Strong restriction on inflationary vacua from the local gauge invariance I: Local gauge invariance and infrared regularity

The primordial perturbation is widely accepted to be generated through the vacuum fluctuation of the scalar field which drives inflation. It is, however, not completely clear what is the natural vacuum in the inflationary universe particularly in the presence of non-linear interactions. In this series of papers, we will address this issue, focusing on the condition required for the removal of the divergence from the infrared (IR) contribution to loop diagrams. We show that requesting the gauge invariance in the local observable universe guarantees the IR regularity of the loop corrections beginning with a simple initial state. In our previous works, the IR regularity condition was discussed using the slow roll expansion, which restricts the background evolution of the inflationary universe. We will show more generally that requesting the gauge invariance/the IR regularity leads to non-trivial constraints on the allowed quantum states.


I. INTRODUCTION
It is widely accepted that primordial curvature perturbations originate from the vacuum fluctuation of the inflaton field in the inflationary universe. Different choices of quantum states lead to different statistics of the primordial fluctuation. Among various options of vacuum states, we usually choose the adiabatic vacuum. A free scalar field can be understood as a set of independent harmonic oscillators. Taking the adiabatic vacuum looks reasonable, because each oscillator tends to have an approximately fixed frequency as the time scale becomes shorter than that of the cosmic expansion. However, once we start to take into account the self-interaction of the field, the reasoning of choosing the adiabatic vacuum becomes obscure. Therefore it is an interesting question whether some physical requirement prohibits taking an arbitrary quantum state or not. This is the question we address in this series of papers.
The key idea is to focus on fluctuations that we can measure in observations. Since the region we can observe is limited to a portion of the whole universe, we need to take into account the local property of the observable fluctuations. In case we can get access to information only on a portion of the whole universe, there appear additional degrees of freedom to choose boundary conditions in fixing the coordinates. As shown in our recent publications [1][2][3][4][5][6] and will be briefly described later, these degrees of freedom in the boundary conditions can be thought of as residual degrees of freedom in fixing the coordinates of the local observable universe. The observable fluctuations should be insensitive to the coordinate choice in the local observable universe. We showed that removing this ambiguity significantly affects the infrared (IR) behaviour of primordial fluctuations. In the conventional gauge-invariant perturbation, in which this ambiguity is not taken care of, it is widely known that loop corrections of massless fields such as the inflaton diverge because of the IR contributions . One may expect that performing the quantization after we remove these residual coordinate degrees of freedom in the local universe can cure the singular behaviour of the IR contributions. However, these residual coordinate degrees of freedom are not present if we deal with the whole universe including the spatial infinity, since what we call the residual coordinate transformation diverges at spatial infinity. In this sense, if we think of the whole universe, they are not gauge degrees of freedom In Ref. [1], we imposed the boundary condition at a finite distance to remove the degrees of freedom in choosing coordinates, but the quantization (or equivalently setting the initial quantum sate) was performed considering the whole spatial section of the universe. In that analysis we concluded that there is no IR divergences irrespective of the initial quantum state. However, there were several overlooked points. After we set the initial quantum state at a finite initial time, we need to translate it into the language written solely in terms of the local quantities. Namely, the Heisenberg operators corresponding to the perturbation variables should be transformed into the ones that satisfy given boundary conditions using an appropriate residual spatial coordinate transformation. Since the non-linear part of this transformation is not guaranteed to be IR regular, it can produce IR divergences.
If we send the initial time to the past infinity, one may think that this problem may disappear because our observable region in comoving coordinates becomes infinitely large in this limit. However, this argument is not so obvious. If we think of, for instance, the conformal diagram of de Sitter space, the causal past of an observer never covers the whole region of a time constant slice in flat chart. Furthermore, in this case we also need to care about the secular growth of fluctuation. In Ref. [1], we discussed the absence of the secular growth, focusing only on modes beyond the horizon length scale. To complete the discussion, we also need to include the vertexes which contain modes which are below the Hubble scale, because these vertexes may yield the secular growth through correlations between sub horizon modes and the super horizon modes.
