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Loop corrections to primordial fluctuations from inflationary phase transitions

We investigate loop corrections to the primordial fluctuations in the single-field inflationary paradigm from spectator fields that experience a smooth transition of their vacuum expectation values. We show that when the phase transition involves a classical evolution effectively driven by a negative mass term from the potential, important corrections to the curvature perturbation can be generated by field perturbations that are frozen outside the horizon by the time of the phase transition, yet the correction to tensor perturbation is naturally suppressed by the spatial derivative couplings between spectator fields and graviton. At one-loop level, the dominant channel for the production of primordial fluctuations comes from a pair-scattering of free spectator fields that decay into the curvature perturbations, and this decay process is only sensitive to field masses comparable to the Hubble scale of inflation.


Introduction
Cosmological correlations in the early universe resolved from the cosmic microwave background and large-scale structure appear to be Gaussian, adiabatic and nearly scale-invariant [1]. These observed correlations are highly consistent with the natural predictions of inflation [2][3][4][5][6] (for a review of inflation, see e.g. [7]) even in a very simple scenario where only one single scalar field is relevant. Albeit at leading order the bilinear average of free fields is sufficient for distinguishing most of the single-field models, Weinberg [8] argued that higher-order corrections to the bilinear correlations from nonlinear interactions of the theory may be as important as the search for non-Gaussian features.
It is now widely accepted that the leading (or tree-graph) contributions to Gaussian or non-Gaussian correlations are generated around the time where perturbations exit the horizon. However, whether higher-order (loop) corrections are naturally suppressed by the same assumption remains unclear, and the consequence of loop corrections to inflationary observables has received much attention [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. For instance, in the single-field inflationary paradigm the curvature perturbation ζ approaches as a constant well outside the horizon, and therefore time-evolving corrections on superhorizon scales, if exsit, could be dominated by loops. Nevertheless the behavior of ζ is strictly constrained by the intrinsic symmetries of inflation so that no time-dependence in the correlation functions can be generated by ζ self-interactions [16][17][18][19][20].
On the other hand, loop corrections contributed by additional light fields can be more revealing, given that isocurvature field perturbations do not necessarily respect to all symmetries in the inflationary background. At least two kinds of isocurvature effects are considered as potential threats to the adiabatic primordial fluctuations. The first concern is the accumulation of isocurvature perturbations on superhorizon scales, which turns into a logarithmic enhancement to the curvature perturbation in terms of the ratio between a suitable infrared size and the wavelength of interest [24,25]. The second worry comes from self-interactions of massless spectator fields, such as quartic self-couplings [8,23]. These self-loop corrections simply result in the power spectrum a logarithmic enhancement in terms of the scale factor during inflation. While the former effect is subject to the issue on the infrared divergence from isocurvature degrees of freedom [17,26,27] (see [28] for a review), the latter effect disappears when all possible loop diagrams are taken into account [16]. In this work, we intend to draw out another kind of temporal enhancement led by isocurvature modes running in loops. (See also [29][30][31][32][33] for yet another enhancing mechanism to primordial fluctuations from generalized spectator fields.) The loop effect of isocurvature fields with non-negligible masses has also been addressed in many previous studies [9,16,34]. Indeed, massive fields exhibit interesting behaviors in both classical and quantum regimes. Isocurvature modes with masses larger than the size of the Hubble scale are oscillating, creating a time-dependent background for their quantum fluctuations that may characterize important informations in the primordial Universe [35,36]. However, with a small mass term the perturbative expansion of the corrections can be summed up into an effective mode function that always decays outside the horizon. As a result, higher-order corrections from a massive field converge to a constant no much different from the value evaluated around horizon exit [9,16], similar to the case of a massless field. This generic conclusion shall hold for massive fields that always stay in one stable vacuum state during inflation.
