Model-independent tests of the random phase hypothesis (Rao's-statistic analyses of the phases of the spherical harmonic coefficients of the CMB temperature field) report statistically significant phase non-uniformities and inter-mode phase correlations at multipoles up to ℓ≤128\ell\le 128 in the foreground-cleaned Planck maps (COMMANDER, SMICA, NILC, SEVEM) and the WMAP 9-year ILC (internal linear combination) map. The physical origin of these phase non-Gaussianities is undetermined: they could be primordial (non-Gaussian inflationary perturbations), residual foreground contamination or unaccounted systematics of component separation (including masking and a posteriori statistical effects), or another cosmological phenomenon. The problem is to determine, with quantitative discriminating evidence, which of these origins produces the reported detections.

Background

The cosmic microwave background (CMB) is the relic radiation field last scattered about 380,000 years after the Big Bang; its temperature anisotropies are conventionally expanded in spherical harmonics, δT(n^)=∑ℓmaℓmYℓm(n^)\delta T(\hat{n})=\sum_{\ell m}a_{\ell m}Y_{\ell m}(\hat{n}), with complex coefficients aℓm=∣aℓm∣eiϕℓma_{\ell m}=|a_{\ell m}|e^{i\phi_{\ell m}} whose arguments ϕℓm\phi_{\ell m} are the spherical harmonic phases. Standard inflationary cosmology predicts that the primordial temperature field is a Gaussian random field (GRF). A foundational property of a GRF on the sphere, dating to Bardeen, Bond, Kaiser and Szalay (1986), is the random phase hypothesis: the phases ϕℓm\phi_{\ell m} are independent and identically distributed uniformly on [0,2π][0,2\pi]. Any phase non-uniformity or phase correlation is therefore a model-independent signature of non-Gaussianity. Detecting non-Gaussianity in the CMB matters because it discriminates among inflationary models; conversely, apparent non-Gaussianity can also arise from Galactic foregrounds (thermal dust, synchrotron, and free-free emission from ionized gas) or from residual systematics of mapmaking and component separation, so its origin must be identified before it can be interpreted. The standard search channel is the bispectrum, the three-point function of the aℓma_{\ell m}, summarized by an amplitude fNLf_{\rm NL}; the Planck collaboration's bispectrum analyses found no significant primordial signal, with fNLlocal=−0.9±5.1f_{\rm NL}^{\rm local}=-0.9\pm 5.1 in the 2018 temperature-and-polarization analysis. Component separation produces the publicly released cleaned maps COMMANDER, SMICA, NILC and SEVEM (Planck), and the ILC (internal linear combination) map (WMAP 9-year), each removing Galactic and extra-galactic emission by different statistical algorithms; residual foregrounds and the Galactic mask used in analysis can themselves correlate or non-uniformize phases. Phase statistics have a smaller literature than the bispectrum. Kovács, Szapudi and Frei (2013) analyzed WMAP 7-year phases and found high-ℓ\ell phase correlations reaching 4.7σ4.7\sigma significance that they attributed to Galactic foreground contamination: after masking contaminated regions no significant non-Gaussianity remained. Kovács, Carron and Szapudi (2013) showed that the generalized phases (a phase-like statistic defined for masked skies) of the WMAP and Planck maps are highly coherent (at 5σ5\sigma for ℓ<700\ell<700), establishing that the phase structure of these maps is real sky signal rather than instrumental noise, but without distinguishing primordial from foreground origin. Chiang and Naselsky (2007) developed phase analysis for incomplete sky coverage and stressed that the sky cut itself can induce apparent phase structure. Planck's own isotropy-and-statistics tests (Planck 2015 results XVI) found the cleaned maps broadly consistent with Gaussianity apart from the well-known large-scale anomalies (cold spot, hemispherical power asymmetry, low variance), whose origin also remains unexplained (Schwarz et al. 2016). Within this landscape, Yadav and Saha (2022) introduced a new model-independent test: Rao's statistic, a non-parametric test of circular uniformity based on the ordered arc-lengths between successive phases of a sample. Applying six test types (uniformity of phases at fixed ℓ\ell or fixed mm; correlations between neighboring and next-to-neighboring mm modes at fixed ℓ\ell; correlations between neighboring and next-to-neighboring ℓ\ell modes at fixed mm) at ℓ≤128\ell\le 128, calibrated against 10000 Monte Carlo simulations of uniform independent phases with significance levels α=0.05\alpha=0.05 and 0.0010.001, they validated the method on Gaussian and non-Gaussian simulations and on foreground-contaminated Planck 100 and 143 GHz channel maps, and then applied it to the cleaned maps. They reported non-uniform or correlated phases for specific multipoles — repeated non-uniformity at ℓ=57\ell=57 across COMMANDER and SMICA, significant correlations at ℓ=17,19,62,109\ell=17,19,62,109 and 13,14,27,47,12813,14,27,47,128, and correlated mode pairs such as (10,11)(10,11), (20,21)(20,21), (88,89)(88,89), (111,112)(111,112) and (12,14)(12,14) — concluding that the presence of these signals violates the random phase hypothesis but that their underlying cause is not established.

Problem Statement

Determine the physical origin of the statistically significant spherical-harmonic-phase non-uniformities and inter-mode phase correlations that Rao's-statistic tests (a non-parametric test of circular uniformity applied to phase sets) of the random phase hypothesis (the Gaussian-field prediction that the phases are uniform and independent) detect at multipoles ℓ≤128\ell\le 128 in the foreground-cleaned Planck CMB temperature maps (COMMANDER, SMICA, NILC, SEVEM) and the WMAP 9-year ILC map (Yadav and Saha 2022). Specifically, establish with quantitative evidence which of the following produces the reported signals — primordial non-Gaussian inflationary perturbations; residual foreground contamination or unaccounted systematic effects of the component separation, including masking/incomplete-sky effects and a posteriori statistical artifacts from the multiplicity of tests; or some other cosmological phenomenon. Any one of these — or a combination of them — counts as a resolution when established with quantitative evidence.

References

  1. Sarvesh Kumar Yadav and Rajib Saha, 'Investigating non-Gaussianity in Cosmic Microwave Background temperature maps using spherical harmonic phases', Journal of Cosmology and Astroparticle Physics 01 (2022) 001, DOI: 10.1088/1475-7516/2022/01/001, arXiv:2001.10960, https://arxiv.org/abs/2001.10960 (source of the problem statement)
  2. J. M. Bardeen, J. R. Bond, N. Kaiser and A. S. Szalay, 'The statistics of peaks of Gaussian random fields', The Astrophysical Journal 304 (1986) 15, DOI: 10.1086/164143, https://doi.org/10.1086/164143
  3. A. Kovács, I. Szapudi and Z. Frei, 'Phase statistics of the WMAP 7 year data', Astronomische Nachrichten 334 (2013) 1020-1023, DOI: 10.1002/asna.201211985, arXiv:1308.0837, https://arxiv.org/abs/1308.0837
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  10. Gabriel Jung, Michele Citran, Bartjan van Tent, Léa Dumilly and Nabila Aghanim, 'Constraints on primordial non-Gaussianity from Planck PR4 data', Astronomy & Astrophysics 702 (2025) A204, DOI: 10.1051/0004-6361/202555283, arXiv:2504.00884, https://arxiv.org/abs/2504.00884