{% 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: "BayeSN-TD: Time Delay and $H_0$ Estimation for Lensed SN H0pe" date: 2025-10-13 categories: papers --- ![AI generated image](/assets/images/posts/2025-10-13-2510.11719.png) M. Grayling Content generated by [gemini-2.5-pro](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/2025-10-13-2510.11719.txt). Image generated by [imagen-4.0-generate-001](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/2025-10-13-2510.11719.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': 'M. Grayling'}). 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 [2510.11719](https://arxiv.org/abs/2510.11719) 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/2510.11719v1 guidislink: true link: https://arxiv.org/abs/2510.11719v1 title: 'BayeSN-TD: Time Delay and $H_0$ Estimation for Lensed SN H0pe' title_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: 'BayeSN-TD: Time Delay and $H_0$ Estimation for Lensed SN H0pe' updated: '2025-10-13T17:59:59Z' updated_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2025 - 10 - 13 - 17 - 59 - 59 - 0 - 286 - 0 - tm_zone: null tm_gmtoff: null links: - !!python/object/new:feedparser.util.FeedParserDict dictitems: href: https://arxiv.org/abs/2510.11719v1 rel: alternate type: text/html - !!python/object/new:feedparser.util.FeedParserDict dictitems: href: https://arxiv.org/pdf/2510.11719v1 rel: related type: application/pdf title: pdf summary: "We present BayeSN-TD, an enhanced implementation of the probabilistic\ \ type Ia supernova (SN Ia) BayeSN SED model, designed for fitting multiply-imaged,\ \ gravitationally lensed type Ia supernovae (glSNe Ia). BayeSN-TD fits for magnifications\ \ and time-delays across multiple images while marginalising over an achromatic,\ \ Gaussian process-based treatment of microlensing, to allow for time-dependent\ \ deviations from a typical SN Ia SED caused by gravitational lensing by stars\ \ in the lensing system. BayeSN-TD is able to robustly infer time delays and produce\ \ well-calibrated uncertainties, even when applied to simulations based on a different\ \ SED model and incorporating chromatic microlensing, strongly validating its\ \ suitability for time-delay cosmography. We then apply BayeSN-TD to publicly\ \ available photometry of the glSN Ia SN H0pe, inferring time delays between images\ \ BA and BC of $\u0394T_{BA}=121.9^{+9.5}_{-7.5}$ days and $\u0394T_{BC}=63.2^{+3.2}_{-3.3}$\ \ days along with absolute magnifications $\u03B2$ for each image, $\u03B2_A =\ \ 2.38^{+0.72}_{-0.54}$, $\u03B2_B=5.27^{+1.25}_{-1.02}$ and $\u03B2_C=3.93^{+1.00}_{-0.75}$.\ \ Combining our constraints on time-delays and magnifications with existing lens\ \ models of this system, we infer $H_0=69.3^{+12.6}_{-7.8}$ km s$^{-1}$ Mpc$^{-1}$,\ \ consistent with previous analysis of this system; incorporating additional constraints\ \ based on spectroscopy yields $H_0=66.8^{+13.4}_{-5.4}$ km s$^{-1}$ Mpc$^{-1}$.\ \ While this is not yet precise enough to draw a meaningful conclusion with regard\ \ to the `Hubble tension', upcoming analysis of SN H0pe with more accurate photometry\ \ enabled by template images, and other glSNe, will provide stronger constraints\ \ on $H_0$; BayeSN-TD will be a valuable tool for these analyses." summary_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: "We present BayeSN-TD, an enhanced implementation of the probabilistic\ \ type Ia supernova (SN Ia) BayeSN SED model, designed for fitting multiply-imaged,\ \ gravitationally lensed type Ia supernovae (glSNe Ia). BayeSN-TD fits for\ \ magnifications and time-delays across multiple images while marginalising\ \ over an achromatic, Gaussian process-based treatment of microlensing, to\ \ allow for time-dependent deviations from a typical SN Ia SED caused by gravitational\ \ lensing by stars in the lensing system. BayeSN-TD is able to robustly infer\ \ time delays and produce well-calibrated uncertainties, even when applied\ \ to simulations based on a different SED model and incorporating chromatic\ \ microlensing, strongly validating its suitability for time-delay cosmography.\ \ We then apply BayeSN-TD to publicly available photometry of the glSN Ia\ \ SN H0pe, inferring time delays between images BA and BC of $\u0394T_{BA}=121.9^{+9.5}_{-7.5}$\ \ days and $\u0394T_{BC}=63.2^{+3.2}_{-3.3}$ days along with absolute magnifications\ \ $\u03B2$ for each image, $\u03B2_A = 2.38^{+0.72}_{-0.54}$, $\u03B2_B=5.27^{+1.25}_{-1.02}$\ \ and $\u03B2_C=3.93^{+1.00}_{-0.75}$. Combining our constraints on time-delays\ \ and magnifications with existing lens models of this system, we infer $H_0=69.3^{+12.6}_{-7.8}$\ \ km s$^{-1}$ Mpc$^{-1}$, consistent with previous analysis of this system;\ \ incorporating additional constraints based on spectroscopy yields $H_0=66.8^{+13.4}_{-5.4}$\ \ km s$^{-1}$ Mpc$^{-1}$. While this is not yet precise enough to draw a meaningful\ \ conclusion with regard to the `Hubble tension', upcoming analysis of SN\ \ H0pe with more accurate photometry enabled by template images, and other\ \ glSNe, will provide stronger constraints on $H_0$; BayeSN-TD will be a valuable\ \ tool for these analyses." tags: - !!python/object/new:feedparser.util.FeedParserDict dictitems: term: astro-ph.CO scheme: http://arxiv.org/schemas/atom label: null published: '2025-10-13T17:59:59Z' published_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2025 - 10 - 13 - 17 - 59 - 59 - 0 - 286 - 0 - tm_zone: null tm_gmtoff: null arxiv_comment: 20 pages, 11 figures, 4 tables. Submitted to MNRAS. BayeSN-TD code will be made public upon acceptance of the paper arxiv_primary_category: term: astro-ph.CO authors: - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: M. Grayling - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: S. Thorp - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: K. S. Mandel - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: M. Pascale - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: J. D. R - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Pierel - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: E. E. Hayes - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: C. Larison - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: A. Agrawal - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: G. Narayan author_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: name: G. Narayan author: G. Narayan ``` 3. **Paper Source (TeX):** ```tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Basic setup. Most papers should leave these options alone. \documentclass[fleqn,usenatbib]{mnras} % MNRAS is set in Times font. If you don't have this installed (most LaTeX % installations will be fine) or prefer the old Computer Modern fonts, comment % out the following line \usepackage{newtxtext,newtxmath} % Depending on your LaTeX fonts installation, you might get better results with one of these: %\usepackage{mathptmx} %\usepackage{txfonts} % Use vector fonts, so it zooms properly in on-screen viewing software % Don't change these lines unless you know what you are doing \usepackage[T1]{fontenc} % Allow "Thomas van Noord" and "Simon de Laguarde" and alike to be sorted by "N" and "L" etc. in the bibliography. % Write the name in the bibliography as "\VAN{Noord}{Van}{van} Noord, Thomas" \DeclareRobustCommand{\VAN}[3]{#2} \let\VANthebibliography\thebibliography \def\thebibliography{\DeclareRobustCommand{\VAN}[3]{##3}\VANthebibliography} %%%%% AUTHORS - PLACE YOUR OWN PACKAGES HERE %%%%% % Only include extra packages if you really need them. Avoid using amssymb if newtxmath is enabled, as these packages can cause conflicts. newtxmatch covers the same math symbols while producing a consistent Times New Roman font. Common packages are: \usepackage{graphicx} % Including figure files \usepackage{amsmath,bm} % Advanced maths commands \usepackage{caption} \usepackage{subcaption} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% AUTHORS - PLACE YOUR OWN COMMANDS HERE %%%%% % Please keep new commands to a minimum, and use \newcommand not \def to avoid % overwriting existing commands. Example: %\newcommand{\pcm}{\,cm$^{-2}$} % per cm-squared \newcommand{\mures}{$\mu_{\rm res}$} \newcommand{\kmsmpc}{km s$^{-1}$ Mpc$^{-1}$} \defcitealias{Grayling24}{G24} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% TITLE PAGE %%%%%%%%%%%%%%%%%%% \title{BayeSN-TD: Time Delay and $H_0$ Estimation for Lensed SN H0pe} % The list of authors, and the short list which is used in the headers. % If you need two or more lines of authors, add an extra line using \newauthor \author[M. Grayling et al.]