Abstract
Abstract. We calculate the trispectrum of the primordial curvature perturbation generated by an epoch of slow-roll inflation in the early universe, and demonstrate that the non-gaussian signature imprinted at horizon crossing is unobservably small, of order τ NL r/50, where r < 1 is the tensor-to-scalar ratio. Therefore any primordial nongaussianity observed in future microwave background experiments is likely to have been synthesized by gravitational effects on superhorizon scales. We discuss the application of Maldacena's consistency condition to the trispectrum.
Introduction
In the inflationary scenario [1, 2, 3, 4, 5, 6, 7] , the primordial curvature perturbation, ζ, is generated via the vacuum fluctuations of one or more light scalar fields [8, 9, 10, 11] . It is the only relevant perturbation in the simplest class of single-field models. More generally, however, ζ is sensitive to the model under consideration. It may be accompanied by correlated or uncorrelated isocurvature perturbations [12, 13] which source its evolution, or it may be generated after the end of inflation by the decay of another field [14, 15, 16, 17] .
Introduction
It has recently become clear that interesting information about the particle physics that drove inflation is encoded in the non-gaussian features of the curvature perturbation [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40] . Although observations of the Cosmic Microwave Background (CMB) anisotropy constrain the spectrum of perturbations to be close to scale-invariance with very nearly gaussian statistics [41, 42, 43, 44] , some non-gaussianity is inevitable due to the universal coupling of all matter fields to gravity [45, 46, 47] . The question remains, however, as to the precise level of non-gaussianity that is generated. This is sensitive to details of the inflationary model, such as the number of fields which contribute to the energy density of the universe during inflation [48, 49, 50, 51, 52] , and also to possible non-canonical structures in the scalar field Lagrangian [53, 54, 55, 56, 57, 58, 59] .
Introduction
If pure gaussian statistics are respected, any n-point correlator of the curvature perturbation ζ · · · ζ can always be expressed in terms of the two-point correlator ζζ . A breakdown of this rule for any n 3 indicates a departure from gaussianity. Since some higher-order correlations may be more easily detectable [60, 61, 62] than others, it is important to quantify the theoretical predictions for as many values of n as possible. To date, the majority of quantitative theoretical studies have focused on the three-point function, or equivalently the 'bispectrum' B ζ ,
Introduction
Precise gaussianity corresponds to B ζ = 0. In practise, the non-gaussianity in the threepoint function generated in a particular model of inflation can be quantified in terms of a dimensionless quantity, f NL , which is defined by [19, 63, 64] ‡
Introduction
where P ζ (k 1 ) is the power spectrum of ζ, such that ζ(k 1 )ζ(k 2 ) = (2π) 3 δ( i k i )P ζ (k 1 ). Current observational bounds require |f NL | 100 [65] and are consistent with f NL = 0, whereas a value of |f NL | ∼ 3 should be detectable in the near future [63] . On the other hand, it is a robust prediction that the non-gaussian signal imprinted at horizon crossing during single-field inflation is |f NL | ∼ 0.01, which is too small to be observable, [19] and the situation is similar in models with more than one field [22, 25, 48, 26, 28, 49, 50] .
Introduction
(For an alternative analysis, see Refs. [66, 67] .)
Introduction
In this paper, we make a comparable quantitative prediction for the primordial four-point function, or equivalently the 'trispectrum' T ζ [68, 69] :
Introduction
This can be quantified by defining a dimensionless 'non-linearity' parameter τ NL such that [32, 24, 28 ]
Introduction
where k ij = k i + k j , and the condition i k i = 0 ensures that one-half of the permutations are equal to the other half, giving twelve distinct terms.
Introduction
At present there is only a weak experimental bound on the non-linearity parameter, which is roughly |τ NL | 10 8 [28] . The WMAP satellite should strengthen this to |τ NL | 2 × 10 4 and the Planck satellite will be sensitive to a value of |τ NL | ∼ 560 [69] . Therefore, there is a pressing need to develop estimates of τ NL in those models of inflation which we wish to confront with experimental data in the near future. An expression for τ NL was presented recently by Alabidi & Lyth [28] (see also [34] ). However, these authors neglected any contributions arising from quantum interactions at horizon crossing. The purpose of the present paper is to provide a more complete expression for the nonlinearity parameter where such quantum effects are taken into account. We consider a general multi-field scenario in which the fields are minimally coupled to gravity with a target space metric δ αβ , and where the slow-roll approximation applies around the time of horizon crossing.
