Edgeworth streaming model for redshift space distortions

Edgeworth streaming model for redshift space distortions

Cora Uhlemann cora.uhlemann@physik.lmu.de Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresienstrasse 37, 80333 Munich, Germany Excellence Cluster Universe, Boltzmannstrasse 2, 85748 Garching, Germany    Michael Kopp kopp.michael@ucy.ac.cy Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresienstrasse 37, 80333 Munich, Germany Excellence Cluster Universe, Boltzmannstrasse 2, 85748 Garching, Germany University Observatory, Ludwig-Maximilians University Munich, Scheinerstrasse 1, 81679 Munich, Germany    Thomas Haugg thomas.haugg@physik.lmu.de Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresienstrasse 37, 80333 Munich, Germany
Abstract

We derive the Edgeworth streaming model (ESM) for the redshift space correlation function starting from an arbitrary distribution function for biased tracers of dark matter by considering its two-point statistics and show that it reduces to the Gaussian streaming model (GSM) when neglecting non-Gaussianities. We test the accuracy of the GSM and ESM independent of perturbation theory using the Horizon Run 2 -body halo catalog. While the monopole of the redshift space halo correlation function is well described by the GSM, higher multipoles improve upon including the leading order non-Gaussian correction in the ESM: the GSM quadrupole breaks down on scales below 30 Mpc whereas the ESM stays accurate to 2% within statistical errors down to 10 Mpc. To predict the scale dependent functions entering the streaming model we employ Convolution Lagrangian perturbation theory (CLPT) based on the dust model and local Lagrangian bias. Since dark matter halos carry an intrinsic length scale given by their Lagrangian radius, we extend CLPT to the coarse-grained dust model and consider two different smoothing approaches operating in Eulerian and Lagrangian space, respectively. The coarse-graining in Eulerian space features modified fluid dynamics different from dust while the coarse-graining in Lagrangian space is performed in the initial conditions with subsequent single streaming dust dynamics, implemented by smoothing the initial power spectrum in the spirit of the truncated Zel’dovich approximation. Finally, we compare the predictions of the different coarse-grained models for the streaming model ingredients to -body measurements and comment on the proper choice of both the tracer distribution function and the smoothing scale. Since the perturbative methods we considered are not yet accurate enough on small scales, the GSM is sufficient when applied to perturbation theory.

I Introduction

Redshift space distortions observed in galaxy surveys provide a unique insight into the build-up of cosmological structure by gravitational clustering of dark matter and its tracers such as halos and galaxies. Indeed, the redshift space two point correlation function carries valuable information on both, the real-space clustering and the peculiar velocity field since the observed redshift depends not only on distance but also on deviations from the overall Hubble flow. Peculiar velocities are generated by and hence correlated with the clustering of matter.

There are two main effects in redshift space, a term introduced in Sargent and Turner (1977), that affect the correlation function on large and small scales, respectively. On large scales the peculiar velocity associated with the coherent infall into overdense regions squashes structures and enhances the correlation function along the line of sight which is captured by linear theory and known as the Kaiser effect Kaiser (1987). On small scales, the elongation of nonlinear structures along the line of sight, the so-called ‘Fingers of God’ effect coined in Tully and Fisher (1978) and first described in Jackson (1972), leads to a suppression of the correlation function. Based on this observation one of the first streaming models was developed in Peebles (1980) by assuming an exponential relative or pairwise velocity distribution with a scale-independent dispersion. Dispersion models Park et al. (1994); Peacock and Dodds (1994) aimed to phenomenologically combine effects of linear clustering and small-scale velocity dispersion which act both multiplicative onto the redshift space power spectrum when their correlation is neglected. However, they have been shown to be unable to properly account for redshift space distortions over a vast range of scales by means of -body simulations Hatton and Cole (1998); Nishimichi and Taruya (2011).

