Statistics of maxima of the linear Gaussian density field

Transcription

1 Chapter 4 Statistics of maxima of the linear Gaussian density field Density maxima define a point process in which the fundamental quantity is the set of positions which are local maxima of the density field. Since the evolved density field is highly nonlinear, the peak constraint is generally applied to the initial Lagrangian Gaussian density field, with the assumption that the most prominent peaks should be in one-to-one correspondence with luminous galaxies or massive halos in the low redshift Universe. 4. Spectral moments and characteristic scales Owing to the peak constraint, the clustering properties of density peaks depend on statistics of the density field δ s x smoothed on scale R s and its first and second derivatives i δ s x and i j δ s x. In what follows, we consider δ s to be an homogeneous Gaussian random field of zero mean. It is convenient to introduce the normalized variables νx =δ s x/σ, η i i δ s x/σ and ζ ij x i j δ s x/σ, where σ n are the spectral moments σ s π dk k n+ P δ kw kr s. 4. Here and henceforth, we will denote the power spectrum of the smoothed, linear density field by P s k =P δ kw kr s. W kr s is a spherically symmetric smoothing kernel. A Gaussian filter is adopted throughout this Section to ensure convergence of all the spectral moments. We can define characteristic scales by taking ratios of spectral moments. For instance, σ n R n 3 4. σ n+ defines an ordered sequence of characteristic lengths R R R The first two scales are the typical separation R between zero-crossings of the density field and the mean distance R between extrema. These are the only scales involved in the calculation of the peak correlation functions. For subsequent use, we also introduce the dimensionless parameters γ n = σ n σ n σ n

2 34CHAPTER 4. STATISTICS OF MAXIMA OF THE LINEAR GAUSSIAN DENSITY FIELD which reflect the range over which k n P s k is large. Note that γ n takes values between zero and unity solely. The analogous quantities to σn at non-zero separation are defined as follows: ξ n l r = π dk k n+ P s k j l kr, 4.4 where j l x are spherical Bessel functions. As l gets larger, these harmonic transforms become increasingly sensitive to the small scale power. The auto- and cross-correlations of the fields νx, η i x, and ζ ij x at different spatial locations can generally be decomposed into components with definite transformation properties under rotations. At a single position x, these simplify to νζ ij = 3 γ δ ij, η i η j = 3 δ ij, ζ ij ζ lm = 5 δ ijδ lm + δ il δ jm + δ im δ jl 4.5 and ν =. All the other correlators vanish. 4.. The Kac-Rice formula Let {x, x,, x p, }be the positions of the extrema maxima, minima and saddle points in a realization of the smoothed density field. The density n ext x of this point process formally reads n ext x = p δ D x x p. 4.6 As shown in Kac 943 and Rice 954, the density n ext x can be entirely expressed in terms of the peak signifiance ν and its derivatives. To see why this is the case, note that, in the neighbourhood of an extremum located at x p, the first derivative η i is approximately η i x 3 R ζ ij x p x x p j. 4.7 j Using the properties of the Dirac delta, the number density of extrema can thus be written as n ext x = p δ D x x p = 33/ R 3 det ζ ij x δ D [η k x], 4.8 provided that the Hessian ζ ij is invertible. The delta function δ D [η k x] ensures that all the extrema are included. This expression, known as the Kac-Rice formula, holds for arbitrary smooth random fields. Since we are interested in counting the maxima solely, we further require that ζ ij x p be negative definite at the extremum position x p. We will also restrict the set to those maxima with a certain threshold height ν. The number density of peaks of height ν thus is n pk x = 33/ R 3 det ζ ij x δ D [η k x] θ H λ 3 δ D [ νx ν ], 4.9 where θ H is the Heaviside step function and λ 3 is the smallest eigenvalue of the matrix ζ ζ ij.

