Non-Gaussianity and Statistical Anisotropy from Vector Field Populated Inflationary Models

Abstract

We present a review of vector field models of inflation and, in particular, of the statistical anisotropy and non-Gaussianity predictions of models with SU(2) vector multiplets. Non-Abelian gauge groups introduce a richer amount of predictions compared to the Abelian ones, mostly because of the presence of vector fields self-interactions. Primordial vector fields can violate isotropy leaving their imprint in the comoving curvature fluctuations 𝜁 at late times. We provide the analytic expressions of the correlation functions of 𝜁 up to fourth order and an analysis of their amplitudes and shapes. The statistical anisotropy signatures expected in these models are important and, potentially, the anisotropic contributions to the bispectrum and the trispectrum can overcome the isotropic parts.

1. Introduction

In the standard cosmological model, at very early times the Universe undergoes a quasi de Sitter exponential expansion driven by a scalar field, the inflaton, with an almost flat potential. The quantum fluctuations of this field are thought to be at the origin of both the Large Scale Structures and the Cosmic Microwave Background (CMB) fluctuations that we are able to observe at the present epoch [1]. CMB measurements indicate that the primordial density fluctuations are of order 10⁻⁵ have an almost scale-invariant power spectrum and are fairly consistent with Gaussianity and statistical isotropy [2–6]. All of these features find a convincing explanation within the inflationary paradigm. Nevertheless, deviations from the basic single-(scalar)field slow-roll model of inflation are allowed by the experimental data. On one hand, it is then important to search for observational signatures that can help discriminate among all the possible scenarios; on the other hand, it is important to understand what the theoretical predictions are in this respect for the different models.

Non-Gaussianity and statistical anisotropy are two powerful signatures. A random field is defined “Gaussian” if it is entirely described by its two-point function, higher order connected correlators being equal to zero. Primordial non-Gaussianity [7, 8] is theoretically predicted by inflation: it arises from the interactions of the inflaton with gravity and from self-interactions. However, it is observably too small in the single-field slow-roll scenario [9–11]. Alternatives to the latter have been proposed that predict higher levels of non-Gaussianity such as multifield scenarios [12–18], curvaton models [19–25], and models with noncanonical Lagrangians [26–30]. Many efforts have been directed to the study of higher order (three and four-point) cosmological correlators in these models [11, 22, 29, 31–47] and towards improving the prediction for the two-point function, through quantum loop calculations [10, 48–55]. From WMAP, the bounds on the bispectrum amplitude are given by −4<𝑓^loc_𝑁𝐿<80 [56] and by −125<𝑓^equil_𝑁𝐿<435 [57] at 95% CL, respectively in the local and in the equilateral configurations. For the trispectrum, WMAP provides −5.6×10⁵<𝑔_𝑁𝐿<6.4×10⁵ [58], 𝑔_𝑁𝐿 being the “local” trispectrum amplitude from cubic contributions (see also [59]), whereas from Large-Scale-Structures data −3.5×10⁵<𝑔_𝑁𝐿<8.2×10⁵ [60], at 95% CL. Planck [61] and future experiments are expected to set further bounds on primordial non-Gaussianity.

Statistical isotropy has always been considered one of the key features of the CMB fluctuations. The appearance of some “anomalies” [62–65] in the observations though, after numerous and careful data analysis, suggests a possible a breaking of this symmetry that might have occurred at some point of the Universe history, possibly at very early times. This encouraged a series of attempts to model this event, preferably by incorporating it in theories of inflation. Let us shortly describe the above mentioned “anomalies”. First of all, the large-scale CMB quadrupole appears to be “too low” and the octupole “too planar”; in addition to that, there seems to exist a preferred direction along which quadrupole and octupole are aligned [3, 62, 66–68]. Also, a “cold spot”, that is, a region of suppressed power, has been observed in the southern Galactic sky [63, 69]. Finally, an indication of asymmetry in the large-scale power spectrum and in higher-order correlation functions between the northern and the southern ecliptic hemispheres was found [64, 65, 70–72]. Possible explanations for these anomalies have been suggested such as improper foreground subtraction, WMAP systematics, statistical flukes; the possibilities of topological or cosmological origins for them have been proposed as well. Moreover, considering a power spectrum anisotropy due to the existence of a preferred spatial direction 𝑛 and parametrized by a function 𝑔(𝑘) as 𝑃(𝑘)=𝑃(𝑘)(1+𝑔(𝑘)(𝑘⋅𝑛)²), (1) the five-year WMAP temperature data have been analyzed in order to find out what the magnitude and orientation of such an anisotropy could be. The magnitude has been found to be 𝑔=0.29±0.031 and the orientation aligned nearly along the ecliptic poles [73]. Similar results have been found in [74], where it is pointed out that the origin of such a signal is compatible with beam asymmetries (uncorrected in the maps) which should therefore be investigated before we can find out what the actual limits on the primordial 𝑔 are.

Several fairly recent works have taken the direction of analysing the consequences, in terms of dynamics of the Universe and of cosmological fluctuations, of an anisotropic preinflationary or inflationary era. A cosmic no-hair conjecture exists according to which the presence of a cosmological constant at early times is expected to dilute any form of initial anisotropy [75]. This conjecture has been proven to be true for many (all Bianchi type cosmologies except for the Bianchi type-IX, for which some restrictions are needed to ensure the applicability of the theorem), but not all kinds of metrics and counterexamples exist in the literature [76–78]. Moreover, even in the event isotropization should occur, there is a chance that signatures from anisotropic inflation or from an anisotropic preinflationary era might still be visible today [79–82]. In the same context of searching for models of the early Universe that might produce some anisotropy signatures at late time, new theories have been proposed such as spinor models [83–86], higher p-forms [87–92] and primordial vector field models (see Section 2 for a quick review).

We are going to focus on statistical anisotropy and non-Gaussianity predictions of primordial vector field models. As mentioned above, there are great expectations that Planck and new experiments will, among other things, shed more light on the level of non-Gaussianity of the CMB fluctuations and on the nature of the unexpected anisotropy features we mentioned (see, e.g., [93]). Models that combine both types of predictions could be more easily testable and, from non-Gaussianity measurement, more stringent statistical anisotropy predictions could be produced or viceversa.

Within vector field models, higher order correlators had been computed in [94–98] and, more recently, in [99, 100] for 𝑈(1) vector fields. We considered 𝑆𝑈(2) vector field models in [101, 102]. Non-Abelian theories offer a richer amount of predictions compared to the Abelian case. Indeed, self-interactions provide extra contributions to the bispectrum and trispectrum of curvature fluctuations that are naturally absent in the Abelian case. We verified that these extra contributions can be equally important in a large subset of the parameter space of the theory and, in some cases, can even become the dominant ones.

This paper is structured as follows: in Section 2 we review some vector field models of inflation; in Section 3 we present the 𝑆𝑈(2) model; in Section 4 we provide the results for the two, three and, four-point functions of the curvature fluctuations; in Section 5 we present the non-Gaussianity amplitudes for the bispectrum and for the trispectrum; in Section 6 we show and discuss their shapes; finally in Section 7 we draw our conclusions.

2. Inflation and Primordial Vector Fields

The attempt to explain some of the CMB “anomalous” features as the indication of a break of statistical isotropy is the main reason behind ours and many of the existing inflationary models populated by vector fields, but not the only one. The first one of these models [103] was formulated with the goal of producing inflation by the action of vector fields, without having to invoke the existence of a scalar field. The same motivations inspired the works that followed [104–106]. Lately, models where primordial vector fields can leave an imprint on the CMB have been formulated as an alternative to the basic inflationary scenario, in the search for interesting non-Gaussianity predictions [94–102]. Finally, vector fields models of dark energy have been proposed [107–111]. All this appears to us as a rich bag of motivations for investigating these scenarios.

Before we quickly sketch some of them and list the results so far achieved in this direction, it is important to briefly indicate and explain the main issues and difficulties that these models have been facing. We will also shortly discuss the mechanisms of production of the curvature fluctuations in these models.

Building a model where primordial vector fields can drive inflation and/or produce the observed spectrum of large scale fluctuations requires a more complex Lagrangian than the basic gauge invariant ℒ_vector=−(√−𝑔/4)𝐹^𝜇𝜈𝐹_𝜇𝜈. In fact, for a conformally invariant theory as the one described by ℒ_vector, vector fields fluctuations are not excited on superhorizon scales. It is then necessary to modify the Lagrangian. For some of the existing models, these modifications have been done to the expense of destabilizing the theory, by “switching on” unphysical degrees of freedom. This was pointed out in [112–114], where a large variety of vector field models was analyzed in which longitudinal polarization modes exist that are endowed with negative squared masses (the “wrong” signs of the masses are imposed for the theory to satisfy the constraints that allow a suitable background evolution). It turns out that, in a range of interest of the theory, these fields acquire negative total energy, that is, behave like “ghosts”, the presence of which is known to be responsible for an unstable vacuum. A related problem for some of these theories is represented by the existence of instabilities affecting the equations of motion of the ghost fields [112–114].

In the remaing part of this section, we are going to present some of these models together with some recent attempts to overcome their limits.

In all of the models we will consider, primordial vector fields fluctuations end up either being entirely responsible for or only partially contributing to the curvature fluctuations at late times. This can happen through different mechanisms. For example, if the vector fields affect the universe expansion during inflation, its contribution 𝜁_𝐴 to the total 𝜁 can be derived from combining the definition of the number of e-foldings (𝑁=∫𝐻𝑑𝑡) with the Einstein equation (𝐻²=(8𝜋𝐺/3)(𝜌_𝜙+𝜌_𝐴), 𝜌_𝐴 being the energy density of the vector field and 𝜌_𝜙 the inflaton energy density) and using the 𝛿N expansion of the curvature fluctuation in terms of both the inflaton and the vector fields fluctuations (see Section 4). To lowest order we have [96] 𝜁_𝐴=𝐴_𝑖2𝑚²_𝑃𝛿𝐴_𝑖, (2) where a single vector field has been taken into account for simplicity (𝑚_𝑃 is the reduced Planck mass, 𝐴 is the background value of the field and 𝛿𝐴 its perturbation). When calculating the amplitude of non-Gaussianity in Section 5, we will refer to this case as “vector inflation” for simplicity.

