{% 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: "Testing Lens Models of PLCK G165.7+67.0 Using Lensed SN H0pe" date: 2025-10-09 categories: papers --- ![AI generated image](/assets/images/posts/2025-10-09-2510.07637.png) Matt Grayling Content generated by [gemini-2.5-pro](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/2025-10-09-2510.07637.txt). Image generated by [imagen-4.0-generate-001](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/2025-10-09-2510.07637.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': 'Aadya Agrawal'}). 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 [2510.07637](https://arxiv.org/abs/2510.07637) 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 stands at the forefront of cosmological exploration, pioneering novel approaches that fuse fundamental physics with the transformative power of artificial intelligence. We are a dynamic team of researchers, including PhD students, postdoctoral fellows, and project students, based at the University of Cambridge. Our mission is to unravel the mysteries of the Universe, from its earliest moments to its present-day structure and ultimate fate. We tackle fundamental questions in cosmology and astrophysics, with a particular focus on leveraging advanced Bayesian statistical methods and AI to push the frontiers of scientific discovery. Our research spans a wide array of topics, including the [primordial Universe](https://arxiv.org/abs/1907.08524), [inflation](https://arxiv.org/abs/1807.06211), the nature of [dark energy](https://arxiv.org/abs/2503.08658) and [dark matter](https://arxiv.org/abs/2405.17548), [21-cm cosmology](https://arxiv.org/abs/2210.07409), the [Cosmic Microwave Background (CMB)](https://arxiv.org/abs/1807.06209), and [gravitational wave astrophysics](https://arxiv.org/abs/2411.17663). ### Our Research Approach: Innovation at the Intersection of Physics and AI At The Handley Research Group, we develop and apply cutting-edge computational techniques to analyze complex astronomical datasets. Our work is characterized by a deep commitment to principled [Bayesian inference](https://arxiv.org/abs/2205.15570) and the innovative application of [artificial intelligence (AI) and machine learning (ML)](https://arxiv.org/abs/2504.10230). **Key Research Themes:** * **Cosmology:** We investigate the early Universe, including [quantum initial conditions for inflation](https://arxiv.org/abs/2002.07042) and the generation of [primordial power spectra](https://arxiv.org/abs/2112.07547). We explore the enigmatic nature of [dark energy, using methods like non-parametric reconstructions](https://arxiv.org/abs/2503.08658), and search for new insights into [dark matter](https://arxiv.org/abs/2405.17548). A significant portion of our efforts is dedicated to [21-cm cosmology](https://arxiv.org/abs/2104.04336), aiming to detect faint signals from the Cosmic Dawn and the Epoch of Reionization. * **Gravitational Wave Astrophysics:** We develop methods for [analyzing gravitational wave signals](https://arxiv.org/abs/2411.17663), extracting information about extreme astrophysical events and fundamental physics. * **Bayesian Methods & AI for Physical Sciences:** A core component of our research is the development of novel statistical and AI-driven methodologies. This includes advancing [nested sampling techniques](https://arxiv.org/abs/1506.00171) (e.g., [PolyChord](https://arxiv.org/abs/1506.00171), [dynamic nested sampling](https://arxiv.org/abs/1704.03459), and [accelerated nested sampling with $\beta$-flows](https://arxiv.org/abs/2411.17663)), creating powerful [simulation-based inference (SBI) frameworks](https://arxiv.org/abs/2504.10230), and employing [machine learning for tasks such as radiometer calibration](https://arxiv.org/abs/2504.16791), [cosmological emulation](https://arxiv.org/abs/2503.13263), and [mitigating radio frequency interference](https://arxiv.org/abs/2211.15448). We also explore the potential of [foundation models for scientific discovery](https://arxiv.org/abs/2401.00096). **Technical Contributions:** Our group has a strong track record of developing widely-used scientific software. Notable examples include: * [**PolyChord**](https://arxiv.org/abs/1506.00171): A next-generation nested sampling algorithm for Bayesian computation. * [**anesthetic**](https://arxiv.org/abs/1905.04768): A Python package for processing and visualizing nested sampling runs. * [**GLOBALEMU**](https://arxiv.org/abs/2104.04336): An emulator for the sky-averaged 21-cm signal. * [**maxsmooth**](https://arxiv.org/abs/2007.14970): A tool for rapid maximally smooth function fitting. * [**margarine**](https://arxiv.org/abs/2205.12841): For marginal Bayesian statistics using normalizing flows and KDEs. * [**fgivenx**](https://arxiv.org/abs/1908.01711): A package for functional posterior plotting. * [**nestcheck**](https://arxiv.org/abs/1804.06406): Diagnostic tests for nested sampling calculations. ### Impact and Discoveries Our research has led to significant advancements in cosmological data analysis and yielded new insights into the Universe. Key achievements include: * Pioneering the development and application of advanced Bayesian inference tools, such as [PolyChord](https://arxiv.org/abs/1506.00171), which has become a cornerstone for cosmological parameter estimation and model comparison globally. * Making significant contributions to the analysis of major cosmological datasets, including the [Planck mission](https://arxiv.org/abs/1807.06209), providing some of the tightest constraints on cosmological parameters and models of [inflation](https://arxiv.org/abs/1807.06211). * Developing novel AI-driven approaches for astrophysical challenges, such as using [machine learning for radiometer calibration in 21-cm experiments](https://arxiv.org/abs/2504.16791) and [simulation-based inference for extracting cosmological information from galaxy clusters](https://arxiv.org/abs/2504.10230). * Probing the nature of dark energy through innovative [non-parametric reconstructions of its equation of state](https://arxiv.org/abs/2503.08658) from combined datasets. * Advancing our understanding of the early Universe through detailed studies of [21-cm signals from the Cosmic Dawn and Epoch of Reionization](https://arxiv.org/abs/2301.03298), including the development of sophisticated foreground modelling techniques and emulators like [GLOBALEMU](https://arxiv.org/abs/2104.04336). * Developing new statistical methods for quantifying tensions between cosmological datasets ([Quantifying tensions in cosmological parameters: Interpreting the DES evidence ratio](https://arxiv.org/abs/1902.04029)) and for robust Bayesian model selection ([Bayesian model selection without evidences: application to the dark energy equation-of-state](https://arxiv.org/abs/1506.09024)). * Exploring fundamental physics questions such as potential [parity violation in the Large-Scale Structure using machine learning](https://arxiv.org/abs/2410.16030). ### Charting the Future: AI-Powered Cosmological Discovery The Handley Research Group is poised to lead a new era of cosmological analysis, driven by the explosive growth in data from next-generation observatories and transformative advances in artificial intelligence. Our future ambitions are centred on harnessing these capabilities to address the most pressing questions in fundamental physics. **Strategic Research Pillars:** * **Next-Generation Simulation-Based Inference (SBI):** We are developing advanced SBI frameworks to move beyond traditional likelihood-based analyses. This involves creating sophisticated codes for simulating [Cosmic Microwave Background (CMB)](https://arxiv.org/abs/1908.00906) and [Baryon Acoustic Oscillation (BAO)](https://arxiv.org/abs/1607.00270) datasets from surveys like DESI and 4MOST, incorporating realistic astrophysical effects and systematic uncertainties. Our AI initiatives in this area focus on developing and implementing cutting-edge SBI algorithms, particularly [neural ratio estimation (NRE) methods](https://arxiv.org/abs/2407.15478), to enable robust and scalable inference from these complex simulations. * **Probing Fundamental Physics:** Our enhanced analytical toolkit will be deployed to test the standard cosmological model ($\Lambda$CDM) with unprecedented precision and to explore [extensions to Einstein's General Relativity](https://arxiv.org/abs/2006.03581). We aim to constrain a wide range of theoretical models, from modified gravity to the nature of [dark matter](https://arxiv.org/abs/2106.02056) and [dark energy](https://arxiv.org/abs/1701.08165). This includes leveraging data from upcoming [gravitational wave observatories](https://arxiv.org/abs/1803.10210) like LISA, alongside CMB and large-scale structure surveys from facilities such as Euclid and JWST. * **Synergies with Particle Physics:** We will continue to strengthen the connection between cosmology and particle physics by expanding the [GAMBIT framework](https://arxiv.org/abs/2009.03286) to interface with our new SBI tools. This will facilitate joint analyses of cosmological and particle physics data, providing a holistic approach to understanding the Universe's fundamental constituents. * **AI-Driven Theoretical Exploration:** We are pioneering the use of AI, including [large language models and symbolic computation](https://arxiv.org/abs/2401.00096), to automate and accelerate the process of theoretical model building and testing. This innovative approach will allow us to explore a broader landscape of physical theories and derive new constraints from diverse astrophysical datasets, such as those from GAIA. Our overarching goal is to remain at the forefront of scientific discovery by integrating the latest AI advancements into every stage of our research, from theoretical modeling to data analysis and interpretation. We are excited by the prospect of using these powerful new tools to unlock the secrets of the cosmos. Content generated by [gemini-2.5-pro-preview-05-06](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/2510.07637v1 guidislink: true link: https://arxiv.org/abs/2510.07637v1 title: Testing Lens Models of PLCK G165.7+67.0 Using Lensed SN H0pe title_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: Testing Lens Models of PLCK G165.7+67.0 Using Lensed SN H0pe updated: '2025-10-09T00:17:20Z' updated_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2025 - 10 - 9 - 0 - 17 - 20 - 3 - 282 - 0 - tm_zone: null tm_gmtoff: null links: - !!python/object/new:feedparser.util.FeedParserDict dictitems: href: https://arxiv.org/abs/2510.07637v1 rel: alternate type: text/html - !!python/object/new:feedparser.util.FeedParserDict dictitems: href: https://arxiv.org/pdf/2510.07637v1 rel: related type: application/pdf title: pdf summary: "Supernova H0pe is a multiply-imaged Type Ia supernova (SN Ia) and the\ \ second lensed SN to yield a measurement of the Hubble constant by the time-delay\ \ cosmography method, finding $H_0 = 75.4^{+8.1}_{-5.5} \\text{km s}^{-1} \\text{Mpc}^{-1}$.\ \ We investigate the seven lens modeling approaches used to derive $H_0$, assessing\ \ their agreement with $\u039B\\text{CDM}$ constraints from SN Ia surveys through\ \ a purely observational comparison. While photometrically derived magnifications\ \ yield distance moduli in line with $\u039B\\text{CDM}$ expectations, our comparison\ \ reveals that lens model predictions, even the most precise ones, consistently\ \ overestimate the magnification, with a offset of $ \u0394\u03BC> 1$ mag. This\ \ known bias, already appreciated by modeling teams, is independently confirmed\ \ through our analysis and highlights the value of lensed SNe as a tool to test\ \ model accuracy. If unaccounted for, such magnification biases can propagate\ \ into uncertainties in derived cosmological parameters, including $H_0$, and\ \ affect the interpretation of future precision measurements. These findings highlight\ \ a critical challenge for precision cosmology using strongly lensed transients.