Lensing amplitude anomaly and varying electron mass alleviate the Hubble and $S_8$ tensions

Abstract

Cosmological measurements have revealed tensions within the standard $\Lambda$CDM model, notably discrepancies in the Hubble constant and $S_8$ parameter. A modified recombination scenario involving a time-varying electron mass has been proposed as a feasible solution to the Hubble tension without exacerbating the $S_8$ tension. Recent observations have further revealed other potential deviations from the $\Lambda$CDM framework, such as non-flat spatial curvature and an anomalous CMB lensing amplitude. In this study, we explore whether introducing a variation in the electron mass $m_e$, allowing non-zero spatial curvature $\Omega_K$, and a free lensing amplitude $A_{\rm lens}$ can resolve these persistent tensions. Using the Planck Public Release (PR) 3 and ACT power spectra, Planck PR4 and ACT lensing maps, together with BAO measurements from DESI DR2, we obtain $H_0 = 69.61^{+0.60}_{-0.55} \rm \, km \, s^{-1} \, Mpc^{-1}$ and $S_8= 0.808\pm0.012$, with $\Delta m_e / m_e = 0.0109^{+0.0068}_{-0.0066}$ and $A_{\rm lens} = 1.030^{+0.039}_{-0.037}$, both exceeding the $\Lambda$CDM expectations. We find no indication of spatial curvature deviating from flatness, even when including the Cosmic Chronometers and SNe Ia samples. However, when adopting the latest Planck power spectra likelihoods, NPIPE and HiLLiPoP, we obtain lower electron masses with $\Delta m_e / m_e = -0.0063^{+0.0095}_{-0.0099}$ and $-0.0095^{+0.0078}_{-0.0079}$, relieving the $S_8$ tension only. The lensing amplitude remains anomalously high, with $A_{\rm lens} = 1.053^{+0.042}_{-0.040}$ and $1.075^{+0.044}_{-0.043}$. Our results point to a promising direction for cosmological models to reconcile the aforementioned discrepancies, although more precise data from future experiments will be necessary to clarify the aforementioned modifications.