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Unbiased analysis of primordial non-Gaussianity: the multipoles of the full relativistic power spectrum

This paper presents a consistent relativistic framework for analyzing the galaxy power spectrum multipoles, demonstrating that neglecting general relativistic effects causes severe biases in measuring primordial non-Gaussianity ( $f_{\rm NL}$ ) but that these can be mitigated and constraints improved by 15–20% through a bright-faint multi-tracer analysis, all implemented in the new code \textsc{CosmoWAP}.

Original authors: Chris Addis, Sêcloka L. Guedezounme, Jessie Hammond, Chris Clarkson, Federico Montano, Stefano Camera, Sheean Jolicoeur, Roy Maartens

Published 2026-06-23

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CC BY 4.0

Original authors: Chris Addis, Sêcloka L. Guedezounme, Jessie Hammond, Chris Clarkson, Federico Montano, Stefano Camera, Sheean Jolicoeur, Roy Maartens

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, three-dimensional ocean of galaxies. For decades, cosmologists have been trying to understand the very first ripples in this ocean—tiny, random fluctuations created just after the Big Bang. These ripples are called Primordial Non-Gaussianity (PNG). If we can measure them perfectly, they tell us exactly what kind of physics happened in the universe's first split second.

However, measuring these ripples is like trying to hear a whisper in a hurricane. The "hurricane" is the way our view of the universe gets distorted as the light from distant galaxies travels to us.

This paper is about cleaning up the "noise" in our measurements so we can finally hear that whisper clearly. Here is the breakdown of what the authors did, using simple analogies.

1. The Problem: The Distorted Mirror

When we look at galaxies, we don't just see them where they are; we see them through a "funhouse mirror" created by the universe itself.

The Standard View: Usually, scientists use a simple map (Newtonian physics) to plot where galaxies are. It's like looking at a flat map of a city.
The Real View: But the universe is curved and expanding. As light travels to us, it gets stretched, bent by gravity (lensing), and delayed by time (time delay). It's like looking at that city through a wavy, moving glass pane.

The authors found that for the next generation of massive galaxy surveys (like Euclid and MegaMapper), this "wavy glass" is no longer a tiny annoyance. It's a massive distortion. If you ignore it, your map of the universe is wrong.

2. The Big Mistake: Ignoring the "Integrated" Effects

The paper focuses on a specific type of distortion called Integrated Effects.

The Analogy: Imagine you are taking a photo of a friend standing on a hill.
- Local effects are things happening right next to your friend (like them waving).
- Integrated effects are things happening along the entire path of the light from your friend to your camera (like the wind bending the light, or the air shimmering).

Previous studies mostly ignored the "path" effects when looking at the 3D map of galaxies. The authors say, "You can't ignore the path!" They developed a new mathematical framework to calculate exactly how the light bends and stretches along the entire journey from the galaxy to Earth.

3. The Shocking Result: A 20-Sigma Error

The authors ran simulations for future telescopes. They asked: "What happens if we ignore these path distortions?"

The Result: The answer was catastrophic.
- For the Euclid telescope, ignoring these effects would make us think the primordial signal is 3 times stronger than it really is (a 3-sigma error).
- For the MegaMapper telescope, the error would be 20 times stronger (a 20-sigma error).

In plain English: If we don't fix our math, we will confidently claim to have discovered a new law of physics, when in reality, we just forgot to account for the fact that light gets bent by gravity on its way to us. It's like blaming a loud noise on a ghost when it's actually just a car backfiring.

4. The Solution: The "Bright-Faint" Split

Even after fixing the math, there's still a problem. The amount of distortion depends on how bright the galaxies are. But we aren't 100% sure exactly how bright galaxies should be (this is called the "luminosity function"). This uncertainty creates a new kind of fog over our measurements.

The Fix: The authors propose a clever trick called a Multi-Tracer Analysis (specifically, a "Bright-Faint Split").

