---
title: "What a telescope adds to the sky"
description: "Calculate Gaussian beam attenuation, follow sky-mask coupling, and distinguish an ideal full-sky likelihood from a foreground-marginalized analysis."
canonical_url: "https://sandboxphysics.com/stories/cosmology/observing-and-likelihood/"
language: "en"
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  zh-CN: "https://sandboxphysics.com/zh/stories/cosmology/observing-and-likelihood/"
last_modified: "2026-10-28"
---

[Cosmology from expansion to inference](https://sandboxphysics.com/stories/cosmology/)

L3 · Chapter 6

# What a telescope adds to the sky

Sandbox Physics · 2026-10-28

A telescope measures a blurred sky with noise and foreground light. To infer the universe, predict those measurements as well as the cosmological signal. A good match to a plotted curve becomes informative only after its uncertainty and the uncertain instrument parameters enter the calculation.

## Before you begin

Read the spherical-harmonic sky and cosmic-variance chapter. You need Gaussian random variables, variance and a definite integral. A covariance matrix lists the variances and correlations of several measurements.

[Review the preceding chapter](https://sandboxphysics.com/stories/cosmology/one-sky-statistics/)

- Calculate how a circular beam suppresses a harmonic mode.
- Explain why masking couples angular scales.
- Choose a likelihood appropriate to the data and carry nuisance uncertainty into inference.

## 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

## Try the prediction

### [Let foreground light change a fitted temperature](https://sandboxphysics.com/labs/cmb-spectrum-thermometer/)

1. Predict whether warm dust makes a blackbody-only temperature too high or too low.
2. Compare the blackbody-only fit with the fit that includes dust.
3. Explain which added frequency information separates the two components.

### [Follow the beam through one saved sky](https://sandboxphysics.com/labs/cmb-sky-angular-spectrum/)

1. Predict which angular scales fade first as the beam widens.
2. Change only the beam width and compare the map with the power from its same harmonic coefficients.
3. Explain why increasing resolution helps at small scales without providing a second independent universe.

## Exercises and explanations

1. If the beam amplitude response is $B_\ell=0.5$, what fraction of signal power remains, and how much does beam deconvolution amplify additive noise power?

   Read the explanation

   One quarter of the signal power remains because power is proportional to amplitude squared. Dividing by that response multiplies the additive noise power by four. This operation restores a mean response and cannot recover information at no cost.

2. A dust amplitude is known only approximately and resembles part of a cosmological signal. Why can fixing its best-fit value give a narrower answer than marginalizing?

   Read the explanation

   Fixing conditions on one dust value and removes its uncertainty. Marginalizing includes the range of cosmological explanations compatible with other allowed dust values. Their overlap carries uncertainty into the cosmological parameter. The size and direction of the change depend on the model and prior.

[Previous: How much can one sky tell us about its power?](https://sandboxphysics.com/stories/cosmology/one-sky-statistics/) [Next: Read a public lensing likelihood](https://sandboxphysics.com/stories/cosmology/lensing-likelihood/)

## Sources and calculation scope

1. [Hivon et al., MASTER of the CMB Anisotropy Power Spectrum, astro-ph/0105302v1 (2001)](https://arxiv.org/abs/astro-ph/0105302v1)

   The pseudo-angular-power estimator and ensemble mask-coupling relation. No real-sky mask inversion is performed by the full-sky teaching experiment.

2. [David Tong, Cosmology, Chapter 3](https://davidtong.org/teaching/cosmology/cosmohtml/S3)

   Structure Formation: linear growth, pressure support, transfer functions and the microwave sky. The pressureless growth example applies inside the Hubble scale, during matter domination.

3. [Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209v4 (2021)](https://arxiv.org/html/1807.06209v4)

   Sections 2.2–2.3 define the data likelihoods, section 3.1 discusses the acoustic scale, and sections 6.2 and 7.3–7.5 separate lensing smoothing, curvature, dark energy and relativistic species. Published in A&A 641, A6 (2020), with the v4 corrigendum. This course does not reproduce the full Planck likelihood or chains.