Foregrounds, detector response and recovered maps

Several physical signals enter each detector channel

A microwave channel contains the cosmic signal together with foreground emission and instrument response. Recovering a CMB map requires assumptions about frequency dependence, calibration, angular resolution and noise.

The CMB response can be made frequency independent in calibrated thermodynamic temperature-fluctuation units using the local Planck derivative δI_ν = (dB_ν/dT)δT about a common blackbody. That statement does not say raw specific intensity is flat with frequency.

Real channels also have bandpasses and beams, and foreground spectra can vary across the sky. A simple linear separation is useful for exposing these ingredients but does not establish that a small number of real channels removes every foreground.

The example below follows one real harmonic coefficient in two exactly calibrated noiseless temperature channels. Every amplitude, foreground response and beam width is supplied, with no measured sky map or residual.

Use one amplitude convention before mixing channels

Choose a CMB coefficient s = 10 μK and a foreground reference coefficient d = 4 μK, both in calibrated thermodynamic fluctuation units. The supplied effective foreground responses are f_1 = 1 and f_2 = 3. Without beam smoothing, the channels would be s + d = 14 μK and s + 3d = 22 μK.

For an ideal narrow circular temperature beam, use B_(i,l) = exp[−l(l + 1)σ_i²/2], with σ_i = θ_FWHM,i/√(8 ln 2). The chosen full widths are 0.01 and 0.02 radians, already expressed in radians rather than an API’s possible arcminute convention.

At the selected l = 100, the beam amplitudes are B_1 ≈ 0.912954 and B_2 ≈ 0.694696. The noiseless measured coefficients are m_i = B_i(s + f_i d).

This is an amplitude model for a temperature coefficient. For an angular power spectrum, the corresponding beam window is B_l². Polarisation beam conventions and arbitrary real detector profiles are outside the supplied calculation.

Unequal smoothing changes an uncorrected mixture

The chosen beams give m_1 ≈ 12.781352 μK and m_2 ≈ 15.283319 μK. Treating those unequally smoothed numbers as the direct common-resolution mixture would give the wrong recovered inputs.

Correct the known nonzero beams first: r_1 = m_1/B_1 = 14 μK and r_2 = m_2/B_2 = 22 μK. Then d = (r_2 − r_1)/2 = 4 μK and s = (3r_1 − r_2)/2 = 10 μK.

If the beam correction is omitted, the same algebra gives a naive CMB coefficient about 11.530368 μK and foreground coefficient about 1.250984 μK. These are deterministic consequences of applying the wrong response model, not observed bias estimates.

The exact recovery depends on two known distinct component responses, exactly known calibration and beams, and no additional components or noise. The response difference f_2 − f_1 = 2 makes this particular two-component system invertible.

A smaller beam amplitude can make inversion fragile

The beam suppresses finer angular structure more strongly in this ideal model. At l = 300, B_1 ≈ 0.442984 while B_2 ≈ 0.0385082. Their power windows are about 0.196235 and 0.00148288, respectively.

An amplitude correction divides by B_i, not B_i². If additive channel noise were present, that division would also multiply its amplitude by 1/B_i. At l = 300 the second channel’s amplification factor would be about 25.9685.

No noise distribution or error bars have been assigned here. Under a separately supplied zero-mean independent channel-noise model, the recovered CMB variance would involve [9σ_n1²/B_1² + σ_n2²/B_2²]/4. Independence and those variances would be additional assumptions.

Thus exact noiseless algebra does not guarantee a useful real high-resolution recovery. Small responses, uncertainty in the beam and errors in the component spectrum can make an inverse problem sensitive even when its matrix is formally invertible.

A recovered map must carry its response and limitations

Real component separation accounts for bandpass integration, calibration, beams, noise and several foreground components. Effective responses can vary with position or physical conditions, so the supplied fixed foreground factor 3 is not a measured universal dust spectrum.

Algorithms can use multiple channels, spatial information and different physical or statistical assumptions. Agreement between methods is informative but does not certify that all shared systematic errors have vanished.

The interpretation of a recovered map or power spectrum therefore needs its unit convention, transfer function, masks, uncertainty model and residual tests. A smooth-looking sky image alone does not supply those checks.

The two-channel example gives a concrete starting rule: put channels into compatible units and response conventions before solving for their components, and retain the sensitivity introduced by any correction.

Worked reasoning

Try the reasoning

1. Is f_2 = 3 a measured dust spectrum?

   Show the answer and reasoning

   No.

   It is a chosen effective thermodynamic channel response for the linear example.

2. Should a temperature harmonic amplitude be divided by B_l² to undo this beam?

   Show the answer and reasoning

   No; its amplitude response is B_l.

   B_l² is the corresponding power-spectrum window.

3. Does exact recovery in this example prove two real channels remove every foreground?

   Show the answer and reasoning

   No.

   Real spectra, extra components, noise, calibration, bandpass and beam uncertainty require a more complete inference model.

What to carry forward

• Component separation needs compatible units and channel response models.
• Known unequal beams must be accounted for before the supplied mixture is inverted.
• Beam amplitude, power window and inverse-noise amplification are different quantities.