Mon. Not. R. Astron. Soc. 000, 000–000 (2012)

Printed 9 December 2012

(MN LATEX style file v2.2)

Pairwise velocity statistics of biased tracers in Convolution Lagrangian Perturbation Theory: Application to Redshift Space Distortions Lile Wang1,2? , Beth Reid1 , and Martin White1,3 1

Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA

2

Department of Physics and Tsinghua Centre for Astrophysics (THCA), Tsinghua University, Beijing 100084, China

3

Departments of Physics and Astronomy, University of California, Berkeley, CA 94720, USA

Accepted 2012...; Received 2012...; in original form 2012

ABSTRACT

An accurate description of the pairwise velocities of biased tracers is an essential ingredient to understanding redshift space distortions and utilizing them as a cosmological probe. In this paper we compute the first and second moments of the pairwise velocity distribution by extending the Convolution Lagrangian Perturbation theory (CLPT) formalism of Carlson et al. 2012. Our predictions outperform standard perturbation theory calculations in many cases when compared with halo pairwise velocities drawn from N -body simulations. We combine the CLPT predictions of real space clustering and velocity statistics in the Gaussian streaming model to obtain predictions for the monopole and quadrupole correlation functions accurate to x% down to scales of 15 (??) h−1 Mpc for biased tracers with b > 1.3 at redshift z = 0.55. Also say something about magnitude of second order bias effects on the velocity statistics 1

INTRODUCTION

The large-scale structure (LSS) as traced by the distribution of galaxies is the focus of several ongoing and upcoming observational campaigns to investigate the fundamental physics of the Universe, including the properties of the initial conditions, the imprint of (massive) neutrinos becoming non-relativistic, and perhaps foremost, the behavior of the mysterious dark energy. Improving our theoretical understanding of the LSS will enhance the scientific return of these projects. In particular, a more detailed understanding of the anisotropy in the observed clustering is of great interest, as the imprint of peculiar velocities in redshift survey maps known as redshift space distortions (RSD) allow a consistency test in general relativity between the expansion history and growth of perturbations; such tests could provide support for modified gravity theories as an explanation for the observed cosmic expansion. Moreover, a precise understanding of the peculiar velocity induced anisotropy in galaxy clustering would improve our ability to measure the geometrically induced anisotropy known as the AlcockPaczynski effect (Alcock & Paczynski 1979), and thus constrain the expansion rate H(z) directly (for further details, see Samushia et al. 2011). N -body simulations are invaluable in LSS investigations, as the physics of self-gravitating media is highly nonlinear up to scales of ∼ 10 h−1 Mpc today. Perturbation c 2012 RAS

theory approaches remain popular for improving physical insight and because the potential cosmological parameter space to be explored is quite large. Within Newtonian gravity, a good approximation of general relativity in most of the matter structure formation study (e.g. Bardeen 1980), several different types of perturbation theories have been developed. On one hand, standard perturbation theory (SPT) adopts Eulerian description of fluids focusing on the velocity field and Eulerian density contrast (e.g. Peebles 1980, for linear theory), and is extended to quasi-linear or higher order using different types of techniques (Makino et al. 1992; Jain & Bertschinger 1994; Valageas 2004; see Carlson et al. 2012 for a comparison of several perturbative techniques). On the other hand, there is a Lagrangian approach to the problem, which instead relies on the analysis of the Lagrangian density contrast and displacement field (Buchert 1992, 1994; Bouchet et al. 1995). Lagrangian perturbation theory (LPT) and SPT give identical results for the power spectrum when expanded to the same order. However, the results are quite different if we consider the so-called Lagrangian resummation theory (LRT), which introduces nonperturbative effects by evaluating bulk effect of resummed terms (Matsubara 2008b,a). A key success of LRT is a very accurate description of the redshift space two-point correlation function of dark matter halos on scales of interest for studying baryon acoustic oscillations (BAO). However, the

2

Wang et al.

