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Register a new userSingle-Field Consistency relation and $delta N$-Formalism
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Added by
Ali Akbar Abolhasani Dr.
Publication date
2018
fields
Physics
and research's language is
English
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According to the equivalence principal, the long wavelength perturbations must not have any dynamical effect on the short scale physics up to ${cal O} (k_L^2/k_s^2)$. Their effect can be always absorbed to a coordinate transformation locally. So any physical effect of such a perturbation appears only on scales larger than the scale of the perturbation. The bispectrum in the squeezed limit of the curvature perturbation in single-field slow-roll inflation is a good example, where the long wavelength effect is encoded in the spectral index through Maldacenas consistency relation. This implies that one should be able to derive the bispectrum in the squeezed limit without resorting to the in-in formalism in which one computes perturbative corrections field-theoretically. In this short paper, we show that the $delta N$ formalism as it is, or more generically the separate universe approach, when applied carefully can indeed lead to the correct result for the bispectrum in the squeezed limit. Hence despite the common belief that the $delta N$ formalism is incapable of recovering the consistency relation within itself, it is in fact self-contained and consistent.
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We develop a theory of nonlinear cosmological perturbations on superhorizon scales for a multi-component scalar field with a general kinetic term and a general form of the potential in the context of inflationary cosmology. We employ the ADM formalism and the spatial gradient expansion approach, characterised by O(epsilon^2), where epsilon=1/(HL) is a small parameter representing the ratio of the Hubble radius to the characteristic length scale L of perturbations. We provide a formalism to obtain the solution in the multi-field case. This formalism can be applied to the superhorizon evolution of a primordial non-Gaussianity beyond the so-called delta N formalism which is equivalent to O(epsilon^0) of the gradient expansion. In doing so, we also derive fully nonlinear gauge transformation rules valid through O(epsilon^2). These fully nonlinear gauge transformation rules can be used to derive the solution in a desired gauge from the one in a gauge where computations are much simpler. As a demonstration, we consider an analytically solvable model and construct the solution explicitly.
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