In this work, we take into account the possibility that some fields may experience a transition of their vacuum expectation values during inflation. The transition can happen instantly due to quantum tunneling or jumping, or it can take place smoothly under a classical evolution. Both kinds of phase transitions are well motivated since they might be the origin of inflation [2][3][4][5][6]. In particular during a smooth transition of the field values one usually finds a period that the dynamics corresponds to a negative effective mass (see for example [37]). Although these fields end up in some stable vacua at most of the times, the phase with negative masses can lead to the growth of field perturbations on superhorizon scales, and thus, also lead to the growth of loop corrections. The aim of this study is to clarify how large could those growing corrections reside in the cosmological observables.
Our calculations must focus on contributions from superhorizon modes of isocurvature perturbations, given that a mass term does not change the behavior of a field with sufficiently large wavenumbers. For our purpose it turns out to be convenient to adopt a specific perturbative expansion of the loop diagrams proposed by Senatore and Zaldarriaga [16], based on the modified Feynman rules for cosmological correlation functions developed earlier by Musso [38]. In this specific arrangement of the loop diagrams we clearly identify the dominant contribution from the mass-induced interactions in the late-time limit where all perturbative fields are sufficiently classicalized.
This article is organized as follows. In section 2, we review loop corrections from massive fields with an extended discussion on the case of negative effective masses. We show the existence of negative masses in the transition phase of field expectation values by a sampling model given in section 3, and in section 4 we calculation the one-loop correction to the curvature perturbation and the tensor perturbation based on the given sample. We discuss the possible contribution from higher-order perturbations beyond one-loop in section 5 and, finally, our conclusion is summarized in section 6.

Loop corrections from massive fields
In this section we investigate loop corrections to the curvature perturbation induced by spectator fields only through gravitational interactions. For massless fields any interaction in the Lagrangian must involve at least one spatial or temporal derivative so that their contribution are suppressed on scales well outside the horizon. As a result the curvature perturbation contributed by massless fields at loop level is always frozen sometime after horizon crossing [16]. The situation beocmes more interesting when a spectator field gains interactions without derivatives from a mass term. Higher-order interactions with no derivative can result in time-evolving corrections on superhorizon scales, essentially sourced by the evolution of the wavefunction of a massive field. However, these loop corrections always decay with a positive mass term, as discussed in [9,16]. Here we consider a simple example to see the time-dependence of the induced curvature perturbation, but treating the mass value as a free parameter with the possibilty to be negative.
The inflationary paradigm. To make a concrete discussion, we introduce two real scalars, namely, the inflaton field φ(x, t) whose quantum fluctuations generate the largescale curvature perturbations ζ(x, t) at the time of horizon exit, and a spectator field σ(x, t) with insignificant density during inflation. The action of our interest is therefore given by where M p is the reduced Planck mass, and we shall specify the potential in the next section. We apply the metric of the Arnowitt-Deser-Minser (ADM) formalism and take the parametrization in the spatially flat slicing to the lapse function, the shift function and the spatial metric as The ADM metric in terms of the conformal time η = − ∞ t dt/a thus reads (2.4) The scale factor during inflation a(η) ∼ −1/(Hη) diverges as η → 0, where the Hubble parameter H = H/a is nearly a constant. Let us decompose the inflaton field into homogeneous parts and pertubations as φ(x, η) = φ(η) + δφ(x, η). Following the computation in [39] and omitting any contribution from σ, we obtain the solutions for linear perturbations