{% 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: "A Bayesian Perspective on Evidence for Evolving Dark Energy"
date: 2025-11-13
categories: papers
---
Content generated by [gemini-2.5-pro](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/2025-11-13-2511.10631.txt).
Image generated by [imagen-4.0-generate-001](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/2025-11-13-2511.10631.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': 'Dily Duan Yi Ong'}). 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 [2511.10631](https://arxiv.org/abs/2511.10631) 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
---
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/2511.10631v1
guidislink: true
link: https://arxiv.org/abs/2511.10631v1
title: A Bayesian Perspective on Evidence for Evolving Dark Energy
title_detail: !!python/object/new:feedparser.util.FeedParserDict
dictitems:
type: text/plain
language: null
base: ''
value: A Bayesian Perspective on Evidence for Evolving Dark Energy
updated: '2025-11-13T18:54:40Z'
updated_parsed: !!python/object/apply:time.struct_time
- !!python/tuple
- 2025
- 11
- 13
- 18
- 54
- 40
- 3
- 317
- 0
- tm_zone: null
tm_gmtoff: null
links:
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
href: https://arxiv.org/abs/2511.10631v1
rel: alternate
type: text/html
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
href: https://arxiv.org/pdf/2511.10631v1
rel: related
type: application/pdf
title: pdf
summary: "The DESI collaboration reports a significant preference for a dynamic\
\ dark energy model ($w_0w_a$CDM) over the cosmological constant ($\u039B$CDM)\
\ when their data are combined with other frontier cosmological probes. We present\
\ a direct Bayesian model comparison using nested sampling to compute the Bayesian\
\ evidence, revealing a contrasting conclusion: for the key combination of the\
\ DESI DR2 BAO and the Planck CMB data, we find the Bayesian evidence modestly\
\ favours $\u039B$CDM (log-Bayes factor $\\ln B = -0.57{\\scriptstyle\\pm0.26}$),\
\ in contrast to the collaboration's 3.1$\u03C3$ frequentist significance in favoring\
\ $w_0w_a$CDM. Extending this analysis to also combine with the DES-Y5 supernova\
\ catalogue, our Bayesian analysis reaches a significance of $3.07{\\scriptstyle\\\
pm0.10}\\,\u03C3$ in favour of $w_0w_a$CDM. By performing a comprehensive tension\
\ analysis, employing five complementary metrics, we pinpoint the origin: a significant\
\ ($\\approx 2.95\u03C3$), low-dimensional tension between DESI DR2 and DES-Y5\
\ that is present only within the $\u039B$CDM framework. The $w_0w_a$CDM model\
\ is preferred precisely because its additional parameters act to resolve this\
\ specific dataset conflict. The convergence of our findings with independent\
\ geometric analyses suggests that the preference for dynamic dark energy is primarily\
\ driven by the resolution of inter-dataset tensions, warranting a cautious interpretation\
\ of its statistical significance."
summary_detail: !!python/object/new:feedparser.util.FeedParserDict
dictitems:
type: text/plain
language: null
base: ''
value: "The DESI collaboration reports a significant preference for a dynamic\
\ dark energy model ($w_0w_a$CDM) over the cosmological constant ($\u039B\
$CDM) when their data are combined with other frontier cosmological probes.\
\ We present a direct Bayesian model comparison using nested sampling to compute\
\ the Bayesian evidence, revealing a contrasting conclusion: for the key combination\
\ of the DESI DR2 BAO and the Planck CMB data, we find the Bayesian evidence\
\ modestly favours $\u039B$CDM (log-Bayes factor $\\ln B = -0.57{\\scriptstyle\\\
pm0.26}$), in contrast to the collaboration's 3.1$\u03C3$ frequentist significance\
\ in favoring $w_0w_a$CDM. Extending this analysis to also combine with the\
\ DES-Y5 supernova catalogue, our Bayesian analysis reaches a significance\
\ of $3.07{\\scriptstyle\\pm0.10}\\,\u03C3$ in favour of $w_0w_a$CDM. By performing\
\ a comprehensive tension analysis, employing five complementary metrics,\
\ we pinpoint the origin: a significant ($\\approx 2.95\u03C3$), low-dimensional\
\ tension between DESI DR2 and DES-Y5 that is present only within the $\u039B\
$CDM framework. The $w_0w_a$CDM model is preferred precisely because its additional\
\ parameters act to resolve this specific dataset conflict. The convergence\
\ of our findings with independent geometric analyses suggests that the preference\
\ for dynamic dark energy is primarily driven by the resolution of inter-dataset\
\ tensions, warranting a cautious interpretation of its statistical significance."
