### 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

Index