180 Cosmic Microwave Background
8.2 Temperature Anisotropies
Dipole Anisotropy. The temperature measurement of Penzias and Wilson’s antenna
was not very precise by today’s standards. Their conclusion about the isotropy of the
CMB was based on an accuracy of only 1.0K. When the measurements improved over
the years it was found that the CMB exhibited adipole anisotropy. The temperature
varies minutely over the sky in such a way that it is maximally blueshifted in one
direction (call it훼) and maximally redshifted in the opposite direction (훼+ 180 ∘). In
a direction훼+휃it is
푇(휃)=푇(훼)( 1 +휈cos휃), (8.17)
where휈is the amplitude of the dipole anisotropy. Although this shift is small, only
휈푇(훼)≈ 3 .35mK, it was measured with an accuracy better than 1% by the Differential
Microwave Radiometer (DMR) instrument on board the COBE satellite [4].
At the end of Chapter 5 we concluded that the hot Big Bang cosmology predicted
that the CMB should be essentially isotropic, since it originated in the LSS, which
has now receded to a redshift of푧≃1080 in all directions. Note that the most distant
astronomical objects known have redshifts of about푧=7. Their distance in time to
the LSS is actually much closer than their distance to us.
In the standard model the expansion is spherically symmetric, so it is quite clear
that the dipole anisotropy cannot be of cosmological origin. Rather, it is well explained
by our motion ‘against’ the radiation in the direction of maximal blueshift with relative
velocity휈.
Thus there is a frame in which the CMB is isotropic—not a rest frame, since radia-
tion cannot be at rest. This frame is then comoving with the expansion of the Universe.
We referred to it in Section 2.2, where we noted that, to a fundamental observer at
rest in the comoving frame, the Universe must appear isotropic if it is homogeneous.
Although general relativity was constructed to be explicitly frame independent, the
comoving frame in which the CMB is isotropic is observationally convenient. The
fundamental observer is at position B in Figure 8.2.
The interpretation today is that not only does the Earth move around the Sun, and
the Solar System participates in the rotation of the Galaxy, but also the Galaxy moves
relative to our Local Galaxy Group, which in turn is falling towards a center behind the
Hydra–Centaurus supercluster in the constellation Virgo. From the observation that
our motion relative to the CMB is 369± 0 .9kms−^1 , these velocity vectors add up to a
peculiar motion of the Galaxy of about 550kms−^1 , and a peculiar motion of the Local
Group of about 630kms−^1 [5]. Thus the dipole anisotropy seen by the Earth-based
observer A in Figure 8.1 tells us that we and the Local Group are part of a larger,
gravitationally bound system.
Multipole Analysis. Temperature fluctuations around a mean temperature푇 0 in a
direction훼on the sky can be analyzed in terms of theautocorrelation function퐶(휃),
which measures the product of temperatures in two directionsm,nseparated by an
angle휃and averaged over all directions훼,
퐶(휃)=⟨
훿푇(m)
푇 0훿푇(n)
푇 0⟩
, m⋅n=cos휃. (8.18)