LRT predictions are inaccurate on scales ∼ 20 − 70Mpc h−1 , where deviations from linear theory are still only ∼ 10%. A recent paper by Carlson et al. (2012) put forward convolution Lagrangian perturbation theory (CLPT) which improves the LRT method by keeping more terms from perturbative expansion (i.e. consider more about the “bulk flow” for non-perturbative effect) and gave dramatically better results in comparison with simulations, particularly for the description of the redshift space clustering of dark matter. Even though it might be undermined by the “shell crossing” effect (e.g. Carlson et al. 2009; Valageas 2011) on small scales (i.e. the Jacobian between Lagrangian and Eulerian coordinates becomes singular), LPT is also of specific interest in modeling the distribution of galaxies in redshift space. Current theories of galaxy formation rely on the cooling of gas within dark matter potential wells to form galaxies. Therefore, like dark matter halos, galaxies are biased tracers of the underlying matter distribution. The Lagrangian local bias model provides a better description of dark matter halos than Eulerian bias (e.g. Roth & Porciani 2011; Baldauf et al. 2012; Chan et al. 2012; Wang & Szalay 2012). However, we terms involving the tidal tensor may also become important for high mass halos (Sheth et al. 2012); such terms would contribute to the quadrupole correlation function. The inclusion of non-linear redshift space distortion effects is also straight-forward in Lagrangian approaches; a time derivative of the original displacement field is simply added in the line-of-sight (LOS) direction. Thus we expect CLPT to provide a better prediction of the statistical description of the dark matter halo and galaxy distributions. The predictions of CLPT are very accurate for the twopoint correlation function of relatively massive halos in the absence of redshift space distortions, and the methodology is easily extendable to compute properties of the pairwise halo velocity distributions that generate redshift space distortions. The primary purpose of this paper is to examine CLPT’s accuracy in predicting these statistics, in comparison with N -body simulations. While we will see that CLPT provides an accurate description of the velocity distributions; unfortunately, the CLPT predictions for the anisotropy in the two-point correlation function measured by the quadrupole are still inaccurate on the quasi-linear scales of interest (Carlson et al. 2012, see Figure 5 of). Therefore, in this paper we reexamine the non-perturbative approach advocated in Reid & White (2011), the scale-dependent Gaussian streaming model. The model convolves the real space two-point correlation function with an approximation to the (scale-dependent) velocity distribution functions to predict redshift space clustering. The primary advantage of this approach is there is no implicit dependence of the theory’s accuracy on the smallness of the difference between the redshift space and real space (or Lagrangian space) pair separations I think this is true in LRT and CLPT with biased tracers..? This paper is structured as follows. In Section 2 we provide some analytic prerequisites for estimating statistic functions with CLPT, and in Section 4 we show the numerical evaluation of those perturbative analytic results. Those CLPT results are taken into the Gaussian streaming model in Section 5 and compared with values obtained directly and indirectly by simulations. Section 6 gives the summary of this article.

2

CLUSTERING AND VELOCITY STATISTICS IN CLPT

In this section we extend the work of Matsubara (2008a) and Carlson et al. (2012) to enable the calculation of moments of the pairwise velocity distribution for tracers that are biased in a local Lagrangian sense. Once the real space two-point correlation function and pairwise velocity distributions are known, they determine the observed two-point clustering in redshift space. 2.1

Preliminaries

Throughout this work we will adopt the “plane-parallel” approximation, so that the line-of-sight (LOS) is chosen along a single Cartesian axis (ˆ z). While wide-angle effects could potentially be important in modern surveys (P´ apai & Szapudi 2008), Samushia et al. (2012) have shown that in practice these effects are small given current errors. The redshiftspace position s of an object differs from its real-space position r due to its peculiar velocity, s = x + vz (x) b z,

(1)

where vz (x) ≡ uz (x)/(aH) is the line-of-sight (LOS) component of object’s velocity (assumed non-relativistic) in units of the Hubble velocity. In linear theory, the peculiar velocity field is assumed curl- free, and its divergence is sourced by the underlying matter fluctuations: ∇ · v = −f δm

(2)

where f ≡ d ln D/d ln a specifies the growth rate of fluctuations. Measurements of two-point clustering as a function of angle with respect to the LOS direction can directly constrain f times the normalization of matter fluctuations (Percival & White 2009). In this paper, we will focus on the prediction of the two-point correlation function: ξ(r) = hδ(x)δ(x + r)i.

(3)

In real space, ξ(r) ≡ ξ(r) is only a function of the separation length, while in redshift space ξ(s) depends on the cosine of the angle between the pair separation vector and the LOS, ˆ. It is convenient and common to condense the µs ≡ ˆs · z information in ξ(s) into Legendre polynomial moments: X ξ(s, µs ) = ξ` (s)L` (µ). (4) `

By symmetry, odd ` moments vanish. In linear theory, only ` = 0, 2, 4 contribute; we will focus our model predictions on those moments. Throughout this paper, we adopt the Einstein summation convention and the following convention of Fourier transform and its inverse (n is the number of dimension, here it is usually 1 or 3): Z Z dn k ˜ n −ikx ˜ F (k) = d xF (x)e , F (x) = F (k)eikx . (5) (2π)n 2.2

Pairwise velocity statistics and the Gaussian streaming model

Eqn. 12 of Scoccimarro (2004) (see also Fisher 1995) relates the observed clustering of a tracer in redshift space to the c 2012 RAS, MNRAS 000, 000–000

Pairwise velocity statistics in CLPT underlying real space clustering and full pairwise velocity distribution function without any approximations: Z ∞ 1 + ξ(rσ , rπ ) = dy[1 + ξ(r)]P(vz = rπ − y, r). (6) −∞

Here rσ is the transverse separation in both real and redshift space, rπ is the LOS pair separation in redshift space, and y is the LOS separation in real space, so that r2 = rσ2 + y 2 . Reid & White (2011) showed that even though the true P is certainly non-Gaussian, approximating it with a Gaussian provides an accurate description of the redshift space correlation function of massive halos: Z dy 1 + ξ s (rσ , rπ ) = [1 + ξ(r)] 2 (r, µ)]1/2 [2πσ12   (7) [rπ − y − µv12 (r)]2 × exp − , 2 2σ12 (r, µ) In the scale-dependent Gaussian streaming model, the Gaussian probability distribution function is centered at µv12 (r), the mean velocity between a pair of tracers as a function of their real space separation: v12 (r)ˆ r=

h[1 + δ(x)][1 + δ(x + r)][v(x + r) − v(x)]i h[1 + δ(x)][1 + δ(x + r)]i

(8)

The factor [1 + δ(x)][1 + δ(x + r)] in the numerator and the denominator specifies that we are computing the average relative velocity over pairs of tracers, rather than over randomly chosen points in space. By symmetry, the mean velocity is directed along the pair separation vector; projecting it onto the LOS brings a factor of µ = y/r in Eqn. 49. Similarly, the width of the velocity PDF is different for components along and perpendicular to the pair separation vector, so the LOS velocity dispersion can be decomposed as a sum with contributions from the one-dimensional velocity dispersions perpendicular and parallel to the LOS. 