Later in Refs. [3,4], leaving the residual coordinate degrees of freedom unfixed, we calculated quantities that are invariant under the residual coordinate transformation in the local universe under the so called slow-roll approximation. Then, we derived the conditions on the initial states that guarantee the IR regularity. These conditions are the ones that guarantee the absence of additional IR divergences originating from the coordinate transformation mentioned above. However, the physical meaning of these conditions is not transparent from the derivation in Refs. [3,4].
In this paper, as in the conventional perturbation theory and also as in Refs. [3,4], we will perform the quantization over the whole universe. Then, calculating observable quantities, which are invariant under the residual coordinate transformation in the local universe, we will derive the necessary conditions for the initial quantum state to be free from IR divergences more generally, removing the limitation to the slow-roll approximation. We will also show that the IR regularity conditions can be thought of as the conditions that request the equivalence between two systems related by means of the dilatation transformation of spatial coordinates. In this paper, we will consider a case in which the interaction is turned on at a finite initial time. In the succeeding paper, we will discuss the case when the initial time is sent to the past infinity. This paper is organized as follows. In Sec. II, we first show a symmetry of the system, which plays an important role in our discussion. Then, following Ref. [3,4], we construct an operator which remains invariant under the residual coordinate transformation. In Sec. III, we address the conditions on the initial states. In Sec. IV, we will summarize the results of this paper and will outline future issues.

II. PREPARATION
In this section, we will introduce basic ingredients to calculate observable fluctuations. In this paper, we consider a standard single field inflation model whose action takes the form where M pl is the Planck mass and φ is a dimensional scalar field divided by M pl . In Sec. II A, we show a symmetry of this system, which plays a key role in our argument. In Sec. II B, after we introduce a variable which preserves invariance regarding the coordinate choice in the local universe, we provide a way of quantizing the system.

A. Symmetry of the system
To fix the time slicing, we adopt the uniform field gauge δφ = 0. Under the ADM metric decomposition, which is given by we take the spatial metric h ij as where e ρ denotes the background scale factor, ζ is the so-called curvature perturbation and δγ ij is traceless: where the spatial index was raised by δ ij . As spatial gauge conditions we impose the transverse conditions on δγ ij : In this paper, we neglect the vector and tensor perturbations. The tensor perturbation, which is a massless field, can also contribute to the IR divergence of loop corrections. We will address this issue in our future publication. Solving the Hamiltonian and momentum constraints, we can derive the action which is expressed only in terms of the curvature perturbation [29]. Since the spatial metric is given in the form e 2ρ e 2ζ dx 2 , we naively expect that the explicit dependence on the spatial coordinates appear in the action for ζ only in the form of the physical distance e ρ e ζ dx. We first examine this property.
Using the Lagrangian density in the physical coordinates L phys , we express the action as We can confirm that the Lagrangian density L phys is composed of terms which are in a covariant form regarding a spatial coordinate transformation such as h ij ∂ i N j and h ij N i ∂ j ζ. Using we can absorb the curvature perturbation ζ without differentiation which appears from the spatial metric, for instance, as After a straightforward repetition, the Lagrangian density can be recast into Here, to stress the fact that all ζs which remain unabsorbed are associated with differentiation ∂ t or ∂/∂x i , we introduced D which denotes differentiations in general. Using this expression of the action, the Hamiltonian constraint is given by 10) and the momentum constraints are given by Perturbing these constraints, we can show that the Lagrange multipliers N andŇ i are given by solutions of the elliptic type equations: To address the system of the whole universe, the integration in the action is assumed to be taken over the whole universe. Then, by assuming the regularity at the spatial infinity, these Poisson equations can be uniquely solved as (2.13) Substituting these expressions of N andŇ i into the action (2.9), we can show that the action takes the form: (2.14) Here, we put the subscript dx on D to specify the spatial coordinates used in D. The equation (2.14) explicitly shows that the action for ζ only depends on the physical spatial distance. Now we are ready to show the dilatation symmetry of the system, which plays a crucial role in discussing the IR regularity. (See also the discussions in [30][31][32][33][34][35][36].) Changing the coordinates in the integral from x to e −s x with a constant parameter s, we can rewrite the action (2.14) as (2. 15) This implies that the action for ζ possesses the dilatation symmetry As long as we consider a theory which preserves the three-dimensional diffeomorphism invariance, the system preserves the dilatation symmetry. It is because the above-mentioned dilatation symmetry can be thought of as one of the spatial coordinate transformations. This symmetry is discusses also in Refs. [37,38].