Introduction
The plan of this paper is as follows. In §2, we outline the δN formalism for computing the trispectrum of the curvature perturbation on super-horizon scales. In §3, we calculate the fourth-order interaction lagrangian for the field perturbations, including both scalar and vector contributions to the action for the first time. Given the form of this lagrangian, the four-point expectation value of the field fluctuations, and hence the momentum-dependence of the trispectrum, can be calculated by employing the Schwinger-Keldysh formalism [70, 71, 72, 73] . This is achieved in §4 and represents one of the main results of the paper. We then derive an upper bound on the magnitude of the trispectrum in §5 and find that τ NL r/50, where r < 1 is the tensorto-scalar ratio. Such a result implies that any non-gaussian signature generated at horizon crossing from the four-point correlator of the curvature perturbation will be unobservably small. We then consider the case of single-field inflation in §6, where the curvature perturbation is automatically conserved on superhorizon scales. This conservation leads to a 'consistency' condition between the correlators of ζ of the type discussed by Maldacena [19] . We conclude with a discussion in §7.
The δN formalism for the trispectrum
The most efficient method for calculating the four-point correlator of the curvature perturbation is to employ the δN formalism introduced by Starobinsky [74, 75] , and generalized to non-linear perturbation theory by Lyth, Malik & Sasaki [76] . § In this approach, it is assumed that the evolution of the universe in causally disconnected regions is like the evolution of separate locally unperturbed universes, where pressure and density can take different values [78, 79, 80] . This implies that ζ = δN [74, 75, 76, 33, 24] , where N(φ, ρ) is the number of e-folds of expansion between an initial spatially flat hypersurface on which the fields have values φ α and a final uniform-density slice on which the energy density is given by ρ. On large scales, ζ is equivalent to the comoving curvature perturbation, R, up to a sign convention.
The δN formalism for the trispectrum
As demonstrated by Lyth & Rodríguez [33] , if slow-roll inflation is valid on the initial flat slice, ζ can be written as a power series in terms of the field perturbations at that time
The δN formalism for the trispectrum
where N ,α = ∂N/∂φ α and α indexes the space of light fields φ α . The perturbations in these fields are denoted by δφ α and are defined on the flat hypersurfaces. The leadingorder relation for the power spectrum of the curvature perturbation is then given by
The δN formalism for the trispectrum
where a subscript ' * ' implies that background quantities should be evaluated at horizon crossing and P * is related to the dimensionless power spectrum, P * , for a massless scalar field by
The δN formalism for the trispectrum
In principle, P ζ receives corrections from higher derivatives of N. We will assume that these corrections can be neglected when computing the leading non-gaussianity. This is reasonable since it is known experimentally that ζ is dominated by a very nearly gaussian contribution. To calculate ζζζζ , one forms the product of four copies of Eq. (5) and takes an expectation value in the initial vacuum state [81] . The lowest-order contribution in derivatives of N is given by
The δN formalism for the trispectrum
where the correlator of field perturbations on the right-hand side is evaluated a few e-foldings after horizon crossing, such that the fields have had sufficient time to become classical but have not evolved significantly. The δφ-correlator in (8) can be separated into two pieces. The first is an irreducible connected part in the sense of Feynman diagrams (see Figure 1(a) ), which we denote by δφ α δφ β δφ γ δφ δ c , and which is absent when the fields perturbations are precisely gaussian. The second contribution is a reducible part and is given by the sum over all ways of combining the four fields into pairs, with each pair yielding a copy of the spectrum. This reducible part is a disconnected contribution, and is always present, even when the δφ α are gaussian. It therefore contains no more information concerning the primordial non-gaussianity than § See also Langlois & Vernizzi [77] for an alternative non-linear generalization.
The δN formalism for the trispectrum
(a) Figure 1 . Connected (a) and disconnected (b) contributions to the four-point function of the {δφ}. The disconnected contribution factorizes into a product of two copies of the power spectrum, and contains no information concerning primordial non-gaussianities beyond those encoded in loop corrections, which are expected to be unobservably small [82] .
The δN formalism for the trispectrum
is already present in the power spectrum, and in the remainder of this paper we consider only connected contributions to the trispectrum. In principle, the contribution to ζζζζ in Eq. (8) is accompanied by terms which are higher-order in the derivatives of N, of which an example was computed by Sasaki, Väliviita & Wands [83] . We will briefly address these contributions in §6, and will return to them in a forthcoming publication.
