Strong restriction on inflationary vacua from the local gauge invariance II: Infrared regularity and absence of the secular growth in Euclidean vacuum

We investigate the initial state of the inflationary universe. In our recent publications, we showed that requesting the gauge invariance in the local observable universe to the initial state guarantees the infrared (IR) regularity of loop corrections in a general single clock inflation. Following this study, in this paper, we show that choosing the Euclidean vacuum ensures the gauge invariance in the local universe and hence the IR regularity of loop corrections. It has been suggested that loop corrections to inflationary perturbations may yield the secular growth, which can lead to the break down of the perturbative analysis in an extremely long term inflation. The absence of the secular growth has been claimed by picking up only the IR contributions, which we think is incomplete because the non-IR modes which are comparable to or smaller than the Hubble scale potentially can contribute to the secular growth. We prove the absence of the secular growth without neglecting these non-IR modes to a certain order in the perturbative expansion. We also discuss how the regularity of the n-point functions for the genuinely gauge invariant variable constrains the initial states of the inflationary universe. These results apply in a fully general single field model of inflation.

and hence the evolution of ζ is described by the above non-local action. Here the inverse Laplacian ∂ −2 is usually supposed to be defined as multiplying the inverse of the eigenvalue of the Laplacian operator by using the harmonic decomposition. When we evaluate the loop corrections to the n-point functions expanding them in terms of the interaction picture field ζ I , we need to evaluate the expectation values such as Inserting the scale invariant spectrum into ζ 2 I leads to the logarithmic divergence as ζ 2 I ∝ d 3 k/k 3 . The second expression of Eq. (1.3), which may arise as a consequence of the operation of ∂ −2 , is more singular as ζ I ∂ −2 ζ I ∝ d 3 k/k 5 , which diverges quadratically. The presence of non-local interactions enhances the long range correlations, and hence the singular behaviour in the IR. When we introduce the IR cutoff, say at the Hubble scale at a particular time t 0 , the variance ζ 2 I shows the logarithmic secular growth as ζ 2 I ∝ aH a0H0 dk/k ∼ log a/a 0 where a 0 and H 0 , respectively, denote the scale factor and the Hubble scale at t = t 0 . If the IR divergence exists, the loop corrections, which are suppressed by an extra power of the amplitude of the power spectrum (H/M pl ) 2 , may dominate in case inflation continues sufficiently long, leading to the break down of perturbation.
The dilatation symmetry as a necessary ingredient for IR regularity. The regularization of the IR contributions has been discussed in a number of publications [25][26][27][28][29][30][31][32][33][34][35][36][37][38]. The important aspect in discussing the long wavelength mode of ζ is the dilatation symmetry of the system. As is expected from the fact that the spatial metric is given in the form a 2 e 2ζ dx 2 , a constant shift of the dynamical variable ζ can be absorbed by the overall rescaling of the spatial coordinates. Hence, the action for ζ preserves the dilatation symmetry: where s is a constant parameter. (There are a number of literatures where this dilatation symmetry is addressed. See for instance, Refs. [39,40] and the references therein.) One may naively expect that we can absorb the IR divergent contribution of ζ using this constant shift. As an example, we set the parameter s to the spatial average of the curvature perturbation within the Hubble patch at t 0 ,ζ(t 0 ), where the size of the Hubble patch in comoving coordinates is given by 1/(a 0 H 0 ). Then, the logarithmically divergent two-point function ζ 2 I seems to be replaced with (ζ I −ζ I ) 2 ∝ aH a0H0 dk/k , which is finite but still grows logarithmically in time. One may think that if the system is described in such a way that the symmetry under the time dependent dilatation transformation is manifest, setting s(t) to the time dependent spatial average in the Hubble patch, the logarithmic growth ofζ(t) might be eliminated. However, the reduced action written in terms of ζ (1.2) does not preserve the invariance under the dilatation transformation with the time dependent parameter s(t). For example, in the recent literature [40], the authors showed that when we consider the whole universe with the infinite spatial volume, the dilatation transformation should be time independent to preserve the action invariant. In addition, the two-point function with ∂ −2 cannot be regularized by considering the dilatation symmetry alone. This quick consideration tells us that the presence of the dilatation symmetry of the system may play an important role in the regularization of the IR contributions but is not sufficient to guarantee the IR regularity and the absence of the secular growth.
Residual gauge degrees of freedom in the local universe. A missing piece in the above discussion is to pay careful attention to what are the quantities we can actually observe. Since our observable region is a limited portion of the whole universe, the observable fluctuations must be composed of local quantities. Furthermore, as the information that we can access is limited to our observable region, there is no reason to request the regularity at the spatial infinity in solving the elliptic constraint equations (1.1). Then, there arise degrees of freedom in choosing the boundary conditions of Eqs. (1.1). The degrees of freedom in solutions of N and N i can be understood as the degrees of freedom in choosing coordinates. As we showed in Refs. [25,26], these residual coordinate transformations are expressed in terms of homogeneous solutions to the Laplace equation as where s i j1···jm (t) are symmetric traceless tensors, which satisfy δ jj ′ s i j1···j···j ′ ···jm (t) = 0. Here, we abbreviated the nonlinear terms in the coordinate transformation. Note that this coordinate transformations include the dilatation transformation with the time dependent function s(t). Since the transformations in Eq. (1.5) are nothing but coordinate transformations, the diffeomorphic invariant action S = d 4 xL[ζ, N, N i ] should preserve the symmetry under these transformations. Thus, when we consider only the local observable region, which is a potion of the whole universe, we find an infinite number of coordinate transformations which keep the action invariant. Considering the dilatation transformation in the whole universe is subtle in the sense that the transformation diverges at the spatial infinity, even if the parameter s is very small. By contrast, restricted to the local region, the magnitude of the coordinate transformations in Eq. (1.5) is kept perturbatively small. In this paper, we refer to the local observable (spacetime) region as O. The size of the observable region on each time slicing is supposed to be of order 1/a(t)H(t) at least in the far past since the past light cone asymptotes to that size. We should note that, once we insert the expressions of N and N i into the action to obtain the action for the curvature perturbation ζ, the symmetry under the residual coordinates transformation is lost, because specific boundary conditions are chosen for N and N i in fixing coordinates.
To emphasize the distinction between the coordinate transformations associated with the change of the boundary conditions and the usual gauge transformation, which keeps the action invariant, we denote the former by the gauge transformation in the italic font.
Removing the residual gauge degrees of freedom. One way to realize the invariance under the gauge transformation is fixing the gauge conditions completely. The residual gauge degrees of freedom introduced above can be also removed by employing additional gauge conditions, i.e., by fixing the boundary conditions of N and N i at the boundary of the local region O. Then, we naturally expect that the IR regularity may be explicitly shown by performing the quantization in this local region, since the wavelengths that fit within this local region O will be bounded by the size of the region. Although the quantization in the local region is an interesting approach, it is not so clear how to perform the quantization after removing the residual gauge degrees of freedom. One of the difficulties is that even the translation symmetry of the quantum state cannot be easily guaranteed in the local system, since it is manifestly broken by introducing the boundary condition at a finite distance. (See also the discussion in Ref. [27]).
As an alternative way, in Ref. [28], we first set the initial state considering the whole universe, and then we performed the residual gauge transformation (1.5) to fix the coordinates so that the IR contributions are absorbed. Through the transformation with s(t) =ζ(t), the curvature perturbation is transmitted as (1.6) Here, ζ(x) is the original curvature perturbation defined in the whole universe and its spatial average over the whole universe is set to 0 as in the conventional cosmological perturbation theory. By contrast, ζ(t, e −ζ(t) x) −ζ(t) is the curvature perturbation relevant to the local universe, and its spatial average over the local region Σ t ∩ O is set to 0, where Σ t is a time constant surface. In Ref. [28] we considered the fluctuation of the inflaton, using the flat gauge, but the same discussion follows also for the curvature perturbation ζ. In the recent publication by Senatore and Zaldarriaga [38], the same degrees of freedom in choosing coordinates are used in a slightly different way to absorb the IR divergent contributions. If the non-linear terms in the residual gauge transformation at the initial time (1.6) did not yield IR divergent contributions, the discussion in Ref. [28] would have proved the absence of IR divergence in general. What was shown there is that once the field operator after the residual gauge transformation is guaranteed to be regular at the initial time, its succeeding evolution does not produce IR divergence. The heart of the proof is that ζ I (x) is replaced with ζ I (x) −ζ I (t) in the expansion of the composite operators in terms of the interaction picture field, after the residual gauge transformation, and hence the IR contributions from ζ I (x) are always canceled by those fromζ I (t). However, the non-linear part of the transformation at the initial time containsζ(t) whose IR contributions logarithmically diverge. The lesson is that it is not straightforward to reformulate the way of quantization so that the IR divergent contributions therein are all absorbed by the residual gauge transformation. (The absorption of the IR modes of the curvature perturbation was intended in other frameworks such as δN formalism [31,32] and the semi-classical approach [33]. ) The secular growth. The appearance of IR divergence due to the residual gauge transformation mentioned above might be evaded by sending the initial time to the past infinity. This is because the size of the local region Σ t ∩ O in comoving coordinates becomes infinitely large in this limit, making the discrepancy between the average in the local region and that in the global universe smaller and smaller. Then it might be effectively unnecessary to perform the residual gauge transformation at the initial time, although this statement is not very rigorous. We should note that when we send the initial time to the past infinity, it is too naive to neglect the non-IR modes which are comparable to or shorter than the Hubble length scale, because all the modes were much shorter than the Hubble length scale in the distant past. This makes the issue regarding the secular growth much more complicated. For instance, once we include the contributions from the non-IR modes, we cannot use the conservation of ζ k in the limit k/aH ≪ 1, where k is the comoving wavenumber of the external leg, relying on the long wavelength approximation such as δN formalism. Here, in a simple example, we show that vertex integrations can yield the apparent secular growth through the non-linear contributions from the modes at around the Hubble scale. Even if the vertex is confined in the region O, the integration region of each vertex is still infinite in the time direction as dtd 3 xa 3 (· · · ) ≃ d(ln a)/H 4 (· · · ), which may cause the secular growth. Roughly speaking, the integrand (· · · ) will be written in terms of the dimensionless time dependent slow roll parameters and the wavenumber of the fields in this vertex k m /aH normalized by the Hubble scale. If we focus on the non-linear interaction composed of the modes with k m /aH of order unity, the integrand (· · · ) are expressed only in terms of the parameters which are supposed to change very slowly in time and then the contribution from the interaction vertex seems to yield the logarithmic growth. This is another origin of the secular growth, which should be distinguished from the one inherited from the IR behavior of (ζ I ) 2 . Of course the above argument is too native, but it shows that the absence of the secular growth from the vertex integration is rather subtle, requiring more careful treatment about the modes around the Hubble scale. Because of this subtlety, introducing the UV cutoff at the length scale longer or equal to the Hubble length scale by hand makes the discussion incomplete. In fact, if it were allowed to simply neglect the short wavelength modes, the discussion in Ref. [28] with the initial time t i sent to −∞ would have given a rough proof of the absence of IR divergence without any limitation to the quantum state by sending the initial time to the past infinity, which contradicts our current claim that the quantum state is restricted in order to avoid IR divergence. Recently, the absence of the secular growth was claimed relying on the conservation of the curvature perturbation in Refs. [37,38], but the aspects mentioned above were not discussed. In addition, even if the conservation of ζ k in the limit k/aH ≪ 1 is proved, the logarithmic enhancement in the form (k/aH) 2 ln(k/a i H i ) may give rise, where a i and H i are the scale factor and the Hubble parameter at the initial time. The factor ln(k/a i H i ) can become large to overcome the suppression by (k/aH) when we send the initial time to the past infinity.