The Analogy: Imagine you are trying to measure the wind speed in a forest.
- If you only look at the tall oak trees, you might get confused because you don't know exactly how tall they should be.
- But, if you look at both the tall oaks and the short shrubs, and compare how they move relative to each other, the confusion disappears. The differences between the two groups cancel out the uncertainty.

By splitting the galaxy survey into a "Bright" group and a "Faint" group and analyzing them together, the authors show we can reduce the uncertainty in our measurements by 15–20%. This allows us to see the primordial signal much more clearly.

5. The Toolkit: CosmoWAP

Finally, the authors didn't just do the math on paper; they built a software tool called CosmoWAP.

Think of this as a new, high-precision GPS for cosmologists. It automatically calculates all these complex "path distortions" and "bright-faint splits" so that other scientists can use it to analyze data from future telescopes without getting lost in the math.

Summary

The Goal: Measure the earliest moments of the universe.
The Obstacle: Light gets distorted on its way to us, and for future telescopes, this distortion is huge.
The Risk: Ignoring this distortion leads to massive, false discoveries (up to 20 times the real signal).
The Fix: Use a new math framework to account for the distortion and split galaxies into "bright" and "faint" groups to cancel out uncertainties.
The Outcome: A clearer, unbiased view of the universe's birth, provided by the new CosmoWAP software.

Technical Summary: Unbiased analysis of primordial non-Gaussianity: the multipoles of the full relativistic power spectrum

Problem Statement
Next-generation spectroscopic galaxy surveys (e.g., Euclid, MegaMapper, DESI) aim to constrain local-type primordial non-Gaussianity (PNG), parameterized by $f_{\text{NL}}$ , through the scale-dependent halo bias in the galaxy power spectrum. However, as these surveys probe larger volumes and ultra-large scales, standard Newtonian treatments of redshift-space distortions (RSD) become insufficient. General relativistic projection effects—specifically integrated contributions (lensing convergence, time delay, and the Integrated Sachs-Wolfe effect) and wide-separation corrections (breaking local translation invariance)—are degenerate with the local PNG signal. Neglecting these effects can lead to biased inferences of $f_{\text{NL}}$ . Furthermore, the uncertainty in the survey luminosity function (evolution and magnification biases) introduces additional degeneracies that degrade constraints.

Methodology
The authors develop a consistent theoretical framework to compute the multipoles of the full relativistic 3D power spectrum, $P(k)$ , including all relevant light-cone effects.

Formalism: Building on the Yamamoto estimator for power spectrum multipoles, the authors define a "local" power spectrum $P_{\text{loc}}(k, \mu; d)$ for a single line-of-sight (LOS) $d$ . They explicitly model the galaxy number counts as a sum of standard Kaiser terms ( $S$ ), local relativistic projection terms ($NI$), and integrated terms ( $I$ ).
Integrated Effects: A primary focus is the derivation of the integrated cross-integrated ( $I \times I$ ) and integrated cross-local ( $I \times S$ ) contributions to the power spectrum. Unlike previous angular analyses, these are computed directly in Cartesian Fourier space. The authors perform a wide-separation expansion (in $\epsilon = x_{12}/d$ ) to handle the breaking of translation invariance caused by integrated effects, which depend on the LOS choice (parameterized by $t$ , where $t=0$ is an endpoint and $t=1/2$ is a midpoint).
Wide-Separation Corrections: The framework includes wide-angle and radial evolution corrections to the covariance matrix, a novel calculation for multi-tracer analyses.
Forecasting: The authors employ Fisher matrix and Markov Chain Monte Carlo (MCMC) analyses to forecast constraints on $f_{\text{NL}}$ for various survey configurations (DESI BGS, Euclid H $\alpha$ , Roman, MegaMapper LBG, SKAO2). They marginalize over nuisance parameters representing uncertainties in the linear bias ( $b_1$ ), evolution bias ( $b_e$ ), and magnification bias ( $Q$ ).
Multi-Tracer Approach: To mitigate degeneracies between luminosity function uncertainties and $f_{\text{NL}}$ , the authors implement a "bright-faint" split, dividing the galaxy sample into two populations based on luminosity to utilize cross-correlations and odd-parity multipoles.
Implementation: All calculations are implemented in the Python code CosmoWAP and companion Mathematica notebooks MathWAP.

