Primordial non-Gaussianity and the inﬂationary Universe

Non-Gaussianity of primordial ﬂuctuations is one of the most important probes of the very early Universe and is now severely constrained by cosmological observations. In this review, we describe the formalism to investigate primordial non-Gaussianities such as the bispectrum and trispectrum, and summarize the current observational constraints on the so-called non-linearity parameters. We also discuss models of the origin of primordial ﬂuctuations, paying particular attention to their non-Gaussian nature. The scale-dependence of non-linearity parameters and non-Gaussianities from isocurvature ﬂuctuations are also discussed. . . . . . .


Introduction
The origin of primordial density fluctuations is one of the most important issues in understanding the phenomena in the very early Universe. Quantum fluctuations of the inflaton field generated during inflation are usually considered as the origin of density fluctuations [1][2][3][4][5]. However, the precise physical mechanism by which these fluctuations are created remains unknown. Outstanding questions include the shape of the inflaton potential, the structure of the kinetic term of the inflaton, the number of fields other than the inflaton and their roles in generating fluctuations, and so on. In models beyond the standard model of particles physics, such as supersymmetry and superstring theory, scalar fields are ubiquitous and, if some scalar field other than the inflaton acquires fluctuations, such a scalar field can also generate a primordial perturbation. Examples of this include the curvaton model [6][7][8], the modulated reheating scenario [9,10], and so on.
On the other hand, the current cosmological observations (e.g., Planck [11]) are so precise that we can obtain various pieces of information on the properties of primordial fluctuations. The primary observables are the power spectra of (scalar) density perturbations and gravitational waves, which are well measured or constrained by the current data [12]. However, it has been argued that non-Gaussianities of primordial fluctuations can also give invaluable information. In particular, the degrees of non-Gaussianities are quite different between primordial fluctuations generated from a standard inflation and other mechanisms, and thus it is very important to investigate the non-Gaussian nature of primordial fluctuations. In fact, recent Planck data put a tight constraint on the size of non-Gaussianities [13], which gives many implications for the origin of primordial fluctuations and the early Universe.
In this review, we discuss models of primordial fluctuations, in particular, in light of the Planck result. We first summarize the probes of non-Gaussianities such as the bi-and trispectra, describing

Probes of non-Gaussianities and observational constraints
The properties of the curvature perturbation ζ can be probed by its statistical measures such as the power spectrum P ζ , bispectrum B ζ , and trispectrum T ζ , which correspond to the 2-, 3-, and 4-point correlation functions of ζ : ζ k 1 ζ k 2 ζ k 3 ζ k 4 c = (2π) 3 T ζ (k 1 , k 2 , k 3 , k 4 )δ (k 1 + k 2 + k 3 + k 4 where the subscript c represents the fact that we take the connected part for the correlation functions. The power spectrum is well measured by cosmological observations such as the cosmic microwave background (CMB) and is usually parametrized as where A s (k ref )(=Ã s ) and n s are the amplitude and the spectral index at the reference scale k ref , respectively. We also introducek = k/k ref for shorthand notation. The Planck team has precisely measured these quantities as ln(10 10Ã s ) = 3.089 +0.024 −0.027 and n s = 0.9603 ± 0.0073 (1σ CL) with k ref = 0.05 Mpc −1 in the framework of the standard flat CDM model [12,14] 1 , using the Planck temperature + WMAP polarization data. 1 In the usual 6-parameter analysis, the spectral index n s is assumed to be constant and the values mentioned here are from such an analysis. However, one can also probe the scale-dependence of n s , the so-called runnings, which are defined as α s ≡ dn s d ln k , β s ≡ dα s d ln k .

Bispectrum
The so-called non-linearity parameter f NL is often used to characterize the amplitude of the bispectrum. Unlike the power spectrum, which depends only on a single wave number, the bispectrum depends on three wave numbers. Depending on the mechanism of how primordial fluctuations are generated, the shape, or the wavenumber-dependence, of the bispectrum changes. This property can be used to differentiate models of the origin of primordial fluctuations. Thus, f NL are defined for some specific "shapes" of the bispectrum. Popular examples include local, equilateral, and orthogonal types. In the following, we describe these types (see also Ref. [13]). Observational constraints on these f NL are summarized in Sect. 2.3.

Local type.
This type of bispectrum can arise when non-linearities in the primordial adiabatic fluctuations are generated in superhorizon scale, and the non-linearities are local in real space.
Local-type non-Gaussianity can also be generated in models with isocurvature fluctuations. However, in such a case, we have to introduce another non-linearity parameter for isocurvature fluctuations. We discuss this issue in Sect. 5.

Equilateral type.
When one considers a general single-field inflation with a non-canonical kinetic term, or interactions with general higher-derivative operators, the shape of the bispectrum becomes the so-called (nearly) equilateral type. Models generating this type include: k-inflation [51,52], Dirac-Born-Infeld (DBI) inflation [53,54], ghost inflation [55][56][57], trapped inflation [58], the Lifshitz scalar [59], and so on. To obtain an observational constraint on this type, a template with the 3/31  following form is adopted [60]: In this form, the amplitude peaks around k 1 k 2 k 3 , which can be seen from Fig. 1.

