1. Motivation

In the last few decades the advances in observational cosmology have led the field to its “golden age.” Cosmologists are beginning to nail down the basic cosmological parameters. We now know that we live in a Universe which is  Gyr old and is spatially flat to about and is made of baryons, dark matter, and remaining in the form of dark energy. Although we know the constituents to high accuracy, we still do not completely understand the physics of the beginning, the nature of dark energy and dark matter. Many upcoming CMB experiments complimented with observational campaign to map 3D structure of the Universe and new particle physics constraints from the Large Hadron Collider will enable us to move beyond the knowledge of what the universe is made of, to why the universe is the way it is. In this paper we focus on learning about the physics responsible for the initial conditions for the universe.

Inflation [1–4] is perhaps one of the most promising paradigms for the early universe, which, apart from solving some of the problems of the Big Bang model like the flatness and horizon problem, also gives a mechanism for producing the seed perturbations for structure formation [5–9] and other testable predictions.

Most observational probes based on 2-point statistics like CMB power spectrum still allow vast number of inflationary models. Moreover, the alternatives to inflation such as cyclic models are also compatible with the data. Characterizing the non-Gaussianity in the primordial perturbations has emerged as powerful probe of the early universe. The amplitude of non-Gaussianity is described in terms of dimensionless nonlinearity parameter (defined in Section 3). Different models of inflation predict different amounts of , starting from to , above which values have been excluded by the WMAP data already. Non-Gaussianity from the simplest inflation models that are based on a slowly rolling scalar field is very small [10–15]; however, a very large class of more general models with, for example, multiple scalar fields, features in inflaton potential, nonadiabatic fluctuations, noncanonical kinetic terms, deviations from Bunch-Davies vacuum, among others [16], for a review and references therein generates substantially higher amounts of non-Gaussianity.

The measurement of the bispectrum of the CMB anisotropies is one of the most promising and “clean” way of constraining . Many efficient methods for evaluating bispectrum of CMB temperature anisotropies exist [17–21]. So far, the bispectrum tests of non-Gaussianity have not detected any significant in temperature fluctuations mapped by COBE [22] and WMAP [18, 23–28]. On the other hand, some authors have claimed non-Gaussian signatures in the WMAP temperature data [29–33]. These signatures cannot be characterized by and are consistent with nondetection of .

Currently the constraints on the come from temperature anisotropy data alone. By also having the polarization information in the cosmic microwave background, one can improve sensitivity to primordial fluctuations [34, 35]. Although the experiments have already started characterizing polarization anisotropies [36–39], the errors are large in comparison to temperature anisotropy. The upcoming experiments such as Planck will characterize polarization anisotropy to high accuracy.

The organization of the paper is as follow. In Section 2 we review the inflationary cosmology focusing on how the microscopic quantum fluctuations during inflation get converted into macroscopic sees perturbations for structure formation, and as CMB anisotropies. In Section 3 we discuss theoretical predictions for non-Gaussianity from the inflationary cosmology. In Section 4 we show how the primordial non-Gaussianity is connected to the CMB bispectrum, and describe/review CMB bispectrum-based estimators to constrain primordial non-Gaussianity (). In Section 5 we discuss the current constraints on by CMB bispectrum and what we can hope to achieve in near future. We also discuss non-primordial sources of non-Gaussianity which contaminate primordial bispectrum signal. In Section 6 we discuss other methods for constraining besides CMB bispectrum. Finally in Section 7 we summarize with concluding remarks.

2. Introduction: The Early Universe

One of the most promising paradigms of the early universe is inflation [1, 2, 4], which, apart from solving the flatness, homogeneity, and isotropy problem, also gives a mechanism for producing the seed perturbations for structure formation and other testable predictions¹ (for a recent review of inflationary cosmology see [40]). During inflation, the universe goes through an exponentially expanding phase. From the Friedman equation, the condition for the accelerated expansion is For both matter and radiation this condition is not satisfied. But it turns out that for a scalar field, the above condition can be achieved. For a spatially homogeneous scalar field, , moving in a potential, , the energy density is given by and the pressure is given by Hence the condition for accelerated expansion of the universe dominated with scalar field is Physically this condition corresponds to situations where kinetic energy of the field is much smaller than its potential energy. This condition is referred to slowly-rolling of the scalar field. During such slow-roll, the Hubble parameter, , is nearly constant in time, and the expansion scale factor is given by This exponential expansion drives the observable universe spatially flat, homogeneous, and isotropic.

