Distribution function approach to redshift space distortions. Part II: N-body simulations

Abstract

Measurement of redshift-space distortions (RSD) offers an attractive method
to directly probe the cosmic growth history of density perturbations. A
distribution function approach where RSD can be written as a sum over density
weighted velocity moment correlators has recently been developed. We use Nbody
simulations to investigate the individual contributions and convergence of this
expansion for dark matter. If the series is expanded as a function of powers of
mu, cosine of the angle between the Fourier mode and line of sight, there are a
finite number of terms contributing at each order. We present these terms and
investigate their contribution to the total as a function of wavevector k. For
mu^2 the correlation between density and momentum dominates on large scales.
Higher order corrections, which act as a Finger-of-God (FoG) term, contribute
1% at k~0.015h/Mpc, 10% at k~0.05h/Mpc at z=0, while for k>0.15h/Mpc they
dominate and make the total negative. These higher order terms are dominated by
density-energy density correlations which contribute negatively to the power,
while the contribution from vorticity part of momentum density auto-correlation
is an order of magnitude lower. For mu^4 term the dominant term on large scales
is the scalar part of momentum density auto-correlation, while higher order
terms dominate for k>0.15h/Mpc. For mu^6 and mu^8 we find it has very little
power for k<0.15h/Mpc. We also compare the expansion to the full 2D P^ss(k,mu)
as well as to their multipoles. For these statistics an infinite number of
terms contribute and we find that the expansion achieves percent level accuracy
for kmu<0.15h/Mpc at 6th order, but breaks down on smaller scales because the
series is no longer perturbative. We explore resummation of the terms into FoG
kernels, which extend the convergence up to a factor of 2 in scale. We find
that the FoG kernels are approximately Lorentzian.

Full text

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arXiv:1109.1609v1 [astro-ph.CO] 8 Sep 2011
Prepared for submission to JCAP
Distribution function approach to
redshift space distortions:
N-body simulations
Teppei Okumura,aUroˇ s Seljak,a,b,c,dPatrick McDonald,b,eand
Vincent Desjacquesd
aInstitute for the Early Universe, Ewha Womans University, Seoul 120-750, S. Korea
bDepartment of Physics and Lawrence Berkeley National Laboratory, University of California, Berke-
ley, California 94720, USA
cDepartment of Astronomy, University of California, Berkeley, California 94720, USA
dInstitute of Theoretical Physics, University of Zurich, 8057 Zurich, Switzerland
ePhysics Dept., Brookhaven National Laboratory, Building 510A, Upton, NY 11973-5000, USA
E-mail: teppei@ewha.ac.kr, useljak@berkeley.edu, pvmcdonald@lbl.gov,
dvince@physik.uzh.ch
Abstract. Measurement of redshift-space distortions (RSD) offers an attractive method to directly
probe the cosmic growth history of density perturbations. A distribution function approach where
RSD can be written as a sum over density weighted velocity moment correlators has recently been de-
veloped. In this paper we use results of N-body simulations to investigate the individual contributions
and convergence of this expansion for dark matter. If the series is expanded as a function of powers
of µ, cosine of the angle between the Fourier mode and line of sight, then there are a finite number
of terms contributing at each order. We present these terms and investigate their contribution to the
total as a function of wavevector k. For µ2the correlation between density and momentum dominates
on large scales. Higher order corrections, which act as a Finger-of-God (FoG) term, contribute 1% at
k ∼ 0.015 h Mpc−1, 10% at k ∼ 0.05 h Mpc−1at z = 0, while for k > 0.15 h Mpc−1they dominate
and make the total negative. These higher order terms are dominated by density-energy density
correlations which contributes negatively to the power, while the contribution from vorticity part of
momentum density auto-correlation adds to the total power, but is an order of magnitude lower. For
µ4term the dominant term on large scales is the scalar part of momentum density auto-correlation,
while higher order terms dominate for k > 0.15 h Mpc−1. For µ6and µ8we find it has very little
power for k < 0.15 h Mpc−1, shooting up by 2-3 orders of magnitude between k < 0.15 h Mpc−1and
k < 0.4 h Mpc−1. We also compare the expansion to the full 2-d Pss(k,µ), as well as to the monopole,
quadrupole, and hexadecapole integrals of Pss(k,µ). For these statistics an infinite number of terms
contribute and we find that the expansion achieves percent level accuracy for kµ < 0.15 h Mpc−1at
6-th order, but breaks down on smaller scales because the series is no longer perturbative. We explore
resummation of the terms into FoG kernels, which extend the convergence up to a factor of 2 in scale.
We find that the FoG kernels are approximately Lorentzian with velocity dispersions around 600km/s
at z = 0.
Keywords: galaxy clustering, power spectrum, redshift surveys
ArXiv ePrint: 1109.XXXX

