This is a pedagogical review on primordial non-Gaussianities from inflation models. We introduce formalisms and techniques that are used to compute such quantities. We review different mechanisms which can generate observable large non-Gaussianities during inflation, and distinctive signatures they leave on the non-Gaussian profiles. They are potentially powerful probes to the dynamics of inflation. We also provide a nontechnical and qualitative summary of the main results and underlying physics.

An ambitious goal of modern cosmology is to understand the origin of our Universe all the way to its very beginning. To what extent this can be achieved largely depends on what type of observational data we are able to get. Thanks to many modern experiments, we are really making progress in this direction.

One of the representative experiments is the Wilkinson Microwave Anisotropy Probe (WMAP) satellite [

There are two amazing facts about the CMB temperature map. On the one hand, it is extremely isotropic, despite the fact that the causally connected region at the time when CMB formed spans an angle of only about

The inflationary scenario [

The inflationary scenario has several generic predictions on the properties of the density perturbations that seed the large scale structures.

They are primordial. Namely, they were laid down at superhorizon scales and entering the horizon after the Big Bang.

They are approximately scale-invariant. This is because, during the 60 e-folds, each mode experiences the similar expansion when they are stretched across the horizon.

They are approximately Gaussian. In simplest slow-roll inflation models, the inflaton is freely propagating in the inflationary background at the leading order. This is also found to be true in most of the other models and for different inflationary mechanisms. So the tiny primordial fluctuations can be treated as nearly Gaussian.

The CMB temperature anisotropy is the ideal data that we can use to test these predictions. The obvious first step is to analyze their two-point correlation functions, that is, the power spectrum. All the above predictions are verified to some extent [

But is this enough?

Experimentally, the amplitude and the scale-dependence of the power spectrum consist of about

Theoretically, inflation still remains as a paradigm. We do not know what kind of fields are responsible for the inflation. We do not know their Lagrangian. We also would like to distinguish inflation from other alternatives. Being our very first data on quantum gravity, we would like to extract the maximum number of information from the CMB map to understand aspects of the quantum gravity. All these motivate us to go beyond the power spectrum.

To give an analogy, in particle physics, two-point correlation functions of fields describe freely propagating particles in Minkowski spacetime. More interesting objects are their higher-order correlations. Measuring these are the goals of particle colliders. Similarly, the power spectrum here describes the freely propagating particles in the inflationary background. To find out more about their interaction details and break the degeneracies among models, we need higher-order correlation functions, namely, non-Gaussianities. So the role non-Gaussianities play for the very early universe is similar to the role colliders play for particle physics.

With these motivations in mind, in this paper, we explore various mechanisms that can generate potentially observable primordial non-Gaussianities, and are consistent with the current results of power spectrum. We will not take the approach of reviewing models one by one. Rather, we divide them into different categories, such that models in each category share the same physical aspect which leaves a unique fingerprint on primordial non-Gaussianities. On the one hand, if any such non-Gaussianity is observed, we know what we have learned concretely in terms of fundamental physics. On the other hand, explicit forms of non-Gaussianities resulted from this exploration provide important clues on how they should be searched in data. Even if the primordial density perturbations were perfectly Gaussian, to test it, we would still go through these analyses until various well-motivated non-Gaussian forms are properly constrained.

The following is the outline of the paper. For readers who would like to get a quick and qualitative understanding of the main results instead of technical details, we also provide a shortcut after the outline.

In Section

In Section

In Section

In Section

Sections

In Section

Here is a road map for readers who wish to have a non-technical explanation and understanding of our main results. After reading the short review on the inflation model and density perturbations in Section

The subject of the primordial non-Gaussianities is a fast-growing one. There exists many nice reviews and books in this and closely related subjects. The introductions to inflation and density perturbations can be found in many textbooks [

In this section, we give a quick review on basic elements of inflation and density perturbations. We consider the simplest slow-roll inflation. The action is

The requirement of having at least

Now let us study the perturbations. To keep things simple but main points illustrated, in this section, we will ignore the perturbations in the gravity sector and only perturb the inflaton,

We now write down the mode function explicitly by solving (

The mode function

If this were the end of story, all the primordial density perturbations would be determined by this two-point function and they are Gaussian. The rest of the paper will be devoted to making the above procedure rigorous and to the calculations of higher-order non-Gaussian correlation functions in this and various other models.