A toy model is shown in Figure 1. In the slow-roll phase, rolls down on slowly, satisfying (4) and hence driving the universe to expand exponentially. Near the minima of the potential, oscillates rapidly and inflation ends. After inflation ends, interactions of with other particles lead to decay with a decay rate of , producing particles and radiation. This is called a reheating phase of the universe, as converts its energy density into heat by the particle production.

Figure 1: A toy scenario for the dynamics of the scalar field during inflation. During the flat part of potential, universe expand exponentially. When field reaches near the minima of the potential, the field oscillates and the radiation is generated.

Not only does inflation solve, the flatness, homogeneity and isotropy problem, but it also gives a mechanism for generating seed perturbations. During inflation the quantum fluctuation in the field exponentially stretched due to the rapid expansion phase. The proper wavelength of the fluctuations stretched out of the Hubble-horizon scale to that time, . Once outside the horizon, the characteristic rms amplitude of these fluctuations is . These fluctuations do not change in time while outside the horizon. After inflation, and reheating, the standard hot-big scenario starts. As the universe decelerates, at some point the fluctuations reenter the Hubble horizon, seeding matter and radiation fluctuations in the universe. Figure 2 summarizes the evolution of characteristic length scales.

Figure 2: Evolution of comoving horizon and generation of perturbations in the inflationary universe. figure is from [40].

2.1. Primordial Perturbations

We use linearly perturbed conformal Friedmann Lematre Robertson Walker (FLRW) metric of the form where all the metric perturbations, , , , and , are 1, and functions of conformal time . The spatial coordinate dependence of the perturbations is described by the scalar harmonic eigenfunctions, , , and , that satisfy , , and . Note that is traceless: .

Let us consider two new perturbation variables [8, 41]; which are Gauge invariant. Here is perturbations in the intrinsic spatial curvature. While reduces to in the spatially flat Gauge (), or to in the comoving gauge (), its value is invariant under any gauge transformation. Similarly , which reduces to in the comoving gauge, and to in the spatially flat gauge, is also gauge invariant. The perturbation variable helps the perturbation analysis not only because of being gauge invariant, but also because it is conserved on super-horizon scales throughout the cosmic evolution.

The quantum fluctuations generate the gauge-invariant perturbation, , that reduces to either or depending on which gauge we use, either the spatially flat gauge or the comoving gauge. Hence, and are equivalent to each other at linear order. The benefit of using is that it relates these two variables unambiguously, simplifying the transformation between and .

The solution for is valid throughout the cosmic history regardless of whether a scalar field, radiation, or matter dominates the universe; thus, once created and leaving the Hubble horizon during inflation, remains constant in time throughout the subsequent cosmic evolution until reentering the horizon. The amplitude of is fixed by the quantum-fluctuation amplitude in This is the spectrum of on super-horizon scales.

2.2. From Primordial Perturbations to CMB Anisotropies

The metric perturbations perturb CMB, producing the CMB anisotropy on the sky. Among the metric perturbation variables, the curvature perturbations play a central role in producing the CMB anisotropy.

As we have shown in the previous subsection, the gauge-invariant perturbation, , does not change in time on super-horizon scales throughout the cosmic evolution regardless of whether a scalar field, radiation, or matter dominates the universe. The intrinsic spatial curvature perturbation, , however, does change when equation of state of the universe, , changes. Since remains constant, it is useful to write the evolution of in terms of and ; however, is not gauge invariant itself, but is gauge invariant, so that the relation between and may look misleading. In 1980, Bardeen et al. [42] introduced another gauge-invariant variable, (or in the original notation), which reduces to in the zero-shear gauge, or the Newtonian gauge, in which . is given by Here, the terms in the parenthesis represent the shear, or the anisotropic expansion rate, of the hypersurfaces. While represents the curvature perturbations in the zero-shear gauge, it also represents the shear in the spatially flat gauge in which . Using , we may write as where the terms in the parenthesis represent the gauge-invariant fluid velocity.

