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arxiv: 2606.12551 · v1 · pith:GJZE2WKInew · submitted 2026-06-10 · 🌌 astro-ph.CO

Complex yet Hermitian: Gaussian covariance of cross-correlation and multi-tracer power spectra

Federico Montano , Stefano Camera , Emiliano Sefusatti , Mohamed Yousry Elkhashab This is my paper

Pith reviewed 2026-06-27 08:23 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords Gaussian covariancemulti-tracer power spectracomplex power spectraparity-odd signalsweighted estimatorLegendre multipolesHermitian matrix
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The pith

A general expression for the Gaussian covariance of multi-tracer power spectra works for both real and complex cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a unified formula for the covariance of power spectrum measurements from multiple tracers. This formula applies whether the spectra are real-valued even-parity signals or complex ones that include odd-parity contributions. Accurate covariances are needed to extract precise cosmological constraints from large surveys. Previous expressions were limited to real spectra, so extending them allows analysis of relativistic effects and cross-correlations without separate derivations. The authors demonstrate the formula on Legendre multipoles and two-dimensional power spectra.

Core claim

We generalise previous theoretical results for the Gaussian covariance of multi-tracer power spectrum measurements, providing a general expression applicable to both real (even-parity) and complex (both even- and odd-parity) power spectra. We focus on a generic weighted estimator at first, and then showcase how our general formalism applies to Legendre power spectrum multipoles and two-dimensional power spectrum, recovering known limits in appropriate cases. We validate our predictions against Gaussian Monte Carlo simulations and investigate the structure of the covariance matrix, including the Hermitian properties of its imaginary part.

What carries the argument

General analytical expression for the covariance of a generic weighted estimator of power spectra, valid for both real and complex (even- and odd-parity) cases.

If this is right

  • The expression recovers all previously known results for real even-parity spectra as special cases.
  • It supplies the full covariance matrix for complex spectra that include parity-odd terms without requiring separate derivations.
  • The imaginary part of the covariance matrix is Hermitian, preserving the required symmetry properties.
  • The same formula applies without change to Legendre multipoles and to two-dimensional power spectra.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Analysts can now include both even- and odd-parity modes inside a single covariance matrix when performing multi-tracer analyses of relativistic signals.
  • The reduction in the need for large simulation suites for covariance estimation becomes more pronounced once odd-parity spectra are routinely measured.
  • The Hermitian structure may simplify numerical inversion or eigenvalue analysis of the full covariance when complex spectra are present.

Load-bearing premise

The Gaussian approximation for the four-point function of the density field remains valid for the weighted estimator under consideration.

What would settle it

A direct numerical comparison in which the analytical covariance matrix differs substantially from the covariance measured in a large ensemble of Gaussian Monte Carlo realizations of the complex multi-tracer power spectra.

read the original abstract

Accurate modelling of the covariance of clustering observables is essential to fully exploit current and future survey data, which is expected to constrain large-scale clustering signals with unprecedented precision. Computational costs of simulation-based estimates motivate analytical approaches, especially in light of the growing interest towards multi-tracer analyses and parity-odd signatures in two-point statistics, which respectively mitigate cosmic variance and probe relativistic projection effects on cosmological scales. In this work, we generalise previous theoretical results for the Gaussian covariance of multi-tracer power spectrum measurements, providing a general expression applicable to both real (even-parity) and complex (both even- and odd-parity) power spectra. We focus on a generic weighted estimator at first, and then showcase how our general formalism applies to Legendre power spectrum multipoles and two-dimensional power spectrum, recovering known limits in appropriate cases. We validate our predictions against Gaussian Monte Carlo simulations and investigate the structure of the covariance matrix, including the Hermitian properties of its imaginary part.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper generalizes previous theoretical results on the Gaussian covariance of multi-tracer power spectrum measurements to a generic weighted estimator that applies to both real (even-parity) and complex (even- and odd-parity) power spectra. It derives the covariance expression, applies the formalism to Legendre multipoles and two-dimensional power spectra (recovering known real-case limits), validates the predictions against Gaussian Monte Carlo simulations, and examines the Hermitian properties of the imaginary part of the covariance matrix.

