Local bias approach to the clustering of discrete density peaks

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Article

Reference

Local bias approach to the clustering of discrete density peaks

DESJACQUES, Vincent

Abstract

Maxima of the linear density field form a point process that can be used to understand the spatial distribution of virialized halos that collapsed from initially overdense regions. However, owing to the peak constraint, clustering statistics of discrete density peaks are difficult to evaluate. For this reason, local bias schemes have received considerably more attention in the literature thus far. In this paper, we show that the two-point correlation function of maxima of a homogeneous and isotropic Gaussian random field can be thought of, up to second order at least, as arising from a local bias expansion formulated in terms of rotationally invariant variables. This expansion relies on a unique smoothing scale, which is the Lagrangian radius of dark matter halos. The great advantage of this local bias approach is that it circumvents the difficult computation of joint probability distributions. We demonstrate that the bias factors associated with these rotational invariants can be computed using a peak-background split argument, in which the background perturbation shifts the corresponding probability distribution [...]

DESJACQUES, Vincent. Local bias approach to the clustering of discrete density peaks.

Physical Review. D , 2013, vol. 87, no. 4

DOI : 10.1103/PhysRevD.87.043505

Available at:

http://archive-ouverte.unige.ch/unige:35139

Disclaimer: layout of this document may differ from the published version.

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Local bias approach to the clustering of discrete density peaks

Vincent Desjacques*

De´partement de Physique The´orique and Center for Astroparticle Physics (CAP), Universite´ de Gene`ve, 24 quai Ernest Ansermet, CH-1211 Gene`ve, Switzerland

(Received 22 November 2012; published 4 February 2013)

Maxima of the linear density field form a point process that can be used to understand the spatial distribution of virialized halos that collapsed from initially overdense regions. However, owing to the peak constraint, clustering statistics of discrete density peaks are difficult to evaluate. For this reason, local bias schemes have received considerably more attention in the literature thus far. In this paper, we show that the two-point correlation function of maxima of a homogeneous and isotropic Gaussian random field can be thought of, up to second order at least, as arising from a local bias expansion formulated in terms of rotationally invariant variables. This expansion relies on a unique smoothing scale, which is the Lagrangian radius of dark matter halos. The great advantage of this local bias approach is that it circumvents the difficult computation of joint probability distributions. We demonstrate that the bias factors associated with these rotational invariants can be computed using a peak-background split argument, in which the background perturbation shifts the corresponding probability distribution functions. Consequently, the bias factors are orthogonal polynomials averaged over those spatial locations that satisfy the peak constraint. In particular, asphericity in the peak profile contributes to the clustering at quadratic and higher order, with bias factors given by generalized Laguerre polynomials. We speculate that our approach remains valid at all orders, and that it can be extended to describe clustering statistics of any point process of a Gaussian random field. Our results will be very useful to model the clustering of discrete tracers with more realistic collapse prescriptions involving the tidal shear for instance.

DOI:10.1103/PhysRevD.87.043505 PACS numbers: 98.80.k, 95.35.+d

I. INTRODUCTION

In the biasing scenario introduced by Kaiser [1], virialized halos form out of initially overdense regions with a linear density (extrapolated to the redshift of interest) equal to c 1:686. Since then, this picture has received con- siderable support from observational data. Even though dark matter halos are extended objects, they form a spatial point process as far as their clustering is concerned.

However, this essential feature has remained elusive in most theoretical descriptions of halo clustering, which assume that halos are a Poisson sampling of a more fundamental, continuous halo density fieldhðxÞ.

The peak formalism first proposed by [2,3] in a cosmo- logical context is interesting because it is a well-behaved point process. In this approach, virialized halos are associated with maxima of the initial density field. The displacement from their initial (Lagrangian) to final (Eulerian) position can be computed upon assuming phase space conservation [4]. Clustering statistics of these discrete density peaks display many of the features present in measurements of halo clustering extracted from N-body simulations. In particular, discrete density peaks exhibit a k-dependent linear bias factor [5,6], small-scale exclusion [7,8], and a linear velocity bias [9]. Some of these predic- tions have recently been tested in numerical simulations [10,11]: peaks of the linear density field appear to provide a

good approximation to the formation sites of dark matter halos withM*M_?.

However, despite recent progress towards the computation of peak clustering statistics [4] and a formulation of peak theory within the excursion set formalism [12,13], discrete density peaks lack a clear connection with the more conven- tional local bias schemes [14], in which halos are approxi- mated as a continuous field. Furthermore, while in the local bias model the computation of halo correlation functions is straightforward (though there are ambiguities regarding the filtering scale etc.); in the peak formalism calculations are particularly tedious owing to the peak constraint [3,4,6,15].

In the most comprehensive analysis thus far, Ref. [4] suc- ceeded in computing the peak two-point correlationpkðrÞ up to second order, including the Zel’dovich displacement.