We use in rest of the paper because it gives the closest analogy to the Newtonian potential, which we have some intuition of. reduces to in the zero-shear gauge (or the Newtonian gauge) in which the metric (6) becomes just like the Newtonian limit of the general relativity.

The gauge-invariant velocity term, , differentiates from . Since this velocity term depends on the equation of state of the universe, , the velocity and change as changes, while is independent of . The evolution of on super-horizon scales in cosmological linear perturbation theory gives the following [43]: for adiabatic fluctuations, and hence in the radiation era (), and in the matter era (). then perturbs CMB through the so-called (static) Sachs-Wolfe effect [44]

At the decoupling epoch, the universe has already been in the matter era in which , so that we observe adiabatic temperature fluctuations of , and the CMB fluctuation spectrum of the Sachs-Wolfe effect, , is By projecting the 3-dimensional CMB fluctuation spectrum, , on the sky, we obtain the angular power spectrum², [45], where and denote the conformal time at the present epoch and at the decoupling epoch, respectively, and is a spectral index which is conventionally used in the literature.

On small angular scales (), the Sachs-Wolfe approximation breaks down, and the acoustic physics in the photon-baryon fluid system modifies the primordial radiation spectrum [46]. To calculate the anisotropies at all the scales, one has to solve the Boltzmann photon transfer equation together with the Einstein equations. These equations can be solved numerically with the Boltzmann code such as CMBFAST [47]. The CMB power spectrum then can be written as

Here is called the radiation transfer function, and it contains all the physics which modifies the primordial power spectrum to generate CMB power spectrum . For the adiabatic initial conditions, in the Sachs-Wolfe limit, . Often in the literature power spectrum, , is used instead of . The two are related as . is called the dimensionless power spectrum.

If were exactly Gaussian, all the statistical properties of would be encoded in the two-point function or in in the spherical harmonic space. Since is directly related to through (11), all the information of is also in-coded in . Although which is related to a Gaussian variable, , through , in the linear order also obeys Gaussian statistics; however the nonlinear relation between and makes (and hence and CMB anisotropies) slightly non-Gaussian. The non-linear relation between and is not the only source of non-Gaussianity in the CMB anisotropies. For example, at the second order, the relationship between and is also non-linear.

2.3. Probes of the Cosmological Initial Conditions

The main predictions of a canonical inflation model are the following: (i)spatial flatness of the observable universe, (ii)homogeneity and isotropy on large angular scales of the observable universe, (iii)seed scalar and tensor perturbation with primordial density perturbations being(a)nearly scale invariant,(b)nearly adiabatic, (c)very close to Gaussian.

At the time of writing, these predictions are consistent with all current observations. This represents a major success for the inflationary paradigm. On the other hand, the inflationary paradigm can be realized by a large “zoo”³ of models. In addition, somewhat surprisingly, there exist scenarios where the Universe first contracts and then expands (such as the ekpyrotic/cyclic model), which (up to theoretical uncertainties regarding the precise mechanics of the bounce) also reproduce Universes with the properties described above. What we would like to do is to find observables that allows us to distinguish between members of the inflationary zoo. The exciting fact is that upcoming experiments will have the sensitivity to achieve this goal. Tilt and Running: Inflationary models very generically predict a slight deviation from completely flat spectrum. If we write the primordial power spectrum as , then correspond to flat spectrum and the quantity is called a tilt, which characterizes the deviation from scale invariant spectrum. Although the deviations from the scale invariance are predicted to be small, the exact amount of deviation depends on the details of the inflationary model. For example in most slow roll models is of order , where is a number of -folds to the end of inflation. Ghost inflation, however, predicts negligible tilt. Hence characterizing the tilt of the scalar spectral index is a useful probe of the early universe. Currently the most stringent constraints on tilt come from the WMAP 5-year data, [48], which already disfavors inflationary models with “blue spectral index” (). The error on will reduce to for upcoming Planck satellite and to for futuristic CMBPol-like satellite [49].

