Determination of the large scale volume weighted halo velocity bias in simulations
aa r X i v : . [ a s t r o - ph . C O ] J un Determination of the large scale volume weighted halo velocity bias in simulations
Yi Zheng,
1, 2
Pengjie Zhang,
3, 4, 1, ∗ and Yipeng Jing
3, 4, † Key Laboratory for Research in Galaxies and Cosmology,Shanghai Astronomical Observatory, 80 Nandan Road, Shanghai, 200030, China Korea Astronomy and Space Science Institute, Daejeon 305-348, Republic of Korea Center for Astronomy and Astrophysics, Department of Physics and Astronomy,Shanghai Jiao Tong University, 955 Jianchuan road, Shanghai, 200240 IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China
A profound assumption in peculiar velocity cosmology is b v = 1 at sufficiently large scales, where b v is the volume weighted halo(galaxy) velocity bias with respect to the matter velocity field. However,this fundamental assumption has not been robustly verified in numerical simulations. Furthermore,it is challenged by structure formation theory (BBKS, 1986, ApJ; Desjacques and Sheth, 2010,PRD), which predicts the existence of velocity bias (at least for proto-halos) due to the fact thathalos reside in special regions (local density peaks). The major obstacle to measure the volumeweighted velocity from N-body simulations is an unphysical sampling artifact. It is entangled in themeasured velocity statistics and becomes significant for sparse populations. With recently improvedunderstanding of the sampling artifact (Zhang, Zheng and Jing, 2015, PRD; Zheng, Zhang andJing, 2015, PRD), for the first time we are able to appropriately correct this sampling artifact andthen robustly measure the volume weighted halo velocity bias . (1) We verify b v = 1 within 2% modeluncertainty at k < ∼ . h/ Mpc and z = 0-2 for halos of mass ∼ -10 h − M ⊙ , and, therefore,consolidates a foundation of the peculiar velocity cosmology. (2) We also find statistically significantsigns of b v = 1 at k > ∼ . h/ Mpc. Unfortunately, whether this is real or caused by residual samplingartifact requires further investigation. Nevertheless, cosmology based on k > ∼ . h/ Mpc velocitydata shall be careful this potential velocity bias.
PACS numbers: 98.80.-k; 98.80.Es; 98.80.Bp; 95.36.+x
INTRODUCTION
Large scale peculiar velocity is maturing as a pow-erful probe of cosmology. Peculiar velocity directly re-sponds to the gravitational pull of all clustered matterand energy, making it a precious tool to study dark mat-ter (DM), dark energy, and the nature of gravity (e.g.[1–5]). Measuring peculiar velocity at cosmological dis-tances with the conventional method of distance indica-tors is challenging, albeit improving (e.g., [6, 7]). Al-ternatively, redshift space distortion (RSD) provides away of measuring peculiar velocity at cosmological dis-tances, free of the otherwise overwhelming contaminationof Hubble flow. It enables ∼
1% accuracy in the velocitypower spectrum measurement at z ∼ b v of galaxies vanishesat large scales ( b v = 1), namely, that the galaxy velocityfield is statistically identical to that of the matter veloc-ity field at large scales. The strong equivalence principlepredicts that galaxies sense the same acceleration as am-bient DM particles. Hence, one would naturally expectstatistically identical velocity for galaxies and DM parti- ∗ Email me at: [email protected] † Email me at: [email protected] cles, at > ∼ /h scales where the only operating forceis gravity. However, a loop hole in this argument is thatgalaxies and their host halos only reside in special regions(local density peaks). The same environmental differenceis known to cause b v < v ∝ f Db v at large scale, uncertainties in b v lead to systematic errorin all existing f D measurements [16], δ ( f D ) f D = 1 − b − v . (1)Here f ≡ d ln D/d ln a and D is the linear density growthfactor. Therefore we have to understand b v to 1% or bet-ter to make the peculiar velocity competitive with otherdark energy probes.A key intermediate step to understand the galaxy ve-locity bias is to understand the halo velocity bias [27].N-body simulations are ideal to robustly clarify this is-sue. What