Likelihoods

ELiCA provides 9 likelihoods, all inheriting from a common base class that extends cobaya’s CMBlikes with the gLoLLi transform and Sellentin-Heavens correction.

Multi-field likelihoods

These operate on the full 3-map system (100GHz, 143GHz, WL) and differ in which spectra enter the chi-squared:

cobaya name	Class	Description
`elica`	`elica.elica`	Flagship hybrid: cross-spectra + WLxWL (4 spectra in chi-squared)
`elica.cross`	`elica.cross`	Cross-spectra only (3 spectra in chi-squared)
`elica.full`	`elica.full`	All 6 auto + cross spectra in chi-squared

All three apply the HL transform to the full 3x3 spectral matrix. The covmat_cl parameter in the dataset file controls which spectra are retained in the data vector after the transform — spectra not listed are effectively marginalized over.

Single-field likelihoods

Each operates on a single map pair with a 1x1 spectral matrix:

cobaya name	Description
`elica.EE_100x100`	100GHz auto-spectrum
`elica.EE_100x143`	100GHz x 143GHz cross-spectrum
`elica.EE_100xWL`	100GHz x WL cross-spectrum
`elica.EE_143x143`	143GHz auto-spectrum
`elica.EE_143xWL`	143GHz x WL cross-spectrum
`elica.EE_WLxWL`	WL auto-spectrum

Method

The likelihood computation follows:

Hamimeche & Lewis (HL) transform — Gaussianizes the power spectrum data via eigendecomposition. ELiCA uses a modified gHL computation following Mangilli et al. (2015) that handles negative eigenvalues via sign(x) * gHL(|x|) instead of clipping them to zero.
Offset — An additive correction baked into the data, noise, and fiducial spectra before the HL transform to ensure positive-definiteness.
Sellentin-Heavens correction — Accounts for the finite number of simulations used to estimate the covariance matrix:

\[\chi^2_{\mathrm{SH}} = N_{\mathrm{sims}} \ln\left(1 + \frac{\chi^2}{N_{\mathrm{sims}} - 1}\right)\]
Spectrum marginalization — The HL transform operates on the full spectral matrix, but only a subset of the transformed spectra enters the chi-squared. This effectively marginalizes over the excluded spectra, following the approach in Galloni et al. (2025).