The study, published in The European Physical Journal C by Yang Han, Lu Chen, Guo-Hao Li and Pei-Yuan Xu, dissects the CMB angular power spectrum into its constituent physical contributions. Rather than treating the cosmic microwave background as a single curve to be fitted, the researchers introduced six phenomenological amplitude parameters, each of which rescales one specific effect: the Sachs–Wolfe effect, the Doppler effect, the early and late Integrated Sachs–Wolfe effects, polarization from anisotropic Thomson scattering, and gravitational lensing by large-scale structure. In the standard Lambda Cold Dark Matter model, every one of these amplitudes should equal exactly one. Any significant departure would signal that the model’s internal accounting of the early Universe is incomplete.

To carry out the test, the team modified the CAMB Boltzmann solver, the standard code for computing CMB power spectra, so that each physical contribution could be scaled independently, and then ran Markov Chain Monte Carlo analyses with the Cobaya sampler. They used two data combinations. The first, dubbed PDP, combines Planck PR3 low-multipole temperature and polarization data, the Planck PR4 CamSpec high-multipole likelihood based on NPIPE maps covering multipoles from 30 to 2500, Planck PR4 lensing reconstruction, BAO measurements from the Dark Energy Spectroscopic Instrument Data Release 2, and the uncalibrated PantheonPlus sample of 1701 spectroscopically confirmed Type Ia supernovae spanning redshifts from 0.01 to 2.26. The second, PADP, swaps the high-multipole Planck likelihood for a cut version retaining multipoles up to 1000 for temperature and 600 for polarization, and adds power spectra from the Atacama Cosmology Telescope Data Release 6, which measures multipoles from 600 up to 8500, together with the joint ACT DR6 plus Planck lensing likelihood extending to lensing multipoles of 1250.

The headline result concerns the lensing amplitude. Gravitational lensing by intervening matter deflects CMB photons on their journey from the last scattering surface, smoothing the acoustic peaks of the power spectra and converting E-mode polarization into B-modes. The parameterization rescales the lensing potential power spectrum, and the analysis found a lensing amplitude of 1.0498, with uncertainties of roughly 0.028, from the PDP combination, and 1.0740 plus or minus 0.0273 from PADP. Those correspond to deviations of 1.77 and 2.71 standard deviations from the expected value of one. The lensing anomaly is not new — the Planck collaboration itself reported a preference for enhanced lensing at about three sigma using only its temperature and polarization spectra, a preference that softened to two-sigma consistency once Planck’s own lensing reconstruction was included — but the new work shows how the tension behaves when modern external datasets enter the fit.

The most striking finding is that the DESI DR2 baryon acoustic oscillation measurements, drawn from bright galaxy samples, luminous red galaxies, emission line galaxies, quasars and Lyman-alpha forests across redshifts from 0.1 to 4.2, are the main driver of the strengthened anomaly. The BAO data pull the matter density parameter toward lower values — in the PADP framework, from 0.3109 plus or minus 0.0075 without BAO to 0.2992 plus or minus 0.0037 with it — and because the lensing amplitude and the matter density are anticorrelated in every data combination the team examined, this shift pushes the preferred lensing amplitude upward. Within PADP, the anomaly stands at 1.45 sigma when only CMB and lensing data are used, but climbs to 2.83 sigma once DESI DR2 is added. CMB lensing measurements, by contrast, pull the anomaly back down, from 1.30 to 0.78 sigma in PDP and from 1.81 to 1.45 sigma in PADP. The PantheonPlus supernovae play only a minor role.

The authors are careful about what this means. The stronger preference in the PADP combination, they note, reflects the interplay of Planck likelihood choices, lensing reconstruction and external datasets rather than anything specific to the ACT data, which on its own measures a lensing amplitude of 1.007 plus or minus 0.057, fully consistent with unity. A joint CMB analysis by Tristram and colleagues similarly reports 1.027 plus or minus 0.058. The picture that emerges is of a lensing anomaly whose strength depends delicately on which datasets are combined, with the DESI galaxy survey now acting as the decisive lever through its tight grip on the matter density.