{ M. Grayling$^1$\thanks{Email: mg2102@cam.ac.uk}, S. Thorp$^{1,2}$, K. S. Mandel$^{1,3}$, M. Pascale$^4$, J. D. R. Pierel$^5$, E. E. Hayes$^{1}$, \newauthor \ C. Larison$^5$, A. Agrawal$^{6, 7}$, G. Narayan$^{6,7}$ \\ $^1$ Institute of Astronomy and Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK \\ $^2$ The Oskar Klein Centre, Department of Physics, Stockholm University, AlbaNova University Centre, SE 106 91 Stockholm, Sweden \\ $^3$ Statistical Laboratory, DPMMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK\\ $^4$ Department of Astronomy, University of California, 501 Campbell Hall 3411, Berkeley, CA 94720, USA \\ $^5$ Space Telescope Science Institute, Baltimore, MD 21218, USA \\ $^6$ Department of Astronomy, University of Illinois Urbana-Champaign, 1002 West Green Street, Urbana, IL 61801, USA \\ $^7$ NSF-Simons AI for the Sky (SkAI) Institute, 875 N. Michigan Ave., Suite 3500, Chicago, IL 60611, USA \\ } % These dates will be filled out by the publisher \date{Accepted XXX. Received YYY; in original form ZZZ} % Enter the current year, for the copyright statements etc. \pubyear{2025} % Don't change these lines \begin{document} \label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \maketitle % Abstract of the paper \begin{abstract} We present BayeSN-TD, an enhanced implementation of the probabilistic type Ia supernova (SN Ia) BayeSN SED model, designed for fitting multiply-imaged, gravitationally lensed type Ia supernovae (glSNe Ia). BayeSN-TD fits for magnifications and time-delays across multiple images while marginalising over an achromatic, Gaussian process-based treatment of microlensing, to allow for time-dependent deviations from a typical SN Ia SED caused by gravitational lensing by stars in the lensing system. BayeSN-TD is able to robustly infer time delays and produce well-calibrated uncertainties, even when applied to simulations based on a different SED model and incorporating chromatic microlensing, strongly validating its suitability for time-delay cosmography. We then apply BayeSN-TD to publicly available photometry of the glSN Ia SN H0pe, inferring time delays between images BA and BC of $\Delta T_{BA}=121.9^{+9.5}_{-7.5}$ days and $\Delta T_{BC}=63.2^{+3.2}_{-3.3}$ days along with absolute magnifications $\beta$ for each image, $\beta_A = 2.38^{+0.72}_{-0.54}$, $\beta_B=5.27^{+1.25}_{-1.02}$ and $\beta_C=3.93^{+1.00}_{-0.75}$. Combining our constraints on time-delays and magnifications with existing lens models of this system, we infer $H_0=69.3^{+12.6}_{-7.8}$ \kmsmpc, consistent with previous analysis of this system; incorporating additional constraints based on spectroscopy yields $H_0=66.8^{+13.4}_{-5.4}$ \kmsmpc. While this is not yet precise enough to draw a meaningful conclusion with regard to the `Hubble tension', upcoming analysis of SN H0pe with more accurate photometry enabled by template images, and other glSNe, will provide stronger constraints on $H_0$; BayeSN-TD will be a valuable tool for these analyses. \end{abstract} % Select between one and six entries from the list of approved keywords. % Don't make up new ones. \begin{keywords} gravitational lensing: strong, supernovae: individual: SN H0pe, methods: statistical \end{keywords} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% BODY OF PAPER %%%%%%%%%%%%%%%%%% \section{Introduction} \label{sec:Intro} Strong gravitational lensing is a phenomenon whereby a massive system, such as a galaxy or galaxy cluster, lies along the line-of-sight between an observer and an astronomical source. The gravitational impact of the lensing system magnifies the background source and causes multiple images of it to appear. In the event of a time-varying event such as a quasar or supernova, multiple images of the system will appear with observable time offsets caused by differences in geometry and gravitational potential between the paths travelled through the lensing system to reach the observer. It was first proposed by \citet{Refsdal64} that the Hubble constant $H_0$ could be measured by combining the time delay between multiple images of a gravitationally lensed supernova (glSN) with a model of the mass distribution of the lensing system. The potential of glSNe to provide an independent measurement of $H_0$ is invaluable given the ongoing tension between early-Universe measurements from the cosmic microwave background \citep[CMB;][]{Planck20} and local measurements based on the distance ladder using SNe Ia \citep[e.g.][]{Reiss22, Li25} (although this is not reflected across all distance ladder measurements, see e.g. \citealp{Freedman24}; for a full review including other distance indicators please see \citealp{Valentino21}). In this work we present BayeSN-TD, an enhanced version of the probabilistic type Ia supernova (SN Ia) SED model, BayeSN \citep{M20, T21, Grayling24}, adapted for fitting light curves of glSNe, and validate the performance of this model through application to simulations. We then apply this model to obtain constraints on the time delays and magnifications of SN H0pe using photometry from \citet{Pierel24}, along with corresponding constraints on $H_0$ by combining these with the lens models of SN H0pe presented in \citet{Pascale25}. Any gravitationally-lensed time-varying event could in principle be used to estimate a time delay and consequently $H_0$; gravitationally-lensed quasars have previously been used to infer $H_0$ in a number of different studies \citep[e.g.][]{Keeton97, Wong20, Birrer20, TDCosmo25} - see \citet{Birrer24} for a recent review. However, glSNe have several advantages over quasars for these analyses. One reason is that SNe fade, allowing for isolated analysis of the lens and host and more accurate photometry using a template, compared with the blended light from a quasar, lens and host \citep{Ding21}. SNe also have simpler light curves with variability over weeks to months, compared with the longer-term stochastic variation from quasars - this simplifies time-delay estimation and means that shorter observing campaigns are required for these measurements. The more compact size of the source for glSNe compared with quasars also reduces the impact of microlensing on time-delay estimates \citep{Tie18, Bonvin19}. For the special case of a gravitationally-lensed type Ia supernova (glSN Ia), we can additionally use the standardisable nature of these events to constrain the absolute magnification of each image, providing additional constraint on the lens model and limiting the uncertainty caused by the mass sheet degeneracy \citep[e.g.][]{Falco85, Kolatt98, Holz01, Oguri03, Nordin14}. However, some of the advantages to using glSNe to constrain $H_0$ also lead to inherent challenges. The fact that these events fade on relatively fast timescales -- in contrast to lensed quasars -- also makes them more difficult to discover, especially considering their rarity. To date, only a small number of glSNe have been observed. The first resolved, multiply-imaged glSN was SN Refsdal, a peculiar type II SN. Analysis of this object led to estimates of $H_0=64.8^{+4.4}_{-4.3}$~\kmsmpc or $H_0=66.6^{+4.1}_{-3.3}$~\kmsmpc depending on the lens model weights \citep{Kelly23a,Kelly23b}. A number of other glSNe have been observed which were not suitable for $H_0$ inference; for example iPTF16geu \citep{Goobar17} and SN Zwicky \citep{Goobar23, Pierel23, Larison25} were spectroscopically-confirmed glSNe with very short time delays of $\sim0.25-2.0$ days which prevented $H_0$ estimates of reasonable precision \citep{Dhawan20, Pierel23}. The first glSN Ia with time delays long enough to enable a competitive $H_0$ analysis was SN H0pe, discovered in March 2023 \citep{Frye23} by the `Prime Extragalactic Areas for Reionization and Lensing Science' \citep[PEARLS; PID 1176,][]{Windhorst23} James Webb Space Telescope (JWST) programme. SN H0pe was followed up in a DDT programme (PID 4446, PI: B. Frye) for two additional epochs of NIRCam photometry and one epoch of NIRSpec spectroscopy. Template photometry of the field was obtained in a recent programme (PID 4744; PIs: B. Frye and J. Pierel). Photometric and spectroscopic analysis, the complete set of lensing evidence, and the first lens model are presented in \citet{Frye24}. \citet{Pierel24} presented time-delay and magnification estimates using photometric observations, \citet{Chen24} presented time-delay analysis using spectroscopic observations and \citet{Pascale25} used these time-delay estimates to infer $H_0$. This analysis led to an inferred value of $H_0=75.4^{+8.1}_{-5.5}$ \kmsmpc when leveraging absolute magnitude information about SNe Ia and $H_0=71.8^{+9.8}_{-7.6}$ \kmsmpc without using this information. Since the discovery of SN H0pe, another glSN Ia with time-delays suitable for $H_0$ inference has been identified; SN Encore \citep{Pierel24b}, which notably occurred in the same galaxy as SN Requiem \citep{Rodney21}. \citet{Pierel25} and \citet{Suyu25} respectively present time-delays and lens models for SN Encore which yield an estimate of $H_0=66.9^{+11.2}_{-8.1}$ \kmsmpc. Given the serendipitous discovery of SN H0pe, photometric data for two of the three images is only available significantly after peak, extending out to a phase of $\sim$ +60 days. This necessitates the use of a SN SED model which extends both to NIR wavelengths and late-time phases, making BayeSN the only viable current SN Ia SED model for this analysis. \citet{Pierel24} applied an extended version of the BayeSN model presented in \citet{W22} which covered phases out to 50 