Primordial Non-Gaussianity in the Cosmic Microwave Background

Abstract

In the last few decades, advances in observational cosmology have given us a standard model of cosmology. We know the content of the universe to within a few percent. With more ambitious experiments on the way, we hope 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 primordial non-Gaussianity as a probe of the physics of the dynamics of the universe at the very earliest moments. We discuss (1) theoretical predictions from inflationary models and their observational consequences in the cosmic microwave background (CMB) anisotropies; (2) CMB-based estimators for constraining primordial non-Gaussianity with an emphasis on bispectrum templates (3) current constraints on non-Gaussianity and what we can hope to achieve in the near future and (4) nonprimordial sources of non-Gaussianities in the CMB such as bispectrum due to second order effects, three way crosscorrelation between primary-lensing-secondary CMB, and possible instrumental effects.

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 13.7±0.1 Gyr old and is spatially flat to about 1% and is made of 4.6±0.1% baryons, 22.8±1.3% dark matter, and remaining 72.6±1.5% 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 𝑓_NL (defined in Section 3). Different models of inflation predict different amounts of 𝑓_NL, starting from 𝑂(1) to 𝑓_NL∼100, 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 𝑓_NL. 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 𝑓_NL 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 𝑓_NL and are consistent with nondetection of 𝑓_NL.

Currently the constraints on the 𝑓_NL 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 (𝑓_NL). In Section 5 we discuss the current constraints on 𝑓_NL 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 𝑓_NL 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 𝜌+3𝑝<0. (1) 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, 𝜙, moving in a potential, 𝑉(𝜙), the energy density is given by 𝜌_𝜙=12(𝑑𝜙)𝑑𝑡²+𝑉(𝜙), (2) and the pressure is given by 𝑝_𝜙=12(𝑑𝜙)𝑑𝑡²−𝑉(𝜙). (3) Hence the condition for accelerated expansion of the universe dominated with scalar field 𝜙 is (𝑑𝜙)𝑑𝑡²<𝑉(𝜙). (4) 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, 𝐻(𝑡)=𝑑ln𝑎/𝑑𝑡, is nearly constant in time, and the expansion scale factor 𝑎(𝑡) is given by 𝑎(𝑡)=𝑎(𝑡₀)exp(^𝑡_𝑡0𝐻(𝑡^′)𝑑𝑡^′)≈𝑎(𝑡₀)exp[𝐻(𝑡)(𝑡−𝑡₀)]. (5) 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, 𝜙 rolls down on 𝑉(𝜙) slowly, satisfying (4) and hence driving the universe to expand exponentially. Near the minima of the potential, 𝜙 oscillates rapidly and inflation ends. After inflation ends, interactions of 𝜙 with other particles lead 𝜙 to decay with a decay rate of Γ_𝜙, producing particles and radiation. This is called a reheating phase of the universe, as 𝜙 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 𝜙 exponentially stretched due to the rapid expansion phase. The proper wavelength of the fluctuations stretched out of the Hubble-horizon scale to that time, 𝐻⁻¹. Once outside the horizon, the characteristic rms amplitude of these fluctuations is |𝜙|_rms∼𝐻/(2𝜋). 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 𝑑𝑠²=𝑎²(𝜏){−(1+2𝐴𝑄)𝑑𝜏²−2𝐵𝑄_𝑖𝑑𝜏𝑑𝑥^𝑖+[(1+2𝐻_L𝑄)𝛿_𝑖𝑗+2𝐻_T𝑄_𝑖𝑗]𝑑𝑥^𝑖𝑑𝑥^𝑗}, (6) where all the metric perturbations, 𝐴, 𝐵, 𝐻_L, and 𝐻_T, are ≪1, and functions of conformal time 𝜏. The spatial coordinate dependence of the perturbations is described by the scalar harmonic eigenfunctions, 𝑄, 𝑄_𝑖, and 𝑄_𝑖𝑗, that satisfy 𝛿^𝑖𝑗𝑄_𝑖𝑗=−𝑘²𝑄, 𝑄_𝑖=−𝑘⁻¹𝑄_𝑖, and 𝑄_𝑖𝑗=𝑘⁻²𝑄_𝑖𝑗+(1/3)𝛿_𝑖𝑗𝑄. Note that 𝑄_𝑖𝑗 is traceless: 𝛿^𝑖𝑗𝑄_𝑖𝑗=0.

Let us consider two new perturbation variables [8, 41]; 𝜙̇𝑢≡𝛿𝜙−𝑎𝐻ℛ,𝜁≡−𝑎𝐻𝜙̇𝑢=ℛ−𝑎𝐻𝜙̇𝛿𝜙, (7) which are Gauge invariant. Here ℛ≡𝐻_L+(1/3)𝐻_T is perturbations in the intrinsic spatial curvature. While 𝑢 reduces to 𝛿𝜙 in the spatially flat Gauge (ℛ≡0), or to −(𝜙̇/𝑎𝐻)ℛ in the comoving gauge (𝛿𝜙≡0), its value is invariant under any gauge transformation. Similarly 𝜁, which reduces to ℛ in the comoving gauge, and to −(𝑎𝐻/𝜙̇)𝛿𝜙 in the spatially flat gauge, is also gauge invariant. The perturbation variable 𝜁 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, 𝑢, that reduces to either 𝛿𝜙 or (𝜙̇/𝑎𝐻)ℛ depending on which gauge we use, either the spatially flat gauge or the comoving gauge. Hence, 𝛿𝜙_flat and (𝜙̇/𝑎𝐻)ℛ_com are equivalent to each other at linear order. The benefit of using 𝑢 is that it relates these two variables unambiguously, simplifying the transformation between 𝛿𝜙_flat and ℛ_com.

The solution for 𝜁 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, 𝜁 remains constant in time throughout the subsequent cosmic evolution until reentering the horizon. The amplitude of 𝜁 is fixed by the quantum-fluctuation amplitude in 𝑢Δ²_𝜁(𝑘)=(𝑎𝐻𝜙̇)²Δ²_𝜙(𝑘)≈(𝑎𝐻²𝜙̇)2𝜋²𝐻=[²2𝜋(𝑑𝜙)]𝑑𝑡². (8) This is the spectrum of 𝜁 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, 𝜁, 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, ℛ, however, does change when equation of state of the universe, 𝑤≡𝑝/𝜌, changes. Since 𝜁 remains constant, it is useful to write the evolution of ℛ in terms of 𝜁 and 𝑤; however, ℛ is not gauge invariant itself, but 𝜁 is gauge invariant, so that the relation between ℛ and 𝜁 may look misleading. In 1980, Bardeen et al. [42] introduced another gauge-invariant variable, Φ (or Φ_H in the original notation), which reduces to ℛ in the zero-shear gauge, or the Newtonian gauge, in which 𝐵≡0≡𝐻_T. Φ is given byΦ=ℛ−𝑎𝐻𝑘𝐻̇(−𝐵−^T𝑘). (9) Here, the terms in the parenthesis represent the shear, or the anisotropic expansion rate, of the 𝜏=constant hypersurfaces. While Φ represents the curvature perturbations in the zero-shear gauge, it also represents the shear in the spatially flat gauge in which ℛ≡0. Using Φ, we may write 𝜁 as𝜁=ℛ−𝑎𝐻𝜙̇𝛿𝜙=Φ−𝑎𝐻𝑘(𝑣_𝜙−𝐻̇_T𝑘), (10) where the terms in the parenthesis represent the gauge-invariant fluid velocity.

We use Φ in rest of the paper because it gives the closest analogy to the Newtonian potential, which we have some intuition of. Φ reduces to ℛ 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, 𝑣−𝑘⁻¹𝐻̇_T, differentiates 𝜁 from Φ. Since this velocity term depends on the equation of state of the universe, 𝑤=𝑝/𝜌, the velocity and Φ change as 𝑤 changes, while 𝜁 is independent of 𝑤. The evolution of Φ on super-horizon scales in cosmological linear perturbation theory gives the following [43]:Φ=3+3𝑤5+3𝑤𝜁, (11) for adiabatic fluctuations, and hence Φ=(2/3)𝜁 in the radiation era (𝑤=1/3), and Φ=(3/5)𝜁 in the matter era (𝑤=0). Φ then perturbs CMB through the so-called (static) Sachs-Wolfe effect [44] Δ𝑇𝑇=−1+𝑤5+3𝑤𝜁. (12)

At the decoupling epoch, the universe has already been in the matter era in which 𝑤=0, so that we observe adiabatic temperature fluctuations of Δ𝑇/𝑇=−(1/3)Φ=−(1/5)𝜁, and the CMB fluctuation spectrum of the Sachs-Wolfe effect, Δ²_SW(𝑘), is Δ²_SW1(𝑘)=9Δ²_Φ1(𝑘)=Δ25^2𝜁(𝑘). (13) By projecting the 3-dimensional CMB fluctuation spectrum, Δ²_SW(𝑘), on the sky, we obtain the angular power spectrum², 𝐶_𝑙 [45],𝐶^SW_𝑙=4𝜋^∞₀𝑑𝑘𝑘Δ²_𝑆𝑊(𝑘)𝑗²_𝑙[𝑘(𝜏₀−𝜏_dec)]=𝐶₂^𝑆𝑊Γ[(9−𝑛_𝑠)/2]Γ[𝑙+(𝑛_𝑠−1)/2]Γ[(𝑛_𝑠+3)/2]Γ[𝑙+(5−𝑛_𝑠,)/2] (14) where 𝜏₀ and 𝜏_dec denote the conformal time at the present epoch and at the decoupling epoch, respectively, and 𝑛_𝑠≡1+[𝑑lnΔ²(𝑘)/𝑑ln𝑘] is a spectral index which is conventionally used in the literature.

On small angular scales (ℓ>10), 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 𝐶_ℓ=4𝜋^∞₀𝑑𝑘𝑘Δ²_Φ(𝑘)𝑔²_Tℓ(𝑘). (15)

Here 𝑔_Tℓ(𝑘) is called the radiation transfer function, and it contains all the physics which modifies the primordial power spectrum Δ_Φ to generate CMB power spectrum 𝐶_ℓ. For the adiabatic initial conditions, in the Sachs-Wolfe limit, 𝑔_T𝑙(𝑘)=−(1/3)𝑗_𝑙[𝑘(𝜏₀−𝜏_dec)]. Often in the literature power spectrum, 𝑃_Φ(𝑘), is used instead of Δ²_Φ(𝑘). The two are related as Δ²_Φ(𝑘)=(2𝜋²)⁻¹𝑘³𝑃_Φ(𝑘). Δ²_Φ(𝑘) is called the dimensionless power spectrum.