Orthogonal type.
In some models, another type of bispectrum can arise, which is orthogonal to the local and equilateral forms and is called the orthogonal type. Its bispectrum form is [61]: This shape gives a positive amplitude at the equilateral limit (k 1 k 2 k 3 ), and negative in the folded (or flattened) limit (k 1 2k 2 2k 3 ). This shape arises in a model with some linear combinations of higher-derivative interactions [61].

Folded (flattened) type.
Usually, constraints on f NL are reported for the local, equilateral, and orthogonal types 2 ; however, the so-called folded or flattened shape (k 1 2k 2 2k 3 ) appears in models with a non-Bunch-Davies vacuum [63][64][65]95], curvature-generated interaction [66], Galilean symmetry [67], and so on. The form of a bispectrum of this type is given by In fact, this form can be written in terms of F (equil) and from which we can compute the trispectrum: where k 13 = |k 1 + k 3 | and τ (local) NL and g (local) NL are non-linearity parameters characterizing the size of the trispectrum (later in this subsection, we omit "(local)" from the non-linearity parameters). When a single field (source) is responsible for the curvature perturbation, τ NL is given by f NL as 3 On the other hand, when multiple sources contribute to the curvature perturbation, τ NL is larger than T. Takahashi mixed inflaton-curvaton (or spectator field 4 ) model [21][22][23][24][25][26], the ungaussiton model [28,29], and so on. Thus, τ NL may serve as a first signature of non-Gaussianity in such a model. Regarding g NL , most models have a particular relation between f NL and g NL . In Ref. [42], localtype models have been classified into three categories by using the f NL -g NL relation: (i) "linear g NL " type: Although, precisely speaking, the relation between f NL and g NL actually depends on the model parameters, the above classification is useful to give an idea of how large g NL can be. In fact, even for the "enhanced g NL " type, the power n in | f NL | n is n = 2 for most models [42]. Thus, g NL is not so enhanced compared to f NL in most cases. However, we remark that, if we allow some level of fine-tuning, large values of g NL are possible even when f NL O(1) [68]. Such examples are the self-interacting curvaton [141][142][143][144][145][146][147], the modulated curvaton [42,43], and modulated decay of the curvaton [45][46][47], in which some cancellation occurs to suppress the amplitude of the bispectrum, while cancellation does not happen in the trispectrum. For details of this issue, see Ref. [68].

Equilateral type.
The trispectrum of general single-field inflation models including a nontrivial kinetic term can be schematically written as [69] Here, T s1,s2,s3 are the trispectra coming from the scalar-exchange diagram and T c1,c2,c3 are those from the contact-interaction diagram (explicit forms of each trispectrum can be found in Ref. [69]). We will also discuss the trispectrum in models with general single-field inflation models in Sect. 3.1. Among the trispectra, T c1 has a diagonal-free form and is strongly correlated with the other trispectra, which motivates us to adopt T c1 as a template to probe the size of the equilateral-type trispectrum. The shape correlation between T c1 and the other functions, and also the trispectra produced in various models such as DBI inflation, ghost inflation, and Lifshitz scalar models are analyzed in Refs. [70][71][72].
To define the non-linearity parameter of the equilateral trispectrum, first we define the estimator t NL [69,70], by taking the RT (regular tetrahedron) limit in the trispectrum as where the RT limit corresponds to k 1 = k 2 = k 3 = k 4 = k 12 = k 14 ≡ k. By putting the c1-type for the trispectrum whose explicit form can be given as PTEP 2014, 06B105 T. Takahashi we define the non-linearity parameter for the equilateral trispectrum t equil NL 5 with Eq. (15). Observational constraints on the non-linearity parameter will be given in the next subsection.

Observational constraints
Here we summarize observational constraints on the non-linearity parameters f NL , τ NL , and g NL from CMB and large-scale structure (LSS) data.

Trispectrum
: τ NL and g NL . Observational constraints on the trispectrum, more specifically, on the non-linearity parameters τ NL and g NL , are less explored compared to f NL . However, there are several constraints available from the actual data, which we list below.

Models of primordial non-Gaussianities
In this section, we discuss models of the origin of primordial fluctuations, paying particular attention to their non-Gaussian nature. In fact, in most models, the predicted value of f NL can vary from , depending on the model parameters. Thus, even with a severe constraint on f NL from Planck, most models are still viable and the model parameters are just tightly constrained.
Although many models have been proposed in the literature, here we discuss some representative ones. 8