is most relevant for cosmology, in particu-lar RSD cosmology, is the volume weighted halo velocitybias at large scales [28]. Unfortunately, measuring thevolume weighted velocity statistics through inhomoge-neously and sparsely distributed particles/halos is highlychallenging, due to a sampling artifact [17–22].This sampling artifact arises from the fact that weonly have information of velocities at positions of par-ticles/halos. Therefore the sampling of the volumeweighted velocity field is incomplete. Even worse, sincethe particle/halo velocity field is correlated with the par-ticle/halo distribution, the sampling of volume weightedvelocity field is imperfect. Such completeness and im-perfection leads to inaccurate measurement of velocitystatistics, which we call the “sampling artifact”. Forsparse populations, it can cause ∼
10% systematic un-derestimation of the velocity power spectrum at k =0 . h/ Mpc [20–22]. Even worse, it also depends on theintrinsic LSS (large scale structure) fluctuation in theparticle distribution and its correlation with velocity [21].This sampling artifact is by itself unphysical, in the sensethat it solely arises from the limitation of robustly mea-suring the volume weighted velocity statistics given theinhomogeneously and sparsely distributed velocity data.Given its existence, the rawly measured velocity biasfrom simulation is a mixture of the real velocity bias andthe sampling artifact in the following form:ˆ b v (wrong) = b v (true) × sampling artifact . (2)Namely, the raw bias measurement ˆ b v is wrong by a mul-tiplicative factor caused by the sampling artifact. With-out rigorous correction, the sampling artifact can be mis-interpreted as a significant velocity bias and mislead thepeculiar velocity cosmology.Therefore, robustly understanding the sampling arti-fact is a prerequisite for reliably measuring the true ve-locity bias. For this purpose, we developed the theoryof the sampling artifact in [20] and rigorously confirmedthe existence of the sampling artifact in simulations [21].We further tested the theory against simulations and im-proved it to 1% accuracy at k = 0 . h/ Mpc for popu-lations with number density > ∼ − (Mpc /h ) − [21]. Inparticular, [21] demonstrates the sharp distinction be-tween a real velocity bias and the sampling artifact, forDM samples. It first constructs samples with a fractionof the simulation DM particles randomly selected fromall the simulation particles. By construction, the veloc-ity statistics of the random samples shall be statisticallyidentical to those of the sample including all simulationDM particles. Namely b v (DM) = 1. However, the rawmeasurement shows ˆ b v (DM) = 1 of high significance [21].The fake ˆ b v (DM) = 1 then clearly demonstrates the sam-pling artifact (Eq. 2).In the current paper, we applied this improved un-derstanding of the sampling artifact [20, 21] to robustlyeliminate it in velocity measurement and correctly deter-mine the true volume weighted halo velocity bias for thefirst time. This differs from existing numerical works onmeasuring velocity bias [12, 13, 22, 23], which either focuson proto-halos, the density weighted halo velocity statis-tics, or the volume weighted halo velocity mixed with thesampling artifact. SIMULATION SPECIFICATIONS
We analyze the same J1200 N-body simulation in [21],run with a particle-particle-particle-mesh (P M) code[24]. It adopts a ΛCDM cosmology with Ω m = 0 . Λ = 0 . b = 0 . σ = 0 . n s = 1 and h = 0 . /h box size, 1024 particles and massresolution of 1 . × M ⊙ /h . The halo catalogue isconstructed by Friends-of-Friends (FOF) method with alinking length b = 0 .
2. Gravitationally unbound “halos”have been excluded from the catalogue. In total we have N h = 6 . × halos with at least 10 simulation par-ticles, at z = 0. We choose the mass weighted center asthe halo center and the velocity averaged over all mem-ber particles as the halo velocity. We try three mass binsdetailed in table I. Set ID mass range h M i N h / n h b h (density) A1 ( z = 0 .
0) 10-3700 39 7 . . . z = 0 . . z = 1 . . z = 2 . . .
76 4.3 A2 ( z = 0 .
0) 1 . . . z = 0 . . z = 1 . . z = 2 . . B ( z = 0 .