If Φ were exactly Gaussian, all the statistical properties of Φ would be encoded in the two-point function or in 𝐶_ℓ in the spherical harmonic space. Since Φ is directly related to 𝜁 through (11), all the information of 𝜁 is also in-coded in 𝐶_ℓ. Although 𝜁 which is related to a Gaussian variable, 𝑢, through 𝜁=−(𝑎𝐻/𝜙̇)𝑢, in the linear order 𝜁 also obeys Gaussian statistics; however the nonlinear relation between 𝜁 and 𝑢 makes 𝜁 (and hence Φ and CMB anisotropies) slightly non-Gaussian. The non-linear relation between 𝜁 and Φ is not the only source of non-Gaussianity in the CMB anisotropies. For example, at the second order, the relationship between Φ and Δ𝑇/𝑇 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 Δ_Φ(𝑘)=𝐴(𝑘₀)(𝑘/𝑘₀)^𝑛𝑠−1, then 𝑛_𝑠=1 correspond to flat spectrum and the quantity |𝑛_𝑠−1| 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 |𝑛−1| is of order 1/𝑁_𝑒, where 𝑁_𝑒∼60 is a number of 𝑒-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, 𝑛_𝑠=0.960^+0.014_−0.013 [48], which already disfavors inflationary models with “blue spectral index” (𝑛_𝑠>1). The 1𝜎 error on 𝑛_𝑠 will reduce to Δ𝑛_𝑠=0.0036 for upcoming Planck satellite and to Δ𝑛_𝑠=0.0016 for futuristic CMBPol-like satellite [49].

Apart from the tilt in the primordial power spectrum, inflationary models also predict 𝑛_𝑠 to be slightly scale dependent. This scale dependence is referred to as “running” of the spectral index 𝑛_𝑠 and is defined as 𝑑𝑛_𝑠/𝑑ln𝑘. The constraints on the running from the WMAP 5-year data are −0.090<𝑑𝑛_𝑠/𝑑ln𝑘<0.0019 [48]. The 1𝜎 error will reduce to Δ(𝑑𝑛_𝑠/𝑑ln𝑘)=0.0052 for upcoming Planck satellite and to Δ(𝑑𝑛_𝑠/𝑑ln𝑘)=0.0036 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 𝐻 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, 𝑟. The limit from WMAP 5-year data is 𝑟<0.22 (2𝜎) [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., ℛ 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 𝑓_NL even if the nature of the non-Gaussianities can be quite different. Different models of inflation predict different amounts of 𝑓_NL, starting from very close to zero for almost Gaussian perturbations, to 𝑓_NL≈100 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 𝑓_NL∼0.05 [15]. In models where higher-order derivatives of the potential are important, the value of 𝑓_NL varies from 𝑓_NL∼0.1 where higher order derivatives are suppressed by a low UV cutoff [80] to 𝑓_NL∼100 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 𝑓_NL∼100 [81]. The additional field models generating inhomogeneities in nonthermal species [82] can generate 𝑓_NL∼5 [83]; while curvaton models, where isocurvature perturbations in second field during the inflation generate adiabatic perturbations after the inflation, can have 𝑓_NL∼10 [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, Φ(𝑘), can be simplified using the translational symmetry to give ⟨Φ(𝐤_𝟏)Φ(𝐤_𝟐)Φ(𝐤_𝟑)⟩=(2𝜋)³𝛿³(𝐤_𝟏+𝐤_𝟐+𝐤_𝟑)𝑓_NL⋅𝐹(𝑘₁,𝑘₂,𝑘₃), (16) where 𝐹(𝑘₁,𝑘₂,𝑘₃) tells the shape of the bispectrum in momentum space while the amplitude of non-Gaussianity is captured dimensionless non-linearity parameter 𝑓_NL. The shape function 𝐹(𝑘₁,𝑘₂,𝑘₃) 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, 𝐹(𝑘₁,𝑘₂,𝑘₃), can be broadly classified into three classes [86]. The local, “squeezed,” non-Gaussianity, where 𝐹(𝑘₁,𝑘₂,𝑘₃) is large for the configurations in which 𝑘₁≪𝑘₂≈𝑘₃. 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 𝐹(𝑘₁,𝑘₂,𝑘₃) is large for the configuration when 𝑘₁≈𝑘₂≈𝑘₃. Finally the folded [105, 106] shape, where 𝐹(𝑘₁,𝑘₂,𝑘₃) is large for the configurations in which 𝑘₁≈2𝑘₂≈2𝑘₃. Figure 3 shows these three shapes.

Figure 3 

Shapes of non-Gaussianity. The shape function 𝐹(𝑘1,𝑘2,𝑘3) forms a triangle in Fourier space. The triangles are parametrized by rescaled Fourier modes, 𝑥2=𝑘2/𝑘1 and 𝑥3=𝑘3/𝑘1. 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]: 𝜁(𝐫)=𝜁_𝐿3(𝐫)+5𝑓_NL(𝜁²_𝐿(𝐫)−⟨𝜁²_𝐿(𝐫)⟩), (17) where 𝜁_𝐿(𝑟) is the linear Gaussian part of the perturbations, and 𝑓_NL characterizes the amplitude of primordial non-Gaussianity. Different inflationary models predict different amounts of 𝑓_NL, starting from 𝑂(1) to 𝑓_NL∼100, 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 𝐹_local(𝑘₁,𝑘₂,𝑘₃)=2Δ²_Φ𝑓_NL[1𝑘^{3−(𝑛𝑠}₁⁻¹⁾𝑘^{3−(𝑛𝑠}₂⁻¹⁾+1𝑘^{3−(𝑛𝑠}₁⁻¹⁾𝑘^{3−(𝑛𝑠}₃⁻¹⁾+1𝑘^{3−(𝑛𝑠}₂⁻¹⁾𝑘^{3−(𝑛𝑠}₃⁻¹⁾], (18) where Δ_Φ 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 𝑘. In the limit in which one of the modes becomes of very long wavelength [111], 𝑘₃→0, (i.e., the other two 𝑘's become equal and opposite), 𝜁_𝑘3 freezes out much before 𝑘₁ and 𝑘₂ and behaves as a background for their evolution. In this limit 𝐹_local is proportional to the power spectrum of the short and long wavelength modes𝐹_local∝1𝑘³₃1𝑘³₁. (19)

As an example, for canonical single field slow-roll inflationary models, the three-point function is given by [15] 𝐹_slow-roll(𝑘₁,𝑘₂,𝑘₃)=18(35)Δ²_Φ1Π𝑘³_𝑖[(3𝜖−2𝜂)_𝑘𝑖𝑘³_𝑖+𝜖_𝑖≠𝑗𝑘_𝑖𝑘²_𝑗∑+8𝜖^𝑖>𝑗𝑘²_𝑖𝑘²_𝑗𝑘₁+𝑘₂+𝑘₃], (20) where 𝜖 and 𝜂 are the usual slow-roll parameters and are assumed to be much smaller than unity. Taking the limit 𝑘₃→0 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 𝑓_NL [12, 15]. However in the slow roll limit 𝜂,𝜖≪1 and hence the amplitude is too small to detect.

(a)

(b)

Figure 4 

Plot of the function 𝐹(1,𝑥2,𝑥3)𝑥22𝑥23 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 𝑥2=𝑥3=1 and set to zero outside the region 1−𝑥2≤𝑥3≤𝑥2. Here 𝑥3≡𝑘3/𝑘1, 𝑥2≡𝑘2/𝑘1, and 𝜖=𝜂=1/30. The figures are taken from Babich et al. 2004 [86].

(a)

(b)

Figure 5 

Plot of the function 𝐹(1,𝑥2,𝑥3)𝑥22𝑥23 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 𝑥2 =𝑥3=1 and set to zero outside the region 1−𝑥2≤𝑥3≤𝑥2. Here 𝑥3≡𝑘3/𝑘1 and 𝑥2≡𝑘2/𝑘1. 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 𝑘 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] 𝐹_equil.(𝑘₁,𝑘₂,𝑘₃)=𝑓^equil._NL⋅6Δ²_Φ1⋅[−𝑘^{3−(𝑛𝑠}₁⁻¹⁾𝑘^{3−(𝑛𝑠}₂⁻¹⁾2+(2perm.)−(𝑘₁𝑘₂𝑘₃)^{2−2(𝑛𝑠−1)/3}+1𝑘^{1−(𝑛𝑠}₁^−1)/3𝑘^{2−2(𝑛𝑠}₂^−1)/3𝑘^{3−(𝑛𝑠}₃⁻¹⁾+(5perm.)]. (21)

The models of this kind have large 𝐹(𝑘₁,𝑘₂,𝑘₃) for the configurations, where 𝑘₁≈𝑘₂≈𝑘₃. 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 𝐹_hd(𝑘₁,𝑘₂,=3𝑘3)Δ40^2Φ𝜙̇²Λ⁴1Π𝑘³_𝑖[1𝑘₁+𝑘₂+𝑘₃×(_𝑖𝑘⁵_𝑖+_𝑖≠𝑗(2𝑘⁴_𝑖𝑘_𝑗−3𝑘³_𝑖𝑘²_𝑗)+_{𝑖≠𝑗≠ℓ}(𝑘³_𝑖𝑘_𝑗𝑘_ℓ−4𝑘²_𝑖𝑘²_𝑗𝑘_ℓ))]. (22) The previous model uses (1/8Λ⁴)(∇𝜙)²(∇𝜙)² as a leading order operator. DBI inflation, which can produce large non-Gaussianity, 𝑓_NL∼100, also has 𝐹(𝑘₁,𝑘₂,𝑘₃) 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]:𝐹_ghost(𝑘₁,𝑘₂,𝑘₃)3=−(5)³⋅2√2𝜋^3/2Γ(1/4)³𝐻⁵𝛽𝑀²(𝑀)𝛼𝐻⁴×1∏_i𝑘³_𝑖^0−∞𝑑𝜂𝜂⁻¹𝐹^∗(𝜂)𝐹^∗(𝑘₂𝑘₁𝜂)𝐹^′∗(𝑘₃𝑘₁𝜂)×𝑘₃(𝑘₁⋅𝑘₂)+symm., (23) where 𝛼 and 𝛽 are free parameters of order unity, and𝐹(𝑥)=𝜋8(−𝑥)^3/2𝐻⁽¹⁾_3/4(𝑥²2). (24) Ghost inflation also produces large non-Gaussianity, 𝑓_NL∼100. Figure 3 shows the shape of non-Gaussianity of equilateral type by showing 𝐹(𝑘₁,𝑘₂,𝑘₃) 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, 𝑘₁≈2𝑘₂≈2𝑘₃, with shape function given as [105, 106, 116] 𝐹_non-BD(𝑘₁,𝑘₂,𝑘₃3)=(5)³⋅1𝑀²_𝑝4∏(2𝑘³_𝑖)𝐻⁶𝜙̇²_𝑗3𝑘²₁𝑘²₂𝑘²₃𝑘²_𝑗𝑘_𝑗×ℛ𝑒(𝛽_𝑘𝑗)(cos(𝑘_𝑗𝜂₀)−1), (25) where 𝛽_𝑘𝑗 are the Bogoliubov coefficients which encode information about the initial conditions, 𝜂₀ is the initial conformal time and 𝑘_𝑗=∑_𝑖𝑘_𝑖−2𝑘_𝑗.