To make use of the dilatation symmetry, we introduce another set of canonical variables than ζ(x) and its conjugate momentum π(x). We can show thatζ satisfy the canonical commutation relations as well as ζ(x) and π(x). Actually, using the commutation relations for ζ(x) and π(x), we can verify and also Using Eq. (2.16), we can show that the Hamiltonian densities expressed in terms of these two sets of the canonical variables are related with each other as where on the second equality, we again changed the coordinates in the integral as x → e −s x. Note that the Hamiltonian density forζ andπ is given by the same functional as the one for ζ and π withζ shifted byζ − s. We define the non-interacting part of the Hamiltonian density forζ andπ by the quadratic part of H in perturbation, assuming that s is as small asζ andπ, as follows,H where H 0 is the free part of the Hamiltonian density for ζ and π. Using Eq.
Note that the Hamiltonian densities H I andH I also take the same functional form except for the constant shift ofζ.

B. Residual coordinate transformations and quantization
In this subsection, we consider a way to calculate observable fluctuations. First, we begin with the classical theory. To obtain the action for ζ by eliminating the Lagrange multipliers N and N i , we need to solve the Hamiltonian and momentum constraint equations. As is schematically expressed in Eqs. (2.12), these constraint equations are given by the elliptic-type equations. Here we note that the region where we can observationally access is restricted to the causally connected region whose spatial volume is bounded at a finite past. As far as we are concerned only with the observable region, boundary conditions for Eqs. (2.12) cannot be restricted from the regularity of the spatial infinity, which is far outside of the observable region. The degrees of freedom for solutions of N and N i can be understood as degrees of freedom in choosing coordinates. Since the time slicing is fixed by the gauge condition δφ = 0, there are remaining degrees of freedom only in choosing the spatial coordinates. In the following, we refer to such degrees of freedom which cannot be uniquely specified without the knowledge from the outside of the observable region as the residual gauge degrees of freedom in the local universe. We write the term gauge in the italic fonts, because changing the boundary conditions for N and N i in the local region modifies the action for ζ obtained after solving constraint equations, while it keeps the action expressed by N , N i , and ζ invariant. In this sense, the change of the boundary condition is distinct from the usual gauge transformation, which keeps the action invariant.
The observable fluctuations should be free from such residual gauge degrees of freedom. Following Refs. [3,4], 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 for spatial coordinate transformations, become genuinely gauge invariant, if we evaluate it in the geodesic normal coordinates span 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. We consider the three-dimensional geodesics whose affine parameter ranges from λ = 0 to 1 with the initial "velocity" given by Here we put the index gl on the global coordinates, using the simple notation x for the geodesic normal coordinates, which will be mainly used in this paper. 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 As long as the deviation from the FRW universe is kept perturbatively small, we can foliate the universe by the geodesic normal coordinates. Next, we quantize the system to calculate quantum correlation functions which become observable. A straightforward and frequently used way to preserve the gauge invariance is to eliminate gauge degrees of freedom by fixing the coordinates completely. In the local observable universe, the complete gauge fixing requires us to fix the boundary conditions in solving the constraint equations. However, to quantize the locally restricted system, we need to abandon several properties which are available in the quantization of the whole universe. One is that the quantization restricted to a local system cannot be compatible with the global translation symmetry at least in a manifest way. Another but related aspect is that basis functions for a mode decomposition become rather complicated (even if it exists) than the Fourier modes. To preserve the global translation symmetry manifestly, we take another way of quantization than the quantization in the completely fixed gauge.
In this paper, we perform quantization in the whole universe with the infinite spatial volume. The global translation symmetry in the spatial directions is then manifestly guaranteed in the sense that a shift of the spatial coordinates x to x + a just changes the overall phase factor in the Fourier mode by e ika . Based on this idea, we consider and calculate observable quantities. Since the gauge invariant variable g R does not include the conjugate momentum of ζ, we can consider products of g R at an equal time without the problem of operator ordering. The n-product of g R, i.e., g R(x 1 ) · · · g R(x n ), preserves the gauge invariance in the local universe. To calculate the n-point functions of g R, we need to specify the quantum state as well. One may think that the quantum state should be also invariant under the residual gauge transformations. However, we cannot directly discuss this invariance as a condition for allowed quantum states in this approach, because the residual gauge degrees of freedom are absent when we quantize fields in the whole universe.