They showed that some of the first- and second-order contributions could be obtained from a peak-background split formulated in terms of conditional mass functions. In con- trast to most analytic models of halo clustering, which assume that the (k-independent) bias coefficients are the peak-background split biases, they derived this equivalence from first principles. However, they could not determine the physical origin of the other second-order contributions.

Moreover, the peak constraint is clearly too simplistic to describe the clustering of low mass halos. In this mass range, one should consider more elaborated constraints involving the tidal shear, etc. In this regard, it would be very desirable to find a simpler way of computing the correlation functions of generic point processes of a (Gaussian) random field.

*[email protected]

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In this paper, we suggest a simple, physically motivated prescription based on the peak-background split to compute the correlation functions of generic point processes driven by homogeneous and isotropic Gaussian random fields. We argue that clustering statistics of such point processes can be reduced to the evaluation of correlators of aneffectivecontinuous overdensity which, in the case of discrete peaks, is a function of the local (smoothed) mass density field and its derivatives. Our approach combines in a single coherent picture peak theory, peak-background split, local bias, and the excursion set framework. For sake of clarity, we will focus on the two-point correlation function of initial density peaks as computed in Ref. [4] to explain the fundamentals of our approach.

The paper is organized as follows. SectionII furnishes a brief summary of clustering in peak theory. SectionIII is the central section of the paper, where we present the connection between rotational invariants, peak- background split, and a local peak bias prescription.

Finally, Sec.IVdiscusses the implications of our findings.

II. CORRELATION FUNCTIONS FOR DENSITY PEAKS

We begin with a short recapitulation of the computation of correlation functions in the peak formalism. Let_s be the linear mass density field smoothed on scaleR_swith a spherically symmetric filter. For convenience, we work with the normalized variables ðxÞ ¹

0_sðxÞ, _iðxÞ

1

₁@isðxÞ, and ijðxÞ ¹

2@i@jsðxÞ. Here,is the peak height or significance, and

²nðRsÞ 1 2²

Z1

0 dkk²^ðⁿ^þ¹^ÞPsðkÞ (1) are moments of the power spectrum PsðkÞ ¼ hjsðkÞj²i.

A Gaussian filter is frequently adopted to ensure convergence of all the spectral moments_n, but one should bear in mind that the peak height associated with dark matter halos is always computed with a tophat filter (see Ref. [13] for details).

The first few spectral moments_n can be combined into a dimensionless spectral width ₁ ¼²₁=ð₀₂Þ that takes values between zero and unity. 1 reflects the range over which the smoothed power spectrum P_sðkÞ is significant, i.e., ₁ 1 for a sharply peaked power spectrum whereas ₁0for a power spectrum that covers a wide range of wave numbers.

Correlations of density maxima of_s can be evaluated using the Kac-Rice formula [16,17]. The trick is to Taylor- expandiðxÞaround the positionxpkof a local maximum. As a result, the number density of (so-called BBKS [3]) peaks of height⁰at positionxin the smoothed density field_scan be expressed in terms of the field_sand its derivatives:

npkð⁰; Rs;xÞ 3³⁼²

R³_? jdetðxÞj_D½ðxÞ _H½₃ðxÞ_D½ðxÞ ⁰; (2)

whereR? ffiffiffi p3

ð1=2Þis the characteristic radius of a peak (and not the interpeak distance). The three-dimensional Dirac distribution DðÞ ensures that all extrema are included. The factors of theta function _Hð3Þ, where3 is the lowest eigenvalue of the shear tensorij, and the Dirac deltaDð⁰Þfurther restrict the set to density maxima of the desired significance⁰.

The (disconnected) N-point correlations ^ð_pk^N^Þ (or joint intensities) of density maxima are defined as the ensemble averages of products ofnpkð; Rs;xÞ,

^ð_pk^N^Þð; R_s;x₁;. . .;x_NÞ

hnpkð; Rs;x1Þ npkð; Rs;xNÞi: (3) For the Gaussian initial conditions considered here, multivariate normal distribution are assumed to perform the ensemble average. In the particular case N¼1, hnpkð; Rs;xÞi ¼npkð; RsÞ is the average, differential number density of peaks of height identified on the filtering scaleR_s[3],

npkð; RsÞ ¼ 1

ð2Þ²R³_?e²⁼²G^ð₀¹^Þð1; 1Þ

¼effiffiffiffiffiffiffi²⁼² p2

1 V_?

G^ð₀¹^Þð₁; ₁Þ: (4) In the last equality, V?¼ ð2Þ³⁼²R³? is the typical three- dimensional extent of a density peak [12]. The functions G^ðn^Þð1; 1Þare defined in AppendixB. In particular, the ratioG^ð1Þ_k =G^ð1Þ₀ is equal to thekth momentu^k of the peak curvature u. Similarly, the reduced two-point correlation function for maxima of a given significance separated by a distancer¼ jrj ¼ jx2x1jis

pkð; Rs; rÞ ¼^ð_pk²^Þð; Rs; rÞ

n²_pkð; R_sÞ 1: (5) Notice that, in ^ð_pk²^Þ, we have ignored the shot-noise term

n_pk_Dðx₂x₁Þthat arises from the self-pairs as it matters only at zero-lag (in the peak power spectrum, however, this contributes a constant Poisson noise 1=npk at all wave numbers).