as (see appendix A) where φ = κ 2 φ 2 /(2H 2 ) and φ ≡ dφ/dη. Inserting the solutions (2.5) back into our theory (2.1), we recover the quadratic action for the curvature perturbation in the single field inflationary scenario The free-field operator (2π) 3 ζ I (x, η) = d 3 k e ik·x ζ k (η) is then defined from the solution of a field equation derived from (2.6), where the mode function ζ k ≡ ζ kâk + ζ * kâ † −k normalized by the usual vacuum in the Minkowski limit is well-known: The massive mode. We now consider the spectator field σ that has only gravitational interaction with φ, which is manifest by the condition ∂ 2 V /(∂φ∂σ) ≡ V φσ = 0. In the ADM formalism (2.2), the action of σ is given by For convenience, we consider that the spectator field has a well-defined decomposition as σ(x, η) = σ(η) + δσ(x, η), where the classical equation of motion for the homogeneous part σ(η) is given in appendix A. We will treat the field potential as a non-perturbative contribution to the free field solutions. In the Fourier space the equation of motion for the field perturbation is then reads Note that we have promoted the theory into 2ν space dimensions as considered in [8], where ν → 3/2 recovers the usual space dimensionality 3. Assuming that M 2 ≡ a 2 V σσ /H 2 = V σσ /H 2 is a constant, the mode function is solved as where H L is the Hankel function of the first kind with L > 0. The free-field operator is then given by (2π) 3 σ I (x, η) = d 3 k e ik·x σ k (η), where σ k ≡ σ kâk + σ * kâ † −k . For imaginary L led by M > ν the perturbation δσ k vanishes soon after horizon exit [9]. Expanding the mode function (2.10) around |kη| → 0, we get (2.11) Since we consider L > 0, the term in the second line of (2.11) dominates as |kη| → 0, so that δσ k ∼ a L−ν at late times. On the other hand we can obtain the late-time behavior of the (retarded) Green function from the expansion (2.11). To the lowest order in |kη| with the condition Similarly, if we ignore the corrections from the slow-roll parameter the late-time behavior of the Green function for ζ, simply replacing σ by ζ with L = ν in (2.12), is estimated as G ζ ∼ a(η 1 ) −2ν when taking η → 0. As shown in [38], the free-field operators ζ I , σ I in the interaction picture and the Green functions G ζ , G σ are the only ingredients that we need to compute n-point correlation functions to all loops. For instance, in 2ν space dimensions the cubic interactions between ζ and σ given by (2.8) are where the inflaton perturbation δφ and the metric pertubations α, β are to be replaced by ζ I according to (2.5). As mentioned, the interaction without field derivatives has the highest powers of a, which we denoted as For |kη| → 0, the dominant one-loop correction to the curvature perturbation induced by L 1 can be schematically given by the Green's function method (we will come back to this point in Sec. 4, see also [16]) Taking the above approximation for G ζ and δσ, we estimate the late-time behavior of the induced curvature perturbation as Therefore with M 2 > 0, we have L < ν such that ζ 1 decays on superhorizon scales. In this case ζ 1 is dominated by the value evaluated around the time of horizon exit, and the contribution from all the other derivative interactions shown in (2.14) should be taken into account. In the massless limit (M → 0) the correction ζ 1 vanishes as V σσ → 0. We remark that the interesting behavior can appear for a massive mode with M 2 < 0, where L > ν and ζ 1 ∼ a 2L−2ν is growing with time. Ignoring for the moment the issue of negative mass states in field theory, we emphasize that the time evolution of ζ 1 on superhorizon is physical and is not cancelled by other loop diagrams, as the corrections induced by massive modes with positive mass square.

Inflationary phase transitions
The nature of inflation remains unknown, and it is believed that there exist a large number of metastable vacua around the energy scale of inflation such that fields may transit from one local minimum to another. The transition could occur instantly through quantum tunneling or jumping, which is subject to first-order phase transitions. On the other hand, the transition could be smooth if one field moves to a new vacuum by (slow-)rolling. The latter case is refered to as second-order phase transitions and will be the main focus of this work.
