⚛️ general relativity

Visualising relativistic effects in redshift space distortions of large scale structure

This paper qualitatively illustrates how higher-order relativistic effects, including Doppler and gravitational redshift, distort large-scale over-dense and under-dense regions in redshift space into asymmetric, egg- or bean-like shapes that break the line-of-sight symmetry typically associated with standard redshift space distortions.

Original authors: Pritha Paul, Chris Clarkson

Published 2026-07-14

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Pritha Paul, Chris Clarkson

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 you are looking at a giant, fluffy cloud of galaxies floating in the universe. In the "real" world, if this cloud is perfectly round, it stays round. But when we look at it through our telescopes, we don't see it in real space; we see it in "redshift space." This is like looking at the cloud through a funhouse mirror that stretches and squashes things based on how fast the galaxies are moving toward or away from us.

For a long time, scientists thought this mirror just turned round clouds into perfect ovals (ellipsoids). If the cloud was collapsing inward, the mirror squashed it flat like a pancake. If it was expanding outward, it stretched it like a rubber band. This was the "Kaiser effect," a well-known rule that treats the universe like a simple, Newtonian playground.

But this paper suggests that the mirror is actually much more complicated. The authors, Pritha Paul and Chris Clarkson, argue that when you zoom out to the very largest scales of the universe, the mirror doesn't just make ovals. It starts to twist the clouds into weird, asymmetrical shapes—like eggs or beans.

The Invisible Push and Pull
Why does this happen? The paper explains that the standard "oval" view only accounts for the Doppler effect (the change in sound pitch as an ambulance passes you, but for light). However, on huge scales, two other subtle effects kick in: the gravitational redshift (light losing energy climbing out of a gravity well) and higher-order Doppler effects.

Think of it like this: The standard oval distortion is like a gentle wind blowing a kite. But these new relativistic effects are like invisible hands grabbing the kite from the side and pulling it into a lopsided shape. The paper calculates that these "invisible hands" are usually very weak—about a factor of $H/k$ smaller than the main wind effect (where $H$ is the expansion rate of the universe and $k$ is the size of the structure). Because they are so small, we usually ignore them. But on the biggest scales, they become important enough to break the symmetry.

The Egg and the Bean
To visualize this, the authors created computer models of two idealized cosmic structures: a giant cluster of galaxies (an over-density) and a massive cosmic void (an under-density, or a giant empty bubble).

The Cluster (Isothermal Sphere): They modeled a dense ball of matter. In the old Newtonian view, this ball would just look like a squashed egg. But when they added the relativistic "invisible hands," the shape changed. One side of the egg became fatter, and the other thinner. It wasn't a symmetrical oval anymore; it looked like a lopsided egg or a bean.
The Void (Cosmic Bubble): They also modeled a giant empty hole in the universe where matter is flowing outward. Again, the old view said it would stretch into a symmetrical oval. The new view showed it twisting into an asymmetrical shape, with one side looking very different from the other.

The paper explicitly rules out the idea that these structures remain perfectly symmetrical along the line of sight once these effects are included. The authors show that while the "Newtonian" part of the distortion creates even-numbered patterns (like perfect ovals), the "relativistic" part creates odd-numbered patterns (like the dipole that makes one side different from the other).

How Sure Are We?
It is important to note that this paper is a theoretical exploration, not a report on a new telescope discovery. The authors didn't go out and photograph these egg-shaped galaxies. Instead, they used mathematical formulas and computer simulations to predict what should happen if we look at these structures with enough precision.

They suggest that if we start stacking many voids together to study them, we might finally see these asymmetries. The paper argues that these distortions are real consequences of Einstein's theory of relativity applied to the expanding universe, but they are currently hidden because they are so subtle compared to the main effects.

The Takeaway
The main finding is that the universe's "funhouse mirror" is more complex than we thought. While we used to think large structures just got squashed into ellipses, the authors suggest that when we account for the full complexity of relativity, those structures actually get twisted into egg-like or bean-like shapes. This isn't just a tiny detail; it changes how we interpret the map of the universe. The authors propose that by looking for these specific asymmetries, especially in the way voids are arranged, we might be able to measure these relativistic effects directly for the first time.

In short: The universe isn't just stretching into ovals; on the biggest scales, it's twisting into weird, asymmetrical shapes, and we might finally be able to see them if we look closely enough.

Technical Summary: Visualising Relativistic Effects in Redshift Space Distortions of Large Scale Structure

Problem Statement
Redshift space distortions (RSD) arise because the observed positions of galaxies are determined by their redshift, which includes contributions from both the Hubble flow and peculiar velocities. In the standard Newtonian approximation (the Kaiser effect), spherical overdensities (clusters) and underdensities (voids) appear distorted into ellipsoids along the line of sight due to coherent infall or outflow. While this effect is well-understood on linear scales, it preserves symmetry along the line of sight, generating only even multipoles in the correlation function.

