Large gauge transformation, Soft theorem, and Infrared divergence in inflationary spacetime

It is widely known that the primordial curvature perturbation $\zeta$ has several universal properties in the infrared (IR) such as the soft theorem, which is also known as the consistency relation, and the conservation in time. They are valid in rather general single clock models of inflation. It has been argued that these universal properties are deeply related to the large gauge transformations in inflationary spacetime. However, the invariance under the large gauge transformations is not sufficient to show these IR properties. In this paper, we show that the locality condition is crucial to show the consistency relation and the conservation of $\zeta$. This argument also can apply to an interacting system with the inflaton and heavy fields which have arbitrary integer spins, including higher spin fields, which may be motivated from string theory. We will also show that the locality condition guarantees the cancellation of the IR divergences in a certain class of variables whose correlation functions resemble cosmologically observable quantities.


Introduction
The presence of a symmetry helps us to systematically understand various properties of a physical system. Understanding the physics based on the symmetry may open up the possibility of understanding a specific example based on a more universal property of physical systems which share similar symmetry groups. It has been suggested that the infrared (IR) properties of the primordial perturbations generated during inflation are deeply related to the large gauge transformations. The large gauge transformation is a local symmetry transformation which does not approach the unity at the infinity of spacetime [1,2].
The large gauge transformations which play an important role in cosmology are those defined at a time constant slicing and diverge at the spatial infinity. In single field models of inflation, the invariance under the dilatation and shear transformations, which are both large gauge transformations, directly ensures the massless property of the curvature perturbation ζ and the gravitational waves γ ij . It is widely known that these massless fields ζ and γ ij have several universal properties in the IR, which are valid in rather general single field models of inflation. First, at the tree-level computation, ζ and γ ij are non-linearly conserved in time in the IR limit [3,4,5,6,7,8,9,10,11]. (The conservation of ζ was discussed by including the radiative corrections of ζ in Refs. [12,13].) Second, the influence from the soft modes of ζ and γ ij on the hard modes are, rather generically, characterized by the so-called consistency relation [14,15], which is an example of the soft theorem in cosmology. It has been suggested, e.g., in Refs. [3,16,17,18,19,20], that these universal properties are both consequences of the invariance under the large gauge transformations. However, it was also revealed that this invariance is not sufficient to derive the consistency relation. In Refs. [18,19,20], the analyticity in the soft limit was additionally imposed to derive the consistency relation.
The consistency relation has been understood as a relation between the n-point correlation function with n hard modes and the (n + 1)-point correlation function with n hard modes and one soft mode. In deriving the consistency relation, usually, the validity of the perturbation theory is presumed. However, this is not trivially guaranteed, because the massless fields ζ and γ ij are not screened in the large scale limit and their radiative corrections can diverge due to their unsuppressed IR contributions. Therefore, unless the IR divergence is regularized without violating the dilatation invariance, the consistency relation cannot be well-defined as a relation between the correlation functions. Notice that introducing a naive IR cutoff which regularizes the IR radiative corrections can violate the dilatation invariance, which is a crucial property for the consistency relation.
The goal of this paper is to clarify the relation between the large gauge transformations and the above-mentioned universal properties of ζ and γ ij in the IR, i.e., the soft theorem (also known as the consistency relation), the conservation in time, and the possible appearance of the IR divergence. In particular, along the line of the argument in Refs. [18,19,20], we will clarify the physical meaning of the analyticity in the soft limit, which is needed to derive the consistency relation. In Ref. [18], it was argued that this condition is related to the locality of theory. In this paper, we scrutinize this argument, clarifying what we need to require as the locality condition, because the locality has a broad meaning. We also discuss what the consistency relation actually describes, taking into account the possibility that the perturbative prediction can be spoiled by the IR divergence.
We will also show that the large gauge transformations play a crucial role also in discussing the radiative corrections from massive fields. Being motivated by string theory, which may predict the presence of higher spin fields in the four dimensional theory obtained after compactification, we consider massive fields with arbitrary integer spins. After deriving the consistency relation for the hard modes of the massive fields with non-zero spins, we discuss the condition that the curvature perturbation ζ stops evolving in time in the soft limit under the influence of the radiative corrections from the massive fields. This is a generalization of our previous study [19], where we considered the radiative corrections from a massive scalar field, to a massive field with a general integer spin. As was argued in Ref. [21], exploring an imprint of the higher spin fields in the primordial non-Gaussianity of ζ may provide a unique probe of string theory. (See also Refs. [22,23,24,25].) Our study provides the conditions for the absence of such an imprint made after the Hubble crossing time of the comoving scale of our interest during inflation.
Recently, the relation among the large gauge transformations, the soft theorem, and the IR divergence has been studied intensively for gauge theories in an asymptotically flat spacetime [26,27,28]. (For a review, see Ref. [29].) In Refs. [26], it was shown that Weinberg's soft theorem [30], which describes the influence of the soft photon and graviton, can be obtained as a Ward-Takahashi identity of the asymptotic symmetries at the null infinity. (The soft theorem for massless higher spin fields in an asymptotically flat spacetime was discussed in Ref. [31].) More recently, (the cancellation of) the IR divergence was discussed, using the Noether charge of the asymptotic symmetries [32]. At first sight, the IR structures for the gauge fields in the asymptotically flat spacetime have similar properties to those for the primordial perturbations ζ and γ ij in cosmology. We give a closer look at this apparent similarity.
This paper is organized as follows. In Sec. 2, considering single field models of inflation, we discuss the relation between the large gauge transformation and the consistency relation for ζ. Here, we seek for a deeper understanding about the locality condition as the necessary condition to derive the consistency relation for ζ. In Sec. 3, we will show that the discussion in Sec. 2 can be straightforwardly extended to an interacting system composed of the inflaton and the massive fields with non-zero spins. We also show that the locality condition implies both the consistency relation for the hard modes of the massive fields and the conservation of ζ. In Sec. 4, we show that the locality condition also ensures the cancellation of the IR divergence for a certain class of variables. In Sec. 5, we briefly show that the discussion for the soft graviton proceeds almost in parallel to the one for the soft modes of ζ, discussed in Sec. 2-4. In Sec. 6, after summarizing our results, we discuss a similarity and an apparent difference between the IR properties for the gauge fields in the asymptotically flat spacetime and those for the primordial perturbations.

Asymptotic symmetry and soft theorem
In this section, we discuss the relation between the large gauge transformations and the soft theorem in cosmology, also known as the consistency relation. We clarify the condition that derives the consistency relation and discuss its physical meaning. Before we start our discussion, we clarify what we mean by "soft" and "hard." The soft modes mean the modes with k/aH → 0 but k = 0, and the hard modes mean the remaining modes, including the super Hubble modes with k/aH < ∼ 1 but excluding the limit k/aH → 0. By contrast, we describe the wave vector of a longer mode and that of a shorter mode as k L and k S , respectively, simply based on the ratio between these wavenumbers, i.e., k L /k S ≪ 1. As we will discuss in Sec. 4, in the limit k/aH ≪ 1, a perturbative expansion can break down in computing some quantity and taking this limit requires a careful treatment.