General single-field inflation models
Non-Gaussianity has been extensively investigated by many authors in various extensions of a standard single-field inflation model. In fact, by using the most general scalar-tensor theory Lagrangian with second-order equation of motion proposed by Horndeski [87], one can treat single-field inflation models in a unified way, and the bispectrum has been calculated in this setup in Refs. [88][89][90][91][92]. Another approach to studying a wide class of general single-field inflation models is with the effective field theory of inflation [93,94].
However, in the following, we consider the Lagrangian of the form [52] where R is the Ricci scalar and X ≡ −(1/2)g μν ∂ μ φ∂ ν φ, with φ being the inflaton. Below, we briefly discuss the power, bi-, and trispectra with this Lagrangian. For a discussion of non-Gaussianity in this type of general single-field inflation model, see Refs. [69,[95][96][97][98][99][100][101]. For general multi-field inflation, see Refs. [102,103]. With this Lagrangian, the power spectrum is given by where H is the Hubble parameter and c s is the sound speed, which is Here we define the slow-roll parameters, including those other than : The bispectrum can be written as [95] where we have explicitly shown only the leading terms in the slow-roll parameters , η, and s. A o, ,η,s are the terms with , η, and s whose explicit forms can be found in Ref. [95]. In the above expression, To describe the amplitude of the bispectrum, one can generalize the definition of the non-linearity parameter f NL as for expression (40), with only the leading-order terms, as [92] The trispectrum can also be computed as [69] T where T c1,c2,c3 are the trispectra from the contact-interaction diagrams and T s1,s2,s3 are those from the scalar-exchange diagrams (for explicit forms of these trispectra, see Ref. [69]). As discussed in the previous section, T c1 is adopted as a template to define the non-linearity parameter for the equilateral trispectrum since it has a diagonal-free form.
Up to now, we have given general formulas for the model described with the Lagrangian (36). Below we discuss some concrete models of single-field inflation.

Slow-roll single-field inflation with a canonical kinetic term.
For a standard slow-roll single-field inflation with a canonical kinetic term, one has P = X − V (φ). The bispectrum for this model has been calculated as [104] where η φ = M 2 pl V φφ /V is the slow-roll parameter defined with the inflaton potential V . The above expression can be derived from A and A η in Eq. (40). We note that η φ here and η H defined in Eq. (39) are different and, for the case of P = X − V (φ), they are related as η H = −2η φ + 4 . In the squeezed limit where k 3 → 0, the above bispectrum reduces to By taking the same limit in B (local) ζ given in Eq. (7), and comparing it with the above expression, we obtain the consistency relation between f and n s as [104] f (local) NL In fact, this relation has been shown to hold for any single-field inflation model regardless of its potential and its kinetic term [105], as long as a Bunch-Davies vacuum and a negligible decaying mode are assumed. Counterexamples of the violation of this consistency relation have been discussed in Refs. [106][107][108][109], where a nonvacuum initial state is considered or the decaying mode is non-negligible. Since current constraints on the spectral index indicate that 1 − n s = O(0.01), single-field inflation models are consistent with the Planck constraint shown in Eq. (17). However, if f

k-inflation. k-inflation is a model where the inflationary expansion is driven by the (nonstandard) kinetic term. A power law k-inflation model is described with
where K (φ) is a function of φ. When one considers the case where the scale factor is given as a(t) ∝ t 2/3γ with γ being a constant, the form of K (φ) becomes Then the sound speed c s and the other parameters such as , λ, and μ are written as where we have used X = (2 − γ )/(4 − 3γ ) in the second equality in c 2 s . This equation can be obtained from the equation of motion. By using Eq. (43), one can find the equilateral f NL for c s 1 as where Eq. (51) is used in the second equality with c s 1. From the Planck constraint given in (17), one can derive the bound on γ . By adopting the prior 0 < γ < 2/3, the bound γ ≥ 0.05 (95% CL) has been obtained [13]. In fact, this constraint is in conflict with the one derived from the constraint on the spectral index n s − 1 = −3γ , which gives the limit as 0.01 ≤ γ ≤ 0.02 [12]. These arguments are a severe test of this model.
The trispectrum in the k-inflation model is given as [69] T where c 2 s 1 is assumed. Thus, the contribution from T s3 is dominant compared to the other ones.
For the DBI inflation model, the form of P(X, φ) is Then we have For the bispectrum, one can find that the term with (1/c 2 s − 1 − 2λ/ ) vanishes in Eq. (40) and the equilateral non-linearity parameter f (equil) NL for the limit of c s 1 is In fact, the shape of the bispectrum in DBI inflation is not exactly the equilateral one given in Eq. (8) 6 . However, the shape correlation between the DBI inflation and the equilateral ones is quite high [118]. For the shape correlation between various models, see Ref. [118]. Nevertheless, an analysis using the bispectrum shape of the DBI inflation has also been done in Ref. [13], in which the non-linearity parameter for DBI inflation is found to be This limit can be translated into the one for the sound speed in the DBI case as c DBI s > 0.07 (95% CL) [13].
For the trispectrum, we obtain, for the limit c 2 Thus the contributions from T c2,c3 are small compared to the other ones.

Multi-field (local-type) models
When the curvature perturbations are generated on superhorizon scales, the shapes of non-Gaussianities become the local type. To describe the curvature perturbation of this type, the δ N formalism [119][120][121][122][123] is useful and is usually adopted. Thus, before looking at some concrete models of the local type, first we briefly summarize the formulas in this formalism.