\ \ With next-generation surveys such as LSST, Roman, and Euclid poised to discover\ \ many more gravitationally lensed supernovae, the development and validation\ \ of robust, accurate lens models will be essential for using these rare events\ \ to probe cosmology." summary_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: "Supernova H0pe is a multiply-imaged Type Ia supernova (SN Ia) and the\ \ second lensed SN to yield a measurement of the Hubble constant by the time-delay\ \ cosmography method, finding $H_0 = 75.4^{+8.1}_{-5.5} \\text{km s}^{-1}\ \ \\text{Mpc}^{-1}$. We investigate the seven lens modeling approaches used\ \ to derive $H_0$, assessing their agreement with $\u039B\\text{CDM}$ constraints\ \ from SN Ia surveys through a purely observational comparison. While photometrically\ \ derived magnifications yield distance moduli in line with $\u039B\\text{CDM}$\ \ expectations, our comparison reveals that lens model predictions, even the\ \ most precise ones, consistently overestimate the magnification, with a offset\ \ of $ \u0394\u03BC> 1$ mag. This known bias, already appreciated by modeling\ \ teams, is independently confirmed through our analysis and highlights the\ \ value of lensed SNe as a tool to test model accuracy. If unaccounted for,\ \ such magnification biases can propagate into uncertainties in derived cosmological\ \ parameters, including $H_0$, and affect the interpretation of future precision\ \ measurements. These findings highlight a critical challenge for precision\ \ cosmology using strongly lensed transients. With next-generation surveys\ \ such as LSST, Roman, and Euclid poised to discover many more gravitationally\ \ lensed supernovae, the development and validation of robust, accurate lens\ \ models will be essential for using these rare events to probe cosmology." 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.GA scheme: http://arxiv.org/schemas/atom label: null published: '2025-10-09T00:17:20Z' published_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2025 - 10 - 9 - 0 - 17 - 20 - 3 - 282 - 0 - tm_zone: null tm_gmtoff: null arxiv_comment: 18 pages, 7 figures. Submitted to ApJ arxiv_primary_category: term: astro-ph.CO authors: - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Aadya Agrawal - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: J. D. R. Pierel - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Gautham Narayan - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: B. L. Frye - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Jose M. Diego - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Nikhil Garuda - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Matthew Grayling - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Anton M. Koekemoer - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Kaisey S. Mandel - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: M. Pascale - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: David Vizgan - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Rogier A. Windhorst author_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Rogier A. Windhorst author: Rogier A. Windhorst ``` 3. **Paper Source (TeX):** ```tex %% Beginning of file 'sample631.tex' %% %% Modified 2022 May %% %% This is a sample manuscript marked up using the %% AASTeX v6.31 LaTeX 2e macros. %% %% AASTeX is now based on Alexey Vikhlinin's emulateapj.cls %% (Copyright 2000-2015). See the classfile for details. %% AASTeX requires revtex4-1.cls and other external packages such as %% latexsym, graphicx, amssymb, longtable, and epsf. Note that as of %% Oct 2020, APS now uses revtex4.2e for its journals but remember that %% AASTeX v6+ still uses v4.1. All of these external packages should %% already be present in the modern TeX distributions but not always. %% For example, revtex4.1 seems to be missing in the linux version of %% TexLive 2020. One should be able to get all packages from www.ctan.org. %% In particular, revtex v4.1 can be found at %% https://www.ctan.org/pkg/revtex4-1. %% The first piece of markup in an AASTeX v6.x document is the \documentclass %% command. LaTeX will ignore any data that comes before this command. The %% documentclass can take an optional argument to modify the output style. %% The command below calls the preprint style which will produce a tightly %% typeset, one-column, single-spaced document. It is the default and thus %% does not need to be explicitly stated. %% %% using aastex version 6.3 \documentclass[twocolumn, times]{aastex631} % \documentclass[manuscript, times]{aastex631} %% The default is a single spaced, 10 point font, single spaced article. %% There are 5 other style options available via an optional argument. They %% can be invoked like this: %% %% \documentclass[arguments]{aastex631} %% %% where the layout options are: %% %% twocolumn : two text columns, 10 point font, single spaced article. %% This is the most compact and represent the final published %% derived PDF copy of the accepted manuscript from the publisher %% manuscript : one text column, 12 point font, double spaced article. %% preprint : one text column, 12 point font, single spaced article. %% preprint2 : two text columns, 12 point font, single spaced article. %% modern : a stylish, single text column, 12 point font, article with %% wider left and right margins. This uses the Daniel %% Foreman-Mackey and David Hogg design. %% RNAAS : Supresses an abstract. Originally for RNAAS manuscripts %% but now that abstracts are required this is obsolete for %% AAS Journals. Authors might need it for other reasons. DO NOT %% use \begin{abstract} and \end{abstract} with this style. %% %% Note that you can submit to the AAS Journals in any of these 6 styles. %% %% There are other optional arguments one can invoke to allow other stylistic %% actions. The available options are: %% %% astrosymb : Loads Astrosymb font and define \astrocommands. %% tighten : Makes baselineskip slightly smaller, only works with %% the twocolumn substyle. %% times : uses times font instead of the default %% linenumbers : turn on lineno package. %% trackchanges : required to see the revision mark up and print its output %% longauthor : Do not use the more compressed footnote style (default) for %% the author/collaboration/affiliations. Instead print all %% affiliation information after each name. Creates a much %% longer author list but may be desirable for short %% author papers. %% twocolappendix : make 2 column appendix. %% anonymous : Do not show the authors, affiliations and acknowledgments %% for dual anonymous review. %% %% these can be used in any combination, e.g. %% %% \documentclass[twocolumn,linenumbers,trackchanges]{aastex631} %% %% AASTeX v6.* now includes \hyperref support. While we have built in specific %% defaults into the classfile you can manually override them with the %% \hypersetup command. For example, %% %% \hypersetup{linkcolor=red,citecolor=green,filecolor=cyan,urlcolor=magenta} %% %% will change the color of the internal links to red, the links to the %% bibliography to green, the file links to cyan, and the external links to %% magenta. Additional information on \hyperref options can be found here: %% https://www.tug.org/applications/hyperref/manual.html#x1-40003 %% %% Note that in v6.3 "bookmarks" has been changed to "true" in hyperref %% to improve the accessibility of the compiled pdf file. %% %% If you want to create your own macros, you can do so %% using \newcommand. Your macros should appear before %% the \begin{document} command. %% \newcommand{\lensedsn}{SN\,H$0$pe} \newcommand{\vdag}{(v)^\dagger} \newcommand\aastex{AAS\TeX} \newcommand\latex{La\TeX} \usepackage{todonotes} \usepackage{multirow} \usepackage{amsmath} % \usepackage{doi} %% Reintroduced the \received and \accepted commands from AASTeX v5.2 %\received{March 1, 2021} %\revised{April 1, 2021} %\accepted{\today} %% Command to document which AAS Journal the manuscript was submitted to. %% Adds "Submitted to " the argument. %\submitjournal{PSJ} %% For manuscript that include authors in collaborations, AASTeX v6.31 %% builds on the \collaboration command to allow greater freedom to %% keep the traditional author+affiliation information but only show %% subsets. The \collaboration command now must appear AFTER the group %% of authors in the collaboration and it takes TWO arguments. The last %% is still the collaboration identifier. The text given in this %% argument is what will be shown in the manuscript. The first argument %% is the number of author above the \collaboration command to show with %% the collaboration text. If there are authors that are not part of any %% collaboration the \nocollaboration command is used. This command takes %% one argument which is also the number of authors above to show. A %% dashed line is shown to indicate no collaboration. This example manuscript %% shows how these commands work to display specific set of authors %% on the front page. %% %% For manuscript without any need to use \collaboration the %% \AuthorCollaborationLimit command from v6.2 can still be used to %% show a subset of authors. % %\AuthorCollaborationLimit=2 % %% will only show Schwarz & Muench on the front page of the manuscript %% (assuming the \collaboration and \nocollaboration commands are %% commented out). %% %% Note that all of the author will be shown in the published article. %% This feature is meant to be used prior to acceptance to make the %% front end of a long author article more manageable. Please do not use %% this functionality for manuscripts with less than 20 authors. Conversely, %% please do use this when the number of authors exceeds 40. %% %% Use \allauthors at the manuscript end to show the full author list. %% This command should only be used with \AuthorCollaborationLimit is used. %% The following command can be used to set the latex table counters. It %% is needed in this document because it uses a mix of latex tabular and %% AASTeX deluxetables. In general it should not be needed. %\setcounter{table}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% The following section outlines numerous optional output that %% can be displayed in the front matter or as running meta-data. %% %% If you wish, you may supply running head information, although %% this information may be modified by the editorial offices. %\shorttitle{AASTeX v6.3.1 Sample article} %\shortauthors{Schwarz et al.} %% %% You can add a light gray and diagonal water-mark to the first page %% with this command: %% \watermark{text} %% where "text", e.g. DRAFT, is the text to appear. If the text is %% long you can control the water-mark size with: %% \setwatermarkfontsize{dimension} %% where dimension is any recognized LaTeX dimension, e.g. pt, in, etc. %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\graphicspath{{./}{figures/}} %% This is the end of the preamble. Indicate the beginning of the %% manuscript itself with \begin{document}. \begin{document} \title{Testing Lens Models of PLCK G165.7+67.0 Using Lensed SN H0pe} \author[0009-0008-1965-9012]{Aadya~Agrawal} \affiliation{Department of Astronomy, University of Illinois Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA} \affiliation{NSF-Simons AI for the Sky (SkAI) Institute, 875 N. Michigan Ave., Suite 3500, Chicago, IL 60611, USA} %% LaTeX will automatically break titles if they run longer than %% one line. However, you may use \\ to force a line break if %% you desire. In v6.31 you can include a footnote in the title. %% A significant change from earlier AASTEX versions is in the structure for %% calling author and affiliations. The change was necessary to implement %% auto-indexing of affiliations which prior was a manual process that could %% easily be tedious in large author manuscripts. %% %% The \author command is the same as before except it now takes an optional %% argument which is the 16 digit ORCID. The syntax is: %% \author[xxxx-xxxx-xxxx-xxxx]{Author Name} %% %% This will hyperlink the author name to the author's ORCID page. Note that %% during compilation, LaTeX will do some limited checking of the format of %% the ID to make sure it is valid. If the "orcid-ID.png" image file is %% present or in the LaTeX pathway, the OrcID icon will appear next to %% the authors name. %% %% Use \affiliation for affiliation information. The old \affil is now aliased %% to \affiliation. AASTeX v6.31 will automatically index these in the header. %% When a duplicate is found its index will be the same as its previous entry. %% %% Note that \altaffilmark and \altaffiltext have been removed and thus %% can not be used to document secondary affiliations. If they are used latex %% will issue a specific error message and quit. Please use multiple %% \affiliation calls for to document more than one affiliation. %% %% The new \altaffiliation can be used to indicate some secondary information %% such as fellowships. This command produces a non-numeric footnote that is %% set away from the numeric \affiliation footnotes. NOTE that if an %% \altaffiliation command is used it must come BEFORE the \affiliation call, %% right after the \author command, in order to place the footnotes in %% the proper location. %% %% Use \email to set provide email addresses. Each \email will appear on its %% own line so you can put multiple email address in one \email call. A new %% \correspondingauthor command is available in V6.31 to identify the %% corresponding author of the manuscript. It is the author's responsibility %% to make sure this name is also in the author list. %% %% While authors can be grouped inside the same \author and \affiliation %% commands it is better to have a single author for each. This allows for %% one to exploit all the new benefits and should make book-keeping easier. %% %% If done correctly the peer review system will be able to %% automatically put the author and affiliation information from the manuscript %% and save the corresponding author the trouble of entering it by hand. %\correspondingauthor{August Muench} %\email{greg.schwarz@aas.org, gus.muench@aas.org} \author[0000-0002-2361-7201]{J.