The calculation of Eq. (5) at second order in the mass correlation and its derivatives is quite tedious [3,4,15]

because one must evaluate the joint probability distribution for the ten-dimensional vector of variables y^> ¼ ðiðxÞ; ðxÞ; AðxÞÞ at two different spatial locations x¼x₁ and x₂, i.e., a total of 20 variables. Here, the components A; A¼1;. . .;6 symbolize the independent entries ij¼11, 22, 33, 12, 13, 23 of _ij. Fortunately, as was shown in Ref. [4], most of the terms nicely combine together, so that the final result can be recast into the compact expression

VINCENT DESJACQUES PHYSICAL REVIEW D87,043505 (2013)

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_pkð; R_s; rÞ ¼ ð~b²_I^ð₀⁰^ÞÞ þ1

2ð^ð₀⁰^Þ~b²_II^ð₀⁰^ÞÞ 3

²₁ð^ð₁¹⁼²^Þb~_II^ð₁¹⁼²^ÞÞ 5

²₂ð^ð₂¹^Þb~_II^ð₂¹^ÞÞ 1þ2

5@lnG^ð₀^Þð₁; ₁Þj_¼₁ þ 5

2⁴₂

ð^ð₀²^ÞÞ²þ10

7 ð^ð₂²^ÞÞ²þ18

7 ð^ð₄²^ÞÞ² 1þ2

5@lnG^ð₀^Þð1; 1Þj¼1

₂ þ 3

2⁴₁½ð^ð₀¹^ÞÞ²þ2ð^ð₂¹^ÞÞ² þ 3

²₁²₂½3ð^ð₃³⁼²^ÞÞ²þ2ð^ð₁³⁼²^ÞÞ²: (6)

The functions ^ð_‘ⁿ^ÞðrÞ are quantities analogous to ²_n but defined for a finite separationr,

^ð_‘ⁿ^ÞðRs; rÞ ¼ 1 2²

Z1

0 dkk²^ðⁿ^þ¹^ÞPsðkÞj‘ðkrÞ; (7) where j_‘ðxÞ are spherical Bessel functions. On the right- hand side of Eq. (6), all the correlations depend on the filtering scale and the separation. However, the first line contains terms involving the first- and second-order peak bias parameters ~bI and ~bII (to be defined shortly), the second line retains adependence through the function 1þ ð2=5Þ@lnG^ð₀^Þð1; 1Þj¼1 solely, whereas the last two terms on the right-hand side depend on the separationr (andRs) only. Hence, unlike standard local bias expansions (Eulerian or Lagrangian), the peak two-point correlation also exhibits quadratic terms linear in the second-order bias ~b_II. These terms involve derivatives of the linear mass correlation ^ð₀⁰^Þ and, therefore, vanish at zero lag.

Clearly, they arise because the peak correlation also depends on the statistical properties of_iand_ij.

Reference [4] also showed that the Lagrangian peak bias factors~b_Nðk₁;. . .; k_NÞcan be constructed upon averaging over the peak curvature products ofbandbu, where

bð; RsÞ ¼ 1 0

₁u 1²₁

; (8)

buð; RsÞ ¼ 1 2

u1 1²₁

: (9) For peak of significanceon the smoothing scaleRs, the first-order bias ~bI is defined as the Fourier space multi- plication (we omit the dependence onRsandfor short- hand convenience) [6]

b~_IðkÞ ¼b₁₀þb₀₁k² whereb₁₀¼b and b₀₁¼b_u: (10) The overline designates the average over the peak curvature.b01 can be quite large for moderate peak heights. In the high peak limit1; however, it is negligible so that

~bIðkÞis nearly scale independent (like in local bias models). Similarly, the Fourier space expression of the second- order peak bias~b_II is [4]

~b_IIðk₁; k₂Þ ¼b₂₀þb₁₁ðk²₁þk²₂Þ þb₀₂k²₁k²₂; (11)

where k₁ and k₂ are wave modes and the k-independent coefficients b₂₀,b₁₁, andb₀₂are

b₂₀ð; R_sÞ b² 1 ²₀ð1²₁Þ

¼ 1 ²₀

²2₁uþ²₁u²

ð1²₁Þ² 1 ð1²₁Þ

;