The δN formalism for the trispectrum
In obtaining a prediction for the non-linearity parameter τ NL from Eq. (8), it proves very useful to decompose the trispectrum into a number of copies of the dimensionless power spectrum, P * . For a target space metric δ αβ , one may define the connected part of the trispectrum in terms of a form-factor M 4 (k 1 , k 2 , k 3 , k 4 ), which parametrizes the momentum-dependence of δφδφδφδφ . More specifically, combining Eqs. (6), (7) and (8) leads to a definition for M 4 given by
The δN formalism for the trispectrum
where i ∈ {1, 2, 3, 4} and the sum over permutations includes all rearrangements of the indices {α, β, γ, δ} which simultaneously permute {k 1 , k 2 , k 3 , k 4 }. Note that the form-factor may depend on the relative orientation of the k i and not merely on their magnitude. Note also that the k i should have approximately equal magnitude in order for the epoch of horizon crossing to be more or less unique. The momentum-dependence of τ NL may then be deduced by comparing Eqs. (8) and (9) with Eq. (4). This is facilitated by rewriting Eq. (4) in the form
The δN formalism for the trispectrum
where we have substituted for the power spectrum (6) . It then follows from Eqs. (8), (9) and (10) that the contribution to τ NL that is generated at horizon crossing is given by
The δN formalism for the trispectrum
where the sum is over all permutations of the {k 1 , · · · , k 4 }.
The δN formalism for the trispectrum
Finally, the non-linearity parameter can be related to the 'tensor-to-scalar' ratio, r, which is defined in terms of the corresponding power spectra such that
The δN formalism for the trispectrum
where Eq. (6) has been employed. Hence, it follows that
The δN formalism for the trispectrum
The magnitude of this contribution can therefore be derived immediately once the formfactor M 4 (k 1 , k 2 , k 3 , k 4 ) has been calculated [25] . This calculation will be the focus of the following two sections.
The fourth-order action for the perturbations
The appropriate formalism for computing expectation values such as δφ α δφ β δφ γ δφ δ is the so-called 'closed time path' integral, which was developed by Schwinger [70] and others [71, 84] , and subsequently extended to cosmology [72, 73, 19, 85, 86, 59, 87] . This integral expresses the four-point correlator as
The fourth-order action for the perturbations
where W is a weight involving the action for the field perturbations δφ α and takes the form
The fourth-order action for the perturbations
where L(δφ) is the lagrangian for the {δφ}. In Eqs. (14)- (15), we have adopted a common time of observation t ′ , and t * is any fiducial time satisfying t ′ < t * . The functional integral is over all 'forward-going' fields δφ + [84] which begin in the vacuum state at t → −∞ together with all 'backward-going' fields δφ − which obey the same boundary condition and coincide with the δφ + at t * . This differs from the conventional Feynman expression in which only the forward-going field is present [88, 89] . This difference arises because the Feynman formula computes a scattering amplitude from the past to the future, whereas the Schwinger formula computes an expectation value.
The fourth-order action for the perturbations
The lagrangian can be naturally expressed as a sum of terms involving a definite power of the perturbation δφ,
The fourth-order action for the perturbations
where L k contains k such powers. Hence, the gaussian free theory is completely specified by L 2 , but the L k terms with k 3 correspond to interactions. In the perturbative régime these can be assumed to be small corrections to the free theory, and generate the non-gaussianities we wish to calculate. In order to compute the n-point expectation value, it is necessary to know the functional forms for all L k with k n. The L 2 and L 3 terms were obtained in Ref. [22] and are sufficient to calculate the three-point expectation value and therefore the bispectrum B δφ . The term L 4 is necessary to determine the four-point expectation value, and hence the corresponding trispectrum T δφ . The scalar part of L 4 has already been derived by Sloth [82] . In the following section we extend this result by incorporating the effect of the vector part of L 4 .
The Lagrangian for field perturbations
We consider the class of inflationary models where Einstein gravity is minimally coupled to a set of scalar fields φ α :
The Lagrangian for field perturbations
where g = det g, R is the Ricci scalar of spacetime and V is the interaction potential for the fields. It is assumed that the potential can drive a phase of slow-roll inflation, but is otherwise left arbitrary.
The Lagrangian for field perturbations
In order to work with the δN formalism, we require the perturbations δφ to be defined on 'flat' spatial hypersurfaces on which the three-metric takes the form
The Lagrangian for field perturbations
, where ∂ i γ ij = 0. These slices are indeed flat in the conventional sense if the tensor mode γ ij is absent, which for simplicity we assume to be the case . It proves convenient to employ the ADM form for the metric,
The Lagrangian for field perturbations
where N(t, x) and N i (t, x) denote the lapse function and shift vector, respectively. The action (17) can be expressed in terms of these degrees of freedom such that ¶
The Lagrangian for field perturbations
where
The Lagrangian for field perturbations
is proportional to the extrinsic curvature of the spatial slices and π α =φ α − N j ∇ j φ α is proportional to the momentum conjugate to φ α . The degrees of freedom N and N i may be integrated out by first extremizing the action with respect to these fields, solving the corresponding algebraic equations of On the basis of the calculations reported by Maldacena in Ref. [19] , any expectation value involving a tensor mode will be smaller than an expectation value involving only scalars. Since the scalar nongaussianity is already hard to detect (and, despite a strong theoretical prejudice in its favour, the tensor contribution has not yet even been confirmed to exist), it seems reasonable to restrict attention to the scalar sector. In that case, γ ij does not enter in the tree-level graphs and can be dropped without loss of generality.
