102 An introduction to the physics of cosmology
From the discussion of the Sachs–Wolfe effect, we know that, on large scales,
the scalar index is the same as index in the matter power spectrum:nS=n=
1 − 6 + 2 η. By the same method, it is easily shown thatnT= 1 − 2 (although
different definitions ofnTare in use in the literature; the convention here is that
n=1 always corresponds to a constantT^2 (")). Finally, then, we can write the
inflationary consistency equation:
CT"
C"S= 6. 2 ( 1 −nT).The slope of the scalar perturbation spectrum is the only quantity that containsη,
and sonSis not involved in a consistency equation, since there is no independent
measure ofηwith which to compare it.
From the point of view of an inflationary purist, the scalar spectrum is
therefore an annoying distraction from the important business of measuring the
tensor contribution to the CMB anisotropies. A certain degree of degeneracy
exists here (see Bondet al1994), since the tensor contribution has no acoustic
peak;C"Tis roughly constant up to the horizon scale and then falls. A spectrum
with a large tensor contribution therefore closely resembles a scalar-only spectrum
with smallerb(and hence a relatively lower peak). One way in which this
degeneracy may be lifted is through polarization of the CMB fluctuations. A non-
zero polarization is inevitable because the electrons at last scattering experience
an anisotropic radiation field. Thomson scattering from an anisotropic source
will yield polarization, and the practical size of the fractional polarizationPis
of the order of the quadrupole radiation anisotropy at last scattering:P&1%.
Furthermore, the polarization signature of tensor perturbations differs from that
of scalar perturbations (e.g. Seljak 1997, Hu and White 1997); the different
contributions to the total unpolarizedC"can in principle be disentangled, allowing
the inflationary test to be carried out.
How do these theoretical expectations match with the recent data, shown in
figure 2.22? In many ways, the match to prediction is startlingly good: there
is a very clear acoustic peak at"220, which has very much the height and
width expected for the principal peak in adiabatic models. As we have seen, the
location of this peak is sensitive to, since it measures directly the angular size
of the horizon at last scattering, which scales as"∝−^1 /^2 for open models.
The cut-off at"1000 caused by last-scattering smearing also moves to higher
"for low;ifwere small enough, the smearing cut-off would be carried to
large", where it would be inconsistent with the upper limits to anisotropies on
10-arcminute scales. This tendency for open models to violate the upper limits
to arcminute-scale anisotropies is in fact a long-standing problem, which allowed
Bond and Efstathiou (1984) to deduce the following limit on CDM universes:
& 0. 3 h−^4 /^3.The known lack of a CMB peak at high"was thus already a very strong argument
for a flat universe (with the caveats expressed in the earlier section on geometrical