Statistical cosmology with quadratic density fields

Mon. Not. R. Astron. Soc. 338, 806–815 (2003)

Statistical cosmology with quadratic density fields

Peter Watts� and Peter Coles�School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD

Accepted 2002 September 27. Received 2002 September 24; in original form 2002 August 15

ABSTRACTPrimordial fluctuations in the cosmic density are usually assumed to take the form of a Gaussianrandom field that evolves under the action of gravitational instability. In the early stages,while they have low amplitude, the fluctuations grow linearly. During this phase, the Gaussiancharacter of the fluctuations is preserved. Later on, when the fluctuations have an amplitude ofthe order of the mean density or larger, non-linear effects cause departures from Gaussianity. InFourier space, non-linearity is responsible for coupling Fourier modes and altering the initiallyrandom distribution of phases that characterizes Gaussian initial conditions. In this paper,we investigate some of the effects of quadratic non-linearity on basic statistical properties ofcosmological fluctuations. We demonstrate how this form of non-linearity can affect asymptoticproperties of density fields such as homogeneity, ergodicity, and behaviour under smoothing.We also show how quadratic density fluctuations give rise to a particular relationship betweenthe phases of different Fourier modes which, in turn, leads to the generation of a non-vanishingbispectrum. We thus elucidate the relationship between higher-order power spectra and phasedistributions.

Key words: methods: statistical – cosmology: theory – large-scale structure of Universe.

1 I N T RO D U C T I O N

In most popular versions of the gravitational instability model forthe origin of cosmic structure, particularly those involving cosmicinflation (Guth 1981; Guth & Pi 1982), the initial fluctuations thatseeded the structure formation process form a Gaussian random field(Adler 1981; Bardeen et al. 1986).

Gaussian random fields are the simplest fully-defined stochasticprocesses, which makes analysis of them relatively straightforward.Robust and powerful statistical descriptors can be constructed thathave a firm mathematical underpinning and are relatively simpleto implement. Second-order statistics such as the ubiquitous powerspectrum (e.g. Peacock & Dodds 1996) furnish a complete descrip-tion of Gaussian fields. They have consequently yielded invaluableinsights into the behaviour of large-scale structure in the latest gen-eration of redshift surveys, such as the 2dF Galaxy Redshift Survey(2dFGRS) (Percival et al. 2001). Important though these methodsundoubtedly are, the era of precision cosmology we are now enter-ing requires more thought to be given to methods for both detectingand exploiting departures from Gaussian behaviour.

The pressing need for statistics appropriate to the analysis of non-linear stochastic processes also suggests a need to revisit some of thefundamental properties cosmologists usually assume when study-ing samples of the Universe. Gaussian random fields have many

�E-mail: [email protected] (PW); [email protected] (PC)

useful properties. It is straightforward to impose constraints that re-sult in statistically homogeneous fields, for example. Perhaps withmore relevance, we can understand the conditions under which av-erages over a single spatial domain are well defined, the constraintof sample homogeneity. The conditions under which such fields canbe ergodic are also well established. It is known that smoothingGaussian fields preserves Gaussianity, and so on. These propertiesare all somewhat related, but not identical. Indeed, as we see in Sec-tion 4, looking at the corresponding properties of non-linear fieldsturns up some interesting results and delivers warnings to be careful.An exploration of these properties is the first aim of this paper.

Even if the primordial density fluctuations were indeed Gaussian,the later stages of gravitational clustering must induce some formof non-linearity. One particular way of looking at this issue is tostudy the behaviour of Fourier modes of the cosmological densityfield. If the hypothesis of primordial Gaussianity is correct thenthese modes began with random spatial phases. In the early stagesof evolution, the plane-wave components of the density evolve in-dependently like linear waves on the surface of deep water. As thestructures grow in mass, they interact with other in non-linear ways,more like waves breaking in shallow water. These mode–mode inter-actions lead to the generation of coupled phases. While the Fourierphases of a Gaussian field contain no information (they are random),non-linearity generates non-random phases that contain much in-formation about the spatial pattern of the fluctuations. Although thesignificance of phase information in cosmology is still not fully un-derstood, there have been a number of attempts to gain quantitativeinsight into the behaviour of phases in gravitational systems. Ryden

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Quadratic density fields 807

& Gramann (1991), Soda & Suto (1992) and Jain & Bertschinger(1998) concentrated on the evolution of phase shifts for individualmodes using perturbation theory and numerical simulations. An al-ternative approach was adopted by Scherrer, Melott & Shandarin(1991), who developed a practical method for measuring the phasecoupling in random fields that could be applied to real data. Mostrecently, Chiang & Coles (2000), Coles & Chiang (2000), Chiang(2001) and Chiang, Coles & Naselsky (2002) have explored theevolution of phase information in some detail.

Despite this recent progress, there is still no clear understand-ing of how the behaviour of the Fourier phases manifests itself inmore orthodox statistical descriptors. In particular, there is muchinterest in the usefulness of the simplest possible generalization ofthe (second-order) power spectrum, i.e. the (third-order) bispectrum(Peebles 1980; Scoccimarro et al. 1998; Scoccimarro, Couchman &Frieman 1999; Verde et al. 2000, 2001, 2002). Since the bispectrumis identically zero for a Gaussian random field, it is generally ac-cepted that the bispectrum encodes some form of phase informationbut it has never been elucidated exactly what form of correlationit measures. Further possible generalizations of the bispectrum areusually called polyspectra; they include the (fourth-order) trispec-trum (Verde & Heavens 2001) or a related but simpler statistic calledthe second spectrum (Stirling & Peacock 1996). The exploration ofthe connection between polyspectra and non-linearly induced phaseassociation is the second aim of this paper.

We investigate both these issues by reference to a particular formof non-Gaussian field that serves both as a kind of phenomenologi-cal paradigm and as a reasonably realistic model of non-linear evo-lution from Gaussian initial conditions. The model involves a fieldwhich is derived by a simple quadratic transformation of a Gaussiandistribution, hence the term quadratic non-linearity. Quadratic fieldshave been discussed before from a number of contexts (e.g. Coles& Barrow 1987; Moscardini et al. 1991; Falk, Rangarajan &Srednicki 1993; Luo & Schramm 1993; Luo 1994; Gangui et al.1994; Koyama, Soda & Taruya 1999; Peebles 1999a,b; Matarrese,Verde & Jimenez 2000; Verde et al. 2000, 2001; Komatsu & Spergel2001; Shandarin 2002; Bartolo, Matarrese & Riotto 2002); for fur-ther discussion see Section 3. Our motivation is therefore very simi-lar to that of Coles & Jones (1991), which introduced the lognormaldensity field as an illustration of some of the consequences of a moreextreme form of non-linearity involving an exponential transforma-tion of the linear density field.