rest-frame days after peak and used linear extrapolation beyond that range. However, for analyses of such events it is desirable to extend the coverage of the model to later phases. Another significant challenge when fitting light curves of glSNe with typical SN models is the presence of microlensing -- lensing caused by small perturbers in the lens plane such as stars -- which can have a time-varying impact across the observed SN light curve. This can significantly affect inferred time delays from glSNe \citep[e.g.][]{Dobler06, Goldstein18, Pierel19} and must be accounted for in such analyses \citep[see e.g. Fig 1 of][]{Hayes24}. Microlensing can cause otherwise typical SNe to appear significantly different with deviations from typical SN SEDs. It is typical to fit SN Ia light curves using empirical SED models such as BayeSN \citep{T21, M20,Grayling24} or SALT \citep{Guy07, Kenworthy21}, trained on populations of non-lensed SNe Ia. Naively applying these models to glSNe without mitigating the impact of microlensing could potentially lead to biased time-delay estimates and underestimated uncertainties. Previous studies have found that, for SNe Ia, microlensing is effectively achromatic for approximately the first 3 weeks after explosion \citep{FoxleyMarrable18, Goldstein18, Huber19, Huber21} - the impact on a SN light curve is the same in each band. Supernova Time Delays (\textsc{SNTD}), presented in \citet{PR19}, mitigates the impact of microlensing by applying SN models to measured colours rather than measured photometry, as colours are insensitive to achromatic microlensing \citep{Goldstein18}. The impact of microlensing is also effectively achromatic during the plateau phase of SNe IIP \citep{Bayer21}, and \citet{Grupa25} analysed colour curves for time-delay inference of simulated SNe IIP light curves. \citet{Huber22, Huber24} followed a different approach, training machine learning methods for time-delay estimation using physical simulations of SN observables which incorporate the effect of chromatic microlensing using intensity profiles of theoretical models. \citet{Hayes24} applied a template-independent method for time-delay inference based on Gaussian processes (GPs), using an analytic, achromatic treatment of microlensing. This has since been enhanced to incorporate templates along with a chromatic, GP-based treatment of microlensing \citep{Hayes25b}. In this work, we present BayeSN-TD, a method for fitting glSNe Ia which combines the probabilistic SN Ia SED model BayeSN with a GP treatment of microlensing to allow for deviations from the SED model. This approach models the underlying SN Ia light curve along with variations between each image; this enables joint inference of typical SN Ia standardisation parameters such as light curve shape, time delays between each image and the impact of microlensing on each image. Performing these fits as a single joint inference ensures that the impact of microlensing on the time delay and overall SN light curve is incorporated robustly and forms part of the statistical error budget when estimating $H_0$. For this work we assume an achromatic treatment of microlensing. We validate the performance of BayeSN-TD through application to simulations of glSNe Ia which incorporate the effect of microlensing, demonstrating that this model is able to robustly infer time delays with well-calibrated uncertainties. We also introduce a new extended version of the BayeSN model with coverage to a phase of +85 days for use in cases where only late-time photometry is available. Having validated the performance of BayeSN-TD through simulations, we apply this new model to the photometry of SN H0pe presented in \citet{Pierel24} to obtain estimates of time delays and magnifications, and combine these with the lens models presented in \citet{Pascale25} to obtain an associated constraint on $H_0$. The structure of this paper is as follows. In Section \ref{sec:method} we detail the BayeSN-TD model and its implementation, as well as the phase-extended version of the BayeSN model used for this work. We then validate the performance of our model on a variety of simulated glSNe in Section \ref{sec:sims}, before applying it to photometry of SN H0pe from \citet{Pierel24} in Section \ref{sec:h0pe} to obtain time-delay and magnifications estimates. We then present corresponding constraints on $H_0$ from our time-delays and magnifications in Section \ref{H0}. Finally, we conclude in Section \ref{sec:conclusions}. \section{Method} \label{sec:method} We begin by giving an overview of the BayeSN model which forms the basis for this work, and detailing how we have extended it with BayeSN-TD for application to strongly lensed SNe Ia. The full description of the BayeSN SED model is presented in \citet{M20}\footnote{Building upon earlier hierarchical Bayesian multi-passband SN Ia light curve models of \citet{Mandel09, Mandel11}}, with further discussions in \citet{T21, TM22, W22, Thorp24, Grayling24, Uzsoy24, Hayes25, GraylingPopovic25}. \subsection{The BayeSN Model} BayeSN is a probabilistic SED model for SNe Ia, with the full time- and wavelength-varying SED given by: \begin{equation} \label{bayesn_equation} \begin{aligned} -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^s_1W_1(t,\lambda_r) + \epsilon^s(t,\lambda_r)+A^s_V\xi\big(\lambda_r;R^{(s)}_V\big) \end{aligned} \end{equation} where $t$ signifies the phase relative to B-band maximum, and $\lambda_r$ denotes the rest-frame wavelength. BayeSN uses the optical-NIR SN Ia SED template from \citet{Hsiao07} as a zeroth-order template, with an arbitrary scaling factor $M_0$ of -19.5\footnote{Note that $M_0$ does not define the absolute magnitude scale of SNe Ia, this is arbitrarily fixed to -19.5 with $W_0$ then defining the mean intrinsic SED for the population.}. Latent variables, with distinct values for each SN, are denoted by superscript $s$, while all other parameters are global hyperparameters that are shared across the population. The individual components constituting the model are detailed below: \begin{itemize} \item The function $W_0(t, \lambda_r)$ warps and normalises the zeroth-order SED template, which establishes a mean intrinsic SED for the SN Ia population. $W_1(t, \lambda_r)$ is a functional principal component (FPC) designed to capture the primary mode of intrinsic SED variation across the population of SNe Ia. These components are both implemented as cubic spline surfaces. \item For each SN, the coefficient $\theta_1^s$ quantifies the impact of the $W_1$ FPC. This coefficient is defined with a Normal prior distribution such that $\theta_1^s \sim N(0, 1)$. When combined, $W_1(t, \lambda_r)$ and $\theta_1^s$ effectively model the `broader-brighter' relationship inherent to SNe Ia, where intrinsically brighter light curves are observed to evolve over more extended timescales around their peak \citep{Phillips93}. \item $\delta M^s$ is an achromatic, time-independent magnitude offset for each SN, drawn from a normal distribution with $\delta M^s \sim N(0, \sigma_0^2)$. The hyperparameter $\sigma_0$, which defines the intrinsic achromatic scatter across the population, is inferred during the model's training phase. \item The term $\epsilon^s(t,\lambda_r)$ is a time- and wavelength-dependent function that describes residual intrinsic colour variations within the SED that are not accounted for by the $\theta_1^sW_1(t, \lambda_r)$ component. This parameter is represented by a cubic spline function over time and wavelength, which is defined by a matrix of knots, $\mathbf{E}^s$. These knots are drawn from a multivariate Gaussian distribution, $\mathbf{e}^s \sim N(0, \mathbf{\Sigma}_\epsilon)$, where $\mathbf{e}^s$ is the vectorised version of the $\mathbf{E}^s$ matrix. The covariance matrix $\mathbf{\Sigma}_\epsilon$ functions as a model hyperparameter that characterises the distribution of this residual scatter across the SN Ia population, and is inferred during training. \item The host galaxy extinction law for each supernova is described by $A_V^s$ and $R_V^{(s)}$. $A_V^s$ represents the total V-band extinction amount, while $R_V^{(s)}$ describes the slope of the \citet{Fitzpatrick99} dust extinction law assumed by the model. $R_V^{(s)}$ can be treated as either a shared parameter for the whole population or as a latent parameter for each supernova drawn from a distribution. For $A_V^s$, an exponential prior is assumed, governed by a scale parameter $\tau_A$, such that $A_V^s \sim \text{Exponential}(\tau_A)$. \end{itemize} An advantage of BayeSN is that it models the physically-distinct effects of intrinsic variations and host-galaxy dust on the supernova SED when fitting SN Ia light curves \citep[e.g.][]{Mandel17}. The rest-frame, host galaxy dust-extinguished SED model $S_s(t, \lambda_r)$ is then scaled based on distance modulus $\mu^s$, redshifted and corrected for Milky Way dust extinction assuming $R_V=3.1$, using a \citet{Fitzpatrick99} dust law with dust maps from \citet{Schlafly11}. Model photometry can be derived by integrating this SED through photometric