(1 + δ(x))(1 + δ(x + r))(v ` (x + r) − v ` (x))2 2 (9) σ12 (r, µ) = h(1 + δ(x))(1 + δ(x + r))i 2 σ12 (r, µ)

=

2 µ2 σk2 (r) + (1 − µ2 )σ⊥ (r)

(10) 2 σ⊥,k (r)

Linear theory expressions for v12 (r) and are given in, e.g. , Fisher (1995); Gorski (1988); Gorski et al. (1989); Reid & White (2011), in which one finds that pairwise mean 2 infall v12 (r) is proportional to bf , while σ12 (r) scales as f 2 with no dependence on the large-scale bias b at linear order. Reid & White (2011) evaluated Eqns. 8 and 9 in standard perturbation theory under the assumption of a linear bias b relating the tracer and matter density fields, δt (x) = bδm (x). The purpose of the present work is to evaluate the same statistics in CLPT. There are several reasons to examine the same statistics within CLPT: • The Lagrangian perturbation theory approach uses displacements between initial and final positions as the perturbative variable; the velocity field is simply a time derivative of the displacement field and thus a natural variable of the theory. • Standard perturbation theory does an unsatisfactory job of describing the smoothing of the BAO features, which are nearly erased in the quadrupole correlation function; as a result, the analysis of Reid (2012) used Lagrangian resummation theory (LRT; Matsubara 2008b) for their model above separations of 70 h−1 Mpc. We expect CLPT to perform as well as LRT on those scales. c 2012 RAS, MNRAS 000, 000–000

3

• The inaccuracy of the streaming model results with standard perturbation theory inputs for the velocity statistics in Reid & White (2011) was traced to the errors in the perturbative calculation of v12 and its derivative, v12 /dr, and was smallest for halos with second-order bias near 0. CLPT naturally includes higher-order bias corrections, and will allow us to quantify the size of the second order contributions to the velocity statistics of interest. 2.3

Resummed Lagrangian Perturbation Theory (LRT): formalism for biased tracers

Lagrangian perturbation theories perform a perturbative expansion in the displacement field Ψ = Ψ(1) + Ψ(2) + Ψ(3) + · · · . Ψ relates the Eulerian coordinates x and Lagrangian coordinates q of a mass element or discrete tracer object: x(q, t) = q + Ψ(q, t) .

(11)

Here q is the initial coordinate, so by definition Ψ(q, 0) = 0. With this in mind, the relation between Eulerian and Lagrangian fields of matter density contrast (δ = ρ/¯ ρ − 1) is given by [1 + δm (x, t)] d3 x = [1 + δm (q, 0)]d3 q = d3 q .

(12)

Matsubara (2008b,a) laid out the formalism for including redshift space distortions and non-linear local Lagrangian biasing within Lagrangian perturbation theory. The density contrast of our tracer field in Lagrangian space, δ(q), is related to the underlying Lagrangian matter density fluctuations smoothed on scale R: 1 + δ(q) = F [δm,R (q)] ,

(13)

where F (δ) is the bias function. Note that the smoothing scale R naturally drops out in the final predictions for all statistics of interest in this paper, but is necessary to keep intermediate quantities well-behaved. Noting the property of Dirac delta function δ D [f (x)] = δ D (x − x0 )/|f 0 (x0 )| (x0 is the root of f (x) = 0), we have Z 1 + δ(x, t) = d3 q F [δR (q)]δ D [x − q − Ψ(q, t)] . (14) After a coordinate transformation {q1 , q2 } → {q = q2 − q1 , Q = (q1 + q2 )/2}, and expressing both F and δ D in Eqn. 14 by their Fourier representations, the two-point correlation function in real space is given by [see also equations (15) through (20) in Carlson et al. (2012)]: Z Z Z d3 k ik·(q−r) dλ1 dλ2 1 + ξ(r) = d3 q e (2π)3 2π 2π (15) D E i(λ1 δ1 +λ2 δ2 +k·∆) ˜ ˜ × F (λ1 )F (λ2 ) e , where δ1,2 = δ(q1,2 ), ∆ = Ψ(q2 ) − Ψ(q1 ), and F˜ (λ) is the Fourier transform of F (δR ) with coordinates pair δR versus λ. 2.4

Velocity moments in CLPT: formalism

The relative peculiar velocity between two tracers at Eulerian coordinates x1 and x2 can be simply expressed in terms of the time derivative of the displacement field Ψ: ˙ u(x2 ) − u(x1 ) = a (x˙ 2 − x˙ 1 ) = a∆.

(16)

4

Wang et al.

