Advances in Astronomy

Volume 2010, Article ID 980523, 64 pages

http://dx.doi.org/10.1155/2010/980523

## Primordial Non-Gaussianity and Bispectrum Measurements in the Cosmic Microwave Background and Large-Scale Structure

^{1}Department of Applied Mathematics and Theoretical Physics, Centre for Theoretical Cosmology, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK^{2}Institut de Physique Théorique, Commissariat à l'Énergie Atomique, 91191 Gif-sur-Yvette, France

Received 5 February 2010; Accepted 12 May 2010

Academic Editor: Dragan Huterer

Copyright © 2010 Michele Liguori et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The most direct probe of non-Gaussian initial conditions has come from bispectrum measurements of temperature fluctuations in the Cosmic Microwave Background and of the matter and galaxy distribution at large scales. Such bispectrum estimators are expected to continue to provide the best constraints on the non-Gaussian parameters in future observations. We review and compare the theoretical and observational problems, current results, and future prospects for the detection of a nonvanishing primordial component in the bispectrum of the Cosmic Microwave Background and large-scale structure, and the relation to specific predictions from different inflationary models.

#### 1. Introduction

The standard inflationary paradigm predicts a flat Universe perturbed by nearly-Gaussian and scale-invariant primordial perturbations. These predictions have been verified to a high degree of accuracy by Cosmic Microwave Background (CMB) and Large-Scale Structure (LSS) measurements, such as those provided by the Wilkinson Microwave Anisotropy Probe (WMAP) [1], the 2dF Galaxy Redshift Survey (2dFGRS) [2], and the Sloan Digital Sky Survey (SDSS) [3]. Despite this success, it has proved to be difficult to discriminate between the vast array of inflationary scenarios that have been proposed by high-energy theoretical investigations or even to rule-out alternatives to inflation. Since most of the present constraints on the Lagrangian of the inflaton field have been obtained from measurements of the two-point function, or power spectrum, of the primordial fluctuations, a natural step to extend the available information is to look at non-Gaussian signatures in higher-order correlators.

The lowest-order additional correlator to take into account is the three-point function or its counterpart in Fourier space, the* bispectrum*. Most models of inflation are characterized by specific predictions for the bispectrum of the primordial perturbations in the gravitational potential . The bispectrum of these perturbations is defined as where we have introduced the notation so that the Dirac delta function here is . Together with the assumption of statistical homogeneity and isotropy for the primordial perturbations, this implies that the bispectrum is a function of the triplet defined by the magnitude of the wavenumbers , and forming a closed triangular configuration. The current constraints that we are able to derive on the bispectrum provide additional information about the early Universe; the possible detection of a non-vanishing primordial bispectrum in future observations would represent a major discovery, especially as it is predicted to be negligible by standard inflation.

The cosmological observable most directly related to the initial curvature bispectrum is given by the bispectrum of the CMB temperature fluctuations, which provide a map of the density perturbations at the time of decoupling, the earliest information we have about the Universe. Current measurements of individual triangular configurations of the CMB bispectrum are, however, consistent with zero. Studies of the primordial bispectrum, therefore, are usually characterized by constraints on a single-*amplitude parameter*, denoted by once a specific model for is assumed. Since most models predict a curvature bispectrum obeying the hierarchical scaling , with being the curvature power spectrum, the non-Gaussian parameter roughly quantifies the ratio defining the “strength” of the primordial non-Gaussian signal. In addition, we can write where encodes the functional dependence of the primordial bispectrum on the specific triangle configurations. For brevity, the characteristic shape-dependence of a given bispectrum is often referred to simply as the * bispectrum shape* (a precise definition of the bispectrum shape function will be given in Section 2.1). Inflationary predictions for both the amplitude and the shape of are strongly model dependent. Notice that the subscript “” stands for “nonlinear”, since a common phenomenological model for the non-Gaussianity of the initial conditions can be written as a simple nonlinear transformation of a Gaussian field. Generically, of course, non-Gaussianity is associated with nonlinearities, such as nontrivial dynamics during inflation, resonant behaviour at the end of inflation (“preheating”), or nonlinear postinflationary evolution. At the very least, future CMB and LSS observations are expected to be able to eventually detect the last of the three effects mentioned above.

Perturbations in the CMB provide a particularly convenient test of the primordial density field because CMB temperature and polarization anisotropies are small enough to be studied in the *linear regime* of cosmological perturbations. Once the effects of foregrounds are properly taken into account, a non-vanishing CMB bispectrum at large scales would be a direct consequence of a non-vanishing primordial bispectrum. As we will see, while other CMB probes of primordial non-Gaussianity are available, such as tests of the topological properties of the temperature map based on Minkowski Functionals or measurements of the CMB trispectrum, the estimator for the non-Gaussian parameter has been shown to be optimal. We will focus mostly on this bispectrum estimator in the section of this paper dedicated to the CMB.

In the standard cosmological model, the large-scale structure of the Universe, that is, the distribution of matter and galaxies on large scales, is the result of the nonlinear evolution due to gravitational instability of the same initial density perturbations responsible for the CMB anisotropies. This is, perhaps, the most important prediction of the inflationary framework which provides a common origin for the CMB and large-scale structure perturbations as the result of tiny quantum fluctuations stretched over cosmological scales during a phase of accelerated expansion. The large-scale structure we observe at low redshift, however, is characterized by large voids and small regions with very large-matter density, and it is therefore a much less direct probe of the initial conditions. The *distribution of matter becomes a highly non-Gaussian field* precisely as a result of the nonlinear growth of structures, * even for Gaussian initial conditions*. This non-Gaussianity is expressed, in particular, by a non-vanishing matter bispectrum at* any* measurable scale, including the largest scales probed by current or future redshift surveys. In this context, the effect of primordial non-Gaussianity, that is, of an initial component in the curvature bispectrum, will constitute a* correction* to the galaxy bispectrum. It follows that the possibility of constraining or detecting this initial component is strictly related to our ability to *distinguish* it from other primary sources of non-Gaussianity, that is, the nonlinear gravitational evolution, and, in the case of galaxy surveys, nonlinear bias.

The study of non-Gaussian initial conditions for large-scale structure has a relatively long history, with important contributions going back to the mid eighties. The standard picture that has been developed over the years assumed that, at large scales, the effect of primordial non-Gaussianity on the galaxy distribution is simply given in terms of an additional component to the galaxy bispectrum. This is obtained, in perturbation theory, as the linearly evolved and linearly biased initial matter bispectrum, related to the curvature bispectrum by the Poisson equation. Such component becomes subdominant as the gravity-induced non-Gaussian contribution grows in time. In this framework, as one can expect, high-redshift and large-volume galaxy surveys would constitute the best probes of the initial conditions. It has been shown, in fact, that proposed and planned redshift surveys, such as those of Euclid [4], should be able to provide constraints on the primordial non-Gaussian parameters comparable to, if not better than, those expected from CMB missions such as those of Planck. What is more important, in the event of a detection by Planck, is that confirmation by large-scale structure observations will be required.

Recent results from N-body simulations with non-Gaussian initial conditions, however, have revealed a more complex picture. The effect of primordial non-Gaussianity at large scales is not limited to an additional contribution to the galaxy bispectrum, but it quite dramatically affects the galaxy bias relation itself, that is, the relation between the matter and galaxy distributions. A surprising consequence is that it induces a large correction even for the galaxy power spectrum. Such an effect has attracted considerable recent attention and, remarkably, has placed constraints on the non-Gaussian parameter from current LSS datasets which already appear to marginally improve on CMB limits. However, from a theoretical point of view, a proper understanding of the phenomenon is not fully developed yet. For example, reliable predictions for the galaxy bispectrum are not yet available. Most importantly, as for general cosmological parameter estimation, a complete likelihood analysis aimed at constraining, or detecting, primordial non-Gaussianity in large-volume redshift surveys should involve* joint* measurements of the galaxy power spectrum and bispectrum, as well as possibly higher-order correlation functions. While we are still far from a proper assessment of what such analysis would be able to achieve, current results in this direction are very encouraging.

This review is divided into four parts. In Section 2 we will first discuss initial conditions as defined in terms of the primordial curvature bispectrum and its phenomenology. We will then review the observational consequences of primordial non-Gaussianity on the CMB bispectrum, Section 3, and on the large-scale structure bispectrum as measured in redshift surveys, Section 4. In both cases we will discuss theoretical models for the observed bispectra and technical problems related to the estimation of the non-Gaussian parameters, with the differences that naturally characterize such distinct observables. We also give an example of joint analysis using both CMB and large-scale structure when we consider the possibility of constraining a strongly scale-dependent non-Gaussian parameter , emerging in some recently proposed inflationary models.

#### 2. Initial Conditions and the Primordial Bispectrum

In this section we will briefly overview the main predictions of inflationary models regarding the non-Gaussianity (NG) of the primordial curvature perturbation field. The link between NG of primordial density fluctuations and NG of CMB and LSS will be shown in following sections. In order to provide a full description of an NG random field, all correlators beyond the 2-point function are in principle necessary. However in this review we will focus on the primordial bispectrum (i.e., three-point function in Fourier space). This is not only justified by the fact that the bispectrum is the first and simplest higher-order correlator to look at, but also by the fact that most models of inflation predict vanishingly small correlators beyond the bispectrum. In Section 2.1 we will introduce the relevant quantities, their mathematical definitions, and we will provide a general overview and classification of the bispectra predicted in different inflationary scenarios (only from a purely mathematical point of view, without linking them to the Physics originating them at this stage). Finally, a useful eigenmode expansion technique for bispectra will be introduced in Section 2.2 and applied to the calculation of correlations between different bispectra in Section 3.4. In the same section we will also show which kinds of bispectra are predicted by different models of inflation.

##### 2.1. The Primordial Bispectrum and Shape Function

The starting point for this discussion is the primordial gravitational potential perturbation which was seeded by quantum fluctuations during inflation or by some other mechanism in the very early Universe (. When characterizing the fluctuations we usually work in Fourier space with the (flat space) transform defined through The primordial power spectrum of these potential fluctuations is found using an ensemble average: where we have assumed that physical processes creating the fluctuations are statistically isotropic so that only the dependence on the wavenumber remains . Recall that, for nearly scale-invariant perturbations, the fluctuation variance on the horizon scale is almost constant , implying that .

The primordial bispectrum is found from the Fourier transform of the three-point correlator as Here, the delta function enforces the triangle condition, that is, the constraint that the wavevectors in Fourier space must close to form a triangle: . Examples of such triangles are shown in Figure 1, illustrating the basic squeezed, equilateral, and flattened triangles to which we will refer later. Note that a specific triangle can be completely described by the three lengths of its sides and so, in the isotropic case, we are able to describe the bispectrum using only the wavenumbers . The triangle condition restricts the allowed wavenumber configurations to the interior of the tetrahedron illustrated in Figure 2.