A local bias approach to the clustering of discrete density peaks

0.3 MB·English

by  Vincent Desjacques

A local bias approach to the clustering of discrete density peaks
Vincent Desjacques1,∗
1D´epartement de Physique Th´eorique and Center for Astroparticle Physics (CAP) Universit´e de Gen`eve,
24 quai Ernest Ansermet, CH-1211 Gen`eve, Switzerland
Maximaofthelineardensityfieldformapointprocessthatcanbeusedtounderstandthespatial
distribution of virialized halos that collapsed from initially overdense regions. However, owing to
thepeak 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
thispaper,weshowthatthe2-pointcorrelationfunctionofmaximaofahomogeneousandisotropic
Gaussian random field can be thought of, up to second order at least, as arising from a local
3
bias expansion formulated in terms of rotationally invariant variables. This expansion relies on a
1
uniquesmoothing scale, which istheLagrangian radius of dark matterhalos. Thegreat advantage
0
of this local bias approach is that it circumvents the difficult computation of joint probability
2
distributions. We demonstrate that thebias factors associated with these rotational invariantscan

n becomputedusingapeak-backgroundsplit argument,inwhichthebackgroundperturbationshifts
a thecorresponding probability distribution functions. Consequently, thebias factors are orthogonal
J
polynomials averaged over those spatial locations that satisfy the peak constraint. In particular,

7 asphericityinthepeakprofilecontributestotheclusteringatquadraticandhigherorder,withbias
1 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
O
with more realistic collapse prescriptions involvingthe tidal shear for instance.
C
. PACSnumbers: 98.80.-k,98.65.-r,95.35.+d,98.80.Es
h
p
-
o I. INTRODUCTION sites of dark matter halos with M &M⋆.
r
t However,despiterecentprogresstowardsthecomputa-
s
a Inthebiasingscenariointroducedby[1],virializedha- tion of peak clustering statistics [4] and a formulation of
[ los form out of initially overdense regions with a linear peak theory within the excursion set formalism [12, 13],

density(extrapolatedtothe redshiftofinterest)equalto discrete density peaks lack a clear connection with the
2
v δc 1.686. Since then, this picture has received con- more conventional local bias schemes [14], in which ha-
8 side≈rable support from observational data. Even though losareapproximatedasacontinuousfield. Furthermore,
2 darkmatterhalosareextendedobjects,theyformaspa- while in the local bias model the computation of halo
1 tial point process as far as their clustering is concerned. correlationfunctionsisstraightforward(thoughthereare
4 However, this essential feature has remained elusive in ambiguitiesregardingthefilteringscaleetc.),inthepeak
1. most theoretical descriptions of halo clustering, which formalism calculations are particularly tedious owing to
1 assume that halos are a Poissonsampling of a more fun- the peak constraint [3, 4, 6, 15]. In the most compre-
2 damental, continuous halo density field δ (x). hensive analysis thus far, ref. [4] succeeded in comput-
h
1 ing the peak 2-point correlation ξ (r) up to second or-
pk
: Thepeakformalismfirstproposedby[2,3]inacosmo-
v der,includingtheZel’dovichdisplacement. Theyshowed
logicalcontext is interesting because it is a well-behaved
i that some of the first- and second-order contributions
X point process. In this approach, virialized halos are as-
could be obtained from a peak-background split formu-
r sociated with maxima of the initial density field. The lated in terms of conditional mass functions. In con-
a displacementfromtheirinitial(Lagrangian)tofinal(Eu-
trast to most analytic models of halo clustering, which
lerian) position can be computed upon assuming phase
assumethatthe(k-independent) biascoefficientsarethe
space conservation [4]. Clustering statistics of these dis-
peak-background split biases, they derived this equiva-
crete density peaks display many of the features present
lence from first principles. However, they could not de-
in measurements of halo clustering extracted from N-
terminethephysicaloriginoftheothersecond-ordercon-
body simulations. In particular, discrete density peaks
tributions. Moreover, the peak constraint is clearly too
exhibitak-dependentlinearbiasfactor[5,6],small-scale
simplistic to describe the clustering of low mass halos.
exclusion[7,8],andalinearvelocitybias[9]etc. Someof
In this mass range, one should consider more elaborated
these predictions have recently been tested in numerical
constraints involving the tidal shear etc. In this regards,
simulations [10, 11]: peaks of the linear density field ap-
it would be very desirable to find a simpler way of com-
pear to provide a good approximation to the formation
putingthecorrelationfunctionsofgenericpointprocesses
of a (Gaussian) random field.
In this paper, we suggest a simple, physically moti-
∗Electronicaddress: [email protected] vated prescription based on the peak-background split
2
to compute the correlation functions of generic point where R √3(σ /σ ) is the characteristic radius of
⋆ 1 2
≡
processes driven by homogeneous and isotropic Gaus- a peak (and not the interpeak distance). The three-
sian random fields. We argue that clustering statistics dimensionalDiracdistribution δ (η) ensuresthat allex-
D
of such point processes can be reduced to the evalua- tremaareincluded. Thefactorsofthetafunctionθ (λ ),
H 3
tion of correlators of an effective continuous overdensity where λ is the lowest eigenvalue of the shear tensor ζ ,
3 ij
which, in the case of discrete peaks, is a function of the andthe Dirac delta δ (ν ν′) further restrictsthe setto
D
local (smoothed) mass density field and its derivatives. density maxima of the de−sired significance ν′.
Ourapproachcombinesinasinglecoherentpicturepeak The(disconnected)N-pointcorrelationsρ(N) (orjoint
theory, peak-background split, local bias and the excur- pk
intensities)ofdensitymaximaaredefinedastheensemble
sion set framework. For sake of clarity, we will focus on
averagesof products of n (ν,R ;x),
the 2-point correlation function of initial density peaks pk s
as computed in [4] to explain the fundamentals of our
approach. ρ(N)(ν,R ,x ,...,x )
pk s 1 N
The paper is organized as follows. Sec. II furnishes a
briefsummaryofclusteringinpeaktheory. Sec.IIIisthe npk(ν,Rs,x1) npk(ν,Rs,xN) . (3)
≡ ×···×
central Section of the paper, where we present the con-
D E
nection between rotational invariants, peak-background
Forthe Gaussianinitialconditions consideredhere,mul-
split and a local peak bias prescription. Finally, Sec.IV
tivariate normal distribution are assumed to perform
discusses the implications of our findings.
the ensemble average. In the particular case N = 1,
n (ν,R ,x) = n¯ (ν,R ) is the average, differential
pk s pk s
h i
number density of peaks of height ν identified on the fil-
II. CORRELATION FUNCTIONS FOR
tering scale R [3],
DENSITY PEAKS s
We begin with a short recapitulation of the computa- n¯ (ν,R )= 1 e−ν2/2G(1)(γ ,γ ν) (4)
pk s (2π)2R3 0 1 1
tion of correlation functions in the peak formalism. Let ⋆
δ be the linear mass density field smoothed on scale e−ν2/2 1
s = G(1)(γ ,γ ν).