The plan is as follows. In the following section, we introduce somefundamental concepts underlying statistical cosmology, more or lessfrom first principles. We do this in order to allow the reader to seeexplicitly what assumptions underlie standard statistical practice. InSection 3 we then look at some of the contexts in which quadraticnon-linearity may arise, either primordially or during the non-lineargrowth of structure from Gaussian fields. In Section 4 we revisitsome of the basic properties used in Section 2 from the point of viewof quadratic non-linearity and we show how some basic implicitassumptions are violated. Then, in Section 5, we explore how phasecorrelations arise in quadratic fields and we relate these to higher-order statistics of quadratic fields. We discuss the lessons learnedfrom this study in Section 6.

2 BA S I C S TAT I S T I C A L C O N C E P T S

We start by giving some general definitions of concepts which welater use in relation to the particular case of cosmological densityfields. In order to put our results in a clear context, we develop the

basic statistical description of cosmological density fields; see also,for example, Peebles (1980).

2.1 Fourier description

We follow standard practice and consider a region of the Universehaving volume V u, for convenience assumed to be a cube of side L �ls, where ls is the maximum scale at which there is significant struc-ture due to the perturbations. The region V u can be thought of asa ‘fair sample’ of the Universe if this is the case. It is possible toconstruct, formally, a ‘realization’ of the Universe by dividing it intocells of volume V u with periodic boundary conditions at the faces ofeach cube. This device is often convenient, but in any case we oftentake the limit V u → ∞. Let us denote by ρ̄ the mean density in avolume V u and take ρ(x) to be the density at a point in this regionspecified by the position vector x with respect to some arbitraryorigin. As usual, we define the fluctuation to be

δ(x) = [ρ(x) − ρ̄]/ρ̄. (1)

We assume this to be expressible as a Fourier series:

δ(x) =∑

k

δk exp(ik · x) =∑

k

δ∗k exp(−ik · x);

δk = 1

Vu

∫Vu

δ(x) exp(−ik · x) dx. (2)

The Fourier coefficients δk are complex quantities, δk = |δk| exp(iφk) with amplitude |δk| and phase φk. The assumption of periodicboundaries results in a discrete k-space representation; the sum istaken from the Nyquist frequency kNy = 2π/L , where V u = L3, toinfinity. Note that as L → ∞, kNy → 0. Conservation of mass in V u

implies δk=0 = 0 and the reality of δ(x) requires δ∗k = δ−k.

If, instead of the volume V u, we choose a different volume V ′u

the perturbation within the new volume would again be representedby a series of the form (2), but with different coefficients δk. Nowconsider a (large) number N of realizations of our periodic volumeand label these realizations Vu1, Vu2, Vu3, . . . , VuN. It is meaningfulto consider the probability distribution P(δk) of the relevant coef-ficients δk from realization to realization across this ensemble. Wetypically assume that the distribution is statistically homogeneousand isotropic, in order to satisfy the cosmological principle, and thatthe real and imaginary parts of δk have a Gaussian distribution andare mutually independent, so that

P(w) = V 1/2u(

2πα2k

)1/2 exp

(−w2Vu

2α2k

), (3)

where w represents either Re[δk] or Im[δk] and α2k = σ 2

k/2; σ 2k is

the spectrum. This is the same as the assumption that the phases φk

in equation (5) are mutually independent and randomly distributedover the interval between φ = 0 and φ = 2π. In this case, the moduliof the Fourier amplitudes have a Rayleigh distribution:

P(|δk|, φk) d|δk| dφk = |δk|Vu

2πσ 2k

exp

(−|δk|2Vu

2σ 2k

)d|δk| dφk. (4)

Because of the assumption of statistical homogeneity and isotropy,the quantityP(δk) depends only on the modulus of the wave vector kand not on its direction. It is fairly simple to show that, if the Fourierquantities |δk| have the Rayleigh distribution, then the probabilitydistribution P(δ) of δ = δ(x) in real space is Gaussian, so that

P(δ) dδ = 1

(2πσ 2)1/2exp

(− δ2

2σ 2

)dδ, (5)

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808 P. Watts and P. Coles

where σ 2 is the variance of the density field δ(x). This is a strict def-inition of Gaussianity. However, Gaussian statistics do not alwaysrequire the distribution (7) for the Fourier component amplitudes.According to its Fourier expansion, δ(x) is simply a sum over a largenumber of Fourier modes whose amplitudes are drawn from somedistribution. If the phases of each of these modes are random, thenthe central limit theorem will guarantee that the resulting superpo-sition will be close to a Gaussian if the number of modes is largeand the distribution of amplitudes has finite variance. Such fieldsare called weakly Gaussian.

2.2 Covariance functions and probability densities

We now discuss the real-space statistical properties of spatial per-turbations in ρ. The covariance function is defined in terms of thedensity fluctuation by

ξ (r ) = 〈[ρ(x) − ρ̄][ρ(x + r ) − ρ̄]〉ρ̄2

= 〈δ(x)δ(x + r )〉. (6)

The angular brackets in this expression indicate two levels of av-eraging: first, a volume average over a representative patch of theuniverse and, second, an average over different patches within theensemble, in the manner of Section 2.1. Applying the Fourier ma-chinery to equation (6) we arrive at the Wiener–Khintchin theorem,relating the covariance to the spectral density function or powerspectrum, P(k):

ξ (r ) =∑

k

⟨|δk|2⟩

exp(−ik · r ). (7)

This, in passing to the limit V u → ∞, becomes

ξ (r ) = 1

(2π)3

∫P(k) exp(−ik · r ) dk. (8)

Averaging equation (7) over r gives

〈ξ (r )〉r = 1

Vu

∑k

⟨|δk|2⟩∫

exp(−ik · r ) dr = 0. (9)

The function ξ (r ) is the two-point covariance function. In ananalogous manner it is possible to define spatial covariance functionsfor N > 2 points. For example, the three-point covariance functionis

ζ (r , s) = 〈[ρ(x) − ρ̄][ρ(x + r ) − ρ̄][ρ(x + s) − ρ̄]〉ρ̄3

(10)

which gives

ζ (r , s) = 〈δ(x)δ(x + r )δ(x + s)〉, (11)

where the spatial average is taken over all the points x and over alldirections of r and s such that |r − s| = t . In other words, over allpoints defining a triangle with sides r, s and t. The generalization ofequation (10) to N > 3 is obvious.

