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Constraints on Phenomenological Amplitudes of CMB Anisotropy with Multi-Datasets

This paper constrains six phenomenological amplitude parameters for CMB anisotropy effects using multi-dataset combinations (Planck, ACT, DESI, PantheonPlus), finding that only the lensing amplitude ( $A_{\rm{L}}$ ) shows a significant deviation from the standard $\Lambda$ CDM model while no extensions notably resolve the Hubble or $\sigma_8$ tensions.

Original authors: Yang Han, Lu Chen, Guo-Hao Li, Pei-Yuan Xu

Published 2026-05-05

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CC BY 4.0

Original authors: Yang Han, Lu Chen, Guo-Hao Li, Pei-Yuan Xu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, ancient drum that was struck at the very moment of its birth. The sound waves rippling across that drum are what we call the Cosmic Microwave Background (CMB). It's the oldest light in existence, a faint glow that fills the entire sky. By studying the tiny ripples and bumps in this light, cosmologists try to understand how the universe works.

For decades, scientists have used a "standard recipe" called the $\Lambda$ CDM model to predict what this drum should sound like. It's a very successful recipe, but recently, the actual sound of the universe has started to sound a little "off-key." There are some persistent notes that don't quite match the recipe, creating what scientists call "tensions" (like the Hubble tension, where different ways of measuring the universe's expansion rate disagree).

The Experiment: Tuning the Dials

In this paper, the researchers asked a simple question: What if we tweak the volume of specific physical effects that shape this cosmic drum?

They identified six key "knobs" or "dials" that control how the CMB light behaves:

Lensing ( $A_L$ ): Like looking at the drum through a wavy glass lens, which distorts the image.
Sachs-Wolfe ( $A_{SW}$ ): The gravitational "redshift" or "blueshift" as light climbs out of deep gravity wells.
Doppler ( $A_{Dop}$ ): The shift in sound caused by the movement of gas clouds (baryons) when the light was emitted.
Early ISW ( $A_{eISW}$ ): A subtle effect from the transition between the radiation-dominated and matter-dominated eras.
Late ISW ( $A_{lISW}$ ): A late-stage effect caused by dark energy stretching space.
Polarization ( $A_{Pol}$ ): The direction in which the light waves vibrate.

The researchers used a super-computer simulation (a modified version of a tool called CAMB) to turn these dials up and down. They then compared their new, tweaked predictions against real data from three major sources:

Planck: A space satellite that mapped the whole sky.
ACT: A telescope on the ground in the Atacama Desert that looks at very small, detailed spots.
DESI & PantheonPlus: Data on the distribution of galaxies and exploding stars (supernovae) to help measure distances.

The Findings: Which Knob is Broken?

After crunching the numbers, here is what they found:

The "Lensing" Knob is Stuck: The only dial that the data strongly suggests needs to be turned up is the Lensing one ( $A_L$ ). The data says the lensing effect is about 3% stronger than the standard recipe predicts. This is a significant finding, suggesting that the way gravity bends light in the early universe might be more intense than we thought.
The Other Knobs are Fine: The other five dials (Sachs-Wolfe, Doppler, etc.) all seem to be set exactly where the standard recipe says they should be. The data doesn't require any changes to them.
The "Late ISW" Knob is Hard to Hear: The effect of the "Late ISW" dial is so faint and only happens on the largest scales that the current data can't really measure it accurately. It's like trying to hear a whisper in a hurricane.

Did It Fix the "Off-Key" Notes?

The researchers hoped that by tweaking these dials, they might finally solve the universe's "off-key" problems, specifically the Hubble tension (disagreement on expansion speed) and the $\sigma_8$ tension (disagreement on how clumpy matter is).

The result? No.
Even with the Lensing dial turned up, the Hubble tension remains. The universe still expands at a rate that doesn't quite match the local measurements. The "clumpiness" of matter is slightly better explained, but not enough to fully resolve the mystery.

The Power of Ground-Based Telescopes

A major highlight of this paper is the comparison between using just the space satellite (Planck) versus adding the ground-based telescope (ACT).

Adding the ACT data was like putting on high-definition glasses. It didn't change the main conclusion (that Lensing is high), but it made the measurement of that Lensing effect much sharper and more certain.
It also drastically improved the measurement of the Polarization knob, reducing the uncertainty by more than ten times. This proves that ground-based telescopes are essential for seeing the fine details of the cosmic drum.

The Bottom Line

The universe's "drum" is mostly playing the standard song, but the Lensing effect is playing slightly louder than the sheet music says it should. While this discovery is interesting, simply turning up that one volume knob doesn't fix the other musical problems (the Hubble and $\sigma_8$ tensions) that have been bothering cosmologists. To solve those, we likely need a completely new instrument or a new song, not just a volume adjustment.