~D.~R.~Pierel} \affiliation{Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA} \author[0000-0001-6022-0484]{Gautham Narayan} \affiliation{Department of Astronomy, University of Illinois Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA} \affiliation{NSF-Simons AI for the Sky (SkAI) Institute, 875 N. Michigan Ave., Suite 3500, Chicago, IL 60611, USA} \author[0000-0003-1625-8009]{B. L. Frye} \affiliation{Department of Astronomy/Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA} \author[0000-0001-9065-3926]{Jose M. Diego} \affiliation{Instituto de Física de Cantabria (CSIC-UC). Avda. Los Castros s/n. 39005 Santander, Spain} \author[0000-0003-3418-2482]{Nikhil Garuda} \affiliation{Department of Astronomy, The University of Texas at Austin, 2515 Speedway Boulevard, Austin, TX 78712, USA} \author[0000-0002-6741-983X]{Matthew~Grayling} \affiliation{Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK} \author[0000-0002-6610-2048]{Anton M. Koekemoer} \affiliation{Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA} \author[0000-0001-9846-4417]{Kaisey~S.~Mandel} \affiliation{Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK} \author[0000-0002-2282-8795]{M.~Pascale} \affiliation{Department of Astronomy, University of California, 501 Campbell Hall \#3411, Berkeley, CA, 94720, USA} \author[0000-0001-7610-5544]{David Vizgan} \affiliation{Department of Astronomy, University of Illinois Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA} \author[0000-0001-8156-6281]{Rogier A. Windhorst} \affiliation{School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287-6004, USA} % \email{Rogier.Windhorst@asu.edu} % \author{Friends} % \affiliation{Many institutions} %% Note that the \and command from previous versions of AASTeX is now %% depreciated in this version as it is no longer necessary. AASTeX %% automatically takes care of all commas and "and"s between authors names. %% AASTeX 6.31 has the new \collaboration and \nocollaboration commands to %% provide the collaboration status of a group of authors. These commands %% can be used either before or after the list of corresponding authors. The %% argument for \collaboration is the collaboration identifier. Authors are %% encouraged to surround collaboration identifiers with ()s. The %% \nocollaboration command takes no argument and exists to indicate that %% the nearby authors are not part of surrounding collaborations. %% Mark off the abstract in the ``abstract'' environment. \begin{abstract} Supernova H0pe is a multiply-imaged Type Ia supernova (SN~Ia) and the second lensed SN to yield a measurement of the Hubble constant by the time-delay cosmography method, finding $H_0 = 75.4^{+8.1}_{-5.5} \text{km s}^{-1} \text{Mpc}^{-1}$ \citep{Pascale_SNH0pe_2025}. We investigate the seven lens modeling approaches used to derive $H_0$, assessing their agreement with $\Lambda \text{CDM}$ constraints from SN~Ia surveys through a purely observational comparison. While photometrically derived magnifications yield distance moduli in line with $\Lambda \text{CDM}$ expectations, our comparison reveals that lens model predictions, even the most precise ones, consistently overestimate the magnification, with a offset of $ \Delta \mu > 1$~mag. This known bias, already appreciated by modeling teams, is independently confirmed through our analysis and highlights the value of lensed SNe as a tool to test model accuracy. If unaccounted for, such magnification biases can propagate into uncertainties in derived cosmological parameters, including $H_0$, and affect the interpretation of future precision measurements. These findings highlight a critical challenge for precision cosmology using strongly lensed transients. With next-generation surveys such as LSST, Roman, and Euclid poised to discover many more gravitationally lensed supernovae, the development and validation of robust, accurate lens models will be essential for using these rare events to probe cosmology. \end{abstract} %% Keywords should appear after the \end{abstract} command. %% The AAS Journals now uses Unified Astronomy Thesaurus concepts: %% https://astrothesaurus.org %% You will be asked to selected these concepts during the submission process %% but this old "keyword" functionality is maintained in case authors want %% to include these concepts in their preprints. \keywords{Supernova, Strong Lensing, Cosmology} %% From the front matter, we move on to the body of the paper. %% Sections are demarcated by \section and \subsection, respectively. %% Observe the use of the LaTeX \label %% command after the \subsection to give a symbolic KEY to the %% subsection for cross-referencing in a \ref command. %% You can use LaTeX's \ref and \label commands to keep track of %% cross-references to sections, equations, tables, and figures. %% That way, if you change the order of any elements, LaTeX will %% automatically renumber them. %% %% We recommend that authors also use the natbib \citep %% and \citet commands to identify citations. The citations are %% tied to the reference list via symbolic KEYs. The KEY corresponds %% to the KEY in the \bibitem in the reference list below. \section{Introduction}\label{sec:intro} \begin{figure*}[!ht] \centering \includegraphics[width=0.85\linewidth]{figs/aadya_SNH0pe_color_image.pdf} \caption{Mosaic of the central region of the PLCK G165.7+67.0 cluster field, covering a width of 1 arcminute, showing the full depth obtained in all 5 epochs for all 6 NIRCam filters F090W, F150W, F200W, F277W, F356W, and F444W (see Section \ref{sec:obs} for further details). The insets show a close up of each of the images of SN H0pe based on the JWST Epoch 1 data. Each of the insets has a partial crosshair that shows the location of the SN.} \label{fig:color_im} \end{figure*} Supernovae have been central to astronomical discovery for centuries, beginning with Tycho Brahe’s 1572 observation of a ``new star'' that overturned the prevailing notion of an immutable sky. Type Ia supernovae (SNe Ia), are thermonuclear explosions of CO white dwarfs and function as ‘standardizable candles’, enabling astronomers to measure the distances to extragalactic sources \citep{standard_candles_1992, Tripp_1998, Guy_2007_SALT,SN_cosmology_review, Pantheon_2022}. SNe~Ia also serve as probes to measure the Hubble constant ($H_0$), a cosmological parameter that quantifies the local rate of expansion of the universe \citep{Riess_1998}. Measuring $H_0$ provides a critical test of the standard $\Lambda \text{CDM}$ cosmological model \citep{LCDM_review_2018, lcdm_H0_review_2022}. The ``local distance-ladder'' method, using Cepheids and SNe~Ia, is one of the primary methods used to measure $H_0$ \citep{Riess_2024_review}. The `Supernova $H_0$ for the Equation of State' (SH0ES) team \citep{Riess_2024} measured a value of $H_0 = 72.6 \pm 2.0$ km s$^{-1}$ Mpc$^{-1}$. This value differs from the value of $H_0 = 67.4 \pm 0.5$ km s$^{-1}$ Mpc$^{-1}$ found by \citet{Planck_2020} using measurements from the Cosmic Microwave Background (CMB) in conjunction with $\Lambda \text{CDM}$. This `Hubble Tension' at the $5\sigma$ level \citep{Verde_2019, Riess_2016}, represents one of the most significant issues in cosmology today. Many solutions have been proposed to resolve the discrepancy \citep{Weinberg_2013}, including using new probes such as redshift-space distortions \citep{RSD_tension}, standard sirens \citep{standard_sirens_1, standard_sirens_aas} and precision gravity tests\citep{MOND}. Assuming that both the SNe~Ia and CMB measurements are accurate, several recent studies have explored modifications to the standard cosmological model--i.e., Early Dark Energy \citep[i.e.,][]{kamionkowski_early_de_2023}, Late Dark Energy \citep[i.e.,][]{late_de}, and Dark Energy models with additional degrees of freedom \citep[i.e.,][]{Di_Valentino_2021}--in an effort to reconcile the differing measurements. Each of these proposed scenarios has significant implications for processes such as star formation, galaxy evolution, and large-scale structure, requiring thorough scrutiny. Another possibility for the source of the tension lies in a systematic error in one of the many probes. Larger surveys and more precise measurements are being taken and planned to determine the cause. One promising approach to addressing the Hubble tension is to leverage strongly gravitationally lensed supernovae (glSNe) for cosmological measurements and probes of $H_0$ \citep{Refsdal_1964, Suyu_24_TDC, Goobar_2025}. Strong gravitational lensing occurs when light from a background source bends around a massive foreground system, causing the light to take different paths to reach the observer and resulting in multiple images of the source \citep{Einstein_1936, Zwicky_lenses_1937, Zwicky_nebulae_1937}. When a SN explodes in a multiply-imaged host galaxy, the light from the SN follows paths of varying length and gravitational potential, resulting in both a geometric and a Shapiro (gravitational) time delay. Together, these effects produce an observable time delay between images, determined by the distribution of the gravitational potential of the foreground lens system \citep{Narayan_Bartelmann_1996, Oguri_2019}. This time delay depends on the angular diameter distances of the source, lens and observer relative to each other. As these distances are dependent on cosmology, measuring the time delays in turn can be used to probe cosmological parameters. When combined with a lens model of the foreground mass distribution, these time delays allow glSNe to serve as one-step probes of $H_0$ \citep{Refsdal_1964} in a method called Time Delay Cosmography. The lens model provides the difference in Fermat potential between the image positions, which (combined with the observed time delays) allows for the inference of the 'time-delay distance,' and thus, $H_0$. Since this distance scales inversely with $H_0$, accurate modeling of the lens potential is essential to extract reliable cosmological constraints. While other transients can also be used for Time Delay Cosmography, SNe are particularly advantageous for this method \citep{Pierel_PhotometricTimedelay_2024a}. As the SN flux fades, images of the host without the SN flux can be taken to get more accurate photometry without host contamination, providing a more precise $H_0$ value \citep{Ding_2021}. Additionally, the non-repeating and non-stochastic light curve of glSNe can be leveraged to break degeneracies in the lens models which is difficult for other lensed objects that permit time delay cosmography e.g., quasars \citep{Birrer_2024_review, Arendse_2024}. A glSN~Ia, in particular, provides even more precision as there exist well-understood models of SN~Ia evolution \citep{Guy_2007_SALT,Kenworthy_2021_SALT3,SALT3NIR_2022,Mandel_2022, Ward_2023_SNIa, grayling_24_mnras}, which makes determining time-delays simpler. Furthermore, a glSN~Ia enables the measurement of the absolute magnifications of its multiple images—an observable that directly informs the lensing geometry and the inferred value of the Hubble constant. As demonstrated by \citet{Liu_Oguri_2025}, combining time-delay measurements with absolute magnification constraints significantly improves the precision of $H_0$ inference by breaking degeneracies in the lens model. This joint analysis not only tightens the posterior on $H_0$ but also helps mitigate biases that arise from uncertainties in the lens potential, especially in regions with limited imaging constraints. \begin{figure}[t] \centering \includegraphics[trim= 0.5cm 1.5cm 0.5cm 1.5cm, clip, width=\linewidth]{figs/Hope_pipeline_publish_ver-Page-5.pdf} \caption{A schematic showing the method used in this analysis. Each step has an associated section (S.), table (T.) or figure (F.) in this paper, that provides more details about the work.