⚛️ general relativity

The Odd-Parity Part of the Observed Galaxy Trispectrum

This paper calculates the leading-order relativistic corrections to the galaxy trispectrum in redshift space, demonstrating that these effects produce a parity-odd signal that must be carefully distinguished from potential intrinsic parity violation before it can be used to probe inflationary physics.

Original authors: Pritha Paul, Chris Clarkson, Roy Maartens

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Pritha Paul, Chris Clarkson, 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, invisible 3D puzzle made of galaxies. For decades, cosmologists have been checking if this puzzle has a hidden "handedness"—a secret preference for being left-handed or right-handed, much like your hands are mirror images that can't be perfectly stacked on top of each other. This property is called parity.

In the standard rules of the game, the universe is supposed to be perfectly symmetrical; if you looked at it in a mirror, it should look exactly the same. But what if it's not? If we find a "handedness" in the way galaxies are arranged, it could be a smoking gun for brand-new physics that happened right after the Big Bang.

The Tetrahedron Test

To catch this cosmic handedness, scientists can't just look at pairs of galaxies (2-point) or even triplets (3-point). Why? Because if you flip a pair or a triangle in 3D space, you can just spin them around to make them look the same again. They are too simple to reveal a secret.

You need a tetrahedron—a pyramid shape made of four galaxies. In 3D space, a tetrahedron is the simplest shape that cannot be superimposed on its mirror image just by spinning it. It's like trying to fit a left-handed glove onto a right-handed hand; no matter how you twist it, it won't fit. This paper focuses on the trispectrum, which is the mathematical way of measuring how these four-galaxy tetrahedrons are arranged.

The Cosmic "Ghost" Signal

The authors of this paper, Pritha Paul, Chris Clarkson, and Roy Maartens, asked a crucial question: If we see a "handed" signal in these tetrahedrons, is it a sign of new physics from the early universe, or is it just an illusion caused by how we look at the universe?

They found that even if the universe is perfectly symmetrical at its core, our view of it is distorted by relativity.

Think of the universe as a vast ocean. When we observe galaxies, we aren't looking at them where they are right now; we are looking at them where they were when their light left them, traveling along our "past light cone." Because galaxies are moving (peculiar velocity) and space itself is stretching (expansion), the way we measure their positions gets warped.

The paper calculates that these relativistic effects act like a cosmic ghost. They create a "parity-odd" signal—a fake handedness—that appears in the data even if the underlying universe is perfectly symmetrical. It's like looking at a straight stick in a pool of water; the stick looks bent because of the water, not because the stick is actually bent.

What They Did (and Didn't) Do

The team didn't go out and measure a new signal in the sky. Instead, they built a detailed mathematical simulation to calculate exactly how strong this "ghost" signal should be. They took the equations of General Relativity and applied them to the trispectrum, calculating the "kernels" (the mathematical building blocks) that describe how density, velocity, and gravity distort our view.

They specifically looked at two types of future galaxy surveys:

Euclid-like: A near-infrared spectroscopic survey.
SKA-like: A radio telescope survey mapping neutral hydrogen.

The Numbers: How Big is the Illusion?

The results of their simulation are quite surprising. They found that these relativistic effects are not tiny, negligible blips.

At the scales where matter and energy are roughly equal (the "equality scales"), these relativistic corrections create a signal that is 10% or more of the total expected signal.
Depending on the angle from which you look at the galaxy tetrahedrons (the "viewing angles"), this fake signal can grow even larger, reaching 20% to 80% of the total signal in certain configurations.

This means that if you ignore relativity, you might think you've discovered a violation of the universe's symmetry, when in reality, you've just been fooled by the water in the pool.

The "Cancellation" Game

The authors also discovered a fascinating "tug-of-war" happening inside the math. The total signal is made of different parts: some come from the first and second orders of calculation, others from the third.

In their simulations, they found that for some survey types (like the SKA-like radio survey), the third-order effects dominate, while for others (like the Euclid-like survey), the second-order effects take the lead. But here's the kicker: these different parts often cancel each other out.

Imagine two people pushing a car in opposite directions. If they push with equal strength, the car doesn't move. The paper shows that the "odd" signal is often the result of these massive forces nearly canceling each other out, leaving a very small, delicate net result. This makes the final signal highly sensitive to the specific shape of the galaxy arrangement and the type of survey being used.

What They Ruled Out

The paper is very clear about what is not happening. They explicitly state that the standard "Newtonian" redshift-space distortions (the usual way we account for galaxies moving toward or away from us) are parity symmetric. In other words, the standard effects we've known about for years do not create this handedness. The "ghost" signal comes strictly from the relativistic corrections (Doppler effects, gravitational redshift, and time delays) that are usually ignored because they are thought to be too small.

The Bottom Line

This paper doesn't claim to have found a violation of parity in the universe. Instead, it acts as a warning label for future experiments.

The authors suggest that as we prepare for massive new surveys like Euclid and SKA, we must account for these relativistic "ghosts." If we don't, we might mistake a mathematical artifact of our own viewing angle for a fundamental law of physics. The paper concludes that these corrections are significant enough that they cannot be neglected in any serious analysis of the trispectrum.

In short: The universe might be perfectly symmetrical, but our view of it is slightly tilted. Before we claim to find new physics, we need to straighten our glasses.

Problem
The standard cosmological model predicts that large-scale structure statistics should be invariant under parity transformations. Detecting a violation of this symmetry would indicate physics beyond the standard model, potentially offering insights into inflationary physics. While parity violation has been searched for in vector and tensor sectors (e.g., CMB B-modes), constructing a parity-sensitive statistic in the scalar sector (galaxy density field) is challenging. Lower-order statistics like the 2-point and 3-point correlation functions are insensitive to parity because their geometric configurations (points and triangles) can be superimposed on their mirror images via rotation. The 4-point correlation function (4PCF), or trispectrum, represents the lowest-order statistic capable of detecting parity violation, as the tetrahedron formed by four galaxies cannot be superimposed on its mirror image.