Large gauge transformations
Likewise in the discussion about the soft photons and gravitons in the asymptotically flat spacetime, a large gauge transformation plays a crucial role for a clear understanding about the soft modes of ζ and γ ij in an inflationary spacetime. In line with Refs. [1,2], we define the large gauge transformation as follows. A local symmetry denotes a symmetry under a transformation which is parametrized by a spacetime dependent function, while a global symmetry denotes a symmetry under a transformation by a spacetime independent function.
Among local symmetry transformations, it is important to make a distinction between small gauge transformation and large gauge transformations. The former becomes the identity at the infinity and the latter does not. In Refs. [26,29], it was shown that the soft theorem for the photons and the gravitons in the asymptotically flat spacetime can be derived from the Ward-Takahashi identities for large gauge transformations which do not vanish on J ± .

Dilatation as a large gauge transformation
First, let us clarify the prescription we adopt. In this paper, we use the ADM form of the line element: where we introduced the lapse function N , the shift vector N i , and the spatial metric h ij . We determine the time slicing, employing the uniform field gauge: We express the spatial metric h ij as where γ ij is set to traceless. As spatial gauge conditions, we impose To discuss the soft modes of the primordial perturbations in the spatially flat FRW background, we consider the large gauge transformations, which do not vanish at the spatial infinity on a time constant surface. This large gauge transformation was first discussed in the context of cosmology by Weinberg in Ref. [3]. In the unitary gauge, where the fluctuation of the inflaton vanishes, we consider, in particular, the dilatation: where s is a constant parameter. Under the dilatation, the curvature perturbation ζ transforms as The change of ζ is given by The classical action in a diffeomorphism (Diff) invariant theory remains invariant under the transformation of ζ given in Eq. (2.6). As one may expect from the fact that the dilatation shifts ζ by −s, the dilatation invariance is related to the massless property of ζ, which implies that ζ is conserved at large scales in single clock inflation.

Two different prescriptions of dilatation
The dilatation invariance may be somehow confusing, because it also appears as a part of the de Sitter invariance by changing the time coordinate simultaneously. The Killing vector which corresponds to this transformation is given by where η denotes the conformal time. The dilatation symmetry in de Sitter group states that the time shift can be compensated by the scale transformation. Since inflation has to end at some point, the time translation symmetry needs to be broken in the context of inflationary scenario.
The time translation symmetry is broken, when the physical frequency ω ph becomes well below Λ b with Λ 4 b ≡φ 2 . In the effective field theory of inflation [33], the Goldstone mode, the pion π, is introduced to restore the invariance under the time reparametrization in the symmetry breaking phase. With this construction, the pion Lagrangian non-linearly preserves the invariance since π appears only in the combination t + π. Through the coupling with the metric perturbations, the pion acquires the mass of m π = O( √ ε 1 H).
The relation between the dilatation discussed in Sec. 2.1.1 and the one discussed here is somewhat puzzling. As we will discuss in the following section, the former is preserved in an arbitrary quasi FRW spacetime, while the latter is a part of the de Sitter symmetry and is broken below the symmetry breaking scale Λ b . Related to this point, preserving the former dilatation invariance directly ensures that ζ should be massless. On the other hand, there is no simple argument which shows the massless property of ζ in the latter prescription, where ζ is related to the Goldstone mode π as ζ = −Hπ. Related to this point, recall that the pion acquires the mass m π through the coupling with the metric perturbation for ω ph < ∼ m π 1 . Therefore, in the regime where ζ approaches a constant value, the pion is no longer massless. The curvature perturbation is sometimes said to be a Goldstone mode, since π = −ζ/H is the Goldstone mode. This statement can cause a confusion, because it may sound as if ζ preserves the shift symmetry, being massless, because it is a Goldstone boson associated with the breaking of the de Sitter symmetry. In the following, by the dilatation, we mean the former one, which is a spatial coordinate transformation without the time coordinate change. Considering this dilatation, we discuss the IR behaviour of ζ such as the consistency relation and the conservation in time. We will emphasize that the invariance under this dilatation, which is a part of the large gauge transformations, is preserved as well in the quantized system and therefore there is no spontaneous symmetry breaking in this prescription.

Dilatation invariance and Noether charge
In this subsection, we discuss several implications of the dilatation invariance in single clock inflation. Following Ref. [16], we define the Noether charge for the dilatation as where π ζ denotes the conjugate momentum of ζ, which satisfies Using the Fourier components of the fields 2 we can rewrite the Noether charge Q ζ as In performing the Fourier transformation, we did not drop the surface term. Therefore, the charge Q ζ given in Eq. (2.9) is identical to the one given in Eq. (2.13). The first term of Q ζ only operates on the k = 0 mode. The Noether charge Q ζ can diverge due to the IR modes, because it is an integral over the infinite spatial volume. In the following, we neglect higher order terms of O(s 2 ).
Equation (2.13) gives the non-perturbative definition of the Noether charge for the dilatation. Since an explicit computation usually relies on perturbative expansion in the interaction picture, one may want to introduce the Noether charge by using the fields in the interaction picture as 3 where ∆ s ζ I (t, x) is given by When we perform the dilatation transformation in the interaction picture or after the perturbative transformation, there is one caveat which should be kept in mind. Changing order of performing the finite dilatation transformation and performing the perturbative expansion leads us to a different answer. In order words, performing the dilatation in the interaction picture (with a use of Q I ζ ) and performing the dilatation in the Heisenberg picture (with Q ζ ) are different, while this discrepancy disappears, when we consider the infinitesimal transformation generated by dQ ζ /ds| s→0 . A lesson from here is that for a legitimate prescription, the dilatation transformation should be performed in the nonperturbative Heisenberg picture. A more detailed discussion can be found in Appendix A.

Revisiting consistency relation
In this subsection, we discuss the condition(s) to derive the consistency relation of ζ. We will see that while the dilatation invariance plays a crucial role, it is neither sufficient nor even necessary to derive the consistency relation.