The other five amplitudes largely behave themselves. The Sachs–Wolfe amplitude came out at 0.9938 from PDP and 0.9973 from PADP, deviations of 1.13 and 0.59 sigma, but the researchers explicitly warn against reading this as a physical anomaly. The Sachs–Wolfe term is the dominant gravitational source at the last scattering surface and affects a broad range of multipoles, making it strongly and negatively degenerate with the combination of the primordial curvature perturbation amplitude and the reionization optical depth, which together set the overall normalization of the CMB spectra. When the Sachs–Wolfe amplitude is freed, that normalization shifts by 1.4 sigma in PDP and 0.5 sigma in PADP, indicating the constraint is partly driven by this interplay. The Doppler amplitude landed at 0.9991 and 1.0034 in the two combinations, the early Integrated Sachs–Wolfe amplitude at 1.0079 and 1.0127, and the polarization amplitude at 1.0019 and 0.9981 — all consistent with the standard prediction.

One parameter proved essentially unconstrainable. The late Integrated Sachs–Wolfe effect, generated by the decay of gravitational potentials as dark energy accelerates cosmic expansion, leaves its imprint only at low multipoles, and its influence at multipoles of 30 and above is negligible. The analysis could only place an upper bound of 0.7508 from PDP and a broad, poorly constrained interval from PADP — less stringent than the Planck collaboration’s own limits, a consequence of the different data combinations employed. The team also verified that their boundary choice separating the early and late ISW regimes, set at redshift 30 following previous studies, leaves the power spectra essentially unchanged when varied between 10 and 50.

Where the ground-based data genuinely earns its keep is polarization. Including the ACT DR6 high-multipole measurements tightened the uncertainty on the polarization amplitude by approximately 35 percent relative to the Planck-only combination, a direct reflection of the parameter’s strong influence on the high-multipole temperature-polarization and polarization-polarization spectra. The authors argue this demonstrates that small-scale measurements from ground-based facilities, including future experiments such as SPT-3G and the Simons Observatory, provide unique information that satellite data alone cannot supply, and should be routinely incorporated in future CMB analyses.

The broader cosmological tensions remain stubborn. Across all the one-parameter extensions, the inferred Hubble constant stayed near 68 kilometers per second per megaparsec, offering no relief from the discrepancy with the locally measured value of roughly 73. For the sigma-8 tension — the roughly two-sigma mismatch between the CMB-predicted amplitude of matter fluctuations and lower values from late-time structure — the lensing extension provided only mild alleviation: the team obtained a structure growth parameter of 0.8028 from both combinations, statistically compatible with the Dark Energy Survey Year 6 cosmic shear result of 0.798, though the authors stress this falls short of a full resolution. In the end, the analysis delivers a nuanced verdict: the standard model’s accounting of CMB physics holds up effect by effect, with the lone exception of a lensing amplitude whose significance now hinges, more than ever, on the galaxy maps used to pin down the matter density of the Universe.

Subject of Research: Constraints on phenomenological amplitude parameters of CMB anisotropy effects using Planck, ACT, DESI and PantheonPlus data

Article Title: Constraints on phenomenological amplitudes of CMB anisotropy from multiple datasets

Article References: Han, Y., Chen, L., Li, G.-H., & Xu, P.-Y. (2026). Constraints on phenomenological amplitudes of CMB anisotropy from multiple datasets. The European Physical Journal C, 86(10), Article 1234. https://doi.org/10.1140/epjc/s10052-026-16506-1

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16506-1

Keywords: cosmic microwave background, lensing anomaly, Lambda CDM, Planck, Atacama Cosmology Telescope, DESI DR2, PantheonPlus, Hubble tension, sigma-8 tension, polarization, Sachs-Wolfe effect, cosmological parameters