The covariance functions are related to the moments of the prob-ability distributions of δ(x). If the fluctuations form a Gaussianrandom field then the N-variate distributions of the set δi ≡ δ(xi )are just multi-variate Gaussians of the form

PN (δ1, . . . , δN ) = 1

(2π)N/2(det C)1/2exp

(−1

2

∑i, j

δi C−1i j δ j

).

(12)

The correlation matrix Cij can be expressed in terms of the covari-ance function

Ci j = 〈δiδ j 〉 = ξ (r i j ). (13)

It is convenient to go a stage further and define the N-point connectedcovariance functions as the part of the average 〈δi . . . δN 〉 that is notexpressible in terms of lower-order functions, e.g.

〈δ1δ2δ3〉=〈δ1〉c〈δ2δ3〉c + 〈δ2〉c〈δ1δ3〉c + 〈δ3〉c〈δ1δ2〉c

+ 〈δ1〉c〈δ2〉c〈δ3〉c + 〈δ1δ2δ3〉c, (14)

where the connected parts are 〈δ1δ2δ3〉c, 〈δ1δ2〉c, etc. Since 〈δ〉 = 0by construction, 〈δ1〉c = 〈δ1〉 = 0. Moreover, 〈δ1δ2〉c = 〈δ1δ2〉 and〈δ1δ2δ3〉c = 〈δ1δ2δ3〉. The second-order and third-order connectedparts are simply the same as the covariance functions. Fourth-orderand higher-order quantities are different, however. The connectedfunctions are just the multi-variate generalization of the cumulantsκ N (Kendall & Stuart 1977). One of the most important properties ofGaussian fields is that all of their N-point connected covariances arezero beyond N = 2, so that their statistical properties are fixed oncethe set of two-point covariances (13) is determined. All large-scalestatistical properties are therefore determined by the asymptotic be-haviour of ξ (r ) as r → ∞. This simplifying property is not sharedby non-Gaussian fields, a fact that we explore in Section 4.

3 QUA D R AT I C N O N - L I N E A R I T Y

In this section we discuss some of the circumstances whereinquadratic non-linearity may arise. At the outset we should admitthat this study is primarily phenomenological and our intention islargely to use this as a model that displays some consequences ofnon-linearity. In the following, we refer to the general class of mod-els incorporating the quadratic transformation of a Gaussian fieldas quadratic models. In subsequent sections we pay closer attentionto the behaviour of two subclasses of the quadratic models, whichwe refer to as the Peebles model, and the perturbative model. Theirdetails are discussed below.

3.1 Phenomenology

As far as we are aware, the first application of quadratic density fieldsin a cosmological setting was by Coles & Barrow (1987) who werestudying the possible sample properties of non-Gaussian tempera-ture fluctuations on the cosmic microwave background (CMB) sky.In fact, they studied a series of models called χ 2

n models obtainedvia the transformation

Y = X 21 + X 2

2 + · · · X 2n, (15)

where n is the order and the Xi are independent Gaussian randomfields with identical covariance functions. Similar models were alsoexplored by Moscardini et al. (1991) using Y to model either theprimordial density field or the primordial gravitation potential; seealso Koyama et al. (1999), Verde et al. (2000), Matarrese et al.(2000), Verde et al. (2001) and Komatsu & Spergel (2001). Therewas no physical motivation for this as a model of CMB temperaturefluctuations; it was used simply because it could be used to calculateanalytical results for such a field to compare with similar results fora Gaussian. Indeed the χ2

n random field has been used as a modelfor non-Gaussian phenomena in a wide range of fields, includingsurface physics and geology (e.g. Adler 1981).

3.2 Inflation

Since the early 1980s (e.g. Guth & Pi 1982) it has been commonlybelieved that the inflationary scenario of the very early Universe re-sults in the imprint of primordial Gaussian fluctuations. Since then,

C© 2003 RAS, MNRAS 338, 806–815


and with the invention of increasingly complicated models of theinflationary process, it has become clearer that inflation can producesignificant levels of non-Gaussianity. The simplest ‘slow-roll’ mod-els of inflation involving a single dominant scalar field do indeedproduce Gaussian fluctuations, but including non-linear terms andback-reaction does produce some element of non-Gaussianity as doextra degrees of freedom (Salopek & Bond 1991; Salopek 1992;Falk et al. 1993; Gangui et al. 1994; Wang & Kamionkowski 2000;Bartolo et al. 2002).

Typically the non-linear contributions arising during inflationmanifest themselves as higher-order contributions to the effectiveNewtonian gravitational potential, �, i.e.

� = φ + α(φ2 − 〈φ2〉)+ . . . , (16)

where φ is a Gaussian field (not the phase) and α is a constantwhich is vanishingly small in most models. The term in 〈φ2〉 isneeded to ensure � has zero mean. Because the mean value of theNewtonian potential is not physically meaningful anyway, this termis not really important. A more radical suggestion for an inflation-induced quadratic model is offered by Peebles (1999a,b). In thePeebles model, the density field is given by

ρ(x) = 1

2m2ψ(x)2, (17)

where ψ is a scalar field and m is an effective mass. We return tothe statistical consequences of this particular model in Section 4.

3.3 Gravitational non-linearity

In a simple perturbative model, the non-linear density contrast at apoint r is given by the relation

δ(x) = δ1(x) + εδ2(x) (18)

where δ1(x) is a Gaussian random field, δ2(x) is a quadratic randomfield derived by squaring δ1, and ε is a small factor that controlsthe degree of non-linearity. To be precise we should also include aconstant term in this expression in order to ensure that 〈δ(x)〉 = 0,but this does not play any role in the following for reasons mentionedabove so we ignore it. Using a constant ε is not rigorous but at leastqualitatively it shows how the lowest non-linear corrections comeinto play. Henceforth we refer to equation (18) as the perturbativemodel. For a detailed discussion of perturbation theory performedproperly see Bernardeau et al. (2002).