Page 2

Contents
1Introduction
1
2Redshift-space distortions from the distribution function
2.1Angular dependence
2
3
3 Numerical analysis
3.1N-body simulations
3.2 Matter power spectrum
3.3Legendre expansion
3
3
4
5
4Fingers-of-God resummation
4.1Testing the Fingers-of-God model
4.2Mass-weighted vs volume-weighted velocities
7
10
12
5Expansion in powers of µ2
5.1µ2terms
5.2µ4terms
5.3Higher order terms
13
14
14
14
6Conclusions14
A Derivatives of FoG kernels from higher-order PLL′’s
16
1Introduction
Galaxy redshift surveys are one of the most powerful tools to probe cosmological models [1]. One way
to extract the information is from the shape of the power spectrum or correlation function, assuming
it traces the underlying dark matter. Another method involves baryonic acoustic oscillations (BAOs),
detected in various redshift surveys, which enables one to measure angular diameter distance and
compare it to the same quantity measured in cosmic microwave background. This in turn probes the
expansion history of the Universe and allows to study the nature of dark energy [e.g., 2, 3]. Third
piece of information in redshift surveys comes from redshift space distortions (RSD): the observed
galaxy distribution is distorted along the line of sight due to the Doppler shifts caused by peculiar
velocities [4–6]. In linear theory this allows one to measure the rate of growth of structure, which
allows for another way to measure the matter content of the universe, including the amount and
nature of dark energy. The last two methods are complementary: cosmological models in different
gravity theories with the same expansion history cannot be distinguished by the distance scales of
BAOs, but can if growth of structure is also measured [e.g., 7–9]. Additional information is obtained
by Alcock-Paczy´ nski test [10–12].
RSD have been analyzed in many galaxy surveys to determine the cosmological models [e.g.,
13–22]. However, it was shown by [23–26] that the growth rate reconstructed from the redshift-space
distortions can have scale dependent biases, which indicate a breakdown of linear theory predictions.
These effects show up on relatively large scales, suggesting one must go beyond the linear theory in
the analysis of RSD. This will become even more important in the future, with several ongoing and
upcoming galaxy surveys that will measure RSD to a high precision [27–31].
Given the high precision of the future surveys, correspondingly more accurate theoretical pre-
dictions become essential for their interpretation. As was emphasized by [32], there are important
nonlinear effects that need to be addressed in order to achieve accurate theoretical predictions. In
order to account for the nonlinearity of the gravitational evolution, standard perturbation theory has
long been used to describe the power spectrum at quasi-nonlinear scales [e.g., 33]. Recently there have
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been many studies to predict the power spectrum in nonlinear regime beyond the framework of the
standard perturbation theory (SPT) [34–39]. These different approaches were compared to the N-
body simulations in [40, 41]. Similarly, initial RSD work was based on the lowest order SPT [42–45].
However, as pointed out by [32, 43], SPT in redshift space breaks down at larger scales than in real
space because of nonlinear redshift distortion effects, sometimes called Finger-of-God (FoG) effect [4].
Recent development using more sophisticated perturbation methods applicable to the redshift-space
power spectrum includes [46–48]. [22] present detailed comparisons of these predictions with the
observed galaxy data.
Recent paper [49] has shown that one can write the Fourier mode of density in redshift space
as a sum over mass weighted moments of radial velocity, which are integrals of powers of velocity
over the momentum part of the phase space distribution function. The corresponding RSD power
spectrum can be written as a sum over auto and cross-correlators of these moments. This series
always converges for sufficiently small expansion parameter defined below. We will use the Fourier
description in this paper and scale is expressed in terms of wavevector amplitude k, while angular
dependence is expressed in terms of µ, cosine of the angle between the line of sight and Fourier mode.
The expansion parameter depends on the product of the two k?= kµ. It has been shown in [49]
that the moments can be decomposed into helicity eigenstates, which are eigenmodes under rotation
around direction of k vector. Only equal helicity eigenstates correlate, leading to a specific angular
structure of the correlators. This analysis shows that if one expands the series into powers of µ2,
a finite number of terms contribute at each (finite) order. This suggests that RSD can be better
understood in terms of this expansion rather than the Legendre moments usually used [49]. On the
other hand, Legendre moments are uncorrelated in real observations, while powers of µ2are not,
leading to correlations between the higher and lower orders. We will pursue both approaches here.
This is the second paper in a series studying the redshift-space distortions based on a distribution
function approach, following the theory and angular decomposition presented in [49]. In this paper
we test the formalism to describe the redshift-space power spectrum in nonlinear regime using a
large set of cosmological N-body simulations, as well as present the individual terms of expansion for
comparison against each other. We focus on the dark matter in this paper, leaving the application
to halos and galaxies to future work. The structure of this paper is as follows. In section 2 we
briefly describe the distribution function approach to RSD. Then we apply it to simulations to test
this expansion and show the contributions from individual terms: in section 3 we first show the
contributions to the 2-d power spectrum in redshift space, then proceed to Legendre moments. We
discuss the FoG modeling in section 4 and present an attempt to compare the expansion to one in
terms of volume weighted quantities. Finally in section 5 we apply the method to powers of µ2
expansion, which we argue is a more natural way to expand 2-d information, showing individual
contributions to the lowest order terms. This is followed by conclusions in section 6.
2Redshift-space distortions from the distribution function
The exact evolution of collisionless particles is described by the Vlasov equation [1]. Following the
discussion by [50], we start from the distribution function of particles f(x,q,t) at phase-space position
(x,q) in order to derive the perturbative redshift-space distortions. Here x is the comoving position
and q = p/a is the comoving momentum (p is the proper momentum). The density field in redshift
space is related to moments of distribution function as
δs(k) =
?
L=0
1
L!
?ik?
H
?L
TL
?(k) ,(2.1)
where H is the Hubble parameter and TL
?(k) is the Fourier transform of TL
?(x), defined as
TL
?(x) =m
¯ ρ
?
d3q f (x,q)uL
?=
?
(1 + δ(x))uL
?(x)
?
x,(2.2)
where u?is the radial comoving velocity, mu?= q?= q· ˆ r, m is the particle mass, ˆ r is the unit vector
pointing along the observer’s line of sight and ¯ ρ is the mean mass density.
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The power spectrum in redshift space is given by [49]
Pss(k) =
∞
?
L=0
1
L!2
?kµ
H
?2L
PLL(k) + 2
∞
?
L=0
?
L′>L
(−1)L
L! L′!
?ikµ
H
?L+L′
PLL′(k) ,(2.3)
where k||/k = cosθ = µ.
It is useful to compare this to Kaiser’s linear theory prediction [5, 32]. Thus we have
Pss
Kaiser(k) =



