Tombs and Christopher~G. Lester. \newblock A method to challenge symmetries in data with self-supervised learning. \newblock \emph{Journal of Instrumentation}, 17\penalty0 (08):\penalty0 P08024, August 2022. \newblock ISSN 1748-0221. \newblock \doi{10.1088/1748-0221/17/08/p08024}. \newblock URL \url{http://dx.doi.org/10.1088/1748-0221/17/08/P08024}. \bibitem[James~Gibbon(2023)]{lester_gibbon2022} (Supervised by Christopher G.~Lester) James~Gibbon. \newblock Detection of parity violation in snails through unsupervised learning, 2023. \newblock URL \url{https://www.hep.phy.cam.ac.uk/~lester/teaching/PartIIIProjects/2023-James-Gibbon-Snails.pdf}. \bibitem[{BOSS collaboration}(2016{\natexlab{b}})]{2016_REID_BOSS} {BOSS collaboration}. \newblock {SDSS-III Baryon Oscillation Spectroscopic Survey Data Release 12: galaxy target selection and large-scale structure catalogues}. \newblock \emph{\mnras}, 455\penalty0 (2):\penalty0 1553--1573, January 2016{\natexlab{b}}. \newblock \doi{10.1093/mnras/stv2382}. \bibitem[{BOSS collaboration}(2013)]{marston_mNRAS_2013} {BOSS collaboration}. \newblock {Stellar masses of SDSS-III/BOSS galaxies at z ∼ 0.5 and constraints to galaxy formation models}. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 435\penalty0 (4):\penalty0 2764--2792, 09 2013. \newblock ISSN 0035-8711. \newblock \doi{10.1093/mnras/stt1424}. \newblock URL \url{https://doi.org/10.1093/mnras/stt1424}. \bibitem[Saito et~al.(2016)Saito, Leauthaud, Hearin, Bundy, Zentner, Behroozi, Reid, Sinha, Coupon, Tinker, White, and Schneider]{10.1093/mnras/stw1080} Shun Saito, Alexie Leauthaud, Andrew~P. Hearin, Kevin Bundy, Andrew~R. Zentner, Peter~S. Behroozi, Beth~A. Reid, Manodeep Sinha, Jean Coupon, Jeremy~L. Tinker, Martin White, and Donald~P. Schneider. \newblock {Connecting massive galaxies to dark matter haloes in BOSS – I. Is galaxy colour a stochastic process in high-mass haloes?} \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 460\penalty0 (2):\penalty0 1457--1475, 05 2016. \newblock ISSN 0035-8711. \newblock \doi{10.1093/mnras/stw1080}. \newblock URL \url{https://doi.org/10.1093/mnras/stw1080}. \bibitem[Leauthaud et~al.(2016)Leauthaud, Bundy, Saito, Tinker, Maraston, Tojeiro, Huang, Brownstein, Schneider, and Thomas]{10.1093/mnras/stw117} Alexie Leauthaud, Kevin Bundy, Shun Saito, Jeremy Tinker, Claudia Maraston, Rita Tojeiro, Song Huang, Joel~R. Brownstein, Donald~P. Schneider, and Daniel Thomas. \newblock {The Stripe 82 Massive Galaxy Project – II. Stellar mass completeness of spectroscopic galaxy samples from the Baryon Oscillation Spectroscopic Survey}. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 457\penalty0 (4):\penalty0 4021--4037, 02 2016. \newblock ISSN 0035-8711. \newblock \doi{10.1093/mnras/stw117}. \newblock URL \url{https://doi.org/10.1093/mnras/stw117}. \bibitem[Bundy et~al.(2017)Bundy, Leauthaud, Saito, Maraston, Wake, and Thomas]{Bundy_2017} Kevin Bundy, Alexie Leauthaud, Shun Saito, Claudia Maraston, David~A. Wake, and Daniel Thomas. \newblock The stripe 82 massive galaxy project. iii. a lack of growth among massive galaxies. \newblock \emph{The Astrophysical Journal}, 851\penalty0 (1):\penalty0 34, December 2017. \newblock ISSN 1538-4357. \newblock \doi{10.3847/1538-4357/aa9896}. \newblock URL \url{http://dx.doi.org/10.3847/1538-4357/aa9896}. \bibitem[{BOSS collaboration}(2017{\natexlab{b}})]{Alam_2017} {BOSS collaboration}. \newblock The clustering of galaxies in the completed sdss-iii baryon oscillation spectroscopic survey: cosmological analysis of the dr12 galaxy sample. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 470\penalty0 (3):\penalty0 2617–2652, March 2017{\natexlab{b}}. \newblock ISSN 1365-2966. \newblock \doi{10.1093/mnras/stx721}. \newblock URL \url{http://dx.doi.org/10.1093/mnras/stx721}. \bibitem[{BOSS collaboration}(2012{\natexlab{b}})]{Anderson_2012_weights} {BOSS collaboration}. \newblock The clustering of galaxies in the sdss-iii baryon oscillation spectroscopic survey: baryon acoustic oscillations in the data release 9 spectroscopic galaxy sample. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 427\penalty0 (4):\penalty0 3435–3467, December 2012{\natexlab{b}}. \newblock ISSN 1365-2966. \newblock \doi{10.1111/j.1365-2966.2012.22066.x}. \newblock URL \url{http://dx.doi.org/10.1111/j.1365-2966.2012.22066.x}. \bibitem[Feldman et~al.(1994)Feldman, Kaiser, and Peacock]{Feldman_1994_fkp} Hume~A. Feldman, Nick Kaiser, and John~A. Peacock. \newblock Power-spectrum analysis of three-dimensional redshift surveys. \newblock \emph{The Astrophysical Journal}, 426:\penalty0 23, May 1994. \newblock ISSN 1538-4357. \newblock \doi{10.1086/174036}. \newblock URL \url{http://dx.doi.org/10.1086/174036}. \bibitem[{Royal Society Events}(2024)]{RS_A} {Royal Society Events}. \newblock {Royal Society Events} challenging the standard cosmological model, 2024. \newblock URL \url{https://royalsociety.org/science-events-and-lectures/2024/04/cosmological-model/}. \bibitem[{Philosophical Transactions of the Royal Society A}(2024)]{PT_RSA} {Philosophical Transactions of the Royal Society A}. \newblock {Philosophical Transactions of the Royal Society A}, 2024. \newblock URL \url{https://royalsocietypublishing.org/journal/rsta}. \bibitem[Ker et~al.(2019)Ker, Singh, Bai, Rao, Lim, and Wang]{med_3dCNN1} Justin Ker, Satya~P. Singh, Yeqi Bai, Jai Rao, Tchoyoson Lim, and Lipo Wang. \newblock Image thresholding improves 3-dimensional convolutional neural network diagnosis of different acute brain hemorrhages on computed tomography scans. \newblock \emph{Sensors}, 19\penalty0 (9), 2019. \newblock ISSN 1424-8220. \newblock \doi{10.3390/s19092167}. \newblock URL \url{https://www.mdpi.com/1424-8220/19/9/2167}. \bibitem[Singh et~al.(2020)Singh, Wang, Gupta, Goli, Padmanabhan, and Gulyás]{singh20203d_CNN} Satya~P. Singh, Lipo Wang, Sukrit Gupta, Haveesh Goli, Parasuraman Padmanabhan, and Balázs Gulyás. \newblock 3d deep learning on medical images: A review, 2020. \end{thebibliography} \begin{thebibliography}{51} \providecommand{\natexlab}[1]{#1} \providecommand{\url}[1]{\texttt{#1}} \expandafter\ifx\csname urlstyle\endcsname\relax \providecommand{\doi}[1]{doi: #1}\else \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi \bibitem[Lue et~al.(1999)Lue, Wang, and Kamionkowski]{PhysRevLett.83.1506.lue} Arthur Lue, Limin Wang, and Marc Kamionkowski. \newblock Cosmological signature of new parity-violating interactions. \newblock \emph{Phys. Rev. Lett.}, 83:\penalty0 1506--1509, Aug 1999. \newblock \doi{10.1103/PhysRevLett.83.1506}. \newblock URL \url{https://link.aps.org/doi/10.1103/PhysRevLett.83.1506}. \bibitem[Kamionkowski and Souradeep(2011)]{Kamionkowski_2011} Marc Kamionkowski and Tarun Souradeep. \newblock Odd-parity cosmic microwave background bispectrum. \newblock \emph{Physical Review D}, 83\penalty0 (2), January 2011. \newblock ISSN 1550-2368. \newblock \doi{10.1103/physrevd.83.027301}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevD.83.027301}. \bibitem[Shiraishi(2016)]{Shiraishi_2016} Maresuke Shiraishi. \newblock Parity violation in the cmb trispectrum from the scalar sector. \newblock \emph{Physical Review D}, 94\penalty0 (8), October 2016. \newblock ISSN 2470-0029. \newblock \doi{10.1103/physrevd.94.083503}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevD.94.083503}. \bibitem[Minami and Komatsu(2020)]{PhysRevLett.125.221301.komatsu} Yuto Minami and Eiichiro Komatsu. \newblock New extraction of the cosmic birefringence from the planck 2018 polarization data. \newblock \emph{Phys. Rev. Lett.