0) 2 . . M ⊙ /h and the halo number density n h has unit of10 − (Mpc /h ) − . h M i is the mean halo mass. N h is the to-tal halo number in one halo mass bin. The density bias b h is averaged around k = 0 . h/ Mpc. The mass bin B at z = 0 has density bias of unity, designed for better control ofthe sampling artifact. CORRECTING THE SAMPLING ARTIFACT
We aim to measure the halo velocity bias defined inFourier space, b v ( k ) ≡ s P vh,E ( k ) P v DM ,E ( k ) . (3)The subscript “E” denotes the gradient (irrotational)part of the velocity, which is most relevant for pecu-liar velocity cosmology. The subscripts“ h ” and “DM”refer to halos and DM simulation particles respectively.Throughout this paper, we restrict ourself to the vol-ume weighted power spectrum. We adopt the NP (Near-est Particle) method [19] to sample the velocity field on256 uniform grids. Before correcting the sampling arti-fact, the measured velocity power spectrum ˆ P E ( k ) differsfrom its true value by a factor C ( k ) ≡ ˆ P vE ( k ) /P vE ( k ). We FIG. 1: The measured velocity bias for mass bin A1 and B atdifferent redshifts (Table 1). The data points connected withsolid lines are the final results, after correcting the samplingartifact . The error bars are the r.m.s dispersions between 10realizations of DM control samples. For comparison, we alsoshow the raw measurements (dashed curves), which are essen-tially the sampling artifact, unrelated to the physical veloc-ity bias. Our correction of the sampling artifact has percentlevel model uncertainty at k = 0 . h/ Mpc and we have high-lighted it with the two dashed straight lines with somewhatarbitrary value 1 ± .
02. After correction, we find b v = 1at k < ∼ . h/ Mpc within 2% model uncertainty and henceconsolidate this fundamental assumption of peculiar velocitycosmology. At k > ∼ . h/ Mpc, there are signs of b v = 1, whichrequire further rigorous investigation/verification. Measuringthe velocity bias to higher accuracy requires improvement overthe existing understanding of the sampling artifact. found that [21] C ( k ) ≃ h e i k · D i e k ξ D ( r = α/k ) / ≡ C T ( k ) . (4)Here, D is the deflection field pointing from a parti-cle used for velocity assignment to the correspondinggrid point to which the velocity is assigned. Reference[20] showed that D fully captures the sampling artifact. ξ D is the spatial correlation of D . For α = 1 / n P ∼ − (Mpc /h ) − , C T ( k ) agrees with the actual C ( k ) to ∼
1% at k ≤ . h/ Mpc [21]. The subscript “T”denotes that it is our theory prediction. We caution thatthe theoretically predicted C T ( k ) may deviate from thetrue C ( k ) since the theory prediction is not exact. Fur-thermore, C (or C T ) of dark matter particles can differfrom that of dark matter halos.We take two steps to correct for the sampling artifact. Step one . We use Eq. 4 to correct for the bulk of the
FIG. 2: Same as Fig. 1, but for mass bin A2 and B (Table 1).The apparent “anti-bias” before correction disappears aftercorrection. Roughly speaking, the corrected velocity bias at k < . h/ Mpc is consistent with unity within 2% model un-certainty. Nevertheless, there are signs of increasing b v withincreasing k and z . sampling artifact. Step two . There are residual sam-pling artifact since our theory is not perfect ( C T = C ).We further correct this residual sampling artifact withthe aid of DM control samples (DMCs). They are con-structed by randomly selecting simulation DM particlesfrom the full simulation sample with the requirement σ D (DMC) = σ D (halo) [29]. σ D ≡ h D i / is the dom-inant factor determining the sampling artifact [20, 21].The halo sample and DMCs have identical sampling ar-tifact at the k → C h,T /C h ≃ C DMC , T /C DMC . The subscripts “ h ”and“DMC” denote properties of halos and DM controlsamples, respectively. We then obtain b v ( k ) ≃ vuut ˆ P vh,E ( k ) P v DM ,E ( k ) s C h,T ( k ) s C DMC , T ( k ) C DMC ( k ) . (5)The terms on the r.h.s. are, respectively, the raw velocitybias measurement without correcting the sampling arti-fact, the step one correction, and the step two correction. P v DM , E is measured from the full J1200 simulation sam-ple, which is essentially free of sampling artifact due toits high ¯ n P [19]. All the correction terms ( C h,T , C DMC , T and C DMC ) are directly calculated from the J1200 simu-lation.The inaccuracy of Eq. 5 increases with k . For peculiarvelocity cosmology to be competitive, at least we shallutilize measurement at k ≤ . h/ Mpc. So we choose k = 0 . h/ Mpc as the pivot scale for quoting the accu-racy. Overall we expect ∼
1% accuracy [30], extrapo-lating from the DM cases. We caution the readers onthis ∼
1% uncertainty in the measured b v ( k ) (Figs. 1 &2). Somewhat arbitrary, we quote the systematic errorin the measured b v as 2% at k < . h/ Mpc. Therefore,only if | b v − | > .
