Table Of ContentChasing Unbiased Spectra of the Universe
Yong-Seon Song1, Takahiro Nishimichi2, Atsushi Taruya3, Issha Kayo4∗
1Korea Astronomy and Space Science Institute, Daejeon 305-348, R. Korea
2Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study,
the University of Tokyo (Kavli IPMU, WPI), Kashiwa, Chiba 277-8583, Japan
3Research Center for Early Universe, School of Science,
University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
4Department of Physics, Toho University, 2-2-1 Miyama, Funabashi, Chiba 274-8510, Japan
(Dated: January 16, 2013)
Thecosmological powerspectrumofthecoherentmatterflowismeasuredexploitinganimproved
3 prescriptionfortheapparentanisotropicclusteringpatterninredshiftspace. Newstatisticalanalysis
1 is presented to provide an optimal observational platform to link the improved redshift distortion
0 theoreticalmodeltofuturerealdatasets. Thestatistical poweraswellasrobustnessofourmethod
2 aretestedagainst60realizations of8h−3 Gpc3 darkmattersimulationmapsmockingtheprecision
n levelofupcomingwide–deepsurveys. Weshowedthatwecanaccuratelyextractthevelocitypower
a spectrum up to quasi linear scales of k ∼ 0.1hMpc−1 at z = 0.35 and up to k ∼ 0.15hMpc−1 at
J higher redshifts within a couple of percentage precision level. Our understanding of redshift space
4 distortionisprovedtobeappropriateforprecisioncosmology, andourstatisticalmethodwillguide
1 usto righteous path to meet thereal world.
] PACSnumbers: draft
O
C
I. INTRODUCTION matter density field [9–24]. This technique relies on the
.
h redshift space distortions seen in galaxy surveys. Even
p though we expect the clustering of galaxies in real space
The emergence of a standard model for the Universe
o- dominated by an unknown substance of dark materials to haveno preferreddirection,galaxymaps producedby
estimating distances from redshifts obtained in spectro-
r has revolutionized our understanding of the Universe.
t scopic surveys reveal an anisotropic galaxy distribution.
s Sincethefirstfirmevidenceofdarkenergyin1998[1,2],
a there has been substantial observational and theoreti- Theanisotropiesarisebecausegalaxyrecessionvelocities,
[ from which distances are inferred, include components
cal research aiming at understanding the true nature of
fromboth the Hubble flow and peculiar velocities driven
1 this phenomenon. In recent years, many authors have
v started exploring the possibility that dark energy, and bytheclusteringofmatter[25–27]. Measurementsofthe
3 the observed acceleration of the expansion of the Uni- anisotropiesallowconstraintsto be placedonthe rateof
3 growth of clustering.
verse, could be the consequence of an incomplete theory
1
of gravity on cosmological scales and may require modi- Measurements of coherent motion field from redshift
3
. fications to Einstein’s theory of General Relativity. The distortion maps have been plagued by systematic uncer-
1
information of underlying science about the Universe is tainties which have made their cosmological constraints
0
given by looking at structure formation on large scales. uncompetitivecomparedtootherprobesoftheUniverse.
3
1 Severalauthorshaveshownthatbycombiningvarious The cosmological density and velocity field couple to-
: probes of the large–scale structure in the Universe, it is gether and evolve nonlinearly. In addition, the mapping
v
possible to testthe relationshipbetweenthese quantities formula between the real and redshift space is intrinsi-
i
X which,inthelinearregime,cangenerallybedescribedby cally nonlinear. These nonlinearities prevent us from
r two functions of time and scale [3–8]. Those can be con- inferring the linear coherent motion from the redshift
a strained through a combination of weak lensing and an space clustering straightforwardly. Recently, an accu-
independentprobeofmatter–energyfluctuations. Thisis ratetheoreticalmodelfortheredshiftdistortionwaspro-
motivated by the fact that the weak lensing experiments posed in [28]. They take into account the fact that the
probe the geometrical potential of combination between linear squeezing and non–linear smearing effects on dis-
curvature perturbation and Newtonian force, which de- tortedmaps are not separable to eachother and develop
termines the trajectories of photons through the Uni- a more elaborate description than simple factorized for-
verse, while matter fluctuation measurements probe the mulation. Thederivedcorrectiontermsatleadinghigher
Newtonianforcealonedetermininglocalinhomogeneities orders assist us to achieve better fit to simulated data.