Rwes wwoitrhkawsitphhetrhiceanlloyrmsyamlimzeedtrvicarfiialtbelre.sFνo(rx)conve1nδien(xce),, √2π (cid:18)V⋆(cid:19) 0 1 1
≡ σ0 s
ηi(x) ≡ σ11∂iδs(x) and ζij(x) ≡ σ12∂i∂jδs(x). Here, ν is In the last equality, V⋆ = (2π)3/2R⋆3 is the typical 3-
the peak height or significance, and
dimensional extent of a density peak [12]. The functions
1 ∞ G(nα)(γ1,γ1ν) are defined in Appendix B. In particular,
σn2(Rs)≡ 2π2 Z0 dkk2(n+1)Ps(k). (1) ptheeakractuiorvGat(ku1)r/eGu(0.1)Siismeiqlauralyl,ttohtehreedkuthcemdo2m-peonitntukcoorfretlhae-
are moments of the power spectrum P (k) = δ (k)2 . tionfunctionformaximaofagivensignificanceseparated
s s
h| | i
A Gaussianfilter is frequently adoptedto ensure conver- by a distance r= r = x x is
2 1
| | | − |
genceofallthespectralmomentsσ ,butoneshouldbear
n
inmindthatthepeakheightassociatedwithdarkmatter ρ(2)(ν,R ,r)
halosisalwayscomputedwithatophatfilter(see[13]for ξ (ν,R ,r)= pk s 1, (5)
pk s n¯2 (ν,R ) −
details). The first few spectral moments σn can be com- pk s
bined into a dimensionless spectral width γ =σ2/(σ σ
1 1 0 1
that takes values between zero and unity. γ1 reflects the Notice that, in ρ(2), we have ignoredthe shot-noise term
pk
range over which the smoothed power spectrum P (k) is
s n¯ δ (x x ) that arises from the self-pairs as it mat-
pk D 2 1
significant, i.e. γ 1 for a sharply peaked power spec- −
1 ≈ ters only at zero-lag (in the peak power spectrum how-
trum whereas γ 0 for a power spectrum that coversa
1 ≈ ever, this contributes a constant Poisson noise 1/n¯pk at
wide range of wavenumbers.
all wavenumbers).
Correlations of density maxima of δ can be evalu-
s The calculation of Eq.(5) at second order in the mass
ated using the Kac-Rice formula [16, 17]. The trick is to
correlation and its derivatives is quite tedious [3, 4, 15]
Taylor-expand η (x) around the position x of a local
i pk because one must evaluate the joint probability distri-
maximum. As a result, the number density of (BBKS)
bution for the 10-dimensional vector of variables y⊤ =
peaks of height ν′ at position x in the smoothed density α
(η (x ),ν(x ),ζ (x )) at two different spatial locations
field δ can be expressed in terms of the field δ and its i α α A α
s s x = x and x , i.e. a total of 20 variables. Here, the
derivatives: α 1 2
components ζ ,A= 1, ,6 symbolize the independent
A
···
n (ν′,R ,x) 33/2 detζ(x) δ [η(x)] θ [λ (x)] (2) entries ij = 11,22,33,12,13,23 of ζij. Fortunately, as
pk s ≡ R3 | | D H 3 was shown in [4], most of the terms nicely combine to-
⋆
gether, so that the final result can be recast into the
δ [ν(x) ν′] ,
× D − compact expression
3
1 3 5 2
ξ (ν,R ,r)= b˜2ξ(0) + ξ(0)˜b2ξ(0) ξ(1/2)˜b ξ(1/2) ξ(1)b˜ ξ(1) 1+ ∂ lnG(α)(γ ,γ ν)
pk s I 0 2 0 II 0 − σ12 1 II 1 − σ22 2 II 2 (cid:18) 5 α 0 1 1 (cid:12)α=1(cid:19)
+(cid:0) 5 (cid:1)ξ(2)(cid:0)2+ 10 ξ(2(cid:1)) 2+ 1(cid:0)8 ξ(2) 2 1+(cid:1)2∂ ln(cid:0)G(α)(γ ,γ ν(cid:1)) 2 (cid:12)(cid:12)
+ 2σ324h(cid:0)ξ0(1)(cid:1)2+27ξ(cid:0)(12) 2(cid:1) + 73(cid:0) 43(cid:1)ξ(i3(cid:18)/2) 2+52αξ(3/02) 21. 1 (cid:12)(cid:12)(cid:12)α=1(cid:19) (6)
2σ4 0 2 σ2σ2 3 1
1 1 2
h(cid:0) (cid:1) (cid:0) (cid:1) i h (cid:0) (cid:1) (cid:0) (cid:1) i
The functions ξ(n)(r) are quantities analogous to σ2 but where k and k are wavemodes and the k-independent
ℓ n 1 2
defined for a finite separation r, coefficients b , b and b are
20 11 02
1 ∞ 1
ξℓ(n)(Rs,r)= 2π2 dkk2(n+1)Ps(k)jℓ(kr), (7) b20(ν,Rs)≡b2ν − σ2(1 γ2) (12)
Z0 0 − 1
1 ν2 2γ νu¯+γ2u2 1
where jℓ(x) are spherical Bessel functions. In the right- = − 1 1
hand side of Eq.(6), all the correlations depend on the σ02 " (1−γ12)2 − (1−γ12)#
filtering scale and the separation. However,the first line
γ2
contains terms involving the first and second order peak b (ν,R ) b b + 1 (13)
bias parameters b˜I and ˜bII (to be defined shortly), the 11 s ≡ ν u σ12(1−γ12)
second line retains a ν-dependence through the function 1 1+γ2 νu¯ γ ν2+u2
t1w+o (t2er/m5)s∂αinlntGhe(0αr)(iγg1h,tγ-h1aνn)|dα=s1idesodleelpye,nwdhoenretahsetsheepalarast- = σ0σ2"(cid:0) 1(cid:1)(1−−γ121)(cid:2)2 (cid:3)
tion r (and Rs) only. Hence, unlike standard local bias + γ1 ,
expansions (Eulerian or Lagrangian), the peak 2-point (1 γ2)
− 1 #
correlation also exhibits quadratic terms linear in the
second-order bias ˜bII. These terms involve derivatives and
(0)
of the linear mass correlation ξ and, therefore, vanish
0 1
at zero lag. Clearly, they arise because the peak correla- b (ν,R ) b2 (14)
02 s ≡ u− σ2(1 γ2)
tion also depends on the statistical properties of ηi and 2 − 1
tζoijrR.seb˜f.[(4k]a,l.s.o.s,hkow)edcatnhabtethceonLsatgruractnegdiaunppoenakavbeiarasgfiancg- = σ122 "u2−(21γ−1νγu¯12+)2γ12ν2 − (1−1γ12)# .
