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Pseudo-Cℓ approach to directional stacking

Lea Harscouet*, Amy Wayland, and David Alonso

  • Department of Physics, Denys Wilkinson Building, University of Oxford, Keble Road, Oxford OX1 3RH, United Kingdom

  • *Contact author: lea.harscouet@physics.ox.ac.uk
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Phys. Rev. D 114, 083516 – Published 9 October, 2026

DOI: https://doi.org/10.1103/bkn8-f1t1

Abstract

Stacking (i.e., averaging) the value of a given astrophysical field around sources allows us to detect new cosmological signatures, such as the kinematic Sunyaev-Zel’dovich, and gain insight on the astrophysical properties of galaxies and their environment. Further information may be gained by orienting these stacks along preferred axes defined by a local directed field, such as the transverse galaxy velocities, galaxy shapes, or the local tidal forces. Examples of this are searches for the moving lens effect, the detection of dipole signatures, or the study of cosmic filaments. Here we show that all directional stacking signals may be reconstructed, without loss of information, in terms of the cross-power spectrum between the quantity of interest and the E and B modes of the spin field used to define the preferred axes weighted by the local galaxy density. The power spectrum approach has several practical advantages, in terms of speed, finite-resolution effects, data visualization, and combination with other cosmological probes. We also argue that, in some cases, such as stacking using velocities or tidal forces reconstructed from the density field, the recovered signal may be dominated by information that is already present in the cross-spectrum between the target field and the galaxy overdensity itself.

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References (54)