Significance. If the central derivation holds, the result supplies an analytical covariance model for multi-tracer analyses that includes parity-odd signals, which is directly useful for current and upcoming surveys where simulation-based covariances are computationally expensive. The explicit validation against Gaussian Monte Carlo realizations and the recovery of known limits for real spectra constitute reproducible checks that strengthen the work.

minor comments (3)
  1. [§1] The abstract and introduction refer to 'previous theoretical results' without citing the specific papers whose expressions are being generalized; adding these references in §1 would clarify the scope of the extension.
  2. [§3] In the section deriving the general covariance for the weighted estimator, the transition from the four-point function to the final expression for complex fields could benefit from an intermediate step showing how the Hermitian conjugate terms arise explicitly.
  3. [Figure 2] Figure captions for the covariance-matrix visualizations should state the exact number of Monte Carlo realizations used and the binning scheme, to allow direct comparison with the analytic prediction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation for minor revision. The report does not list any major comments requiring point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives a general expression for the Gaussian covariance of a weighted multi-tracer estimator that applies to both real and complex power spectra. It explicitly recovers known limits for Legendre multipoles and 2D spectra as special cases, and validates the analytic result against independent Gaussian Monte Carlo realizations. No load-bearing step reduces by construction to a fitted parameter, a self-citation chain, or an ansatz imported from the authors' prior work; the central claim is a direct algebraic generalization whose correctness is checked externally rather than assumed.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.1-grok · 5706 in / 918 out tokens · 14662 ms · 2026-06-27T08:23:26.772334+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

69 extracted references · 24 linked inside Pith

  1. [1]

    Mellier, Abdurro’uf, J.A

    Euclid Collaboration, Y. Mellier, Abdurro’uf, J.A. Acevedo Barroso, A. Ach´ ucarro, J. Adamek et al., Euclid: I. overview of the euclid mission, A&A697(2025) A1 [2405.13491]

    arXiv 2025

  2. [2]

    Abareshi, J

    DESI Collaboration, B. Abareshi, J. Aguilar, S. Ahlen, S. Alam, D.M. Alexander et al., Overview of the instrumentation for the dark energy spectroscopic instrument, AJ164(2022) 207 [2205.10939]

    Pith/arXiv arXiv 2022

  3. [3]

    Mainieri, R.I

    V. Mainieri, R.I. Anderson, J. Brinchmann, A. Cimatti, R.S. Ellis, V. Hill et al., The wide-field spectroscopic telescope (wst) science white paper, arXiv e-prints (2024) arXiv:2403.05398 [2403.05398]. – 19 –

    arXiv 2024

  4. [4]

    Schlegel, J.A

    D.J. Schlegel, J.A. Kollmeier, G. Aldering, S. Bailey, C. Baltay, C. Bebek et al., The megamapper: A stage-5 spectroscopic instrument concept for the study of inflation and dark energy, arXiv e-prints (2022) arXiv:2209.04322 [2209.04322]

    arXiv 2022

  5. [5]

    Bacon, R.A

    Square Kilometre Array Cosmology Science Working Group, D.J. Bacon, R.A. Battye, P. Bull, S. Camera, P.G. Ferreira et al., Cosmology with phase 1 of the square kilometre array red book 2018: Technical specifications and performance forecasts, PASA37(2020) e007 [1811.02743]

    arXiv 2018

  6. [6]

    Bartolo, E

    N. Bartolo, E. Komatsu, S. Matarrese and A. Riotto, Non-gaussianity from inflation: theory and observations, Phys. Rep.402(2004) 103 [astro-ph/0406398]

    Pith/arXiv arXiv 2004

  7. [7]

    Yoo, General relativistic description of the observed galaxy power spectrum: Do we understand what we measure?, Physical Review D82(2010)

    J. Yoo, General relativistic description of the observed galaxy power spectrum: Do we understand what we measure?, Physical Review D82(2010)

    2010

  8. [8]

    Challinor and A

    A. Challinor and A. Lewis, Linear power spectrum of observed source number counts, Physical Review D84(2011)

    2011

  9. [9]