N 1 N
over the peak curvature products of bν and bu, where By definition, ˜bm acts on the functions ξ(n1)(r) and
II ℓ1
1 ν γ1u ξ(n2)(r) as follows:
bν(ν,Rs)= σ 1− γ2 , (8) ℓ2
0 (cid:18) − 1 (cid:19)
bu(ν,Rs)= σ12 (cid:18)u1−−γγ112ν(cid:19) . (9) ξℓ(1n1)b˜mIIξℓ(2n2) ≡ 4π14 Z0∞dk1Z0∞dk2k12(n1+1)k22(n2+1)
(cid:0) b˜m(k(cid:1),k )P (k )P (k )j (k r)j (k r). (15)
For peak of significance ν on the smooting scale Rs, the × II 1 2 s 1 s 2 ℓ1 1 ℓ2 2
firstorderbiasb˜ isdefinedastheFourierspacemultipli-
I As pointed out by [4], the piece b˜2ξ(0)+(1/2)ξ(0)b˜2ξ(0)
cation (we omit the dependence on R and ν for short- I 0 0 II 0
s
can be thought of as arising from the continuous, deter-
hand convenience) [6]
ministic, local bias relation
b˜ (k)=b +b k2 (10)
I 10 01 1
where b10 =¯bν and b01 =¯bu . δpk(x)=b10δs(x)−b01∇2δs(x)+ 2b20δs2(x) (16)
The overlinedesignates the averageoverthe peak curva- b11δs(x) 2δs(x)+ 1b02 2δs(x) 2 ,
− ∇ 2 ∇
ture. b can be quite large for moderate peak heights.
01 (cid:2) (cid:3)
In the high peak limit ν 1 however, it is negligible so where the bias factorsb arepeak-backgroundsplit bias
ij
that b˜ (k) is nearly scale≫-independent (like in local bias factors that follow from expanding the conditional peak
I
models). Similarly, the Fourier space expression of the number density in a series in the small background den-
second order peak bias ˜bII is [4] sity perturbation δl. This expansion is local in the sense
that, exceptfor the filtering, it involvesquantities evalu-
b˜ (k ,k )=b +b k2+k2 +b k2k2 , (11) ated at x solely. However, an essential difference with
II 1 2 20 11 1 2 02 1 2
(cid:0) (cid:1)
4
the widespread local bias model [14] is the fact that, η(±1) can be obtained by projecting out the scalar and
when computing the ensemble average δ (x )δ (x ) , vector parts of the correlation of the Cartesian compo-
pk 1 pk 2
h i
we must ignore all powers of zero-lag moments (such nentsη withtheprojectionoperatorP =e e +e
i + + −
tahs,e ela.gtt.,erσd04oiens nhδos2t(xex1h)δibs2(itx2s)uic)htcoonretrciobvuetrioξnpsk((rt)hissin’ncoe ew−h.erTehηe(s)ru=leηo(sf)t(hxum),bvaisnitshhatunhlηe1(sss)ηs2(s′)is′==h⊗0η.1(sW)ηe2(−fisn′)d⊗i,
zero-lag requirement’ also arises in the derivation of the α α −
’renormalized’ bias parameters of [18]). All the terms
in Eq.(16) are of course invariant under rotations since η(0)η(0) = 1 ξ(1) 2ξ(1) (19)
δpk(x) transforms as a scalar under rotations. Clearly h 1 2 i 3σ12 0 − 2
however, this series expansion is not the most generic 1 (cid:0) (cid:1)
Lagrangianexpansion we may conceive of (see, e.g., [19] hη1(±1)η2(±1)i= 3σ2 ξ0(1)+ξ2(1)
non nonlocal Lagrangianbias). 1
Notwithstanding these results, [4] did not succeed in η(±1)η(∓1) =0. (cid:0) (cid:1)
h 1 2 i
findingaphysicalinterpretationoftheothersecond-order
terms in the right-hand side of Eq.(6), even though it
Here and henceforth, the subscripts “1” and “2” will de-
was pretty clear that they – at least partially – arise
note variables evaluated at position x and x for short-
from coupling involving the components of the gradient 1 2
hand convenience. Similarly, the symmetric tensor ζ
η and the hessian ζ . ij
i ij can be decomposed into its trace and traceless compo-
nents,
III. A INTUITIVE INTERPRETATION OF ξpk(r)
1
ζ uδ +ζ˜ (20)
ij ij ij
InthisSection,weproposeanintuitive,physicallymo- ≡−3
tivated explanation of Eq. (6) that is grounded in the 1 2
ζ˜ =S ζ(S)+ ζ(V)ˆr +ζ(V)ˆr + ζ(T) .
peak-background split argument [1]. We begin with a ij ij 3 i j j i 3 ij
r r
brief introduction to the helicity basis, which was used (cid:16) (cid:17)
in [4] to compute probability distributions ofthe density
fieldanditsderivativesattwodifferentspatiallocations. The variables u ≡ −trζ = −ζii and ζ(S) ≡ ζ(0) are
the longitudinal and transverse helicity-0 modes, ζ(V)
i
are the components of a transverse vector, ζ(V) ˆr = 0,
A. Probability density in the helicity basis whereas ζ(T) is a symmetric, traceless, transver·se ten-
ij
sor, δijζ(T) = ζ(T)ˆrj = 0. Explicit expressions for these
The2-pointcorrelationfunctionofinitialdensitypeaks ij ij
variables are
is the ensemble average of n (y )n (y ) over the joint
pk 1 pk 2
probability density P (y ,y ;r), where y y(x ) are
2 1 2 α α
≡ 3 1
the values of the field and its derivatives at position ζ(S) Slmζ = 3ˆrlˆrm δlm ζ (21)
lm lm
x . In what follows, n (y) will also designate Eq.(2). ≡ 2 2 −
α pk
(Fηol(lxow)in,νg(x[4]),,ζwe(xca)n)tdheactomapppoesaertinhethveajroiainbtlepsroybα⊤abi=l- ζi(V) ≡√3Vilmζlm =√(cid:0)3 δil−ˆriˆrl (cid:1)ˆrmζlm (22)
i α α A α
ity density P2(y1,y2;r) in the helicity basis (e+,ˆr,e−), ζ(T) 3Tlmζ (cid:0) (cid:1) (23)
where ij ≡r2 ij lm
3 1
e+ ≡ ieˆφ√−2eˆθ, ˆr≡r/r, e− ≡ ieˆφ√+2eˆθ (17) =r2(cid:18)PilPjm− 2PijPlm(cid:19)ζlm ,
and ˆr, eˆ , and eˆ are orthonormal vectors in spherical
θ φ where S , Vbc and Tcd are the scalar,vectorand tensor
coordinates(r,θ,φ). Theorthogonalityrelationsbetween ab a ab
projections operators (see, e.g., [20]). We have intro-
thesevectorsaree e =ˆrˆr=1ande e =e ˆr=0,
±· ± · +· − ±· duced factors of 1/3 and 2/3 in the decomposition
where the inner product between two vectors u and v is
Eq.(20)suchthatthezero-pointmomentsofthehelicity-
defined as u·v ≡ uivi ≡ uivi. Unless otherwise stated, 0, -1 and -2 variapbles all equpal 1/5 (see Eq. (24) below).
an overline will denote complex conjugation throughout
Thehelicity-1componentsofζ(V)andtheircomplexcon-
Sec. IIIA.