  1. M. Li, R. E. Angulo, S. D. M. White, and J. Jasche, Matched filter optimization of kSZ measurements with a reconstructed cosmological flow field, Mon. Not. R. Astron. Soc. 443, 2311 (2014).
  2. E. Schaan et al. (ACTPol Collaboration), Evidence for the kinematic Sunyaev-Zel’dovich effect with the Atacama Cosmology Telescope and velocity reconstruction from the Baryon Oscillation Spectroscopic Survey, Phys. Rev. D 93, 082002 (2016).
  3. E. Schaan et al. (Atacama Cosmology Telescope Collaboration), Atacama cosmology telescope: Combined kinematic and thermal Sunyaev-Zel’dovich measurements from BOSS CMASS and LOWZ halos, Phys. Rev. D 103, 063513 (2021).
  4. M. Mallaby-Kay et al., Kinematic Sunyaev-Zel’dovich effect with ACT, DES, and BOSS: A novel hybrid estimator, Phys. Rev. D 108, 023516 (2023).
  5. B. Hadzhiyska et al., Evidence for large baryonic feedback at low and intermediate redshifts from kinematic Sunyaev-Zel’dovich observations with ACT and DESI photometric galaxies, Phys. Rev. D 112, 083509 (2025).
  6. B. Ried Guachalla et al., Backlighting extended gas halos around luminous red galaxies: Kinematic Sunyaev-Zel’dovich effect from DESI Y1 and ACT data, Phys. Rev. D 112, 103512 (2025).
  7. Y.-C. Cai, J. A. Peacock, A. de Graaff, and S. Alam, Detection of cosmological dipoles aligned with transverse peculiar velocities, Mon. Not. R. Astron. Soc. 541, 2093 (2025).
  8. S. C. Hotinli, E. Pierpaoli, S. Ferraro, and K. Smith, Transverse velocities and matter gradient correlations: A new signal and a new challenge to moving-lens analyses, Phys. Rev. D 108, 083508 (2023).
  9. A. Beheshti, E. Schaan, and A. Kosowsky, Moving lens effect: Simulations, forecasts, and foreground mitigation, Phys. Rev. D 111, 043510 (2025).
  10. S. C. Hotinli, K. M. Smith, and S. Ferraro, First detection of the moving lens effect with ACT and DESI LS (to be published).
  11. S. G. Gallego, S. Cantalupo, S. Lilly, R. A. Marino, G. Pezzulli, J. Schaye, L. Wisotzki, R. Bacon, H. Inami, M. Akhlaghi, S. Tacchella, J. Richard, N. F. Bouche, M. Steinmetz, and M. Carollo, Stacking the Cosmic Web in fluorescent Ly α emission with MUSE, Mon. Not. R. Astron. Soc. 475, 3854 (2018).
  12. T. Vernstrom, G. Heald, F. Vazza, T. J. Galvin, J. L. West, N. Locatelli, N. Fornengo, and E. Pinetti, Discovery of magnetic fields along stacked cosmic filaments as revealed by radio and X-ray emission, Mon. Not. R. Astron. Soc. 505, 4178 (2021).
  13. T. Hodgson, F. Vazza, M. Johnston-Hollitt, and B. McKinley, Stacking the synchrotron cosmic web with FIGARO, Pub. Astron. Soc. Aust. 39, e033 (2022).
  14. B. Hadzhiyska, S. Ferraro, and R. Zhou, Tracing cosmic gas in filaments and halos: Low-redshift insights from the kinematic Sunyaev-Zel’dovich effect, Phys. Rev. D 111, 023534 (2025).
  15. H. Tanimura, G. Hinshaw, I. G. McCarthy, L. Van Waerbeke, N. Aghanim, Y.-Z. Ma, A. Mead, A. Hojjati, and T. Tröster, A search for warm/hot gas filaments between pairs of SDSS Luminous Red Galaxies, Mon. Not. R. Astron. Soc. 483, 223 (2019).
  16. A. de Graaff, Y.-C. Cai, C. Heymans, and J. A. Peacock, Probing the missing baryons with the Sunyaev-Zel’dovich effect from filaments, Astron. Astrophys. 624, A48 (2019).
  17. M. Lokken et al., Superclustering with the Atacama Cosmology Telescope and Dark Energy Survey. I. Evidence for thermal energy anisotropy using oriented stacking, Astrophys. J. 933, 134 (2022).
  18. M. Lokken et al., Superclustering with the Atacama Cosmology Telescope and Dark Energy Survey: II. Anisotropic large-scale coherence in hot gas, galaxies, and dark matter, Astrophys. J. 982, 186 (2025).
  19. E. J. Baxter, B. D. Sherwin, and S. Raghunathan, Constraining the rotational kinematic Sunyaev-Zel’dovich effect in massive galaxy clusters, J. Cosmology Astropart. Phys. 06 (2019) 001.
  20. S. Goldstein and J. C. Hill, Evidence of galaxy cluster rotation in the cosmic microwave background, arXiv:2512.10951.
  21. T. Yang, Y.-C. Cai, W. Cui, J. A. Peacock, R. Davé, H. Mo, H. Wang, and X. Yang, A measurement of gas rotation in galaxy groups via the kinetic Sunyaev-Zeldovich effect, arXiv:2603.14494.
  22. M. Nikjoo, J. Zuntz, and B. Moews, Weak lensing by photometric density ridges, arXiv:2603.04025.
  23. P. A. R. Ade et al. (Planck Collaboration), Planck 2015 results. XVI. Isotropy and statistics of the CMB, Astron. Astrophys. 594, A16 (2016).
  24. N. Padmanabhan, X. Xu, D. J. Eisenstein, R. Scalzo, A. J. Cuesta, K. T. Mehta, and E. Kazin, A 2 per cent distance to z =0.35 by reconstructing baryon acoustic oscillations-I. Methods and application to the Sloan Digital Sky Survey, Mon. Not. R. Astron. Soc. 427, 2132 (2012).
  25. L. Harscouet, K. Wolz, A. Wayland, D. Alonso, and B. Hadzhiyska, kSZ for everyone: The pseudo-Cl approach to stacking, arXiv:2512.14625.
  26. F. J. Qu et al., Precision kinematic Sunyaev–Zel’dovich measurements across halo mass and redshift with DESI DR2 and ACT DR6: Part I. Luminous red galaxies, arXiv:2604.19744.