of matter–energy. They also include non–linear corrections formulated us-
The coherent motion of the galaxies opens a unique ing the closure approximation to predict the non–linear
opportunity to access the fluctuations of the underlying growthindensity–density,density–velocityandvelocity–
velocityspectra. Inthiswork,weassumeaperfectcross–
correlation between density and velocity fields at linear
level to decompose the coherent motion spectra. Thus
∗Electronicaddress: [email protected];[email protected];[email protected];[email protected];htoh-ue.alci.njpear den-
2
sity power spectrum and its velocity counterpart. but related as
Despitealltheoreticalefforts,theexactformofFoGef-
fectisunknown. Weadoptmostcommonfunctionalform PδliΘn(k) = f(z)Pδliδn(k), (2)
of FoG effect, such as Gaussian or Lorentzian, and pa- Plin(k) = f2(z)Plin(k), (3)
ΘΘ δδ
rameterizeFoGeffectusingone-dimensionalvelocitydis-
persionrepresentingtherandomnessofthemotionwhich wheref(z)=dlnD+(z)/dlnaisthegrowthrateparam-
erases the correlation structure on small scales. The pa- eter with D+ being the linear growth rate. In this case,
rameter space is extended to include this uncertainty of allthevelocityinformationiscontainedintheparameter,
FoGeffectinadditiontothelinearspectraofdensityand f(z), which is constant over wavenumber k.
velocityfields,whichwereoriginallyproposedby[29,30]. However,the planned/ongoinggalaxyredshift surveys
We run MCMC routine to find best set of spectra and aswellastheexistinglargesurveysmainlytargetweakly
FoG parameter, and we find that coherent motion spec- nonlinearscale,wherethe feature ofBAOsis prominent.
tra are measurable at linear regime with good precision. Moreover,byappropriatelymodelingthisregime,wecan
Coherentmotionspectraaremeasuredatacoupleofper- inprincipleenlargetherangeofwavenumbertobetaken
centage accuracy with appropriate k cut–off. intoaccountinthe analysis,andimprovethe constraints
The paper is organizedas follows. In Sec. II, we begin on the gravitational law. Thus, we have to somehow
by introducing suggested theoretical models of redshift incorporate nonlinearity to make maximum use of these
distortions. We then describe the statistical method to surveys.
extractthe coherentmotionspectra. Using this method, First of all, the cross- and auto-power spectra of the
Sec. III presents our main results on the measurements density and the velocity fields are naturally expected to
of linear density and velocity spectra. In Sec. IV, the receive nonlinear corrections. Another important effect
impact of wrong prior assumption on the decomposition arisesfromtherandommotionofgalaxies. Bycombining
of power spectra is discussed. Finally, we conclude in thesetwoeffects,Scoccimarro[27]proposesthefollowing
Sec. V. formula
P˜(k,µ)= Pδδ(k)+2µ2PδΘ(k)+µ4PΘΘ(k) GX(kµ),(4)
II. METHODOLOGY wherethee(cid:8)ffectofrandommotioniscapture(cid:9)dbythefac-
tor GX(kµ). Note that we have replacedthe three linear
We first highlight analytical models for the redshift- spectra in Eq. (1), Piljin(k), with their nonlinear counter-
space power spectrum. We then show our methodology parts, Pij(k). In the original paper by Scoccimarro [27],
of reconstructing the linear density and velocity power the factor GX(k) is designed so that it accounts for the
spectra from the two-dimensional power spectrum ob- randommotionofthegalaxiesatlargescale,andhecon-
served in redshift space. siders the Gaussian shape for this factor:
GX(kµ)=GGAU(kµ)=exp −(kµσv)2 , (5)
(cid:8) (cid:9)
A. Analytical models for the power spectrum in where σv denotes the dispersion of the one-point PDF
redshift space of the velocity in one-dimension. At smaller scale, in-
side the cluster of galaxies, the virial motion of galaxies
At large scale, we expect that the density field as well alsogiveasuppressionofthe powerspectruminredshift
asthevelocityfieldaresmallperturbationstothehomo- space. ThiseffectiscalledFinger-of-God(FoG),andcan
geneous universe. When the higher-order contributions also be approximatelydescribed by multiplication of the
arenegligiblysmall,thetwo-dimensionalpowerspectrum factorGX(kµ). ALorentzianformofthisfactorhasbeen
in redshift space, P˜(k,µ), can be expressed as frequently adopted based on the results of N-body sim-
ulations:
P˜(k,µ)=Pδliδn(k)+2µ2PδliΘn(k)+µ4PΘlinΘ(k), (1) GX(kµ)=GLOR(kµ)= 1 . (6)
1+(kµσv)2
where we denote by δ and Θ the density contrast and
the velocity divergence, with the latter defined by Θ ≡ For convenience, we do not distinguish between these
−(1+z)∇v/H. The auto- and cross-power spectra of damping effects, andsimply callthem asFoG inthis pa-
the two fields in linear theory are expressed as Plin(k), per, although, strictly speaking, the former one at large
ij
with i and j being either δ or Θ. This formula describes scale has a different origin.