?1 + fµ2?2Plin(k);linear,
P00+ 2fµ2?
ik
Hµf
?
P01+ f2µ4?
k
Hµf
?2
P11 ;nonlinear,
(2.4)
These terms will in general have nonlinear corrections, so we call this approximation the nonlinear
Kaiser order approximation. Replacing these lowest 3 moments with the standard linear theory we
obtain the original linear Kaiser model of equation 2.4. Here we want to view this series simply as a
series in k?, investigating the convergence as more terms are added.
Note that the calculations never require anything but simple power spectra of mass-weighted
powers of velocity to be computed from the simulations. As we will compare RSD power spectrum to
the sum from individual terms there should not be much sampling variance in the comparison, because
both are calculated from the same simulation, so the large-scale fluctuations will be the same. The
order of k?= kµ needed for convergence to a given level of accuracy will inevitably increase as one goes
to increasingly small scales, with the whole expansion eventually breaking down once kµσ/H > 1,
where σ is a typical (comoving) velocity of the system and H is the Hubble parameter.
2.1Angular dependence
By performing helicity decomposition [49] show that the power spectrum can be written as
PLL′(k) =
?
(l=L,L−2,..)
?
(l′=L′,L′−2,..; l′≥l)
l?
m=0
PL,L′,m
l,l′
(k)Pm
l(µ)Pm
l′ (µ),(2.5)
where Pm
the spherical harmonics. There are 5 numbers that describe these objects: L and L′describe the power
of two velocity moments we are correlating, l, l′describe the rank of the object, for example l = 1 is
rank-1, which is a 3-d vector, l = 2 is a 3-d tensor etc. Finally, m is the helicity eigennumber, which
ranges between 0 and l (l ≤ l′). Only equal helicity components of expansion have a non-vanishing
correlator. There is a close relation between the order of the moments and their angular dependence.
The lowest contribution in powers of µ to Pss(k) is µL+L′if L + L′is even or µL+L′+1if L + L′is
odd, and the highest is µ2(L+L′). Thus for P00(k) the only angular term is isotropic term (µ0), for
P01(k) the only angular term is µ2, P11(k) and P02(k) contain both µ2and µ4etc. Note that only
even powers of µ enter in the final expression, as required by the symmetry. We can thus write
l(µ) are the associated Legendre polynomials, which determine the angular dependence of
Pss(k) =
∞
?
L=0
1
L!2
?k
H
?2L
4L
?
j=2L
P(j)
LL(k)µj+2
∞
?
L=0
?
L′>L
(−1)L
L! L′!
?ik
H
?L+L′
2(L+L′)
?
j=(L+L′)or(L+L′+1)
P(j)
LL′(k)µj,
(2.6)
so that terms P(j)
The j index has to be even, so the lowest order is either L+L′or L+L′+1, whichever is even. These
terms can be uniquely extracted from simulations from angular dependence of PLL′ terms and so we
will focus on them, although sometimes it is useful to decompose them into the individual helicity
eigenstates instead.
LL′ are coefficients in expansion in powers of µjof contributions of L,L′terms to Pss.
3Numerical analysis
3.1
N-body simulations
The power spectra of the derivative expansion are all from mass-weighted velocity moments and
thus can be straightforwardly measured from simulations. We use a series of N-body simulations
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Page 5

of the ΛCDM cosmology seeded with Gaussian initial conditions, which is an updated version of
[51]. The primordial density field is generated using the matter transfer function by CMBFAST [52].
We adopt the standard ΛCDM model with the mass density parameter Ωm = 0.279, the baryon
density parameter Ωb= 0.0462, the Hubble constant h = 0.7, the spectral index ns= 0.96, and a
normalization of the curvature perturvations ∆2
the density fluctuation amplitude σ8≈ 0.81, which are the best-fit parameters in the WMAP 5-year
data [53]. We employ 10243particles of mass 3.0×1011h−1M⊙in a cubic box of side 1600 h−1Mpc.
The positions and velocities of all the dark matter particles are output at z = 0, 0.509, 0.989, and
2.070, which are quoted as z = 0, 0.5, 1, and 2 in what follows for simplicity. We use 12 independent
realizations in order to reduce the statistical scatters. For the detail of the simulations see [51].
Next we describe how we measure the power spectra from our simulation samples. We assign
the density field and the mass-weighted velocity moments in real space on 10243grids using a cloud-
in-cell interpolation method according to the positions of particles. To directly measure Pss(k) we
also need the density field in redshift space. In measuring the redshift-space density field, we distort
the positions of particles along the line-of-sight according to their peculiar velocities before we assign
them to the grid. We regard each direction along the three axes of simulation boxes as the line of
sight and the statistics are averaged over three projections of all realizations for a total of 36 samples.
We use a fast Fourier transform to measure the Fourier modes of the density fields in real space
δ(k) and in redshift space δs(k), as well as the mass-weighted velocity moment fields in real space
TL
mass-weighted velocity moments PLL′(k), are measured by multiplying the modes of the two fields (or
squaring in case of auto-correlation) and averaging over the Fourier modes. Throughout this paper,
we neglect shot noise because we have sufficient number of dark matter particles and such an effect
is thus negligibly small. Error bars in the following results are estimated from bootstrap resampling.
The dispersion in power spectra measurements is large on large scales because of sampling variance,
but it is mostly eliminated by taking the ratio of the two spectra obtained from the same set of
realizations (e.g. [54]).
R= 2.21 × 10−9(at k = 0.02 Mpc−1) which gives
?(k). Finally, the power spectrum in redshift space, Pss(k), as well as the power spectra of the
3.2Matter power spectrum
In this subsection we first measure the redshift-space power spectrum, Pss(k,µ), directly in redshift
space, which we call the “reference” power spectrum. The reference power spectrum in redshift space
is shown as functions of (k,µ) at z = 0 and 2 as the red lines in figure 1. We adopt the constant µ
binning into five bins between 0 ≤ µ ≤ 1, but only three µ bins among the five are plotted. In figure 1
we also show contributions of the terms of PLL′ for (0 ≤ L+L′≤ 4) to Pss(k,µ) computed from the
mass-weighted velocity moments. At µ ∼ 0 contributions from the higher order power spectra of the
velocity moments are small and Pss≃ P00because each PLL′ is multiplied by a factor of (kµ)L+L′.
On large scales one expects P00to be followed by the other two linear order terms, which are P01
and the scalar part of P11, i.e. P110
nonlinear terms, the vector part of P11, P111
both helicity 0, one from energy density correlated with the density P020
stress density correlated with density, P020
sufficiently low µ these nonlinear terms dominate over the linear term in P110
scales. For example, for µ = 0.1 at z = 0 we see that P02dominates over P11on all scales probed,
despite the fact that P11contains a linear order term. We also see that P11does not follow the linear
theory on all but the largest scales, but instead has the shape dependence similar to P02, characteristic
of the nonlinear terms. As pointed out in [49], the nonlinear helicity 1 (vector or vorticity) terms
in P11 are closely connected to P02 and partially cancel each other. This angular decomposition is
discussed further below in section 5, where we present the individual helicity terms separately.
Because of (kµ)L+L′weight the higher-order terms scale more rapidly with k, and dominate
on small scales: this is the region where RSD are dominated by FoG effects. One needs to take into
account more and more higher-order terms in order to make the expansion (equation 2.3) valid at such
smaller scales. This effect is more significant at z = 0 due to higher velocities. One can see that the
higher order terms cross the lower order terms at kµ ∼ 0.2 h Mpc−1(z = 0) and kµ ∼ 0.4 h Mpc−1
11. Note however that the latter scales as µ4, while there are two
11, and the scalar part of P02, which itself has two terms,
00, and one from anisotropic
02, that contain terms which scale as µ2. As a result, for
11
even on very large
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Page 6