In the time-independent approximation to the perturbative kernels where Ψ(k) ∝ Dk for D the linear growth function (Eqn. 46 of Matsubara 2008b), ˙ (k) = kHf Ψ(k) . Ψ

(17)

Thus we have a perturbative expansion for the Cartesian components of vn (adopting the units of Eqn. 1) in terms of the components of ∆n vn (x2 ) − vn (x1 ) =

X k

kf ∆(k) n =

˙n ∆ . H

The velocity generating function Z(r, J) is given by Z Z Z dλ1 dλ2 d3 k ik·(q−r) e Z(r, J) = d3 q (2π)3 2π 2π E D ˙ i λ δ +λ δ +k·∆+J· ∆/H ) . × F˜ (λ1 )F˜ (λ2 ) e ( 1 1 2 2

(18)

(19)

Note that ξ(r) = Z(r, 0) − 1 (Eqn. 15), and derivatives of Z give the pairwise velocity moments of interest (i.e., the numerators in Eqns. 8 and 10):

Z

h[1 + δ(x)][1 + δ(x + r)][Πpk=1 (vik (x + r) − vik (x))]i = Z Z d3 k ik·(q−r) dλ1 dλ2 d3 q e (2π)3 2π 2π * !! + ˙i ∆ p i(λ δ +λ δ +k·∆) k 1 1 2 2 × F˜ (λ1 )F˜ (λ2 ) Πk=1 e H   ∂ Z(r, J) = Πpk=1 −i ∂Jik J=0 (20)

Here the set {ik } specifies the Cartesian coordinate direction for each derivative with respect to Jik . Before proceeding to evaluate Eqn. 20, we generalize the definitions of the functions K, L, and M in Carlson et al. (2012) to include J. These functions are convenient shorthand for intermediate results. ˙ X = λ1 δ1 + λ2 δ2 + k · ∆ + J · ∆/H (21) Kp,{i1 ..ip } (q, k, λ1 , λ2 ) =  p  ∂ −i eiX ∂Jik J=0 Z dλ1 dλ2 Lp,{i1 ..ip } (q, k) = Kp,{i1 ..ip } (q, k, λ1 , λ2 ) 2π 2π Z d3 k ik·(q−r) Mp,{i1 ..ip } (r, q) = e Lp,{i1 ..ip } (q, k) (2π)3 p h[1 + δ(x)][1 + δ(x + r)][Πk=1 (vik (x + r) − vik (x))]i = Z d3 qMp,{i,..,i} (r, q)

(22) (23)

3.1

makes the evaluation of K tractable; here hX N ic is the N th cumulant of the random variable X. Taylor expanding the exponential on the right hand side of Eqn. 26 only for terms that vanish in the limit |q| → ∞ and keeping only terms up to O(PL2 ), Carlson et al. (2012), Sec. 4 obtains D E K0 = eiX J=0  2 2 1 −(1/2)Aij ki kj −(1/2)(λ2 2 1 +λ2 )σR 1 − λ1 λ2 ξL + λ21 λ22 ξL =e e 2 1 i − (λ1 + λ2 )Ui ki + (λ1 + λ2 )2 Ui Uj ki kj − Wijk ki kj kk 2 6 i + λ1 λ2 (λ1 + λ2 )ξL Ui ki − (λ1 + λ2 )A10 ij ki kj 2  i + (λ21 + λ22 )Ui20 ki − iλ1 λ2 Ui11 ki + O(PL3 ) . 2 (27) In equation (27), we adopt the following short-hand definitions: 2 hδ12 ic = hδ22 ic = σR ,

hδ1 δ2 ic mn m n Ui = hδ1 δ2 ∆i ic , Amn ij = mn Wijk = hδ1m δ2n ∆i ∆j ∆k ic , Ui = Ui10 ,

Aij = A00 ij ,

= ξL (q) , hδ1m δ2n ∆i ∆j ic ,

(25)

EVALUATING THE CLPT PREDICTIONS Evaluating ξ(r) in CLPT

To begin, we review the calculation of the real space twopoint correlation function ξ(r) in CLPT, first presented in

(28)

00 Wijk = Wijk .

Subsequently we apply the identity of integration of F˜ (λ), which reads (see Matsubara 2008a; Carlson et al. 2012) Z 2 2 dλ ˜ F (λ)eλ σR /2 (iλ)n = hF (n) i , (29) 2π where hF (n) i is the expectation value of the nth derivative of F (δR ). We can now evaluate L0 analytically:  1 2 L0 = e−(1/2)Aij ki kj 1 + hF 0 i2 ξL + hF 00 i2 ξL + 2ihF 0 iUi ki 2 i − [hF 00 i + hF 0 i2 ]Ui Uj ki kj − Wijk ki kj kk − hF 0 iA10 ij ki kj 6 + 2ihF 0 ihF 00 iξL Ui ki + ihF 00 iUi20 ki  − ihF 0 i2 Ui11 ki + O(PL3 ) .

(24)

The first subscript of these functions indicates the number of derivative terms p, and the second is a list containing the Cartesian indices of the p derivatives.

3

Carlson et al. (2012). The cumulant expansion theorem, # " ∞ X iN N iX hX ic , (26) he i = exp N! N =1

(30) The integration with respect to k is then conducted, using some basic relations for Gaussian integration (see appendix C of Carlson et al. 2012), which gives d3 k ik·(q−r) e L0 (q, k) (2π)3 −1 1 = e−(1/2)(A )ij (qi −ri )(qj −rj ) (2π)3/2 |A|1/2  1 1 2 × 1 + hF 0 i2 ξL + hF 00 i2 ξL − 2hF 0 iUi gi + Wijk Γijk 2 6 Z

M0 =

− [hF 00 i + hF 0 i2 ]Ui Uj Gij − hF 0 i2 Ui11 gi − hF 00 iUi20 gi  3 − 2hF 0 ihF 00 iξL Ui gi − hF 0 iA10 G + O(P ) . ij ij L c 2012 RAS, MNRAS 000, 000–000

Pairwise velocity statistics in CLPT (31)

5

where we define (up to desired order)

Here we define gi = (A−1 )ij (qj − rj ) ,

Gij = (A−1 )ij − gi gj ,

Γijk = (A−1 )ij gk + (A−1 )ki gj + (A−1 )jk gi − gi gj gk .