A different fluctuation production process is the curvaton mechanism which was initially formulated for scalar theories but it is also applicable to vectors [96, 115, 116]. Specifically, inflation is driven by a scalar field, whereas the curvaton field(s) (now played by the vectors), has a very small (compared to the Hubble rate) mass during inflation. Towards the end of the inflationary epoch, the Hubble rate value starts decreasing until it equates the vector mass; when this eventually happens, the curvaton begins to oscillate and it will then dissipate its energy into radiation. The curvaton becomes responsible for a fraction of the total curvature fluctuation that is proportional to a parameter (𝑟) related to the ratio between the curvaton energy density and the total energy density of the universe at the epoch of the curvaton decay [96] 𝜁_𝐴=𝑟3𝛿𝜌_𝐴𝜌_𝐴, (3) where 𝑟≡3𝜌_𝐴/(3𝜌_𝐴+4𝜌_𝜙). Anisotropy bounds on the power spectrum favour small values of 𝑟.

From (2) and (3) we can see that, dependending on which one of these two mechanisms of production of the curvature fluctuations is considered, different coefficients will result in the 𝛿N expansion (see (22)).

In this section, we will describe both models where inflation is intended to be vector field driven and those models in which, instead, the role of the inflaton is played by a scalar field, whereas the energy of the vector gives a subdominant contribution to the total energy density of the universe during the entire inflationary phase.

2.1. Self-Coupled Vector Field Models

A pioneer work on vector field driven inflation was formulated by Ford [103], who considered a single self-coupled field 𝐴_𝜇 with a Lagrangian 𝐿_vector1=−4𝐹_𝜇𝜈𝐹^𝜇𝜈+𝑉(𝜓), (4) where 𝐹_𝜇𝜈≡𝜕_𝜇𝐵_𝜈−𝜕_𝜈𝐵_𝜇 and the potential 𝑉 is a function of 𝜓≡𝐵_𝛼𝐵^𝛼. Different scenarios of expansion are analyzed by the author for different functions 𝑉. The universe expands anisotropically at the end of the inflationary era and this anisotropy either survives until late times or is damped out depending on the shape and the location of the minima of the potential.

The study of perturbations in a similar model was proposed by Dimopoulos in [115] where he showed that for a Lagrangian 𝐿_vector1=−4𝐹_𝜇𝜈𝐹^𝜇𝜈+12𝑚²𝐵_𝜇𝐵^𝜇, (5) and for 𝑚²≃−2𝐻², the transverse mode of the vector field is governed by the same equation of motion as a light scalar field in a de Sitter stage. A suitable superhorizon power spectrum of fluctuations could therefore arise. In order to prevent production of large scale anisotropy, in this model the vector field plays the role of the curvaton while inflation is driven by a scalar field.

2.2. Vector-Field Coupled to Gravity

The Lagrangian in (5) may be also intended, at least during inflation, as including a nonminimal coupling of the vector field to gravity; indeed the mass term can be rewritten as 𝐿_vector⊃12(𝑚²₀+𝜉𝑅)𝐵_𝜇𝐵^𝜇, (6) where, for the whole duration of the inflationary era, the bare mass 𝑚₀ is assumed to be much smaller than the Hubble rate and the Ricci scalar 𝑅=−6[𝑎/̈𝑎+(𝑎/̇𝑎)²] can be approximated as 𝑅≃−12𝐻². For the specific value 𝜉=1/6, (5) is retrieved.

For the Lagrangian just presented, Golovnev et al. [104] proved that the problem of excessive anisotropy production in the case where inflation is driven by vector fields can be avoided if either a triplet of mutually orthogonal or a large number 𝑁 of randomly oriented vector fields is considered.

The Lagrangian (6) with 𝜉=1/6 was also employed in [116], where inflation is scalar field-driven and a primordial vector field affects large-scale curvature fluctuations and, similarly, in [117], which includes a study of the backreaction of the vector field on the dynamics of expansion, by introducing a Bianchi type-I metric.

2.3. Ackerman-Carroll-Wise (ACW) Model

A model was proposed in [118, 119] where Lagrange multipliers (𝜆) are employed to determine a fixed norm primordial vector field 𝐵_𝜇𝐵^𝜇=𝑚²𝐿_vector⊃𝜆(𝐵^𝜇𝐵_𝜇−𝑚²)−𝜌_Λ, (7) where 𝜌_Λ is a vacuum energy. The expansion rate in this scenario is anisotropic: if we orient the 𝑥-axis of the spatial frame along the direction determined by the vector field, we find two different Hubble rates: along the 𝑥-direction it is equal to 𝐻²_𝑏=𝜌_Λ𝑚²_𝑃1,𝑃(𝜇) (8) and it is given by 𝐻_𝑎=(1+𝑐𝜇²)𝐻_𝑏 along the orthogonal directions; 𝜇≡𝑚/𝑚_𝑃, 𝑃 is a polynomial function of 𝜇 and 𝑐 is a parameter appearing in the kinetic part of the Lagrangian that we omitted in (7) (see [118, 119] for its complete expression). As expected, an isotropic expansion is recovered if the vev of the vector field is set to zero.

2.4. Models with Varying Gauge Coupling

Most of the models mentioned so far successfully solve the problem of attaining a slow-roll regime for the vector-fields without imposing too many restrictions on the parameters of the theory and of avoiding excessive production of anisotropy at late times. None of them though escapes those instabilities related to the negative energy of the longitudinal modes (although a study of the instabilities for fixed-norm field models was done in [120] where some stable cases with non-canonical kinetic terms were found). As discussed in [112–114], in the self-coupled model a ghost appears at small (compared to the horizon) wavelengths; in the non-minimally coupled and in the fixed-norm cases instead the instability concerns the region around horizon crossing.

Models with varying gauge coupling can overcome the problem of instabilities and have recently attracted quite some attention. In [94], the authors consider a model of hybrid inflation [121–124] with the introduction of a massless vector field 1𝐿⊃2(𝜕_𝜇𝜙𝜕^𝜇𝜙+𝜕_𝜇𝜒𝜕^𝜇1𝜒)−4𝑓²(𝜙)𝐹_𝜇𝜈𝐹_𝜇𝜈+𝑉(𝜙,𝜒,𝐵_𝜇), (9) where 𝜙 is the inflaton and 𝜒 is the so-called “waterfall” field. The potential 𝑉 is chosen in such a way as to preserve gauge invariance; this way the longitudinal mode disappears and instabilities are avoided.

Similarly, Kanno et al. in [125] (see also [96, 126]) consider a vector field Lagrangian of the type 𝐿_vector1=−4𝑓²(𝜙)𝐹^𝜇𝜈𝐹_𝜇𝜈, (10) but in a basic scalar field-driven inflation model. Very recently, in [127] the linear perturbations in these kind of models have been investigated.

Finally, in [96, 100] varying mass vector field models have been introduced 𝐿_vector1=−4𝑓²(𝜙)𝐹^𝜇𝜈𝐹_𝜇𝜈+12𝑚²𝐵_𝜇𝐵^𝜇, (11) where 𝑓⋍𝑎^𝛼 and 𝑚⋍𝑎 (𝑎 is the scale factor and 𝛼 is a numerical coefficient). The special cases 𝛼=1 and 𝛼=−2 are of special interest. In fact, introducing the fields 𝐴_𝜇 and 𝐴_𝜇, related to one another by 𝐴_𝜇≡𝑓𝐵_𝜇=𝑎𝐴_𝜇 (𝐴_𝜇 and 𝐴_𝜇 are, resp., the comoving and the physical vectors), it is possible to verify that the physical gauge fields are governed by the same equations of motion as a light scalar field in a de Sitter background. Vector fields in this theory can then generate the observed (almost) scale-invariant primordial power spectrum.

3. 𝑆𝑈(2) Vector Model: Equations of Motion for the Background and for Linear Perturbations

Let us consider some models where inflation is driven by a scalar field in the presence of an 𝑆𝑈(2) vector multiplet [101, 102]. A fairly general Lagrangian can be the following: 𝑆=𝑑⁴𝑥√𝑚−𝑔[^2𝑃𝑅2−𝑓²(𝜙)4𝑔^𝜇𝛼𝑔^𝜈𝛽_𝑎=1,2,3𝐹^𝑎_𝜇𝜈𝐹^𝑎_𝛼𝛽−𝑀²2𝑔^𝜇𝜈_𝑎=1,2,3𝐵^𝑎_𝜇𝐵^𝑎_𝜈+𝐿_𝜙], (12) where 𝐿_𝜙 is the Lagrangian of the scalar field and 𝐹^𝑎_𝜇𝜈≡𝜕_𝜇𝐵^𝑎_𝜈−𝜕_𝜈𝐵^𝑎_𝜇+𝑔_𝑐𝜀^𝑎𝑏𝑐𝐵^𝑏_𝜇𝐵^𝑐_𝜈 (𝑔_𝑐 is the 𝑆𝑈(2) gauge coupling). Both 𝑓 and the effective mass 𝑀 can be viewed as generic functions of time. The fields 𝐵^𝑎_𝜇 are comoving and related to the physical fields by 𝐴^𝑎_𝜇=(𝐵^𝑎₀,𝐵^𝑎_𝑖/𝑎). The free field operators can be Fourier expanded in their creation and annihilation operators 𝛿𝐴^𝑎_𝑖(=𝑑𝑥,𝜂)₃𝑞(2𝜋)³𝑒^{𝑖𝑥𝑞⋅}×_{𝜆=𝐿,𝑅,long}[𝑒^𝜆_𝑖(𝑞)𝑎_𝑞^𝑎,𝜆𝛿𝐴^𝑎_𝜆(𝑞,𝜂)+𝑒_𝑖^∗𝜆(−𝑞)(𝑎_−𝑞^𝑎,𝜆)^†𝛿𝐴_𝜆^∗𝑎(𝑞,𝜂)], (13) where the polarization index 𝜆 runs over left (𝐿), right (𝑅) and longitudinal (long) modes and [𝑎_𝑘^𝑎,𝜆,(𝑎^{𝑎′,𝜆′}_𝑘′)^†]=(2𝜋)³𝛿_𝑎,𝑎′𝛿_𝜆,𝜆′𝛿⁽³⁾(𝑘−𝑘^′). (14) Here, 𝜂 the conformal time (𝑑𝜂=𝑑𝑡/𝑎(𝑡)). Once the functional forms of 𝑓 and 𝑀 have been specified, the equations of motion for the vector bosons can be written. For the most part, the calculations are quite general in this respect. In fact, the expression of all correlation functions, prior to explicitating the wavefunction for the gauge bosons, apply to any 𝑆𝑈(2) theory with an action as in (12), both for what we will call the “Abelian” and for the “non-Abelian” contributions. In particular, the structure of the interaction Hamiltonian is independent of the functional dependence of 𝑓 and 𝑀 and determines the general form of and the anisotropy coefficients appearing in the final “non-Abelian” expressions (see Section 4). When it comes to explicitate the wavefunctions, a choice that can help keeping the result as easy to generalize as possible is the following: 𝛿𝐵^𝑇√=−𝜋𝑥2√𝑘[𝐽_3/2(𝑥)+𝑖𝐽_−3/2(𝑥)], (15) for the transverse mode and 𝛿𝐵^‖=𝑛(𝑥)𝛿𝐵^𝑇, (16) for the longitudinal mode (𝑛 is a unknown function of 𝑥≡−𝑘𝜂) [101, 102]. Let us see why. As previously stated, for 𝑓≃𝑎^𝛼 and with 𝛼=0,1,−2, it is possible to verify that the (physical) transverse mode behaves exactly like a light scalar field in a de Sitter background. Considering the solution (15) then takes into account at least these special cases. As to the longitudinal mode, a parametrization was adopted as in (16) in order to keep the analysis more general and given that, because of the instability issues, introducing this degree of freedom into the theory requires special attention. We are going to keep the longitudinal mode “alive” in the calculations we present, by considering a nonzero function 𝑛(𝑥), and focus on the simplest case of 𝑓=1. This case is known to be affected by quantum instabilities in the longitudinal mode, anyway we choose 𝑓=1 for the sake of simplicity in our presentation. The results can be easily generalized to gauge invariant models (please refer to [102] for a sample generalization of the calculations to massless 𝑓≃𝑎^(1,−2) models).