Key Contributions

Derivation of $I \times I$ in 3D Power Spectrum: The paper provides the first consistent calculation of the $I \times I$ contribution (lensing-lensing, time-delay-time-delay, etc.) directly within the 3D Fourier power spectrum multipoles, extending beyond the Limber approximation.
LOS Dependence and Odd Multipoles: The authors demonstrate that integrated contributions break local translation invariance, leading to a dependence on the LOS choice ( $t$ ). Crucially, for endpoint LOS choices ( $t \neq 1/2$ ), these terms generate non-zero, imaginary, odd-parity multipoles (e.g., the dipole) in the auto-power spectrum, a novel result in the 3D context.
Bias Quantification: The study quantifies the bias introduced in $f_{\text{NL}}$ inference when relativistic and wide-separation effects are neglected.
Novel Covariance Calculation: The authors present a new analytic calculation of the full multi-tracer power spectrum multipole covariance, including wide-separation corrections and integrated effects, which are essential for accurate error estimation on large scales.
Multi-Tracer Mitigation: The work demonstrates that a bright-faint multi-tracer analysis can partially break the degeneracy between $f_{\text{NL}}$ and luminosity function uncertainties, improving constraints.

Results

Bias on $f_{\text{NL}}$ : Ignoring integrated and relativistic effects leads to significant biases in the best-fit $f_{\text{NL}}$ . For a Euclid-like H $\alpha$ survey, the bias is approximately $3\sigma$ , while for a MegaMapper-like Lyman-break galaxy (LBG) survey, it reaches $\sim 20\sigma$ . The bias is driven primarily by the large positive contribution of the $I \times I$ term, which boosts power on large scales, counteracting the negative suppression from local relativistic terms.
Non-linear Effects: The $I \times I$ contribution is sensitive to non-linear power spectrum modeling even at mildly non-linear scales ( $k \sim 0.05\, h/\text{Mpc}$ ), showing a $\sim 10\%$ boost compared to linear theory, which increases the bias on $f_{\text{NL}}$ if unaccounted for.
Luminosity Function Uncertainty: When including relativistic effects, the uncertainty in the luminosity function (marginalized over $b_e$ and $Q$ ) increases the error on $f_{\text{NL}}$ by $\approx 30\%$ for Euclid and $\approx 20\%$ for MegaMapper compared to a fixed-bias scenario.
Multi-Tracer Improvement: Utilizing a bright-faint split reduces the uncertainty in $f_{\text{NL}}$ by $15\text{--}20\%$ compared to a single-tracer analysis with the same relativistic modeling. This improvement arises because the odd-parity multipoles in the cross-correlation help constrain the evolution and magnification biases, breaking the degeneracy with $f_{\text{NL}}$ .
Detectability: The lensing contribution to the power spectrum is forecast to be detectable in Euclid, SKAO2, and MegaMapper surveys. However, the time delay and ISW contributions are unlikely to be detectable in the 3D Fourier power spectrum multipoles.

Significance
The paper establishes that for Stage-IV and future galaxy surveys, a "non-Newtonian" approach is mandatory for an unbiased analysis of primordial non-Gaussianity. The authors argue that neglecting integrated relativistic effects and wide-separation corrections leads to false detections or severe biases in $f_{\text{NL}}$ . Furthermore, they show that while these effects introduce new degeneracies with the luminosity function, these can be effectively managed using multi-tracer techniques. The work provides a complete, public computational framework (CosmoWAP) for incorporating these complex light-cone effects into standard power spectrum analyses, ensuring that future constraints on early-universe physics are robust against relativistic systematics.