} \label{fig:schematic} \vspace{-2mm} \end{figure} A handful of glSNe have been discovered in recent years. SN Refsdal, the first multiply-imaged SN, enabled a measurement of the time delay, and from it, a value for $ H_0 = 66.6^{+ 4.1}_{-3.3}$ km s$^{-1}$ Mpc$^{-1}$ by the method of time-delay cosmography \citep{Kelly_2023_a, Kelly_2023_b}. Subsequently discovered glSNe such as SN Requiem \citep{Rodney_2021a}, AT2022riv \citep{AT2022riv} and C22 \citep{Chen_2022} were either detected too late to observe multiple images or found in archival imaging, limiting opportunities for follow-up. However, SN Requiem may offer another opportunity, with a predicted reappearance expected around ~2027 \citep{Suyu_2025_Encore_Requiem}. SN Zwicky \citep{Goobar_2023_zwicky, Pierel_2023_zwicky, Larison_2025_zwicky} and iPTF16geu \citep{Goobar_2017}, two glSNe~Ia identified in galaxy-scale lenses, were ultimately too compact to provide meaningful constraints on $H_0$. \lensedsn~was the first glSN~Ia with sufficient data and time delays long enough to facilitate the measurement of $H_0$ \citep{Frye_2023, Frye_2024}. This SN was triply-imaged, enabling the measurement of the two time delays by a photometric approach \citep{Pierel_PhotometricTimedelay_2024a}, and a rare {\it spectroscopic} one \citep{chen_SN_2024}. Owing to the nature of this SN as a standard candle, absolute magnifications were also measured for all three SN appearances as a part of these photometric and spectroscopic programs, and this set of five observables was inducted into the $H_0$ inference \citep{Pascale_SNH0pe_2025}. Seven lens models were constructed from seven different approaches, and the results of all groups were double-blinded by strictly enforced protocols. On scaling the lens-predicted values for the observables to the measured ones, a value of $H_0 = 75.4^{+8.1}_{-5.5} \text{km s}^{-1} \text{Mpc}^{-1}$ was inferred \citep{Pascale_SNH0pe_2025}. This result highlights the potential for using such systems to address the Hubble Tension. The most recent glSN~Ia to be observed is SN Encore, incredibly found in the same galaxy as SN Requiem, with the probability of $\lesssim 3\%$ for detecting two SNe-Ia in this system in the last decade \citep{Pierel_Encore_2024}. SN Encore found an measurement of $H_0 = 66.9^{+11.2}_{-8.1} \text{km s}^{-1} \text{Mpc}^{-1}$ in a double blind analysis \citep{Pierel_2025_Encore, Suyu_2025_Encore_Requiem}. Lensed SNe~Ia also present a unique opportunity to test existing lens models \citep{Rodney_testing, Patel_2014}. Among different lens modeling methods, one of the most common ones is based on the assumption that light traces mass or `LTM' \citep{Zitrin_2009}. This means that the galaxies in a cluster act as tracers of the dark matter distribution. This method is well-tested and has proven to result in robust lens models that can replicate observations well. However, when predicting the reappearance of SN Refsdal, \citet{Zitrin_2021} discovered a minor but pivotal numerical artifact that had perviously gone unnoticed in the Time Delay calculation. While this particular issue could also have been revealed through other checks, it illustrates a broader benefit: lensed transients provide a direct way to evaluate the underlying methodology and assumptions of lens models. Despite the low probability of source, lens, and observer alignments, the \textit{James Webb Space Telescope (JWST)} has enabled the observation of many multiply-imaged systems. In the coming years, facilities like the \textit{Vera C. Rubin Observatory}, \textit{Nancy Grace Roman Telescope}, and others are expected to observe $\sim 100$ glSNe \citep{LSST_Roman_rates, Arendse_2024, Pierel_Roman_rates}. The dramatic increase in the number of observed glSNe systems necessitates robust lens models. % In order to make full use of these systems, robust lens models are required. % GlSNe observed with these facilities will be located at higher redshift than the majority of other observed SNe due to lensing magnification. To this end, we tested the lens models of PLCK G165.7+67.0 described in \citet{Pascale_SNH0pe_2025} using observations of \lensedsn. Figure \ref{fig:color_im} shows the full-depth mosaic from combining all 5 epochs of the PLCK G165.7+67.0 cluster field, along with insets showing close-up images of \lensedsn~as observed in Epoch 1. Figure \ref{fig:schematic} shows the workflow and data used for this analysis. The figure includes information on section, table or figure where the elements are described in this paper which is organized as follows. In Section \ref{sec:obs} we describe the observations and data used in this analysis. A brief description of the lens models taken from the \citet{Pascale_SNH0pe_2025} are given in Section \ref{sec:lens_models}. In Section \ref{sec: sed modeling}, we detail the methods used to conduct the analysis. This analysis relies heavily on SN~Ia SED models and thus, under Section \ref{sec: sed modeling}, Subsection \ref{subsec: sed models} describes each model and Subsection \ref{subsec: getting mu} describes the methods used to calculate the distance fitted parameters from each of the models. In Section \ref{sec:results} we state the results of the analysis which are further discussed in Section \ref{sec: concl}. \begin{table*}[!ht] \caption{Photometry of \lensedsn~ as given in \citet{Pierel_PhotometricTimedelay_2024a}} \centering \begin{tabular*}{\linewidth}{@{\extracolsep{\stretch{1}}}*{7}{c}} \toprule MJD & Filter & Exp Time (s) & $m_{2a}$ & $m_{2b}$ & $m_{2c}$ \\ \hline 60033 & F090W & 2490 & $30.55 \pm 0.47$ & $25.53 \pm 0.05$ & $ 28.71 \pm 0.11$ \\ 60033 & F115W & 2490 & $28.35 \pm 0.11$ & $24.96 \pm 0.06$ & $ 26.16 \pm 0.07$ \\ 60033 & F150W & 1889 & $26.50 \pm 0.08$ & $23.93 \pm 0.07$ & $ 24.95 \pm 0.07$ \\ 60033 & F200W & 2104 & $26.45 \pm 0.08$ & $24.21 \pm 0.08$ & $ 25.25 \pm 0.08$ \\ 60033 & F277W & 2104 & $25.72 \pm 0.14$ & $24.93 \pm 0.14$ & $ 25.58 \pm 0.14$ \\ 60057 & F090W & 1245 & $>29.71 $ & $26.44 \pm 0.07$ & $ 30.23 \pm 0.74$ \\ 60057 & F150W & 858 & $27.06 \pm 0.10$ & $24.03 \pm 0.07$ & $ 25.45 \pm 0.08$ \\ 60057 & F200W & 1760 & $26.82 \pm 0.09$ & $24.35 \pm 0.08$ & $ 25.06 \pm 0.08$ \\ 60057 & F277W & 1760 & $26.07 \pm 0.15$ & $25.20 \pm 0.14$ & $ 25.37 \pm 0.14$ \\ 60074 & F090W & 1417 & $>29.48 $ & $27.21 \pm 0.09$ & $ 30.45 \pm 0.81$ \\ 60074 & F150W & 1245 & $27.37 \pm 0.12$ & $24.42 \pm 0.07$ & $ 25.94 \pm 0.08$ \\ 60074 & F200W & 1760 & $26.74 \pm 0.09$ & $24.81 \pm 0.08$ & $ 25.24 \pm 0.08$ \\ 60074 & F277W & 1760 & $26.28 \pm 0.15$ & $25.21 \pm 0.14$ & $ 25.18 \pm 0.14$ \\ \hline \end{tabular*} \label{tab:phot_table} \end{table*} \section{Data and Observations} \label{sec:obs} This Section summarizes: (1) the discovery and observations of \lensedsn~and (2) the procedures used to extract photometry for the three appearances \lensedsn. \subsection{Summary of Observations} \lensedsn~ was discovered in JWST imaging of the PLCK G165.7+67.0 cluster field, taken on 30 March 2023 (MJD 60033) in eight NIRCam filters as part of the Prime Extragalactic Areas for Reionization and Lensing Science (PEARLS) program (PID 1176, PI: R~Windhorst). For details on the reduction and analysis of this dataset, we refer the reader to \citet{Windhorst_2023} and \citet{Frye_2023, Frye_2024}. In the NIRCam F150W image, three transient sources were identified by comparing to archival F160W Hubble Space Telescope (HST) imaging of the same field, acquired on 30 May 2016 (MJD 57538) under an unrelated program (GO-14223, PI: B.~Frye; \citet{Frye_2023}). The similarity in wavelength coverage and throughput between the two filters enabled this comparison. The identification of the three sources in the JWST data, the relatively high brightness of the most luminous source at approximately 24.3 (AB) mag, and the close proximity of all three to a galaxy designated as ``Arc 2'' led to the application of an estimator to the initial three light curve points, yielding a 95\% probability that the event was a Type Ia supernova \citep{Frye_2023}. This led to a JWST DDT proposal to monitor the SN by photometry and spectroscopy (PID 4446, PI: B. Frye), and two ground-based proposals to obtain redshifts of the SN host galaxy using the Large Binocular Telescope (LBT; PI: B. Frye), and the MMT (B. Frye). The LBT observations enabled the measurement of the spectroscopic redshift of the SN host galaxy to be $z = 1.783 \pm 0.002$ \citep{Polletta_2023}, consistent with a prior photometric redshift estimate \citep{Pascale_2022}. The JWST follow-up observations provided two additional epochs of NIRCam imaging in six filters (F090W, F150W, F200W, F277W, F356W, and F444W), along with NIRSpec Prism, G140M, and G235M spectroscopy of all three SN appearances and two images of the SN host galaxy. The NIRSpec spectroscopy enabled a redshift measurement for the third image of the host galaxy and refined redshift estimates for Arc 2a and Arc 2c to $z = 1.7833 \pm 0.0010$ and $ z = 1.7834 \pm 0.0005$, respectively \citep{chen_SN_2024, Frye_2024}. The follow-up imaging was taken on 22 April 2023 (MJD 60056) and 9 May 2023 (MJD 60073), resulting in three observing epochs spaced by approximately one rest-frame week, or about three observer-frame weeks. Two additional observing epochs were obtained as part of a JWST Cycle 3 program, which provides the first template image of Arc 2 (PID 4744, PIs: B.~Frye and J.~Pierel). % \vspace{} \subsection{Photometry} The steps taken to reduce the data set and perform the photometry are described in detail by \citet{Windhorst_2023} and \citet{Pierel_PhotometricTimedelay_2024a}. Briefly, the observations of \lensedsn~were calibrated using the STScI JWST Pipeline\footnote{\url{https://github.com/spacetelescope/jwst}} (Version 1.12.5; \citet{JWSTCalibrationPipeline_2024}) and were available as multiple data products. The products relevant for the photometric measurements were the level 2 ``CAL" images, which are individual exposures that have been calibrated with the pipeline, bias-subtracted, dark-subtracted, flat-fielded, but have not been corrected for geometric distortion and the level 3 drizzled \citep{multidrizzle_2003} images that combine the individual exposures \citep[common when analyzing dithered exposures; see][]{Windhorst_2023}\footnote{This data was obtained from the Barbara A. Mikulski Archive for Space Telescopes (MAST) at the STScI (accessible via \doi{10.17909/zk1p-2q51})}. Measuring the position and brightness of a SN is understood to yield the best results when done on the CAL images as this processing preserves the PSF structure better than the drizzled images \citep{Rigby_JWST_2023}, but the host galaxy brightness relative to the SN made this impractical for longer wavelengths. Photometry was therefore performed on the F150W level 2 images to achieve a baseline for comparison, but then a drizzled image PSF fitting routine was used to measure the final photometry, while the F150W CAL image photometry ensured accuracy. Another consideration when measuring SN photometry is separating the host flux from the SN flux. Due to the lack of a template image at the time, a lens model was used to create a host surface brightness profile. Subtracting this profile from the observed flux isolated the SN flux. We adopt the photometry reported in Table 2 of \citet{Pierel_PhotometricTimedelay_2024a}, reproduced here in Table \ref{tab:phot_table}. The photometry is reported in AB magnitudes and the upper limits for the F090W filter for two of the epochs are $3~\sigma$. A complete reanalysis that includes the fifth observing epoch (shown in Figure \ref{fig:color_im}) will appear in a future paper (Agrawal et al. 2025, \textit{in prep.