3 4.. THE AVERAGE PEAK NUMBER DENSITY The average peak number density Owing to the constraints on the derivatives of the density field, the calculation the N-point correlation function of peaks of a 3-dimensional random field requires the evaluation of high-dimensional integrals over a joint probability distribution in N variables. Therefore, even the evaluation of the -point correlation or average density of density maxima n pk proves difficult. The average number density of peaks of a 3-dimensional Gaussian random field was first calculated in Bardeen et al Substituting the Kac-Rice relation, the average number density of peaks of height ν reads n pk ν,r s 33/ R 3 = 33/ R 3 det ζ δ 3 [η] θλ 3 θν ν d 6 ζ det ζ θλ 3 P η k =,ν,ζ A, 4. where P y is the -point probability distribution for the -dimensional vector y η k x,νx,ζ A x, andζ A are the six independent components of the matrix ζ = ζ ij. We follow the notation of Bardeen et al. 986 and associate the components ζ A, A =,...,6 to the entries ij =,, 33,, 3, 3 of ζ. λ 3 is the smallest eigenvalue of the matrix ζ. Finally, d 6 ζ = i j dζ ij is the usual Lebesgue measure on the six-dimensional space of symmetric matrices. P y is a multivariate Gaussian whose covariance matrix M has dimensions. This covariance matrix can be decomposed into four block matrices of size 4 and 6, where M = 3 I 3 3 M= M M 3 M 3 M, M = 5 A I, γ, M 3 = Here, I is the 3 3 identity matrix, z n m is a n m matrix with all entries equal to z, and A encodes the covariances between the components ζ A with A =,, 3, A= It is pretty clear that P y should preserve its functional form under the action of the rotation group, so it must involves rotational invariants such as scalar products of vectors or matrix traces. In fact, because the representations of SO3 transform independently under rotations, P y factorizes into the product P y =P ν, up η k P ζ A, 4.4 where ν and u = trζ are scalars l =, η k transform as the components of a vector η l =, and ζ A are the independent entries of the traceless symmetric matrix ζ = ζ +/3u l =.

4 36CHAPTER 4. STATISTICS OF MAXIMA OF THE LINEAR GAUSSIAN DENSITY FIELD The variable u = δ s /σ is the curvature of the density field. The probability densities for the l =, and variables are [ P ν, u = π exp ν + u ] γ νu γ γ / P η k = exp 3 π η η [ P ζ A = π 5/ 5 exp 5 ζ ] 4 tr 4.7 With the peak constraint, P η k reduces to a trivial multiplicative factor of 3/π 3/. Taking into account the fact that ζ A are the components of a 3 3 symmetric matrix, the latter can be diagonalized and d 6 ζ A can be recast into the product of two Haar measures one for diagonal matrices and the second for orthogonal matrices d 6 ζ =8π λ d 3 λdr. 4.8 where d 3 λ = dλ dλ dλ 3, λ = i<j λ i λ j is the Vandermonde determinant and λ λ λ 3 are the three ordered eigenvalues of the matrix ζ. dr is the Haar measure for the Euler angles for example on the group SO3 normalised to dr=. To integrate over the eigenvalues of ζ, we transform to the new set of variables {u, v, w}, where u = λ + λ + λ 3, v = λ λ 3, w = λ λ + λ The variables v and w are shape parameters that characterize the asymmetry of the density profile in the neighborhood of density maxima. Note that our choice of ordering impose the constraints v and v w v. With these new coordinates, the volume element d 3 λ becomes d 3 λ =/3dudvdw and the Vandermonde determinant simplifies to λ =vv w, while tr ζ =/33v + w. Following Bardeen et al. 986, we introduce the auxiliary function F u, v, w 33 det ζ λ =u w [ u + w 9v ] v v w. 4. F u,v,w measures the degree of asphericity expected for a peak and, therefore, can be used to determine the probability distribution of ellipticity v/u and prolateness w/u. It scales as F u 3 in the limit u i.e., it is proportional to the three-dimensional extent of a peak. In the variables Eq.4.9, the peak number density becomes where 3 5 5/ n pk ν, R s = π 5 R 3 γ dudvdw F u, v, wθλ 3 e Q 8π dr, 4. Q = ν + u γ ν γ +5 3v + w. 4. The integral over the Haar measure dr of the orthogonal group runs over all the possible ortientation of the orthonormal triad. Clearly, this is some fraction of the volume 8π of the SO3 manifold. The actual result is π as we don t care whether the axes are directed towards the positive

5 4.. THE AVERAGE PEAK NUMBER DENSITY 37 Figure 4.: Differential number density of peaks n pk ν, R s for several power spectrum shapes parametrized by γ =.,.3,...,.9 curves from left to right. The filter is a Gaussian window of characteristic radius R s =4h Mpc adapted from Peacock & Heavens 985 and Bardeen et al 986.