To reunite the two disparate results for large and small scales, given by the linear theory Kaiser (1987) and the streaming model for nonlinear scales Peebles (1980), the so-called Gaussian streaming model (GSM) was introduced in Fisher (1995). To obtain the GSM, the matter correlation function in redshift space was derived by considering the joint probability distribution of density and velocity. Assuming that the density is a Gaussian random field and the velocity is related to density as in linear perturbation theory one obtains a simple expression for the redshift space correlation function. It is given by a convolution of the real space correlation function and an approximately Gaussian pairwise velocity distribution whose mean and variance are given by the scale-dependent mean and variance of the pairwise velocity. The GSM obtained via this approach can be understood as generalization of the streaming model originally introduced in Peebles (1980) to a scale-dependent rather than constant velocity dispersion which correctly reproduces the linear theory result Kaiser (1987). The GSM, derived for the special case of Gaussian fluctuations in Fisher (1995), has been generalized to fully non-Gaussian fields in Scoccimarro (2004). Furthermore, therein a connection between the redshift-space clustering and the pairwise velocity moments has been established.

Furthermore, it has been shown recently in Bianchi et al. (2015) that the assumption that the pairwise velocity distribution is locally Gaussian, with its mean and variance themselves Gaussian distributed allows to accurately recover the non-Gaussian pairwise velocity distribution measured in simulations. This approach is different from the one presented here, where we assume that the mean and variance are not random variables but functions of separation that are either determined from data or inferred from theory.

We start from a phase space distribution function for dark matter or its tracers, similar as done in Vlah et al. (2012, 2013). Indeed, our formulation relates the distribution function approach studied in Fourier space in Vlah et al. (2012, 2013) and the Gaussian streaming model for redshift space distortions operating in configuration space. We decide to work in configuration rather than Fourier space. A practical reason is that our formulation of perturbation theory will naturally produce expressions in real space. Another argument is the fact that small spatial scales in the correlation function can be strongly affected by late-time baryonic physics, while large scales, most importantly the baryon acoustic oscillations (BAO) peak, are not affected, see Angulo et al. (2014). Therefore although late-time baryonic physics is confined to small , it appears spread out in Fourier space. A generalized dispersion model, taking nonlinear couplings between density and velocity fields into account, has been proposed in Taruya et al. (2013) to provide consistent predictions for power spectra and correlation functions at the same time.

To predict halo correlation functions in redshift space the GSM has been combined with perturbation theory to extract the streaming model ingredients, namely the real space correlation and the mean pairwise velocity and its dispersion, in Reid and White (2011); Wang et al. (2014); White (2014). A test of different analytic and phenomenological streaming models combined with perturbation theory, performed in White et al. (2015), showed that they reasonably fit the simulations on intermediate scales while all models fail at small scales with Lagrangian schemes having the best performance around the scale of BAO.

It is well known that no perturbative framework is able to accurately describe the fully nonlinear regime of structure formation. Fortunately, dark matter halos and their progenitors, which we denote by proto-halos, can themselves be treated as large cold dark matter (CDM) particles and therefore described by a pressureless dust fluid. The motion of these proto-halos is mostly determined by the large scale gravitational field and therefore much better describable with perturbation theory. The pressureless CDM fluid is described by a coupled system of differential equations consisting of continuity, Euler and Poisson equations. These equations can be solved perturbatively – either in the Eulerian frame (SPT)Bernardeau et al. (2002) where everything is expanded in terms of density and velocity or in the Lagrangian frame (LPT) Buchert (1992) where fluid-trajectories or displacement fields are considered. It is clear that the fluid description should be applied only on scales larger than the particle size, in case of proto-halo “particles” this is the Lagrangian size of the halos. Therefore it is natural to implement the Lagrangian halo size as a physically meaningful coarse-graining scale into the fluid description for (proto-)halos Bond and Myers (1996). This approach is to be seen in contrast to the so-called effective field theory of LSS Porto et al. (2014) for dark matter where the dependence of dark matter properties on the smoothing scale is unphysical and removed through renormalization. In order to model the trajectories of proto-halos we study in this paper a coarse-grained dust model in terms of the displacement field within Lagrangian perturbation theory (LPT).