02 at k < . h/ Mpc, are we capable ofdetecting a non-unity velocity bias. More accurate mea-surement of velocity bias requires better correction of thesampling artifact to below 1%, either by improved mod-elling of the sampling artifact, or by improved velocityassignment method (e.g. [25]).
NO VELOCITY BIAS AT k < ∼ . h/ Mpc
Figs. 1 & 2 show b v for all mass bins listed in Table1. The raw measurements suggest an “anti-bias”, unan-imous for all mass bins at all redshifts. This is most sig-nificant for the more massive bin A
1, reaching b v ∼ . k = 0 . h/ Mpc (Fig. 1). Possibly by coincidence,this “anti-bias” agrees well with theoretical predictionsof proto-halos based on linear/Gaussian statistics. How-ever, we have solid evidences that it is essentially an il-lusion caused by the sampling artifact. The sampling ar-tifact causes systematic suppression of P v [20–22], mim-icking an anti-bias. In another word, the apparent “anti-bias” is unreal in the sense that is unrelated to the truevelocity statistics of halos and is irrelevant for cosmology.Theoretically, we expect the sampling artifact to ex-ist for any populations of inhomogeneously distributedobjects, and its impact to be significant for sparse popu-lations [20]. It has been robustly detected for the case ofDM simulation particles [19, 21]. Therefore, it must alsoexist for DM halos [20]. Fig. 3 further consolidates thistheoretical prediction. It shows that the DM control sam-ples containing a fraction of DM simulation particles havesmaller P v than the full DM sample at k < ∼ . h/ Mpc.Furthermore, P v decreases with decreasing number den-sity. If the number densities of these DM control samplesmatch those of halo samples, their P v match each otherclosely, especially at k < . h/ Mpc. This behavior holdsfor all three mass bins and four redshifts investigated.The DM control samples are constructed by randomlyselecting DM simulation particles, so by construction thedifference in P v to the full DM sample should not existand any difference must be caused by the sampling arti-fact. The similarities between DM control samples andhalo samples then strongly suggest that the “anti-bias”implied by the raw measurement is merely the samplingartifact and is therefore unrealistic . The bin A M > M ⊙ /h is a factor of ∼
10 more sparse than A M < M ⊙ /h , so it suffers from a larger sampling FIG. 3: The sampling artifact in the velocity power spectrummeasured in N-body simulations, which causes systematic un-derestimation at k < ∼ . h/ Mpc. (1) The halo velocity powerspectra (dash lines) are lower than measured from all DMsimulations particles (dash-dot line). (2) The velocity powerspectra of DM control samples containing a fraction of allsimulation particles are also lower. Member particles in thecontrol samples are randomly selected from the full simulationparticles and hence must have statistically identical velocitypower spectra. Therefore the observed deficit in the DM ve-locity power spectrum is caused by the sampling artifact [21].(3) When the number density of DM control samples and halosamples are identical, they have similar (but not identical) ve-locity power spectra and similar deficit with respect to the fullDM sample. These are solid evidences of significant samplingartifact in the measured halo velocity power spectrum. Themost crucial step in measuring the halo velocity bias is to un-derstand and correct this sampling artifact. This is the solepurpose of our two preceding works [20, 21], which show thatthe sampling artifact depends on not only the number density,but also on the intrinsic LSS fluctuation and its correlationwith the velocity field. artifact, ∼