Condition for consistency relation
The consistency relation for ζ was first derived by Maldacena from an explicit computation of the bi-spectrum of ζ [14]. Afterwards, the consistency relation was derived in a rather general setup of single field models of inflation [15], while several examples that do not satisfy the consistency relation were also reported [34,35,36] (see also Refs. [37,38]).
In Ref. [14], it was argued that the bi-spectrum in the squeezed limit, which describes the correlation between the long mode k L and short mode k S , can be computed by considering the influence of the long mode on the short mode. This influence can be described as the dilatation k S → e −ζ k L k S . While ζ is not invariant under the dilatation transformation 4 , we can construct another variable which is invariant under the dilatation transformation [39,40]. In Ref. [41], we showed that the leading contributions to the bi-spectrum 3 Notice that since the free Hamiltonian H0 is not invariant under the dilatation, i.e., H0, Q I ζ = 0 , (2.14) in contrast to Q ζ , which is time independent, the "charge" defined in the interaction picture Q I ζ varies in time. 4 The curvature perturbation ζ is invariant under the small gauge transformations but not under the large gauge transformations.
of such an invariant variable in the squeezed limit are canceled, choosing the adiabatic vacuum. In this computation, the squeezed bi-spectrum of ζ satisfies Maldacena's consistency relation, which is the key to ensure the cancellation.
The consistency relation does not hold for an arbitrary quantum state even in single field models. In Refs. [17,18,20], it has been directly and indirectly suggested that the dilatation invariance of the quantum state: is crucial to derive the consistency relation. In Ref. [17], it was shown that the invariance of the 1PI effective action for ζ under the dilatation leads to a set of identities which relate the n-point function and the (n + 1)-point function for ζ. What we directly obtain from these identities, which are the so called Ward-Takahashi (WT) identity, is "the consistency relation," which relates the n-point function with n hard modes to the (n+1)-point function with the homogeneous mode ζ k=0 in addition to the n hard modes. The only difference from the consistency relation is in that the inserted mode is not the soft mode ζ k with k = 0. Along this line, in Refs. [20,42,43], "the consistency relation" was derived starting with the dilatation invariance of the wave function in the ζ representation, Ψ[ζ] ≡ ζ|Ψ , where | ζ is the normalized eigenstate of ζ, i.e., which directly follows from the invariance of the quantum state under the dilatation (2.17) 5 . If and only if the WT identity which describes the insertion of ζ k=0 can be extended to the relation which describes the insertion of the soft mode ζ k with k = 0, we obtain the consistency relation: where C (n) denotes the n-point function of ζ with the momentum conservation factor removed. Since we additionally need to impose that this extension is possible, requesting the dilatation invariance of the quantum state (2.17) is not enough to derive the consistency relation. 5 For instance, at O(s), Eq. (2.18) gives which can be obtained from 0 = ζ(x)|Q ζ |ψ by inserting the expression of the Noether charge, given in Eq. (2.9). Here, we used ζ|π ζ |ψ ∝ δψ[ζ]/δζ. Now, the question is "What is the physical meaning of the additional condition that allows the WT identity to be smoothly extended to the consistency relation, which describes the insertion of the soft mode?" This issue was first addressed in Ref. [18], where it was argued that this condition is related to the locality of the theory. Even if the original theory is local, the Lagrangian density for ζ in the unitary gauge becomes non-local due to the presence of the Lagrange multipliers, the lapse function and the shift vector, which are given by solving the elliptic equations. For instance, at the linear order in perturbation, the shift vector N i includes a contribution given by which introduces non-local interaction vertices. In the standard slow-roll inflation,ζ k is suppressed in the limit k → 0 asζ k = O(k p ζ k ) with p ≥ 1. However, in the absence of this suppression, the coefficients of the non-local interaction vertices in the Fourier space can be singular in the limit k → 0. Then, N i should be determined discontinuously at k = 0 in order to avoid the singular behaviour.
Along this line, our purpose of this section is to sharpen the relation between the condition for the locality and the condition of being able to extend the "consistency relation" with the insertion of the homogeneous mode to the consistency relation with the soft mode k = 0, clarifying the physical meaning of the condition. As was mentioned above, the Lagrangian density for ζ (and also for the gravitational waves) is non-local in the sense that the Lagrangian density cannot be solely determined by the dynamical fields at each spacetime point. For a classical theory, the condition for the smooth extension of the homogeneous mode k = 0 to the soft modes k = 0 with a suitable fall-off at the spatial infinity is nothing but the one to pick up the Weinberg's adiabatic mode [3]. We elaborate the physical meaning of this condition for a quantum theory by using the Noether charge Q ζ . As will be discussed in the next section, with the use of the Noether charge, a generalization to the case with higher spin fields proceeds straightforwardly.