3.4 Bias

Attempts to confront theories of cosmological structure formationwith observations of galaxy clustering are complicated by the un-certain and possibly biased relationship between galaxies and thedistribution of gravitating matter. A particular simple and useful wayof modelling this relationship is through the idea of a local bias. Insuch models, the propensity of a galaxy to form at a point where thetotal (local) density of matter is ρ is taken to be some function f (ρ)(Coles 1993; Fry & Gaztanaga 1993). This amounts to a statementof the form

δg(x) = f [δ(x)], (19)

where δg is the density contrast inferred from galaxy counts or otherclustering statistics. In the simplest local bias models, f (δ) is a con-stant usually called b. Clearly a linear bias of this form simply scalesthe variance, covariance functions and power spectra of the under-lying field but has no effect on the detailed form of the statistical

distribution. Models where the bias is non-linear (but still local) areuseful as they subject constraints on the effect that the bias mayhave on galaxy clustering statistics, without making any particu-lar assumption about the form of f (Coles 1993). Fry & Gaztanaga(1993) discussed the implications of bias with the form

f (δ) =∞∑

n=0

bnδn, (20)

in which bn cannot all be chosen independently because the meanof δg must again be zero. On a scale where the density field islinear, we can therefore see that a non-linear bias with b2 �= 0 willresult in quadratic contributions to δg even if they do not contributesignificantly to δ.

4 A S Y M P TOT I C P RO P E RT I E S

In developing the statistical background in Section 2, especially theFourier description of random fluctuating fields, we have made anumber of assumptions along the way that are necessary in order forthe resulting descriptors to be well defined. The terminology relat-ing to these assumptions is often used very loosely in cosmology, atleast partly because they bear a relatively simple relationship to eachother when the fields being dealt with are Gaussian. However, in thegeneral case of non-Gaussian fields many subtleties arise relatingto the presence of higher-order statistical correlations. It is espe-cially important in the current era of high-precision developmentsin statistical cosmology also to be precise about the foundations.

In the following, we discuss some of the large-scale properties ofrandom fields in a formal fashion, with particular reference to thequadratic model. The behaviour of covariance functions on largescales will turn out to be very important so, to take the simplestexample, consider the Peebles model from Section 3.2. In this casewe basically have a quadratic density field δ = ψ2 − 〈ψ2〉 whereψ is assumed to be a statistically homogeneous random field andwe have subtracted the mean value of ψ2 to ensure that δ has zeromean.

Let us suppose that ψ has a well-behaved covariance function�(r ) and that the covariance function of δ is, as usual, ξ (r ). It istrivial to show that that

ξ (r ) = 2�2(r ) (21)

so that ξ (r ) must be positive for all r (Adler 1981). Note that addingconstant terms to δ would not alter this behaviour. In the perturbativecase with a field of the form δ = ψ + αψ2, such as the examplesgiven in equations (16) and (18), the resulting covariance functionhas a behaviour of the form

ξ (r ) = �(r ) + α2�2(r ). (22)

In this case the covariance function of the resulting field wouldcontain terms of order �(r ), the corresponding covariance functionof the underlying Gaussian field. If �(r ) < 0 on some scale thenas long as α is small, the resulting ξ (r ) need not be positive in thiscase.

4.1 Statistical homogeneity

The formal definition of strict statistical homogeneity for a randomfield (also called stationarity) is that the set of finite-dimensionaljoint probability distributions, which we called PN (δ1, . . . , δN ) inSection 2.3, must be invariant under spatial translations, i.e.

PN [δ(x1), . . . , δ(xN )]= PN [δ(x1+ x), . . . , δ(xN + x)] (23)

C© 2003 RAS, MNRAS 338, 806–815


for any x. This must be true for all orders N. For a Gaussian randomfield in which the form of PN (δ1, . . . , δN ) is given by equation (23),a necessary and sufficient condition for δ(x) to be strictly homo-geneous is that the covariance function 〈δ(x1)δ(x2)〉 is a functionof x1 − x2 only (Adler 1981). Statistical isotropy can be added byrequiring rotation invariance. We can define weaker versions of ho-mogeneity and isotropy according to which only the moments ofthe distribution are translation invariant. For example, second-orderhomogeneity and isotropy (all that is required for the analysis ofpower spectra or two-point covariance functions) basically meansthat the function ξ (r ) does not depend on either the origin or thedirection of r , but only on its modulus. Since the properties of aGaussian random field depend only on second-order properties, thisweaker condition is sufficient to require condition (23) in this case.

This does not mean that any function satisfying translation androtation invariance is necessarily the covariance function of a ho-mogeneous random field (in either the strict or second-order sense).For one thing, the power spectrum must be positive (or zero) forall k, which places a constraint on the shape of any possible ξ (r ) –which must be convex. The result (16) also implies that

limr→∞

1

r 3

∫ r

0

ξ (r ′)r ′2 dr ′ = 0 (24)

for such fields. A perfectly homogeneous distribution would haveP(k) ≡ 0 and ξ (r ) would be identically zero for all r. Note, however,that it is possible for fields obeying either equations (21) or (22) tobe statistically homogeneous.

4.2 Sample homogeneity

Statistical homogeneity plays a vital role in both the analysis of clus-tering and the formal development of the theory of cosmological per-turbation growth. Unfortunately, the use of the word ‘homogeneity’in this context leads to a confusion regarding the more fundamentaluse of this word in cosmology. Standard cosmologies are based onthe cosmological principle, which requires our Universe to be ho-mogeneous and isotropic on large scales. More loosely, it needs to besufficiently homogeneous and isotropic that the Robertson–Walkermetric and the Friedmann equations furnish an adequate approxi-mation to the evolution of the Universe. Statistical homogeneity asdescribed above is a much weaker requirement than the requirementthat one realization from a probability ensemble (our Universe) hasasymptotically small fluctuations when smoothed on a sufficientlylarge scale. In fact, the analogous relation to equation (24) in thecase of sample homogeneity is far stronger

limr→∞

∫ r

0

ξ (r ′)r ′2 dr ′ = 0 (25)

as this requires the real fluctuations in density to be asymptoticallysmall within a single realization. Notice that the requirement forsample homogeneity is such that, in general, the covariance functionmust change sign, from positive at the origin, where ξ (0) = σ 2 �0, to negative at some r to make the overall integral (24) convergein the correct way.

It is clear from this discussion that statistical homogeneity doesnot require sample homogeneity. Revisiting the quadratic modelnow reveals another interesting point: the Peebles model (21) cannotbe sample homogeneous, even if it is statistically homogeneous. Ifwe want δ to have a covariance function matching observations, sayξ (r ) ∼ r−2, then the underlying Gaussian field must have �(r ) ∼r−1 which violates the constraint for it to be sample homogeneous.