4. Correlation Functions of Curvature Fluctuations: Analytic Expressions

We are now ready to review the computation of the power spectrum, bispectrum, and trispectrum for the curvature fluctuations 𝜁 generated during inflation ⟨𝜁_𝑘1𝜁_𝑘2⟩=(2𝜋)³𝛿⁽³⁾(𝑘₁+𝑘₂)𝑃_𝜁(𝑘),⟨𝜁_𝑘1𝜁_𝑘2𝜁_𝑘3⟩=(2𝜋)³𝛿⁽³⁾(𝑘₁+𝑘₂+𝑘₃),×𝐵_𝜁(𝑘₁,𝑘₂,𝑘₃),⟨𝜁_𝑘1𝜁_𝑘2𝜁_𝑘3𝜁_𝑘4⟩=(2𝜋)³𝛿⁽³⁾(𝑘₁+𝑘₂+𝑘₃+𝑘₄),×𝑇_𝜁(𝑘₁,𝑘₂,𝑘₃,𝑘₄). (17)(18)(19) Notice that, on the right-hand side of (17) through (19), we indicated a dependence from the direction of the wavevectors; in models of inflation where isotropy is preserved, the power spectrum and the bispectrum only depend on the moduli of the wave vectors. This will not be the case for the 𝑆𝑈(2) model.

The 𝛿N formula [31, 128–130] will be employed 𝜁(𝑥,𝑡)=𝑁(𝑥,𝑡^∗,𝑡)−𝑁(𝑡^∗,𝑡)≡𝛿𝑁(𝑥,𝑡), (20) which holds if times 𝑡^∗ and 𝑡 are chosen, respectively, on a flat and on a uniform density temporal slices (𝑁 is the number of e-foldings of inflation occurring between these two times). We employ a spatial metric 𝑔_𝑖𝑗=𝑎²(𝑡)𝑒^−2Ψ(𝑒^𝛾)_𝑖𝑗, and at linear level the curvature perturbation corresponds to 𝜁≡−Ψ+𝐻𝛿𝑢, where 𝛿𝑢 is the fluctuation in the total energy density and Ψ is a scalar metric fluctuation. In the presence of a single scalar field 𝜙, (20) is further expandable as 𝜁(𝑥,𝑡)=_𝑛𝑁^(𝑛)(𝑡^∗,𝑡)𝑛!(𝛿𝜙(𝑥,𝑡^∗))^𝑛, (21) where 𝑁^(𝑛) is the partial derivative of the e-folding number w.r.t. 𝜙 on the initial hypersurface 𝑡^∗.

If we apply (21) to the inflaton +𝑆𝑈(2) vector model, we have 𝜁(𝑥,𝑡)=𝑁_𝜙𝛿𝜙+𝑁^𝜇_𝑎𝛿𝐴^𝑎_𝜇+12𝑁_𝜙𝜙(𝛿𝜙)²+12𝑁^𝜇𝜈_𝑎𝑏𝛿𝐴^𝑎_𝜇𝛿𝐴^𝑏_𝜈+𝑁^𝜇_𝜙𝑎𝛿𝜙𝛿𝐴^𝑎_𝜇+1𝑁3!^𝜙𝜙𝜙(𝛿𝜙)³+1𝑁3!^{𝜇𝜈𝜆𝑎𝑏𝑐}𝛿𝐴^𝑎_𝜇𝛿𝐴^𝑏_𝜈𝛿𝐴^𝑐_𝜆+12𝑁^𝜇_𝜙𝜙𝑎(𝛿𝜙)²𝛿𝐴^𝑎_𝜇+12𝑁^𝜇𝜈_𝜙𝑎𝑏𝛿𝜙𝛿𝐴^𝑎_𝜇𝛿𝐴^𝑏_𝜈+1𝑁3!^{𝜙𝜙𝜙𝜙}(𝛿𝜙)⁴+1𝑁3!^{𝜇𝜈𝜆𝜂𝑎𝑏𝑐𝑑}𝛿𝐴^𝑎_𝜇𝛿𝐴^𝑏_𝜈𝛿𝐴^𝑐_𝜆𝛿𝐴^𝑑_𝜂+⋯, (22) where now 𝑁_𝜙≡(𝜕𝑁)𝜕𝜙_𝑡∗,𝑁^𝜇_𝑎≡(𝜕𝑁𝜕𝐴^𝑎_𝜇)_𝑡∗,𝑁^𝜇_𝜙𝑎𝜕≡(²𝑁𝜕𝜙𝜕𝐴^𝑎_𝜇)_𝑡∗ (23) and so on for higher order derivatives.

On a general ground, given the fact that the temporal modes can always be expressed in terms of the longitudinal modes, (22) can be rewritten by retaining only the spatial part for the vector field fluctuations (see also [96]).

Our plan is to show the derivation the correlation functions of 𝜁 from the ones of 𝛿𝜙 and 𝛿𝐴^𝑎_𝑖, after a replacement of the 𝛿N expansion (22) in (17) through (19).

The correlation functions can be evaluated using the Schwinger-Keldysh formula [48–52] ⟨Ω|Θ(𝑡)|Ω⟩=⟨0|[𝑇(𝑒^{𝑖∫𝑡0𝐻𝐼(𝑡′)𝑑𝑡′})]Θ_𝐼(𝑡)[𝑇(𝑒^{−𝑖∫𝑡0𝐻𝐼(𝑡′)𝑑𝑡′})]|0⟩, (24) where, on the left-hand side, the operator Θ and the vacuum Ω are in the interacting theory whereas, on the right-hand side, all operators are in the so-called “interaction picture”, that is, they can be treated as free fields (the Fouries expansion in (13) thus apply), and |0⟩ is the free theory vacuum.

When calculating the spectra of 𝜁, the perturbative expansions in (22) and (24) will be carried out to only include tree-level contributions, neglecting higher order “loop” terms, either classical, that is, from the 𝛿N series, or of quantum origin, that is, from the Schwinger-Keldysh series. Assuming that the 𝑆𝑈(2) coupling 𝑔_𝑐 is “small” and that we are dealing with “small” fluctuations in the fields and given the fact that a slow-roll regime is being assumed, it turns out that it is indeed safe for the two expansions to be truncated at tree-level.

The correlation functions of 𝜁 will then result as the sum of scalar, vector and (scalar and vector) mixed contributions. As to the vector part, this will be made up of terms that are merely generated by the 𝛿N expansion, that is, they only include the zeroth order of the in-in formula (we call these terms “Abelian”, being them retrievable in the 𝑈(1) case), and by (“non-Abelian”) terms arising from the Schwinger-Keldysh operator expansion beyond zeroth order, that is, from the gauge fields self-interactions.

Let us now discuss the level of generality of the results we will present in the next sections, w.r.t. the choice of a specific Lagrangian.

The expression for the Abelian contributions provided in Sections 4.1 and 4.2.1 apply to any 𝑆𝑈(2) model of gauge interactions with no direct coupling between scalar and vector fields (extra terms would be otherwise needed in (33) and (34)). The next stage in the Abelian contributions computation would be to explicitate the derivatives of the e-foldings number and the wavefunctions of the fields: they both depend on the equations of motion of the system, therefore the fixing of a specific model is required at this point. As to the non-Abelian contributions, the results in (42) and (43) are completely general except for assuming, again, that no direct vector-scalar field coupling exists. The structure of (52) and (55) is instead due to the choice of a non-Abelian gauge group. The expressions of the anisotropy coefficients 𝐼_𝑛 and 𝐿_𝑛 in (52) and (55) depend on the specific non-Abelian gauge group (for 𝑆𝑈(2) one of the 𝐼_𝑛 is given in (54)). Finally, the specific expressions of the isotropic functions 𝐹_𝑛 (a sample of which is shown in (53)) and 𝐺_𝑛 were derived considering the Lagrangian (12) with 𝑓=1 and the eigenfunctions for the vector bosons provided in (15) and (16).