In this reference frame,we decompose the firstderiva- jugates are given by ζ(±1) ≡ ζ(V) · e± = √3ei±ˆrjζij
tives as and ζ(±1) ζ(V) e = √3ei ˆrjζ , whereas ζ(±2)
≡ · ± ± ij ≡
ζ(T)ei ej = 3/2ei ej ζ and ζ(±2) ζ(T)ei ej =
η η(0)ˆr+η(+1)eˆ +η(−1)e . (18) ij ± ± ± ± ij ≡ ij ± ±
≡ + − 3/2ei ej ζ are the two independent helicity-2 modes
± ± ijp
Here,η(0) η ˆrandη(±1) η e arethehelicity-0and (polarizations) and their complex conjugates, respec-
-1 compon≡ents·. The corre≡latio·n±properties of η(0) and tpively. Hereafter designating ζ(s)(x ) as ζ(s), the cor-
α α
5
relation properties of these variables are the following: latter is obtained upon setting ξ(n) 0 whenever ℓ = 0
ℓ ≡ 6
intheexpressionofB(r). Asaconsequence,C(r)retains
1 1 2 18
ζ(0)ζ(0) = ξ(2) ξ(2)+ ξ(2) (24) the correlations ν1ν2 , ν1u2 , u1u2 and parts of the
h 1 2 i σ22 (cid:18)5 0 − 7 2 35 4 (cid:19) covariances η(m)hη(m)iahnd ζi(m)hζ(m)i. Howebver, [4] did
(±1) (±1) 1 1 (2) 1 (2) 12 (2) not provideha1conv2inciing exhpl1anat2ionifor their observa-
ζ ζ = ξ ξ ξ
h 1 2 i σ2 5 0 − 7 2 − 35 4 tion. We shall do it now.
2 (cid:18) (cid:19)
1 1 2 3 Firstly, it is pretty clear that, since the peak 2-point
ζ(±2)ζ(±2) = ξ(2)+ ξ(2)+ ξ(2) ,
h 1 2 i σ2 5 0 7 2 35 4 correlation is invariant under rotations of the reference
2 (cid:18) (cid:19) frame, it should be possible to express it in terms of ro-
and ζ(s)ζ(s′) = ζ(s)ζ(−s′) . All the other correlations tational invariants constructed from the variables ν, ηi
h 1 2 i h 1 2 i and ζ . The peak significance ν and the trace u are two
vanish. Note that the covariances are real despite the ij
obvious candidates, but they are not the only ones. The
fact that the helicity-1 and -2 variables are complex.
vector η of first derivatives and the traceless matrix ζ˜
While the average peak number density only depends
yield two additional invariants, i.e. the square modulus
on the matrix M of covariances at the same location,
η η and the trace tr(ζ˜2). In the helicity basis, these
the computation of the peak 2-point correlation func- ·
invariants can be written
tion ξ (r) and higher-order clustering statistics from
pk
Eq.(3)generallyinvolvecovariancesoftherandomfields
at different locations. For ξ (r), the covariance matrix η η =η(0)η(0)+η(+1)η(+1)+η(−1)η(−1) (25)
pk ·
C(r) yy† , where y = (y ,y ), is a 20-dimensional and
1 2
≡ h i
matrix that may be partitioned into four 10 10 block
×
matrices: the zero-point contribution M in the top left 2
tr ζ˜2 = ζ(0)ζ(0)
and bottom right corners, and the cross-correlation ma-
3
trix B(r) and its transpose in the bottom left and top (cid:20)
(cid:0) (cid:1)
right corners, respectively. Expressions for M and B(r) + ζ(+s)ζ(+s)+ζ(−s)ζ(−s) . (26)
in the helicity basis canbe found in the Appendix of [4]. sX=1,2(cid:16) (cid:17)(cid:21)
The 3 3symmetric matrixζ actually providesa third
ij
B. Rotational invariants ×
invariantwithrespecttorotations: thedeterminantdetζ.
However, as we shall see below in the discussion of the
Translational and rotational invariance implies that peak-backgroundsplit,becausethisdeterminantonlyen-
n¯pk does not depend onspatial position, and that ξpk(r) tersthepeaknumberdensity npk(y) andnotthe 1-point
be a function of the distance r solely. In this regards,[4] multivariate normaldistribution P (y) (the argumentof
1
noted that, although the covariance matrix C(r) in the theexponentialisquadraticinthe variables),itdoesnot
helicity basis (17) does not depend on the direction ˆr of contribute directly to the peak bias. This suggests that
the separationvectorr, itis notequaltothe angularav- we look at the covariances of η2(x) and ζ2(x), where
erage covariance matrix C(r) (1/4π) dΩ C(r). The these are defined as
ˆr
≡
R
b
1 d3k d3k
η2(x) η(x) η(x)= 1 2 δ k k δ (k )δ (k )ei(k1+k2)·x , (27)
≡ · −σ2 (2π)3 (2π)3 ij 1i 2j s 1 s 2
1 Z Z
3 3 d3k d3k 1
ζ2(x) tr ζ˜2(x) = 1 2 δ δ δ δ k k k k δ (k )δ (k )ei(k1+k2)·x , (28)
≡ 2 2σ2 (2π)3 (2π)3 il jm− 3 ij lm 1i 1j 2l 2m s 1 s 2
h i 2 Z Z (cid:18) (cid:19)
where δ (k) are the Fourier modes of the smoothed den- A furnishes details of the calculation of η2η2 ):
s h 1 2i
sity field. As we shall see in Sec. IIIC, the variables
3η2 and 5ζ2 are distributed as chi-squared (χ2) variates η2η2 =1+ 2 ξ(1) 2+2 ξ(1) 2 (29)
with 3 and 5 degrees of freedom, respectively. Using ei- h 1 2i 3σ4 0 2
1
ther the Fourier space expression of η2 η2(x ) (not 2 h(cid:0) (cid:1) (cid:0) (cid:1) i
α ≡ α η2ν2 =1+ ξ(1/2)ξ(1/2) . (30)
to be confonded here with a cartesian component of the h 1 2i σ2σ2 1 1
0 1
vector η) or the fact that only components of identical
helicity correlate, it is straightforward to compute the Ignoring the zero-lag contributions, these terms corre-
followingcorrelators(forillustrativepurposes,Appendix spond exactly (up to a sign factor) to some of those en-
tering the second-order contribution of ξ (r) in Eq.(6),
pk
6
with (ξ(1/2)b ξ(1/2)) being proportional to η2ν2 in Owing to rotational invariance, this probability density
particul1ar. T20he1 computations of correlatorshin1vo2lviing is a function ofν, u, η2 and ζ2 solely (see, e.g., [21] for a
ζ2 ζ2(x ) proceeds in a similar way although, in this systematic analysis of distribution functions of homoge-
α ≡ α
case, it is much easier to sum the correlations among neous and isotropic random fields). The quadratic form
equal helicity components. For instance, Q (y) that appears in the exponential factor reads
1
ν2+u2 2γ νu 3 5
