Inflationary spectral tilts as a result of the dilatation symmetry breaking

We derive the spectral indices and their runnings of single inflation models by a new approach. We perform a dilatation transformation to the linear cosmological perturbations and derive a current (non-)conservation law. Using it, we construct a dilatation charge and a Ward-Takahashi identity for the two-point correlators, and derive two \textit{exact} expressions for the tree-level spectral indices. First, we apply the slow roll expansion to one of the exact expressions. We calculate the spectral indices and their runnings up to the second and the third order of slow roll parameters respectively, with use of the"horizon crossing formalism". By construction, our results are more rigorous and generic than the previous works. Then, we analyze another exact expression to understand how the perturbations and the slow roll parameters contribute to the spectral indices. By a numerical calculation, we confirm that only the behaviors of the slow roll parameters during a few e-folds around the horizon crossing affect significantly to the values of spectral indices. The analysis in this article indicates that if one cannot use the slow roll parameters, regardless of their values, as the expansion parameters around the horizon crossing, then one can no longer apply the slow roll expansion to the spectral indices, and it is thus necessary to apply the more generic method introduced here.


I. INTRODUCTION
Inflation is the phase of accelerated expansion in the early universe [1][2][3]. It can resolve the three big problems of the big bang universe and can create the seed of the large scale structure by the quantum fluctuation around the quasi de Sitter background. Inflation is supposed to end and transit to the big bang universe accompanied with the reheating process. The observation of Cosmic Microwave Background (CMB) anisotropy [4,5] indicates that the temperature fluctuations of CMB are essentially gaussian, which is in concordance with the quantum fluctuations predicted by the inflationary theory [6][7][8][9]. Thus, at the present time, inflation is considered as the standard theory of the early universe.
The observations of CMB anisotropy have imposed significant constraints to the numerous inflationary models. Initiated by the COBE project [10], WMAP [11], PLANCK [4,5], Keck/BICEP [12] projects have clarified that the primordial E-mode power spectrum of the temperature fluctuations is red-tilted in the Fourier space, and there is only a tiny fraction of the B-mode power spectrum. Mainly by the PLANCK, the changing ratio of E-mode spectral tilt and the size of non-gaussianities of E-mode correlations are also constrained. While some inflationary models have been excluded by these observational facts, some other models still survive. Not only the potential-driven inflation [13][14][15][16], but also the kinetically driven inflation [17][18][19] and their hybrid type mod- * Electronic address: rsaito@ntu.edu.tw els [20][21][22] can still accord with the present observational constraints. To further distinguish and exclude them, we need more precise and accurate observations. It is expected that the next generation projects, such as PRISM, Lite-BIRD, CMB-S4, AliCPT, would obtain much more precise constraints to the scalar spectral tilt, its running and the ratio of E-mode and B-mode spectra. This, in turn, requires to derive more precise theoretical predictions for those observables.
From the theoretical side, the inflationary observables are derived mainly by using the slow roll parameters. If and only if the slow roll parameters are small enough during the inflation, we can use them as expansion parameters for the observables. Among the observables, the spectral indices and their runnings have been derived up to the second order of the slow roll parameters [30][31][32][33]. The previous results are, however, restricted to the potential-driven inflation models only. Further, in [30], the authors imposed an ansatz for the model parameters to derive the power spectrum. In [32,33], the authors performed an uncertain Taylor expansion for the power spectra, where the convergence of the expansion is not guaranteed. Moreover, their predictions rely on the slow roll expansion, so that we cannot apply them to a class of models where the slow roll parameters are not necessarily small throughout the inflation. In fact, many models are reported, in which the predictions of the slow roll expansion crucially deviates from the numerical calculation (e.g. [36][37][38][39]). Therefore, the previous methods according to the slow roll expansion are far from rigorous and generic.
On the other hand, there is another expansion method for the inflationary fluctuations, known as δN formalism [40][41][42][43]. This method is based on the leading order gradi-arXiv:1812.08981v1 [hep-th] 21 Dec 2018 ent or momentum expansion parametrized by = k/aH, where k is the spatial momentum of the fluctuations, a is the scale factor of a local homogeneous universe and H is the Hubble parameter of the local universe. The formalism is reliable up to the first order of , and it is applicable only for the fluctuations on the super-horizon scales where 1. Thus, we cannot use it for the models in which O( 2 ) terms induce significant effects to the fluctuations. Then, the gradient expansion has been recently extended up to the next-to-leading order for limited cases only [44]. Therefore, at the present, we cannot deal strictly with various models which can potentially survive the observational constraints. To face with the coming high precision observations, it is necessary to derive more rigorous and versatile theoretical predictions for the observables.