    Bonvin and R

    C. Bonvin and R. Durrer, What galaxy surveys really measure, Physical Review D84(2011)

    2011

  10. [10]

    Castorina and M

    E. Castorina and M. White, Beyond the plane-parallel approximation for redshift surveys, MNRAS476(2018) 4403 [1709.09730]

    Pith/arXiv arXiv 2018

  11. [11]

    Noorikuhani and R

    M. Noorikuhani and R. Scoccimarro, Wide-angle and relativistic effects in fourier-space clustering statistics, Physical Review D107(2023)

    2023

  12. [12]

    McDonald, Gravitational redshift and other redshift-space distortions of the imaginary part of the power spectrum, J

    P. McDonald, Gravitational redshift and other redshift-space distortions of the imaginary part of the power spectrum, J. Cosmology Astropart. Phys.2009(2009) 026 [0907.5220]

    Pith/arXiv arXiv 2009

  13. [13]

    Beutler and E

    F. Beutler and E. Di Dio, Modeling relativistic contributions to the halo power spectrum dipole, J. Cosmology Astropart. Phys.2020(2020) 048 [2004.08014]

    arXiv 2020

  14. [14]

    O. Umeh, K. Koyama and R. Crittenden, Testing the equivalence principle on cosmological scales using the odd multipoles of galaxy cross-power spectrum and bispectrum, J. Cosmology Astropart. Phys.2021(2021) 049 [2011.05876]

    arXiv 2021

  15. [15]

    Seljak, Extracting primordial non-gaussianity without cosmic variance, Phys

    U. Seljak, Extracting primordial non-gaussianity without cosmic variance, Phys. Rev. Lett. 102(2009) 021302 [0807.1770]

    Pith/arXiv arXiv 2009

  16. [16]

    McDonald and U

    P. McDonald and U. Seljak, How to evade the sample variance limit on measurements of redshift-space distortions, J. Cosmology Astropart. Phys.2009(2009) 007 [0810.0323]

    Pith/arXiv arXiv 2009

  17. [17]

    Abramo and K.E

    L.R. Abramo and K.E. Leonard, Why multitracer surveys beat cosmic variance, MNRAS432 (2013) 318 [1302.5444]

    Pith/arXiv arXiv 2013

  18. [18]

    Fonseca, S

    J. Fonseca, S. Camera, M.G. Santos and R. Maartens, Hunting down horizon-scale effects with multi-wavelength surveys, ApJ812(2015) L22 [1507.04605]

    Pith/arXiv arXiv 2015

  19. [19]

    Alonso and P.G

    D. Alonso and P.G. Ferreira, Constraining ultralarge-scale cosmology with multiple tracers in optical and radio surveys, Phys. Rev. D92(2015) 063525 [1507.03550]

    Pith/arXiv arXiv 2015

  20. [20]

    Borzyszkowski, D

    M. Borzyszkowski, D. Bertacca and C. Porciani, LIGER: mock relativistic light-cones from Newtonian simulations, Monthly Notices of the Royal Astronomical Society471(2017) 3899

    2017

  21. [21]

    Percival, O

    W.J. Percival, O. Friedrich, E. Sellentin and A. Heavens, Matching bayesian and frequentist coverage probabilities when using an approximate data covariance matrix, MNRAS510(2022) 3207 [2108.10402]

    arXiv 2022

  22. [22]

    Eisenstein and M

    D.J. Eisenstein and M. Zaldarriaga, Correlations in the spatial power spectra inferred from angular clustering: Methods and application to the automated plate measuring survey, ApJ 546(2001) 2 [astro-ph/9912149]

    Pith/arXiv arXiv 2001

  23. [23]

    Gazta˜ naga and R

    E. Gazta˜ naga and R. Scoccimarro,The three-point function in large-scale structure: redshift distortions and galaxy bias, MNRAS361(2005) 824 [astro-ph/0501637]. – 20 –

    Pith/arXiv arXiv 2005

  24. [24]

    Farina, M

    A. Farina, M. Guidi, A. Veropalumbo and C. Guida, Denoising clustering covariance matrices with rotational invariant estimators, 2026