B. Summary of upcoming results
Short summary of the results. In this subsection, we summarize what we will show in this paper. Taking account of the current status of IR issues mentioned above, we will establish the following three statements in this paper: 1. There is an alternative equivalent Hamiltonian that describes the quantum dynamics of our interest and whose interaction part is solely composed of the IR irrelevant operators(, which mean the field operators associated with the operations that manifestly suppress the IR contribution such as ∂ i /aH and ∂ t /H).
2. The Euclidean vacuum state, which is specified by the regularity when the time coordinates in the n-point functions are analytically continued to the imaginary in the complex plane, is physically the same both in the alternative description mentioned in item 1, and in the original description.
3. The n-point functions in the Euclidean vacuum state respect the spatial translation invariance and are regular in the IR. The secular growth is absent, even if we include the vertices with non-IR modes, as long as very high order of loop corrections are not concerned.
Below we add a little more detailed explanations about the above three items. Gauge issue. In this paper, the quantization and fixing the initial quantum state as a starting point of our discussion is performed in the original system which describes the whole universe, where the residual gauge degrees of freedom are left unfixed. Then, following Refs. [25-27, 30, 41], we introduce a field operator which preserves the invariance under any spatial coordinates transformations, including residual gauge transformations. We refer to such an operator as a genuine gauge invariant operator. As a representative, we consider a genuine gauge invariant curvature perturbation, g R. As long as the expectation values of such genuine gauge invariant operators are concerned, we can perform the residual gauge transformation without affecting the results of computations. We will show that, using this residual gauge transformation, the boundary conditions of the non-local operator ∂ −2 in the action can be modified to be regular in the IR.
Requirement of the gauge invariance in quantum state. To calculate the n-point functions which preserve the invariance under the residual gauge transformations, the initial state should be also specified in a genuinely gauge invariant manner. However, when we perform the quantization considering the whole universe, preserving the residual gauge invariance becomes obscure, because these residual gauge degrees of freedom are not present as long as we deal with the whole universe. In our previous paper [27], we discovered a correspondence between the IR regularity and the invariance under the residual gauge transformations, which will provide an important clue to the guiding principle in choosing the genuinely gauge invariant initial state. To discuss this point, aside from the original canonical variables ζ(x) and its conjugate momentum π(x), whose evolution is governed by the action (1.2), we introduced another set of the canonical variables corresponding to the description in the coordinates shifted by a constant dilatation transformation: where s is a time independent c-number andπ(x) is the conjugate momentum ofζ(x). In Ref. [27], we showed that requesting the equivalence between the two quantum systems described by {ζ, π} and {ζ,π} guarantees the IR regularity of loop corrections. Here, the equivalence of two quantum systems means that the same iteration scheme (or formally the same initial condition of the interacting system) gives physically the same quantum state in both systems related to each other by the dilatation transformation. Namely, all the expectation values evaluated in both systems are equivalent if we take into account how they transform under dilatation transformation. Requesting this equivalence will be thought of as the invariance of the initial state under the dilatation transformation. In Ref. [27], we employed the iteration scheme in which the interaction is turned on at a finite past. Then, it turned out that the IR regularity/gauge invariance condition cannot be consistently imposed. In the present paper we will set the initial quantum state at the infinite past. We will show that the above transformation can be extended to allow a time dependence of the parameter s. As we described in the previous section, this extension plays a crucial role in discussing the absence of the secular growth. The Euclidean vacuum. The second and third items are related with each other, once we establish the correspondence between the gauge invariance and the IR regularity. We will show that the two quantum systems described by {ζ, π} and {ζ,π} are equivalent if we choose the Euclidean vacuum, which is defined by requesting the regularity of the n-point functions at the distant past with the time path rotated toward the complex plane. To be more specific, as the second item, we will show that the n-point functions for ζ(x) calculated by the canonical variables {ζ, π} with the boundary condition of the Euclidean vacuum agrees with the n-point functions forζ(t, e s(t) x) calculated by the canonical variables {ζ,π} under formally the same boundary condition, i.e., ζ(t, x 1 )ζ(t, x 2 ) · · · ζ(t, x n ) {ζ,π} = ζ (t, e s(t) x 1 )ζ(t, e s(t) x 2 ) · · ·ζ(t, e s(t) x n ) {ζ,π} . (1.8) Combined with the previously mentioned technique to deal with the gauge issue, we will show that when we choose the Euclidean vacuum, the Hamiltonian density for {ζ,π}, can be expressed only in terms of the IR irrelevant operators. The IR regularity and the absence of the secular growth. As for the third item, we evaluate the n-point function of the genuinely gauge invariant operator. Performing the quantization in the canonical system of {ζ,π}, we will show that the IR contributions do not diverge and that the secular growth is suppressed. We carefully investigate the contributions from the modes which are comparable to or less than the Hubble scale, i.e., k aH, without employing the asymptotic expansion with respect to k/aH. As is stressed at the end of the preceding subsection, this point is one of the necessary ingredients to show the absence of the secular growth. One may naively expect that the UV modes with k/aH 1 will not effectively contribute to the vertex integration because of the oscillatory behaviour. A more careful consideration tells us that this naive expectation is not necessarily correct. In general, vertex integrations become a mixture of the positive and negative frequency mode functions, which yields the phase in the UV limit e iη(k1−k2+k3−··· ) where η represents the conformal time which runs from −∞ to 0. Then, the phase does not necessarily exhibit the rapid oscillation even for the modes with k m /aH ≃ −k m η 1, where m = 1, 2, · · · , which can be a cause of secular growth. Intriguingly, choosing the Euclidean vacuum plays a crucial role not only in the IR limit but also in the UV limit. One can show that there is no mixing between the positive and the negative frequency modes, if we choose the Euclidean vacuum. Therefore, secular growth is evaded in this case.
The outline of the paper. The outline of this paper is as follows. In Sec. II, we will briefly review the way to construct the genuinely gauge invariant operator g R, following Refs. [25,26]. Then, we will introduce the canonical variables {ζ,π} and will derive the Hamiltonian for these variables. In Sec. III, we will discuss the items 1 and 2 that we mentioned above. In Sec. III A, we will describe the boundary conditions of the Euclidean vacuum and will prove Eq. (1.8), which implies that the boundary conditions of the Euclidean vacuum select the same ground state both in {ζ, π} and {ζ,π}. In Sec. III B and Sec. III C, we will formulate the canonical quantization in terms of {ζ,π} and will show that the interacting vertices for these canonical variables consist only of the IR irrelevant operators. Particularly in Sec. III C, we will show that using the residual gauge degrees of freedom, the non-local operator ∂ −2 can be made IR regular. In Sec. IV, we will discuss the item 3. In Sec. IV A, we will show that the boundary condition of the Euclidean vacuum leads to the so-called iǫ prescription in a perturbative expansion. In Sec. IV B, we will calculate the Wightman propagator, by which the n-point functions are expanded. Then, in Sec. IV C, we explicitly evaluate n-point functions to investigate the IR regularity and the secular growth. In Sec. V, as concluding remarks, we discuss another possibility of the initial state which satisfies the IR regularity/gauge invariance conditions. We will also mention the related papers to clarify what is new in this paper.
The advantage of the in-in formalism. In our previous publications [27][28][29], in calculating n-point functions, we used the retarded Green function to solve the non-linear Heisenberg equation. This is because we thought that using the retarded Green function, whose Fourier mode is regular in the IR limit, makes the proof of the IR regularity transparent. However, the perturbative expansion using the retarded Green function is not suitable for the present purpose, because the positive and negative frequency modes are mixed in the vertex integrations once the retarded Green function is used. Therefore, the boundary conditions of the Euclidean vacuum does not guarantee the convergence of the time integrations for all the vertices. By contrast, when we calculate the n-point functions in the in-in formalism, all vertex integrals can be made manifestly convergent by adopting the boundary conditions of the Euclidean vacuum (see Sec. III A). Since the n-point functions obtained from the solution written in terms of the retarded Green function agree with those obtained in the in-in formalism, the vertices which do not converge should vanish in the final result of the n-point functions. However, the cancellation is obscured in an explicit perturbative expansion. Therefore, in this paper, we calculate the n-point function totally based on the in-in formalism, without using the retarded Green function.