}) % \vspace{1em} \section{Lens Models}\label{sec:lens_models} The seven lens models analyzed in this paper are described in detail by previous works \citep{Frye_2024, Kamieneski_2024, Pascale_SNH0pe_2025}. Briefly, there were seven lens modeling teams that analyzed the cluster and obtained the model-predicted time delays and absolute magnifications. All seven lens modeling subgroups agreed in advance to use a consistent set of 21 image systems (including five with spectroscopic redshifts). They also settled on the positions, F200W brightnesses, and morphological parameters of 161 galaxies present in the cluster, along with any known galaxy interlopers. The details are specified in \citet{Frye_2024}. Each subgroup conducted a double-blinded analysis without knowledge of the spectroscopic/photometric time delays or absolute magnifications or other lens modeling teams’ analyses that was strictly enforced. All models assumed a fiducial cosmology of $\Omega_m = 0.3$, $\Omega_\Lambda = 0.7$ and an $H_0 = 70 \text{ km s}^{-1} \text{Mpc}^{-1}$. A range of approaches from parametric to non-parametric were used to build the seven lens models with one even combining strong and weak lensing. Parametric models use a superposition of analytic mass density profiles and assume they characterize the entire mass distribution, with a large cluster scale halo described by a large-scale profile and smaller halos for individual galaxies. In contrast, non-parametric models describe the mass distribution using a flexible grid with few assumptions about the dark matter profile of cluster halo. Semi-parametric models use a combination of the flexible grid-based mass distribution with analytic halo profiles. \begin{table*}[!ht] \caption{Predicted magnifications of \lensedsn~per image as given in and weights for each model as stated in Tables 1 and 2 in \citet{Pascale_SNH0pe_2025} } \centering \begin{tabular*}{\linewidth}{@{\extracolsep{\stretch{1}}}*{7}{c}} \toprule \# & Reference & $|\mu_a|$ & $|\mu_b|$ & $|\mu_{c}|$ & $\text{Weight}_{\text{TD-only}}$ \\ \hline 1 & \textsc{GLAFIC} \citep{Oguri_2019} & $8.02_{-0.57}^{+0.64}$ & $12.23_{-1.51}^{+1.30}$ & $9.32_{-0.87}^{+0.80}$ & 0.21 \\ 2 & \textsc{ Zitrin-Analytic} \citep{Zitrin_2009} & $11.25_{-0.90}^{+1.05}$ & $16.03_{-1.81}^{+1.77}$ & $14.48_{-1.65}^{+1.91}$ & 0.16\\ 3 & \textsc{LENSTOOL} \citep{kneib_2011} & $6.67_{-0.14}^{+0.11}$ & $9.82_{-0.31}^{+0.23}$ & $8.94_{-0.29}^{+0.21}$ & 0.16 \\ 4 & \textsc{ MARS} \citep{MARS_2022} & $6.82_{-0.44}^{+0.50}$ & $9.17_{-0.72}^{+0.80}$ & $7.55_{-0.70}^{+0.86}$ & 0.15 \\ 5 & \citep{chen_2020} & $6.52_{-0.22}^{+0.24}$ & $10.35_{-0.41}^{+0.46}$ & $6.68_{-0.22}^{+0.24}$ & 0.27 \\ 6 & \textsc{ WSLAP+} \citep{Diego_2005} & $28.37_{-7.15}^{+6.34}$ & $63.70_{-17.34}^{+16.70}$ & $36.04_{-9.52}^{+9.57}$ & 0.03 \\ 7 & \textsc{ Zitrin-LTM} \citep{Zitrin_2009} & $5.66_{-0.14}^{+0.15}$ & $9.77_{-0.51}^{+0.47}$ & $8.80_{-0.46}^{+0.67}$ & 0.03 \\ \hline Photometry & \citet{Pierel_PhotometricTimedelay_2024a} & $4.43_{-1.60}^{+1.52}$ & $8.00_{-2.34}^{+3.42}$ & $6.43_{-1.13}^{+1.25}$ & - \\ Spectroscopy & \citet{chen_SN_2024} & $10.93_{-5.16}^{+8.63}$ & $13.22_{-2.33}^{+7.49} $ & $7.14_{-1.66}^{+1.553}$ & - \\ \hline \end{tabular*} \label{tab:models} \end{table*} The seven models are listed below with a few key points for each model summarized from the appendix of \citet{Pascale_SNH0pe_2025}. \begin{itemize} \item{Model 1 is a parametric model built using \textsc{GLAFIC} \citep{glafic_oguri_2010, Oguri_2019, Oguri_2021} with five Navarro-Frank-White (NFW) \citep{NFW_1997} profiles to model the dark-matter halos. Cluster galaxies were modeled using pseudo-Jaffe ellipsoid profiles scaled by the F200W flux of each galaxy. The model increases computational efficiency by modifying resolution based on magnification gradients using an adaptive-mesh grid. The model also includes an external shear component for flexibility with a Markov Chain Monte Carlo method of $\sim 10^4$ steps used for optimization.} \item{ Model 2 is also a parametric approach built using a modified version of the method used in \citet{Zitrin_2015} (further details about the new version are given in \citet{Furtak_2023}). The cluster galaxies were modeled as double pseudo-isothermal elliptical mass-density distributions (dPIE, \citet{eliasdottir_2007}) while the cluster-scale dark matter halo was described by a pseudo-isothermal elliptical mass- density distribution (PIEMD; \citet{kassiola_1993}). Flexibility was added by allowing independent scaling of four central galaxies. Circular profiles were assumed for all cluster galaxies. This model also used MCMC for optimization and uncertainty calculation.} \item{Model 3 is another parametric approach built using \textsc{LENSTOOL}\footnote{\url{1 https://projets.lam.fr/projects/lenstool/wiki}} \citep{kneib_2011}, which uses the locations of the multiply-imaged systems to constrain the mass profiles. The member galaxies were assigned masses scaled to their F200W fluxes, relative to a characteristic $L^*$ galaxy ($m_{F200W} = 17.0$ AB mag) while the dominant merging components were modeled using two cluster-scale PIEMD halos. MCMC sampling with 10 chains of 1000 steps was used to fit the many free parameters of the model. Full details of the modeling approach are provided in \citet{Kamieneski_2024}.} \item{Model 4 was built using the \textsc{MARS} algorithm \citep{MARS_2022, MARS_2024}, a free-form, grid-based lens modeling method incorporating both strong lensing (SL) and weak lensing (WL), contrasting the parametric approaches taken in Models 1, 2 and 3. The \textsc{MARS} team provided two models to predict the magnifications and time delays: an SL-only model on a $100 \times 100$ grid ($90^" \times 90^"$), and a combined SL+WL model on a $400 \times 400$ grid ($360^" \times 360^"$). The SL-only model takes in a chi-squared minimization of multiple SL images of the source plane and a regularization term based on maximum cross entropy, which helps suppress noise and promotes a stable, quasi-unique mass reconstruction. The SL+WL model uses an additional term minimizing the difference between the observed and predicted reduced shear values.} \item{Model 5 is a semi-parametric approach and was originally built for a blind prediction of the reappearance of SN Refsdal \citep{chen_2020}. This model combines a non-parametric at the cluster-scale with analytic profiles at the individual galaxy-scale similar to parametric methods. This model uses symmetric analytic NFW profiles with the same scale radius and masses scaled using stellar flux (measured from JWST F090W imaging) to parameterize the galaxy dark matter halos. The model flexibly maps the cluster-scale dark matter distribution by applying smooth perturbations to the lensing potential without assuming symmetry. It minimizes offsets between lensed images using source-plane positions but is optimized in the image plane.} \item{Model 6 is a hybrid model that was built using the \textsc{WSLAP+} software \citep{Diego_2005, Sendra_2014}. The model uses the light distribution of individual galaxies to trace the small-scale halos while describing the large-scale components using a grid of Gaussians. A uniform grid was recursively built and an adaptive grid was derived from it. This adaptive grid was then used to generate different models of the system and the time delay and magnifications. Limited constraints in certain regions of the cluster resulted in a large range of predicted magnifications. A system of two linear equations per lensing constraint are used to optimize the model. The model predicted that image C of \lensedsn~ would be the last to arrive (inconsistent with observations) and hence did not have any weight in the $H_0$ inference done in \citet{Pascale_SNH0pe_2025}.} \item{ Model 7 is semi-parametric and uses a modified version of Zitrin-LTM \citep{Zitrin_2009, Zitrin_2015, Broadhurst_2005} originally developed for time-delay cosmography of SN Refsdal. It uses the F200W brightness of the cluster-member galaxies as weights when estimating the stellar mass with a power-law mass-surface-density profile. These profiles are smoothed with a Gaussian kernel to approximate the cluster dark matter distribution. Systematics are accounted for by varying an external shear component. Individual galaxies like the BCG are fitted freely. Spectroscopic redshifts are used where available and photometric redshifts are input as best guesses for the remaining systems. An MCMC method is used to infer the model parameters in the image plane.} \end{itemize} Table \ref{tab:models} lists the predicted magnifications for each image of the SN for each model (extracted from \citet{Pascale_SNH0pe_2025}, their Table 1). In \citet{Pascale_SNH0pe_2025}, the value of $H_0$ is inferred from the time delays using a Bayesian framework and equations \ref{eq: h0_rescale} \& \ref{eq:prob_h0} are drawn from Equations 4 and 5 of \citet{Pascale_SNH0pe_2025}. Each of the lens model predicted time delays assumed a fiducial $H_0 = 70 \text{km s}^{-1} \text{Mpc}^{-1}$. The fiducial time delays are scaled as a function of $H_0$ to match the observed photometric time delays, as shown in Equation \ref{eq: h0_rescale}. \begin{equation} \label{eq: h0_rescale} \Delta t_{i,j}^{pred} (H_0) = \Delta t_{i,j}^{fid} \times \frac{70 \text{ km s}^{-1}\text{Mpc}^{-1}}{H_0} \end{equation} The probability distribution of $H_0$ is obtained by marginalizing over all the lens models, weighting each by its ability to reproduce the observed ratios of time delays. The set of observables $\mathcal{O}: \{ \Delta t_{a,b},\Delta t_{b,c} \}$, each have measurements from the LC and predictions from each model $M_I$ assuming a fiducial $H_0$. Equation \ref{eq: h0_rescale} is used to rescale the fiducial $H_0$ till the predictions match the LC measurements. The probability for the value of $H_0$ from a single lens model $M_1$ with n number of observables using Bayes' theorem is: \begin{equation} \label{eq:prob_h0} \begin{split} p_m(M_l;H_0|LC)\hspace{2mm} \propto &\hspace{2mm} P(H_0) P(M_l) \smallint P(\mathcal{O}|M_l;H_0) \\ &\times P(\mathcal{O}|LC)d\mathcal{O}_1....d\mathcal{O}_n \end{split} \end{equation} The weighted sum of N individual lens model likelihoods is the overall posterior: \begin{equation} \label{eq:likelihood_h0} \begin{split} P(H_0|LC) \hspace{2mm} \propto \hspace{2mm} & P(H_0)\sum_1^N \smallint P(\mathcal{O}|M_l;H_0) \\ & \times P(\mathcal{O}|LC)d\mathcal{O}_1....d\mathcal{O}_n \end{split} \end{equation} The weights for each model are calculated using Equation \ref{eq:weights} and then normalized based on the sum for all the models. \begin{equation}\label{eq:weights} \begin{split} w_l = \int dH_0 \int d\theta P(\theta |LC) P(\theta| H_0, M_l)P(H_0) P(M_l) \end{split} \end{equation} % \begin{equation}\label{eq:weights} % \begin{split} % w_l = \int dH_0 \frac{\int d\theta P(\theta|LC) P(\theta| H_0, M_l)P(H_0) P(M_l)}{\sum_{l=1}^N \int dH_0 P(\theta |LC)P(\theta|H_0,M_l)P(H_0)P(M_l)} % \end{split} % \end{equation} This assumes partial independence among the lens models and uses only the photometric light curve data. Further formalism and methods detailing the $H_0$ inference is in \citet{Pascale_SNH0pe_2025}. They also performed a joint analysis including spectroscopic and magnification constraints; however, our analysis uses only weights calculated based on the ability of the models to reproduce the ratios of the photometric time delays (shown in Equation \ref{eq:weights}), with the values of the weights listed in Table \ref{tab:models}. The values in Table \ref{tab:models} are originally presented in Table 1 and Table 2 from \citet{Pascale_SNH0pe_2025} and included here for completeness. \section{Modeling the Intrinsic SN~Ia Light Curve}\label{sec: sed modeling} \subsection{Photometric Corrections} \label{subsec: de-lensing} % \subsection{De-lensing a lensed supernova}\label{subsec: de-lensing} In order to test the lens models, the \lensedsn~data had to be corrected to describe the intrinsic \lensedsn~light curve. The hypothesis behind this testing is that if the lens models are correct, removing the magnification from \lensedsn~based on the model predicted values should yield fluxes that are consistent with a typical unlensed SN~Ia at the same redshift (z = 1.78). To further ensure that the data is as close to intrinsic SN~Ia light curves, the light curves for each image are time-shifted based on the inferred photometric time delay to result in a single composite light curve. To create an intrinsic SN light curve, the time delay calculated in \citet{Pierel_PhotometricTimedelay_2024a} for each image was added to the respective epochs for all the three images. Image 2a measured a delay of $-116.6^{+10.8}_{-9.3}$ days while Image 2c measured a delay of $-48.6^{+3.6}_{-4.0}$ days with respect to Image 2b. This time-delay corrected data was then demagnified based on the individual models' predicted magnification for each image (specified in Table \ref{tab:models}. Spectroscopic magnifications were also measured for \lensedsn~as stated in \citet{chen_SN_2024} but not used for this analysis. \subsection{The SN~Ia SED Models} \label{subsec: sed models} \subsubsection{BayeSN}\label{subsubsec:bayesn} The first SED model used here is \textsc{BayeSN} \citep{Mandel_2022, Thorp_bayesn, grayling_24_mnras, grayling_bayesn}, a hierarchical Bayesian SED model for SNe~Ia. This framework uses probabilistic generative modeling to account for intrinsic and dust effects underlying the data. The full time- and wavelength-varying SED is described by Equation 1 from \citep{Thorp_bayesn}: \begin{equation} \label{eqn: bayesn} \begin{split} - 2.5 \log_{10} [S_s &(t, \lambda_r)/ S_0 (t, \lambda_r)] = \\ M_0 + &W_0 (t, \lambda_r) + \delta M^{s} + \theta_1^s W_1 (t, \lambda_r) \\ + \epsilon^{s} & (t, \lambda_r) + A_{V}^{s} \xi (\lambda_r ; R_{V}^{(s)}) \end{split} \end{equation} where $t$ is the rest-frame phase relative to B-band maximum and $\lambda_r$ is rest-frame wavelength. $S_0(t,\lambda_r)$ is the fixed baseline zeroth-order optical-NIR SN~Ia SED template of \citet{Hsiao_2007} along with the arbitrary normalization constant $M_0$ which is set to -19.5 \footnote{Note that $W_0$ ultimately sets the absolute magnitude of the mean intrinsic SED}. Parameters denoted with $s$ are latent SN parameters and have unique values for each SN $s$. All other parameters are global hyper-parameters shared across the population. They are described in more detail below. % \todo{figure out how to make this S smaller???