In this article, we derive the tree-level spectral indices exactly for single field inflation models by a new approach. By definition, the indices are intrinsically the scaling dimensions of the two-point correlators in the Fourier space. Hence, if we perform a dilatation transformation to the two-point correlators, we should naturally obtain exact expressions of the spectral indices. According to this idea, we define the dilatation transformation to the linear cosmological perturbation theory and construct the dilatation charge. Then, by using the Ward-Takahashi (WT) identity involved with the charge, we derive two expressions of the spectral indices for the scalar and the tensor perturbations without any ansatzes nor approximations. By construction, both expressions are exact and applicable to all wavelengths and all times for generic models. One of the two expressions is suitable for the slow roll expansion, while the other is suitable for numerical calculations. Since we ignore the quantum effects, our results remain at the classical level. It is still enough, however, to compare with the previous results for linear perturbations. We note that the dilatation transformation and the charge here are entirely different from those which are used to relate the three-point correlators with the two-point correlators [45][46][47][48]. In the previous works, the authors performed large gauge transformations that are included into the diffeomorphism, which were applied to the soft modes of perturbations only. Here, we perform instead the dilatation transformation that is included into the conformal transformation regardless of the wavelengths of the perturbations.
This article is organized as follows. In Sec. II, firstly, we give generic actions for the linear perturbations. Next, we perform the dilatation transformation, and derive the current (non-)conservation law and the charge. Then, constructing the WT identity, we derive the exact expressions for the spectral indices. In Sec. III, we expand the spectral indices by the slow roll parameters up to the second order. We derive rigorous and generic expressions for the spectral indices and their runnings, and compare the results with the previous ones. In Sec. IV, we analyze one of the exact expressions in detail and perform numerical calculations of the scalar spectral index for the Starobinsky model [2]. We compare the result with the slow roll expansion and the 1/N * expansion. Sec. V is devoted to the summary. We add two appendices. In Appendix A, we deform one of the exact expressions for the indices to the other exact expression. In Appendix B, we consider two specific models, power-law inflation [49] and ultra slow roll inflation [50], for the slow roll expansion. We use the unit 8πG = c = = 1 throughout this article.

II. WARD-TAKAHASHI IDENTITY
We consider single field inflation models in a generic way. Instead of specifying a model, we just give a generic action S[φ, g µν ], which consists of the scalar field φ(x) and the metric field g µν (x). We assume that the given theory possesses the 4-dimensional diffeomorphism invariance and has the flat Friedmann-Lemaître-Robertson-Walker background solution. For the single field inflation models, the background dynamics is governed by the homogeneous part of the scalar field φ(t).
To consider the cosmological perturbation theory around the background, we use the Arnowitt-Deser-Misner formalism and take the comoving gauge as follows: ζ is the curvature perturbation and γ ij is the tracelesstransverse modes of tensor perturbation. In this gauge, the generic action reduces to where K ij and 3 R ij are the extrinsic and intrinsic curvature, respectively, and the dots denote irrelevant higher derivative terms. Following the derivation in [51,52], we can naturally obtain the generic free actions of scalar perturbation and tensor perturbation: where we restrict the perturbations to possess the Lorentz-invariant scaling. These generic free actions include all of Horndeski theory and a part of Degenerate-Higher-Order-Scalar-Tensor theories. We define the speeds of sound, the conformal times and the canonical perturbations for the scalar and the The free actions of canonical fields have the same forms, so that we can analyze their evolutions in the same manner.
