Abstract
We consider the superpotential formalism to describe the evolution of D scalar fields during inflation, generalizing it to include the case with non-canonical kinetic terms. We provide a characterization of the attractor behaviour of the background evolution in terms of first and second slow-roll parameters (which need not be small). We find that the superpotential is useful in justifying the separate universe approximation from the gradient expansion, and also in computing the spectra of primordial perturbations around attractor solutions in the δN formalism. As an application, we consider a class of models where the background trajectories for the inflaton fields are derived from a product separable superpotential. In the perspective of the holographic inflation scenario, such models are dual to a deformed CFT boundary theory, with D mutually uncorrelated deformation operators. We compute the bulk power spectra of primordial adiabatic and entropy cosmological perturbations, and show that the results agree with the ones obtained by using conformal perturbation theory in the dual picture.
Highlights
In ref. [25], considering the renormalisation group (RG) flow with two fixed points (a fixed point (FP) is a point where the beta function vanishes), which corresponds to the time evolution in cosmology from one de Sitter to another de Sitter, it was shown that with the choice of eq (5.6), the power spectrum of the curvature perturbation ζ in single field models is conserved at large scales so that the holographic computation gives a result consistent with the standard cosmological perturbation theory
We reviewed the superpotential formalism for multi-field inflation, and extended it to include the case of non-minimal kinetic terms
The logarithm of the superpotential plays an interesting role in the dual description of inflation, as the c-function for the RG flow in the boundary theory, whose gradient is related to the beta functions
Summary
Consider a scalar field theory in a (d + 1)-dimensional spacetime, with an action of the form. If the gradients of all fields are aligned in the same time-like direction, the energy momentum tensor (2.2) has the form of a perfect fluid, with pressure P and energy density given by ρ = 2PIJ XIJ − P. The perfect fluid form will be valid for our background solution, where all fields depend only on time. When the field space metric is non-trivial, one may wish to write the equation of motion (2.5) in a covariant form [52,53,54] which is manifestly independent of the choice of field space coordinates φI. We note that the H-J formulation, given by equations (2.12) and (2.13), involves only first partial derivatives of scalar functions in field space, and so it is automatically covariant
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