δ N formalism.
In the δ N formalism, the superhorizon curvature perturbation ζ on the uniform (total) energy density hypersurface at some final time t f is given by fluctuations in the e-folding number as where andN is calculated for the background expansion. t * is the initial time and is usually taken to be just after the cosmological scale crosses the horizon during inflation. The initial hypersurface is taken to be a flat slice. Then the curvature perturbation ζ can be expressed as where δφ a * is a fluctuation of a scalar field φ a at the time of horizon crossing. Here we label the scalar field by a and N a = d N/dφ a * and so on. Then the power spectrum is given as where H * is the Hubble parameter at the time of horizon crossing and P δφ is the power spectrum of the scalar field fluctuation δφ: PTEP 2014, 06B105 T. Takahashi Non-linearity parameters for the bi-and trispectra are calculated as [124][125][126] With these formulas, we can explicitly write down the expressions for the non-linearity parameters given a concrete model. From Eqs. (66) and (67), one can derive an important relation between f NL and τ NL . With the use of the Cauchy-Schwartz inequality, one finds which is called the Suyama-Yamaguchi inequality [127]. Although this inequality was originally derived at the tree level, some papers have investigated it including loop corrections [42,128]. In particular, even if we include all the loop contributions, the above formula holds [129]. At the tree level, the equality holds only when a single field is responsible for primordial fluctuations. However, when a loop contribution becomes sizable, the equality is not necessarily satisfied even for a singlefield model [129,130]. Nevertheless, as long as the loop contribution is not significant, the above inequality can still be used to probe the multi-field nature of the model. Now, in the following, we discuss some representative models generating local-type non-Gaussianity. For a more exhaustive discussion of local-type models, see Ref. [42].

Curvaton model.
When there exists another light scalar field other than the inflaton during inflation, such a scalar field also acquires quantum fluctuations. If the energy density of such a scalar field is negligible during inflation, its fluctuations act as isocurvature ones. However, they can be converted into adiabatic ones after inflation when the scalar field decays into radiation. Such a scenario is called the curvaton mechanism (the scalar field that is responsible for the fluctuations is called the curvaton) [6][7][8].
In this model, since the curvature perturbation ζ is generated on superhorizon scales, local-type non-Gaussianity is generated, which has been much investigated in the literature [124, T. Takahashi and the non-linearity parameters f NL and g NL are given by [124,133,135] where r dec roughly corresponds to the energy density of the curvaton at the time of its decay and is defined as with ρ σ and ρ r being the energy densities of the curvaton and radiation, respectively. σ osc is the value of σ at the beginning of its oscillation and σ osc = dσ osc /dσ * with σ * being the value of σ at the horizon exit. When we assume a quadratic potential for the curvaton, σ osc and σ osc vanish and the terms with σ osc in Eqs. (71) and (72) become irrelevant. However, when one considers a curvaton model with a selfinteraction [141][142][143][144][145][146][147] and axion (pseudo-Nambu-Goldstone) type [148][149][150][151], the derivatives of σ osc can give a significant contribution, which may drastically affect the prediction for the non-linearity parameters, depending on the model parameters in the potential.
However, for the case with a quadratic potential, f NL and g NL are determined by a single parameter r dec , and, in turn, the Planck constraint on f NL can give a severe bound on r dec 7 as [13] r dec ≥ 0.15 (95% CL).
Thus, the curvaton should almost dominate at its decay to satisfy the constraint. Furthermore, sizable isocurvature fluctuations can also be generated in the curvaton model, depending on when and how CDM or the baryon is created, which also gives a severe constraint on the model [131,[138][139][140]154,155].
• Multi-curvaton model A model with multiple curvatons has also been investigated in Refs. [35][36][37]42]. Even with two curvatons, its predictions for the curvature perturbation generally become very complicated compared to those for a (standard) single curvaton case (for full expressions of ζ up to third order, see Ref. [42]). Unlike the single curvaton case, the value of f NL from the multi-curvaton case can be large as f NL O(1) or of order unity even when both curvatons are dominant or subdominant at 7 In addition to the constraint from f NL , one should also consider those from the spectral index n s , which is also now severely constrained. For spectator field models including the curvaton model, the modulated reheating scenario, and so on, the spectral index is given by [121,152,153] where is defined in Eq. (39) and with U (σ ) being the potential for the spectator field. To be consistent with the observed value (n s 0.96), one needs to assume (i) a large field inflation model (large ) or (ii) a large negative mass squared for the spectator field. For further discussion, see Ref. [152].