4.1. The Power Spectrum

The power spectrum of 𝜁 can be straightforwardly derived at tree-level, using the 𝛿N expansion (22), from the inflaton and the vector fields power spectra 𝑃_𝜁(𝑘)=𝑃^iso(𝑘)[1+𝑔^𝑎𝑏(𝑘⋅𝑁_𝑎)(𝑘⋅𝑁_𝑏)+𝑖𝑠^𝑎𝑏𝑘⋅(𝑁_𝑎×𝑁_𝑏)], (25) which generalizes the results of [96] for the 𝑈(1) case.

The isotropic part of the previous expression has been factorized in 𝑃^iso(𝑘)≡𝑁²_𝜙𝑃_𝜙(𝑘)+(𝑁_𝑐⋅𝑁_𝑑)𝑃₊^𝑐𝑑, (26) where we have defined the following combinations 𝑃_±^𝑎𝑏1≡(2)(𝑃_𝑅^𝑎𝑏±𝑃_𝐿^𝑎𝑏), (27) from the power spectra for the right, left and longitudinal polarization modes 𝑃_𝑅^𝑎𝑏≡𝛿_𝑎𝑏𝛿𝐴^𝑎_𝑅(𝑘,𝑡^∗)𝛿𝐴_𝑅^𝑏∗(𝑘,𝑡^∗𝑃),_𝐿𝑎𝑏≡𝛿_𝑎𝑏𝛿𝐴^𝑎_𝐿(𝑘,𝑡^∗)𝛿𝐴_𝐿^𝑏∗(𝑘,𝑡^∗𝑃),_𝑎𝑏long≡𝛿_𝑎𝑏𝛿𝐴^𝑎_long(𝑘,𝑡^∗)𝛿𝐴^𝑏∗_long(𝑘,𝑡^∗). (28) The anisotropic parts are weighted by the coefficients 𝑔^𝑎𝑏≡𝑁^𝑎𝑁^𝑏(𝑃^𝑎𝑏_long−𝑃₊^𝑎𝑏)𝑁²_𝜙𝑃_𝜙+(𝑁_𝑐⋅𝑁_𝑑)𝑃₊^𝑐𝑑,𝑠^𝑎𝑏≡𝑁^𝑎𝑁^𝑏𝑃₋^𝑎𝑏𝑁²_𝜙𝑃_𝜙+(𝑁_𝑐⋅𝑁_𝑑)𝑃₊^𝑐𝑑, (29) (where a sum is intended over indices 𝑐 and 𝑑 but not over 𝑎 and 𝑏). Equation (26) can also be written as 𝑃^iso(𝑘)=𝑁²_𝜙𝑃_𝜙[1+𝛽_𝑐𝑑𝑃₊^𝑐𝑑𝑃_𝜙], (30) after introducing the parameter 𝛽_𝑐𝑑≡𝑁_𝑐⋅𝑁_𝑑𝑁²_𝜙. (31) Notice that what when we say “isotropic”, as far as the expression for the power spectrum is concerned, we simply mean “independent” of the direction of the wave vector. In this case instead, the vector bosons introduce three preferred spatial directions: the r.h.s. of (25) depends on their orientation w.r.t. the wave vector.

As expected, the coefficients 𝑔^𝑎𝑏 and 𝑠^𝑎𝑏 that weight the anisotropic part of the power spectrum are related to 𝛽_𝑐𝑑, that is, to the parameters that quantify how much the expansion of the universe is affected by the vector bosons compared to the scalar field.

Assuming no parity violation in the model, we have 𝑠^𝑎𝑏=0; the parameters 𝑔^𝑎𝑏 and 𝛽_𝑎𝑏 are instead unconstrained. In the 𝑈(1) case and for parity conserving theories, (25) reduces to [96] 𝑃_𝜁(𝑘)=𝑃^iso_𝜁(𝑘)[1+𝑔(𝑘⋅𝑛)], (32) where 𝑛 indicates the preferred spatial direction; also one can check that in this simple case, if 𝑃₊⋍𝑃_𝜙 and 𝑃_long=𝑘𝑃₊ (𝑘≢1), the relation 𝑔=(𝑘−1)𝛽/(1+𝛽) holds, where 𝛽≡(𝑁_𝐴/𝑁_𝜙)² (the anisotropy coefficient 𝑔 is not to be confused with the 𝑆𝑈(2) coupling constant 𝑔_𝑐). If it is safe to assume |𝑔|≪1 (see discussion following (1) and references [73, 74]), a similar upper bound can also be placed on 𝛽.

In the case where more than one special directions exists, as in the 𝑆𝑈(2) model, no such analysis on the anisotropy data has been so far carried out, the 𝑔^𝑎 parameters cannot then be constrained, unless assuming that the three directions converge into a single one; in that case a constraint could be placed on the sum |𝑔|≡|∑_𝑎𝑔^𝑎|, where 𝑎=1,2,3 and 𝑃_𝜁(𝑘)=𝑃^iso_𝜁(𝑘)[1+𝑔^𝑎(𝑘⋅𝑛_𝑎)].

4.2. Higher-Order Correlators

We will present the results for the tree-level contributions to the bispectrum and to the trispectrum of 𝜁.

These can be classified in two categories, that we indicate as “Abelian” and “non-Abelian”. The former are intended as terms that merely arise from the 𝛿N expansion and are thus retrievable in the Abelian case; the latter are derived from the linear and quadratic expansions (in terms of the gauge bosons interaction Hamiltonian) of the Schwinger-Keldysh formula and are therefore peculiar to the non-Abelian case.

We are going to provide both types of contributions, in preparation for discussing and comparing their magnitudes later on in Section 5.

4.2.1. Abelian Contributions

By plugging the 𝛿N expansion (22) in (18) and (19), we have 𝐵_𝜁(𝑘₁,𝑘₂,𝑘₃)⊃12𝑁²_𝜙𝑁_𝜙𝜙[𝑃_𝜙(𝑘₁)𝑃_𝜙(𝑘₂+1)+perms.]2𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘𝑙_𝑐𝑑[Π^𝑎𝑐_𝑖𝑘(𝑘₁)Π^𝑏𝑑_𝑗𝑙(𝑘₂+1)+perms.]2𝑁_𝜙𝑁^𝑖_𝑎𝑁^𝑗_𝜙𝑏[𝑃_𝜙(𝑘₁)Π^𝑎𝑏_𝑖𝑗(𝑘₂)+perms.]+𝑁_𝜙𝑁²_𝜙𝐵_𝜙(𝑘₁,𝑘₂,𝑘₃), (33) for the bispectrum (see also [96]) and 𝑇_𝜁(𝑘₁,𝑘₂,𝑘₃,𝑘₄)⊃𝑁⁴_𝜙𝑇_𝜙(𝑘₁,𝑘₂,𝑘₃,𝑘₄)+𝑁³_𝜙𝑁_𝜙𝜙[𝑃_𝜙(𝑘₁)𝐵_𝜙(|𝑘₁+𝑘₂|,𝑘₃,𝑘₄)+perms.]+𝑁²_𝜙𝑁^𝑖_𝑎𝑁^𝑗_𝜙𝑏[𝑃^𝑎𝑏_𝑖𝑗(𝑘₃)𝐵_𝜙(𝑘₁,𝑘₂,|𝑘₃+𝑘₄|)+perms.]+𝑁²_𝜙𝑁²_𝜙𝜙[𝑃_𝜙(𝑘₁)𝑃_𝜙(𝑘₂)𝑃_𝜙(|𝑘₁+𝑘₃|)+perms.]+𝑁³_𝜙𝑁_𝜙𝜙𝜙[𝑃_𝜙(𝑘₁)𝑃_𝜙(𝑘₂)𝑃_𝜙(𝑘₃)+perms.]+𝑁²_𝜙𝑁^𝑖_𝜙𝑎𝑁^𝑗_𝜙𝑏[𝑃^𝑎𝑏_𝑖𝑗(𝑘₁+𝑘₃)𝑃_𝜙(𝑘₁)𝑃_𝜙(𝑘₂)+perms.]+𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘_𝜙𝑐𝑁^𝑙_𝜙𝑑×[𝑃^𝑎𝑐_𝑖𝑘(𝑘₁)𝑃^𝑏𝑑_𝑗𝑙(𝑘₂)𝑃_𝜙(|𝑘₁+𝑘₃|)+perms.]+𝑁²_𝜙𝑁^𝑖_𝑎𝑁^𝑗_𝜙𝜙𝑏[𝑃_𝜙(𝑘₁)𝑃_𝜙(𝑘₂)𝑃^𝑎𝑏_𝑖𝑗(𝑘₃)+perms.]+𝑁_𝜙𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘𝑙_𝜙𝑐𝑑[𝑃^𝑎𝑐_𝑖𝑘(𝑘₁)𝑃^𝑏𝑑_𝑗𝑙(𝑘₂)𝑃_𝜙(𝑘₃)+perms.]+𝑁_𝜙𝜙𝑁_𝜙𝑁^𝑖_𝜙𝑎𝑁^𝑗_𝑏×[𝑃_𝜙(𝑘₂)𝑃_𝜙(|𝑘₁+𝑘₂|)𝑃^𝑎𝑏_𝑖𝑗(𝑘₄)+perms.]+𝑁^𝑖𝑗_𝑎𝑏𝑁^𝑘_𝑐𝑁^𝑙_𝜙𝑑𝑁_𝜙[𝑃^𝑖𝑘_𝑎𝑐(𝑘₂)𝑃^𝑗𝑙_𝑏𝑑(𝑘₁+𝑘₂)𝑃_𝜙(𝑘₄)+perms.]+𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘𝑙_𝑐𝑑𝑁^𝑚𝑛_𝑒𝑓×[𝑃^𝑎𝑐_𝑖𝑘(𝑘₁)𝑃^𝑏𝑒_𝑗𝑚(𝑘₂)𝑃^𝑑𝑓_𝑙𝑛(𝑘₁+𝑘₃)+perms.]+𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘_𝑐𝑁^𝑙𝑚𝑛_𝑑𝑒𝑓[𝑃^𝑎𝑑_𝑖𝑙(𝑘₁)𝑃^𝑏𝑒_𝑗𝑚(𝑘₂)𝑃^𝑐𝑓_𝑘𝑛(𝑘₃)+perms.], (34) for the trispectrum (see also [97]).

