Beyond δN formalism
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Beyond δN formalism Atsushi Naruko , ∗ Yu-ichi Takamizu , † and Misao Sasaki ‡ APC (CNRS-Universit´e Paris 7), 10 rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France Yukawa Institute for Theoretical Physics Kyoto University, Kyoto 606-8502, Japan (Dated: September 14, 2018)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 contextof inflationary cosmology. We employ the ADM formalism and the spatial gradient expansionapproach, characterised by O ( ǫ ), where ǫ = 1 / ( HL ) is a small parameter representing the ratioof the Hubble radius to the characteristic length scale L of perturbations. We provide a formalismto obtain the solution in the multi-field case. This formalism can be applied to the superhorizonevolution of a primordial non-Gaussianity beyond the so-called δN formalism which is equivalent to O ( ǫ ) of the gradient expansion. In doing so, we also derive fully nonlinear gauge transformationrules valid through O ( ǫ ). These fully nonlinear gauge transformation rules can be used to derivethe solution in a desired gauge from the one in a gauge where computations are much simpler. Asa demonstration, we consider an analytically solvable model and construct the solution explicitly. PACS numbers: 98.80.-k, 98.90.Cq
I. INTRODUCTION
Recent observations of the cosmic microwave background anisotropy show a very good agreement of the observationaldata with the predictions of conventional, single-field slow-roll models of inflation, that is, adiabatic Gaussian randomprimordial fluctuations with an almost scale-invariant spectrum [1]. Nevertheless, possible non-Gaussianities frominflation has been a focus of much attention in recent years, mainly driven by recent advances in cosmologicalobservations. In particular, the PLANCK satellite [2] is expected to bring us preciser data and it is hoped thata small but finite primordial non-Gaussianity may actually be detected.To study possible origins of non-Gaussianity, one must go beyond the linear perturbation theory. An observationallydetectable level of non-Gaussianity cannot be produced in the conventional, single-field slow-roll models of inflation,since the predicted magnitude is extremely small, suppressed by the slow-roll parameters. Then a variety of ways togenerate a large non-Gaussianity have been proposed. (See e.g. a focus section in CQG [3] and references thereinfor recent developments.) They may be roughly classified into two; multi-field models where non-Gaussianity canbe produced classically on superhorizon scales, and non-canonical kinetic term models where non-Gaussianity can beproduced quantum mechanically on subhorizon scales. In particular, in the former case, the δN formalism turned outto be a powerful tool for computing non-Gaussianities thanks to its full non-linear nature.On the superhorizon scales, one can employ the spatial gradient expansion approach [4–30]. It is characterisedby an expansion parameter, ǫ = 1 / ( HL ), representing the ratio of the Hubble radius to the characteristic lengthscale L of the perturbation. In the context of inflation, based on the leading order in gradient expansion, the δN formalism [6, 11, 12] or the separate universe approach [16] was developed. It is valid when local values of the inflatonfield at each local point (averaged over each horizon-size region) determine the evolution of the universe at each point.This leading order in the gradient expansion provides a general conclusion for the evolution on superhorizon scalesthat the adiabatic growing mode is conserved on the comoving hypersurface [17].In this paper, we consider the curvature perturbation on superhorizon scales up through next-to-leading order ingradient expansion, that is, to O ( ǫ ). To make our analysis as general as possible, we extend the δN formalism in thefollowing two aspects: One is to go beyond the single-field assumption, and the other is to go beyond the slow-rollcondition. While in the case of single-field inflation, the curvature perturbation remains constant as mentioned above,the superhorizon curvature perturbation can change in time in the case of multi-field inflation. Furthermore, even forsingle-field inflation, the time evolution can be non-negligible due to a temporal violation of the slow-roll condition.In order to study such a case, the δN formalism is not sufficient since the decaying mode cannot be neglected any ∗ Email : naruko˙at˙apc.univ-paris7.fr † Email : takamizu˙at˙yukawa.kyoto-u.ac.jp ‡ Email : misao˙at˙yukawa.kyoto-u.ac.jp longer, which usually appears at O ( ǫ ) of gradient expansion and is known to play a crucial role already at the levelof linear perturbation theory [31, 32], not to mention the case of nonlinear perturbation theory [20, 21, 28, 29].Multi-field inflation may be motivated in the context of supergravity since it suggests the existence of many flatdirections in the scalar field potential. In multi-field inflation, a non-slow-roll stage may appear when there is a changein the dominating component of the scalar field. For example, one can consider a double inflation model in which aheavier component dominates the first stage of inflation but damps out when the Hubble parameter becomes smallerthan the mass, while a lighter component is negligible at the first stage but dominates the second stage of inflationafter the heavier component has decayed out [24, 34, 35]. However, these previous analyses are essentially based onthe δN formalism and it is in general necessary to extend it to O ( ǫ ), that is, to the beyond δN formalism. We focuson the case of a multi-component scalar field. As for a single scalar field, it has been developed in [28].We mention that a multi-scalar case in the gradient expansion approach was studied previously [25, 26]. However,it turns out to be valid only for a restricted situation (discussed later). Here we develop a general framework for fullynonlinear perturbations and present a formalism for obtaining the solution to O ( ǫ ). Then as an example we considera specific model which allows an analytical treatment of the equations of motion.This paper is organised as follows. In Sec. II, we introduce a multi-component scalar field and derive basic equations.We compare several typical time-slicing conditions and mention the differences of them from the single-field case. InSec. III, we develop a theory of nonlinear cosmological perturbations on superhorizon scales. We formulate it onthe uniform e -folding slicing (which is defined later). In Sec. IV, as a demonstration of our formalism, we consideran analytically solvable model and give the solution explicitly. Sec. V is devoted to a summary and discussions.Some details are deferred to Appendices. In Appendix A, the coincidence between some of time slicing conditionsis discussed by using the Einstein equations. In Appendix B, we write down the basic equations on the uniformexpansion slicing, and study the behaviour of the curvature perturbation in this slicing. In Appendix C, we givegeneral nonlinear gauge transformation rules valid to next-to-leading order in gradient expansion. In Appendix D, weverify our formalism in a single-field model. Finally, in Appendix E, we discuss the structure of the Hamiltonian andmomentum constraint equations in the gradient expansion. II. BASICSA. The Einstein equations
We develop a theory of nonlinear cosmological perturbations on superhorizon scales. For this purpose we employthe ADM formalism and the gradient expansion approach. In the ADM decomposition, the metric is expressed as ds = g µν dx µ dx ν = − α dt + ˆ γ ij (cid:0) dx i + β i dt (cid:1)(cid:0) dx j + β j dt (cid:1) , (2.1)where α is the lapse function, β i is the shift vector and Latin indices run over 1,2 and 3. We introduce the extrinsiccurvature K ij defined by K ij = 12 α (cid:16) ∂ t ˆ γ ij − ˆ D i β j − ˆ D j β i (cid:17) , (2.2)where ˆ D is the covariant derivative with respect to the spatial metric ˆ γ ij . In addition to the standard ADM decom-position, the spatial metric and the extrinsic curvature are further decomposed so as to separate trace and trace-freeparts ˆ γ ij = a ( t ) e ψ γ ij ; det γ ij = 1 , (2.3) K ij = a ( t ) e ψ (cid:18) Kγ ij + A ij (cid:19) ; γ ij A ij = 0 , (2.4)where a ( t ) is the scale factor of a fiducial homogeneous Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) spacetimeand the determinant of γ ij is normalised to be unity and A ij is trace free. The explicit form of K is given by K ≡ ˆ γ ij K ij = 1 α h (cid:0) H + ∂ t ψ (cid:1) − ˆ D i β i i , (2.5) See, however, a special case of single-field inflation studied recently in [33] where the would-be decaying mode of the comoving curvatureperturbation happens to be rapidly growing outside the horizon and an extended version of the δN formalism remains to be validalthough the curvature perturbation is no longer conserved. where H is the Hubble parameter defined by H ( t ) ≡ da ( t ) dt . a ( t ).As for a matter field, let us focus on a minimally-coupled multi-component scalar field, S m = Z d x √− g P ( X IJ , φ K ) ; X IJ ≡ − g µν ∂ µ φ I ∂ ν φ J , (2.6)where I , J and K run over 1 , , · · · , M with φ K denoting the K -th component of the scalar field. Note that we donot assume a specific form of both the kinetic term and potential, which are arbitrary functions of X IJ and φ K . Thistype of Lagrangian can be applied to, for example, multi-field DBI inflationary models. For the calculation of theirnon-Gaussianities, see, e.g. [36, 37] and also [38, 39] for recent developments.The equation of motion for the scalar field is given by2 √− g ∂ µ (cid:16) √− gP ( IJ ) g µν ∂ ν φ J (cid:17) + P I = 0 , (2.7)where the subscript I in P I represents a derivative with respect to φ I and P ( IJ ) is defined as P ( IJ ) = 12 (cid:18) ∂P∂X IJ + ∂P∂X JI (cid:19) . (2.8)The energy-momentum tensor is T µν = 2 P ( IJ ) ∂ µ φ I ∂ ν φ J + P g µν . (2.9)Notice that this energy-momentum tensor cannot be written in the perfect fluid form any more, which is one of maindifferences from the single-field case.All the independent components of the energy-momentum tensor are conveniently expressed in terms of E and J i as E ≡ T µν n µ n ν , J i ≡ − T iµ n µ , (2.10)and T ij , where n µ is the unit vector normal to the time constant surfaces and is given by n µ dx µ = − αdt , n µ ∂ µ = 1 α ( ∂ t − β i ∂ i ) . (2.11)For convenience, we further decompose T ij in the same way as Eq. (2.4), T ij = a ( t ) e ψ (cid:18) Sγ ij + S ij (cid:19) ; S ≡ γ ij T ij . (2.12)Now we write down the Einstein equations. In the ADM decomposition, the Einstein equations are separated intofour constraints, the Hamiltonian constraint and three momentum constraints, and six dynamical equations for thespatial metric. The constraints are1 a e ψ h R − (cid:16) D ψ + 2 D i ψD i ψ (cid:17)i + 23 K − A ij A ij = 2 E , (2.13)23 ∂ i K − e − ψ D j (cid:16) e ψ A ji (cid:17) = J i , (2.14)where R ≡ R [ γ ] is the Ricci scalar of the normalised spatial metric γ ij , D i is the covariant derivative with respect to γ ij , D ≡ γ ij D i D j , γ ij is the inverse of γ ij , and the spatial indices are raised or lowered by γ ij and γ ij , respectively.As for the dynamical equations for the spatial metric, we rewrite Eq. (2.2) as ∂ ⊥ ψ = − Hα + 13 (cid:18) K + ∂ i β i α (cid:19) , (2.15) ∂ ⊥ γ ij = 2 A ij + 1 α (cid:16) γ ik ∂ j β k + γ jk ∂ i β k (cid:17) T F . (2.16)The equations for the extrinsic curvature ( K , A ij ) are given by ∂ ⊥ K = − K − A ij A ij + 1 a e ψ α (cid:16) D α + D i αD i ψ (cid:17) −
12 ( S + E ) , (2.17) ∂ ⊥ A ij = − KA ij + 2 A ik A kj + 1 α (cid:18) A ik ∂ j β k + A jk ∂ i β k − A ij ∂ k β k (cid:19) − a e ψ (cid:20) R ij + D i ψD j ψ − D i D j ψ − α (cid:16) D i D j α − D i αD j ψ − D j ψD i α (cid:17)(cid:21) T F + S ij , (2.18)where ∂ ⊥ ≡ n µ ∂ µ , and we have introduced the trace-free projection operator [ ... ] T F defined for a tensor Q ij as Q T Fij ≡ Q ij − γ ij γ kl Q kl . (2.19)Finally, the equations of motion for the scalar field (2.7) are ∂ ⊥ (cid:16) P ( IJ ) ∂ ⊥ φ J (cid:17) + KP ( IJ ) ∂ ⊥ φ J − αa e ψ ∂ i (cid:16) αae ψ P ( IJ ) γ ij ∂ j φ J (cid:17) − P I = 0 . (2.20) B. Gradient expansion and assumption
In the gradient expansion approach we suppose that the characteristic length scale L of a perturbation is longerthan the Hubble length scale 1 /H of the background, i.e. HL ≫