A big advantage of Lagrangian schemes Buchert (1992) is the clearer physical picture they offer for the study of halo correlation functions, which are a key ingredient of the halo model Cooray and Sheth (2002) that is widely used in the analysis of galaxy, cluster and lensing surveys. In order to understand halo correlations one needs to understand the bias between the halo field and the underlying dark matter field. But halo bias is best understood using the spherical collapse model and excursion set theory Bond et al. (1991); Lacey and Cole (1993), both of which operate in the initial conditions and therefore in Lagrangian space, where they locally identify proto-halos within the initial density field and assign mass and collapse time to them. Therefore once the clustered or biased field of proto-halos is known it can be propagated to Eulerian space using a Lagrangian method. Another advantage of Lagrangian methods concerns the convergence properties and the accuracy of the correlation function on the scales of interest, like the BAO scale or the mildly nonlinear scales. It is known that LPT performs much better on those scales, see the first Figure of Tassev (2014); a higher precision is achieved with a smaller order in perturbation theory. The better convergence properties of the LPT displacement field compared to standard perturbation theory (SPT) in Eulerian space are mainly due to fact that the relation between the density contrast and the displacement field is nonlinear and can be handled nonperturbatively.

In first order LPT it is possible to analytically compute the density correlation function from the first order displacement field in a nonperturbative fashion which is called Zel’dovich approximation (ZA), see Zel’dovich (1970). In the ZA particles are displaced along straight trajectories, parametrized by the linear growth function, in a direction determined by their initial velocity. Despite its simplicity, the ZA is capable of accurately describing gravitational dynamics over a surprisingly wide range of scales Coles et al. (1993); Tassev (2014). In Coles et al. (1993) the so-called truncated Zel’dovich approximation (TZA) was proposed as phenomenological method to improve the agreement between Zel’dovich and proper -body simulations by artificially smoothing the initial power spectrum at the nonlinear scale of the final time of the simulation. The effect of the smoothing is to decrease the velocity in high density regions thereby reducing the amount of shell-crossing events and subsequent erasure of overdensities. Therefore, counterintuitively, smoothing the initial power spectrum, which reduces the initial power on small scales, actually can increase the final power on those scales. Focusing on statistical properties of the nonlinearly evolved density field like the power spectrum, the TZA amounts to smoothing the linear initial power spectrum without affecting the dynamics itself. A detailed study and comparison between different filters in Melott et al. (1994) revealed that a Gaussian filtering scheme leads to best agreement with -body data and considerable improvement over sharp k-truncation as originally suggested in Coles et al. (1993) and top-hat filtering as studied in Pietroni et al. (2012).

It is known that the Post-Zel’dovich approximation (PZA), where the displacement fields are calculated from second order LPT, improves over the ZA. Accordingly, the truncated Post-Zel’dovich approximation (TPZA) with a smoothed initial power spectrum performs even better than TZA, compare Buchert et al. (1994); Weiss et al. (1996). We apply the framework of Convolution Lagrangian perturbation theory (CLPT) developed in Carlson et al. (2013) which recovers the ZA at lowest order while providing an approximation to PZA at higher order. CLPT can be understood as a partial resummation of the formalism presented in Matsubara (2008) providing a nonperturbative resummation of LPT that incorporates nonlinear halo bias. We will compare two different smoothing approaches within CLPT, namely a coarse-graining in Eulerian space (cgCLPT) with a coarse-graining in Lagrangian space implemented by smoothing the initial power spectrum in the spirit of the truncated Zel’dovich approximation (TCLPT). Those two procedures are distinct since a coarse-graining in Eulerian space also modifies the underlying dynamics becoming manifest beyond linear order in Lagrangian space, see Uhlemann and Kopp (2015), while our coarse-graining in Lagrangian space only affects the initial conditions.

Structure

This paper is organized as follows: In Sec. II we derive the Edgeworth streaming model (ESM) for the redshift space correlation function starting from an arbitrary distribution function for biased tracers of dark matter by considering its two-point statistics and show that it reduces to the Gaussian streaming model (GSM) when neglecting non-Gaussianities in the pairwise velocity distribution. We then demonstrate the accuracy of the GSM and ESM on the basis of -body simulations employing the Horizon Run 2 halo catalog. In Sec. III we built up on existing work and describe how the ingredients of the streaming models can be inferred from the dust model and propose two different coarse-grained generalizations of the fluid description. In Sec. IV we compute the real-space halo correlation function and the halo velocity statistics for the dust model employing Convolution Lagrangian Perturbation Theory (CLPT) with two different coarse-graining schemes, an Eulerian (cgCLPT) and a Lagrangian (TCLPT) one. We conclude and describe possible further interesting lines of study in Sec. V. A list of abbreviations commonly used within this work can be found in App. A.