tags:
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
term: astro-ph.CO
scheme: http://arxiv.org/schemas/atom
label: null
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
term: astro-ph.IM
scheme: http://arxiv.org/schemas/atom
label: null
published: '2025-11-13T18:54:40Z'
published_parsed: !!python/object/apply:time.struct_time
- !!python/tuple
- 2025
- 11
- 13
- 18
- 54
- 40
- 3
- 317
- 0
- tm_zone: null
tm_gmtoff: null
arxiv_comment: 5 pages, 1 figure, 1 table
arxiv_primary_category:
term: astro-ph.CO
authors:
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Dily Duan Yi Ong
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: David Yallup
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Will Handley
author_detail: !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Will Handley
author: Will Handley
```
3. **Paper Source (TeX):**
```tex
\documentclass[%
aps,
prl,
twocolumn,
superscriptaddress,
amsmath,amssymb,
nofootinbib,
natbib,
]{revtex4-2}
\usepackage[hidelinks]{hyperref}
\usepackage{graphicx}
\usepackage{dcolumn}
\usepackage{bm}
\usepackage{booktabs}
\usepackage[english]{babel}
\begin{document}
\title{A Bayesian Perspective on Evidence for Evolving Dark Energy}
\author{Dily Duan Yi Ong}
\email{dlo26@cam.ac.uk}
\affiliation{Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.}
\affiliation{Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge, CB3 0HE, U.K.}
\affiliation{Newnham College, Sidgwick Avenue, Cambridge, CB3 9DF, U.K.}
\author{David Yallup}
\affiliation{Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.}
\affiliation{Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.}
\author{Will Handley}
\affiliation{Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.}
\affiliation{Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.}
\date{\today}
\begin{abstract}
The DESI collaboration reports a significant preference for a dynamic dark energy model ($w_0w_a$CDM) over the cosmological constant ($\Lambda$CDM) when their data are combined with other frontier cosmological probes. We present a direct Bayesian model comparison using nested sampling to compute the Bayesian evidence, revealing a contrasting conclusion: for the key combination of the DESI DR2 BAO and the Planck CMB data, we find the Bayesian evidence modestly favours $\Lambda$CDM (log-Bayes factor $\ln B = -0.57{\scriptstyle\pm0.26}$), in contrast to the collaboration's 3.1$\sigma$ frequentist significance in favoring $w_0w_a$CDM. Extending this analysis to also combine with the DES-Y5 supernova catalogue, our Bayesian analysis reaches a significance of $3.07{\scriptstyle\pm0.10}\,\sigma$ in favour of $w_0w_a$CDM. By performing a comprehensive tension analysis, employing five complementary metrics, we pinpoint the origin: a significant ($2.95{\scriptstyle\pm 0.04}\,\sigma$), low-dimensional tension between DESI DR2 and DES-Y5 that is present only within the $\Lambda$CDM framework. The $w_0w_a$CDM model is preferred precisely because its additional parameters act to resolve this specific dataset conflict. The convergence of our findings with independent geometric analyses suggests that the preference for dynamic dark energy is primarily driven by the resolution of inter-dataset tensions, warranting a cautious interpretation of its statistical significance.
\end{abstract}
\maketitle
\section{\label{sec:introduction}Introduction}
The DESI DR2 data release~\cite{desi2025} reports up to 4.2$\sigma$ preference for dynamical dark energy ($w_0w_a$CDM) over $\Lambda$CDM based on a frequentist hypothesis test derived from a likelihood ratio based test statistic. This strong evidence comes from analysing the combination of DESI DR2 BAO data, the Planck 2018 CMB measurements~\citep{Planck2018params} and the DES-Y5 supernova catalogue~\citep{descollaboration2025darkenergysurveycosmology}. This result has generated a great deal of interest in Cosmology, signalling a potential deviation from the standard cosmological model~\citep{CosmoVerseNetwork:2025alb}. Recent independent analyses have questioned these claims from a variety of angles, with particular focus being paid to the robustness of the DES-Y5 supernova catalogue~\citep{efstathiou_evolving_2025,vincenzi2025comparingdessn5yrpantheonsn} or by examining consistency between DESI data releases~\cite{Efstathiou2025BAO}. Despite persistent questions about combining late and early time probes, the community consensus holds that early-time probes alone—combining Planck CMB with DESI BAO data—show a 3$\sigma$ frequentist significance favouring evolving dark energy over $\Lambda$CDM. By established heuristics for interpreting frequentist results, this exceeds the threshold for evidence of new physics~\cite{Lyons:2013yja}.