10-2
105
10-1
100
101
102
103
104
105
Pss(k,µ)
µ=0.1
z=0
100
105
101
102
103
104
Pss(k,µ)
µ=0.5
101
102
103
104
0.01 0.1
Pss(k,µ)
k [h Mpc-1]
µ=0.9
µ=0.1
z=2 Pss
ref
P00
P01
-P02
P11
-P03
-P21
P04
-P13
P22
µ=0.5
0.01 0.1
k [h Mpc-1]
µ=0.9
Figure 1. Power spectra measured in redshift space Pss(k,µ) and individual contributions to it from the
terms of the moments expansion up to 4-th order at z = 0 (left) and z = 2 (right). The width of µ bin is 0.2
centered around the values shown in each panel. The solid and dashed lines show the positive values, while
the dotted lines the negative values.
(z = 2). This is where the perturbative parameter kµσ/H becomes of order unity and the perturbative
approach breaks down. At that point higher order terms dominate over the lower order terms and we
no longer have a convergence. This can be seen in figure 1: while for high k the redshift space power
spectrum Pss(k,µ) decreases in power relative to the real space case P00(k), the individual terms in
the series expansion increase due to their (kµ)L+L′. This suggests that a non-perturbative approach is
needed in this regime: we will explore the so-called FoG resummation in section 4. Figure 1 suggests
that the typical velocity σ in the expansion is about 500km/s at z = 0 and 250km/s at z = 2. We
confirm these numbers in a more detailed FoG analysis below.
3.3Legendre expansion
We can compare the agreement between moments expansion and the full Pssas a function of the
order in the series (L,L′). It is customary to expand the redshift-space power spectrum in terms of
Legendre multipole moments [e.g., 22, 24, 55–57]. The motivation for this expansion is that if one
uses full angular information the Legendre moments are uncorrelated. Using Legendre polynomials
Pl(µ), we have [58]
Pss(k,µ) =
?
The multipole moments, Pss
l, are obtained by inversion of this relation,
?+1
l=0,2,4,···
Pss
l(k)Pl(µ) , (3.1)
Pss
l(k) =2l + 1
2
−1
Pss(k,µ)Pl(µ)dµ .(3.2)
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Page 7

102
103
k Pss
0,sum(k)
Monopole
z=0
ref
linear
FoG
z=0.5
NL Kaiser
2nd
3rd
z=1
4th
6th
z=2
-0.04
0
0.04
0.01 0.1
Pss
0,sum/Pss
0,ref –1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
Figure 2. Upper panels: we show monopole moments of power spectrum in redshfit space Pss. The vertical
axis is multiplied by k to clearly illustrate departures from a reference power spectrum. The green, blue,
yellow, magenta and cyan lines respectively show our model prediction up to nonlinear Kaiser, 2nd, 3rd, 4th
and 6th order corrections, measured from the simulations. The black lines are linear theory prediction. The
black points with errorbars show the reference power spectrum. The red lines show our FoG model (section
4). Lower panels: Error between the summed power spectrum and the reference spectrum. The meaning of
the color of each line is the same as that of the upper panels. For reference errorbars are shown for the result
of our FoG model.
Most of the studies on modeling redshift-space distortions focus on the monopole (l = 0) and
quadrupole (l = 2), although hexadecapole (l = 4) also contains some cosmological information
[56].
In figure 2 we show the monopole power spectrum at z = 0, 0.5, 1, and 2 summed up to nonlinear
Kaiser, 2nd, 3rd, 4th, and 6th order approximations in kµ expansion. Here we denote the summation
at a given order as including all terms that have the same L+L′: hence 2nd order includes all 3 Kaiser
terms P00, P01and P11, as well as P02, while the nonlinear Kaiser model includes only the first three.
The lower panels show the error for a given order, Pss
with the input cosmological parameters of our simulations are also plotted for comparison. The power
spectrum of the nonlinear Kaiser model starts to deviate from the reference spectrum at very large
scales, k ≃ 0.05 h Mpc−1. However, adding the term P02, which has the same order contribution as
P11, to the nonlinear Kaiser model, improves the accuracy. Adding the higher order terms continues
to improve the accuracy down to smaller and smaller scales, but the gains decrease as we approach
the scale k = σ−1∼ 0.2 h Mpc−1(z = 0), where the perturbative expansion breaks down. Our
formula for the redshift-space monopole spectrum Pss
accurate within a few percent accuracy at k ≃ 0.2 h Mpc−1at z = 0 and at k ≃ 0.4 h Mpc−1at
z = 2. It predicts not only the overall shape of the redshift-space power spectrum up to these scales
but also baryon acoustic oscillations (BAO): to see this more clearly we show the summed power
spectra divided by the smoothed no-wiggle spectrum [59] in figure 3.
Figure 4 is the same as figure 2, but shows the results for the quadrupole spectra Pss
the nonlinear quadrupole spectra crosses zero at high-k, there exists a singularity point for the ratio of
summed and reference spectra at small scales. The predictions for the quadrupole moment reproduce
the reference spectrum within a few percents up to the scales of the singular point, k ≃ 0.15 h Mpc−1
at z = 0 and k ≃ 0.3 h Mpc−1at z = 2. Figure 5 shows the results for the hexadecapole spectrum.
We adopt broader k binning for the hexadecapole moment at k < 0.1 h Mpc−1and put artificial
cuts for the plots of kPss
4
because of large sampling variance. We do not show the results obtained
from the nonlinear Kaiser, 2nd and 3rd order approximations in lower panels because they strongly
0,sum/Pss
0,ref−1. The linear theory power spectra
0,sum, summed up to or more than 6th order, is
2. Because
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Page 8