  27. B. Hadzhiyska et al., Precision kinematic Sunyaev–Zel’dovich measurements across halo mass and redshift with DESI DR2 and ACT DR6: Part II. Bright galaxy survey and emission-line galaxies, arXiv:2604.19745.
  28. A. Baleato Lizancos and M. White, Harmonic analysis of discrete tracers of large-scale structure, J. Cosmology Astropart. Phys. 05 (2024) 010.
  29. K. Wolz, D. Alonso, and A. Nicola, Catalog-based pseudo-Cℓs, J. Cosmology Astropart. Phys. 01 (2025) 028.
  30. M. Reinecke, S. Belkner, and J. Carron, Improved cosmic microwave background (de-)lensing using general spherical harmonic transforms, Astron. Astrophys. 678, A165 (2023).
  31. D. Alonso, B. Hadzhiyska, and M. A. Strauss, Recovering the tidal field in the projected galaxy distribution, Mon. Not. R. Astron. Soc. 460, 256 (2016).
  32. J. N. Goldberg, A. J. Macfarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan, Spin-s spherical Harmonics and ð, J. Math. Phys. (N.Y.) 8, 2155 (1967).
  33. D. Alonso, J. Sanchez, A. Slosar (LSST Dark Energy Science Collaboration), A unified pseudo-Cℓ framework, Mon. Not. R. Astron. Soc. 484, 4127 (2019).
  34. D. Alonso, Pseudo-C ℓ  s for spin- s fields with component-wise weighting, Open J. Astrophys. 8, 53 (2025).
  35. A. Nicola, C. García-García, D. Alonso, J. Dunkley, P. G. Ferreira, A. Slosar, and D. N. Spergel, Cosmic shear power spectra in practice, J. Cosmology Astropart. Phys. 03 (2021) 067.
  36. C. Ramírez-Pérez, J. Sanchez, D. Alonso, and A. Font-Ribera, CoLoRe: Fast cosmological realisations over large volumes with multiple tracers, J. Cosmology Astropart. Phys. 03 (2022) 002.
  37. E. Paillas et al., Optimal reconstruction of baryon acoustic oscillations for DESI 2024, J. Cosmology Astropart. Phys. 01 (2025) 142.
  38. X. Chen et al., Extensive analysis of reconstruction algorithms for DESI 2024 baryon acoustic oscillations, J. Cosmology Astropart. Phys. 05 (2026) 001.
  39. A. S. Bolton et al., Spectral classification and redshift measurement for the SDSS-III baryon oscillation spectroscopic Survey, Astron. J. 144, 144 (2012).
  40. N. Aghanim et al. (Planck Collaboration), Planck 2018 results. I. Overview and the cosmological legacy of Planck, Astron. Astrophys. 641, A1 (2020).
  41. N. Aghanim et al. (Planck Collaboration), Planck 2018 results. VIII. Gravitational lensing, Astron. Astrophys. 641, A8 (2020).
  42. N. Aghanim et al. (Planck Collaboration), Planck 2015 results. XXII. A map of the thermal Sunyaev-Zeldovich effect, Astron. Astrophys. 594, A22 (2016).
  43. W. Coulton et al., Atacama cosmology telescope: High-resolution component-separated maps across one third of the sky, Phys. Rev. D 109, 063530 (2024).
  44. T. McClintock et al. (DES Collaboration), Dark Energy Survey Year 1 results: Weak lensing mass calibration of redMaPPer galaxy clusters, Mon. Not. R. Astron. Soc. 482, 1352 (2019).
  45. A. Porredon et al., Dark Energy survey year 3 results: Optimizing the lens sample in combined galaxy clustering and galaxy-galaxy lensing analysis, Phys. Rev. D 103, 043503 (2021).
  46. H. Park, P. R. Shapiro, E. Komatsu, I. T. Iliev, K. Ahn, and G. Mellema, The Kinetic Sunyaev-Zel’dovich Effect as a probe of the physics of cosmic reionization: The effect of self-regulated reionization, Astrophys. J. 769, 93 (2013).
  47. H. Park, E. Komatsu, P. R. Shapiro, J. Koda, and Y. Mao, The impact of nonlinear structure formation on the power spectrum of transverse momentum fluctuations and the kinetic Sunyaev-Zel’dovich effect, Astrophys. J. 818, 37 (2016).
  48. A. Wayland, D. Alonso, and A. L. Posta, Detailed theoretical modelling of the kinetic Sunyaev-Zel’dovich stacking power spectrum, J. Cosmology Astropart. Phys. 01 (2026) 015.
  49. M. Lokken, J. R. Bond, R. Hložek, N. J. Carlson, Z. Li, and A. van Engelen, The superclustering of hot gas: Cosmological sensitivity in the Websky simulations, J. Cosmol. Astropart. Phys. 05 (2026) 011.
  50. C. García-García, D. Alonso, and E. Bellini, Disconnected pseudo-Cl covariances for projected large-scale structure data, J. Cosmology Astropart. Phys. 11 (2019) 043.
  51. I. Duncan, D. Alonso, A. Slosar, and K. Storey-Fisher, Can we constrain structure growth from galaxy proper motions?, Open J. Astrophys. 7, 9 (2024).
  52. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
  53. E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Expanded and Improved ed., Pure and Applied Physics; Vol. 5 (Academic Press, New York, 1959).
  54. G. Chon, A. Challinor, S. Prunet, E. Hivon, and I. Szapudi, Fast estimation of polarization power spectra using correlation functions, Mon. Not. R. Astron. Soc. 350, 914 (2004).

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Phys. Rev. D 114, 083516– Published 9 October, 2026
  • Received 27 May 2026
  • Accepted 2 September 2026
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