Apart from the tilt in the primordial power spectrum, inflationary models also predict to be slightly scale dependent. This scale dependence is referred to as “running” of the spectral index and is defined as . The constraints on the running from the WMAP 5-year data are [48]. The error will reduce to for upcoming Planck satellite and to for a fourth-generation satellite such as CMBPol [49].

Primordial Gravitational Waves
Inflation also generates tensor perturbations (gravitational waves), which although small compared to scalar component, are still detectable, in principle. So far primordial gravitational waves have not been detected. There are upper limits on their amplitude; see [50] for a current observational bounds on the level for primordial gravitational waves. Detection of these tensor perturbations or primordial gravitational waves is considered a “smoking gun” for the inflationary scenario. In contrast to inflation, ekpyrotic (cyclic) models predict an amount of gravitational waves that is much smaller than polarized foreground emission would allow us to see even for an ideal CMB experiment. Primordial scalar perturbations create only E-modes of the CMB⁴, while primordial tensor perturbations generate both parity even E-modes and parity odd B-modes polarization [51–53]. The detection of primordial tensor B-modes in the CMB would confirm the existence of tensor perturbations in the early universe. This primordial B-mode signal is directly related to the Hubble parameter during inflation, and thus a detection would establish the energy scale at which inflation happened. Various observational efforts are underway to detect such B-mode signal of the CMB [54]. Search for primordial B-modes is challenging. Apart from foreground subtraction challenges, and the challenge of reaching the instrumental sensitivity to detect primordial B-modes, there are several nonprimordial sources such as weak lensing of CMB by the large-scale structure [55, 56], rotation of the CMB polarization [57], and instrumental systematics that generate B-modes which contaminate the inflationary signal [58, 59]. The amplitude of gravitational waves is parametrized as the ratio of the amplitude of tensor and scalar perturbations, . The limit from WMAP 5-year data is () [48].

Isocurvature Modes
Inflationary models with a single scalar field predict primordial perturbations to be adiabatic. Hence detection of isocurvature density perturbations is a “smoking gun” for multifield models of inflation. A large number of inflationary models with multiple scalar fields predict some amount of isocurvature modes [60–72]. For example, curvaton models predict the primordial perturbations to be a mixture of adiabatic and isocurvature perturbations. Isocurvature initial conditions specify perturbations in the energy densities of two (or more) species that add up to zero. It does not perturb the spatial curvature of comoving slice (i.e., is zero, hence the name isocurvature). In general, there can be four types of isocurvature modes, namely: baryon isocurvature modes, CDM isocurvature modes, neutrino density isocurvature modes, and neutrino velocity isocurvature modes. These perturbations imprint distinct signatures in the CMB temperature and E-polarization anisotropies [73]. The contribution of isocurvature modes is model dependent, and different models predict different amounts of it. There exists an upper limit on the allowed isocurvature modes using CMB temperature anisotropies [74, 75] a characterization (or detection of any) of isocurvature modes has a potential of discriminating between early Universe models.

Primordial Non-Gaussianity
Canonical inflationary models predict primordials perturbations to be very close to Gaussian [5–9], and any non-Gaussianity predicted by the canonical inflation models is very small [14, 15]. However models with nonlinearity [10, 13, 76], interacting scalar fields [12, 77], and deviation from ground state [78, 79] can generate large non-Gaussian perturbations. The amplitude of the non-Gaussian contribution to the perturbation is often referred to as even if the nature of the non-Gaussianities can be quite different. Different models of inflation predict different amounts of , starting from very close to zero for almost Gaussian perturbations, to for large non-Gaussian perturbations. For example, the canonical inflation models with slow roll inflation, where only a couple of derivatives of potential, are responsible for inflationary dynamics, predict [15]. In models where higher-order derivatives of the potential are important, the value of varies from where higher order derivatives are suppressed by a low UV cutoff [80] to based on Dirac-Born-Infeld effective action. Ghost inflation, where during inflation, the background has a constant rate of change as opposed to the constant background in conventional inflation, is also capable of giving [81]. The additional field models generating inhomogeneities in nonthermal species [82] can generate [83]; while curvaton models, where isocurvature perturbations in second field during the inflation generate adiabatic perturbations after the inflation, can have [84].