(32)

Finally, the desired correlation function is given by the q integration of M0 : Z

d3 q M0 (r, q) .

1 + ξ(r) =

(33)

mn In order to evaluate M0 , we expand Uimn , Amn and Wijk ij (n) in (28) with respect to ∆ , i.e.

mn(p)

(p)

mn(pq)

= hδ1m δ2n ∆i ic ,

Ui

mn(pqr)

Wijk

(p)

Aij (q)

(p)

(q)

= hδ1m δ2n ∆i ∆j ic ,

(r)

= hδ1m δ2n ∆i ∆j ∆k ic ,

˙ ni hδ1 ∆ U˙ n = = Un(1) + 3Un(3) + · · · , f ˙ ni hδ 2 ∆ U˙ n20 = 1 = Un20(2) + · · · , f ˙ ni hδ1 δ2 ∆ U˙ n11 = = Un11(2) + · · · , f ˙ ni h∆i ∆ (11) (13) (31) (22) A˙ in = = Ain + 3Ain + Ain + 2Ain + · · · , f ˙ ni hδ1 ∆i ∆ 10(12) 10(21) = 2Ain + Ain + ··· , A˙ 10 in = f ˙ ˙ ijn = hδ1 ∆i ∆j ∆n i = 2W (112) + W (121) + W (211) + · · · . W ijn ijn ijn f (38)

(34) Integrate with respect to λ1 and λ2 , we have (see the appendices of Carlson et al. 2012)

and up to desired order, we have (1)

Ui = Ui

(3)

+ Ui

20(2)

Ui20 = Ui

(11)

Aij = Aij

+ ··· ,

Wijk =

(22)

+ Aij

10(12)

A10 ij = Aij

+ ··· ,

(112) Wijk

+

11(2)

Ui11 = Ui (13)

+ Aij

10(21)

+ Aij

Z (31)

+ Aij

L1,n =

+ ··· ,

2

+ ··· ,

(35)

+ ··· ,

(121) Wijk

(211)

+ Wijk

+ ··· .

We refer the readers to appendices B through C in Carlson et al. (2012) for details of evaluating those correlators.

3.2

dλ1 dλ2 ˜ F (λ1 )F˜ (λ2 )K1,n (λ1 , λ2 , k, q) 2π 2π

The mean pairwise velocity in CLPT

2

1 0 00 ˙ ijn + 2ihF 0 iki A˙ 10 ˙ − ki kj W in + 2hF ihF iξL Un 2 + 2i[hF 00 i + hF 0 i2 ]ki Ui U˙ n + ihF 0 i2 ξL ki A˙ in  − 2hF 0 iki kj Ui A˙ in + O(PL3 ) .

(39)

Then evaluate the integration over k, we have

To compute the mean pairwise velocity in CLPT, we first evaluate K1,n , again making use of the cumulant expansion theorem: K1,n (λ1 , λ2 , k, q) = #" ∞ " ∞  # X iN  ∆ X iN D N E ˙n N X X = exp N! N! H c c N =0 N =0

2

= f e−(1/2)Aij ki kj e−(1/2)(λ1 +λ2 )σR  × 2hF 0 iU˙ n + iki A˙ in + hF 00 iU˙ n20 + hF 0 i2 U˙ n11

Z M1,n =

d3 k ik·(q−r) e L1,n (q, k) (2π)3

−1 f2 e−(1/2)(A )ij (qi −ri )(qj −rj ) (2π)3/2 |A|1/2  × 2hF 0 iU˙ n − gi A˙ in + hF 00 iU˙ n20 + hF 0 i2 U˙ 11n

=



.

J=0

(36) Up to the second order of the linear power spectrum [i.e. O(PL2 )], equation (36) is recast as

(40)

1 0 00 ˙ ijn − 2hF 0 igi A˙ 10 ˙ Gij W in + 2hF ihF iξL Un 2 − 2[hF 00 i + hF 0 i2 ]gi Ui U˙ n − hF 0 i2 ξL gi A˙ in  0 3 ˙ − 2hF iGij Ui Ain + O(PL ) , −

K1,n (λ1 , λ2 , k, q) 2

2

2

= f e−(1/2)Aij ki kj e−(1/2)(λ1 +λ2 )σR  1 × i(λ1 + λ2 )U˙ n + iki A˙ in − (λ21 + λ22 )U˙ n20 2 1 ˙ ijn − (λ1 + λ2 )ki A˙ 10 − λ1 λ2 U˙ n11 − ki kj W in 2 2 − iλ1 λ2 (λ1 + λ2 )ξL U˙ n − i(λ1 + λ2 ) ki Ui U˙ n − iλ1 λ2 ξL ki A˙ in − i(λ1 + λ2 )ki kj Ui A˙ in + O(PL3 ) c 2012 RAS, MNRAS 000, 000–000

and finally, (37)

 .

v12,n (r) = [1 + ξ(r)]−1

Z

d3 q M1,n (r, q) .