filters, which can then be compared with observed photometry to compute a likelihood. BayeSN training involves jointly inferring all global and latent parameters across a population of SNe Ia (for a complete discussion on model training, see \citealp{M20, T21}). We marginalise over all latent parameters and obtain estimates of global parameters from the posterior distributions. Once trained, BayeSN can be applied to fit light curves of individual SNe by inferring posteriors of the latent parameters for each SN conditional on the fixed population-level parameters inferred during model training. \subsection{Improving Phase Coverage of BayeSN} \label{extended_bayesn} As discussed in \citet{Pierel24}, one challenge faced by analysis of SN H0pe and other strongly lensed SNe discovered by JWST is a lack of coverage of SN Ia SED models at later phases, especially in NIR wavelengths. \citet{Pierel24} applied a phase-extended version of the BayeSN model presented in \citet{W22}, which was defined up to 50 rest-frame days after peak, and utilised linear extrapolation beyond this phase. As part of this work, we train a new BayeSN model with later phase coverage extending out to 85 rest-frame days after peak. We apply this model within BayeSN-TD for time delay estimation of SN H0pe in this work and make it publicly available for future analyses. This phase-extended BayeSN model has been used as the basis for fitting another glSN Ia, SN Encore, as presented in \citet{Pierel25}. Previous trained BayeSN models were presented in \citet{M20}, \citet{T21} and \citet{W22}. \citet{M20} trained on a compilation of local SNe Ia presented in \citet{Avelino19}, while \citet{T21} trained on a sample of SNe Ia from Foundation DR1 \citet{Foley18} and \citet{W22} trained on the combination of those two datasets. In addition, \citet{TM22} applied the model presented in \citet{M20} to a sample of SNe Ia exclusively from CSP-I \citep{Krisciunas17}, selecting a sample from that presented in \citet{Uddin20}. Within this work, we train on a combination of all SNe across these analyses, yielding a total training set of 278 SNe Ia. Of these, those from Foundation have only optical photometry while the rest also have NIR (YJH bands). In terms of technical implementation, this model is very similar to that of \citet{W22} except for the addition of extra spline knots at later phases of +55, +70 and +85 days when defining $W_0(t,\lambda_r)$, $W_1(t,\lambda_r)$ and $\mathbf{\Sigma}_\epsilon$ to allow the later phase coverage. One other difference is that we include U-band data in the training set, unlike \citet{M20} and \citet{W22} but similarly to the phase-extended model applied in \citet{Pierel24}. We make this choice to include F090W data within our analysis, as F090W data for SN H0pe covers rest-frame U-band. Full details around training the BayeSN model are presented in \citet{M20}, while the code used for training the model is described in \citet{Grayling24}. This new, phase-extended BayeSN model can be found at \url{https://github.com/bayesn/bayesn-model-files/tree/main}, and is incorporated within the public BayeSN code available here: \url{https://github.com/bayesn/bayesn}. For more discussion of this new BayeSN model, please see Appendix \ref{appendix:extended_bayesn}. \subsection{Fitting Multiple Images Using BayeSN-TD} When applying a trained BayeSN model to fit a single SN Ia light curve, a number of different latent SN parameters are inferred: the `shape' parameter $\theta_1^s$, $V$-band host galaxy dust extinction $A_V^s$ (and, optionally, the slope of the dust extinction law $R_V^s$), the distance modulus $\mu^s$, the residual intrinsic colour surface $\epsilon^s(t,\lambda_r)$ and finally the time of B-band maximum $t^s_\text{max}$. When fitting a multiply-imaged type Ia supernova, many of these parameters are treated as being shared across each image i.e. we are seeing multiple images of the same intrinsic SN light curve. However, separate parameters are included for different images of the same SN to account for time delays and magnification. A full description of the parameters which are shared between images and those which vary between images is given below. Please note that the index $s$ denotes parameters shared across all images of a SN $s$, while the index $i$ denotes parameters which differ between separate images of the same SN. \begin{itemize} \item \textbf{Parameters shared between images} \begin{itemize} \item Light curve shape $\theta_1^s$ \item Host galaxy dust extinction parameters $A_V^s$ and $R_V^s$ \item Residual intrinsic colour $\epsilon^s(t,\lambda_r)$ \end{itemize} \item \textbf{Parameters varying between images} \begin{itemize} \item Time of B-band maximum $t_\text{max}^{si}$ \item Distance modulus $\mu^{si}$ is treated separately for each image to allow for differences in magnification. \item BayeSN-TD incorporates an analytic treatment for the effect of microlensing using a GP, as outlined in Section \ref{gibbs}. Each image of a SN receives its own GP hyperparameters and corresponding microlensing curve. \end{itemize} \end{itemize} In future, further complexity could be incorporated in the model. For example, we could account for differences in the dust properties in the lens along the line-of-sight to each of the images along with the effect of dust extinction in the host galaxy of the SN; this would require $A_V^s$ and $R_V^s$ parameters for host galaxy extinction for the SN along with separate $A_V^{si}$ and $R_V^{si}$ parameters for each image to capture the separate effect of dust extinction in the lens for each image. \subsection{Incorporating Microlensing} \label{gibbs} The multiple images of a lensed source result from different paths taken by the light from each image through the lens system; light for each image therefore passes through a unique star field, each with its own lensing magnification map. These maps vary on the scale of microarcseconds, comparable to typical physical sizes of the photospheres of SNe. Over time, as the photosphere of a SN expands it passes over an increasing number of microlens caustics. This causes a time-varying magnification which is unique for each image \citep[e.g.][]{Dobler06, Bagherpour06, FoxleyMarrable18}. This effect, microlensing, can have a significant impact on time-delay measurements \citep[e.g.][]{Goldstein18, Pierel19, Hayes24}, and any time-delay analysis of glSNe must consider this impact. In the standard BayeSN model, variation around the population mean intrinsic SED for SNe Ia is governed by a functional principal component $\theta_1^s$ along with the impact of dust extinction and residual intrinsic scatter $\epsilon^s(t,\lambda_r)$ corresponding to the distribution of intrinsic SN colours. However, when applying BayeSN to strongly lensed supernovae we additionally allow for deviations from the SED model as a result of microlensing. In this work we opt for a flexible treatment of microlensing using Gaussian processes (GPs). GPs have been used extensively in transient astronomy, for example to model light curves \citep[e.g.][]{Grayling21, Grayling23, Revsbech18, aigrain23}. GPs have also been used extensively for time-delay cosmography, applied to strongly-lensed quasars \citep{Hojjati14, Tak17, Hu20, Meyer23} and SNe \citep{Kelly23b, Hayes24}. For our microlensing treatment, we assume a zero mean function $E[\delta\beta(t)] = 0$, treating the impact of microlensing as a perturbation around the BayeSN SED model. The choice of covariance function impacts the characteristic scale over which the function being modelled varies. In this work we use a Gibbs kernel \citep{Gibbs97}, first proposed for a treatment of microlensing in \citet{Hayes24}. The Gibbs kernel is non-stationary and allows the length scale parameter to vary with time. This quality is well suited for modelling microlensing as it allows for faster evolution as the SN photosphere crosses a stellar caustic and slower evolution elsewhere. The Gibbs kernel describing the covariance of the Gaussian process between two phases $t$ and $t'$ is given by \begin{equation} \label{gibbs_equation} k^{\text{Gibbs}}_{\bm{\Lambda}}(t, t') = A\Bigg(\frac{2l(t)l(t')}{l^2(t)+l^2(t')}\Bigg)^{0.5}\exp\Bigg(-\frac{(t-t')^2}{l^2(t) + l^2(t')}\Bigg) \end{equation} where $l(t)$ is a variable length-scale function given by \begin{equation} l(t) = \lambda(1-p\phi_{(\tau_{\rm ML}, \eta)}(t)). \end{equation} The parameters $\bm{\Lambda} = \{A, \lambda, p, \tau_{\rm ML}, \eta\}$ are tuning parameters, which are not directly physically interpretable but would relate to the amplitude and size of a microlensing caustic as well as the SN ejecta velocity, since these would determine the size and timescale of the microlensing magnification. $\phi_{(\tau_{\rm ML}, \eta)}(t)$ is a Gaussian probability density function with mean $\tau_{\rm ML}$ and standard deviation $\eta$, while $A$ is an amplitude parameter. These GP parameters are independent for each image $i$, given that microlensing will impact each image differently. We refer to the set of