Large non-Gaussianities in the Effective Field Theory Approach to Single-Field Inflation:the Trispectrum

# Large non-Gaussianities in the Effective Field Theory Approach to Single-Field Inflation: the Trispectrum

Nicola Bartolo, Matteo Fasiello, Sabino Matarrese and Antonio Riotto Dipartimento di Fisica “G. Galilei”, Università degli Studi di Padova, via Marzolo 8, I-35131 Padova, Italy INFN, Sezione di Padova, via Marzolo 8, I-35131 Padova, Italy Dipartimento di Fisica “G. Occhialini”, Università degli Studi di Milano Bicocca and INFN, Sezione di Milano Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy CERN, Theory Division, CH-1211 Geneva 23, Switzerland

DFPD-2010-A-13   CERN-PH-TH/2010-146
###### Abstract

We perform the analysis of the trispectrum of curvature perturbations generated by the interactions characterizing a general theory of single-field inflation obtained by effective field theory methods. We find that curvature-generated interaction terms, which can in general give an important contribution to the amplitude of the four-point function, show some new distinctive features in the form of their trispectrum shape-function. These interesting interactions are invariant under some recently proposed symmetries of the general theory and, as shown explicitly, do allow for a large value of the trispectrum.

, , and

## 1 Introduction

Inflation [1] is one of the central pillars of modern cosmology. Not only it provides a natural solution to the flatness, horizon and monopole problems of standard Big-Bang cosmology, but can also explain the production of density perturbations in the early Universe which then lead to LSS [2, 3, 4, 5, 6] in the distribution of galaxies and temperature anisotropies in the CMB [7, 8, 9, 10, 11, 12].
Besides the simplest single-field slow roll inflationary model, many other inflationary mechanisms have been proposed since inflation was first introduced, and all are compatible with the CMB and LSS observations. In order to probe deeper into the dynamics of inflation and to remove the degeneracies generated by the many models so far proposed, one might study observable quantities which are sensitive to deviations from Gaussianity [13]: starting from the three-point function [14, 15], one then considers the trispectrum [16] and in general higher-order correlators as well as loop effects in the power spectrum [17]. Such non-Gaussian features will depend on the various interactions characterizing any given inflationary model in the form of self-interactions of the inflaton, its coupling with gravity and interactions with other fields in the case of multi-field inflation (see, e.g., [18, 19, 20] for comprehensive and updated reviews). These investigations are spurred by the fact that the continued analysis of WMAP data [12] and the recent launch of the Planck satellite [21, 22] provide the exciting opportunity to actually test the predictions of this zoo of models at the level of bispectrum and trispectrum of curvature perturbations. A very useful tool in analyzing the possible signatures of the different inflationary models is given by the effective field theory approach to inflation recently introduced in [23] and further expanded in [24, 25, 26, 27]. There are various advantages in employing this formalism. Indeed, it provides a unifying perspective on inflation in that it automatically accounts for many known inflationary mechanisms. To each set of interactions for a given inflationary model there corresponds in the effective Lagrangian a linear combination of operators obtained by turning on and off some specific coefficients that regulate the weight of the operators (we introduce these coeffcients later on and refer to them as ’s). The unifying power of the effective field theory approach is quite manifest in that, in principle, it allows these coefficients considerable more freedom than what they are granted in any specific inflationary model. In fact, by being for the most part free parameters (a couple of these coefficients are to obey some inequalities if one wants, as we do, the generalized speed of sound to be smaller than unity), the ’s allow for the description of known interactions with relative weights which would otherwise be fixed, so by employing effective field theory one enlarges the region of the parameters space than can be spanned. Besides that, in the effective Lagrangian some of the ’s multiply curvature-generated operators (see below Eq. (2)) that are sometimes neglected but should in principle be studied as, in fact, their contribution can be relevant [28] and increase the dimension of the parameters space of the theory.
In this paper we use effective field theory techniques to study the trispectrum of a very general theory for single field inflation. In particular, we concentrate on the contributions of many novel curvature-generated terms as well as interactions that characterize Ghost inflation [29]. As an ordering principle among the numerous interactions such a general approach comprises, we employ two additional symmetries of the action recently introduced in [30, 31] (see also [32]) and examine the cases when one or both of these requirements are imposed on the theory. Following [33], we analyze the shapes of these terms in four different configurations so as to identify distinctive effects in the trispectrum shape-function from the various interactions. We first analyze the contribution to the scalar exchange diagram due to a curvature-related term which generates an interesting flat shape for the bispectrum [28]. This term produces a shape function which is, in some configurations, different from all the shapes due to leading interactions in general single-field inflation models [33].111In these models the inflaton Lagrangian is an arbitrary function of the inflaton and its first derivative. The theories we will consider here further generalize these models. We calculate and plot the contributions to the contact interaction diagram by several interactions: we rediscover the shape-function due to the leading fourth-order interaction in ghost inflation first obtained in [30] and plot it in new configurations; an analysis of other interesting term is also performed allowing us to identify novel distinctive features of curvature-related interactions.
The paper is organized as follows. In section 2 we build on [24] to introduce the very general effective theory we employ in all subsequent calculations. The reader who is familiar with this procedure might want to skip this part and start directly from Eq. (LABEL:h4). In section 3 we characterize the various interaction terms of the theory according to their behaviour under the action of two specific symmetries. Afterwards, in section 4, we proceed by briefly outlining the tools of the IN-IN formalism successively employed in the trispectrum calculations. A separate analysis of the scalar exchange and contact interaction diagrams contributions to the trispectrum is performed. Section 5 contains a summary of the findings and comments on further work. In Appendix A we report the details of the scalar exchange diagram calculations. In Appendix B we show the reliability of our simplifying assumption on the classical solution to the equations of motion for the general theory considered in this paper.
For the sake of clarity, we stress here that whenever we refer to general single-field inflation models, often when elaborating on the results of [33], we are dealing with theories which account for a great fraction of known inflationary models (DBI, K-inflation etc.), but still miss an important subset including Ghost inflation and curvature-generated interactions in general. On the other hand, the effective approach of [24] we employ here also covers the latter.

