Introduction to Cosmology

Temperature Anisotropies 181

A

B

Figure 8.2The observer A in the solar rest frame sees the CMB to have dipole anisotropy—the
length of the radial lines illustrate the CMB intensity—because he is moving in the direction
of the arrow. The fundamental observer at position B has removed the anisotropy.

For small angles(휃)the temperature autocorrelation function can be expressed as a
sum ofLegendre polynomials푃퓁(휃)of order퓁,thewavenumber, with coefficients or
powers푎^2 퓁,

퐶(휃)=^1

4 휋

∑∞

퓁= 2

푎^2 퓁( 2 퓁+ 1 )푃퓁(cos휃). (8.19)

All analyses start with the quadrupole mode퓁=2 because the퓁=0 monopole mode
is just the mean temperature over the observed part of the sky, and the퓁=1 mode is
the dipole anisotropy. As a rule of thumb, higher multipoles correspond to fluctuations
on angular scales

휃≃^60

∘

퓁

.

In the analysis, the powers푎^2 퓁are adjusted to give a best fit of퐶(휃)to the observed
temperature. The resulting distribution of푎^2 퓁 values versus퓁is called thepower
spectrumof the fluctuations. The higher the angular resolution, the more terms of