microlensing parameters for each image collectively as $\bm{\Lambda}_{si} = \{ A_{si}, \lambda_{si}, p_{si}, \tau_{{\rm ML},si}, \eta_{si} \}$. The microlensing curve for each image $i$, $\delta\beta_{si}(t)$, is modelled as a realisation of an independent Gaussian process: \begin{equation} \delta\beta_{si}(t) \sim \mathcal{GP}(0, k^{\text{Gibbs}}_{\bm{\Lambda}_{si}}(t, t')). \end{equation} Given a set of observed phases of image $i$ of SN $s$, $\bm{t}_{si}$, the vector of microlensing curve values evaluated at these phases, $\bm{\delta\beta}_{si}$, thus has a multivariate Gaussian prior distribution: \begin{equation} \label{eqn:gp_mvn} \bm{\delta\beta}_{si} \sim N(0, \bm{K}^{\text{Gibbs}}_{\bm{\Lambda}_{si}}(\bm{t}_{si},\bm{t}_{si}) ), \end{equation} where $\bm{K}^{\text{Gibbs}}_{\bm{\Lambda}_{si}}(\bm{t}_{si},\bm{t}_{si})$ is the covariance matrix with elements populated by the Gibbs kernel evaluated on all pairs of observed phases in $\bm{t}_{si}$. It is important to note that for this work, we make the simplifying assumption of achromatic microlensing i.e. the same microlensing curve applies to all bands of a given image's light curve. As mentioned previously, for SNe Ia this approximation is valid for approximately the first 3 weeks after explosion, covering optical peak luminosity, but does not hold to later times \citep{FoxleyMarrable18, Goldstein18, Huber19, Huber21}. The SNTD method presented in \citet{PR19}, and applied for the SN H0pe analysis in \citet{Pierel24}, involves directly fitting colour curves which are insensitive to achromatic microlensing; the impact of chromatic microlensing is then considered as part of the systematic error budget. \citet{Grupa25} also analysed colour curves to remove the impact of achromatic microlensing. \citet{Hayes24} included an analytic, achromatic treatment of microlensing. A chromatic treatment of microlensing would be of interest to explore in future work but would add significant additional complexity to the model, though since the development of BayeSN-TD, \citet{Hayes25b} has developed a GP-based treatment of chromatic microlensing. In this work we test the robustness of time-delay estimation with an achromatic microlensing treatment to the impact of chromatic microlensing. \subsection{Priors} \label{priors} In this section we detail the priors included when fitting multiply-imaged glSNe Ia with BayeSN-TD, which are outlined in Table \ref{priors_table}. \begin{table} \centering \caption{Priors on BayeSN-TD parameters when fitting light curves of glSNe Ia. \newline $^* TN(\mu, \sigma^2, a, b)$ denotes a Truncated normal distribution with mean and variance $\mu$ and $\sigma^2$ prior to truncation, lower truncation bound $a$ and upper truncation bound $b$.} \begin{tabular}{cc} Parameter & Prior \\ \hline $A_V$ & $A_V\sim \text{Exp}(0.32 \text{ mag})$\\ $R_V$ & $R_V\sim TN^*(2.51, 0.65^2, 1.2, \infty)$\\ $\theta_1$ & $\theta_1\sim N(0, 1)$ \\ $A$ & $A\sim \text{Half}-N(0.1)$ \\ $\lambda$ & $\lambda\sim U(10, 150)$ \\ $p$ & $p\sim U(0, 1)$ \\ $\tau_\text{ML}$ & $\tau_\text{ML}\sim U(-10, 85)$ \\ $\eta$ & $\eta\sim U(1, 40)$ \\ \end{tabular} \label{priors_table} \end{table} Each BayeSN model is defined over a specific phase range. For example, the models presented in \cite{M20}, \cite{T21} and \cite{W22} are all defined from $-10$ days to $+40$ days in the rest-frame relative to B-band maximum. However, given that time-of-maximum is not perfectly known a priori when fitting a SN light curve, it is important that the model has some ability to extrapolate beyond this phase range. This allows for the time-of-maximum to be sampled during light curve fitting without data falling in and out of phase coverage as the sampler explores possible $t_\text{max}$ values. In practice that corresponds to linear extrapolation of $W_0(t, \lambda_r)$, $W_1(t, \lambda_r)$ and $\epsilon^s(t,\lambda_r)$. As a result, when fitting SN light curves with BayeSN, the following procedure is followed: \begin{enumerate} \item Each SN requires a fiducial estimate for the time of maximum, $T_\text{max}$ - we will refer to this fiducial value as $\text{T}_\text{max}^\text{fid}$. This can be based on some simple algorithm, a previous SALT fit or a maximum a posteriori (MAP) estimate from the BayeSN model. \item This fiducial value is used to convert observer-frame MJDs into rest-frame phases relative to peak. \item Data is selected based on the rest-frame phase coverage of the model being used - data points outside of this phase range are discarded. \item When the light curve is fit, the parameter $t_\text{max}$ is treated as being a rest-frame shift to the fiducial value. We use a uniform prior on this shift such that $t_\text{max} \sim U(-10 \text{ days}, +10 \text{ days})$. This is equivalent to an observer-frame prior of $T_\text{max} \sim U(T_\text{max}^\text{fid} - 10\times(1+z_\text{hel}^s), T_\text{max}^\text{fid} + 10\times(1+z_\text{hel}^s))$, where $z_\text{hel}^s$ is the heliocentric redshift of a SN $s$. \item After fitting, the posterior distribution on $t_\text{max}$ can be used alongside the fiducial value $T_\text{max}^\text{fid}$ to obtain a posterior distribution on the observer-frame time-of-maximum. \end{enumerate} The prior window of 10 rest-frame days either side of $T_\text{max}^\text{fid}$ is imposed to prevent the model linearly extrapolating far beyond the range over which the model is defined. In particular, when allowed to extrapolate far beyond its specified range $\epsilon^s(t,\lambda_r)$ can exhibit some unphysical behaviour. This prior width far exceeds typical uncertainties on time-of-maximum. When using our model for time-delay estimation, we fit for a separate $T_\text{max}^i$ for each image $i$. \subsubsection{Priors on Microlensing} The priors on the parameters in the Gibbs kernel which we use to model microlensing, described in Equation \ref{gibbs_equation}, are detailed in Table \ref{priors_table}. In general, we choose broad, uninformative priors for these kernel parameters. The exception to this is the case of the amplitude parameter $A$, for which we use a half-Normal prior with a scale factor of 0.1. This is chosen to reflect the typical scale of microlensing deviations while avoiding imposing a hard upper-limit to ensure that more extreme microlensing events can still be fit. While these priors are ultimately arbitrary, when validating our model on simulations which incorporate a realistic treatment of microlensing we find that our model produces well-calibrated uncertainties and can capture deviations from a base SN Ia SED model caused by microlensing. This demonstrates the priors that we have used within our model are suitable. \subsection{Full Posterior} We now define the full posterior of the BayeSN-TD model. We define the complete set of parameters to be inferred as follows: \begin{itemize} \item $\bm{\Theta}_s$: The set of parameters shared across all images, describing the physical properties of the SN and the impact of host galaxy dust extinction. \[ \bm{\Theta}_s = \{ \theta_1^s, A_V^s, R_V^s, \epsilon^s \} \] \item $\bm{\Phi}_s$: The set of parameters that are specific to each of the $I$ lensed images. This is a collection of parameter sets, one for each image $i \in \{1, ..., I\}$. \[ \bm{\Phi}_s = \{ \bm{\Phi}_{s1}, \bm{\Phi}_{s2}, ..., \bm{\Phi}_{sI} \}, \quad \text{where} \quad \bm{\Phi}_{si} = \{ T_{\text{max}}^{si}, \mu^{si}, \bm{\Lambda}_{si} \} \] \end{itemize} Here, $T_{\text{max}}^{si}$ is the observer-frame time of B-band maximum, $\mu^{si}$ is the apparent distance modulus (capturing both cosmological distance and magnification), and $\bm{\Lambda}_{si}$ represents the set of hyperparameters for the microlensing GP for image $i$. Let $\bm{\mathcal{F}} = \{ \bm{\mathcal{F}}_1, ..., \bm{\mathcal{F}}_I \}$ be the full set of observed photometric light curve data for all images, $\bm{\mathcal{T}} = \{ \bm{\mathcal{T}}_1, ..., \bm{\mathcal{T}}_I \}$ be the full set of times of observation for all images, and $\bm{\mathcal{H}}$ be the set of fixed, pre-trained population hyperparameters from the base BayeSN model. Finally, $z_s$ is the spectroscopic redshift of SN $s$, which is used purely for time dilation and spectral redshifting, not for constraining distance. The full joint posterior distribution for all unknown parameters conditional on the observed data is given below, factorised to explicitly show the contributions from each image and the shared properties of the source: \begin{multline} P(\bm{\Theta}_s, \bm{\Phi}_s, \{\bm{\delta\beta}_{si} \}\mid \bm{\mathcal{F}}, \bm{\mathcal{T}}, \bm{\mathcal{H}}, z_s) \propto \\ \left[ \prod_{i=1}^{I} P(\bm{\mathcal{F}}_i \mid \bm{\mathcal{T}}_i, \bm{\Theta}_s, \bm{\Phi}_{si}, \bm{\delta\beta}_{si}, z_s) \times P(\bm{\delta\beta}_{si} \mid \bm{\Lambda}_{si}) \times P(\Phi_{si}) \right] \\ \times P(\bm{\Theta}_s \mid \bm{\mathcal{H}}) \label{eq:bayeSN_TD_posterior} \end{multline} This expression comprises four key components: the data likelihood $P(\bm{\mathcal{F}}_i \mid \bm{\mathcal{T}}_i, \bm{\Theta}_s, \bm{\Phi}_{si},\bm{\delta\beta}_{si}, z_s)$, the