(12) b11ð; RsÞ bbuþ ²₁

²₁ð1²₁Þ

¼ 1 02

ð1þ²₁Þu₁½²þu²

ð1²₁Þ² þ ₁ ð1²₁Þ

; (13) and

b02ð; RsÞ b²u 1 ²₂ð1²₁Þ

¼ 1 ²₂

u²21uþ²₁²

ð1²₁Þ² 1 ð1²₁Þ

: (14) By definition,~b^mII acts on the functions^ð_‘ⁿ₁¹^ÞðrÞand^ð_‘ⁿ₂²^ÞðrÞ as follows:

ð^ð_‘ⁿ¹^Þ

1

~b^mII^ð_‘ⁿ²^Þ

2 Þ 1

4⁴ Z1

0 dk1

Z1

0 dk2k²₁^ðⁿ¹^þ¹^Þk²₂^ðⁿ²^þ¹^Þ ~b^m_IIðk₁; k₂ÞP_sðk₁ÞP_sðk₂Þj_‘₁ðk₁rÞj_‘₂ðk₂rÞ:

(15) As pointed out by Desjacqueset al. [4], the pieceb~²_I^ð₀⁰^Þþ ð1=2Þ^ð₀⁰^Þ~b²II^ð₀⁰^Þ can be thought of as arising from the continuous, deterministic, local bias relation

_pkðxÞ ¼b₁₀_sðxÞ b₀₁r²_sðxÞ þ1

2b₂₀²_sðxÞ b11sðxÞr²sðxÞ þ1

2b02½r²sðxÞ²; (16) where the bias factors bij are peak-background split bias factors that follow from expanding the conditional peak number density in a series in the small background density perturbation l. This expansion is local in the sense that, except for the filtering, it involves quantities evaluated at x solely. However, an essential difference with the

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widespread local bias model [14] is the fact that, when computing the ensemble average h_pkðx₁Þ_pkðx₂Þi, we must ignore all powers of zero-lag moments [such as, e.g.,⁴₀inh²sðx1Þ²sðx2Þi] to recoverpkðrÞsince the latter does not exhibit such contributions (this ‘‘no zero-lag requirement’’ also arises in the derivation of the ‘‘renor- malized’’ bias parameters of Schmidtet al. [18]). All the terms in Eq. (16) are of course invariant under rotations sincepkðxÞtransforms as a scalar under rotations. Clearly, however, this series expansion is not the most generic Lagrangian expansion we may conceive of (see, e.g., Ref. [19] for nonlocal Lagrangian bias).

Notwithstanding these results, Desjacqueset al. [4] did not succeed in finding a physical interpretation of the other second-order terms on the right-hand side of Eq. (6), even though it was pretty clear that they—at least partially—

arise from coupling involving the components of the gra- dientiand the Hessianij.

III. AN INTUITIVE INTERPRETATION OFpkðrÞ In this section, we propose an intuitive, physically motivated explanation of Eq. (6) that is grounded in the peak-background split argument [1]. We begin with a brief introduction to the helicity basis, which was used in Ref. [4] to compute probability distributions of the density field and its derivatives at two different spatial locations.

A. Probability density in the helicity basis The two-point correlation function of initial density peaks is the ensemble average ofnpkðy1Þnpkðy2Þ over the joint probability density P₂ðy₁;y₂;rÞ, where yyðxÞ are the values of the field and its derivatives at positionx. In what follows, npkðyÞ will also designate Eq. (2).

Following [4], we can decompose the variables y^> ¼ ð_iðxÞ; ðxÞ; _AðxÞÞthat appear in the joint probability densityP₂ðy₁;y₂;rÞin the helicity basisðeþ;r;^ eÞ, where

eþieê^ ffiffiffi2

p ; r^r=r; eie^þe^ ffiffiffi2

p ; (17) and r^, e^ , and e^ are orthonormal vectors in spherical coordinatesðr; ; Þ. The orthogonality relations between these vectors areee¼r^r^¼1andeþe¼e r^¼0, where the inner product between two vectorsuand v is defined as uvu_iv_iu_ivⁱ. Unless otherwise stated, an overline will denote complex conjugation throughout Sec.III A.

In this reference frame, we decompose the first derivatives as

^ð⁰^Þr^þ^ðþ¹^Þe^þþ^ð¹^Þe: (18) Here,^ð⁰^Þr^and^ð¹^Þeare the helicity0 and1components. The correlation properties of^ð⁰^Þand ^ð1Þ can be obtained by projecting out the scalar and

vector parts of the correlation of the Cartesian components _i with the projection operatorP¼eþ eþþe e. The rule of thumb is that h^ð₁^s^Þ^ð₂^s⁰^Þi ¼ h^ð₁^s^Þ^ð₂ ^s⁰^Þi, where ^ð^s^Þ¼^ðsÞðxÞ, vanish unlessss⁰ ¼0. We find

h^ð₁⁰^Þ^ð₂⁰^Þi ¼ 1

3²₁ð^ð₀¹^Þ2^ð₂¹^ÞÞ; h^ð₁ ¹^Þ^ð₂ ¹^Þi ¼ 1

3²₁ð^ð₀¹^Þþ^ð₂¹^ÞÞ; h^ð₁ ¹^Þ^ð₂ ¹^Þi ¼0:

(19) Here and henceforth, the subscripts ‘‘1’’ and ‘‘2’’ will denote variables evaluated at positionx1 andx2 for short- hand convenience. Similarly, the symmetric tensorijcan be decomposed into its trace and traceless components,

ij

1

3uijþ~ij; ~ij ¼Sij^ð^S^Þþ

ffiffiffi 1 3 s

ð_i^ð^V^Þr^jþ_j^ð^V^ÞrîÞ þ ffiffiffi 2 3 s

_ij^ð^T^Þ: (20) The variables u tr ¼ _iⁱ and ^ð^S^Þ^ð⁰^Þ are the longitudinal and transverse helicity0 modes, _i^ð^V^Þ are the components of a transverse vector, ^ð^V^Þr^¼0, whereas _ij^ð^T^Þ is a symmetric, traceless, transverse tensor, ^ij_ij^ðTÞ¼_ij^ðTÞr^^j¼0. Explicit expressions for these variables are

^ð^S^Þ3

2S^lmlm¼1

2ð3 ^r^lr^^m^lmÞlm; (21) _i^ð^V^Þ ffiffiffi

3 p

V_i^lm_lm¼ ffiffiffi 3

p ð^l_ir^_ir^^lÞr^^m_lm; (22)

_ij^ð^T^Þ ffiffiffi 3 2 s

T_ij^lmlm¼ ffiffiffi 3 2 s

ðP^l_iP^m_j 1

2PijP^lmÞlm; (23) where Sab,Va^bc, andT_ab^cd are the scalar, vector, and tensor projections operators (see, e.g., Ref. [20]). We have introduced factors of ffiffiffiffiffiffiffiffi

p1=3

and ffiffiffiffiffiffiffiffi p2=3

in the decomposition Eq. (20) such that the zero-point moments of the helicity 0, 1, and 2 variables all equal 1=5 [see Eq. (24) below]. The helicity1 components of ^ð^V^Þ and their complex conjugates are given byffiffiffi ^ð¹^Þ^ð^V^Þe¼ p3

eⁱr^^jij and ^ð¹^Þ^ð^V^Þe ¼ ffiffiffi p3

eⁱr^^jij, whereas ^ð²^Þ_ij^ð^T^Þeⁱe^j¼ ffiffiffiffiffiffiffiffi

p3=2

eⁱe^jij and ^ð²^Þ_ij^ð^T^Þeⁱe^j

¼ ffiffiffiffiffiffiffiffi p3=2

eⁱe^jij are the two independent helicity2 modes (polarizations) and their complex conjugates, respectively. Hereafter designating ^ðsÞðxÞ as ^ð^s^Þ, the correlation properties of these variables are the following:

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h₁^ð⁰^Þ₂^ð⁰^Þi ¼ 1 ²₂

1

5^ð₀²^Þ2

7^ð₂²^Þþ18 35^ð₄²^Þ

; h₁^ð¹^Þ^ð₂ ¹^Þi ¼ 1

²₂ 1

5^ð₀²^Þ1

7^ð₂²^Þ12 35^ð₄²^Þ

; h₁^ð2Þ^ð2Þ₂ i ¼ 1

²₂ 1

5^ð2Þ₀ þ2

7^ð2Þ₂ þ 3 35^ð2Þ₄

; (24)

andh₁^ð^s^Þ₂^ð^s⁰^Þi ¼ h₁^ð^s^Þ^ð₂ ^s⁰^Þi. All the other correlations vanish. Note that the covariances are real despite the fact that the helicity1and2variables are complex.

While the average peak number density only depends on the matrix M of covariances at the same location, the computation of the peak two-point correlation function _pkðrÞ and higher-order clustering statistics from Eq. (3) generally involve covariances of the random fields at different locations. ForpkðrÞ, the covariance matrixCðrÞ hyy^yi, wherey¼ ðy₁;y₂Þ, is a 20-dimensional matrix that may be partitioned into four1010block matrices: the zero-point contribution M in the top left and bottom right corners and the cross-correlation matrixBðrÞand its trans- pose in the bottom left and top right corners, respectively.

Expressions for M and BðrÞ in the helicity basis can be found in the Appendix of Ref. [4].

B. Rotational invariants

Translational and rotational invariance implies thatnpk

does not depend on spatial position, and that _pkðrÞ is a function of the distancersolely. In this regard, Desjacques et al. [4] noted that, although the covariance matrixCðrÞin the helicity basis (17) does not depend on the direction r^ of the separation vector r, it is not equal to the angular average covariance matrix ^CðrÞ ð1=4ÞR

d^rCðrÞ. The latter is obtained upon setting^ð_‘ⁿ^Þ0whenever‘0in the expression ofBðrÞ. As a consequence, ^CðrÞretains the correlations h12i, h1u2i, hu1u2i, by only parts of the covariances h^ð₁^m^Þ^ð₂^m^Þi and h₁^ð^m^Þ^ð₂^m^Þi. However, Ref. [4]

did not provide a convincing explanation for their obser- vation. We shall do it now.