(41)

Typically v12,n is projected along the direction of pair separation vector, i.e. v12 = v12,n rˆn .

6

Wang et al.

3.3

The pairwise velocity dispersion in CLPT

The integration kernel for the velocity dispersion tensor is  D  E ∂ ∂ K2,nm (λ1 , λ2 , k, q) = −i −i eiX(J) ∂Jm ∂Jn #( " ∞ # J→0 " ∞ E X iN D X iN D N E ˙ n∆ ˙ mX N X ∆ = exp N ! N ! c c N =0 N =0 " ∞ #" ∞ #) E E X iN D X iM D ˙ nX N ˙ mX M + ∆ ∆ , N! M! c c N =0 M =0 (42) Expand equation (42) to the second order of the linear power spectrum (i.e. to the order of O(PL2 )): K2,nm (λ1 , λ2 , k, q) 2

2

2

= f 2 e−(1/2)Aij ki kj e−(1/2)(λ1 +λ2 )σR  × (λ1 + λ2 )2 U˙ n U˙ m − (λ1 + λ2 )(A˙ in ki U˙ m + A˙ im ki U˙ n ) − A˙ im ki A˙ jn kj + [1 − λ1 λ2 ξL − (λ1 + λ2 )Ui ki ]A¨nm 3 ¨ + i(λ1 + λ2 )A¨10 nm + iWinm ki + O(PL ) , (43) where we define (up to desired order) ˙ n∆ ˙ mi h∆ (13) (31) (22) = A(11) A¨nm = nm + 3Anm + 3Anm + 4Anm , f2 ˙ n∆ ˙ mi hδ1 ∆ A¨10,nm = = 2A10(12) + 2A10(21) , nm nm 2 f ˙ ˙ ¨ inm = hδ1 ∆i ∆n ∆m i = 2W (112) + 2W (121) + W (211) . W inm inm inm f2 (44) Then evaluate the integration with respect to λ1 , λ2 , we have Z dλ1 dλ2 ˜ L2,nm = F (λ1 )F˜ (λ2 )K2,nm (λ1 , λ2 , k, q) (2π) (2π)  = 2[hF 0 i2 + hF 00 i]U˙ n U˙ m + 2ihF 0 i(A˙ in ki U˙ m + A˙ im ki U˙ n ) − A˙ im ki A˙ jn kj + [1 + hF 0 i2 ξL + 2ihF 0 iUi ki ]A¨nm 3 2 −(1/2)Aij ki kj ¨ + 2hF 0 iA¨10 , nm + iWinm ki + O(PL ) f e (45) and then with respect to k: Z d3 k ik·(q−r) M2,nm = e L2,nm (q, k) (2π)3 =

f2 (2π)3/2 |A|1/2

e−(1/2)(A

−1

lg (M/M )

hF 0 i

hF 00 ip−b

1

< 12.483

0.3438

−0.555

2

12.484 − 12.784

0.4377

−0.542

3

12.785 − 13.085

0.6556

−0.433

4

13.086 − 13.386

0.9671

−0.097

Seq. No.

2 the tensor σ12,nm . In order to obtain the velocity dispersion components parallel to and perpendicular to the pairwise 2 separation unit vector rˆ, we project σ12,nm into different directions:

4

)ij (qi −ri )(qj −rj )

(46) 2 σ12,nm

can be obtained by Z 2 σ12,nm (r) = [1 + ξ(r)]−1 d3 q M2,nm (r, q) .

Table 1. Bias parameters obtained by fitting the real space correlation functions for different halo mass bins. Please note that we set hF 00 i = 0 empirically, althogh we show the values of hF 00 ip−b here.

2 σk2 = σ12,nm rˆn rˆm ,

 × 2[hF 0 i2 + hF 00 i]U˙ n U˙ m − 2hF 0 i(A˙ in gi U˙ m + A˙ im gi U˙ n ) − A˙ im A˙ jn Gij + [1 + hF 0 i2 ξL − 2hF 0 iUi gi ]A¨nm 3 ¨ + 2hF 0 iA¨10 nm − Winm gi + O(PL ) . Finally,

Figure 1. Real space correlation function fitted to simulations. CLPT results are given by heavy solid curves for hF 00 i = 0 and light solid curves for hF 00 i = hF 00 ip−b (see Table 1); simulation results are presented by shade bands showing its error range. Mass ranges are indicated by labels.

(47)

Desired component of pairwise velocity dispersion can be obtained by different components or kinds of contractions of

2 2 K σ⊥ = (σ12,nm δnm − σk2 )/2 .