    2026

  25. [25]

    Pope and I

    A.C. Pope and I. Szapudi, Shrinkage estimation of the power spectrum covariance matrix, MNRAS389(2008) 766 [0711.2509]

    Pith/arXiv arXiv 2008

  26. [26]

    O’Connell, D

    R. O’Connell, D. Eisenstein, M. Vargas, S. Ho and N. Padmanabhan, Large covariance matrices: smooth models from the two-point correlation function, MNRAS462(2016) 2681 [1510.01740]

    Pith/arXiv arXiv 2016

  27. [27]

    Pearson and L

    D.W. Pearson and L. Samushia, Estimating the power spectrum covariance matrix with fewer mock samples, MNRAS457(2016) 993 [1509.00064]

    Pith/arXiv arXiv 2016

  28. [28]

    Friedrich and T

    O. Friedrich and T. Eifler, Precision matrix expansion – efficient use of numerical simulations in estimating errors on cosmological parameters, Monthly Notices of the Royal Astronomical Society473(2017) 4150–4163

    2017

  29. [29]

    Hall and A

    A. Hall and A. Taylor, A bayesian method for combining theoretical and simulated covariance matrices for large-scale structure surveys, MNRAS483(2019) 189 [1807.06875]

    Pith/arXiv arXiv 2019

  30. [30]

    Friedrich, F

    O. Friedrich, F. Andrade-Oliveira, H. Camacho, O. Alves, R. Rosenfeld, J. Sanchez et al., Dark energy survey year 3 results: covariance modelling and its impact on parameter estimation and quality of fit, MNRAS508(2021) 3125 [2012.08568]

    arXiv 2021

  31. [31]

    Fumagalli, A

    Euclid Collaboration, A. Fumagalli, A. Saro, S. Borgani, T. Castro, M. Costanzi et al., Euclid preparation. xxxv. covariance model validation for the two-point correlation function of galaxy clusters, A&A683(2024) A253 [2211.12965]

    arXiv 2024

  32. [32]

    Feldman, N

    H.A. Feldman, N. Kaiser and J.A. Peacock, Power-spectrum analysis of three-dimensional redshift surveys, The Astrophysical Journal426(1994) 23

    1994

  33. [33]

    Scoccimarro, M

    R. Scoccimarro, M. Zaldarriaga and L. Hui, Power spectrum correlations induced by nonlinear clustering, The Astrophysical Journal527(1999) 1–15

    1999

  34. [34]

    Meiksin and M

    A. Meiksin and M. White, The growth of correlations in the matter power spectrum, Monthly Notices of the Royal Astronomical Society308(1999) 1179–1184

    1999

  35. [35]

    Hamilton, C.D

    A.J.S. Hamilton, C.D. Rimes and R. Scoccimarro, On measuring the covariance matrix of the non-linear power spectrum from simulations, MNRAS371(2006) 1188 [astro-ph/0511416]

    Pith/arXiv arXiv 2006

  36. [36]

    Lacasa and R

    F. Lacasa and R. Rosenfeld, Combining cluster number counts and galaxy clustering, J. Cosmology Astropart. Phys.2016(2016) 005 [1603.00918]

    Pith/arXiv arXiv 2016

  37. [37]

    Grieb, A.G

    J.N. Grieb, A.G. S´ anchez, S. Salazar-Albornoz and C. Dalla Vecchia, Gaussian covariance matrices for anisotropic galaxy clustering measurements, MNRAS457(2016) 1577 [1509.04293]

    Pith/arXiv arXiv 2016

  38. [38]

    Howlett and W.J

    C. Howlett and W.J. Percival, Galaxy two-point covariance matrix estimation for next generation surveys, MNRAS472(2017) 4935 [1709.03057]

    Pith/arXiv arXiv 2017

  39. [39]

    Lacasa, Covariance of the galaxy angular power spectrum with the halo model, A&A615 (2018) A1 [1711.07372]

    F. Lacasa, Covariance of the galaxy angular power spectrum with the halo model, A&A615 (2018) A1 [1711.07372]

    Pith/arXiv arXiv 2018

  40. [40]