To derive the spectral indices in the Fourier space, first, we consider a dilatation transformation defined in the coordinate space as where the label I = {s, t}, x µ I = (τ I , x) and α is a global constant parameter. Note that even if we redefine τ I by shifting a finite constant from the present definitions (5) and (7), we can always obtain the same transformation as Eq. (12) by acting a time translation to the redefined τ I . Thus, the definition of τ I never affects the results below. Under the dilatation transformation, the Lie derivatives of the field and the Lagrangian density become where ∂ Iµ := ∂/∂x µ I . We do not take a contraction for the label I. We define (non-)conserved current as By using the Euler-Lagrange equation for u I (x I ), we can easily find a current (non-)conservation law θ I represents the breaking size of dilatation invariance of the free action for u I (x I ). By integrating Eq. (16) on the spacetime manifold, we can derive a relation between the dilatation charge Q I and the breaking term where the spatial integration covers all of the spatial hypersurface. τ I0 is an arbitrary time so that the τ I0 dependence of the charge and of the last line of Eq. (18) will cancel out each other. This implies that we can take τ I0 to −∞ and extrapolate the behaviors of the perturbations and the background before the inflation to both sides of Eq. (18) even if we do not know the actual behavior of the whole of system before the inflation. The contribution from the extrapolated part will cancel each other out on both sides. We will extrapolate the field behaviors as in the Minkowski spacetime at the past infinity as usual. Now, we canonically quantize the system. We define the conjugate momentum as and introduce the canonical commutation relations at the same time Then, we perform the Fourier expansion for the field The Heisenberg equation for π I require the mode function u Ik to be satisfied with the Mukhanov-Sasaki(MS) equation By imposing a normalization for the Wronskian W , we can normalize the commutation relations between the creation and annihilation operators as We define the free vacuum |0 as In Eq. (23), if the time-dependent mass term dominates over the oscillation term, namely, k 2 |m 2 I |, the mode function has the general asymptotic solutions A I and B I are the normalization constants dependent on the spatial momentum k only. We may determine them by requiring the free vacuum corresponds to the Bunch-Davies vacuum. We call the former solution z I and the latter solution z I τ I dτ I /z 2 I as the growing mode and the decaying mode respectively.
Then, using Eq. (18), we write a WT identity for the two point function as where we take τ I0 = −∞. In general, of course, the above identity does not hold for an interacting field since the naive relation (18) will be changed by quantum corrections. However, at least for the free field, we can still use the classical one (18) as a relation between the quantum operators. We can calculate both sides of identity straightforwardly. The left hand side becomes We discarded the boundary terms of integration. On the other hand, after several integration by parts, the first term of the right hand side becomes where the quantities with the subscript 0 are evaluated at the time τ I0 = −∞. We extrapolate the mode function as it obeys Eq. (23) even before the inflation. We also extrapolate the mass at the past infinity as it approaches to zero faster than 1/τ I based on the estimation m 2 We note that these extrapolations do not need to follow the actual history of the universe before the inflation since the τ I0 dependence will cancel each other on both sides of WT identity. It is sufficient to fix the mass and the mode function at past infinity, with which one can perform the calculation straight-forwardly. Then, we find the first term of the right hand side becomes zero: The breaking term becomes Equating the both sides, we obtain the WT identity in the Fourier space as We define the tree level power spectra of the scalar perturbation and the tensor perturbation as The scalar and tensor spectral indices are defined as These relations imply To derive the explicit expressions of the spectral indices, we substitute Eqs. (39) and (40) to the WT identity (35). Then, we finally obtain (42) where N s = 1, N t = 0. We can deform n I into another form which is useful for a numerical calculation. Performing several integrations by parts with use of Eqs. (23) and (24), we obtain where we take the contour the same as the iε prescription. See Appendix A for the technical detail of this deformation.
Here, we stress that all of the above results (41)-(43) are exact expressions of the tree level spectral indices without any ansatzs nor approximations. Basically, we can use them for all time and all wavelengthes, not restricted to the super-horizon scales. We can apply Eqs. (41)- (43) to all of the single field inflation models which possess the Lorentz-invariant scaling even if the background and the perturbations evolve in highly nontrivial ways.