(48)

REAL SPACE STATISTICS OBTAINED BY CLPT

We present the CLPT prediction of different types of statistics in this section, which serve as the input of streaming redshift space distortion model. All calculations are completed with pertinent results in Section 2 in mind. First we fit our results to the N -body simulation results (Reid & White 2011; White & et al. 2011). We tried to take both hF 0 i and hF 00 i as free parameters, but the outcomes seem to suggest that hF 00 i should be some value that is very close to zero if it is non-zero. Therefore, we empirically set hF 00 i = 0 for simplicity. In order to justify this empirical choice, we also adopt the hF 00 i values obtained by c 2012 RAS, MNRAS 000, 000–000

Pairwise velocity statistics in CLPT

Figure 2. Pairwise infall velocity. Our CLPT results are shown by heavy solid curves, compared with simulations shown by shade bands showing its error range, and quasi-linear results in Reid & White (2011) shown by light dotted curves. Similar to Figure 1, different mass bins are labelled in each panel.

peak-background relation (e.g. Matsubara 2008a), which are denoted by hF 00 ip−b . Comparing the hF 00 ip−b results with hF 00 i = 0 ones in Figure 1, we As we will see in the following context, this assumption gives rather good results. Subsequently, the only free parameter, hF 0 i, is adjusted to the values specified in Table 1 so that the theoretical predictions are nicely overlapped with simulation results. In Figure 1 we present the comparison for different halo mass bins. Note that the consistency between CLPT results and simulation is almost perfect at all scales, ranging from ∼ 2 Mpc h−1 through the BAO scale (∼ 100 Mpc h−1 ), even though hF 00 i related terms are actually not adopted. As long as the bias parameter hF 0 i are obtained by fitting, we assume that those values are still valid for evaluating time derivatives. Take them back into equations (40) and (46), and conduct pertinent integration, we obtain the scale dependence of pairwise infall velocity. The results (divided by the linear theory) are shown in Figure (2) by heavy dashed curves, compared with shaded error range plot showing the simulation values. For comparison purpose, we also introduce corresponding results in Reid & White (2011) as light dotted curves. Except the highest mass bin, our theory shows very good consistency with the simulation, while c 2012 RAS, MNRAS 000, 000–000

7

Figure 3. Pairwise velocity dispersion. For clearer presentation 2 2 /σ 2 we show the results of σk2 /σk,sim and σ⊥ ⊥,sim + 1/2, where those quantities with subscript “sim” indicates the results coming from the simulations. CLPT results are shown by heavy 2 /σ 2 dash-dotted curves (σ⊥ ⊥,sim + 1/2) and heavy dashed curves 2 ), and quasi-linear results in Reid & White (2011) are (σk2 /σk,sim 2 /σ 2 presented by light solid curves (σ⊥ ⊥,sim +1/2) and light dotted 2 curves (σk2 /σk,sim )

dashed curves (Reid & White 2011) deviates from simulation systematically at lowest scales. This improvement is expected to produce better estimation of redshift space results, which will be discussed in Section 5. CLPT gives quite good prediction about ξ(r) and v12 (r) in real space, though, it is not so ideal in predicting the velocity dispersion. In of Figure 3, we observe quite significant deviation of velocity dispersion calculation at small scales between CLPT results. The prediction is even worse than Reid & White (2011). As always, we add a constant to the CLPT results and Reid & White (2011) results of σk2 (r) 2 and σ⊥ so that they take the same value as simulation at 2 r = 130 Mpc h−1 . The constants for σk2 (r) and σ⊥ are almost the same, with only ∼ 1 per cent relative difference.

8

Wang et al.

Figure 4. Redshift space statistics obtained by Gaussian streams . ing model specified in Reid & White (2011), showing ξ0s /ξ0,lin Shaded bands present simulation values (showing the error range) and our CLPT values are presented by heavy solid curves.

5 5.1

REDSHIFT SPACE DISTORTION AND STATISTICS Gaussian streaming model of redshift space distortions

Although in Carlson et al. (2012) the authors provided redshift space version of correlation function and its multipole moments, here in this paper we adopt a different scheme, viz. the Gaussian streaming model developed in Reid & White (2011). This model assumes that the distribution of pairwise infall velocity along the line of sight can be described by a Gaussian distribution whose mean and variance are given by 2 2 (r) respectively. µv12 (r) and σ12 (r, µ) = µ2 σk2 (r)+(1−µ2 )σ⊥ By this assumption the redshift correlation function is given by Z dy 1 + ξ s (rσ , rπ ) = [1 + ξ(r)] 2 [2πσ12 (r, µ)]1/2   (49) [rπ − y − µv12 (r)]2 , × exp − 2 2σ12 (r, µ) where we take r2 = rσ2 + y 2 and µ = y/r. We intend to 2 evaluate the scale-dependent functions v12 (r), σk2 (r) and σ⊥ by perturbation calculations elaborated in Sections 2 and 4. 2 measured In addition, we will also take v12 (r), σk2 (r) and σ⊥ from simulation as a comparison.

5.2

Multipole moments of redshift space correlation function given by CLPT

Since the redshift space correlation functions are directiondependent, we can apply multipole expansion to them, which reads X ξ s (s, µs ) = Pl (µs )ξls (s) , l

ξls (s) =

2l + 1 2 (rσ2

Z

(50)

1

dµs Pl (µs )ξls (s) , −1

where s = + rπ2 )1/2 is the redshift space distance and µs = rπ /s. Generally we are most interested in the lowest non-zero moments, i.e. l = 0, 2. In Figure 4 and 5, we present the lowest two non-zero multipole moments divided by linear theory (see Fisher 1995; Reid & White 2011, for linear theory).

s . Labels and curve Figure 5. Redshift space results of ξ2s /ξ2,lin indications are identical to Figure 4.