The Lagrangian for field perturbations
¶ In what follows we employ a useful summation convention for the spatial indices, which are labelled {i, j, · · ·}. When these appear as a raised/lowered pair, it is to be understood that they are contracted with the 3-metric, h ij . On the other hand, if a pair appear with both indices lowered, they are contracted with the flat Euclidean metric δ ij . In other words,
The Lagrangian for field perturbations
motion, and then substituting the solutions back into the action. This results in an action that can be expressed entirely in terms of propagating fields. Although the solutions to the constraint equations are in general non-local, which is signalled (for example) by the presence of formal operators such as the inverse Laplacian ∂ −2 , this will not be problematic provided we remain interested only in momentum-space expectation values. On the other hand, the continuation of such expectation values back to real space may not necessarily be a straightforward procedure.
The constraints The constraint equation for the lapse function is given by
and the corresponding expression for the shift vector takes the form
The constraints The constraint equation for the lapse function is given by
Eqs. (21) and (22) can be solved perturbatively by writing
The constraints The constraint equation for the lapse function is given by
where β j is divergenceless, i.e. ∇ j β j = 0. It is then assumed that the variables α, ϑ and β j can be expanded as a power series in the field perturbations:
The constraints The constraint equation for the lapse function is given by
where a subscript m denotes the number of powers of δφ that are present in each term.
The constraints The constraint equation for the lapse function is given by
As shown in Ref. [54] , although in principle we would need to include terms in the sum up to m = 4 in order to obtain L 4 , in practise the terms with m = 3 and m = 4 cancel out of the final answer. Let us first consider the N i constraint. We find from the O(δφ) term that
The constraints The constraint equation for the lapse function is given by
whereas α 2 and β 2j are obtained from the O(δφ 2 ) term:
The constraints The constraint equation for the lapse function is given by
In principle, the O(δφ 3 ) term could be used to find α 3 and β 3j , but these contributions are not needed.
The constraints The constraint equation for the lapse function is given by
With regard to the lapse function, the O(1) term in the constraint equation (21) is the Friedmann equation:
The constraints The constraint equation for the lapse function is given by
The O(δφ) term yields
The constraints The constraint equation for the lapse function is given by
and the term ϑ 2 follows from the O(δφ 2 ) equation:
The constraints The constraint equation for the lapse function is given by
Likewise, the O(δφ 3 ) term can be used to obtain ϑ 3 . Note that since ϑ 2 = 0 and it remains finite asφ α → 0, some metric back-reaction from the coupling to the scalar fields will still be present even in the extreme de Sitter limit.
The constraints The constraint equation for the lapse function is given by
The second-order perturbation theory of multiple scalar fields has previously been studied by Malik in the large-scale limit [90] . Compared to the lower-order perturbation theory considered in Refs. [19, 53, 22] , the most distinctive new feature of the above analysis is the combination of α 2 and β 2j appearing in Eq. (26) . These terms can be separated using the divergenceless condition on β j together with the reasoning that is employed in the proof of the Helmholtz Theorem [91, 92] . For this purpose, it is convenient to recall the elementary identity
The constraints The constraint equation for the lapse function is given by
for any vector field W. A given vector F may be decomposed into its incompressible and irrotational pieces by writing F = grad ϕ + curl A and using Eq. (30) to identify
The constraints The constraint equation for the lapse function is given by
After applying this construction to Eq. (26), it follows that α 2 can be expressed in the form
The constraints The constraint equation for the lapse function is given by
where
The constraints The constraint equation for the lapse function is given by
Likewise, it can be shown that to leading-order, β 2j satisfies
The constraints The constraint equation for the lapse function is given by
where '≃' denotes equality only at leading-order in the slow-roll expansion.
The constraints The constraint equation for the lapse function is given by
+ The exact expression is 1
The constraints The constraint equation for the lapse function is given by
It is important that β 2j cannot be written as a total gradient. Similar terms were identified in Refs. [23, 77] , and in Ref. [77] it was shown that such quantities decay like a −3 in an expanding universe. This is in agreement with non-perturbative arguments which predict the decay of shear during inflation [93] , and is essential to maintain the relationship between R and ζ on large scales. We thank F. Vernizzi for emphasizing this point to us.