Conditions on the homogeneous mode and hard modes
As was mentioned in Sec. 2.3.1, the validity of the consistency relation is deeply related to the invariance of the wave function or the effective action under the dilatation, which is a large gauge transformation. Maldacena derived the consistency relation for the tree-level bi-spectrum in the squeezed limit, i.e., (2.19) with n = 3, choosing the adiabatic vacuum (or the Euclidean vacuum) [14]. This vacuum also can be defined non-perturbatively by requesting the regularity of correlation functions in the limit t → −∞(1 ± iǫ). Here, the time path is rotated towards the imaginary axis in the distant past. This serves one of the examples of the quantum state |Ψ , which preserves the dilatation invariance, since this definition does not artificially introduce any specific scale. At perturbative level, a correlation function for the Euclidean vacuum can be calculated by adopting the iǫ prescription. Here, the (free) mode function should be chosen to be the one for the adiabatic vacuum (a.k.a, the Bunch-Davies vacuum in de Sitter limit) [44].
In order to study the condition for the dilatation invariance of the quantum state (2.17), we decompose the wave function in terms of the eigenstates of the spatial average of ζ all over a time constant slicing, e.g., at the end of inflation, The eigenstate |ζ c satisfiesζ whereζ c is a c-number eigenvalue. In order to distinguish the eigenvalues ofζ, which are c-numbers, from the operatorζ, we put the index c on the eigenvalues. In Eq. (2.21), we factorized the wave function ofζ, ψ(ζ c ), from ζ c |Ψ , while absorbing the phase into | Ψ ζc , as From the normalization condition for | Ψ ζc , the amplitude of the wave function |ψ(ζ c )| is unambiguously defined. Since the quantum state | Ψ also includes the inhomogeneous modes with k = 0, ζ c |Ψ should be understood as a vector in infinite dimensional Hilbert space. We express the normalized quantum state for all the modes with k = 0 obtained by the projection of | Ψ to the eigen state |ζ c as | Ψ ζc . Using Eq. (2.12), we obtain Since an operation of e iQ ζ shifts the eigenvalue ofζ by s, i.e., Using this condition, we can express the dilatation invariance condition for the quantum state | Ψ , (2.17), as Operating ζ c ′ | on Eq. (2.27), the real and imaginary parts, respectively, give and The first condition (2.28) requires that the amplitude of the wave function ψ(ζ c ), which represents the probability distribution ofζ, should be flat in the direction ofζ in the Hilbert space. The condition (2.28) is satisfied, e.g., for a Gaussian wave function whose variance blows up in the limit of the homogeneous mode, which is the case for a nearly scale invariant power spectrum. The condition (2.28) requires that the probability distribution should be non-perturbatively flat in the direction of the homogeneous modeζ. The second condition may be a little more non-trivial. The quantum state of the inhomogeneous modes generically changes under a variation of the homogeneous modeζ. The condition (2.29) restricts how the quantum state of the inhomogeneous modes should respond to the change of the homogeneous mode, which is expressed by the operation of Q ζ . Since the consistency relation describes an insertion of an inhomogeneous mode with k = 0 that suitably falls off at the spatial infinity, we need to extend the above argument to inhomogeneous modes. For this purpose, we introduce the generator of a spatial dependent dilatation given by where W (x) denotes a smooth window function which is normalized as d 3 x W (x) = 1 and which vanishes at |x| > ∼ L. Here, we set L to be of order of 1/k L . The generator Q W ζ (x) induces the dilatation only for the fields at x ′ with |x − x ′ | < ∼ L. The Noether charge (2.9) is defined by the integral all over the time constant slicing and hence the integral does not converge in general. The introduction of the window function can make the integral converge. Since the domain of integration is finite, the generator Q W ζ (x) depends on where we choose the center of the integral domain, x. Performing the Fourier transformation, we obtain whereŴ (k) denotes the Fourier mode of the window function, which is normalized as lim k→0Ŵ (k) = 1 , and vanishes for k ≫ 1/L. Here, we introduced We focus on the field within a large volume of O(L 3 c ). We set L c to be much larger than all the wavelengths, i.e., L c ≫ 1/k L , 1/k S , and we will send L c → ∞ after our computation. Next, we introduce a smeared field in momentum space defined bỹ ζ k L describes a collective mode with the representative wavenumber k L and when we evaluateζ in the position space by performing the inverse Fourier transformation, it decays outside the local volume of O(L 3 c ). The fluctuations outside the local volume can be described as the modes which are orthogonal to the collective modeζ k L . Since the correlations between the fluctuations for |x| ≪ L c and those outside the local volume of O(L 3 c ) are negligibly small, in what follows, we neglect the fluctuations outside the local volume, which will disappear after taking the limit L c → ∞.
The commutation relation of Q W ζ with the long mode ζ k L is given by 3 . We will find that the factor (2πL c ) 3 , which blows up in the limit L c → ∞, is cancelled out in the final expression of the consistency relation as it should be. Whilst, the commutation relation with the short mode ζ p S is given by where we approximated k S + k L as k S . Repeating the argument around Eqs. (2.21)-(2.23) except that the homogeneous modē ζ is now replaced with a collective inhomogeneous soft modesζ k L , we expand the quantum state | Ψ in terms of the orthonormal basis {|ζ c p L }, which are the eigenstates ofζ p L , as where we factorized the amplitude of the wave function, |ψ(ζ c k L )|, from ζ c k L |Ψ , as before. Using Eq. (2.35), we obtain As we have already discussed, the dilatation invariance requires the conditions (2.28) and (2.29). In particular, the second condition (2.29) restricts how the inhomogeneous modes respond to the insertion of the homogeneous modeζ. In the following, we will show that when this condition can be extended to the soft mode with k L = 0, i.e., is fulfilled, we can derive the consistency relation for ζ as shown below.
In order to show the consistency relation by using the condition (2.39), we evaluate in two ways: first by operating iQ W ζ (k L ) on the quantum state | Ψ and second by considering the change of the short wavelength modes ζ k S under the inhomogeneous dilatation, expressed by the commutation relation with iQ W ζ (k L ). When the condition (2.39) is satisfied, using Eq. (2.38), we obtain When we neglect the non-linear contributions of the soft modes 6 , the wave function of ζ c k L is given by the Gaussian distribution function. Since the square of |ψ(ζ c k L )| gives the probability distribution, we can express the amplitude of the Gaussian wave function as where P ζ (k L ) denotes the power spectrum of ζ. This is because the variance ofζ c k L is given by where the last equality is exact in the limit L c → ∞. Remember thatζ c k L is not an independent variable fromζ c −k L , sinceζ c k L =ζ c * −k L holds from the reality of ζ. Using Eq. (2.42), we obtain Here and hereafter, we use s.l.
≈ to express that we approximate the wave function of the soft mode by the above Gaussian distribution function. Replacing the eigenvalueζ c k L with the operatorζ k L , which commutes with the integral overζ c k L , we obtain Using this expression, we arrive at Meanwhile, using Eq. (2.36), we obtain (2.47) 6 Here, all the interaction vertexes connected to more than two soft modes are neglected.
Equating these two expressions derived from the two different ways, we have The ordinary form of the consistency relation (2.19) is the one obtained by removing the delta function, which describes the momentum conservation, from the expression If we take the average of Eq.(2.49) operating L 3 c d 3 p LŴ (k L − p L ), we recover Eq. (2.48). After we take the limit L c → ∞, which is automatically required when we take the soft limit k L → 0, the averaging window becomes infinitesimally narrow. Therefore, we can conclude that Eq. (2.48) is equivalent to the ordinary consistency relation.
Under the Gaussian approximation of | ψ(ζ c k L ) |, the only assumption imposed to derive the consistency relation (2.19) is Eq. (2.39), which states that the influence of inhomogeneous dilatation is identical to shifting, by +(2πL c ) 3 s k L , the collective soft modeζ k L which interacts with the hard modes. (The factor (2πL c ) 3 here is an artifact caused by discussing the Fourier space collective mode.) The consistency relation can be obtained, when the condition (2.29), which was required to preserve the dilatation invariance of the quantum state, can be extended to the inhomogeneous soft mode with k L = 0. In this sense, the condition (2.39) can be understood as a quantum version of the condition for Weinberg's adiabatic mode. The condition (2.39) cannot be satisfied, in case the (linear) soft mode ζ k L does not stop evolving in time in the limit k L → 0. In fact, the condition (2.39) states that performing the inhomogeneous dilatation transformation that induces the time-independent shift of ζ k L , is equivalent to shifting (the eigen value of) ζ k L at the evaluation time, which implies that the inserted soft mode should be dominated by a constant contribution. Now, we show that the condition (2.39) can be indeed understood as the locality condition. For this purpose, we consider a set of eigenstates for the Fourier mode ζ k L which satisfies ζ k L | ζ c L = ζ c k L | ζ c L instead of the eigenstates for the collective modeζ k L . Using the complete set of | ζ c L , we decompose the quantum state | Ψ as where Dζ L represents the functional integral over all soft modes. The extension of Eq. (2.39) to the case of continuous modes will be Using this relation, we obtain where is the coarse grained field corresponding to the degrees of freedom of soft modes. Let us imagine a set of the separate universes whose sizes are of O(L). The operator Q W ζ (x) induces the dilatation only within the separate universe centered at x. The condition (2.52) states that the impact of the soft mode on the quantum state of short modes is limited only to the influence which is equivalent to the inhomogeneous dilatation in the separate universe.
Notice that since (the amplitude of) the wave function is not flat in the direction of the inhomogeneous mode of ζ, i.e., ∂|ψ(ζ k L )|/∂ζ k L = 0 , as seen in Eq. (2.44), the quantum state | Ψ does not remain invariant under the inhomogeneous dilatation Q W ζ (k L ), i.e., As shown in Eq. (2.45), operating the generator of the inhomogeneous dilatation inserts the soft modeζ k L with k L = 0, which changes the quantum state, even if the state is invariant under the homogeneous dilatation. As will be discussed in Sec. 4, because of that, choosing a quantum state which is invariant under the dilatation is not enough to guarantee the IR regularity. Let us emphasize that there is no spontaneous symmetry breaking for the large gauge transformations: the symmetry under the dilatation, generated by Q ζ , is preserved also after the quantization, while the inhomogeneous dilatation, generated by Q W ζ , is not a symmetry of the classical action.
Although we assumed that the wave function of the soft mode, |ψ(ζ c k L )|, is given by the Gaussian distribution, the non-linear interactions of the hard modes are fully kept. Therefore, the consistency relation thus derived can apply to a much more general setup compared to the original one by Maldacena in Ref. [14]. For instance, the wavelengths of the hard modes can be arbitrary as far as k S L ≃ k S /k L ≫ 1, i.e., they are not necessarily in super Hubble scales. In addition, obviously the same argument as above can be applied, even if the time coordinates of the short modes ζ k Si (t i ) with i = 1, · · · , n are different among them.
Because of the Gaussian approximation for ψ(ζ c k L ), the consistency relation derived here do not contain non-linear interactions of the soft modes. Notice that since there is no approximation for the soft modes in the locality condition (2.39) or (2.52), when we include the non-linear contributions of the soft modes, the same condition leads to the consistency relation 7 . However, the non-linear interactions of the soft modes in general yield the IR divergences through their radiative corrections. Therefore, this extension requires a more careful consideration. This issue will be discussed in Sec. 4.