GP prior (Eq. \ref{eqn:gp_mvn}) on the microlensing for each image $P(\bm{\delta\beta}_{si} \mid \bm{\Lambda}_{si})$, the prior on the parameters unique to each image $P(\bm{\Phi}_{si})$, and the prior on the shared SN parameters $P(\bm{\Theta}_s \mid \bm{\mathcal{H}})$. The prior terms are outlined in Section \ref{priors}. Assuming independent photometric measurements with Gaussian measurement uncertainties, the likelihood for the data of a single image $i$ of a SN $s$ is the product of the probabilities of each individual flux measurement. Note that we define the set of flux measurements $\mathcal{F}_i = \{ f_{ij} \}$. The likelihood for image $i$ is conditional on both the shared source parameters $\Theta_s$ and its own unique lensing parameters $\Phi_{si}$, such that: \begin{multline} P(\bm{\mathcal{F}}_i \mid \bm{\mathcal{T}}_i, \bm{\Theta}_s, \bm{\Phi}_{si}, \bm{\delta\beta}_{si}, z_s) = \\ \prod_{j} \mathcal{N}\left(\hat{f}_{sij} \mid f_{sij}(T_{ij}, \delta\beta_{si}(t_{ij}), \bm{\Theta}_s, \bm{\Phi}_{si}, z_s), \sigma_{ij}^2\right) \end{multline} where $\hat{f}_{ij}$ and $\sigma_{ij}$ are the observed flux and its uncertainty for the $j$-th observation of image $i$, $T_{ij}$ is the time of this observation and $t_{ij}$ is the rest-frame phase of this observation. Including distance, magnification and microlensing effects, the model SED of each image becomes: \begin{equation} F_{si}(t, \lambda_r) = S_s(t, \lambda_r) \times 10 ^{- 0.4[\mu_{si} + \delta\beta_{si}(t)]} \end{equation} where $S_s(t, \lambda_r)$ is the BayeSN model from equation \ref{bayesn_equation} and $\delta\beta_{si}(t)$ is the microlensing curve for image $i$ of supernova $s$, which is defined in magnitude space\footnote{Any overall magnification caused by microlensing will be degenerate with the combined distance-magnification parameter $\mu_{si}$, our microlensing treatments will capture relative changes in magnification during the light curve of each image.}. The model flux $f_{ij}$ is then defined by integrating this SED, redshifted to the observer-frame, through the filter of each observation $j$, with the likelihood evaluated in flux space. %In \citet{M20}, this likelihood was evaluated in magnitude space as the model was applied to high signal-to-noise data where the assumption of symmetric uncertainties in log space was valid. In this work, we evaluate the likelihood in flux space as we do not apply BayeSN-TD to consistently high signal-to-noise observations. \subsection{Obtaining Posteriors on Time Delay} One of the main goals when fitting multiply-imaged glSNe is to estimate the time delay between different images. BayeSN-TD does not directly fit for a time-delay parameter but does allow for posteriors on the time delay to be easily obtained. The model directly samples the time of maximum, implemented as a rest-frame shift from a fiducial value $\text{T}_\text{max}^\text{fid,i}$ as outlined in Section \ref{priors} - each image $i$ has an associated $t_\text{max}^i$ parameter. We can use the posteriors on $t_\text{max}^i$ to derive posteriors on $\text{T}_\text{max}^i$ by simply converting to observer frame, \begin{equation} T_\text{max}^{i} = T_\text{max}^{\text{fid},i} + (1+z_\text{hel}^s) \times t_\text{max}^i \end{equation} for each posterior sample of $t_\text{max}^i$. After this, we obtain posteriors on the time delay between images $i$ and $k$, $\Delta T_{ik}$, by evaluating, \begin{equation} \Delta T_{ik} = T_\text{max}^i - T_\text{max}^k \end{equation} for each step along our MCMC chains. \subsection{Implementation of BayeSN-TD} BayeSN-TD is a modification of the BayeSN code presented in \citet{Grayling24}, developed based on \textsc{numpyro} and \textsc{jax}. As a result, it shares the same advantages; the code is designed for GPU-acceleration and can perform Bayesian inference quickly and efficiently. Although the very limited samples of real observed glSNe mean that high computational performance is not essential---unlike regular SNe Ia---this does enable us to apply this complex model to large samples of simulated data to assess performance. \section{Validation on Simulations} \label{sec:sims} We begin by assessing the performance of BayeSN-TD on simulated populations of lensed SNe Ia. To date, such simulations have generally used the SALT SED model for SNe Ia \citep{Guy07, Kenworthy21}. As a result, the application of our model to these simulations results in an inherent model misspecification. BayeSN has previously proven robust when inferring population-level properties from simulated data sets using SALT \citep{GraylingPopovic25}, but there will still be differences between the two. In our case, this is a valuable test given that in reality the empirical models we use to simulate and perform inference will never perfectly match the true properties of SNe Ia. By applying our model to these simulations, we can assess whether our results are robust when applied to data simulated from a different model. \subsection{Roman Simulations} \label{roman} We first assess the performance of BayeSN-TD at recovering time delays from simulated glSNe from the \textit{Nancy G.\ Roman Space Telescope} presented in \cite{Pierel21}. These simulations were based on the "All-z" Roman SN observing strategy described in \cite{Hounsell18}, with a few modifications to reflect more recent survey updates. This pipeline used the extended SALT2 model presented in \citet{Pierel18} to simulate SNe Ia. These simulations incorporate the effect of microlensing based on 12 different microlensing maps; however, this treatment assumes achromatic microlensing similarly to the BayeSN-TD model. These simulations therefore provide an opportunity for testing our achromatic GP treatment of microlensing on realistic achromatic microlensing simulations. The results of time-delay recovery with BayeSN-TD are summarised in Table \ref{sim_results}. Despite the difference between the model used to simulate the light curves and the model used for inference, BayeSN-TD performs well at recovering the true simulated time delays. Overall, the bias between true and inferred time delays is 0.09 days, negligible when considering that across all simulated glSNe the mean of all posterior uncertainties is 3.07 days. The posterior uncertainties from BayeSN-TD are also well-calibrated - the true simulated time delays lie within the 68 and 95 per cent credible intervals in 67.7 and 93.5 per cent of cases respectively, quantities we will refer to as $f_{68}$ and $f_{95}$ hereafter. The top panel of Fig. \ref{roman_sim_plots} shows the distribution of residuals for inferred time delays, $\Delta T_\text{fit} - \Delta T_\text{true}$, and the bottom panel shows the cumulative density function of $(\Delta T_\text{fit} - \Delta T_\text{true}) / \sigma_{\Delta T}$ (the `pull') compared with a Gaussian CDF. This bottom panel further demonstrates that BayeSN-TD produces well-calibrated posteriors, with this distribution closely following a Gaussian. \begin{figure} \centering \includegraphics[width = \linewidth]{figures/roman_sim_results.pdf} \caption{\textbf{Upper:} Histogram showing distribution of time-delay residuals relative to true simulated values when applying BayeSN-TD to simulations of glSNe Ia observed by Roman presented in \citet{Pierel21}. \textbf{Lower:} Cumulative density of time-delay residual normalised by posterior uncertainties (dashed) shown alongside the expected cumulative density function for a Normal distribution represented by the shaded region. This demonstrates that BayeSN-TD produces well-calibrated uncertainties for these simulations.} \label{roman_sim_plots} \end{figure} An example of a BayeSN-TD fit to one of the simulated Roman light curves is shown in Fig. \ref{Roman_example_good}. The left panels show the simulated photometry compared with the model fit for each image, while the right panels show the simulated achromatic microlensing curves compared with the posterior from BayeSN-TD. With this dataset, it was not possible to access the raw simulated microlensing curves - instead, the influence of microlensing can be determined by comparing the `observed' simulated magnitudes with the true simulated magnitudes before microlensing, albeit this is after the effect of measurement noise. This is why each point in the simulated microlensing curves has a corresponding uncertainty. Despite the mismatch between the models being used for simulation and for inference, it is clear that BayeSN-TD is able to closely match the simulated photometry. In addition, the lower right panel demonstrates that our model is able to successfully match the deviations from typical SEDs of SNe Ia as a result of microlensing. In some cases the posterior distribution will be centred around zero where the data does not provide any constraint on microlensing, such as in the upper right panel. However, the model is able to constrain cases of significant microlensing. Note that any overall magnification as a result of microlensing - a shift in the y-axis of the right panels - will be captured by BayeSN-TD's distance parameters, and these plots represent relative changes in microlensing magnification across the duration of the SN. \begin{figure*} \centering \includegraphics[width = \linewidth]{figures/Ia_715065_320990_1.pdf} \caption{\textbf{Left panels}: Simulated 2-image Roman glSN Ia light curve from \citet{Pierel24} along with associated BayeSN-TD fits. \textbf{Right panels}: Plotted data points represent simulated deviation from model light curves as a result of microlensing, with associated uncertainties from measurement noise as true simulated values post-microlensing, without noise, are not available. Plotted line and shaded region represent the posterior mean and standard deviation on microlensing from BayeSN-TD, demonstrating that with Roman simulations the model is able to constrain the deviation away from a typical SN Ia template as a result of microlensing. Note that these simulations, along with BayeSN-TD, assume achromatic microlensing.