}, 125:\penalty0 221301, Nov 2020. \newblock \doi{10.1103/PhysRevLett.125.221301}. \newblock URL \url{https://link.aps.org/doi/10.1103/PhysRevLett.125.221301}. \bibitem[Philcox(2023)]{philcox2023cmb} Oliver H.~E. Philcox. \newblock Do the cmb temperature fluctuations conserve parity?, 2023. \bibitem[Philcox and Shiraishi(2024)]{philcox2024testing} Oliver H.~E. Philcox and Maresuke Shiraishi. \newblock Testing parity symmetry with the polarized cosmic microwave background, 2024. \bibitem[Saito et~al.(2007)Saito, Ichiki, and Taruya]{Saito_2007} Shun Saito, Kiyotomo Ichiki, and Atsushi Taruya. \newblock Probing polarization states of primordial gravitational waves with cosmic microwave background anisotropies. \newblock \emph{Journal of Cosmology and Astroparticle Physics}, 2007\penalty0 (09):\penalty0 002–002, September 2007. \newblock ISSN 1475-7516. \newblock \doi{10.1088/1475-7516/2007/09/002}. \newblock URL \url{http://dx.doi.org/10.1088/1475-7516/2007/09/002}. \bibitem[Yunes et~al.(2010)Yunes, O’Shaughnessy, Owen, and Alexander]{Yunes_2010} Nicolás Yunes, Richard O’Shaughnessy, Benjamin~J. Owen, and Stephon Alexander. \newblock Testing gravitational parity violation with coincident gravitational waves and short gamma-ray bursts. \newblock \emph{Physical Review D}, 82\penalty0 (6), September 2010. \newblock ISSN 1550-2368. \newblock \doi{10.1103/physrevd.82.064017}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevD.82.064017}. \bibitem[Wang et~al.(2013)Wang, Wu, Zhao, and Zhu]{Wang_2013} Anzhong Wang, Qiang Wu, Wen Zhao, and Tao Zhu. \newblock Polarizing primordial gravitational waves by parity violation. \newblock \emph{Physical Review D}, 87\penalty0 (10), May 2013. \newblock ISSN 1550-2368. \newblock \doi{10.1103/physrevd.87.103512}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevD.87.103512}. \bibitem[Orlando et~al.(2021)Orlando, Pieroni, and Ricciardone]{Orlando_2021} Giorgio Orlando, Mauro Pieroni, and Angelo Ricciardone. \newblock Measuring parity violation in the stochastic gravitational wave background with the lisa-taiji network. \newblock \emph{Journal of Cosmology and Astroparticle Physics}, 2021\penalty0 (03):\penalty0 069, March 2021. \newblock ISSN 1475-7516. \newblock \doi{10.1088/1475-7516/2021/03/069}. \newblock URL \url{http://dx.doi.org/10.1088/1475-7516/2021/03/069}. \bibitem[Jenks et~al.(2023)Jenks, Choi, Lagos, and Yunes]{jenks2023parameterized} Leah Jenks, Lyla Choi, Macarena Lagos, and Nicolás Yunes. \newblock Parameterized parity violation in gravitational wave propagation, 2023. \bibitem[Cahn et~al.(2021)Cahn, Slepian, and Hou]{cahn2021test} Robert~N. Cahn, Zachary Slepian, and Jiamin Hou. \newblock A test for cosmological parity violation using the 3d distribution of galaxies, 2021. \bibitem[{BOSS collaboration}(2017{\natexlab{a}})]{10.1093/mnras/stx721} {BOSS collaboration}. \newblock {The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample}. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 470\penalty0 (3):\penalty0 2617--2652, 03 2017{\natexlab{a}}. \newblock ISSN 0035-8711. \newblock \doi{10.1093/mnras/stx721}. \newblock URL \url{https://doi.org/10.1093/mnras/stx721}. \bibitem[{BOSS collaboration}(2011)]{2011AJ....142...72E} {BOSS collaboration}. \newblock {SDSS-III: Massive Spectroscopic Surveys of the Distant Universe, the Milky Way, and Extra-Solar Planetary Systems}. \newblock \emph{\aj}, 142\penalty0 (3):\penalty0 72, September 2011. \newblock \doi{10.1088/0004-6256/142/3/72}. \bibitem[{BOSS collaboration}(2012{\natexlab{a}})]{Dawson_2013} {BOSS collaboration}. \newblock The baryon oscillation spectroscopic survey of sdss-iii. \newblock \emph{The Astronomical Journal}, 145\penalty0 (1):\penalty0 10, dec 2012{\natexlab{a}}. \newblock \doi{10.1088/0004-6256/145/1/10}. \newblock URL \url{https://dx.doi.org/10.1088/0004-6256/145/1/10}. \bibitem[Philcox et~al.