Testing the Gaussianity and Statistical Isotropy of the Universe
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Primordial NonGaussianity 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 nonGaussianity 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) CMBbased estimators for constraining primordial nonGaussianity with an emphasis on bispectrum templates (3) current constraints on nonGaussianity and what we can hope to achieve in the near future and (4) nonprimordial sources of nonGaussianities in the CMB such as bispectrum due to second order effects, three way crosscorrelation between primarylensingsecondary 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 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 2point 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 nonGaussianity in the primordial perturbations has emerged as powerful probe of the early universe. The amplitude of nonGaussianity 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. NonGaussianity 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 BunchDavies vacuum, among others [16], for a review and references therein generates substantially higher amounts of nonGaussianity.
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 nonGaussianity 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 nonGaussian 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 nonGaussianity from the inflationary cosmology. In Section 4 we show how the primordial nonGaussianity is connected to the CMB bispectrum, and describe/review CMB bispectrumbased estimators to constrain primordial nonGaussianity (). 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 nonprimordial sources of nonGaussianity 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 predictions^{1} (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 slowlyrolling of the scalar field. During such slowroll, 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 slowroll 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.
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 Hubblehorizon 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 hotbig 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.
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 superhorizon scales throughout the cosmic evolution.
The quantum fluctuations generate the gaugeinvariant 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 quantumfluctuation amplitude in This is the spectrum of on superhorizon 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 gaugeinvariant perturbation, , does not change in time on superhorizon 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 gaugeinvariant variable, (or in the original notation), which reduces to in the zeroshear 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 zeroshear 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 gaugeinvariant 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 zeroshear gauge (or the Newtonian gauge) in which the metric (6) becomes just like the Newtonian limit of the general relativity.
The gaugeinvariant 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 superhorizon 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 socalled (static) SachsWolfe 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 SachsWolfe effect, , is By projecting the 3dimensional CMB fluctuation spectrum, , on the sky, we obtain the angular power spectrum^{2}, [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 SachsWolfe approximation breaks down, and the acoustic physics in the photonbaryon 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 SachsWolfe 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 twopoint function or in in the spherical harmonic space. Since is directly related to through (11), all the information of is also incoded 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 nonGaussian. The nonlinear relation between and is not the only source of nonGaussianity in the CMB anisotropies. For example, at the second order, the relationship between and is also nonlinear.
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 5year 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 CMBPollike 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 5year data are [48]. The error will reduce to for upcoming Planck satellite and to for a fourthgeneration 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 Emodes of the CMB^{4}, while primordial tensor perturbations generate both parity even Emodes and parity odd Bmodes polarization [51–53]. The detection of primordial tensor Bmodes in the CMB would confirm the existence of tensor perturbations in the early universe. This primordial Bmode 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 Bmode signal of the CMB [54]. Search for primordial Bmodes is challenging. Apart from foreground subtraction challenges, and the challenge of reaching the instrumental sensitivity to detect primordial Bmodes, there are several nonprimordial sources such as weak lensing of CMB by the largescale structure [55, 56], rotation of the CMB polarization [57], and instrumental systematics that generate Bmodes 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 5year 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 Epolarization 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 NonGaussianity
Canonical inflationary models predict primordials perturbations to be very close to Gaussian [5–9], and any nonGaussianity 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 nonGaussian perturbations. The amplitude of the nonGaussian contribution to the perturbation is often referred to as even if the nature of the nonGaussianities 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 nonGaussian 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 higherorder 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 DiracBornInfeld 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 nonGaussianity, 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 NonGaussianity
Large primordial nonGaussianity 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 BunchDavies vacuum state before the quantum fluctuation were generated.
To characterize the nonGaussianity one has to consider the higher order moments beyond twopoint function, which contains all the information for Gaussian perturbations. The 3point function which is zero for Gaussian perturbations contains the information about nonGaussianity. The 3point 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 nonGaussianity is captured dimensionless nonlinearity parameter . The shape function correlates fluctuations with three wavevectors and form a triangle in Fourier space. Depending on the physical mechanism responsible for the bispectrum, the shape of the 3point function, , can be broadly classified into three classes [86]. The local, “squeezed,” nonGaussianity, where is large for the configurations in which . Most of the studied inflationary and Ekpyrotic models produce nonGaussianity of local shape (e.g., [82, 84, 87–104]). Second, the nonlocal, “equilateral,” nonGaussianity 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.
NonGaussianity of Local Type
The local form of nonGaussianity may be parametrized in real space as^{5} [13, 107, 108]:
where is the linear Gaussian part of the perturbations, and characterizes the amplitude of primordial nonGaussianity. 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 nonlinear 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 nonGaussianity of local type.
Being local in real space, nonGaussianity 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 slowroll inflationary models, the threepoint function is given by [15] where and are the usual slowroll 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 nonGaussianity shape for local type and for slowroll model. Although in this limit, slowroll models do predict nonGaussianity of local type but as evident from (20), the bispectrum of inflaton perturbations yields a nontrivialscale dependence of [12, 15]. However in the slow roll limit and hence the amplitude is too small to detect.
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NonGaussianity of Equilateral Type
While vast numbers of inflationary models predict nonGaussianity of local type, this model, for instance, fails completely when nonGaussianity 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 noncanonical kinetic terms such as the DiracBornInfeld (DBI) action [112], the ghost condensation [81], or any other singlefield 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 DiracBornInfeld (DBI) action produce a bispectrum of the form The previous model uses as a leading order operator. DBI inflation, which can produce large nonGaussianity, , 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 nonGaussianity, . Figure 3 shows the shape of nonGaussianity of equilateral type by showing for ghost inflation and for a model with a higher derivative term.
Folded Shape
So far the 3point functions were calculated assuming the regular BunchDavis vacuum state, giving rise to either local or equilateral type nonGaussianity. However if the bispectrum is calculated by dropping the assumption of BunchDavis 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 groundbased 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 nonGaussianities are expected to be small. With high quality data on the way, the field of nonGaussianity is taking off. CMB bispectrum contains information which is not present in the powerspectrum 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 Epolarization () 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 nonGaussianity present in the primordial perturbations gets transferred to the observed CMB via (26). The most common way to look for nonGaussianity in the CMB is to study the bispectrum, the threepoint function of temperature and polarization anisotropies in harmonic space. The CMB angular bispectrum is defined as and the angularaveraged 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 threepoint function as defined in (16).
To forecast constraints on nonGaussianity 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 nonGaussianity 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: where the functions and are given by Recently, a new bispectrum template shape, an orthogonal shape, has been introduced [120] which characterizes the size of the signal () which peaks both for equilateral and flattriangle configurations. The shape of nonGaussianities associated with is orthogonal to the one associated to . The bispectrum for orthogonal shape can be written as [120]
4.1. Estimator
An unbiased bispectrumbased minimum variance estimator for the nonlinearity parameter in the limit of full sky and homogeneous noise can be written as [17, 20, 25] where is angle averaged theoretical CMB bispectrum for the model in consideration. The normalization can be calculated to require the estimator to be unbiased, . If the bispectrum is calculated for then the normalization takes the following form: The estimator for nonGaussianity, (36), can be simplified using (26) to yield where is a shape of 3point function as defined in (16). Given the shape , one is interested in, it is conceptually straightforward to constrain the nonlinearity parameter from the CMB data. Unfortunately the computation time for the estimate scales as , which is computationally challenging as even for the WMAP data the number of pixels, is of order . The scaling can be understood by noting that each spherical harmonic transform scales as and the estimator requires number of spherical harmonic transforms.
The computational cost decreases if the shape can be factorized as with which the estimator simplifies to and computational cost now scales as . For Planck () this translates into a speedup by factors of millions, reducing the required computing time from thousands of years to just hours and thus making 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 in a computationally efficient way [121].
The extension of the estimator of 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 operations in comparison to the full bispectrum calculation which takes operations.
For a given shape, the estimator for nonlinearity parameter can be written as , where for the equilateral, local and orthogonal shapes, the can be written as with and is a fraction of sky. Index and can either be or
Indices and can either be or . Here, is 1 when , 6 when , and 2 otherwise: is the theoretical bispectrum for [21].
It has been shown that the previous estimators defined in (42) are minimum variance amongst bispectrumbased 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 : . 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 by where and are the and 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 by Note that , 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, , and take the Monte Carlo average, . Let us suppose that contains terms in the form of , where , , and are some filtered maps. Use the Wick's theorem to rewrite the average of a cubic product as . Finally, remove the MC average from single maps and replace maps in the product with the simulated maps . 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 skycut induces a monopole contribution outside the mask. In the analysis, one subtracts the monopole from outside the mask before measuring , 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 .)
The estimator is still computationally efficient, taking only (times the sampling, which is of order 100) operations in comparison to the full bispectrum calculation which takes operations. Here refers to the total number of pixels. For Planck, , 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 nonGaussianity analysis. Over the course of WMAP operation the field of nonGaussianity has made vast progress both in terms of theoretical predictions of nonGaussianities 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 suboptimal estimator, one could not use the entirety of WMAP data and only the data up to were used to obtain the constraint [23]. These limits were around 30 times better than the previous constraints of 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 3year data up to , this estimator had yielded the tightest constraint at the time on as [26]. This estimator was further generalized to utilize both the temperature and 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 blowup in the variance at high multipoles^{6}. The first analysis using this estimator shows an evidence of nonGaussianity of local type at around in the WMAP 3year data. Independent analysis shows the evidence of nonGaussianity at lower significance, around (see Table 1).
 
The difference between the results by [28, 122] for WMAP 3year data using the Kp0 mask can be a result of different choices of weighting in the nearoptimal estimator. The optimal estimator has a unique weighting scheme. 
By the time of the third WMAP data release (with 5year obsevational data) in 2008 the 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 3year data, there is an evidence for nonGaussianity of local type at around level [122]. However with WMAP 5year data the significance goes down from to [122]. Table 2 compares the constraints obtained by different groups using WMAP 3year and WMAP 5year data. Figure 6 shows this comparison in more detail, showing the constraints also as a function of maximum multipole used in the analysis. Few comments are in place: () constraints on from WMAP 3year data as a function of show a trend where the mean value rises at around , below which data is consistent with Gaussianity and above which there is deviation from Gaussianity at above . The result becomes roughly independent of with evidence for nonGaussianity at around level () independent analysis and using different estimators (optimal and nearoptimal with linear term) see this deviation from nonGaussianity at around in WMAP 3year data, () significance of nonGaussianity goes down to around with WMAP 5year data. The drop in the mean value between WMAP 3year and 5year data can be attributed to statistical shift.

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The best constraints on the equilateral and orthogonal shape of nonGaussianity using the WMAP 5year data are and respectively [120].
As we were completing this paper, the WMAP 7year data was released, with constraints , and [123].
5.2. Future Prospects
Now we discuss the future prospects of using the bispectrum estimators for constraining the nonlinearity parameter for local and equilateral shapes. We compute the Fisher bounds for three experimental setups, () cosmic variance limited experiment with perfect beam (ideal experiment hereafter), () Planck satellite with and noise sensitivity Karcmin and beam FWHM , and () a futuristic CMBPollike satellite experiment with noise sensitivity Karcmin and beam FWHM (CMBPol hereafter). Beside we fix all the other cosmological parameters to a standard fiducial model with a flat cosmology, with parameters described by the best fit to WMAP 5year results [48], given by and . We calculate the theoretical CMB transfer functions and power spectrum from publicly available code CMBFAST [47]. We also neglect any nonGaussianity which can be generated during recombination or there after. We discuss the importance and effect of these nonprimordial nonGaussianities in the next section.
The scaling of signaltonoise as a maximum multipole for the local [124, 125] and equilateral model [126] are In principle one could go to arbitrary high but in reality secondary signals will certainly overwhelm primary signal beyond we restrict to the analysis to . In Figure 7, we show the Fisher bound as function of maximum multipole , for local and equilateral type of nonGaussianity. For both local and equilateral cases, we show the Fisher bound for the analysis using only the CMB temperature information (), only the CMB polarization information (), and the combined temperature and polarization analysis. Note that by having both the temperature and polarization information one can improve the sensitivity by combining the information. Apart from combining the and signal, one can also do crosschecks and diagnostics by independently analysing the data. Temperature and polarization will have different foregrounds and instrumental systematics.
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A CMBPollike experiment will be able to achieve the sensitivity of for nonGaussianity of local type and for nonGaussianity of equilateral type. For the local type of nonGaussianity, 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 ISWlensing and SZlensing has been ignored.
Running NonGaussianity
The primordial nonGaussian parameter has been shown to be scaledependent in several models of inflation with a variable speed of sound, such as DiracBornInfeld (DBI) models. Starting from a simple ansatz for a scaledependent amplitude of the primordial curvature bispectrum for primordial nonGaussianity,
where and is a pivot point. The primordial bispectrum is therefore determined in terms of two parameters: the amplitude and the new parameter 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 that quantifies the running of for current and future CMB missions such as WMAP, Planck, and CMBPol. We will consider some nonzero as our fiducial value for the Fisher matrix evaluation. Clearly, in order to be able to constrain a scaledependence of , its amplitude must be large enough to produce a detection. If is too small to be detected ( 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 large enough to enable a detection. Figure 8 shows the joint constraints on and . In the event of a significant detection of the nonGaussian component, corresponding to for the local model and for the equilateral model of nonGaussianity, is able to determine with a uncertainty of and , 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].
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5.3. Contaminations
A detection of nonGaussianity has profound implications on our understanding of the early Universe. Hence it is important to know and quantify all the possible sources of nonGaussianities in the CMB. Here we highlight some sources of nonGaussianities due to secondorder 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 NonGaussianities
Current analysis of the CMB data ignore the contributions from the secondary nonGaussianities. 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 [108, 128]. However on smaller angular scales several effects start to kick in, for example, () the bispectrum contribution due to unresolved point source like thermal SunyaevZeldovich clusters or standard radio sources, () threeway correlations between primary CMB, lensed CMB, and secondary anisotropies. We will refer to the bispectrum generated due to these threeway correlations as , where some secondaries are, the integrated SachsWolfe (ISW) , SunyaevZeldovich signal and ReesSciama [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: The amplitude of bispectrum due to primarylensingsecondary crosscorrelation is proportional to the product of primary CMB powerspectrum and power spectrum of crosscorrelation between secondary and lensing signals.
The reduced bispectrum from the residual point sources (assuming Poisson distributed) is constant, that is, 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, ISWlensing 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 ISWlensing contribution will bias the estimate for the local by around [19]. The bispectrum contribution from primarylensingReesSciama signal also peaks at squeezed limit and contribute to effective local [132]. For Planck sensitivity the point source will contamination the local nonGaussianity by around [133]. A recent analysis of the full secondorder Boltzmann equation for photons [134] claims that second order effects add a contamination .
The generalization of the Fisher matrix given by (30) to include multiple bispectrum contribution is