Recent analyses of the BOSS dataset using the 4PCF have reported varying levels of evidence for parity-odd signals, ranging from significant detections to null results after covariance recalculations. A critical issue is distinguishing intrinsic parity violation from observational contaminants. While standard Newtonian redshift-space distortions (RSD) are parity symmetric, the full relativistic description of observed galaxy number counts is not. Relativistic projection effects, including Doppler terms, gravitational redshift, and lensing, introduce asymmetries along the line of sight. These effects, suppressed by a factor of $H/k$ relative to standard RSD, generate an imaginary part in the power spectrum and bispectrum, and potentially a parity-odd component in the trispectrum. Before claiming intrinsic parity violation, these relativistic contaminants must be exhaustively characterized.

Methodology
The authors calculate the observed galaxy trispectrum in redshift space, explicitly including leading-order relativistic effects. The methodology involves:

Perturbative Expansion: The galaxy number contrast $\Delta_g$ is expanded to third order in perturbation theory. The authors derive the general relativistic expression for galaxy number counts, incorporating density fluctuations, peculiar velocities, and metric potentials.
Kernel Derivation: The authors derive the Fourier space kernels for the number counts up to third order. They separate these into Newtonian ( $K_N$ $K_{N}$ ) and General Relativistic ( $K_{GR}$ $K_{GR}$ ) components.
- The Newtonian kernels contain only even powers of $\mu = \hat{k} \cdot \hat{n}$ (where $\hat{n}$ is the line of sight), making them parity-even.
- The relativistic kernels contain odd powers of $\mu$ arising from terms like $v \cdot n$ and radial derivatives of potentials, which break parity symmetry.
Trispectrum Construction: Using Wick's theorem, the connected trispectrum is constructed from the product of first, second, and third-order kernels. The total trispectrum is decomposed into parity-even ( $T_{even}$ ) and parity-odd ( $T_{odd}$ ) parts based on the transformation $k \to -k$ .
Numerical Evaluation: The authors numerically compute the ratio $|T_{odd}/T_{even}|$ for specific survey configurations. They analyze two distinct survey types: an Euclid-like near-infrared spectroscopic survey and an SKA-like 21cm intensity mapping survey. The analysis varies scale ( $k$ ), configuration angles (defining the tetrahedron shape), and viewing angles (orientation relative to the line of sight). They also investigate the contributions of different perturbative orders ( $T_{1113}$ and $T_{1122}$ ) and the specific roles of the first, second, and third-order relativistic kernels.

Key Contributions

Derivation of Third-Order Kernels: The paper provides a detailed derivation of the third-order Newtonian and relativistic kernels in Fourier space, extending previous work on power spectra and bispectra to the trispectrum level.
Quantification of Relativistic Contamination: The authors demonstrate that relativistic projection effects inherently generate a parity-odd trispectrum, even if the primordial universe is parity-symmetric.
Parameter Space Exploration: The study systematically maps the dependence of the parity-odd signal on survey parameters (bias, evolution bias, magnification bias), scale, and geometric configurations.
Survey-Specific Analysis: The paper explicitly calculates these effects for Euclid and SKA-like surveys, showing how different bias parameters influence the magnitude of the relativistic corrections.

Results

Magnitude of Corrections: For equality scales ( $k \approx 0.01 h \text{Mpc}^{-1}$ ), the relativistic corrections to the trispectrum are significant. Depending on the configuration and viewing angles, the ratio $|T_{odd}/T_{even}|$ can reach 10% or more, and in specific angular configurations, can exceed 80%.
Order Contributions: The relative dominance of second-order ( $T_{1122}$ ) versus third-order ( $T_{1113}$ ) contributions varies significantly between survey types and configurations. For Euclid-like surveys, the second-order contribution often dominates, whereas for SKA-like surveys, the third-order contribution can be dominant.
Cancellations: There are significant cancellations between the different perturbative contributions ( $T_{1113}$ and $T_{1122}$ ), which can suppress the total parity-odd amplitude ( $T_{odd}$ ) relative to the individual terms. These cancellations are highly sensitive to viewing angles and configuration choices.
Kernel Dependence: The first-order relativistic kernel ( $K^{(1)}_{GR}$ ) often dominates the signal for Euclid-like surveys, while higher-order kernels play a larger role for SKA-like surveys.
Redshift Evolution: The signal's dependence on redshift is complex and sensitive to the choice of evolution and magnification biases. For some bias configurations, the signal is suppressed at higher redshifts, while for others, it remains significant.

Significance
The paper argues that the relativistic corrections to the trispectrum are non-negligible and constitute a major source of "contamination" when searching for intrinsic parity violation. The authors state that while standard RSD terms are parity symmetric, the relativistic terms introduce a dipole-like parity-odd contribution that becomes dominant for surveys probing large fractions of the sky and high redshifts.

The significance of this work lies in its cautionary role: previous claims of parity violation in the BOSS data (which averaged over lines of sight) may need re-evaluation in light of these relativistic effects, or at least, these effects must be rigorously modeled to isolate any intrinsic signal. The authors conclude that future analyses of the trispectrum, particularly with upcoming large-scale surveys like Euclid and SKA, must account for these relativistic projection effects to ensure accurate constraints on inflationary physics and tests of General Relativity. They suggest that joint analyses with the bispectrum and trispectrum could provide tighter constraints on these relativistic corrections.