Technical Summary: Constraints on Phenomenological Amplitudes of CMB Anisotropy with Multi-Datasets

Problem Statement
While the standard $\Lambda$ CDM model successfully describes cosmic evolution, it faces persistent internal inconsistencies, notably the Hubble tension (discrepancy between local $H_0$ measurements and CMB-inferred values) and the $\sigma_8$ tension (discrepancy in matter fluctuation amplitudes). Additionally, Planck data has shown a mild preference for an enhanced gravitational lensing effect ( $A_L > 1$ ). To systematically investigate the consistency of the $\Lambda$ CDM model and the specific physical mechanisms contributing to Cosmic Microwave Background (CMB) anisotropies, this paper employs a model-independent approach. It tests whether rescaling the amplitudes of individual physical effects—specifically Lensing, Sachs–Wolfe (SW), Doppler, early Integrated Sachs–Wolfe (eISW), late Integrated Sachs–Wolfe (lISW), and Polarization—can resolve these tensions or reveal deviations from standard predictions.

Methodology
The authors modified the CAMB Boltzmann solver to introduce six phenomenological amplitude parameters, denoted as $A_{new}$ (where $new \in \{L, SW, Dop, eISW, lISW, Pol\}$ ). Each parameter rescales the contribution of its corresponding physical effect to the CMB angular power spectra ($TT$, $TE$, $EE$). In the standard $\Lambda$ CDM model, all $A_{new}$ are expected to equal 1.

The study constrained these parameters within the $\Lambda$ CDM+ $A_{new}$ framework using two distinct data combinations:

PDP: Planck (PR3/PR4) + DESI DR2 (BAO) + PantheonPlus (SNIa).
PADP: Planck + ACT DR6 (high-resolution ground-based CMB) + DESI DR2 + PantheonPlus.

Markov Chain Monte Carlo (MCMC) simulations were performed using the Cobaya package. The analysis focused on seven free parameters: $\{\Omega_c h^2, \Omega_b h^2, A_s, n_s, \tau, \theta_{MC}, A_{new}\}$ . The statistical preference of the extended models over the base $\Lambda$ CDM model was evaluated using the Akaike Information Criterion (AIC).

Key Contributions and Results

Lensing Anomaly ( $A_L$ ): The $\Lambda$ $Λ$ CDM+ $A_L$ $A_{L}$ model is the only extended scenario favored by the data according to the AIC.
- From the PDP dataset: $A_L = 1.0656^{+0.0304}_{-0.0303}$ ( $2.16\sigma$ deviation from 1).
- From the PADP dataset (including ACT DR6): $A_L = 1.0795^{+0.0260}_{-0.0289}$ ( $3.06\sigma$ deviation from 1).
- The inclusion of ACT DR6 data significantly tightens the constraint on $A_L$ and strengthens the preference for $A_L > 1$ , suggesting the actual gravitational lensing effect may be stronger than predicted by standard $\Lambda$ CDM.
Sachs-Wolfe and Doppler Effects:
- $A_{SW}$ shows deviations of $1.21\sigma$ (PDP) and $1.96\sigma$ (PADP) from unity.
- $A_{Dop}$ remains consistent with the standard value of 1 within $68\%$ confidence limits.
Integrated Sachs-Wolfe (ISW) Effects:
- $A_{eISW}$ is consistent with 1, with uncertainties reduced by approximately 13% compared to previous Planck-only analyses.
- $A_{lISW}$ is poorly constrained ( $A_{lISW} < 0.7577$ for PDP; large error bars for PADP) because the lISW effect has negligible influence at multipoles $\ell \geq 30$ .
Polarization ( $A_{Pol}$ ):
- $A_{Pol}$ is consistent with 1. However, the inclusion of ACT DR6 data reduces the uncertainty on $A_{Pol}$ by more than one order of magnitude (from $\sim 0.06$ to $\sim 0.0046$ ), highlighting the critical role of high- $\ell$ ground-based data in constraining polarization effects.
Cosmological Tensions:
- Hubble Tension: The introduction of single-parameter extensions ( $A_{new}$ ) does not alleviate the Hubble tension; $H_0$ remains stable at $\approx 68$ km s $^{-1}$ Mpc $^{-1}$ across all models.
- $\sigma_8$ Tension: The $\Lambda$ CDM+ $A_L$ model yields a mild alleviation of the $\sigma_8$ tension, reducing $\sigma_8$ to $\approx 0.796$ (a $\sim 1.5\sigma$ tension with KiDS-1000), but other extended models show no significant improvement.

Significance and Claims
The paper claims that while the standard $\Lambda$ CDM model remains robust, the data—particularly when augmented with high-resolution ground-based observations from ACT DR6—favors a stronger gravitational lensing effect ( $A_L > 1$ ) at a significance of over $3\sigma$ . The study demonstrates that high- $\ell$ measurements are essential for constraining phenomenological parameters related to lensing and polarization, reducing uncertainties by orders of magnitude compared to space-based data alone.

However, the authors modestly conclude that these single-parameter extensions do not resolve the fundamental Hubble or $\sigma_8$ tensions. The work underscores the importance of incorporating ground-based high- $\ell$ data (such as from ACT, SPT-3G, and the Simons Observatory) in future CMB analyses to better understand the physical mechanisms driving anisotropies and to rigorously test the internal consistency of the $\Lambda$ CDM model.