III. SLOW ROLL EXPANSION
In this section, we expand the exact expression (41) by the slow roll parameters up to the second order and compare the results with the previous works. First, we introduce the e-folding number n as where a in is an initial finite value of the scale factor at a certain time, and thus n in = 0. If the universe always expands, we can use n as a clock. Next, we define the generic slow roll parameters following [23] as where H := d lna/dt is the Hubble parameter and F I and G I are the coefficient functions appearing in the original free actions (3) and (4). By using n and the slow roll parameters, we can expand the conformal time as where ', n' denotes the differentiation with respect to n. The subscript i on δ Ii corresponds to the order of slow roll parameters. Using them, up to the second order, the breaking size θ I becomes We denoted higher order terms of the slow roll parameters than the second order by O( 3 i ). To evaluate Θ I (42) up to the second order of slow roll parameters, we expand the mode function also as follows: where u (i) Ik is the i-th order quantity of slow roll parameters. Since θ I is already of the second order, we need the zeroth order term of the mode function u (0) Ik only. In the slow roll expansion, the zeroth order term of mode function in the quasi-de Sitter spacetime must be given by that in the exact de Sitter spacetime, so we can replace the mode function in Θ I with Substituting Eqs. (50) and (52) into Eq. (42), we obtain where γ E is the Euler-Mascheroni constant and C ≈ −0.729637. In the calculation, we treated the second order product of slow roll parameters as a constant since its variation becomes of the third order. Now, we focus on the case where the asymptotic solutions of mode function (28) are governed by the growing mode 1 In this case, ψ Ik = u Ik /z I and the spectra (36) depend on the spatial momentum only, which implies that they almost conserve on the superhorizon scales. The asymptotic values of spectral indices become We note that although Eq. (56) seems to depend on the time, actually it does not since the asymptotic values of spectra depend on the spatial momentum only. Therefore, we can change the time in Eq. (56) to an arbitrary time. If we change it to the horizon crossing time for the mode k * , we finally obtain the asymptotic values of spectral indices 1 For other specific cases, see Appendix B.
in a generic form as All of the slow-roll parameters are evaluated at the horizon crossing time for the mode k * when k * = a * H * (n * ). The logarithmic term represents the deviation between the modes k and k * , and it disappears if we choose k = k * . We can safely ignore the higher order terms denoted by O( 3 i * ) for the models that all of the slowroll parameters are small enough at the horizon crossing time. We stress that we evaluate the asymptotic values of spectral indices, say, the values at the end of inflation, by using the slow roll parameters evaluated at the horizon crossing time. This is the "horizon crossing formalism" itself.
To derive the runnings of spectral indices defined as we choose k = k * , so that we can relate the spatial momentum to the horizon crossing time through k = k * = a * H * (n * ). By this, we get a relation between differentiations Then, we can obtain the asymptotic values of the runnings of spectral indices up to the third order where all of the slow roll parameters are evaluated at the horizon crossing time for the mode k = k * .
As the simplest example, we consider the following case which is obtained for the canonical scalar field models with the potential. Using Eq. (58) and (61) for this case, up to the second order, we obtain These correspond to the previous results [30,33]. In our formalism, however, the results are more rigorous and generic. While the previous results rely on an ansatz for z I or the Wentzel-Kramers-Brillouin approximation for the mode function, our slow roll expansion does never need any ansatzes nor approximations. Also, we can easily derive the spectral indices and their runnings for the generic models without deriving the power spectra themselves. Further, in [33], they perform an uncertain Taylor expansion for the asymptotic values of spectral indices to evaluate them by the slow roll parameters at the horizon crossing time. We showed explicitly, however, that we do not need such an uncertain Taylor expansion and can evaluate the spectral indices just by applying the horizon crossing formalism. We derived the runnings of spectral indices up to the third order also. We comment on tiny effects coming from around the end of inflation. In typical scenarios, the inflation ends when the slow roll parameters rapidly grow and reach to O(1) values. Thus, around the end, the slow roll expansion would break down. However, this hardly affects the final results since in the case the perturbations freeze out on super-horizon scales, the mode function remains almost the same as the asymptotic value (54). On the other hand, the exact ending time of the inflation differs from |kτ I | = 0, and thus the asymptotic values of spectral indices have errors, which are represented by the O(kτ I ) term in Eq. (53). Those errors are, however, extremely small since at the end of inflation, |kτ I | ∼ e −50 for the observational window of the mode k. Therefore, in substance, we can use the asymptotic values of the spectral indices (58) and their runnings (61) all the way to the end of the inflation.
We cannot apply, however, the slow roll expansion to the models in which the slow roll parameters grow above a certain limit nearby the horizon crossing time firstly, and then turn to small values again on the super-horizon scales. In these models, the mode function around the horizon crossing time behaves in a highly non-trivial way, and the contributions from Θ I and/or τ I deviate from the simple slow roll expansion significantly. We can further appreciate this by looking at the other expressions for the index (43). We study their behaviors in the next section.

IV. NUMERICAL ANALYSIS
In this section, we perform a numerical analysis for the Starobinsky model. First, we give a numerical setup for the background and the perturbation of the model. We focus on the scalar perturbation only since at the present, the tensor spectral index is not yet observed. Next, we compare the numerical result of slow roll expansion with the 1/N * expansion for the model. Then, we analyze the behavior of integration in Eq. (43), quoting the perturbation on the exact de Sitter background as a schematic example. We compare the numerical result of Eq. (43) for the scalar perturbation with the results of slow roll expansion and 1/N * expansion also.