Ii Edgeworth Streaming Model

In order to infer predictions for the halo correlation function in redshift space we use the Gaussian streaming model, originally derived in Fisher (1995) and studied in Reid and White (2011) for the dust model. Starting from an arbitrary distribution of proto-halos we present a self-contained derivation of the Gaussian streaming model (GSM) from general assumptions which allows to include non-Gaussian corrections leading to the Edgeworth streaming model (ESM). We test the accuracy of the GSM and ESM using -body simulation data from the Horizon Run 2 (HR2) Kim et al. (2009, 2011) independent of perturbation theory. We then describe in Sec. III how the ingredients of the streaming models can be inferred from the dust model and it’s coarse-grained generalization and present the CLPT computation and results in Sec. IV.

ii.1 Derivation of the ESM

Let the phase space distribution function of dark matter tracers (like galaxies, clusters or halos) be given by . In this section we do not make any assumptions about its dynamics or statistical properties apart from that it is spatially statistically homogeneous

(1)

In addition we assume that the tracer density field

(2)

and higher moments are statistically homogeneous and isotropic.

The observed position of a tracer – its angle on the sky and its observed redshift – corresponds to a point on the observer’s past light cone. As a first step towards calculating tracer correlations on the past light cone, we will make two common simplifying assumptions. First, since we are interested in equal-time correlation functions, we will approximate the light cone in the neighbourhood of by the =const slice. Secondly, we use the distant observer approximation, where the line of sight is assumed to be a fixed direction which is without loss of generality chosen as the direction of the axis, to relate the observed redshift-space position of a dark matter tracer to its real-space position . Those approximations are despite their simplicity sufficient even for modern wide-area surveys within the level of current error bars, see e.g. Fig. 10 in Samushia et al. (2012). For a general definition of redshift space and a discussion of wide-angle effects in linear perturbation theory we refer to Matsubara (2000). In the distant observer approximation the observed comoving distance in redshift space is affected by the peculiar velocity of the tracer along the line of sight via

(3a)
where and . The observed position of the tracer perpendicular to the line of sight remains unaffected if we neglect gravitational lensing. In contrast, its coordinate parallel to the line of sight depends on the peculiar velocity
(3b)

Since objects cannot disappear going from real space to redshift space (assuming that all objects remain observable) we have the following relation between the densities in real and redshift space

(4)

Although the correction to the real space position in redshift space is very small , the clustering is affected considerably since the change of volume measure between real and redshift space, given by the Jacobian between and , involves the gradient of in linear perturbation theory Kaiser (1987). In the distant observer approximation, the tracer density fluctuation in redshift space (4) can be equivalently written as

(5)

which holds even for the case where the tracer velocity is not a single valued function of but instead has multiple streams or a continuous distribution, see also Seljak and McDonald (2011). Later, in Sec. III, we will consider the special case of single streaming tracers described by the dust model for which this relation simplifies to (26a).

We are interested in the redshift space two-point correlation function

(6)

where . By inserting (5) in (6) and re-expressing the delta functions in Fourier space and integrating over and one momentum variable the correlation function can be brought into the following form

(7a)
(7b)

where is the pairwise generating function. Next we Taylor expand around

(8)

Keeping only the terms up to third order we obtain

(9a)
(9b)

with the cumulants as expansion coefficients

(10a)
(10b)
(10c)
(10d)

where . Since we have to evaluate all expressions at we project the cumulants onto the line of sight (13). Expanding in (9a) up to second order in implies that all redshift space distortion induced clustering is encoded in the scale dependent mean and variance given by the pairwise velocity and its dispersion . As we will shortly see, this corresponds to the Gaussian streaming model (GSM). Since the GSM is known to be a good approximation, we will perform an expansion around this Gaussian

This approach is similar to the idea behind Convolution Lagrangian perturbation theory (CLPT), see Carlson et al. (2013). To obtain the Gaussian streaming model it is crucial to expand in cumulants and keep the pairwise velocity mean and dispersion in the exponent, corresponding to specific resummation of moments. Within the distribution function approach to redshift space distortions developed in Seljak and McDonald (2011); Vlah et al. (2012, 2013) a moment expansion without such an resummation was performed such that the connection to the Gaussian streaming model is not manifest and has not been discussed.