The fourth-order action
After use of the constraint equations (21) and (22) as outlined above, the fourth-order action is found to be given by
The fourth-order action
When appropriately truncated in powers of slow-roll, the scalar sector of this action is equivalent to that presented in Appendix A of Ref. [82] . However no truncation in slow-roll has been made in Eq. (36), which is exact. The leading-order, slow-roll terms in Eq. (36) arise from α 2 , ϑ 2 , and β 2j [via Eqs. (32), (29) and (35)]. These all contain terms of O(ǫ 0 ), where ǫ ≡ −Ḣ/H 2 parametrizes the degree of departure from a pure de Sitter expansion. Extracting these contributions from Eq. (36) then leads us to the fourth-order action for the perturbations that we have been seeking:
The fourth-order action
The form of the lagrangian L 4 then follows immediately from Eq. (37).
Loop corrections
The presence of interaction terms in L 4 which are 'unprotected,' in the sense that they are not suppressed by a positive power of the slow-roll parameters, might give rise to concerns that the two-point function δφ α δφ β also receives large, unprotected loop corrections. If this were the case, it would seriously impair our ability to make predictions about the early universe from models of inflation. Fortunately, however, the calculations reported in Ref. [82] indicate that these unprotected loop corrections occur at about 1% of the tree-level, which is not large enough to harm our predictivity. On the other hand, it should be remarked that it has yet to be verified that such corrections are never large at any order in the interaction of δφ.
Loop corrections
Given the interaction lagrangian (37) , it only remains to apply Eqs. (14)- (15) in order to obtain the four-point expectation value of the field fluctuations and therefore the momentum-dependence of the form-factor M 4 and the non-linearity parameter τ NL . This will be focus of the following section.
The trispectrum form-factor M 4

The momentum-dependence of M 4
It is convenient to calculate the four-point expectation value in three separate stages by considering the blocks of terms in Eq. (37) which consist of either two, one or zero copies of the pure vector β 2j .
The momentum-dependence of M 4
• Vector × vector. This is the term containing β 2j ∂ 2 β 2j . It can be calculated following the general principles outlined in §3 and described in more detail in Refs. [19, 85, 53, 22, 86, 82, 72, 73, 84] . Before proceeding with the calculation, however, it will be helpful to introduce some new notation. In particular, we define a vector Z ij which is a function of two momenta k i and k j , such that
The momentum-dependence of M 4
where σ ij is the scalar combination
The momentum-dependence of M 4
The vector × vector term then provides a contribution to the four-point expectation value δφ
The momentum-dependence of M 4
where 'c.c.' denotes the complex conjugate of the preceding term, the index i ranges over {1, 2, 3, 4}, the conformal time equivalents of (t ′ , t * ) are given by (η ′ , η * ) and the total scalar momentum is defined by k t = k 1 + k 2 + k 3 + k 4 . The sum includes all permutations of the indices {α, β, γ, δ} which simultaneously permute the momentum labels {1, 2, 3, 4}. The time of observation, η ′ , should be chosen to be a few e-foldings after the modes k i have crossed the horizon, so that it can be safely assumed that the fluctuations δφ have become classical but have not undergone significant evolution. At the level of accuracy to which we are working, this will be the case if the condition |k t η ′ | ∼ O(ǫ) ≪ 1 is satisfied. When this condition holds, we may further specify H(η ′ ) = H(η) = H * to a good approximation, since the integral over η will receive only negligible contributions from times long before the epoch of horizon crossing. Likewise, extending the upper limit η * in Eq. (40) to the infinite future (η * → 0 − ) introduces no significant error. This implies that the integral (40) can then be evaluated analytically and we find that the contribution of this piece to the formfactor M 4 is given by
The momentum-dependence of M 4
where we have defined
The momentum-dependence of M 4
Since Eq. (42) is purely real, the addition of its complex conjugate has been absorbed into an overall factor of 2.
The momentum-dependence of M 4
• Scalar × vector. The same principles apply to the scalar × vector sector, which comprises the term aδφ α β 2j ∂ j δφ α . Its contribution to the four-point expectation value is given by
The momentum-dependence of M 4
Since the integral is equivalent to W 24 , we may immediately deduce the corresponding contribution of this term to the form-factor is
The momentum-dependence of M 4
• Scalar × scalar. The final piece is comprised of aϑ 2 Σ + (3/4)a 3 ∂ −2 Σ∂ −2 Σ and is independent of β 2 i . A loop integral involving this term was calculated in Ref. [82] . After truncating ϑ 2 to leading order, using Eq. (29), the scalar piece can be rewritten
The momentum-dependence of M 4
The first of these terms makes a contribution of the form
The momentum-dependence of M 4
It proves useful when evaluating this integral to define a three-momentum generalization of W ij ,
The momentum-dependence of M 4
Hence, the contribution of Eq. (46) to the form-factor is
The momentum-dependence of M 4
Finally, the second term, −(3/4)a 3 ∂ −2 Σ∂ −2 Σ, yields
The momentum-dependence of M 4
and provides the contribution
The momentum-dependence of M 4
to the form-factor.
The momentum-dependence of M 4
It follows, therefore, that the complete momentum-dependence of the trispectrum can be determined by assembling Eqs. (41), (44), (48) and (50) . Specifically, we find that the form-factor is given by
The momentum-dependence of M 4
and Eq. (51) represents the main result of the paper. The form-factor has dimensions of three powers of the momentum, which is the same behaviour as the corresponding formfactor M 3 which determines the bispectrum. However, the momentum-dependence of M 4 is considerably more complicated than that of M 3 and, in particular, it depends on the products k i · k j and not merely the magnitudes k i .
The limit of equal momentum
If two of the momenta, say k 1 and k 2 , have equal magnitude k 1 = k 2 ≡ p, and are also equal and opposite, k 1 + k 2 = 0, it follows from momentum conservation that the other two momenta must also be equal and opposite, albeit with a possibly different common magnitude, i.e., k 3 = k 4 ≡ q. On such configurations, M 4 appears to be pathological, since the factors (k 1 + k 2 ) −1 and (k 3 + k 4 ) −2 naïvely diverge. If this were the case it would be disastrous, since the non-gaussianity associated with the four-point function is presumably bounded from above by the WMAP data.
The limit of equal momentum
Despite appearances, the form-factor M 4 does not diverge on these degenerate quadrilaterals. This can be verified by introducing a simple specification for the k i . We define a matrix of angles θ ij ∈ [0, π] which satisfy
The limit of equal momentum
This matrix obeys the obvious symmetry constraint θ ij = θ ji , and therefore has 6 components, which cannot all be chosen independently. To fully specify the {k i }, it is necessary to include two additional angles which determine the absolute orientation of one of the k i , which we take to be k 4 . However, these angles do not contribute to M 4 and we need not be explicit about their precise assignment. There is another angular degree of freedom corresponding to rigid rotations which leave k 4 fixed, which does not change the intrinsic geometry of the {k i }. In total, therefore, the magnitudes of the four momenta, the two angles which determine the orientation of k 4 , the angular degree of freedom corresponding to rotations which leave k 4 fixed, and the five independent θ ij comprise the twelve degrees of freedom which are required to completely specify four vectors in R 3 . Momentum conservation supplies three constraints which eliminate three of these degrees of freedom. These can be chosen to be three of the angles θ ij . It is convenient to work with the angles {θ r = θ r4 }, in terms of which the others can be written as
The limit of equal momentum
We wish to re-express the form-factor (51) in terms of p, q and θ r . This requires explicit expressions for W 24 and W 124 , which can be shown to be
The limit of equal momentum
We also have the additional relation σ 12 = σ 34 = q 2 (1+cos θ 3 ) and the quantities Z 12 ·Z 34 and k 2 · Z 34 can be reduced to
The limit of equal momentum
respectively. It is now relatively straightforward to rewrite expression (51) given Eqs. (56)- (59) and we find after some algebra that
The limit of equal momentum
The naïve divergence associated with the limit k 1 → −k 2 reappears as θ 3 → π. However, Eq. (60) should not be considered in isolation, but only after symmetrization over the labels {1, 2, 3, 4}. Indeed, after symmetrization over the exchange 1 ⇌ 2 we find
The limit of equal momentum
and it follows that Eq. (61) possesses a finite limit as θ 3 → π which is not spoilt by further symmetrization. Hence, as anticipated, Eq. (61) demonstrates that M 4 does not diverge on degenerate quadrilaterals.
How large can the trispectrum be?
We now proceed to estimate the magnitude of the non-linearity parameter |∆τ NL | that arises from Eq. (51). This is important, since the non-gaussianity associated with the trispectrum may be observable in the near future if |∆τ NL | 560 [69] . The form-factor M 4 is formally valid only when k i ≈ k j for all i and j. We will show in §6 that, at least in the single-field case, M 4 must become of order O(ǫ) in the limit where one k i becomes much smaller than the others. In this limit one cannot trust (51) to give the dominant contribution to M 4 , but it is consistent with the explicit momentum dependence in (51) , which approaches zero in the formal limit where any k i → 0. This provides evidence that the largest contribution to τ NL from the connected four-point function of the {δφ α } arises on equilateral configurations, although a more complete treatment of this question would be desirable. In this section, we assume that |∆τ NL | is maximized on equilateral configurations of the {k i }.
How large can the trispectrum be?
The relevant expressions for the equilateral configuration follow after we have specified p = q = k in Eqs. (53)- (59) . The equilateral condition is not independent of momentum conservation. In practise this means that the angles θ r cannot be chosen freely, but must instead obey the constraint r cos θ r = −1.
