Astronomy and Astrophysics – Astrophysics
Scientific paper
1994-02-08
Phys.Rev.D50:758,1994; Erratum-ibid.D54:2980,1996
Astronomy and Astrophysics
Astrophysics
22 pages, LaTeX plus two uuencoded postscript figures, revised version incorporates a new section on the philosophical and pra
Scientific paper
10.1103/PhysRevD.50.758 10.1103/
To first order in the deviation from scale invariance the inflationary potential and its first two derivatives can be expressed in terms of the spectral indices of the scalar and tensor perturbations, $n$ and $n_T$, and their contributions to the variance of the quadrupole CBR temperature anisotropy, $S$ and $T$. In addition, there is a ``consistency relation'' between these quantities: $n_T= -{1\over 7}{T\over S}$. We discuss the overall strategy of perturbative reconstruction and derive the second-order expressions for the inflationary potential and its first two derivatives and the first-order expression for its third derivative, all in terms of $n$, $n_T$, $S$, $T$, and $dn/d\ln k$. We also obtain the second-order consistency relation, $n_T =-{1\over 7}{T\over S}[1+0.11{T\over S} + 0.15 (n-1)]$. As an example we consider the exponential potential, the only known case where exact analytic solutions for the perturbation spectra exist. We reconstruct the potential via Taylor expansion (with coefficients calculated at both first and second order), and introduce the Pad\'{e} approximant as a greatly improved alternative.
Liddle Andrew R.
Turner Michael S.
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