Testing scale-dependent primordial non-Gaussianity as the origin of the large-scale CMB anomalies: a joint bispectrum and power-modulation analysis beyond the Sachs-Wolfe approximation

The hemispherical power asymmetry and the low-multipole power deficit of the cosmic microwave background (CMB) persist at the 2-3 sigma level across WMAP and Planck. Adhikari, Shandera and Erickcek (Phys. Rev. D 93, 023524 (2016)) showed, in the Sachs-Wolfe approximation, that scale-dependent local primordial non-Gaussianity - a bispectrum amplitude $f_{\mathrm{NL}}(k)$ that grows toward large scales - coupled to our particular realization of superhorizon modes generates exactly the observed kind of dipole (and monopole) power modulation, without invoking any exotic superhorizon fluctuation. They posed, but left open, the question of whether actual CMB data favor this explanation. Answering it requires going beyond their simplified setting: a joint analysis on real data that constrains the non-Gaussianity amplitude and tilt, together with the number of superhorizon e-folds, simultaneously from CMB bispectrum estimators and from measurements of the dipole and monopole power modulation as functions of scale, using realistic temperature transfer functions (with polarization included where informative), noise and sky masks, with the non-Gaussian covariance among these estimators included. No such analysis exists: Planck's running non-Gaussianity constraints use the bispectrum alone, and Bayesian treatments of the asymmetry use the modulation alone. The problem is to construct and execute this joint analysis, and thereby decide whether the large-scale anomalies are evidence for scale-dependent non-Gaussianity or a new bound on it.

Background

The cosmic microwave background (CMB) temperature fluctuations are statistically isotropic to high precision, but on the largest angular scales several departures from the simplest expectations persist at the 2-3 sigma level in data from both the WMAP and Planck CMB satellite missions: a relative lack of power at multipoles $\ell \lesssim 40$ compared with the best-fit Lambda-CDM cosmology, and a hemispherical power asymmetry - the fluctuation amplitude in one half of the sky exceeds that in the other by roughly 5-7% on large scales, conventionally modeled as a dipole modulation $\Delta T/T \rightarrow (1 + A(\ell)\,\hat{n}\cdot\hat{p})\,\Delta T/T$ with amplitude $A \approx 0.06$ that weakens toward small scales, plus an associated monopole (direction-independent) modulation $A_0(\ell)$ of the power amplitude.

Primordial non-Gaussianity (NG) is commonly parameterized by the local ansatz $\Phi = \phi + f_{\mathrm{NL}}(\phi^2 - \langle\phi^2\rangle)$, where $\Phi$ is the Bardeen gravitational potential (the Newtonian-gauge metric perturbation whose fluctuations source the CMB anisotropies) and $\phi$ a Gaussian field; the resulting bispectrum (three-point function in harmonic space) is strongest in squeezed configurations (one wavelength much longer than the other two). Planck bounds the scale-independent local amplitude at its pivot scales to $f_{\mathrm{NL}}^{\mathrm{local}} = -0.9 \pm 5.1$; the constraint from the lowest multipoles alone is far weaker, allowing $|f_{\mathrm{NL}}|$ of order a few hundred on large scales. Erickcek, Kamionkowski and Carroll (Phys. Rev. D 78, 123520 (2008)) showed that single-field slow-roll inflation cannot generate the observed asymmetry without violating the homogeneity of the universe, and proposed instead a large superhorizon perturbation (wavelength exceeding the observable universe) in a curvaton-like field (a light field distinct from the inflaton that can generate the observed curvature perturbations); such mechanisms are constrained by the quadrupole and octupole they induce through the Grishchuk-Zel'dovich effect (the CMB anisotropy generated by gradients of superhorizon modes).

