We present a compressed Gaussian likelihood for the Planck cosmic microwave background (CMB) low-$\ell$ E-mode polarization data, constructed from the SRoll2 likelihood which provides a state-of-the-art constraint on the reionization optical depth $\tau$ to date. The non-Gaussian form of CMB low-$\ell$ temperature (TT) and E-mode polarization likelihoods makes them incompatible with Fisher matrix analyses that require an analytic Gaussian ${\chi }^2$, such as the Fisher-bias formalism and Fisher forecasts. We show that the ${\chi }^2$ of an offset log-normal likelihood takes a Gaussian form in the log-transformed power spectrum amplitudes, and can therefore be used directly in Fisher matrix analyses without any explicit change of variables. Building on this, we compress the SRoll2 likelihood into a small number of piecewise offset log-normal functions and validate it against the full SRoll2 likelihood via Markov chain Monte Carlo (MCMC) combined with Planck and ACT DR6 data, finding excellent agreement across all lambda cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$) parameters and in extended cosmological models. We further demonstrate that Fisher matrix uncertainty estimates from our compressed likelihood agree well with the full MCMC posteriors. We release our compressed likelihood planck-gaussian-lowl, a lightweight Python package incorporating the compressed low-$\ell$ TTlikelihood from previous work, allowing a straightforward incorporation of the Planck CMB low-$\ell$ data into any Gaussian-likelihood-based analysis.

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• Cosmology