In this section, we review the in-in formalism and the related techniques that are used to calculate the correlation functions in time-dependent background. The main procedure is summarized in the last subsection.

We start with the in-in formalism [

We are interested in the correlation functions of superhorizon primordial perturbations generated during inflation. So our goal is to calculate the expectation value of an operator

We start by looking at how the time-dependence in

The Hamiltonian of the system

We consider a time-dependent background,

Using (

To have a systematic scheme to do the perturbation theory, we split

So the idea is to encode the leading kinematic evolution in terms of the interaction picture fields, and calculate the effects of the interaction through the series expansion in terms of powers of

In summary, the expectation value (

The perturbation theory is also often done in terms of the Lagrangian formalism. In the following, we show that they are equivalent. In the above, we perform perturbations on the Hamiltonian, and define

The Hamiltonian

Being the solutions of the second-order differential equation, generally the mode function is a linear superposition of two independent solutions. So we need to specify the initial condition. For inflation models, as long as the field theory applies, one can always find an early time at which the physical momentum of the mode is much larger than the Hubble parameter and study a time interval much less than a Hubble time. Under these conditions, the equations of motion approach to those in the Minkowski limit, in which the mode function is a linear superposition of two independent plane waves, one with positive frequency and another negative. The ground state in the Minkowski spacetime is the positive one. The mode function which approaches this positive frequency state in the Minkowski limit is called the Bunch-Davies state. In physical coordinates, this limit is proportional to

We also would like to write the vacuum of the interacting theory (

The integrand

When evaluating (

To evaluate the integrand, one can shift around the orders of fields in that product, following the rules of the commutation relations. A

In the following, we demonstrate this using a simple example. We consider a field

An example of Feynman diagram.

In Figure

Note that all terms are contracted. The result can be further evaluated using (

Now we deal with the time ordered integrals in the series expansion. There are two ways to expand (

In the first form, we simply expand the exponential in (

In the second form, we rearrange the factorized form so that all the time variables are time-ordered, and all the integrands are under a unique integral. The

Each representation has its computational advantages and disadvantages.

The factorized form is most convenient to achieve the UV (

The commutator form is most convenient to get the correct leading order behavior in the IR. The mutual cancellation between different terms are made explicit in terms of the nested commutators, before the multiple integral is performed. However, such a regrouping of integrands makes the UV convergence very implicit. Recall that the contour deformation is made to damp the oscillatory behavior in the infinite past. In the commutator form, for any individual term in the integrand, we can still generically choose a unique convergence direction in terms of contour deformation. Although the directions are different for different terms, they can be separately chosen for each of them. But now the problem is, if these different terms have to be grouped as in the nested commutator so that the IR cancellation is explicit, the two directions get mixed. Hence, the explicit IR cancellation is incompatible with the explicit UV convergence in this case.

To take advantage of both forms, we introduce a cutoff

We will not always encounter all these subtleties in every model, but there does exist such interesting examples, as we will see in Section

To end this section, we summarize the procedure that we need to go through to calculate the correlation functions in the in-in formalism.

Starting with the Lagrangian

Work out the Hamiltonian in terms of

Choose appropriate forms in Section

Simplest inflation models generate negligible amount of non-Gaussianities that are well below our current experimental abilities [

single scalar field inflation

with canonical kinetic term

always slow-rolls

in Bunch-Davies vacuum

in Einstein gravity.

This list is extracted based on Maldacena's computation of three-point functions in an explicit slow-roll model [

The Lagrangian for the

The inflaton starts near the top of the potential and slowly rolls down. As we have reviewed in Section

To study the perturbation theory, it is convenient to use the ADM formalism, in which the metric takes the form

We plug (

We restrict to the case where the slow-roll parameters are

If we choose the spatially flat gauge, we make

We quantize the field as

We next look at the cubic action. For single field models,

To estimate the order of magnitude of the bispectrum, we only need to keep track of the factors of

The slow-roll parameters are of order

The following are two examples of slow-roll potentials in the simplest inflation models that we studied in Section

However, when it comes to the more realistic model building in a UV complete setup, such as in supergravity and string theory, situations get much more complicated. For example, it is natural that we encounter multiple light and heavy fields, and the potentials for them form a complex landscape. These multiple fields live in an internal space, whose structure can be very sophisticated. In string theory, some of them manifest themselves as extra dimensions and can have intricate geometry and warping. All these elements have to coexist with the inflationary background that introduces profound back-reactions.