The potential V (φ, σ) given in (2.1) is at least composed by an inflaton potential V 0 (φ) and a subdominant potential V 1 (σ) of the field σ as where the background density is attributed to V 0 (φ) ≈ 3M 2 p H 2 . To our purpose it is convenient to consider a quasi de Sitter expansion charaterized by the slow-roll parameter the Hubble parameter H ∼ a − H decays slowly with time, and so as the background density V 0 (φ). For simplicity, let us introduce a specific potential of the form where Θ(σ) is a step function to make that V 1 = λv 4 /4 is a constant for σ < 0, as seen by Fig. 1. We also introduce a dimensionless mass parameter We assume the initial condition σ < 0 so that σ is effectively a massless field at the beginning. During this phase the behavior of σ is govern by its quantum fluctuation with a typical size ∆σ q ∼ H 2π . The field value at the next moment has no predictability, but there is a global minimum at σ = v. Suppose that σ lands around the origin (σ → 0 + ) accidentally at a moment η i , it suddenly feels a mass M 2 ≈ −λv 2 /H 2 i < 0, where H i is the Hubble parameter at η = η i . This negative mass tend to drag σ towards the global minimum. As long as the classical deviation ∆σ c dominates the quantum fluctuation ∆σ q , the behavior of the σ field can be described by the classical equation of motion (A.4), and for σ > 0 we have where N ≡ ln a is the e-folding number. At some value σ = σ i > 0, the classical deviation ∆σ c start to surpass the quantum effect ∆σ q and the field will start to roll down the plateau. Since the transition is smooth for |M 2 | ∼ O(1), we neglect the term d 2 σ/dN 2 so that (3.4) becomes where i is the value of H evaluated at η = η i , the solution of (3.4) is given by The value of σ i can be estimated by the classical deviation within one e-fold as ∆σ c ∼ λv 2 3H 2 σ, and the condition ∆σ c ≥ H 2π gives σ i ≥ 3H 3 i /(2πλv 2 ). Once σ reaches the value v/ √ 3, the mass square M 2 starts to change from negative to positive. We can estimate the duration of the slow roll in a pure de Sitter expansion by taking i → 0, where (3.6) shows The phase of slow roll ends at If the field mass is small compared with the Hubble parameter (|M 2 | 1), quantum effects from short wavelength fluctuations become important. In this case the behavior of σ is usually approximated by the stochastic approach [40,41] where the quantum noise f (N ) satifies the correlation f ( As the field is slowly rolling, the equation motion of σ 2 derived from (3.9) shows where the term σ 4 becomes less important as the slow-roll phase starts from σ → σ i near the origin [37]. In a pure de Sitter expansion H → H i we can easily find the solution 11) and the duration of slow roll is obtained by using

One-loop corrections
To calculate loop corrections on superhorizon scales it is convenient to adopt the diagrammatic expansion scheme performed in [16,18], based on the modified Feynman rules developed in [38]. In this scheme, higher-order perturbations are always resolved into solutions of the free fields by virtue of the Green's method, and therefore correlation functions are simply resulted from convolutions of free fields with retarded Green functions. For example, the ζ two-point function is given by where |Ω is the vacuum of the interacting theory and |0 is the vacuum of the free theory. The −∞ + implies a replacement of η 1 by η 1 (1 + iε) when taking η 1 → −∞, such that the time-integration is projected from |0 to |Ω . Equation (4.1) is then interpreted as the two-point correlation of ζ I evolved from η 1 to η after projecting the free vacuum to the interacting vacuum with the operator that satisfies U I (η, η 1 )U † I (η, η 1 ) = 1. The Taylor expansion of U I up to second order in H I leads to [16]: where CIM denotes the contribution from the cut-in-the-middle diagrams and CIS denotes those from the cut-in-the-side diagrams. In particular, these one-loop diagrams are given by the cubic interactions H As seen by (2.5), interactions with each one α is suppressed by one factor φ H , and interactions involved with β are higher-order in the slow-roll parameters. To the leading order in φ , H (3) I derived from (2.14) comprises a ζ field and a pair of σ fields, while H (4) I includes a pair of ζ fields and a pair of σ fields. In this case there is only one corresponding CIM diagram, where two free σ fields scatter to generate a propagating ζ at second order and ζ 2 CIM is the correlation of two of these propagating ζ fields. There are two kinds of CIS diagrams. The first one is a propagating second-order σ field generated by a scattering of a free ζ with a free σ. This propagating σ scatters with a free σ field again and results in a propagating ζ at third order. ζ 2 CIS,1 is then the correlation of a third-order ζ with a free ζ field. Similarly, ζ 2 CIS,2 is also a correlation of a third-order ζ with a free ζ field, where the third-order ζ field is generated by the scattering of a free ζ with a pair of free σ fields through the interaction H

Linear perturbations with phase transitions
We have seen that σ can obtain a time-dependent mass square M 2 = V σσ /H 2 during a transition to another local minimum, and that M 2 can become negative during phase transitions. To specify the discussion, we consider a transition driven by the type of potentials as (3.2). We can solve the mode function u k = aδσ k with a simplified equation of motion given by where M 2 − = |M 2 | is the phase that M 2 < 0, and therefore l > 3/2 for η i < η ≤ η f and 0 < l < 3/2 for η > η f . Without the loss of generality, σ is assumed to be massless by η i .