Later when testing the accuracy of this model, we will restrict ourselves to the leading order non-Gaussian term. However one can systematically expand the exponential of the non-Gaussian contributions to in an Edgeworth expansion Bernardeau and Kofman (1995); Juszkiewicz et al. (1995) around a Gaussian pairwise velocity probability distribution. The Edgeworth series is an asymptotic expansion to approximate a probability distribution using its cumulants . With the Gaussian distribution as reference function it can be written as, see Eq. (43) in Blinnikov and Moessner (1998),

(11a)
where are the normalized and rescaled cumulants
(11b)
the Bell polynomials
(11c)
and the probabilists’ Hermite polynomials
(11d)

In the following we perform the Edgeworth expansion up to explicitly taking into account the first non-Gaussian correction given by the pairwise velocity skewness . Kurtosis would arise in the next order but won’t be considered in this paper.

We can now plug the Edgeworth expansion (11a) of according to Eq. (9) into the correlation function Eq. (7a). In the course of the calculation we will use cylindrical coordinates

since does not depend on the angle . Performing five of the six integrals in Eq. (7a) we obtain the Gaussian streaming model (GSM) Eq. (12a) at second order in the cumulant expansion and the leading and up to order corrections of the Edgeworth streaming model (ESM) (12b), (12c) at third and -th order, respectively

(12a)
(12b)
(12c)

In more detail, the integral in Eq. (7a) introduces the pairwise probability distribution multiplied by , while the trivial integrals enforce . The integral ensures , while the -integral gives a factor of . We defined

(13a)
(13b)
(13c)
(13d)

In a previous study of the GSM Reid and White (2011), the following formula, inspired by the exact result from Fisher (1995) for the case where both density and velocity fields are Gaussian and related to one another as in linear theory, was suggested to calculate Gaussian streaming redshift space distortions

(14)

where our and corresponds to and used in Reid and White (2011); Wang et al. (2014), respectively. Note that, (II.1) corresponds to (12a) when the variance, given by the second pairwise velocity moment , is replaced by the pairwise velocity dispersion . The two quantities are related via Eq. (13c) such that and . By expanding one obtains the GSM (12a) with the second cumulant as variance whereas when expanding one obtains the GSM (II.1) with the second moment as variance. When linearized, both expressions (12a) and (II.1) agree, because is second order, and correctly reproduce the Kaiser formula as shown in Fisher (1995); Reid and White (2011).

Figure 1: Comparison between the Horizon Run 2 measurement (data points) of the multipoles of the redshift space correlation function defined in (15) and the GSM (12a) using (thick dashed) or (thin dashed) related via (13c). upper panel The quadrupole shifted by where labels the mass bin. middle panel The hexadecapole shifted by . lower panel The hexacontatetrapole (64-pole) shifted by .

It is natural to follow an expansion in and to keep only the Gaussian part in the exponential in case the pairwise velocity distribution is close to a Gaussian. On the other hand the moment expansion of Seljak and McDonald (2011) is natural from a perturbation theory perspective, in which only moment spectra are kept that are nonzero up to certain order in perturbation theory.

That the pairwise distribution function is indeed approximately Gaussian with a variance given by rather than becomes clear in Fig. 1, where we compare the GSM with the second cumulant (12a) to the GSM with the second moment (II.1) as the variance of the Gaussian. The exact definition of the redshift space multipoles depicted in Fig. 1 and the reason for their normalization will be given in the next subsection. As we can clearly see the use of the second cumulant significantly improves the agreement for the redshift space distribution function with the -body simulation compared to the model where the second moment is used. In Reid and White (2011) it has been phenomenologically accounted for that difference by subtracting the square of the mean infall from to get the dispersion about the mean. We leave it for future work to directly compare the ESM to the distribution function approach Vlah et al. (2013).