Massive particles with arbitrary spins
In this section, we consider an influence of heavy fields with arbitrary spins, which interact with the inflaton directly or indirectly through the gravitational interaction. The argument in this section is a generalization of the one in Ref. [19], which showed the consistency relation for a heavy scalar field and the conservation of ζ, taking into account radiative corrections of the heavy scalar field. In the previous section, we derived the condition for the consistency relation, using the generator(s) of the inhomogeneous dilatation. This argument can be extended to the cases with the radiative corrections of massive non-zero spin fields straightforwardly. In this section and in Appendix B, to keep generality, we consider a (d + 1)-dimensional spacetime.

Setup of the problem
In this section, we consider a heavy field whose mass is M S and spin is S ≥ 0, including higher spin fields. We only consider the mass range where there is no instability [21,45], e.g., M 2 ≥ 2H for a spin 2 field. For our purpose, we do not need to specify the detail of the interaction for the interacting system with the inflaton and the massive fields. We simply express the action as where χ and O iα denote a set of heavy fields χ I with I = 1, 2, · · · and a composite operator of χ, respectively, and δg denotes the set of the metric perturbations, N , N i , ζ, and γ ij . Here, the action S ad [δg] only includes the metric perturbations in the unitary gauge defined by the condition δφ = 0 and it is identical to the action in single field models of inflation. The heavy fields and the inflaton φ also can interact directly. For the present purpose, we do not need to specify the composite operators O iα . We only need to specify their scaling dimensions ∆ α , i.e., they transform as under the dilatation transformation x → x s = e s x, where i α denotes tensor indices. While the heavy fields can be a fermion with a half integer spin, we assume that δg interacts with χ only through the composite operators which transform as tensors (with an integer spin) under coordinate transformations. A composite operator with n tensor (lower) indices has a scaling dimension n, e.g., ∆ = 0 for a scalar composite operator and ∆ = 1 for a vector one.

Soft theorem for heavy fields with non-zero spins
In this subsection, we derive the consistency relation or the soft theorem in the presence of the heavy fields, extending the discussion in the previous section. In the previous section, expanding the quantum state | Ψ as in Eq. (2.21), we derived the conditions (2.28) and (2.29) by requesting the invariance of the quantum state under the dilatation. Here, repeating the same argument except that now the quantum state | Ψ also includes the heavy fields in addition to the inhomogeneous modes of ζ, we obtain the same conditions as Eqs. (2.28) and (2.29) from the dilatation invariance of | Ψ . Since | Ψ ζc also includes the degrees of freedom for the heavy fields, the condition (2.29) restricts how both of the inhomogeneous modes of ζ and the heavy fields should respond to the dilatation transformation. Similarly, when the condition (2.29) can be extrapolated to the inhomogeneous dilatation with the fall-off at the spatial infinity by replacing the homogeneous modeζ c in (2.29) with the soft modeζ k L , i.e., when the locality condition (2.39) is fulfilled also in the interacting system with the inflaton and the heavy fields, we can derive the soft theorem, which describe the influence of the soft mode.
Recall that the consistency relation can be derived by evaluating the change of the quantum state | Ψ and the change of the operators for the heavy fields. In order to derive the soft theorem for the heavy fields, we consider Likewise in the discussion for the single field case, whether the dilatation parameter is homogeneous or inhomogeneous does not affect the transformation of the short modes. Using this expression, we obtain Equating these two expressions and sending L c to the infinity, we obtain the consistency relation for the heavy fields as where we defined ∆ ≡ n i=1 ∆ α i . We put a prime to denote the correlation functions without the multiplicative factor (2π) d and the delta function which expresses the momentum conservation.

Effective action
In order to show that the radiative corrections of the heavy fields do not induce any time evolution of ζ at large scales when the condition (2.39) is satisfied, we compute the effective action for ζ by integrating out the heavy fields χ in the closed time path (or the in-in) formalism. In particular, the contributions of the heavy fields χ are described by the Feynman and Vernon's influence functional [46,47]. In this subsection, we briefly summarize the way to calculate the influence functional and the effective action. We will see that now the argument to show the conservation proceeds almost in parallel to the one for the heavy scalar field, discussed in Ref. [19].

Influence functional
Performing the path integral along the closed time path, the n-point function of the curvature perturbation ζ is given by where we double the fields: δg + and χ + denote the fields defined along the path from the past infinity to the time t and δg − and χ − denote the fields integrated from the time t to the past infinity. Since N and N i are the Lagrange multiplies, which are eliminated by solving the constraint equations, we perform the path integral only regarding the dynamical degrees of freedom δg dy ≡ (ζ, γ ij ) and χ. An insertion of δg + (x) into the path integral in the numerator as above gives a correlation function in the time ordering, expressed by T , while an insertion of δg − (x) gives a correlator in the anti-time ordering, expressed byT . Separating the part which describes the radiative corrections of the heavy fields as we can express the n-point function for ζ superficially as if there are only the metric perturbations and the inflaton as (3.11) The effective action is recast into 12) where S ′ eff is the so-called influence functional, given by where we factorized S ad [δg ± ] which commutes with the path integral over χ ± . Here, we only consider the correlation functions for ζ, but the effective action S eff [δg + , δg − ] describes the evolution of both ζ and γ ij affected by the quantum fluctuations of the heavy fields χ. For our later use, we introduce the correlation functions of χ computed in the absence of the metric perturbations as (3.14) Expanding S ′ eff in terms of the metric perturbations δg = (δN, N i , ζ, γ ij ), we obtain where S ′ eff(n) denotes the terms which include n δg α s, given by with the non-local interaction vertices induced by the heavy fields: In Eq. (3.16), each δg am with m = 1, · · · , n should add up all the metric perturbations δN am , N i,am , ζ am , γ ij am . Here and hereafter, for notational brevity, we omit the summation symbol over δg unless necessary. Inserting Eq. (3.13) into Eq. (3.17), we can express the non-local interaction vertices by using the correlators for χ. These expressions are summarized in Appendix B. Once we expand the effective action as in Eq. (3.16), the shift symmetry is lost at each order in perturbation about δg. In the following, we will show that we can rewrite the effective action in such a way that the shift symmetry is manifestly preserved by using the consistency relation.