Adhikari, Shandera and Erickcek (2016) observed that local-type NG offers a generic alternative that needs no exotic superhorizon fluctuation: with local NG, small-scale modes are coupled to longer-wavelength modes, so within any finite sub-volume a particular realization of modes with wavelengths larger than the sub-volume modulates the observed small-scale power, even though the universe as a whole is statistically isotropic. Our observable volume is one such realization, so there is a computable 'non-Gaussian cosmic variance' probability of a dipole and monopole power modulation. If the NG is scale-dependent, $f_{\mathrm{NL}}(k) = f_{\mathrm{NL}}^0 (k/k_0)^{n_{f_{\mathrm{NL}}}}$ with a red tilt $n_{f_{\mathrm{NL}}} < 0$ (their example: $f_{\mathrm{NL}}(\ell) = 50\,(\ell/60)^{-0.64}$), the modulation is confined to the lowest multipoles as observed while Planck's small-scale bispectrum bounds are respected; an amplitude allowed by the weak large-scale-only constraint then suffices to make the observed asymmetry statistically unremarkable. The number of superhorizon e-folds $N_{\mathrm{extra}}$ (how far the primordial spectrum extends beyond our horizon) controls the variance of the superhorizon realization and hence the predicted spread of modulation amplitudes.

The 2016 paper derived the probability distributions of $A$ and $A_0$ analytically, confirmed them with numerical CMB maps, and performed Bayesian parameter estimation and model comparison. But all of this was done in the Sachs-Wolfe approximation ($\Delta T/T = -\Phi/3$, valid only at the very lowest multipoles), with schematic likelihoods built from single measured values of $A$ and $A_0$ rather than from the actual sky maps, and without realistic transfer functions (the functions relating primordial perturbations to the observed CMB temperature and E-mode polarization anisotropies across all multipoles, where E-mode polarization is the curl-free component of the CMB polarization pattern), instrumental noise, or sky masks. Whether the actual data favor the scale-dependent NG explanation is therefore still unknown.

Problem Statement

Determine, by a full realistic analysis of CMB data, whether scale-dependent local primordial non-Gaussianity can account for the large-scale CMB anomalies. Concretely: extend the Adhikari-Shandera-Erickcek framework beyond the Sachs-Wolfe approximation - using the observed CMB temperature transfer functions (and E-mode-polarization transfer functions where informative) over the full range of measured multipoles, with realistic instrumental noise and sky masks - and use it to constrain the parameters of scale-dependent local non-Gaussianity (amplitude $f_{\mathrm{NL}}^0$ and spectral index $n_{f_{\mathrm{NL}}}$, together with the number of superhorizon e-folds $N_{\mathrm{extra}}$ that controls the variance of the superhorizon realization) through a joint likelihood that combines bispectrum-estimator measurements of $f_{\mathrm{NL}}(k)$ with measurements of the dipole power modulation $A(\ell)$ and the monopole power modulation $A_0(\ell)$, including the non-Gaussian covariance among these observables induced by the same superhorizon realization, and marginalizing over that realization. The required deliverables, which jointly support this single objective, are: (i) validated analytic and/or Monte-Carlo expressions for the joint statistics of the modulation and bispectrum estimators with full transfer functions, replacing the Sachs-Wolfe-only formulas; (ii) posterior constraints on $(f_{\mathrm{NL}}^0, n_{f_{\mathrm{NL}}}, N_{\mathrm{extra}})$ from actual CMB temperature data (with polarization included where informative), or - if current data are shown to be insufficient - a demonstrably sufficient forecast establishing what data quality would be required; and (iii) the Bayesian evidence for the scale-dependent non-Gaussian model relative to the Gaussian, statistically isotropic Lambda-CDM model. The outcome either establishes the first observational link between the hemispherical asymmetry / low-multipole power deficit and primordial non-Gaussianity, or converts the anomalies into a sharp joint bound on strongly tilted $f_{\mathrm{NL}}(k)$ at the largest scales.