(2021)Philcox, Hou, and Slepian]{philcox2021detection} Oliver H.~E. Philcox, Jiamin Hou, and Zachary Slepian. \newblock A first detection of the connected 4-point correlation function of galaxies using the boss cmass sample, 2021. \bibitem[Hou et~al.(2023)Hou, Slepian, and Cahn]{Hou_2023} Jiamin Hou, Zachary Slepian, and Robert~N Cahn. \newblock Measurement of parity-odd modes in the large-scale 4-point correlation function of sloan digital sky survey baryon oscillation spectroscopic survey twelfth data release cmass and lowz galaxies. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 522\penalty0 (4):\penalty0 5701–5739, May 2023. \newblock ISSN 1365-2966. \newblock \doi{10.1093/mnras/stad1062}. \newblock URL \url{http://dx.doi.org/10.1093/mnras/stad1062}. \bibitem[Philcox(2022)]{Philcox_2022} Oliver~H.E. Philcox. \newblock Probing parity violation with the four-point correlation function of boss galaxies. \newblock \emph{Physical Review D}, 106\penalty0 (6), September 2022. \newblock ISSN 2470-0029. \newblock \doi{10.1103/physrevd.106.063501}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevD.106.063501}. \bibitem[{BOSS collaboration}(2016{\natexlab{a}})]{Rodr_guez_Torres_2016} {BOSS collaboration}. \newblock The clustering of galaxies in the sdss-iii baryon oscillation spectroscopic survey: modelling the clustering and halo occupation distribution of boss cmass galaxies in the final data release. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 460\penalty0 (2):\penalty0 1173–1187, April 2016{\natexlab{a}}. \newblock ISSN 1365-2966. \newblock \doi{10.1093/mnras/stw1014}. \newblock URL \url{http://dx.doi.org/10.1093/mnras/stw1014}. \bibitem[Kitaura et~al.(2016)Kitaura, Rodríguez-Torres, Chuang, Zhao, Prada, Gil-Marín, Guo, Yepes, Klypin, Scóccola, Tinker, McBride, Reid, Sánchez, Salazar-Albornoz, Grieb, Vargas-Magana, Cuesta, Neyrinck, Beutler, Comparat, Percival, and Ross]{Kitaura_2016} Francisco-Shu Kitaura, Sergio Rodríguez-Torres, Chia-Hsun Chuang, Cheng Zhao, Francisco Prada, Héctor Gil-Marín, Hong Guo, Gustavo Yepes, Anatoly Klypin, Claudia~G. Scóccola, Jeremy Tinker, Cameron McBride, Beth Reid, Ariel~G. Sánchez, Salvador Salazar-Albornoz, Jan~Niklas Grieb, Mariana Vargas-Magana, Antonio~J. Cuesta, Mark Neyrinck, Florian Beutler, Johan Comparat, Will~J. Percival, and Ashley Ross. \newblock The clustering of galaxies in the sdss-iii baryon oscillation spectroscopic survey: mock galaxy catalogues for the boss final data release. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 456\penalty0 (4):\penalty0 4156–4173, January 2016. \newblock ISSN 1365-2966. \newblock \doi{10.1093/mnras/stv2826}. \newblock URL \url{http://dx.doi.org/10.1093/mnras/stv2826}. \bibitem[{Hou} et~al.(2022){Hou}, {Cahn}, {Philcox}, and {Slepian}]{2022PhRvD.106d3515H} Jiamin {Hou}, Robert~N. {Cahn}, Oliver H.~E. {Philcox}, and Zachary {Slepian}. \newblock {Analytic Gaussian covariance matrices for galaxy N -point correlation functions}. \newblock \emph{\prd}, 106\penalty0 (4):\penalty0 043515, August 2022. \newblock \doi{10.1103/PhysRevD.106.043515}. \bibitem[Krolewski et~al.(2024)Krolewski, May, Smith, and Hopkins]{krolewski2024evidenceparityviolationboss} Alex Krolewski, Simon May, Kendrick Smith, and Hans Hopkins. \newblock No evidence for parity violation in boss, 2024. \newblock URL \url{https://arxiv.org/abs/2407.03397}. \bibitem[Taylor et~al.