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Downloaded from https://academic.oup.com/ptep/article-abstract/2014/6/06B105/1559723 by guest on 28 July 2018 PTEP 2014, 06B105 T. Takahashi their decays. Assuming a simple quadratic potential for the curvatons, the non-linearity parameters can be given by the ratio of their energy densities to the total one at the first and second curvaton decays, which are the counterparts of the r dec parameter in the single curvaton case, as well as by the initial amplitudes for the curvatons, which can in turn be constrained by the Planck f NL limit. Interestingly, it has been argued in Ref. [37] that the curvature perturbation ζ can be temporarily enhanced if multiple curvatons dominate the Universe at different epochs. Furthermore, if both curvatons contribute to the final curvature perturbation, it has also been discussed that non-Gaussian signature may come from the trispectrum rather than from the bispectrum. The gravitational waves generated by the temporal enhancement of ζ from the second-order perturbation have also been investigated in Ref. [37].

Modulated reheating scenario.
When the decay rate of the inflaton φ depends on a scalar field σ and if σ is light enough to acquire quantum fluctuations during inflation, the timing of the decay varies from place to place, which generates fluctuations in the reheating temperature and hence the curvature perturbation ζ . This kind of model is called modulated reheating or inhomogeneous reheating [9,10] and a field σ responsible for density fluctuations in this model is called the "modulus" or "modulaton." A sizable local-type non-Gaussianity can be generated in this model and has been studied in Refs. [27,127,156,157].
By adopting the δ N formalism, one can write down the curvature perturbation in this model as [127] where is the decay rate of the inflaton and σ = d /dσ * . Here we assume that the inflaton oscillates under a quadratic potential (hence its energy density decreases as ρ φ ∝ a −3 ) 8 and the decay rate of the inflaton is much smaller than the Hubble parameter at the end of inflation. We note that the second assumption does not necessarily hold [158]. We also assume that the scalar field σ does not move significantly during the reheating stage and its dynamics hardly affects the density perturbations. However, in some cases, the dynamics of σ does affect the prediction of ζ ; such cases are discussed in Ref. [157]. From Eq. (77), one can easily derive the non-linearity parameters as From these equations, one finds that the predictions for f NL and g NL strongly depend on the functional form of . To discuss this in a quantitative way, here we consider the following form for : PTEP 2014, 06B105 T. Takahashi where we Taylor-expand in σ/M and truncate at the second order with M corresponding to some energy scale. The coefficients α and β are the parameters of order unity, which depend on some concrete models. Then the non-linearity parameters are given as Then, the Planck constraint on f NL can be translated into that for the combination of α and β as −0.9 < β/α 2 < 1.4 (95% CL) 9 [68]. In this model, the constraint from f NL restricts the functional form of and the parameters in .
Similarly to the case with the curvaton, isocurvature fluctuations can also arise in some cases [140,159,160], which can also give a severe constraint on the model. Furthermore, we note that the modulaton can be produced from inflaton decay because the modulaton has to couple to the inflaton in this model, which may give a dark radiation component [161].
In the above, we have assumed that the decay rate of the inflaton depends on another scalar field. However, some variants of this model have also been discussed in the literature. We briefly describe some of them below.
• Modulated decay of the curvaton It is also possible that the decay rate of the curvaton σ depends on another scalar field χ . Then the decay rate of the curvaton fluctuates to generate the curvature perturbation [45][46][47]. The non-linearity parameters in this model can be written as [45] where r dec is the ratio defined for the curvaton energy density given in Eq. (73) and χ = d /dχ * and so on. As seen from the expressions above, this model shares some similarities with the curvaton and modulated reheating models.
• Modulated curvaton A modulaton in the modulated reheating model can also act as a curvaton after it generates a curvature perturbation via the modulated reheating mechanism. This kind of model is called the "modulated curvaton" [42,43]. In this model, a single scalar field acts as both the curvaton and modulaton; thus, this is different from the "modulated decay of the curvaton" discussed above, where the modulaton is different from the curvaton. Thus the curvature perturbation in this model is obtained from the sum 9 Here we have adopted the 95% CL constraint from Planck: −8. T. Takahashi of ζ curvaton and ζ modulated given in Eqs. (70) and (77), respectively, as from which one can calculate f NL and g NL .

Inhomogeneous end of hybrid inflation.
The curvature perturbation can also be generated at the end of inflation if its end is controlled by some light scalar field that fluctuates. This kind of situation can arise in a hybrid inflation model. Here let us consider the potential for hybrid inflation of the form [18]: where φ and χ are an inflaton and a waterfall field, respectively. σ is a light field that acquires fluctuations and couples to χ . m φ and m σ are the masses for φ and σ . λ, g, and f are coupling constants and v corresponds to the vacuum expectation value. The effective mass squared of the waterfall field χ is written as and when m 2 χ becomes negative, inflation ends due to the tachyonic instability, which occurs when φ reaches a critical value: Here it should be noticed that the critical value depends on a fluctuating scalar field σ , thus φ cr also fluctuates, which generates the adiabatic fluctuations [16][17][18][19].
The non-linearity parameters in this model can be calculated as [18,42] where | cr and η φ | cr are the slow-roll parameters evaluated at the time when φ reaches φ cr and a prime represents the derivative with respect to σ . In the above expressions, we have neglected slowroll suppressed terms. The Planck constraint on f (local) NL can be translated into that for a particular combination of parameters as −10.7 < η cr v 2 /( f 2 σ 2 ) < 17.2 (95% CL) [68].

