Astrophysics (Index)About

angular power spectrum

(power spectrum)
(method of characterizing the CMB)

An angular power spectrum (often shortened to power spectrum, but this latter phrase is also used for other types of power spectra) is a type of characterization of a function on the surface of a sphere, or analogously, all directions in space from a point. It captures variance as a function of scale. An angular two-point correlation does 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). For each l, it is the average of the squares of the spherical harmonic coefficients associated with the value l (i.e., the square of their RMS). The spherical harmonic coefficients associated with l specify the placement of l equal-sized regions over the sphere.

The cosmic microwave background (CMB) is often characterized by such an angular power spectrum of its anisotropy, showing the temperature variation for 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, the whole spectral energy distribution is considered). The most commonly cited CMB angular power spectrum is of its temperature, but similar spectrums of CMB polarization and/or gravitational lensing are also used. 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.

The angular power spectrum can be calculated for any function of a sphere, including any type of intensity map over the celestial sphere: it is the CMB's that have been found to be useful, but others might be useful to cosmology as well.

Further reading:

Referenced by pages:
CMB anisotropies
CMB lensing
CMB polarization
diffusion damping
Hellings and Downs curve
initial fluctuation spectrum
magnetic field
N-point function