A. Equations
We consider the Starobinsky model which has the following potential in the Einstein frame. M is a model parameter which has the mass dimension 1. The Friedmann equations and the equation of motion of the scalar field are expressed as where the dot denotes the derivative with respect to the cosmic time t. If we use the e-folding number (45) as a clock, we can rewrite Eqs. (65) as Combining Eqs. (66)-(68) gives the following single variable equation: We will use Eq. (69) as a background equation. The behavior of the Hubble parameter will be obtained by substituting the solution of Eq. (69) into Eq. (66). The relevant slow roll parameters for this model are given as While, the dynamics of the scalar perturbation ψ sk = u sk /z s is governed by the following equation: which can be reduced from Eq. (23). Here, z s for the model was given by z s = a √ 2 1 . Then, we define the end of inflation as the time when and we consider the scalar perturbation ψ sk which leave the horizon 54 e-folds before the end of inflation. Namely, we determine the wavenumber k of perturbation as k * = a * H * (n * ), where n * is satisfied with Initial Conditions We give the initial conditions and the mass parameter M for the background as so that we get The above initial conditions yield large e-folding number of the slow roll regime; n end 119.897. The background approaches rapidly to the slow roll attractor regime whereφ −V ,φ /3H, so that after a few e-folds, the dynamics of φ(t) is hardly affected by the choice of initial values.
For the scalar perturbation, because of the problem of calculation costs, we will take n pin as an initial e-folding number which satisfies n end − n pin = 64. Then, we will redefine n as n − n pin , so that in the new definition, we obtain n pin = 0, n * = 10 and n in −55.897. At the time n pin = 0, the perturbation which leave the horizon at n * = 10 stays deep inside the horizon yet, so we approximate its initial values by those on the exact de Sitter background. Using Eq. (52) with τ s −1/aH, we give the initial values of scalar perturbation at n pin as ψ sk,n (n pin ) π sk − z s,n z s + 1 where π sk and σ sk are the real and the imaginary part of ψ sk respectively. We set the initial phase of the mode function u (0) sk to be zero for simplicity, which can be absorbed into the initial phase of the vacuum.

B. Slow roll expansion vs 1/N * expansion
Using the slow roll expansion, the spectral index n s for the Starobinsky model is derived as Eq. (63), and its numerical value gives Whereas, the large N * expansion up to O(1/N 2 * ) for Eq. (63) gives the following expression: These two expansions differ slightly within the range of 1 − 2 . Since 1/N * expansion here is just an approximation up to O(1/N 2 * ) and the definition of N * is entirely based on the slow roll approximation 2 , the result based on the slow roll expansion (80) would be more reliable than that of the 1/N * expansion (81). If the sensitiv- scalar spectral index, it would be better to use numerical results of the slow roll expansion as the theoretical prediction for a fixed N * .

C. Analysis of the other expression for the spectral index
Here, we use the expression (43) for the scalar perturbation (82) to evaluate the scalar spectral index n s .
First, as a schematic example, we consider the above expression for the perturbation on the exact de Sitter background. In that case, we obtain the exact solution for the mode function as Eq. (52), and we give z s = a. Using a new variable x s := kτ s and Eq. (44), the expression (82) reduces to Note that x s is always a negative number. We can easily perform the integration and get the exact solution which is consistent to a direct calculation from Eq. (52), (36) and (37). Then, we consider Eq. (83) for the time |x send | 1. Up to the second order of x send , we obtain This is also consistent to the exact solution (84) up to the second order. We find that only the imaginary part of the above integration contributes to the spectral index at While, the e-folding number from the end to the horizon crossing time is defined as which does not include theφ dependence of N * .