ii.2 Accuracy of the GSM and ESM

Figure 2: The redshift-space multipoles , and at predicted by the GSM (12a) (thin dashed) and ESM (12b) (thick dashed) using the HR2 data compared to the direct HR2 measurement (data points) normalized with respect to their bias factors for the different mass bins. Similar to previous plots we added a mass bin dependent constant to all curves for better visibility. upper left panel The monopole shifted by . upper middle and right panel The quadrupole shifted by . lower panel The hexadecapole shifted by .
Figure 3: The ratio of the GSM (12a) (lower panel) and ESM (12b) (upper panel) prediction using the HR2 data and the direct HR2 measurement (data points) for the lowest three redshift-space multipoles. left panel The monopole middle panel The quadrupole . right panel The hexadecapole .

In the following, we assess the accuracy of the GSM (12a) and the leading order of the ESM (12b) by comparing the results of the corresponding integrals (12) with the directly measured redshift space halo correlation function . This is done by inserting the real space correlation , the pairwise velocity and velocity dispersion measured in an -body simulation into Eq. (12a) and additionally measuring the skewness and plugging it into Eq.  (12b).

The Horizon Run 2 (HR2) -body simulation Kim et al. (2009, 2011) has an enormous size of 7200 Mpc/ and consists of particles of mass . For the mass units we use the notation . We measured halo correlation functions and velocity statistics from large galaxy-sized haloes to cluster-sized halos at the redshift . In an accompanying work Kopp et al. (2015) we describe in detail how the correlation functions and Gaussian streaming ingredients haven been determined from the HR2 halo catalog. In order to evaluate and compare from Eq. (12) to simulations it is useful to expand into Legendre polynomials using and

(15a)
(15b)

vanishes for all odd . In linear perturbation theory, the only non-zero multipoles are the monopole , quadrupole and hexadecapole and even in the nonlinear regime the magnitude of rapidly decreases with . The linear results go back to Hamilton (1992) and are given in Reid and White (2011) for the case of Eulerian bias as

(16)

where is the linear growth rate and are the spherical Bessel functions. We use their prefactors, given in terms of linear local Eulerian bias determined from the best fitting mass for the real space correlation function, see Tab. 1, as a normalization when plotting multipoles. In Fig. 2 we compare the redshift space halo correlation function predicted from the GSM (12a) and ESM (12b), by measuring their ingredients from the HR2 data, to the direct measurements within HR2 for the redshift-space monopole , quadrupole and hexadecapole . We find that the ESM (12b) clearly improves the quadrupole and hexadecapole on small scales compared to the GSM (12a). As evident from Fig. 3 the quadrupole predicted by ESM (12b) is accurate to 2% within statistical errors down to in contrast to the GSM (12a) which breaks down below . A similar trend can be observed for the hexadecapole whereas the monopole is less sensitive to non-Gaussian terms. Apparently smaller halos are more sensitive to non-Gaussian corrections which is in line with the expectation that smaller objects are more affected by nonlinear dynamics.

We conclude that the GSM is a very accurate model for the multipoles , of the redshift space halo correlation function on scales larger than while the ESM stays accurate down to . Our result is consistent with the previous finding that the GSM monopole is accurate on the percent level down to and the quadrupole down to , compare Fig. 6 in Reid and White (2011). This shows that the expansion of around was justified and that halos over a wide range of masses can indeed be reasonably described by the GSM/ESM (12a/12b).

Having established the range of validity of the streaming models GSM and ESM, we can use them as a basis for the theoretical modeling of redshift space halo correlation functions being aware of their limitations. As a next step, accurate theoretical predictions for the streaming model ingredients, , , and are needed. In the following two sections we will combine the GSM/ESM with perturbation theory employing that halos can be treated as single-streaming objects when the fluid description is only applied on scales larger than their size, given by the Lagrangian radius. More precisely, we will calculate the streaming model ingredients from Convolution Lagrangian Perturbation Theory (CLPT) based on the dust model and extend it to include a coarse-graining scale chosen to be the Lagrangian radius.

ii.3 Pairwise generating and tracer cumulants

By performing a cumulant expansion we can relate the term contained in the exponential of the generating function from (7b) to the cumulants of the tracer distribution function . Therefore we introduce the moment generating functional

(17)

which allows to compute the cumulants of the distribution function according to

(18)

This can be used to re-express from (7b)

(19)

where , and .