the effect of coherent velocity flow towards overdensity Recently, Taruya, Nishimichi & Saito [28] proposed a
at large scale (Kaiser effect). Because of this effect, the more accurate model for the redshift-space distortion.
clusteringpatterninredshiftspaceisenhancedalongthe MotivatedbythefactthattheKaiserandtheFoGeffects
line-of-sightdirection. Ifthegravitationallawfollowsthe can not be separated, and should not be described as a
generalrelativity,the three spectraare notindependent, factorisable form as in Eq. (4), they derived correction
3
terms which have long been overlooked. Their formula three spectra, P (k). In doing so, we simply assume the
ij
reads fiducial GR cosmology used in running the simulations.
We pre-compute the nonlinear corrections to the three
P˜(k,µ)= Pδδ(k)+2µ2PδΘ(k)+µ4PΘΘ(k) spectra up to the second-order in the Born approxima-
(cid:8) +A(k,µ)+B(k,µ)}GX(kµ), (7) tion:
δPfid(k)=Pfid(k)−Pfid,lin(k), (11)
where the full expressions for the terms A(k,µ) and ij ij ij
B(k,µ) based on perturbation theory can be found in while we allow to vary the linear part according to
the Appendix A of that paper [28]. Let us note some Eq. (10). By adding up linear and nonlinear parts, we
important features in the new terms. First, they include have
higher-order polynomials in µ and f: the A-term has a g (k) g (k)
term which scales as f3µ6, while we have a f4µ8 con- Pij(k)= gfiid(k)gfijd(k)Pifijd,lin(k)+δPifijd(k). (12)
tribution in the B-term. Thus, they become relatively i j
importantatµ≃1. Anotherpointisthatastheseterms Strictly speaking, the second term depends on the cos-
arise as a non-linear coupling between the density and mological model as well as the gravitational law. As a
the velocity fields, they are of the order O({Plin(k)}2). first trial, however, we simply let this term unchanged
ij
We thus may able to omit them on linear scales. from its fiducial value. In Sec. IV, we will relax the as-
sumptiontoexecute amoregeneralanalysis: weadopta
wrong cosmologicalmodel as the fiducial model, and see
B. Decomposition strategy how well we can recover the true spectra.
Wealsoscalethecorrectionterms,A(k,µ)andB(k,µ)
We now describe how we decompose the power spec- as follows. In every step of the fitting, given set of gδ(k)
truminredshiftspaceintospectraofdensityandvelocity. and gΘ(k), we compute their simple arithmetic means:
Before that, let us introduce two useful quantities con-
1 1
trolling the amplitude of the linear power spectra. We g¯δ = gδ(ki), g¯Θ = gΘ(ki), (13)
Nbin Nbin
define i i
X X
where the subscripti runs overk-bins up to a maximum
Pδliδn(k,z)=gδ2(k,z)Pδliδn(k,zlss), (8) wavenumber,kmax,andwedenotethenumberofthebins
PΘlinΘ(k,z)=gΘ2(k,z)PΘlinΘ(k,zlss), (9) byNbin. Again,wepre-computethecorrectiontermsfor
the fiducial cosmology using the standard perturbation
where zlss stands for the redshift at the time of the theory, which we denote Afid and Bfid, and scale them
last scattering. The parameters, gδ and gΘ, describe according to the average values of g and g :
δ θ
the growth rate of the density and the velocity fields
from that epoch. Since the spectrum Pδliδn(k,zlss) is well A(k,µ) = Afid k,µ; g¯δ , g¯θ ,
constrained by observations of the CMB temperature g¯fid g¯fid
(cid:18) δ θ (cid:19)
anisotropy, the parameters solely capture the properties g¯ g¯
ofthegravitationallaw,andareexpectedtobefreefrom B(k,µ) = Bfid k,µ; δ , θ . (14)
g¯fid g¯fid
the assumptions in the initial condition. (cid:18) δ θ (cid:19)
In reconstructing the spectra, we first bin the power In the above, the terms originated from δ (Θ) are multi-
spectrummeasuredfromsimulationsintobinsofkandµ. plied by g¯δ/g¯δfid (g¯Θ/g¯Θfid).