6 38CHAPTER 4. STATISTICS OF MAXIMA OF THE LINEAR GAUSSIAN DENSITY FIELD or negative direction. Furthermore, the condition that the density extrema be maxima θ H λ 3, i.e. all three eigenvalues of the Hessian are negative, enforces the constraint u + w 3v. Another condition, u, should also be applied if one is interested in selecting maxima with positive threshold height. Therefore, the integration domain is the interior of the triangle bounded by the points,, u/4, u/4 and u/,u/. On defining the integral over the asymmetry parameters as fu 3 5 5/ { u/4 +v u/ v } dv dw + dv dw F u, v, we 3v 5 +w 4.3 π v u/4 3v u = u 3 3u { [ ] [ ]} 5 5 Erf u u +Erf [ 3u u e 5u /8 + 5π ] e 5u / 5 the peak number density eventually reads where n pk ν, R s = π R 3 G γ,γ ν e ν / G n γ,ω=, 4.4 du u n fu e u ω / γ π. 4.5 γ The integration over u must generally be done numerically. It is worthwhile noticing that, while the exponential exp[ u ω / γ ] decays rapidly to zero, un fu are monotonically and rapidly rising. As a result, the functions G n γ,w are sharply peaked around their maximum. For large values of ω, we find that G and G asymptote to G γ,ω ω 3 3γ ω + B γ ω e Aγ ω 4.6 G γ,ω ω 4 +3ω γ + B γ ω 3 e Aγ ω. 4.7 The coefficients Aγ, B γ and B γ are obtained from the asymptotic expansion of the Error function that appears in eq We have explicitly A = 5/ 9 5γ, B = 43 π 9 5γ 5/, B = 4B 9 5γ. 4.8 Fig.4. shows the differential peak number density n pk ν, R s for various <γ <. The value of ν at which n pk ν, R s peaks strongly depends on the value of γ. For small γ.3, the peak is at ν, reflecting the fact that there is significant power on all scales. For large γ.7, most of the power is in a narrow range of wavenumber, so that density maxima can reach heights well above σ fluctuations ν =. 4.3 The peak -point correlations For a point process such as density maxima, the expectation value of the product n pk x n pk x is see Eq..5 n pk x n pk x = n pk δ D x x + n pk[ +ξpk r ]. 4.9

7 4.3. THE PEAK -POINT CORRELATIONS 39 Ignoring self-pairs, the connected -point correlation function of density peak ξ pk r is defined as +ξ pk r = n pk x n pk x / n pk 4.3 for r = x x >. Let P be the joint probability for the density and its first and second derivatives at positions x and x. Furthermore, let x 3 and y 3 be the smallest eigenvalues of the 3 3 symmetric tensors ζ and ζ, where ζ i = ζx i. The -point correlation between peaks of height ν at x and ν and x can now be written as +ξ pk r = 3 3 n d 6 ζ d 6 ζ det ζ det ζ θ H x 3 θ H y pk νr6 P η =,ν,ζ, η =,ν,ζ ; r, where, for shorthand convenience, subscripts attached to ν and η also designate quantities evaluated at different Lagrangian positions. The joint probability distribution of the density fields together with its first and second derivatives, P η,ν,ζ, η,ν,ζ ; r, is given by a multivariate Gaussian whose covariance matrix C has dimensions. This matrix may be partitioned into four block matrices, M= y y = y y in the top left corner and bottom right corners, B= y y and its transpose in the bottom left and top right corners, respectively. Here, y i yx i is the previously defined -dimensional vector y at position x i. Unlike M=M, M 3 ;M 3, M, which describes the covariances at a single position, the cross-covariance matrix B generally is a function of the separation vector r. As shown in Desjacques et al., B can be constructed upon transforming η i, ν and ζ A to an helicity basis, in which it becomes a function of r only. We will not give the explicit expression of B because we will only work out in details the leading order contribution to the peak correlation which involves a reduced cross-covariance matrix Desjacques 8. As a result, on expanding the joint density P y, y ; r in the small perturbation B, one obtains Desjacques 8 ] P y, y ; r =P y P y [+y M BM y + OB. 4.3 Setting u i = trζ i and ζ i = ζ i +/3u i,wearriveat { y M BM ν γ u ν γ u y = σ γ γ ξ r+ u γ ν σ γ u γ ν γ ξ r+ γ [ ν γ u u γ ν σ γ γ ] } u γ ν ν γ u + γ γ ξ r + 5 σ tr ζ ζ ξ r + terms linear in ζ A x and ζ A x. At this point, we shall remember that the principal axes of ζ and ζ are not necessarily aligned with those of the coordinate frame. Without loss of generality, we can write ζ = R XR and ζ = R ỸR, where R i are orthogonal matrices and X, Ỹ are the diagonal matrices consisting of the three ordered eigenvalues x i + u /3 and y i + u /3 of ζ and ζ. Expressing the volume

