CMB Likelihoods and Parameter Inference

A CMB parameter constraint is not a property of a plotted spectrum alone. It is the output of a versioned data vector, bandpower windows, covariance, instrument and foreground model, prior measure, and sampling calculation. This page develops that statistical object and then binds it to the Planck 2018 and ACT DR6.02 likelihoods. The source and release cutoff for the observational statements is 2026-08-19.

Required background. Probability and Inference for Gravity supplies Probability and Statistical Inference for correlated Gaussian data, priors, marginalization, and convergence tests.

Helpful background. The Einstein Equation supplies Einstein Equations for the theory spectra; Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects foreground physics; Relativistic Cosmology supplies the background and perturbation parameters; and The Wave Equation and Characteristics supplies Waves, Green Functions, and Geometric Optics for beams and transfer functions.

Bandpowers are a calibrated data vector

Write $D_\ell^{XY}=\ell(\ell+1)C_\ell^{XY}/(2\pi)$ for $XY\in\{TT,TE,EE\}$. A survey does not compare a theoretical value at the center of a bin directly with a measured point. For a declared bandpower normalization, its prediction is

$$m_b^{XY}(\boldsymbol\theta,\boldsymbol\eta) =\sum_\ell W_{b\ell}^{XY}(\boldsymbol\eta)D_\ell^{XY}(\boldsymbol\theta) +F_b^{XY}(\boldsymbol\theta,\boldsymbol\eta),$$

where $W_{b\ell}$ includes the published bin window and any declared multiplicative calibration or response, while $F_b$ contains additive foreground and leakage contributions. The latter can depend on the theory spectrum as well as nuisance parameters; a gain cannot generally be modeled as a cosmology-independent additive offset. The data vector $d$ is an ordered stack of frequency, detector, spectrum, and bin labels. Its covariance must use that same order. Rebinning either object without applying the corresponding linear transformation to the other changes the likelihood.

Absolute and relative gains multiply the spectrum model; a temperature gain error therefore enters a TT auto-spectrum twice. Polarization efficiency, polarization angle, beam eigenmodes, T-to-P leakage, bandpass shifts, mapmaking transfer functions, masks, and pseudo-$C_\ell$ mode coupling also belong to the forward model. Cross-spectra remove a mean auto-noise bias, not the noise contribution to their covariance. These details are why a released plot or diagonal error bar is not a likelihood.

Gaussian likelihood and posterior measure

Let $r=d-m(\boldsymbol\theta,\boldsymbol\eta)$. A normalized multivariate Gaussian has

$$-2\ln\mathcal L =r^{\mathsf T}C^{-1}r+\ln\det C+N\ln(2\pi).$$

If $C$ is fixed, the final two terms are constant for parameter estimation and may be omitted. If $C=C(\boldsymbol\theta,\boldsymbol\eta)$, dropping $\ln\det C$ changes the answer: the quadratic term alone would reward models that inflate their covariance. A full-sky, noise-free $\widehat C_\ell$ has a Wishart rather than Gaussian sampling law; the bandpower Gaussian is an approximation justified only after binning, masking, filtering, and covariance validation.

For temperature alone, a noise-free full-sky Gaussian field provides $2\ell+1$ independent real degrees of freedom at each $\ell$. Therefore

$$\frac{(2\ell+1)\widehat C_\ell^{TT}}{C_\ell^{TT}} \sim\chi^2_{2\ell+1}, \qquad \operatorname{Var}(\widehat C_\ell^{TT}) =\frac{2(C_\ell^{TT})^2}{2\ell+1}.$$

This is cosmic variance: observing the same sky with a quieter detector does not provide more independent realizations of its primordial modes. At low $\ell$ the distribution is visibly skewed, which explains why a symmetric Gaussian error bar can be misleading even before masking or foregrounds are introduced. The joint temperature–polarization covariance has the matrix Wishart generalization.

With cosmological parameters $\boldsymbol\theta$ and nuisance parameters $\boldsymbol\eta$,

$$p(\boldsymbol\theta,\boldsymbol\eta\mid d) =\frac{\mathcal L(d\mid\boldsymbol\theta,\boldsymbol\eta) \pi(\boldsymbol\theta,\boldsymbol\eta)} {\int d\boldsymbol\theta\,d\boldsymbol\eta\, \mathcal L\pi}.$$

The prior is therefore part of the reported inference, especially for weakly measured calibration or foreground directions. Profiling a nuisance parameter, fixing it, and marginalizing it are different operations. A credible chain report gives the sampler, proposal or adaptation, independent seeds, burn-in rule, effective sample sizes and split-chain convergence, then checks residuals or replicated data; $R-1\lt 0.01$ alone is not a posterior predictive test.