Even with varieties of model building ingredients, it has been proven to be very subtle to construct an explicit and self-consistent inflation model. Indeed various problems have been noticed over the years in the course of the inflation model building. For example, consider the following problems.

As we have seen, in order to have slow-roll inflation [

DBI inflation [

Large field inflation models require the field range to be much larger than

Even in cases where there is no fundamental restriction on the excursion of fields, one encounters problems constructing the large field inflationary potential. Large field potentials that arise from a fundamental theory take the following general from:

None of the arguments in the above list is meant to show that the specific type of inflation is impossible. In fact, these have been the driving forces for the ingenuity and creativity in the field of inflation model building. This list is used to demonstrate some typical examples of complexities in reality. Often times, solving one problem will be companied by other structures that make the model step beyond the simplest one. So we may want to keep an open mind that the algebraic simplicity may not mean the simplicity in Nature.

Following is a partial list of possibilities that allow us to go beyond the no-go theorem in Section

Instead of single field inflation, we can consider quasisingle field or multifield inflation models (Sections

Instead of canonical kinetic terms, there are models where the higher derivative kinetic terms dominate the dynamics (Section

Instead of following the attractor solution such as the slow-roll precisely, features can be present in the potentials or internal space, that temporarily break the attractor solution, or cause small but persistent perturbations on the background evolution (Sections

Instead of staying in the Bunch-Davies vacuum, other excitations can exist due to, for example, boundary conditions or low scales of new physics (Section

Although strong constraints, from experimental results and theoretical consistencies, exist on non-Einstein gravities, early universe may provide an opportunity for their appearance. We use this category to include a variety of possibilities, such as modified gravities, noncommutativity, nonlocality and models beyond field theories.

There are also strong motivations from data analyses for us to search and study different large non-Gaussianities. The signal-to-noise ratio in the CMB data is not large enough for us to detect primordial non-Gaussianities model-independently. A well-established method is to start with a theoretical non-Gaussian ansatz, and construct optimal estimators that compare theory and data by taking into accounts all momenta configurations. This then gives constraints on the parameters characterizing the theoretical ansatz. Therefore, the following two important possibilities exist. First, the primordial non-Gaussianities exist in data could be missed if we did not start with a right theoretical ansatz. Second, even if a non-Gaussian signal was detected with one ansatz, it does not mean that we have found the right one. So different well-motivated non-Gaussian templates are needed for clues on how corresponding data analyses should be formed. From a different perspective, even if the primordial density perturbations were Gaussian, we would still do the similar amount of work and reach the conclusion after various well-motivated non-Gaussian forms are properly constrained.

In this paper, we will be mainly interested in the three-point correlation functions of the scalar primordial perturbation

The three-point function is a function of three momenta,

Under different circumstances, different properties of

The dependence of

One is called the

Momentum configurations: (a) equilateral, (b) squeezed, and (c) folded.

Another is called the

For bispectra that are approximately scale invariant, the shape is a more important property [

There are also cases where the running becomes the most important property, while the shape is relatively less important [

The above dissection will become less clean for cases where both properties become important.

One thing is clear. The

It is useful to quantify the correlations between different non-Gaussian profiles, because as we mentioned in data analyses an ansatz can pick up signals that are not completely orthogonal to it. In real data analyses this is performed in the CMB

In typical data analyses [

In this section, we relax several restrictions of the no-go theorem on single field inflation models and study how large non-Gaussianities can arise. We present the formalisms and compute the three-point functions. We emphasize how different physical processes during inflation are imprinted as distinctive signatures in non-Gaussianities. Obviously, any mechanism that works for single field inflation can be generalized to multifield inflation models.