0.9
1
1.1
Monopole
z=2
ref
0.9
1
1.1
z=1
FoG
linear
0.9
1
1.1
Pss
0,sum(k)/Pss
0,no-wiggle(k)
z=0.5
NL Kaiser
2nd
0.9
1
1.1
0 0.05 0.1 0.15 0.2 0.25 0.3
k [h Mpc-1]
z=0
6th
0.6
0.8
1
Quadrupole
z=2
0.6
0.8
1
z=1
0.6
0.8
1
Pss
2,sum(k)/Pss
2,no-wiggle(k)
z=0.5
0.6
0.8
1
0 0.05 0.1 0.15 0.2 0.25 0.3
k [h Mpc-1]
z=0
Figure 3. Redshift-space power spectrum divided by the no-wiggle approximation [59], monopole (left) and
quadrupole (right). The green, blue and cyan lines show our model prediction up to nonlinear Kaiser, 2nd
and 6th order corrections. The dotted black lines are linear theory prediction, while the solid black lines the
reference power spectrum. The red lines show our FoG model.
deviate from the reference power spectrum (as shown in the upper panels of figure 5). Although the
measurement of the hexadecapole moment of the redshift-space power spectrum is very noisy, the
higher order expansion predictions give a good agreement if we consider summation to 6th order.
4Fingers-of-God resummation
It is clear from the results in previous section that while for kµσ/H < 1 we have a convergence
and only a finite number of terms need to be considered, there is no convergence for kµσ/H > 1:
individual terms become larger and larger as we go to higher orders, yet the total sum in Pss(k,µ)
remains well behaved. This suggests we need to explore ways to resum the terms.
While it is difficult to make progress in general terms, there are specific situations that can be
controlled. We are interested in a situation where pieces of terms disconnect within the correlation
function. For example, in P02term we correlate δ with (1+δ)u2
scales, as will be the case in systems with large velocity dispersion caused by nonlinear gravitational
collapse, then on large scales this term becomes P02 ∼ P00σ2, where σ2= ?u2
tribution to Pssis −P00(kµσ/H)2. We see that this term scales as the linear order term, but has
opposite sign to it, i.e. This term suppresses power and this suppression scales as k2. The long range
correlation is entirely in the density field. While this analysis suggests the mean square velocity field
enters as the physical parameter, as discussed in [49], any bulk velocity that displaces particles as a
?, and if u2
?is dominated by the small
?? and the total con-
– 7 –

Page 9

102
103
k Pss
2,sum(k)
Quadrupole
z=0
ref
linear
FoG
z=0.5
NL Kaiser
2nd
3rd
z=1
4th
6th
z=2
-0.04
0
0.04
0.01 0.1
Pss
2,sum/Pss
2,ref –1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
Figure 4. Same as figure 2, but for the quadrupole. The dashed lines at the top panels show positive values
while the dotted lines show negative values.
101
0.1
102
k Pss
4,sum(k)
Hexadecapole
z=0
ref
linear
FoG
z=0.5
NL Kaiser
2nd
3rd
z=1
4th
6th
z=2
-0.1
0
0.01 0.1
Pss
4,sum/Pss
4,ref –1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
Figure 5. Same as figure 2, but for the hexadecapoles. We adopt the different bin sizes in logarithmic spacing
at k ≤ 0.1 h Mpc−1and k ≥ 0.1 h Mpc−1. Artificial cuts are put for the plots of the hexadecapoles at low k
because of large sampling variance.
solid body will not contribute to FoG. This cancellation shows up in part of P11cancelling P02, such
that only the dispersion part enters into the σ2term, while the bulk part cancels.
One can identify similar terms at higher order, for example P04 ∼ P00?u4
?u4
??c is the connected part of the curtosis. There will also be a term
P22∼ P00σ4. The total contribution to Pssfrom these terms is thus
?? and one can write
?? = 3σ4+ ?u4
??c, where ?u4
Pss(k,µ) = P00[(1 − (kµσ/H)2+ (kµσ/H)4/2 + 2(kµ/H)4?u4
??c/4!···]
= P00e−(kσµ/H)2+2(kµ/H)4?u4
??c/4!···.(4.1)
The same calculation can be done at the field level. We are interested in the situation where the pieces
– 8 –