This model behaves in a similar way to fractal models of the type

discussed, for example, by Coleman & Pietronero (1992). Mentionof fractal models tends to send mainstream cosmologists scream-ing into the hills, but the lack of sample homogeneity they describeneed not be damaging to standard methods. To give a prominent il-lustration, consider the behaviour of the gravitational potential fielddefined by a Gaussian random field with the Harrison–Zel’dovichspectrum. In such a case, the fluctuations will always be small (of or-der 10−5 to be consistent with observations) but they are independentof scale, and thus there is never a scale at which sample homogeneityis reached. It is not particularly important for the purposes of galaxyclustering studies that the Universe obeys the property of samplehomogeneity. What is more important is that estimates of statisti-cal properties obtained from different samples vary with respect tothe ensemble-averaged property in a fashion which is under controlfor large samples. This does not require asymptotic convergence tohomogeneity.

Finally, note that the perturbative quadratic model with equa-tion (22) can be sample homogeneous if �(r ) obeys condition (25)and α is sufficiently small. Perturbative corrections, such as thosedescribed in Section 3.3, do not therefore necessarily induce sampleinhomogeneity.

4.3 Ergodicity and fair samples

We have already introduced the idea of a ‘fair sample hypothesis’,which is basically that averages over finite patches of the Universecan be treated as averages over some probability ensemble. Peebles(1980), for example, gives the definition of a fair sample hypothesisin a number of ways. First, he states

‘. . . the fair sample hypothesis is taken to mean that the Universe is sta-tistically homogeneous and isotropic.’

Later we find

‘Samples from well-separated spots are uncorrelated, and the collectionof such samples is a statistical ensemble generated by many independentapplications . . . ’

This second definition is close to that we have used in Section 2,but it is clear that it is stronger than the first definition. Related tothe fair sample hypothesis, but not identical to it, is the so-called er-godic property, which is that averages over an infinite domain withina single realization can be treated as averages over the probabilityensemble. The second definition of a fair sample is a stronger state-ment than the ergodic property, as it involves the properties of finitepatches rather than an infinite domain within a single realizationfrom the probability ensemble.

Ergodic properties are extremely difficult to prove, but results doexist for Gaussian random fields (Adler 1981). Intriguingly, in thiscase the result is extremely simple. The necessary and sufficientcondition for a statistically homogeneous Gaussian random fieldto be ergodic is that its power spectrum (defined above) should becontinuous. Continuity of the power spectrum leads, by standardFourier analysis, to the result that

limr→∞

1

r 3

∫ r

0

[ξ (r ′)]2r ′2 dr ′ = 0. (26)

This requires the covariance function to be decreasing. In fact, anystatistically homogeneous Gaussian field will be ergodic if ξ (r ) → 0as r → ∞. Notice then that a Gaussian random field can be ergodicwithout being sample homogeneous.

A general form of this ergodic theorem does not exist for arbitrarynon-Gaussian random fields, but fortunately this does not matter.

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What is needed for statistical cosmology is not an ergodic propertybut something closer to a version of the fair sample hypothesis.The ergodic property does not require the fair sample property, as itinvolves averages over infinite domains of a single realization. Thefair sample hypothesis does not require ergodicity, either. From thiswe can conclude that the ergodic property is irrelevant and use ofthis term should be avoided.

There is, however, one particularly neat relationship between er-godicity and statistical homogeneity for the Peebles model. Noticethat, if we take δ ∝ ψ2 and require ψ to be a Gaussian random fieldwith covariance function ξ (r ), then the covariance function of δ isjust ξ 2(r ), exactly the form that appears in equation (26). If we takeψ to be an ergodic Gaussian random field then this guarantees theresulting quadratic field must be at least second-order statisticallyhomogeneous.

4.4 Asymptotic independence and smoothing

The considerations we have discussed above generally lead to re-quirements that the correlations between points become small asthe separation between the points grows large. For a Gaussian field,if ξ (r ) → 0 as r → ∞ then the probability distributions tend toan asymptotically independent form. This must be the case becausesuch a field contains only second-order correlations. For example,in the limit that the correlation matrix Cij becomes diagonal, thetwo-point Gaussian probability density P2(δ1, δ2) → P1(δ1)P1(δ2),consistent with the requirement of independence, i.e. P(A, B) =P(A)P(B). Similar results will hold for higher-order N-point dis-tributions. This result means that, for Gaussian fields, absence of(second-order) correlation, i.e.

〈X1 X2〉 = 〈X1〉〈X2〉, (27)

means independence. Lack of correlation only requires full inde-pendence for Gaussian fields. Independence always implies lack ofcorrelation whether the field is Gaussian or not.

We can see then that even though a non-Gaussian field, such asa quadratic model, may be uncorrelated on large scales, consistentwith the requirements above, this does not necessarily mean thatpoints are asymptotically independent.

The reason for discussing this is that it is very relevant to whathappens to a random field as it is smoothed on successively largerscales. This smoothing is equivalent to filtering the field with a lowpass filter. The filtered field, δ(x; Rf), may be obtained by convo-lution of the ‘raw’ density field with some function F having acharacteristic scale Rf:

δ(x; Rf) =∫

δ(x′)F(|x − x′|; Rf) dx′. (28)

The filter F has the following properties: F = constant � R−3f if

|x − x′| � Rf, F � 0 if |x − x′| � Rf,∫

F(y; Rf) dy = 1.If the underlying density field is Gaussian then the filtered field

will also be Gaussian. This is a result of the fact that filtering essen-tially constructs a weighted average of the underlying field and anysum of Gaussian variates is itself a Gaussian variate (e.g. Kendall& Stuart 1977). According to the central limit theorem, the sumof a large number of independent variates drawn from a distribu-tion with finite variance also tends to a Gaussian distribution. Wewould imagine, therefore, that if distant points were asymptoticallyindependent then the effect of filtering on a non-Gaussian field is to‘Gaussianize’ it. In fact, this is assumed in standard statistical cos-mology. We know the small-scale distribution is non-Gaussian, butaveraging over sufficiently large smoothing windows is assumed to

recover the linear field, or something close to it. But, for a generalnon-Gaussian field, how quickly do points have to become indepen-dent in order for filters to Gaussianize the distribution?

