Cosmic background fluctuations 101V(φ). As well as the characteristic energy density of inflation,V, this can be
characterized by two dimensionless parameters
≡
m^2 P
16 π(V′/V)^2
η≡m^2 P
8 π(V′′/V),
wheremPis the Planck mass,V′ = dV/dφ, and all quantities are evaluated
towards the end of inflation, when the present large-scale structure modes were
comparable in size to the inflationary horizon. Prior to transfer-function effects,
the primordial fluctuation spectrum is specified by a horizon-scale amplitude
(extrapolated to the present)δHand a slopen:
^2 (k)=δ^2 H(
ck
H 0) 3 +n
.The inflationary predictions for these numbers are
δH∼V^1 /^2
m^2 P^1 /^2
n= 1 − 6 + 2 η,which leaves us in the unsatisfactory position of having two observables and three
parameters.
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). We
therefore distinguish explicitly between scalar and tensor contributions to the
CMB fluctuations by using appropriate subscripts. The former category are those
described by the Sachs–Wolfe effect, and are gravitational potential fluctuations
that relate directly to mass fluctuations. The relative amplitude of tensor and
scalar contributions depended on the inflationary parameteralone:
C"T
CS" 12. 4 6 ( 1 −n).The second relation to thetilt(which is defined to be 1−n) is less general,
as it assumes a polynomial-like potential, so thatηis related to.Ifwemake
this assumption, inflation can be tested by measuring the tilt and the tensor
contribution. For simple models, this test should be feasible:V =λφ^4 implies
n 0 .95 andCT"/CS" 0 .3. To be safe, we need one further observation, and
this is potentially provided by the spectrum ofC"T. Suppose we write separate
power-law index definitions for the scalar and tensor anisotropies:
CS"∝"nS−^3 , CT"∝"nT−^3.