Page 10

of a term like δ(x)u2
?(x) are disconnected within a correlation calculation, e.g.,
?
X(y)δ(x)u2
?(x)
?
=
?X(y)δ(x)?
?
u2
?
?
+ other terms. We can re-sum the pieces into a FoG factor
G1/2(kµ) = exp
??
L=1
1
L!
?ikµ
H
?L
?uL
??c
?
,(4.2)
where ?uL
lowest order term here is just the usual Gaussian kernel exp?−k2µ2σ2/2H2?. After this re-summation
?
L=1
L!
??cis the connected part of ?uL
?? and note that the odd L terms are zero by symmetry. The
we can write the redshift-space density field as
δs(k) = G1/2(kµ)δ(k) +
?
1
?ikµ
H
?L?
(1 + δ(x))
?
uL
?
?
c(x)
?
k
?
, (4.3)
where
motivates us to write
?
uL
?
?
cis understood to be uL
?minus all possible internal averages of any number of u’s. This
Pss(k) = G([kµσ/H]2)PKaiser(k), (4.4)
where PKaiseraccount for the lowest 3 terms given by equation 2.4 and where G(x) is exponential in
the simplest case where higher order reduced moments can be ignored, while more generally it is a
function with alternating signs of coefficients in Taylor expansion. We have written G(x) in terms
of x = [kµσ/H]2only: if curtosis is present then we can either write it by adding additional (kµ)4
terms to the exponential, exp[−(kσµ/H)2+ 2(kµ/H)4?u4
more general functional form of G(x) than an exponential.
Note that in the simplest form this “derivation” gives exactly the exponential FoG form proposed
in the literature [10, 32, 60–62]. Other forms for G(x) have been proposed in the literature, e.g.
??c/4!···], or, equivalently, we allow for a
G(x = (kµσ/H)2) =
?(1 + x)−1Lorentzian,
exp(−x) Gaussian,
(4.5)
where σ was treated as a free parameter. See [26, 63] for the studies to adopt more than one free
parameter for the FoG term. These empirical forms all behave qualitively in the same manner, i.e.
to first order of Taylor expansion in (kµ)2they give G = 1 − (kµσ/H)2+ O([kµσ/H]4) and higher
order terms alternate in the signs.
The effective velocity dispersions can be evaluated from our expression of the redshift-space
power spectrum (equation 2.3) and uniquely resummed into the FoG kernels because the higher-order
terms of the expansion contain all information. We know that a Taylor series of the FoG kernels like
in equation 4.5 produce positive and negative terms alternatively, just like the terms in equation 2.3.
However, one needs three independent FoG kernels for the three lowest-order PLL′’s, i.e., P00, P01,
and P11. These three terms need to be individually multiplied by something which corresponds to
the generalized FoG kernels.
To formalize this, we can write
Pss(k) = G00
?[kµσ00/H]2?P00+ 2G01
?[kµσ01/H]2?ikµ
HP01+ G11
?[kµσ11/H]2?(kµ)2
H2P11.(4.6)
We have defined three different velocity dispersions σ00, σ01, and σ11. The expressions of the velocity
dispersions in the FoG kernels can be uniquely derived from equation 2.3 as
σ2
00(k) =P02
P00,
P03+ 3P21
P01
P13
P11.
(4.7)
σ2
01(k) =1
6
,(4.8)
σ2
11(k) =1
3
(4.9)
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Page 11

To go beyond that and determine the form of FoG kernel we expand the FoG terms as a Taylor
series in terms of (kµ)2. Because all the phenomenological FoG models have the first derivative equal
to −1, we define dGLL′(x)/dx|x=0= −1. Now let us consider the ansatz for the FoG model,
GLL′ (xLL′;αLL′) =
?
1 +xLL′
αLL′
?−αLL′
, (4.10)
where xLL′ = (kµσLL′/H)2. Each FoG kernels contains two parameters, σLL′ and αLL′. These
can reproduce the functional forms in previous studies: Lorentzian for αLL′ = 1 and Gaussian for
αLL′ = ∞. The FoG parameter αLL′ is related to n-th derivative of GLL′ as (n ≥ 2)
dn
dxnGLL′(x)|x=0= (−1)n
n−1
?
m=1
?
1 +
m
αLL′
?
.(4.11)
We use the αLL′’s determined from the lowest contribution at second order, as
αLL′(k) =
?
G
′′
LL′(k) − 1
?−1
.(4.12)
Note that the expressions for our FoG models of equation 4.6 preserve full generality up to order
(kµ/H)5. The quality of the ansatz can be investigated by looking at higher order terms: the com-
parison of the derivatives of the FoG kernels up to 4th order with the expansion terms is given in
appendix A.
The FoG model in equation 4.10 is not the only option to choose. As discussed above, the
resummation can be expressed more elegantly in terms of connected moments, where the cumulant
theorem naturally leads to an expression of the form
GLL′ = exp[−(kµσ/H)2+ (kµτ/H)4+ ···],(4.13)
where the lowest two cumulants are variance σ2= ?u2
part of ?u4
??. We can generalize this expression and introduce 3 variance and 3 curtosis terms separately
for 00, 01, and 11 terms, and we can relate the kurtosis terms τLL′’s to the parameters αLL′’s defined
above as 2(τLL′/σLL′)4= 1/αLL′.
Let us summarize this discussion: the decoupling of small scale velocity dispersion like terms from
the long range correlations motivates a resummation of the terms into the so called FoG kernels, which
multiply the long range correlation terms contained in density-density (P00), density-momentum (P01)
and momentum-momentum (P11) correlators. Only a portion of the terms can be motivated in such
a way, while other terms are simply nonlinear couplings that do not reduce to linear order correlation
on large scales. It is therefore difficult to provide a formal justification for this resummation, but
it is worth analyzing to what extent this approach is useful. Here we use the distribution function
expansion in equation 2.3 to formally define FoG kernel parameters: up to (kµ)5this is equivalent
to the exact expansion, but requires 3 different FoG kernels. An additional point that needs to be
emphasized is that these quantities as we defined are a function of angle and scale, i.e. they are µ
and k dependent. Below we use numerical simulations to compare the original expansion in equation
2.3 to the FoG resummation version to see to what extent FoG approach is useful for the general
treatment of RSD.
?? and kurtosis τ4= ?u4
??c, which is the connected
4.1Testing the Fingers-of-God model
In this subsection we compare the FoG resummation of the higher order terms discussed above to
numerical simulations. The upper panels of figure 6 show velocity dispersions determined from our
simulations using equations 4.7 – 4.9. For clarity we plot the spherically-averaged velocity dispersions,
i.e., those obtained from the monopole spectra. While the 3 velocity dispersion terms are similar to
each other, they differ in the amplitude and scale dependence, suggesting that it is important to
independently consider the FoG kernel for each spectrum in the nonlinear Kaiser formula. Note that
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Page 12