First, it is pretty clear that, since the peak two-point correlation is invariant under rotations of the reference frame, it should be possible to express it in terms of rotational invariants constructed from the variables,i, and ij. The peak significanceand the traceuare two obvious candidates, but they are not the only ones. The vectorof first derivatives and the traceless matrix~yield two addi- tional invariants, i.e., the square modulus and the tracetrð~²Þ. In the helicity basis, these invariants can be written

¼^ð⁰^Þ^ð⁰^Þþ^ðþ¹^Þ^ðþ¹^Þþ^ð¹^Þ^ð¹^Þ; (25) and

trð~²Þ ¼2 3

^ð⁰^Þ^ð⁰^Þþ X

s¼1;2

ð^ðþ^s^Þ^ðþ^s^Þþ^ð^s^Þ^ð^s^ÞÞ : (26) The 33 symmetric matrixij actually provides a third invariant with respect to rotations: the determinant det. However, as we shall see below in the discussion of the peak-background split, because this determinant only enters the peak number densitynpkðyÞand not the one-point multivariate normal distributionP₁ðyÞ(the argument of the exponential is quadratic in the variables), it does not contribute directly to the peak bias. This suggests that we look at the covariances of²ðxÞand²ðxÞ, where these are defined as ²ðxÞ ðxÞ ðxÞ

¼ 1 ²₁

Z d³k1

ð2Þ³

Z d³k2

ð2Þ³ijk1ik2jsðk1Þ

sðk₂Þeⁱ^ð^k¹^þ^k²^Þ^x; (27)

²ðxÞ 3

2tr½~²ðxÞ

¼ 3 2²₂

Z d³k1

ð2Þ³

Z d³k2

ð2Þ³

iljm1 3ijlm

k_1ik_1jk_2lk_2m_sðk₁Þ_sðk₂Þeⁱ^ð^k¹^þ^k²^Þ^x; (28) where_sðkÞare the Fourier modes of the smoothed density field. As we shall see in Sec.III C, the variables3²and5² are distributed as chi-squared (²) variates with 3 and 5 degrees of freedom, respectively. Using either the Fourier space expression of ²²ðxÞ (not to be confounded here with a Cartesian component of the vector) or the fact that only components of identical helicity correlate, it is straightforward to compute the following correlators (for illustrative purposes, Appendix A furnishes details of the calculation ofh²₁²₂i):

h²₁²₂i ¼1þ 2

3⁴₁½ð^ð₀¹^ÞÞ²þ2ð^ð₂¹^ÞÞ²; (29) h²₁²₂i ¼1þ 2

²₀²₁^ð₁¹⁼²^Þ^ð₁¹⁼²^Þ: (30) Ignoring the zero-lag contributions, these terms correspond exactly (up to a sign factor) to some of those entering the second-order contribution of pkðrÞ in Eq. (6), with ð^ð₁¹⁼²^Þb₂₀^ð₁¹⁼²^ÞÞbeing proportional toh²₁²₂i, in particular.

The computations of correlators involving ² ²ðxÞ proceeds in a similar way although, in this case, it is much easier to sum the correlations among equal helicity components. For instance,

h₁²₂²i ¼1þ2

h₁^ð⁰^Þ₂^ð⁰^Þi²þ2X

s¼1;2

ðh₁^ðþ^s^Þ^ðþ₂ ^s^Þi² þ h₁^ð^s^Þ^ð₂ ^s^Þi²Þ

; (31)

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where we used the fact that h₁^ð^s^Þ₂^ð^s^Þi ¼ h₁^ð^s^Þ^ð₂^s^Þi. After some algebra, we find

h₁²₂²i ¼1þ 2 5⁴₂

ð^ð₀²^ÞÞ²þ10

7 ð^ð₂²^ÞÞ²þ18 7 ð^ð₄²^ÞÞ²

; (32) and

h₁²²₂i ¼1þ 2

5²₁²₂½3ð^ð₃³⁼²^ÞÞ²þ2ð^ð₁³⁼²^ÞÞ²; (33) h₁²²₂i ¼1þ 2

²₀²₂^ð₂¹^Þ^ð₂¹^Þ: (34) The cross-correlations of²₁ and₁² withu₂ in place of₂ are identical except for the superscriptðnÞ, which should be replaced by ðnþ1Þ. Again, all these terms can be found among the second-order contributions on the right-hand side of Eq. (6).