(2023)Taylor, Craigie, and Ting]{taylor2023} Peter~L. Taylor, Matthew Craigie, and Yuan-Sen Ting. \newblock Unsupervised searches for cosmological parity-violation: An investigation with convolutional neural networks, 2023. \newblock URL \url{https://arxiv.org/abs/2312.09287}. \bibitem[Craigie et~al.(2024)Craigie, Taylor, Ting, Cuesta-Lazaro, Ruggeri, and Davis]{craigie2024unsupervisedsearchescosmologicalparity} Matthew Craigie, Peter~L. Taylor, Yuan-Sen Ting, Carolina Cuesta-Lazaro, Rossana Ruggeri, and Tamara~M. Davis. \newblock Unsupervised searches for cosmological parity violation: Improving detection power with the neural field scattering transform, 2024. \newblock URL \url{https://arxiv.org/abs/2405.13083}. \bibitem[{DESI collaboration}(2016)]{2016arXiv161100036D} {DESI collaboration}. \newblock {The DESI Experiment Part I: Science,Targeting, and Survey Design}. \newblock \emph{arXiv e-prints}, art. arXiv:1611.00036, October 2016. \newblock \doi{10.48550/arXiv.1611.00036}. \bibitem[{DESI Collaboration Et Al}(2023)]{https://doi.org/10.5281/zenodo.7964161} {DESI Collaboration Et Al}. \newblock The early data release of the dark energy spectroscopic instrument, 2023. \newblock URL \url{https://zenodo.org/record/7964161}. \bibitem[{Euclid Collaboration}(2011)]{2011_EUCLID_def} {Euclid Collaboration}. \newblock {Euclid Definition Study Report}. \newblock \emph{arXiv e-prints}, art. arXiv:1110.3193, October 2011. \newblock \doi{10.48550/arXiv.1110.3193}. \bibitem[{Euclid collaboration}(2020)]{2020_Euclid} {Euclid collaboration}. \newblock Euclid preparation: Vii. forecast validation for euclid cosmological probes. \newblock \emph{Astronomy \& Astrophysics}, 642:\penalty0 A191, October 2020. \newblock ISSN 1432-0746. \newblock \doi{10.1051/0004-6361/202038071}. \newblock URL \url{http://dx.doi.org/10.1051/0004-6361/202038071}. \bibitem[{Spergel} et~al.(2015){Spergel}, {Gehrels}, {Baltay}, {Bennett}, {Breckinridge}, {Donahue}, {Dressler}, {Gaudi}, {Greene}, {Guyon}, {Hirata}, {Kalirai}, {Kasdin}, {Macintosh}, {Moos}, {Perlmutter}, {Postman}, {Rauscher}, {Rhodes}, {Wang}, {Weinberg}, {Benford}, {Hudson}, {Jeong}, {Mellier}, {Traub}, {Yamada}, {Capak}, {Colbert}, {Masters}, {Penny}, {Savransky}, {Stern}, {Zimmerman}, {Barry}, {Bartusek}, {Carpenter}, {Cheng}, {Content}, {Dekens}, {Demers}, {Grady}, {Jackson}, {Kuan}, {Kruk}, {Melton}, {Nemati}, {Parvin}, {Poberezhskiy}, {Peddie}, {Ruffa}, {Wallace}, {Whipple}, {Wollack}, and {Zhao}]{2015Roman} D.~{Spergel}, N.~{Gehrels}, C.~{Baltay}, D.~{Bennett}, J.~{Breckinridge}, M.~{Donahue}, A.~{Dressler}, B.~S. {Gaudi}, T.~{Greene}, O.~{Guyon}, C.~{Hirata}, J.~{Kalirai}, N.~J. {Kasdin}, B.~{Macintosh}, W.~{Moos}, S.~{Perlmutter}, M.~{Postman}, B.~{Rauscher}, J.~{Rhodes}, Y.~{Wang}, D.~{Weinberg}, D.~{Benford}, M.~{Hudson}, W.~S. {Jeong}, Y.~{Mellier}, W.~{Traub}, T.~{Yamada}, P.~{Capak}, J.~{Colbert}, D.~{Masters}, M.~{Penny}, D.~{Savransky}, D.~{Stern}, N.~{Zimmerman}, R.~{Barry}, L.~{Bartusek}, K.~{Carpenter}, E.~{Cheng}, D.~{Content}, F.~{Dekens}, R.~{Demers}, K.~{Grady}, C.~{Jackson}, G.~{Kuan}, J.~{Kruk}, M.~{Melton}, B.~{Nemati}, B.~{Parvin}, I.~{Poberezhskiy}, C.~{Peddie}, J.~{Ruffa}, J.~K. {Wallace}, A.~{Whipple}, E.~{Wollack}, and F.~{Zhao}. \newblock {Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report}. \newblock \emph{arXiv e-prints}, art. arXiv:1503.03757, March 2015. \newblock \doi{10.48550/arXiv.1503.03757}. \bibitem[{WFIRST collaboration}(2019)]{2019Roman} {WFIRST collaboration}. \newblock {The Wide Field Infrared Survey Telescope: 100 Hubbles for the 2020s}. \newblock \emph{arXiv e-prints}, art. arXiv:1902.05569, February 2019. \newblock \doi{10.48550/arXiv.1902.05569}. \bibitem[Lee and Yang(1956)]{PhysRev.104.254-weaktheory} T.