What a telescope adds to the sky

Write an equation for the measurement

Let $a_{\ell m}$ be a temperature-sky coefficient in microkelvin. For a full-sky map and an axisymmetric beam, the observed coefficient $b_{\ell m}$ is the beam-filtered sky plus instrument noise $n_{\ell m}$ and effective foreground $f_{\ell m}$ . The foreground here already includes the response of the observing channel.

A circular Gaussian beam has a full width at half maximum $\theta_{\mathrm{FWHM}}$ . Convert it to radians before calculating its standard deviation $\sigma_b$ . Each harmonic amplitude is multiplied by $B_\ell$ ; a power spectrum is multiplied by its square. A wider beam therefore removes fine angular structure much more strongly than broad patches.

b_{\ell m}=B_\ell a_{\ell m}+n_{\ell m}+f_{\ell m},\qquad B_\ell=\exp\!\left[-\frac{\ell(\ell+1)\sigma_b^2}{2}\right]

This diagonal harmonic response assumes the same circular beam in every direction. Asymmetric beams and direction-dependent processing require a more general response.

\sigma_b=\frac{\theta_{\mathrm{FWHM}}}{\sqrt{8\ln2}}

The square-root factor converts the width between half-height points into the standard deviation of a Gaussian.

Reading: Planck 2018 results. VI. Cosmological parameters

A ten-arcminute beam at multipole one thousand

Take an illustrative beam width of $10\,\mathrm{arcmin}$ and inspect $\ell=1000$ . Converting angular units gives $\theta_{\mathrm{FWHM}}=10\pi/(180\times60)=0.00291\,\mathrm{rad}$ . Dividing by the Gaussian width factor gives $\sigma_b=0.00124\,\mathrm{rad}$ . Substitution yields $B_{1000}=0.466$ and $B_{1000}^2=0.217$ . Only about twenty-two percent of this mode’s original power remains.

Dividing observed power by $B_\ell^2$ restores the mean signal response, but also amplifies noise power. In this example the factor is $1/B_{1000}^2=4.61$ . The lost resolution therefore has a statistical cost even when the beam is known exactly. An uncertain beam adds another source of uncertainty.

A masked sky mixes harmonic modes

Galactic emission can overwhelm the cosmological signal. An analysis often multiplies the map by a weight $W(\hat{\boldsymbol n})$ that suppresses contaminated directions. Multiplication on the sky mixes spherical-harmonic coefficients. The power calculated directly from the weighted map is called a pseudo-spectrum.

Its ensemble mean involves a coupling matrix $M_{\ell\ell'}$ determined by the mask. Noise and foreground contributions must also be treated. The retained sky fraction alone cannot encode which modes are coupled. A full analysis computes the response and the covariance for its actual weighting, filtering and binning.

This also explains why neighbouring bandpowers can have correlated errors. A plot that shows only separate vertical error bars hides part of the information needed for fitting.

\left\langle\widetilde C_\ell\right\rangle=\sum_{\ell'}M_{\ell\ell'}B_{\ell'}^2C_{\ell'}+\left\langle\widetilde N_\ell\right\rangle

This relation is written after foreground treatment for a statistically isotropic signal and a known mask. Extra filtering and pixel response would supply additional factors.

Reading: Hivon et al.

Turn a probability model into an inference

Begin with an ideal Gaussian, foreground-free, full-sky temperature field and isotropic Gaussian noise. Write $S_\ell=B_\ell^2C_\ell+N_\ell$ for its total observed power. The measured power $\widehat C^{\mathrm{obs}}_\ell$ is a sum of squares, so its scaled sampling distribution is chi-squared. At low multipoles this distribution is visibly skewed.

When many modes contribute to each bandpower, a Gaussian approximation can become useful. Its covariance must still describe the mask, noise and any processing. Planck uses different low- and high-multipole likelihood constructions; the ideal full-sky formula is a useful starting example.

Foreground amplitudes, calibration and beam parameters are nuisance parameters: they affect the measurement although cosmology is the main question. Fixing them selects one conditional result. Marginalizing integrates over their allowed values with their prior $\pi(\boldsymbol\eta)$ . If they can imitate a cosmological change, carrying their uncertainty usually broadens the inferred range. Maximizing over them instead is profiling, a different operation.

The next chapter performs a restricted public bandpower analysis. Its frozen template and covariance make every step reproducible, and define exactly which inference is being made.

\frac{(2\ell+1)\widehat C^{\mathrm{obs}}_\ell}{S_\ell}\sim\chi^2_{2\ell+1},\qquad -2\ln\mathcal L=\sum_\ell(2\ell+1)\left[\ln S_\ell+\frac{\widehat C^{\mathrm{obs}}_\ell}{S_\ell}\right]+\mathrm{const}

This exact likelihood assumes independent Gaussian full-sky harmonic modes. The constant is independent of the model power. A fixed reference unit is implicit in the logarithm.

\mathcal L_{\mathrm{marg}}(\boldsymbol\vartheta)=\int\mathcal L(d\mid\boldsymbol\vartheta,\boldsymbol\eta)\,\pi(\boldsymbol\eta)\,d\boldsymbol\eta

The data are d, cosmological parameters are vartheta and nuisance parameters are eta. A posterior also includes the prior on the cosmological parameters.

Reading: David Tong · Planck 2018 results. VI. Cosmological parameters

What a telescope adds to the sky

Write an equation for the measurement

A ten-arcminute beam at multipole one thousand

A masked sky mixes harmonic modes

Turn a probability model into an inference

Try the prediction

Let foreground light change a fitted temperature

Follow the beam through one saved sky

Exercises and explanations

Sources and calculation scope