Constraining Primordial Power Asymmetry from Galaxy Clustering and Peculiar Velocity Information

Abstract: Primordial power asymmetry would probe departures from statistical isotropy, offering a clue to the physics of the primordial Universe. We investigate how such asymmetries can be tested with galaxy surveys and quantify how peculiar-velocity information can improve these tests. We develop a unified bipolar spherical harmonic (BipoSH) analysis of the auto- and cross-power spectra of galaxy density and line-of-sight peculiar velocity, and perform Fisher forecasts combining a Euclid-like spectroscopic galaxy survey with peculiar velocities reconstructed from Simons Observatory-like CMB maps through the kinetic Sunyaev-Zel'dovich effect. For dipolar asymmetry, we consider both scale-independent and scale-dependent modulations proportional to $k<sup>{-0.5}$. Galaxy clustering provides most of the constraining power in both cases, while peculiar velocities add only modest information. For quadrupolar asymmetry, we account for the recently identified anisotropic galaxy-bias response and marginalize over its amplitude. We find that galaxy clustering alone suffers from a degeneracy between the primordial quadrupolar modulation and anisotropic galaxy bias. This degeneracy can be substantially broken by adding peculiar-velocity information: the density-velocity cross-spectrum provides complementary information to the galaxy auto-spectrum, while the velocity auto-spectrum constrains the primordial modulation independently of anisotropic galaxy bias. The velocity information becomes increasingly effective for more negative scale dependence, yielding constraints tighter than those from BOSS for scale dependences proportional to $k<sup>{-1}$ and $k<sup>{-2}$. Our results demonstrate that peculiar-velocity information provides a complementary avenue for testing primordial power asymmetry with large-scale structure, particularly for quadrupolar asymmetry in the presence of anisotropic galaxy bias.

• The paper develops a unified BipoSH Fisher forecast showing that galaxy clustering primarily constrains dipolar asymmetry, while peculiar velocities are crucial for breaking anisotropic-bias degeneracies in quadrupolar searches.
• The joint analysis finds quadrupolar constraints improve substantially for red-tilted models, reaching a marginalized error of about 1.0 × 10⁻² at α = −1, while velocity-only data can outperform existing BOSS clustering bounds at α = −2.
• The results establish peculiar-velocity measurements as a bias-independent cross-check and probe for primordial asymmetry, while emphasizing that nonlinear effects, kSZ uncertainties, survey geometry, and tracer-bias modeling require validation.

Overview

This paper presents a Fisher-matrix forecast for constraining two forms of primordial power asymmetry—dipolar and quadrupolar modulations of the primordial power spectrum—using a joint analysis of galaxy clustering (gg), the galaxy–velocity cross-spectrum (gv), and the peculiar-velocity auto-spectrum (vv). The analysis is built on a unified BipoSH decomposition of redshift-space power spectra within linear perturbation theory and the local plane-parallel approximation, applied to a Euclid-like spectroscopic survey with kSZ-reconstructed peculiar velocities from Simons Observatory-like CMB maps. The central result is that peculiar-velocity information plays qualitatively different roles in the two cases: it adds only modest statistical information for dipolar asymmetry, but becomes essential for quadrupolar asymmetry, where anisotropic galaxy bias introduces a degeneracy that velocity data can break.

Motivation and model setup

The paper is motivated by CMB anomalies: the reported hemispherical power asymmetry at roughly $3\sigma$ with amplitude $A\simeq 0.06$–$0.07$ [Gordon:2006ag; Hoftuft:2009rq; Aiola:2015rqa], and the absence of detected quadrupolar asymmetry, constrained at the $|g_*|\lesssim 10^{-2}$ level by the CMB. LSS offers an independent probe with three-dimensional mode access. A key theoretical input is the recent identification of an anisotropic halo-bias response: for quadrupolar asymmetry, biased tracers acquire a correction proportional to $g_{2M}b_1^{(2)}$, confirmed in $N$-body simulations to be negative and mass-dependent [Shiraishi:2023zda; Masaki:2024hzn]. Because this contribution occupies the same $L=2$ BipoSH sector as the direct primordial modulation, galaxy clustering alone suffers from a partial degeneracy between $g_{2M}$ and $b_1^{(2)}$. Peculiar velocity, which traces gravitational motion without bias parameters and carries an additional $k^{-1}$ factor in its response kernel ($A\simeq 0.06$0), provides a probe free of this contamination and weighted toward large scales.

The authors adopt phenomenological scale dependences: $A\simeq 0.06$1 for the dipole, motivated by the CMB fit $A\simeq 0.06$2 to $A\simeq 0.06$3 of Aiola et al., and $A\simeq 0.06$4 with $A\simeq 0.06$5 for the quadrupole. For the fiducial anisotropic-bias relation they use $A\simeq 0.06$6, extrapolated from cluster-scale halo simulations—an extrapolation the authors explicitly flag, though they marginalize over its overall amplitude so the forecast does not require its normalization a priori.