In this subsection, we study large non-Gaussianities generated by noncanonical kinetic terms in general single field inflation models, following [

Consider the following action for the general single field inflation [

It is a nontrivial question which forms of

Following the same procedure that is outlined in Section

To calculate the bispectrum, we look at the cubic action. In the following, we list three terms that are most interesting for this subsection,

The order of magnitude contribution from these three terms can be estimated similarly as we did in (

The full results we obtained can be used in different regimes.

If we look at the limit,

If we take the opposite, slow-roll limit,

We can also look at the intermediate parameter space. In slow-roll inflation models, one can also add higher derivative terms [

The other terms that we did not list in (

In the rest of this subsection, we focus on the first case.

In Figures

Shape of

Shape of

An ansatz (

The shape of

The scale dependence in

The underlying physics of the equilateral shape can be readily understood in terms of their generation mechanism. In single field inflation, the long wavelength mode that exits the horizon are frozen and can have little interaction with modes within the horizon. The large interaction only occurs among modes that are crossing the horizon at about the same time. These modes then have similar wavelengths. This is why the shape of the non-Gaussianity peaks at the equilateral limit in momentum space.

This physical origin also suggests the caveat that, as long as there are large interactions involving modes with similar wavelengths, an equilateral-like shape may arise. For example, such cases can happen in multifield models where there are particle creation [

An explicit example of the above general results is the DBI inflation [

The physical consequence is now easy to obtain using the general results in this subsection. In our notation the Lagrangian is

DBI inflation is still driven by the potential energy. The general single field inflation models also include the k-inflation [

Multifield generalization have been studied in [

The current CMB constraint on the equilateral ansatz (

Although various slow-variation parameters in (

As an example, we study a sharp feature in the slow-roll potential. The fact that a sharp feature in potential can enhance non-Gaussianities has long been anticipated and qualitative estimates have been made by different methods [

We start by studying the behavior of the slow-roll parameters. We use a small step in potential as an example and will ignore numerical coefficients. We use

So

With these qualitative behavior in mind, we now study the three-point function. An important fact of the formalisms in Sections

In all terms in the cubic expansion (

For long wavelength modes that already crossed the horizon at the time of the sharp feature,

Behavior of the slow-roll parameters for a step (solid line,

This ansatz describes the most important running behavior of this bispectrum. Notice that the oscillatory frequency in the

In practice, (

Numerical result (dashed line) for the bispectrum running for a sharp step (

A numerical result with

Sharp features can also appear elsewhere instead of potentials, for example, in the internal warped space for DBI inflation [

Non-attractor initial conditions can be included as a case of sharp features, except that we only observe the relaxation part.

In this subsection, we consider a different type of features. These features may or may not be sharp, but the most important property is their periodicity. Such features will induce an oscillatory component to the background evolution, in particular, to the couplings in the interaction terms. We denote this oscillatory frequency as

We now study the properties of such a non-Gaussianity, following reference [

To estimate the integral, we use the unperturbed mode function. Similar to the sharp feature case, we get

First, let us look at its oscillatory running in

Next, let us look at the size of the non-Gaussianity. Each

Summarizing both the running behavior and the amplitude, we get the following ansatz for the bispectrum:

As mentioned, we have derived this ansatz from the last term in (

The running and shape of the resonance bispectrum (

More arbitrary scale-dependence can be introduced if the features are applied over a finite range, or with varying periodicity and amplitude.

As a useful comparison, the resonant running here and sinusoidal running that we studied in the last subsection are clearly distinguishable from each other observationally. The resonant running oscillates with periods that are always much smaller than the local scale,

As an illustration, we look at an example,

A numerical example is shown in Figure

Numerical result (solid line) of the bispectrum running for the example (

In this subsection, we study the effect of nonstandard vacuum on the primordial non-Gaussianities. We consider a different wave-function from the Bunch-Davies vacuum when modes are well within the horizon. To start, let us first discuss several motivations for this case.