ζ2ζ2 =1+2 ζ(0)ζ(0) 2 (31) Q (y)= − 1 + η2+ ζ2 , (36)
h 1 2i h 1 2 i 1 2(1 γ2) 2 2
(cid:26) − 1
+2 ζ(+s)ζ(+s) 2+ ζ(−s)ζ(−s) 2 , sothatexp[ Q (y)] retainsfactorizationwithrespectto
sX=1,2(cid:16)h 1 2 i h 1 2 i (cid:17)(cid:27) (aνn,duζ),(sη)2(wanit−dhζs12=. Fu1r,the2r)maorreec,osminpcleexη(rsa)n(dwoimthvsar=ia±bl1es)
whereweusedthefactthat ζ(s)ζ(−s) = ζ(s)ζ(s) . After with mean 0 and v±aria±nce 1/3 and 1/5 respectively and
some algebra, we find h 1 2 i h 1 2 i since νζ(0) = νη(0) = 0 and η(0)ζ(0) = 0 at the
h i h i h i
same spatial location, the quantities 3η2(x) and 5ζ2(x)
ζ2ζ2 =1+ 2 ξ(2) 2 are independent χ2-distributed variables with 3 and 5
h 1 2i 5σ4 0 degrees of freedom, respectively (similar conclusions can
2
h(cid:0) +(cid:1)10 ξ(2) 2+ 18 ξ(2) 2 , (32) bdeefodrrmawatniofnortetnhseord,isstereib[u1t9i,o2n2o])f.tThehecroemfopreo,ntehnets1-opfotihnet
7 2 7 4
probability density can also be written as
(cid:0) (cid:1) (cid:0) (cid:1) i
and
P (y)d10y= (ν,u)dνdu
hζ12η22i=1+ 5σ212σ22 3 ξ3(3/2) 2+2 ξ1(3/2) 2 (33) 1 N×χ23(3η2)d(3η2)χ25(5ζ2)d(5ζ2), (37)
2 h (cid:0) (cid:1) (cid:0) (cid:1) i
ζ2ν2 =1+ ξ(1)ξ(1) . (34) where (ν,u) is the bivariate normal
h 1 2i σ2σ2 2 2 N
0 2
1 ν2+u2 2γ νu
1
Thecross-correlationsofη12 andζ12 withu2 inplaceofν2 N(ν,u)= 2π 1 γ2 exp − 2(1 −γ2) , (38)
areidenticalexceptforthe superscript(n), whichshould − 1 (cid:20) − 1 (cid:21)
be replaced by (n+1). Again, all these terms can be and p
foundamongthesecond-ordercontributionsintheright-
1
hand side of Eq. (6). χ2(x)= xk/2−1e−x/2 (39)
Therefore, the actual dependence of ξ (r) on the in- k 2k/2Γ(k/2)
pk
variants η2(x) and ζ2(x), whose covariances involve the
correlation functions ξ(n) with ℓ = 0, is the fundamen- isaχ2-distributionwithk degreesoffreedom. Notethat
ℓ 6 the distribution of ζ2 is coupled with the last rotational
talreasonforC(r)beingdifferentfromtheangleaverage
invariant detζ [23]. However, it can be easily checked
C(r). Thosecorrelations,whichariseuponexpandingthe
that, upon integrating overthe (uniform) distribution of
joint probability density at second order, eventually all detζ, we obtain the χ2-distribution χ2(5ζ2).
nbicely combine together(and with terms proportionalto Ref. [4] discussed how the peak b5ias factors b can
(ξ0(1))2 and (ξ0(2))2) to yield the second-order correlators be derived from a peak-background split. They airjgued
hη12ν22i, hζ12η22i etc. that, while the k-dependent piece bn0 is related to the
The question then arises of the calculation of the co- nth order derivative of the differential number density
efficients of these quadratic terms in the peak 2-point n¯ , derivatives cannot produce the bias factors b , b
pk 01 11
correlation function. We already know that the coeffi- etc. multiplying the k-dependent terms. Forthis reason,
cients multiplying products of the form ξ(n)ξ(n′) are the they considered a second implementation of the peak-
0 0
quadraticpeak-backgroundsplit biases associatedto the backgroundsplit [24, 25] in which the dependence of the
scalars ν and u. Does this hold also for the coefficients mass function on the overdensity of the background is
multiplying η2η2 , ζ2ν2 etc. ? derivedexplicitly. However,it is possible to write all the
h 1 2i h 1 2i
peakbiasfactorsb asderivativesof (ν,u)ratherthan
ij
N
n¯ . More precisely, the b are the bivariate Hermite
pk ij
C. Generalizing the peak background-split polynomials
i j
The probability distribution P1(y) that is needed to H (ν,u)= (ν,u)−1 ∂ ∂ (ν,u) (40)
ij
computen¯ isamultivariateGaussianofcovariancema- N −∂ν −∂u N
pk (cid:18) (cid:19) (cid:18) (cid:19)
trix M,
relative to the weight (ν,u), further averagedover the
N
1 peak curvature u. Therefore, they are peak-background
P (y)d10y= e−Q1(y)d10y. (35)
1 (2π)5 detM1/2 split biases in the sense that they can be derived from
| |
7
the transformation ν ν +ǫ and u u+ǫ , where Taking into account the peak constraint, the first-order
1 2
→ →
ǫ and ǫ are long-wavelength background perturbations bias factors thus is
i 2
uncorrelated with the (small-scale) fields ν(x) and u(x).
1 3 3
Ref. [4] did not express the peak-background split this χ = d10yn (y) η2 P (y) (45)
10 σ2n¯ pk 2 − 2 1
way because they considered the effect of a background 1 pk Z (cid:18) (cid:19)
perturbation after the integration over the peak curva- 3
= , (46)
ture. In terms of the rotational invariants introduced −2σ2
1
above, we can write the b as
ij
which is precisely what we were expecting. Similarly, we
1 define the biasparameterχ associatedtothe invariant
σiσjb = d10yn (y)H (ν,u)P (y), (41) 0k
0 2 ij n¯ pk ij 1 ζ2(x) as the ensemble average of the Laguerre polyno-
pk Z mialL(3/2)(5ζ2/2)orthogonalwith respectto the weight
k
whereitisunderstoodthatP1(y)takestheformEq.(37) χ2(5ζ2). Namely,
and d10y = dνdud(3η2)d(5ζ2). Factors of 1/σ and 5
0
1/σ are introduced because bias factors are ordinarily ( 1)k 5ζ2
2 σ2kχ − d10yn (y)L(3/2) P (y) (47)
dtievfienserdatrheelartitvheantotthheenpohrymsiaclaizlefidelvdaδrisa(bxl)esa.ndInitpsrdaecrtiivcae-, 2 0k ≡ n¯pk Z pk k (cid:18) 2 (cid:19) 1
the integral is most easily performed upon transforming Note that, although we use a single symbol χij to des-
the 5 degrees of freedom attached to 5ζ2 to the shape ignatethe bias factorsderivedfromthe χ2-distributions,
parametersv andw andthe 3Euleranglesthatdescribe thevariablesη2 andζ2,unlikeν andu,areuncorrelated.
theorientationofthe principalaxisframe(seeAppendix The first-order bias factor thus is
B for details).
1 5 5
In [23], it was noticed that, in the presence of non- χ = d10yn (y) ζ2 P (y). (48)
01 σ2n¯ pk 2 − 2 1
Gaussianity,the1-pointprobabilitydensityP1(y)canbe 2 pk Z (cid:18) (cid:19)
expandedinthesetoforthogonalpolynomialsassociated
To evaluate the integral, we first express the measure
to the weight provided by P1(y) in the Gaussian limit. d(5ζ2) in terms of the ellipticity v and prolateness w,
The same logicapplies to the peak bias factors. Namely, so that ζ2 can be written as ζ2 = 3v2 + w2 (see Ap-
the bij are drawn from the orthogonal polynomials as- pendix B for details). A multiplicative factor of ζ2 will
sociated to (ν,u), i.e. bivariate Hermite polynomials. arise upon, e.g., taking the derivative of exp( 5αζ2/2)
N
Therefore, we expect that η2(x) and ζ2(x) also gener- −
with respect to α. In the notation of [4], our fac-
ate bias parameters, and that these are drawn from the tor of ζ2 precisely corresponds to their derivative term
orthogonalpolynomialsassociatedwithχ2-distributions, (2/5)∂ lnG(α)(γ ,γ ν) evaluated at α = 1 (see Ap-
i.e. generalizedLaguerrepolynomials. These are defined − α 0 1 1
pendixB).TakingintoaccountthefactorofG(1)(γ ,γ ν)
as 0 1 1
in the denominator, χ can eventually be written
01
x−αex dn
L(nα)(x)= n! dxn e−xxn+α (42) χ = 5 1+ 2∂ lnG(α)(γ ,γ ν) . (49)
and are orthogonal over [0, [ w(cid:0)ith respec(cid:1)t to the χ2- 01 −2σ22 (cid:18) 5 α 0 1 1 (cid:12)α=1(cid:19)
∞ (cid:12)
distribution with k = 2(α+1) degrees of freedom. The The physical interpretation of this result(cid:12) is straightfor-
orthogonality relation can be expressed as ward: ζ2(x) is a scalar that describes the asymmetry of
the peak density profile. In the high peak limit, χ 0
01
∞dxxαe−xL(α)(x)L(α)(x)= Γ(n+α+1)δ . reflecting the fact that the most prominent peaks→are
n m n! mn nearly spherical (see Fig.9 of [4]).
Z0
(43) Thephysicaloriginfortheappearanceoftheseorthog-
ThefirstgeneralizedLaguerrepolynomialsareL(α)(x)= onal polynomials can be found in the peak-background
0
1 and L(α)(x)= x+α+1. split. Long-wavelengthbackgroundperturbationslocally
1 − modulate the mean of the distributions (ν,u), χ2(3η2)
Given the correlator Eq. (29), the term proportional andχ2(5ζ2). The resulting non-centraldNistribution3s can
to 1/σ4 in the right-hand side of Eq.(6) indicates that 5
1 then be expanded in the appropriate set of orthogonal
thefirst-orderbiasparameterχ associatedwiththein-
10
polynomials. In practice, it is convenient to introduce a
variant σ2η2(x) = ( δ )2(x) is χ = 3/(2σ2). The
1 ∇ s 10 − 1 shift or translation operator Tˆ to describe the action of
aforementioned considerations suggest that we define ǫ
a background perturbation on the distribution of rota-
the kth-order bias factor as the Laguerre polynomial
tional invariants. For the scalars ν and u, we define the
( 1)kL(1/2)(3η2/2) averaged over all the possible peak
− k shift operator as
configurations, i.e.
Tˆ exp( ǫ ∂ ǫ ∂ ) , (50)
( 1)k 3η2 ǫ ≡ − 1 ν − 2 u
σ2kχ − d10yn (y)L(1/2) P (y).
1 k0 ≡ n¯ pk k 2 1 where ǫ and ǫ are small perturbations to the peak sig-
pk Z (cid:18) (cid:19) 1 2
(44) nificance and the peak curvature, i.e. ν ν +ǫ and
1
→
8
u u+ǫ . The action of Tˆ on the probability density degrees of freedom is
2 ǫ
→
aNn(dν,uǫ) ,isretospsehctifitvetlhye(t(hzeerore)amsoenanfoorfthνeamndinuusbsyign−ǫi1s Tˆǫχ2k(x) (56)
2
that−Hermite polynomials include a factor of ( 1)i+j). 1 ∞ ǫ j x α
A straightforwardcalculation gives − = 2 : j!Γ−1k2 +j (∂xx)j: 2 e−x/2
Xj=0 (cid:0) 2 (cid:1) (cid:16) (cid:17)
(ν ǫ ,u ǫ ) 1 ∞ ǫ(cid:0) j (cid:1) x α+j
N(ν,u)−1TˆǫN(ν,u)= N −(ν1,u)− 2 (51) = 2 j!Γ−1k2 +j ∂jx2 2 e−x/2
N Xj=0 (cid:0) 2 (cid:1) (cid:20)(cid:16) (cid:17) (cid:21)
=f(ǫ1,ǫ2)eǫ1σ0bν+ǫ2σ2bu , e−x/2 x(cid:0)α ∞ (cid:1) ǫ j x
= −2 L(α)
2 2 Γ 1k+j j 2
whereb andb aregiveninEqs.(8)and(9),andf(ǫ ,ǫ ) (cid:16) (cid:17) Xj=0 (cid:0)2 (cid:1) (cid:16) (cid:17)
iνstheǫexνapnodneuntuialǫfa.ctTohreinlaNst(νex,up)rewssitiohnthisearegpelnaceeram1teinn2gt =χ2k(x) ∞ ΓΓ1k21+k j(cid:0) −2ǫ(cid:1)jL(jα) x2 .
fun→ctio1nofbiva→riat2eHermitepolynomials. Onexpanding Xj=0 2(cid:0) (cid:1) (cid:16) (cid:17) (cid:16) (cid:17)
it in the small parameters ǫ and ǫ , This is precisely the (cid:0)Laguerr(cid:1)e series expansion of a non-
1 2
central χ2-variate derived in [26] (see Appendix C).
Therefore,
f(ǫ1,ǫ2)eǫ1σ0bν+ǫ2σ2bu pk χ2(x) −1Tˆ χ2(x)= χ′k2(x;ǫ) ,
D = (cid:12)(cid:12)(cid:12)∞ Eσ0iσ2jbij ǫii1! ǫjj2! , (52) where χ′2(x;(cid:2)λ)kis a(cid:3) nonǫ-cekntral χ2χ-d2ki(sxt)ribution with k
i,j=0 (cid:18) (cid:19)(cid:18) (cid:19) k
X degrees of freedom and non-centrality parameter λ ǫ.
The latter is defined as the sum of squaresλ= k ≡µ2,
i=1 i
we recover the bias factors b once the results are aver- where µ are the means of the random variables. We
ij i
P
aged over all locations that satisfy the peak constraint. can now read off the bias factors from the expansion of
Note that the bias parameter b∇2δ defined in [18] bears [χ2k]−1Tˆǫχ2k in generalized Laguerre polynomials. For in-
the same physical meaning as our b : both represent stance,
01
the leading-order response of the tracer abundance to a χ′2(5ζ2;ǫ ) ∞ Γ 5 ǫ j
uniform shift in the curvature of the density field. 5 5 pk = 2 5 σ2jχ . (57)
χ2(5ζ2) Γ 5 +j 2 2 0j
Forthequadraticvariablesη2 andζ2,Eq.(42)suggests (cid:28) k (cid:12)(cid:12) (cid:29) Xj=0 2(cid:0) (cid:1) (cid:16) (cid:17)
that we express the shift operator in terms of both x Note that the m(cid:12)(cid:12) ore common(cid:0)gener(cid:1)ating function
and∂ . Thedefinitionissomewhatcumbersomebecause
x ∞
we must take into account not only the ordering of x (1 ǫ)−α−1exp xǫ = ǫnL(α)(x) (58)
and ∂x, but also the factor of Γ(n + α + 1)/n! in the − "2 1−ǫ # n=0 n
orthogonality relation Eq.(43). A sensible definition of X
appears to bear little (cid:0)conne(cid:1)ction with the non-central
Tˆ for the variable x=3η2 and 5ζ2 is
ǫ χ2-distribution.
To better understand the reason why the peak-
∞ Γ 1k ǫ j backgroundsplitgeneratesanon-centralχ2-distribution,
Tˆ : 2 ∂ x : (53) we note that, owing to the relation H (x) L (x2) be-
ǫ ≡ j!Γ 1k+j −2 x 2k ∼ k
Xj=0 (cid:0)2 (cid:1) (cid:16) (cid:17) tween Hermite and Laguerre polynomials, we could also
=Γ 1k :(cid:0)Iα −(cid:1)2ǫ∂xx : , (54) whahveeredeyfinisednχow10tahnedvχe0c1toars(sνe,cuon,ηd,dζ˜eri)vaotfiviensdeopfePn1d(eyn)t,
(cid:0)2 (cid:1) −(cid:0)p2ǫ∂xx α/2(cid:1) normal random variables such thait ηA2 = 3i=1ηi3 and
(cid:0) (cid:1) ζ2 5 ζ˜2. A little algebra shows that
≡ A=1 A P
whereI (x)isamodifiedBesselfunctionofthefirstkind
α P 1 3 1 ∂ 2
and the symbol :: of normal ordering is borrowed from χ d10yn (y) P (y), (59)
10 pk 1
quantum field theory. In the presentdiscussion,the nor- ≡ n¯pk Z j=1(cid:18)σ1∂ηj(cid:19)
X
mal ordering is defined as
Another way of writing this formula would be to absorb
a factor of 1/√2 in the definition of η(±1) (and thus ex-
:(∂ x)n: f(x) ∂n xnf(x) , (55) plicitly deal with complex normal distributions). Analo-
x ≡ x gously, we have
(cid:0) (cid:1)
5 2
where f(x) is some test function. With this definition, χ 1 d10yn (y) 1 ∂ P (y). (60)
the action of Tˆǫ on a χ2-distribution with k = 2(α+1) 01 ≡ n¯pk Z pk A=1(cid:18)σ2∂ζ˜A(cid:19) 1
X
9
This suggests that the effect of a background pertur- itly that each of the combinations (ν,u), η2 and ζ2 of
bation on η2 and ζ2 can also be thought of as shift- rotational invariants generates a set of orthogonal poly-
ing the components of the first derivatives according to nomials which, upon taking the ensemble average over
η η +ǫ ,andthoseofζ˜accordingtoζ˜ ζ˜ +ǫ . all the possible peak configurations, yields a set of bias
i i 3i A A 5A
→ →
The small perturbations ǫ and ǫ need not be the factors. Furthermore,weshowedthatthesebiasparame-
3i 5A
same for distinct i and/or A. However, owing to in- terscanbeconstructedfromasuitableapplicationofthe
variance under rotations, only the length of the vector peak-background split to the probability densities char-
3 (ǫ )2 ǫ and 5 (ǫ )2 ǫ matter. This is acterizing the invariants. We will now demonstrate that
i=1 3i ≡ 3 A=1 5A ≡ 5
the reason why the background perturbation effectively we can interpret the peak 2-point correlation as arising
sPhifts the respective χP2-distributions to non-central χ2- from a functional relation of the form δ = (δ ,...).
pk s
F
distributions, with non-centrality parameter λ = ǫ and
3
λ=ǫ . Note that it should be possible to formulate this
5
peak-backgroundsplitwiththeconditionalpeaknumber
density n¯pk(ν,Rs|δl,Rl) in a large-scale region of over- D. A local bias approach to ξpk(r)
density δ , like in [4]. However,one should then consider
l
two long-wavelength perturbations η2 and ζ2 in order
l l First, let us make sure that [4] obtained the correct
to describe the effect of the backgroundperturbation on
η2(x) and ζ2(x), in addition to δ (which suffices to de- expression for ξpk(r). Adding all the second-order con-
l tributions induced by η2, ζ2 and their cross-correlations
scribe the effect of the background wave for both ν and
with ν and u, our result differs from theirs in that the
u since these variables are correlated).
lasttermintheright-handsideofEq.(6)appearstomiss
To conclude, [27] also pointed out that, even though
there is no functional relation npk = F(δ) for dis- a multiplicative factor of 1+(2/5)∂αlnG(0α)(γ1,γ1ν)|α=1.
crete density peaks, it is nevertheless possible to define Checkingthecalculationof[4],wefoundthemissingmul-
renormalized bias parameters as the expectation values tiplicative factor in their Eqs. (A50) and (A51), in the
c . However, he did not compute them explic- form of q(r)tr(ζ˜2). This term was fortuitously omitted
n ∼ hFi i
itly, nor specified what is (though it is pretty clear intheirfinalexpressionofξ (r). Consequently,the cor-
pk
F
that it is related to n ). Here, we demonstrated explic- rect answer is
pk
1 3 5 2
ξ (ν,R ,r)= b˜2ξ(0) + ξ(0)˜b2ξ(0) ξ(1/2)˜b ξ(1/2) ξ(1)b˜ ξ(1) 1+ ∂ lnG(α)(γ ,γ ν)
pk s I 0 2 0 II 0 − σ12 1 II 1 − σ22 2 II 2 (cid:18) 5 α 0 1 1 (cid:12)α=1(cid:19)
+(cid:0) 5 (cid:1)ξ(2)(cid:0)2+ 10 ξ(2(cid:1)) 2+ 1(cid:0)8 ξ(2) 2 1+(cid:1)2∂ ln(cid:0)G(α)(γ ,γ ν(cid:1)) 2 (cid:12)(cid:12) (61)
+ 2σ324h(cid:0)ξ0(1)(cid:1)2+27ξ(cid:0)(12) 2(cid:1) + 73(cid:0) 43(cid:1)ξ(i3(cid:18)/2) 2+52αξ(3/02) 21 11+ (cid:12)(cid:12)(cid:12)2α∂=1l(cid:19)nG(α)(γ ,γ ν) .