Therefore, the actual dependence of_pkðrÞon the invariants ²ðxÞ and ²ðxÞ, whose covariances involve the correlation functions ^ð_‘ⁿ^Þ with ‘0, is the fundamental reason for CðrÞ being different from the angle average

^CðrÞ. Those correlations, which arise upon expanding the joint probability density at second order, eventually all nicely combine together [and with terms proportional to ð^ð₀¹^ÞÞ² and ð^ð₀²^ÞÞ²] to yield the second-order correlators h²₁²₂i,h₁²²₂ietc.

The question then arises of the calculation of the coefficients of these quadratic terms in the peak two-point correlation function. We already know that the coefficients multiplying products of the form^ð₀ⁿ^Þ^ð₀ⁿ⁰^Þare the quadratic peak-background split biases associated with the scalars andu. Does this hold also for the coefficients multiplying h²₁²₂i,h₁²²₂i, etc.?

C. Generalizing the peak-background split The probability distribution P₁ðyÞ that is needed to compute n_pk is a multivariate Gaussian of covariance matrix M,

P1ðyÞd¹⁰y¼ 1

ð2Þ⁵jdetMj¹⁼²e^Q¹^ð^y^Þd¹⁰y: (35) Owing to rotational invariance, this probability density is a function of,u,², and²solely (see, e.g., Ref. [21] for a systematic analysis of distribution functions of homogeneous and isotropic random fields). The quadratic form Q1ðyÞthat appears in the exponential factor reads

Q1ðyÞ ¼²þu²21u 2ð1²₁Þ þ3

2²þ5

2²; (36) so that exp½Q1ðyÞ retains factorization with respect to ð; uÞ,², and². Furthermore, since^ð^s^Þ (withs¼ 1) and^ðsÞ(withs¼ 1,2) are complex random variables with mean 0 and variance1=3 and1=5, respectively, and

sinceh^ð⁰^Þi ¼ h^ð⁰^Þi ¼0andh^ð⁰^Þ^ð⁰^Þi ¼0at the same spatial location, the quantities 3²ðxÞand 5²ðxÞ are independent²-distributed variables with 3 and 5 degrees of freedom, respectively (similar conclusions can be drawn for the distribution of the components of the deformation tensor; see Refs. [19,22]). Therefore, the one-point probability density can also be written as

P₁ðyÞd¹⁰y¼Nð; uÞddu

²₃ð3²Þdð3²Þ²₅ð5²Þdð5²Þ; (37) whereNð; uÞis the bivariate normal

Nð; uÞ ¼ 1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1²₁

q exp

²þu²21u 2ð1²₁Þ

; (38) and

²_kðxÞ ¼ 1

2^k=2ðk=2Þx^k=2¹e^x=2 (39) is a²distribution withkdegrees of freedom. Note that the distribution of²is coupled with the last rotational invari- antdet[23]. However, it can be easily checked that, upon integrating over the (uniform) distribution of det, we obtain the² distribution²₅ð5²Þ.

Reference [4] discussed how the peak bias factors bij

can be derived from a peak-background split. They argued that, while the k-dependent piecebn0 is related to thenth order derivative of the differential number density npk, derivatives cannot produce the bias factors b₀₁, b₁₁, etc.

multiplying the k-dependent terms. For this reason, they considered a second implementation of the peak- background split [24,25] in which the dependence of the mass function on the overdensity of the background is derived explicitly. However, it is possible to write all the peak bias factorsbijas derivatives ofNð; uÞrather than

n_pk. More precisely, the b_ij are the bivariate Hermite polynomials

Hijð; uÞ ¼Nð; uÞ¹ @

@ _i

@

@u _j

Nð; uÞ (40) relative to the weight Nð; uÞ, further averaged over the peak curvature u. Therefore, they are peak-background split biases in the sense that they can be derived from the transformation!þ1andu!uþ2, whereiand 2are long-wavelength background perturbations uncorrelated with the (small-scale) fields ðxÞ and uðxÞ.

Reference [4] did not express the peak-background split this way because they considered the effect of a background perturbation after the integration over the peak curvature. In terms of the rotational invariants introduced above, we can write thebijas

ⁱ₀^j₂bij¼ 1 n_pk

Z

d¹⁰yn_pkðyÞHijð; uÞP1ðyÞ; (41)

(8)

where it is understood thatP₁ðyÞtakes the form of Eq. (37) andd¹⁰y¼ddudð3²Þdð5²Þ. Factors of1=₀and1=₂ are introduced because bias factors are ordinarily defined relative to the physical field_sðxÞand its derivatives rather than the normalized variables. In practice, the integral is most easily performed upon transforming the 5 degrees of freedom attached to5² to the shape parametersvandw and the three Euler angles that describe the orientation of the principal axis frame (see AppendixBfor details).

In Ref. [23], it was noticed that, in the presence of non- Gaussianity, the one-point probability densityP1ðyÞcan be expanded in the set of orthogonal polynomials associated with the weight provided byP1ðyÞin the Gaussian limit.