~D. Lee and C.~N. Yang. \newblock Question of parity conservation in weak interactions. \newblock \emph{Phys. Rev.}, 104:\penalty0 254--258, Oct 1956. \newblock \doi{10.1103/PhysRev.104.254}. \newblock URL \url{https://link.aps.org/doi/10.1103/PhysRev.104.254}. \bibitem[Wu et~al.(1957)Wu, Ambler, Hayward, Hoppes, and Hudson]{PhysRev.105.1413-weakexp} C.~S. Wu, E.~Ambler, R.~W. Hayward, D.~D. Hoppes, and R.~P. Hudson. \newblock Experimental test of parity conservation in beta decay. \newblock \emph{Phys. Rev.}, 105:\penalty0 1413--1415, Feb 1957. \newblock \doi{10.1103/PhysRev.105.1413}. \newblock URL \url{https://link.aps.org/doi/10.1103/PhysRev.105.1413}. \bibitem[Sakharov(1967)]{Sakharov:1967dj} A.~D. Sakharov. \newblock {Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe}. \newblock \emph{Pisma Zh. Eksp. Teor. Fiz.}, 5:\penalty0 32--35, 1967. \newblock \doi{10.1070/PU1991v034n05ABEH002497}. \bibitem[Carroll(1998)]{Carroll_1998} Sean~M. Carroll. \newblock Quintessence and the rest of the world: Suppressing long-range interactions. \newblock \emph{Physical Review Letters}, 81\penalty0 (15):\penalty0 3067–3070, October 1998. \newblock ISSN 1079-7114. \newblock \doi{10.1103/physrevlett.81.3067}. \newblock URL \url{http://dx.doi.org/10.1103/PhysRevLett.81.3067}. \bibitem[Hilbert(1915)]{Hilbert1915} D.~Hilbert. \newblock Die grundlagen der physik . (erste mitteilung.). \newblock \emph{Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse}, 1915:\penalty0 395--408, 1915. \newblock URL \url{http://eudml.org/doc/58946}. \bibitem[Lester and Schott(2019)]{Lester_2019} Christopher~G. Lester and Matthias Schott. \newblock Testing non-standard sources of parity violation in jets at the lhc, trialled with cms open data. \newblock \emph{Journal of High Energy Physics}, 2019\penalty0 (12), December 2019. \newblock ISSN 1029-8479. \newblock \doi{10.1007/jhep12(2019)120}. \newblock URL \url{http://dx.doi.org/10.1007/JHEP12(2019)120}. \bibitem[Lester and Tombs(2022)]{lester2022using} Christopher~G. Lester and Rupert Tombs. \newblock Using unsupervised learning to detect broken symmetries, with relevance to searches for parity violation in nature. (previously: ``{Stressed GANs} snag desserts''), 2022. \newblock ISSN 2835-8856. \newblock URL \url{https://openreview.net/forum?id=QFJ3gtbwHR}. \newblock Published in ``Transactions on Machine Learning Research''. \bibitem[Tombs and Lester(2022)]{Tombs_2022} Rupert Tombs and Christopher~G. Lester. \newblock A method to challenge symmetries in data with self-supervised learning. \newblock \emph{Journal of Instrumentation}, 17\penalty0 (08):\penalty0 P08024, August 2022. \newblock ISSN 1748-0221. \newblock \doi{10.1088/1748-0221/17/08/p08024}. \newblock URL \url{http://dx.doi.org/10.1088/1748-0221/17/08/P08024}. \bibitem[James~Gibbon(2023)]{lester_gibbon2022} (Supervised by Christopher G.~Lester) James~Gibbon. \newblock Detection of parity violation in snails through unsupervised learning, 2023. \newblock URL \url{https://www.hep.phy.cam.ac.uk/~lester/teaching/PartIIIProjects/2023-James-Gibbon-Snails.pdf}. \bibitem[{BOSS collaboration}(2016{\natexlab{b}})]{2016_REID_BOSS} {BOSS collaboration}. \newblock {SDSS-III Baryon Oscillation Spectroscopic Survey Data Release 12: galaxy target selection and large-scale structure catalogues}. \newblock \emph{\mnras}, 455\penalty0 (2):\penalty0 1553--1573, January 2016{\natexlab{b}}. \newblock \doi{10.1093/mnras/stv2382}. \bibitem[{BOSS collaboration}(2013)]{marston_mNRAS_2013} {BOSS collaboration}. \newblock {Stellar masses of SDSS-III/BOSS galaxies at z ∼ 0.5 and constraints to galaxy formation models}. