**4.6. The primordial power spectrum**

Summarizing the results of the last section, inflation predicts not only a flat, smooth universe, but also provides a natural mechanism for the production of primordial density and gravitational wave fluctuations. The scalar, or density fluctuation amplitude when a mode crosses the horizon is given by

(102) |

and the gravitational wave amplitude is given by

(103) |

for each of the two polarization modes for the gravitational wave. These
are the amplitudes for
a single mode when its wavelength (which is changing with time due to
expansion) is equal to the horizon size. In the case of slow roll,
with
small and *H* slowly varying, modes of different wavelengths will
have approximately the same amplitudes, with slow variation as a
function of scale. If we define the
power spectrum as the variance per logarithmic interval,

(104) |

inflation generically predicts a power-law form for
*P*_{S}(*k*),

(105) |

so that the *scale invariant spectrum*, one with equal amplitudes
at horizon crossing,
is given by *n* = 1. The current observational best fit for the
spectral index *n* is
[32]

(106) |

The observations are in agreement with inflation's prediction of a nearly (but not exactly) scale-invariant power spectrum, corresponding to a slowly rolling inflaton field and a slowly varying Hubble parameter during inflation. One can also consider power spectra which deviate from a power law,

(107) |

but inflation predicts the variation in the spectral index
*dn* / *d* log*k* to be small,
and we will not consider it further here. Similarly, the tensor
fluctuation spectrum in inflationary models is a power-law,

(108) |

(Note the unfortunate convention that the scalar spectrum is defined as
a power law with
index *n* - 1 while the tensor spectrum is defined as a power law
with index *n*_{T},
so that the scale-invariant limit for tensors is *n*_{T} = 0!)

It would appear, then, that we have four independent observable
quantities to work with:
the amplitudes of the scalar and tensor power spectra at some fiducial
scale *k*_{*}:

(109) |

and the spectral indices *n*,*n*_{T} of the power
spectra. (Generally, the scale *k*_{*} is
taken to be about the scale of the horizon size today, corresponding to
the CMB quadrupole.) In fact, at least within the
context of inflation driven by a single scalar field, not all of these
parameters are independent. This is because the tensor spectral index is
just given by the equation of
state parameter ,

(110) |

However, from Eqs. (102) and (103), and from the equation of motion in the slow-roll approximation (86), we have a simple expression for the ratio between the amplitudes of tensor and scalar fluctuations:

(111) |

so that the tensor/scalar ratio and the tensor spectral index are not
independent parameters,
but are both determined by the equation of state during inflation, a
relation known as the
*consistency condition* for slow-roll inflation.
^{(10)}
It is conventional to define the tensor scalar ratio as the ratio
*r* of the contributions of
the modes to the CMB quadrupole, which adds roughly a factor of 10
[33]:

(112) |

Similarly, the scalar spectral index can be expressed in terms of a second slow roll parameter ,

(113) |

where depends on the second derivative of the potential,

(114) |

So any simple inflation model gives us three independent parameters to
describe the primordial
power spectrum: the amplitude of scalar fluctuations
*A*_{S}, the tensor/scalar ratio *r*,
and the scalar spectral index *n*. The important point is that
these are *observable*
parameters, and will allow us to make contact between the physics of
very high energies and
the world of observational cosmology, in particular the cosmic microwave
background. In the
next section, we will see in detail how to accomplish this for a simple
model.

^{10} In the case of multi-field
inflation, this condition relaxes to an inequality,
*P*_{T} / *P*_{S}
- 2*n*_{T}
[34].
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