1. Therefore, ǫ ≡ / ( HL ) is regarded as a smallparameter and we can systematically expand equations in the order of ǫ , identifying a spatial derivative is of order ǫ , ∂ i Q = O ( ǫ ) Q . To clarify the order of gradient expansion, we introduce the superscript ( n ). For example, (2) α meansthe lapse function at second order in gradient expansion.As a background spacetime, we consider a FLRW universe. At O ( ǫ ) of the gradient expansion, there is apparentlyno spatial gradient and the universe is locally homogeneous and isotropic. This leads to the following condition onthe spatial metric: ∂ t γ ij = O ( ǫ ) . (2.21)Since we adopt this assumption, the spatial metric at leading order is given by an arbitrary spatial function of thespatial coordinates, (0) γ ij = f ij ( x k ) , (2.22)under the condition that the eigenvalues of f ij are all positive definite everywhere. From the definition of A ij ,Eq. (2.16), the above assumption implies A ij = O ( ǫ ) . (2.23)Throughout this paper, in order to simplify the equations, we set the shift vector to zero up to second order ingradient expansion, β i = O ( ǫ ) . (2.24)Let us call this choice of the spatial coordinates as the time-slice-orthogonal threading . Here we mention that theabove condition does not completely fix the spatial coordinates. As discussed later, one can actually make an arbitrarycoordinate transformation of the form, x i → ¯ x i = f i ( x j ). C. Leading order in gradient expansion
In this subsection, we study the leading order gradient expansion and make clear the correspondence between theleading order equations and background equations. This correspondence can be used to construct the solution atleading order in gradient expansion in terms of the background solution.At leading order in gradient expansion, the Einstein equations are13 K = 2 P ( IJ ) ∂ τ φ I ∂ τ φ J − P , (2.25) ∂ i K = − P ( IJ ) ∂ i φ I ∂ τ φ J , (2.26) ∂ τ K = − P ( IJ ) ∂ τ φ I ∂ τ φ J , (2.27)and the scalar field equation is ∂ τ (cid:16) P ( IJ ) ∂ τ φ J (cid:17) + KP ( IJ ) ∂ τ φ J − P I = 0 , (2.28)where we have introduced the proper time τ by τ ( t, x i ) ≡ Z x i = const. α ( t ′ , x i ) dt ′ . (2.29)In terms of τ , the expression of K in Eq. (2.5) is simplified under the time-slice-orthogonal threading condition, K = 1 α ∂ t (cid:0) a e ψ (cid:1) a e ψ = 3 ∂ τ (cid:0) ae ψ (cid:1) ae ψ . (2.30)Under the identifications, ae ψ ⇔ a , and τ ⇔ t , (2.31)one also has the correspondence, K ⇔ H . This means that the basic equations at leading order, Eqs. (2.27)and (2.28), take exactly the same form as those in the background modulo above identifications. Namely, given abackground solution, φ I ( t ) (cid:12)(cid:12)(cid:12) background = φ I BG h t, φ I ( t ) i , (2.32)one can construct the solution at leading order in gradient expansion as φ I ( t, x i ) (cid:12)(cid:12)(cid:12) gradient = φ I BG h τ, φ I ( τ ) i . (2.33)All the information of inhomogeneities is contained in the initial condition as well as in the proper time τ throughEq. (2.29). Thus it is sufficient to solve the background equations to obtain the solution at leading order in gradientexpansion.In passing, we note that the e -folding number is often used as the time coordinate to describe the backgroundevolution. For convenience, we define it as the number of e -folds counted backward in time from a fixed final time.That is, N ( t ) = Z t t H ( t ′ ) dt ′ . (2.34)Accordingly, the scale factor is expressed as a ( N ) = a e − ( N − N ) . (2.35)By replacing t with τ and H with K/ e -fold number to the one defined locally in space as N ( t, x i ) ≡ Z t t dt ′ α ( t ′ , x i ) K ( t ′ , x i ) (cid:12)(cid:12)(cid:12) x i = const. . (2.36)Again one can check the validity of the above correspondence by rewriting Eqs. (2.25), (2.27) and (2.28) in terms of N as the time coordinate. D. Various slicings and their coincidences
One needs to specify the gauge condition to study perturbations in perturbation theory or in gradient expansion.Since spatial coordinates have been already fixed by the time-slice-orthogonal threading, one has to determine thetime-slicing condition. Here, let us list various slicings and their definitions,Comoving ; J i = 0 , (2.37)Uniform expansion ; K ( t, x i ) = 3 H ( t ) , (2.38)Uniform energy ; E ( t, x i ) = E ( t ) , (2.39)Synchronous ; α ( t, x i ) = 1 , (2.40)Uniform e -folding number ; N ( t, x i ) = N ( t ) . (2.41)Hereinafter, we call the uniform expansion, uniform energy and uniform e -folding number slicings as the uniform K ,uniform E and uniform N slicings, respectively.We mention that there is a remaining gauge degree of freedom in the synchronous or uniform N slicing, while thetime slices are completely fixed in the uniform K and uniform E slicings. As for the uniform N slicing, the gaugecondition demands ∂ t ψ to vanish from Eqs. (2.30) and (2.36). This means one can freely choose the initial value of ψ (and hence its spatial configuration at any later time because ψ is conserved). This corresponds to the freedom inthe choice of the initial time-slice as we see later. Utilising this freedom, we can make a scalar quantity, one of scalarfields φ I or K for example, homogeneous on the initial slice. E. Towards the next-to-leading order in gradient expansion
As we have seen in subsection II C, the leading order solutions are given by functions of τ in terms of the backgroundsolutions. At next-to-leading order in gradient expansion, terms with spatial derivatives of the leading order solutionappear in the evolution equations. To evaluate those terms, one needs to calculate the spatial derivative of the lapsefunction, for example in ∂ i φ , ∂ i φ BG ( τ ) = ∂ τ φ BG ( τ ) ∂ i τ = ∂ τ φ BG ( τ ) Z ∂ i α dt . (2.42)However the leading order (0) α is in general given explicitly only after solving the following equation for α : α = f h t, φ ( τ ) i = f (cid:20) t, φ (cid:18)Z α dt (cid:19)(cid:21) . (2.43)As a demonstration, the analysis on the uniform K slices is performed in Appendix B, and it is clearly shown that itis almost impossible to solve this equation, at least in an analytical way.This problem did not appear in the single-field case. It is because one can show that various different slicings becomeidentical at leading order in gradient expansion. In particular, all the slicings listed in subsection II D coincide witheach other as shown in Appendix A:comoving = uniform K = uniform E −→ L99 synchronous = uniform N , (2.44)where → means the left ones imply the right ones and L99 means it holds when one chooses the initial slice to be thecomoving, uniform K or uniform E slicing by using the remaining gauge degree of freedom. Thus the lapse functionis homogeneous in all the slicings in the above, and we may set α = 1 if desired.On the other hand, one has to face this problem in the case of multi-field inflation. We overcome this problemby choosing the synchronous slicing or uniform N slicing, which gives us a homogeneous time coordinate. On theseslicings, one can evaluate the spatial derivatives of the leading order solution which appear as source terms andconstruct a solution to next-to-leading order in gradient expansion by integrating those terms.There are two necessary steps before reaching the goal. Once we have a solution, it is necessary to construct aconserved quantity out of it that can be directly related to observable quantities. It is widely known that the comovingcurvature perturbation eventually become conserved in a single-field model in linear theory. In non-linear theory, thereexists a corresponding quantity, ψ on the comoving, uniform K or uniform E slicing, which is conserved at leadingorder in gradient expansion [17]. Even in the multi-field case, the system effectively reduces to a single-field systemafter the so-called non-adiabatic pressure has died out, that is, when the adiabatic limit is reached. Therefore it isnecessary to perform a nonlinear gauge transformation from the uniform N slicing to one of those three slicings. Thisis one of the steps. Since the comoving slicing is not well defined in general in the multi-field case [12], we choose theuniform K slicing as the target gauge.The other step to be taken is related to the definition of the curvature perturbation at next-to-leading order. Inlinear theory the curvature perturbation is named so because it determines the three-dimensional Ricci scalar, and ψ can be called so to full nonlinear order in the context of the leading order in gradient expansion. At next-to-leadingorder, however, ψ itself is no longer adequate to be called the curvature perturbation [28]. One needs to add thecontribution from part of γ ij , which we call χ , to obtain a properly defined curvature perturbation conserved through O ( ǫ ). Therefore, after transforming from the uniform N slicing to the uniform K slicing, one has to evaluate thecombination, R K ≡ ψ K + χ K /
3. This is the
Beyond δN formalism .Before concluding this section, we mention the difference between our work and that of Weinberg [25, 26]. There itwas assumed that the lapse function can be chosen to be equal to unity at leading order in gradient expansion, henceall the scalar fields are homogeneous. This severely constrains the class of scalar field models as well as the initialcondition because the curvature perturbation must be always conserved at leading order in gradient expansion. Herewe do not impose such assumptions and perform a completely general analysis. III. BEYOND δN FORMALISM
Let us first summarise the five steps in the
Beyond δN formalism .1. Write down the basic equations (the Einstein equations and scalar field equation) in the uniform N slicingwith the time-slice-orthogonal threading. For convenience let us call the choice of the coordinates in which oneadopts the uniform X slicing with the time-slice-orthogonal threading the X gauge. So the above choice is the N gauge. In this gauge the metric components at leading order are trivial since both ψ and γ ij are independentof time.2. First solve the leading order scalar field equation under an appropriate initial condition and then the next-to-leading order scalar field equation which involves spatial gradients of the leading order solution.3. Solve the next-to-leading order Einstein equations for the metric components and their derivatives.4. Determine the gauge transformation from the N gauge to the K gauge and apply the gauge transformationrules to obtain the solution in the K gauge.5. Evaluate the curvature perturbation R = ψ + χ/ K gauge, where χ is to be extracted from γ ij .In what follows, we describe these steps in detail but only formally. An example in which these steps can be computedanalytically will be discussed in Sec. IV.Step 1:First, we rewrite the uniform N slicing condition from Eqs. (2.30) and (2.36) as α ( t, x i ) K ( t, x i ) = 3 H ( t ) ⇔ ∂ t ψ ( t, x i ) = 0 . (3.1)Hence ψ is constant in time and is given by a function of the spatial coordinates alone, ψ ( t, x i ) = ψ ( x i ) ≡ C ψ ( x i ) . (3.2)In the N gauge, the Einstein equations are reduced to the following equations. The constraints are1 a e C ψ h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17)i + 23 K = 4 K P ( IJ ) ∂ N φ I ∂ N φ J − P , (3.3)23 ∂ i K − e − C ψ D j (cid:16) e C ψ A ji (cid:17) = 2 K P ( IJ ) ∂ N φ I ∂ i φ J . (3.4)The evolution equations for K , A ij and γ ij are ∂ N K = KP ( IJ ) ∂ N φ I ∂ N φ J − a e C ψ K h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17)i − a e C ψ (cid:20) D (cid:18) K (cid:19) + D i (cid:18) K (cid:19) D i C ψ (cid:21) + 3 a e C ψ K P ( IJ ) D i φ I D i φ J , (3.5) ∂ N A ij = 3 A ij + 3 a e C ψ K (cid:16) R ij + D i C ψ D j C ψ − D j D i C ψ (cid:17) T F − a e C ψ (cid:20) D i D j (cid:18) K (cid:19) − D i (cid:18) K (cid:19) D j C ψ − D j (cid:18) K (cid:19) D i C ψ + 1 K P ( IJ ) D i φ I D j φ J (cid:21) T F , (3.6) ∂ N γ ij = − K A ij . (3.7)The scalar field equation is K ∂ N (cid:18) K P ( IJ ) ∂ N φ J (cid:19) − K P ( IJ ) ∂ N φ J − Ka e C ψ ∂ i e C ψ K P ( IJ ) γ ij ∂ j φ J ! − P I = 0 . (3.8)We rewrite Eq. (3.8) by eliminating ∂ N K with Eq. (3.5) as K ∂ N (cid:16) P ( IJ ) ∂ N φ J (cid:17) + K (cid:16) P ( KL ) ∂ N φ K ∂ N φ L − (cid:17) P ( IJ ) ∂ N φ J − P I = Ka e C ψ ( e C ψ ∂ i e C ψ K P ( IJ ) γ ij ∂ j φ J ! + 13 (cid:20) D (cid:18) K (cid:19) + D i (cid:18) K (cid:19) D i C ψ (cid:21) P ( IJ ) ∂ N φ J ) + 16 a e C ψ h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) − P ( KL ) γ ij ∂ i φ K ∂ j φ L i P ( IJ ) ∂ N φ J . (3.9)Thus once K is expressed in terms of the scalar field and its derivatives, the above equation gives a closed scalarfield equation. An explicit derivation of the closed scalar field equation is possible only after we specify the explicitform of P ( X IJ , φ ) as a function of X IJ and φ . Here we describe generally but formally the procedure to obtain K as a function of the scalar field.First, we separate the term with time-derivatives in X IJ and denote it by K Y IJ where Y IJ ≡ ∂ N φ I ∂ N φ J / X IJ = K Y IJ − a e C ψ γ ij ∂ i φ I ∂ j φ J . (3.10)At leading order, we can neglect the spatial-derivative term in Eq. (3.10). Then, P and P ( IJ ) are given by P ( K Y IJ , φ I ) and P ( IJ ) ( K Y IJ , φ I ), respectively. To next-to-leading order, P and P ( IJ ) can be expanded as P ( X IJ , φ K ) = P ( K Y IJ , φ K ) − a e C ψ γ ij ∂ i (0) φ I ∂ j (0) φ J ∂P∂X IJ + O ( ǫ ) , (3.11) P ( IJ ) ( X KL , φ M ) = P ( IJ ) ( K Y KL , φ M ) − a e C ψ γ ij ∂ i (0) φ K ∂ j (0) φ L ∂P ( IJ ) ∂X KL + O ( ǫ ) . (3.12)Inserting these expressions into Eq. (3.3), one obtains an algebraic equation for K . Solving it gives an expression of K in terms of the scalar field. Then a closed equation for the scalar field is obtained by plugging it into Eq. (3.9).Step 2:Although we can keep our discussion completely general, below we focus on the case of a multi-component canonicalscalar field, P = 12 δ IJ X IJ − V ( φ , · · · , φ M ) and P ( IJ ) = 12 δ IJ . (3.13)This choice is taken purely for the sake of simplicity and clarity, because the expansion K can be explicitly expressedin terms of the scalar field in this case. In general, one cannot obtain an explicit expression of K in terms of thescalar field unless the form of P is explicitly specified. Nevertheless, the discussion below also applies to the generalcase perfectly.From Eq. (3.3) we find K (cid:18) − ∂ N φ I ∂ N φ I (cid:19) − (cid:26) V − a e C ψ h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) + D i φ I D i φ I i(cid:27) . (3.14)Inserting this into Eq. (3.9), one obtains the following closed equation: (cid:18) − ∂ N φ J ∂ N φ J (cid:19) − ∂ N φ I − ∂ N φ I + 3 V I V = Ka e C ψ V ( e C ψ ∂ i e C ψ K γ ij ∂ j φ I ! + (cid:20) D (cid:18) K (cid:19) + D i (cid:18) K (cid:19) D i C ψ (cid:21) ∂ N φ I ) − D i φ J D i φ J a e C ψ V ∂ N φ I + 12 a e C ψ V h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) + D i φ J D i φ J i "(cid:18) − ∂ N φ K ∂ N φ K (cid:19) − ∂ N φ I − ∂ N φ I . (3.15)Since each term in the right-hand side of the equation involves two spatial derivatives, we can neglect them at leadingorder. At next-to-leading order, they can be understood as source terms whose time evolution have been alreadydetermined from the leading order solution. As noted in the above, although one cannot obtain an equation like theabove explicitly for general P , one can always derive a closed equation for the scalar field once a specific form of P isgiven.Solving the closed equation for the scalar field, the solution is formally obtained as (0) φ N = (0) φ N h N ; (0) φ , (0) ∂ N φ , C ψ , (0) γ ij i , (2) φ N = (2) φ N h N ; (2) φ , (2) ∂ N φ , D ( (0) φ , (0) ∂ N φ , C ψ , (0) γ ij ) i , (3.16)where the subscript N indicates the N gauge. In the arguments on the right-hand side, the subscript 0 denotes theinitial value, and D ( · · · ) the spatial derivatives of the quantities inside the parentheses.Step 3:Once the solution for the scalar field is obtained, one can solve Eqs. (3.5), (3.6) and (3.7) to obtain the metricquantities. As for K , however, it is simpler and in fact better to use Eq. (3.14), which is essentially the Hamiltonianconstraint, since the integral constants appearing from integrating Eq. (3.5) are not freely specifiable but must satisfythe Hamiltonian constraint.Step 4:Given the solution in the N gauge, the next step is to find the gauge transformation from the N gauge to the K gauge. It can be determined as follows. As noted above, the expression for K N in terms of the scalar field is obtainedfrom Eq. (3.14). Since the leading order and next-to-leading order scalar field solutions are expressed as Eqs. (3.16),the same is true for K N , (0) K N = (0) K N h N ; (0) φ , (0) ∂ N φ , C ψ , (0) γ ij i , (2) K N = (2) K N h N ; (2) φ , (2) ∂ N φ , D ( (0) φ , (0) ∂ N φ , C ψ , (0) γ ij ) i . (3.17)Let the transformation from the uniform N slicing to the uniform K slicing be given by N → ˜ N = N + n ( N, x i ) orconversely ˜ N + ˜ n ( ˜ N , ˜ x i ) = N , where the uniform K slice is given by ˜ K =const.. This nonlinear gauge transformationis discussed in detail in Appendix C. The nonlinear gauge transformation generator ˜ n ( ˜ N , x i ) from the N gauge to K gauge is then determined by the condition that K is spatially homogeneous on the uniform K slice by definition: (0) K K ( ˜ N , ˜ x i ) = (0) K N ( ˜ N + (0) ˜ n, x i ) = 0 , (3.18) (2) K K ( ˜ N , ˜ x i ) = (2) K K h ˜ N + (2) ˜ n, D ( (0) K N , (0) ˜ n ) i = 0 . (3.19)Then the other quantities in the K gauge are obtained by applying the above gauge transformation. In particular,the determinant of the spatial metric ψ K in the K gauge is obtained from Eqs. (C28) and (C33), and the unimodularpart of the spatial metric γ ij K from Eqs. (C29) and (C34).0Step 5:We are to construct the nonlinear curvature perturbation, R K = ψ K + χ K /
3, where χ K is the scalar part of themetric γ ij K . The scalar part of γ ij is defined in the same way as the single-scalar case [27], χ ≡ − △ − n ∂ i e − ψ ∂ j h e ψ (cid:0) γ ij − δ ij (cid:1)io , (3.20)where △ − is the inverse Laplacian operator on the flat background. Extracting χ K from γ ij K , and combining ψ K and χ K , we finally obtain the nonlinear curvature perturbation R K = ψ K + χ K / K gauge. IV. SOLVABLE EXAMPLE
In this section, we demonstrate how to obtain the solution up to next-to-leading order in gradient expansion byapplying our formalism to a specific, analytically solvable model.
A. Model and equations
For simplicity, we consider a canonical scalar field with exponential potential [23], P = 12 X IJ − V ( φ I ) , V ( φ I ) = W exp "X J m J φ J , (4.1)where W is a constant. The leading order scalar field equation is given by (cid:18) − ∂ N (0) φ J ∂ N (0) φ J (cid:19) − ∂ N (0) φ I − ∂ N (0) φ I + 3 m I = 0 , (4.2)where we have omitted a summation symbol over the field component indices, J . Hereafter summation is impliedover repeated component indices.Further we assume the two slow-roll conditions on the leading order equation. The first one is that we can neglectthe “kinetic energy” in the energy density of the scalar field, ∂ N (0) φ I ∂ N (0) φ I ≪ . (4.3)The second one is that we can neglect the “acceleration”, ∂ N (0) φ I ≪ ∂ N (0) φ I . (4.4)It is important that we apply these assumptions only to (0) φ . We do not impose the slow-roll conditions on (2) φ . Sowe can rewrite Eq. (4.2) as ∂ N (0) φ I − m I = 0 . (4.5)At next-to-leading order, we have ∂ N (2) φ I − ∂ N (2) φ I + ( ∂ N (0) φ I − m I ) ∂ N (0) φ J ∂ N (2) φ J = S φI a e C ψ (0) V , (4.6)where S φI = 3 (0) Ke C ψ ∂ i e C ψ (0) K D i (0) φ I ! + (0) K (cid:20) D (cid:18) (0) K (cid:19) + D i (cid:18) (0) K (cid:19) D i C ψ (cid:21) ∂ N (0) φ I − h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) + 2 D i (0) φ J D i (0) φ J i ∂ N (0) φ I . (4.7)Inserting the leading order equation (4.5) into (4.6), it gives e N ∂ N (cid:16) e − N ∂ N (2) φ I (cid:17) = S φI a e C ψ (0) V . (4.8)This is the basic equation at next-to-leading order.1
B. Solution
We can easily solve the leading order and next-to-leading order scalar field equations. The solution of Eq. (4.5) isgiven by (0) φ I ( N ) = C φI + m I (cid:0) N − N (cid:1) , (4.9)and the solution of Eq. (4.8) is obtained as (2) φ I = 13 D φI h e N − N ) − i + Z NN dN ′ e N ′ Z N ′ N dN ′′ e − N ′′ S φI a e C ψ (0) V ! , (4.10)where N is an initial time and C φI and D φI represent the initial values of the scalar field and its time derivative. Notethat the solution (0) φ I satisfies the slow-roll conditions (4.3) and (4.4) if all the masses are small, M ≡ X I m I ≪ . (4.11)Here let us show the time-independence of S φI , which is given by Eq. (4.7). From Eq. (3.14) we have (0) K = 3 (0) V . (4.12)The leading order potential (0) V is given by (0) V = W exp "X I m I (0) φ I = C V exp h M (cid:0) N − N (cid:1)i , (4.13)where C V is the initial value of the potential, C V ( x i ) ≡ (0) V ( N ) = W exp "X I m I C φI . (4.14)Substituting the above solution into Eq. (4.7) gives S φI = 3 √ C V e C ψ ∂ i e C ψ √ C V D i C φI ! − (cid:20) D (cid:0) log C V (cid:1) − D i (cid:0) log C V (cid:1) D i (cid:0) log C V (cid:1) + D i (cid:0) log C V (cid:1) D i C ψ (cid:21) m I − h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) + 2 D i C φJ D i C φJ i m I . (4.15)The time-independence of S φI is now clear since it is expressed solely in terms of C ψ , C V , C φI and (0) γ ij which are alltime-independent functions. Therefore we can rewrite the second order solution (4.10) in a simpler manner as (2) φ I = 13 D φI h e N − N ) − i + S φI I φ ( N ) a e C ψ (0) V , (4.16)where I φ ( N ) a e C ψ (0) V = 1 e C ψ Z NN dN ′ a − Z N ′ N dN ′′ a (0) V ! . (4.17)Given the solution for the scalar field up to second order in gradient expansion, we can now obtain those for K , A ij and γ ij . At leading order, K is expressed in terms of the scalar field through Eq. (4.12) as (0) K = p (0) V = p C V exp (cid:20) M (cid:0) N − N (cid:1)(cid:21) . (4.18)Because of the assumption (2.21), (0) γ ij and (0) A ij are trivial, (0) γ ij = C γij , (0) A ij = 0 . (4.19)2At next-to-leading order, Eq. (3.14) becomes (2) K = p (0) V (cid:26) (2) V (0) V + 16 m I ∂ N (2) φ I − a e C ψ (0) V h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) + D i φ I D i φ I i(cid:27) , (4.20)and Eq. (3.6) reduces to e N ∂ N (cid:16) e − N A ij (cid:17) = 1 a e C ψ (0) K S
Aij , (4.21)where S Aij is independent of time. Specifically, S Aij = 3 (cid:16) R ij + D i C ψ D j C ψ − D i D j C ψ (cid:17) T F − (cid:20) − D i (cid:16) D j (cid:0) log C V (cid:1)(cid:17) + 12 D i (cid:0) log C V (cid:1) D j (cid:0) log C V (cid:1)(cid:21) T F − h D i (cid:0) log C V (cid:1) D j C ψ + D j (cid:0) log C V (cid:1) D i C ψ + D i C φI D j C φI i T F . (4.22)Substituting the solution of the scalar field into Eq. (4.20), we obtain (2) K as (2) K (0) K = 16 X I m I D φI h e N − N ) − i + 1 a e C ψ (0) V S K , (4.23)where S K = 12 m I S φI I φ + 16 m I S φI (0) Va Z NN dN ′ a (0) V − h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) + D i C φI D i C φI i . (4.24)As for A ij , we obtain it by integrating Eq. (4.21), A ij = C Aij e N − N ) + S Aij a e C ψ Z NN dN ′ a (0) K . (4.25)Finally, (2) γ ij is given by solving Eq. (3.7), (2) γ ij = − C Aij Z NN dN ′ e N ′ − N )(0) K − S Aij e ψ Z NN dN ′ a K Z N ′ N dN ′′ a (0) K ! . (4.26)Before concluding this subsection, let us fix the remaining gauge degree of freedom on the uniform N slicingmentioned in subsection II D. We can make (0) K spatially homogeneous on the initial slice by using this gauge degreeof freedom, C V ( x i ) = (0) V = const . , (4.27)where (0) V is a pure constant independent of both space and time. With this choice, we can substantially simplifythe above expressions of the solution because the spatial derivative of C V vanishes.To summarise, thus obtained solution is φ I = C φI + m I ( N − N ) + 13 D φI h e N − N ) − i + S φI I φ ( N ) a e C ψ (0) V , (4.28) K = p (0) V (cid:26) m I D φI h e N − N ) − i + S K ( N, x i ) a e C ψ (0) V (cid:27) , (4.29) γ ij = C γij − √ C Aij √ (0) V e N − N ) I γ ( N ) − S Aij J γ ( N ) a e C ψ (0) V , (4.30) A ij = C Aij e N − N ) + S Aij I A ( N ) √ a e C ψ √ (0) V , (4.31)3where (0) V = (0) V exp (cid:2) M ( N − N ) (cid:3) and C φI , D φI , C γij and C Aij are initial values of φ , ∂ N φ , γ ij and A ij , respectively.The source functions S φI and S Aij are time-independent whose explicit forms are S φI = 3 e C ψ ∂ i (cid:16) e C ψ D i C φI (cid:17) − h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) + 2 D i C φJ D i C φJ i m I , (4.32) S Aij = 3 (cid:16) R ij + D i C ψ D j C ψ − D i D j C ψ − D i C φI D j C φI (cid:17) T F , (4.33)and the functions I φ , S K , I A , I γ and J γ are given by I φ ( N ) = a V Z NN dN ′ a Z N ′ N dN ′′ a (0) V , (4.34) S K ( N, x i ) = 12 m I S φI I φ ( N ) + 16 m I S φI (0) Va Z NN dN ′ a (0) V − h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) + D i C φI D i C φI i , (4.35) I γ ( N ) = a p (0) V Z NN dN ′ a √ (0) V , (4.36) J γ ( N ) = a V Z NN dN ′ a √ (0) V Z N ′ N dN ′′ a √ (0) V ! , (4.37) I A ( N ) = √ (0) Va Z NN dN ′ a √ (0) V . (4.38)