A non-Bunch-Davies vacuum can actually occur much more simply than it might sound like. Any deviation from the attractor solution of the inflaton generically generates a component of non-Bunch-Davies vacuum. This is because a general mode function is a superposition of two components,

In inflationary background, modes can be quantized in terms of time-dependent creation and annihilation operators,

There are inflation models where the scale of new physics can be very low. In particular, in warped space it is proportional to the exponentially small warp factor. In some DBI inflation models [

After these discussions, let us now focus on a specific simple problem [

So the mode function is

The shape of

Shape of

The case for slow-roll inflation is qualitatively similar, and more examples of the bispectra shapes and the observational prospects are discussed in [

In order to facilitate the data analyses, a simple ansatz has been proposed in [

Two ansatz for the folded shape. (a) Equation (

Another type of non-Bunch-Davies vacuum, namely, an

Having considered single field inflation, we now relax the condition on the number of fields. At least during inflation, we only need to consider quantum fluctuations of light fields, since if the mass of fields are very heavy, (here the relevant scale is

Therefore, as a natural step beyond the single field, let us consider slow-roll models with one inflationary direction, and one or more other directions that have mass neither much heavier nor much lighter than

Note that the thematic order in this paper is not chronological. The non-Gaussianities in this type of models were not computed until very recently for a couple of reasons. If the mass of particles is of order

There are potentially different ways massive isocurvatons can be coupled to the inflaton. We currently do not have a general approach in terms of model building. So what we will do is to first study this problem through a simple example, and then discuss the features of the results that can be regarded as generic signatures of this class of models [

We consider the case where the inflaton is turning constantly by going around (a fraction of) a circle with radius

Quasi-single field inflation with turning trajectory. The field

To study the perturbation theory, we perturb the fields in the spatially flat gauge,

There are several important points for this Hamiltonian.

First, the kinematic Hamiltonian (

Second, there is a sharp contrast between the

Third, the coupling between the isocurvaton and inflaton appears as a form of a two-point vertex operator in (

Feynman diagrams for the transfer vertex (a), corrections to the power spectrum from isocurvature modes (b), and the leading bispectrum (c).

We calculate correlation functions corresponding to the Feynman diagrams Figure

In the IR (

In the UV (

To take advantage of both forms, we introduce a cutoff

Numerical results for the shapes of bispectra with intermediate forms. We plot

To better understand the shapes analytically, we can work out the squeezed limit (

Recall that the squeezed limit of

The physical origin of such shapes can be understood as follows, and should be a generic signature for the quasisingle field inflation models. As we have seen, the large equilateral non-Gaussianity arises because the interacting modes cross the horizon around the same time. The shape of bispectrum peaks at the equilateral limit where the modes all have comparable wavelengths. As we will see in Section

In this massless limit, an infrared cutoff to the integrals are necessary. Otherwise the transfer will last forever for the constant turn case. The cutoff corresponds to the ending of the turning. Let us discuss the following two cases. First, we still keep

To connect with data analyses, guided by the numerical results and analytical squeezed limit, we can use the following ansatz to describe the full family of shapes:

Shape ansatz (

The fluctuations of more massive (

As we have seen in Section

We recall that, in single field inflation, if we use the uniform inflaton gauge where there are no fluctuations in the inflaton field, the scalar perturbation

We would like to generalize this picture to the multifield case in the following

We consider a set of scalars

We pick an initial spatially flat slice, on which there is no scalar fluctuations in the metric and all the fluctuations are in the scalar fields

We pick the final uniform density slices. Relative to the unperturbed and perturbed initial spatially flat slices, we have, respectively, the unperturbed and perturbed final uniform density slices. For single field inflation, these two final surfaces are the same. For multifield models, they are generally different. Such final slices have the properties that the universe has the same energy densities and field configurations everywhere on them. They can be chosen during either the inflation or the reheating. After that, every separated universe will have the same evolution. The only difference is the scale factor. This is the analogy of the uniform inflaton gauge in single field inflation. We study the cases where such slices exist.