## 2 The Hamiltonian up to fourth order

We will follow the effective theory approach first introduced in Ref. [24] in order to write down the complete theory of single-field models of inflation up to fourth order in perturbations, subjected to the sole requirement of an approximate shift symmetry, , on the scalar degree of freedom . Let us give a brief account on how to obtain the main formulas. We start with the scalar field responsible for inflation, which is split as an unperturbed part plus the fluctuation:

 ϕ(→x,t)=ϕ0(t)+δϕ(→x,t). (1)

For reasons that will soon become clear, one chooses here to work in the the comoving (or unitary) gauge for which [23]. As a result, the action will no longer be invariant under full spacetime diffeomorphisms (diffs) but only under the spatial reparametrizations. This is the starting point to write the most general unitary gauge space diffs invariant Lagrangian at the desired order in perturbation theory [24]:

 S=∫d4x√−gF(Rμνρσ,g00,Kμν,∇μ,t), (2)

where is the extrinsic curvature and the indices on the metric entries are free indices. Taking into account the fluctuations around a FRW background, one obtains the following action

 S = ∫d4x√−g[12M2PlR+M2Pl˙Hg00−M2Pl(3H2+˙H)+ (3) ∑n≥2F(n)(g00+1,δKμν,δRμνρσ;∇μ;t)],

where the fluctuations contained in are at least second order. The next step is restoring full spacetime diffs invariance. To see how it works, we take from [24] the following sample terms in the action:

 ∫d4x√−g[A(t)+B(t)g00(x)] . (4)

Consider the time reparametrization: ; under its action (and after a simple variable redefinition) Eq. (4) reads:

 ∫d4x√−g(x)[A(t−ξ0(x))+B(t−ξ0(x))∂(t−ξ0(x))∂xμ∂(t−ξ0(x))∂xνgμν(x)]. (5)

