Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

Inflation with a blue eigenvalue spectrum 191
range Since this corresponds to a simple
inversion of the integral (4) and subsequent insertion into (5) leads to
the already known potential
of power–law inflation, where is an integration constant.
Similarly, for the extreme blue spectrum (12) the graceful exit function
reads yielding via (4) and (5) the
reconstructed potential
where is a constant. According to its shape,
cf. Ref. [21], it interpolates between the power–law inflation in the
limit and intermediate inflation. Its form is similar to the exact
solution of Easther [23] except that the latter yields the too large value
for the spectral index.
For some of the new blue spectrum (17) we have reconstructed the
potential numerically in Ref. [2]. Their form resembles those of
Refs. [23, 16].
The range of for which inflation occurs depends also on the (so-far
missing) information on the spectral index of tensor perturbations, cf.
Eq. (8). Since all our new solutions are ‘deformations’ of the constant
solution we expect the first order consistency relation of powerlaw
inflation to hold to some approximation, i.e.
[24]. In second order, it is found [2] that for our solutions is always
negative.
Of particular importance [24] is also the squared ratio of the two further
observables, the amplitude of gravitational waves, versus its
counter part the amplitude of primordial scalar density perturbations:
where the approximation is valid in first order. The numerical results
are displayed in Fig. 5 and 6 of [2]. From first order approximations, we
expect the range of this ratio within ~ [0,0.2). For several well-known
inflationary models (power-law and polynomial chaotic, e.g.) the predicted
ratio can be found in Table 2 of [24]. Recently, assisted inflation
was able to produce a scalar spectral index closer to scale-invariance
(but for ) because of the large number of non-inflationary real
scalar fields present in this theory [7].