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 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).

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.

An 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 are commonly cited, but others might be useful to cosmology as well.

(harmonics,cosmology,CMB,function)
Further reading:
http://en.wikipedia.org/wiki/Cosmic_microwave_background
https://www.roe.ac.uk/ifa/postgrad/pedagogy/2006_tojeiro.pdf
https://cds.cern.ch/record/605007/files/0302217.pdf
https://www.astro.umd.edu/~miller/teaching/astr422/lecture21.pdf
https://sci.esa.int/web/planck/-/51555-planck-power-spectrum-of-temperature-fluctuations-in-the-cosmic-microwave-background
https://www.colorado.edu/studentgroup/ostem/outreach/workshop-understanding-cosmic-microwave-background/week-4-cmb-workshop

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

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