Dipolar constraints

For both scale-independent and $A\simeq 0.06$7-modulated dipoles, galaxy clustering dominates the constraining power. At $A\simeq 0.06$8, the joint error on $A\simeq 0.06$9 ($0.07$0 scale-independent; $0.07$1 for the tilted case) is essentially identical to the gg-only result, with gv and vv errors larger by factors of roughly 2–4. Since the dipolar model introduces no bias-like contaminant into the observables, there is no degeneracy for velocity data to break; the practical value of gv and vv here is as consistency checks against probe-specific systematics, since a coherent signal across all three spectra would be difficult to attribute to a single observational artifact.

Quadrupolar constraints and the role of anisotropic bias

The quadrupolar results constitute the main contribution. With $0.07$2 marginalized, the gg-only constraint degrades severely—for example, from $0.07$3 unmarginalized to $0.07$4 marginalized at $0.07$5, a penalty factor of about 4–5. Three structural findings emerge:

• Complementary degeneracy directions: the gv cross-spectrum has a different degeneracy direction in the $0.07$6 plane than gg, and the combination $0.07$7 already recovers most of the full BipoSH constraining power.
• Bias-independent constraint: the vv auto-spectrum yields an approximately vertical band in the $0.07$8 plane, constraining $0.07$9 independently of $|g_*|\lesssim 10^{-2}$0. This also makes vv robust to mis-modeling of the tracer response: combining it with density data can reveal a shifted inference rather than merely reducing error bars.
• Scale-dependence dependence: negative tilts enhance low-$|g_*|\lesssim 10^{-2}$1 modes where velocity sensitivity peaks. Changing $|g_*|\lesssim 10^{-2}$2 from $|g_*|\lesssim 10^{-2}$3 to $|g_*|\lesssim 10^{-2}$4 improves the marginalized joint constraint from $|g_*|\lesssim 10^{-2}$5 to $|g_*|\lesssim 10^{-2}$6 (a factor of 3.0), while the fixed-$|g_*|\lesssim 10^{-2}$7 error improves only by about 1.3—evidence that velocity information specifically drives the improvement under marginalization.

A headline quantitative result is that the vv-only constraint beats the BOSS DR12 clustering bound (corrected for the anisotropic-bias contribution but without marginalization) for red-tilted models: nominally tighter at $|g_*|\lesssim 10^{-2}$8 ($|g_*|\lesssim 10^{-2}$9 vs. $g_{2M}b_1^{(2)}$0) and substantially tighter at $g_{2M}b_1^{(2)}$1 ($g_{2M}b_1^{(2)}$2 vs. $g_{2M}b_1^{(2)}$3). These forecasts remain weaker than ideal cosmic-variance-limited CMB temperature-plus-polarization forecasts by one to two orders of magnitude, a comparison the authors state plainly rather than downplay.

Dependence on survey parameters

Varying $g_{2M}b_1^{(2)}$4 shows distinct behavior: for $g_{2M}b_1^{(2)}$5 errors keep improving with $g_{2M}b_1^{(2)}$6, whereas for $g_{2M}b_1^{(2)}$7 constraints saturate at a few times $g_{2M}b_1^{(2)}$8, dominated by cosmic variance at low $g_{2M}b_1^{(2)}$9. In the latter case the marginalized and unmarginalized joint errors nearly coincide ($N$0 vs. $N$1), showing that low-$N$2 velocity information removes most of the anisotropic-bias degradation. Noise scans indicate that even with reconstruction noise pushed to $N$3, the marginalized vv constraint remains tighter than marginalized gg for $N$4; conversely, reducing $N$5 below 10 yields little gain because the forecast is then cosmic-variance limited. Increasing the Euclid–CMB overlap fraction $N$6 from 0.3 to 0.7 improves the joint error modestly (e.g., $N$7 at $N$8), with diminishing returns toward full overlap.

Limitations and open questions

The forecast rests on several assumptions the authors identify explicitly: linear perturbation theory, local plane-parallel geometry, Gaussian covariance, statistically independent redshift bins, and a scale-independent velocity-noise model with $N$9. The $L=2$0 fitting relation is calibrated for cluster-scale halos and extrapolated to Euclid-like samples without simulation support at those masses. Realistic analyses must additionally handle kSZ noise calibration, optical-depth uncertainties, survey windowing, foregrounds, reconstruction-induced correlations, and wide-angle, relativistic, and nonlinear corrections. An open question left by the paper is whether the forecasted complementarity survives once these systematic effects are incorporated and validated against mock catalogs; the assumed primordial scale dependence $L=2$1 itself must be tested separately rather than assumed.

Conclusion

This work establishes a unified BipoSH Fisher framework demonstrating that peculiar velocities are a targeted remedy for the specific pathology of quadrupolar power-asymmetry searches—anisotropic galaxy bias—while contributing mainly systematic cross-checks for the dipolar case. The most consequential claim is that kSZ-reconstructed velocity auto-spectra alone can outperform existing BOSS clustering bounds for red-tilted quadrupolar models, entirely independent of assumptions about anisotropic bias. Given recent detections of galaxy–velocity and velocity–velocity power spectra with ACT DR6 and DESI DR2, the proposed analysis path is observationally grounded, though its quantitative accuracy awaits treatment of the nonlinear and observational effects enumerated above.