the end |x send | 1. For a further analysis, we separate the integration in (85) as follows: where x s> and x s satisfy x s is the time when sin 2x s is at any of extrema, and thus the third part of integration becomes zero: This implies roughly that from the sub horizon scales, there is no contribution to the spectral index at the time x send since the integrand oscillates periodically. We take x s as the time corresponding to the last extremum of the sine function after the horizon crossing. The first part of integration (86) is the contribution to n dS s (x send ) from the super-horizon scales where |x s | 1, and the second part is that from only a few e-folds around the horizon crossing. Up to the second order of x s> , the contribution from the first part becomes Im x send xs> e −2ixs dx s = Im where we ignore O(x 2 send ) terms. This shows clearly that there is less contributions to n dS s (x send ) from the superhorizon scales, which matches to the conservation of the perturbation (see Fig. 1). In the following, we ignore the contribution from the super-horizon scales. Eventually, in Eq. (85), we can reduce the integration region to around the horizon crossing time only and extract the imaginary part of the integrand: The accuracy of integration is up to O(x 2 s> ). We expect a similar behavior of integrand for the generic case also. The situation, however, changes importantly because the slow roll parameters will deform the integrand from the exact de Sitter case. We can understand the effect of slow roll parameters by looking at the MS equation (23) for the generic cases In the sub horizon scales where c s k aH, the oscillation term dominates over the mass term if the slow roll parameters remain not significantly big. In such a region, the mode function does not feel its mass during one period. Thus, as in Fig. 1, the cancellation by the oscillation takes place in the deep sub horizon scales even if the slow roll parameters deform the mass. While, in the super-horizon scales where c s k aH, the mass term dominates over the oscillation term and the mode function u sk rapidly approaches to the asymptotic value (28). Even if the slow roll expansion breaks down at the region c s k aH, the asymptotic behavior of mode function pursues as long as the mass term dominates enough over the oscillation term. Thus, there is less contribution from the superhorizon scales also even for the generic cases. The breakdown of slow roll expansion on the super-horizon scales implies a deformation of just the tiny fraction between x s> and x send in Fig. 1, which remains still negligible. On the other hand, the mode function can be significantly deformed around the horizon crossing time, since it can feel the deformation of mass from the exact de Sitter case during a few e-folds around the horizon crossing time. Therefore, unlike the de Sitter case, we need to choose x s as the time when the imaginary part of integrand (82) is at the extremum a few e-folds before the horizon crossing time for the generic cases.
Following the speclation in the above, we reduce the integration region of Eq. (82) in terms of n as where n > is a time a few e-folds after the horizon crossing and n is a time when the imaginary part of integrand is at the extremum. With use of the decomposition ψ sk = π sk + iσ sk , we write the prefactor of integration as In the de Sitter example, we can safely ignore the imaginary part of the prefactor since π sk for the de Sitter case approaches to zero on the super-horizon scales. However, it is not for the present case because we changed the argument between π sk and σ sk to perform the numerical calculation. Thus, to apply the analysis obtained from the de Sitter case for the present case, we need to tune the phase of ψ sk so that the real part of ψ sk (n end ) vanishes. We can always perform this procedure without a loss of generality by adding the phase factor to ψ sk as ψ ϑ sk = π ϑ sk + iσ ϑ sk := ψ sk e iϑ , which leads to π ϑ sk (n end ) = 0. Then, using ψ ϑ sk , we can rewrite Eq. (92) as where π ϑ sk,n σ ϑ sk,n = (π 2 sk,n − σ 2 sk,n )cosϑ sinϑ + π sk,n σ sk,n (cos 2 ϑ − sin 2 ϑ) . We multiplied aH to the integrand since here, we change the measure as dn = aHdτs. Similarly to the de Sitter case, the integrand (multiplied aH) almost behaves as the sine function.  deformation of the integrand from the exact de Sitter case causes the red spectral tilt, and n s | end seems to converge into the value 0.9646. This result is consistent with both of the slow roll expansion method and 1/N * expansion method within the range of 1 differences. The slight difference would originate from the approximations for the integration region and for the initial values of the mode function (78). As we expected, the numerical value of spectral index from Eq. (95) almost corresponds to that of horizon crossing formalism based on the slow roll expansion. Thus, the analysis we performed for the integration region is legitimate. To calculate the spectral index, it is the most important to know the behaviors of the mode function and the slow roll parameters around the horizon crossing time. In turn, even if the slow roll expansion breaks down around the horizon crossing time only, we can no longer apply the horizon crossing formalism to the spectral index. This is because the mass around the crossing will be strongly deformed from the de Sitter case. This interpretation is consistent with the well-known result of the Starobinsky model with the linear scalar potential [36]. We note that even if all the slow roll parameters are smaller than O(1) values, the slow roll expansion can break when we cannot use the slow roll parameters as the expansion parameters. Hence, if the potential has a bump and/or a cliff and the slow roll expansion breaks there, we need to perform the numerical analysis for the spectral index as here instead of using the horizon crossing formalism. We can apply the same analysis to the tensor perturbation also.

