Atomic Orbitals, Music, and the Early Universe
If you've seen pictures of atomic orbitals in chemistry, the first panel should look familiar. It shows a single spherical harmonic as a three-dimensional wave pattern including radial displacement from the origin, analogous to the probability of finding an electron some distance from the nucleus. The numbers ℓ and m determine the pattern's complexity and orientation. Move the sliders and see what happens. To see what superposition does, check "Superpose second harmonic" and select a second pair of ℓ and m values.
The second panel shows a simulated Cosmic Microwave Background (CMB) sky, using observed angular spectra from Planck collaboration data. Only the angular coordinates are kept and the value is represented using color across the surface of a sphere, projected onto a flat canvas using a Mollweide projection.
Unchecking Show only current (ℓ, m) displays a 'heat map' formed by superposing many spherical harmonics with different amplitudes up to the maximum ℓ you selected with the first slider. Increasing the maximum ℓ adds progressively finer wave patterns, revealing more detailed structure in the simulated universe.
Pressing Build Universe starts an animation that adds modes up to the current maximum ℓ value on the slider. Notice how the largest features appear first, while smaller details emerge as higher-order harmonics are added. This is the same mathematical framework cosmologists use to describe the temperature fluctuations of the real CMB.
Maybe you've heard someone online break down a song into “stems.” Software can separate the bass, guitar, keyboards, and vocals. These programs use Fourier methods to decompose complex sounds into simpler components that can then be recombined to reproduce the song.
Just as a musical chord can be decomposed into individual notes, any sufficiently smooth function on a sphere, such as the temperature pattern of the CMB, can be decomposed into spherical harmonics. The real CMB contains significant power well beyond ℓ = 200. Reproducing those finer details requires the superposition of tens of thousands of spherical harmonic modes—far more computation than is practical in a web browser.