} The function, $W_0 (t. \lambda_r)$ manipulates the zeroth-order SED template into a mean intrinsic SED that represents the population. The $W_1 (t, \lambda_r)$ component describes the first mode of the intrinsic SED variation for SNe~Ia. The $\theta_1^s$ coefficient combines with the $W_1 (t, \lambda_r)$ component to represent the `broader-brighter' relation observed in SNe~Ia described in \citet{Phillips_1993}. The relation states that intrinsically brighter light curves evolve over longer timescales around peak brightness. An achromatic, time-independent offset in magnitude is described by $\delta M^s$ for each SN from a normal distribution inferred during model training. The component $\epsilon^{s} (t, \lambda_r)$ accounts for time-varying intrinsic color variations that are not captured by $\theta_1^s W_1 (t, \lambda_r)$. The total V-band extinction is $A_{V}^{s}$ that accounts for one aspect of the host galaxy extinction law. Finally, $R_{V}^{(s)}$ makes up the second part of the host galaxy extinction law and parametrizes the \citet{Fitzpatrick_1999} dust extinction law. The model described above gives a rest-frame, host-galaxy dust-extinguished SED model $S_s (t, \lambda_r)$ based on Equation \ref{eqn: bayesn}. This $S_s (t, \lambda_r)$ is scaled based on distance modulus $\mu^s$ before being redshifted and corrected for Milky Way dust extinction to produce an observer-frame SED. This SED is integrated through photometric filters to produce model photometry which is compared with observed photometry to compute a likelihood. For the complete formalism, refer to \citet{Mandel_2022}. During training, the model produces posterior estimates for a set of hyperparameters (e.g., spectral weights, dust parameters, and intrinsic scatter), which are marginalized over all latent variables. For simplicity, the posterior means of these hyperparameters are used as point estimates. In distance-fitting mode, the posterior distribution of a supernova’s latent parameters and distance modulus is conditioned on these fixed hyperparameters. This distribution incorporates priors on light-curve shape, color, intrinsic scatter, and host dust. When fitting an individual supernova, the time of maximum light is also treated as a free parameter. By sampling the joint posterior, the marginal posterior distribution of the distance modulus can be approximated. Importantly, this distribution may be asymmetric due to the non-negativity of the dust effects and is not assumed to be Gaussian. \subsubsection{SALT3-NIR}\label{subsubsec:salt} The second SED model used here is \textsc{SALT3-NIR} which is based on a framework that characterizes the spectral flux as a function of wavelength and rest-frame phase. Given below is a brief description of the model based on \citet{SALT3NIR_2022}. \begin{equation} \begin{split} F(p, \lambda) = &x_0 [M_0(p,\lambda) + x_1 M_1(p,\lambda)] \\ &\cdot \exp (c \cdot C L(\lambda)) \end{split} \end{equation} Three components are determined from the training process characterize the SNe population, with three parameters used to fit each individual SN light curve. The first component, $M_0(p, \lambda)$, represents a baseline SN~Ia; the second, $M_1(p, \lambda)$, provides a first-order linear correction; and the third, $CL(\lambda)$, is a color law accounting for intrinsic color and dust effects based on apparent color. The three fit parameters used for each individual SNe are: $x_0$ (flux normalization), $x_1$ (amplitude of $M_1$, analogous to the stretch parameter $\theta_1^S$ in \textsc{BayeSN}), and $c$ (SN~Ia color). These three fit parameters are used in the Tripp equation \citep{Tripp_1998}, \begin{equation} \mu = -2.5\log_{10} (x_0) + \alpha x_1 - \beta c - M_0 , \end{equation} to determine the luminosity distances of SNe~Ia. The $\alpha$ and $\beta$ are global parameters that relate luminosity to stretch and color, respectively, while $M_0$ is the absolute magnitude of normal SNe~Ia. Typically, the \textsc{SALT2mu} method described in \citet{Marriner_2011} is used to determine the $\alpha$, $\beta$ and $M_0$ parameters while calculating the distance moduli by minimizing the Hubble residuals in different redshift bins. This leads to a cosmology-independent distance determination. For our analysis, we used $\alpha = 0.14 \pm 0.003$ and $\beta = 3.12 \pm 0.017$ to use the \textsc{SALT2mu} method to determine the distance modulus for each predicted magnification. A key difference to note between \textsc{SALT3-NIR} and \textsc{BayeSN} is how they treat dust during the SED fitting process. The \textsc{SALT3-NIR} has a global dust model that is fit after the light curve of each SN is fit in a two step process. In contrast, \textsc{BayeSN} fits the dust to each SN individually while fitting the light curve allowing for distance inference in a single step. \begin{figure}[h!] \centering \includegraphics[trim= 0.9cm 1.5cm 1.4cm 1.9cm, clip,width=\linewidth]{figs/stacked_lightcurves_pub.pdf} \caption{Light curves for \lensedsn~photometry, de-magnified using absolute magnifications as stated in \citet{Pascale_SNH0pe_2025} as predicted by the lens modeling approach of \citet{chen_2020}, are shown fitted with both SED models. The top panel shows \textsc{BayeSN} fit, with the residuals shown in the second panel. The third panel shows the fit using \textsc{SALT3-NIR} with the corresponding residuals in bottom panel. Light curves are color-coded by filter, with vertical offsets (indicated in the legend) applied for clarity. The shaded regions represent the $2 \sigma$ uncertainty intervals for each light curve. Photometric data are overplotted with error bars, also color-coded by filter. Each SN image is represented by a different marker: circles for Image 2a, squares for 2b, and triangles for 2c. } \label{fig:lightcurves} \end{figure} \subsection{Extracting Distance Moduli from the Intrinsic Light Curves} \label{subsec: getting mu} The `de-magnified' photometry described in Subsection \ref{subsec: de-lensing} was fitted for the distance modulus using the two SN~Ia spectral-energy distribution(SED) models - \textsc{BayeSN} \citep{Mandel_2022} and \textsc{SALT3-NIR} \citet{SALT3NIR_2022} described in Sections \ref{subsubsec:bayesn} and \ref{subsubsec:salt}. \textsc{BayeSN} uses a pre-trained model as a baseline SED to do the fitting. For our analysis, we used a extended phase NIR model (Grayling et. al \textit{in prep}) which helped account for the longer rest frame timescale as well as the longer NIR wavelengths of the observations. As an example, Figure \ref{fig:lightcurves} shows the fit and residuals of the observed light curves after de-magnification using the lens modeling approach of \citet{chen_2020} with both SN~Ia SED models. We choose the \citet{chen_2020} model because it receives the highest weight in \citet{Pascale_SNH0pe_2025} based on its accuracy in predicting time delays between images. The figure demonstrates that both SED models yield comparable performance, despite differences in their methodology, indicating that the choice of SED model is not a significant source of bias in this analysis. Each lens model predicts a magnification for each image of \lensedsn~along with the associated uncertainties as specified in Table \ref{tab:models}. For our analysis, the uncertainties here constitute \emph{systematic} shifts in the data rather than random errors. For example, a magnification value higher than the mean for Image 2a would result in all of the photometry for Image 2a shifting together during the demagnification rather than a random scatter about the mean. Thus, these uncertainties in the predicted magnifications cannot be accounted for by simple error propagation. To account for the systematic uncertainties in the predicted magnifications of each image for all the models, the photometry was de-magnified using a range of predicted magnifications for each image (specified by the value itself, and the upper and lower $1 \sigma$ limit of expected magnification) for every lens model. Each iteration of the modified image was then fitted with both SN~Ia SED models to ensure that the full parameter space was explored. The resulting distribution of distance modulus values are shown in Figure \ref{fig:models_error}. This procedure provides the most conservative and rigorous bound on the systematic impact of an error in the magnification. Finally, to test the lens model predictions, the 'de-magnified' distance moduli were compared to the expected distance moduli for the different inferred cosmologies at $z=1.783$. We use cosmological parameters inferred by the \citet{Pantheon_2022} $\text{Flat}\Lambda\text{CDM}$ as well as \citet{DES_5Y} $\text{Flat}\Lambda\text{CDM}$ and $\text{Flat}w_aw_0\text{CDM}$ shown in Figure\ref{fig:models_error}. All of the included cosmological parameters were inferred using SN only and were not combined with any other probes. This ensures that the $3\sigma$ error regions given for each cosmology includes the intrinsic SN scatter. The final aspect to consider is the method used by the SN~Ia fitters to obtain a distance modulus. This analysis doesn't use any absolute calibrators (eg: Cephieds etc), without which SN~Ia SED model fitters can only provide relative distance estimates. They adopt a fiducial value of $H_0$ to put photometric distance estimates on a distance scale. Thus, for a self-consistent analysis, the distance can and must be trivially rescaled to ensure that a single $H_0$ value is being used for both the distance estimates and the cosmological model. \textsc{BayeSN} uses an assumed value of $H_0 = 73.24$ while \textsc{SALT3-NIR} uses the $H_0$ value from the Pantheon+ cosmological constraints \citep{Pantheon_2022}. In order to compare the extracted distance moduli with the cosmological constraints from Pantheon+ and DES 5Y survey results, they need to be rescaled to the same $H_0$ value, else the differing values cause a persistent shift in the inferred distance moduli. For our analysis we use the Pantheon+ value of $H_0 = 73.6 \pm 1.1$ \citep{Pantheon_2022} for all the cosmological constraints, with the rescaling based on the ratio of the new $H_0$ value to the old assumed $H_0$. \begin{figure*}[h] \centering \includegraphics[trim= 0.2cm 0.2cm 0.2cm 0.2cm, clip,width=0.80\linewidth]{figs/ridgeline_plot_with_vals_pub_new.pdf} \caption{Distance modulus ($\mu$) measurements for each of the lens models described in Section \ref{sec:lens_models} along with the photometric and spectroscopic measurements of magnification from \citep{Pierel_PhotometricTimedelay_2024a} and \citet{Pascale_SNH0pe_2025}. The purple regions show the normalized distribution of distance moduli found using \textsc{SALT3-NIR} while the pink depicts the same found using \textsc{BayeSN}. The expected distance modulus for Flat$\Lambda$CDM (Dotted Orange) and Flat$w_aw_0$CDM (Dot-Dashed yellow) as described in \citet{DES_5Y} at $z=1.783$ are shown as well. Expected distance modulus at $z=1.783$ for Flat$\Lambda$CDM parameters given in \citet{Pantheon_2022} is shown in solid green. Each of the cosmological models has a $5\sigma$ uncertainty region shown in the respective colors as well.