At this stage the procedure we will adopt consists in promoting to a field, and requiring the following gauge transformation rule: on . With this assumption in place, the above action is invariant under full spacetime diffeomorphisms. The scalar degree of freedom makes its appearance in the time dependence of the coefficients and in the transformed metric. This procedure is essentially the same as the one of standard gauge theory: a Goldstone boson which transforms non-linearly under the gauge transformation provides the longitudinal component of a massive gauge boson. For high enough energies, the Goldstone becomes the only relevant degree of freedom. This is the so-called equivalence theorem. The same is true for our case: for sufficiently high energy the mixing with gravity becomes irrelevant and the scalar becomes the only relevant mode in the dynamics (decoupling regime). One then needs to identify the scale of the energy above which this approximation holds (on the other side of the energy range one always keeps in mind the upper energy threshold that comes with the use of effective theory). This procedure puts some bounds on the values of some of the coefficients that drive quadratic operators in the action [24, 28]. We will not be concerned with these issues because most of the interaction we will be analyzing start at third order in perturbations. It suffices here to say that, since one is concerned with correlators just after horizon crossing, the decoupling procedure works as long as the decoupling energy is smaller than the Hubble rate .
From now on we will work in the decoupling regime. In considering the terms of Eq. (3), we will therefore use only the unperturbed entries of the metric tensor. To write the effective Lagrangian up to third order, we start from Eq. (3) and follow the algorithm given in [24]. Fluctuations are encoded in the terms. In order to be as general as possible, one must also include all possible contributions coming from extrinsic curvature terms.
Following the procedured outlined above, the third and fourth-order Lagrangian is obtained. We will use the third-order expression to calculate the contribution to the trispectrum of curvature perturbations arising from the scalar exchange diagram [28]:

 S3 = ∫d4x√−g[M2Pl˙H(∂μπ)2+M2(t)4(2˙π2−2˙π(∂iπ)2a2)−43M3(t)4˙π3 (6) − ¯M1(t)32(−2H(∂iπ)2a2+(∂iπ)2∂2jπa4)−¯M2(t)22((∂2iπ)(∂2jπ)+H(∂2iπ)(∂jπ)2a4 + 2˙π∂2i∂jπ∂jπa4)−¯M3(t)22((∂2iπ)(∂2jπ)+2H(∂iπ)2∂2iπ+2˙π∂2ijπ∂jπa4) − 23¯M4(t)31a2˙π2∂2iπ+¯M5(t)23˙πa4(∂2iπ)2+¯M6(t)23˙πa4(∂ijπ)2−¯M7(t)3!(∂2iπ)3a6 − ¯M8(t)3!∂2iπa6(∂jkπ)2−¯M9(t)3!1a6∂ijπ∂jkπ∂kiπ].

The above action and its extension at higher perturbative orders covers many inflationary theories, providing a unifying perspective which is hard to obtain without an effective approach. Indeed, by switching on and off a single coefficient one has control over all corresponding operators in the action (the bar on the ’s signals that these are curvature-generated terms). The hope is to be able to identify distinctive features for as many as possible different combinations of the ’s in the form of specific patterns they produce in the shapes of the various correlators of curvature perturbations. The degeneracies among the results for different inflationary mechanisms that will inevitably arise might be removed by a joint analysis of the different n-point functions, starting with the bispectrum, the trispectrum, loop corrections to the power spectrum and so on. Let us briefly go through some of the main features of the third order effective action above. All the comments can be straightforwardly extended to the fourth-order expression as well.

• Consider only the quadratic terms: for one recovers the usual quadratic Lagrangian for the fluctuations, with sound speed . Switching on corresponds to allowing models with sound speed smaller than unity, , which are often linked to a high level of primordial non-Gaussianity [35, 24] as for DBI inflation. Further allowing for a non-zero in the de Sitter limit, one recovers Ghost Inflation [29]. Similarly, having all ’s set to zero, and going to third and higher order with the ’s, one retrieves the interactions characterizing DBI inflation [34, 35, 33] and K-inflation [36, 37] theories and others. 222See [38, 39] for some recently introduced examples that require, in order to be recovered in the effective field theory approach, that one relaxes the implicit assumption of a shift symmetry for the scalar .

• The action in Eq. (6) contains in principle additional terms but, being interested in those generating large non-Gaussianities, a selection has been made. Specifically, at every order in fluctuations and for each coefficient, only leading terms are considered. One starts from the realization that, even for the most generic quadratic action the following estimates hold at horizon crossing [28]:

 ˙π∼Hπ,∇π∼H√α0+√α20+8β0π≡H/~csπ (7)

where is a sort of generalized speed of sound.
Also, as mentioned before, in the action the scalar appears only trough its derivatives. When faced with a given multiplying terms of the same perturbative order with a given number of derivatives one therefore knows that the leading term will be the one with the most spatial derivatives.
There is also the comparison between the same perturbative order but different terms to be made. All non-zero coefficients in front of the various operators might be assumed to be of the same order [29]; interestingly, in [27], employing renormalization and unitarity arguments, a natural (relative) value of was obtained for the coefficients. In general, we shall not restrict ourselves to these situations. Considering theories with a speed of sound different from unity and allowing for ’s of different orders greatly increases the number of viable terms for large non-Gaussianities. Let us consider an illustrative example. Take the interaction terms

 −2M2(t)4˙π(∂iπ)2a2;−¯M7(t)3!(∂2iπ)3a6

At horizon crossing, the region from which we expect the main contribution to n-point functions, the comparison reads like . In a Lorentz invariant theory with coefficients of the same order the first term would clearly prevail. Allowing makes the comparison less obvious and an further strengthens this point. Here a word of caution is in order: from simple dimensional analysis a term with ever increasing spatial derivatives will have an coefficient with smaller and smaller exponent (counterbalanced by an higher exponent for ) and must eventually be subleading with respect to the contributions with fewer derivatives. This is because in the effective theory approach one is roughly making an expansion (with being the scale of the underlying theory) and, although one can fully employ the freedom to have ’s of different size up to some perturbative order in order to resuscitate interesting contributions to the correlators, the -driven contribution must eventually (from some on) cease to be relevant.

Employing the same calculational algorythm first introduced in [24] and used in writing the complete third order action above, we obtain the most general fourth-order action in this set up:

 S4=∫d4x√−g[12!M2(t)4(∂iπ)4a4+2M3(t)4˙π2(∂iπ)2a2+23M4(t)4˙π4
 −¯M1(t)34(H(∂iπ)4a4−2˙π(∂iπ)2∂2jπa4)−¯M2(t)22((∂jπ)2(∂2iπ)2a6+2∂2kπ∂iπ∂ijπ∂jπa6)
 −¯M3(t)22((∂ijπ)2(∂kπ)2a6+2∂iπ∂ijπ∂jkπ∂kπa6)+23¯M4(t)3˙π(∂iπ)2∂2jπa4
 −¯M6(t)23!(∂kπ)2(∂ijπ)2a6−¯M7(t)3!(32(∂2iπ)2H(∂jπ)2a6+6˙π∂2kπ(∂j∂2iπ)∂jπa6)
 −¯M8(t)3!(H(∂iπ)2(∂2jπ)2a6+H(∂iπ)2(∂jkπ)22a6−2H∂2kπ∂iπ∂ijπ∂jπa6+2˙π∂2kπ∂2i∂jπ∂jπa6
 +2˙π∂2k∂iπ∂ijπ∂jπa6+2˙π∂ij∂ijk∂kπa6)−¯M5(t)23!(∂iπ)2(∂2jπ)2a6
 −¯M9(t)2(H∂2kπ(∂ijπ)22a6−H∂iπ∂ijπ∂jkπ∂kπa6+˙π∂ijπ∂ijkπ∂kπa6+˙π∂2i∂jπ∂jkπ∂kπa6)
 +¯M310(t)3˙π3∂2iπa2−¯M211(t)3!˙π2(∂2iπ)2a4−¯M212(t)3!˙π2(∂ijπ)2a4+¯M13(t)4!2˙πa6(∂2iπ)3
 +¯M14(t)4!2˙π∂2kπ(∂ijπ)2a6+¯M15(t)4!2˙π∂ijπ∂jkπ∂kiπa6−¯N1(t)4!(∂2iπ)4a8−¯N2(t)4!(∂2kπ)2(∂ijπ)2a8
 −¯N3(t)4!∂2ρπ∂ijπ∂jkπ∂kiπa8−¯N4(t)4!(∂ijπ)4a8−¯N5(t)4!∂ijπ∂jkπ∂kρπ∂ρiπa8] (8)

Note that, as pointed out in [40], starting at fourth order in perturbations, one cannot immediately read off the Hamiltonian from the expression of the Lagrangian, in other words does not hold here. We use the results one obtains by adopting the correct procedure which was outlined in detail in [40].
Let us split the interaction Hamiltonian we will be concerned with as ; one can prove that the overall interaction Hamiltonian is then:

 Hint=−L3−L4+∫d3x√−g[12M42+H2ϵM2P−3H¯M31((∂iπ)4M82a4+4˙π2(∂iπ)2M42M43a2 +4˙π4M83+(∂kπ)2∂2i∂jπ∂jπM42¯M22a6+2˙π2∂2i∂jπ∂jπM43¯M22a4+(∂2i∂jπ∂jπ)2¯M424a8 +(∂kπ)2∂2i∂jπ∂jπM42¯M23a6+2˙π2∂2i∂jπ∂jπM43¯M23a4+(∂2i∂jπ∂jπ)2¯M22¯M232a8+(∂2i∂jπ∂jπ)2¯M434a8 +4˙π(∂kπ)2∂2iπM42¯M343a4+8˙π3∂2iπM43¯M343a2+2˙π∂2kπ∂2i∂jπ∂jπ¯M22¯M343a6+2˙π∂2kπ∂2i∂jπ∂jπ¯M23¯M343a6 +4˙π2(∂2kπ)2¯M649a4−(∂iπ)2(∂2kπ)2M42¯M253a6−2˙π2(∂2iπ)2M43¯M253a4−(∂2kπ)2∂2i∂jπ∂jπ¯M22¯M256a8 −(∂2kπ)2∂jπ∂jπ¯M23¯M256a8−2˙π(∂2kπ)3¯M34¯M259a6+(∂2kπ)4¯M4536a8−(∂kπ)2(∂ijπ)2M42¯M263a6 −2˙π2(∂ijπ)2M43¯M263a4−(∂klπ)2∂2i∂jπ∂jπ¯M22¯M266a8(2M42+H2ϵM2P−3H¯M31)−(∂klπ)2∂2i∂jπ∂jπ¯M23¯M266a8(2M42+H2ϵM2P−3H¯M31) −2˙π∂2kπ(∂ijπ)2¯M34¯M269a6+(∂2kπ)2(∂ijπ)2¯M25¯M2618a8+(∂ijπ)4¯M4636a8⎞⎠]

where the above terms besides are all at fourth order in perturbations.

## 3 Symmetries

Having written the complete Hamiltonian, we now proceed to calculate the four-point function contributions arising from interaction terms at third and fourth order. We employ here the IN-IN formalism [42, 43, 44, 45] and conveniently split the contributions to the four-point function as the ones arising from terms that make up the contact interaction diagram and the ones that generate the scalar exchange diagram as in the figure below.

It is useful at this stage to offer some comments on the calculations we are going to present. As mentioned, the literature already contains a thorough analysis of the trispectra for general single-field inflation models, see for example [33]. Work on the four-point function for ghost inflationary models has recently been presented [32, 30]. Our starting point, being based on a comprehensive effective theory, clearly encompasses all these models. Working with the effective Hamiltonian above translates into many immediate advantages as listed before but, on the other hand, in calculating the resulting four point function, one faces a substantial number of terms and it is therefore natural to look for some ordering principle which would single out some contributions to the trispectrum as the leading ones and allow us to concentrate on them only. In this context employing a symmetry for the whole theory can prove very useful. Indeed in [31, 30] the authors consider only those allowed by a particular (approximate in [31]) symmetry of the action, respectively:

 S1:π→−π;S2:π→−πandt→−t. (10)