V. SUMMARY
In this article, we have directly derived the two exact expressions of the spectral indices from the Ward-Takahashi identity associated with the dilatation charge. We use generic free actions for the scalar and the tensor perturbations, so that we can apply the expressions to all of the single field inflation models which possess the Lorentz-invariant scaling. In the slow roll expansion, we have used one of the expressions (41) and have derived the spectral indices up to the second order of the slow roll parameters. Applying our expression to the canonical scalar model with the potential, the result is consistent with the previous works. Our formalism is, however, more rigorous and generic since we do not need any ansatzes nor approximations to the models and is therefore applicable to many more models than before. We have revealed that in the slow roll expansion, to evaluate the final values of the spectral indices at the end of inflation, we just need the horizon crossing formalism in which we change the time in the final values to the time when a * H * = k * , since the final values do not depend on the time actually. We have also derived the runnings of the spectral indices up to the third order.
To understand how the perturbations and the back-ground contribute to the spectral indices, we have analyzed the other expression of the spectral indices (43). First, we analyzed the scalar perturbation on the exact de Sitter background as a schematic example. Then, we applied the same method to the Starobinsky model and performed the numerical calculation. We found that the contributions to the spectral indices are only from around the horizon crossing time, and there is less contribution from the sub and the super-horizon scales. The spectral tilts are intrinsically connected to the masses of perturbations, deformed by the slow roll parameters, around the horizon crossing time. This is because only around that time, the perturbations can feel the deformation of masses from the exact de Sitter case. In turn, this result indicates that even if the slow roll expansion breaks down around the horizon crossing time only, we can no longer apply the slow roll expansion since the mode functions around the horizon crossing time significantly deviate from those of the slow roll expansion. In such a case, the approximate expression (95) we have derived will be a powerful tool to evaluate the spectral indices. In Sec. IV, we have evaluated the values of the scalar spectral index in several methods. It was shown that there is 1 discrepancy between horizon crossing formalism and 1/N * expansion in Starobinsky's model. In fact, 1/N * expansion only has 1 level precision, therefore, this is an expected result. On the other hand, our method is consistent with both the horizon crossing formalism and the 1/N * expansion in 1 accuracy. Therefore, our method may be regarded as a useful indicator of n s , similar to that using the 1/N * expansion.
We take the following contour for the integration in Eq. (A4): where we apply the iε prescription so as to calculate the above integration easily. We are not aware of the implication of this prescription. However, we do not need to take care about it since the extrapolated mode function before the inflation does not need to follow the actual history of the universe. It is enough to set the contour at the past infinity calculable. We extrapolate the mode function at the second contour as the same form as Eq. (32). Then, the integration from the second contour gives where we used the Wronskian (24) at the second equality. Introducing ψ Ik = u Ik /z I and performing a integration by parts, we obtain the spectral index in the other generic form (43) n I = N I + 1 − 4Im ψ * 2 Ik |ψ Ik | 2 τ I −∞(1+iε) z 2 I (∂τ I ψ Ik ) 2 dτ I .
Appendix B: Specific models for the slow roll expansion

Power-law inflation
We first consider the power-law inflation model as one of specific models. The scale factor and the coefficients in the free action are given by In this case, for the scalar and the tensor perturbations, the conformal times and the masses become the same ones , We can find that the breaking size θ I becomes exactly equal to zero Thus, for the power-law inflation model, the free actions preserve the dilatation symmetry. The same is true for the free scalar field on the exact de Sitter background. The asymptotic behavior of the mode function is governed by the growing mode u Ik → A I z I + B I z I τ I dτ I z 2 I → A I z I when |kτ I | → 0 , Consequently, we can obtain the asymptotic values of the spectral indices for the power-law inflation as n s − 1| |kτ I |→0 = n t | |kτ I |→0 = − 2 1 1 − 1 . (B6)

Ultra slow roll inflation
As another specific case, we consider the ultra slow roll inflation model. For this model, we cannot take the comoving gauge at the asymptotic future |kτ I | = 0, but before that time, we can still work on the comoving gauge. The coefficients in the free action are given by In this model, the slow roll parameter 1 rapidly decays in −6 powers of the scale factor, and thus other slow roll parameters become f s1 = g s1 = 2 = −6 , others = 0 .