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 ℓ_max≈2500. The CMB power spectrum is obtained by reducing all the information of 𝑁_pix (∼10⁶ for WMAP and ∼10⁷ 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 𝑎_𝑙𝑚=𝑇⁻¹∫𝑑²𝐧Δ𝑇(𝐧)𝑌^∗_ℓ𝑚 can be related to the primordial fluctuation Φ as𝑎^𝑝_ℓ𝑚=𝑏_ℓ4𝜋(−𝑖)^ℓ𝑑³𝑘(2𝜋)³Φ(𝐤)𝑔^𝑝_ℓ(𝑘)𝑌^∗_ℓ𝑚(𝐤)+𝑛_ℓ𝑚, (26) where Φ(𝐤) is the primordial curvature perturbations, for a comoving wavevector 𝐤, 𝑔^𝑝_ℓ(𝑟) is the radiation transfer function, where the index 𝑝 refers to either temperature (𝑇) or E-polarization (𝐸) of the CMB. A beam function 𝑏_ℓ and the harmonic coefficient of noise 𝑛_ℓ𝑚 are instrumental effects. Equation (26) is written for a flat background, but can easily be generalized.

Any non-Gaussianity present in the primordial perturbations Φ(𝐤) 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 𝐵_ℓ^𝑝𝑞𝑟_{1ℓ2ℓ3,𝑚1𝑚2𝑚3}≡⟨𝑎^𝑝_ℓ1𝑚1𝑎^𝑞_ℓ2𝑚2𝑎^𝑟_ℓ3𝑚3⟩, (27) and the angular-averaged bispectrum is 𝐵_ℓ^𝑝𝑞𝑟_1ℓ2ℓ3=_{𝑚1𝑚2𝑚3}ℓ₁ℓ₂ℓ₃𝑚₁𝑚₂𝑚₃𝐵^pqr_{ℓ1ℓ2ℓ3,𝑚1𝑚2𝑚3}, (28) where the matrix is the Wigner 3J symbol imposing selection rules which makes bispectrum zero unless (i)ℓ₁+ℓ₂+ℓ₃ = integer, (ii)𝑚₁+𝑚₂+𝑚₃=0, (iii)|ℓ_𝑖−ℓ_𝑗|≤ℓ_𝑘≤ℓ_𝑖+ℓ_𝑗 for 𝑖,𝑗,𝑘=1,2,3.

Using (26) the bispectrum can be written as𝐵_ℓ^𝑝𝑞𝑟_1ℓ2ℓ3=(4𝜋)³(−𝑖)^{ℓ1+ℓ2+ℓ3}_{𝑚1𝑚2𝑚3}ℓ₁ℓ₂ℓ₃𝑚₁𝑚₂𝑚₃×𝑑³𝑘₁(2𝜋)³𝑑³𝑘₂(2𝜋)³𝑑³𝑘₃(2𝜋)³𝑌^∗_ℓ1𝑚1(𝐤₁)𝑌^∗_ℓ2𝑚2(𝐤₂)×𝑌^∗_ℓ3𝑚3(𝐤₃)𝑔^𝑝_ℓ1(𝑘₁)𝑔^𝑞_ℓ2(𝑘₂)𝑔^𝑟_ℓ3(𝑘₃)×⟨Φ(𝐤₁)Φ(𝐤₂)Φ(𝐤₃)⟩, (29) where ⟨Φ(𝐤₁)Φ(𝐤₂)Φ(𝐤₃)⟩ 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 𝑝_𝑎 can be written as [20, 34, 108] ℱ_𝑎𝑏=_{{𝑖𝑗𝑘,𝑝𝑞𝑟}ℓ1≤ℓ2≤ℓ3}1Δ_ℓ1ℓ2ℓ3𝜕𝐵_ℓ^𝑝𝑞𝑟_1ℓ2ℓ3𝜕𝑝_𝑎(Cov⁻¹)_{𝑖𝑗𝑘,𝑝𝑞𝑟}𝜕𝐵_ℓ^𝑖𝑗𝑘_1ℓ2ℓ3𝜕𝑝_𝑏. (30) The indices 𝑎 and 𝑏 run over all the parameters bispectrum depends on that we will assume all the cosmological parameters except 𝑓_NL to be known. Indices 𝑖𝑗𝑘 and 𝑝𝑞𝑟 run over all the eight possible ordered combinations of temperature and polarization given by 𝑇𝑇𝑇,𝑇𝑇𝐸,𝑇𝐸𝑇,𝐸𝑇𝑇,𝑇𝐸𝐸,𝐸𝑇𝐸,𝐸𝐸𝑇 and 𝐸𝐸𝐸, the combinatorial factor Δ_ℓ1ℓ2ℓ3 equals 1 when all ℓ's are different, 6 when ℓ₁=ℓ₂=ℓ₃, and 2 otherwise. The covariance matrix Cov is obtained in terms of 𝐶_ℓ^𝑇𝑇, 𝐶_ℓ^𝐸𝐸, and 𝐶_ℓ^𝑇𝐸 (see [21, 34]) by applying Wick's theorem.

For non-Gaussianity of the local type, for which the functional form 𝐹(𝑘₁,𝑘₂,𝑘₃), is given by (18), we have 𝜕𝐵_ℓ^𝑖𝑗𝑘_1ℓ2ℓ3𝜕𝑓_NL=(2ℓ₁+1)(2ℓ₂+1)(2ℓ₃+1)ℓ4𝜋¹ℓ₂ℓ₃000×2^∞₀𝑟²𝑑𝑟[−𝛼^𝑖_ℓ1𝛽^𝑗_ℓ2𝛽^𝑘_ℓ3+2perm.], (31) where the functions 𝛼 and 𝛽 are given by𝛼^𝑖_ℓ2(𝑟)≡𝜋𝑑𝑘𝑘²𝑔^𝑖_ℓ(𝑘)𝑗_ℓ𝛽(𝑘𝑟),_𝑖ℓ2(𝑟)≡𝜋𝑑𝑘𝑘⁻¹𝑔^𝑖_ℓ(𝑘)𝑗_ℓ(𝑘𝑟)Δ_Φ𝑘^𝑛𝑠−1. (32) In the previous expression we use the dimensionless power spectrum amplitude Δ_Φ, which is defined by 𝑃_Φ(𝑘)=Δ_Φ𝑘^{−3+(𝑛𝑠−1)}, where 𝑛_𝑠 is the tilt of the primordial power spectrum. One can compute the transfer functions 𝑔^𝑇_ℓ(𝑘) and 𝑔^𝐸_ℓ(𝑘) 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: 𝜕𝐵_ℓ^𝑖𝑗𝑘_1ℓ2ℓ3𝜕𝑓_NL=(2ℓ₁+1)(2ℓ₂+1)(2ℓ₃+1)ℓ4𝜋¹ℓ₂ℓ₃000×6𝑟²𝑑𝑟[−𝛼^𝑖_ℓ1𝛽^𝑗_ℓ2𝛽^𝑘_ℓ3+2perm.+𝛽^𝑖_ℓ1𝛾^𝑗_ℓ2𝛿^𝑘_ℓ3+5perm.−2𝛿^𝑖_ℓ1𝛿^𝑗_ℓ2𝛿^𝑘_ℓ3], (33) where the functions 𝛿 and 𝛾 are given by 𝛾^𝑖_ℓ2(𝑟)≡𝜋𝑑𝑘𝑘𝑔^𝑖_ℓ(𝑘)𝑗_ℓ(𝑘𝑟)Δ_Φ^1/3𝑘^{(𝑛𝑠−1)/3},𝛿^𝑖_ℓ2(𝑟)≡𝜋𝑑𝑘𝑔^𝑇_ℓ(𝑘)𝑗_ℓ(𝑘𝑟)Δ_Φ^2/3𝑘^{2(𝑛𝑠−1)/3}. (34) Recently, a new bispectrum template shape, an orthogonal shape, has been introduced [120] which characterizes the size of the signal (𝑓^ortho_NL) which peaks both for equilateral and flat-triangle configurations. The shape of non-Gaussianities associated with 𝑓^ortho_NL is orthogonal to the one associated to 𝑓^equil_NL. The bispectrum for orthogonal shape can be written as [120] 𝜕𝐵_ℓ^𝑖𝑗𝑘_1ℓ2ℓ3𝜕𝑓^ortho_NL=(2ℓ₁+1)(2ℓ₂+1)(2ℓ₃+1)ℓ4𝜋¹ℓ₂ℓ₃000×18𝑟²𝑑𝑟[−𝛼^𝑖_ℓ1𝛽^𝑗_ℓ2𝛽^𝑘_ℓ3+2perm.+𝛽^𝑖_ℓ1𝛾^𝑗_ℓ2𝛿^𝑘_ℓ32+5perm.−3𝛿^𝑖_ℓ1𝛿^𝑗_ℓ2𝛿^𝑘_ℓ3]. (35)

4.1. Estimator

An unbiased bispectrum-based minimum variance estimator for the nonlinearity parameter in the limit of full sky and homogeneous noise can be written as [17, 20, 25]𝑓_NL=1𝑁_{ℓ𝑖𝑚𝑖}ℓ₁ℓ₂ℓ₃𝑚₁𝑚₂𝑚₃𝐵_ℓ1ℓ2ℓ3𝐶_ℓ1𝐶_ℓ2𝐶_ℓ3𝑎_ℓ1𝑚1𝑎_ℓ2𝑚2𝑎_ℓ3𝑚3, (36) where 𝐵_ℓ1ℓ2ℓ3 is angle averaged theoretical CMB bispectrum for the model in consideration. The normalization 𝑁 can be calculated to require the estimator to be unbiased, ⟨𝑓_NL⟩=𝑓_NL. If the bispectrum 𝐵_ℓ1ℓ2ℓ3 is calculated for 𝑓_NL=1 then the normalization takes the following form:𝑁=_ℓ𝑖(𝐵_ℓ1ℓ2ℓ3)²𝐶_ℓ1𝐶_ℓ2𝐶_ℓ3. (37) The estimator for non-Gaussianity, (36), can be simplified using (26) to yield𝑓_NL=1𝑁⋅_{𝑙𝑖𝑚𝑖}𝑑²𝑛𝑌_𝑙1𝑚1(𝑛)𝑌_𝑙2𝑚2(𝑛)𝑌_𝑙3𝑚3(×𝑛)_∞0𝑟²𝑑𝑟𝑗_𝑙1(𝑘₁𝑟)𝑗_𝑙2(𝑘₂𝑟)𝑗_𝑙3(𝑘₃𝑟)𝐶_𝑙⁻¹₁𝐶_𝑙⁻¹₂𝐶_𝑙⁻¹₃×2𝑘²₁𝑑𝑘₁𝜋2𝑘²₂𝑑𝑘₂𝜋2𝑘²₃𝑑𝑘₃𝜋𝐹(𝑘₁,𝑘₂,𝑘₃)Δ^𝑇_𝑙1(𝑘₁)×Δ^𝑇_𝑙2(𝑘₂)Δ^𝑇_𝑙3(𝑘₃)𝑎_𝑙1𝑚1𝑎_𝑙2𝑚2𝑎_𝑙3𝑚3, (38) where 𝐹(𝑘₁,𝑘₂,𝑘₃) is a shape of 3-point function as defined in (16). Given the shape 𝐹(𝑘₁,𝑘₂,𝑘₃), one is interested in, it is conceptually straightforward to constrain the non-linearity parameter from the CMB data. Unfortunately the computation time for the estimate scales as 𝑁^5/2_pix, which is computationally challenging as even for the WMAP data the number of pixels, is of order 𝑁_pix∼10⁶. The scaling can be understood by noting that each spherical harmonic transform scales as 𝑁^3/2_pix and the estimator requires ℓ²(∝𝑁_pix) number of spherical harmonic transforms.

The computational cost decreases if the shape can be factorized as𝐹(𝑘₁,𝑘₂,𝑘₃)=𝑓₁(𝑘₁)𝑓₂(𝑘₂)𝑓₃(𝑘₃), (39) with which the estimator simplifies to𝑓_NL=1𝑁⋅^∞₀𝑑²𝑛^∞₀𝑟²𝑑𝑟³_{𝑖=1𝑙𝑖𝑚𝑖}×2𝑘²𝑑𝑘𝜋𝑗_𝑙𝑖(𝑘𝑟)𝑓_𝑖(𝑘)Δ^𝑇_𝑙𝑖(𝑘)𝐶_𝑙⁻¹_𝑖𝑎_{𝑙𝑖𝑚𝑖}𝑌_{𝑙𝑖𝑚𝑖}(𝑛), (40) and computational cost now scales as 𝑁^3/2_pix. For Planck (𝑁_pix∼5×10⁷) this translates into a speed-up by factors of millions, reducing the required computing time from thousands of years to just hours and thus making 𝑓_NL estimation feasible for future surveys. The speed of the estimator now allows sufficient number of Monte Carlo simulations, to characterize its statistical properties in the presence of real world issues such as instrumental effects, partial sky coverage, and foreground contamination. Using the Monte Carlo simulations it has been shown that estimator is indeed optimal, where optimality is defined by saturation of the Cramer Rao bound, if noise is homogeneous. Note that even for the nonfactorizable shapes, by using the flat sky approximation and interpolating between the modes, one can estimating 𝑓_NL in a computationally efficient way [121].

The extension of the estimator of 𝑓_NL from the temperature data [17] to include both the temperature and polarization data of the CMB is discussed in Babich and Zaldarriaga [34] and Yadav et al. [20, 21, 35]. Summarizing briefly, we construct a cubic statistic as a combination of (appropriately filtered) temperature and polarization maps which is specifically sensitive to the primordial perturbations. This is done by reconstructing a map of primordial perturbations and using that to define a fast estimator. We also show that this fast estimator is equivalent to the optimal estimator by demonstrating that the inverse of the covariance matrix for the optimal estimator [34] is the same as the product of inverses we get in the fast estimator. The estimator still takes only 𝑁^3/2_pix operations in comparison to the full bispectrum calculation which takes 𝑁^5/2_pix operations.

For a given shape, the estimator for non-linearity parameter can be written as 𝑓_NL=𝑆_shape/𝑁_shape, where for the equilateral, local and orthogonal shapes, the 𝑆_shape can be written as𝑆_equilateral=3𝑓_sky𝑟²𝑑𝑟𝑑²𝑛2×[𝐵(𝑛,𝑟)𝐵(𝑛,𝑟)𝐴(𝑛,𝑟)+3𝐷(𝑛,𝑟)³𝑆−2𝐵(𝑛,𝑟)𝐶(𝑛,𝑟)𝐷(𝑛,𝑟)],_local=1𝑓_sky𝑟²𝑑𝑟𝑑²𝑆𝑛𝐵(𝑛,𝑟)𝐵(𝑛,𝑟)𝐴(𝑛,𝑟),_orthogonal=9𝑓_sky𝑟²𝑑𝑟𝑑²𝑛8×[𝐵(𝑛,𝑟)𝐵(𝑛,𝑟)𝐴(𝑛,𝑟)+9𝐷(𝑛,𝑟)³−2𝐵(𝑛,𝑟)𝐶(𝑛,𝑟)𝐷(𝑛,𝑟)], (41)(42) with𝐵(𝑛,𝑟)≡_{𝑖𝑝𝑙𝑚}(𝐶⁻¹)^𝑖𝑝𝑎^𝑖_ℓ𝑚𝛽^𝑝_ℓ(𝑟)𝑌_ℓ𝑚(𝑛),𝐶(𝑛,𝑟)≡_{𝑖𝑝𝑙𝑚}(𝐶⁻¹)^𝑖𝑝𝑎^𝑖_ℓ𝑚𝛽^𝑝_ℓ(𝑟)𝑌_ℓ𝑚(𝑛),𝐴(𝑛,𝑟)≡_{𝑖𝑝𝑙𝑚}(𝐶⁻¹)^𝑖𝑝𝑎^𝑖_ℓ𝑚𝛼^𝑝_ℓ(𝑟)𝑌_ℓ𝑚(𝑛),𝐷(𝑛,𝑟)≡_{𝑖𝑝𝑙𝑚}(𝐶⁻¹)^𝑖𝑝𝑎^𝑖_ℓ𝑚𝛽^𝑝_ℓ(𝑟)𝑌_ℓ𝑚(𝑛), (43) and 𝑓_sky is a fraction of sky. Index 𝑖 and 𝑝 can either be 𝑇 or 𝐸∶𝑁=_{𝑖𝑗𝑘𝑝𝑞𝑟2≤ℓ1≤ℓ2≤ℓ3}1Δ_ℓ1ℓ2ℓ3×𝐵^{𝑝𝑞𝑟,prim}_ℓ1ℓ2ℓ3(𝐶⁻¹)_ℓ^𝑖𝑝₁(𝐶⁻¹)_ℓ^𝑗𝑞₂(𝐶⁻¹)_ℓ^𝑘𝑟₃𝐵^{𝑖𝑗𝑘,prim}_ℓ1ℓ2ℓ3. (44)

Indices 𝑖,𝑗,𝑘,𝑝,𝑞 and 𝑟 can either be 𝑇 or 𝐸. Here, Δ_ℓ1ℓ2ℓ3 is 1 when ℓ₁≠ℓ₂≠ℓ₃, 6 when ℓ₁=ℓ₂=ℓ₃, and 2 otherwise: 𝐵^{𝑝𝑞𝑟,prim}_ℓ1ℓ2ℓ3 is the theoretical bispectrum for 𝑓_NL=1 [21].

It has been shown that the previous estimators defined in (42) are minimum variance amongst bispectrum-based estimators for full sky coverage and homogeneous noise [21]. To be able to deal with the realistic data, the estimator has to be able to deal with the inhomogeneous noise and foreground masks. The estimator can be generalized to deal with partial sky coverage as well as inhomogeneous noise by adding a linear term to 𝑆_prim: 𝑆_prim→𝑆_prim+𝑆^linear_prim. For the temperature only case, this has been done in [25]. Following the same argument, we find that the linear term for the combined analysis of CMB temperature and polarization data is given by𝑆^linear_prim=−1𝑓_sky𝑟²𝑑𝑟𝑑²𝑛{2𝐵(𝑛,𝑟)⟨𝐴_sim(𝑛,𝑟)𝐵_sim(𝑛,𝑟)⟩_MC+𝐴(𝑛,𝑟)⟨𝐵²_sim(𝑛,𝑟)⟩_MC}, (45) where 𝐴_sim(𝑛,𝑟) and 𝐵_sim(𝑛,𝑟) are the 𝐴 and 𝐵 maps generated from Monte Carlo simulations that contain signal and noise, and ⟨⋯⟩ denotes the average over the Monte Carlo simulations.

The generalized estimator is given by𝑓_NL=𝑆_prim+𝑆^linear_prim𝑁. (46) Note that ⟨𝑆^linear_prim⟩_MC=−⟨𝑆_prim⟩_MC, and this relation also holds for the equilateral shape. Therefore, it is straightforward to find the generalized estimator for the equilateral shape: first, find the cubic estimator of the equilateral shape, 𝑆_equil., and take the Monte Carlo average, ⟨𝑆_equil.⟩_MC. Let us suppose that 𝑆_equil. contains terms in the form of 𝐴𝐵𝐶, where 𝐴, 𝐵, and 𝐶 are some filtered maps. Use the Wick's theorem to rewrite the average of a cubic product as ⟨𝐴𝐵𝐶⟩_MC=⟨𝐴⟩_MC⟨𝐵𝐶⟩_MC+⟨𝐵⟩_MC⟨𝐴𝐶⟩_MC+⟨𝐶⟩_MC⟨𝐴𝐵⟩_MC. Finally, remove the MC average from single maps and replace maps in the product with the simulated maps ⟨𝐴⟩_MC⟨𝐵𝐶⟩_MC+⟨𝐵⟩_MC⟨𝐴𝐶⟩_MC+⟨𝐶⟩_MC⟨𝐴𝐵⟩_MC→𝐴⟨𝐵_sim𝐶_sim⟩_MC+𝐵⟨𝐴_sim𝐶_sim⟩_MC+𝐶⟨𝐴_sim𝐵_sim⟩_MC. This operation gives the correct expression for the linear term, both for the local form and the equilateral form.

The main contribution to the linear term comes from the inhomogeneous noise and sky cut. For the temperature only case, most of the contribution to the linear term comes from the inhomogeneous noise, and the partial sky coverage does not contribute much to the linear term. This is because the sky-cut induces a monopole contribution outside the mask. In the analysis, one subtracts the monopole from outside the mask before measuring 𝑆_prim, which makes the linear contribution from the mask small [25]. For a combined analysis of the temperature and polarization maps, however, the linear term does get a significant contribution from a partial sky coverage. Subtraction of the monopole outside of the mask is of no help for polarization, as the monopole does not exist in the polarization maps by definition. (The lowest relevant multipole for polarization is 𝑙=2.)

The estimator is still computationally efficient, taking only 𝑁^3/2_pix (times the 𝑟 sampling, which is of order 100) operations in comparison to the full bispectrum calculation which takes 𝑁^5/2_pix operations. Here 𝑁_pix refers to the total number of pixels. For Planck, 𝑁_pix∼5×10⁷, and so the full bispectrum analysis is not feasible while our analysis is.

5. Constraints from the CMB Bispectrum

5.1. Current Status

Currently the the Wilkinson Microwave Anisotropy Probe (WMAP) satellite provides the “best” (largest number of signal dominated modes) CMB data for non-Gaussianity analysis. Over the course of WMAP operation the field of non-Gaussianity has made vast progress both in terms of theoretical predictions of non-Gaussianities from inflation and improvement in the bispectrum- based estimators. At the time of WMAP's first data release in 2003 the estimator was suboptimal in the presence of partial sky coverage and/or inhomogeneous noise. With the sub-optimal estimator, one could not use the entirety of WMAP data and only the data up to 𝑙_max350 were used to obtain the constraint 𝑓^local_NL=38±96(2𝜎) [23]. These limits were around 30 times better than the previous constraints of |𝑓_NL|<1.5×10³ from the Cosmic Background Explorer (COBE) satellite [22].

By the time of second WMAP release in 2007, the estimator was generalized by adding a linear to the KSW estimator which allows to deal with partial sky coverage and inhomogeneous noise. The idea of adding a linear term to reduce excess variance due to noise inhomogeneity was introduced in [25]. Applied to a combination of the Q, V, and W channels of the WMAP 3-year data up to 𝑙_max∼400, this estimator had yielded the tightest constraint at the time on 𝑓_NL as −36<𝑓_NL<100(2𝜎) [26]. This estimator was further generalized to utilize both the temperature and 𝐸-polarization information in [20], where it was pointed out that the linear term had been incorrectly implemented in 30 of [25]. Using MonteCarlo simulations it has been shown that this corrected estimator is nearly optimal and enables analysis of the entire WMAP data without suffering from a blow-up in the variance at high multipoles⁶. The first analysis using this estimator shows an evidence of non-Gaussianity of local type at around 2.8𝜎 in the WMAP 3-year data. Independent analysis shows the evidence of non-Gaussianity at lower significance, around 2.5𝜎 (see Table 1).

Table 1 

Hint of local non-Gaussianity at 2𝜎 level.

By the time of the third WMAP data release (with 5-year obsevational data) in 2008 the 𝑓_NL estimation technique was improved further by implementing the covariance matrix including inhomogeneoous noise to make the estimator completely optimal [122]. Using the optimal estimator and using the entirety of WMAP 3-year data, there is an evidence for non-Gaussianity of local type at around 2.5𝜎 level 𝑓_NL≈58±23(1𝜎) [122]. However with WMAP 5-year data the significance goes down from ∼2.5𝜎 to ∼1.8𝜎 [122]. Table 2 compares the constraints obtained by different groups using WMAP 3-year and WMAP 5-year data. Figure 6 shows this comparison in more detail, showing the constraints also as a function of maximum multipole 𝑙_max used in the analysis. Few comments are in place: (1) constraints on 𝑓_NL from WMAP 3-year data as a function of 𝑙_max show a trend where the mean value rises at around 𝑙_max=450, below which data is consistent with Gaussianity and above which there is deviation from Gaussianity at above 2𝜎. The result becomes roughly independent of ℓ_max>550 with evidence for non-Gaussianity at around 2.5𝜎 level (2) independent analysis and using different estimators (optimal and near-optimal with linear term) see this deviation from non-Gaussianity at around 2.5𝜎 in WMAP 3-year data, (3) significance of non-Gaussianity goes down to around 2𝜎 with WMAP 5-year data. The drop in the mean value between WMAP 3-year and 5-year data can be attributed to statistical shift.

Table 2 

Summary of constraints on local non-Gaussianity.

(a)

(b)

Figure 6 

(a) Constraints on local 𝑓NL using WMAP 3-year data as a function of maximum multipole ℓmax used in the analysis. The red circles are the results obtained using near optimal estimator by Yadav and Wandelt [28]. The green triangles are using the the near-optimal estimator by Smith et al. [122]. The blue square results are obtained using either the optimal estimator by Smith et al. [122]. For all the three analysis Kp0 mask was used. (b) Comparison between the 5-year results (optimal estimator, raw maps) reported in Komatsu et al. [48] and results obtained using the optimal or suboptimal estimator by Smith et al. [122].

The best constraints on the equilateral and orthogonal shape of non-Gaussianity using the WMAP 5-year data are 𝑓^equil_NL=155±140(1𝜎) and 𝑓^orthog._NL=149±110(1𝜎) respectively [120].

As we were completing this paper, the WMAP 7-year data was released, with constraints 𝑓^local_NL=32±21(1𝜎),𝑓^equil._NL=26±140(1𝜎), and 𝑓^orthog_NL=−202±104(1𝜎) [123].

5.2. Future Prospects

Now we discuss the future prospects of using the bispectrum estimators for constraining the non-linearity parameter 𝑓_NL for local and equilateral shapes. We compute the Fisher bounds for three experimental setups, (1) cosmic variance limited experiment with perfect beam (ideal experiment hereafter), (2) Planck satellite with and noise sensitivity Δ_𝑝=56𝜇K-arcmin and beam FWHM 𝜎=7^′, and (3) a futuristic CMBPol-like satellite experiment with noise sensitivity Δ_𝑝=1.4𝜇K-arcmin and beam FWHM 𝜎=4^′ (CMBPol hereafter). Beside 𝑓_NL we fix all the other cosmological parameters to a standard fiducial model with a flat Λ𝐶𝐷𝑀 cosmology, with parameters described by the best fit to WMAP 5-year results [48], given by Ω_𝑏=0.045,Ω_𝑐=0.23,𝐻₀=70.5,𝑛_𝑠=0.96,𝑛_𝑡=0.0, and 𝜏=0.08. We calculate the theoretical CMB transfer functions and power spectrum from publicly available code CMBFAST [47]. We also neglect any non-Gaussianity which can be generated during recombination or there after. We discuss the importance and effect of these nonprimordial non-Gaussianities in the next section.

The scaling of signal-to-noise as a maximum multipole 𝑙_max for the local [124, 125] and equilateral model [126] are(𝑆𝑁)_local∝lnℓ_max𝑆,(𝑁)_equil∝ℓ_max. (47) In principle one could go to arbitrary high 𝑙_max but in reality secondary signals will certainly overwhelm primary signal beyond 𝑙_max>3000∶ we restrict to the analysis to 𝑙_max=3000. In Figure 7, we show the 1𝜎 Fisher bound as function of maximum multipole ℓ_max, for local and equilateral type of non-Gaussianity. For both local and equilateral cases, we show the Fisher bound for the analysis using only the CMB temperature information (𝑇𝑇𝑇), only the CMB polarization information (𝐸𝐸𝐸), and the combined temperature and polarization analysis. Note that by having both the temperature and 𝐸-polarization information one can improve the sensitivity by combining the information. Apart from combining the 𝑇 and 𝐸 signal, one can also do cross-checks and diagnostics by independently analysing the data. Temperature and polarization will have different foregrounds and instrumental systematics.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7 

Fisher predictions for minimum detectable 𝑓NL (at 1 𝜎) as a function of maximum multipole ℓmax. Upper panels are for the local model while ((d),(e), and (f)) are for the equilateral model. ((a) and (d)) show an ideal experiment, ((b) and(e)) are for CMBPol like experiment with noise sensitivity Δ𝑝=1.4𝜇K-arcmin and beam FWHM 𝜎=4′ and ((c) and (f)) are for Planck- like satellite with and noise sensitivity Δ𝑝=40𝜇K-arcmin and beam FWHM 𝜎=5′. In all the panels, the solid lines represent temperature and polarization combined analysis; dashed lines represent temperature only analysis; dot-dashed lines represent polarization only analysis.

A CMBPol-like experiment will be able to achieve the sensitivity of Δ𝑓^local_NL≃2(1𝜎) for non-Gaussianity of local type and Δ𝑓^equil._NL≃13(1𝜎) for non-Gaussianity of equilateral type. For the local type of non-Gaussianity, this amounts to an improvement of about a factor of 2 over the Planck satellite and about a factor of 12 over current best constraints. These estimates assume that foreground cleaning can be done perfectly that is, the effect of residual foregrounds has been neglected. Also the contribution from unresolved point sources and secondary anisotropies such as ISW-lensing and SZ-lensing has been ignored.

Running Non-Gaussianity
The primordial non-Gaussian parameter 𝑓_NL has been shown to be scaledependent in several models of inflation with a variable speed of sound, such as Dirac-Born-Infeld (DBI) models. Starting from a simple ansatz for a scale-dependent amplitude of the primordial curvature bispectrum for primordial non-Gaussianity, 𝑓_NL→𝑓_NL(𝐾𝑘_𝑝)^𝑛NG, (48) where 𝐾≡(𝑘₁𝑘₂𝑘₃)^1/3 and 𝑘_𝑝 is a pivot point. The primordial bispectrum is therefore determined in terms of two parameters: the amplitude 𝑓_NL and the new parameter 𝑛_NG quantifying its running. One can generalize the Fisher matrix analysis of the bispectra of the temperature and polarization of the CMB radiation and derive the expected constraints on the parameter 𝑛_NG that quantifies the running of 𝑓_NL(𝑘) for current and future CMB missions such as WMAP, Planck, and CMBPol. We will consider some nonzero 𝑓_NL as our fiducial value for the Fisher matrix evaluation. Clearly, in order to be able to constrain a scaledependence of 𝑓_NL, its amplitude must be large enough to produce a detection. If 𝑓_NL is too small to be detected (𝑓_NL<2 is a lowest theoretical limit even for the ideal experiment), we will obviously not be able to measure any of its features, either. The following we will then always consider a fiducial value of 𝑓_NL large enough to enable a detection. Figure 8 shows the 1−𝜎 joint constraints on 𝑓_NL and 𝑛_NG. In the event of a significant detection of the non-Gaussian component, corresponding to 𝑓_NL=50 for the local model and 𝑓_NL=100 for the equilateral model of non-Gaussianity, is able to determine 𝑛_NG with a 1−𝜎 uncertainty of Δ𝑛_NG≃0.1 and Δ𝑛_NG≃0.3, respectively, for the Planck mission and a factor of two better for CMBPol. In addition to CMB, one can include the information of the galaxy power spectrum, galaxy bispectrum, and cluster number counts as a probe of nonGaussianity on small scales to further constrain the two parameters [127].

(a)

(b)

(c)

(d)

Figure 8 

1-𝜎 constraints on 𝑓NL and 𝑛NG for local (a) and (b) and equilateral model (c) and (d) assuming 𝑘𝑝=0.04 Mpc−1 and fiducial values 𝑓NL=50, 𝑛NG=0 for the local case and 𝑓NL=100,𝑛NG=0 for the equilateral case. Dashed lines correspond to the limits from the temperature information alone, dotted lines to polarization (𝐸𝐸𝐸), while the continuous lines correspond to all bispectrum combinations. We consider Planck (a) and (c) and CMBPOl-like (b) and (d) CMB experiment.

5.3. Contaminations

A detection of non-Gaussianity has profound implications on our understanding of the early Universe. Hence it is important to know and quantify all the possible sources of non-Gaussianities in the CMB. Here we highlight some sources of non-Gaussianities due to second-order anisotropies after last scattering surface and during recombination. The fact that Gaussian initial conditions imply Gaussianity of the CMB is only true at linear order. We will also discuss the effects of instrumental effects and uncertainties in the cosmological parameters on the bispectrum estimate.

5.3.1. Secondary Non-Gaussianities

Current analysis of the CMB data ignore the contributions from the secondary non-Gaussianities. For WMAP resolutions it may not be a bad approximation. Studies of the dominant secondary anisotropies conclude that they are negligible for the analysis of the WMAP data for 𝑙_max<800 [108, 128]. However on smaller angular scales several effects start to kick in, for example, (1) the bispectrum contribution due to unresolved point source like thermal Sunyaev-Zeldovich clusters or standard radio sources, (2) three-way correlations between primary CMB, lensed CMB, and secondary anisotropies. We will refer to the bispectrum generated due to these three-way correlations as 𝐵^{secondary-𝜅}, where some secondaries are, the integrated Sachs-Wolfe (ISW) 𝐵^ISW-𝜅, Sunyaev-Zeldovich signal and Rees-Sciama [23, 108, 128–131].

For Future experiments such as Planck and CMBPOl the joint estimation of primordial and secondary bispectrum will be required. The observed bispectrum in general would take the following form:𝐵^obs_ℓ1ℓ2ℓ3=𝑓_NL𝐵^prim_ℓ1ℓ2ℓ3+𝑏_ps𝐵^ps_ℓ1ℓ2ℓ3+𝐴_𝑆𝑍𝐵^SZ-𝜅_ℓ1ℓ2ℓ3+𝐴_ISW𝐵^ISW-𝜅_ℓ1ℓ2ℓ3+⋯. (49) The amplitude of bispectrum due to primary-lensing-secondary cross-correlation is proportional to the product of primary CMB power-spectrum and power spectrum of cross-correlation between secondary and lensing signals.

The reduced bispectrum from the residual point sources (assuming Poisson distributed) is constant, that is, 𝑏^ps_ℓ1ℓ2ℓ3=constant. The value of the constant will depend on the flux limit at which the point source can be detected and on assumed flux and frequency distribution of the sources.

Depending on the shape of primordial bispectrum in consideration, some secondary bispectra are more dangerous than others. For example, ISW-lensing 𝐵^ISW-𝜅 peaks at the “local” configurations hence,it is more dangerous for local primordial shape than the equilateral primordial shape. For example for the Planck satellite if the secondary bispectrum is not incorporated in the analysis, the ISW-lensing contribution will bias the estimate for the local 𝑓_NL by around Δ𝑓^local_NL≈10 [19]. The bispectrum contribution from primary-lensing-Rees-Sciama signal also peaks at squeezed limit and contribute to effective local 𝑓^local_NL≈10 [132]. For Planck sensitivity the point source will contamination the local non-Gaussianity by around Δ𝑓^local_NL∼1 [133]. A recent analysis of the full second-order Boltzmann equation for photons [134] claims that second order effects add a contamination Δ𝑓_NL∼5.

The generalization of the Fisher matrix given by (30) to include multiple bispectrum contribution is ℱ^(𝑋𝑌)_𝑎𝑏=_{{𝑖𝑗𝑘,pqr}ℓ1≤ℓ2≤ℓ3}1Δ_ℓ1ℓ2ℓ3𝜕𝐵^pqr_ℓ^,(𝑋)_1ℓ2ℓ3𝜕𝑝_𝑎×(Cov⁻¹)_{𝑖𝑗𝑘,pqr}𝜕𝐵_ℓ^{𝑖𝑗𝑘,(𝑌)}_1ℓ2ℓ3𝜕𝑝_𝑏, (50) where the additional 𝑋 and 𝑌 indices denote a component such as primordial, point-sources, and ISW-lensing,and so forth. For fixed cosmological parameters, the signal to noise (𝑆/𝑁)_𝑖 for the component 𝑖 is (𝑆𝑁)_𝑖=1ℱ^(𝑖𝑖). (51)

5.3.2. Non-Gaussianities from Recombination

Non-Gaussianities can be generated during recombination. One requires to solve second-order Boltzmann and Einstein equations to evaluate the effect. The second-order effect on CMB is an active field of study [135–153] see [154] for a recent review. The non-Gaussianities produced during recombination comprise of various effects, for example, see [135].

The dominant bispectrum due to perturbative recombination comes from perturbations in the electron density. The amplitude of perturbations of the free electron density 𝛿_𝑒 is around a factor of 5 larger than the baryon density perturbations [155]. The bispectrum generated due to 𝛿_𝑒 peaks around the “local” configuration with corresponding effective non-linearity amplitude 𝑓_NL∼few [156, 157].

The bispectrum contribution due to second-order terms which are the products of the first-order perturbations is calculated in [158]. The bispectrum contribution from these terms which also peak for the squeezed triangles is small and can be neglected in the analysis. For example, the signal to noise is about 0.4 at 𝑙_max=2000 for a full-sky, cosmic variance limited experiment.

Another contribution to bispectrum which peaks for the equilateral configurations comes from the nonlinear evolution of the second order gravitational potential. Because of this effect, the minimum detectable non-Gaussianity parameter 𝑓^equil._NL changes by Δ𝑓^equil._NL=𝒪(10) for Planck-like experiment [126]. The bispectrum peaks for the equilateral shape because the growth of potential happens on scales smaller than the horizon size.

On large scales, in the absence of primordial non-Gaussianities and assuming matter domination (so that the early and late ISW can be neglected), it has been shown in [151] that for the squeezed limit the effective 𝑓_NL generated by second-order gravitational effects on the CMB 𝑓_NL=−1/6−cos(2𝜃) (also see [141, 152, 153]). Here 𝜃 is the angle between the short and the long modes. The angle dependent contribution comes from lensing.

5.3.3. Effect of Cosmological Parameter Uncertainties

Impact of uncertainties on the cosmological parameters effect, the error bar on 𝑓_NL. The effect of cosmological parameters have been discussed in [25, 26, 28, 159]. The cosmological parameters are determined using the 2-point statistics of the CMB, and therefor we expect the largest effect of 𝑓_NL would come from those parameters which leave the CMB power spectrum unchanged while change the bispectrum. The expectation value of the estimator ⟨𝑓_NL1⟩=𝑁_{ℓ1,ℓ2,ℓ3}𝐵_ℓ1ℓ2ℓ3𝐵_ℓ1ℓ2ℓ3𝐶_ℓ1𝐶_ℓ2𝐶_ℓ3, (52) changes with the change in cosmological parameters. Here 𝐵_ℓ1ℓ2ℓ3 is the true CMB bispectrum. When changing the parameters the normalization 𝑁 should be changed to make the estimator unbiased. In general for a set of cosmological paramerets {𝑝_𝑖}, the error in 𝑓_NL is given by [159]𝛿𝑓_NL=_𝑖𝑗𝜕𝑓_NL𝜕𝑝_𝑖|_{𝑝𝑖=𝑝𝑖}𝜕𝑓_NL𝜕𝑝_𝑗|_{𝑝𝑗=𝑝𝑗}Cov(𝑝_𝑖,𝑝_𝑗). (53) Here the average parameter values 𝑝_𝑖 and their covariance matrix Cov(𝑝_𝑖,𝑝_𝑗) can be determined using CMB-likelihood analysis tools.

If the parameters are allowed to vary in the analysis then for WMAP this increases the 1𝜎 uncertainty in 𝑓_NL by 𝛿𝑓^local_NL/𝑓_NL≈16% for the local shape and 𝛿𝑓^equil_NL/𝑓_NL≈14% for the equilateral shape. For Planck experiment, the increases in 1𝜎 uncertainity are 𝛿𝑓^local_NL/𝑓_NL≈5% for local shape and 𝛿𝑓^equil._NL/𝑓_NL≈4% for the equilateral shape. Most of the contribution to the error comes from three cosmological parameters, the amplitude of scalar perturbations Δ_Φ, the tilt of the power spectrum of the saclar perturbations 𝑛_𝑆, and re-ionization optical depth 𝜏.

For modes inside the horizon during reionization, the reionization optical depth 𝜏 appears as a multiplicative factor 𝑒^−𝜏 in front of transfer function 𝑔^𝑖_ℓ. For local model one of the mode is outside so the effect on bispectrum 𝑏̃^local_ℓ1ℓ2ℓ3=exp(−2𝜏)𝑏^local_ℓ1ℓ2ℓ3 and for equilateral model all the modes are inside the horizon so 𝑏̃^equil._ℓ1ℓ2ℓ3=exp(−3𝜏)𝑏^equil._ℓ1ℓ2ℓ3. This reduces to 𝛿𝑓^local_NL≃−2𝑓_NL𝜏 for local model and 𝛿𝑓^equil._NL≃−3𝑓_NL𝜏 for equilateral model.

The effect of amplitude of perturbations can be seen by noting that the level of non-Gaussianity is given by 𝑓_NL⋅Δ_Φ^1/2. Hence the decreases (increase) in the amplitude of perturbations relax (tighten) the constraints on 𝑓_NL. The effect of red tilt (𝑛_𝑠<1) can be thought of as a reduction in power on at scales shorter than first peak and enhancement of power on larger scales. The effect of blue tilt is just opposite of red tilt. For local shape, the limit on 𝑓_NL becomes tighter proportional to Δ^1/2_long [26]. Note that [159] shows that the effect of cosmological parameters is negligible if the parameters are allowed to vary in the analysis and then marginalize over.

5.3.4. Instrumental Effects and Distortions along the Line of Sight

Here we point out that any cosmological or instrumental effect that can be modelled as a line- of sight CMB distortions of the primary CMB doesnot generate new bispectrum contribution. Although they can modify the primordial bispectrum. A general model of line-of-sight distortions of the primary CMB are described in [58, 59, 160], where the changes in the Stokes parameter of the CMB due to distortions along the line-of-sight can be written as 𝛿[𝑄±𝑖𝑈](𝐧)=[a±𝑖2𝑤](𝐧)[𝑄±𝑖𝑈](𝐧)+[𝑓₁±𝑖𝑓₂](𝐧)[𝑄∓𝑖𝑈](𝐧)+[𝛾₁±𝑖𝛾₂](𝐧)𝑇(𝐧)+𝜎𝑝(𝐧)⋅∇[𝑄±𝑖𝑈](𝐧;𝜎)+𝜎[𝑑₁±𝑖𝑑₂](𝐧)[𝜕₁±𝑖𝜕₂]𝑇(𝐧;𝜎)+𝜎²𝑞(𝐧)[𝜕₁±𝑖𝜕₂]²𝑇(𝐧;𝜎)+⋯. (54) The first line captures the distortions in a single perfectly known direction 𝐧. The distortions in second line capture mixing of the polarization fields in a local region of length scale 𝜎 around 𝐧. We Taylor expand the CMB fields 𝑄,𝑈, and 𝑇 around the point 𝐧 and consider the leading-order terms. Here 𝑄,𝑈, and 𝑇 stand for primordial (undistorted) CMB fields. Since (𝑄±𝑖𝑈)(𝐧) is spin ±2 field, 𝑎(𝐧) is a scalar field that describes modulation in the amplitude of the fields in a given direction 𝐧; 𝜔(𝐧) is also a scalar field that describes the rotation of the plane of polarization, (𝑓₁±𝑖𝑓₂) are spin ±4 fields that describe the coupling between two spin states (spin-flip), and (𝛾₁±𝑖𝛾₂)(𝐧) are spin ±2 fields that describe leakage from the temperature to polarization (monopole leakage hereon). Distortions in the second line of (54), (𝑝₁±𝑝₂),(𝑑₁±𝑑₂), and 𝑞 are measured in the units of the length scale 𝜎. The field (𝑝₁±𝑖𝑝₂)(𝐧) is a spin ±1 field and describes the change in the photon direction; we will refer to it as a deflection field. Finally (𝑑₁±𝑑₂)(𝐧) and 𝑞(𝐧) describe leakage from temperature to polarization, (𝑑₁±𝑑₂)(𝐧) is spin ±1 field, and we will refer to it as dipole leakage; 𝑞(𝐧) is a scalar field that we will call quadrupole leakage.

These distortions can be produced by various cosmological processes such as weak gravitational lensing of the CMB, screening effects from patchy reionization, rotation of the plane of polarization due to magnetic fields or parity violating physics, and various instrumental systematics such as gain fluctuations, pixel rotation, differential gain, pointing, differential ellipticity are also captured via line-of-sight distortions. All these distortions modify the primordial bispectrum as 𝐵_{(ℓ1,ℓ2,ℓ3)}=𝐵_{(ℓ1,ℓ2,ℓ3)}+𝑑²ℓ^′(2𝜋)²𝐶_𝑙^𝒟𝒟_′×[𝐵_{(ℓ1,ℓ2−ℓ′,ℓ3+ℓ′)}𝑊^𝒟(ℓ₁−ℓ^′)𝑊^𝒟(ℓ₂−ℓ^′′)+perm.], (55) where 𝑊 is a window which depends on the type of distortion in consideration and tells how the primordial CMB bispectrum modes are coupled to the distortion field power spectrum 𝐶_ℓ^𝒟𝒟. The effect of the distortions on the bispectrum is to smooth out the acoustic features. These effects for the case of lensing have been shown to be small and can be neglected [161, 162].

In [163], the impact of the 1/𝑓 noise and asymmetric beam on local 𝑓^local_NL has been found insignificant in the context of a Planck-like experiment.

6. Other Probes of Non-Gaussianity in the CMB

Although using the full bispectrum is the most sensitive cubic statistic other statistical methods may be sensitive to different aspects of non-Gaussianity and, more importantly, different methods have different systematic effects. Therefore it is important to study various probes. In this section we will discuss some of the methods which have been recently used or developed to test for primordial non-Gaussianities in the CMB.

Trispectrum
The four-point function in harmonic space is called trispectrum, which can be written as ⟨𝑎_𝑙1𝑚1𝑎_𝑙2𝑚2𝑎_𝑙3𝑚3𝑎_𝑙4𝑚4⟩=_𝐿𝑀(−1)^𝑀ℓ₁ℓ₂𝐿𝑚₁𝑚₂𝑚₃ℓ₃ℓ₄𝐿𝑚₁𝑚₂𝑚₃𝑇^𝑙1𝑙2_𝑙3𝑙4(𝐿), (56) where 𝑇^𝑙1𝑙2_𝑙3𝑙4(𝐿) is the angular averaged trispectrum, 𝐿 is the length of a diagonal that forms triangles with 𝑙₁ and 𝑙₂ and with 𝑙₃ and 𝑙₄, and the matrix is the Wigner 3-𝑗 symbol. The trispectrum contains unconnected part, 𝑇_𝐺, 𝑇_𝐺^𝑙1𝑙2_𝑙3𝑙4(𝐿)=(−1)^𝑙1+𝑙3(2𝑙₁+1)(2𝑙₃+1)𝐶_𝑙1𝐶_𝑙3𝛿_𝑙1𝑙2𝛿_𝑙3𝑙4𝛿_𝐿0+(2𝐿+1)𝐶_𝑙1𝐶_𝑙2[(−1)^{𝑙1+𝑙2+𝐿}𝛿_𝑙1𝑙3𝛿_𝑙2𝑙4+𝛿_𝑙1𝑙4𝛿_𝑙2𝑙3]. (57) which comes from the Gaussian part of the perturbations, and the connected part 𝑇_𝑐 which contains non-Gaussian signatures. Using permutation symmetry, one may write the connected part of the trispectrum as 𝑇_𝑐^𝑙1𝑙2_𝑙3𝑙4(𝐿)=𝑃^𝑙1𝑙2_𝑙3𝑙4(𝐿)+(2𝐿+1)_𝐿′×[(−1)^𝑙2+𝑙3ℓ₁ℓ₂𝐿ℓ₄ℓ₃𝐿^′𝑃^𝑙1𝑙3_𝑙2𝑙4(𝐿^′)+(−1)^𝐿+𝐿′ℓ₁ℓ₂𝐿ℓ₃ℓ₄𝐿^′𝑃^𝑙1𝑙4_𝑙3𝑙2(𝐿^′)]. (58) where 𝑃^𝑙1𝑙2_𝑙3𝑙4(𝐿)=𝑡^𝑙1𝑙2_𝑙3𝑙4(𝐿)+(−1)^{2𝐿+𝑙1+𝑙2+𝑙3+𝑙4}𝑡^𝑙2𝑙1_𝑙4𝑙3(𝐿)+(−1)^{𝐿+𝑙3+𝑙4}𝑡^𝑙1𝑙2_𝑙4𝑙3(𝐿)+(−1)^{𝐿+𝑙1+𝑙2}𝑡^𝑙2𝑙1_𝑙3𝑙4(𝐿). (59) Here, the matrix is the Wigner 6-𝑗 symbol, and 𝑡^𝑙1𝑙2_𝑙3𝑙4(𝐿) is called the reduced trispectrum, which contains all the physical information about non-Gaussianities. For non-Gaussianity of local-type for which Φ(𝑥)=Φ_𝐿(𝑥)+𝑓_NL[Φ²_𝐿−⟨Φ²_𝐿⟩]+𝑔_NLΦ³_𝐿, (60) both 𝑓_NL and 𝑔_NL contribute to the trispectrum, but only 𝑓_NL contributes to the bispectrum. Tripectrum-based estimators for measuring 𝑓_NL and 𝑔_NL have been developed [164–168]. For local template, the bispectrum nearly contains all the information on 𝑓_NL [166] however if the non-Gaussianity is seen in bispectrum, trispectrum can serve as a important cross-check. Generically for single field slow-roll models the trispectrum is small and unobservable [169] however for more general single field models whenever the equilateral bispectrum is large, the trispectrum is large as well [170–172]. For example for equilateral non-Gaussianity [173] study how to tune the model parameters to get large trispectrum and small bispectrum. For multifield inflation one can construct models that predicts small 𝑓_NL but large 𝑔_NL, for example, [174] discussed the local form from a multi-field inflation and briefly mentioned the condition in their class of models to get the large trispectrum and small bispectrum. Joint constraints on both 𝑓_NL and 𝑔_NL have the potential to add to the specificity of the search for primordial non-Gaussianity. For a given model, these two numbers will often be predicted in terms of a single model parameter, such as a coupling constant see, for example, [175] for the case of ekyprotic models. Using WMAP 5-year data, the constraints on 𝑔_NL using the trispectrum are −7.4<𝑔_NL/10⁵<8.2 at 2𝜎 [176].

Minkowski Functionals
Minkowski Functionals (MFs) describe morphological properties (such as area, circumference, and Euler characteristic) of fluctuating fields [177–180]. For a 𝑑-dimensional fluctuating field, 𝑓, the 𝑘th Minkowski Functionals of weakly non-Gaussian fields in, 𝑉_𝑘^(𝑑)(𝜈) for a given threshold 𝜈=𝑓/𝜎₀ can be written as [181, 182] 𝑉_𝑘^(𝑑)=1(𝜈)(2𝜋)^(𝑘+1)/2𝜔_𝑑𝜔_𝑑−𝑘𝜔_𝑘(𝜎₁√𝑑𝜎₀)^𝑘𝑒^−𝜈2/2×{𝐻_𝑘−11(𝜈)+[6𝑆⁽⁰⁾𝐻_𝑘+2𝑘(𝜈)+3𝑆⁽¹⁾𝐻_𝑘+(𝜈)𝑘(𝑘−1)6𝑆⁽²⁾𝐻_𝑘−2(𝜈)]𝜎₀+𝒪(𝜎²₀)}, (61) where 𝜎₀≡⟨𝑓²⟩^1/2 is the variance of the fluctuating field, 𝐻_𝑛(𝜈) are the Hermite polynomials, 𝜔_𝑘≡𝜋^𝑘/2/Γ(𝑘/2+1), and finally 𝑆^(𝑖) are the “skewness parameters” defined as 𝑆⁽⁰⁾≡⟨𝑓³⟩𝜎⁴₀,𝑆⁽¹⁾3≡−4⟨𝑓²(∇²𝑓)⟩𝜎²₀𝜎²₁,𝑆⁽²⁾≡−3𝑑2(𝑑−1)⟨(∇𝑓)⋅(∇𝑓)(∇²𝑓)⟩𝜎⁴₁, (62) which characterize the skewness of fluctuating fields and their derivatives. Here 𝜎_𝑖 characterizes the variance of the fluctuating field and is given by 𝜎²_𝑖=14𝜋_𝑙(2ℓ+1)[ℓ(ℓ+1)]^𝑖𝐶²_ℓ. (63) For CMB, for which 𝑑=2 and 𝑓=Δ𝑇/𝑇, the skewness parameters are [183] 𝑆⁽⁰⁾=14𝜋𝜎⁴_{0𝑙𝑖𝑚𝑖}𝐵^{𝑚1𝑚2𝑚3}_{𝑙1𝑙2𝑙3}𝒢^{𝑚1𝑚2𝑚3}_{𝑙1𝑙2𝑙3}𝑊_𝑙1𝑊_𝑙2𝑊_𝑙3,𝑆⁽¹⁾=316𝜋𝜎²₀𝜎²_{1𝑙𝑖𝑚𝑖}𝑙₁(𝑙₁+1)+𝑙₂(𝑙₂+1)+𝑙₃(𝑙₃+1)3×𝐵^{𝑚1𝑚2𝑚3}_{𝑙1𝑙2𝑙3}𝒢^{𝑚1𝑚2𝑚3}_{𝑙1𝑙2𝑙3}𝑊_𝑙1𝑊_𝑙2𝑊_𝑙3,𝑆⁽²⁾=38𝜋𝜎⁴_{1𝑙𝑖𝑚𝑖}{[𝑙₁(𝑙₁+1)+𝑙₂(𝑙₂+1)−𝑙₃(𝑙₃+1)]3×𝑙₃(𝑙₃+1)+(𝑐yc.)}𝐵^{𝑚1𝑚2𝑚3}_{𝑙1𝑙2𝑙3}𝒢^{𝑚1𝑚2𝑚3}_{𝑙1𝑙2𝑙3}×𝑊_𝑙1𝑊_𝑙2𝑊_𝑙3, (64) where 𝐵_{𝑙1𝑙2𝑙3} is the CMB bispectrum, 𝑊_𝑙 represents a smoothing kernel which depends on the experiment beam and 𝒢^{𝑚1𝑚2𝑚3}_{𝑙1,𝑙2,𝑙3} is the usual Gaunt function.

Since MFs can be determined as weighted sum of the bispectrum, they contain less information than the bispectrum. MFs can still be useful because they perhaps suffer from different systematics, though they are less specific to primordial non-Gaussianity since they measure a smaller number of independent bispectrum modes. Also, the bispectrum is defined in Fourier (or harmonic) space while the MFs are defined in real space. Limits on non-Gaussianity of local-type from the MFs of the WMAP 5-year temperature data are −70<𝑓_NL<91(2𝜎) [188]. The MFs from the Planck temperature data should be sensitive to 𝑓_NL∼20 at 1𝜎 level [193] in contrast to bispectrum which is sensitive to 𝑓_NL∼5 at 1𝜎 level. Note that polarization data further improves the sensitivity.

Wavelets
Several studies have used wavelet representations of the WMAP maps to search for a non-Gaussian signal [194–200]. In most of these studies, wavelets were used as a tool for blind searches of non-Gaussian anomalies in a basis with resolution in both scale and location. However, in some more recent studies, wavelets were tuned to look for non-Gaussianity of a particular type. In the context of searches for primordial non-Gaussianity of local type, wavelet-based estimators for 𝑓_NL have been built by extracting a signature of local non-Gaussianity that is cubic in the wavelet coefficients from simulations of non-Gaussian skies and searching for this signature in data. This ability to calibrate on a set of simulations makes the wavelet approach very flexible. While not optimal in a least-squared sense, using a wavelet representation can be thought of as a generalized cubic statistic with a different weighting scheme to the optimal bispectrum estimator. Using such estimators therefore provides a useful exploration of nearly optimal cubic estimators similar to the full bispectrum estimator. Any believable detection of non-Gaussianity should be robust to such changes in the analysis. Similarly, contaminating non-Gaussianity from astrophysical and instrumental systematics will propagate through the analysis in a different way to the bispectrum-based analysis.

There are several constraints on local 𝑓_NL using wavelet-based estimators. For example, using the COBE data the constraints are |𝑓_NL|<2200(1𝜎) [201]. Using an estimator based on the skewness of the wavelet coefficients, Mukherjee and Wang constrain the 𝑓_NL value for WMAP 1-yr data obtaining 𝑓_NL=50±160(2𝜎) [29]. Using an extension of the previous estimator by combining wavelet coefficients at different contiguous scales, Curto et al. obtain −8<𝑓_NL<111(2𝜎) [202]. Recently, using a generalized third-order estimator based on the wavelet coefficients, Curto et al. obtain −18<𝑓_NL<80(2𝜎) [190].

Needlet Bispectrum
Needlets are a family of spherical wavelets which are localized and asymptotically uncorrelated [203, 204]. The needlet-based statistics has been considered for testing Gaussianity and isotropy (see, e.g., [205–211]). Using the bispectrum of needlet coefficient, the constraints on non-Gaussianity of local-type using WMAP 5-year data yield 𝑓_NL=73±62(2𝜎) [212, 213]. As is clear, the needlet- based bispectrum is not as sensitive as the CMB bispectrum discussed in Section 4; however again in the event of detection the needlet based methods can be calibrated on simulations and represent a different weighting scheme for handling the sky mask and anisotropic noise. Finally, needlets and wavelets allow for the possibility to analyze spatially localized regions in the sky.

Probing Non-Gaussianity Using Bayesian Statistics
A somewhat different approach to searching for non-Gaussianity is provided by the Bayesian approach. Here, the starting point is an explicit physical or statistical model for the data and the goal is to evaluate the posterior density of the parameters of the model and/or the relative probability of the Gaussianity and non-Gaussianity.

On large scales, in the Sachs-Wolfe regime, one can simplify the Bayesian approach by modeling directly the temperature anisotropy. Rocha et al. (2001) [214] discuss a Bayesian exploration of a model, where each spherical harmonic coefficient is drawn from a non-Gaussian distribution. In this regime, the simple form of the non-Gaussian potential for the local model (17) also translates into a simple model for the temperature anisotropy. Reference [215] develop several results for it, including an analytical expression of the evidence ratio of the Gaussian and non-Gaussian models. At the level of current data, this approximation is too restrictive, since most of the information about 𝑓_NL is contained near the resolution limit of the experiment, where most of the measured perturbation modes are concentrated.

A full implementation of a physical non-Gaussian model must include the effect of Boltzmann transport. In the context of local non-Gaussianity, the model equation (17) suggests that a full Bayesian treatment may be feasible. At the time of writing, no fully Bayesian analysis for local 𝑓_NL has been published. The effort has focused on developing approximations to the full Bayesian problem.

Using a perturbative analysis, [216] relates the frequentist bispectrum estimator to moments of the Bayesian posterior distribution. Reference [217] described approximations to the full Bayesian treatment that simplify the analysis for high signal-to-noise maps and compared these to the full Bayesian treatment for a simple 1D toy model of non-Gaussian sky where this analysis is feasible.

7. Summary

The physics of the early universe responsible for generating the seed perturbations in the CMB is not understood. Inflation which is perhaps the the most promising paradigm for generating seed perturbations allows for vast number of inflationary models that are compatible with data based on 2-point statistics like CMB power spectrum. 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 probe for discriminating between different models of the early universe. Models based on slowly rolling single field produce undetectable amount of non-Gaussianity. Single field models without the slow roll can generate large (detectable with future experiments) non-Gaussianities but (1) cannot produce large non-Gaussianity of local type unless inflation started with excited vacuum state, (2) if non-Gaussianity is produced it would naturally be as bispectrum while higher order as trispectrum can be generated, it requires fine tuning.

The bispectrum of the CMB is one of the most promising tool for connecting the non-Gaussianities in the cosmic microwave background and the models of inflation. Bispectrum-based estimator which saturates Cramer-Rao bound has been developed and well characterised using non-Gaussian MonteCarlos. Other statistics although not as sensitive to non-Gaussianity as an optimally weighted bispectrum estimator do provide independent checks and have different systematics. While Bayesian analysis has been applied in the context of nonGaussianity analysis, this still appears to be an open area for fruitful investigation.

Given the importance of detecting primordial non-Gaussianity, it is crucial to characterise any nonprimordial sources of non-Gaussianities. We describe several sources of non-Gaussianities such as from second-order anisotropies after last scattering surface and during recombination.

With Planck launched and taking data, we look forward to the next few years as an exciting time in the exploration of primordial non-Gaussianity in the cosmic microwave background.

Acknowledgment

A. P.S. Yadav gratefully acknowledges support from the IBM Einstein Fellowship.

Endnotes

1. Although inflation is the most popular theory for the early universe, other mechanisms, for example, ekpyrotic models [218] and cyclic models [219, 220] have been proposed for generating nearly scale invariant Gaussian perturbations, while retaining homogeneity and flatness. In the cyclic universe, there is no beginning of time, and our expansion of the universe is one out of the infinite number of such cycles. Each cycle consists of the following phases. (1) A hot big bang phase, during which a structure formation takes place. (2) An accelerated expansion phase which dilutes the matter and radiation energy density. Since observations suggest that our universe is going through an accelerated expansion phase, in the cyclic model interpretation, we are presently going through this phase. (3) A decelerating phase, which makes the universe flat, and generates nearly Gaussian and scale invariant density perturbations. (4) A big crunch/bang transition phase during which matter and radiation is created. Although the mechanism is different, the outcome of phase (3) of the cyclic model is in some sense analogous to a slow-roll expansion phase of inflation; and phase (4) will correspond with the reheating phase in the inflationary scenario. As we will discuss in the next section these two scenarios can be distinguished by their different predictions about the gravitational waves, and non-Gaussianity. Cyclic models predict negligible contribution of gravitational waves while inflationary models can produce large gravitational wave contribution, which can be detected by next generation experiments. Second, cyclic models produce much larger non-Gaussianity (of local type) in comparison to the standard slow-roll inflationary scenario.
2. For the scale invariant (𝑛=1) case, 𝐶^SW_𝑙=[𝑙(𝑙+1)]⁻¹6𝐶^SW₂.
3. Example of some inflationary models are: eternal inflation, hybrid inflation, chaotic, Ghost inflation, Tilted Ghost inflation, DBI inflation, brane inflation, N-flation, bubble inflation, extended inflation, false vacuum inflation, power law inflation, k-inflation, hyperextended inflation, supersymmetric inflation, Quintessential inflation, Natural inflation, Super inflation, Supernatural inflation, D-term inflation, B -inflation, Thermal inflation, discrete inflation, Assisted inflation, Polar cap inflation, Open inflation, Topological inflation, Double inflation, Multiple inflation, Induced-gravity inflation, Warm inflation, stochastic inflation, Generalized assisted inflation, self-sustained inflation, Graduated inflation, Local inflation, Singular inflation, Slinky inflation, locked inflation, Elastic inflation, Mixed inflation, Phantom inflation, Boundary inflation, Non-commutative inflation, Tachyonic inflation, Tsunami inflation, Lambda inflation, Steep inflation, Oscillating inflation, Mutated Hybrid inflation, intermediate inflation, and Inhomogeneous inflation.
4. To first order in perturbations, primordial scalar perturbations do not generate B-modes of CMB. However at second (and higher) order in perturbations, scalar perturbations do produce B-modes [135, 221]. The B-modes generated from higher order perturbations are expected to be smaller than the tensor B-mode levels that the upcoming and future experiments (like CMBPol) are sensitive too.
5. Or equivalently Φ(𝐫)=Φ_𝐿(𝐫)+𝑓_NL(Φ²_𝐿(𝐫)−⟨Φ²_𝐿(𝐫)⟩).
6. We will refer to the estimator in [25] as a near-optimal-v1, while the corrected estimator of [28] as near-optimal.