Likelihood Theory

LiLit implements multiple likelihood approximations for CMB analysis. This section provides detailed formulations and implementation notes.

Overview

LiLit supports both single-field and multi-field likelihood calculations with the following approximations:

  • Exact likelihood: Based on Hamimeche & Lewis (2008)

  • Gaussian likelihood: Standard Gaussian approximation with proper multi-field covariance

  • Correlated Gaussian likelihood: Accounts for multipole correlations (single-field only, under development)

Each approximation handles field-specific multipole ranges (\(\ell_{\rm min}\), \(\ell_{\rm max}\)) and sky fractions (\(f_{\rm sky}\)) appropriately.

Exact Likelihood

The exact likelihood approximation follows Hamimeche & Lewis (2008).

Single-field Case

For a single field, the log-likelihood is:

\[\log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)f_{\rm sky}\left[\frac{C_{\ell}^{\rm obs}}{C_{\ell}^{\rm th}}-\log\left(\frac{C_{\ell}^{\rm obs}}{C_{\ell}^{\rm th}}\right)-1\right]\]

Multi-field Case

For \(N\) fields, the formula becomes:

\[\log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)f_{\rm sky}^{\rm eff}\left[\text{Tr}\left(\mathcal{C}_{\rm obs}\mathcal{C}^{-1}_{\rm th}\right) - \log\left|\mathcal{C}_{\rm obs}\mathcal{C}^{-1}_{\rm th}\right| - N\right]\]

where the covariance matrix has the structure:

\[\begin{split}\mathcal{C}_{\rm obs} = \left(\begin{array}{ccc} C_{\ell}^{XX} & C_{\ell}^{XY} & C_{\ell}^{XZ}\\ C_{\ell}^{YX} & C_{\ell}^{YY} & C_{\ell}^{YZ}\\ C_{\ell}^{ZX} & C_{\ell}^{ZY} & C_{\ell}^{ZZ} \end{array}\right)\end{split}\]

Each entry includes signal and noise contributions:

\[C_{\ell}^{XX} = C_{\ell}^{\rm CMB} + C_{\ell}^{\rm FGs} + N_{\ell}^{X} + \ldots\]

Implementation Details

Multipole Ranges: The multipole range for cross-correlations between two fields is determined by the intersection of their individual ranges.

Sky Fraction: For multiple \(f_{\rm sky}\) values, an effective value is computed as the geometric mean: \(f_{\rm sky}^{\rm eff} = \sqrt[N]{\prod_i f_{\rm sky}^i}\).

Singular Matrices: When multipole cuts or excluded probes make the covariance matrix singular, LiLit automatically identifies and removes null diagonal entries along with their corresponding rows and columns.

Excluded Probes: Specific cross-correlations can be excluded (e.g., excluded_probes = ["xz"]) by setting the corresponding covariance entries to zero.

Gaussian Likelihood

The Gaussian approximation assumes normally distributed power spectra with known covariance.

Single-field Case

\[\log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)f_{\rm sky}\left[\frac{\left(C_{\ell}^{\rm obs} - C_{\ell}^{\rm th}\right)^2}{\sigma^{2}_{\ell}}\right]\]

where the variance is:

\[\sigma^{2}_{\ell} = \frac{2}{(2\ell+1)f_{\rm sky}}\left(C_{\ell}^{\rm obs}\right)^2\]

Multi-field Case

The data vector is formed from the upper triangular part of the covariance matrix:

\[X_\ell = \left(C_{\ell}^{XX}, C_{\ell}^{XY}, C_{\ell}^{XZ}, C_{\ell}^{YY}, C_{\ell}^{YZ}, C_{\ell}^{ZZ}\right)\]

The covariance matrix for this data vector is:

\[\text{Cov}^{\rm ABCD}_{\ell} = \frac{1}{(2\ell+1)f_{\rm sky}^{AB}f_{\rm sky}^{CD}}\left( \sqrt{f_{\rm sky}^{AC}f_{\rm sky}^{BD}}C_\ell^{AC}C_\ell^{BD} + \sqrt{f_{\rm sky}^{AD}f_{\rm sky}^{BC}}C_\ell^{AD}C_\ell^{BC} \right)\]

where cross-correlation sky fractions are: \(f_{\rm sky}^{XY} = \sqrt{f_{\rm sky}^{XX}f_{\rm sky}^{YY}}\).

The likelihood is then:

\[\log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)\left[X_\ell \cdot \text{Cov}^{-1}_{\ell} \cdot X_\ell^{\rm T}\right]\]

Implementation Differences from Exact Case

Multipole Ranges: Cross-correlation ranges use geometric mean: \(\ell_{\rm max}^{XY} = \sqrt{\ell_{\rm max}^{XX}\ell_{\rm max}^{YY}}\).

Masking: The covariance matrix is computed for the full range, then masked entries are removed before inversion. The data vector is masked consistently.

Sky Fractions: Each field retains its individual \(f_{\rm sky}\) factor, unlike the exact case which uses an effective value.

Correlated Gaussian Likelihood

This approximation accounts for correlations between different multipoles (currently single-field only).

Formulation

\[\log\mathcal{L} = -\frac{1}{2}\left[\left(\vec{C^{\rm obs}} - \vec{C^{\rm th}}\right) \cdot \text{Ext}^{-1} \cdot \left(\vec{C^{\rm obs}} - \vec{C^{\rm th}}\right)^{\rm T}\right]\]

where \(\vec{C^{\rm obs}}\) and \(\vec{C^{\rm th}}\) are vectors over the multipole range, and \(\text{Ext}\) is an externally provided covariance matrix.

Requirements

  • External covariance matrix must exclude \(\ell = 0, 1\)

  • Covariance matrix should already account for \(f_{\rm sky}\) effects

  • Multi-field extension is under development

Sky Cut Approximations

All likelihood implementations make approximations regarding sky cuts:

  1. Mode Counting: The factor \((2\ell+1)f_{\rm sky}\) approximates the reduction in available modes due to masking

  2. No Mode Coupling: Correlations between different multipoles induced by the mask are neglected

  3. Effective Sky Fraction: Multi-field cases use geometric averaging of individual sky fractions

These approximations are valid when:

  • Sky cuts are not too severe (\(f_{\rm sky} \gtrsim 0.3\))

  • Mask boundaries are not too complex

  • Cross-correlation regions have substantial overlap

For more accurate treatment of sky cuts, external covariance matrices accounting for mode coupling should be used with the correlated Gaussian likelihood.

References

  • Hamimeche, S. & Lewis, A., 2008, Likelihood analysis of CMB temperature and polarization power spectra, arXiv:0801.0554

  • Campeti, P. et al., 2020, Measuring the spectrum of primordial gravitational waves with CMB, PTA and Laser Interferometers, arXiv:2007.04241

  • Allys, E. et al., 2022, Probing cosmic inflation with the LiteBIRD cosmic microwave background polarization survey, arXiv:2202.02773