200
400
600
σLL’ (k) [km s-1]
z=0
00
01
11
LL’
z=0.5z=1z=2
0
1
0.01 0.1
1/αLL’(k) or 2(τLL’/σLL’)4(k)
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
Figure 6. Spherically averaged FoG parameters, velocity dispersions (upper panels) and the power law index
of FoG kernel (lower panels). The circles/triangles have been respectively offset in the negative/positive
direction for clarity.
these values are higher than those determined by previous studies (e.g., figure 7 of [47]). This is
because our quantities are based on the mass-weighted velocities in contrast to the volume-weighted
velocities discussed in [47]. We discuss the latter in section 4.2.
The lower panels in figure 6 show the inverses of the second parameters for our FoG model,
1/αLL′(k) defined by equation 4.12. As we have discussed above, the parameters are equivalent to
2(τLL′/σLL′)4when we adopt the kurtosis as the second parameters. Note that for a Lorentzian
model and a Gaussian model we have 1/αLL′ = 1 and 1/αLL′ = 0, respectively. The impression from
the figure is that α parameters have strong k dependences and behave differently from each other at
large scales. However, the FoG terms have negligibly small effects on the shape of the power spectrum
at large scales. On the other hand, at lower redshifts the α values converge to nearly a constant at
k > 0.1 h Mpc−1where the FoG effect starts to play an important role, with typical values between
1 < αLL′ < 2, i.e. Lorentzian FoG kernel is a better approximation than Gaussian. The convergence
of αLL′ to a single value is worse as we go to higher redshifts and the difference between the values of
αLL′’s remains large. In order to see the angular dependence of our FoG parameters, we show σLL′
and 1/αLL′ at z = 0 as functions of k and µ at the upper and lower panels in figure 7, respectively.
Now we focus on the power spectrum Pssusing the FoG model discussed above. In figures 2, 4
and 5 the resulting power spectra with the FoG kernels are shown as the red points. As we have seen
in section 3.2, there were certain scales at which the power spectrum from our perturbative expansion
approach breaks down and diverse, k ≃ 0.2 h Mpc−1at z = 0 and k ≃ 0.4 h Mpc−1at z = 2 for the
monopole spectrum even though we sum up to the 6th order terms. Our FoG model dramatically
improves the results. The accuracy of a few percent is achieved up to k ≃ 0.4 h Mpc−1at z = 0 and
even up to to k < 1.0 h Mpc−1at z = 0.5 and 1. On the other hand, the improvement at z = 2 is not
so much better than the lower redshfits because of a strong suppression by the FoG. Our FoG model
works for the quadrupole spectrum as well as for the monopole spectrum: it predicts the quadrupole
spectrum down to very small scales (see the upper panels of figure 4). Even the accuracy of the
hexadecapole spectrum of our FoG model is quite good, at the level of 10%, and it reproduces the
shape over all the scales probed (see upper panel of figure 5).
– 11 –

Page 13

400
600
σLL’ (k,µ) [km s-1]
(L,L’)=(0,0)
0.1
0.5
0.9
µ
(L,L’)=(0,1)(L,L’)=(1,1)
0
0.01
1
0.1
1/αLL’(k,µ)
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
0.01 0.1
k [h Mpc-1]
Figure 7. Angular dependence of FoG parameters at z = 0. The width of µ bin is 0.2 centered around the
values shown in the top left panel.
4.2Mass-weighted vs volume-weighted velocities
It is worth comparing our approach based on the power spectra of mass-weighted velocity moments
with those of volume-weighted velocity moments, which are commonly used to model nonlinear power
spectra (e.g., [25, 32, 47, 62, 63]). The difference comes from the fact that the L-th moment of the
mass-weighted moments TL
?in equation 2.1 contains contributions not only from L-th
order in perturbation theory, but also (L+1)-th order because of the term δ(x)uL
equation 2.1 with the same order term, we obtain
?= (1 + δ)uL
?(x). By regrouping
δs(k,µ) = δ(k,µ) +ik?
Hu?(k,µ) +
∞
?
L=2
?
1
(L − 1)!
?ik?
H
?L−1?
δ(x)uL−1
?
(x)
?
k+1
L!
?ik?
H
?L
uL
?(k)
?
,
(4.14)
where the first and the second terms of the right-hand side are respectively the zeroth and first order
terms, while the bracketed terms correspond to the L-th order terms (L ≥ 2). By squaring equation
4.14, we obtain the same equation as equation 2.3, but the terms regrouped into the same order in
SPT,
Pss(k) = P00+ 2ikµ
HP0u1
?+(kµ)2
H2Pu1
?u1
?+
?
2ikµ
HP0w1
?−(kµ)2
H2P0u2
?
?
+ ···(4.15)
where wL
?= TL
As many previous studies have already discussed (e.g. [63]), measuring the power spectrum of
volume-weighted velocity moments is not as straight-forward as measuring mass-weighted velocity
moments used in our formalism. In order to measure the moments of volume-weighted velocities, we
divide the interpolated moments of mass-weighted velocities by the interpolated density before the
field is Fourier-transformed TL
?. This can be noisy: some points on the grid may not
have any particles, so a sufficiently coarse grid is needed. More importantly, the results depend on
the grid size, i.e. on smoothing (see [32, 64] for a detailed discussion of how to measure the volume-
weighted velocities from N-body simulations). Following the same process as described in section
?− uL
?= [δuL
?]k, thus P0wL
?= P0L− P0uL
?and similarly for higher orders.
?/(1 + δ) = uL
– 12 –