Soft theorem and effective action
In order to show the conservation of ζ in the presence of the radiative corrections of the heavy fields, here we rewrite the consistency relation (3.8). As was mentioned in the previous section, the time coordinates of the hard modes can be different among different composite operators. Taking an appropriate ordering of the composite operators, i.e., we can put the index a i = ± on each composite operator O {iα i }k Si in the consistency relation (3.8). In the following, we use the prescription introduced in the previous subsection (see also Appendix B). In particular, all the correlation functions should be understood as being computed in the path ordering of the closed time path with the distinction of ±.
Employing the Gaussian approximation for the soft mode of ζ again, we can compute the correlation function in the first line of Eq. (3.8) as where | Ψ, 0 ζ denotes the quantum state | Ψ with ζ being in non-interacting vacuum.
Here, S int χ denotes a set of the interaction vertexes which include only one ζ without derivative and the massive fields χ and is given by We express the interaction action S int χ with the heavy fields on the paths ± as S int ± χ , respectively. Factoring out the power spectrum of ζ, we obtain where the power spectrum of ζ is canceled between the numerator and the denominator.
In the first line of Eq. (3.21), we omitted the time coordinate of the soft mode ζ k L , since it should be constant in time to satisfy the condition (2.39). (If one wants to specify the time coordinate of ζ k L , we can place it along the closed time path at the end of inflation where ζ + k L = ζ − k L .) Expressing the composite operators in the position space, we obtain Since we neglected the higher order contributions of the soft modes using the approximation of s.l.
≈, we replaced the quantum state | Ψ in the first line with | Ψ, 0 ζ .

Conservation of ζ
For the purpose of showing the conservation of ζ, let us introduce Since the metric perturbations transform as for each f {iα} in the action (3.2), we rewrite the action for the heavy fields as Sincef {ia} transforms as a scalar under the dilatation, the metric perturbations included inf {ia} can be expressed only in terms of δĝ(x), which also transform as a scalar.
Taking the first and the second variations of S χ with respect to ζ, we obtain Since we expressed the action as in Eq. (3.27), the terms in which the derivative operates on f {iα} vanish. Variations with respect to the other metric perturbations, δN, e −ζ N i , e −2ζ γ ij , which are in the combination of δĝ, yield and so on. Multiplying a d (t)∂f {iα} /∂g(x)| δg=0 on Eq. (3.22) with n = 1 and all the remaining metric perturbations set to 0 and taking summation over α, we obtain δg ± ζ + (x, y) + W δg ± ζ − (x, y) (3.32) for δg = ζ, δN, N i , γ ij . Notice that the derivative with respect to e −ζ N i and the one with respect to N i give the same answer after setting δg to 0. The same story also follows for γ ij . Here, we expressed δS χ /δζ| ζ=0 (t, k L ) in Eq. (3.22) in the position space. In deriving Eq. (3.22), we used the explicit forms of W (2) δga 1 δga 2 (x 1 , x 2 ), summarized in Appendix B. Multiplying δg ± on Eq. (3.32) and integrating over x µ , we finally obtain the key formula to show the presence of the constant solution as where we performed integration by parts. Meanwhile, multiplying e −ik ′ L ·x on Eq. (3.32) and integrating over x µ , we obtain (3.34) By adding the left hand side of Eq. (3.33) multiplied by a constant parameter −s and Eq. (3.34) with δg ± = ζ ± multiplied by −s 2 /2, the linear and the quadratic terms in the effective action can be given by δga (x) where δg s denote the metric perturbations δg after the inhomogeneous dilatation. For notational brevity, here we used the same notation as those for the global dilatation, given in Eqs. (3.25) and (3.24). Here, each δg i,a (i = 1, 2) sums over δN a,s , N i,a,s , ζ a,s , and γ ij,a,s . In deriving Eq. (3.35), we used The first term in Eq. (3.33) changes the argument of the metric perturbations in the linear term of the metric perturbations in Eq. (3.35). We also changed the arguments of the quadratic terms, taking into account that the modification appears only in higher orders of perturbation. The tadpole contributions, which are the terms in the second line of Eq. (3.35) should vanish by using the the background equation of motion. (See Ref. [19] for a discussion about the heavy scalar field.) Equation (3.35) shows that with the use of the consistency relation, δg α (x) in S ′ eff can be replaced with δg α, s (x). Since the rest of the effective action, S ad , is simply the classical action for the single field model, it also should be invariant under this replacement.
Therefore, when the locality condition (2.39) holds, the total effective action S eff preserves the invariance under the inhomogeneous dilatation with the suitable fall-off , i.e., under the change of δg α to δg s,α . Since Eqs. (3.33) and (3.34) also hold for the soft modes not only for the homogeneous mode with k = 0, we can shift ζ(x) by an inhomogeneous but time-independent function instead of the homogeneous constant parameter. The invariance of the effective action, which also includes the radiative corrections from the non-zero spin massive fields, under the shift of the soft mode ζ k L directly implies the existence of the constant solution for ζ 8 . We assumed the locality condition (2.39) to derive the consistency relation, which was used to show the invariance under the replacement of ζ with ζ s in the effective action. As was argued in Sec. 2.3.2, the validity of Eq. (2.39) requires the linear perturbation of the soft mode ζ k L to be time independent. For that, the decaying mode of ζ k L should die off sufficiently fast after the Hubble crossing. This happens when the background trajectory is on an attractor, e.g., when the heavy fields do not alter this nature of the background classical trajectory. As far as the radiative corrections of the heavy fields χ do not turn the "decaying" mode of ζ k L into a growing mode, the existing constant solution should be the dominant solution of ζ in the large scale limit. This should be the case, when the radiative corrections of the heavy fields remain perturbative. In Ref. [19], we listed the condition for the conservation of ζ in the presence of the radiative corrections of a massive (scalar) field as • The radiative corrections of the heavy field are perturbatively small.
• The background trajectory is on an attractor.
• The quantum system preserves the dilatation invariance. Now the second and third conditions are rephrased by the single condition: • The locality condition (2.39) is satisfied.
This guarantees the presence of the constant solution also for the soft modes ζ k L with k L = 0, not only for the homogeneous mode. The current argument also can apply to massive fields with arbitrary spins, including higher spin fields.