In the following we will see that non-Gaussianity, far from being merely a test of standard inflation, may reveal detailed information about the state and physics of the very early Universe, if it is present at the level suggested by the theoretical arguments above.

3. Primordial Non-Gaussianity

Large primordial non-Gaussianity can be generated if any of the following condition is violated [85]. (i)Single Field. Only one scalar field is responsible for driving the inflation and the quantum fluctuations in the same field is responsible for generating the seed classical perturbations. (ii)Canonical Kinetic Energy. The kinetic energy of the field is such that the perturbations travel at the speed of light. (iii)Slow Roll. During inflation phase the field evolves much slowly than the Hubble time during inflation. (iv)Initial Vacuum State. The quantum field was in the Bunch-Davies vacuum state before the quantum fluctuation were generated.

To characterize the non-Gaussianity one has to consider the higher order moments beyond two-point function, which contains all the information for Gaussian perturbations. The 3-point function which is zero for Gaussian perturbations contains the information about non-Gaussianity. The 3-point correlation function of Bardeen's curvature perturbations, , can be simplified using the translational symmetry to give where tells the shape of the bispectrum in momentum space while the amplitude of non-Gaussianity is captured dimensionless non-linearity parameter . The shape function correlates fluctuations with three wave-vectors and form a triangle in Fourier space. Depending on the physical mechanism responsible for the bispectrum, the shape of the 3-point function, , can be broadly classified into three classes [86]. The local, “squeezed,” non-Gaussianity, where is large for the configurations in which . Most of the studied inflationary and Ekpyrotic models produce non-Gaussianity of local shape (e.g., [82, 84, 87–104]). Second, the nonlocal, “equilateral,” non-Gaussianity where is large for the configuration when . Finally the folded [105, 106] shape, where is large for the configurations in which . Figure 3 shows these three shapes.

Figure 3: Shapes of non-Gaussianity. The shape function forms a triangle in Fourier space. The triangles are parametrized by rescaled Fourier modes, and . Figure is from [40].

Non-Gaussianity of Local Type
The local form of non-Gaussianity may be parametrized in real space as⁵ [13, 107, 108]: where is the linear Gaussian part of the perturbations, and characterizes the amplitude of primordial non-Gaussianity. Different inflationary models predict different amounts of , starting from to , beyond which values have been excluded by the Cosmic Microwave Background (CMB) bispectrum of WMAP temperature data. The bispectrum in this model can be written as where is the amplitude of the primordial power spectrum.

The local form arises from a non-linear relation between inflaton and curvature perturbations [10, 11, 13], curvaton models [84], or the New Ekpyrotic models [109, 110]. Models with fluctuations in the reheating efficiency [9, 10] and multifield inflationary models [17] also generate non-Gaussianity of local type.

Being local in real space, non-Gaussianity of local type describes correlations among Fourier modes of very different . In the limit in which one of the modes becomes of very long wavelength [111], , (i.e., the other two 's become equal and opposite), freezes out much before and and behaves as a background for their evolution. In this limit is proportional to the power spectrum of the short and long wavelength modes

As an example, for canonical single field slow-roll inflationary models, the three-point function is given by [15] where and are the usual slow-roll parameters and are assumed to be much smaller than unity. Taking the limit gives the local form as in (19). To show this point, Figure 4 compares the non-Gaussianity shape for local type and for slow-roll model. Although in this limit, slow-roll models do predict non-Gaussianity of local type but as evident from (20), the bispectrum of inflaton perturbations yields a non-trivial-scale dependence of [12, 15]. However in the slow roll limit and hence the amplitude is too small to detect.

Figure 4: Plot of the function for the Slow-Roll inflation as given by (20) (a) and the local distribution as given by (18) (b). The figures are normalized to have value 1 for equilateral configurations and set to zero outside the region . Here , , and . The figures are taken from Babich et al. 2004 [86].

Figure 5: Plot of the function for the inflation with higher derivatives as given by (22) (a) and the ghost inflation as given by (23) (b). The figures are normalized to have value 1 for equilateral configurations and set to zero outside the region . Here and . The figures are taken from Babich et al. 2003 [86].