C. Solution on the uniform K slice In this subsection, we derive the solution on the K gauge by applying a gauge transformation to the solution onthe N gauge. To do so, we first need to determine the generator of the gauge transformation between the two slices, N → ˜ N = N + n ( N, x i ) or conversely ˜ N + ˜ n ( ˜ N , ˜ x i ) = N . The nonlinear gauge transformation is discussed in detailin Appendix C.As is clear from Eq. (4.18) with the initial condition C V = (0) V =const., Eq. (4.27), the uniformity of K onthe uniform N slices is kept in this model at leading order if it is chosen so on the initial slice. Thus no gaugetransformation is necessary at leading order.At next-to-leading order, the gauge transformation of K is from Eq. (C35) given by (2) ˜ K = (2) K + (2) ˜ n∂ N (0) K . (4.39)Since (2) ˜ K vanishes on the uniform K slice, (2) ˜ n is determined from Eq. (4.20) as (2) ˜ n = − M (cid:26) m I D φI h e N − N ) − i + 1 a e C ψ (0) V S K (cid:27) . (4.40)The solution for ψ is obtained from Eq. (C33) as˜ ψ = ψ − (2) ˜ n + O ( ǫ )= C ψ + 2 M (cid:26) m I D φI h e N − N ) − i + 1 a e C ψ (0) V S K (cid:27) , (4.41)where C ψ is given in (3.2) and that for γ ij is same as γ ij on the N gauge from Eq. (C34),˜ γ ij = γ ij + O ( ǫ ) . (4.42) D. The curvature perturbation
At next-to-leading order in gradient expansion, we have to extract the scalar component χ from γ ij to obtain thecurvature perturbation R . In linear theory, the variable χ reduces to the traceless scalar-type component H Lin T , and4the curvature perturbation is given by R Lin = ( H Lin L + H Lin T / Y , where ψ → H Lin L in the linear limit, and we havefollowed the notation of [40].Neglecting the contribution of gravitational waves and focusing on that of the scalar-type perturbations, we canassume that (0) γ ij in the K gauge approaches the flat metric at late times when the adiabatic limit is reached, (0) γ ij → δ ij (cid:0) N → (cid:1) . (4.43)This condition completely fixes the remaining spatial gauge degrees of freedom, say, within the class of the time-slice-orthogonal threading.Under this condition, we rewrite γ ij on the K gauge given in Eq. (4.30) to manifest its time and spatial dependence, γ ij = δ ij + 2 √ C Aij ( x i ) f A (0) + C χij ( x i ) f χ (0) − √ C Aij ( x i ) f A ( N ) − C χij ( x i ) f χ ( N ) , (4.44)where C γij = δ ij from Eq. (4.43), and C χij is given by C χij = 6 e − C ψ (cid:18) ∂ i C ψ ∂ j C ψ − ∂ i ∂ j C ψ − ∂ i C φI ∂ j C φI (cid:19) T F . (4.45)The time-dependent functions f A and f χ are given by f A ( N ) = e N − N ) √ (0) V I γ ( N ) , f χ ( N ) = 1 a V J γ ( N ) , (4.46)where the functions I γ and J γ are defined in Eqs. (4.36) and (4.37), respectively.Now we extract the scalar part from C Aij and C χij by using the definition of χ given by Eq. (3.20). We note thatthis definition is unique in the sense that it reduces to the standard scalar part in the limit of linear theory and givesthe O ( ǫ ) correction unambiguously.The contribution of C Aij to χ may be determined by evaluating Eq. (3.4) on the initial slice with the help ofEqs. (4.28) and (4.31), e − C ψ ∂ j (cid:16) e C ψ C Aij (cid:17) = 23 ∂ i (2) K ( N ) − r (0) V D φI ∂ i C φI , (4.47)where (2) K ( N ) = p (0) V m I D φI + 4 △ C ψ + 2 ∂ i C ψ ∂ i C ψ − ∂ i C φI ∂ i C φI a ( N ) e C ψ (0) V ! . (4.48)Applying the definition of χ in Eq. (3.20) to the above, we find the contribution from C Aij is χ A = −√ (2) K ( N ) (cid:16) f A (0) − f A ( N ) (cid:17) + 32 p (0) V △ − ∂ i (cid:16) D φI ∂ i C φI (cid:17)(cid:16) f A (0) − f A ( N ) (cid:17) . (4.49)As for the contribution of C χij to χ , we can make use of the relation,16 △ R h e C ψ δ ij i = ∂ i n e − C ψ ∂ j h e C ψ (cid:16) ∂ i C ψ ∂ j C ψ − ∂ i ∂ j C ψ (cid:17) T F io , (4.50)where the left-hand side is the Ricci scalar of e C ψ δ ij . Thus the contribution from C χij is χ χ = − R h e C ψ δ ij i(cid:16) f χ (0) − f χ ( N ) (cid:17) + 94 △ − n ∂ i e − ψ ∂ j h e C ψ (cid:16) ∂ i C φI ∂ j C φI (cid:17) T F io(cid:16) f χ (0) − f χ ( N ) (cid:17) . (4.51)Adding both contributions together, we obtain χ = χ A + χ χ , (4.52)where χ A and χ χ are given, respectively, by Eqs. (4.49) and (4.51).5Finally summing all the contributions to the curvature perturbation R = ψ K + χ K /
3, where ψ K and χ K are givenby Eqs. (4.41) and (4.52), respectively, we obtain R K = C ψ + 2 M (cid:26) m I D φI h e N − N ) − i + 1 a e C ψ (0) V S K (cid:27) − √ (2) K ( N ) (cid:16) f A (0) − f A ( N ) (cid:17) − R h e C ψ δ ij i(cid:16) f χ (0) − f χ ( N ) (cid:17) + p (0) V △ − ∂ i (cid:16) D φI ∂ i C φI (cid:17)(cid:16) f A (0) − f A ( N ) (cid:17) + 34 △ − n ∂ i e − ψ ∂ j h e C ψ (cid:16) ∂ i C φI ∂ j C φI (cid:17) T F io(cid:16) f χ (0) − f χ ( N ) (cid:17) . (4.53)What we need to know is the final value of R at sufficiently late times, N → a → a e N ). We take N to be atime around which the scales relevant to cosmological observations crossed the Hubble horizon, hence N &
50. Inthis case, at N = 0, the curvature perturbation reduces to R K ( N = 0) ≈ (0) C ψ + (2) C ψ − m I M D φI . (4.54)The first term, (0) C ψ represents the leading order curvature perturbation obtainable in the usual δN formalism,and the remaining terms represent O ( ǫ ) contributions, the calculation of which is the main purpose of the beyond δN formalism. As a check, we have confirmed that the above result is consistent with the one obtained in linearperturbation theory with the same background solution.It should be noted that there is no contribution from χ K at N = 0 by definition. However, this does not meanthe evaluation of χ K was meaningless. In general it can make an important contribution to R around the horizoncrossing time N ∼ N , and by matching our R with that evolved from inside the horizon at N ∼ N , the final value R ( N = 0) is determined both by the values of ψ K and χ K and by their derivatives at N ∼ N .It may be noted that the last term proportional to D φI comes the time derivative of φ I at N = N , ∂ N φ I ( N ) = D φI = O ( ǫ ). Although this is generically small by construction, the contribution of the term itself can become largesince it is proportional to m I /M ∼ /M where M ≪ V. SUMMARY AND DISCUSSION
In this paper, we developed a theory of nonlinear cosmological perturbations on superhorizon scales in the context ofinflationary cosmology. We considered a multi-component scalar field with a general kinetic term and a general formof the potential. To discuss the superhorizon dynamics, we employed the ADM formalism and the spatial gradientexpansion approach.Different from the single-field case, there is a difficulty in solving the equations in the multi-field case. At leading-order, the equations take the same form as those for the homogeneous and isotropic FLRW background with suitableidentifications of variables. In particular, there are correspondences between the cosmic time in the background andthe proper time along each comoving trajectory, τ ⇔ t , and the scale factor with the one defined locally ae ψ ⇔ a .This implies that given the background solution, the solution at leading-order in gradient expansion is automaticallyobtained.In the single-field case, one can show the coincidence among the comoving, uniform expansion, and synchronousslicings at leading order. This allows us to set the lapse function to unity, and replace the proper time by the cosmictime in the solution for the scalar field. Then the metric is expressed in terms of the scalar field easily, and the next-to-leading order equations can be solved straightforwardly because the space-time dependence of the source termsconsisting of the leading order quantities is explicitly known.In cosmological perturbation theory, the most important quantity to be evaluated is the curvature perturbationon the comoving slices which is conserved on superhorizon scales after the universe has reached the adiabatic limit.This quantity accurate to next-to-leading order may be relatively easily obtained in the single-field case because ofthe above mentioned coincidence among the comoving, uniform expansion and synchronous slicings.On the other hand, in the multi-field case, such a coincidence between different slicings does not hold. This impliesthe following. One can express the lapse function as a function of the scalar field, but the scalar field is also a functionof the proper time. Thus one has the equation, α = f h t, φ ( τ ) i = f (cid:20) t, φ (cid:18)Z α dt (cid:19)(cid:21) . (5.1)To go beyond the leading order, one needs to solve this equation for α , but it seems almost impossible.6In this paper, we developed a formalism to go beyond the leading order which avoids the above problem. Namely,we first solve the field equations in a slicing in which the lapse function is trivial. The synchronous slicing is one ofsuch slicings, but we adopt the uniform e -folding number slicing in which the time slices are chosen in such a waythat the number of e -folds along each orbit orthogonal to the time slices, N , is spatially homogeneous on each timeslice. In other words, the uniform N slicing is a synchronous slicing if N is used as the time coordinate.In this slicing we can solve the equations to next-to-leading order without encountering the above mentionedproblem. After the solution to next-to-leading order is obtained, we transform it to the one in the uniform expansionslicing which is known to be identical to the comoving slicing on superhorizon scales in the adiabatic limit. Thus thegauge transformation laws play an essential role in our formalism. We derived them which are accurate to next-to-leading order. Note that they are fully nonlinear in nature in the language of the standard perturbation approach.They are summarised in Appendix C.As a demonstration of our formalism, we considered an analytically solvable model and constructed the explicitform of the solution. Namely, we considered a multi-component canonical scalar field with exponential potential, V ( φ I ) = W exp[ P J m J φ J ]. Following the general procedure discussed above, we first solved for the scalar field andthe metric in the uniform N slicing. Then using a remaining gauge degree of freedom of this slicing, we set the initialcondition such that it coincides with the uniform expansion slicing at leading order. In this slicing with this initialcondition, the next-to-leading order solution was straightforwardly found. Then we applied the derived nonlineargauge transformation to it to obtain the solution in the uniform expansion slicing. Finally, from thus obtained spatialmetric which takes the form, e ψ γ ij where γ ij is a unimodular metric, we constructed the generalised, nonlinearcurvature perturbation R defined by R = ψ + χ χ ≡ − △ − n ∂ i e − ψ ∂ j h e ψ (cid:0) γ ij − δ ij (cid:1)io . (5.2)By inspecting the form of the final value of the curvature perturbation, we argued that the decaying modes of thescalar field may give rise to non-negligible contribution to the final, conserved curvature perturbation. To quantifythe effect, it is necessary to match our solution on super-horizon scales with the one solved from sub-horizon scales.The evaluation of these next-to-leading order corrections to the spectrum as well as to the bispectrum of the curvatureperturbation is left for future study. Acknowledgments
AN is grateful to Shinji Mukohyama and Yuki Watanabe for valuable comments and fruitful discussions. YT wouldlike to thank Ryo Saito, Teruaki Suyama, Jun’ichi Yokoyama, Kiyoung Choi, Seoktae Koh and Shuichiro Yokoyamafor their comments and discussions on this work. This work is supported in part by Monbukagaku-sho Grant-in-Aidfor the Global COE programs, “The Next Generation of Physics, Spun from Universality and Emergence” at KyotoUniversity. AN is partly supported by Grant-in-Aid for JSPS Fellows No. 21-1899 and JSPS Postdoctoral Fellowshipsfor Research Abroad. YT also wishes to acknowledge financial support by the RESCEU, University of Tokyo andby JSPS Grant-in-Aid for Young Scientists (B) No. 23740170 and for JSPS Postdocoral Fellowships. The work ofMS is supported by JSPS Grant-in-Aid for Scientific Research (A) No. 21244033, and by Grant-in-Aid for CreativeScientific Research No. 19GS0219.
