Almanac: MCMC-based signal extraction of power spectra and maps on the sphere

TL;DR

Almanac introduces a Bayesian field-level inference framework on the sphere that jointly infers full-sky maps and their angular power spectra for multiple spin fields using Hamiltonian Monte Carlo. By adopting curved-sky spin-weight formalism and flexible Cholesky-based parameterisations, it robustly handles extremely high-dimensional posteriors, mitigates E/B leakage, and provides cosmology-independent posterior data products under statistical isotropy. The method is demonstrated on CMB-like simulations, showing accurate recovery of $T$, $E$, and $B$ spectra and maps, with rigorous convergence diagnostics (FMI, Hanson statistic, ESS) confirming reliable sampling. Almanac thus offers a principled, all-sky alternative to traditional two-point estimators, enabling analyses that preserve higher-order information and are sensitive to potential systematics or new physics, while remaining compatible with future large-scale surveys.

Abstract

Inference in cosmology often starts with noisy observations of random fields on the celestial sphere, such as maps of the microwave background radiation, continuous maps of cosmic structure in different wavelengths, or maps of point tracers of the cosmological fields. Almanac uses Hamiltonian Monte Carlo sampling to infer the underlying all-sky noiseless maps of cosmic structures, in multiple redshift bins, together with their auto- and cross-power spectra. It can sample many millions of parameters, handling the highly variable signal-to-noise of typical cosmological signals, and it provides science-ready posterior data products. In the case of spin-weight 2 fields, Almanac infers $E$- and $B$-mode power spectra and parity-violating $EB$ power, and, by sampling the full posteriors rather than point estimates, it avoids the problem of $EB$-leakage. For theories with no $B$-mode signal, inferred non-zero $B$-mode power may be a useful diagnostic of systematic errors or an indication of new physics. Almanac's aim is to characterise the statistical properties of the maps, with outputs that are completely independent of the cosmological model, beyond an assumption of statistical isotropy. Inference of parameters of any particular cosmological model follows in a separate analysis stage. We demonstrate our signal extraction on a CMB-like experiment.

Figures (18)

• Figure 1: Directed acyclic graph showing Almanac's Bayesian hierarchical model for the inference of maps and angular power spectra. The angular power spectra ${\sf{C}}$ are drawn from a prior distribution $\pi$, spherical harmonic coefficients $\boldsymbol{a}$ for the fields are then drawn from a gaussian prior $\mathcal{G}$, and the fields are then transformed to configuration space, where noise $\boldsymbol{n}$ (drawn from a likelihood $\mathcal{L}$ with noise covariance ${\sf{N}}$) is added to yield the data $\boldsymbol{d}$. The beam (for the CMB) is not shown.
• Figure 2: Number of free parameters in a Euclid-like experiment that are jointly estimated by Almanac: for a given $\ell_{\rm max}$, all power spectra $C_\ell$ up to $\ell_{\rm max}$ are estimated. Additionally, all $a_{\ell m}$-modes up to $\ell_{\rm max}$ of the underlying (noise-free) field are inferred. The colours indicate the number of redshift bins. This figure assumes both $E$ and $B$-modes are estimated for spin-weight 2 fields (subplot a), and that there is only a single tracer of the spin-weight 0 field in subplot (b), as in galaxy clustering. For a CMB experiment, the spin-weight 0 field would be observed at different frequencies, hence contributing multiple fields. As can be seen, several hundred millions of parameters are to be jointly inferred, implying a requirement for a sampler that can handle this dimensionality very efficiently.
• Figure 3: Same as Fig. \ref{['Fig:Numpar_euclid']} but for a Planck-like CMB experiment measuring a $TQU$ field, but typically observed in a number of different wavelengths. The primary CMB temperature and polarization maps, together with their power spectra then yield several ten millions of parameters to be jointly inferred, independently of the number of frequencies.
• Figure 4: Comparison between the negative log-posterior (potential ${\psi}$) of two different coordinate systems: (top) Cholesky parameterisation $\{\boldsymbol{x}, {\sf{K}} \}$ and the (bottom)$\{\boldsymbol{a}, \ln({\sf{C}}) \}$ parameterisation applied for the same spin-weight 2 lower dimensional test case ($\ell_{\rm max} = 32$).
• Figure 5: Comparison of the impact of the different phases and methods of HMC tuning for (a) a spin-weight 0 lower dimensional case and (b) a spin-weight 2 lower dimensional case -- both with $\ell_{\rm max} = 32$. (First row) shows the case for a manually tuned HMC, i.e., when the user finds an apparently optimal set of HMC parameters by trial and error. (Second row) Only the step-size tuning is performed by using the standard deviation of the post burn-in samples as a proxy for the step-sizes. (Third row) The Hessian is kept as the step-sizes and only tuning of the leap-frog parameters is performed, finding the factor $\tilde{g}$, which re-scales the step-sizes. (Fourth row) All tuning stages are applied as described in Sect. \ref{['sec:HMCtuning']}.