Euclid preparation. XCVIII. Cosmology Likelihood for Observables in Euclid (CLOE). 5: Extensions beyond the standard modelling of theoretical probes and systematic effects

This paper details the extension and validation of the Euclid Cosmology Likelihood for Observables in Euclid (CLOE) pipeline to accommodate beyond-standard-model cosmologies, including magnification bias, massive neutrinos, and modified gravity, while outlining future improvements for enhanced efficiency and flexibility.

The Gist

Imagine the Euclid space telescope as a giant, ultra-precise camera sent into space to take a massive portrait of the universe. Its job is to map billions of galaxies to understand the invisible forces holding the cosmos together: Dark Matter and Dark Energy.

To make sense of these billions of data points, scientists need a sophisticated "calculator" or software pipeline. In this paper, the authors describe how they upgraded this calculator, which they call CLOE (Cosmology Likelihood for Observables in Euclid). They didn't just tweak the settings; they rewired the engine to handle more complex theories about how the universe works.

Here is a breakdown of the three major upgrades they made, explained with simple analogies:

1. The "Magnifying Glass" Effect (Magnification Bias)

The Problem:
Imagine you are counting birds in a forest. Usually, you just count what you see. But, imagine that gravity acts like a giant, invisible magnifying glass. If a massive object (like a cluster of dark matter) sits between you and the birds, it bends the light.

• The Distortion: This bending stretches the area you are looking at, making the birds look more spread out (fewer per square inch).
• The Hidden Bonus: However, because the light is magnified, some birds that were too faint to see before suddenly become visible.
• The Result: You end up with a confusing mix: the birds look more spread out, but there are also more of them than expected because you can see the faint ones.

The Upgrade:
Previously, the CLOE calculator mostly ignored this "magnifying glass" effect for the specific type of data Euclid gets from spectroscopy (which measures the speed of galaxies). The authors added a new feature to CLOE that accounts for this distortion.

• Why it matters: They found that if you ignore this effect, your final calculation of the universe's expansion speed (the Hubble constant) and how clumpy matter is (sigma-8) will be slightly wrong—off by about half a standard deviation. It's like trying to measure a room with a ruler that has stretched rubber bands on it; you need to correct for the stretch to get the true size.

2. The "Universal Translator" for Gravity Theories (The Weyl Potential)

The Problem:
The standard model of physics (General Relativity) says that gravity works in a specific way. But some scientists think gravity might work differently on cosmic scales (Modified Gravity).
To test these new theories, scientists usually use two different "languages" or calculators:

1. Solver A: Calculates how matter grows and clumps together.
2. Solver B: Calculates how light bends (lensing) around that matter.
   The problem is that these two calculators often speak different languages. To make them talk to each other, scientists had to manually translate the results, which is slow, clunky, and prone to errors. It's like trying to have a conversation between a person speaking French and a person speaking Japanese by writing everything down on a piece of paper and translating it word-by-word.

The Upgrade:
The authors built a "Universal Translator" directly into CLOE. Instead of forcing the two calculators to speak different languages, they created a new way to define the "lensing signal" that works directly with the output of the gravity solver.

• The Benefit: Now, CLOE can instantly test complex theories about how gravity might be broken or modified without needing a clumsy manual translation step. It allows them to plug in new theories of gravity and immediately see how they would look in Euclid's data.

3. The "Ghost Particles" (Massive Neutrinos)

The Problem:
Neutrinos are tiny, ghostly particles that zip through the universe at near-light speed. Even though they are tiny, they have a tiny bit of mass. Because they move so fast, they don't like to clump together like regular matter (like stars or dark matter).

• The Effect: When neutrinos zoom past, they smooth out the "clumps" of matter in the universe. This changes the pattern of how galaxies are arranged.
• The Complication: In the past, the calculator treated all matter as if it were the same "soup." But because neutrinos are so fast, they need to be treated as a separate ingredient in the recipe. If you don't separate them, you get the wrong recipe for how the universe evolved.

The Upgrade:
The authors updated CLOE to treat neutrinos as a distinct ingredient. They created a new "filter" that separates the "cold" matter (which clumps) from the "hot" neutrinos (which zoom).

• The Benefit: This allows the calculator to accurately predict how the presence of heavy neutrinos would change the map of the universe. They tested this against another famous calculator (MontePython) and confirmed that their new method produces the same accurate results, ensuring they can trust the data when Euclid starts sending back real numbers.

The Bottom Line

The authors tested these three upgrades using "fake" data (simulations) that looked exactly like what Euclid will see.

• They proved that ignoring the magnifying glass effect leads to wrong answers.
• They proved that the Universal Translator works perfectly for testing new gravity theories.
• They proved that the Neutrino Filter accurately accounts for ghost particles.

By making these changes, the CLOE pipeline is now ready to handle the most complex questions about the universe. It ensures that when Euclid finally takes its photos, the scientists will be able to read the results correctly, distinguishing between the standard model of the universe and exciting new physics that might be hiding in the data.