Keeping the theory calculation, likelihood, and sampler as separate modules makes these choices inspectable. This architecture is described in Torrado and Lewis 2021, § I, printed pp. 1–2, PDF.

Planck 2018 as a versioned likelihood

The Planck 2018 baseline combines distinct statistical regimes. Commander provides TT for $2\le\ell\lt 30$, SimAll provides EE over the same range (lowE), and Plik supplies high-multipole TT from about 30 to 2500 and TE/EE from about 30 to 2000. Plik jointly models the 100, 143, and 217 GHz spectra, masks, beams, calibration, foregrounds, and covariance. Planck Collaboration 2020 V, §§2–3.4, printed pp. 4–53 (Open PDF) defines these pieces; its §§3.6, 3.8, and 3.9, printed pp. 58–73, give consistency tests, priors, and cuts.

As one dated marginal check, the Planck TT,TE,EE+lowE+lensing/base-$\Lambda$CDM Plik chain gives $H_0=67.36\pm0.54\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $100\theta_*=1.04110\pm0.00031$. These numbers depend on the six-parameter model, the neutrino and recombination assumptions, the lensing likelihood, and the named 2018 products. They are not calibration-independent constants. The associated sampled parameters and systematic comparison with CamSpec are given in Planck Collaboration 2020 VI, §3.1 and Table 2, printed pp. 14–16 (Open PDF).

ACT DR6.02 full and compressed likelihoods

The ACT DR6.02 release on NASA LAMBDA provides maps, beams, passbands, masks, spectra, windows, SACC vectors, likelihood software, and chains. Its multifrequency likelihood is LAT MFLike 1.0.0 with fgspectra 1.3.0. The 1651-point TT/TE/EE vector uses five temperature array-bands and four polarization array-bands. PA4 f150 is excluded; PA4 f220 has no polarization. Baseline minima are $\ell=1000$ for PA4 f220, PA5 f090, and PA6 f090, 800 for PA5 f150, and 600 for PA6 f150, with $\ell_{\max}=8500$. These selections follow the null tests rather than an assumption that every released mode is usable.

The selection is therefore explicit: it fixes array-band membership, sky and source masks, polarization availability, and every multipole threshold before the cosmological fit.

The model has 15 foreground parameters, of which 14 vary independently, plus six gain-calibration parameters, four polarization efficiencies, and five bandpass shifts. Gains and bandpasses carry Gaussian priors; polarization efficiencies have bounded uniform priors. Cobaya calls CAMB with theory $\ell_{\max}=9000$, and the released analysis required $R-1\lt 0.01$. Louis et al. 2025, §§6.1–7.3, printed pp. 22–29, eqs. (16)–(35) and Tables 3–4 (Open PDF) gives the data model, fixed-covariance likelihood, priors, sampler settings, simulation recovery, and frequency nulls.

ACT-lite instead contains 45 foreground-marginalized CMB bandpowers in each of TT, TE, and EE. Its 135-vector and covariance are approximated as Gaussian; only an overall calibration and a polarization efficiency remain. The exact DR6.02 SACC product page identifies dr6_data.fits for MFLike and dr6_data_cmbonly.fits for ACT-lite, while the released-chain index distinguishes actbase, actlite, actext-tau, Planck-only, and P-ACT data tags. A likelihood name without its tag and external low-$\ell$ choice is not enough to reproduce a posterior.

Marginal, residual, and cut diagnostics

For the public actbase combination—ACT DR6.02 MFLike plus the Planck Sroll2 large-scale EE likelihood—the baseline marginal is

$$H_0=66.11\pm0.79\ \mathrm{km\,s^{-1}\,Mpc^{-1}},$$

with $\Omega_bh^2=0.02259\pm0.00017$ and $n_s=0.9666\pm0.0077$. The maximum-posterior multifrequency residual gives $\chi^2=1590.91$ for 1651 data points and 34 fitted model parameters, hence 1617 degrees of freedom and a reported PTE of 67%. The compressed ACT-lite fit gives $-2\ln\mathcal L=142.2$ for 128 degrees of freedom and PTE 18%; its cosmological marginals agree with MFLike within $0.1\sigma$. These are three different checks—a marginal, a residual statistic, and a compression test—not three estimates of the same quantity. They are reported in Louis et al. 2025, §8.1 and Table 5, printed pp. 30–32, and Appendix F, printed pp. 64–65 (Open PDF).