Here we note that even though the operator g R is not affected by the residual gauge degrees of freedom, this does not imply that the n-point functions of g R are uncorrelated to the fields in the causally disconnected region. To explain this aspect more clearly, let us consider the n-point function of g R at t = t f whose vertexes are located within the observable region O f . For a later use, we refer to the spacetime region which is causally connected to spacetime points in O f as the observable region O. After expanding of the operator g R in terms of ζ I , the interaction picture field of ζ, even if interaction vertexes which affect g R(t f , x) are confined within the observable region O, the n-point functions of g R have correlation to the outside of O through the Wightman function of ζ I , which can be expressed in the Fourier space as where v k is the mode function for ζ I . Since all the vertexes are confined within O, the spatial distance |x 1 − x 2 | is bounded from above. However, the IR modes with k ≤ |x 1 − x 2 | −1 are not suppressed and let the Wightman function G + diverge for scale-invariant or red-tilted spectrums. This long-range correlation becomes the origin of the IR divergence of loop contributions. Since the observable quantities should take a finite value, we request the IR regularity of observable fluctuations, which is achieved only when the quantum state is selected so that the long-range correlation is properly isolated from the observable quantities. In the following section, we will show that requesting the absence of the IR divergence in fact constrains the quantum state of the inflationary universe.

III. IR REGULARITY CONDITION AND THE RESIDUAL GAUGE INVARIANCE
A simple way to address the evolution of a non-linear system is to solve the Heisenberg equation perturbatively by assuming that the interaction is turned on at the initial time. In this section, taking this setup, we calculate the correlation function of g R up to one-loop order and investigate the IR behaviour.

A. Specifying the iteration scheme
For notational convenience, we introduce the horizon flow functions as with n ≥ 1, but without assuming these functions are small we leave the background inflation model unconstrained.
The assumption that the interaction is turned on at the initial time t i requests where π I is the conjugate momentum defined from the non-interacting action: as π I := 2M 2 pl e 3ρ ε 1ζI . Employing the initial conditions (3.2), we can relate the Heisenberg picture field ζ to the interaction picture field ζ I as where U I is the unitary operator given by Since the Heisenberg fields are related to the interaction picture fields by the unitary operator, the canonical commutation relations for the canonical variables ζ and π guarantee those for ζ I and π I as As we showed in Appendix A, the n-point functions for ζ given in Eq. (3.4) agree with those given by the solution of ζ written in terms of the retarded Green function. We express the equation of motion for ζ schematically as where L denotes the differential operator and the left-hand side of Eq. (3.7) is the same equation of motion as is derived from the non-interacting part of the action (3.3). By using the retarded Green function: which satisfies , the solution of the equation of motion with the initial condition (3.2) is given by where the non-linear term is given by We expand the interaction picture field ζ I , which satisfies as follows (3.14) The mode function v k satisfies L k v k = 0 where The mode function is normalized as where the Klein-Gordon inner product is defined by With this normalization, we obtain the commutation relations for the creation and annihilation operators as Inserting Eq. (3.14) into Eq. (3.9), we can rewrite the retarded Green function as where R k (t, t ′ ) is given by We calculate the n-point function of g R, setting the initial state to the vacuum defined by Next, we explicitly calculate non-linear corrections. Since we are interested in the IR divergence, employing the iteration scheme of L −1 R , we pick up only the terms which can contribute to the IR divergences. For the convenience, we here introduce the symbol " IR ≈" as in [4] to denote an equality under the neglect of the terms unrelated to the IR divergences. The IR divergences mean the appearance of the factor ζ 2 I . For the scale invariant spectrum, this variance diverges logarithmically. Once a temporal or spatial differentiation acts on one of two ζ I s in ζ 2 I , the variance no longer diverges. In this sense we keep only terms which can yield ζ 2 I . When we write down the Heisenberg operator ζ in terms of the interaction picture field ζ I , at the one loop level, this is equivalent to keep only the terms without temporal and/or spatial differentiations and the terms containing only one interaction picture field with differentiation.
