F-X1b · A wider beam erases smaller patches

CMB Sky and Angular Power

Why does a broad telescope beam remove small sky patterns before large ones? A broad beam averages neighbouring sky directions. Small hot and cold patches cancel within it, so their angular power falls quickly. Large patches change little across the same beam and keep more of their contrast.

Drag the receiver footprint across a synthetic all-sky temperature map. Widen the beam and compare the remaining patterns with the angular power computed from the same sky.

Original sky RMS $35\,\mathrm{\mu K}$

Blurred sky RMS $30\,\mathrm{\mu K}$

Power retained $80\%$

This synthetic full-sky field contains real orthonormal spherical harmonics of degrees two through twelve. Independent fixed-seed Gaussian coefficients have a prescribed teaching spectrum; the small-scale multiplier changes degrees seven through twelve. The mean temperature and dipole are removed. An ideal circular Gaussian beam attenuates each coefficient by its harmonic window function; the original and blurred power spectra are calculated from those same coefficients. The Hammer map uses bilinear interpolation of a latitude–longitude grid for display, while numerical readouts use the coefficients exactly. These large patches illustrate beam smoothing, not the measured CMB acoustic peaks or cosmological parameter inference. No instrument noise, foregrounds, sky mask, polarization or evolving universe is modeled. The beam circle is a local schematic footprint; its drawn radius represents half the beam width near the map centre.

Synthetic model data

Power in large and small patches

Dashed blue: original synthetic sky. Teal: the same coefficients after beam smoothing. Each point is one angular degree; larger degrees describe smaller patches.

\ell,\qquad D_\ell=\frac{\ell(\ell+1)}{2\pi}C_\ell\;[\mathrm{\mu K}^2]

Sources and definitions

Power follows the orthonormal spherical-harmonic convention. The beam is an ideal circular Gaussian. The finite harmonic field is a teaching example, and its two spectra are evaluated directly from its coefficients. HEALPix conventions · Gaussian beam

Read the discoveries behind these models →

Start with the essentials

Focus question: Why does a broad telescope beam remove small sky patterns before large ones?
One-sentence intuition: Beam width changes the sky we measure, with the strongest loss at small angular scales.

Core mathematical model

Build the temperature map from the same modes

\Delta T(\hat n)=\sum_{\ell=2}^{12}\sum_{m=-\ell}^{\ell}a_{\ell m}Y_{\ell m}(\hat n)

Each spherical harmonic is a pattern on the whole sphere. The real coefficients set its strength, in microkelvin. The teaching sky has neither a mean offset nor a dipole.

Power at one angular degree

C_\ell=\frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}|a_{\ell m}|^2

The power averages the squared coefficients at a given degree. The chart multiplies it by degree times degree plus one, divided by twice pi.

A circular Gaussian beam

b_\ell=e^{-\ell(\ell+1)\sigma_b^2/2},\qquad C_\ell^{\rm beam}=b_\ell^2C_\ell,\qquad\sigma_b=\frac{\theta_{\rm FWHM}}{\sqrt{8\ln2}}

Use radians for the beam width. Smoothing multiplies each coefficient by the beam window, so power is multiplied by its square.

Common difficulties

A measurement to interpret carefully

Typical misconceptionA low high-degree power measurement must mean the original sky had little small-scale structure.

Better mental modelA broad receiver beam also suppresses that power. Interpreting an observed spectrum requires the beam response as well as the measured map.

Run the experiment

01
Compare the two sky views
Choose Sharp sky, then switch between Through the beam and Original sky.
What to observe: With zero beam width the maps and their power curves coincide.
02
Broaden the receiver footprint
Choose Broad beam. Switch sky views without changing the colour scale.
What to observe: Small hot and cold patches fade together. The higher-degree end of the teal power curve falls most strongly.
03
Give the sky smaller patches
Choose Small patches, then increase Beam width slowly.
What to observe: The synthetic sky starts with more power in degrees seven through twelve, and the beam removes much of that extra detail.
04
Look at another realization
Choose Another sky, then compare the map and power curve.
What to observe: New random coefficients change the patch positions and their actual power. Both displays update from that same new field.

Core mathematical model

Build the temperature map from the same modes

Power at one angular degree

A circular Gaussian beam

Common difficulties

A measurement to interpret carefully

Run the experiment

Compare the two sky views

Broaden the receiver footprint

Give the sky smaller patches

Look at another realization