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190 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
constraint
Together with (9) this implies that the spectral index of our new class
of solutions can only adopt the following discrete eigenvalues
For solutions with large we find asymptotically
which reveals that the scale invariant Harrison–Zel‘dovich solution with
index is the limiting point of our new class of solutions. Moreover,
our new discrete spectrum approaches it from the blue side, which
previously was considered rather difficult to achieve.
In order to correlate this with observational restrictions, let us display
the highest eigenvalues for For we have
and, because of this implies Then
Eq. (9) delivers
for the value of the scalar spectral index. The integration constant
remains arbitrary. The fourth order establishes the constant as an
algebraic function of and similar for higher orders.
For the fourth order constraint is
while now plays the role of a free constant. Then, zeroth order
vanishes if
which is very close to the experimental value from COBE.
Furthermore, it is revealed that
and so on (where are some algebraic relations). The higher order
coefficients of the series can be derived recursively via MATHEMATICA,
similarly as in the case of the Bartnik–McKinnon solution, cf. Ref. [22].
4. COMPLETE EIGENVALUE SPECTRUM
In cosmological applications, the energy density in our second order
reconstruction has only a limited range. For the deceleration parameter
to become negative, as necessary for inflation,
our solutions are constrained by In the case of the constant
solution merely the spectral index is restricted to the continuous