IR divergences of inflationary correlators
In the previous two sections, we showed that the locality condition (2.39) leads to the consistency relation and the conservation of ζ. In this section, as another related subject, we show that the condition (2.39) also plays an important role for the cancellation of the IR divergent contributions.

Overview of IR divergence problem
It is widely known that the loop corrections of a massless perturbation mode such as the curvature perturbation ζ can yield various IR enhancements. (See, e.g., Refs. [48,49,50,51,52,53,54,55,56].) In this section, following Ref. [57], we briefly summarize the IR enhancements. When we perform the perturbative expansion in terms of the interaction picture field ζ I , an interaction vertex which includes ζ I without derivative can yield the radiative correction which is proportional to ζ 2 I . The super Hubble modes with k < ∼ aH contribute to ζ 2 I as aH 0 d 3 k/k 3 , yielding the logarithmic enhancement. We distinguish the divergent contribution due to the modes 0 ≤ k ≤ k c (IRdiv) from the convergent but secularly growing one (∝ ln a) due to the modes k c ≤ k ≤ aH (IRsec). Here, k c denotes an IR cutoff. The IRsec, which cannot be removed by introducing the comoving IR cutoff, originates from the accumulation of the super Hubble modes. The accumulation of the super Hubble modes enhances the time integral at each interaction vertex (SG), introducing another secularly growing term proportional to ln a or increasing the power of ln a included in the integrand. These IR enhancements also can be introduced by the soft graviton.

Cancellation of IR divergence and the locality condition
In the series of papers [39,40,44,57,58,59,60,61,62], we showed that the IR enhancements, i.e., IRdiv, IRsec, and SG, are due to the influences from the outside of the observable region. What we observe in cosmological measurements corresponds to a quantity evaluated in a completely fixed gauge. Fixing the gauge conditions eliminates the influence of the gauge degrees of freedom. However, it is not possible to determine the gauge condition outside the observable region, even if we completely specify the way of observation. Therefore, unless the causality is manifestly ensured as in the harmonic gauge, the degrees of freedom outside the observable region can affect the boundary conditions of the observable region. As was argued in Refs. [39,40], changing the boundary conditions corresponds to changing the spatial coordinates in the local observable region. Such spatial coordinate transformations are the large gauge transformations. We dubbed a variable which is independent of those boundary conditions of the local observable region as a genuine gauge invariant variable.
In Refs. [39,40,44,57,58,61,62], we showed that all the IR enhancements are cancelled out, when we calculate a correlation function of a genuine gauge invariant variable for a specific initial state such as the adiabatic vacuum. In Ref. [61], we argued that whether the IR enhancements in the correlation function of the genuine gauge invariant operator disappear or not depends on the choice of the initial states. In this subsection, scrutinizing the condition on the quantum state to ensure the cancellation of the IR enhancements, we show that when the locality condition (2.39) is satisfied, the IRdiv, which comes from the momentum integral, is cancelled in the correlation functions of the invariant variable under the large gauge transformations. In the following, we denote a genuine gauge invariant variable as g R(x) without specifying it. (One way to construct g R(x) was discussed in Refs. [39,40].) One important property of g R is being constructed only by local quantities such that commute with the soft modes and remains invariant under the inhomogeneous dilatation, i.e., As in Eq. (2.37), but focusing on a single soft mode, we expand the correlation function of g R as where we noted that the soft modeζ k L commutes with the genuine gauge invariant variable. When the "locality" condition holds, since g R commutes with Q W ζ (k L ) we obtain Since the correlator in Eq. (4.3) is independent of the soft modeζ c k L , it commutes with the integral overζ c k L . Then, the divergent integral dζ c k L |ψ(ζ c k L )| 2 in Eq. (4.2) simply becomes the normalization factor, which should be canceled in computing connected diagrams. Here, we picked up a certain wavenumber k L . However, repeating the same procedure for the whole soft modes, we find that all the soft modes which correlate with the hard modes are canceled out. This cancellation yields a suppression of the soft modes which interact with the hard modes and ensures the absence of the IRdiv in the correlation function of g R. In this way, we find that while the quantum state | Ψ is not invariant under the inhomogeneous dilatation, which inserts the soft mode ζ k L , the correlation function of g R for the quantum state | Ψ is insensitive to the insertion. In Ref. [41], this cancellation of the correlation between the soft modes and the hard modes was presented by considering the squeezed bi-spectrum.
Here, let us further discuss the relation between the genuine gauge invariance and the absence of the IRdiv. Changing the boundary condition at the edge of the observable region, we can alter the spatial average of the curvature perturbation ζ [58,39,40]. This can be expressed as the dilatation whose constant parameter s is given by the spatial average of ζ in the observable region. This dilatation changes the constant part of all the modes with k L < ∼ 1/L O , where L O is the size of the observable region, not only the homogeneous mode with k = 0. Therefore, the genuine gauge invariance requires being insensitive to the excitation of the constant soft modes ζ k L . As we argued in Sec. 2.3.2 (see around Eq. (2.54)), quantum states which satisfy the locality condition do not preserve the genuine gauge invariance in the sense that it is not insensitive to the insertion of the constant part of the soft modes ζ k L . What preserves the genuine gauge invariance is the correlation function of the genuine gauge invariant operator evaluated for such quantum states 9 . While the relation between the dilatation invariance and the cancellation of the IRdiv has been discussed in a number of literatures, e.g., in Refs. [39,40,44,58,61,63,64,65,66,67], this aspect has not been clearly described elsewhere.
By contrast, since the curvature perturbation ζ k is not a genuine gauge invariant operator, the correlation functions which includes the operator ζ k suffers from the IRdiv (and also IRsec and SG). Because of that, the correlation functions ζ k L O k S1 · · · O k Sn diverge due to the accumulation of the soft modes. Here, O k S is either ζ k S or Oy k S . In order to make these correlation functions finite, we need to somehow introduce IR regularization. However, recall that a naive introduction of the IR cutoff violates the dilatation invariance, which was the starting point of the discussion about the consistency relation. Therefore, to be precise, we should not understand the consistency relation as the relation between the correlation functions for the Heisenberg operators ζ k L O k S1 · · · O k Sn and O k S1 · · · O k Sn , since they are not well-defined. Instead, the consistency relation we discussed in this paper should be understood as the relation between the "correlation function" for the hard modes O k S1 · · · O k Sn without any propagation of the soft modes and the one with the additional insertion of the free soft mode as an external leg. Then, both of them do not contain the loop corrections of the soft modes, which can lead to the IR enhancements. This is the reason why we needed to employ the approximation s.l.
≈ in deriving the consistency relation.

