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Constraining Primordial Power Asymmetry from Galaxy Clustering and Peculiar Velocity Information

This paper presents a unified bipolar spherical harmonic analysis demonstrating that while galaxy clustering dominates constraints on dipolar primordial power asymmetry, incorporating peculiar velocity information is crucial for breaking degeneracies with anisotropic galaxy bias to significantly improve constraints on quadrupolar asymmetry.

Original authors: Keita Minato, Atsushi Taruya, Teppei Okumura, Maresuke Shiraishi

Published 2026-08-17

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CC BY 4.0

Original authors: Keita Minato, Atsushi Taruya, Teppei Okumura, Maresuke Shiraishi

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, cosmic ocean. For decades, scientists have believed this ocean is perfectly calm and uniform, like a smooth sheet of glass stretched across the sky. This idea, called "statistical isotropy," suggests that no matter which direction you look, the Universe looks the same on large scales. It's a comforting thought, but it's also a big assumption. If we could find a ripple, a tilt, or a "preferred direction" in this cosmic ocean, it would be a massive clue about how the Universe began, potentially revealing physics that our current textbooks don't even know about yet.

To find these ripples, astronomers usually look at the Cosmic Microwave Background (CMB), which is the oldest light in the Universe, like a baby photo of the cosmos. But there's another way to look: by studying the "Large Scale Structure," which is the vast web of galaxies and empty space that makes up the Universe today. Think of galaxies as islands in our cosmic ocean. By mapping where these islands are and how they move, we can test if the ocean is truly smooth or if there's a hidden current pushing things one way. The big challenge is that our tools to measure these movements aren't perfect, and sometimes the "islands" themselves (the galaxies) might be biased, making them look like they're moving differently than the water they float in.

This paper is like a detective story where the authors try to solve the mystery of whether the Universe has a hidden tilt. They focus on two types of "tilts": a dipole (where one side of the sky is stronger than the other, like a lopsided balloon) and a quadrupole (where the strength changes in a four-lobed pattern, like a four-leaf clover). The authors ask a crucial question: Can we use the movement of galaxies—called "peculiar velocity"—to get a better answer than just looking at where the galaxies are?

The team, led by Keita Minato and colleagues, built a sophisticated mathematical model to combine two types of data: the positions of galaxies (like taking a snapshot of the islands) and their speeds (like measuring the current of the water). They simulated a future survey, similar to the Euclid mission, which will map millions of galaxies, combined with velocity data reconstructed from the Cosmic Microwave Background. Their main finding is that for the "dipole" tilt, the galaxy positions do most of the heavy lifting, and the speed data acts mostly as a helpful second opinion to check for errors. However, for the "quadrupole" tilt, the story changes dramatically.

Here is the twist: when looking for the four-lobed pattern, the galaxies themselves are tricky. Because galaxies are biased tracers (they don't perfectly represent the underlying matter), their positions create a confusing mix of signals that makes it hard to tell if the tilt is real or just an artifact of how galaxies form. It's like trying to hear a whisper in a noisy room; the noise (galaxy bias) drowns out the signal. The authors found that adding the velocity data acts like a noise-canceling headphone. Since the speed of the galaxies depends on gravity and not on how "biased" the galaxies are, the velocity data cuts through the confusion.

Specifically, the paper shows that while galaxy positions alone struggle to separate the real cosmic tilt from the noise of galaxy bias, the velocity data provides a clean, independent view. When they combined the position data with the velocity data, the uncertainty in measuring the quadrupole tilt dropped significantly, especially for models where the tilt is stronger on the largest scales. In fact, for certain types of large-scale tilts, the velocity data alone could provide constraints that are tighter than what was achieved by previous major galaxy surveys like BOSS.

The authors are careful to note that these are forecasts based on simulations and theoretical models, not a discovery of a tilt itself. They haven't found the tilt yet; they have simply proven that if it exists, combining galaxy maps with velocity measurements is the best way to find it. They also point out that this method is particularly powerful because it doesn't rely on knowing the exact "bias" of the galaxies, which is often a source of error. By using the "speed" of the galaxies as a cross-check, they can break the degeneracy that has stumped previous attempts.

In short, this paper suggests that to hear the faint whispers of the early Universe, we shouldn't just listen to where the galaxies are; we need to listen to how fast they are moving, too. It's a promising new strategy that could help us finally determine if the Universe is truly uniform or if it holds a secret directional bias from its very birth.

Technical Summary: Constraining Primordial Power Asymmetry from Galaxy Clustering and Peculiar Velocity Information

Problem Statement
The standard cosmological model assumes the Universe is statistically homogeneous and isotropic on large scales. However, testing this assumption is crucial for identifying physics beyond simple isotropic early-Universe scenarios. While the Cosmic Microwave Background (CMB) has reported large-scale anomalies suggesting dipolar power asymmetry and placed constraints on quadrupolar asymmetry, Large Scale Structure (LSS) offers a complementary, low-redshift probe with access to a vastly larger number of Fourier modes.

A significant challenge in using LSS to constrain primordial power asymmetry, particularly for the quadrupolar case, is the degeneracy between the primordial signal and the anisotropic response of biased tracers (galaxies). Recent theoretical and simulation work indicates that the primordial quadrupolar modulation couples to the tidal field in the tracer-density relation, generating an anisotropic galaxy-bias contribution. This contribution occupies the same angular sector as the direct primordial modulation, creating a strong degeneracy in galaxy clustering analyses that has not been fully accounted for in previous constraints.