} \label{Roman_example_good} \end{figure*} As mentioned above these simulations are based on the extended SALT2 model presented in \citet{Pierel18}, which extended the coverage of the default SALT2 template further into near-ultraviolet and near-infrared wavelengths using sophisticated extrapolation techniques. This extrapolation technique was applied for wavelengths less than 3500 \AA. It should be noted that these extrapolations were intended to enable simulations in these wavelength regimes, but not to make SALT2 capable of fitting light curves at these wavelengths. When applying BayeSN-TD to simulations based on rest-frame wavelengths significantly less than 3500 \AA, we found that differences between our BayeSN model and the extrapolated SALT2 model in this wavelength regime led to poor quality fits to simulated $Z$-band (F087 band) light curves. Differences between the models will be particularly prevalent at these wavelengths given SALT2 is based on extrapolation. To avoid this issue, we exclude $Z$-band data when fitting simulated glSNe where the observer-frame $Z$-band probes wavelengths bluer than 3000 \AA. This is done purely because BayeSN does not match SALT2 extrapolation in this region, and does not mean that BayeSN should not be applied to real data at these wavelengths. \begin{table*} \centering \caption{Summary of performance of BayeSN-TD when applied to Roman simulations of glSNe Ia from \citet{Pierel24} which incorporate achromatic microlensing, along with performance when applied to LSST simulations from \citet{Arendse24} which incorporate chromatic microlensing. $f_{68}$ details the percentage of simulations where the true simulated value was within the 68 per cent credible interval of the posterior, while $f_{95}$ is the same but for the 95 per cent credible interval.} \begin{tabular}{cccccccc} Simulation & $N_\text{SN}$ & |$\Delta T_\mathrm{fit} - \Delta T_\mathrm{true}$| & |$\Delta T_\mathrm{fit} - \Delta T_\mathrm{true}$| & |$\Delta T_\mathrm{fit} - \Delta T_\mathrm{true}$| & Median & $f_{68}$ & $f_{95}$ \\ & & < 1 day & < 3 days & < 5 days & $\Delta T_\mathrm{fit} - \Delta T_\mathrm{true}$ / days & \\ \hline Roman & 1000 & 0.417 & 0.792 & 0.906 & 0.09 & 67.7\% & 93.5\% \\ LSST & 1134 & 0.386 & 0.746 & 0.884 & -0.08 & 68.2\% & 90.2\% \end{tabular} \label{sim_results} \end{table*} \subsection{LSST Simulations} \begin{figure} \centering \includegraphics[width = \linewidth]{figures/lsst_sim_results.pdf} \caption{As Fig. \ref{roman_sim_plots} but for LSST simulations of glSNe Ia presented by \citet{Arendse24} which incorporate a chromatic effect of microlensing.} \label{lsst_sim_plots} \end{figure} We next explore realistic simulations of glSNe observed by LSST as presented in \cite{Arendse24}, using the `lensed Supernova Simulation Tool` (\textsc{lensedSST}). This pipeline uses the SALT3 model \citep{Kenworthy21} for the SEDs of glSNe Ia, simulated using \textsc{sncosmo} \citep{Barbary25}. Unlike the Roman simulations explored in Section \ref{roman}, these simulations do incorporate a chromatic treatment of microlensing. This provides an ideal opportunity for us to test whether our achromatic treatment of microlensing enables us to obtain robust time delay constraints of glSNe Ia which are impacted by chromatic microlensing. Microlensing is accounted for in the simulations using SN Ia explosion models from \textsc{ARTIS} \citep{Kromer09} combined with microlensing maps from \textsc{Gerlumph} \citep{Vernardos14a, Vernardos14b, Vernardos15}. There are a number of simulated glSNe for which resolved\footnote{Cases where each image has individually-resolved photometry as opposed to blended photometry of both images combined.} photometry was not available for both images. These simulations aimed to provide a general, realistic data set for analysis of lensed supernovae including those without resolved photometry. However, in our case BayeSN-TD is aimed for application to SNe with resolved photometry of each image such as SN H0pe. For this analysis, we select only simulated SNe with at least 10 data points for each image, across all bands. This data quality cut requires only a small number of data points per photometric band. Out of 5000 total simulated glSNe Ia available, we apply BayeSN-TD to 1134 objects. The results of time-delay recovery with BayeSN-TD for these LSST simulations are summarised in Table \ref{sim_results}. Compared with the Roman simulations with achromatic microlensing, the inclusion of chromatic microlensing in the simulations makes only a small impact to model performance. Most notably, this does not lead to a bias in the inferred time delays - the median deviation from the truth across all 1134 simulated SNe that were fit was just -0.08 days, negligible compared to the 3.08 day mean posterior uncertainty across all SNe. Even with chromatic microlensing, the uncertainties remain well-calibrated with $f_{68}=68.2\%$. There is a small decrease in $f_{95}$ compared with the Roman simulations, from 93.5\% to 90.2\%, suggesting that chromatic microlensing is causing a larger fraction of outliers. This is unsurprising, and it remains reassuring that there is a negligible overall bias and $f_{95}$ remains close to 95\%. Fig. \ref{lsst_sim_plots} is similar to Fig. \ref{roman_sim_plots} but shows results for these LSST simulations rather than the Roman simulations. The bottom panel of Fig. \ref{lsst_sim_plots} further demonstrates that BayeSN-TD can provide well-calibrated uncertainties on time delays; the distribution of $(\Delta T_\text{fit} - \Delta T_\text{true}) / \sigma_{\Delta T}$ closely follows a Gaussian with just a small number of outliers in the tails. An example of a BayeSN-TD fit to one of these simulated LSST light curves is shown in Fig. \ref{LSST_example}. The left panels show the model fits along with the simulated photometry for each image, while the right panels compare the posterior distributions on microlensing curves with the impact of microlensing on the simulated data. As with the Roman simulations, the plotted data shows the deviation from the model on the simulated light curves as a result of microlensing, incorporating the effect of measurement noise. These plots show the different impacts of microlensing in each band along with the posterior distribution we obtain on the microlensing curve from our achromatic treatment. This example demonstrates that our GP treatment of microlensing is able to capture deviations around the template of typical SNe Ia, even considering the mismatch between the SALT models used for the simulations and the BayeSN model used for inference. The lower right panel shows the posterior distribution on microlensing curve closely tracing the impact of microlensing on the simulated light curve. A further example is shown in Fig. \ref{LSST_example_extreme}, where the BayeSN-TD model has been able to capture an extreme microlensing event and infer that the very brightest points of the SN light curve are driven by microlensing rather than SN luminosity. This perfectly demonstrates the ability of our GP-based microlensing treatment to realistically capture the varied impact of microlensing. Overall, our achromatic treatment of microlensing seems to roughly average over the impact of microlensing in each band. \begin{figure*} \centering \includegraphics[width = \linewidth]{figures/67.pdf} \caption{\textbf{Left panels}: Simulated 2-image LSST glSN Ia light curve from \citet{Arendse24} along with associated BayeSN-TD fits. \textbf{Right panels}: Plotted data points represent simulated deviation from model light curves as a result of microlensing, with associated uncertainties from measurement noise as true simulated values post-microlensing, without noise, are not available. These simulations include chromatic microlensing, therefore different filters are differently impacted. Plotted line and shaded region represent the posterior mean and standard deviation on microlensing from the achromatic treatment included in BayeSN-TD.