| Astrophysics (Index) | About |
An angular power spectrum (often shortened to the phrase, power spectrum, but which is also used for other types of "non-angular" power-related spectra) is a type of characterization of a function on the surface of a sphere, or analogously, on all directions in space from a point. It captures how much difference there is between the function values at different points as a function of the angular distance between them. An angular two-point correlation would do this, showing a correlation for each possible angle, but the spectrum more often termed the angular power spectrum is the correlation over each multipole moment (i.e., spherical harmonic order), l of a multipole expansion (analogous to a Fourier series expansion, but for functions over the surface of a sphere). The multipole moments of each order (l) are separated by the same angular distance, and the angular distance decreases as l increases. For each order l, the power spectrum is the average of the squares of the spherical harmonic coefficients associated with that l (i.e., the square of their RMS). Each order l has l spherical harmonic coefficients, each representing a division of the surface of the sphere (for example, there are three coefficients that have order l=3). This allows characteristics of the scale of the variations in a function over a sphere to be displayed in a two-dimensional graph.
The cosmic microwave background (CMB) is often characterized by such an angular power spectrum of its anisotropies (CMB anisotropies) that show its temperature variation, using coefficients of a multipole expansion of the temperature over the celestial sphere. The temperature is calculated (in principle) by Wien's displacement law from the received EMR (in practice, more of its spectral energy distribution is considered). This is the most-commonly-cited CMB angular power spectrum, but analogous spectrums of CMB polarization and gravitational lensing are also often cited. From the angular power spectrum, scales in the early universe can be determined, and values such as the six parameters of the Lambda-CDM model can be checked for consistency with the CMB.
An angular power spectrum can be calculated for any function of a sphere, such as any type of intensity map over the celestial sphere as well as densities of some type of astronomical object over the celestial sphere; it is the CMB's that are commonly cited, but others are used in cosmology as well.