Methodology
The authors develop a unified framework to analyze dipolar and quadrupolar power asymmetries using both galaxy density and line-of-sight peculiar velocity fields. The methodology involves:

Unified BipoSH Formalism: The authors employ the Bipolar Spherical Harmonic (BipoSH) decomposition to characterize the angular dependence of the auto- and cross-power spectra of galaxy density ( $g$ ) and peculiar velocity ( $v$ ). This formalism naturally handles the dependence on both the wavevector direction ( $\hat{k}$ ) and the line-of-sight direction ( $\hat{n}$ ).
Modeling Power Asymmetry:
- Dipolar ( $n=1$ ): Modeled as a modulation proportional to $Y_{1M}(\hat{n})$ , motivated by CMB hemispherical asymmetry. The modulation amplitude is tested for both scale-independent and scale-dependent ( $k^{-0.5}$ ) forms.
- Quadrupolar ( $n=2$ ): Modeled as a modulation proportional to $Y_{2M}(\hat{k})$ . Crucially, the authors include the anisotropic-bias correction to the galaxy kernel, proportional to the product of the primordial quadrupole ( $g_{2M}$ ) and the anisotropic-bias coefficient ( $b^{(2)}_1$ ).
Fisher Forecasting: The authors perform Fisher matrix forecasts for a joint analysis of a Euclid-like spectroscopic galaxy survey and peculiar velocities reconstructed from Simons Observatory-like CMB maps via the kinetic Sunyaev–Zel'dovich (kSZ) effect.
- The analysis marginalizes over the anisotropic-bias parameter $b^{(2)}_1$ (treated as a nuisance parameter) for the quadrupolar case, while fixing isotropic parameters ( $b_1, f$ ).
- Three power spectra are utilized: galaxy auto-spectrum ($gg$), density-velocity cross-spectrum ($gv$), and velocity auto-spectrum ($vv$).
- Survey specifications follow Euclid (four redshift bins) with kSZ velocity reconstruction noise characterized by a ratio $R_N$ and a sky overlap fraction $\eta_{cov}$ .

Key Contributions

Unified Treatment of Bias and Asymmetry: The paper explicitly incorporates the anisotropic galaxy-bias response ( $b^{(2)}_1$ ) into the Fisher analysis for quadrupolar asymmetry, addressing a known degeneracy that limits the constraining power of galaxy clustering alone.
Role of Peculiar Velocities: The authors quantify how peculiar velocity information complements galaxy clustering. They demonstrate that while the velocity field is not a biased tracer, its $k^{-1}$ dependence makes it particularly sensitive to large-scale modes where power asymmetry signals are often enhanced.
Decomposition of Constraining Power: The study distinguishes the specific roles of the $gg$, $gv$, and $vv$ spectra. It identifies that the $gv$ cross-spectrum helps break the degeneracy between $g_{2M}$ and $b^{(2)}_1$ due to a different degeneracy direction in the parameter space, while the $vv$ auto-spectrum provides a constraint on $g_{2M}$ that is entirely independent of $b^{(2)}_1$ .

Results

Dipolar Asymmetry: For both scale-independent and scale-dependent ( $k^{-0.5}$ ) models, galaxy clustering ($gg$) provides the dominant constraining power. Peculiar velocity information ($gv$ and $vv$) adds only modest improvement to the joint constraint but serves as a valuable consistency check with different observational systematics.
Quadrupolar Asymmetry:
- Degeneracy Breaking: Galaxy clustering alone suffers from a strong degeneracy between the primordial modulation and anisotropic bias. Adding peculiar velocity information substantially breaks this degeneracy.
- Scale Dependence: The effectiveness of velocity information increases with more negative scale dependence (e.g., $k^{-1}$ and $k^{-2}$ ). For $k^{-2}$ dependence, the velocity auto-spectrum ($vv$) alone yields constraints tighter than previous BOSS DR12 results (which did not marginalize over $b^{(2)}_1$ ).
- Joint Constraints: For scale-dependent models, the joint analysis ($gg+gv+vv$) achieves marginalized 1 $\sigma$ errors on $g_{2M}$ significantly tighter than galaxy clustering alone. For $\alpha = -2$ , the joint error is $\sim 1.4 \times 10^{-3}$ , compared to $\sim 6.4 \times 10^{-3}$ for $gg$ alone.
- Noise and Coverage: Reducing velocity-reconstruction noise ( $R_N$ ) and increasing the CMB-Euclid sky overlap ( $\eta_{cov}$ ) improve constraints, with the latter having a more pronounced effect for large-scale-enhanced models.

Significance
The paper claims that peculiar-velocity information provides a complementary and essential avenue for testing primordial power asymmetry, particularly for the quadrupolar case in the presence of anisotropic galaxy bias. By utilizing the velocity auto-spectrum, which is independent of the anisotropic-bias parameter, the analysis offers a direct constraint on the primordial modulation that is robust against uncertainties in the tracer response. The authors emphasize that while their forecasts are based on linear theory and Gaussian covariance assumptions, the qualitative conclusion—that velocity fields break the bias-asymmetry degeneracy—remains a critical insight for future large-scale structure analyses. The results suggest that upcoming surveys combining galaxy clustering with kSZ-derived velocities (such as Euclid and Simons Observatory) can significantly improve upon current limits on primordial power asymmetry, offering a pathway to probe the physics of the early Universe with systematics distinct from the CMB.

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