Mixed inflaton and spectator model.
Even if we consider models with the curvaton, modulated reheating, and so on, where a spectator field is assumed to be responsible for the generation of the curvature perturbation, we still need the inflaton to drive the inflationary expansion. In general, the inflaton would also acquire quantum fluctuations during inflation and it can contribute to the curvature perturbation ζ as well. In such a case, multiple sources can be responsible for the final ζ . This T. Takahashi kind of model is called the mixed inflaton and spectator field model [165]; some explicit examples are discussed in the literature, such as the mixed inflaton and curvaton model [21][22][23][24][25], the mixed inflaton and modulated reheating model [27], and so on.
Here we briefly discuss a general mixed inflaton and spectator field model [165], where fluctuations from the inflaton φ and the spectator field σ such as the curvaton and the modulaton both contribute to the curvature perturbation. By using the formulas given in Eqs. (66), (67), and (68), the non-linearity parameters in this model are given by 10 where we have defined the ratio of the power spectra from the inflaton P (φ) ζ and the spectator P In addition, f NL ) corresponds to f NL for the case when only the inflaton (the spectator field) is responsible for the curvature perturbation. In general, non-Gaussianities from the inflaton are small compared to those from the spectator field, and thus, in practice, we can neglect the contributions from the inflaton f (φ) NL and g (φ) NL in the above formulas. In such a case, from Eqs. (90) and (91), one finds that τ NL and f NL are related as Thus, even if f NL is small, τ NL can be large when the fractional contribution of the spectator field to the power spectrum is small.
Here we also remark that, in the mixed model, the non-linearity parameter for the spectator sector f (σ ) NL can be large by having a small value of R to satisfy the Planck constraint on f NL . However, τ NL becomes large instead in this case, which can be seen from Eq. (94). This shows that the trispectrum would still be useful to test this kind of scenario 11 . 10 Here we neglect the contribution from cross terms like N φσ , which can be justified when we consider a quadratic potential for the curvaton and a slow-rolling inflaton [166]. 11 Constraints on the spectral index and the tensor-to-scalar ratio are also important to test the model. In the mixed model, they are given by The implications of the constraints on n s and r from the Planck results for the mixed inflaton and spectator field model are discussed in Ref. [165].

Ungaussiton model.
When there are two scalar fields that both contribute to the curvature perturbation, one can consider a situation where one field only contributes to the linear order, but the other only gives the second-order contribution, in which ζ can be written as where φ can be regarded as the inflaton and σ is some light scalar field. This kind of model has been discussed in Refs. [28,29] and dubbed the "ungaussiton" model [29]. In this model, the non-linearity parameters are dominated by the so-called loop contributions and can be written as [29] where L corresponds to the box size in which the Fourier modes are taken and ln(k L) is usually assumed to be O(1). From the above expressions, one can derive the relation between f NL and τ NL as where we set log(k L) to be unity and P ζ ∼ 10 −9 is used in the last equality. Hence, in this model, τ NL can be large even when f NL ∼ O(1).