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 435\penalty0 (4):\penalty0 2764--2792, 09 2013. \newblock ISSN 0035-8711. \newblock \doi{10.1093/mnras/stt1424}. \newblock URL \url{https://doi.org/10.1093/mnras/stt1424}. \bibitem[Saito et~al.(2016)Saito, Leauthaud, Hearin, Bundy, Zentner, Behroozi, Reid, Sinha, Coupon, Tinker, White, and Schneider]{10.1093/mnras/stw1080} Shun Saito, Alexie Leauthaud, Andrew~P. Hearin, Kevin Bundy, Andrew~R. Zentner, Peter~S. Behroozi, Beth~A. Reid, Manodeep Sinha, Jean Coupon, Jeremy~L. Tinker, Martin White, and Donald~P. Schneider. \newblock {Connecting massive galaxies to dark matter haloes in BOSS – I. Is galaxy colour a stochastic process in high-mass haloes?} \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 460\penalty0 (2):\penalty0 1457--1475, 05 2016. \newblock ISSN 0035-8711. \newblock \doi{10.1093/mnras/stw1080}. \newblock URL \url{https://doi.org/10.1093/mnras/stw1080}. \bibitem[Leauthaud et~al.(2016)Leauthaud, Bundy, Saito, Tinker, Maraston, Tojeiro, Huang, Brownstein, Schneider, and Thomas]{10.1093/mnras/stw117} Alexie Leauthaud, Kevin Bundy, Shun Saito, Jeremy Tinker, Claudia Maraston, Rita Tojeiro, Song Huang, Joel~R. Brownstein, Donald~P. Schneider, and Daniel Thomas. \newblock {The Stripe 82 Massive Galaxy Project – II. Stellar mass completeness of spectroscopic galaxy samples from the Baryon Oscillation Spectroscopic Survey}. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 457\penalty0 (4):\penalty0 4021--4037, 02 2016. \newblock ISSN 0035-8711. \newblock \doi{10.1093/mnras/stw117}. \newblock URL \url{https://doi.org/10.1093/mnras/stw117}. \bibitem[Bundy et~al.(2017)Bundy, Leauthaud, Saito, Maraston, Wake, and Thomas]{Bundy_2017} Kevin Bundy, Alexie Leauthaud, Shun Saito, Claudia Maraston, David~A. Wake, and Daniel Thomas. \newblock The stripe 82 massive galaxy project. iii. a lack of growth among massive galaxies. \newblock \emph{The Astrophysical Journal}, 851\penalty0 (1):\penalty0 34, December 2017. \newblock ISSN 1538-4357. \newblock \doi{10.3847/1538-4357/aa9896}. \newblock URL \url{http://dx.doi.org/10.3847/1538-4357/aa9896}. \bibitem[{BOSS collaboration}(2017{\natexlab{b}})]{Alam_2017} {BOSS collaboration}. \newblock The clustering of galaxies in the completed sdss-iii baryon oscillation spectroscopic survey: cosmological analysis of the dr12 galaxy sample. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 470\penalty0 (3):\penalty0 2617–2652, March 2017{\natexlab{b}}. \newblock ISSN 1365-2966. \newblock \doi{10.1093/mnras/stx721}. \newblock URL \url{http://dx.doi.org/10.1093/mnras/stx721}. \bibitem[{BOSS collaboration}(2012{\natexlab{b}})]{Anderson_2012_weights} {BOSS collaboration}. \newblock The clustering of galaxies in the sdss-iii baryon oscillation spectroscopic survey: baryon acoustic oscillations in the data release 9 spectroscopic galaxy sample. \newblock \emph{Monthly Notices of the Royal Astronomical Society}, 427\penalty0 (4):\penalty0 3435–3467, December 2012{\natexlab{b}}. \newblock ISSN 1365-2966. \newblock \doi{10.1111/j.1365-2966.2012.22066.x}. \newblock URL \url{http://dx.doi.org/10.1111/j.1365-2966.2012.22066.x}. \bibitem[Feldman et~al.(1994)Feldman, Kaiser, and Peacock]{Feldman_1994_fkp} Hume~A. Feldman, Nick Kaiser, and John~A. Peacock. \newblock Power-spectrum analysis of three-dimensional redshift surveys. \newblock \emph{The Astrophysical Journal}, 426:\penalty0 23, May 1994. \newblock ISSN 1538-4357. \newblock \doi{10.1086/174036}. \newblock URL \url{http://dx.doi.org/10.1086/174036}. \bibitem[{Royal Society Events}(2024)]{RS_A} {Royal Society Events}. \newblock {Royal Society Events} challenging the standard cosmological model, 2024. \newblock URL \url{https://royalsociety.org/science-events-and-lectures/2024/04/cosmological-model/}. \bibitem[{Philosophical Transactions of the Royal Society A}(2024)]{PT_RSA} {Philosophical Transactions of the Royal Society A}. \newblock {Philosophical Transactions of the Royal Society A}, 2024. \newblock URL \url{https://royalsocietypublishing.org/journal/rsta}. \bibitem[Ker et~al.