Cosmic microwave background anisotropies are direction-dependent perturbations of the relic radiation temperature and polarization.
Thomson scattering of a local radiation quadrupole generates linear cosmic microwave background polarization. Its acoustic phase follows the photon velocity and is shifted by one quarter-period relative to the monopole temperature oscillation.
The photon monopole is the direction-averaged fractional temperature perturbation. In Newtonian gauge its gravitationally observable effective temperature is the Sachs-Wolfe combination .
In Newtonian gauge, combines the intrinsic photon temperature perturbation with the gravitational redshift at emission. With constant potentials and negligible baryon loading it oscillates as for adiabatic initial conditions.
The photon dipole represents the bulk velocity of the radiation. In tight coupling to baryons, under the convention used here.
The photon quadrupole is the leading anisotropy that sources linear polarization through Thomson scattering.
Before recombination, frequent Thomson scattering couples photons, electrons, and baryons into an acoustic fluid.
The tight-coupling approximation expands the photon-baryon Boltzmann equations in powers of and , where is the Thomson scattering rate.
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There is a slight variation in temperature of CMB across the sky of the order of 200 microKelvin. It is small to the ~2.7 K average temperature, but it can be measured.
If the initial conditions of the Big Bang and the laws of physics were perfectly symmetric, then we could expect the universe to just be one perfectly uniform boring soup.
But instead some asymetry made all the fun weird things we see today happen eventually, like galaxies and life.
And the cosmic microwave background serves as a way for us to look back in time to the early conditions of the universe, as it was set in stone as soon as the universe became transparent to this light during recombination.
Or if you want to get poetic, it is the closest we can ever get to listening to the original word of God when he setup the initial conditions of the universe.
The ansiotropies of CMB is the ultimate astronomical compass we will ever have, as it is the thing with the least proper motion.