In this paper, we refer to a term which does (not) contribute to the IR divergences simply as an IR (ir)relevant term. In the following, we will use where R is a derivative operator which suppresses the IR modes of ζ I such as ∂ ρ and ∂ i /ρe ρ . Equation (3.22) can be shown as follows. The Fourier transformation of L −1 R ζ I Dζ I is proportional to where ζ I k and (Rζ I ) k denote the Fourier modes of ζ I and Rζ I . Since (Rζ I ) k−p (t ′ ) − (Rζ I ) k (t ′ ) is suppressed and ζ I,p becomes time independent in the limit p → 0, the IR relevant piece of the integrand of the momentum integral can be replaced by We will also use Keeping only the IR relevant terms, the action for ζ is simply given by where we used the lapse function N and the shift vector N i , given by solving the constraint equations as follows, The action (3.25) can be easily derived from the action for the non-interacting theory. Extending the action (3.3) to a nun-linear expression which preserves the dilatation symmetry, we obtain where the abbreviated terms do not appear in the non-interacting action. The abbreviated terms should have more than two fields with differentiation. Such terms are IR irrelevant up to the one-loop order. Thus, we obtain Eq. (3.25).
Taking the variation of the action with respect to ζ, we obtain the evolution equation as By expanding ζ as ζ = ζ I + ζ 2 + ζ 3 · · · , the equation of motion is recast into where we defined where we noted the properties of L −1 R given in Eqs. (3.22) and (3.24).

B. Calculating the gauge invariant operator
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 Using the geodesic normal coordinates, the gauge-invariant curvature g R is expressed as Here, we introduce the spatial average of ζ at the local observable region whose spatial scale is L t in the geodesic normal coordinates as where W Lt (x) is the window function which specifies our observable region. We approximate this averaging scale at each time t by the horizon scale, i.e., L t ≃ {e ρ(t)ρ (t)} −1 . Using gζ (t), we decompose the spatial coordinates as Using the window function W Lt (x), we define a local average of an operator O(t, x) as We note that the factor e −{ζ(t,e −ζ x)− gζ (t)} , which can be expanded in perturbation as yields only IR regular contributions up to the one-loop order such as (ζ I −ζ I )ζ I and ζ I x · ∂ x ζ I . The latter one also becomes finite because of the derivative suppression and the finiteness of |x| 1 . For the reason mentioned above, neglecting the factor e −{ζ(t,e −ζ x)− gζ (t)} , we rewrite the gauge invariant operator g R as Since the spatial curvature s R is given by using the curvature perturbation in the geodesic normal coordinates: we can describe the gauge-invariant spatial curvature g R as at least up to the third order in perturbation. At the second equality, we again used the fact that the exponential factor e −(ζ− gζ ) do not give IR relevant contributions. As is pointed out by Tsamis and Woodard in Ref. [39] and also in Ref. [40], using the geodesic normal coordinates can introduce an additional origin of UV divergence. We suppose that replacing ζ in the geodesic normal coordinates with gζ , which is smoothed by the window function, can moderate the singular behaviour. Inserting the solution of ζ given in Eqs. (3.33) and (3.34) into Eq. (3.42), we obtain In deriving the expression of g ζ 3 , we used We can verify that the term with (1 − L −1 R L) in Eq. (3.45) vanishes. Since this term is eliminated by operating L, it always vanishes if its value and its first time derivative are both zero at the initial time, which is automatically satisfied by the definition of the retarded integral.