II. CONSTRUCTING THE GAUGE INVARIANT QUANTITY
In this paper, as an explicit model of inflation, we consider a standard single field inflation model whose action takes the form where M pl is the Planck mass and we set φ to a dimensionless scalar field, dividing it by M pl . However, as long as we consider a scalar field with the second-order kinetic term, an extension proceeds in a straightforward way. In Sec. II A, we will construct the genuine gauge invariant operator corresponding to the spatial curvature of a φ-constant surface. In Sec. II B, we will introduce the canonical system {ζ,π} whose Hamiltonian density is composed only of the IR irrelevant operators.

A. Gauge invariant operator and quantization
We fix the time slicing by adopting the uniform field gauge δφ = 0. Under the ADM metric decomposition, which is given by we take the spatial metric h ij as where a := e ρ is the scale factor, ζ is the so-called curvature perturbation and δγ ij is a traceless tensor: As spatial gauge conditions we impose the transverse conditions on δγ ij : Since the time slicing is fixed by the gauge condition δφ = 0, there are remaining residual gauge degrees of freedom only in choosing the spatial coordinates. In this paper, we neglect the vector and tensor perturbations. The tensor perturbation, which is massless, can also contribute to the IR divergence of loop corrections. We will address this issue in our future publication. Following Refs. [25,26], we construct a genuine gauge invariant operator, which preserves the gauge invariance in the local observable universe. For the construction, we note that the scalar curvature s R, which transforms as a scalar quantity under spatial coordinate transformations, becomes genuinely gauge invariant, if we evaluate it in the geodesic normal coordinates on each time slice. The geodesic normal coordinates are introduced by solving the spatial three-dimensional geodesic equation: where s Γ i jk is the Christoffel symbol with respect to the three dimensional spatial metric on a constant time hypersurface and λ is the affine parameter. Here we put the index gl on the global coordinates, to reserve the simple notation x for the geodesic normal coordinates, which will be mainly used in this paper. We consider the three-dimensional geodesics whose affine parameter ranges from λ = 0 to 1 with the initial "velocity" given by A point x i in the geodesic normal coordinates is identified with the end point of the geodesic, x i gl (x, λ = 1) in the original coordinates. Using the geodesic normal coordinates x i , we perturbatively expand x i gl as x i gl = x i + δx i (x). Then, we can construct a genuinely gauge invariant variable as where t denotes the cosmological time.