We evolve the unperturbed

We expand

We have assumed that the statistics of the

Most generally, one identifies

So far we have not used the condition that the isocurvatons are massless (

Now let us consider the Gaussian fluctuations

Shape of the local form (

The physics of this shape can be understood from the derivation above. As explicitly demonstrated in (

If the perturbation

We use the curvaton model [

In this model, we assume that during inflation there is another light field

Another assumption of the curvaton model is that the primordial fluctuations in the inflaton field is much smaller than what is needed to achieve

At the initial spatially flat slice

The large local form has been studied most extensively in the past. Variety of possibilities exist. They all share the common feature that non-Gaussianities are generated by some massless isocurvaton fields which acquire the superhorizon evolution during the inflation. For example, in multifield slow-roll inflation a turning trajectory [

Local form is also found in different contexts, such as models with special types of massive gauge fields that acquire superhorizon evolution [

The current CMB constraint on the local bispectrum is

In this subsection, we summarize the main results of Sections

In single field inflation, the long wavelength modes that already exited the horizon are frozen. They cannot have large interactions with short wavelength modes that are still within the horizon. When modes are well within the horizon, they oscillate and the contributions to non-Gaussianities average out. Therefore the only chance to have large interaction is when all modes have similar wavelengths and exit the horizon at about the same time. Theories with higher derivative kinetic terms provide such interaction terms. This is why the resulting bispectrum shape peaks at the equilateral limit in the momentum space. It drops to zero at the squeezed limit

A sharp feature, in a potential or internal field space, introduces a sharp change in the slow-roll parameters, or the generalized slow-variation parameters. This can boost the magnitudes of time-derivatives of some parameters by several orders of magnitude while still keep the power spectrum viable. These time-derivatives act as couplings in the interaction terms. So they enhance the non-Gaussianities among the modes which are near the horizon-exit. How deep they affect the modes inside the horizon depends on how sharp the changes are.

The changes in these parameters can be roughly approximated as delta-functions in time. Correlation functions involve integrations of products of the slow-variation parameters and the mode functions. The latter contain oscillatory behavior

The periodic features do not have to be sharp. They introduce a small background oscillatory component in the slow-variation parameters. On the other hand, the mode functions are also oscillatory before they exit the horizon. Their frequencies are high when they lie deep inside the horizon and become lower as their wavelengths get stretched by the inflation. They are frozen after the wavelengths become comparable with the horizon size

The periodicity of the features leads to a periodic-scale-invariance in density perturbations. Namely, they are scale invariant if we rescale all momenta by a discrete e-fold

The usual mode function of the Bunch-Davies vacuum has the positive energy mode

All mechanisms discussed so for single field inflation apply to multifield inflation. We now consider new effects caused by introducing more fields to inflation models. These extra fields are called isocurvatons.

Since light fields typically acquire a mass of order

Unlike multifield slow-roll inflation, where each flat direction only has small nonlinear terms in order to satisfy the slow-roll conditions, massive directions are not inflationary direction and are free to have large nonlinear self-interactions. These nonlinear interactions can be transferred to the curvature mode through couplings and source the large non-Gaussianity.

The massive isocurvaton eventually decays after horizon exit simply because they are diluted by the expansion. How fast it decays depends on its mass. If the mass is heavier,

The numerical results of these shapes are presented in Figure

The fluctuation amplitudes of massless scalars do not decay after the horizon exit, and therefore this opens up a multifield space for the superhorizon evolution. For superhorizon modes, we can use the separate universe picture and study the classical behavior of different patches of universe. These patches are separated by horizons and evolve independently of each other. So the evolution is local in space.

Non-Gaussianities are generated when this multifield evolution is nonlinear, and any nonlinearity arising in the separate universe picture should also be local in space. A locality in position space translates to a nonlocality in momentum space. This is why the resulted local shape bispectrum peaks at the squeezed limit. The behavior is

In all cases, the power spectra are either approximately scale-invariant so indistinguishable from the simplest slow-roll models, or modified with features that can be made small enough to satisfy the current observational constraints.

Large bispectra generically implies large trispectra, that is, the four-point correlation functions. But trispectra contain more information and can be large even if bispectra are small. Experimentally, trispectra are more difficult to detect, but contain much more shape configurations. Each category above should have interesting extensions to trispectra. See [

It is certain that this list will grow in future works, providing more refined and diverse connections between theories and experiments.