} \label{fig:models_error} \end{figure*} \section{Testing Lens Model Systematics} \label{sec:results} We analyzed the predicted magnifications of all the seven lens models used in \citep{Pascale_SNH0pe_2025} as well as the magnifications measured directly from the photometric and spectroscopic data. Photometric magnifications were estimated by comparing the apparent magnitude of \lensedsn~ to that predicted from the observed population of field SNe~Ia at $z=1.783$ \citep{Pierel_PhotometricTimedelay_2024a}. A similar approach was adopted to measure the magnification from the spectroscopic data \citep{chen_SN_2024}. Using only the lens model–predicted magnifications, we find weighted mean distance moduli of $46.35^{+0.47}_{-0.32}$ with \textsc{SALT3-NIR} and $46.33^{+0.49}_{-0.26}$ with \textsc{BayeSN}. These values were computed using the TD-only weights from \citet{Pascale_SNH0pe_2025}. We emphasize that these weighted mean values only include the lens models from Section \ref{sec:lens_models}, and explicitly do not include the distance moduli extracted from the magnifications directly measured from photometry and spectroscopy, which are reported separately in Table~\ref{tab:models}. Figure \ref{fig:SVB_LCDM} shows that the weighted mean distance modulus is systematically higher than expected from SN~Ia–based cosmological constraints, indicating that the lens models for \lensedsn~over-predict magnifications. This trend persists regardless of which SN~Ia SED model is used to fit the light curves. As shown in Figure \ref{fig:lightcurves}, both SED models yield similar residuals, demonstrating that the over-prediction does not arise from the light curve fitting itself. To place these results in context, we compare the lens model–predicted distance moduli with the values expected at $z=1.783$ under Flat$\Lambda$CDM and Flat$w_aw_0$CDM cosmologies. Specifically, we adopt the Flat$\Lambda$CDM model from \citet{Pantheon_2022} (hereafter Pantheon+ Flat$\Lambda$CDM) and both Flat$\Lambda$CDM and Flat$w_aw_0$CDM from \citet{DES_5Y} (hereafter DES5Y Flat$\Lambda$CDM and DES5Y Flat$w_aw_0$CDM). Since these cosmological constraints are derived solely from SN datasets, the $5\sigma$ uncertainty regions in Figure \ref{fig:SVB_LCDM} naturally encompass the intrinsic scatter of SNe~Ia. % As the weighted mean distance moduli lie beyond the error regions of the cosmological parameters, it is evident that this effect cannot be explained by the intrinsic SNe scatter. % However, weighted averages do not provide a complete picture of how the individual lens models compare to these cosmological models. Figure \ref{fig:models_error} shows the probability density functions (PDF) of the distance moduli at $z = 1.783$ for each lens model, as well as the photometric and spectroscopic analyses, for both SED models. These PDFs incorporate the systematic uncertainties associated with the predicted magnifications of each of the three Images of \lensedsn~(2a, 2b, 2c) for every individual lens model. \textsc{SALT3-NIR} values are shown in purple while \textsc{BayeSN} values are in pink. The orange dotted line marks the distance modulus predicted by the DES5Y Flat$\Lambda$CDM model \citep{DES_5Y} at $z=1.783$, with its $5\sigma$ uncertainty shown by the orange shaded band. The yellow dash-dotted line indicates the DES5Y Flat$w_aw_0$CDM prediction \citep{DES_5Y}, with the corresponding $5\sigma$ region shaded in yellow. The solid green line shows the Pantheon+ Flat$\Lambda$CDM prediction \citep{Pantheon_2022} at the same redshift, with the green shaded region highlighting the $5\sigma$ uncertainty. It is evident from Figure~\ref{fig:models_error} that the lens model–predicted magnifications (as seen in Table\ref{tab:models}) are systematically biased high when compared to the predictions of both the Pantheon+ and DES-5Y cosmological models. In contrast, the photometric estimate of $\mu$ agrees well with these fiducial cosmologies, independent of the SED model used. The spectroscopically derived $\mu$ also shows tension with the fiducial cosmologies, but its much larger uncertainty compared to the lens models reduces the significance of this offset. A similar trend is suggested in Figure~3 of \citet{Pascale_SNH0pe_2025}, though it is not a central focus of that analysis. Here, we explicitly highlight and confirm this behavior through our independent methodology. \begin{figure}[t] \centering \includegraphics[trim= 0.3cm 0.3cm 0.3cm 0.3cm, clip,width=\linewidth]{figs/SVB_weighted_LCDM_pub.pdf} \caption{The weighted distance modulus for \textsc{SALT3-NIR} (purple) vs \textsc{BayeSN} (pink). The error bars for the two points show the $5~\sigma$ spread for the distribution. The distance modulus - redshift relation for three cosmological constraints are plotted as well. The DES5Y Flat$\Lambda$CDM is depicted in a orange dotted line. The DES5Y Flat$w_0w_a$CDM is shown in dotted-dashed yellow. The green solid line is the Pantheon+ Flat$\Lambda$CDM cosmology. Each cosmological model is accompanied by $5\sigma$ error regions in the respective colored shaded regions. } \label{fig:SVB_LCDM} \end{figure} To quantify the offset, we define $\Delta \mu$ as the difference between the SN-inferred and cosmology-predicted distance modulus: $\Delta\mu \equiv \mu_{\rm SN} - \mu_{\Lambda{\rm CDM}}$, measured at $z=1.783$, across the seven lens model for both SED models and three cosmological baselines. For \textsc{SALT}, the pooled (``grand'') mean $\Delta\mu$ is $1.073 \pm 0.085$~mag under Pantheon+ flat $\Lambda$CDM, $1.102 \pm 0.082$~mag under DES5Y flat $\Lambda$CDM, and $1.235 \pm 0.149$~mag under DES5Y $w_0w_a$CDM. For \textsc{BayeSN}, we find similarly elevated values of $1.037 \pm 0.071$~mag (Pantheon+), $1.066 \pm 0.067$~mag (DES5Y Flat$\Lambda$CDM), and $1.199 \pm 0.141$~mag (DES5Y Flat$w_0w_a$CDM). These pooled means reflect consistent overestimation of the lensing magnification relative to the cosmological expectation, regardless of SED model. However, there is also substantial variation between individual lens models. For \textsc{SALT}, the per-model mean $\Delta\mu$ values span $0.622$–$2.391$~mag (Pantheon+), $0.652$–$2.421$~mag (DES5Y Flat$\Lambda$CDM), and $0.784$–$2.553$~mag (DES5Y $w_0w_a$CDM). For \textsc{BayeSN}, the corresponding ranges are $0.640$–$2.199$~mag (Pantheon+), $0.669$–$2.229$~mag (DES5Y Flat$\Lambda$CDM), and $0.802$–$2.361$~mag (DES5Y $w_0w_a$CDM). These wide ranges suggest that while the overall bias is robust, it is not uniform across all lens model predictions. In all cases, the quoted ``$\pm$'' values denote the pooled per-point $1\sigma$ dispersion, calculated across all lens model data points per cosmology and SED model, and should not be interpreted as the uncertainty on the mean. For comparison, the uncertainty on each per-model mean—combining both photometric scatter and the uncertainty in the reference cosmological distance modulus—is typically $\sim0.039$–$0.042$~mag for Pantheon+ Flat$\Lambda$CDM, $\sim0.034$–$0.039$~mag for DES5Y Flat$\Lambda$CDM, and $\sim0.094$–$0.096$~mag for DES5Y Flat$w_0w_a$CDM, for both \textsc{SALT} and \textsc{BayeSN}. Figure \ref{fig:models_error} also illustrates slight differences in distance modulus estimates between \textsc{SALT3-NIR} and \textsc{BayeSN}, which arise from differences in how each SED model treats different aspects during fitting. In \textsc{SALT3-NIR}, uncertainties are evaluated in two stages: first in fitting the light-curve parameters, and second in deriving the distance modulus with the \textsc{SALT2mu} method. By contrast, \textsc{BayeSN} infers the distance modulus in a single step that jointly accounts for correlations among light-curve shape, color, and luminosity distance. This contrast is further illustrated in Figure \ref{fig:SVB_LCDM}, which compares the weighted average $\mu$ values from each SED model. The \textsc{SALT3-NIR} estimate (purple) lies in slightly lower tension with the cosmological baselines than the \textsc{BayeSN} estimate (pink), though the difference is not statistically significant. Another factor that can influence the distance inference is biases introduced in the light curve fitting. Selection effects and inference priors in SN Ia light curve fitting can introduce systematic biases in distance modulus of order $ \sim 0.05 - 0.10$ mag. While we do not perform detailed simulations to correct for these biases in this work, we qualitatively assess their likely direction based on previous studies. For \textsc{SALT3-NIR} and \textsc{BayeSN}, a negative bias is expected, particularly at higher redshift or lower signal-to-noise, where selection effects favor intrinsically brighter SNe Ia, leading to underestimated distances \citet{Pantheon_2022, Hounsell_2018}. Nevertheless, given that the cosmologies used in this comparison were derived using the same SED models applied to large samples of Type Ia supernovae from DES5Y and the Pantheon analysis, we should expect that the Type Ia SN H0pe would be consistent with the same fiducial cosmologies if the lens models describe the system accurately. Figure \ref{fig:Lens_slopes} shows the magnification of each SN image as a function of the local radial slope of the potential at the SN image locations for each lens model defined as: % \vspace{-1em} \begin{equation} |\gamma| = |- d\ln \psi/d \ln r| \end{equation} The figure also shows the weighted and unweighted linear fits of the magnification as a function of the radial log slope, with Pearson $r = -0.230$ (weighted) and $r = -0.321$ (unweighted). Weights used for the fit were the inverse square of the magnification errors reported for each model. Based on the negative correlation above, we can speculate that the true slope of the potential may be steeper than the one from current lens models. \begin{figure*}[t] \centering \includegraphics[trim= 0.0cm 0.0cm 0.0cm 0.0cm, clip,width=0.8\linewidth]{figs/lens_potential_log_slope_correlation_.pdf} \caption{Magnification ($\mu$) of each SN image 2a, 2b, 2c (circle, square, triangle) for all the seven lens model vs the local radial slope of the lensing potential $|\gamma| = |- d \ln \psi / d \ln r|$ at those positions. Each marker is colored according to the lens models. The error bars are $2 \sigma$ uncertainty regions as reported in \citep{Pascale_SNH0pe_2025}. The dashed line shows a weighted linear fit in $\mu$ versus $\gamma$ (r = -0.230) while the dotted line shows the unweighted linear fit (r=-0.321). The negative correlation here suggests that a steeper slope leads to a lower magnification value. } \label{fig:Lens_slopes} \end{figure*} % \vspace{-1em} \section{Discussion and Conclusion} \label{sec: concl} Strongly-lensed transients are a new avenue to understanding many different aspects of astrophysics - dark matter, dark energy, high-$z$ SNe, and more. To leverage the full potential of these unique systems, analyses must be robust and lens models must be well-tested. Here, we test the robustness of the lens models used in the cosmological analysis of \lensedsn. We de-magnified the \lensedsn~photometry using the absolute magnifications predicted by each of the lens models. We combined this with the time-delays measured in \citet{Pierel_PhotometricTimedelay_2024a} to produce an ``intrinsic'' SN light curve from the de-magnified photometry. Next, we used existing Type Ia SED models to estimate the distance modulus ($\mu$) for each predicted magnification. Finally, we compared these $\mu$ values with those expected from $\Lambda$CDM to assess the performance of the lens models. Our analysis shows that the calculated distance moduli are systematically higher than expected at $z=1.783$ as seen in \ref{fig:models_error}, suggesting that the lens models overestimate the absolute magnifications at the $\sim 4.5\sigma$ level. For each of the seven lens models as well as the spectroscopic and photometric methods of measuring magnifications, the derived distance moduli are greater than the value expected for each of the cosmological models at $z=1.783$. To quantify this offset, we define $\Delta \mu \equiv \mu_{\text{SN}} - \mu_{\Lambda\text{CDM}}$, the difference between the SN-inferred and cosmology-predicted distance moduli. Across all lens models, SED models and cosmological baselines, we consistently find $\Delta \mu > 1$~mag. For \textsc{SALT}, the pooled mean offset is $1.073 \pm 0.085$~mag under the Pantheon+ flat $\Lambda$CDM, $1.102 \pm 0.082$~mag under DES5Y flat $\Lambda$CDM, and $1.235\pm0.149$~mag under DES5Y $w_0w_a$CDM. Similarly, for \textsc{BayeSN}, we find offsets of $1.037 \pm 0.071$, $1.066 \pm 0.067$ and $1.199 \pm 0.141$~mag under the same three cosmologies, respectively. These results suggest that current lens-model magnifications may be systematically overestimated, independent of the assumed SN~Ia SED model or cosmological baseline. Rather than undermining their utility, this highlights how cluster-lensed SNe provide a rare and powerful testbed to identify and correct systematics in lens modeling. Although the degree of offset varies across models, all predict values higher than expected at this redshift. This is surprising because the magnifications of each model were determined in a blinded analysis (as specified in \citet{Pascale_SNH0pe_2025}) and used a variety of techniques to construct the lens models. Considering the different methods combined with the blinded analysis, some scatter about the true value is normal and would be expected. However, none of the models/methods lead to an underestimation of the distance modulus. This systematic shift toward higher distance modulus values suggests an underlying cause beyond random scatter. We also measured the logarithmic slopes of the lensing potential for each lens model at the locations of the \lensedsn~images. Figure \ref{fig:Lens_slopes} shows that shallower models predict higher magnifications while steeper models predict lower magnifications. This inverse correlation is modest but consistent, with a Pearson correlation coefficient of $r = -0.230$ when weighting by the uncertainties in magnification, and $r = -0.321$ when unweighted. This trend suggests that the true potential of the G165 system may be steeper than predicted by any current lens models, and potentially contributing to the systematic bias seen in Figure \ref{fig:models_error}. The magnification overestimates were first seen in Figure 3 of \citet{Pascale_SNH0pe_2025}. Although the overestimates were recognized, they could not be corrected due to the strictly enforced post-unblinding rules agreed upon beforehand \citep{Pascale_SNH0pe_2025}. Revised lens models will reduce this bias (Pascale et al. 2025, \textit{in prep}), which could be indicative of a systematic error that we have been unaware of due to the lack of avenues for testing these lens models. One possible explanation is that the discrepancy arises from the cosmological models themselves. However, the cosmological models have held up well against various testing methods until now and been proven to match observations well. In contrast, there have been few avenues to test cluster scale lens models until now. This system is notable in that it includes a transient and multiple image systems at different spectroscopically confirmed redshifts—features that help break the mass sheet degeneracy and enable the construction of more robust lens models. We refer the reader to \citet{Pascale_SNH0pe_2025} for a more thorough discussion on the impact of mass-sheet and more generally mass-slope degeneracy in this field. This does not mean that the $\Lambda \text{CDM}$ cosmologies are infallible, but merely that we must conduct our due diligence and ensure that other sources of error have been accounted for. Another avenue that could be affecting this analysis is the photometry. \lensedsn~ did not have a template image to get precise SN flux measurements at the time of this analysis and a lens model was used to subtract the host flux. Our initial calculations suggest that the improvement expected in photometric data from the inclusion of a template will not be enough to account for the overestimated distance moduli, though it may be a contributing factor. We plan to address this further in our future work, now that a template image has been obtained in a JWST Cycle 3 program. An important next step is to consider how these results impact the broader goal of using glSNe to measure $H_0$. Assuming fixed time delays, modifying the predicted absolute magnifications to bring the lens model–based distance modulus into agreement with current cosmological constraints would result in a higher inferred $H_0$. In the case of \lensedsn, such a revision could shift the inferred value further from the CMB-based measurement, potentially altering the degree of Hubble tension--at least for this individual object. However, it is important to note that magnification predictions are inversely correlated with time-delay predictions. If the lens models are improved to have lower magnifications to resolve the overestimation, we would be sampling larger time delays when measuring $H_0$. This is seen in the $H_0$ inference done in \cite{Pascale_SNH0pe_2025}, where the TD-only weighted $H_0$ value is lower than those weighted using magnifications. It is important to note that these findings are based on the currently available lens models and photometric data. Both are expected to improve, with new lens model reconstructions currently underway (Pascale et al. 2025, \textit{in prep.}) and updated measurements of the SN photometry and time delays made possible by a recently obtained template image (Agrawal et al. 2025, \textit{in prep}). These updates will provide a more robust basis for interpreting the magnification predictions and their cosmological implications of \lensedsn. There are several methods to mitigate lens modeling systematics in $H_0$ inference. \cite{Liu_Oguri_2025} demonstrates that additional lensing evidence such as additional multiply lensed systems/events leads to improvements in lens modeling. The paper also highlights the correlation between the Hubble constant and magnification, which motivates implementing a magnification-weighting scheme—that is, properly sampling the posterior (e.g., \cite{Pascale_SNH0pe_2025}). Another option is to use the magnifications as a constraint in the lens modeling itself, as was done for the unblinded models in \cite{Pascale_SNH0pe_2025}. % The systematic overestimation of absolute magnifications by lens models leads to a lower inferred $H_0$ value. Cluster-scale lens models are inherently complex, built on current understanding of dark matter and gravity together with simplifying assumptions. Until now, there have been limited opportunities to test these methods. Our analysis shows that lensed supernovae are useful probes of lensing at these scales. It is yet unclear if this overestimation is due to an underlying assumption made when building the lens model or some new underlying physics. Both possibilities present exciting avenues forward and may be key to improving our models to leverage them for time delay cosmography. It is clear that a larger sample of cluster-lensed SNe~Ia is needed to disentangle the true underlying cause and will lead to a substantial improvement in our lens modeling. Our analysis also underscores the importance of using magnification constraints in lens modeling for time delay cosmography as well as using multiple lens models for $H_0$ inference. Figure \ref{fig:models_error} highlights how reliance on a single lens model, without independent avenues for testing, can allow hidden systematics to propagate into the final result. Improving these lens models is a crucial step towards high-precision cosmology with lensed phenomena. \lensedsn~highlights the power of cluster-lensed SNe to reveal systematics and improve the robustness of $H_0$ inference. \section{Future Work} \label{sec:future work} We are investigating why all lens models consistently over predict the magnifications. Our first step is to repeat the analysis using updated SN photometry corrected for host flux contamination, thanks to a new template image obtained in a JWST Cycle 3 program. Next, we will probe the discrepancy between lens-model predictions and cosmological values at $z=1.783$, to determine whether it stems from modeling issues or potentially points to new physics. We are also applying this method to other lensed SNe, particularly SN Requiem and SN Encore—two events in the same source-lens system expected to provide tighter constraints—and will extend the analysis to future strongly lensed supernovae. \section{Acknowledgments} \label{sec:Acknow} This paper is based in part on observations with the NASA/ESA Hubble Space Telescope obtained from the Mikulski Archive for Space Telescopes at STScI. We thank the DDT and JWST scheduling team at STScI for extraordinary effort in getting the DDT observations used here scheduled quickly. AA acknowledges support from AST-2206195 (P.I. Narayan), to develop anomaly detection methods to identify lensed supernovae, and HST-GO-17128 (PI: R. Foley) to adapt BayeSN to model \emph{JWST} and \emph{HST} observations of type Ia supernovae. AA gratefully acknowledge support from NSF AST-2421845 and support from the Simons Foundation as part of the NSF-Simons SkAI Institute as a 2026 SkAI Graduate Fellow at UIUC. JDRP is supported by NASA through a Einstein Fellowship grant No. HF2-51541.001 awarded by the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. GN gratefully acknowledges NSF support from NSF CAREER grant AST-2239364, supported in-part by funding from Charles Simonyi, to model type Ia supernovae with ground- and space-based data, AST 2206195 to develop anomaly detection methods to identify lensed supernovae, aand DOE support through the Department of Physics at the University of Illinois, Urbana-Champaign (Grant No. 13771275) to deploy the lensed SN modeling pipeline from this work for the Vera C. Rubin Observatory. GN also gratefully acknowledge support from NSF AST-2421845 and support from the Simons Foundation as part of the NSF-Simons SkAI Institute, and NSF OAC-1841625, OAC-1934752, OAC-2311355, AST-2432428 as part of the Scalable Cyberinfrastructure for Multi-messenger Astrophysics (SCIMMA) team. MG and KSM are supported by the European Union’s Horizon 2020 research and innovation programme under European Research Council Grant Agreement No 101002652 (BayeSN; PI K. Mandel) and Marie Skłodowska-Curie Grant Agreement No 873089 (ASTROSTAT-II). This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 21-46756. RAW acknowledges support from NASA JWST Interdisciplinary Scientist grants NAG5-12460, NNX14AN10G and 80NSSC18K0200 from GSFC. %% For this sample we use BibTeX plus aasjournals.bst to generate the %% the bibliography. The sample631.bib file was populated from ADS. To %% get the citations to show in the compiled file do the following: %% %% pdflatex sample631.tex %% bibtext sample631 %% pdflatex sample631.tex % \pdflatex{sample631.tex} % \newpage \bibliography{references}{} \bibliographystyle{aasjournal} %% This command is needed to show the entire author+affiliation list when %% the collaboration and author truncation commands are used. It has to %% go at the end of the manuscript. %\allauthors %% Include this line if you are using the \added, \replaced, \deleted %% commands to see a summary list of all changes at the end of the article. %\listofchanges % \section*{APPENDIX I: Handling Dust for a high-z reddened SNe } %This section will only be added if a dust corection is done. Since I'm using the values without a dust correction, it's not needed. % \section*{APPENDIX I} \appendix \vspace{-2em} \section{BayeSN Fits and Corner Plots} \label{app:bayesn_corner} The \textsc{BayeSN} model described in Section \ref{subsubsec:bayesn} were used to fit the de-lensed light curves for each of the seven independent cluster lens models, as well as for both the photometrically and spectroscopically measured magnifications. This ensures that the inference of the intrinsic SN~Ia parameters is consistently propagated across the full range of lensing scenarios explored in this work. Figure~\ref{fig:corner} shows the corner plot of the posterior distributions for the key \textsc{BayeSN} light-curve parameters obtained when adopting the mean magnifications of the three SN images predicted by the \citet{chen_2020} lens model. The fitted parameters include the host-galaxy dust extinction $A_V$, the light-curve shape $\theta$, the time of maximum $t_{\max}$, the dstance modulus $\mu$ and the intrinsic luminosity scatter $\Delta M$. The two-dimensional contours highlight the covariance between these quantities, while the marginalized histograms along the diagonal display the recovered uncertainties. As seen in Figure~\ref{fig:corner}, the posteriors are unimodal and well-sampled, with clear $68\%$ and $95\%$ credible intervals and no evidence for unconstrained directions in parameter space. The inferred parameter values are consistent with expectations from the \textsc{BayeSN} training set and remain stable when alternative lens models are used. \begin{figure*}[!ht] \centering \includegraphics[trim= 0.9cm 0.9cm 0.9cm 0.9cm, clip,width=0.81\linewidth]{figs/Corner_pub.pdf} \caption{Corner plot for the posterior distributions of the \textsc{BayeSN} fit parameters using the mean magnifications from the \citet{chen_2020} lens model. The contours show the $68\%$ and $95\%$ credible intervals, while the diagonal panels display the marginalized distributions for $A_V$, $\theta$, $t_{\max}$, $\mu$ and $\Delta M$.} \label{fig:corner} \end{figure*} \end{document} % End of file `sample631.tex'. ``` 4. **Bibliographic Information:** ```bbl \begin{thebibliography}{} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \providecommand{\url}[1]{\href{#1}{#1}} \providecommand{\dodoi}[1]{doi:~\href{http://doi.org/#1}{\nolinkurl{#1}}} \providecommand{\doeprint}[1]{\href{http://ascl.net/#1}{\nolinkurl{http://ascl.net/#1}}} \providecommand{\doarXiv}[1]{\href{https://arxiv.org/abs/#1}{\nolinkurl{https://arxiv.org/abs/#1}}} \bibitem[{{Adi} \& {Kovetz}(2021)}]{MOND} {Adi}, T., \& {Kovetz}, E.~D. 2021, Phys. Rev. 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Rev., 51, 679, \dodoi{10.1103/PhysRev.51.679} \bibitem[{{Zwicky}(1937{\natexlab{b}})}]{Zwicky_nebulae_1937} ---. 1937{\natexlab{b}}, Phys. Rev., 51, 290, \dodoi{10.1103/PhysRev.51.290} \end{thebibliography} ``` 5. **Author Information:** - Lead Author: {'name': 'Aadya Agrawal'} - Full Authors List: ```yaml Aadya Agrawal: {} J. D. R. Pierel: {} Gautham Narayan: {} B. L. Frye: {} Jose M. Diego: {} Nikhil Garuda: {} Matt Grayling: postdoc: start: 2025-10-01 thesis: null image: /assets/group/images/matt_grayling.jpg Anton M. Koekemoer: {} Kaisey S. Mandel: {} M. Pascale: {} David Vizgan: {} Rogier A. Windhorst: {} ``` 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 [2510.07637](https://arxiv.org/abs/2510.07637) is featured in the first sentence. Generate only the final Markdown output that meets all these requirements. {% endraw %}