} \label{LSST_example} \end{figure*} \begin{figure*} \centering \includegraphics[width = \linewidth]{figures/1055.pdf} \caption{As Fig. \ref{LSST_example} but for a particularly extreme example of microlensing. In this case, BayeSN-TD is able to identify that the peak of this light curve is driven by microlensing rather than the luminosity of the SN, and recover the true time delay within 1 day.} \label{LSST_example_extreme} \end{figure*} \section{Application to SN H0pe} \label{sec:h0pe} Having established that our model is able to robustly infer time delays for SNe Ia which are impacted by microlensing in Section \ref{sec:sims}, we now apply BayeSN-TD to real photometry of SN H0pe to estimate time delays and magnifications. \subsection{Light Curve Fits} We begin by fitting the observed photometry of SN H0pe with BayeSN-TD, presented in Table 2 of \citet{Pierel24}. As discussed in Section \ref{priors}, BayeSN-TD fits for time-of-maximum of each image relative to some fiducial peak phase. The prior on $t_\text{max}$ is then a uniform distribution 10 rest-frame days either side of this fiducial peak phase for each image. Considering the high redshift of SN H0pe, this means that in the observer-frame the priors on the peak MJD of each image are broad, uninformative uniform distributions such that \begin{equation*} \begin{aligned} \text{MJD}_\text{max,2a} \sim 59924 + U(-27.8, 27.8)\hspace{2pt} \\ \text{MJD}_\text{max,2b} \sim 60033 + U(-27.8, 27.8)\hspace{2pt} \\ \text{MJD}_\text{max,2c} \sim 59989 + U(-27.8, 27.8). \end{aligned} \end{equation*} Our BayeSN-TD fits to the light curve of SN H0pe are shown in Fig. \ref{h0pe_fit_plot}, and fit parameters are shown in Table \ref{h0pe_fit_params}. Fig. \ref{h0pe_corner} shows joint posteriors on our fit parameters. We infer $\theta=-1.27\pm0.29$, which corresponds to a B-band 15-day decline $\Delta m_{15,\text{B}}\approx0.91$ mag. We find that SN H0pe has a large amount of host-galaxy dust reddening, with $A_V=0.95\pm0.14$ and a relatively low $R_V=1.80\pm0.28$. A low value of $R_V$ is fairly typical for more highly-reddened SNe Ia \citep[e.g.][]{TM22, Burns14, Amanullah14}. Unsurprisingly, as shown in Fig. \ref{h0pe_corner}, there is a large degree of covariance between $R_V$ and $A_V$. \begin{table} \centering \caption{Summary of parameter estimates inferred for SN H0pe when fit with BayeSN-TD, including time delays $\Delta T_{ij}$ and flux space magnifications $\beta_i$. Values quoted as $X\pm Y$ represent posterior means and standard deviations, while values quoted as $X^{+Y}_{-Z}$ represent posterior medians and 68 per cent credible intervals.} \begin{tabular}{cc} Parameter & Value \\ \hline $\theta$ & $-1.27\pm0.29$ \\ $A_V$ & $0.95\pm0.14$ \\ $R_V$ & $1.80\pm0.28$ \\ $\Delta T_{BA}$ & $121.9^{+9.5}_{-7.5}$ days \\ $\Delta T_{BC}$ & $63.2^{+3.2}_{-3.3}$ days \\ $\beta_A$ & $2.38^{+0.72}_{-0.54}$ \\ $\beta_B$ & $5.27^{+1.25}_{-1.02}$ \\ $\beta_C$ & $3.93^{+1.00}_{-0.75}$ \\ \end{tabular} \label{h0pe_fit_params} \end{table} \begin{figure} \centering \includegraphics[width = \linewidth]{figures/sn_h0pe_fit.pdf} \caption{BayeSN-TD fits to each image of SN H0pe. Lines and hashed regions represent the posterior mean and standard deviation on the light curve fits.} \label{h0pe_fit_plot} \end{figure} \begin{figure*} \centering \includegraphics[width = \linewidth]{figures/h0pe_param_corner.pdf} \caption{Corner plot showing joint and marginal distributions on $A_V$, $R_V$, $\theta_1$ and time delays $\Delta T_{BA}$ and $\Delta T_{BC}$ when fitting SN H0pe with BayeSN-TD. Quoted values represent posterior medians and 68 per cent credible intervals.} \label{h0pe_corner} \end{figure*} \subsubsection{Microlensing of SN H0pe} Given that BayeSN-TD incorporates a GP-based treatment of microlensing, we can also examine our posteriors on microlensing magnification to see if the model predicts significant microlensing for the light curve of SN H0pe. Fig. \ref{h0pe_ml_constraints} shows the posterior mean and standard deviation for the microlensing magnification at each epoch of photometry for each image. For images A and B, this line is effectively flat, consistent with no time-varying impact of microlensing. Note that this does not rule out that the observed light curves are influenced by microlensing, simply showing that we cannot constrain its impact with the available data. Image C seems to qualitatively show a weak upward trend in magnification between the available epochs of photometry, but considering the size of this change relative to the posterior uncertainties we cannot make a strong conclusion. Unlike with the very well-sampled Roman light curves shown in Section \ref{roman}, for SN H0pe we cannot obtain good constraints on microlensing from the available data by considering deviations away from typical SN Ia templates. \begin{figure} \centering \includegraphics[width = \linewidth]{figures/h0pe_ml_constraints.pdf} \caption{Posterior mean and standard deviation of microlensing magnification, shown for each image against MJD.} \label{h0pe_ml_constraints} \end{figure} \subsubsection{Differences with Previous Results} \label{td_differences} There are some notable differences between some of the parameters inferred in this work and those from \citet{Pierel24}, most notably the time delay between images B and C, $\Delta T_{BC}$. There are a number of methodological differences between these two works, which are: \begin{enumerate} \item In this work we apply the model directly to photometry, which contains colour information. In contrast, \citet{Pierel24} applies SNTD to fit the observations specifically in colour space rather than in light curve space. \item We marginalise over an achromatic treatment of microlensing when fitting SN light curves. \citet{Pierel24} does not explicitly consider the impact of microlensing on top of the SN Ia SED model; instead, by fitting colours rather than photometry, the method employed by \citet{Pierel24} is insensitive to achromatic microlensing. \item We use a different BayeSN model, training a new model with extended phase coverage and more SNe in the training set; \citet{Pierel24} used a phase-extended version of the BayeSN model presented in \citet{W22}. \item We incorporate $\epsilon^s(t,\lambda_r)$, detailed in Section \ref{bayesn_equation}, within the model when fitting, marginalising over the distribution of residual intrinsic SN colours. \citet{Pierel24} does not marginalise over this distribution when fitting, instead considering the impact of $\epsilon^s(t,\lambda_r)$ as part of the systematic error budget. \end{enumerate} To investigate what is driving the difference in $\Delta T_{BC}$, we did explore modifying our analysis to remove some of these differences; we repeated our analysis using the same BayeSN model as in \citet{Pierel24}, disabling our GP microlensing treatment and fixing $\epsilon^s(t,\lambda_r)=0$ rather than marginalising over the distribution of residual intrinsic scatter. However, we found that these modifications did not make significant differences to our results and did not reconcile the differences between $\Delta T_{BC}$ inferred this work and \citet{Pierel24}. Overall, the exact cause of this discrepancy remains uncertain; the full reanalysis of SN H0pe to be presented in \textcolor{blue}{Agrawal et al. in prep.}, with higher accuracy photometry from new template images, will illuminate the root cause. \subsection{Magnification} As mentioned previously, one of the key advantages of using strongly-lensed SNe Ia for $H_0$ inference is our ability to standardise them and probe absolute magnifications, which can break the mass sheet degeneracy and provide further information about the lens. Please note, we hereafter refer to the absolute magnification of an image $i$ as $\beta_i$. BayeSN fits for a distance modulus $\mu^s$ jointly with all other parameters in a light curve fit. For BayeSN-TD, in practice the magnification will be degenerate with distance - in this case we fit for a distance modulus $\mu^s_i$ for each image $i$, where this single parameter captures both effects. One approach to estimate the magnification would be to evaluate the distance modulus at the redshift of the SN under an assumed cosmological model, $\mu_\text{cosmo}(z_s)$, and obtain a posterior on the magnification of each image by evaluating absolute magnification $\beta_i =10^{-0.4(\mu^s_i - \mu_\text{cosmo}(z_s))}$, giving a flux space magnification for each step along the chain. However, this relies on an assumed cosmological model including an assumed value of $H_0$. \citet{Pierel24} instead estimates a cosmology-independent magnification by comparing the apparent magnitude of SN H0pe to predictions of the apparent magnitude of non-lensed SNe Ia at the redshift of SN H0pe based on fits to the Pantheon+ sample \citep{Brout22}. We follow a similar approach here to infer an absolute magnification without relying on an assumed cosmology. To do this, we fit all 39 SNe Ia with a redshift $z>1$ in the Pantheon+ \citep{Brout22} sample with the BayeSN model we present in this work to estimate $\mu^s$ for each SN $s$. For this redshift range ($1