B. Dilatation symmetry in the global universe
The focus of this subsection is on the dilatation transformation, shifting to the rescaled spatial coordinates: Solving the Hamiltonian and momentum constraint equations, we can derive the action that is expressed only in terms of the curvature perturbation ζ(x), which is schematically written as Using the curvature perturbation ζ and the conjugate momentum defined by π := δL/δ(∂ t ζ), the Hamiltonian density is given by the Legendre transform as What is important here is only the fact that the curvature perturbation ζ appear in the action either with differentiation or in the form of the combination of the physical distance e ρ+ζ dx [27]. In the new coordinates (2.9), the physical distance is written as e ρ+ζ(t,x)−s(t) dx, with the definition of a new variablẽ Thus, if the field ζ(x) is replaced withζ(t,x)−s(t) under the change of the coordinates from x tox, the action basically remains invariant. To express ∂ t ζ(x) in terms of the new variableζ, we denote the partial differentiation with the spatial coordinates x fixed as (∂ tζ (t,x)) x . The subscript associated with the parentheses specifies the spatial coordinates that we fix in taking the partial differentiation. Then, we have For brevity, when the fixed spatial coordinates are identical to the ones in the argument of the variable, we simply use ∂ t . Then, we can establish an identity (2.14) Recalling the relation between x andx (2.9), this equality also means the equality at the level of Lagrangian density, We introduce the canonical conjugate momentum corresponding toζ(t,x) in the standard way as Noticing the relation As is expected, using the commutation relations for ζ and π together with Eqs. (2.12) and (2.17), we can verify as well as The Hamiltonian density forζ(x) andπ(x) is obtained in the standard way as where in the equality on the second line we used Eq. (2.16). The last equality is exactly the same Legendre transformation as in the original system and therefore we can use the same functional form of the Hamiltonian density H.
Assuming that s(t) is as small asζ(x) andπ(x), we decompose the Hamiltonian densities H andH into the non-interacting parts, which include only the quadratic terms, and the interacting parts as In the above we used the coordinates x instead ofx for the {ζ,π} system, but it will not cause any confusion after the relations between the {ζ, π} and {ζ,π} systems have been established. Here, we replaced Remarkably, the non-interacting part of the Hamiltonian density does not change at all under the dilatation transformation. Using Eq. (2.20), we find that the interaction Hamiltoniañ H I [ζ,π] is given byH In this way, we can write downH I only in terms ofζ(x) − s(t),ζ with differentiation,π andṡ(t). In Ref. [27], we introduced the two sets of the canonical conjugate variables which are connected by the dilatation transformation with a constant parameter s. When we take the limit where s(t) is constant, the Hamiltonian densityH(x) takes the same functional form as H(x) except for the constant shift ofζ(x) by −s. It is because, without modifying the gauge condition, we can perform the dilatation transformation with the constant parameter s also in the whole universe. Then the action which preserves the diffeomorphic invariance becomes invariant under the change from ζ(x) to ζ(t, e −s x) − s. Here we have extended the argument in Ref. [27] to allow s to depend on time. As we mentioned in Sec. I, this extension plays the crucial role in our discussion about the secular growth. In the next section, we will show that all the interaction vertices in the canonical system {ζ,π} are composed only of the IR irrelevant operator.

III. INTERACTION HAMILTONIAN WITH THE IR IRRELEVANT OPERATORS
In this section, we describe the first two of the three items we raised in Sec. I. In the preceding section, we derived the Hamiltonian for the canonical variablesζ(x) andπ(x). Since {ζ, π} and {ζ,π} are connected by the canonical transformation, if we choose the same initial state in both of the two canonical systems, the n-point functions for the same operator, for instance g R, calculated in these canonical systems should agree with each other. However, even if we adopt operationally the same scheme to select the initial state in these two systems, it does not guarantee that the selected initial states are the same. In Sec. III A, after we describe the definition of the Euclidean vacuum, we will show that the condition of the Euclidean vacuum operationally selects the same quantum state irrespective of the choice of the canonical variables. This ensures the equivalence of these two canonical systems including the choice of the initial quantum state, which we mentioned in the item 2. In Sec. III B, we will perform the quantization using the canonical variables {ζ,π}. As we will show in Sec. III C, by virtue of the equivalence between the two canonical systems, the interaction vertices for {ζ,π} can be expressed in terms of operator products composed only of the IR irrelevant operators.

A. Euclidean vacuum and its uniqueness
In the case with a massive scalar field in de Sitter spacetime, the boundary condition specified by rotating the time path in the complex plane can be understood as requesting the regularity of correlation functions on the Euclidean sphere which can be obtained by the analytic continuation from the ones on de Sitter spacetime. The vacuum state thus defined is called Euclidean vacuum state. Because of the similarity, here we also refer to the state which is specified by a similar boundary condition as the Euclidean vacuum. To be more precise, we define the Euclidean vacuum as follows. In the in-in formalism, the insertion of interaction vertices is ordered along the closed time path. By rotating the time path toward the imaginary plane, the forward time evolution begins at η(t i ) = −∞(1 − iǫ) and ends at the final time t f and the backward time evolution begins at t f and ends at η(t i ) = −∞(1 + iǫ). Here we set ǫ to a small positive number. Since rotating the time path can be better understood by using the conformal time η, we introduced the conformal time η as .
We define the Euclidean vacuum, requesting the regularity of the n-point functions with an arbitrary natural number n in the limit of η(t i ) → −∞(1 ± iǫ), i.e., where a = 1, · · · n and T c denotes the time ordering along the closed time path. We first show that the n-point functions of ζ are uniquely fixed by requesting the condition (3.2). In this paper, for simplicity, we assume that e ρρ (ρ) is rapidly increasing in time so that Next, we show that the boundary condition of the Euclidean vacuum uniquely determines the n-point functions F n (x 1 , · · · x n ). We schematically describe the Heisenberg equation for ζ(x) as

4)
where L is the second-order differential operator: (3.5) For notational convenience, we introduced the horizon flow functions, with n ≥ 2, but we do not assume that these functions are small to keep the background evolution unconstrained except for requesting Eq. (3.3), which is valid, for instance, when ε n are constant in time. Using the Heisenberg equation (3.4), we can obtain the evolution equation of the path-ordered n-point functions F n (x 1 , · · · x n ) as where L xa is the derivative operator L given in Eq.
where f n (x 1 , · · · , x n ) is a homogeneous solution, while we assume that the specific solution satisfies the regularity condition in the limits η(t a ) → −∞(1±iǫ). Now the question is whether the boundary condition (3.2) allows us to add any homogeneous solutions. In the Fourier space, f n can be expanded by e −ikη(ta) or e ikη(ta) in the limits η(t a ) → −∞(1 ± iǫ). The regularity at η(t a ) → −∞(1 + iǫ) accepts e −ikη(ta ) only, while the regularity at η(t a ) → −∞(1 − iǫ) accepts the other. Thus the regularity condition in the two limits does not allow to add any homogeneous solutions f n , which implies that the n-point functions F n (x 1 , · · · , x n ) are uniquely fixed by the boundary condition of the Euclidean vacuum.
Next, we show that this uniqueness is ensured independent of whether we use the canonical variables {ζ, π} or {ζ,π}. We employ the boundary condition of the Euclidean vacuum for the canonical variableζ as well, requesting Then, we can show that the path-ordered n-point functions F n (x 1 , · · · x n ) := T cζ (t 1 , e s(t1) x 1 ) · · ·ζ(t n , e s(tn) x n ) {ζ,π} , (3.10) agree with the n-point functions F n (x 1 , · · · x n ) = T c ζ(x 1 ) · · · ζ(x n ) {ζ,π} fixed by the boundary condition (3.2), i.e., Here putting the suffixes {ζ, π} or {ζ,π}, we denote the canonical variables used in imposing the boundary condition explicitly. We again schematically describe the Heisenberg equation forζ as Since ζ(x) andζ(x) are connected by the canonical transformation, the equation of motion obtained by operating L on can be recast into Eq. (3.4) by using Eq. (3.12). A similar argument follows for the equations of motion for the correlation functions F n andF n . Using the equation of motion for the n-point functions ofζ(x), which can be derived from Eq. (3.12), we can confirm that an operation of L xa oñ F n (x 1 , · · · x n ) = T cζ (x 1 ) · · ·ζ(x n ) {ζ,π} + s(t 1 ) T c x 1 · ∂ x1ζ (t 1 , x 1 ) · · ·ζ(t n , x n ) {ζ,π} + · · · (3.14) leads to This equation takes the same form as the equation of motion (3.7). We also note that the boundary condition of the Euclidean vacuum (3.9) impliesF The equivalence (3.11) is now transparent, because the equations of motion (3.7) and (3.15), and the boundary conditions (3.2) and (3.16) are the same, and the latter specify the solutions of the former uniquely. This equivalence is a distinctive property of the Euclidean vacuum 1 . Here we took the boundary conditions for n-point functions as the definition of the Euclidean vacuum state, assuming the existence of such a quantum state. In Sec. IV A, we explain such a Euclidean vacuum, if exists, should be the one given by the ordinary iǫ prescription.