    Lacasa, The impact of braiding covariance and in-survey covariance on next-generation galaxy surveys, A&A634(2020) A74 [1909.00791]

    F. Lacasa, The impact of braiding covariance and in-survey covariance on next-generation galaxy surveys, A&A634(2020) A74 [1909.00791]

    arXiv 2020

  41. [41]

    Bayer, J

    A.E. Bayer, J. Liu, R. Terasawa, A. Barreira, Y. Zhong and Y. Feng, Super-sample covariance of the power spectrum, bispectrum, halos, voids, and their cross covariances, Phys. Rev. D108 (2023) 043521 [2210.15647]

    arXiv 2023

  42. [42]

    Sugiyama, S

    N.S. Sugiyama, S. Saito, F. Beutler and H.-J. Seo, Perturbation theory approach to predict the covariance matrices of the galaxy power spectrum and bispectrum in redshift space, Monthly Notices of the Royal Astronomical Society497(2020) 1684–1711. – 21 –

    2020

  43. [43]

    Wadekar and R

    D. Wadekar and R. Scoccimarro, Galaxy power spectrum multipoles covariance in perturbation theory, Physical Review D102(2020)

    2020

  44. [44]

    Philcox, D.J

    O.H.E. Philcox, D.J. Eisenstein, R. O’Connell and A. Wiegand, Rascalc: a jackknife approach to estimating single- and multitracer galaxy covariance matrices, MNRAS491(2020) 3290 [1904.11070]

    arXiv 2020

  45. [45]

    Philcox and D.J

    O.H.E. Philcox and D.J. Eisenstein, Computing the small-scale galaxy power spectrum and bispectrum in configuration space, MNRAS492(2020) 1214 [1912.01010]

    arXiv 2020

  46. [46]

    Wadekar, M.M

    D. Wadekar, M.M. Ivanov and R. Scoccimarro, Cosmological constraints from boss with analytic covariance matrices, Phys. Rev. D102(2020) 123521 [2009.00622]

    arXiv 2020

  47. [47]

    Forero-S´ anchez, M

    D. Forero-S´ anchez, M. Rashkovetskyi, O. Alves, A. de Mattia, N. Padmanabhan, H. Seo et al., Analytical and ezmock covariance validation for the desi 2024 results, J. Cosmology Astropart. Phys.2025(2025) 055 [2411.12027]

    arXiv 2024

  48. [48]

    Smith, Covariance of cross-correlations: towards efficient measures for large-scale structure, Monthly Notices of the Royal Astronomical Society400(2009) 851–865

    R.E. Smith, Covariance of cross-correlations: towards efficient measures for large-scale structure, Monthly Notices of the Royal Astronomical Society400(2009) 851–865

    2009

  49. [49]

    Blake, P

    C. Blake, P. Carter and J. Koda, Power spectrum multipoles on the curved sky: an application to the 6-degree field galaxy survey, MNRAS479(2018) 5168 [1801.04969]

    Pith/arXiv arXiv 2018

  50. [50]

    Abramo, J.V.D

    L.R. Abramo, J.V.D. Ferri and I.L. Tashiro, Fisher matrix for multiple tracers: the information in the cross-spectra, Journal of Cosmology and Astroparticle Physics2022(2022) 013

    2022

  51. [51]

    Barreira, On the impact of galaxy bias uncertainties on primordial non-gaussianity constraints, Journal of Cosmology and Astroparticle Physics2020(2020) 031–031

    A. Barreira, On the impact of galaxy bias uncertainties on primordial non-gaussianity constraints, Journal of Cosmology and Astroparticle Physics2020(2020) 031–031

    2020

  52. [52]

    Dournac, A

    Euclid Collaboration, F. Dournac, A. Blanchard, S. Ili´ c, B. Lamine, I. Tutusaus et al., Euclid preparation. xlvii. improving cosmological constraints using a new multi-tracer method with the spectroscopic and photometric samples, A&A690(2024) A30 [2404.12157]

    arXiv 2024

  53. [53]

    Beutler and E.D

    F. Beutler and E.D. Dio, Modeling relativistic contributions to the halo power spectrum dipole, Journal of Cosmology and Astroparticle Physics2020(2020) 048–048