C. Conditions for the absence of IR divergence
We next discuss the condition that the IR divergence does not arise in the expectation values of the gauge invariant variable g R. Then, using the above expression, two point function of g R up to the one loop order is obtained as where we used an abbreviated notation ∂ 2 (a) := ∂ 2 xa and ∆ (a) := (e ρρ ) −2 ∂ 2 xa for a = 1, 2. As we mentioned at the end of the preceding section, the correlation function of g R can contain IR divergences. One may think that the IR regularity is guaranteed if (2L −1 R ∆ + x·∂ x )ζ I = 0 is imposed. However, this condition is in conflict with the use of the ordinary retarded integral in the iteration process. In fact, one can calculate (2L −1 R ∆ + x·∂ x )ζ I as where L −1 R,k is the Fourier mode of L −1 R . The requirement that this expression should identically vanish leads to (3.49) The first term is independent of x, while the second one manifestly depends on x. This shows that the condition (2L −1 R ∆ + x· ∂ x )ζ I = 0 is incompatible with the use of the retarded integral.
In place of this naive condition, an alternative possibility one can think of is to impose where φ(k) is an arbitrary real function. This condition continues to hold once it and its time derivative are satisfied on a certain time slice because both sides of this equation vanish under an operation of the second order differential operator L. The condition can be rewritten as a condition on mode functions which is not contradictory as in the case of Eq. (3.49). With these conditions, the expectation value in Eq. (3.47) can be summarized in total derivative form as where we used k 3/2 D = e −iφ(k) k · ∂ k k 3/2 e iφ(k) and dΩ k denotes the integration over the angular directions of k. Since this integral of total derivative vanishes, the IR divergence disappears. Although requesting the condition (3.52) can make the IR divergence vanish, a bit more careful thought rules out this possibility. Since the left hand side of (3.52) and its time derivative vanish at the initial time, the condition (3.52) requests Dv k (t i ) = Dv k (t i ) = 0. Operating the differentiation k · ∂ k on the normalization condition of the mode functions, leads to a contradiction. The right hand side trivially vanishes after the operation of k · ∂ k , while the left hand side gives 3. Even though the IR regularity condition (3.52) cannot be compatible with the initial conditions (3.2), it is still instructive to give an alternative interpretation of the condition (3.52). We adopted the initial condition (3.2) for {ζ, π}, which identifies the Heisenberg fields with the corresponding interaction picture fields at the initial time and selected the vacuum state for the free field at the initial time. We denote a set of operations which specify the interacting quantum state by an iteration scheme. In the canonical system of {ζ, π}, we fixed the iteration scheme by Eq. (3.2) and Eq. (3.21). Then, when we take the same iteration scheme in the canonical system {ζ,π}, it is not obvious whether the same vacuum state as the one in the canonical system {ζ, π} is picked up or not. Before closing this section, we show that the condition (3.52) is identical to the condition that these two vacua are equivalent.
Since the transformation from {ζ, π} to {ζ,π} is a canonical transformation, the correlation functions for the same initial state calculated in these two canonical systems should agree with each other, i.e., To employ the same iteration scheme, we request that both of ζ andζ are solved by using L −1 R by identifying the Heisenberg fields with the interaction picture fields at the initial time. We expand the interaction picture fields for ζ andζ, which we denote as ζ I andζ I , respectively, in terms of the same mode function v k as Further, we select the vacuum states that are erased under the action of a k andã k in the respective systems. We denote these vacuum states specified by a k andã k as | 0 and |0 , respectively. Now we show that the equivalence between the two point functions, i.e., 0 |ζ(x 1 )ζ(x 2 )| 0 = 0 |ζ(t, e s x 1 )ζ(t, e s x 2 )|0 , (3.57) yields the condition (3.52). We expandζ(t, e s x i ) as taking into account that the interaction Hamiltonian is shifted by −s. Then, the right-hand side of Eq. (3.57) gives 0 |ζ(t, e s x 1 )ζ(t, e s x 2 )|0 where "· · · " denotes higher order terms in perturbation. Since the terms in the second line of Eq. (3.59) agree with the left-hand side of Eq. (3.57), the other terms on the right-hand side of Eq. (3.59) should vanish to satisfy Eq. (3.57). The remaining terms of order s at the leading order inζ I in Eq. (3.59) is given by where we performed integration by parts. Now it is clear that requesting Eq. (3.54) gives the same condition as requesting the IR regularity (3.52), except for the irrelevant k-dependence of the phase of the mode functions.

