A representation of a signal's strength as a function of frequency, revealing periodicities.The power spectrum is a mathematical tool that decomposes a signal — whether it's a time series of ocean wave heights, a sound recording, or the brightness of a star over time — into its constituent frequencies. It tells you how much of the signal's total 'energy' or 'power' is concentrated at each frequency. A signal that repeats regularly, like a pendulum swinging, will show a sharp peak in its power spectrum at the corresponding frequency; a random, noisy signal will show a broad, flat distribution. This makes the power spectrum an essential first step in identifying hidden periodicities in data that might otherwise look chaotic.
The concept originates from Joseph Fourier's work in the early 19th century on the decomposition of periodic functions into sums of sines and cosines. The term 'power spectrum' was formalized later, particularly through the work of Norbert Wiener in the 20th century, who applied it to stochastic processes.
The power spectrum is typically computed using the Fourier transform, which converts a signal from the time domain (amplitude vs. time) into the frequency domain (amplitude vs. frequency). The squared magnitude of the Fourier transform at each frequency gives the power at that frequency. For discrete data, a fast Fourier transform (FFT) algorithm efficiently performs this calculation. The result is a plot or array showing how power is distributed across frequencies, allowing you to spot dominant oscillations, trends, or noise characteristics.
Power spectra are used everywhere from audio engineering (equalizers) to astrophysics (detecting exoplanet orbital periods from stellar brightness variations) and climate science (identifying El Niño cycles in temperature records).
The power spectrum of a perfect sine wave is a single spike at its frequency, but the power spectrum of a single short pulse is a flat distribution across all frequencies — which is why a handclap sounds like a 'white noise' burst.
| r | variable | tensor-to-scalar ratio — dimensionless ratio |
| P_t | variable | power spectrum of tensor perturbations — power spectrum |
| P_s | variable | power spectrum of scalar perturbations — power spectrum |
| P(k) | variable | Perturbed power spectrum — power spectrum cubic metre |
| P_0(k) | variable | Background power spectrum — power spectrum cubic metre |
| A | variable | Amplitude of the bump — amplitude |
| k | variable | Wavenumber — wavenumber per metre |
| k_0 | variable | Central wavenumber of the bump — wavenumber per metre |
| σ | variable | Width of the bump in log-k — width |
| exp | operator | Exponential function — exponential |
| log | operator | Natural logarithm — logarithm |
| P(k) | variable | Suppressed power spectrum — power spectrum cubic metre |
| P_0(k) | variable | Unsuppressed power spectrum — power spectrum cubic metre |
| k | variable | Wavenumber — wavenumber per metre |
| k_fs | variable | Free-streaming wavenumber — wavenumber per metre |
| exp | operator | Exponential function — exponential |
| n_s | variable | scalar spectral index — scalar spectral index |
| P_R | variable | power spectrum of curvature perturbations — power spectrum |
| k | variable | wavenumber — wavenumber reciprocal metre |
| d/d ln k | operator | derivative with respect to natural logarithm of wavenumber — logarithmic derivative |
| C^ell | variable | Angular power spectrum at multipole ell — angular power spectrum |
| z | variable | Redshift — redshift |
| H(z) | variable | Hubble parameter at redshift z — hubble parameter kilometre per second per megaparsec |
| chi(z) | variable | Comoving distance to redshift z — comoving distance metre |
| W(z) | variable | Window function (selection function) at redshift z — window function |
| P(k,z) | variable | Three-dimensional power spectrum at wavenumber k and redshift z — power spectrum cubic metre |
| k | variable | Wavenumber — wavenumber reciprocal metre |
| ell | variable | Multipole moment — multipole moment |
| c | constant | Fundamental constant relating distance and time in relativity — speed of light = 299792458 metre per second |
| ∫ dz | operator | Sum over all redshifts along the line of sight — integral operator |
| P(k) | variable | Dimensionless power spectrum — power spectrum |
| k | variable | Wavenumber — wavenumber reciprocal metre |
| k_* | variable | Pivot wavenumber — wavenumber reciprocal metre |
| n_s | variable | Spectral index — spectral index |
| A_s | constant | Amplitude of primordial scalar perturbations at the pivot scale — scalar amplitude = 2.1e-9 |