    2020

  54. [54]

    Sefusatti, M

    E. Sefusatti, M. Crocce, S. Pueblas and R. Scoccimarro, Cosmology and the bispectrum, Physical Review D74(2006)

    2006

  55. [55]

    Dodelson and F

    S. Dodelson and F. Schmidt, 14 - analysis and inference, in Modern Cosmology (Third Edition), S. Dodelson and F. Schmidt, eds., pp. 401–435, Academic Press (2026), DOI

    2026

  56. [56]

    Addis, S.L

    C. Addis, S.L. Guedezounme, J. Hammond, C. Clarkson, F. Montano, S. Camera et al., Unbiased analysis of primordial non-gaussianity: the multipoles of the full relativistic power spectrum, arXiv e-prints (2025) arXiv:2511.09466 [2511.09466]

    Pith/arXiv arXiv 2025

  57. [57]

    White, Y.-S

    M. White, Y.-S. Song and W.J. Percival, Forecasting cosmological constraints from redshift surveys, Monthly Notices of the Royal Astronomical Society397(2009) 1348–1354

    2009

  58. [58]

    G.-B. Zhao, Y. Wang, A. Taruya, W. Zhang, H. Gil-Mar´ ın, A. de Mattia et al., The completed sdss-iv extended baryon oscillation spectroscopic survey: a multitracer analysis in fourier space for measuring the cosmic structure growth and expansion rate, Monthly Notices of the Royal Astronomical Society504(2021) 33–52

    2021

  59. [59]

    Dlamini, S

    S. Dlamini, S. Jolicoeur and R. Maartens, Constraining the growth rate on linear scales by combining skao and desi surveys, The European Physical Journal C84(2024)

    2024

  60. [60]

    Barberi-Squarotti, S

    M. Barberi-Squarotti, S. Camera and R. Maartens, Radio-optical synergies at high redshift to constrain primordial non-gaussianity, Journal of Cosmology and Astroparticle Physics2024 (2024) 043

    2024

  61. [61]

    Jolicoeur, R

    S. Jolicoeur, R. Maartens and S. Dlamini, Constraining primordial non-gaussianity by combining next-generation galaxy and 21 cm intensity mapping surveys, The European Physical Journal C83(2023) . – 22 –

    2023

  62. [62]

    Kopana, S

    M. Kopana, S. Jolicoeur and R. Maartens, Multi-tracing the primordial universe with future surveys, The European Physical Journal C84(2024)

    2024

  63. [63]

    Montano and S

    F. Montano and S. Camera, Detecting relativistic doppler in galaxy clustering with tailored galaxy samples, Physics of the Dark Universe46(2024) 101570

    2024

  64. [64]

    Montano and S

    F. Montano and S. Camera, Detecting relativistic doppler by multi-tracing a single galaxy population, Physics of the Dark Universe46(2024) 101634

    2024

  65. [65]

    Karagiannis, R

    D. Karagiannis, R. Maartens, J. Fonseca, S. Camera and C. Clarkson, Multi-tracer power spectra and bispectra: formalism, Journal of Cosmology and Astroparticle Physics2024(2024) 034

    2024

  66. [66]

    Abramo and D

    L.R. Abramo and D. Bertacca, Disentangling the effects of doppler velocity and primordial non-gaussianity in galaxy power spectra, Phys. Rev. D96(2017) 123535 [1706.01834]

    Pith/arXiv arXiv 2017

  67. [67]

    Jeong and F

    D. Jeong and F. Schmidt, Parity-odd galaxy bispectrum, Phys. Rev. D102(2020) 023530 [1906.05198]

    arXiv 2020

  68. [68]

    P. Paul, C. Clarkson and R. Maartens, Apparent parity violation in the observed galaxy trispectrum, Phys. Rev. Lett.133(2024) 121001 [2402.16478]

    arXiv 2024

  69. [69]

    Aghanim, Y

    Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi et al., Planck 2018 results. vi. cosmological parameters, A&A641(2020) A6 [1807.06209]. – 23 –

    Pith/arXiv arXiv 2018