(2019)Ker, Singh, Bai, Rao, Lim, and Wang]{med_3dCNN1} Justin Ker, Satya~P. Singh, Yeqi Bai, Jai Rao, Tchoyoson Lim, and Lipo Wang. \newblock Image thresholding improves 3-dimensional convolutional neural network diagnosis of different acute brain hemorrhages on computed tomography scans. \newblock \emph{Sensors}, 19\penalty0 (9), 2019. \newblock ISSN 1424-8220. \newblock \doi{10.3390/s19092167}. \newblock URL \url{https://www.mdpi.com/1424-8220/19/9/2167}. \bibitem[Singh et~al.(2020)Singh, Wang, Gupta, Goli, Padmanabhan, and Gulyás]{singh20203d_CNN} Satya~P. Singh, Lipo Wang, Sukrit Gupta, Haveesh Goli, Parasuraman Padmanabhan, and Balázs Gulyás. \newblock 3d deep learning on medical images: A review, 2020. \end{thebibliography} ``` 5. **Author Information:** - Lead Author: {'name': 'Samuel Hewson'} - Full Authors List: ```yaml Samuel Hewson: partiii: start: 2023-10-01 end: 2024-06-01 supervisors: - Will Handley thesis: Galaxies, snails and parity violation summer: start: 2024-06-24 end: 2024-08-01 supervisors: - Chris Lester thesis: null image: /assets/images/user.png Will Handley: pi: start: 2020-10-01 thesis: null postdoc: start: 2016-10-01 end: 2020-10-01 thesis: null phd: start: 2012-10-01 end: 2016-09-30 supervisors: - Anthony Lasenby - Mike Hobson thesis: 'Kinetic initial conditions for inflation: theory, observation and methods' original_image: images/originals/will_handley.jpeg image: /assets/group/images/will_handley.jpg links: Webpage: https://willhandley.co.uk Chris Lester: coi: start: 2023-10-01 thesis: null image: https://www.phy.cam.ac.uk/sites/default/files/styles/leading/public/media/profile/lesterc.jpg?itok=QNLbVvIS links: Department webpage: https://www.phy.cam.ac.uk/directory/lesterc ``` This YAML file provides a concise snapshot of an academic research group. It lists members by name along with their academic roles—ranging from Part III and summer projects to MPhil, PhD, and postdoctoral positions—with corresponding dates, thesis topics, and supervisor details. Supplementary metadata includes image paths and links to personal or departmental webpages. A dedicated "coi" section profiles senior researchers, highlighting the group’s collaborative mentoring network and career trajectories in cosmology, astrophysics, and Bayesian data analysis. ==================================================================================== Final Output Instructions ==================================================================================== - Combine all data sources to create a seamless, engaging narrative. - Follow the exact Markdown output format provided at the top. - Do not include any extra explanation, commentary, or wrapping beyond the specified Markdown. - Validate that every bibliographic reference with a DOI or arXiv identifier is converted into a Markdown link as per the examples. - Validate that every Markdown author link corresponds to a link in the author information block. - Before finalizing, confirm that no LaTeX citation commands or other undesired formatting remain. - Before finalizing, confirm that the link to the paper itself [2410.16030](https://arxiv.org/abs/2410.16030) is featured in the first sentence. Generate only the final Markdown output that meets all these requirements. {% endraw %}