It is obvious from Figure 4 and 5 that the agreement is satisfactory for monopole and quadrupole moments be−1 fore the theory breaks down at r < ∼ 10 Mpc h . For the quadrupole moment (l = 2), we observe that CLPT model gives good overlap with simulations throughout moderate −1 < scales (10 Mpc h−1 < ∼ s ∼ 120 Mpc h ). Meanwhile, even if our CLPT theory breaks down at s ∼ 10 Mpc h−1 , it correctly implies the trend of “falling down” at even smaller s (although the location of falling is not correct). The results for hexadecapole (l = 4) are worse than lower order moments, yet we suggest that the agreement is still satisfactory at such a high order. It is a little bit strange that a worse input (compared 2 measured from simulation, see with v12 (r), σk2 (r) and σ⊥ Figure 3 and its caption) for equation (49) generates even better results (esp. the quadrupole and hexadecapole moments, l = 2 and 4). We find a reasonable explanation to this qualitatively. It is easy to observe from Figure 3 that the velocity dispersion tends to be underestimated at small 2 scales if we combine the error of σk2 (r) and σ⊥ qualitatively, 2 especially on smaller scales. A smaller σ12 gives a “narrower” distribution in equation (49), which means the intergrand is more concerntrated at some specific |y|, or equivalently |µ|, or approximately |µs |. This causes the fact that the intergrand in equation (50) for l = 2 is less likely to be overlapped with the lowest value of P2 (µs ), which makes ξ2s (s) to take a smaller absolute value (esp. on smaller scales). If we note that ξ2s is always negative, it well explains why the trend of overestimation is “compensated”, at least partially, at the lowest s. However, we still cannot explain why the “compensation” eliminates the deviation almost exactly. We expect to address this problem in our future research.

6

SUMMARY

By introducing an auxiliary term J in the generating function, we generalize the CLPT scheme elaborated in Carlson et al. (2012), which is used to estimate pairwise infall velocity and velocity dispersion in real space as functions of pair separation. We note that, besides better results for the correlation function ξ(r), it also gives better estimates for the magnitude of pairwise infall velocity v12 (r). Compared with some previous results (e.g. Matsubara 2008b,a; Reid & White 2011), CLPT contains nonc 2012 RAS, MNRAS 000, 000–000

Pairwise velocity statistics in CLPT perturbative evaluation of its exponential term, which does not split the bulk effect of the different terms and could converge better at lower order. Our work shows that this assertion is true for not only ξ(r), but also for v12 (r) as 2 well. However, to the second order of velocity moment, σ12 is not estimated as well as we expect. In future works, we expect to address this problem more directly. Although the improvement of CLPT is not so effective for some statistical quantities, we still take it into the Gaussian streaming model (e.g. Reid & White 2011), which generates better estimation of monopole and especially quadrupole correlation function in redshift space. Those results stem on better evaluation of v12 (r) for sure, 2 yet the error of σ12 offsets other types of error on the other hand (see subsection 5.2). We expect to figure out what exactly contribute to this offset, and whether this almost-exact offset is a kind of coincidence in future works.

REFERENCES Alcock C., Paczynski B., 1979, Nature, 281, 358 Baldauf T., Seljak U., Desjacques V., McDonald P., 2012, ArXiv e-prints Bardeen J. M., 1980, Phys Rev D, 22, 1882 Bouchet F. R., Colombi S., Hivon E., Juszkiewicz R., 1995, A&A, 296, 575 Buchert T., 1992, MNRAS, 254, 729 Buchert T., 1994, MNRAS, 267, 811 Carlson J., Reid B., White M., 2012, ArXiv e-prints Carlson J., White M., Padmanabhan N., 2009, Phys Rev D, 80, 043531 Chan K. C., Scoccimarro R., Sheth R. K., 2012, Phys Rev D, 85, 083509 Fisher K. B., 1995, ApJ, 448, 494 Gorski K., 1988, ApJL, 332, L7 Gorski K. M., Davis M., Strauss M. A., White S. D. M., Yahil A., 1989, ApJ, 344, 1 Jain B., Bertschinger E., 1994, ApJ, 431, 495 Makino N., Sasaki M., Suto Y., 1992, Phys Rev D, 46, 585 Matsubara T., 2008a, Phys Rev D, 78, 083519 Matsubara T., 2008b, Phys Rev D, 77, 063530 P´ apai P., Szapudi I., 2008, MNRAS, 389, 292 Peebles P. J. E., 1980, The large-scale structure of the universe. Princeton University Press, Princeton, NJ Percival W. J., White M., 2009, MNRAS, 393, 297 Reid B. A., White M., 2011, MNRAS, 417, 1913 Reid B. A. e. a., 2012, MNRAS, 426, 2719 Roth N., Porciani C., 2011, MNRAS, 415, 829 Samushia L., Percival W. J., Guzzo L., Wang Y., Cimatti A., Baugh C., Geach J. E., Lacey C., Majerotto E., Mukherjee P., Orsi A., 2011, MNRAS, 410, 1993 Samushia L., Percival W. J., Raccanelli A., 2012, MNRAS, 420, 2102 Scoccimarro R., 2004, Phys. Rev. D, 70, 083007 Sheth R. K., Chuen Chan K., Scoccimarro R., 2012, ArXiv e-prints Valageas P., 2004, A&A, 421, 23 Valageas P., 2011, A&A, 526, A67 Wang X., Szalay A., 2012, Phys Rev D, 86, 043508 White M., et al. 2011, ApJ, 728, 126

c 2012 RAS, MNRAS 000, 000–000

9