Let us now provide some definition for the quantities introduced in the previous equations Π^𝑎𝑏_𝑖𝑗(𝑘)≡𝑇^even_𝑖𝑗(𝑘)𝑃₊^𝑎𝑏+𝑖𝑇^odd_𝑖𝑗(𝑘)𝑃₋^𝑎𝑏+𝑇^long_𝑖𝑗(𝑘)𝑃^𝑎𝑏_long, (35)

where 𝑇^even_𝑖𝑗(𝑘)≡𝑒^𝐿_𝑖(𝑘)𝑒_𝑗^∗𝐿(𝑘)+𝑒^𝑅_𝑖(𝑘)𝑒_𝑗^∗𝑅(𝑇𝑘),_odd𝑖𝑗(𝑘)≡𝑖[𝑒^𝐿_𝑖(𝑘)𝑒_𝑗^∗𝐿(𝑘)−𝑒^𝑅_𝑖(𝑘)e_𝑗^∗𝑅(𝑇𝑘)],_long𝑖𝑗(𝑘)≡𝑒^𝑙_𝑖(𝑘)𝑒_𝑗^∗𝑙(𝑘). (36) The polarization vectors are 𝑒^𝐿(𝑘)≡(1/√2)(cos𝜃cos𝜙−𝑖sin𝜙,cos𝜃sin𝜙+𝑖cos𝜙,−sin𝜃), 𝑒^𝑅(𝑘)=𝑒^∗𝐿(𝑘) and 𝑒^𝑙(𝑘)=𝑘=(sin𝜃cos𝜙,sin𝜃sin𝜙,cos𝜃), from which we have 𝑇^even_𝑖𝑗(𝑘)=𝛿_𝑖𝑗−𝑘_𝑖𝑘_𝑗,𝑇^odd_𝑖𝑗(𝑘)=𝜖_𝑖𝑗𝑘𝑘_𝑘,𝑇^long_𝑖𝑗(𝑘)=𝑘_𝑖𝑘_𝑗. (37) The purely scalar terms in (33)-(34) are already known from the literature: in single-field slow-roll inflation 𝑃_𝜙=𝐻²_∗/2𝑘³, where 𝐻_∗ is the Hubble rate evaluated at horizon exit; the bispectrum and the trispectrum of the scalar field (𝐵_𝜙 and 𝑇_𝜙) can be found in [9–11, 38, 131] (they were also reported in [102, (11) and (12)]). As to the mixed (scalar-vector) terms, they can be ignored if one considers a Lagrangian where there is no direct coupling between the inflaton and the gauge bosons and where slow-roll assumptions are introduced for the fields (see [102, Section 4] for a complete discussion on this). Let us then look at the (purely) vector part. Its anisotropy features can be stressed by rewriting them as follows (see also [96, 97]) 𝐵_𝜁(𝑘₁,𝑘₂,𝑘₃)⊃12𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘𝑙_𝑐𝑑Π^𝑎𝑐_𝑖𝑘(𝑘₁)Π^𝑏𝑑_𝑗𝑙(𝑘₂)=𝑀^𝑐_𝑘𝑁^𝑘𝑙_𝑐𝑑𝑀^𝑑_𝑙𝑇_𝜁(𝑘₁,𝑘₂,𝑘₃,𝑘₄)⊃𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘l_𝑐𝑑𝑁^𝑚𝑛_𝑒𝑓𝑃^𝑎𝑐_𝑖𝑘(𝑘₁)𝑃^𝑏𝑒_𝑗𝑚(𝑘₂)×𝑃^𝑑𝑓_𝑙𝜂(𝑘₁+𝑘₃)+𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘_𝑐𝑁^𝑙𝑚𝑛_𝑑𝑒𝑓×𝑃^𝑎𝑑_𝑖𝑙(𝑘₁)𝑃^𝑏𝑒_𝑗𝑚(𝑘₂)𝑃^𝑐𝑓_𝑘𝜂(𝑘₃)=𝑀^𝑐_𝑖𝐿^𝑖𝑗_𝑐𝑒𝑀^𝑒_𝑗+𝑀^𝑓_𝑖𝑀^𝑒_𝑗𝑀^𝑑_𝑘𝑁^𝑖𝑗𝑘_𝑓𝑒𝑑, (38)(39) where 𝑀^𝑐_𝑘(𝑘)≡𝑁^𝑖_𝑎𝑃^𝑎𝑐_𝑖𝑘(𝑘)=𝑃₊^𝑎𝑐(𝑘)[𝛿_𝑖𝑘𝑁^𝑖_𝑎+𝑝^𝑎𝑐(𝑘)𝑘_𝑘(𝑘⋅𝑁_𝑎)+𝑖𝑞^𝑎𝑐(𝑘)(𝑘×𝑁_𝑎)_𝑘𝐿],_{𝑗𝑙𝑐𝑒}(𝑘)≡𝑁^𝑗𝑖_𝑐𝑑𝑃^𝑑𝑓_𝑖𝑘(𝑘)𝑁^𝑘𝑙_𝑓𝑒=𝑃₊^𝑑𝑓(𝑘)[𝑁^𝑗_𝑐𝑑⋅𝑁^𝑙_𝑒𝑓+𝑝^𝑑𝑓(𝑘)(𝑘⋅𝑁^𝑗_𝑐𝑑)(𝑘⋅𝑁^𝑙_𝑒𝑓)+𝑖𝑞^𝑑𝑓(𝑘)𝑘⋅𝑁^𝑗_𝑐𝑑×𝑁^𝑙_𝑒𝑓]. (40) In the previous equations, we defined 𝑝^𝑎𝑐𝑃(𝑘)≡^𝑎𝑐long−𝑃₊^𝑎𝑐𝑃₊^𝑎𝑐,𝑞^𝑎𝑐𝑃(𝑘)≡^−𝑎𝑐𝑃₊^𝑎𝑐, (41) with 𝑁_𝑎≡(𝑁¹_𝑎,𝑁²_𝑎,𝑁³_𝑎) and 𝑁^𝑗_𝑐𝑑≡(𝑁^𝑗1_𝑐𝑑,𝑁^𝑗2_𝑐𝑑,𝑁^𝑗3_𝑐𝑑).

Notice that, as for the power spectrum (25), also in (38)-(39) the anisotropic parts of the expressions are weighted by coefficients that are proportional either to 𝑃₋ or to (𝑃_long−𝑃₊). When these two quantities are equal to zero, the (Abelian) bispectrum and trispectrum are therefore isotropized. 𝑃₋=0 in parity conserving theories, like the ones we have been describing. According to the parametrization (16) of the longitudinal mode, we have 𝑃_long−𝑃₊=(|𝑛(𝑥)|²−1)𝑃₊.

4.2.2. Non-Abelian Contributions

We list the non-Abelian terms for the bispectrum 𝐵_𝜁(𝑘₁,𝑘₂,𝑘₃)⊃𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘_𝑐𝐵^𝑎𝑏𝑐_𝑖𝑗𝑘(𝑘₁,𝑘₂,𝑘₃), (42) and for the trispectrum 𝑇_𝜁(𝑘₁,𝑘₂,𝑘₃,𝑘₄)⊃𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘_𝑐𝑁^𝑙_𝑑𝑇^{𝑎𝑏𝑐𝑑}_{𝑖𝑗𝑘𝑙}(𝑘₁,𝑘₂,𝑘₃,𝑘₄)+𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁_𝜙𝑁^𝑘_𝜙𝑐[𝑃_𝜙(𝑘₃)𝐵^𝑎𝑏𝑐_𝑖𝑗𝑘(𝑘₁,𝑘₂,𝑘₃+𝑘₄)+perms.]+𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘_𝑐𝑁^𝑙𝑚_𝑑𝑒×[𝑃^𝑎𝑑_𝑖𝑙(𝑘₁)𝐵^𝑏𝑐𝑒_𝑗𝑘𝑚(𝑘₁+𝑘₂,𝑘₃,𝑘₄)+perms.]. (43) The computation of the vector bosons spectra ⟨𝛿𝐴^𝑎_𝑖𝛿𝐴^𝑏_𝑗𝛿𝐴^𝑐_𝑘⟩=𝛿⁽³⁾(𝑘₁+𝑘₂+𝑘₃)𝐵^𝑎𝑏𝑐_𝑖𝑗𝑘,⟨𝛿𝐴^𝑎_𝑖𝛿𝐴^𝑏_𝑗𝛿𝐴^𝑐_𝑘𝛿𝐴^𝑑_𝑙⟩=𝛿⁽³⁾(𝑘₁+𝑘₂+𝑘₃+𝑘₄)𝑇^{𝑎𝑏𝑐𝑑}_{𝑖𝑗𝑘𝑙}, (44) will be reviewed in this section. This requires the expansion of the in-in formula up to second order in the interaction Hamiltonian ⟨Θ(𝜂^∗)⟩⊃𝑖⟨𝑇[Θ^𝜂∗_−∞𝑑𝜂^′(𝐻⁺_int(𝜂^′)−𝐻⁻_int(𝜂^′+))]⟩(−𝑖)²2⟨𝑇[Θ^𝜂∗_−∞𝑑𝜂^′(𝐻⁺_int(𝜂^′)−𝐻⁻_int(𝜂^′×))_{𝜂∗−∞}𝑑𝜂^′′(𝐻⁺_int(𝜂^′′)−𝐻⁻_int(𝜂^′′))]⟩. (45) The interaction Hamiltonian needs to be expanded up to fourth order in the fields fluctuations, that is, 𝐻_int=𝐻⁽³⁾_int+𝐻⁽⁴⁾_int, where 𝐻⁽³⁾_int=𝑔_𝑐𝜀^𝑎𝑏𝑐𝑔^𝑖𝑘𝑔^𝑗𝑙(𝜕_𝑖𝛿𝐵^𝑎_𝑗)𝛿𝐵^𝑏_𝑘𝛿𝐵^𝑐_𝑙+𝑔²_𝑐𝜀^𝑒𝑎𝑏𝜀^𝑒𝑐𝑑𝑔^𝑖𝑘𝑔^𝑗𝑙𝐵^𝑎_𝑖𝛿𝐵^𝑏_𝑗𝛿𝐵^𝑐_𝑘𝛿𝐵^𝑑_𝑙,𝐻⁽⁴⁾_int=𝑔²_𝑐𝜀^𝑒𝑎𝑏𝜀^𝑒𝑐𝑑𝑔^𝑖𝑗𝑔^𝑘𝑙𝛿𝐵^𝑎_𝑖𝛿𝐵^𝑏_𝑘𝛿𝐵^𝑐_𝑗𝛿𝐵^𝑑_𝑙. (46) To tree-level, the relevant diagrams are pictured in Figures 1 and 2. By looking at (46), we can see that there is a bispectrum diagram that is lower in terms of power of the 𝑆𝑈(2) coupling (∼g_c) compared to the trispectrum (∼g_c²); as a matter of fact, for symmetry reasons that we are going to discuss later in this section, 𝑔²_𝑐 interaction terms are needed to provide a nonzero contributions to the bispectrum.