The answer to this question is given by Fan & Bardeen (1995).It depends on a function called the Rosenblatt dependence rate,which governs the rate at which the maximum value of |P(AB)− P(A)P(B)| tends to zero at large separations (A and B are anycombination of values of δi in different locations). The rate at whichasymptotic independence is reached is called the mixing rate. Thisis quite a technical issue, and we leave the details to Fan & Bardeen(1995). On the other hand, when the field in question is a localtransformation of a Gaussian random field (such as in the quadraticmodel) then there is a simple result for the mixing rate, namely thatif the covariance function of the underlying field falls off as a power,i.e. as 1/rq as r → ∞, then q � 3. This is sometimes referred to asthe requirement for pseudo-Markov behaviour (Adler 1981). If thiscriterion is satisfied then all local transformations of the underlyingfield satisfy the mixing rate condition and consequently becomeGaussian if smoothed on sufficiently large scales.

Let us look at this issue in the light of the Peebles model.Peebles (1999b) has shown that this model is fully self-similar.When smoothed on larger and larger scales the distribution func-tion does not tend to a Gaussian but retains the same (χ 2) form.At first sight this looks extremely surprising. Suppose we imagine asimple version of a χ2 random field built upon discrete cells of somesize R. Suppose the field were uniform within each cell but that ad-jacent cells were generated independently from the χ2 distribution.Filtering on a scale Rf in this case simply corresponds to addingneighbouring (uncorrelated) cells. This would produce somethinglike a sum of N = (Rf/R)3 independent values from a quadraticmodel. This would produce a resulting field of the form (15). As Rf

becomes larger, N increases and the resulting distribution changes toa distribution of χ2

n of larger and larger n. It is well known that, as n→ ∞, the distribution of Yn is asymptotically close to the Gaussianas expected from the central limit theorem.

In the case of the Peebles model, however, the mixing rate con-dition is not satisfied. Although distant points are asymptoticallyindependent, the rate at which they tend to independence is not suf-ficient to produce a Gaussian distribution. Regardless of the scaleof smoothing, as long as the covariance function is chosen to bescale-free, the density field retains a distribution having the sameshape. This demonstrates that this model is, in fact, a kind of fractalas discussed above.

Notice that in the perturbative case where we take the quadraticcontribution to represent only the non-linear effect on initiallyGaussian fluctuations then it is guaranteed to satisfy the mixing ratecondition as long as the Gaussian field does that generates it. It istherefore justified to assume that the model described in Section 3.3does become Gaussian when smoothed on large scales.

5 QUA D R AT I C P H A S E C O U P L I N G

In Section 2 we pointed out that a convenient definition of aGaussian field could be made in terms of its Fourier phases, whichshould by independent and uniformly distributed on the interval[0, 2π]. A breakdown of these conditions, such as the correlationof phases of different wave modes, is a signature that the field hasbecome non-Gaussian. In terms of cosmic large-scale structure for-mation, non-Gaussian evolution of the density field is symptomaticof the onset of non-linearity in the gravitational collapse process,suggesting that phase evolution and non-linear evolution are closelylinked. A relatively simple picture emerges for models where the

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primordial density fluctuations are Gaussian and the initial phasedistribution is uniform. When perturbations remain small, evolu-tion proceeds linearly, individual modes grow independently and theoriginal random phase distribution is preserved. However, as pertur-bations grow large their evolution becomes non-linear and Fouriermodes of different wavenumbers begin to couple together. This givesrise to phase association and consequently to non-Gaussianity. It isclear that phase associations of this type should be related in someway to the existence of the higher-order connected covariance func-tions, which are traditionally associated with non-linearity and arenon-zero only for non-Gaussian fields. In this section, such a re-lationship is explored in detail using an analytical model for thenon-linearly evolving density fluctuation field. Phase correlationsof a particular form are identified and their connection to the covari-ance functions is established.

5.1 A simple non-linear model

We adopt the simple perturbative model of equation (18) in order toapproximate the non-linear evolution of the density field. Althoughthe equivalent transformation in formal Eulerian perturbation theoryis a good deal more complicated, the kind of phase associations thatwe deal with here are precisely the same in either case. In terms ofthe Fourier modes, in the continuum limit, we have for the first-orderGaussian term

δ1(x) =∫

d3k|δk| exp[iφk] exp [ik · x] (29)

and for the second-order perturbation

δ2(x) = [δ1(x)]2 =∫

d3k d3k ′|δk||δk′ | exp[i(φk + φk′ )]

× exp [i(k + k′) · r ]. (30)

The quadratic field, δ2, illustrates the idea of mode coupling asso-ciated with non-linear evolution. The non-linear field depends on aspecific harmonic relationship between the wavenumber and phaseof the modes at k and k′. This relationship between the phases inthe non-linear field, i.e.

φk + φk′ = φk+k′ , (31)

where the right-hand side represents the phase of the non-linearfield, is termed quadratic phase coupling.

5.2 The two-point covariance function

The two-point covariance function can be calculated using the def-initions of Section 2, namely

ξ (r ) = 〈δ(x)δ(x + r )〉. (32)

Substituting the non-linear transform for δ(x) [equation (18)] intothis expression gives four terms

ξ (r ) = 〈δ1(x)δ1(x + r )〉 + ε〈δ1(x)δ2(x + r )〉+ ε〈δ2(x)δ1(x + r )〉 + ε2〈δ2(x)δ2(x + r )〉. (33)

The first of these terms is the linear contribution to the covariancefunction whereas the remaining three give the non-linear correc-tions. We shall focus on the lowest order term for now.

As we outlined in Section 2, the angular brackets 〈〉 in these ex-pressions are expectation values, formally denoting an average overthe probability distribution of δ(x). Under the fair sample hypothe-sis, we replace the expectation values in equation (32) with averagesover a selection of independent volumes so that 〈〉 → 〈〉vol,real. The

first average is simply a volume integral over a sufficiently largepatch of the Universe. The second average is over various realiza-tions of the δk and φk in the different patches. Applying these rules tothe first term of equation (33) and performing the volume integrationgives

ξ11(r ) =∫

d3k d3k ′〈|δk||δk′ | exp [i(φk + φk′ )]〉real

×δD(k + k′) exp [ik′ · r ], (34)

where δD is the Dirac delta-function. The above expression is sim-plified given the reality condition

δk = δ∗−k, (35)

from which it is evident that the phases obey

φk + φ−k = 0 mod[2π]. (36)

Integrating equation (34) we therefore find that

ξ11(r ) =∫

d3k⟨|δk|2

⟩real

exp [−ik · r ] (37)

so that the final result is independent of the phases. Indeed, this isjust the Fourier transform relation between the two-point covariancefunction and the power spectrum we derived in Section 2.1.