Given that σ starts to roll down from very close to the origin, we may neglect the change of σ in (3.3) so that M 2 ≈ −λv 2 /H 2 is a good approximation in the first few e-foldings, where the time dependence in M is mild with a sufficently small H . To reach a good understanding on the behavior of u k , it is enough to consider a pure de Sitter expansion by taking H = 0, and thus H = H i , η = −1/H = −1/(H i a). Using a new time variable z = kη, the solution of (4.7) with a constant M 2 − = λv 2 /H 2 i and a constant M 2 where z i ≡ kη i and J l (Y l ) is the Bessel fuction of the first (second) kind. Let us first focus on the time interval z i < z ≤ z f . If M 2 − 1, the mode function δσ k with k < k i ≡ −1/η i is nearly a constant by z = z i . In the limit z → 0, δσ k → iH/ √ 2k 3 in (4.8) while J l (−z) automatically drops out in (4.9), and one can match the solution at z = z i to find that b 2 = −π(−z i ) l−3/2 /( √ 2 2 l Γ(l)), where On the other hand, δσ k with k > k i leaves the horizon at some epoch η k > η i , where η k −1/k. Although b 1 , b 2 can be solved by matching the value of δσ k and δσ k at z = z i , in the limit z i 1 we find that which are independent of z i . This implies that mode functions with k k i cannot feel the phase transition, and the vacuum state in the Minkowski regime δσ k ≈ −(H/ √ 2k 3 ) × ze −iz is almost unchanged after z i . Therefore, for k > k i we shall match |δσ k | between the vacuum state with the late-time growing mode Y l in (4.9) at z = z k ≡ kη k , which leads to For |M 2 | > 1, the time evolution of δσ k becomes important after the phase transition. As a simple example, let us consider a special case l = 5/2 (that is M 2 − = 4 ), where (4.9) has an analytical expression based on J 5/2 (−z) and Y 5/2 (−z). By matching δσ k and δσ k at z = z i , we can solve the coefficients as and now (4.9) is given by Considering the limit |z| |z i | ≤ 1 for mode functions with k ≤ k i , (4.16) is reduced as This solution agrees with (4.11) for l = 5/2 up to a constant factor 4/5. 1 Similarly, one can solve c 1 , c 2 in (4.10) by matching the solutions at z = z f . For k < k i , we can simply match (4.11) with (4.10) in the limit z → 0 and find that where l ± ≡ 9/4 ∓ M 2 ± . Since l + < 3/2, the mode function decays for z > z f as the case of a massive field with a positive mass square [9].

Induced curvature perturbations
We now calculate the one-loop corrections to the ζ two-point function induced by the σ field which undergoes a phase transition as (4.7). The late-time contribution from ζ 2 CIM involves two propagators of each G ζ ∼ a −3 , four free fields of each δσ ∼ a l−3/2 and two vertices. For the non-derivative cubic interaction (2.15) each vertex brings in an integration ∼ φ V σσ dηa 4 (η), and we can find from (2.17) that ζ 2 CIM ∼ a 4l−6 , which is growing during η i < η ≤ η f where l = l − > 3/2. On the other hand, in the ζ 2 CIS,1 correlation two 1 In the limit zi ≤ z −1, temporarily growing δσ fields are replaced by one propagator G σ . The internal propagator G σ (η 1 ; η 2 ) effectively contributes a −3 to the two-point function after both η 1 and η 2 are integrated out. In ζ 2 CIS,2 the quartic interactions can provide at best two free δσ's with one non-derivative vertex ∼ 2 φ V σσ dηa 4 (η). In both CIS diagrams one of the external field is a free ζ whose contribution can be locally absorbed into the homogeneous background through a rescaling of the scale factor a [16]. This is the essential difference between CIS diagrams and the CIM one. As a result one concludes that the CIM diagram is the dominant one-loop corrections on superhorizon scales.