Bayesian model comparison, well-established in cosmology, offers a complementary approach by calculating the Bayesian evidence—which naturally penalises model complexity through prior volume integration~\cite{2008ConPh..49...71T} (Bayesian Occam's razor). Using both frequentist and Bayesian perspectives has become increasingly common~\citep{Herold:2025hkb}, and for claims as significant as evolving dark energy, employing these established frameworks~\citep{Adame_2025} is essential.
In this letter we present a Bayesian analysis of DESI DR2 using nested sampling with \texttt{PolyChord}~\cite{Handley2015PolychordI,Handley2015PolychordII}. Full details are in our upcoming companion paper~\cite{UnimpededPaper}. Here we focus on comparing $\Lambda$CDM and $w_0w_a$CDM for the dataset combinations in DESI's Table VI~\cite{desi2025}. We find that when running this well established Bayesian Model Comparison pipeline, \emph{Bayesian} evidence for evolving dark energy is significantly reduced, with the combination of CMB and BAO data favouring the $\Lambda$CDM model. We conclude with some discussion on the origin of the different conclusions reached by the two methodologies.
\section{\label{sec:methods}Methods}
\begin{table*}
\centering
\begin{tabular}{@{}l@{\hspace{12pt}}r@{\hspace{8pt}}r@{\hspace{16pt}}r@{\hspace{8pt}}r@{}}
\toprule
& \multicolumn{2}{c@{\hspace{16pt}}}{This Work (Bayesian)} & \multicolumn{2}{c}{DESI Collab. (Frequentist)} \\
\cmidrule(lr){2-3} \cmidrule(l){4-5}
Dataset & $\ln B$ & Significance & $\Delta\chi^2_{\mathrm{MAP}}$ & Significance \\
\midrule
\multicolumn{5}{l}{\textit{Individual Datasets}} \\
DESI DR2 & $-1.47{\scriptstyle\pm 0.11}$ & n/a & $-4.7$ & 1.7$\sigma$ \\
DESI DR1 & $-1.64{\scriptstyle\pm 0.10}$ & n/a & --- & --- \\
\midrule
\multicolumn{5}{l}{\textit{Pairwise Combinations}} \\
DESI DR2 + CMB (no lensing) & $-0.38{\scriptstyle\pm 0.25}$ & n/a & $-9.7$ & 2.7$\sigma$ \\
DESI DR1 + CMB (no lensing) & $-0.50{\scriptstyle\pm 0.25}$ & n/a & --- & --- \\
DESI DR2 + CMB & $-0.57{\scriptstyle\pm 0.26}$ & n/a & $-12.5$ & 3.1$\sigma$ \\
DESI DR1 + CMB & $-0.38{\scriptstyle\pm 0.26}$ & n/a & --- & --- \\
DESI DR2 + Pantheon+ & $-2.77{\scriptstyle\pm 0.12}$ & n/a & $-4.9$ & 1.7$\sigma$ \\
DESI DR1 + Pantheon+ & $-2.98{\scriptstyle\pm 0.11}$ & n/a & --- & --- \\
DESI DR2 + Union3 & $+0.25{\scriptstyle\pm 0.12}$ & $1.39{\scriptstyle\pm 0.31}\,\sigma$ & $-10.1$ & 2.7$\sigma$ \\
DESI DR1 + Union3 & $+0.42{\scriptstyle\pm 0.11}$ & $1.59{\scriptstyle\pm 0.10}\,\sigma$ & --- & --- \\
DESI DR2 + DES-Y5 & $+1.56{\scriptstyle\pm 0.12}$ & $2.33{\scriptstyle\pm 0.06}\,\sigma$ & $-13.6$ & 3.3$\sigma$ \\
DESI DR1 + DES-Y5 & $+0.84{\scriptstyle\pm 0.11}$ & $1.92{\scriptstyle\pm 0.07}\,\sigma$ & --- & --- \\
\midrule
\multicolumn{5}{l}{\textit{Triplet Combinations}} \\
DESI DR2 + CMB + Pantheon+ & $-1.70{\scriptstyle\pm 0.26}$ & n/a & $-10.7$ & 2.8$\sigma$ \\
DESI DR2 + CMB + Union3 & $+1.37{\scriptstyle\pm 0.27}$ & $2.23{\scriptstyle\pm 0.15}\,\sigma$ & $-17.4$ & 3.8$\sigma$ \\
