Constraints on Phenomenological Amplitudes of CMB Anisotropy with Multi-Datasets

Cosmic microwave background anisotropies encode crucial information about the early Universe and fundamental cosmological physics. Although the standard $\Lambda$CDM model provides a successful description of cosmic evolution, persistent cosmological tensions and subtle small-scale anomalies still challenge its internal consistency. In this paper, we investigate six phenomenological amplitude parameters $A_{\rm{new}}$ (new=L, SW, Dop, eISW, lISW, Pol) corresponding to the key effects related to CMB anisotropy: the Lensing, Sachs-Wolfe, Doppler, early Integrated Sachs-Wolfe, late Integrated Sachs-Wolfe, and Polarization effects, respectively. Using modified CAMB and Cobaya packages, we constrain the $\Lambda$CDM$+A_{\rm{new}}$ models with two data combinations: Planck+DESI+PantheonPlus (PDP) and Planck+ACT+DESI+PantheonPlus (PADP). Only the $\Lambda$CDM+$A_{\rm{L}}$ is favored by AIC, with $A_{\rm{L}}=1.0656_{-0.0303}^{+0.0304}$ from PDP and $A_{\rm{L}}=1.0795_{-0.0289}^{+0.0260}$ from PADP, which implies 2.16$\sigma$ and 3.06$\sigma$ deviation from the $\Lambda$CDM model; values of $A_{\rm{SW}}$ show 1.21$\sigma$ and 1.96$\sigma$ deviations to 1; $A_{\rm{lISW}}$ is poorly constrained because the lISW effect has negligible influence at $\ell \geq 30$; and others are consistent with the $\Lambda$CDM model. Moreover, no noticeable improvement on the Hubble and $\sigma_8$ tensions is found within these one-parameter extended scenarios. ACT DR6 high-$\ell$ data strengthens the $\Lambda$CDM$+A_{\rm{L}}$ preference over the $\Lambda$CDM model, and reduces $A_{\rm Pol}$ uncertainty by more than one order of magnitude, highlighting the importance of ground-based high-$\ell$ observations for future CMB analyses.