Appendix A: Coincidence of four slices in single scalar case
In the single-field case, it can be shown that the four slicings, the comoving, uniform K , synchronous and uniform N slicings, coincide with each other at leading order in gradient expansion. Here we demonstrate it. For simplicity,we consider a canonical scalar field, but the generalisation is straightforward.At leading order in gradient expansion, the Hamiltonian and momentum constraints are written as13 K = 12 (cid:0) ∂ τ φ (cid:1) + V ( φ ) , (A1) ∂ i K = − ∂ τ φ∂ i φ , (A2)where τ is defined in terms of the lapse function as τ ( t, x i ) ≡ Z x i = const. α ( t ′ , x i ) dt ′ . (A3)7Here let us choose the uniform K slicing, K ( t, x i ) = K ( t ) . (A4)Then Eq. (A2) immediately implies that the scalar field is uniform on this slice, φ ( t, x i ) → φ = φ ( t ) . (A5)This shows the uniform K slice coincides with the comoving slice.Now since the left-hand side of Eq. (A1) as well as the potential term on the right-hand side is homogeneous, itfollows that the kinetic term is homogeneous. This means ∂ τ φ = unif orm → α = α ( t ) . (A6)Therefore the uniform K slicing coincides with the synchronous slicing. Then the spatial homogeneity of both K and α implies that of the number of e -folds N by definition, Eq. (2.36). Thus we have shown the coincidence of these fourslicings at leading order in gradient expansion.As we noticed in subsection II D, there is a remaining gauge degree of freedom in the synchronous and uniform N slicings. In the above proof, this freedom is tacitly fixed since we took the uniform K slicing as a starting point. Inother words, the uniform K (or comoving) slicing implies it is synchronous and uniform N . However, if one startsfrom the synchronous or uniform N slicing, we have to set the initial data such that K or φ is uniform on the initialslice in order to show the coincidence. In fact, if we use this remaining degree of freedom to set a different conditionon the initial data, for example ψ = 0, the synchronous or uniform N slicing will not coincide with the uniform K orcomoving slicing.In passing let us also comment on the conservation of the curvature perturbation. From the definition of K ,Eq. (2.5), one has ∂ t ψ ( t, x i ) = − H ( t ) + 13 α ( t, x i ) K ( t, x i ) . (A7)This shows how the inhomogeneous part on the right-hand side contributes to the evolution of the curvature pertur-bation. As we have discussed above, there is no such inhomogeneities in the uniform K or comoving slicing in thesingle-scalar case. This is a proof of the conservation of the nonlinear curvature perturbation on super-horizon scales.On the other hand, in the case of multi-field, the corresponding scalar component on the right-hand side of themomentum constraint, Eq. (A2), becomes uniform on the uniform K slice. However this scalar component does notappear in Eq. (A1) in general, and hence one cannot show the homogeneity of the lapse function as in the single-fieldcase.We note that this does not mean that the Hamiltonian and momentum constraints are not related. Actually theleading order momentum constraint reduces to the spatial derivative of the Hamiltonian constraint in the slow-rolllimit as shown in Appendix E 2 a. For a non-slow roll case, this relation between the Hamiltonian and momentumconstraints holds only if we take into account the next-to-leading order corrections properly (see Appendix E 2 b). Appendix B: Uniform K slicing In this Appendix, we derive the basic equations on the uniform K slicing, K ( t, x i ) = 3 H ( t ), and discuss a difficultyin constructing the solution at next-to-leading order in gradient expansion.In the uniform K slicing and time-slice-orthogonal threading, the constraint equations are given by1 a e ψ h R − (cid:16) D ψ + 2 D i ψD i ψ (cid:17)i + 6 H = 2 E , (B1) e − ψ D j (cid:16) e ψ A j i (cid:17) = 2 P ( IJ ) ∂ τ φ I ∂ i φ J . (B2)The evolution equation for K and A ij are3 ∂ τ H = −
32 ( E + P ) − a e ψ h R − (cid:16) D ψ + 2 D i ψD i ψ (cid:17)i + 1 a e ψ α (cid:16) D α + D i αD i ψ − αP ( IJ ) D i φ I D i φ J (cid:17) , (B3) ∂ τ A ij = − HA ij − a e ψ (cid:16) R ij + D i ψD j ψ − D i D j ψ (cid:17) T F + 1 a e ψ α (cid:16) D i D j α − D i αD j ψ − D j αD i ψ − αP ( IJ ) D i φ I D j φ J (cid:17) T F , (B4)8where E = 2 P ( IJ ) ∂ τ φ I ∂ τ φ J − P , (B5)and τ is defined in terms of the lapse function as τ ( t, x i ) ≡ Z x i = const. α ( t ′ , x i ) dt ′ , (B6)and the equation for the scalar field is ∂ τ (cid:16) P ( IJ ) ∂ τ φ J (cid:17) + 3 HP ( IJ ) ∂ τ φ J − a αe ψ ∂ i (cid:16) αe ψ P ( IJ ) D i φ J (cid:17) − P I = 0 . (B7)First, let us consider the leading order. The Hamiltonian constraint (B1) reduces to3 H = (0) E + O ( ǫ ) , (B8)hence (0) E is homogeneous on this slice. This means the solution for (0) E exactly coincides with the backgroundsolution E BG . As for P , the leading order solution is easily constructed in terms of the background solution once itis known as discussed in Sec. II C. To be specific, if the background pressure is expressed as a function of t , P BG ( t ) = P BG h t, P ( t ) i , (B9)the leading order solution for P is given by (0) P ( t, x i ) = P BG h τ, P ( τ ) i . (B10)The lapse function can be obtained from Eq. (B3) as (0) α = − dH ( t ) dt (0) E ( t ) + (0) P ( t, x i ) + O ( ǫ ) . (B11)Thus the lapse function becomes inhomogeneous if (0) P is so. From Eq. (2.5), the evolution of ψ is related with theinhomogeneity of (0) α in this slicing as ∂ t ψ = H ( α − , (B12)which reflects the well-known fact that the curvature perturbation evolves if a non-adiabatic pressure exists. Howeverthere is a profound problem in Eq. (B11).Although it looks like Eq. (B11) gives the solution for (0) α , it is not so because the right-hand side depends also on (0) α . In reality, Eq. (B10) gives (0) P as a function of τ , which is given by the time integration of the lapse function.Therefore Eq (B11) can be schematically expressed as α = f h t, P ( τ ) i = f (cid:20) t, P (cid:18)Z α dt (cid:19)(cid:21) . (B13)In general it is very difficult to solve this equation, at least analytically.As clear from Eq. (B3), one needs to evaluate the spatial derivatives of α for example at next-to-leading order. Butthis will be almost impossible. This is the main problem in constructing a solution at next-to-leading order in gradientexpansion on the uniform K slicing. This kind of problem arises also on other slicings except for the synchronous oruniform N slicings. This is why the uniform N slicing is used in our formalism.As a closing remark of this Appendix, we emphasise that the above difficulty becomes actually problematic only atnext-to-leading order. It is not a problem at leading order. This is because the solution of the “curvature perturbation” (0) ψ is given by a time integration which depends only on the initial and final values of N . From Eq. (B12) we have (0) ψ ( t, x i ) − (0) ψ ( t , x i ) = Z tt dt ′ (cid:0) (0) α − (cid:1) H = N ( t, x i ) − N ( t , x i ) − N ( t ) + N ( t ) . (B14)Thus, at leading order in gradient expansion, as far as we are interested in the curvature perturbation there is no needto know the solution for (0) α . This is why the δN formalism works well despite the presence of the above problem.9 Appendix C: Nonlinear gauge transformation
We derive the gauge transformation rules for the metric, its derivative ( K and A ij ) and the scalar field. We considera nonlinear gauge transformation from a coordinate system with vanishing shift vector β i = 0, to another coordinatesystem in which the new shift vector also vanishes, ˜ β i = 0. We note that once the time slicing is changed, the shiftvector appears in the new slicing in general. So the spatial coordinates also need to be changed to eliminate thusappeared shift vector.We use the background e -folding number N as the time coordinate and define the temporal and spatial shift, n and L i , respectively, N → ˜ N = N + n ( N, x i ) ,x i → ˜ x i = x i + L i ( N, x i ) , or conversely ˜ N + ˜ n ( ˜ N , ˜ x i ) = N , (C1)˜ x i + ˜ L i ( ˜ N , ˜ x i ) = x i . (C2)Under the change of the coordinates, the line element should remain invariant, ds = − α H ( N ) dN + a ( N ) e ψ γ ij dx i dx j = − ˜ α H ( ˜ N ) d ˜ N + a ( ˜ N ) e ψ ˜ γ ij d ˜ x i d ˜ x j . (C3)Equating the coefficients of d ˜ N , d ˜ N d ˜ x i , and d ˜ x i d ˜ x j on both sides of the above, we obtain˜ α H ( ˜ N ) = α H ( N ) (cid:0) ∂ ˜ N ˜ n (cid:1) − a ( N ) e ψ γ ij ∂ ˜ N ˜ L i ∂ ˜ N ˜ L j , (C4)0 = α H ( N ) (cid:0) ∂ ˜ N ˜ n (cid:1) ∂ ˜ i ˜ n − a ( N ) e ψ γ kj ∂ ˜ N ˜ L j (cid:0) δ ki + ∂ ˜ i ˜ L k (cid:1) , (C5) a ( ˜ N ) e ψ ˜ γ ij = a ( N ) e ψ γ kl (cid:0) δ ki + ∂ ˜ i ˜ L k (cid:1)(cid:0) δ lj + ∂ ˜ j ˜ L l (cid:1) − α H ( N ) ∂ ˜ i ˜ n∂ ˜ j ˜ n . (C6)
1. Leading order gauge transformation
First we derive the leading order gauge transformation. To begin with, we note the spatial shift ˜ L i is O ( ǫ ) fromEq. (C5), since it is proportional to the spatial derivative of ˜ n , ∂ ˜ N ˜ L i ∼ γ ij ∂ ˜ i ˜ n = O ( ǫ ) . (C7)Hence neglecting ∂ ˜ N ˜ L i in Eq. (C4), we have α H ( N ) (cid:0) ∂ ˜ N ˜ n (cid:1) = ˜ α H ( ˜ N ) + O ( ǫ ) . (C8)This gives the leading order gauge transformation of the lapse function,˜ α ( ˜ N , ˜ x i ) = H ( ˜ N ) H ( N ) (cid:0) ∂ ˜ N ˜ n (cid:1) α ( N, x i ) + O ( ǫ )= H ( ˜ N ) H ( ˜ N + ˜ n ) (cid:0) ∂ ˜ N ˜ n (cid:1) α ( ˜ N + ˜ n, ˜ x i ) + O ( ǫ ) . (C9)The gauge transformation rules for the spatial metric components, ψ and γ ij , are derived from Eq. (C6). Takingthe determinant of Eq. (C6) and neglecting ∂ ˜ N ˜ L i and ∂ ˜ i ˜ n , we obtain a e ψ = a ( ˜ N ) e ψ + O ( ǫ ) . (C10)0This gives the leading order gauge transformation of ψ as˜ ψ ( ˜ N , ˜ x i ) = ψ ( N, x i ) − ˜ n + O ( ǫ )= ψ ( ˜ N + ˜ n, ˜ x i ) − ˜ n + O ( ǫ ) . (C11)As is clear from Eq. (C6), γ ij is invariant at this order,˜ γ ij ( ˜ N , ˜ x i ) = γ ij ( N, x i ) + O ( ǫ )= γ ij ( ˜ N + ˜ n, ˜ x i ) + O ( ǫ ) . (C12)Under the gauge transformation, the scalar field is essentially invariant except for the change of the arguments,˜ φ ( ˜ N , ˜ x i ) = φ ( N, x i ) . (C13)Then, the leading order gauge transformation is˜ φ ( ˜ N , ˜ x i ) = φ ( ˜ N + (0) ˜ n, ˜ x i ) + O ( ǫ ) . (C14)Here we note that what we call a gauge transformation should perhaps be called a coordinate transformation as faras the change of time slicing is concerned. For example, for a scalar quantity Q , the gauge transformation under thetemporal shift ( t → t + T ) is given by f δQ ( t, x i ) = δQ ( t, x i ) − T ∂ t Q , (C15)which is derived from the invariance of Q as a scalar under the time coordinate transformation,˜ Q (˜ t, x i ) = Q ( t, x i ) . (C16)In the context of gradient expansion, we define the nonlinear gauge transformation of a scalar quantity by Eq. (C16)not by Eq. (C15), since the gradient expansion is a completely nonlinear approach. If we were to expand perturbativelythe left-hand side of Eq. (C16), we would obtain an infinite number of terms, which is definitely something to beavoided. On the other hand, as we see shortly below, to the accuracy of our interest, i.e. to O ( ǫ ), the spatialcoordinate transformation can be linearised. Hence we interpret it as the standard gauge transformation and comparethe quantities at the same coordinate values.Once we have the gauge transformation rules for the metric components, we can readily obtain those for the extrinsiccurvature components, K and A ij . Since A ij vanishes at leading order, we only have to consider the transformationof the expansion K . From its definition, Eq. (2.5), and using the transformation rules for the metric componentsderived above, we find it is invariant at leading order,˜ K ( ˜ N , ˜ x i ) = K ( ˜ N + (0) ˜ n, x i ) + O ( ǫ ) . (C17)Again, as discussed around Eq. (C16), we note that K is invariant only in the sense of the nonlinear gauge transfor-mation we have defined.
