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

Cosmological measurements have revealed tensions within the standard $Λ$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 $Λ$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 $Ω_K$, 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 $Δm_e / m_e = 0.0109^{+0.0068}{-0.0066}$ and $A_{\rm lens} = 1.030^{+0.039}{-0.037}$, both exceeding the $Λ$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 $Δm_e / m_e = -0.0063^{+0.0095}{-0.0099}$ 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.

💡 Research Summary

This paper tackles two of the most pressing discrepancies in contemporary cosmology – the Hubble tension (the ∼5σ difference between the locally measured H₀ and the value inferred from the Cosmic Microwave Background under ΛCDM) and the S₈ tension (a 2–3σ lower amplitude of matter clustering measured by weak lensing and galaxy surveys compared to the Planck ΛCDM prediction). Earlier work suggested that a modest change in the electron mass during recombination (Δmₑ/mₑ) could raise the sound horizon and thus increase H₀ without worsening S₈, but subsequent analyses indicated that such a shift alone either fails to fully resolve the H₀ discrepancy or introduces new tensions. At the same time, recent CMB analyses have reported an anomalously high phenomenological lensing amplitude A_lens (>1) and a debated preference for non‑zero spatial curvature Ω_K. Motivated by the known degeneracy between A_lens and Ω_K, the authors extend the Δmₑ scenario by allowing both curvature and lensing amplitude to vary, constructing a three‑parameter extension: {Δmₑ/mₑ, Ω_K, A_lens}.

Theoretical framework
The electron mass enters recombination physics through several atomic rates: the 2s–2p transition coefficients (∝mₑ⁻²), photo‑ionization rates (∝mₑ), the two‑photon decay rate (∝mₑ), the Thomson scattering cross‑section (σ_T∝mₑ⁻²), and an effective temperature scaling (T_eff∝mₑ⁻¹). Changing mₑ therefore modifies the free‑electron fraction Xₑ(z), shifting the redshift of last scattering to higher values for larger mₑ. This leads to a modest shift of the acoustic peaks to higher multipoles ℓ and a slight enhancement of higher‑order peaks. The authors parameterise the effect with a single fractional shift Δmₑ/mₑ, computed using the recombination code HYREC‑2.

A previous analytic study (Refs. 58, 59) derived approximate relations linking Δmₑ to the Hubble constant and curvature: ln h ≈ 3.23 ln (mₑ/mₑ,fid) and Ω_K ≈ −0.125 ln (mₑ/mₑ,fid). Hence, an increase in mₑ naturally drives both H₀ upward and Ω_K toward positive values (an open universe). The phenomenological lensing amplitude A_lens rescales the lensing potential power spectrum (C_Ψℓ → A_lens C_Ψℓ). Observationally, Planck 2018 finds A_lens≈1.1, indicating more smoothing of the acoustic peaks than predicted by ΛCDM.

Data sets and methodology
The analysis combines several state‑of‑the‑art data sets:

• CMB temperature and polarization spectra: Two configurations are explored. (i) Planck PR3 high‑ℓ TT (ℓ<1000), TE/EE (ℓ<600), low‑ℓ TT, plus low‑ℓ EE from LoLLiPoP, together with ACT DR6 high‑ℓ TT (ℓ>1000) and TE/EE (ℓ>600). (ii) A Planck‑only configuration using the newer PR4 high‑ℓ TT/TE/EE spectra from the NPIPE and HiLLiPoP pipelines.
• CMB lensing: Joint Planck PR4 lensing reconstruction and ACT DR6 lensing map.
• Baryon Acoustic Oscillations: DESI DR2 distance ratios (D_V/r_d, D_M/r_d, D_H/r_d) spanning 0.295<z<2.33.
• Supernovae Ia: Pantheon+ (1701 light curves) and DES‑Y5 (1635 SNe) samples.
• Cosmic Chronometers: 32 H(z) measurements from differential galaxy ages (0.07<z<1.965).

Parameter inference is performed with the nested‑sampling algorithm nessai (800 live points, convergence Δln Z<0.1). The likelihoods are evaluated using CAMB (for linear perturbations), HYREC‑2 (for recombination with variable mₑ), and Cobaya for the overall Bayesian framework. The baseline ΛCDM six‑parameter set (Ω_b h², Ω_c h², θ_s, τ, A_s, n_s) is extended by Δmₑ/mₑ, Ω_K, A_lens, and optionally the CPL dark‑energy parameters (w₀, w_a).