Page 14

-0.04
0.04
0
0.04
Pss
0,sum/Pss
0,ref –1
Monopole
z=0
NL Kaiser
2nd
3rd
-0.04
0
0.01 0.1
Pss
2,sum/Pss
2,ref –1
k [h Mpc-1]
Quadrupole
z=0.5
4th
6th
0.01 0.1
k [h Mpc-1]
z=1
0.01 0.1
k [h Mpc-1]
z=2
0.01 0.1
k [h Mpc-1]
Figure 8. Summed power spectrum at a given order in terms of volume weighted moment expansion compared
to the reference power spectrum. Lines are the same as the lower panels of figures 2 and 4.
3.1, we compute the redshift-space power spectrum based on the power spectra of volume-weighted
velocity moments up to a given level of accuracy.
Figure 8 shows the ratio of the summed power spectra of equation 4.15 to the corresponding
reference spectra in redshift space. Because equation 4.15 is essentially the same as equation 2.3,
here we want to see the convergence of these expressions. At the order of first 3 (Kaiser) terms, the
expansion with volume-weighted velocities is somewhat closer to the reference spectrum than that
with mass-weighted velocities, but both approximations are bad. At higher orders the convergence is
faster with mass weighted defined quantities: one can predict well the redshift-space power spectrum
at k = 0.1 h Mpc−1by including the 2nd and 3rd order corrections at z = 0 and z = 2, respectively,
for our power spectrum of mass-weighted velocity moments, while one needs to include the 4th order
corrections for the power spectrum of volume-weighted velocity moments. Both approaches break
down once one enters the non-perturbative regime (k > 0.2 h Mpc−1at z = 0). We conclude that
in terms of rate of convergence, there is no advantage in defining volume weighted quantities. If one
works with galaxies and halos number density weighting becomes essential, since it is impossible to
define volume weighted velocity moments in a sparsely sampled system [63]. In that situation our
approach is the only meaningful way to define physical quantities that enter in RSD description.
5Expansion in powers of µ2
As we have seen in previous two sections the series expansion of equation 2.3 is convergent on large
scales, but not on small scales. For sufficiently high k any finite order summation fails drastically.
FoG resummation approach fares better, but even that fails for high k. One can sidestep these issues
by considering an alternative expansion in terms of powers of µ2: as discussed in section 2.1 and in
[49] for any finite power of µ2there is a finite number of PLL′ terms contributing to it. For µ0only
P00contributes, for µ2P01, P11and P02, for µ4P11, P02, P03, P12, P04, P13and P22etc.
Only these 3 lowest terms, µ0, µ2, and µ4, contain cosmological information at the linear order,
so in principle these are the only relevant terms. However, if we expand the full Pss(k,µ) into powers
of µ2and try to determine the coefficients from the data, the resulting coefficients will be correlated:
only Legendre expansion assures uncorrelated values. As a result there will be mixing of higher powers
of µ2into lower powers if they are not accounted for in the fits, or there will be strong degeneracies
and the fits will be unstable if all the coefficients are accounted for but we allow them to take any
value. Typically one solves this by regularizing the expansion, i.e. by constraining them to a certain
range of values. In this paper we will not focus on methods how to determine the coefficients of such
expansion from the data, but we will show µ6and µ8expansion terms to develop some understanding
of their scale dependence and amplitude.
– 13 –

Page 15

5.1
µ2terms
In the top panels of figure 9 we show these individual term contributions to the lowest order powers
of µ (we do not show µ0term, which is just the usual real space power spectrum P00). For µ2we see
that the P01dominates for low k, as expected since that is the only term that does not vanish in linear
theory. This term follows linear theory prediction for low k, while for k > 0.1 h Mpc−1it exceeds it,
just like it happens for the dark matter power spectrum P00itself. This is not surprising: as shown
in [49] we can write Pss
dlna, so this term is given by the time derivative of the dark matter
power spectrum and has a similar scale dependence relative to the linear power spectrum.
The next term in terms of relevance is P02. This term has contributions from the correlation
between the energy density and the density P020
00, as well as from the scalar part of the anisotropic
stress correlated with the density P020
02, both helicity 0 scalars. As discussed in section 4, we expect
the first term to be dominant and scale as −k2P00σ2/H2, hence to dominate over P01for kσ/H > 1.
This gives σ = 500km/s at z = 0, decreasing to 200km/s for z = 2. As expected we see this term is
always negative.
The third term contributing to µ2is the vector part (helicity 1) of auto-correlation of momentum
density with itself P111
11. This term is always positive and partially cancels P02. As discussed in [49],
it cancels all the bulk motion contributions to σ2. While this term scales in a similar way as P02, it
is 3-10 times lower in amplitude, so it cancels only a small part of P02.
We see that the total sum never exceeds the linear power spectrum and becomes negative for
k ∼ 0.17 h Mpc−1at z = 0 and k ∼ 0.5 h Mpc−1at z = 2. For scales smaller than that the µ2term
is negative as a consequence of a strong FoG effect.
01(µ2) =dP00(k)
5.2
µ4terms
This term receives contributions from 7 different terms, P11, P02, P03, P12, P04, P13and P22. They
are shown at the middle panels in figure 9. On large scales the dominant term is P11which contains
a linear order contribution. This term agrees with linear theory prediction for k < 0.1 h Mpc−1and
is above that for k > 0.1 h Mpc−1, just like in the case of P00and P01.
The next order term in significance should be P02. We see this term is relatively small and
does not dominate anywhere. P02contribution to µ4arises entirely from from the scalar part of the
anisotropic stress correlated with the density P020
P020
00
does not. Physically one expects the small scale velocity dispersion to be relatively isotropic,
hence anisotropic stress density should be small compared to the energy density.
Next two terms in terms of L + L′are P03 and P12. These two terms provide the dominant
correction to P11on intermediate and large scales k < 0.1 h Mpc−1. On very small scales terms P04,
P22and P13dominate. As discussed in FoG section we expect the first two terms to scale as 3k4σ4P00
and be equal in amplitude, while P13 should be negative and cancel out the bulk flow part of the
other two terms. We see that this expectation is borne out in simulations: in total these terms add
power on small scales. The transition happens at a similar scale as for µ2term, k ∼ 0.17 h Mpc−1at
z = 0 and k ∼ 0.5 h Mpc−1at z = 2.
02, which contains µ4term contribution to Pss, while
5.3Higher order terms
At order higher than µ4we do not have any linear order contributions, so we expect these terms to be
small on large scales. There are many terms that contribute. At the bottom panels in figure 9 we show
the total contributions to µ6and µ8terms. We can see that these terms are indeed negligibly small
at k < 0.1 h Mpc−1at z = 0 and k < 0.3 h Mpc−1at z = 2. At smaller scales these contributions
increase with the scale dependences of k6and k8, respectively.
6Conclusions
In this paper we used numerical simulations to investigate the distribution function expansion ap-
proach to the redshift space distortion power spectrum [49]. The power spectrum in redshift space can
be written as a sum over correlators between mass-weighted velocity moments. We analyzed a large
– 14 –