B. Rewriting the n-point functions
In this subsection, we rearrange the expression for the n-point functions of the genuinely gauge invariant variable g R into a more suitable form to examine the regularity of the IR contributions. First, solving the three dimensional geodesic equations, we obtain the relation between the global coordinates x i gl and the geodesic normal coordinates x i as where the ellipsis means the terms which vanish when ζ(x) is spatially homogeneous, i.e., the terms suppressed in the IR limit. Note that changing the spatial coordinates into the geodesic normal coordinates also modifies the UV contributions. Tsamis and Woodard [45] showed that using the geodesic normal coordinates can introduce an additional origin of UV divergence, which may not be able to be renormalized by local counter terms [46]. It should be clarified whether this issue is a serious problem or not, but we defer it to a future study. Instead, to keep the UV contributions under control, we replace ζ(x) in Eq. (3.17) with the smeared curvature perturbation gζ (t), i.e., where W Lt (x) is a window function which is non-vanishing only in the local region Σ t ∩ O. We approximate the averaging scale at each time t by the Hubble scale, i.e., L t ≃ 1/{e ρ(t)ρ (t)}. Although gζ appears on the right-hand side of Eq. (3.19), gζ is defined iteratively at each order of the perturbation. We calculate the n-point functions of R x g ζ(t, x), instead of g R, with g ζ(t, x) := ζ(t, e − gζ (t) x) . (3.20) Here, R x denotes the IR suppressing operator such as where x is the spacetime coordinates of the field on which these operators act. Although R x g ζ(t, x) is not genuinely gauge invariant, it is still invariant under the dilatation transformation, which is associated with the dominant IR contributions. In fact, since the smeared curvature perturbation gζ (t) transforms into gζ (t) − f under the dilatation transformation: x → e −f x with a constant f , R x g ζ(x) is kept invariant under this transformation. By contrast, the constant part of g ζ(x) can be modified under the dilatation transformation as g ζ(x) → g ζ(x) − f . Since the genuine gauge invariant variable g R(x) should not be affected by the dilatation transformation, which is a part of the residual gauge transformations, g ζ(x) appears only in the form of R x g ζ(x) when we express g R(x) in terms of g ζ(x). As we can compute g R(x) from R x g ζ(x), our goal is to prove that the expectation values of products of R x g ζ(x) are IR regular. First, we calculate the n-point functions of g ζ without the IR suppressing operator R x : Using the eigenstates of gζ (t f ), | s H which satisfy gζ (t f )| s H = s| s H , we can construct a unit operator Inserting it into the expression for the n-point functions, we obtain In the first line we could simply replace gζ (t f ) with s, because gζ (t f ) and ζ(t f , x) commute with each other. Since the Heisenberg picture fieldζ(t, x) is related to the interaction picture fieldζ I (t, x) as where the unitary operatorŨ I (t) is given bỹ Thus, the n-point function can be rewritten as whereT denotes the anti time-ordered product. Notice that the interaction HamiltonianH I does not contain the second or higher derivative of s(t). We construct unit operators, using the eigenstates |s(t) and |ṡ(t) which satisfy gζ I (t)|s(t) = s(t)|s(t) , is the smeared interaction picture field. We next replace all s(t) andṡ(t) with gζ I (t) and gζ I (t), respectively, by inserting the unit operators; To perform this replacement without ambiguity, we fix the operator ordering inH I to the Weyl ordering, in whichζ I (x)−s(t) and π I (x) are symmetrized. Instead of considering the explicit form of the interaction Hamiltonian, we use a schematic expression ofH I which is expanded in a power series ofṡ(t) as although α is at most 1. Here, we stress that the perturbationsζ I (x) and s(t) appear in the Hamiltonian densityH I only in the form ofζ I (x) − s(t) or its spatial differentiations. Inserting the unit operators, we obtaiñ

t) s(t) ṡ(t) ṡ(t) . (3.32)
After we replace s(t) with gζ I (t), gζ I (t) is located next to the operator |s(t) s(t)|. Noticing the fact that s(t)| s(t) can be expressed as where in the second equality, we replaced gζ I (t) in the argument of ζ I with s(t), we use gζ I (t) expressed as gζ instead of the expression given in Eq. (3.29), when we replace s(t) with gζ I (t). Using the formula we replace (ζ I (x) − s(t)) with (ζ I (x) − gζ I (t)) one by one. By induction, the operator A is supposed to be composed of ζ I (x) − s(t) andπ I (x). Since gζ I (t) commute withζ I (x) − s(t), the non-vanishing commutation relation is only the following: gζ where we used Since the commutator including gζ I (t) yields only a local function, we can conclude that operators left after exchanging s(t) with gζ I (t) are also composed ofζ I (x) − s(t) andπ I (x). Repeating this procedure, we can replace all s(t) with gζ I (t) as where to denote the modification after the replacement of s(t) with gζ I (t), we put ′ on the interaction Hamiltonian. Replacinġ s(t) with gζ (t), we obtaiñ We repeat this procedure for all integrating Hamiltonian densities which appear in the perturbative expansion of the n-point functions (3.27). After these replacements, the possible dependence of the n-point functions on s(t) andṡ(t) remains only in |s(t) s(t)| and |ṡ(t) ṡ(t)|. Since requesting the Euclidean vacuum uniquely determines the initial state independent of s(t) andṡ(t), we can remove the identity operators ds(t) |s(t) s(t)| and dṡ(t) |ṡ(t) ṡ(t)| as long as we choose the Euclidean vacuum. (From the same argument, we can remove the identity operator ds |s H H s|.) Then, the Hamiltonian density is recast intoH Note that we can express gζ I (t) as gζ , (3.41) where in the last equality, we inserted 0 = gζ I (t) ∂ t d 3 xW Lt (x)/ d 3 xW Lt (x) and the last term in the last line can be written in terms ofπ I (x).
In this way, we can show that allζ I s in the interaction vertices are multiplied by an IR suppressing operator R x . Notice that, replacing the c-number parameter s(t) with the operator gζ I (t), we rewrote the Hamiltonian density as in Eq. (3.40). In this procedure, we used the fact that the initial state specified by the boundary condition of the Euclidean vacuum does not depend on the choice of the canonical variables. We should emphasize that if this equivalence of the initial state were not guaranteed, we could not express the interaction Hamiltonian only in terms ofζ I s with an IR suppressing operator.