References

1. S. Adhikari, S. Shandera, A. L. Erickcek, 'Large-scale anomalies in the cosmic microwave background as signatures of non-Gaussianity', Phys. Rev. D 93, 023524 (2016), DOI 10.1103/PhysRevD.93.023524, https://arxiv.org/abs/1508.06489
2. Planck Collaboration: Y. Akrami, F. Arroja, M. Ashdown, et al., 'Planck 2018 results. IX. Constraints on primordial non-Gaussianity', arXiv:1905.05697 (2019), https://arxiv.org/abs/1905.05697
3. C. T. Byrnes, D. Regan, D. Seery, E. R. M. Tarrant, 'The hemispherical asymmetry from a scale-dependent inflationary bispectrum', JCAP 06 (2016) 025, DOI 10.1088/1475-7516/2016/06/025, https://arxiv.org/abs/1511.03129
4. C. T. Byrnes, D. Regan, D. Seery, E. R. M. Tarrant, 'Implications of the CMB power asymmetry for the early universe', arXiv:1601.01970 (2016), https://arxiv.org/abs/1601.01970
5. S. Adhikari, A.-S. Deutsch, S. Shandera, 'Statistical anisotropies in temperature and polarization fluctuations from a scale-dependent trispectrum', Phys. Rev. D 98, 023520 (2018), DOI 10.1103/PhysRevD.98.023520, https://arxiv.org/abs/1805.00037
6. J. Muir, S. Adhikari, D. Huterer, 'Covariance of CMB anomalies', Phys. Rev. D 98, 023521 (2018), DOI 10.1103/PhysRevD.98.023521, https://arxiv.org/abs/1806.02354
7. S. Shaikh, S. Mukherjee, S. Das, B. D. Wandelt, T. Souradeep, 'Joint Bayesian analysis of large angular scale CMB temperature anomalies', arXiv:1902.10155 (2019), https://arxiv.org/abs/1902.10155
8. A. L. Erickcek, M. Kamionkowski, S. M. Carroll, 'A Hemispherical Power Asymmetry from Inflation', Phys. Rev. D 78, 123520 (2008), DOI 10.1103/PhysRevD.78.123520, https://arxiv.org/abs/0806.0377
9. I. Agullo, D. Kranas, V. Sreenath, 'Large scale anomalies in the CMB and non-Gaussianity in bouncing cosmologies', Class. Quantum Grav. 38, 065010 (2021), DOI 10.1088/1361-6382/abc521, https://arxiv.org/abs/2006.09605
10. F. K. Hansen, T. Trombetti, N. Bartolo, U. Natale, M. Liguori, A. J. Banday, K. M. Górski, 'Isotropic non-Gaussian g_NL-like toy models that reproduce cosmic microwave background anomalies', Astron. Astrophys. 626, A13 (2019), DOI 10.1051/0004-6361/201833698, https://arxiv.org/abs/1806.08531
11. D. J. Schwarz, C. J. Copi, D. Huterer, G. D. Starkman, 'CMB Anomalies after Planck', Class. Quantum Grav. 33, 184001 (2016), DOI 10.1088/0264-9381/33/18/184001, https://arxiv.org/abs/1510.07929
12. L. Herold, G. E. Addison, C. L. Bennett, H. C. Nofi, J. L. Weiland, 'Nearly Full-sky Low-multipole Cosmic Microwave Background Temperature Anisotropy. III. Cosmic Microwave Background Anomalies', Astrophys. J. 1005, 146 (2026), DOI 10.3847/1538-4357/ae6860, https://arxiv.org/abs/2509.03720
13. G. Jung, N. Aghanim, J. G. Sorce, B. Seidel, K. Dolag, M. Douspis, 'Revisiting the CMB large-scale anomalies: The impact of the Sunyaev-Zeldovich signal from the Local Universe', Astron. Astrophys. 692, A180 (2024), DOI 10.1051/0004-6361/202451238, https://arxiv.org/abs/2406.11543
14. B. Li, Z. Chen, Y.-F. Cai, Y. Mao, 'Testing the scale-dependent hemispherical asymmetry with the 21-cm power spectrum from the epoch of reionization', Mon. Not. R. Astron. Soc. 487, 5564 (2019), DOI 10.1093/mnras/stz1619, https://arxiv.org/abs/1904.04683
15. J. I. Cayuso, M. C. Johnson, 'Towards testing CMB anomalies using the kinetic and polarized Sunyaev Zel'dovich effects', Phys. Rev. D 101, 123508 (2020), DOI 10.1103/PhysRevD.101.123508, https://arxiv.org/abs/1904.10981
16. C. Jung, M. Citran, B. van Tent, L. Dumilly, N. Aghanim, 'Constraints on primordial non-Gaussianity from Planck PR4 data', Astron. Astrophys. 702, A204 (2025), DOI 10.1051/0004-6361/202555283, https://doi.org/10.1051/0004-6361/202555283