Iii Determining Streaming model ingredients based on the dust model

In this Section we describe how the scale-dependent functions entering the streaming model can be determined once a phase-space distribution of the tracers is specified. To draw conclusions based on theoretical modeling it is due to connect the (proto-)halo distribution to an underlying dark matter distribution whose dynamics is known to be governed by the Vlasov-Poisson equation. Halos are biased tracers of dark matter, since according to spherical collapse and excursion set theory Bond et al. (1991); Mo and White (1996); Catelan et al. (1998), the probability of forming a halo depends on the initial density field. Therefore, there are two steps for determining streaming model ingredients:

  1. Choose a model for the distribution function of dark matter that reasonably approximates Vlasov dynamics.

  2. Specify a bias model in order to relate the (proto-)halo cumulants to the ones of dark matter .

In the following we employ the pressureless fluid model as standard model for cold dark matter and discuss different possibilities to incorporate a coarse-graining in this fluid picture. For relating the halo to the dark matter density we use local Lagrangian bias with zero velocity bias and present two possibilities to generalize this notion to higher cumulants.

iii.1 The single-stream case: dust model

In the context of analytical modellng CDM dynamics, usually the dark matter distribution is assumed to be described by the pressureless fluid (dust) model

(20)

which encodes all properties in terms of a number density and a single-streaming and curl-free velocity fulfilling the coupled continuity, Euler and Poisson equations Bernardeau et al. (2002). The cumulants of the dust model are

(21)

which displays that the dust model is entirely described by density and velocity and all higher cumulants such as velocity dispersion vanish identically. Although the dust model is an exact solution of the Vlasov equation, its applicability is limited to the single-stream regime. It does not allow to describe the nonlinear stage of structure formation during which higher cumulants are sourced by the occurence of shell-crossing after which multiple streams form. For (proto-)halos this limitation is not as severe since they approximately behave as single-streaming objects even though a large fraction of dark matter particles resides in halos where it is multi-streaming and not accessible by the dust model. Hence, the proto-halos can also be described in terms of a single-streaming dust fluid

(22)

To connect the density of halos to the dark matter density, we assume local Lagrangian bias

(23)

This equality states that proto-halos identified in the linear initial conditions, depending only on the smoothed initial linear density field , are conserved until they form a proper halo at time . The proto-halo initial density field is assumed to be a local function of the initial linear density field smoothed over some scale related to the Lagrangian size of the proto-halo by applying a window function in Fourier space

(24)

The choice of the appropriate smoothing scale will be elaborated in more detail in a forthcoming paper Kopp et al. (2015). Note that for the computations in both iPT and CLPT this smoothing scale is effectively removed by setting the window function to unity. In Matsubara (2008) this is justified by claiming that the large-scale clustering of biased objects should not depend on the artificial choice of to define the background field and seconded by the assertion that this is demanded by consistency with the approximation being valid only on scales larger than the smoothing radius . In Wang et al. (2014) it is furthermore argued that naturally drops out in the final statistics of interest and is only necessary to keep intermediate quantities well-behaved. We will preserve the smoothing and see in Sec. IV, in particular Figs.  4 and 5 how large the effect on the baryon acoustic peak is when a smoothing at the Lagrangian scale is performed compared to the case where the smoothing is dropped.

The tracer velocity field displaces the proto-halos to their halo virialization sites . We assume zero velocity bias such that proto-halos move along dust fluid trajectories with the dust velocity

(25a)
Local Lagrangian bias (23) allows us to relate the densities in real and Lagrangian space in the following way
(25b)

Due to the single-valuedness of the velocity we recover from inserting (25a) into (4) a simpler relation between densities in real and redshift space

(26a)
We could combine both relations in a single expression by expressing the velocity in terms of the displacement
(26b)

Note however that we will not use formulas (26) explicitly. Instead we will rely on the GSM (12) to go from real space to redshift space and Eq. (28) below, to go from Lagrangian to Eulerian space.