How large can the trispectrum be?
We may choose only two of the θ r arbitrarily. These two angles suffice to parametrize the family of distinct equilateral configurations. If k 4 is chosen to point alongẑ, and we take (62) to determine θ 3 in terms of {θ 1 , θ 2 }, then this family can be written
How large can the trispectrum be?
where cos θ 3 = −1 − cos θ 1 − cos θ 2 , and the angles α and β obey
How large can the trispectrum be?
, and tan
How large can the trispectrum be?
given that γ m is a combination of the θ m ,
How large can the trispectrum be?
The extra degree of freedom, φ 1 , corresponds to rigid rotations of the {k 1 , k 2 , k 3 } which fix k 4 ∝ẑ. One can verify explicitly that Eqs. (53)- (55) and Eq. (62) are obeyed by (63) , irrespective of the value of φ 1 . In the equilateral limit, W 24 and W 124 reduce to
How large can the trispectrum be?
respectively, and this implies that Eq. (60) .
How large can the trispectrum be?
We may then substitute this result into Eq. (13) . Summing over the possible permutations of labels and using Eq. (62) to eliminate the sum r cos θ r , we deduce that
How large can the trispectrum be?
The magnitude of τ NL which is generated depends on the angles {θ r }. However, this dependence is in some sense artificial since the totally permuted form factor, perms M 4 , contains no angular dependence in the equilateral limit. The dependence on the {θ r } which appears in Eq. (69) arises solely from the definition of τ NL which we adopted in Eq. (4) . |∆τ NL | is maximized on the configuration cos θ r = −1/3, which gives
How large can the trispectrum be?
Since the current experimental upper bound on the tensor-to-scalar ratio is r < 0.55 (at 95% confidence) [65] , Eq. (70) immediately implies that the primordial non-gaussianity in the trispectrum that is generated at horizon crossing will be unobservably small. Consequently, as in the case of the bispectrum, if a non-trivial primordial trispectrum is observed in a future CMB experiment, its origin is unlikely to arise from quantummechanical interference around the time of horizon crossing. We observe that (70) is not zero. Taken together with the explicit family of equilateral configurations (63) , which are all allowed by momentum conservation, this removes any worry that the complicated expression obtained for M 4 in Eq. (51) might be constrained to zero by purely kinematical considerations.
How large can the trispectrum be?
Thus far, our discussion has considered the multi-field inflationary scenario. A number of simplifications arise when the analysis is restricted to single-field inflation and we proceed to discuss these in the following section.
Single-field Inflation and the Maldacena consistency relation
In single field inflation, ζ is conserved to all orders on superhorizon scales [78, 76] . This has the immediate consequence that no evolution in τ NL is possible after horizon exit, and Eq. (70) represents the largest possible non-gaussian signal which can be visible in the trispectrum. We conclude that it is too small ever to be detected.
Single-field Inflation and the Maldacena consistency relation
A second special feature of single-field inflation is the existence of a consistency relation between the correlators ζ(k 1 ) · · · ζ(k n ) . In the 'squeezed' limit, where k 1 → 0, the ζ(k 1 ) mode crosses the horizon at a much earlier epoch than the remaining modes {k 2 , . . . , k n }. By the time these modes eventually exit the horizon, the gravitational background has been deformed due to the presence of the ζ(k 1 ) mode. As first pointed out by Maldacena [19] , the k 1 mode and the {k 2 , · · · , k n } modes no longer interfere when they cross the horizon, which implies that the only correlation between them is the one imposed by this gravitational relationship. In the case of the three-point function, this yields the relation
Single-field Inflation and the Maldacena consistency relation
where n s is the spectral index of the scalar perturbation power spectrum. It was later emphasized by Creminelli & Zaldarriaga [94] that the limit (71) is purely kinematical and applies for any metric theory of gravity. This implies that it can be calculated on the basis of gravitational physics alone [27] . Maldacena's arguments apply for any correlation function of ζ, including the fourpoint function. In this case, the consistency relationship is given by
Single-field Inflation and the Maldacena consistency relation
To gain further insight into the nature of Eq. (72), it is instructive to count powers of slow-roll. In general, the derivatives of the number of e-foldings, N, are controlled by the slow-roll parameter ǫ = −Ḣ/H 2 . If ǫ is almost constant over the range of e-folds under consideration, it follows that N ≃ −ǫ −1 ln H. This implies that each derivative of N with respect to the scalar field generates an extra power of
Single-field Inflation and the Maldacena consistency relation
By counting powers of slow-roll in Eq. (72), it may be verified that this behaviour produces a four-point function of order O(ǫ −1 ). This can be compared to the horizon-crossing contribution, Eq. (8), which is O(ǫ −2 ). Eq. (72) implies that (51) becomes at most of O(ǫ) when any of the k i → 0, at least in the single field case. Therefore, the leading slow-roll term in the four-point function of ζ will be given by possible subdominant corrections to (51) which do not vanish in the squeezed limit, together with the terms which contribute to the four-point function at next-order in the slow-roll parameter ǫ, i.e., the terms of order O(ǫ −1 ). The first such term is
Single-field Inflation and the Maldacena consistency relation
where the sum contains twenty-four terms that are obtained by rearrangements of the momenta, and B is the δφ-bispectrum. The momentum conservation condition implies that one-half of the permuations are equal to the other half, giving twelve distinct terms. The second relevant term is
Single-field Inflation and the Maldacena consistency relation
At the tree-level, only this term was included in the estimate for τ NL given by Alabidi & Lyth [28] and Lyth [34] , although these authors also included a one-loop diagram whose contribution we assume to be negligible in comparison with the tree-level. * * Owing to a typographical error, the τ NL which follows from (74) appeared incorrectly in the journal version of Ref. [34] . The correct expression was given in both v1 and v2 of Ref. [28] 
Single-field Inflation and the Maldacena consistency relation
An expression equivalent to this was given in Ref. [83] in the context of the curvaton scenario.