A consequential cut test raises the polarization minima from the pre-unblinding extended value $\ell_{\min}=500$ to the array-dependent baseline minima above. With the final baseline setup, the f090-versus-f150 parameter-difference PTE is 50% for TT/TE/EE and 53% for TE/EE. The pre-unblinding setup gives 0.8% and 1.7%. This comparison is not a pure multipole-cut derivative: that earlier setup also lacked the final tSZ-shape and chromatic-beam treatments. Appendix B of the same paper, printed pp. 49–53, separates the sequence and shows that the polarization cut dominates the shift in $n_s$. Calling the whole change a foreground discovery would therefore be unjustified.

Dataset overlap, systematics, and reproduction boundary

The P-ACT construction retains Planck PR3 high-$\ell$ information only below $\ell=1000$ in TT and below $\ell=600$ in TE/EE, then adds ACT and the Planck low-$\ell$ likelihoods. It neglects the ACT–Planck TT cross-covariance over $600\lt \ell\lt 1000$; simulations estimate that a full treatment would improve an example $N_{\mathrm{eff}}$ error by less than 5%, not that the covariance is exactly zero. Adding a lensing likelihood or external BAO data defines a new data combination and cannot silently inherit the CMB-only label.

A reproducible rerun should download the two named SACC data products and window functions, verify their hashes, pin MFLike 1.0.0 and fgspectra 1.3.0, and pin the theory and sampler environment. For example, a new recipe can pin Cobaya 3.6.2 at commit 899f30a; the ACT paper itself does not state exact Cobaya and CAMB versions, so the versions used in the published chains remain an explicit provenance gap. The rerun should recover the actbase $H_0$ marginal and best-fit residual above, then repeat with actext-tau or a foreground-model variation while preserving all other settings. It must also inspect whitened, spectrum-resolved residuals rather than only total $\chi^2$.

No new high-dimensional chain was executed for this page. The public software repository, SACC data products, released chains, exact selections, and numerical targets have been identified, but the quoted marginal and residual remain checks against the collaboration release. That boundary is important: covariance estimation, low-$\ell$ approximation, beam and calibration uncertainty, foreground degeneracy, prior-volume effects, parameter-dependent covariance, and shared-sky overlap can all move an apparently precise posterior.

Exercises

A one-bin experiment measures $d=A+g+n$, where $n\sim\mathcal N(0,\sigma^2)$ and the additive nuisance offset has prior $g\sim\mathcal N(0,s_g^2)$. With a flat prior on $A$, marginalize over $g$ and compare the result with fixing $g=0$.

Solution

The marginalized likelihood is the convolution of two Gaussians:

$$\mathcal L(d\mid A) =\int dg\,\mathcal N(d;A+g,\sigma^2)\mathcal N(g;0,s_g^2) =\mathcal N(d;A,\sigma^2+s_g^2).$$

Thus the posterior mean is $A=d$, while its standard deviation is $\sqrt{\sigma^2+s_g^2}$. Fixing $g=0$ leaves the same mean but reports only $\sigma$, understating uncertainty whenever $s_g\gt 0$. In a multi-frequency CMB likelihood the analogous nuisance directions can also correlate with cosmological parameters, so fixing them may shift the mean as well as narrow the interval.

References

• ACT Collaboration, “ACT DR6.02 Data Release” (2025), NASA LAMBDA.
• Louis, T., et al. (ACT Collaboration), “The Atacama Cosmology Telescope: DR6 Power Spectra, Likelihoods and ΛCDM Parameters,” arXiv:2503.14452v2 (2025), Open PDF, data products.
• Planck Collaboration, “Planck 2018 Results. V. CMB Power Spectra and Likelihoods,” Astronomy & Astrophysics 641 (2020) A5, DOI, Open PDF.
• Planck Collaboration, “Planck 2018 Results. VI. Cosmological Parameters,” Astronomy & Astrophysics 641 (2020) A6, DOI, Open PDF.
• Torrado, J., and A. Lewis, “Cobaya: Code for Bayesian Analysis of Hierarchical Physical Models,” Journal of Cosmology and Astroparticle Physics 2021(05), 057 (2021), DOI, Open PDF.