DESI DR2 + CMB + DES-Y5 & $+3.32{\scriptstyle\pm 0.27}$ & $3.07{\scriptstyle\pm 0.10}\,\sigma$ & $-21.0$ & 4.2$\sigma$ \\
\bottomrule
\end{tabular}
\caption{Comparison of Bayesian and frequentist model comparison for $w_0w_a$CDM vs $\Lambda$CDM. DESI results from Table VI of Ref.~\cite{desi2025}. Negative $\ln B$ favours $\Lambda$CDM; negative $\Delta\chi^2_{\mathrm{MAP}}$ favours $w_0w_a$CDM. Bayesian significances are only computed when $\ln B > 0$ (favouring $w_0w_a$CDM); n/a indicates cases where the Bayes factor favours $\Lambda$CDM.}
\label{tab:comparison}
\end{table*}
We analyse DESI DR2 BAO data~\cite{desi2025,BAOData} combined with Planck 2018 CMB (CamSpec likelihood~\cite{CamSpec2021}) and Type Ia supernovae (Pantheon+~\cite{Scolnic2018}, Union3~\citep{rubin2025unionunitycosmology2000}, DES-Y5~\citep{descollaboration2025darkenergysurveycosmology}). We use the \texttt{PolyChord}~\cite{Handley2015PolychordI,Handley2015PolychordII} nested sampling algorithm via \texttt{Cobaya}~\cite{Torrado2021Cobaya,cobayaascl} and CAMB~\cite{Lewis:1999bs}. For $w_0w_a$CDM, we adopt DESI's priors ($w_0 \in [-3, 1]$, $w_a \in [-3, 2]$ with $w_0 + w_a < 0$), except setting a 0.06~eV lower bound on neutrino mass from oscillation experiments, versus DESI's 0~eV lower limit.
We compute Bayesian evidence $\mathcal{Z} = P(D|\mathcal{M})$ and log Bayes factor $\ln B = \ln \mathcal{Z}_{w_0w_a\mathrm{CDM}} - \ln \mathcal{Z}_{\Lambda\mathrm{CDM}}$. Following Trotta~\cite{2008ConPh..49...71T}, we convert Bayes factors to Gaussian significance via (i) Bayes factor to $p$-value using the following relation between an upper bound on the Bayes factor $\bar{B}$ and the $p$-value~\citep{sellke_calibration_2001},
\begin{equation}
B \leq \bar{B} = -\frac{1}{e p \ln p} \quad \text{for } p \le e^{-1},
\end{equation}
and (ii) $p$-value to significance via
\begin{equation}
\sigma = \Phi^{-1}(1-p/2),
\end{equation}
where $\Phi^{-1}$ is the inverse normal cumulative distribution function. Following this inversion we can convert our derived Bayes factors into \emph{upper bounds} on significances for comparison with the frequentist results. Converting from Bayes factors to significances allows direct comparison between frequentist and Bayesian hypothesis testing frameworks, however this should be done with considerable caution~\citep{berger_testing_1987,sellke_calibration_2001, kipping_exoplaneteers_2025}. We have chosen a common and conservative approach to this conversion that ensures we quote a reasonable upper bound on the significance it should be possible to obtain. In a Bayesian setting the best practices for interpreting a Bayes factor are to quote the \emph{betting-odds}, and leave it up to the reader to decide if they would \emph{take the bet}. Despite the subtleties of this conversion and some genuine philosophical differences between the two frameworks, one would broadly expect the result of both hypothesis tests to agree on the preferred model.
