**7.2. Inflationary predictions**

The most commonly-discussed mechanism for generating the inhomogeneities
that act as the source for
*T* / *T* is
inflation. Of course, CMB anisotropies
were calculated in largely the modern way well
before inflation was ever considered, by
Peebles & Yu (1970).
The standard approach involves super-horizon fluctuations, which must
be generated by some acausal process. Inflation achieves this -
but we cannot claim that detection of super-horizon modes amounts
to a proof of inflation. Rather, we need some more
characteristic signature of the specific process used
by inflation: amplified quantum fluctuations (see e.g. chapter 11 of
Peacock 1999 or
Liddle & Lyth 2000
for details).

In the simplest models, inflation is driven by a scalar field
, with a potential
*V*(). As
well as the characteristic energy density
of inflation, *V*, this can be characterized by two
parameters, &
, which are
dimensionless
versions of the first and second derivatives of *V* with respect to
.
In these terms, the inflationary predictions for the perturbation
index is

(142) |

Since inflation continues while
&
are small,
some tilt is expected (| *n* - 1| ~ 0.01 to 0.05 in simple models).

The critical ingredient for testing inflation by making further predictions is the possibility that, in addition to scalar modes, the CMB could also be affected by gravitational waves (following the original insight of Starobinsky 1985). The relative amplitude of tensor and scalar contributions depended on the inflationary parameter alone:

(143) |

The second relation to the tilt is less general, as it assumes
a polynomial-like potential, so that
is related to
.
For example, *V* =
^{4}
implies *n*_{S}
0.95 and
*C*_{}^{T} /
*C*_{}^{S}
0.3.
To be safe, we need one further observation, and this
is potentially provided by the spectrum of
*C*_{}^{T}.
Suppose we write separate power-law index definitions for the
scalar and tensor anisotropies:

(144) |

For the scalar spectrum, we had
*n*_{S} = *n* = 1 -
6 +
2;
for the tensors, *n*_{T} = 1 -
2 [although
different definitions of *n*_{T} exist; the convention
here is that *n* = 1 always corresponds to a constant
^{2}()]. Thus, a knowledge of
*n*_{S}, *n*_{T} and the
scalar-to-tensor ratio would overdetermine the model and allow
a genuine test of inflation.