However, on very large scales, relativistic effects—specifically the Doppler effect and gravitational redshift—become relevant. These effects, suppressed by a factor of $H/k$ relative to the leading Newtonian terms, introduce asymmetric distortions that break line-of-sight symmetry. While these relativistic contributions have been studied analytically in the context of correlation functions and higher-order statistics (producing odd multipoles), there has been a lack of detailed investigation into how these effects visually distort idealised structures in redshift space. This paper addresses the gap in visualising how over-dense and under-dense regions deform when higher-order relativistic corrections are included.

Methodology
The authors adopt a perturbative approach to investigate the galaxy number count fluctuations, $\Delta(z, \mathbf{n})$ , up to second order in the $H/k$ expansion. Rather than perturbing individual galaxy positions, the study analyzes the density field directly.

Formalism: The galaxy number count fluctuation is expanded into Newtonian ( $\Delta_N$ $Δ_{N}$ ) and relativistic ( $\Delta_R$ $Δ_{R}$ ) components at first and second order.
- First Order: The Newtonian term includes the density contrast and the Kaiser term ( $-\partial_\chi v_\parallel$ ). The relativistic term is dominated by the Doppler effect, proportional to the line-of-sight velocity $v_\parallel$ and bias parameters (evolution and magnification).
- Second Order: The expansion includes non-linear couplings between density and velocity fields, as well as terms involving the gravitational potential $\psi$ . The relativistic second-order terms introduce odd multipoles (dipole, octupole) that break line-of-sight symmetry, whereas Newtonian terms generate even multipoles.
Idealised Models: Two spherically symmetric profiles are modeled to visualize these distortions:
- Isothermal Sphere: Used to model an overdense cluster. The density profile follows a pseudo-isothermal form, and the velocity field is derived from the gravitational potential via the Euler equation, representing infall.
- Cosmic Void: Modeled using a standard void density profile with a central underdensity and a compensating overdense shell. The velocity field represents outflow.
Visualization: The authors compute the line-of-sight derivatives of the velocity and density fields for these profiles. They generate 2D density plots slicing through the center of symmetry, visualizing:
- Individual first-order Newtonian and relativistic terms.
- Individual second-order Newtonian and relativistic terms.
- Cumulative contributions, where relativistic terms are artificially magnified (by factors of 50 or 200) to make the sub-dominant effects visible against the dominant Newtonian background.

Key Contributions and Results
The paper provides a qualitative and visual decomposition of how relativistic terms modify the appearance of large-scale structures in redshift space:

Symmetry Breaking: The primary result is the demonstration that while Newtonian RSD preserves line-of-sight symmetry (producing ellipsoidal shapes), relativistic terms introduce distinct asymmetries.
- First Order: The relativistic Doppler term creates a dipolar asymmetry. For an infalling cluster, one side appears more overdense than the other, shifting the centroid. For an outflowing void, a similar dipolar pattern emerges.
- Second Order: The relativistic contributions introduce higher-order odd multipoles (e.g., octupoles). The combination of even (Newtonian) and odd (relativistic) multipoles transforms the elliptical shapes into "egg-like" or "bean-like" structures, depending on the sign and magnitude of the velocity and bias parameters.
Isothermal Sphere Results:
- The Newtonian contribution compresses the sphere into an ellipse.
- The addition of the first-order Doppler term breaks this symmetry.
- The cumulative second-order relativistic terms (magnified) show pronounced asymmetry in the contour lines, with the radial profile becoming distinctly non-symmetric.
Cosmic Void Results:
- Similar to the cluster, the Newtonian term creates a symmetric, elongated underdensity surrounded by overdense lobes.
- The relativistic terms introduce strong asymmetries. The void profile exhibits "tighter" contour lines and more pronounced egg-like features compared to the cluster case, driven by the specific shape of the void's velocity profile (which peaks and then falls off).
- The total relativistic contribution is dominated by terms involving the gravitational potential and velocity gradients ( $v_\parallel \partial^2_\chi \psi$ and transverse Doppler terms).

Significance and Claims
The paper claims that visualizing these effects provides an intuitive understanding of the nature of relativistic corrections beyond the simple linear Newtonian approximation. The authors emphasize that:

Interpretation of Stacking: A direct application of these findings is in the analysis of stacked voids. Since relativistic effects break radial symmetry, stacked voids in redshift space are no longer radially symmetric. This asymmetry could serve as a direct observable to measure relativistic effects in Large Scale Structure (LSS).
Multipole Structure: The work reinforces the theoretical understanding that relativistic effects generate odd multipoles (dipole, octupole, etc.) in the density perturbation, distinguishing them from the even multipoles of the Newtonian Kaiser effect.
Bias Dependence: The magnitude and even the sign of the relativistic distortions depend on evolution and magnification biases. This implies that the visual appearance of these distortions is sensitive to the specific galaxy populations being observed.

The authors conclude that while the effects are sub-dominant on small scales, they are increasingly important on large scales and cannot be neglected for precision cosmology. They suggest that future work should extend this analysis to higher-order statistics (bispectrum, trispectrum) and compare these theoretical predictions with numerical simulations that include relativistic effects.

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