\section{\label{sec:results}Results}
Table~\ref{tab:comparison} compares the DESI collaboration's frequentist tests with our Bayesian model comparison for $w_0w_a$CDM versus $\Lambda$CDM. From a Bayesian perspective, the DESI DR2 data alone penalises the complexity of the $w_0w_a$CDM model, favouring the simpler $\Lambda$CDM with a log-Bayes factor of $\ln B = -1.47{\scriptstyle\pm 0.11}$. This preference for $\Lambda$CDM remains when CMB data are added. The combination of DESI DR2 + CMB, which the DESI collaboration highlights as a key result, yields $\ln B = -0.57{\scriptstyle\pm 0.26}$ in our analysis. This stands in contrast to the frequentist finding of a 3.1$\sigma$ preference for $w_0w_a$CDM, setting the stage for a methodological comparison. A similar pattern is observed for DESI DR1 data, which also favours $\Lambda$CDM both individually and in combination with the CMB.
A direct comparison between DR1 and DR2 reveals that while the overall Bayesian landscape remains similar, the improved precision of DR2 acts to sharpen existing trends. Individually, both datasets show a consistent, moderate preference for $\Lambda$CDM ($\ln B_{\mathrm{DR2}} = -1.47{\scriptstyle\pm 0.11}$ vs. $\ln B_{\mathrm{DR1}} = -1.64{\scriptstyle\pm 0.10}$). This stability extends to combinations with CMB and Pantheon+ data, where the evidence favouring $\Lambda$CDM changes only marginally between the two releases. The most significant evolution is in combinations with supernova catalogues that prefer dynamical dark energy. In particular, the evidence from the DESI + DES-Y5 combination is amplified in the new data, with the log-Bayes factor in favour of $w_0w_a$CDM strengthening from $\ln B = +0.84{\scriptstyle\pm 0.11}$ for DR1 to $+1.56{\scriptstyle\pm 0.12}$ for DR2. This suggests that the primary impact of the transition from DR1 to DR2 is not a fundamental shift in the BAO data's model preference, but rather an enhancement of the tensions and synergies observed when combined with external datasets, particularly DES-Y5.
The model preference depends critically on the choice of supernova catalogue. Pairwise combinations with Pantheon+ data strengthen the Bayesian evidence for $\Lambda$CDM ($\ln B = -2.77{\scriptstyle\pm 0.12}$). Conversely, the DES-Y5 catalogue moderately favours $w_0w_a$CDM (pairwise $\ln B = +1.56{\scriptstyle\pm 0.12}$, or $2.33{\scriptstyle\pm 0.06}\,\sigma$). The synergy between probes becomes evident in the triplet combinations: the DESI DR2 + CMB + DES-Y5 dataset yields strong evidence for dynamical dark energy ($\ln B = +3.32{\scriptstyle\pm 0.27}$, or $3.07{\scriptstyle\pm 0.10}\,\sigma$), reversing the conclusion from DESI+CMB alone. The strong preference for $w_0w_a$CDM emerges exclusively with DES-Y5, aligning with ongoing debates about systematic differences between these datasets~\cite{desi2025}. Crucially, even where both methodologies agree on the preferred model (DES-Y5 combination), the Bayesian significance ($3.07{\scriptstyle\pm 0.10}\,\sigma$) remains substantially weaker than the frequentist claim (4.2$\sigma$). Figure~\ref{fig:posterior_comparison} shows the full cosmological parameter space constraints for these three triplet combinations in $w_0w_a$CDM, illustrating how the choice of supernova catalogue affects not only the dark energy parameters but also the broader cosmological constraints.
To quantify the origin of these differing model preferences, we analyse the statistical consistency between datasets using the suite of metrics provided by the unimpeded evidence framework~\cite{UnimpededPaper,UnimpededSoftware}. For the DESI DR2 + DES-Y5 pair within the $\Lambda$CDM model, a significant conflict emerges at $\sigma = 2.95 \pm 0.04\sigma$, exceeding our $2.88\sigma$ look-elsewhere threshold. This is the only pairwise combination to yield a negative evidence ratio ($\log R \approx -0.17$), indicating the datasets are mutually inconsistent. A strongly negative suspiciousness ($\log S = -3.83 \pm 0.03$) confirms a direct likelihood conflict, which the Bayesian dimensionality metric diagnoses as a highly localised, low-dimensional conflict ($d_G = 0.989 \pm 0.073$). This specific tension is substantially resolved in $w_0w_a$CDM: $\sigma$ drops to $1.56 \pm 0.03\sigma$, $\log R$ becomes strongly positive ($\approx +3.5$), and the conflict becomes higher-dimensional ($d_G = 3.54 \pm 0.14$), suggesting it is diffused across the larger constrained parameter space. In stark contrast, the DESI DR2 + Pantheon+ pair exhibits only mild tension across all metrics in $\Lambda$CDM ($\sigma = 1.65 \pm 0.03\sigma$, $\log R > 0$) that is not significantly alleviated in the extended model.