Let us compute ζ 2 CIM . The leading interactions of O( φ ) from (2.14) are where after integrating the first term with time derivatives by parts and replacing it by the equation of motion (2.9), we can recast the interactions (4.19) as By virtue of the Green's method for cosmological correlators [38], we treat the δσ fields as additional sources to the ζ equation of motion as where we define The correlation function due to the CIM diagram is then given by where 1 ≡ (η 1 ) and G ζ k is the Fourier transform of the Green function G ζ . The unequaltime correlation function is The ζ Green function is obtained according to (2.7) as where˜ φ is evaluated atη. To simplify the calculation, we assume that the slow-roll parameter is nearly a constant throughout the time of our interest. By using the symmetry between the pairs of momentums (p, q) ↔ (P, Q) and the symmetry between time integrations (η 1 ↔η 2 ), we can rearrange the integration as where the pair of momentums (P, Q) have been integrated out. Note that the disconnected diagram ζζ dc ≡ ζ 2 has been excluded in (4.26) as it only contributes to the zero-mode correlation k = K = 0. 2 It is useful to reparametrize the internal wave numbers with respect to the external wave number as x = q/k, y = p/k. For k < k i (or z i < 1), we shall compute the time from z = z i and put the cutoff y i ≡ k i /k to the internal wave numbers so that the latetime approximation (4.11) can be applied. After applying these cutoffs we extract only the contribution from superhorizon modes since the start of the phase transition at η = η i . As we will see that the choice of the UV cutoff y i does not affect the results. The correlation (4.26) is then where y 0 = k 0 /k, and k 0 is a suitable cutoff for the largest cosmological scale (see the discussion in [13,42]). Here, we define a useful function . (4.28) In the limit |z| ≤ |z| 1, the time integration is simplified as (4.29) Taking the late-time approximation (4.29) into (4.27), we find that the dominant contribution comes from the IR cutoff y 0 , where and the integral factor in the limit z → 0 reads We put a conservative upper bound M − ≤ 2 to justify the assumption (4.11). The logarithmic scale-dependence ln(k/k 0 ) in (4.30) is subject to the projection effect between two superhorizon modes, which should have no importance for cosmological observables [17]. The study of this issue is beyond the scope of the current paper.
Since δσ k starts to decay after the mass square changes to be positive, the source S ζ has a maximum value around η = η f . We may estimate the value of Π ζ (z f ) from (3.8) in a pure de Sitter expansion by taking M 2 − = M 2 i . Suppose that the energy scale of inflation is H 2 i /M 2 p ∼ i × 10 −10 , then the result in Figure 3 shows that the one-loop correction ζ 2 CIM can have a value around 2 i M 4 Π ζ × 10 −10 times smaller than the tree expectation value ζ 2 tree ∼ H 2 i /( i M 2 p ). With a self-coupling λ ∼ 10 −8 one may realize ζ 2 CIM / ζ 2 tree ∼ O(1), where < 0.0068 according to [1]. This loop correction can be much larger than the usual expectation from massless fields or massive spectator fields that always live in some stable vacuum states [9,16]. Note that the decay of δσ k at η > η f only makes S ζ decay in (4.21) but the curvature perturbation is always enhanced during the phase transitions. As S ζ → 0, ζ converges to a constant again and the ζ 2 CIM correction enhanced by temporarily growing perturbations becomes nearly frozen from η = η f to the end of inflation.