2. Next-to-leading order gauge transformation
Now we derive the gauge transformation at next-to-leading order. To do so, we first have to fix the spatial coordinatetransformation. It is determined by Eq. (C5), which results from the requirement that the shift vector should vanishto O ( ǫ ). In terms of ˜ n , the gauge transformation generator ˜ L i is given as˜ L i = l i (˜ x i ) + Z ˜ N ˜ N dN ′ α (cid:0) ∂ N ′ ˜ n (cid:1) a ( N ) e ψ H ( N ) γ ij ∂ ˜ j ˜ n + O ( ǫ )= l i (˜ x i ) + Z ˜ N ˜ N dN ′ α ( N ′ + ˜ n, ˜ x i ) (cid:0) ∂ N ′ ˜ n (cid:1) a ( N ′ + ˜ n ) e ψ ( N ′ +˜ n, ˜ x i ) H ( N ′ + ˜ n ) γ ij ( N ′ + ˜ n, ˜ x i ) ∂ ˜ j ˜ n + O ( ǫ ) , (C18)where l i is a time-independent spatial vector which represents the remaining gauge degrees of freedom in the spatialcoordinates in the time-slice-orthogonal threading.1To consider the gauge transformation at next-to-leading order, we expand the nonlinear gauge transformationgenerator ˜ n in Eq. (C1) to the leading order term, the next-to-leading order term and so on,˜ n ≡ (0) ˜ n + (2) ˜ n + O ( ǫ ) . (C19)As for the spatial shift ˜ L i , it is unnecessary to expand it, since it is O ( ǫ ) already and the next-to-leading order termin ˜ L i is O ( ǫ ) which only affects the gauge transformation at O ( ǫ ).First we consider the metric components. From Eq. (C4), the next-to-leading order gauge transformation for thelapse function is determined as (2) ˜ α ( ˜ N , ˜ x i ) = αH ( ˜ N ) H ( (0) ˜ N ) (cid:0) ∂ ˜ N (0) ˜ n (cid:1) (2) αα + (2) ˜ n ∂ N αα − (2) ˜ n ∂ N H ( (0) ˜ N ) H ( (0) ˜ N ) + ∂ ˜ N (2) ˜ n ∂ ˜ N (0) ˜ n ! − α H ( ˜ N ) (cid:0) ∂ ˜ N (0) ˜ n (cid:1) a ( (0) ˜ N ) e ψ H ( (0) ˜ N ) γ ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n + ˜ L i ( ∂ ˜ i α ) H ( ˜ N ) H ( (0) ˜ N ) (cid:0) ∂ ˜ N (0) ˜ n (cid:1) + O ( ǫ ) , (C20)where ∂ ˜ i = ∂/∂ ˜ x i and (0) ˜ N ( N, x i ) ≡ N + (0) ˜ n ( N, x i ). Note that all the quantities on the right-hand side are thoseevaluated at N = (0) ˜ N and x i = ˜ x i . Taking the determinant of Eq. (C6), a ( ˜ N ) e ψ = a ( N ) e ψ (cid:0) ∂ ˜ i ˜ L i (cid:1) − a ( N ) e ψ α H ( N ) γ ij ∂ ˜ i ˜ n∂ ˜ j ˜ n + O ( ǫ ) , (C21)we obtain the gauge transformation for (2) ψ , (2) ˜ ψ ( ˜ N , ˜ x i ) = (2) ψ − (2) ˜ n + (2) ˜ n∂ N ψ + ˜ L i ∂ ˜ i ψ + 13 ∂ ˜ i ˜ L i − α a ( (0) ˜ N ) e ψ H ( (0) ˜ N ) γ ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n + O ( ǫ ) . (C22)Then using Eq. (C6) again, the gauge transformation for γ ij is obtained as (2) ˜ γ ij ( ˜ N , ˜ x i ) = (2) γ ij + (2) ˜ n∂ ˜ N γ ij + ˜ L k ∂ ˜ k γ ij + γ jk ∂ ˜ i ˜ L k + γ ik ∂ ˜ j ˜ L k − ∂ ˜ i ˜ L i − α a ( (0) ˜ N ) e ψ H ( (0) ˜ N ) (cid:18) ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n − γ kl ∂ ˜ k (0) ˜ n∂ ˜ l (0) ˜ nγ ij (cid:19) + O ( ǫ ) . (C23)Next we consider the extrinsic curvature. At next-to-leading order, Eq. (2.5) gives (2) ˜ K ( ˜ N , ˜ x i ) = (2) K + (2) ˜ n∂ N (0) K + ˜ L i ∂ ˜ i (0) K + 3 H ( (0) ˜ N ) (cid:0) ∂ ˜ N (0) n (cid:1) α (cid:18) ∂ ˜ N (0) ˜ n (2) αα − ∂ ˜ N (2) ˜ n ∂ ˜ N (0) ˜ n (cid:19) + 3 αγ ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n (cid:0) ∂ ˜ N (0) n (cid:1) a ( (0) ˜ N ) e ψ H ( (0) ˜ N ) + ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n αa ( (0) ˜ N ) H ( (0) ˜ N ) ∂ N (cid:18) α e ψ γ ij (cid:19) + 3 H ( (0) ˜ N ) (cid:0) ∂ ˜ N (0) n (cid:1) α (cid:20) ∂ ˜ N (2) ˜ n (cid:0) − ∂ N ψ (cid:1) − (cid:0) ∂ ˜ N ˜ L i (cid:1) ∂ ˜ i ψ − ∂ ˜ i ∂ ˜ N ˜ L i (cid:21) + H ( (0) ˜ N ) αγ ij (cid:0) ∂ ˜ N (0) n (cid:1) e ψ ∂ ˜ N ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ na ( (0) ˜ N ) H ( (0) ˜ N ) ! + O ( ǫ ) . (C24)And also Eq. (2.16) determines the gauge transformation for A ij as˜ A ij ( ˜ N , ˜ x i ) = A ij − H ( (0) ˜ N )2 (cid:0) ∂ ˜ N (0) ˜ n (cid:1) α (cid:16) ∂ ˜ N ˜ L k ∂ ˜ k γ ij + γ jk ∂ ˜ i ∂ ˜ N ˜ L k + γ ik ∂ ˜ j ∂ ˜ N ˜ L k (cid:17) T F − H ( (0) ˜ N )2 (cid:0) ∂ ˜ N (0) ˜ n (cid:1) α ∂ ˜ N (cid:20) α a ( (0) ˜ N ) e ψ H ( (0) ˜ N ) (cid:16) ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n (cid:17) T F (cid:21) + O ( ǫ ) . (C25)Finally, the next-to-leading order gauge transformation for the scalar field is given from Eq. (C13) as (2) ˜ φ ( ˜ N , ˜ x i ) = (2) φ ( ˜ N + (0) ˜ n, ˜ x i ) + (2) ˜ n∂ N (0) φ ( ˜ N + (0) ˜ n, ˜ x i ) + ˜ L i ∂ ˜ i (0) φ ( ˜ N + (0) ˜ n, ˜ x i ) + O ( ǫ ) . (C26)2To conclude this Appendix, let us summarise the nonlinear gauge transformation derived above. The leading ordertransformation rules are ˜ α ( ˜ N , ˜ x i ) = H ( ˜ N ) H ( (0) ˜ N ) (cid:0) ∂ ˜ N (0) ˜ n (cid:1) α + O ( ǫ ) , (C27)˜ ψ ( ˜ N , ˜ x i ) = ψ − (0) ˜ n + O ( ǫ ) , (C28)˜ γ ij ( ˜ N , ˜ x i ) = γ ij + O ( ǫ ) , (C29)˜ K ( ˜ N , ˜ x i ) = K + O ( ǫ ) , (C30)˜ φ ( ˜ N , ˜ x i ) = φ + O ( ǫ ) . (C31)The next-to-leading order transformation rules are (2) ˜ α ( ˜ N , ˜ x i ) = αH ( ˜ N ) H ( (0) ˜ N ) (cid:0) ∂ ˜ N (0) ˜ n (cid:1) (2) αα + (2) ˜ n ∂ N αα − (2) ˜ n ∂ N H ( (0) ˜ N ) H ( (0) ˜ N ) + ∂ ˜ N (2) ˜ n ∂ ˜ N (0) ˜ n ! − α H ( ˜ N ) (cid:0) ∂ ˜ N (0) ˜ n (cid:1) a ( (0) ˜ N ) e ψ H ( (0) ˜ N ) γ ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n + ˜ L i (cid:0) ∂ ˜ i α (cid:1) H ( ˜ N ) H ( (0) ˜ N ) (cid:0) ∂ ˜ N (0) ˜ n (cid:1) + O ( ǫ ) , (C32) (2) ˜ ψ ( ˜ N , ˜ x i ) = (2) ψ − (2) ˜ n + (2) ˜ n∂ N ψ + ˜ L i ∂ ˜ i ψ + 13 ∂ ˜ i ˜ L i − α γ ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ na ( (0) ˜ N ) e ψ H ( (0) ˜ N ) + O ( ǫ ) , (C33) (2) ˜ γ ij ( ˜ N , ˜ x i ) = (2) γ ij + (2) ˜ n∂ ˜ N γ ij + ˜ L k ∂ ˜ k γ ij + γ jk ∂ ˜ i ˜ L k + γ ik ∂ ˜ j ˜ L k − ∂ ˜ k ˜ L k γ ij − α a ( (0) ˜ N ) e ψ H ( (0) ˜ N ) (cid:18) ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n − γ kl ∂ ˜ k (0) ˜ n∂ ˜ l (0) ˜ nγ ij (cid:19) + O ( ǫ ) , (C34)and (2) ˜ K ( ˜ N , ˜ x i ) = (2) K + (2) ˜ n∂ N (0) K + ˜ L i ∂ ˜ i (0) K + 3 H ( (0) ˜ N ) (cid:0) ∂ ˜ N (0) n (cid:1) α (cid:18) ∂ ˜ N (0) ˜ n (2) αα − ∂ ˜ N (2) ˜ n ∂ ˜ N (0) ˜ n (cid:19) + 3 αγ ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n (cid:0) ∂ ˜ N (0) n (cid:1) a ( (0) ˜ N ) e ψ H ( (0) ˜ N ) + ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n αa ( (0) ˜ N ) H ( (0) ˜ N ) ∂ N (cid:18) α e ψ γ ij (cid:19) + 3 H ( (0) ˜ N ) (cid:0) ∂ ˜ N (0) n (cid:1) α (cid:20) ∂ ˜ N (2) ˜ n (cid:0) − ∂ N ψ (cid:1) − (cid:0) ∂ ˜ N ˜ L i (cid:1) ∂ ˜ i ψ − ∂ ˜ i ∂ ˜ N ˜ L i (cid:21) + H ( (0) ˜ N ) αγ ij (cid:0) ∂ ˜ N (0) n (cid:1) e ψ ∂ ˜ N ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ na ( (0) ˜ N ) H ( (0) ˜ N ) ! + O ( ǫ ) , (C35)˜ A ij ( ˜ N , ˜ x i ) = A ij − H ( (0) ˜ N )2 (cid:0) ∂ ˜ N (0) ˜ n (cid:1) α (cid:16) ∂ ˜ N ˜ L k ∂ ˜ k γ ij + γ jk ∂ ˜ i ∂ ˜ N ˜ L k + γ ik ∂ ˜ j ∂ ˜ N ˜ L k (cid:17) T F − H ( (0) ˜ N )2 (cid:0) ∂ ˜ N (0) ˜ n (cid:1) α ∂ ˜ N (cid:20) α a ( (0) ˜ N ) e ψ H ( (0) ˜ N ) (cid:16) ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n (cid:17) T F (cid:21) + O ( ǫ ) , (C36) (2) ˜ φ ( ˜ N , ˜ x i ) = (2) φ + (2) ˜ n∂ N (0) φ + ˜ L i ∂ ˜ i (0) φ + O ( ǫ ) , (C37)where ˜ L i is given by ˜ L i = l i (˜ x i ) + Z ˜ N ˜ N dN ′ α (cid:0) ∂ N ′ ˜ n (cid:1) a ( (0) ˜ N ) e ψ H ( (0) ˜ N ) γ ij ∂ ˜ j ˜ n + O ( ǫ ) . (C38) Appendix D: Consistency checking
