The curvature perturbation at second order

Abstract

We give an explicit relation, up to second-order terms, between scalar-field fluctuations defined on spatially-flat slices and the curvature perturbation on uniform-density slices. This expression is a necessary ingredient for calculating observable quantities at second-order and beyond in multiple-field inflation. We show that traditional cosmological perturbation theory and the 'separate universe' approach yield equivalent expressions for superhorizon wavenumbers, and in particular that all nonlocal terms can be eliminated from the perturbation-theory expressions. IntroductionAccording to our current ideas, structure in the universe was seeded by quantum fluctuations which were amplified during an inflationary epoch. During inflation these fluctuations dominate the variation in energy density from place to place and therefore generate a gravitational response which can be probed by cosmological observations. Inflationary amplification is believed to occur for any sufficiently light degree of freedom, in the sense that its mass m was substantially less than the Hubble rate H while scales of interest were being carried beyond the horizon. Models motivated by modern concepts in high-energy physics often invoke many light fields, and therefore can be tested only if we have an understanding of their effects. The literature surrounding calculations of the inflationary density perturbation is now very mature-often with agreement on subtle effects to second-or even third-order in perturbation theory-which allows these effects to be predicted in some detail. But despite this maturity it is remarkable that no completely explicit formula has been given for the uniform-density gauge curvature perturbation in an inflationary model with an arbitrary number of fields. 1 A formula of this type would give the next-order term in the classic result ζ = −φ α δφ α /2M 2 P Hǫ which has long been known at first order. It is a key element in computing non-Gaussian signatures in the statistics of the inflationary density perturbation. Here and below, ǫ ≡ −Ḣ/H 2 is the usual slow-roll parameter and δφ α labels the species of light fields.In this paper we supply the missing formula, valid for an arbitrary number of canonical fields and without using the slow-roll approximation. We perform the calculation using two independent methods: traditional 'cosmological perturbation theory', which is an expansion in the amplitude of small fluctuations around a Robertson-Walker background, and the 'separate universe approach', which is an expansion in the amplitude of gradients of the perturbations. In practice (although not required in principle), separate-universe calculations often invoke a second expansion in the amplitude of the fluctuations, after which the two methods should agree for any Fourier mode much larger than the cosmological horizon. For a mode of wavenumber k this requires k/aH ≪ 1, making spatial gradients negligible. We verify that the two approaches give equivalent answers and clarify some issues regarding nonlocal terms which ...

Numerical evaluation of inflationary 3-point functions on curved field space—with the transport method & CppTransport

“…We can also use the above expression to find the density perturbation in the uniformdensity gauge, δρ(u), by employing the δN formula [58] to identify ζ with δρ(u) and substitutė ρ →Ṅ = H withρ →N =Ḣ to give δρ(u) = δρ +ρξ 0 +δρξ 0 +ρ 2 ξ 0ξ0 +ρ 2 ξ 0 2 , (A.90) where spatial gradients have been dropped. Equations (A.87)-(A.90) were first found in [41] and still apply in the non-trivial field space used in our calculations. Eq.…”

“…mTransport uses the 'local' form of ζ defined in Ref. [41], whereas CppTransport and PyTransport uses the 'simple' form (which agrees with Eqs. (2.28a)-(2.28b)).…”

“…Using the methods of Ref. [41] we find that the density fluctuation on spatially flat slices can be written in terms of the covariant perturbations Q I ,…”

“…As explained in Ref. [41], the curvature perturbation ζ does not have a unique definition on subhorizon scales, and the precise value we assign depends which k-dependent terms are kept. mTransport uses the 'local' form of ζ defined in Ref.…”

“…We would like to switch from the spatially-flat gauge used in our calculations so far to the uniform density gauge mainly because ζ is a quantity that is conserved to all orders in perturbation theory [58,59] and can then be used to calculate the power spectrum and bispectrum for an inflation model. As in [41], we use an exponential mapping of the Lie derivative that is used to change gauges, The Lie derivative is performed along a vector, ξ, which is given by…”