As we have seen, in single field inflation, the mode that has exited the horizon is frozen. This is characterized by a constant

In the squeezed limit

To connect the averages we used here with the correlation functions that we defined in previous sections, we need to restore the phase factors. Here the two-point average

Although originally derived for slow-roll inflation, the only assumption is the single field. So this applies to any single field inflation models and has important physical implications that we discuss shortly [

There are three types of interesting corrections to the condition (

Firstly, as mentioned, the right-hand side of (

Secondly, when we assume that the only effect of the frozen superhorizon mode on the much shorter scale is a constant background rescaling, we assume that there is no interaction when these modes are all within the horizon (I would like to thank Yi Wang for helpful discussions on this point). However, large subhorizon interaction is possible in some cases, such as in Sections

Thirdly, even after the long wavelength mode exits the horizon, as long as

The consistency condition (

Besides providing consistency checks for analytical computations, the condition also has interesting physical implications. In the following, we discuss the scale invariant cases [

This consistency relation implies that the tree-level bispectrum in the squeezed limit is determined by the power spectrum and spectral index. We distinguish the following two cases. For the scale-invariant case,

For the loop diagrams, in the scale-invariant case, these terms are suppressed by higher-orders of slow-variation parameters from, for example,

In summary, a detection of an approximately scale-invariant local non-Gaussianity in the infinitely squeezed triangle limit with

In experiments, however, the triangle cannot be perfectly squeezed. So it is an important question how squeezed it should be to achieve the above goal. For example, in the third type of corrections we discussed previously in this subsection, we need

Different shapes and runnings of non-Gaussianities can be superimposed in inflation models. For example, consider the following.

It is possible that different non-Gaussianity generation mechanisms are from different components in a model, or at different stages during inflation. So two or more different shapes can get mixed, and the final shape can be rather different. For example, in Figure

A mixing of the equilateral (Figure

The shapes can also be mixed with runnings. Same as the power spectrum, the non-Gaussianities generically have some mild scale dependence. But a more dramatic case is the superposition with a strong running, such as the sinusoidal or resonant running. For example, an inflaton passing through features frequently and turning constantly at the same time on a potential landscape can generate a bispectrum which is a superposition of the resonant running and intermediate shape, as we illustrate in Figure

A mixing of an intermediate shape [

If a non-Gaussianity is the linear superposition of several base components, one can generally perform a change of bases to make the new bases orthogonalized. For example, as we have seen in Section

Orthogonalization of two shapes in Section

An ansatz

For known examples of general single field inflation, such as the DBI and k-inflation, we generically get equilateral shapes. This is also clear from their physical origin that we have emphasized. The orthogonal shape relies on a delicate cancellation between the two generic shapes. In principle, one can do this since the required parameter space is allowed in our effective field theory of general single field inflation in Section

Let us do a more data-analysis-oriented exercise. We would like to construct an ansatz that is orthogonal to both local and equilateral ansatz, since both were well constrained by data. (Note that

Another factorizable orthogonal ansatz (

One can perform a similar orthogonalization for the two shapes in (

The field of primordial non-Gaussianity is growing rapidly in recent years, with simultaneous progress from the experimental results, data analyses methods, nonlinear cosmology theories, physical model buildings, computational techniques, and theoretical formalisms. The progress that we have seen so far is no doubt just a beginning.

In this paper, we have studied the primordial non-Gaussianities coming from the inflation models, especially various mechanisms that can produce observable large non-Gaussianities with viable power spectra. We emphasized the fingerprints that different underlying physics leave on non-Gaussian profiles, which break the degeneracy of model building. We described the physical pictures and presented their effective Lagrangians to the extent that they can be recognized when encountered in the inflation model building in a more fundamental theory. We also derived the resulting bispectra and represented them in terms of simple ansatz to the extent that they can be useful to data analyses. With the current rapid progress, we anticipate much more future developments along these lines through refinements and discoveries in both theories and experiments.

The standard model of cosmology—the Big Bang theory with

The author would like to thank Rachel Bean, Richard Easther, Girma Hailu, Bin Hu, Min-xin Huang, Shamit Kachru, Eugene Lim, Hiranya Peiris, Sash Sarangi, Gary Shiu, Henry Tye, Yi Wang, and Jiajun Xu for valuable collaborations and sharing their insights on the works reviewed here. The author would also like to thank James Fergusson, Michele Liguori, David Lyth, David Seery, Paul Shellard, and David Wands for very helpful discussions. The author was supported by the Stephen Hawking advanced fellowship.