2σ14h(cid:0) 0 (cid:1) (cid:0) 2 (cid:1) i σ12σ22h (cid:0) 3 (cid:1) (cid:0) 1 (cid:1) i(cid:18) 5 α 0 1 1 (cid:12)(cid:12)α=1(cid:19)
(cid:12)
We note that this omission has an impact only on the provided that we ignore all the contributions involving
small-scale (r . 20 h−1Mpc) peak correlation displayed moments at zero lag. Nonlocality enters through the fil-
inFig.2oftheirpaper. Theirresultsconcerningthepeak- tering solely (which is the reason why we still call it a
backgroundsplit, the gravitationalevolutionofξ (r) or localexpansion). Asemphasizedin[4],itisimportantto
pk
thescale-dependenceofbiasaroundtheBaryonAcoustic realize that δ is not a count-in-cell quantity. Counts-
pk
Oscillation are unaffected. in-cells can generally be constructed using the void gen-
Even though we cannot write down a relation of the erating function, see e.g. [28, 29], but it is beyond the
form n = (δ), the peak correlation function up to scope ofthis paper to compute moments of the peak fre-
pk X
second-orderFcan nonetheless be thought of as arising quency distribution function. Since all the bias factors
from a local bias expansion δ = (δ ,...), i.e. are peak-background split biases obtained from a suit-
pk X s
F able averageof orthogonalpolynomials, one could try to
1 write down an expansion in terms of orthogonal poly-
δpk(x)=b10δs(x)−b01∇2δs(x)+ 2b20δs2(x) (62) nomials in the variables δs, ∇2δs etc. such that all the
contributions involving moments at zero lag cancel out.
b11δs(x) 2δs(x)+ 1b02 2δs(x) 2 Wewillexplorethispossibilityinfuturework. Notethat
− ∇ 2 ∇
this idea was put forward for the first time by [30], who
+χ ( δ )2(x)+ 1χ 3∂(cid:2) ∂ δ δ(cid:3) 2δ (2x), considered correlations of regions above threshold as a
10 s 01 i j s ij s
∇ 2 − ∇
(cid:2) (cid:3)
10
proxy for luminous tracers. More recently, [31] proposed nential factor, we find
an algorithm based on Hermite polynomials to extract
σ 2
k-dependentbiasfactorsfromcross-correlationsbetween δ b = 0 ν2 δ b (67)
c 01 c 10
the halo and Hermite-transformed mass density field. (cid:18)σ1(cid:19) −
Tomakeconnectionwiththeformalismof[27](seealso (cid:0) (cid:1)
atthefirstorderwhereas,atthe secondorder,weobtain
[32]), we define the Fourier space peak bias parameters
cn(k1,...,kn) as the sum over all the contributions to δ2b =δ2b b + δc2γ12 (68)
δpk(x) from a given order. We thus have c 11 c ν u (1 γ2)
− 1
and c1(k)≡(cid:0)b10+b01k2(cid:1)W(kRs) (63) =(cid:18)σσ10(cid:19)2"δc2(cid:18)σν0(cid:19)bν −δc2b2ν + (1γ−12νγ212)#
2
σ
= 0 ν2+ν2δ b δ2b ,
c2(k1,k2)≡ b20+b11 k12+k22 +b02k12k22 (64) (cid:18)σ1(cid:19) (cid:16)− c 10− c 20(cid:17)
(cid:26)
(cid:0) (cid:1) and
2χ (k k )+χ 3(k k )2
− 10 1· 2 01(cid:20) 1· 2 δc2b02 =δc2b2u− σ2(1δc2 γ2) (69)
k2k2 W(k R )W(k R ), 2 − 1
− 1 2 1 s 2 s σ 4 ν γ2ν2
(cid:21)(cid:27) = 0 ν4 2δ2 b +b2 1
whereW(kRs)isthesmoothingkernel. Thesedefinitions (cid:18)σ1(cid:19) " − c(cid:18)σ0(cid:19) ν ν − (1−γ12)#
are consistent with those of the “renormalized” bias pa- σ 4
rameters introduced by [27] who argued that, owing to = σ0 ν2 ν2+1 −2ν2δcb10+δc2b20 .
rotational symmetry, the peak bias parameters should (cid:18) 1(cid:19) h (cid:0) (cid:1) i
take the above functional form. In particular, the cor- At least for n = 1,2, b with 1 l k n can be ex-
kl
≤ ≤ ≤
respondence between the bias factors associated with η2 pressed as a linear combination of bk0, 1 k n, plus a
≤ ≤
and ζ2 and those of [27] is E = 3χ , C = 2χ and polynomialinν. Thisagreeswiththefindingsof[31]ob-
2 01 2 10
−
D = b χ . We stress, however, that the peak bias tainedwithin the excursionsetapproach[31] (except for
2 02 01
factors di−scussed in this work have not been obtained the multiplicative factorsof (σ /σ )2l). Similar relations
0 1
by means of a renormalization procedure. We speculate should hold at any order in the peak bias parameters
that the local bias expansion Eq.(62) can be extended b . This structure arises from the fact that the peak-
ij
to all orders to match the exact peak 2-pointcorrelation backgroundsplit acts on probability densities, which are
function at all separations. Namely, continuous functions of space. For discrete tracers such
as density maxima, the background perturbation affects
∞ 1 d3k d3k the fields appearing in the Gaussian multivariate P (y),
δ (x)= 1 ... n c (k ,...,k ) (65) 1
pk n! (2π)3 (2π)n n 1 n butnotthoseenteringtheexpressionofthepeaknumber
nX=1 Z density npk(y). The peak constraint weights the peak-
δ(k1)...δ(kn)ei(k1+···+kn)·x , backgroundsplitseriesexpansionsuchthatthepeakbias
×
factors are recovered.
wherec (k ,...,k )isasumoverallthepossiblecombi-
n 1 n
nations of rotational invariants involving exactly n pow-
ers of the linear density field δs and/or its derivatives. E. Correlation functions for excursion set peaks
Our peak-background split approach provides a sim-
plewayofpredictingthebiascoefficientsassociatedwith
To make connection with the clustering of dark mat-
any rotational invariant quantity. The Hermite weight-
ter halos, we must ensure that the density in a tophat
ing scheme introduced by [31] furnishes a practical way
region centered at the peak location never reaches the
of measuring the biases b from simulations. Clearly,
ij collapse threshold (δ ) on any smoothing scale R > R .
c s
theirschemecouldbeextendedtoalsomeasurethebiases
Ref.[33]showedthatenforcingtheconditionsδ(R )>δ
s c
χ . However, because discrete density maxima require
ij andδ(R +∆R )<δ asin[34]providesaverygoodap-
s s c
a somewhat more sophisticated treatment of counts-in-
proximation to the first-crossing distribution when the
cells, we leave this for future work. Here, we merely
stochastic walks generated from the variation of R are
s
establish a recursion relation between the b by consid-
ij strongly correlated. An important consequence of this
ering either the property
resultis the possibility of restricting the excursionsetto
those locations that meet the peak constraint [12]. The
∂ ∂
+γ1 e−Q1(y) = νe−Q1(y) , (66) number density of dark matter halos per unit mass and
∂ν ∂u −
(cid:18) (cid:19) volume is usually written
or the generating function in Eq.(51). In the latter case, ρ¯ dν
upon substituting b =(σ /σ )2(ν/σ b ) in the expo- n¯(M)= f(ν) , (70)
u 0 1 0 ν M dM
−