CMBComp: A Simple and Accurate Compressed CMB Likelihood for Dark Energy, Curvature, and Massive Neutrinos

Technical Summary of "CMBComp: A Simple and Accurate Compressed CMB Likelihood for Dark Energy, Curvature, and Massive Neutrinos"

Problem Statement
The full Cosmic Microwave Background (CMB) likelihood is a computationally expensive and structurally complex tool, making it difficult to deploy in exploratory analyses, particularly for late-time phenomenological cosmologies. This limitation hinders fast scans across large model spaces and joint analyses with low-redshift geometric probes, such as Baryon Acoustic Oscillations (BAO). While the dominant CMB information for late-time inference is conceptually understood to be encoded in a small number of geometric quantities (the angular size of the sound horizon, the comoving distance to the last-scattering surface, and physical density parameters), a practical, model-adapted, and accurate compressed likelihood has been needed to replace the full likelihood without sacrificing precision.

Methodology
The authors present CMBComp, a set of compact, Gaussian compressed CMB likelihoods derived from the combined SPA dataset (SPT-3G D1 + ACT DR6 + Planck PR3 primary anisotropies + Planck PR4/NPIPE CMB lensing). The compression strategy relies on the fact that for smooth dark-energy and geometry-driven extensions of the standard model, the CMB primarily acts as a high-redshift geometric anchor.

The framework constructs five distinct compressed likelihoods, each tailored to a specific cosmological model space. The core geometric observables are the shift parameter $R$ and the acoustic angular scale $\ell_a$. Depending on the model, these are augmented by one or two physical density parameters to form the data vector $\mathbf{v}$. The likelihood is defined as:
$$-2 \ln \mathcal{L}_{\text{comp}} = (\mathbf{v}_{\text{th}} - \bar{\mathbf{v}})^\top \mathbf{C}^{-1} (\mathbf{v}_{\text{th}} - \bar{\mathbf{v}})$$
where $\bar{\mathbf{v}}$ and $\mathbf{C}$ are the mean vector and covariance matrix extracted from SPA chains for the specific model, and $\mathbf{v}_{\text{th}}$ is the theoretical prediction computed via CAMB.

The five specific compressions are:

1. CMB-3: A three-parameter compression $(R, \ell_a, \Omega_b h^2)$ for the spatially flat $\Lambda$CDM model.
2. CMB-3w: The same three-parameter vector applied to the dynamical dark-energy $w_0 w_a$CDM model, with mean and covariance re-extracted from $w_0 w_a$CDM chains to account for the expanded parameter space.
3. CMB-4$\nu$: A four-parameter compression $(R, \ell_a, \Omega_b h^2, \Omega_c h^2)$ for the massive-neutrino $\nu\Lambda$CDM model. The inclusion of $\Omega_c h^2$ is necessary to correctly track the matter budget when the sum of neutrino masses ($\sum m_\nu$) varies, avoiding the incorrect treatment of massive neutrinos as cold matter at recombination.
4. CMB-4k: A four-parameter compression $(R, \ell_a, \Omega_b h^2, \Omega_\Lambda)$ for the non-flat $o\Lambda$CDM model. Crucially, for this model, the redshift of last scattering $z_*$ is evaluated self-consistently from CAMB at each MCMC step rather than being fixed, to account for curvature-induced shifts in the sound horizon mapping.
5. CMB-4$\nu w$: A four-parameter compression $(R, \ell_a, \Omega_b h^2, \Omega_c h^2)$ for the joint massive-neutrino and dynamical-dark-energy $\nu w_0 w_a$CDM model, with statistics re-extracted from the corresponding joint chains.

Key Contributions

• Model-Adapted Likelihoods: The paper provides five specific, ready-to-use compressed likelihoods (mean vectors and covariance matrices) derived from the most recent generation of CMB data (SPA).
• Implementation Independence: The authors provide a standalone, lightweight Python implementation (CMBComp) that does not require a full Boltzmann solver (like CAMB) or a complex inference framework (like Cobaya or CosmoMC) to run. It includes a custom background evolution calculator and an adaptive Metropolis–Hastings sampler.
• Validation: The compressed likelihoods are rigorously validated by combining them with the DESI DR2 BAO likelihood and comparing the resulting marginalized posteriors against reference chains generated using the full SPA CMB likelihood.

Results
The validation demonstrates that the compressed likelihoods reproduce the marginalized constraints and 2D contour shapes of the full CMB analysis with high precision across all five model classes:

• $\Lambda$CDM: The compressed posterior shows a small residual shift ($\sim 0.6\sigma$) in $H_0$ and $\Omega_m$ relative to the full CMB. The authors attribute this to the inability of the 3-parameter compression to fully model CMB lensing effects, which preferentially shift the Hubble constant downwards in the full likelihood.
• $w_0 w_a$CDM: The agreement is excellent. The additional dark energy parameters absorb the biases seen in $\Lambda$CDM, resulting in negligible offsets ($< 0.3\sigma$) for all parameters.
• $\nu\Lambda$CDM: The geometric parameters ($\Omega_m, H_0$) are reproduced with errors $< 0.12\sigma$. The neutrino mass upper limit is slightly weaker ($\sim 13\%$) than the full CMB reference, reflecting reduced leverage on $\sum m_\nu$ through the indirect matter-budget relation, but the qualitative structure is preserved.
• $o\Lambda$CDM: The agreement is essentially exact, with all mean shifts below $0.13\sigma$. The self-consistent calculation of $z_*$ and the inclusion of $\Omega_\Lambda$ eliminate the geometric leakage observed in the $\Lambda$CDM case.
• $\nu w_0 w_a$CDM: The dark energy and geometric parameters agree within $0.09\sigma$. The neutrino mass constraint is somewhat broader ($\sim 14\%$) than the reference due to the additional degeneracy directions opened by the dark energy parameters, but the posterior structure is faithfully reproduced.

Significance and Claims
The paper claims that CMBComp is particularly well-suited for cosmological inference in models that modify the late-time expansion history. It enables accurate CMB constraints to be incorporated into new analyses with minimal computational overhead and without reliance on a full Boltzmann-solver-based pipeline.

The authors emphasize that these compressions are valid for models where effects are confined to the late-time expansion history and background evolution, preserving standard early-Universe physics. They explicitly caution that models introducing new early-time physics, modifying recombination, or significantly altering the primordial perturbation sector should still utilize the full CMB likelihood. The work provides a practical, accessible tool for researchers, students, and educators to perform high-precision cosmological inference using a simplified, transparent, and computationally efficient framework.