C. Restricting the interaction vertices to the local region
In the above discussion, we found that the interaction picture fields which appear in the interaction vertices can be expressed only in terms ofπ I (x) andζ I (x) − gζ I (t). Now, we can verify the item 1 presented in Sec. I, which claims that the interaction vertices are constructed only from the IR irrelevant operators. As we showed in the previous subsection, all the interaction picture fields are associated with an IR suppressing operator R x , which increases the power law index with respect to the wavenumber k in the IR limit. To complete the proof of the argument given in the item 1, we need to show that the inverse Laplacian ∂ −2 , which appears in solving the constraint equations to obtain the lapse function and the shift vector, does not reduce the power law index with respect to k in the IR limit. The potential danger can be understood as follows. When we choose the boundary condition specified by the regularity at the spatial infinity following the standard procedure, the action of the operator ∂ −2 yields a multiplicative factor 1/k 2 . This IR singular behavior arises because the information from the outside of our observable region is used to determine the lapse function and the shift vector.
To remove this potential IR singular behavior originating from the inverse Laplacian, we need to discuss the causality. The causality is basically maintained even at the quantum level in the sense that the interaction vertices located outside our observable region O are decoupled in the in-in formalism. In the ordinary field theory with a local interaction, this can be shown by systematically replacing the Wightman function G + with the retarded Green function plus G − (see Appendix of Ref. [27]). However, when the gravitational perturbation is taken into account, it becomes less transparent whether the causality is maintained owing to the issue of the lapse function and the shift vector mentioned above.
Here, we should recall that what we really need to evaluate is the expectation values of genuinely gauge invariant variables, which do not depend on the choice of the residual gauge degrees of freedom. As we explicitly showed in Appendix A, using the residual gauge degrees of freedom, we can modify the boundary conditions of the lapse function N and the shift vector N i so that the terms associated with ∂ −2 are completely specified by the fields within the local region O. Then, the operation of the non-local operator ∂ −2 no longer reduces the power law index with respect to k. In this way, using the degrees of freedom in the choice of boundary conditions, we can localize all the interaction vertices within the causally connected local region O. Since

IV. THE IR REGULARITY AND THE ABSENCE OF THE SECULAR GROWTH
In this section, we will calculate the n-point functions of R x g ζ(t f , x), properly taking into account not only the IR-modes but also the modes with k|η(t)| 1. As stressed in Sec. I, to prove the absence of the secular growth, we need to evaluate the contribution of the latter modes carefully. In the preceding section, we showed that, using the canonical variablesζ(x) andπ(x), we can expand the n-point functions of R x g ζ(x) for the Euclidean vacuum only in terms of the IR irrelevant operators. In this section, based on the perturbative expansion in the {ζ,π} system, we will discuss the IR regularity and the absence of secular growth in the n-point functions.
For our current discussion, the explicit form of the interaction Hamiltonian densityH I is not necessary. We use a formal expressionH where λ(t) is an O(1) dimensionless time dependent function which can be expressed only in terms of the horizon flow functions.
To discriminate different IR suppressing operators, we associate a superscript (m) on R x .

A. Euclidean vacuum as is obtained by iǫ prescription
In the preceding section, we introduced the Euclidean vacuum as a vacuum state which satisfies the boundary condition (3.2)/(3.9). Here we show that this condition forces us to adopt the ordinary perturbative description of the iǫ prescription. We expand the curvature perturbationζ I (x) asζ whereã k is the annihilation operator, which satisfies The mode function v k (t) should satisfy Since the boundary condition (3.2)/(3.9) should hold at the tree level, the asymptotic form of the positive frequency mode where we introduced as an approximate amplitude of the fluctuation. The function f k (t) satisfies the regular second order differential equation with the boundary condition Since both the differential equation and the boundary condition of f k (t) are analytic in k for any t, the resulting function should be analytic as well. Namely, f k (t) does not have any singularity such as a pole on the complex k-plane. We suppose that a positive frequency function for a general vacuum except for the Euclidean vacuum is given by a linear combination of v k and v * k with the Bogoliubov coefficients which have some nontrivial structure of singularities in the complex k-plane or diverge at infinity. The only exception to evade the singularity is setting the Bogoliubov coefficients to constants, but then the UV behavior does not agree with the one in the Minkowski vacuum.
On the other hand, in the limit −kη(t k ) ≪ 1, the function f k (t) is proportional to A(t k )/A(t), where t k is the Hubble crossing time defined by −kη(t k ) = 1, because the curvature perturbation should be constant in this limit. Hence, the expansion for small k is in general given by (4.8) By using Eq. (4.5), the Wightman function is given by Using the in-in formalism, the n-point functions can be expanded by the Wightman function. At this point, the vertex integrals should start with η = −∞ to be able to impose the boundary condition of the Euclidean vacuum (3.2)/(3.9). Although the integrands of the vertex integrals are infinitely oscillating in the limit η → −∞, the time integration can be made convergent by adding a small imaginary part to the time coordinate, which is nothing but the ordinary iǫ prescription. To see the convergence of the time integration more explicitly, we first consider the integral for the vertex which is closest to the past infinity η → −∞(1 − iǫ) (see Fig. 1). The interaction picture fieldsζ I (x) included in this vertex are contracted withζ I (x m ) contained in vertices labelled by m = 1, 2, · · · , n, and give the Wightman function G + (x m , x). Then, the vertex integration with n interaction picture fields is given by The Euclidean vacuum condition (3.2)/(3.9) requires the convergence of this integral when we send η(t i ) → −∞. Since the Wightman functions contain the exponential factor e iη(t) m km , the integral can be made convergent by changing the integration contour as shown in the left panel of Fig. 1, which is exactly what is known as the iǫ prescription. The vertex integration next to the closest to the past infinity can be done in a similar manner, where n ′ is the number of propagators connecting between this second vertex and the vertices other than the first one. If we assume the integration over the time coordinate of the first vertex t up to t ′ , the exponential factor in G + (x m , x) can be replaced as e ikm(η(t)−η(tm)) → e ikm(η(t ′ )−η(tm)) .

(4.12)
Therefore all the Wightman functions connecting the vertices at t ′ or before t ′ with the vertices after t ′ give an exponential factor which is suppressed by adding +iǫ to η. This is again consistent with the boundary condition of the Euclidean vacuum. The same argument can be made for the other vertices as well.

