Cross-spectra likelihood for robust $\tau$ constraints from all satellite polarisation data

Abstract

The Thomson scattering optical depth to reionisation, $\tau$, one of the six parameters of the $\Lambda$CDM model, is primarily constrained by the large-scale E-mode polarisation of the Cosmic Microwave Background (CMB). In this work, we present the E-mode Likelihood for Cross-Analysis (elica), a multi-frequency, harmonic-space likelihood that combines all currently available large-scale satellite polarisation data, namely the Planck LFI 70 GHz channel, the Planck HFI 100 and 143 GHz channels processed with the SRoll2 map-making algorithm, and the WMAP Ka, Q, and V bands. The likelihood is built on an extension of the Hamimeche-Lewis formalism to multi-field partial-sky observations. We validate the pipeline using 500 realistic simulations and find that retaining all cross-spectra and the WMAP-LFI auto-spectrum eliminates the significant bias present when all spectra are retained, while preserving comparable uncertainties in the recovered value of $\tau$. From the low-$\ell$ E-mode power spectrum alone, we obtain $\tau = 0.0575_{-0.0058}^{+0.0048}$ (68% CL). Combining elica with the Planck low-$\ell$ temperature likelihood and the CamSpec high-$\ell$ likelihood, we find $\tau = 0.0581_{-0.0059}^{+0.0048}$ and $\ln(10^{10}A_{\mathrm{s}}) = 3.048_{-0.012}^{+0.011}$. Including ACT{} DR6 + Planck CMB lensing and DESI DR2 BAO measurements, we derive an upper bound on the total neutrino mass of $\sum m_\nu < 0.069$ eV (95% CL). Our results, obtained through careful cross-validation of all available large-scale polarisation datasets, robustly confirm that the optical depth remains relatively low. This severely constrains the possibility of explaining, or even significantly reducing, the tension between DESI-BAO and CMB observations with a high value of $\tau$. The elica likelihood is publicly available.