Scale-dependence of non-Gaussianity
Usually, the non-linearity parameters such as f NL , τ NL , and g NL are assumed to be constant. However, they can be (strongly) scale-dependent in some cases and may serve as a useful probe of models of primordial fluctuations. Furthermore, when multiple observations are made on different scales, and if those measure different values for the non-linearity parameters, it may be explained by the scale-dependence of non-Gaussianity, which we discuss in this section. A general discussion on the scale-dependence of non-linearity parameters is given in Refs. [168][169][170].
The scale-dependence of f NL is defined in the same manner as the spectral index for the power spectrum as Explicit examples of models producing strong scale-dependence have been discussed in the literature, for the local type, such as a self-interacting curvaton [166,171,172], a pseudo-Nambu-Goldstone curvaton [166,173], the mixed inflaton and curvaton model [166,171], the multi-curvaton model [166,171], a model with a DBI isocurvature field [174], the axion N-flation model [175], and, for the equilateral type, the DBI inflation model [110].
Here we focus on the scale-dependence of the local-type f NL to discuss some explicit models. For the local type, f NL can be strongly scale-dependent when: (i) multiple sources simultaneously contribute to density perturbation, or (ii) the third derivative of the potential of a scalar field is nonvanishing. Examples of the former case (i) include the mixed inflaton and curvaton and the multicurvaton models. First, let us consider the mixed inflaton and curvaton model. When the inflaton PTEP 2014, 06B105 T. Takahashi φ and the curvaton σ are simultaneously responsible for the density perturbation, we can explicitly write down the expression for n f NL as [166,168] where R is the ratio defined in Eq. (93) and is the slow-roll parameter defined with the inflaton potential V (φ). Here we have assumed a quadratic potential for the curvaton. In fact, this formula can be applied to general inflaton and spectator field models. A concrete analysis of a mixed inflaton and curvaton model has been done in Ref. [166] and it has been shown that n f NL can be sizable for some parameter ranges.
When the scalar fields responsible for density fluctuations are both subdominant during inflation (i.e., act as isocurvature fields), such as in the case of the multi-curvaton model, n f NL is given by where "a" and "b" denote curvaton fields and K a and K b are defined as η a and η b are the slow-roll η φ parameters defined for the curvatons "a" and "b". From the above formula, one can see that, when η a = η b , the scale-dependence of f NL can be large. Next we consider case (ii). When one assumes a potential for the curvaton that deviates from a quadratic form, the scale-dependence of f NL appears and n f NL is given by Thus a self-interacting curvaton and pseudo-Nambu-Goldstone curvaton can produce (strongly) scale-dependent f NL and some explicit analysis of this case has been done in Refs. [166,171,172]. In Refs. [176][177][178], the expected constraints on n f NL have been investigated for CMB and several LSS surveys. Future observations of CMB μ-distortion may also be able to probe the scale-dependence [179]. The current limit on n f NL from the WMAP 7-year data is [180] We should note here that the observability of n f NL depends on the value of f NL . From the Planck results, f (local) NL on CMB scales is now tightly constrained. However, f NL can also be probed with observations of small scales such as void and halo abundances [181]. If f NL is strongly scale-dependent, such observations might see a signature of non-Gaussianity even if one does not see any on large scales. T. Takahashi already put severe constraints on the size of isocurvature fluctuations, some fractional contribution is still allowed. Furthermore, even when the isocurvature contribution is small at linear level, it does not necessarily mean that non-Gaussianity in isocurvature fluctuations is also small compared to the adiabatic one. Even if we do not find any non-Gaussian signatures from an isocurvature mode, the constraints from non-Gaussianity may give information different from that of the power spectrum. Therefore, isocurvature non-Gaussianity would be worth investigating in light of the above considerations.

Non-
First, we briefly describe the formalism to discuss isocurvature non-Gaussianity. The non-linear isocurvature perturbations are defined for a component i, with respect to radiation, as where ζ i is the curvature perturbation on the uniform energy density hypersurface of a component i and is given as Here w i is an equation of state for a component i and is assumed to be constant.ρ i (t) is the background energy density of the component i. Then, in the same manner as the curvature perturbation for the local type, we can define the non-linearity parameter for isocurvature fluctuations f (iso) NL as where S ig is the linear part for S i . Since, in general, adiabatic and isocurvature fluctuations coexist, a 3-point correlation function can be written as where X I = ζ or S i and the bispectrum B I J K (k 1 , k 2 , k 3 ) is given by . P I J (k) is the power spectrum defined as For the case with X I = X J = ζ , the definition of the power spectrum matches that of Eq. (1). When isocurvature fluctuations exist, we can also define the power spectra for the isocurvature mode and its cross correlation with the adiabatic one as P S (≡ P S i S i ) and P ζ S i , which correspond to S i S i and ζ S i , respectively. The correlated part P ζ S i may vanish, depending on the model. To parametrize the correlation between the adiabatic and isocurvature modes, it is conventional to define the correlation angle as follows: When cos = +1 and −1, we denote them as positively and negatively correlated modes, respectively. The case of cos = 0 corresponds to the uncorrelated mode. 21 As mentioned above, current cosmological observations already give a severe constraint on the size of the isocurvature power spectrum. To investigate the constraint, it is customary to use the ratio of the power spectrum of the isocurvature mode to the adiabatic one or to the total one, which are defined as where k ref is the scale at which the ratio is evaluated. In the literature, either one is used to characterize the size of the isocurvature contributions. We note that, as far as α, β 1, these ratios are almost equivalent.
The Planck team has reported the constraints on β [12]. Depending on the degree of correlation, the constraints are different. Although, in general, the spectral index for the isocurvature power spectrum n iso − 1 ≡ d log P s /d log k is regarded as a free parameter, for some models, such as the axion (uncorrelated mode) and the curvaton (correlated mode), n iso would be almost scale invariant for the former, and the same as the adiabatic mode for the latter. For these particular cases, the constraints are [12] β 0 (n iso = 1) < 0.036, β +1 (n iso = n s ) < 0.0025, β −1 (n iso = n s ) < 0.0087 (95% CL), where the ratio is defined at k ref = 0.002 Mpc −112 and the subscripts 0, +1, and −1 indicate that the quantities are for uncorrelated, positively (fully) correlated, and negatively (fully) correlated, respectively. Constraints on non-Gaussianities from isocurvature fluctuations have also been investigated using the WMAP data [184][185][186] and also for expected future data [187,188]. In general, there are 6 nonlinearity parameters defined as in Eq. (111) where we have used the ratio α here to represent the isocurvature contribution. Observational constraints from the WMAP data obtained in Ref. [186] are summarized in Tables 1 and 2  12 In Ref. [12], constraints on the ratio β are reported for those at three different scales: k low = 0.002 Mpc −1 , k mid = 0.05 Mpc −1 , and k high = 0.10 Mpc −1 . Here we show the values for k low . We note that, when the spectral index for the isocurvature mode is fixed as stated in the text, the constraints on different scales are almost the same. However, when one varies n iso freely in the analysis, the ratio on small scales (or blue-tilted n iso ) is less constrained by the data, in particular for the case with uncorrelated modes [182,183].