Non-Gaussianity of Equilateral Type
While vast numbers of inflationary models predict non-Gaussianity of local type, this model, for instance, fails completely when non-Gaussianity is localized in a specific range in space, the case that is predicted from inflation models with higher derivative terms [81, 106, 112–115]. In these models the correlation is among modes with comparable wavelengths which go out of the horizon nearly at the same time. The shape function for the equilateral shape can be written as [25]

The models of this kind have large for the configurations, where . The equilateral form arises from non-canonical kinetic terms such as the Dirac-Born-Infeld (DBI) action [112], the ghost condensation [81], or any other single-field models in which the scalar field acquires a low speed of sound [106, 115].

As an example, models with higher derivative operators in the usual inflation scenario and a model of inflation based on the Dirac-Born-Infeld (DBI) action produce a bispectrum of the form The previous model uses as a leading order operator. DBI inflation, which can produce large non-Gaussianity, , also has of a similar form.

Ghost inflation, where an inflationary de Sitter phase is obtained with a ghost condensate, produces a bispectrum of the following form [81]: where and are free parameters of order unity, and Ghost inflation also produces large non-Gaussianity, . Figure 3 shows the shape of non-Gaussianity of equilateral type by showing for ghost inflation and for a model with a higher derivative term.

Folded Shape
So far the 3-point functions were calculated assuming the regular Bunch-Davis vacuum state, giving rise to either local or equilateral type non-Gaussianity. However if the bispectrum is calculated by dropping the assumption of Bunch-Davis initial state gives rise to bispectrum shape which peaks for the folded shape, , with shape function given as [105, 106, 116] where are the Bogoliubov coefficients which encode information about the initial conditions, is the initial conformal time and .

4. The Cosmic Microwave Background Bispectrum

Since the discovery of CMB by Penzias and Wilson in 1965 [117] and the first detection of CMB temperature anisotropies on large scales by the COBE DMR [118], the space satellite WMAP and over a dozens of balloon and ground-based experiments have characterized the CMB temperature anisotropies to a high accuracy and over a wide range of angular scales. The space satellite Planck which launched in 2009 will soon characterize the temperature anisotropies to even higher accuracy up to angular scales of . The CMB power spectrum is obtained by reducing all the information of ( for WMAP and for Planck). Such reduction is justified to obtain a fiducial model, given that the non-Gaussianities are expected to be small. With high quality data on the way, the field of non-Gaussianity is taking off. CMB bispectrum contains information which is not present in the power-spectrum and as we say in the previous section, is a unique probe of the early universe.

The harmonic coefficients of the CMB anisotropy can be related to the primordial fluctuation as where is the primordial curvature perturbations, for a comoving wavevector , is the radiation transfer function, where the index refers to either temperature () or E-polarization () of the CMB. A beam function and the harmonic coefficient of noise are instrumental effects. Equation (26) is written for a flat background, but can easily be generalized.

Any non-Gaussianity present in the primordial perturbations gets transferred to the observed CMB via (26). The most common way to look for non-Gaussianity in the CMB is to study the bispectrum, the three-point function of temperature and polarization anisotropies in harmonic space. The CMB angular bispectrum is defined as and the angular-averaged bispectrum is where the matrix is the Wigner 3J symbol imposing selection rules which makes bispectrum zero unless (i) = integer, (ii), (iii) for .

Using (26) the bispectrum can be written as where is the primordial curvature three-point function as defined in (16).

To forecast constraints on non-Gaussianity using CMB data, we will perform a Fisher matrix analysis. The Fisher matrix for the parameters can be written as [20, 34, 108] The indices and run over all the parameters bispectrum depends on that we will assume all the cosmological parameters except to be known. Indices and run over all the eight possible ordered combinations of temperature and polarization given by and , the combinatorial factor equals when all 's are different, when , and otherwise. The covariance matrix Cov is obtained in terms of , , and (see [21, 34]) by applying Wick's theorem.

For non-Gaussianity of the local type, for which the functional form , is given by (18), we have where the functions and are given by In the previous expression we use the dimensionless power spectrum amplitude , which is defined by , where is the tilt of the primordial power spectrum. One can compute the transfer functions and using publicly available codes such as CMBfast [47] and CAMB [119]

In a similar way, from (21), one can derive the following expressions for the bispectrum derivatives in the equilateral case: