1. Introduction

According to our current understanding of the Universe, the morphology of the cosmic microwave background (CMB) temperature field, as well as all cosmological structures that are now visible, like galaxies, clusters of galaxies, and the whole web of large-scale structure, are probably the descendants of quantum process that took place some seconds after the Big Bang. In the standard lore, the machinery responsible for these processes is termed cosmic inflation and, in general terms, what it means is that microscopic quantum fluctuations pervading the primordial Universe are stretched to what correspond, today, to cosmological scales (see [1–3] for comprehensive introductions to inflation.) These primordial perturbations serve as initial conditions for the process of structure formation, which enhance these initial perturbations through gravitational instability. The subsequent (classical) evolution of these instabilities preserves the main statistical features of these fluctuations that were inherited from their inflationary origin–provided, of course, that we restrain ourselves to linear perturbation theory.

However, given that matter has a natural tendency to cluster, and this inevitably leads to nonlinearities (not to mention the sorts of complications that come with baryonic physics), the structures which are visible today are far from ideal probes of those statistical properties. CMB photons, on the other hand, to an excellent approximation experience free streaming since the time of decoupling () and are therefore exempt from these non-linearities (except, of course, for secondary anisotropies such as the Rees-Sciama effect or the Sunyaev-Zel'dovich effect), which implies that they constitute an ideal window to the physics of the early Universe—see, for example, [4–6]. In fact, we can determine the primary CMB anisotropies as well as most of the secondary anisotropies on large scales, such as the Integrated Sachs-Wolfe effect, completely in terms of the initial conditions by means of a linear kernel as follows: where is conformal time, and denote the initial conditions of all matter and metric fields (as well as their time derivatives, if the initial conditions are nonadiabatic). Here is a linear kernel, or a retarded Green's function, that propagates the radiation field to the time and place of its detection, here on Earth. Since that kernel is insensitive to the statistical nature of the initial conditions (which can be thought of as constants which multiply the source terms), those properties are precisely transferred to the CMB temperature field

The statistical properties of the primordial fluctuations are, to lowest order in perturbation theory, quite simple; because the quantum fluctuations that get stretched and enhanced by inflation are basically harmonic oscillators in their ground state, the distribution of those fluctuations is Gaussian, with each mode an independent random variable. The Fourier modes of these fluctuations are characterized by random phases (corresponding to the random initial values of the oscillators), with zero mean, and variances which are given simply by the field mass and the mode's wavenumber . The presence of higher-order interactions (which exist even for free fields, because of gravity) changes this simple picture, introducing higher-order correlations which destroy gaussianity—even in the simplest scenario of inflation [7–9]. However, since these interactions are typically suppressed by powers of the factor , where is Newton's constant and the Hubble parameter during inflation, the corrections are small—but, at least in principle, detectable [10–12].

Since these statistical properties are a generic prediction of (essentially) all inflationary models, they can also be inferred from two ingredients that are usually assumed as a first approximation to our Universe. First, since inflation was designed to stretch our Universe until it became spatially homogeneous and isotropic, it is reasonable to expect that all statistical momenta of the CMB should be spatially homogeneous and rotationally invariant, regardless of their general form. Second, in linear perturbation theory [13] where we have a large number of cosmological fluctuations evolving independently, we can expect, based on the central limit theorem, that the Universe will obey a Gaussian distribution.

The power of this program lies, therefore, in its simplicity: if the Universe is indeed Gaussian, homogeneous, and statistically isotropic (SI), then essentially all the information about inflation and the linear (low redshift) evolution of the Universe is encoded in the variance, or two-point correlation function, of large-scale cosmological structures and/or the CMB. As it turns out, the five-year dataset from the Wilkinson Microwave anisotropy probe (WMAP) strongly supports these predictions [11, 14]. Moreover, the measurements of the CMB temperature power spectrum by the WMAP team, alongside measurements of the matter power spectrum from existing survey of galaxies [15, 16] and data from type Ia supernovae [17–19], have shown remarkable consistency with a concordance model (CDM), in which the cosmos figures as a Gaussian, spatially flat, approximately homogeneous, and statistically isotropic web of structures composed mainly of baryons, dark matter, and dark energy.

However, while the detection of a nearly scale-invariant and Gaussian spectrum is a powerful boost to the idea of inflation, just knowing the variance of the primordial fluctuations is not sufficient to single out which particular inflationary model was realized in our Universe. For that, we will need not only the 2-point function, but the higher momenta of the distribution as well. Therefore, in order to break this model degeneracy, we must go beyond the framework of the CDM, Gaussian, spatially homogeneous, and statistically isotropic Universe.

Reconstructing our cosmic history, however, is not the only reason to explore further the statistical properties of the CMB. The full-sky temperature maps by WMAP [11, 20] have revealed the existence of a series of large-angle anomalies—which, incidentally, were (on hindsight) already visible in the lower-resolution COBE data [21]. These anomalies suggest that at least one of our cherished hypotheses underlying the standard cosmological model might be wrong—even as a first-order approximation. Perhaps the most intriguing anomalies (described in more detail in other review papers in this volume) are the low value of the quadrupole and its alignment of the quadrupole () with the octupole () [22–27], the sphericity [26] (or lack of planarity [28]), of the multipole and the north-south asymmetry [29–33]. In the framework of the standard cosmological model, these are very unlikely statistical events, and yet the evidence that they exist in the real data (and are not artifacts of poorly subtracted extended foregrounds—e.g., [34]) is strong.

Concerning theoretical explanations, even though we have by now an arsenal of ad hoc models designed to account for the existence of these anomalies, none has yet quite succeeded in explaining their origin. Nevertheless, they all share the point of view that the detected anomalies might be related to a deviation of gaussianity and/or statistical isotropy.

In this paper, we will describe, first, how to characterize, from the point of view of the underlying spacetime symmetries, both non-gaussianity and statistical anisotropy. We will adopt two guiding principles. The first is that gaussianity and SI, being completely different properties of a random variable, should be treated separately, whenever possible or practical. Second, since there is only one type of gaussianity and SI but virtually infinite ways away from them, it is important to try to measure these deviations without a particular model or anomaly in mind–although we may eventually appeal to particular models as illustrations or as a means of comparison. This approach is not new and, although not usually mentioned explicitly, it has been adopted in a number of recent papers [35, 36].

One of the main motivations for this model-independent approach is the difficult issue of aprioristic statistics; one can only test the random nature of a process if it can be repeated a very large (formally, infinite) number of times. Since the CMB only changes on a timescale of tens of millions of years, waiting for our surface of last scattering to probe a different region of the Universe is not a practical proposition. Instead, we are stuck with one dataset (a sequence of apparently random numbers), which we can subject to any number of tests. Clearly, by sheer chance, about 30% of the tests will give a positive detection with 70% confidence level (CL), 10% will give a positive detection with 90% CL, and so on. With enough time, anyone can come up with detections of arbitrarily high significance—and ingenuity will surely accelerate this process. Hence, it would be useful to have a few guiding principles to inform and motivate our statistical tests, so that we do not end up shooting blindly at a finite number of fish in a small wheelbarrow.

This paper is divided into two parts. We start Part I by reviewing the basic statistical framework behind linear perturbation theory (Section 2). This serves as a motivation for Section 3, where we discuss the formal aspects of non-Gaussian and statistically isotropic models (Section 2.1), as well as Gaussian models of statistical anisotropy (Section 2.2). Part II is devoted to a discussion on model-independent cosmological tests of non-gaussianity and statistical anisotropy and their application to CMB data. We focus on two particular tests, namely, the multipole vectors statistics (Section 3) and functional modifications of the two-point correlation function (Section 4). After discussing how such tests are usually carried out when searching for anomalies in CMB data (Section 6.1), we present a new formalism which generalizes the standard procedure by including the ergodicity of cosmological data as a possible source of errors (Section 6.2). This formalism is illustrated in Section 7, where we carry a search of planar-type deviations of isotropy in CMB data. We then conclude in Section 8.

Part I: The Linearized Universe

2. General Structure

We start by defining the temperature fluctuation field. Since the background radiation is known to have an average temperature of 2.725 K, we are interested only in deviations from this value at a given direction in the CMB sky. So let us consider the dimensionless function on as follows: where K is the blackbody temperature of the mean photon energy distribution—which, if homogeneity holds, is also equal to the ensemble average of the temperature.

In full generality, the fluctuation field is not only a function of the position vector , but also of the time in which our measurements are taken. In practice, the time and displacement of measurements vary so slowly that we can ignore these dependences altogether. Therefore, we can equally well consider this function as one defined only on the unit radius sphere , for which the following decomposition holds: Since the spherical harmonics obey the symmetry , the fact that the temperature field is a real function implies the identity . This means that each temperature multipole is completely characterized by real degrees of freedom.

2.1. From Inhomogeneities to Anisotropies: Linear Theory

The ultimate source of anisotropies in the Universe is the inhomogeneities in the baryon-photon fluid, as well as their associated spacetime metric fluctuations. If the photons were in perfect equilibrium with the baryons up to a sharply defined moment in time (the so-called instant recombination approximation), their distribution would have only one parameter (the equilibrium temperature at each point), so that photons flying off in any direction would have exactly the same energies. In that case, the photons we see today coming from a line-of-sight would reflect simply the density and gravitational potentials (the “sources”) at the position , where is the radius to that (instantaneous) last scattering surface. Evidently, multiple scatterings at the epoch of recombination, combined with the fact that anisotropies themselves act as sources for more anisotropies, complicate this picture, and in general the relationship of the sources with the anisotropies must be calculated from either a set of Einstein-Boltzmann equations or, equivalently, from the line-of-sight integral equations coupled with the Einstein, continuity, and Euler equations [6].

Assuming for simplicity that recombination was instantaneous, at a time , the linear kernels of (1) reduce to where and are constant coefficients. The photon distribution that we measure on Earth would therefore be given by We can also express this result in terms of the Fourier spectrum of the sources as follows: Now we can use what is usually referred to as “Rayleigh's expansion” (though Watson, in his classic book on Bessel functions, attributes this to Bauer, J. f. Math. LVI, 1859) as follows: where are the spherical Bessel functions. Substituting (6) into (5) we obtain that where we have loosely collected the sources into the term . This expression conveys well the simple relation between the Fourier modes and the spherical harmonic modes. Therefore, up to coefficients which are known given some background cosmology, the statistical properties of the harmonic coefficients are inherited from those of the Fourier modes of the underlying matter and metric fields. Notice that the properties of the s under rotations, on the other hand, have nothing to do with the statistical properties of the fluctuations; they come directly from the spherical harmonic functions .

2.2. Statistics in Fourier Space

The characterization of the statistics of random variables is most commonly expressed in terms of the correlation functions. The two-point correlation function is the ensemble expectation value, In the absence of any symmetries, this would be a generic function of the arguments and , with only two constraints: first, because is a real function, , hence, in our definition ; second, due to the associative nature of the expectation value, . It is obvious how to generalize this definition to 3, 4, or an arbitrary number of fields at different s (or “points”).

Let us first discuss the issue of gaussianity. If we say that the variables are Gaussian random numbers, then all the information that characterizes their distribution is contained in their two-point function . The probability distribution function (pdf) is then formally given by In this case, all higher-order correlation functions are either zero (for odd numbers of points) or they are simply connected to the two-point function by means of Wick's Theorem as follows: where the sum runs over all permutations of the pairs of wave vectors and are weights.

Second, let us consider the issue of homogeneity. A field is homogeneous if its expectation values (or averages) do not dependent on the spatial points where they are evaluated. In terms of the -point functions in real space, we should have the following Writing this expression in terms of the Fourier modes, we get the following Homogeneity demands that the expression in (12) is a function of the distances between spatial points only, not of the points themselves. Hence, the expectation value in Fourier space on the right-hand side of this expression must be proportional to In other words, the hypothesis of homogeneity constrains the -point function in Fourier space to be of the following form: Notice that the “-spectrum” in Fourier space, , can still be a function of the directions of the wavenumbers (it will be, in fact, a function of such vectors, due to the global momentum conservation expressed by the -function.) Models which realize the general idea of (13) correspond to homogeneous but anisotropic universes [37–40].

There is a useful diagrammatic illustration for the -point functions in Fourier space that enforce homogeneity. Notice that we could use the -function in (13) to integrate out any one of the momenta in (12). Let us instead rewrite the -functions in terms of triangles, so for the 4-point function we have whereas for the 5-point function we have and so on, so that the -point -function is reduced to triangles with “internal momenta” (the idea is nicely illustrated in Figure 1.) Substituting the expression for the -point -function into (12) and integrating out all external momenta but the first () and last (), the result is as follows This expression shows explicitly that the real-space -point function above does not depend on any particular spatial point, only on the intervals between points.

Figure 1: Diagrammatic representation of the 2, 3, 4, and 5-point correlation functions in Fourier space. The dashed lines represent internal momenta.

Finally, what are the constraints imposed on the -point functions that come from isotropy alone? Clearly, no dependence on the directions defined by the points, , can arise in the final expression for the -point functions in real space, so from (12) we see that the -point function in Fourier space should depend only on the moduli of the wavenumbers—up to some momentum-conservation -functions, which naturally carry vector degrees of freedom.

In this paper, we will mostly be concerned with tests of isotropy given homogeneity (but not necessarily Gaussianity), so in our case we will usually assume that the -point function in Fourier space assumes the form given in (13).

2.3. Statistics in Harmonic Space

In the previous Section, we characterized the statistics of our field in Fourier space, which in most cases is most easily related to fundamental models such as inflation. Now we will change to harmonic representation, because that is what is most directly related to the observations of the CMB, , which are taken over the unit sphere . We will discuss mostly the two-point function here, and we defer a fuller discussion of -point functions in harmonic space to Section 3.

From (7), we can start by taking the two-point function in harmonic space, and computing it in terms of the two-point function in Fourier space as follows Under the hypothesis of homogeneity, this expression simplifies considerably, leading to If, in addition to homogeneity, we also assume isotropy, then , and the integration over angles factors out, leading to the orthogonality condition for spherical harmonics as follows and as a result the covariance of the s becomes diagonal as follows where we have defined the usual temperature power spectrum in the middle line, and the angular power spectrum in the last line of (20). As a pedagogical note, let us recall that the power spectrum basically expresses how much power the two-point correlation function has per unit as follows:

In an analogous manner to what was done above, we can also construct the angular two-point correlation function in harmonic space as follows: The hypothesis of homogeneity by itself does not lead to significant simplifications, but isotropy leads to a very intuitive expression for the angular two-point function as follows: Clearly, not only is this expression the analogous in of (21), but in fact the Fourier power spectrum and the angular power spectrum are defined in terms of each other as indicated in (23) as follows: Now, using the facts that the spherical Bessel function of order peaks when its argument is approximately given by , and that , we obtain the following (this is one type of what has become known in the literature as Limber's approximations ) Incidentally, from this expression it is clear why it is customary to define

Using (12), we can easily generalize the results of this subsection to -point functions in and in harmonic space, however, the assumption of isotropy alone does very little to simplify our life. The hypothesis of homogeneity, on the other hand, greatly simplifies the angular -point functions, and most of the work in statistical anisotropy of the CMB that goes beyond the two-point function assumes that homogeneity holds. Notice that the issue of gaussianity is, as always, confined to the question of whether or not the two-point function holds all information about the distribution of the relevant variables and is therefore completely separated from questions about homogeneity and/or isotropy.

Also notice that the separable nature of the definition (22) implies here as well, like in Fourier space, a reciprocity relation for the correlation function This symmetry must hold regardless of underlying models and is important in order to analyze the symmetries of the correlation function, as we will see later.

Before we move on, it is perhaps important to mention that the decomposition (22) is not unique. In fact, instead of the angular momenta of the parts, , we could equally well have used the basis of total angular momentum and decomposed that expression as where are known as the bipolar spherical harmonics, defined by [41] where and are the eigenvalues of the total and azimuthal angular momentum operators, respectively. This decomposition is completely equivalent to (22), and we can exchange from one decomposition to another by using the relation where the matrices above are the well-known 3-j coefficients. At this point, it is only a matter of mathematical convenience whether we choose to decompose the correlation function as in (22) or as in (28). Although the bipolar harmonics behave similarly to the usual spherical harmonics in many aspects, the modulations of the correlation function as described in this basis have a peculiar interpretation. We will not go further into detail about this decomposition here, as it is discussed at length in another review article in this volume.

2.4. Estimators and Cosmic Variance

Returning to the covariance matrix (20), we see that, if we assume gaussianity of the s, then the angular power spectrum suffices to describe statistically how much the temperature fluctuates in any given angular scale; all we have to do is to calculate the average (20). This can be a problem, though, since we have only one Universe to measure, and therefore only one set of s. In other words, the average in (20) is poorly determined.

At this point, the hypothesis that our Universe is spatially homogeneous and isotropic at cosmological scales comes not only as simplifying assumption about the spacetime symmetries, but also as a remedy to this unavoidable smallness of the working cosmologist's sample space. If isotropy holds, different cosmological scales are statistically independent, which means that we can take advantage of the ergodic hypothesis and trade averaging over an ensemble for averaging over space. In other words, for a given we can consider each of the real numbers in as statistically independent Gaussian random variables, and define a statistical estimator for their variances as the average The smaller the angular scales ( bigger), the larger the number of independent patches that the CMB sky can be divided into. Therefore, in this limit we should have On the other hand, for large angular scales (small 's), the number of independent patches of our Universe becomes smaller, and (31) becomes a weak estimation of the s. This means that any statistical analysis of the Universe on large scales will be plagued by this intrinsic cosmic sample variance. Notice that this is an unavoidable limit as long as we have only one observable Universe.

Finally, it is important to keep in mind the clear distinction between the angular power spectrum and its estimator (31). The former is a theoretical variable which can be calculated from first principles, as we have shown in Section 2.1. The latter, being a function of the data, is itself a random variable. In fact, if the s are Gaussian, then we can rewrite expression (31) as where is a chi-square random variable with degrees of freedom. According to the central limit theorem, when , approaches a standard normal variable (A standard normal variable is a Gaussian variable with zero mean and unit variance. Any other Gaussian variable with mean and variance can be obtained from through .) which implies that will itself follow a Gaussian distribution. Its mean can be easily calculated using (20) and (31) and is of course given by which shows that the s are unbiased estimators of the s. It is also straightforward to calculate its variance (valid for any ) Because this estimator does not couple different cosmological scales, it has the minimum cosmic variance we can expect from an estimator due to the finiteness of our sample—so it is optimal in that sense. is therefore the best estimator we can build to measure the statistical properties of the multipolar coefficients when both statistical isotropy and gaussianity hold.

In later Sections, we will explore angular or harmonic -point functions for which the assumption of isotropy does not hold. However, it is important to remember at all times that we have only one map, which means one set of s. The estimator for the angular power spectrum, , takes into account all the s by dividing them into the different s and summing over all . Clearly, it will inherit a sample variance for small s, when the s can only be divided into a few “independent parts”. As we try to estimate higher-order objects such as the -point functions, we will have to subdivide the 's into smaller and smaller subsamples, which are not necessarily independent (in the statistical sense) of each other. So, the price to pay for aiming at higher-order statistics is a worsening of the cosmic sample variance.

2.5. Correlation and Statistical Independence

The covariance given in (20) has two distinct, important properties. First, note that its diagonal entries, the 's, are -independent coefficients; this is crucial for having statistical isotropy, as we will show latter. Second, statistical isotropy at the Gaussian level implies that different cosmological “scales” (understood here as meaning the modes with total angular momentum and azimuthal momentum ) should be statistically independent of each other—and this is represented by the Kronecker deltas in (20).

In fact, statistical independence of cosmological scales is a particular property of Gaussian and statistically isotropic random fields and is not guaranteed to hold when gaussianity is relaxed. We will see in the next Section that the rotationally invariant 3-point correlation function (and in general any correlation function) couples to the three scales involved. In particular, if it happens that the Gaussian contribution of the temperature field is given by (20), but at least one of its non-Gaussian moments are nonzero, then the fact that a particular correlation is zero, like for example , does not imply that the scales and are (statistically) independent. This is just a restatement of the fact that, while statistical independence implies null correlation, the opposite is not necessarily true. This can be illustrated by the following example: consider a random variable distributed as Let us now define two other variables and . From these definitions, it follows that and are statistically dependent variables, since knowledge of the mean/variance of automatically gives the mean/variance of . However, these variables are clearly uncorrelated Although correlations are among cosmologist's most popular tools when analyzing CMB properties, statistical independence may turn out to be an important property as well, specially at large angular scales, where cosmic variance is more of a critical issue.

3. Beyond the Standard Statistical Model

Until now we have been analyzing the properties of Gaussian and statistically isotropic random temperature fluctuations. This gives us a fairly good statistical description of the Universe in its linear regime, as confirmed by the astonishing success of the CDM model. This picture is incomplete though, and we have good reasons to search for deviations of either gaussianity and/or statistical isotropy. For example, the observed clustering of matter in galactic environments certainly goes beyond the linear regime where the central limit theorem can be applied, therefore leading to large deviations of gaussianity in the matter power spectrum statistics. Besides, deviations of the cosmological principle may leave an imprint in the statistical moments of cosmological observables, which can be tested by searching for spatial inhomogeneities [42] or directionalities [43].

But how do we plan to go beyond the standard model, given that there is only one Gaussian and statistically isotropic description of the Universe, but infinite possibilities otherwise? This is in fact an ambitious endeavor, which may strongly depend on observational and theoretical hints on the type of signatures we are looking for. In the absence of extra input, it is important to classify these signatures in a general scheme, differentiating those which are non-Gaussian from those which are anisotropic. Furthermore, given that the signatures of non-gaussianity may in principle be quite different from that of statistical anisotropy, such a classification is crucial for data analysis, which requires sophisticated tools capable of separating these two issues.(Although gaussianity and homogeneity/isotropy are mathematically distinct properties, it is possible for a Gaussian but inhomogeneous/anisotropic model to look like an isotropic and homogeneous non-gaussian model. See, e.g., [44].)

We therefore start Section 3.1 by analyzing deviations of gaussianity when statistical isotropy holds. In Section 3.2, we keep the hypothesis of gaussianity and analyze the consequences of breaking statistical rotational invariance.

3.1. Non-Gaussian and SI Models

3.1.1. Rotational Invariance of -Point Correlation Functions

We turn now to the question of non-Gaussian but statistically isotropic probabilities distributions. We will keep working with the -point correlation function defined in harmonic space, since knowledge of these functions enables one to fully reconstruct the CMB temperature probability distribution. Specifically, we would like to know the form of any -point correlation function which is invariant under arbitrary 3-dimensional spatial rotations. When rotated to a new (primed) coordinate system, the -point correlation function transforms as where the s are the coefficients of the Wigner rotation-matrix, which depend on the three Euler-angles , and characterizing the rotation. Notice that in this notation the primed (rotated) system is indicated by the primed s. For the 2-point correlation function, we have already seen that the well-known expression does the job Note the importance of the angular spectrum, being an -independent function.

What about the 3-point function? In this case, the invariant combination is found to be which can be verified by straightforward calculations. Again, the nontrivial physical content of this statistical moment is contained in an arbitrary but otherwise -independent function: the bispectrum [45–47]. As we anticipated in Section 2, rotational invariance of the 3-point correlation is not enough to guarantee statistical independence of the three cosmological scales involved in the bispectrum, although in principle a particular model could be formulated to ensure that , at least for some subset of a general geometric configuration of the 3-point correlation function.

These general properties hold for all the -point correlation function. For the 4-point correlation function, for example, Hu [48] has found the following rotationally invariant combination where the function is known as the trispectrum, and is an internal angular momentum needed to ensure parity invariance. In a likewise manner, it can be verified that the following expression gives the rotationally invariant quadrispectrum .

The examples above should be enough to show how the general structure of these functions emerges under SI; apart from an -independent function, every pair of momenta in these functions are connected by a triangle, which in turn connects itself to other triangles through internal momenta when more than 3 scales are present. In Figure 2, we show some diagrams representing the functions above.

Figure 2: Diagrammatic representation of the 2, 3, 4, and 5-point correlation functions in harmonic space. Here actually represents the pair .

Figure 3: Geometrical representation of the 2pcf in terms of two unit vectors.

Although we have always shown -point functions which are rotationally invariant, the procedure used for obtaining them was rather intuitive, and therefore does not offer a recipe for constructing general invariant correlation functions. Furthermore, it does not guarantee that this procedure can be extended for arbitrary s. Here we will present a recipe for doing that, which also guarantees the uniqueness of the solution.

The general recipe for obtaining the rotationally invariant -point function is as follows: from the expression (39) above, we start by contracting every pairs of Wigner functions, where by “contracting” we mean using the identity and where is a shortcut notation for the three Euler angles. Once this contraction is done, there will remain -functions, which can again be contracted in pairs. This procedure should be repeated until there is only one Wigner function left, in which case we will have an expression of the following form: Now, we see that the only way for this combination to be rotationally invariant is when the remaining function above does not depend on , that is, . Once this identity is applied to the geometrical factors, we are done, and the remaining terms inside the primed -summation will give the rotationally invariant -spectrum.

As an illustration of this algorithm, let us construct the rotationally invariant spectrum and bispectrum. For the 2-point function, there is only one contraction to be done, and after we simplify the last Wigner function, we arrive at where, of course is the well-known definition of the temperature angular spectrum. For the 3-point function, there are two contractions, and the simplification of the last Wigner function gives From this expression and the ortoghonality of the 3-js symbols (see the Appendix), we can immediately identify the definition of the bispectrum as follows:

It should be mentioned that this recipe not only enables us to establish the rotational invariance of any -point correlation function, but it also furnishes a straightforward definition of unbiased estimators for the -point functions. All we have to do is to drop the ensemble average of the primed 's. So, for example, for the 2- and 3-point functions above, the unbiased estimators are given, respectively, by

Notice that isotropy plays the same role, in , that homogeneity plays in . What enforces homogeneity in is the Fourier-space -functions, as in the discussion around Figure 1. However, in the equivalents of the Fourier modes are the harmonic modes, for which there is only a discrete notion of orthogonality—and no Dirac -function. What we found above is that the Wigner 3-j symbols play the same role as the Fourier space -functions; they are the enforcers of isotropy (rotational invariance) for the -point angular correlation function. Hence, the diagrammatic representations of the constituents of the -point functions in Fourier (Figure 1) and in harmonic space (Figure 2) really do convey the same physical idea—one in , the other in .

3.2. Gaussian and Statistically Anisotropic Models

In the last Section, we have developed an algorithm which enables one to establish the rotational invariance of any -point correlation function. As we have shown, this is also an algorithm for building unbiased estimators of non-Gaussian correlations. In this Section, we will change the perspective and analyze the case of Gaussian but statistically anisotropic models of the Universe.

There are many ways in which statistical anisotropy may be manifested in CMB. From a fundamental perspective, a short phase of inflation which produces just enough e-folds to solve the standard Big Bang problems may leave imprints on the largest scales of the Universe, provided that the spacetime is sufficiently anisotropic at the onset of inflation [39]. Another source of anisotropy may result from our inability to efficiently clean foreground contaminations from temperature maps. Usually, the cleaning procedure involves the application of a mask function in order to eliminate contaminations of the galactic plane from raw data. As a consequence, this procedure may either induce, as well as hide some anomalies in CMB maps [28].

It is important to mention that these two examples can be perfectly treated as Gaussian: in the first case, the anisotropy of the spacetime can be established in the linear regime of perturbation theory and therefore will not destroy gaussianity of the quantum modes, provided that they are initially Gaussian. In the second case, the mask acts linearly over the temperature maps, therefore preserving its probability distribution [49].

3.2.1. Primordial Anisotropy

Recently, there have been many attempts to test the isotropy of the primordial Universe through the signatures of an anisotropic inflationary phase [38–40, 50, 51]. A generic prediction of such models is the linear coupling of the scalar, vector, and tensor modes through the spatial shear, which is in turn induced by anisotropy of the spacetime [38]. Whenever that happens, the matter power spectrum, defined in a similar way as in (13), will acquire a directionality dependence due to this type of see-saw mechanism. This dependence can be accommodated in a harmonic expansion of the form where the reality of requires that . Given that temperature perturbations are real, their Fourier components must satisfy the relation . This property taken together with the definition (13) implies that