8 4CHAPTER 4. STATISTICS OF MAXIMA OF THE LINEAR GAUSSIAN DENSITY FIELD elements as the product of Haar measures d 6 ζ =8π x d 3 xdr, we can average over distinct orientations of the principal axes to obtain 8π [ 5 ] dr dr σ tr ζ ζ + terms linear in ζ A x and ζ A x =, 4.33 since SO3 drtr XR ỸR = tr XtrỸ= We have used the property that tr ζ ζ =tr XRỸR with R=R R. In Eq.4.3, the term P y P y will eventually contribute a factor of n pk ν. Therefore, at leading order, the peak correlation is proportional to an integral of y M BM y weighted by P y P y. Since we must distinghuish between variables at x and x this is expecially important when performing integrals over the peak curvature, we will first compute the correlation between two populations of density maxima of height ν and ν identified on the same smoothing scale R s. Transforming to the variables {u, v, w, i =, }, the peak -point correlation function reads ξ pk ν,ν,r s,r= π 6 R 6 γ n pk ν,r s n pk ν,r s Φ ν,ν,u,u,r e Q, i=, { } du i dv i dw i F u i,v i,w i where { ν γ u ν γ u Φ ν,u,ν,u ; r = σ γ γ ξ r+ u γ ν σ γ u γ ν γ ξ r+ γ [ ν γ u u γ ν σ γ γ ] } u γ ν ν γ u + γ γ ξ r 4.35 is Eq.?? averaged over the relative orientation of the frames spanned by the eigenvectors of the matrices ζ and ζ. Φ depends on the separation r through the correlation functions ξ r, ξ r and ξ r only. Moreover, the quadratic form Q is Q = ν + u γν γ +5 3v + w in the variables 4.9. The rest of the computation is easily accomplished. Introducing the mean peak curvature ū i γ,γ ν i = G γ,γ ν i G γ,γ ν i, 4.37 the -point cross-correlation between peaks of height ν and ν simplifies to Desjacques 8 ξ pk ν,ν,r s,r=b ν,r s b ν,r s ξ [b r+ ν,r s b ν,r s 4.38 ] + b ν,r s b ν,r s ξ r+b ν,r s b ν,r s ξ r.

9 4.3. THE PEAK -POINT CORRELATIONS 4 Figure 4.: First-order Lagrangian peak bias factors as a function of peak height for a filtering radius R s =.9 h Mpc or, equivalently, a mass scale M =3 3 M /h. At redshift z =.3, this corresponds to σ peaks. The shape parameter is γ.65 adapted from Desjacques 8.