B. The IR/UV suppressed Wightman function
Since allζ I (x)s in the interaction Hamiltonian are multiplied by the IR suppressing operators R x , the n-point function of R x g ζ(x) can be expanded by the Wightman function R x R x ′ G + (x, x ′ ) and its complex conjugate R x R x ′ G − (x, x ′ ). In this subsection, we calculate the Wightman functions multiplied by the IR suppressing operator, R x R x ′ G + (x, x ′ ) for t > t ′ . After integration over the angular part of the momentum, the Wightman function R x R x ′ G + (x, x ′ ) can be expressed as where we introduced We first show the regularity of the k integration in Eq. (4.13). Since the function f k (t) is not singular, the regularity can be verified if the integration converges both in the IR and UV limits. The regularity in the IR limit is guaranteed by the presence of the IR suppressing operator. The IR suppressing operators R x add at least one extra factor of k|η(t)| or eliminate the leading t-independent term in the IR limit, and yield where we have introduced the spectral index n s − 1 := d log(|A(t k )| 2 )/d log k. Thus, the operation of R x makes the k integration in Eq. (4.13) regular in the IR limit. Next, we consider the convergence in the UV limit. In Eq. (4.13), the integration contour of k should be appropriately modified at k → ∞ so that the integral becomes convergent. This modification of the integration contour can be also understood as a part of the iǫ prescription, because adding a small imaginary part to all the time coordinates as η → η × (1 − iǫ) leads to the replacement η(t ′ ) − η(t) → η(t ′ ) − η(t) + iǫ, where we note η(t ′ ) − η(t) < 0, and hence to introducing an exponential suppression factor for large k. This UV regulator makes the integral finite for the large k contribution except for the case σ ± (x, x ′ ) = 0, where x and x ′ are mutually light-like. Since the expression of the Wightman function obtained after the k integration is independent of the value of ǫ, the regulator makes the UV contributions convergent even after ǫ is sent to zero. For σ ± (x, x ′ ) = 0, the integral becomes divergent in the limit ǫ → 0, but the divergence related to the behavior of the Wightman functions in this limit is to be interpreted as the ordinary UV divergences, whose contribution to the vertex integrals must be renormalized by introducing local counter terms. Thus, the Wightman function R x R x ′ G ± (x, x ′ ) is now shown to be a regular function.
Since the amplitude of the Wightman function with the IR suppressing operator is bounded from above, we can show the regularity of the n-point functions, if the non-vanishing support of the integrands of the vertex integrals is effectively restricted to a finite spacetime region. Since the causality has been established with the aid of the residual gauge degrees of freedom, the question to address is whether vertexes at the distant past is shut off or not. To address the presence of such a long-term correlation, we discuss the asymptotic behavior of the Wightman function R x R x ′ G ± (x, x ′ ), sending t ′ to a distant past. Recall that when σ ± (x, x ′ ) = 0, we can rotate the integration contour with respect to k even toward the direction parallel to the imaginary axis. Rotating the direction of the path appropriately depending on the sign of σ ± (x, x ′ ), the integrand becomes an exponentially decaying function of k. This rotation of the integration contour can be done without hitting any singularity in the complex k-plane, because the function f k (t) is guaranteed to be analytic by construction. If we choose other vacua, this operation induces extra contributions from singularities. Since we send t ′ to the past infinity, assuming |η(t ′ )| ≫ |η(t)|, σ ± (x, x ′ ) is O(|η(t ′ )|), except for the region where the two points are mutually light-like 2 . Then, the integration of k on the right-hand side of Eq. (4.13) is totally dominated by the wavenumbers with k 1/|η(t ′ )| ≪ 1/|η(t)|. Using Eq. (4.14) which gives the asymptotic expansion in the limit k|η(t)| ≪ 1, we obtain where on the second equality, we performed the k integration, rotating the integration contour. We should emphasize that we did not employ the long wavelength approximation regarding the Hubble scale at t ′ to properly evaluate the modes k of O(1/|η(t ′ )|) as well.

C. The secular growth
In this subsection, focusing on the long-term correlation, we discuss the convergence of the vertex integrals of the n-point functions for the Euclidean vacuum. We start with the integration of the n-point interaction vertex which is the closest to given in Eq. (4.15) into Eq. (4.10), the vertex integral V (1) can be estimated as (4.16) As we have explained in Sec. III C, the interaction vertices are confined within the observable region, i.e., the non-vanishing support of the integrand is bounded by |x| L t ≃ |η(t)|. Thus, we obtain (4.17) As we have performed momentum integral first, the exponential suppression for large |η| is not remaining any more. However, picking up η-dependence of the integrand of Eq. (4.17), we still find that the contribution from the distant past is suppressed if Therefore, when a Wightman propagator is connected to a vertex located in the future of x ′ , i.e., when t m > t ′ , the t-integration yields the suppression factor {η(t m )/η(t ′ )} ns +1 2 . We denote the number of such propagators byñ. Similarly, we can evaluate the amplitude of V (2) as Extracting the η ′ -dependent part in the above expression, we obtain Notice that all the Wightman propagators which are connected to the fieldζ I located in the future of x ′ yield the suppression factor |η(t ′′ )| − ns+1 2 . Now the generalization becomes easy. For the N v -th vertex, the temporal integration becomes (4.22) where N f denotes the number ofζ I s contained in the vertices up to the N v -th, M denotes the number of the Wightman propagators connected to a vertex with η > η Nv , andλ denotes the product of the interaction coefficient up to the N v -th vertex. Thus, the convergence condition is given by Since all interaction vertices have at least one Wightman propagator connected with their future vertices, M should satisfy M ≥ 1.
As a simple example, we consider the case where ε 1 is constant. In this case,λ is expressed only in terms of ε 1 and takes a constant value. By assuming M = 1 and using n s − 1 = −2ε 1 , the convergence condition yields 24) with N := N f − 2N v . In the slow roll limit ε 1 ≪ 1, the above condition is recast into The intuitive understanding of the above suppression mechanism is as follows. In the Euclidean vacuum case, only the contributions around the Hubble scale at each time are left unsuppressed (as shown in Fig. 2). When only the modes around the Hubble scale, i.e., k|η| ≃ k/e ρρ = O(1), are relevant, the Wightman function R x R x ′ G + (x, x ′ ) is necessarily suppressed when η(t)/η(t ′ ) ≪ 1. This is because if x and x ′ are largely separated in time, any Fourier mode in the Wightman function cannot be of order of the Hubble scale simultaneously at t and t ′ . When we consider the contribution of vertices located far in the past, at least one Wightman function should satisfy η(t)/η(t ′ ) ≪ 1, and therefore it is suppressed. However, when we consider a diagram for which a cluster of vertices in a distant past is connected to the vertices around the observation time by a single propagator, i.e., in the case with M = 1, the IR suppression comes only from this propagator. When the number of operators in the cluster of vertices in the past is sufficiently large, the suppression due to this propagator can be overwhelmed by the large amplitude of the fluctuation, which increases as the energy scale of inflation increases in the past direction. This corresponds to the case when the condition (4.23) is broken. However, we should also stress that the contributions from the distant past are suppressed and the secular growth never appear in the slow roll inflation, unless the order of perturbative expansion N takes an extremely large value such as 1/ε 1 ≃ O(10 2 ). When the convergence condition (4.23) is satisfied, all the time integrations are dominated by the contributions near its upper end. The order of magnitude of the n-point functions of R x g ζ(t f , x) is then given by (4.26)