5.3 The three-point covariance function

Using the same arguments outlined above it is possible to calculatethe three-point connected covariance function, which is defined as

ζ (r , s) = 〈δ(x)δ(x + r )δ(x + s)〉c. (38)

Making the non-linear transform of equation (18) we find the fol-lowing contributions

ζ (r , s) = 〈δ1(x)δ1(x + r )δ1(x + s)〉c

+ ε〈δ1(x)δ1(x + r )δ2(x + s)〉c+ perms(121, 211)

+ ε2〈δ1(x)δ2(x + r )δ2(x + s)〉c+ perms(212, 221)

+ ε3〈δ2(x)δ2(x + r )δ(x + s)〉c. (39)

Again we consider first the lowest-order term. Expanding in terms ofthe Fourier modes and once again replacing averages as prescribedby the fair sample hypothesis gives

ζ111(r , s) =∫

d3k d3k ′ d3k ′′

× 〈|δk||δk′ ||δk′′ | exp [i(φk + φk′ + φk′′ )]〉real

× δD(k + k′ + k′′) exp [ik′ · r ] exp [ik′′ · s]. (40)

Recall that δ1 is a Gaussian field so that φk, φk′ and φk′′ are inde-pendent and uniformly random on the interval [0, 2π]. Upon inte-gration over one of the wave vectors, the phase term is modifiedso that its argument contains the sum (φk + φk′ + φ−k−k′′ ), or apermutation thereof. Whereas the reality condition of equation (35)implies a relationship between phases of anti-parallel wave vectors,no such conditions hold for modes linked by the triangular con-straint imposed by the Dirac delta-function. In other words, exceptfor serendipity,

φk + φk′ + φ−k−k′′ �= 0. (41)

In fact, due to the circularity of phases, the resulting sum is stilljust uniformly random on the interval [0, 2π] if the phases are ran-dom. Upon averaging over sufficient realizations, the phase termwill therefore cancel to zero so that the lowest-order contribution

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to the three-point function vanishes, i.e. ζ111(r , s) = 0. This is nota new result, but it does explicitly illustrate how the vanishing ofthe three-point connected covariance function arises in terms of theFourier phases.

Next we consider the first non-linear contribution to the three-point function given by

ζ112(r , s) = ε〈δ1(x)δ1(x + r )δ2(x + s)〉, (42)

or one of its permutations. In this case, one of the arguments in theaverage is the field δ2(x), which exhibits quadratic phase couplingof the form (31). Expanding this term to the point of equation (40)using the definition (30), we obtain

ζ112(r , s) =∫

d3k d3k ′ d3k ′′ d3k ′′′〈|δk||δk′ ||δk′′ ||δk′′′ |

× exp[i(φk + φk′ + φk′′ + φk′′′ )]〉real

× δD(k + k′ + k′′ + k′′′) exp [ik′ · r ]

× exp [i(k′′ + k′′′) · s].(43)

Once again, the Dirac delta-function imposes a general constraintupon the configuration of wave vectors. Integrating over one of kgives k′′′ = −k − k′ − k′′ for example, so that the wave vectorsmust form a closed loop. This general constraint, however, doesnot specify a precise shape of loop, instead the remaining integralsrun over all of the different possibilities. At this point, we mayconstrain the problem more tightly by noting that most combinationsof k will contribute zero to ζ (112). This is because of the circularityproperty of the phases and equation (41). Indeed, the only non-zerocontributions arise where we are able to apply the phase relationobtained from the reality constraint, equation (36). In other words,the properties of the phases dictate that the wave vectors must alignin anti-parallel pairs: k = −k′, k′′ = −k′′′ and so forth.

There is a final constraint that must be imposed upon k ifζ is the connected three-point covariance function. In a graphtheoretical sense, the general (unconnected) N-point function〈δl1 (x1)δl2 (x2) . . . δlN (xN )〉 can be represented geometrically by asum of tree diagrams. Each diagram consists of N nodes of orderl i, representing δli (xi ), and a number of linking lines denoting theircorrelations; see Fry (1984) or Bernardeau (1992) for more detailedaccounts. Every node is made up of l i internal points, which repre-sent a factor δk = |δk| exp (i φk) in the Fourier expansion. Accordingto the rules for constructing diagrams, linking lines may join oneinternal point to a single other, either within the same node or in anexternal node (see Fig. 1). The connected covariance functions arerepresented specifically by the subset of diagrams for which everynode is linked to at least one other, leaving none completely iso-lated (Fig. 1; right). This constraint implies that certain pairings ofwave vectors do not contribute to the connected covariance function(Fig. 2).

The above constraints may be inserted into equation (43) by re-writing the Dirac delta-function as a product over delta-functionsof two arguments, appropriately normalized. There are only twoallowed combinations of wave vectors so we have

δD(k + k′ + k′′ + k′′′) → 1

2Vu[δD(k + k′′)δD(k′′′ + k′′′)

+ δD(k + k′′′)δD(k′ + k′′)]. (44)Integrating over two of the k and using equation (36) eliminates thephase terms, leaving the final result:

ζ112(r , s) = 1

Vu

∫d3k d3k ′ ⟨|δk|2|δk′ |2⟩

real

× exp [ik′ · r ] exp [−i(k + k′) · s]. (45)

δ(k ) δ(k )

δ(k) δ(k )

δ(k )

δ(k)

δ(k )

δ(k )

Figure 1. Two of the diagrammatic representations of the three-point func-tion 〈δ1(x)δ1(x + r )δ2(x + s)〉. Internal points (black dots) represent factorsof δk in the Fourier expansion, which are joined up in different configura-tions by the black linking lines. The diagram on the right is connected, in thesense that all nodes (blue circles) are joined together in a tree. The diagramon the left does not contribute to the connected function because it has anisolated node.

k k

kk

k

kk

k

Figure 2. Possible wave vector configurations for the diagrams in Fig. 1constructed under the constraints imposed by the Fourier phases. On the left,the wave vectors associated with the single second-order node form an anti-parallel pair, as do those from the two first-order nodes. This configurationdoes not contribute to the connected covariance function. The diagram onthe right is that for the connected tree in Fig. 1. The wave vectors associatedwith the second-order node align with each of the wave vectors from thefirst-order nodes.

The existence of this quantity has therefore been shown to dependon the quadratic phase coupling of Fourier modes. The relationshipbetween modes and the interpretation of the tree diagrams is alsodictated by the properties of the phases.