10 4CHAPTER 4. STATISTICS OF MAXIMA OF THE LINEAR GAUSSIAN DENSITY FIELD The factors b and b are the linear bias coefficients of density peaks of significance ν i, b ν i,r s = νi γ ū i σ γ 4.39 b ν i,r s = ūi γ ν i σ γ. 4.4 While b is dimensionless, b has units of length, where the length is the scale over which δ s changes significantly. For peaks of significances ν = ν = ν, the -point correlation is ξ pk ν, R s,r= b I ξ r, where b I is equivalent to the Fourier space multiplication by b I k =b + b k. 4.4 To gain some insight into the behaviour of the peak correlation function ξ pk ν, R s,r, Fig. 4. displays the linear peak bias factors b and b as a function of the peak height. For illustration, the density field is smoothed on scale R f =5h Mpc with a Gaussian filter. Note that, whereas b becomes vanishingly small in the high peak limit ν, it cannot be neglected when the peak height does not exceed 4. Therefore, retaining the correlation ξ r of the mass density field solely is not a reasonable approximation for realistic values of the peak height. Fig. 4.3 illustrates the sharpening of the baryon acoustic oscillation due to the presence of a b k piece at the linear level. The additional contributions b b ξ and b ξ to the linear peak -point correlation restore the signature of the acoustic peak, otherwise smeared out in b ξ by the large filtering. The contrast of the acoustic peak can even be enhanced relative to that of the unsmoothed R f =.h Mpc linear density correlation dotted-dashed line. The computation of the second order contribution to the peak -point correlation, which is proportional to B, is quite involved. The result can be cast into the form ξ pk ν, R s,r= b I ξ where + ξ b IIξ + terms linear in bii and independent of b II, 4.4 b II k,k =b + b k + k + b k k 4.43 is the second-order peak bias in Fourier space, and b, b, b are the quadratic bias coefficients [ ] b ν, R s = ν γ νu + γ u σ γ γ 4.44 b ν, R s = +γ νu γ ν + u σ σ γ + γ γ and b ν, R s = σ [ ] u γ νū + γ ν γ γ Here, u n G n γ,γ ν/g γ,γ ν is the nth-order moment of the peak curvature at a given significance ν. For sake of completeness, ξ b II ξ is defined as the Fourier space multiplication ξ n l b II ξ n l = 4π dk dk k n + k n + b II k,k P k P k j l k rj l k r, 4.47

11 4.3. THE PEAK -POINT CORRELATIONS 43 Figure 4.3: A comparison between the unsmoothed linear mass density correlation ξr black, dotted-dashed and the -point correlation ξ pk ν, R s,r of discrete density peaks red, solid around the baryon acoustic oscillation. To obtain the peak correlation, the linear density field was smoothed with a Gaussian filter on mass scale M = 3 3. The dotted-long dashed, short-dashed and long-dashed curves represent the individual contributions b ξ, b b ξ and b ξ to the leading order peak correlation Eq adapted from Desjacques 8, Desjacques et al..

12 44CHAPTER 4. STATISTICS OF MAXIMA OF THE LINEAR GAUSSIAN DENSITY FIELD When l = l =, the real-space counterpart of b II is readily obtained upon making the replacement k. Peaks are not the only example of biased tracers that yields k-dependent bias factors. Excursion set approaches can produce k-dependent bias factors as soon as the filter is not a sharp k-space filter. In any case, the form of the bias parameters should be dictated by spherical symmetry. 4.4 Peak bias factors and peak-background split 4.4. An effective bias relation Although density peaks form a well-behaved point process, the first two terms in the right-hand side of Eq.4.4 can be thought of as arising from the continuous, deterministic bias relation δ pk ν, x =b δ s x b δ s x+ b δ sx b δ s x δ s x+ b [ δ s x], 4.48 which is nonlocal only because the mass density field has been filtered on scale R s. δ pk x is not a count-in-cell peak overabundance, but merely some effective continuous overdensity field that can be used to compute ξ pk ν, r. Therefore, even though δ pk x could be less than - in deep voids, this would not create any problem because δ pk x is not an observable quantity A peak-background split derivation of the k-dependent peak bias factors There are two ways in which the peak-background split is implemented. In the first approach, the Nth-order bias parameter b N ν is related to the Nth-order derivative of the differential number density nν of virialized objects according to b N ν N nν N [ nν] σ ν N, 4.49 Rs where it is understood that the derivative is taken at a fixed smoothing scale since we consider a long-wavelength perturbation to the density field. This relation, which assumes a universal mass function, can be extended to discrete density peaks even though their average number density is a function of both ν and R s : b N ν, R s σ N n pk ν, R s N [ n pk ν, R s ] ν N. 4.5 This means that the large scale, constant and deterministic bias factor returned by the peakbackground split argument is exactly the same as in our approach, when we are on large enough scales that the k-dependence associated with the b k term can be ignored. It turns out that Eq.?? generalizes as follows: higher order derivatives of the peak number density?? with respect to ν result in the large scale, k-independent peak bias coefficients, i.e. b N ν, R S b N ν, R s. 4.5 However, derivatives of Eq.?? cannot produce the k-dependent bias terms like b, b etc., which arise owing to the constraints imposed by derivatives of the mass density field.