Technical Summary

Technical Summary: Euclid Preparation XCVIII – CLOE 5: Extensions beyond the standard modelling of theoretical probes and systematic effects

Problem Statement
The Euclid mission aims to constrain extensions beyond the standard $\Lambda$CDM cosmological model by measuring the positions and shapes of billions of galaxies. However, achieving state-of-the-art constraints requires statistical tools capable of testing alternative theories, such as modified gravity (MG) and massive neutrino models, while rigorously accounting for systematic effects. Current analysis pipelines often rely on approximations that may fail under Euclid's precision, such as neglecting magnification bias in spectroscopic clustering or requiring cumbersome, multi-code workflows to incorporate modified gravity effects (e.g., separating the calculation of the matter power spectrum from the lensing window function). Furthermore, the scale-dependent bias introduced by massive neutrinos requires specific treatment to avoid biases in parameter inference. This paper addresses the need to extend the Euclid Cosmology Likelihood for Observables in Euclid (CLOE) pipeline to accommodate these complexities in a self-consistent, validated manner.

Methodology
The authors describe three specific user cases where CLOE was extended to handle non-standard modelling:

1. Magnification Bias in Spectroscopic Clustering:
   The authors implemented the contribution of gravitational lensing magnification to the spectroscopic galaxy clustering two-point correlation function (2PCF). Using a flat-sky Limber approximation, the 2PCF multipoles were modified to include density-magnification ($\xi_{g\mu}$) and magnification-magnification ($\xi_{\mu\mu}$) cross-terms. These terms depend on the local count slope ($s_{magn}$) of the galaxy sample. The implementation was validated against the external code COFFE (COrrelation Function Full-sky Estimator) using linear perturbation theory.

2. Modified Gravity and the Weyl Potential:
   To handle modified gravity theories where the equality between the Weyl potential ($\Phi + \Psi$) and the Newtonian potential ($\Psi$) is broken, the authors proposed a new implementation strategy. Instead of modifying CLOE and the Boltzmann solver separately, they introduced a conversion factor, $\Gamma(z)$, defined as the ratio of the Weyl power spectrum to the matter power spectrum ($\Gamma^2 = P_{\Upsilon\Upsilon}/P_m$). This factor is computed within a modified Boltzmann solver (e.g., MGCAMB) and passed to CLOE. CLOE then redefines the lensing window function to depend only on geometric quantities, while the power spectrum terms are scaled by $\Gamma(z)$ (or $\Gamma^2(z)$ for shear). This allows the pipeline to use standard observables while capturing MG effects through the power spectrum scaling.

3. Massive Neutrinos and Scale-Dependent Bias:
   The authors implemented the treatment of massive neutrinos as outlined in Euclid Collaboration: Archidiacono et al. (2025) (EP$\nu$). Recognizing that neutrinos do not cluster on small scales, the galaxy bias is defined relative to the cold dark matter and baryon (cb) field rather than the total matter field. The pipeline was updated to compute the non-linear cb power spectrum ($P_{cb}$) from the total matter power spectrum ($P_m$) by subtracting the linear neutrino contributions. A new class, NonlinearNeutrinoApprox, was introduced to handle the combination of non-linear boosts (e.g., baryonic feedback) and neutrino approximations consistently.

Key Contributions and Results

• Magnification Bias Validation: The implementation of magnification bias in the spectroscopic 2PCF was validated against COFFE, showing relative differences of less than 0.2% for the density-magnification term and less than 2% for the magnification-magnification term. Forecasts on synthetic data demonstrated that neglecting this effect leads to systematic shifts in cosmological parameters, specifically underestimating $\sigma_8$ and $H_0$ by approximately $0.4\sigma$ to $0.7\sigma$ in the $\Lambda$CDM model. The model including magnification bias was strongly favored in Bayesian evidence comparisons.
• Modified Gravity Integration: The new $\Gamma(z)$-based approach was validated in the $\Lambda$CDM limit, showing that the modified and standard implementations yield identical results (differences $< 1\%$). Forecasts in the $w_0w_a$CDM model confirmed that the new flag (use_Weyl) does not alter constraints when MG effects are absent. When applied to MG models (parameterized by $\mu_0$ and $\Sigma_0$), the pipeline successfully reproduced the expected degeneracy breaking between growth ($\mu$) and lensing ($\Sigma$) effects when combining galaxy clustering with weak lensing.
• Neutrino Implementation: The neutrino implementation was validated against the MontePython code. While small discrepancies (up to 5%) existed due to different underlying Einstein-Boltzmann solvers (CAMB vs. CLASS) and numerical precision, the relative response of the power spectrum to changes in neutrino mass matched the EP$\nu$ predictions. Forecasts for the 3$\times$2pt probe with massive neutrinos and varying $\Delta N_{eff}$ showed that the pipeline could recover fiducial values, though marginalization over $\Delta N_{eff}$ induced projection effects on $H_0$ and $\omega_b$. The inclusion of a BBN prior on $\omega_b$ significantly tightened constraints on the neutrino mass sum ($\sum m_\nu$) and $\Delta N_{eff}$.

Significance and Future Outlook
The paper concludes that CLOE has been successfully extended to test a broader range of cosmological models and incorporate critical relativistic effects. The modifications ensure that the pipeline is robust against systematic biases (such as magnification bias) and capable of handling the complex parameter spaces of extended models (MG and massive neutrinos) without requiring disjointed coding efforts.

The authors modestly note that while the current implementation covers linear scales and specific non-linear approximations, further work is required for small-scale screening mechanisms in MG theories and the consistent combination of MG with massive neutrinos. Looking forward, the paper identifies the need to integrate emulators (e.g., EuclidEmulator2, ReACT) to accelerate non-linear predictions and to adopt simulation-based inference (SBI) and Hamiltonian Monte Carlo methods to handle the computational demands of high-dimensional parameter spaces. These advancements are presented as necessary steps to fully exploit the statistical power of the upcoming Euclid Data Release 1.