A. Euclidean vacuum satisfies the strong constraint on the initial states
In this paper, we showed that when we choose the Euclidean vacuum as the initial state, the vertex integration in the npoint functions for the genuinely gauge invariant curvature perturbation is regular unless a very high order in the perturbative expansion is concerned. Figure 3 shows the outline of the proof. We should emphasize that the regularity of the n-point functions in the limits η → −∞(1 ± iǫ) plays a crucial role in the proof: (i) Requesting this regularity guarantees the equivalence between two quantum systems, i.e., the original system in which the Hamiltonian contains the IR relevant operators and the quantum system in which the Hamiltonian is totally composed of IR irrelevant operators. (ii) It guarantees the analyticity of the mode function v k (t) with respect to the wavenumber k for arbitrary t. By virtue of the aspect (i), we can rewrite the n-point functions of g ζ into those expressed in {ζ,π}, in which all the field operators are manifestly associated with the IR suppressing operators, R x . The aspect (ii) leads to the exponential suppression in the UV so that the non-vanishing support of the k-integration is restricted to −kη O(1). It might be intriguing that choosing the Euclidean vacuum plays the crucial role in discussing the suppressions both in the IR and UV components. Since these suppressions make the Wightman function (in the position space) associated with an IR suppressing operator regular everywhere except for the light cone limit, the missing piece to prove the regularity of the n-point functions is to show that the integration region of each vertex integral is effectively confined to a finite portion of the spacetime. Using the residual gauge degrees of freedom, we can confine the interaction vertices within the past light cone. Since the long-term correlation is shut off because of the suppression both in the IR and UV, the integration region of the vertex integrals is ensured to be effectively finite. Therefore, the n-point functions for the Euclidean vacuum are expressed by integrals whose integrand and integration region are both finite, and hence they are manifestly regular. Thus, we conclude that the Euclidean vacuum is a suitable initial state of the universe which is free from the IR pathology even in the presence of non-linear interactions.
In this section, we further address the converse question; "When we request that the n-point functions are finite and free from the secular growth, is the Euclidean vacuum the unique possible initial quantum state?" To be precise, the condition we impose here is the regularity of n-point functions on the real time axis including the distant past, i.e., −η ≫ 1. We naively expect that in this case, the Euclidean vacuum is the unique possibility. Since any excitations are blue shifted at an earlier time, any small deviation from the Euclidean vacuum state at a finite time will lead to some singular behavior in the limit −η ≫ 1. However, we do not have any rigorous proof about this argument yet. There might be a fundamental obstacle when we try to make this statement precise. When we trace back the history of the universe, it should inevitably enter the regime in which the background energy density and hence the amplitude of the vacuum fluctuation are so high that the perturbative analysis would not make sense any more.
As an alternative setup of the problem is to require the regularity of the n-point functions just for η > η i with a certain initial time, η i . The relaxed requirement of the regularity allows us to take other states, if correlation functions for these states can be reinterpreted as correlation functions for the Euclidean vacuum. We introduce a new operator with an arbitrary choice of the IR suppression operator R (m) where m is just the label for distinction. Then, we can define the 1 particle state by | 1 (m) := NA † (m) |0 with an appropriate normalization factor N. The n-point functions of R x g ζ(x) at the initial time η = η i for the 1 particle state | 1 (m) defined at the initial time can be expressed in terms of the (n + 2)-point functions for the products of R x g ζ(x) for the Euclidean vacuum. When the initial distribution is regular, as we showed in Ref. [28], the distribution at late times will be kept regular as well. Similarly, we can construct excited states with plural particles. (Similar excited states are discussed in de Sitter spacetime in Ref. [44].) However, here the allowed number of inserted operators might be bounded because our proof of regularity does not apply when the order of perturbation becomes very high.
To extend the above discussion to the n-point functions at a later time, we only need to show the regularity of the n-point functions which are defined as the expectation values of the path ordered products of g ζ and g π, for the Euclidean vacuum without the restriction that all the arguments are at the equal time, which will be a straight forward extension. In this manner, one can construct various excited states that are IR regular and free from the secular growth.

B. Comparison to the recent publications
In the recent papers [38,42], the absence of the secular growth is also claimed. It would be profitable to give a comparison between these works and our current work. First, in these papers, the item 1 raised in Sec. I, i.e., the presence of the canonical system, which is equivalent to the original canonical system and whose interaction Hamiltonian is composed only of the IR irrelevant operators, is postulated, while this is not automatically guaranteed from the symmetry of the classical system. Second, in these papers, the mode function in de Sitter spacetime, whose amplitude is given by a constant Hubble parameter, is used in proving the conservation of the curvature perturbation. This leads to the quantitative discrepancy in the evaluation of the secular growth from ours. For instance, in Ref. [38], the locality of the solutionζ L (x, t) given in Eq. (22) of the paper is crucial in their proof. However, the locality is not necessarily valid, once we take into account the fact that in the chaotic inflation, the amplitude of the fluctuation becomes larger and larger in the distant past asρ ∝ e − dρε1 . When we neglect this effect by setting (ρ/ √ ε 1 ) N in Eq. (4.23) to constant, the convergence condition is always satisfied (unless the interaction coefficient λ, composed of the horizon flow functions, varies rapidly). Therefore our result does not contradict to the conservation of the curvature perturbation that they claimed. The third point is about the treatment of the UV contributions. In this paper, we have not directly discussed about the UV renormalization. We simply assumed that the UV divergent contributions, which are shown to be localized to the region where the two arguments of the Wightman functions are mutually almost light-like, can be renormalized by introducing the local counter terms. As long as the renormalization does not break the dilatation symmetry of the classical action, our discussion can hold. Recently, an interesting investigation about the UV renormalization is pursued in Ref. [42]. It is claimed that a decaying composite operator in the free theory is kept decaying also after the renormalization of loops. Although the non-trivial assumptions such as the locality must be removed or verified, if this statement is correct, the conservation of the curvature perturbation can be shown also in the presence of the loop corrections. We should, however, emphasize that the conservation of the curvature perturbation does not prohibit the appearance of the logarithmic amplification, as we mentioned in Sec. I.
Finally, we also make a comment on the recent progress regarding the IR issues of a test field in the exact de Sitter spacetime, which can be interpreted as an approximation to the entropy mode. The regularity of the loop corrections for the Euclidean vacuum is shown for the massive scalar field by S. Hollands [47] and independently by D. Marolf and I. Morrison [48][49][50]. By contrast, for a massless scalar field, the IR regularity has not been shown and the absence of the secular growth is unclear [51][52][53][54] (see also Ref. [55]). Although the adiabatic curvature perturbation is a sort of massless field in the sense that the Wightman function G + (x, x ′ ) possesses the IR divergence and the long term correlation, the operation of the IR suppressing operators R x , which appear by virtue of the residual gauge symmetry and by choosing the Euclidean vacuum, cures the singular behaviour. Hence, it would be intriguing to discuss a massless field with the exact shift symmetry in the de Sitter spacetime, in comparison with the case of the adiabatic mode. and the momentum constraints yield VÑ n − 3ρζ n + e −2ρ ∂ 2ζ n +ρe −2ρ ∂ iÑ i,n = H n , (A1) 4∂ i ρÑ n −ζ n − e −2ρ ∂ 2Ñ i,n + e −2ρ ∂ i ∂ jÑ j,n = M i,n , where H n and M i,n include n interaction picture fieldsζ I in the combinationζ I − s or with differentiation. EliminatingÑ n from these constraint equations, we obtain where we defined Operating ∂ i on Eq. (A3), we obtain We solve this equation as follows, where G L n (x) is an arbitrary solution of the Laplace equation, i.e., ∂ 2 G L n (x) = 0. Inserting this solution into Eq. (A3), we obtain Again, introducing an arbitrary solution of the Laplace equation G i,n (x), we solve Eq. (A7) as Comparing the expression obtained by operating ∂ i on Eq. (A8) with Eq. (A7), we obtain Using this expression, we rewrite the longitudinal part of G i,n as Inserting Eq. (A10) into Eq. (A8), we obtaiñ When we perform quantization in the whole universe, it is natural to request the regularity of the perturbation at the spatial infinity. This requirement uniquely fixes G L n and the transverse part of G i,n . Then, the shift vector depends on the curvature perturbationζ of the whole universe. To show the IR regularity, here we employ another boundary condition which requests that the integration region of the inverse Laplacian ∂ −2 is confined to around the local observable region O. As is shown in Refs. [25,26], the degrees of freedom in changing the boundary condition can be understood as the gauge degrees of freedom in the local universe. Therefore, the operator g R is invariant under the change of the boundary condition.