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Downloaded from https://academic.oup.com/ptep/article-abstract/2014/6/06B105/1559723 by guest on 28 July 2018 PTEP 2014, 06B105 T. Takahashi   Table 1. Constraints on f NL and f (iso) NL for uncorrelated isocurvature fluctuations [186]. Results with template marginalization of the Galactic foregrounds are given. The values without parentheses are the constraints obtained by fixing the other non-linearity parameter to zero. For the values with parentheses, they are obtained by marginalizing over the other non-linearity parameter. In the table, constraints on two different spectral indices n iso are given. For details of the analysis, see Ref. [186].  Table 2. Constraints on f NL and f (iso) NL for correlated isocurvature fluctuations [186]. Results with template marginalization of the Galactic foregrounds are given. The values without parentheses are the constraints obtained by fixing the other non-linearity parameter to zero. For the values with parentheses, they are obtained by marginalizing over the other non-linearity parameter. In the table, constraints on two different spectral indices n iso are given. For details of the analysis, see Ref. [186]. NL , which includes the correlation angle.
We do not find any non-Gaussianities from isocurvature fluctuations. However, we can use these bounds to obtain constraints on some explicit models such as the axion and the curvaton.
In addition to the bispectrum, the trispectrum from isocurvature fluctuations has also been investigated [189,190].

Implications for concrete models
As previously mentioned, the amplitude of isocurvature fluctuations has already been severely constrained by cosmological observations. However, some fractional contribution to the total density fluctuations is still allowed. Isocurvature fluctuations arise in association with CDM, baryons, neutrinos, and also for dark radiation. Therefore the study of isocurvature mode may give useful information on these sectors.
Furthermore, in the curvaton model, correlated CDM or baryon isocurvature fluctuations can be generated depending on when and how CDM and the baryon are produced [131,[138][139][140]154,155], and its non-Gaussianity has been studied in Refs. [186,190,[204][205][206]. When there are non-zero neutrino chemical potentials, neutrino density isocurvature fluctuations can be generated in the curvaton model [155,207,208]. Correlated isocurvature fluctuations can also arise in a double inflation model [209,210], the non-Gaussian nature of which has been investigated in Ref. [204]. The issues of isocurvature fluctuations in dark radiation have been investigated in Refs. [211,212].
Constraints on non-Gaussianity in isocurvature fluctuations described in the previous subsection may probe some aspects of models that cannot be obtained from the power spectrum. For the implications of observational constraints given in Tables 1 and 2, we refer the reader to Refs. [185,186].

Conclusion
In this review, we have discussed primordial non-Gaussianities as a probe of the inflationary Universe. Now Planck has provided severe constraints on the non-linearity parameters, especially for f NL , given in Sect. 2. These constraints have many implications for models of primordial fluctuations. The simplest single-field inflation models are consistent with the current observational bounds on non-Gaussianities. However, many other models are also consistent with the current bounds, since f NL ∼ O(1) can be achieved by choosing suitable model parameters. Nevertheless, the current bounds restrict the parameter space of these more complicated models.
Compared to the bispectrum, the trispectrum is currently less constrained by the data. Although the non-linearity parameters in the trispectrum, τ NL and g NL , in most models are of the same order of magnitude as f NL , in some cases the trispectrum parameters can be much larger than f NL (such examples are discussed in Ref. [68]). Furthermore, the non-linearity parameters may be strongly scale-dependent in some models; thus, even if the size of non-Gaussianities is small on CMB scales, they may be large, in particular, on small scales that can be probed with small-scale observations, e.g., void and halo abundances [181]. In addition, as discussed in this review, isocurvature non-Gaussianity may still give constraints on dark matter, baryogenesis, and so on.
In the future, CMB experiments could give constraints on f NL at the level of O(1) (see, e.g., Ref. [213]). Observations of the 21 cm line of neutral hydrogen may potentially be able to reach down to the level of f NL ∼ O(0.01) [214] (see also Ref. [215]). This level of precision can differentiate whether the origin of primordial fluctuations is a standard slow-roll single-field inflation or not, since the single-field inflation predicts f NL ∼ 1 − n s O(0.01), which is a firm prediction because the current observations precisely measure n s as n s − 1 O(0.01). On the other hand, alternative models to inflation generally give f NL ∼ O(1) (unless some fine-tuning is assumed).
In addition to the bispectrum and the trispectrum, which have been well investigated so far, even higher-order statistics such as 5-or 6-point correlation functions might have the potential to give insights into the origin of primordial fluctuations [216][217][218][219]. To conclude, further study of primordial non-Gaussianity would pin down the origin of density fluctuations, which in turn will give us a precise picture of the very early Universe.