Then,forthei-thbinofwavenumberk,whichwedenote Wefinallyexplainourstrategyforthe FoGfactor. We
k , we estimate Plin(k ), Plin(k ) and Plin(k ) based on try both Gaussian and Lorentzian functions, and we let
i δδ i δΘ i ΘΘ i
the µ dependence. We assumethat linearδ andlinear Θ σ as afreeparameter. This parameteris determinedby
v
areperfectlycorrelated,Plin = PlinPlin,andtreattwo fitting globally the broadband shape of P˜(k,µ).
δΘ δδ ΘΘ In summary, our reconstruction strategy is as follows.
parameters,gδ(ki)andgΘ(ki),aqsfreeparametersforthe Wemodeltheredshift-spacepowerspectrum,P˜(k,µ),as
k -bin. Infitting P˜(k ,µ),wescalethe spectraaccording
i i
2
to the parameters gδ and gΘ: P˜(k,µ) = gδ(k) Pfid,lin(k)+δPfid(k)
Plin(k)= gi(k) gj(k) Pfid,lin(k). (10) ("(cid:18)gδfid(k)(cid:19) δδ δδ #
ij gifid(k)gjfid(k) ij +µ2 gδ(k) gΘ(k)Pfid,lin(k)+δPfid(k)
gfid(k)gfid(k) δΘ δΘ
(cid:20) δ Θ (cid:21)
where quantities with subscript “fid” are computed for
2
the fiducial cosmologicalmodel. +µ4 gΘ(k) Pfid,lin(k)+δPfid(k)
When we restrict the analysis to linear regime, the "(cid:18)gΘfid(k)(cid:19) ΘΘ ΘΘ #
aboveprocedureisexpectedtoworkproperly. Wefurther
elaborate the procedure to correctly handle the nonlin- +Afid k,µ; g¯δ , g¯Θ +Bfid k,µ; g¯δ , g¯Θ
g¯fid g¯fid g¯fid g¯fid
earity. We adopt the closure approximation (Taruya & (cid:18) δ Θ (cid:19) (cid:18) δ Θ (cid:19)(cid:27)
Hiramatsu [31]) to predict the nonlinear growth in the ×GX(kµ;σ ), (15)
v
4
fast Fourier transformation (FFT) using the cloud-in-
cell (CIC) method. We use bins in k and µ for the
following analysis. k is divided in ∆k = 0.01hMpc−1
linearly equally spaced bins from k = 0.01hMpc−1 to
0.2hMpc−1 and µ is in 20 linear-bins from 0 to 1 with
equalspacing. Theaveragesofmeasured2Dpowerspec-
train(k,µ)coordinateareshowninFig.1. TheGaussian
varianceisusedtoderiveerrorsforeachbinshownaser-
ror bars in Fig. 1, σ[P˜ob(k,µ)] = P˜ob(k,µ) 2/N(k,µ)
where N(k,µ) is number of modes in 20483(h−1Mpc)3
p
in Fourier space.