We plan here to employ our general effective theory to show that, allowing some freedom on the coefficients that modulate the various terms in the third and fourth order action, within each one of the two distinct and quite restrictive symmetry requirements above there are novel curvature-generated terms in the action that should not be disregarded as negligible and that, furthermore, show some distinctive features in the shapes of the trispectrum. We will also describe terms allowed by both the symmetries in Eq. (10) combined. Of course, one need not employ symmetries to switch on or off any specific operator in the action. Most of the contributions are indeed freely adjustable by the correspondent coefficient, a procedure which is, in principle, legitimate since the underlying theory is unknown. We choose here to restrict ourselves to considering only symmetry-abiding terms. Let us comment on each one of the symmetries.
S1 is built upon the following considerations. Often the same coefficients multiply terms of different perturbative orders; consequently the amplitude of the 3-point function will be related to the amplitude of higher order correlators, notably to , the amplitude for the four-point function. Whenever the leading part of the trispectrum is generated by these types of ’s one can estimate that for its effect to be observable has to be five orders of magnitude larger than [31], which leaves little room for feasible models. On the other hand, one quickly realizes those ’s whose first term starts only at the fourth perturbative order (in Eq. (8)) are not plagued by this problem. This then represents a natural way to obtain inflationary models which allow a large, detectable trispectrum untied to the interactions which make up the bispectrum (which might well be small now).333One needs also to check that the interactions driven by coefficients that multiply also third order fluctuations do not become important in the form radiative corrections to the bispectrum. This check is done in [31] and ensures that loop corrections of those terms are not relevant. Indeed, in [31] the authors investigate on the size of all the interactions driven by the 444In the same spirit of the analysis done in [28] for all curvature-generated terms at third order, the authors of [31] consider in the v2 of their paper some extrinsic-curvature terms generated at fourth order. They also comment on their importance in near de Sitter limit and their conclusions apply to our parameters. coefficients in Eq. (8) and show that the leading interactions driven by these parameters are all consistent with the prescription and are expected to give a comparable signal 555It would be interesting to understand to what kind of models, in terms of the fundamental scalar field, the simple resulting effective Lagrangian corresponds in this case. . By construction then, the terms in the interaction Hamiltonian that are going to contribute to the trispectrum and be consistent with the reasoning that inspired the S1 symmetry are only some of the ones that will make up the contact interaction diagram, namely those whose lowest order interaction is already at fourth order. This limits us to the contributions regulated by the following coefficients: .
S2 symmetry, on the other hand, does not prohibit third order interactions, indeed in [30] the interaction is considered and, by inspection of Eq. (6), one can see that also other terms are allowed, the one regulated by and, notably, the term. The -driven term is particularly interesting because its contribution to the bispectrum calculations of [28] generates an interesting flat shape. The scalar exchange diagram will then be built out of the third order S2-obeying terms in the action. In particular, inspired by previous findings, we are going to give a detailed account of the contribution.
If both S1 and S2 are to be enforced one must also exclude from the list of S1-abiding interactions the ones multiplied by . A more clear picture of the situation concerning the various symmetries is presented in Table 1 below.

Table 1

 Coefficients S1 S2 Coefficients S1 S2 M2 M3 M4 ¯M1 ¯M2 ¯M3 ¯M4 ¯M5 ¯M6 ¯M7 ¯M8 ¯M9 X X ✓ X X X X X X X X X ✓ ✓ ✓ X X X X ✓ ✓ X X X ¯M10 ¯M11 ¯M12 ¯M13 ¯M14 ¯M15 ¯N1 ¯N2 ¯N3 ¯N4 ¯N5 / ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ X ✓ ✓ X X X ✓ ✓ ✓ ✓ ✓

The Coefficients marked with “ ✓” in correspondence of a given symmetry S are S-invariant, those marked with “X” violate the S symmetry.

Note that each coefficient might multiply many interactions at each perturbative orders and therefore we mark the coefficient as invariant under a symmetry when all the leading interactions it multiplies are invariant under S1 or S2. Determing the properties of the coefficients in the second row requires no effort, as one can easily verify these ’s first appear in the action as multipliers of fourth-order terms. Things are less linear with the coefficients in the first row (except for ) as they appear at fourth order both multiplying bare interaction terms and multiplying other coefficients as well as interaction terms (for an example of the latter case see the terms written explicitly in Eq. (LABEL:h4)). They also appear at third and some also at second order in perturbations. One then must carefully check that, given a particular coefficient , in none of the interactions it multiplies at any order the leading terms violate the symmetry. For in the first row one can verify after some checks that these terms all parametrize indeed approximately invariant interactions upon requiring the coefficient to be much smaller than the typical such as . This is because in the fourth-order Hamiltonian in Eq. (LABEL:h4) there are terms of the form

 ∝1M4¯M34×{M42,M43,¯M25,¯M26}×(S2−violating\,\, interaction) (11)

which one then assumes to be subleading. We stress this point because it emerges clearly and naturally in the effective theory approach.

## 4 Trispectrum

### 4.1 IN-IN Formalism

We are going to employ the IN-IN formalism to calculate the four point function of curvature perturbation. The most general and compact expression for such a quantity is:

 ⟨Ω|ζk1ζk2ζk3ζk4(t)|Ω⟩=⟨0|¯T{ei∫t0−∞d3xdt′H(x)}ζk1ζk2ζk3ζk4(t)T{e−i∫t0−∞d3x′dt′′H(x)}|0⟩ , (12)

where and indicate respectively anti-time order and time order operations, and stand for the vacuum of the free and interacting theory.
Expanding both the exponentials in Eq. (12), we single out the first non vanishing terms that will contribute to the scalar exchange and contact interaction diagrams.