This pattern is the same when CMB data are included. The triplet combination of DESI DR2 + CMB + DES-Y5 sustains a significant tension of $\sigma = 3.00 \pm 0.08\sigma$ in $\Lambda$CDM. The nature of the conflict evolves from a low- to a high-dimensional disagreement ($d_G$ increases from $\approx 1$ to $>3$), and the likelihood-level conflict becomes exceptionally severe ($\log S = -5.56 \pm 0.09$). While the overall evidence ratio remains positive due to the CMB's constraining power, the suite of metrics points to a profound internal inconsistency. Yet, even this more complex tension is reduced in $w_0w_a$CDM, with $\sigma$ dropping to $1.50 \pm 0.05\sigma$ as the conflict is diffused across a larger parameter subspace ($d_G = 5.98 \pm 0.45$). This demonstrates that the strong Bayesian preference for a dynamic dark energy model is driven by its unique capacity to resolve a specific and significant tension introduced by the DES-Y5 dataset, a tension which persists and becomes more systemic in $\Lambda$CDM when combined with other cosmological probes.
\begin{figure*}
\centering
\includegraphics[width=0.48\textwidth]{posterior_comparison_5_datasets.pdf}
\hfill
\includegraphics[width=0.48\textwidth]{posterior_comparison_3_datasets.pdf}
\caption{Posterior comparisons in $w_0w_a$CDM showing the full cosmological parameter space. \textit{Left:} DESI DR2 alone (black dashed) and pairwise combinations with CMB (purple), Pantheon+ (blue), Union3 (orange), and DES-Y5 (green). \textit{Right:} Triplet combinations with DESI BAO + CMB combined with Pantheon+ (blue), Union3 (orange), and DES-Y5 (green). The differing constraints on $w_0$ and $w_a$ reflect the varying levels of tension between DESI BAO and each additional dataset. Figures produced with \texttt{anesthetic}~\cite{anesthetic}.}
\label{fig:posterior_comparison}
\end{figure*}
Our finding that DESI DR2 in combination with CMB is consistent with $\Lambda$CDM finds independent support from the geometric analysis of Efstathiou~\cite{Efstathiou2025BAO}. Reports of a large discrepancy between Bayesian and frequentist model comparison warrant careful scrutiny. For this class of problems, mature Bayesian workflows are expected to yield results that closely parallel frequentist constructions; in particular, the underlying calculations should be closely related~\citep{bayer_look-elsewhere_2020}. Moreover, frequentist hypothesis testing, when properly calibrated, has underpinned landmark discoveries, including the Higgs boson discovery~\cite{collaboration_observation_2012}. Outside of well-known pathological regimes~\citep{robert_jeffreys-lindley_2014}, broad claims that either inferential paradigm is fundamentally deficient are difficult to justify.
Standard practice in particle physics is therefore to validate the asymptotic approximations used to translate test statistics into significances~\citep{cowan_asymptotic_2011} by means of extensive Monte Carlo pseudo-experiments, despite the considerable computational cost. This validation strategy is conceptually similar to the nested-sampling Monte Carlo machinery employed here~\citep{Fowlie:2021gmr}. In light of the persistent magnitude of the discrepancy across all combinations that include DESI DR2, and given the relatively small effective size of that data set (a 13-component compressed data vector), it is reasonable to question not the frequentist framework \textit{per se}, but the applicability of asymptotic formulae in this instance. A systematic investigation of this possibility is left to future work.
\section{\label{sec:conclusions}Conclusions}
Our Bayesian analysis of DESI BAO reveals a different picture from the frequentist claims of strong evidence for dynamic dark energy. For the key DESI BAO + CMB combination, we find the evidence modestly favours $\Lambda$CDM, in contrast to the reported 3.1$\sigma$ preference for $w_0w_a$CDM. We confirm that the preference for $w_0w_a$CDM is not a feature of the BAO data itself, but emerges exclusively when combined with the DES-Y5 supernova catalogue. Our tension analysis, using five complementary metrics, pinpoints the origin of this preference: a significant ($2.95{\scriptstyle\pm 0.04}\,\sigma$), low-dimensional tension between DESI BAO and DES-Y5 within the $\Lambda$CDM framework. The $w_0w_a$CDM model is preferred with this dataset combination precisely because its additional degrees of freedom are effective at resolving this specific conflict, a tension not present with other supernova catalogues like Pantheon+.