The overall amplitude of P˜(k ,µ ) at µ → 0 is solely
i j
determined by Plin(k ). And the running of P˜(k ,µ )
δδ i i j
along µ direction is determined by Plin(k ) about the
ΘΘ i
pivot point of P˜(k ,µ = 0). Those distinct contribu-
i j
tions of Plin and Plin to the observedspectra lead us to
δδ ΘΘ
simultaneously decompose both through data fitting to
the observed P˜(k ,µ ) in k and µ dimension. Addition-
i j
ally, non–perturbative effect is externally parameterized
using σ appearing in GX, while the higher order loop
v
corrections are expressed using the given linear spectra
parameters of Plin and Plin. Here parameterised one-
FIG. 1: The observed spectra of P˜(k,µ) are presented at scales δδ ΘΘ
0.03hMpc−1 < k < 0.11hMpc−1. Blue triangles represent av- dimensional velocity dispersion σv is set to be a scale-
eraged P˜(k,µ) at given k–µ bins from 60 realizations. The error independent free parameter.
bars are dispersions of measured values of 60 realizations divided We find best-fit parameter space of Plin(k ), Plin(k )
gg i ΘΘ i
by√60. and σ by minimizing [29],
v
imax 20 20
where factors with the subscript “fid” are pre-computed χ2 = [P˜ob(ki,µp)−P˜fit(ki,µp)]
for the fiducial cosmology, and gδ(ki) and gΘ(ki) as well i=XiminXp=1Xq=1
as σv are free parameters. We show results for various × Cov−pq1(ki)[P˜ob(ki,µq)−P˜fit(ki,µq)], (16)
cases in what follows. We include/exclude A(k,µ) and
B(k,µ), we adopt Gaussian and Lorentzian for the FoG where kmin is fixed to be kmin = 0.01hMpc−1, and best
damping factor, and we vary the maximum wavenumber kmax is determined in the following subsections. Off di-
included in the analysis, kmax. agonalelements of the covariancematrix are nearly neg-
ligible and those diagonal elements are written as
1
III. MEASUREMENTS OF LINEAR SPECTRA Cov−1(k )= . (17)
pp i σ[P˜ob(ki,µp)]2
A. The observed spectra P˜(k,µ) from mock We repeat this procedure for each 60 realization, and
catalogues report the averagesof best-fit values.
Forsubsequentanalysis,weusethedarkmatterdistri-
butions created by the simulations in Ref. [32]. The vol- B. Decomposition of linear density–density spectra
ume size of the N-body simulations is (2048h−1Mpc)3,
and we have 60 independent snapshots at each of the We present the result of decomposition of Plin. Lin-
δδ
four output redshifts, z = 0.35, 1, 2 and 3. The fiducial ear spectra of Plin are determined at a few first bins
δδ
cosmological parameters of the simulation are given by about µ = 0 in which the orientation of correlated two
(Ωm = 0.279,Ωb = 0.165Ωm,Ωk = 0,h = 0.701,σ8 = point pairs is transverse. The observed P˜ob at those µ
0.816,ns = 0.96). The distribution of dark matter par- bins are nearly equivalent to density–density spectra it-
ticles is modified according to their peculiar velocity to self. The procedure to decompose Plin is immune from
δδ
incorporate the redshift distortion effect. We adopt the allline-of-sightcontaminationdescribedinSec.II.There-
distant-observer approximation and measure the power fore density–density spectra are measured in high preci-
spectrum in (k ,k ) space, where subscripts ‘⊥’ and sion at arbitrary scale of k. But what we are interested
⊥ k
‘k’ denote perpendicular and parallel components to the in is measuring linear spectra of Plin. Unless the non–
δδ
line-of-sight. linearcontributioninthe decomposedP areseparated,
δδ
The density fluctuation field is constructed by assign- cross–correlation between density and velocity fields is
ing the dark matter particles to 10243 grids for the not guaranteed to be perfect.
5
FIG.2: DecomposedPδliδnarepresentedwithkmax=0.2hMpc−1. FIG.3: DecomposedPΘlinΘarepresentedwithvaryingkmax. (upper
(upperpanel)DashedcurverepresentsfiducialPδliδn. Bluetriangles panel)SolidcurverepresentsfiducialPΘlinΘ. Bluetrianglesrepresent
representfittingresultsusingtheoreticalformulationinEq.15,and fittingresultsusingkmax=0.11hMpc−1,andblackcirclesrepre-
blackcirclesrepresentfittingresultswithoutnon–linearcorrection sent fitting results using kmax = 0.15hMpc−1. (bottom panel)
ftoerrmmseiansuErqe.d1P5δ.liδn(biontttohmepuapnpeelr)pTahneelf.ractionalerrorsarepresented pTahneefl.racTthioinnablleurerocrusravreesprreepsreensteendtftohremeesatismuraetdedPΘelirnΘroirnstuhseinugpptheer
Fishermatrixanalysis.
