Foregrounds, Beams, and Calibration

The CMB has a blackbody spectrum, but a blackbody frequency law does not by itself deliver a clean CMB map. Galactic dust and synchrotron emission, unresolved galaxies, radio sources, the thermal and kinetic Sunyaev–Zeldovich effects, beams, bandpasses, and calibration all enter the same measured modes. Separation is possible only after the frequency response, spatial selection, and covariance are made part of one statistical model. The ACT DR6.02 release provides a concrete example and fixes the evidence status here as of 2026-08-19.

Required background. Probability and Inference for Gravity supplies Probability and Statistical Inference for correlated spectra, priors, and goodness-of-fit tests.

Helpful background. The Einstein Equation supplies Einstein Equations for secondary sources; Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects emission processes; Relativistic Cosmology supplies the primary CMB; and The Wave Equation and Characteristics supplies Waves, Green Functions, and Geometric Optics for beams and polarization.

Frequency separation is an instrument-weighted inverse problem

For array-band $a$ and Stokes-derived field $X$, a useful map model is

$$d_a^X(\mathbf n)=g_a\,\mathcal F_a \left[B_a*\left(s_{\mathrm{CMB}}^X +\sum_c f_{c,a}(\boldsymbol\beta_c)a_c^X\right)\right](\mathbf n) +n_a^X(\mathbf n).$$

Here $g_a$ is calibration, $B_a$ the beam, $\mathcal F_a$ the mapmaking transfer operator, $a_c$ a component field, and $n_a$ noise. A quoted central frequency is not enough. The response to component $c$ is a passband average,

$$f_{c,a}(\boldsymbol\beta_c)= \frac{\int d\nu\,\tau_a(\nu) f_c(\nu;\boldsymbol\beta_c)G_a(\nu)} {\int d\nu\,\tau_a(\nu)G_a(\nu)},$$

Here $\tau_a$ is the measured bandpass. In one explicit convention, $f_c(\nu;\boldsymbol\beta_c)$ is the component’s thermodynamic-temperature SED relative to a reference frequency, and $G_a(\nu)=\partial B_\nu(T)/\partial T\rvert_{T=T_{\rm CMB}}$ converts that temperature to specific intensity before the detector integrates it. The denominator calibrates a unit CMB temperature fluctuation. If the supplied SED is already in intensity units, it must instead be integrated without a second multiplication by $G_a$. Color corrections therefore depend on the source spectrum. Treating intensity units as thermodynamic temperature, or evaluating a steep dust spectrum at a nominal band center, biases the mixing matrix.

The separable map expression also assumes that a common effective beam is adequate within each band. A frequency-dependent beam across a broad passband can couple the component SED to angular response; a precision calculation then integrates the beam and spectrum together.

The components are not interchangeable nuisance terms:

Component	Frequency and angular signature	Important model boundary
Galactic dust	Modified blackbody; power-law angular templates	Spatially varying index or decorrelation can defeat a single SED.
Synchrotron	Falling radio-frequency power law; can be strongly polarized	Curvature and Faraday effects matter outside a narrow band.
Radio sources	Approximately Poisson, so $D_\ell\propto\ell^2$	Mask threshold and source variability change the residual.
Cosmic infrared background	Poisson plus clustered dust emission	Clustered and Poisson SEDs need not share one index.
Thermal SZ	Non-blackbody inverse-Compton spectrum	Relativistic temperature corrections and profile choice matter.
Kinetic SZ	Blackbody to leading order	Frequency cleaning cannot distinguish it from primary CMB.

The last row is the key identifiability warning: frequency coverage cannot separate two components with the same spectral vector without spatial, angular, or external information.

Masks, beams, and cross-spectra define the measured quantity

A masked, filtered map produces a pseudo-spectrum rather than the sky spectrum. Schematically,

$$\widetilde C_\ell^{ab,XY}= \sum_{\ell'}M_{\ell\ell'}^{ab,XY} B_{\ell'}^aB_{\ell'}^bF_{\ell'}^aF_{\ell'}^b C_{\ell'}^{ab,XY}+\widetilde N_\ell^{ab,XY}.$$

The mode-coupling matrix $M$ depends on the exact sky and source masks; $B_\ell$ and $F_\ell$ encode beam and filtering response. Cross-spectra of independent data splits remove the mean auto-noise bias, but their covariance still contains noise. Beam eigenmodes, bandpass shifts, absolute and relative gain, polarization efficiency and angle, temperature-to-polarization leakage, and mask coupling must therefore propagate into either the data covariance or explicit nuisance parameters.

ACT DR6.02 uses the curved-sky MASTER pseudo-$C_\ell$ method and simulations to validate this deconvolution; Louis et al. 2025, §3.2–§3.4, printed pp. 7–11 (Open PDF) gives the simulations, estimator, masks, cuts, and null-based selections. Its final set contains 1640 simulations per split and array-band.

The ACT DR6.02 data and calibration boundary

The named dataset is ACT DR6.02 on NASA LAMBDA, inspected on 2026-08-19. It covers the 2017–2022 seasons at nominal 90, 150, and 220 GHz (approximately 98, 150, and 220 GHz effective channels), about $19{,}400\ \mathrm{deg}^2$, and a typical combined depth near $9.6\ \mu\mathrm K$-arcmin. The standard map stem is act_dr6.02_std_AA_night. The release separately identifies frequency maps, inverse-variance maps, null splits, beams including T-to-P leakage profiles, passbands, masks, simulations, PSPIPE spectra/covariances, SACC likelihood vectors, and chains. Næss et al. 2025, §§2–5, printed pp. 3–30 and Apps. B–E, printed pp. 42–61 (Open PDF) documents the map production and response.