 ⟨Ω|ζk1ζk2ζk3ζk4(t)|Ω⟩= ⟨0|¯T{i∫t0−∞d3xdt′H3(x)}ζk1ζk2ζk3ζk4(t)T{−i∫t0−∞d3x′dt′′H3(x′)}|0⟩ +⟨0|¯T{i22∫∫d3xdt′d3x′dt′′H3(x)H3(x′)}ζk1ζk2ζk3ζk4(t)|0⟩ +⟨0|ζk1ζk2ζk3ζk4(t)T{(−i)22∫t0−∞∫t0−∞d3xdt′d3x′dt′′H3(x)H3(x′)}|0⟩ +⟨0|¯T{i∫t0−∞d3xdtH4(x)}ζk1ζk2ζk3ζk4(t)|0⟩ +⟨0|ζk1ζk2ζk3ζk4(t)T{−i∫t0−∞d3x′dt′′H4(x′)}|0⟩+... (13)

where are the third and fourth-order Hamiltonian in the interaction picture. The latter two terms make up the contact interaction diagram, the rest is responsible for the scalar exchange. Let us also remind the reader that the gauge invariant observable is, at first approximation, linearly related to the scalar via . Also, already at this stage one can see that the result of the four point function is going to depend on six variables. All wavefunctions, once in Fourier space, depend only on the magnitude of their momenta. There are at most ten fields involved in the contractions, eight of which will always depend on the magnitude of the four external momenta (). We are left with one last contraction between two fields depending on the magnitude of one vector which, by construction, is going to be the sum of two external momenta. It turns out that, employing the overall momentum conservation, two variables are sufficient to describe any of these linear combinations, we choose , giving a total of six variables. As clear from above, the -driven third-order interaction we are going to consider further depends on scalar products between the various momenta but, as one can easily verify, these can all be fully specified by using the six variables introduced above. All the variables we will employ are represented in the figure below.

In order to get a tetrahedron as the one in Fig B one must enforce the following inequalities:

 cos(α−β)≥cos(γ)≥cos(α+β) (14)

with

 cos(α)=k21+k214−k242k1k14;cos(β)=k22+k214−k232k2k14;cos(γ)=k21+k22−k2122k1k2. (15)

From one also obtains the usual triangles inequalities. Here we single out some of the inequalities which we are going to use in what follows:

 k212+k214≤4;√1+k24−2k4≤k14≤√1+k24+2k4. (16)

In order to have a visual intuition and understanding of the result, once the calculation of the several contributions to the trispectrum is performed one needs to set up a number of configurations in which four out of the six variables are held fixed. Having more than one configuration also increases one’s ability to distinguish the signatures of different interactions. Following [33], we adopt the set up described below:

• Equilateral configuration: all the external momenta have the same magnitude ; the two variables left are plotted as . Note that when plotting in this configuration we will use the first inequality in Eq. (16). Incidentally, this is the only configuration for which exact calculations for the trispectrum in ghost inflation have been presented (see [30]) so far. Note also that for the equilateral as well as for the other configurations, one conveniently plots the result of the calculations in Eq. (13) for any specific interaction term multiplied by a factor of . It is done also because this factor is generally common to all the contributions and so removing it sharpens the differences between the plots of each interaction term.

• Folded configuration: here one has as well as and . The second and third inequalities in Eq. (16) must be enforced in this case. The variables and are the ones plotted in this configuration.

• Specialized planar limit configuration: in this case we have as well as:

 k12=[k21+k2k42k21(k2k4+√(4k21−k22)(4k21−k24))]1/2. (17)

The variables plotted are going to be and .

• Near double squeezed limit configuration: the tetrahedron is now a planar quadrangle and . The region of interest is in particular the one for which where the following relation holds:

 k2=√k21(−k212+k23+k24)−k2s1k2s2+k212k214+k212k24+k214k24−k214k23−k44+k23k24√2k4

with

 k2s1=2√(k1k4+k1⋅k4)(k1k4−k1⋅k4)k2s2=√(k3k4+