Relevance and irrelevance of soft graviton insertion
The curvature perturbation ζ and the graviton γ ij are both massless fields and they have similar IR behaviours. In this section, we briefly show that the discussion about the consistency relation and the IR divergence for the graviton proceed almost in parallel to those for the curvature perturbation ζ. For this purpose, we consider the shear transformation, which is a large gauge transformation: where S ij is a constant symmetric and traceless tensor S ij . Under this large gauge transformation, the spatial metric transforms as At the linear perturbation, γ ij is shifted asγ ij = γ ij − S ij . The classical action is invariant under the large gauge transformation (5.1).
Similar to the dilatation, we define the Noether charge for the large gauge transformation (5.1) as where π ij γ denotes the conjugate momentum of γ ij and The invariance of the quantum state under this transformation requires Repeating a similar argument, we find that the invariance is preserved, when the following conditions ∂ ∂γ c |ψ(γ c )| = 0 , are satisfied. Here,γ ij denotes the homogeneous mode of γ ij and | Ψ γ c denotes the projected quantum state into the eigenstate ofγ c ij . Inserting the window function W (x) into the integrand of the Noether charge Q γ and performing the Fourier transformation, we define Q W γ (k L ), which inserts the soft graviton γ ij,k L . Again, we find that when the condition (5.7) can be extended to the soft modes with k L = 0, i.e., with S ij k L ≡Ŵ (k L )S ij , we obtain the consistency relation which relates the correlation functions for the hard modes and those with single insertion of the Gaussian soft graviton (see, e.g., Refs. [14,68]). The quantum state | Ψ changes due to the single insertion of the soft graviton, i.e., because the wave function is not completely flat in the direction of the soft graviton γ ij k L with k L = 0 in contrast to the homogeneous mode, whose wave function is completely flat, satisfying Eq. (5.6). Let us emphasize again that this is not a spontaneous symmetry breaking, because the large gauge transformation (5.1) with an inhomogeneous S ij is not a symmetry of the classical action. The correlation function of the genuine gauge invariant operator g R is insensitive to the insertion of the soft graviton. This ensures the absence of the IRdiv due to the soft graviton in the correlation functions of g R evaluated for | Ψ . By contrast, when we evaluate a correlation function for an operator which does not preserve the genuine gauge invariance, the insertion of the soft graviton changes the correlation function and this can lead to a break down of the perturbative expansion [69,71]. (See also Ref. [72].)

Concluding remarks
The relation among the large gauge transformations, the consistency relations for the soft modes of ζ and γ ij , their conservation in time, and the IR enhancements has been discussed in a number of literatures. However, as far as we understand, this relation was not fully clarified and it was sometimes understood in a misleading manner. The purpose of this paper is to sharpen the argument about the relation among these four. The invariance of the quantum state | Ψ under the dilatation and shear transformations can be preserved, when the following two conditions are fulfilled. First, the amplitude of the wave function ψ should be flat towards the directions for the homogeneous modes of ζ and γ ij . Second, operating the Noether charges Q ζ and Q γ on the quantum state of hard modes is equivalent to additively and time-independently shifting the homogeneous modesζ andγ ij , which interact with the hard modes as described in Eqs. (2.29) and (5.7).
The invariance under these large gauge transformations, just itself, leads to neither the consistency relation nor the absence of IRdiv. The additional conditions (2.29) and (5.7) are the non-trivial extensions to those which describe the insertion of the soft modes k L ( = 0), which are not always satisfied. These conditions can be interpreted as the locality condition, which states that the inhomogeneous dilatation and shear transformations only change the values of ζ and γ ij within each local universe. Since the wave function is not flat in the directions of the soft modes for the curvature perturbation and the graviton, the quantum state | Ψ changes due to the insertion of these soft modes. When the locality conditions (2.39) and (5.8) are satisfied, the influence of these soft modes are described by the wellknown consistency relations under the Gaussian approximation of the wave function for the soft modes. This argument also applies in deriving the consistency relation for massive fields. We also showed that the locality condition (2.39) implies the conservation of ζ in the soft limit within the perturbation theory. The same argument also applies to γ ij .
The final issue is the IR enhancements due to the soft modes of ζ and γ ij . In contrast to the correlation functions for ζ and γ ij , which are not genuinely gauge invariant, the correlation functions for a genuine gauge invariant operator remain invariant under the insertion of the soft modes for the curvature perturbation ζ and the graviton γ ij , when the locality conditions hold. This ensures the absence of the IRdiv due to the soft modes of ζ and γ ij . In this paper, we did not discuss the IRsec and the SG, which yields the secular growth. When the locality conditions hold at each time slicing, repeating the same argument as in Refs. [44,62], we can show the absence of the IRsec and SG. In fact, this is the case when we choose the adiabatic vacuum (or the Euclidean vacuum) as the initial state of the universe.
Recently, the relation among the asymptotic symmetry, the soft theorem, and the IR divergence was discussed about gauge theories in asymptotically flat spacetime [26,27,28,32]. In Ref. [26], it was shown that the Weinberg's soft theorem for the soft photons and gravitons can be derived as the Ward-Takahashi identities for the asymptotic symmetry. (For a recent review, see Ref. [29].) In Ref. [32], the relation between the asymptotic symmetry and the IR divergence of the QED was discussed. About the IR divergence in QED, Faddeev and Kulish showed that the IR finiteness can be guaranteed, when we consider the dressed charged particles by soft photon clouds. (See also Refs. [74,75].) While there is a qualitative difference between the in-out formalism in QED and the in-in formalism in cosmology, our genuine gauge invariant operator g R, whose correlators can be IR finite with an appropriate choice of the initial state, also dresses the clouds of the soft ζ and γ ij as external legs.
At first glance, the IR structures for the gauge fields in the asymptotically flat spacetime have a certain similarity to those for the primordial perturbations ζ and γ ij . However, a closer look may also reveal some differences in these two cases. For the gauge theories in the asymptotically flat spacetime, we are interested in the transition at the asymptotic infinity between before and after the propagations of the soft photons and gravitons. On the other hand, for the primordial perturbations, the asymptotic infinity is the spatial infinity and is out of reach in a causal evolution. Instead, what we are interested in is a locally defined quantity like an actually observable quantity. As is summarized above, in order to derive the consistency relation, the soft theorem for ζ and γ ij , we need to assume the locality condition, which is not trivially satisfied even in a Diff invariant theory, at least in the current gauge choice (the Maldacena gauge [14]). This gives a qualitative difference from the gauge theories in the asymptotically flat spacetime, which trivially satisfies the locality. and ζ s (t, x) = U I † s (t) ζ I s (t, x) U I s (t) , (A.3) where U I and U I s denote the unitary operators which relate the Heisenberg and interaction picture fields for before and after the dilatation transformation. Using the interaction Hamiltonian H I ≡ H − H 0 , the unitary operator is given by Since the free Hamiltonian H 0 changes due to the dilatation transformation while the total Hamiltonian does not, the interaction Hamiltonian H I and U I also change through the dilatation, i.e.,