The closure approximation in Eq. (11) is applied for
extracting linear information out of measured P . This locity fields is not parameterized in this paper. Instead,
δδ
we claim to probe linear spectra of density and veloc-
approximation breaks down at specific scale of k. The
upper bound of k is investigated to uncover the limit of ity fields using this closure approximation. Our test for
linearity of measured density spectra in this subsection
the closure approximation for density fields.
We use measured P˜ob(k,µ) up to k = 0.2hMpc−1 at is important to proceed next step of probing coherent
z =0.35. Thenthereare400measuredP˜ob(k,µ) ateach moAtidodnitsiponecatllrya,.measuring Plin(k) is also precious infor-
realization of simulated maps. Those measured data are δδ
fitted by the parameter space of (20Plin, 20Plin, σ ). mationtoprobedistancemeasures. Theimportantinfor-
δδ ΘΘ v mation about the evolution of the universe is imprinted
In order to handle a large parameter space, we adapt
on the large-scale structure of the universe. The broad-
MarkovchainMonte Carlo(MCMC) methods which are
bandshape ofthe powerspectrumprovidesthe informa-
a class of algorithms for sampling from probability dis-
tion about the horizon scale at the epoch of the matter–
tributions around equilibrium points.
radiation equality [33–37]. Although much of this infor-
Dashed curve in Fig. 2 represents the fiducial Plin(k)
δδ mationhasbeenfadedawayduetothenonlinearprocess,
in logarithmic scale. Black circle points in the top panel
the possible extension of distance measure detectability
represent measured P (k ) when non–linear correction
δδ i at bigger k is worth being investigated in the following
terms of δP in Eq. (11) are nullified. Fractional errors
ij works.
against linear spectra of density fields are shown in the
bottom panel of Fig. 2. Non–linearity of density fields
in measured P (k ) is observed from k = 0.1hMpc−1.
δδ i
Measured Plin(k ) deviates from Pfid(k) by 10% at k = C. Decomposition of coherent motion spectra
δδ i δδ
0.2hMpc−1. Blue triangle points represent measured
Pδδ(ki) including δPij. Seen at fractional error bars Results of PΘlinΘ decomposition are presented in this
in the bottom panel, linear spectra of Pδliδn(k) are well subsection. The running of P˜ob along the µ direction is
reproduced within a couple of percentage uncertainties caused by PΘlinΘ, and pivoted from P˜ob at µ = 0. The
through k =0.2hMpc−1. The closure approximation in observedP˜ob aremaximally affectedby peculiar velocity
Eq. (11) for density fields is proved to be trustable at when the orientation of correlation is radial, while mini-
least by this limit of k. mallyaffectedattransverseorientationofcorrelation. In
Cross–correlation coefficient between density and ve- the limit of µ → 1, P˜ob is significantly contaminated by
6
FIG. 5: WetestthecontributionofhigherordertermsofA(k,µ)
oFfIGk.=40:.1C0ohrrMelaptci−on1,buestiwnegeonnPeΘlrinΘealaiznadtiσovniosfpsrimesuenlatteeddactattahleogbuine a0n.1d1hBM(kp,µc−)1in. DEaqs.h1e5d.cTurhveesuprepperresbeonutntdhseofifdkuciisafilxPeΘldinΘa,taknmdabxlu=e
out of 60. The upper bound of k is kmax = 0.11hMpc−1. It trianglesarefittingresultsusingtheoreticalformulationinEq.15.
Black circles in the upper panel represents fitting results without
correspondstothelastbinofbluetrianglepointsinFig.3.
bi–spectral higher order terms of A(k,µ) in Eq. 15, black circles
in the bottom panel represents fitting results without quadratic
higherordertermsofB(k,µ)inEq.15.
allnon–linearsmearingeffectsdescribedinSec.II.Hence,
we develop an appropriate statistical tool to bridge the
improved theoretical model to real datasets.
