Non-Gaussianity of superhorizon curvature perturbations beyond δ N formalism

Abstract

We develop a theory of nonlinear cosmological perturbations on superhorizon scales for a single scalar field with a general kinetic term and a general form of the potential. We employ the ADM formalism and the spatial gradient expansion approach, characterised by O(ǫ m ), where ǫ = 1/(HL) is a small parameter representing the ratio of the Hubble radius to the characteristic length scale L of perturbations. We obtain the general solution for a full nonlinear version of the curvature perturbation valid up through second-order in ǫ (m = 2). We find the solution satisfies a nonlinear second-order differential equation as an extension of the equation for the linear curvature perturbation on the comoving hypersurface. Then we formulate a general method to match a perturbative solution accurate to n-th-order in perturbation inside the horizon to our nonlinear solution accurate to second-order (m = 2) in the gradient expansion on scales slightly greater than the Hubble radius. The formalism developed in this paper allows us to calculate the superhorizon evolution of a primordial non-Gaussianity beyond the so-called δN formalism or separate universe approach which is equivalent to leading order (m = 0) in the gradient expansion. In particular, it can deal with the case when there is a temporary violation of slow-roll conditions. As an application of our formalism, we consider Starobinsky's model, which is a single field model having a temporary non-slow-roll stage due to a sharp change in the potential slope. We find that a large non-Gaussianity can be generated even on superhorizon scales due to this temporary suspension of slow-roll inflation.PACS numbers: 98.80.-k, 98.90.Cq

Wiggly whipped inflation

“…Primordial features and non-Gaussianities generated by this inflaton potential have been discussed widely in the literature [38][39][40][41][42][43]. Since BICEP2 data indicated a large tensor-to-scalar ratio, we revisit similar transition in the context of Whipped Inflation.…”

Application of the beyondδNformalism: Varying sound speed

“…The gradient expansion technique has been applied up to second order in the expansion to a universe dominated by a single [13][14][15][16] and multi-scalar field [18], yielding the formalism "beyond δN ". The formulae have been also extended to be capable of a universe filled with a most generic non-canonical scalar field [19], which can give the so-called G-inflation.…”

“…The gradient expansion approach [7][8][9][10][11][12][13][14][15][16][17][18][19] to discuss the evolution of nonlinear curvature perturbation on superhorizon scales is a powerful tool on calculation as well as * Electronic address: takamizu˙at˙yukawa.kyoto-u.ac.jp, yt313˙at˙cam.ac.uk the second-order perturbation theory [20,21]. The lowest order in the expansion is the so-called δN -formalism [9,12].…”

“…Therefore we focus on the nonlinear theory valid to the next-leading order in the expansion. It called the beyond δN -formalism and it is able to give us not only the local, but also the equilateral shape of bispectrum in contrast to δN -formalism, even though the expansion technique is taken on superhorizon scales [16,18]. Our nonlinear theory of the next-leading order in the expansion includes such subhorizon effect corresponding to the equilateral shape by matching a superhorizon curvature perturbation and subhorizon one suitably.…”

A general proof of the equivalence between the δN and covariant formalisms

“…The δN formalism corresponds to the leading order approximation of the spatial gradient expansion approach [7,[18][19][20][21][22][23][24][25][26][27]. In the gradient expansion approach, the field equations are expanded in powers of spatial gradients and hence it is applicable only to perturbations on very large spatial scales.…”