For k f > k > k i , we have z i > 1 and the time integration is chosen to start at some epoches z k −1 where all internal modes p ≤ k with the cutoff y = 1 have left the horizon. In this case the correlation function (4.26) with the solution (4.13) is again dominated by the contribution around the IR cutoff y 0 as for 3 2 < l ≤ 5 2 in the limit z → 0. The factor z 6−4l i = (k/k i ) 6−4l is introduced by the scale-dependent cutoff z = z k , which suppressed the CIM contribution to modes that exit the horizon during the period η f < η < η i .

Second-order gravitational waves
It is also interesting to ask how large the corrections to the tensor perturbation h ij are induced by the (temporarily) growing field perturbations δσ during phase transitions. We are in particular interested in the contribution from the CIM diagram, which is the dominant effect from massive spectator fields. The tensor perturbation induced by the CIM diagram is in fact a classical process that generates second-order gravitational waves through a pair of scalar perturbations. Unlike the curvature perturbation, the tree-level tensor mode h does not suppress by the slow-roll parameter φ and therefore to find the leading contributions one can neglect any term involves with α, β or δφ. For cubic terms the only interaction that is not suppressed by the slow-roll parameter reads [33,[43][44][45][46] L hσσ = a 2 2 h ij ∂ i δσ∂ j δσ. (4.35) This interaction contributes to the tensor perturbation at second order according to the equation of motion where ⊥ lm ij is the projection operator that extracts the transverse and traceless part of the source generated by (4.35), that is S ij (x, η) = κ 2 ∂ i δσ(x, η)∂ j δσ(x, η). (4.37) The Fourier transform of the tensor perturbation is defined as where we introduce two unit vectors e 1 , e 2 that form orthonormal basis together with another unit vector k/k, and such that Therefore the equation of motion for h + k or h × k takes the form where the source term is led by Applying the same method as used for the curvature perturbation, we can compute the tensor two-point function h k h K from the unequal-time correlation function In terms of the reparametrized wave numbers x = q/k, y = p/k, the projection e ij p i q j = −e ij p i p j is simply where ϕ is the azimuthal angle of p on the (e 1 , e 2 ) plane. The power spectrum for either polarization (h + k or h × k ) is given from the definition For tensor modes that exit the horizon by η = η i where z i = k/k i < 1, we find no IR divergence at zero mode so that the integration can run from 0 to y i to give where F n (z) has been given in (4.28) and the late-time approximation (4.29) is used for the second equation. The integral factor is defined as (4.49) We find that the second-order tensor perturbation induced by superhorizon modes (δσ p with p < k i ) is dominated by the UV cutoff y = y i , and thus is suppressed by a factor (k/k i ) 3 . This is due to the fact that the leading interaction (4.35) is a derivative coupling with two spatial derivatives. We therefore conclude that massive fields cannot contribute large corrections to the tensor perturbation even if their perturbations are growing after horizon exit due to negative masses.
(c) Figure 4: Two-loop diagrams of the cut-in-the-middle (CIM) type. Solid (dotted) lines are external (internal), ζ and δσ are free fields, arrows are propagators, black dots are vertices, and ζ (i) is a propagating ζ field at i-th order in perturbations. In each diagram one dotted circle must correlates with one solid circle in order to get non-vanished contributions, and there is no correlation between ζ and δσ fields.
fields are running in loops. However, as long as a higher loop diagram is of the CIS type, similar to figure 2b or 2c where one of the external field is a free ζ, the contribution from such a diagram is suppressed in the late-time regime due to a rescaling of the local background [16]. Therefore we only need to consider the non-CIS type diagrams. Up to quartic order in perturbations there are two kinds of non-derivative interactions H in fact a function of the mass. In the case of a positive mass square (l < 3/2), higher-order terms decay rapidly so that loop corrections have no effects on the physical observable. With a phase of a smooth transition of the field expectation value, there can be a period of M 2 < 0 (l > 3/2) where loop corrections grow with time and the duration of the growing phase is also mass-dependent. In the scenario (3.2) we always find a maximum correction around |M 2 | = 2 with a large enhancing factor M 4 Π ζ (M ) that depends on the model parameters. We can constrain the value of λ from the condition M 4 Π ζ (M ) < M 2 p /( H 2 ) for the breakdown of perturbative expansion. For a large mass |M 2 | = 4 we find λ > 7 × 10 −9 with = 0.01 from (3.8). On the other hand, the condition λ 4 v 4 3M 2 p H 2 that the massive field has a subdominant density during inflation gives a milder constraint λ 4 3 × 10 −10 for an effective mass λv 2 = 4H 2 .