the measuredPlin(k)becomesunderestimated[30]. The
The statistical methodology to treat non–linear cor- ΘΘ
black circle points in Fig. 3 represents decomposed
rection terms is described in Sec. IIB in detail. In this
method,the higherorderloopcorrectiontermsaregiven PΘlinΘ(k) with kmax = 0.15hMpc−1. This test indicates
that the unknown higher order terms of FoG effect be-
theoretically, but the uncertainty due to FoG effect is
come dominating above k = 0.11hMpc−1. Beyond
phenomenologically parameterized. At scales in which
thefirstordertermofFoGdominates,theassumptionof kmax = 0.11hMpc−1, the first order approximation of
FoGeffectisnotvalid,andthedecomposedPlin startto
FoG effect is valid through fitting σ with datasets. But ΘΘ
v be biased.
beyond this quasi–linear cut–off, our statistical model is
broken down. The decomposition of PΘlinΘ in our analy- We compare fractional errors of decomposed PΘlinΘ(k)
sis is limited by the uncertainty of FoG effect at higher withthetheoreticalestimationusingFishermatrixanal-
orders. ysis [13]. We do not marginalize the Fisher matrix
Using the simulated maps at z = 0.35, we find this with FoG effect. The estimated errors using kmax =
0.11hMpc−1 are presented as thin blue curves in Fig. 3.
upper bound of k scales in which our assumption is
When Plin(k) is not correlated much with FoG effect at
valid. We test two different upper bounds of kmax = ΘΘ
0.11hMpc−1 and0.15hMpc−1. Thebluetrianglepoints smaller k of k < 0.05hMpc−1 , the observed and esti-
in Fig. 3 represent the decomposed PΘlinΘ using kmax = mated errors agree to each other. But when PΘlinΘ(k) is
0.11hMpc−1. The fiducial spectra are well decomposed. affectedmuchby FoGeffectatk →0.1hMpc−1,the ob-
The best fit σv is 3.7h−1Mpc with kmax =0.11hMpc−1 servedfractionalerrorsincrease. Thecorrelationbetween
which corresponds to about 10% non–perturbative con- PΘlinΘ(k¯ =0.10hMpc−1) and σv is presented in Fig 4 us-
tribution to P˜ob at µ = 1. The detailed functional form ingonerealizationofsimulatedmapsoutof60. Thishigh
of FoG is not crucial in our statistical method, because correlation causes the increasing fractional errors about
most FoG functions agree at the first order approxima- a factor of 3.
tion. ThedecomposedPΘlinΘ(ki)isnotmuchdependenton Results in Fig. 5 present the contribution of higher–
types of FoG function, such as Gaussian or Lorentzian, order polynomials in Eq. (15). The decomposed Plin
ΘΘ
in this test. Hereafter, Gaussian function is chosen for will be overestimated or underestimated without A or
describing FoG effect. B. The fractional errors in the bottom panel of Fig. 5
When we increase kmax beyond kmax = 0.11hMpc−1, showthe results,when higher–orderpolynomials arenot
7
fully specify the cosmological model. This is indeed one
of the main reasons why the precision measurements of
the redshift-space distortionsand/orbaryonacoustic os-
cillations are highly desired to unlock the nature of late-
time cosmic acceleration. Hence, a large statistical error
ofthecosmologicalparameterswouldcauseanerroneous
estimation of the non-linear corrections that potentially
leads to a biased decomposition of the coherent motion
spectra. To quantify the size of this, we consider wrong
cosmological models, in which the density parameter of
the dark energy Ω differs from the fiducial value by 3,
v
5 and 10%, keeping the equation-of-state parameter of
darkenergyfixed. Tobeprecise,weadoptslightlylarger
values of Ω , while we fix the spatial curvature Ω , the
v k
spectral index n , the normalization of fluctuation am-
s
plitude atCMB scalesA ,andsomecombinationsofthe
s
parameters,Ωmh2 andΩbh2, since these areexpected to
be tightly constrained by the CMB observations. Then,
we compute the non-linear corrections to the redshift-
space power spectrum, and repeat the same analysis as
examined in previous section.
