Background radiation and foreground removal

Name the measured product before interpreting it

Background radiation is not a detector label that automatically separates distant cosmological emission from everything along the sightline. An instrument records their combined radiation through its response. Foreground and background identify roles in an inference, with the desired component determined by the scientific question.

An absolute spectrum concerns calibrated intensity as a function of frequency. A map of anisotropy concerns variation across directions, in temperature-equivalent intensity or polarization quantities under specified conventions. The average spectral shape and the spatial pattern are related scientific subjects, but are different measurement products.

FIRAS illustrates the importance of absolute calibration. Its differential optical system compared signals, while an external blackbody calibrator supplied the fundamental absolute comparison with the sky. Instrumental differences and reference properties remained part of that comparison; a fitted blackbody spectrum was not simply a raw detector voltage.

A frequency sky map also contains emission from dust, gas, compact sources and other contributions, as well as instrument effects and noise. Removing a modelled component changes the product and its uncertainty. A map called cleaned must therefore be accompanied by the method and conditions that define what has been removed.

Comparable maps need comparable responses

A broad frequency channel integrates a source spectrum over a measured bandpass. Its reported brightness depends on a reference spectrum and unit convention. A colour correction accounts for a different spectral shape; the channel label alone does not specify its response to every component.

Angular beams, pixelization, offsets and calibration also enter. Before combining maps, their differing responses must be represented or brought into a suitable common form. The Planck component-separation paper explicitly models frequency scaling, projection or beam response, calibration and noise as parts of its data relationship.

For the original arithmetic example below, stipulate four invented bands already expressed at one common angular beam, for one pixel. All values use a linearized signed thermodynamic-CMB-equivalent perturbation unit, microK_CMB. This means equivalent intensity changes relative to a reference CMB spectrum, not absolute temperatures of the emitting foreground material.

Let c, a and b be component amplitudes in that convention, with dimensionless band scalings listed below. The model in each band is its row multiplied by (c, a, b), plus additive noise. No row is an actual Planck or FIRAS frequency response, and the two foreground templates are invented mathematical shapes.

Isolation has an uncertainty cost

Use only the first three bands to estimate c with weights (−2, 5, −2). Their sum is 1, so they preserve the CMB column. Applied to template a, they give −2 × 1 + 5 × 2 − 2 × 4 = 0. Applied to template b, they give −2 × 8 + 5 × 4 − 2 × 2 = 0.

Consequently chat = −2d1 + 5d2 − 2d3 = −118 + 270 − 112 = 40 microK_CMB for the noiseless supplied values. Negative weights are permitted in a linear estimator. They do not imply a negative absolute temperature or that a foreground has physically been subtracted from the sky.

Now stipulate independent, zero-mean additive noise with standard deviation 1 microK_CMB in each of those three bands, and treat the template scalings and weights as exact. The variance of chat is the sum of squared weights times the individual variances: 4 + 25 + 4 = 33 (microK_CMB)².

The resulting standard uncertainty is √33 ≈ 5.744562647 microK_CMB, larger than the noise in any one band. A component-isolating combination can amplify noise. Correlated noise would require the covariance expression wᵀVw, and uncertain calibration or template shapes would add further terms outside this deliberately restricted calculation.

An unused measurement can expose some failures

Solving the first three noiseless equations recovers c = 40, a = 3 and b = 2. Those fitted amplitudes predict band 4 as 40 + 8 × 3 + 2 = 66 microK_CMB. Because the fourth band was not used in this fit, comparing it with that prediction supplies an additional check.

For one separate error scenario, add 4 microK_CMB only to band 4 while leaving the first three unchanged. The observed vector becomes (59, 54, 56, 70). The fitted prediction is still 66, so the fourth-band residual, defined as data minus prediction, is +4 microK_CMB.

That residual exposes disagreement with the stated exact mixture, but does not identify its cause. A band-specific calibration or offset, an omitted component, response error or additive noise could affect the comparison. Statistical significance would require the residual covariance, including uncertainty in the prediction.

Keeping a band out of a fit is useful only within the scientific design. If it is repeatedly used to select templates, it has become development information. Further claims of independent predictive performance then need a new reserved test or an explicitly accounted selection process.

A parallel component passes this check

Consider a different scenario: an unmodelled contribution adds 4 microK_CMB equally to all four bands. The data become (63, 58, 60, 70). Its spectral vector is exactly parallel to the retained CMB column (1, 1, 1, 1). This is a stipulated mathematical ambiguity, not a claim that a particular physical foreground has that spectrum.

The first three bands now fit c = 44, a = 3 and b = 2. Their fourth-band prediction is 44 + 24 + 2 = 70, exactly matching the altered fourth value. The isolated CMB estimate is biased by +4 relative to the originally supplied c, while every residual in this exact model is zero.

Adding more observations with the same indistinguishable response cannot by itself label the two parallel contributions separately. Additional spectral, spatial, temporal or external information would need a justified relationship that distinguishes them. A prior preference can select an allocation, but is an added assumption rather than new identifying data.

These two error scenarios show the scope of residual checks. One mismatch is detectable by the unused band, whereas an exactly parallel contribution is not. Neither a small residual nor agreement between algorithms proves that all shared calibration or modelling assumptions are correct.

Component maps precede conditional physical interpretation

Real analyses use many pixels and bands, masks, noise models and multiple component-separation methods. Data splits and alternative processing help examine robustness to particular instrument and foreground assumptions. The Planck authors also distinguish tasks for which different releases are appropriate; a later release should not be assumed to improve every foreground product.

FIRAS documentation retains calibrated spectra, covariance information and model-selection error products as well as derived maps. That separation makes clear that a high-level component map is accompanied by an observation and processing history. Correlations introduced during processing are part of its uncertainty, not disposable detail.

The Foundations acoustic mechanism describes a conditional physical account of CMB angular structure involving coupled photons and baryons, scattering history and geometry. It does not turn a cleaned pixel into a cosmological parameter reading. Parameter inference further compares statistics with predictions under a specified model and likelihood.

Retain frequency responses, beams, units, calibration, noise and covariance, masks, templates, processing choices and residual tests when passing on a background-radiation result. The following lesson assembles that observational handoff while leaving detailed cosmic models and their parameter inference to the independent Cosmology course.

Worked examples

Try the reasoning

1. Do the invented foreground amplitudes in microK_CMB give the physical temperatures of their emitters?

   Show the answer and reasoning

   No.

   They are linearized CMB-equivalent intensity perturbations in the declared unit convention.

2. Does a negative cleaning weight imply negative absolute sky temperature?

   Show the answer and reasoning

   No.

   It is a coefficient in an estimator of signed perturbation amplitudes.

3. What fourth-band residual follows when only that datum rises from 66 to 70 while the fit remains unchanged?

   Show the answer and reasoning

   +4 microK_CMB.

   Residual means data minus the prediction of 66.

4. Does zero residual rule out an unmodelled contribution exactly parallel to the CMB template?

   Show the answer and reasoning

   No.

   It can be absorbed into the fitted CMB amplitude without changing the residual.

What to carry forward

• An absolute spectrum and an anisotropy map are distinct calibrated products.
• Component isolation depends on responses, units, templates and covariance, and can amplify noise.
• Residual tests constrain specified failures but cannot identify contributions with exactly indistinguishable responses.