Before closing the discussion, we briefly comment the consequence due to the selfcoupling of the massive field. For example, a self-interaction L ∼ a 4 λσ 4 as a part of the potential may allow us to perform arbitrary 1PI insertion to any internal σ-field line with the introduction of two more free δσ fields. In this case the breakdown of perturbative expansion would request λ < e ∆N (3−2l) 7×10 −6 with ∆N given by (3.8) at |M 2 | = 2. However the temporal divergence due to the self-coupling of a massive field is not specific for a negative mass square and is also found in the case with a positive mass square [34]. We believe that part of the temporal divergence should be removed by some renormalizations of the theory, as the finding from a resummation of the perturbative masses [9]. A resummation of the quartic self-corrections of a massive field to the propagator in the M 2 1 limit has been performed through the dynamical renormalization group method [47,48], and the question that how are the self-corrections behave after these treatments is left for future efforts.

Conclusions
The transition of vacuum expectation values of scalar fields may play an essential role in cosmic inflation. In this work we have studied loop corrections to primordial fluctuations from a kind of phase transition during inflation that massless fields randomly found their global minima purely triggered by quantum fluctuations. The transitional phase involves with a classical evolution effectively driven by a negative mass term where field perturbations start to grow on superhorizon scales.
At one-loop level, we found that important corrections are only generated by field perturbations that have been frozen outside the horizon by the starting time of phase transition. The dominant diagram at this level is pratically a classical process where a second-order perturbation is created by the scattering of two linear perturbations. We also found that the resulting loop corrections are only sensitive to those field masses comparable to the Hubble scale of inflation. However, such a kind of phase transition cannot generate significant tensor mode perturbations at second order, given that the leading interaction involved with the graviton coupling must contain spatial derivatives.
Based on the above findings we can gain some knowledge of loop effects induced by other kinds of phase transitions. For example a time-dependent transition of the global minimum similar to that of the Higgs mechanism has been considered in [37]. In this kind of models there also exists a period of negative-mass evolution, yet no field perturbations can survive on superhorizon scales by the time of the growing phase. As a result loop corrections are only generated by those modes that exit the horizon during the negativemass evolution, which effectively invokes a scale-dependence in the calculation and thus suppresses the final contribution.
Though the specific scenario investigated in this paper only realizes a large enough correction to the observables in a certain parameter range, we find our results interesting because in reality the physics relevant to inflation should be much more complicated. It is unclear whether there could be a more efficient way for spectator fields to affect primordial fluctuations during inflation, but the current conclusion indicates that loop corrections residing in the observed cosmological correlations may be more important than they were usually expected. By using the above equations we can derive α = κ 2 φ 2H δφ + σ 2H δσ , (A.10) We are interested in the case V (φ, σ) = V 0 (φ) + V 1 (σ) = V 0 (1 + f ), where f = V 1 /V 0 is the fraction of the potential energy of σand φ. Therefore it is convenient to define φ = κ 2 φ 2 /(2H 2 ), σ = κ 2 σ 2 /(2a 2 V 1 ), ζ φ = −Hδφ/φ , ζ σ = −Hδσ/σ , such that (A.10) and (A.11) become Then the result of the single field inflation is recovered in the limit f → 0.
Next, we bring in the tensor perturbation h ij with linear scalar perturbations. At second order, the only non-vanished equation of motion for the tensor perturbation reads (A.14)