We may apply the same rules to the higher-order terms in equa-tion (39). It is immediately clear that the ζ 122 terms are zero becausethere is no way to eliminate the phase term exp[i(φk + φk′ + φk′′+φk′′′ + φk′′′′ )], a consequence of the property equation (41). Diagram-matically, this corresponds to an unpaired internal point within oneof the nodes of the tree. The final, highest-order contribution to thethree-point function is found to be

ζ222(r , s) = 1

V 2u

∫d3k d3k ′ d3k ′′ ⟨|δk|2|δk′ |2|δk′′ |2⟩

real

× exp [i(k − k′) · r ] exp [i(k′ − k′′) · s], (46)

where the phase and geometric constraints allow 12 possible com-binations of wave vectors.

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5.4 Cubic non-linearity and higher order

The above ideas extend simply to higher order where the non-linearfield is represented by a perturbation series that does not truncateat the quadratic term. At the next highest-order, for example, theseries includes δ3 = δ3

1, which introduces a cubic phase couplingin the Fourier expansion. Although quadratic phase coupling is es-sential as a minimum requirement for the three-point covariancefunction, cubic phase coupling is not the minimum requirementfor the next highest-order covariance function. Indeed, quadraticcoupling is sufficient to provide contributions to all of the n-pointcovariance functions, because of the way the phases dictate thatwave vectors must arrange themselves into anti-parallel pairs. Forthe four-point covariance function, the cubic term allows for a dif-ferent diagrammatic representation: a star as opposed to a snaketopology. However, in terms of the constituent wave vectors, loops(as in Fig. 2) contributing to the star topologies are a symmetricsubset of those contributing to the snake topologies.

5.5 Power spectrum and bispectrum

The formal development of the relationship between covariancefunctions and power spectra developed above suggests the useful-ness of higher-order versions of P(k). It is clear from the argumentsof Section 5.2 that a more convenient notation for the power spec-trum than that introduced in Section 2.1 is

〈δkδk′ 〉 = (2π)3 P(k)δD(k + k′). (47)

The connection between phases and higher-order covariance func-tions obtained in Section 5.3 also suggests defining higher-orderpolyspectra of the form

〈δkδk′ . . . δk(n) 〉 = (2π)3 Pn

(k, k′, . . . k(n)

)δD

(k + k′ + . . . k(n)

)(48)

where the occurrence of the delta-function in this expression arisesfrom a generalization of the reality constraint given in equation (36);see, for example, Peebles (1980). Conventionally the version of thiswith n = 3 produces the bispectrum, usually called B(k, k′, k′′)which has found much effective use in recent studies of large-scalestructures (Peebles 1980; Scoccimarro et al. 1998, 1999; Verde et al.2000, 2001, 2002). It is straightforward to show that the bispec-trum is the Fourier transform of the (reduced) three-point covariancefunction by following similar arguments as in Section 5.2; see, forexample, Peebles (1980).

Note that the delta-function constraint requires the bispectrumto be zero except for k-vectors (k, k′, k′′) that form a triangle in k-space. From Section 5.3 it is clear that the bispectrum can only benon-zero when there is a definite relationship between the phasesaccompanying the modes whose wave vectors form a triangle. More-over, the pattern of phase association necessary to produce a realand non-zero bispectrum is precisely that which is generated byquadratic phase association. This shows, in terms of phases, whyit is that the leading-order contributions to the bispectrum emergefrom second-order fluctuations of a Gaussian random field. The bis-pectrum measures quadratic phase coupling.

Three-point phase correlations have another interesting property.While the bispectrum is usually taken to be an ensemble-averagedquantity, as defined in equation (46), it is interesting to considerproducts of terms δkδk′δk′′ obtained from an individual realization.According to the fair sample hypothesis discussed above, we wouldhope appropriate averages of such quantities would yield an estimateof the bispectrum. Note that

δkδk′δk′′ = δkδk′δ−k−k′ = δkδk′δ∗k+k′ ≡ β(k, k′), (49)

using the requirement (36) and the triangular constraint we havediscussed above. Each β(k, k′) will carry its own phase, say φk,k′ ,which obeys

φk,k′ = φk + φk′ − φk+k′ . (50)

It is evident from this that it is possible to recover the completeset of phases φk from the bispectral phases φk,k′ , up to a constantphase offset corresponding to a global translation of the entire struc-ture (Chiang & Coles 2000). This furnishes a conceptually simplemethod of recovering missing or contaminated phase informationin a consistent way, an idea which has been exploited, for example,in speckle interferometry (Lohmann, Weigelt & Wirnitzer 1983). Inthe case of quadratic phase coupling, described by equation (31), theleft-hand side of equation (50) is identically zero leading to a par-ticularly simple approach to this problem, which we shall explorein future work.

6 D I S C U S S I O N

In this paper we have addressed two main issues, using the quadraticmodel as an illustrative example. First we have shown explicitly howthe non-Gaussian Peebles model has properties that contradict stan-dard folklore based on the assumption of Gaussian fluctuations. Wehave used this model to distinguish carefully between various inter-related concepts such as sample homogeneity, statistical homogene-ity, asymptotic independence, ergodicity, and so on. We have shownthe conditions under which each of these is relevant and we havedeployed the quadratic model for particular examples in which theyare violated.

We then used a perturbative quadratic model to show, for the firsttime, how in non-linear processes phase association arises whichhas exactly the correct form to generate non-zero bispectra andthree-point covariance functions. The magnitude of these statisticaldescriptors is, of course, related to the magnitude of the Fouriermodes, but the factor that determines whether they are zero or non-zero is the arrangement of the phases of these modes.

The connection between polyspectra and phase information is animportant one and it opens up many lines of future research, such ashow phase correlations relate to redshift distortion and bias. Also, wehave assumed throughout this study that we can straightforwardlytake averages over a large spatial domain to be equal to ensembleaverages. Using small volumes, of course, leads to sampling uncer-tainties which are quite straightforward to deal with in the case of thepower spectra but more problematic for higher-order spectra suchas the bispectrum. Understanding the fluctuations about ensembleaverages in terms of phases could also lead to important insights.

AC K N OW L E D G M E N T S

PC acknowledges useful discussions with John Peacock, SimonWhite and Sabino Matarrese about matters related to Section 4of this paper. This work was supported by PPARC grant numberPPA/G/S/1999/00660.

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Statistical cosmology with quadratic density fields

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