IV. SUMMARY AND DISCUSSIONS
We have focused on observable quantities which are invariant under the residual gauge degrees of freedom. These degrees of freedom are left unfixed in the ordinary discussion of cosmological perturbation. The key issue for the correlation functions of the gauge invariant operator to be IR regular is shutting off the long range correlation between observable quantities and the fluctuation outside the observable region. This is the matter of how we choose the initial quantum state. The leading effect of the long range correlation on local observables is the constant shift of the so-called curvature perturbation (= the trace part of spatial metric perturbation). The constant shift of the curvature perturbation can be absorbed by the dilatation transformation of the spatial coordinates. Assuming that the interactions are shut off before the initial time, we investigated the conditions that guarantee the equivalence between two systems mutually related by the dilatation, and found that these conditions also guarantee the IR regularity of observable quantities. Therefore, we can think of the IR regularity condition (3.52) as requesting the invariance under the dilatation, which is one of the residual gauge transformation.
We also found that these conditions are not compatible with the prescription where the vacuum state is set to the one for the corresponding non-interacting theory at an initial time. In this setup, the initial time is very particular time, at which the Heisenberg picture fields agree with the interaction picture fields. Then, the curvature scale at the initial time becomes distinguishable from other scales. It is, therefore, natural that the presence of the particular initial time breaks the invariance under the scale transformation. One possibility to avoid breaking the dilatation symmetry is sending the initial time to the infinite past. When we send the initial time to the past infinity, the IR regularity no longer requests the condition (3.52), because Eq. (3.22) does not hold in this limit. At Eq. (3.23), we took ζ p out of the time integration, because ζ p becomes constant in time in the limit p/e ρρ ≪ 1. However, since all the modes become the sub Hubble modes at the distant past, the same argument does not follow when we send the initial time to the past infinity. Thus, when we keep the interaction turned on from the past infinity, our claim in this paper does not prohibit the presence of initial states which guarantee the IR regularity.
Then, the question is whether there is an initial state (or an iteration scheme) which guarantees the IR regularity/the gauge invariance in the local observable universe. In our previous papers [3,4], we found that when we specify the relation between the Heisenberg picture field ζ and the interaction picture field ζ I in a non-trivial way and choose the adiabatic vacuum as the vacuum state for the non-interacting theory, the IR contributions in the two-point functions of g R are regularized. The result of the present paper shows that the relation between ζ and ζ I imposed in Ref. [3,4] is different from the relation fixed by the initial condition (3.2), because the mode equation for the adiabatic vacuum does not satisfy Eq. (3.52). We naturally expect that the IR regular vacuum we found in Ref. [3,4] corresponds to the iteration scheme where the interaction has been active from the past infinity.
In general, when we keep the iteration turned on from the past infinity, the time integration at each interaction vertex does not converge. The iǫ prescription provides a noble way to make the time integration converge. The adiabatic vacuum is the vacuum state which is selected by the iǫ prescription. There is another advantage to fix the iteration scheme by the iǫ prescription. The correspondence between the IR regularity and the gauge invariance provides an important clue to prove the IR regularity. Our result in the simple iteration scheme suggests that the IR regularity of the loop corrections may be ensured, if we employ the iteration scheme which satisfies Ω |ζ(x 1 )ζ ( x 2 ) · · · ζ(x n )| Ω = Ω |ζ(t 1 , e −s x 1 )ζ(t 1 , e −s x 2 ) · · ·ζ(t 1 , e −s x n )|Ω , (4.1) where | Ω and |Ω are the initial states selected by the iteration scheme in the two canonical systems {ζ, π} and {ζ,π}, respectively. Since the iǫ prescription can be shown to select the unique state, which becomes the ground state when the Hamiltonian is conserved in time, we expect that the condition (4.1) can be satisfied if we fix the integration scheme by the iǫ prescription. In our succeeding paper [41], we will verify this expectation and will show that the IR regularity and the absence of the secular growth can be ensured if we employ the iteration scheme with the iǫ prescription.
where we used the abbreviated notation x N := (t N , x) and x α := (t, x α ) for α = 1, · · · n. We defined the operator C as Repeating this procedure, we can show that the n-point functions (A4) agree with the n-point functions for ζ(x) which is iteratively solved by using the retarded Green function. Here, for illustrative purpose, we considered a simple case, but this argument can be generalized in a straightforward manner.