13 4.4. PEAK BIAS FACTORS AND PEAK-BACKGROUND SPLIT 45 Therefore, Desjacques et al. considered the second implementation of the peak background split in which the dependence of the mass function on the overdensity of the background is derived explicitly. The ratio of this conditional mass function to the universal one is then expanded in powers of the background density. The bias factors are the coefficients of this expansion. The key quantity is the average number density of peaks identified on scale R s as a function of the overdensity δ l defined on the large smoothing scale R l R s. This conditional peak number density is n pk ν, R s δ l,r l = G γ, γ ν exp[ ν ɛν l / ɛ ] π 3/ R 3, 4.5 π ɛ where see equation E5 of bardeen/etal:986 ν l δ l, νν l ɛ = σ, uν l γ ɛr, r k σ l σ s σ l k = σ /σ s s u ν, ν l γ ν = γ ν ɛ r ɛ Varu ν, ν l γ γ r γ ɛ [+ɛ r ɛ ɛν l, ], σ, 4.53 /σ s Here, the spectral moments σ ns and σ nl are evaluated on the filtering scale R s and R l, respectively, whereas σn is a cross-correlation between the small and the large smoothing scale, σ n π dk k n+ P δ k W kr s W kr l In addition, note that ɛν l =δ l /σ s σ /σ l does not vanish in the limit R l R s, in which the ratio Σ l σ /σ l is of order unity there is a form factor that depends on the exact shape of the filter. At this point, one shall expand the ratio n pk ν, R s δ l,r l / n pk ν, R s in powers of δ l. This ratio is then interpreted as representing the average overabundance of peaks in regions which have mass overdensity δ l although, strictly speaking, it is a statement about cells of overdensity δ l that have a peak at their center. Setting ɛ but keeping the r dependence means that γ γ but γ ν γ ν[ rδ l /δ s Σ l ]. As a result, derivatives of G with respect to δ l will introduce terms which depend on r. These are terms which could not have been obtained by differentiating the unconditional mass function. At first order in δ l, one obtains δ pk δ B = σ σ l b δ l + σ σ l b δ l 4.55 where b and b were defined in Eq.??. Therefore, the cross correlation between the average overdensity of peaks defined on scale R s and the mass overdensity on scale R l is δ pk δ l R l =σ b + σ b on large scales R l R s Note that this final expression is the Fourier transform of b + b k P δ kw kr s W kr l. This explicitly demonstrates that a conditional mass function approach can produce the k- dependent bias parameters that show up in the peak correlation function. The higher order bias factors are derived in an analogous way. Each term of order δl N will include terms of order r m

14 46CHAPTER 4. STATISTICS OF MAXIMA OF THE LINEAR GAUSSIAN DENSITY FIELD with m N. Terms proportional to r give b, b, etc.; terms proportional to r give b, b, etc. This analysis thus provides a simple way of determining all the additional k-dependent higher-order bias terms that arise in the peaks model. References - J. A. Peacock, A. F. Heavens, The statistics of maxima in primordial density perturbations, Mon. Not. R. Astron. Soc., 7, J. M. Bardeen, J. R. Bond, N. Kaiser, A. S. Szalay, The statistics of peaks of Gaussian random fields, Astrophys. J.,34, V. Desjacques, Baryon acoustic signature in the clustering of density maxima, Phys. Rev. D., 78, V. Desjacques, M. Crocce, R. Scoccimarro, R. K. Sheth, Modeling scale-dependent bias on the baryonic acoustic scale with the statistics of peaks of Gaussian random fields, Phys. Rev. D, 8, 359