The same logic applies to the peak bias factors. Namely, the bij are drawn from the orthogonal polynomials associated withNð; uÞ, i.e., bivariate Hermite polynomials.

Therefore, we expect that ²ðxÞ and ²ðxÞ also generate bias parameters, and that these are drawn from the orthogonal polynomials associated with²distributions, i.e., generalized Laguerre polynomials. These are defined as

L^ðn^ÞðxÞ ¼xe^x n!

dⁿ

dxⁿðe^xxⁿ^þÞ (42) and are orthogonal over ½0;1½ with respect to the ² distribution with k¼2ðþ1Þ degrees of freedom. The orthogonality relation can be expressed as

Z1

0 dxxe^xL^ðÞn ðxÞL^ðÞm ðxÞ ¼ðnþþ1Þ

n! mn: (43) The first generalized Laguerre polynomials areL^ð₀^ÞðxÞ ¼1 andL^ð₁^ÞðxÞ ¼ xþþ1.

Given the correlator Eq. (29), the term proportional to 1=⁴₁ on the right-hand side of Eq. (6) indicates that the first-order bias parameter10associated with the invariant ²₁²ðxÞ ¼ ðr_sÞ²ðxÞis10¼ 3=ð2²₁Þ. The aforemen- tioned considerations suggest that we define thekth-order bias factor as the Laguerre polynomialð1Þ^kL^ð_k¹⁼²^Þð3²=2Þ averaged over all the possible peak configurations, i.e.,

^2k₁ k0 ð1Þ^k npk

Z d¹⁰ynpkðyÞL^ð_k¹⁼²^Þ 3²

2

P1ðyÞ: (44) Taking into account the peak constraint, the first-order bias factor thus is

10¼ 1 ²₁npk

Z

d¹⁰ynpkðyÞ 3 2²3

2

P1ðyÞ (45)

¼ 3

2²₁; (46)

which is precisely what we were expecting. Similarly, we define the bias parameter _0k associated with the invariant ²ðxÞ as the ensemble average of the Laguerre polynomial L^ð_k³⁼²^Þð5²=2Þ orthogonal with respect to the weight²₅ð5²Þ. Namely,

^2k₂ 0k ð1Þ^k n_pk

Z

d¹⁰yn_pkðyÞL^ð3=2Þ_k 5²

2

P1ðyÞ: (47) Note that, although we use a single symbolijto designate the bias factors derived from the ² distributions, the variables ² and ², unlike and u, are uncorrelated.

The first-order bias factor thus is ₀₁¼ 1

²₂n_pk

Z d¹⁰yn_pkðyÞ 5 2²5

2

P₁ðyÞ: (48) To evaluate the integral, we first express the measure dð5²Þ in terms of the ellipticity v and prolateness w, so that ² can be written as ²¼3v²þw² (see Appendix B for details). A multiplicative factor of ² will arise upon, e.g., taking the derivative of expð5²=2Þ with respect to . In the notation of Desjacqueset al. [4], our factor of²precisely corresponds to their derivative term ð2=5Þ@lnG^ð₀^Þð1; 1Þ evaluated at ¼1 (see Appendix B). Taking into account the factor of G^ð₀¹^Þð1; 1Þ in the denominator, 01 can eventually be written

01¼ 5 2²₂

1þ2

5@lnG^ð₀^Þð1; 1Þj_¼₁

: (49) The physical interpretation of this result is straightforward:

²ðxÞis a scalar that describes the asymmetry of the peak density profile. In the high peak limit,₀₁ !0reflecting the fact that the most prominent peaks are nearly spherical (see Fig. 9 of Ref. [4]).

The physical origin for the appearance of these orthogonal polynomials can be found in the peak-background split.

Long-wavelength background perturbations locally modu- late the mean of the distributions Nð; uÞ, ²₃ð3²Þ and ²₅ð5²Þ. The resulting noncentral distributions can then be expanded in the appropriate set of orthogonal polynomials.

In practice, it is convenient to introduce a shift or trans- lation operator T^ to describe the action of a background perturbation on the distribution of rotational invariants. For the scalarsandu, we define the shift operator as

T^expð₁@₂@_uÞ; (50) where ₁ and₂ are small perturbations to the peak significance and the peak curvature, i.e.,!þ₁andu! uþ2. The action of T^ on the probability density Nð; uÞ is to shift the (zero) mean of and u by 1

and2, respectively [the reason for the minus sign is that Hermite polynomials include a factor of ð1Þⁱ^þ^j]. A straightforward calculation gives

Nð; uÞ¹T^Nð; uÞ ¼Nð1; u2Þ Nð; uÞ

¼fð1; 2Þe¹⁰^b^þ²²^b^u; (51) wherebandbuare given in Eqs. (8) and (9), andfð1; 2Þ is the exponential factor inNð; uÞwith the replacement