Single-field Inflation and the Maldacena consistency relation
In principle, therefore, the next-order contribution to the four-point function can be obtained by summing Eqs. (73)- (75) . However, when performing such a sum, selfconsistency would require that the next-order versions of the two-and three-point functions of the field fluctuations δφ and the derivatives of N be employed. To date, the required expressions for these quantities have yet to be calculated since presumably they are unobservably small. Consequently, we should not expect the ǫ and η dependence of Eqs. (73)- (75) to match the right-hand side of Eq. (72), where η = V ,φφ /V is O(ǫ) in the slow-roll expansion. On the other hand, the next-order corrections to the scale factor, a(t), the wavefunctions of the δφ and the third-order action are not expected to contain any intrinsically second-or higher-order slow-roll parameters, which involve the third-or higher-derivatives of the potential. Therefore, summing Eqs. (73)- (75) with the leading-order expressions for N, the scale factor a, and the two-and three-point functions of the field flucations δφ should match the coefficient of V ,φφφ in Eq. (72) . We have verified that this is the case.
Conclusions
In this paper, we have derived the trispectrum of the primordial curvature perturbation imprinted at horizon crossing during inflation. In particular, we have estimated the expected magnitude of the non-linearity parameter, τ NL , which is generated by this process. We find that it is bounded by about r/50, and therefore is too small to be observable by the Planck satellite or other future CMB experiments. In single-field models, ζ is conserved to all orders on superhorizon scales. Therefore, no subsequent generation of non-gaussianity is possible, and the primordial signal will be too small ever to be detected.
Conclusions
One unexpected feature of the four-point action is that it contains no powers of slow-roll parameters. In an earlier publication [22] , two of us (DS and JEL) suggested that the nth order action would contain essentially ⌊n/2⌋ powers ofφ/H, which would imply that the coupling to the spacetime metric (and all backreaction from it) switches off order-by-order in the limitφ/H → 0, and therefore that the field fluctuations are purely Gaussian in the de Sitter limit. Our result for M 4 shows that this does not happen: the scalar fields and the metric remain coupled even in exact de Sitter space, and the fluctuations are therefore non-gaussian. Intuitively, this occurs because the size of the non-gaussianities in the δφ is controlled by the strength of the gravitational interaction, rather thanφ/H, even though powers of slow-roll appear once one has changed variable to the comoving curvature perturbation, ζ. We note, however, that ζ is not an appropriate choice of variable to discuss the limitφ/H → 0, since it becomes singular there.
Conclusions
In the case of single-field inflation, we have applied Maldacena's consistency argument to the trispectrum. For the first time, this has employed the third -order term in the expansion of δN in terms of initial field values. Only the linear and quadratic terms in the expansion have previously been required. In the case of the trispectrum, the consistency relationship arises as a next-order effect in the slow-roll parameter ǫ, in contrast to that of the bispectrum.
Conclusions
We have also derived expressions in the spatially flat gauge for the second-order metric perturbation sourced by multiple scalar fields. These expressions show explicitly that a metric perturbation continues to exist at this order even when the background is pure de Sitter space, corresponding to ǫ → 0. At first-order, the metric perturbation disappears in this limit.

{'paper_id': '8208513', 'title': 'The inflationary trispectrum', 's2_url': 'https://api.semanticscholar.org/CorpusID:8208513', 'source': 'S2ORC'}