In this Appendix, we apply our formalism to the case of a single scalar field and verify the consistency. One way todo this is just to compare the obtained solutions in this formalism with the results obtained previously, for example,in [28]. Here, however, we take another, direct way. Namely, we consider the solution in the comoving gauge andtransform it to the one in the N gauge. Then we insert thus derived solutions for the metric and the scalar field intothe basic equations in the N gauge and show that they are satisfied.3
1. Solution in the uniform N gauge First we construct the solution in the N gauge by transforming it from the one in the comoving gauge. As we haveseen in Appendix A, the leading order solution in the comoving gauge is (0) α c = f α ( N ) , (0) ψ c = C ( x i ) , (0) γ ijc = C ij ( x i ) , K c = f K ( N ) , φ c = f φ ( N ) . (D1)Here f α , f K and f φ are functions of only time and have no spatial dependence. On the other hand, C and C ij arefunctions of only spatial coordinates and have no time dependence. Because of the definition of K , Eq. (2.5), thefunctions f K are f α are related as f K = 3 Hf α . (D2)At next-to-leading order, however, all the quantities listed in Eq. (D1) become space-time dependent except for thescalar field (2) φ , which is by definition a function of only time. Hence without loss of generality, we may absorb it inthe leading order scalar field (0) φ and set (2) φ = 0.To perform the transformation from the comoving slicing to the uniform N slicing, N = ˜ N + ˜ n ( ˜ N , ˜ x i ), one needs todetermine ˜ n . In the uniform N slicing, ψ is time-independent by definition, Eq. (2.36). Considering the transformationof ψ , this gives at leading order, ψ N (˜ x i ) = C ( x i ) − (0) ˜ n + O ( ǫ ) → (0) ˜ n = C (˜ x i ) − ψ N (˜ x i ) + O ( ǫ ) . (D3)At the next-leading-order Eq. (C33) gives0 = (2) ψ c − (2) ˜ n + ˜ L i ∂ ˜ i C + 13 ∂ ˜ i ˜ L i − f α C ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ na ( (0) ˜ N ) e C H ( (0) ˜ N ) + O ( ǫ ) → (2) ˜ n = (2) ψ c + ˜ L i ∂ ˜ i C + 13 ∂ ˜ i ˜ L i − f α C ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ na ( (0) ˜ N ) e C H ( (0) ˜ N ) + O ( ǫ ) , (D4)where ˜ L i is given by Eq. (C38), which is derived from the time-slice-orthogonal threading condition, and the fact that (2) ψ N = 0 is used since it may be absorbed into (0) ψ N . Note that from Eq. (D3), (0) ˜ n is independent of time whichsignificantly simplifies the analysis below.Performing the gauge transformation, we we obtain at leading order, α N ( ˜ N , ˜ x i ) = H ( ˜ N ) H ( (0) ˜ N ) f α ( (0) ˜ N ) + O ( ǫ ) , (D5) γ ij N ( ˜ N , ˜ x i ) = C ij + O ( ǫ ) , (D6) K N ( ˜ N , ˜ x i ) = f K ( (0) ˜ N ) + O ( ǫ ) , (D7) φ N ( ˜ N , ˜ x i ) = f φ ( (0) ˜ N ) + O ( ǫ ) , (D8)and at the next-to-leading order, (2) α N ( ˜ N , ˜ x i ) = f α H ( ˜ N ) H ( (0) ˜ N ) (2) α c f α + (2) ˜ n ∂ N f α f α − (2) ˜ n ∂ N H ( (0) ˜ N ) H ( (0) ˜ N ) + ∂ ˜ N (2) ˜ n ! − f α C ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ nH ( ˜ N ) a ( (0) ˜ N ) e C H ( (0) ˜ N ) + O ( ǫ ) , (D9) (2) γ ij N ( ˜ N , ˜ x i ) = (2) γ ijc + ˜ L k ∂ ˜ k C ij + C jk ∂ ˜ i ˜ L k + C ik ∂ ˜ j ˜ L k − ∂ ˜ k ˜ L k C ij − f α a ( (0) ˜ N ) e C H ( (0) ˜ N ) (cid:18) ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n − C kl ∂ ˜ k (0) ˜ n∂ ˜ l (0) ˜ n C ij (cid:19) + O ( ǫ ) , (D10)4and (2) K N ( ˜ N , ˜ x i ) = (2) K c + (2) ˜ n∂ N f K + (cid:18) f α + ∂ N f α (cid:19) C ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ na ( (0) ˜ N ) e C H ( (0) ˜ N ) − H ( (0) ˜ N ) f α (cid:20)(cid:0) ∂ ˜ N ˜ L i (cid:1) ∂ ˜ i C + 13 ∂ ˜ i ∂ ˜ N ˜ L i (cid:21) + H ( (0) ˜ N ) f α C ij ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n e C ∂ ˜ N (cid:18) a ( (0) ˜ N ) H ( (0) ˜ N ) (cid:19) + O ( ǫ ) , (D11) A ij N ( ˜ N , ˜ x i ) = A ijc − H ( (0) ˜ N )2 f α (cid:16) ∂ ˜ N ˜ L k ∂ ˜ k C ij + C jk ∂ ˜ i ∂ ˜ N ˜ L k + C ik ∂ ˜ j ∂ ˜ N ˜ L k (cid:17) T F + H ( (0) ˜ N )2 f α e C (cid:16) ∂ ˜ i (0) ˜ n∂ ˜ j (0) ˜ n (cid:17) T F ∂ ˜ N (cid:18) f α a ( (0) ˜ N ) H ( (0) ˜ N ) (cid:19) + O ( ǫ ) , (D12) (2) φ N ( ˜ N , ˜ x i ) = (2) ˜ n∂ N f φ + O ( ǫ ) , (D13)where ∂ ˜ i = ∂/∂ ˜ x i and ˜ L i is given explicitly by˜ L i = l i (˜ x i ) + C ij ∂ ˜ j (0) ˜ ne C Z ˜ N ˜ N dN ′ f α a ( (0) ˜ N ) H ( (0) ˜ N ) + O ( ǫ ) . (D14)After some manipulations, Eqs. (D9), (D11) and (D12) may be re-expressed as α N α (0) N − α c f α = ∂ ˜ N (2) ˜ n − (2) ˜ n∂ N log f K ( (0) ˜ N ) − D ˜ i (0) ˜ nD ˜ i (0) ˜ na ( (0) ˜ N ) e C f K ( (0) ˜ N ) , (D15) K N − K c = (2) ˜ n∂ N f K ( (0) ˜ N ) + 3 (cid:18)
12 + ∂ N log f K ( (0) ˜ N ) (cid:19) D ˜ i (0) ˜ n∂ ˜ j (0) ˜ na ( (0) ˜ N ) e C f K ( (0) ˜ N ) − D ˜ n + D ˜ i (0) ˜ nD ˜ i C a ( (0) ˜ N ) e C f K ( (0) ˜ N ) , (D16) A ij N − A ijc = − a ( (0) ˜ N ) e C f K ( (0) ˜ N ) h D ˜ i D ˜ j (0) ˜ n − (cid:16) D ˜ i (0) ˜ nD ˜ j C + D ˜ j (0) ˜ nD ˜ i C (cid:17)i T F − a ( (0) ˜ N ) e C f K ( (0) ˜ N ) (cid:16) − ∂ N log f K ( (0) ˜ N ) (cid:17)(cid:16) D ˜ i (0) ˜ nD ˜ j (0) ˜ n (cid:17) T F , (D17)where D ˜ i is a covariant derivative with respect to (0) γ ij N (or equivalently to C ij ). These expressions are convenientfor later use.
2. Consistency check for A ij Here we explicitly show that the solution for A ij obtained in the previous subsection satisfies the evolution equationin the N gauge. This verifies the consistency of the derived gauge transformation.In the N gauge, the evolution equations for A ij is ∂ ˜ N A ij N = 3 A ij N + 3 a ( ˜ N ) e ψ N K N (cid:16) R ij + D ˜ i ψ N D ˜ j ψ N − D ˜ i D ˜ j ψ N (cid:17) T F − a ( ˜ N ) e ψ N K N (cid:20) K N D ˜ i D ˜ j (cid:18) K N (cid:19) + D ˜ i log K N D ˜ j ψ N + D ˜ j log K N D ˜ i ψ N + D ˜ i φ N D ˜ j φ N (cid:21) T F , (D18)while in the comoving gauge it is ∂ N A ijc = 3 A ijc + 3 a ( N ) e C f K ( N ) (cid:16) R ij + D i C D j C − D i D j C (cid:17) T F . (D19)We rewrite the second line inside the brackets of Eq. (D18) as(2nd line) = (cid:0) ∂ ˜ N log f K (cid:1) ( (0) ˜ N ) h D ˜ i D ˜ j (0) ˜ n − (cid:16) D ˜ i (0) ˜ nD ˜ j ψ N + D ˜ j (0) ˜ nD ˜ i ψ N (cid:17)i + h ∂ N log f K − (cid:0) ∂ ˜ N log f K (cid:1) − ∂ ˜ N log f K i ( (0) ˜ N ) D ˜ i (0) ˜ nD ˜ j (0) ˜ n , (D20)5where we have used the leading order Einstein equation,2 ∂ ˜ N log K N = (cid:0) ∂ ˜ N φ N (cid:1) . (D21)By subtracting Eq. (D19) from Eq. (D18), we find with the aide of Eq. (D20), ∂ ˜ N A ij (cid:12)(cid:12)(cid:12) N c = 3 A ij (cid:12)(cid:12)(cid:12) N c + 3 1 + ∂ ˜ N log f K ( (0) ˜ N ) a ( (0) ˜ N ) e C f K ( (0) ˜ N ) h D ˜ i D ˜ j (0) ˜ n − (cid:16) D ˜ i (0) ˜ nD ˜ j C + D ˜ j (0) ˜ nD ˜ i C (cid:17)i T F + 3 a ( (0) ˜ N ) e C f K ( (0) ˜ N ) h ∂ N log f K − (cid:0) ∂ N log f K (cid:1) i ( (0) ˜ N ) (cid:16) D ˜ i (0) ˜ nD ˜ j (0) ˜ n (cid:17) T F , (D22)where Q (cid:12)(cid:12) N c = Q N − Q c .On the other hand, taking the time derivative of Eq. (D17), which we have obtained by the gauge transformation,we obtain ∂ ˜ N A ij (cid:12)(cid:12)(cid:12) N c = 3 A ij (cid:12)(cid:12)(cid:12) N c + 3 1 + ∂ ˜ N log f K ( (0) ˜ N ) a ( (0) ˜ N ) e C f K ( (0) ˜ N ) h D ˜ i D ˜ j (0) ˜ n − (cid:16) D ˜ i (0) ˜ nD ˜ j C + D ˜ j (0) ˜ nD ˜ i C (cid:17)i T F + 3 a ( (0) ˜ N ) e C f K ( (0) ˜ N ) h ∂ N log f K − (cid:0) ∂ N log f K (cid:1) i ( (0) ˜ N ) (cid:16) D ˜ i (0) ˜ nD ˜ j (0) ˜ n (cid:17) T F . (D23)Comparing Eqs. (D22) and (D23), we see the precise coincidence between the two.