Fig.7 shows the decompositionresults ofthe coherent
FIG.6: MeasuredspectraofPΘlinΘ(ki)arepresentedathigherred- motion spectra at z =0.35. Here, we plot the fractional
shift at z = 1, z = 2 and z = 3 in the top, middle and bottom
errorsoftheresultantspectrainthecasesadoptinglarger
panelsrespectively.
values of Ω . As anticipated, there appears a clear sys-
v
tematictrendthatasincreasingΩ ,thecoherentmotion
v
consideredatall. Itisthecaseofcommonpracticeusing spectratendto deviatefromthe fiducialspectrum. Nev-
thesimplecombinationofKaiser’smodelandFoGeffect. ertheless, apart from k ∼ 0.1hMpc−1, the resultant size
We show that it is not sufficient for the future precision of the bias is rather small. This is because we are basi-
wide–deep surveys. cally looking at the scales where the contribution of the
Additionally, we test our method against higher red- corrections terms is small, and even the 10% change of
shift data at z =1, 2 and3 in Fig. 6. The upper bounds the cosmological parameter gives a little effect on the
of kmax for these redshifts are larger than the low red- non-linear corrections. Also, a slight mismatch of the
shift of z =0.35 as expected, since the damping scale of non-linear corrections can be partly absorbed into the
FoG effect becomes rather milder at higher redshift. We FoG damping factor, which further reduces the impact
found that the linear velocity spectra Plin are precisely ofwrongpriorassumptions. Since the uncertaintyofthe
ΘΘ
measured up to kmax = 0.12, 0.14 and 0.18hMpc−1 at future constraint on Ωv will not be as large as 10%, the
results shown in Fig. 7 may be regarded as a very good
z =1, 2 and 3, respectively.
news,suggestingthatthecoherentmotionspectrawould
be decomposed successfully in a less biased manner.
IV. DISCUSSION
The decomposition results shown in previous section V. CONCLUSION
rely on an idealistic assumption that the nonlinear cor-
rectionsto the redshift-space powerspectra, δP , A and We have presented an improved prescription to recon-
ij
B, are known a priori for the fiducial cosmology. Since struct coherent motion spectra from the matter power
thesecorrectionsmustbecomputedforagivensetofcos- spectrum in redshift space, properly taking account of
mologicalparameters,they mightbe sensitive to the un- the non–linear effects of both the structure growth and
derlying cosmological model and the decomposed power redshift distortions. Statistical analysis is presented to
spectracouldbebiasedifweadoptthewrongcosmologi- bridge theoretical models of redshift distortion to real
calpriors. Inthissection,wediscusstheimpactofwrong dataset. Based on the perturbation theory treatment,
cosmological assumption on the decomposition of power non–linear correction terms of higher-order polynomials
spectra. and non–linear growth functions are appropriately in-
In principle, the on-going and upcoming CMB exper- cluded in the reconstruction analysis. Those contribu-
iments will provide a way to precisely determine a set tions are proved to be influential even at linear scales,
of cosmologicalparameters, from which we can compute andmeasurementsofcoherentmotionspectraaremisled
the non-linearcorrections. Because of the parameter de- without it. On the other hand, non–perturbative cor-
generacies, however, the CMB observation alone cannot rection term such as FoG effect is parameterized. Those
8
allknowns andunknowns in our analysissuccessfully re-
produce fiducial spectra of coherent motion up to some
limited scales. Although it is still steps away to achieve
completeandpracticalobservationaltoolsmeasuringco-
herent motion spectra, results show that our analysis
method righteous path to be developed to meet real
world.
Acknowledgments
Numericalcalculationswereperformedbyusingahigh
performance computing cluster in the Korea Astronomy
and Space Science Institute, Cray XT4 at Center for
Computational Astrophysics, CfCA, of National Astro-
nomical Observatory of Japan, and under the Interdis-
ciplinary Computational Science Program in Center for
Computational Sciences, University of Tsukuba. This
work is supported in part by a Grant-in-Aid for Sci-
entific Research from the Japan Society for the Pro-
motion of Science (JSPS) (No. 24740171 for I.K and
FIG.7: FractionalerrorsofmeasuredspectraofPΘlinΘ(ki)arepre- No. 24540257 for A.T). T. N. is supported by a Grant-
sentedatz=0.35usingdiversetemplatesatdifferentcosmological
models. Results with templates of Ωv larger by 3%, 5% and 10% in-AidforJSPSFellows(PD:22-181)andbyWorldPre-
thantrueΩv areshownattop,middleandbottom panelsrespec- mier International Research Center Initiative (WPI Ini-
tively. tiative), MEXT, Japan.
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