(a)

(b)

Figure 1 

Diagrammatic representations of the tree-level contributions to the vector fields bispectrum.

(a)

(b)

Figure 2 

Diagrammatic representations of the tree-level contributions to the vector fields trispectrum: vector-exchange (a) and contact-interaction (b) diagrams.

The propagators for “plus” and “minus” fields are 𝛿𝐵_𝑖^𝑎,+(𝜂^′)𝛿𝐵_𝑗^𝑏,+(𝜂^′′)=Π^𝑎𝑏_𝑖𝑗(𝜂^′,𝜂^′′)Θ(𝜂^′−𝜂^′′)+Π^𝑎𝑏_𝑖𝑗(𝜂^′,𝜂^′′)Θ(𝜂^′′−𝜂^′),𝛿𝐵_𝑖^𝑎,+(𝜂^′)𝛿𝐵_𝑗^𝑏,−(𝜂^′′)=Π^𝑎𝑏_𝑖𝑗(𝜂^′,𝜂^′′),𝛿𝐵_𝑖^𝑎,−(𝜂^′)𝛿𝐵_𝑗^𝑏,+(𝜂^′′)=Π^𝑎𝑏_𝑖𝑗(𝜂^′,𝜂^′′),𝛿𝐵_𝑖^𝑎,−(𝜂^′)𝛿𝐵_𝑗^𝑏,−(𝜂^′′)=Π^𝑎𝑏_𝑖𝑗(𝜂^′,𝜂^′′)Θ(𝜂^′−𝜂^′′)+Π^𝑎𝑏_𝑖𝑗(𝜂^′,𝜂^′′)Θ(𝜂^′′−𝜂^′), (47) or Π^𝑎𝑏_𝑖𝑗(𝑘)≡𝑇^even_𝑖𝑗(𝑘)𝑃₊^𝑎𝑏+𝑖𝑇^odd_𝑖𝑗(𝑘)𝑃^𝑎𝑏_𝑖𝑗+𝑇^long_𝑖𝑗(𝑘)𝑃^𝑎𝑏_𝑖𝑗,Π_𝑖^𝑎𝑏_j(𝑘)≡𝑇^even_𝑖𝑗(𝑘)𝑃₊^𝑎𝑏+𝑖𝑇^odd_𝑖𝑗(𝑘)𝑃^𝑎𝑏_𝑖𝑗+𝑇^long_𝑖𝑗(𝑘)𝑃^𝑎𝑏_𝑖𝑗, (48) in Fourier space. In the previous equations we set 𝑃_±^𝑎𝑏≡(1/2)(𝑃_𝑅^𝑎𝑏±𝑃_𝐿^𝑎𝑏), 𝑃_𝑅^𝑎𝑏=𝛿_𝑎𝑏𝛿𝐵_𝑅^𝑎𝑏(𝑘,𝜂^∗)𝛿𝐵_𝑅^∗𝑎𝑏(𝑘,𝜂) and 𝑃_±^𝑎𝑏=(𝑃_±^𝑎𝑏)^∗ (similar definitions apply for 𝑃_𝐿^𝑎𝑏 and 𝑃^𝑎𝑏_long).

We are now ready to show the computation of the following contributions to the bispectrum and trispectrum of 𝜁⟨𝜁_𝑘1𝜁_𝑘2𝜁_𝑘3⟩⊃𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘_𝑐⟨𝛿𝐴^𝑎_𝑖(𝑘₁)𝛿𝐴^𝑏_𝑗(𝑘₂)𝛿𝐴^𝑐_𝑘(𝑘₃)⟩,⟨𝜁_𝑘1𝜁_𝑘2𝜁_𝑘3𝜁_𝑘4⟩⊃𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘_𝑐𝑁^𝑙_𝑑×⟨𝛿𝐴^𝑎_𝑖(𝑘₁)𝛿𝐴^𝑏_𝑗(𝑘₂)𝛿𝐴^𝑐_𝑘(𝑘₃)𝛿𝐴^𝑑_𝑙(𝑘₄)⟩. (49)(50) Equation (49) becomes ⟨𝜁_𝑘1𝜁_𝑘2𝜁_𝑘3⟩⊃𝑁^𝑖_𝑎𝑁^𝑗_𝑏𝑁^𝑘_𝑐𝛿⁽³⁾(𝑘₁+𝑘₂+𝑘₃)𝑎³(𝜂^∗)×[𝑑𝜂𝑎⁴(𝜂)Π_𝑖𝑚(𝑘₁)Π^𝑙_𝑗(𝑘₂)Π^𝑚_𝑘(𝑘₃)×(𝑔_𝑐𝜀^𝑎𝑏𝑐𝑘_1𝑙+𝑔²_𝑐𝜀^𝑒𝑑𝑎𝜀^𝑒𝑏𝑐𝐵^𝑑_𝑙)]+perms.+𝑐.𝑐. (51) Even before performing the time integration, one realizes that, because of the antisymmetric properties of the Levi-Civita tensor, the ∼g_c contribution on the r.h.s. of (51) is equal to zero once the sum over all the possible permutations has been performed. The vector bosons bispectrum is therefore proportional to 𝑔²_𝑐. The final result from (51) has the following form⟨𝜁_𝑘1𝜁_𝑘2𝜁_𝑘3⟩⊃(2𝜋)³𝛿⁽³⁾(𝑘₁+𝑘₂+𝑘₃)𝑔²_𝑐𝐻²_∗×_𝑛𝐹_𝑛(𝑘_𝑖,𝜂^∗)𝐼_𝑛(𝑘_𝑖⋅𝑘_𝑗,𝐴_𝑖⋅𝐴_𝑗,𝑘_𝑖⋅𝐴_𝑗), (52) where the 𝐹_𝑛's are isotropic functions of time and of the moduli of the wave vectors (𝑖=1,2,3) and the 𝐼_𝑛's are anisotropic coefficients. The sum in the previous equation is taken over all possible combinations of products of three polarization indices, that is, 𝑛∈(𝐸𝐸𝐸,𝐸𝐸𝑙,𝐸𝑙𝐸,…,𝑙𝑙𝑙), where 𝐸 stands for “even”, 𝑙 for “longitudinal”. The complete expressions for the terms appearing in the sum are quite lengthy (see [101, Section 4.2]). As an example, we report one of these terms 𝐹_𝑙𝑙𝑙=−𝑛⁶(𝑥^∗)124𝑘⁶𝑘²₁𝑘²₂𝑘²₃𝑥^∗2×[𝐴_𝐸𝐸𝐸+(𝐵_𝐸𝐸𝐸cos𝑥^∗+𝐶_𝐸𝐸𝐸sin𝑥^∗)𝐸_𝑖𝑥^∗𝐼],_𝑙𝑙𝑙=𝜀^{𝑎𝑎′𝑏′}𝜀^{𝑎𝑐′𝑒}×[((𝑘₁⋅𝑁^𝑎′)(𝑘₃⋅𝑁^𝑏′)(𝑘₂⋅𝑁^𝑐′)(𝑘₁⋅𝑘₂)×(𝑘₃⋅𝐴^𝑒)−(𝑘₃⋅𝑁^𝑎′)(𝑘₂⋅𝑁^𝑏′)×(𝑘₁⋅𝑁^𝑐′)(𝑘₁⋅𝑘₂)(𝑘₃⋅𝐴^𝑒))+(1⟷3)+(2⟷3)], (53)(54) where 𝐴_𝐸𝐸𝐸, 𝐵_𝐸𝐸𝐸 and 𝐶_𝐸𝐸𝐸 are functions of 𝑥^∗ and of the momenta 𝑘_𝑖≡|𝑘_𝑖| (they are all reported in [101, Appendix C]), 𝐸_𝑖 is the exponential-integral function and 𝑖↔𝑗 means “exchange 𝑘_𝑖 with 𝑘_𝑗”. As we will discuss in more details in Section 6, one of the more interesting features of these models is that the bispectrum and the trispectrum turn out to have an amplitude that is modulated by the preferred directions that break statistical isotropy.

Let us now move to the trispectrum. Again, we count two different kinds of contributions, the first from ∼g_c and the second from ∼g_c² interaction terms, respectively in 𝐻⁽³⁾_int and 𝐻⁽⁴⁾_int. The former produce vector-exchange diagrams, the latter are represented by contact-interaction diagrams (see Figure 2). Their analytic expressions are different, but they both have a structure similar to (52) ⟨𝜁_𝑘1𝜁_𝑘2𝜁_𝑘3𝜁_𝑘4⟩⊃(2𝜋)³𝛿⁽³⁾(𝑘₁+𝑘₂+𝑘₃+𝑘₄)𝑔²_𝑐𝐻²_∗×_𝑛𝐺_𝑛(𝑘_𝑖1,𝑘₂1,𝑘₄,𝜂^∗)𝐿_𝑛(𝑘_𝑖⋅𝑘_𝑗,𝐴_𝑖⋅𝐴_𝑗,𝑘_𝑖⋅𝐴_𝑗), (55) where we define 𝑘1₂≡|𝑘₁+𝑘₂| and 𝑘1₄≡|𝑘₁+𝑘₄| (see [102, Sections 5.2.1 and 5.2.2] for the explicit expressions of the functions 𝐺_𝑛 and 𝐿_𝑛).

5. Amplitude of non-Gaussianity: 𝑓_𝑁𝐿 and 𝜏_𝑁𝐿

In this paper, we use the following definitions for the non-Gaussianity amplitudes65𝑓_𝑁𝐿=𝐵_𝜁(𝑘₁,𝑘₂,𝑘₃)𝑃^iso(𝑘₁)𝑃^iso(𝑘₂,𝜏)+perms._𝑁𝐿=2𝑇_𝜁(𝑘₁,𝑘₂,𝑘₃,𝑘₄)𝑃^iso(𝑘₁)𝑃^iso(𝑘₂)𝑃^iso1(𝑘₄.)+23perms. (56) The choice of normalizing the bispectrum and the trispectrum by the isotropic part of the power spectrum, instead of using its complete expression 𝑃_𝜁, is motivated by the fact that the latter would only introduce a correction to the previous equations proportional to the anisotropy parameter 𝑔, which is a small quantity.

The parameters 𝑓_𝑁𝐿 and 𝜏_𝑁𝐿 receive contributions both from scalar (“𝑠”) and from vector (“𝑣”) fields 𝑓_𝑁𝐿=𝑓^(𝑠)_𝑁𝐿+𝑓^(𝑣)_𝑁𝐿,𝜏_𝑁𝐿=𝜏^(𝑠)_𝑁𝐿+𝜏^(𝑣)_𝑁𝐿. (57)(58) The latter can again be distinguished into Abelian (𝐴) and non-Abelian (𝑁𝐴) 𝑓^(𝑣)_𝑁𝐿=𝑓^(𝐴)_𝑁𝐿+𝑓^(𝑁𝐴)_𝑁𝐿,𝜏^(𝑣)_𝑁𝐿=𝜏^(𝐴1)_𝑁𝐿+𝜏^(𝐴2)_𝑁𝐿+𝜏^(𝑁𝐴1)_𝑁𝐿+𝜏^(𝑁𝐴2)_𝑁𝐿. (59)(60) The contribution 𝑓^(𝐴)_𝑁𝐿 comes from (38), 𝑓^(𝑁𝐴)_𝑁𝐿 from (52), 𝜏^(𝐴1)_𝑁𝐿 and 𝜏^(𝐴2)_𝑁𝐿 from (39), finally 𝜏^(𝑁𝐴1)_𝑁𝐿 from (55) and 𝜏^(𝑁𝐴2)_𝑁𝐿 from the last line of (43).

In order to keep the vector contributions manageable and simple in their structure, all gauge and vector indices will be purposely neglected at this point and so the angular functions appearing in the anisotropy coefficients will be left out of the final amplitude results. This is acceptable considering that these functions will in general introduce numerical corrections of order one. Nevertheless, it is important to keep in mind that the amplitudes also depend on the angular parameters of the theory.

We will now focus on the dependence of 𝑓_𝑁𝐿 and 𝜏_𝑁𝐿 from the nonangular parameters of the theory and quickly draw a comparison among the different contributions listed in (57) through (60).

The expression of the number of e-foldings depends on the specific model and, in particular, on the mechanism of production of the fluctuations. Two possibilities have been described in Section 2. For vector inflation we have 𝑁^𝑖_𝑎=𝐴^𝑎_𝑖2𝑚²_𝑃,𝑁^𝑖𝑗_𝑎𝑏=𝛿_𝑎𝑏𝛿^𝑖j2𝑚²_𝑃, (61) (see [101, Appendix B] for their derivation). In the vector curvaton model the same quantities become [96, 101] 𝑁^𝑖_𝑎=23𝑟𝐴^𝑎_𝑖∑_𝑏|𝐴^𝑏|²,𝑁^𝑖𝑗_𝑎𝑏=13𝑟𝛿_𝑎𝑏𝛿^𝑖𝑗∑_𝑐|𝐴^𝑐|². (62) Neglecting tensor and gauge indices, the expressions above can be simplified as 𝑁_𝐴⋍𝐴/𝑚²_𝑃 and 𝑁_𝐴𝐴⋍1/𝑚²_𝑃 in vector inflation, 𝑁_𝐴⋍𝑟/𝐴 and 𝑁_𝐴𝐴⋍𝑟/𝐴² in the vector curvaton model. Also we have 𝑁_𝐴𝐴𝐴=0 in vector inflation and 𝑁_𝐴𝐴𝐴⋍𝑟/𝐴³ in vector curvaton.

We are now ready to provide the final expressions for the amplitudes: in Table 1 we list all the contributions to 𝑓_𝑁𝐿, Table 2 includes the vector contributions to 𝜏_𝑁𝐿, the scalar contributions being given by 𝜏^(𝑠)_𝑁𝐿=𝜖_𝜙(1+𝛽)³+𝜖²_𝜙(1+𝛽)³. (63) In the expressions appearing in the tables, numerical coefficients of order one have not been reported. Also, 𝑚 is by definition equal to 𝑚_𝑃 in vector inflation and to 𝐴/√𝑟 in the vector curvaton model; 𝑁_𝜙⋍(𝑚_𝑃𝜖_𝜙)⁻¹ and 𝑁_𝜙𝜙⋍𝑚_𝑃⁻², with 𝜖_𝜙≡(𝜙̇²)/(2𝑚²_𝑃𝐻²).

Table 1 

Order of magnitude of 𝑓_𝑁𝐿 in different scenarios.

Table 2 

Order of magnitude of the vector contributions to 𝜏_𝑁𝐿 in different scenarios.

The quantities involved in the amplitude expressions are 𝑔, 𝛽, 𝑟, 𝜖_𝜙, 𝑔_𝑐, 𝑚_𝑃/𝐻, 𝐴/𝑚_𝑃 and 𝐴/𝐻. We already know that 𝑔 and 𝛽 are to be considered smaller than one (see discussion after (32)). Similarly, as mentioned after (3), 𝑟 has to remain small at least until inflation ends so as to attain an “almost isotropic” expansion. The slow-roll parameter 𝜖_𝜙 and the 𝑆𝑈(2) coupling 𝑔_𝑐 are small, respectively, to allow the inflaton to slowly roll down its potential and for perturbation theory to be valid. The ratio 𝑚_𝑃/𝐻 is of order 10⁵ (assuming 𝜖_𝜙∼10⁻¹). Finally, 𝐴/𝑚_𝑃 and 𝐴/𝐻 have no stringent bounds. A reasonable choice could be to assume that the expectation value of the gauge fields is no larger than the Planck mass, that is, 𝐴/𝑚_𝑃≤1. As to the 𝐴/𝐻 ratio, different possibilities are allowed, including the one where it is of order one (see [99, Section 6 and Appendix A] for a discussion on this).

Let us now compare the different amplitude contributions. The ratios between scalar and vector contributions are shown in Table 3 for the bispectrum and Table 4 for the trispectrum. We can observe that the dominance of a given contribution w.r.t. another one very much depends on the selected region of parameter space. It turns out that it is allowed for the vector contributions to be larger than the scalar ones and also for the non-Abelian contributions to be larger than the Abelian ones. This is discussed more in details in [99, Section 6]. An interesting point is, for instance, the following: ignoring tensor and gauge indices, the ratio 𝑔_𝑐𝐴/𝐻, that appears in many of the Tables entries, is a quantity smaller than one; if we consider the different configurations identified by gauge and vector indices, we realize that this is not always true, in fact the value of this ratio can be ≫1 in some configurations.

Table 3 

Order of magnitude of the ratios 𝑓^𝑣_𝑁𝐿/𝑓^𝑠_𝑁𝐿 in different scenarios.

Table 4 

Order of magnitude of the ratios 𝜏^𝑣_𝑁𝐿/𝜏^𝑠_𝑁𝐿 in different scenarios.

Finally, it is interesting to compare bispectrum and trispectrum amplitudes (see Table 5). Again, it is allowed for the ratios appearing in Table 5 to be either large or small, depending on the specific location within the parameter space of the theory. For instance, the combination of a small bispectrum with a large trispectrum is permitted. The latter is an interesting possibility: if the bispectrum was observably small, we could still hope the information about non-Gaussianity to be accessible thanks to the trispectrum.

Table 5 

Order of magnitude of the ratios 𝜏^𝑣_𝑁𝐿/(𝑓^𝑁𝐴_𝑁𝐿)² in different scenarios.

Another interesting feature of this model is that the bispectrum and the trispectrum depend on the same set of quantities. If these correlation functions were independently known, that information could then be used to test the theory and place some bounds on its parameters.

6. Shape of Non-Gaussianity and Statistical Anisotropy Features

Studying the shape of non-Gaussianity means understanding the features of momentum dependence of the bispectrum and higher order correlators. If they also depend on variables other than momenta, it is important to determine how these other variables affect the profiles for any given momentum set-up. This is the case as far as the bispectrum and the trispectrum of the gauge fields are concerned, given the fact that they are functions, besides of momenta, also of a large set of angular variables (see (52) and (55)).

6.1. Momentum Dependence of the Bispectrum and Trispectrum Profiles

We show the study of the momentum dependence of the 𝐹_𝑛 and 𝐺_𝑛 functions in (52) and (55) first and then analyze the angular variables dependence of the spectra, once the momenta have been fixed in a given configuration. A natural choice would be to consider the configuration where the correlators are maximized.

The maxima can be easily determined for the bispectrum by plotting the isotropic functions 𝐹_𝑛 and 𝐺_𝑛 in terms of two of their momenta. These plots are provided in Figure 3, where the variables are 𝑥₂≡𝑘₂/𝑘₁ and 𝑥₃≡𝑘₃/𝑘₁. Each one of the plots corresponds to a single isotropic functions of the sum in (52). It is apparent that the maxima are mostly located in the so-called local region, that is, for 𝑘₁∼𝑘₂≫𝑘₃; three out of the eight graphs do not have their peaks in this configuration but, at the same time, they show negligible amplitudes compared to the “local” peaked graphs.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3 

Plot of 𝑟𝑛≡Θ(𝑥2−𝑥3)Θ(𝑥3−1+𝑥2)𝑥22𝑥23𝑅𝑛(𝑥2,𝑥3), where we define 𝑅𝑛=𝑘61𝐹𝑛. The Heaviside step functions Θ help restricting the plot domain to the region (𝑥2,𝑥3) that is allowed for the triangle 𝑘1+𝑘2+𝑘3=0 (in particular, we set 𝑥3<𝑥2). We also set 𝑥∗=1.