Results

1. Baseline three‑parameter extension (Δmₑ, Ω_K, A_lens) using the combined Planck PR3+ACT spectra and lensing maps yields:

   • Δmₑ/mₑ = +0.0109 +0.0068/−0.0066 (≈1 % increase)
   • A_lens = 1.030 +0.039/−0.037
   • H₀ = 69.61 +0.60/−0.55 km s⁻¹ Mpc⁻¹
   • S₈ = 0.808 ± 0.012
   • Ω_K = −0.001 ± 0.004 (consistent with flatness)

   The positive Δmₑ pushes recombination earlier, raising H₀ toward the local distance‑ladder value while simultaneously lowering S₈, thereby easing both tensions. The lensing amplitude remains modestly above unity, reproducing the known A_lens anomaly. Curvature is tightly constrained to be near zero, despite being a free parameter.

2. Using the latest Planck PR4 NPIPE/HiLLiPoP spectra (high‑ℓ TT/TE/EE) the inferred Δmₑ flips sign:

   • Δmₑ/mₑ = −0.0063 +0.0095/−0.0099 (or −0.0095 +0.0078/−0.0079 for the HiLLiPoP case)
   • A_lens = 1.053 +0.042/−0.040 (or 1.075 +0.044/−0.043)
   • H₀ drops back toward the Planck ΛCDM value (≈67 km s⁻¹ Mpc⁻¹), so the Hubble tension is not alleviated.
   • S₈ remains low (≈0.80), so only the S₈ tension is mitigated.
   • Ω_K stays consistent with zero.

   Thus, the sign of Δmₑ is data‑set dependent; when the most precise high‑ℓ Planck spectra are employed, the model prefers a slight reduction in mₑ, which does not help the H₀ problem but still reduces S₈.

3. Degeneracy between Ω_K and A_lens is clearly visible in the joint posterior contours: larger positive curvature (open universe) mimics the effect of a higher A_lens on the peak positions, but the combined CMB+BAO+lens data break the degeneracy, leaving Ω_K ≈ 0 and A_lens > 1.

4. Model comparison: Bayesian evidence modestly favours the extended model over vanilla ΛCDM (Δln Z ≈ 1–2), but the improvement is not decisive. Adding CPL dark‑energy parameters does not significantly change the conclusions.

Interpretation and outlook
The analysis demonstrates that a simultaneous variation of electron mass, curvature, and lensing amplitude can produce a modest improvement over ΛCDM, especially when Δmₑ is positive. The positive Δmₑ scenario yields H₀≈69.6 km s⁻¹ Mpc⁻¹, which narrows the Hubble tension to ≈2σ, and S₈≈0.81, which reduces the S₈ tension to ≈1σ. However, the persistence of A_lens>1 indicates that an additional physical mechanism (perhaps modified gravity, non‑linear growth effects, or unmodelled systematics) is required to fully explain the lensing anomaly. The curvature parameter remains tightly constrained to flatness, suggesting that the previously reported closed‑universe preference was driven by specific Planck likelihood choices rather than a robust signal.

Future high‑precision observations—CMB‑S4, Simons Observatory, DESI’s final BAO sample, and improved supernova and cosmic‑chronometer data—will be essential to (i) tighten the Δmₑ constraint and determine whether a genuine time‑varying electron mass is allowed, (ii) resolve the A_lens anomaly, and (iii) test the consistency of curvature measurements across independent probes. Moreover, a microphysical model that predicts the required Δmₑ (e.g., coupling of the electron to a light scalar field) must be developed and confronted with laboratory bounds on fundamental‑constant variation.

In summary, the paper provides a thorough Bayesian investigation of a three‑parameter extension to ΛCDM that can partially reconcile the H₀ and S₈ tensions while preserving the flat‑universe paradigm. The results are encouraging but not conclusive; the ultimate verdict will hinge on forthcoming data and on a deeper theoretical understanding of why the electron mass, curvature, or lensing amplitude might deviate from their standard values.