% This work defines, using well established tools in cosmology, a principled point of reference for the \emph{Bayesian} evidence for $w_0w_a$CDM using a variety of probes. In doing so, we highlight a strong tension between reported Bayesian and frequentist hypothesis test results, of a magnitude that is not reconcilable with appeals to philosophical differences in the approaches. We speculate that the reliance on asymptotic formulae in the DESI collaboration's frequentist analysis may be a contributing factor, given the comparatively low dimensionality of the DESI DR2 dataset. Both methodologies should penalise the additional complexity of the $w_0w_a$CDM model, but they do so in fundamentally different ways. Crucially, our conclusion that the DESI DR2 + CMB data are fully consistent with $\Lambda$CDM is independently corroborated by the geometric analysis of Efstathiou~\cite{Efstathiou2025BAO}, whose approach provides an alternative, principled framework for assessing model consistency. The convergence of Bayesian model comparison, our detailed tension diagnostics, and independent geometric analyses provides a robust statistical foundation for the conclusion that claims of evolving dark energy from DESI DR2 are overstated and driven by tensions with a single external dataset. All analysis products are available via the \texttt{unimpeded} framework~\cite{UnimpededPaper,UnimpededSoftware}.
Using established cosmological data analysis methodology, we provide a principled benchmark for the \emph{Bayesian} evidence of $w_0w_a$CDM across multiple probes. This exercise reveals a discrepancy between our Bayesian results and the published frequentist test results, of a magnitude that cannot reasonably be ascribed to philosophical distinctions between the paradigms. We suggest that reliance on asymptotic formulae alone may be a contributing factor, given the modest effective dimensionality of the DESI BAO data set. Although both approaches penalize the additional flexibility of $w_0w_a$CDM, they do so via distinct mechanisms. Importantly, our finding that DESI BAO + CMB data are statistically consistent with $\Lambda$CDM is independently corroborated by the geometric analysis of Efstathiou~\cite{Efstathiou2025BAO}, which offers an alternative, principled framework for assessing model consistency. The convergence of Bayesian model comparison, targeted tension diagnostics, and independent geometric analyses provides a robust statistical basis for the conclusion that claims of evidence for evolving dark energy are overstated and largely driven by tension with a single external data set. All analysis products are available via the \texttt{unimpeded} framework~\cite{UnimpededPaper,UnimpededSoftware}.
\section{Acknowledgments}
The authors were supported by the research environment and infrastructure of the Handley Lab at the University of Cambridge.
The computations were conducted on DiRAC at the Cambridge Service for Data Driven Discovery (CSD3), operated by the University of Cambridge Research Computing on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). DiRAC is funded by BEIS via STFC capital grants ST/P002307/1 and ST/R002452/1, and operations grant ST/R00689X/1. W.H. is supported by a Royal Society University Research Fellowship.
\bibliography{biblio}
\end{document}
```
4. **Bibliographic Information:**
```bbl
```
5. **Author Information:**
- Lead Author: {'name': 'Dily Duan Yi Ong'}
- Full Authors List:
```yaml
Dily Ong:
phd:
start: 2023-10-01
supervisors:
- Will Handley
thesis: null
original_image: images/originals/dily_ong.jpg
image: /assets/group/images/dily_ong.jpg
David Yallup:
postdoc:
start: 2021-01-10
thesis: null
original_image: images/originals/david_yallup.jpg
image: /assets/group/images/david_yallup.jpg
links:
ORCiD: https://orcid.org/0000-0003-4716-5817
linkedin: https://www.linkedin.com/in/dyallup/
Will Handley:
pi:
start: 2020-10-01
thesis: null
postdoc:
start: 2016-10-01
end: 2020-10-01
thesis: null
phd:
start: 2012-10-01
end: 2016-09-30
supervisors:
- Anthony Lasenby
- Mike Hobson
thesis: 'Kinetic initial conditions for inflation: theory, observation and methods'
original_image: images/originals/will_handley.jpeg
image: /assets/group/images/will_handley.jpg
links:
Webpage: https://willhandley.co.uk
```
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 [2511.10631](https://arxiv.org/abs/2511.10631) is featured in the first sentence.
Generate only the final Markdown output that meets all these requirements.
{% endraw %}