For spectra, PA4 f150 was excluded after multiple array null failures. PA4 f220 enters TT but not polarization. The baseline has $\ell_{\max}=8500$ and minimum multipoles 1000 for PA4 f220, PA5 f090, and PA6 f090; 800 for PA5 f150; and 600 for PA6 f150. Sources brighter than 15 mJy at 150 GHz are masked, and bins have width at least 50. These are analysis selections, not completeness-free statements about the sky.

The released maps already include the final Planck-referenced gains. Examples from Table 8 of the spectrum paper are 1.0111 (PA5 f090), 1.0086 (PA6 f090), 0.9861 (PA5 f150), 0.9702 (PA6 f150), and 1.0435 (PA4 f220). Polarization efficiencies remain distinct factors. The measured T-to-P leakage is about 0.3%; ignoring its correction shifts $\Omega_bh^2$ by about $1.7\sigma$ and $n_s$ by about $1\sigma$. A fitted global rotation $0.20^\circ\pm0.08^\circ$ is calibration-sensitive and must not be relabeled as cosmic birefringence without a separate instrument-angle model.

Multifrequency likelihood and foreground stress tests

For bandpower vector $d$ the DR6 MFLike 1.0.0 model uses

$$D_b^{\mathrm{th},aX,bY}=D_b^{\mathrm{CMB},XY} +D_b^{\mathrm{fg},aX,bY}, \qquad -2\ln\mathcal L=(d-m)^{\mathsf T}C^{-1}(d-m)+\mathrm{const},$$

with a fixed covariance $C$. The 1651-point multifrequency vector includes TT, TE, and EE, while the foreground model contains radio, Poisson and clustered CIB, tSZ, kSZ, tSZ×CIB, and Galactic dust terms. The fit includes 15 foreground parameters, 14 of them independently varying, plus six calibration parameters, four polarization efficiencies, and five bandpass shifts, with the priors and component formulae specified by Louis et al. 2025, §6.1, printed pp. 22–27, eqs. (16)–(29) and Tables 3–4, PDF. The baseline ACT fit reports $\chi^2_{\mathrm{MFLike}}=1590.91$ for 1651 data points; the point count is not itself the number of degrees of freedom.

One component-level target is the tSZ amplitude at $\ell=3000$ and 150 GHz. The nominal public analysis finds $a_{\mathrm{tSZ}}=3.4\pm0.4\ \mu\mathrm K^2$ for ACT and $3.3\pm0.4\ \mu\mathrm K^2$ for P-ACT. Allowing the clustered and Poisson CIB terms to have different spectral indices moves the ACT value to $3.0\pm0.4\ \mu\mathrm K^2$; the change is roughly one statistical standard deviation and exposes foreground-model dependence. The nominal P-ACT slope is $\alpha_{\mathrm{tSZ}}=-0.6\pm0.2$; in the ACT split-CIB-index variant it is $-0.7^{+0.3}_{-0.2}$, so the preference for a negative slope remains. These values and the correlated kSZ/CIB alternatives appear in Louis et al. 2025, §8.6, printed pp. 38–40, eqs. (47)–(48) and Table 7, PDF.

Reproduction path and honest boundary

The compact PSPIPE spectra-and-covariance release provides calibrated array-band spectra with columns for data, uncertainty, best-fit total model, and best-fit foreground, plus frequency-combined and foreground-subtracted spectra. The likelihood-binning archive is the 2025-03-18 act_dr6.02_spectra_and_cov_binning_50.tar.gz; the corresponding multifrequency SACC product is dr6_data.tar.gz. A full reproduction uses MFLike 1.0.0, fgspectra 1.3.0, the released covariance, the cuts above, and the priors in Tables 3–4, then repeats the fit after changing the mask, removing one frequency, raising $\ell_{\min}$, and splitting the CIB spectral indices. The software repository and machine-readable data product are both public; neither the paper’s plotted curves nor its rounded table entries substitute for those inputs.

This page has independently checked the product identities, selections, equations, and published component shifts, but does not claim a new MFLike chain run. The numerical values above are therefore a source-backed reproduction target, not a replacement posterior. A credible rerun must also recover the released foreground-subtracted cross-spectra and show that beam, bandpass, leakage, calibration, filtering, mask, noise, and simulation changes remain smaller than the declared statistical and model-dependent errors.

Exercises

Two calibrated channels obey $d_1=s+f_1a+n_1$ and $d_2=s+f_2a+n_2$, where $s$ is CMB and $a$ one foreground. Find weights that preserve $s$ while nulling $a$, and identify when the separation is unstable.

Solution

Require $w_1+w_2=1$ for unit CMB response and $w_1f_1+w_2f_2=0$ for foreground cancellation. Solving gives $w_1=f_2/(f_2-f_1)$ and $w_2=-f_1/(f_2-f_1)$. If $f_1\simeq f_2$, both weights become large: noise is amplified and a small bandpass or spectral-law error leaves a large residual. The same conditioning problem appears in the multicomponent mixing matrix, which is why frequency count alone does not guarantee robust separation.

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.
• Næss, S., et al. (ACT Collaboration), “The Atacama Cosmology Telescope: DR6 Maps,” arXiv:2503.14451v1 (2025), Open PDF, data products.