Shedding light on Dark Energy with Weak Lensing and Hybrid Statistics

T. Lucas Mäkinen Imperial College London | DAMTP, University of Cambridge

New Frontiers in Cosmology · A Coruña
14 August 2026
arXiv:2606.11309 · Williamson & Mäkinen et al.

The team

with important contributions from Marco Gatti, Lorne Whiteway, Ben Wandelt, Judit Prat, Ofer Lahav and the DES Collaboration

information extraction

Science Goal

Cosmic shear distorts distant galaxies by ~1%. That distortion is sensitive to the growth rate of structure and to the distance–redshift relation — so it constrains

But the two-point function is not the whole story:

the fields are evolved into a non-Gaussian state — the extra information lives in higher orders
systematics and selection effects are complex
the likelihood for any statistic is hard or impossible to write down

The goal: extract more information than the power spectrum — and be able to say how much more.

Weak lensing light cone — Credit: CEA-ASp

the obstacle

Simulation-based inference

Keep only those simulations that look like the real Universe — none do exactly.

the data space is enormous

and the simulations are expensive

→ we need extreme data compression.
Which few numbers should we keep?

Finding an Objective · compression = intelligence

CLAUDE SHANNON

“Information theory and the study of communication…”

JOHN VON NEUMANN

“…statistical and quantum mechanics.”

LATENT

A SHARED SCIENCE OF ENTROPY AND RANDOMNESS

“Compression is intelligence”

A model that predicts well is a model that compresses well.

…and in cosmology

UNIVERSE 1

UNIVERSE 2

existing compression
of the data

θ⁽¹⁾

θ⁽²⁾

We need encodings of the data that are informative about theory parameters and well-understood observables.

an optimization problem

Mutual information

ρ = 0.00

0.000NATS

ρ = 0. The joint factorises, p(x,y) = p(x)p(y). Knowing Y tells you nothing about X.

ρ = 0.5. The conditional narrows — Y carries some information about X.

ρ = 0.9. The conditional is far tighter than the marginal. Y pins X down.

ρ → 1. All the mass collapses onto a line. I → ∞: X is determined by Y.

MI is a highly nonlinear function of the distribution (cf. correlation) — and hard to estimate from samples.

↑ ↓ TO SCRUB ρ

(Conditional) Mutual Information

objective(cosmology)

unknown informationoptimize this!

known information(large scales, Gaussian)

Information extraction becomes an optimization problem: find the compression F that maximises what we learn about θ, given what the power spectrum already tells us.

hierarchical hybrid statistics

100 million galaxies → 7 numbers

Power spectrum C_ℓ — information from all scales, compressed to 10 summaries
CNN on patch A — the small-scale, non-Gaussian information the survey is too big to ingest at once
Add patches B and C and repeat, maximising the conditional MI at every step

The chain rule lets us add information sequentially, never double-counting what an earlier summary already captured.

10power-spectrum summaries

+(4 CNN summaries) × 3 patches

7final data points

hierarchical hybrid compression schematic

Williamson & Mäkinen et al. (2026), Fig. 1

data · simulations · validation

Validation & Inference Calibration

DES Year 3

5,000 deg², 4 tomographic bins, 100 million galaxies. Kaiser–Squires convergence maps.

JEFFREY ET AL. 2021

Gower Street simulations

wCDM, 1080³ particles, 12,600 mock surveys — with intrinsic alignments, shear bias, source clustering and n(z) errors forward-modelled.

Maps subject to DES ℓ < 1024 scale cut.

JEFFREY & GATTI ET AL. 2021 · 2025

Coverage

Credible regions are unbiased — the posteriors mean what they say.

Baryons

Re-analysed with baryonified CosmoGridV1: posteriors shift by < 0.3σ.

results · DES Y3, wCDM

Results

S₈= ± 0.017

Ω_m= ± 0.024

w<

68% MARGINAL CREDIBLE INTERVALS

Precise agreement with Planck in both S₈ and Ω_m, and fully consistent with ΛCDM.

Cosmic shear ξ± (two-point)

C_ℓ×CNN (Jeffrey et al.)

Planck

Hybrid statistics — this work

summary

Compression, done properly, is worth a survey upgrade

Most precise cosmology from cosmic shear alone, in precise agreement with Planck
100 million galaxies compressed to 7 data points by maximising conditional mutual information
Not a black box — SBI does not care how the summaries were obtained. The result is a principled Bayesian one
Hybrid by design: physics where physics is strong, neural networks where it is not
Scalable to Euclid and LSST, and applicable to other probes

arXiv:2606.11309 · Williamson & Mäkinen et al.

Alan Heavens · Natalia Porqueres · Josh Williamson · Niall Jeffrey — with Marco Gatti, Lorne Whiteway, Ben Wandelt, Judit Prat, Ofer Lahav and the DES Collaboration

S₈= 0.808 ± 0.017

Ω_m= 0.325 ± 0.024

w< −0.766

Thank you

arXiv:2606.11309 · Williamson & Mäkinen et al.

supplementary · comparison with the state of the art

The most precise weak-lensing cosmology to date

FIGURE OF MERIT (Ω_m, S₈) — HIGHER IS BETTER

DES Y3 two-point ξ±SECCO ET AL.

Betti + 2nd momentsPRAT ET AL.

2nd+3rd mom. + ST + WPHGATTI ET AL.

C_ℓ × CNN map-levelJEFFREY ET AL.

Hybrid statisticsTHIS WORK

1,000 2,000 3,000

×2.9 over the two-point analysis of the same data, and +39% over the previous map-level compression.

+60% in the full (Ω_m, S₈, w) combination — over the previous state of the art.

The most precise joint constraints on (Ω_m, S₈, w) from weak gravitational lensing alone, of any survey to date.

supplementary

Same data, same simulations — different summaries

posteriors from four DES Y3 higher-order-statistics analyses

All four analyses use the same DES Y3 data, the same Gower Street simulations, the same forward model and the same analysis priors. The spread between these contours is therefore only the effect of changing the summary statistic.

FOM(Ω_m, S₈)

2nd + 3rd moments, scattering transform, WPH GATTI ET AL. 2024 1,725

Betti numbers + 2nd moments PRAT ET AL. 2025 1,542

C_ℓ × CNN, field level (MSE) JEFFREY ET AL. 2024 1,885

Hybrid statistics THIS WORK 2,614