The situation is much more complex for the trispectrum, being the number of momentum variables larger than three (𝑘₁, 𝑘₂, 𝑘₃, 𝑘₄, 𝑘1₂ and 𝑘1₄). The momentum dependence of the isotropic functions can be studied by selecting different configurations for the tetrahedron made up by the four momentum vectors, in such a way as to narrow the number of independent momentum variables down to two. A list of possible configurations was presented in [43]. We consider two of them, the “equilateral” and the “specialized planar”.

In the equilateral configuration the four sides of the tetrahedron have the same length (𝑘₁=𝑘₂=𝑘₃=𝑘₄), therefore 1𝑥≡𝑘₂/𝑘₁ and 1𝑦≡𝑘₄/𝑘₁ can be chosen as variables for the plots. The plots of the isotropic functions of contact interaction and vector exchange contributions are provided in Figure 4. The former (𝑐.𝑖.) shows a constant behaviour in this configuration, being independent of 𝑘1₂ and 𝑘1₄. The latter (𝑣.𝑒.(𝐼), 𝑣.e.(𝐼𝐼) and 𝑣.𝑒.(𝐼𝐼𝐼)) diverge as 𝑘⁻³1_𝑖 (𝑖=1,2,and3, resp., for the three plots) in the limit of a flat tetrahedron, that is, 1(𝑘_𝑖/𝑘₁)→0.

(a)

(b)

(c)

(d)

Figure 4 

Plots of the isotropic functions appearing in the vector fields trispectrum (from (55)): 𝑐.𝑖. is the contribution from contact-interaction diagrams, 𝑣.𝑒.(𝐼), 𝑣.𝑒.(𝐼𝐼), and 𝑣.𝑒.(𝐼𝐼𝐼) are the contributions from the vector-exchange diagrams. The equilateral configuration has been considered in this figure.

In the specialized planar configuration, the tetrahedron is flattened and, in addition to that, three of the six momentum variables are set equal to one another (𝑘₁=𝑘₃1=𝑘₄); this leaves two independent variables, which can be 𝑥≡𝑘₂/𝑘₁ and 𝑦≡𝑘₃/𝑘₁. There is a double degeneracy in this configuration, due to the fact that the quadrangle can have internal angles larger than or smaller/equal to 𝜋, as we can see from the plus and minus signs in the expressions for 𝑘1₂ and 𝑘1₃ [43] 𝑘1₂𝑘₁=𝑥1+²𝑦²2±𝑥𝑦2(4−𝑥²)(4−𝑦²𝑘1),₃𝑘₁=𝑥²+𝑦²−𝑥²𝑦²2∓𝑥𝑦2(4−𝑥²)(4−𝑦²). (64) The two cases are plotted in Figures 5 and 6. Notice that divergences generally occur as 𝑥,𝑦→0, as 𝑥→𝑦 and (𝑥,𝑦)→(2,2).

(a)

(b)

(c)

(d)

Figure 5 

Plots of the contact interaction and of the vector-exchange contributions in the specialized planar configuration (plus sign).

(a)

(b)

(c)

(d)

Figure 6 

Plots of the contact interaction and of the vector-exchange contributions in the specialized planar configuration (minus sign).

6.2. Features and Level of Anisotropy

Statistical homogeneity and isotropy are considered characterizing features of the CMB fluctuations distribution, if one ignores the issues raised by the “anomalous” detections we presented in the introduction.

Homogeneity of the correlation functions equates translational invariance and hence total momentum conservation, as enforced by the delta functions appearing on the left-hand sides of (17) through (19). This invariance property can then be pictured as the three momentum vectors forming a closed triangle for the bispectrum and the four momenta arranged in a tetrahedron for the trispectrum (see Figure 7).

(a)

(b)

Figure 7 

Representation of momentum conservation for the bispectrum (the three momenta form a closed triangle) and for the trispectrum (the momenta form a tetrahedron).

Statistical isotropy corresponds to invariance w.r.t. rotations in space of the momentum (for the power spectrum) and of the triangle or tetrahedron made up by the momenta, respectively for the bispectrum and the trispectrum. This symmetry can be broken, as it for example happens in the 𝑆𝑈(2) case, by assuming the existence of preferred spatial directions in the early universe that might be revealed in the CMB observations. When this happens, the correlation functions are expected to be sensitive to the spatial orientation of the wave number or of the momenta triangles and tetrahedrons w.r.t. these special directions. Analitically, the bispectrum and the trispectrum will depend on the angles among the vector bosons and the wave vectors (besides the angles among the gauge bosons themselves), as shown in the coefficients appearing in (52) and (55). This implies that both the amplitude and the shape of bispectrum and trispectrum will be affected by these mutual spatial orientations. The modulation of the shapes by the directions that break statistical anisotropy was discussed with some examples both for the bispectrum and the trispectrum in our previous papers [101, 102]. These examples are here reported in Figures 8 and 9. In Figure 8 we show the plot of the vector contribution to the bispectrum of 𝜁, properly normalized in the configuration 𝑁₃=𝑁_𝐴𝑁(0,0,1),₁=𝑁₂=𝑁_𝐴(sin𝜃cos𝜙,sin𝜃sin𝜙,cos𝜃), (65) where, the (𝑥,𝑦,𝑧) coordinate frame is chosen to be 𝑘₃=𝑥 and 𝑘₁=𝑘₂=𝑧 and 𝛿 is the angle between 𝑁_1,2 and 𝑘₃.

Figure 8 

Plot of 𝑓(𝜃,𝛿)≡[(𝐵𝜁(𝜃,𝛿,𝑥∗,𝑥2,𝑥3)𝑥22𝑥23𝑘61)/(𝑔2𝑐𝐻2𝑚2𝑁4𝐴)] evaluated at (𝑥∗=1,𝑥2=0.9,𝑥3=0.1) in a sample angular configuration. See [99, Appendix D] for its complete analytic expression.

Figure 9 

Plot of the anisotropic part of the trispectrum from the contribution due to vector-exchange diagrams in a sample angular configuration. See [100, Section 8] for its analytic expression.

In Figure 9 we provide a similar plot for the trispectrum, but in a different configuration 𝑁₂⋅𝑘_𝑖𝑁=0(𝑖=1,…,4)₁⋅𝑘₁=cos𝛿,𝑁₁⋅𝑘₂𝑁=0,₃⋅𝑘₂=cos𝜃,𝑁₃⋅𝑘₁=0. (66) In both examples, it is assumed for simplicity that the 𝑁_𝑎 have the same magnitude 𝑁_𝐴 for all 𝑎=1,2,3.

Another comment should be added concerning statistical anisotropy in the model. Notice that both the bispectrum and the trispectrum can be written as the sum of a purely isotropic and an anisotropic parts. The orders of magnitude of these two parts can, for instance, be read from Table 2 for the trispectrum: each one among 𝜏^𝑁𝐴2_𝑁𝐿, 𝜏^𝐴1_𝑁𝐿 and 𝜏^𝐴2_𝑁𝐿 provide the order of magnitude of the level of both their isotropic and anisotropic contributions, which are therefore comparable; 𝜏^𝑁𝐴1_𝑁𝐿 instead quantifies a purely anisotropic contribution which, as discussed in Section 5, can be comparable to the other three parts, if not the dominant one. A similar discussion applies to the bispectrum (see 𝑓^𝐴_𝑁𝐿 and 𝑓^𝑁𝐴_𝑁𝐿 in Table 1). We can then conclude that for the three and for the four point function, there is room in the parameter space of the theory for the anisotropic contributions to be as large as, or even larger than, the isotropic ones.

7. Conclusions

Motivated by the interest in models that combine non-Gaussianity and statistical anisotropy predictions for the CMB fluctuations, we have considered models of inflation where primordial vector fields effectively participate in the production of the curvature perturbations 𝜁. More specifically, we have reviewed the computation of the correlation functions up to fourth order, considering an 𝑆𝑈(2) vector multiplet. The 𝛿N formalism was employed to express 𝜁 in terms of the quantum fluctuations of all the primordial fields. The Schwinger-Keldysh formula was also used in evaluating the correlators.

The correlation functions result as the sum of scalar and vector contributions. The latter are of two kinds, “Abelian” (i.e., arising from the zeroth order terms in the Schwinger-Keldysh expansion) and “non-Abelian” (i.e., originating from the self-interactions of the vector fields). The bispectrum and the trispectrum final results are presented as a sum of products of isotropic functions of the momenta, 𝐹_𝑛 and 𝐺_𝑛 in (52) and (55), multiplied by anisotropy coefficients, 𝐼_𝑛 and 𝐿_𝑛 in (52) and (55), which depend on the angles between the (gauge and wave) vectors.

The amplitude of non-Gaussianity has been presented through the parameters 𝑓_𝑁𝐿 and 𝜏_𝑁𝐿; in particular we have show the dependence of these functions from the nonangular parameters of the theory. We have provided the comparisons among the different (scalar versus vector, Abelian versus non-Abelian) contributions to 𝑓_𝑁𝐿 and 𝜏_𝑁𝐿, noticing that any one of them can be the dominant contribution depending on the selected region of parameter space. In particular, we have stressed how the anisotropic contributions to the bispectrum and the trispectrum can overcome the isotropic parts. An interesting feature of these models is that the bispectrum and the trispectrum depend on the same set of parameters and their amplitudes are therefore strictly related to one another.

We have presented the shapes of both the bispectrum and the trispectrum. The isotropic functions appearing in their final expressions had been analyzed separately from their anisotropy coefficients. The bispectrum isotropic functions had been found to preferably show a local shape. The trispectrum ones had been plotted selecting equilateral and specialized planar configurations. The full expressions (i.e., complete of their anisotropy coefficients) of bispectrum and trispectrum have been presented in specific momenta configuration, in order to provide a hint of the modulation of shapes and amplitudes operated by anisotropy.

We have reviewed old and recent vector field models, indicating both their limits and achievements. We would like to stress that, in our view, the most promising features of these models consists in the possibility of providing both non-Gaussianity and statistical anisotropy predictions that are related to one another because of the fact that they share the same underlying theory. This might, at some point in the future, become a great advantage: measurements of non-Gaussianity could be used to constrain statistical anisotropy or vice versa.

Acknowledgments

This research has been partially supported by ASI Contract I/016/07/0 “COFIS”. A. Riotto acknowledges support by the EU Marie Curie Network UniverseNet (HPRNCT2006035863).