3. Consistency check for K In the N gauge, the evolution equation for K is ∂ ˜ N K N = 3 H ( ˜ N )2 α N (cid:0) ∂ ˜ N φ N (cid:1) − a ( ˜ N ) e ψ N K N h R − (cid:16) D ψ N + 2 D ˜ i ψ N D ˜ i ψ N (cid:17)i − a ( ˜ N ) e ψ N K N (cid:20) K N D (cid:18) K N (cid:19) − D ˜ i log K N D ˜ i ψ N − D ˜ i φ N D ˜ i φ N (cid:21) , (D24)and that in the comoving gauge is ∂ N K c = 3 H ( N )2 α c (cid:0) ∂ N f φ (cid:1) ( N ) − a ( N ) e ψ c f K ( N ) h R − (cid:16) D C + 2 D i C D i C (cid:17)i . (D25)Again we rewrite the second line inside the brackets of Eq. (D24) as(2nd line) = h(cid:0) ∂ ˜ N log f K (cid:1) − ∂ N log f K i ( (0) ˜ N ) D ˜ i (0) ˜ nD ˜ i (0) ˜ n − ∂ ˜ N log f K ( (0) ˜ N ) (cid:16) D ( (0) ˜ n ) + D ˜ i (0) ˜ nD ˜ i C (cid:17) , (D26)where we have used Eq. (D21).By subtracting Eq. (D25) from Eq. (D24), we find with the aide of Eqs. (D15) and (D26), ∂ ˜ N K (cid:12)(cid:12)(cid:12) N c = ∂ ˜ N (cid:16) (2) ˜ n∂ N f K ( ˜ N ) (cid:17) + 3 ∂ ˜ N log f K ( (0) ˜ N ) − a ( (0) ˜ N ) e C f K ( (0) ˜ N ) (cid:16) D ( (0) ˜ n ) + D ˜ i (0) ˜ nD ˜ i C (cid:17) + 3 D ˜ i (0) ˜ nD ˜ i (0) ˜ na ( (0) ˜ N ) e C f K ( (0) ˜ N ) (cid:20) ∂ N log f K ( (0) ˜ N ) + ∂ N log f K ( (0) ˜ N ) − (cid:0) ∂ ˜ N log f K (cid:1) ( (0) ˜ N ) (cid:21) . (D27)On the other hand, taking the time derivative of Eq. (D16), which has been obtained by the gauge transformation,we obtain ∂ ˜ N K (cid:12)(cid:12)(cid:12) N c = ∂ ˜ N (cid:16) (2) ˜ n∂ N f K ( ˜ N ) (cid:17) + 3 ∂ ˜ N log f K ( (0) ˜ N ) − a ( (0) ˜ N ) e C f K ( (0) ˜ N ) (cid:16) D ( (0) ˜ n ) + D ˜ i (0) ˜ nD ˜ i C (cid:17) + 3 D ˜ i (0) ˜ nD ˜ i (0) ˜ na ( (0) ˜ N ) e C f K ( (0) ˜ N ) (cid:20) ∂ N log f K ( (0) ˜ N ) + ∂ N log f K ( (0) ˜ N ) − (cid:0) ∂ ˜ N log f K (cid:1) ( (0) ˜ N ) (cid:21) . (D28)We see that Eqs. (D27) and (D28) coincide precisely with each other.6 Appendix E: Note on constraints
In general relativity, one cannot freely choose the initial values of the metric and matter field variables. They mustbe chosen so as to satisfy the Hamiltonian and momentum constraints. In this Appendix we make some commentson these constraint equations in the context of the spatial gradient gradient expansion.Let us consider M scalar fields minimally coupled with Einstein gravity and count the number of degrees of freedomin this system. In the ADM (Hamiltonian) formalism, the dynamical geometrical degrees of freedom are six spatialmetric components ( ψ, γ ij ) and their time derivatives ( K, A ij ). In addition, we have 2 M dynamical degrees of freedomassociated with the scalar fields and their derivatives. To summarise, C ψ : 1 , C γij : 5 , C K : 1 , C Aij : 5 , C φI : M , D φI : M , (E1)that is, we have 12 + 2 M (= 6 + 6 + M + M ) degrees of freedom. The Hamiltonian and momentum constraintequations give four constraints among them. And there are also four gauge degrees of freedom. Thus there are4 + 2 M (= 12 + 2 M − (4 + 4)) independent physical degrees of freedom left, which represent 2 × M × N gauge, where we take the uniform N slicing andthe time-slice-orthogonal threading. The 2 M scalar field degrees are represented by C φI and D φI . The gravitationalwave degrees of freedom are contained in among 5 degrees of freedom in C γij and C Aij , respectively, 2 of each describethe gravitational degrees of freedom. The remaining 3 degrees of freedom in C γij are fixed by the threading condition β i = 0 together with purely spatial coordinate transformation degrees of freedom (see e.g., Eq. (4.43) in the example ofa canonical scalar field), and those 3 in C Aij are fixed by the momentum constraint (4.47). The Hamiltonian constraintis used to determine C K .Now we are left with C ψ . This may be regarded as the remaining gauge degree of freedom in the N slicing, sincethe slicing condition implies ∂ t ψ = 0. But once the scalar field configuration on the initial slice is fixed, it shouldnot be freely specifiable, since if it were it would contradict with the total number of the physical degrees of freedomcounted above for the general case. In subsection E 1, we find there is indeed a constraint equation that must besatisfied by C ψ . Thus we recover the total number of the true physical degrees of freedom correctly.Recently the validity of the δN formalism was studied by Sugiyama et al. [30], and they found the violation ofthe momentum constraint in a non-slow-roll inflation case. In subsection E 2, we show how the consistency with themomentum constraint is recovered in the gradient expansion.
1. “Hidden” Hamiltonian constraint
In general the Hamiltonian constraint gives a non-trivial relation among the initial values of the metric components,their time derivatives and the matter field variables. However, in Sec. IV, we have used the Hamiltonian constraintto solve for K . Thus it seems it will not give any more constraint. Then where is a constraint corresponding to theHamiltonian constraint? The answer is that it is hided in the evolution equation for K .In order to see it explicitly, let us consider the canonical scalar field model discussed in Sec. IV. We first solve theevolution equation for K , Eq. (3.5), and then compare its solution with the solution obtained form the Hamiltonianconstraint.Let us consider the evolution equation for (2) K in the N gauge. Using the leading order solution and (2) φ given byEq. (4.10), we find ∂ N (cid:18) (2) K √ (0) V (cid:19) = m I D φI e N − N ) + S φI a e C ψ Z dN ′ a (0) V ! − a e C ψ (0) V h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) − D i C φI D i C φI i . (E2)This is easily integrated to give (2) K tmp ( N ) √ (0) V = (2) K tmp ( N ) p (0) V + 13 m I D φI h e N − N ) − i + m I S φI I φ ( N ) a e C ψ (0) V − R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) − D i C φI D i C φI a e C ψ (0) V (cid:18) a V Z dN ′ a V (cid:19) , (E3)7where the suffix tmp indicates that it is obtained from the temporal evolution equation.Now let us carefully examine the solution (E3) in comparison with that obtained in the text, Eq. (4.29). First wecompare the initial values. This determines the initial value of (2) K tmp as (2) K tmp ( N ) p (0) V = 16 m I D φI − a e C ψ (0) V h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) + D i C φI D i C φI i . (E4)Next we subtract the solution (E3) together with thus obtained initial data (E4) from the solution (4.29), (2) K − (2) K tmp √ (0) V = m I S φI a e C ψ (0) V (0) Va Z NN dN ′ a (0) V − I φ ( N ) ! + 14 e C ψ (cid:18) a V − a V (cid:19) h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) + D i C φI D i C φI i + R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17) − D i C φI D i C φI a e C ψ (0) V (cid:18) a V Z dN ′ a V (cid:19) . (E5)After integration parts and some manipulations, we obtain (2) K − (2) K tmp √ (0) V = 12 a e C ψ (0) V (cid:26) m I e C ψ D i (cid:16) e C ψ D i C φI (cid:17) − M + 66 D i C φI D i C φI + M h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17)i(cid:27) (cid:18) a V Z dN ′ a V (cid:19) . (E6)The right-hand side of the above equation must vanish at all times. This demands that the term inside the squarebrackets should vanish, m I e C ψ D i (cid:16) e C ψ D i C φI (cid:17) − M + 66 D i C φI D i C φI + M h R − (cid:16) D C ψ + 2 D i C ψ D i C ψ (cid:17)i = 0 . (E7)Apparently this gives a non-trivial constraint among initial data, C ψ and C φI . This is the “hidden” constraintcorresponding to the Hamiltonian constraint.
2. Consistency with the momentum constraint
Below we show that the momentum constraint is automatically satisfied if the Hamiltonian constraint as well asthe scalar field equation is satisfied. In the case of slow-roll inflation, the leading order Hamiltonian constraint issufficient to show it. For general (non-slow-roll) inflation, one needs the next-leading order Hamiltonian constraint. a. Slow-roll case
Under the slow-roll approximation, the Hamiltonian and momentum constraints reduce respectively to23 K = − P (cid:0) X IJ ( φ K ) , φ L (cid:1) , (E8)23 ∂ i K = 2 K P ( IJ ) ∂ N φ I ∂ i φ J , (E9)and the scalar field equation becomes − K P ( IJ ) ∂ N φ J − P I = 0 . (E10)Now we multiply both sides of the momentum constraint (E9) by 2 K , and substitute the scalar field equation (E10)into the right-hand side of it. This gives(left − handside) = 43 K∂ i K = ∂ i (cid:18) K (cid:19) , (E11)(right − handside) = − P I ∂ i φ I = ∂ i (cid:0) − P (cid:1) . (E12)8One can easily see that these are merely a spatial derivative of the Hamiltonian constraint (E8). Thus we concludethat, at leading order in gradient expansion, the momentum constraint is automatically satisfied if the Hamiltonianconstraint and the scalar field equations are satisfied. b. Non slow-roll case Here we show that in general the leading order momentum constraint is automatically satisfied once the Hamiltonianconstraint and the energy conservation equations are satisfied to next-leading order in gradient expansion.From Eqs. (2.13) and (2.14) with A ij = O ( ǫ ), the Hamiltonian and momentum constraint equations are1 a e ψ h R − (cid:16) D ψ + 2 D i ψD i ψ (cid:17)i + 23 K = 2 E + O ( ǫ ) , (E13)23 ∂ i K = J i + O ( ǫ ) . (E14)The energy conservation law, n ν ∇ ν T µν = 0, in the N gauge is ∂ N E = 3 (cid:0) E + P (cid:1) + 3 Ka e ψ D i (cid:18) e ψ K J i (cid:19) + 2 a e ψ P ( IJ ) D i φ I ∂ j φ J . (E15)Now let us multiply the evolution equation for K , Eq. (3.5), by 2 K/ ∂ N (cid:18) K − E (cid:19) = − a e ψ h R − (cid:16) D ψ + 2 D i ψD i ψ (cid:17)i − Ka e ψ (cid:20) D (cid:18) K (cid:19) + D i (cid:18) K (cid:19) D i ψ (cid:21) − Ka e ψ D i (cid:18) e ψ K J i (cid:19) . (E16)Substituting the Hamiltonian constraint (E13) into the left-hand side of the above equation, we find ∂ N (cid:18) K − E (cid:19) = − ∂ N (cid:26) a e ψ h R − (cid:16) D ψ + 2 D i ψD i ψ (cid:17)i(cid:27) = − a e ψ h R − (cid:16) D ψ + 2 D i ψD i ψ (cid:17)i . (E17)Comparing the above two equations, we obtain0 = − e ψ (cid:20) D (cid:18) K (cid:19) + D i (cid:18) K (cid:19) D i ψ (cid:21) − D i (cid:18) e ψ K J i (cid:19) = D i (cid:20) e ψ K (cid:18) ∂ i K − J i (cid:19)(cid:21) . (E18)Again one can see that the leading order momentum constraint (E14) holds automatically if the next-to-leading orderHamiltonian constraint (E13) holds.What does this mean? In the gradient expansion, we have assumed A ij does not contribute to the leading orderdynamics. This corresponds to neglecting the adiabatic decaying mode in linear theory. In a single field model, it iswell known this decaying mode appears only at the next-leading order both in linear theory [31] and in non-lineartheory [28, 29]. We also mention the work [41] in which they discussed the behavior of decaying modes in differentchoices of gauge. The absence of the decaying mode at leading order in gradient expansion should hold also in thecase of multi-field inflation, at least as long as the background homogeneous solution is stable against a homogeneousbut anisotropic perturbation.To summarise, the point is that the initial data for the scalar field and its time derivatives are not freely specifiablein general, and in particular we need to take into account the next-leading order terms in the Hamiltonian constraintfor the non-slow-roll case. Once we take into account the next-leading order Hamiltonian constraint, the leading ordermomentum constraint is automatically satisfied. As for the next-to-leading order momentum constraint, it constrainsthe initial value of A ij as in Eq. (4.47). [1] WMAP, E. Komatsu et al. , Astrophys. J. Suppl. , 18 (2011), arXiv:1001.4538. [2] [Planck Collaboration], arXiv:astro-ph/0604069.[3] M. Sasaki and D. Wands, Classical and Quantum Gravity , 120301 (2010).[4] E. M. Lifshitz and I. M. Khalatnikov, Adv. Phys. , 185 (1963).[5] V. a. Belinsky, I. m. Khalatnikov, and E. m. Lifshitz, Adv. Phys. , 639 (1982).[6] A. A. Starobinsky, JETP Lett. , 152 (1985).[7] J. M. Bardeen, Phys. Rev. D22 , 1882 (1980).[8] D. S. Salopek and J. R. Bond, Phys. Rev.
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