Derivation

We already showed in (12a) that one can obtain the GSM (II.1) from quite general assumptions, in particular that no assumptions about tracer dynamics and bias are required. Now, we will specialize from (19) to the dust ansatz (25) for the tracer phase-space distribution combined with local Lagrangian bias (23) as considered in Wang et al. (2014).

First, we use the dust model cumulants (21) applied to the proto-halo distribution from CLPT (25) and plug them into the general expression for in terms of tracer cumulants (19) obtaining

We then switch to Lagrangian space making use of local Lagrangian bias (25b) and express the bias function

(27)

as well as the delta function in Fourier space. Next we replace the single streaming velocity by the derivative of the displacement field and integrate over to obtain

(28a)
with
(28b)

where , and .

Previous studies

Originally, in Carlson et al. (2013), Eq. (26b) was used to derive an expression for the two-point correlation function

(29)

which was then evaluated within CLPT to obtain a Post-Zel’dovich approximation for biased tracers in redshift space. The formula (II.1) was suggested in Reid and White (2011) to calculate Gaussian streaming redshift space distortions, following the idea of Fisher (1995) to reconcile the streaming model Peebles (1980) for nonlinear scales with linear theory Kaiser (1987) by considering a scale-dependent variance. In Reid and White (2011), the pairwise velocity mean and second moment entering the streaming model (II.1) were calculated from SPT with linear bias while the real space correlation was inferred from LPT with local Lagrangian bias (25b). Later on, in Wang et al. (2014), the real space correlation and velocity statistics were treated on the same footing and determined within CLPT Carlson et al. (2013) together with local Lagrangian bias. Note that (29) involves a three-dimensional -integral which needs to be evaluated numerically within CLPT Carlson et al. (2013). Studying the expression (28) for in CLPT, the streaming model ingredients can be calculated according to (10) and involve at most two-dimensional numerical integrals Wang et al. (2014). This a practical reason to chose to perform an Edgeworth expansion of to obtain the Gaussian streaming model (12a) and its non-gaussian generalization – the Edgeworth streaming model – whose numerical evaluation is more efficient than the full CLPT expression (29).

iii.2 Beyond single-stream: coarse-graining the dust model

In the following we compare several distinct approaches of coarse-graining a dust fluid, namely a coarse-graining in Eulerian space (cgCLPT) and a coarse-graining in Lagrangian space implemented by smoothing the initial power spectrum in the spirit of the truncated Zel’dovich approximation (TCLPT). A key question is how to generalize the biasing scheme employed for CLPT based on the dust model to the coarse-grained case. So far we assumed local Lagrangian bias for the density (25b) and zero velocity bias. This might be generalized by (a) assuming zero velocity bias and that higher cumulants for the tracer vanish identically motivated by the fact that proto-halos can be described well by single-stream physics such that, in analogy to the CLPT case,

(30a)
or (b) assuming that tracers and dark matter are only biased with respect to density such that all higher tracer cumulants are identical to those of dark matter and
(30b)

Note that in order to write the biasing in analogy to the dust case (28a) it is necessary that is independent of the bias function which is achieved by both relations (30). Then, the redshift space correlation takes the form

(31a)
with
(31b)

where . Hence, the ESM ingredients are still computed according to Eqs. (10) with from (28) replaced by from (31). If we consider the GSM, expanding up to second order in , we see that the first cumulant corresponds to the term that is also present in the single streaming Gaussian streaming model (28b) and contributes both to the mean and variance of the Gaussian. In contrast, the second cumulant is conceptually new and contributes only to the variance of the Gaussian, whereas all higher cumulants are irrelevant for the GSM but only contribute to the ESM.

iii.2.1 Coarse-graining in Eulerian space (cgCLPT)

Coarse-graining the dust model on a length scale in Eulerian space and a velocity scale gives rise to the so-called coarse-grained dust model as described in detail in Uhlemann and Kopp (2015). We shortly recap the main results that are of direct relevance here. The coarse-grained dust model is defined as a smoothing of the dust phase space distribution with a Gaussian filter of width and in and space, respectively

(32)

If and are the (minimal) scales of interest we have to ensure that and in order to be able to resolve these scales. The coarse-grained dust model features higher cumulants which are absent in the pressureless fluid case and given by

(33a)
(33b)