10% at k = 0 . h/ Mpc.Hence, it is essential to correct for the sampling arti-fact. After applying the two step corrections (Eq. 5), the“anti-bias” disappears and we find b v ( k ≤ . h/ Mpc) = 1within 2 σ statistical uncertainty, for the A1 bin at all red-shifts. Taking the extra 2% systematic uncertainty intoaccount, we find no evidence on a non-unity velocity biasat k ≤ . h/ Mpc for halos bigger than 10 M ⊙ /h . At k ≤ . h/ Mpc, b v of bin A b v > b v = 1at k ≤ . h/ Mpc (perhaps except z = 2).How solid are these results? To check it, we constructa mass bin B with M > . × M ⊙ /h . It has identicallarge scale LSS fluctuation as DMCs, so we can betterhandle its sampling artifact by comparing with DMCs.Thus we treat the b v measurement of bin B as the mostaccurate halo velocity bias measurement that we canachieve. Again we find b v = 1 at k ≤ . h/ Mpc. There-fore we conclude that b v = 1 at k < . h/ Mpc within 2%model uncertainty. Settling the issue of whether b v = 1at greater accuracy requires further improvement overexisting understanding of the sampling artifact [21] orbetter velocity assignment method.The vanishing velocity bias ( b v = 1) within 2% modeluncertainty at k < . h/ Mpc verifies a fundamental as-sumption in peculiar velocity cosmology. However, fromthe theoretical viewpoint, this result is quite surpris-ing, as linear theory predicts b v ( k = 0 . h/ Mpc) ≃ . ∼ M ⊙ /h proto-halos (peaks in initial/linearlyevolved density field)[9–11, 13, 14]. The predicted b v < VELOCITY BIAS b v = 1 AT k > ∼ . h/ Mpc ? On the other hand, at k > ∼ . h/ Mpc there are signs of b v = 1 and signs of mass and redshift dependences. (1)For mass bin A
2, the data suggest that b v > b v − k and z . The excess is statisti-cally significant at z = 2 and k ≥ . h/ Mpc. (2) In con-trast, bin A > M ⊙ /h ) has b v ( k > . h/ Mpc) < b v − k > . h/ Mpc andthe difference reaches 10% at k ∼ . h/ Mpc. If thisdifference is indeed intrinsic, instead of a residual sam-pling artifact, it could be caused by different environ-ments where different halos reside. Small halos tend tolive in filaments and have extra infall velocity with re-spect to large halos. The infall velocity has a correlationlength of typical filament length of tens of Mpc, and, hence, shows up at k > ∼ . h/ Mpc. Unfortunately, ourunderstanding of the sampling artifact at k > . h/ Mpcis considerably poorer [21]. Therefore we are not ableto draw decisive conclusions, other than that cosmologybased on the peculiar velocity at k > ∼ . h/ Mpc mustkeep caution on this potential velocity bias.
CONCLUSIONS AND DISCUSSIONS
This paper presents the first determination of volumeweighted halo velocity bias through N-body simulations.The raw measurements suffer from a severe sampling ar-tifact which could be misinterpreted as a significant “ve-locity bias.” We are able to appropriately correct thesampling artifact following our previous works [20, 21]and measure the physical velocity bias. Two major find-ings are as follows: • b v = 1 at k ≤ . h/ Mpc within 2% model uncer-tainty. It consolidates the peculiar velocity cosmol-ogy; • Signs of b v = 1 at k > ∼ . h/ Mpc and signs that b v − k > ∼ . h/ Mpc velocity data toconstrain cosmology.Accurate measurement of the velocity bias in simula-tions heavily relies on robust correction of the samplingartifact. The sampling artifact depends on not only thehalo number density, but also on the intrinsic LSS fluc-tuation of the halo distribution and its correlation withthe halo velocity field [21]. It is for this reason that ourunderstanding of the sampling artifact at k ∼ . h/ Mpcso far is no better than 1%. Therefore, we caution thereaders that the measured velocity bias has ∼
1% (orsomewhat arbitrarily 2%) model uncertainty (systematicerror). For a similar reason, we cannot fully quantifythe accuracy of Eq. 5 and the accuracy in the samplingartifact corrected b v [31]. We know it is more accuratethan Eq. 4 and have estimated its accuracy by extrap-olating from the DM cases [21]. Nevertheless, this am-biguity prevents us from drawing an unambiguous con-clusion on whether the found b v = 1 at k > . h/ Mpcis real, or whether b v deviates from unity by less than1% at k < . h/ Mpc. Therefore a major future workwill be to improve understanding of the sampling artifact(e.g., discussions in the Appendix of [21]). Furthermore,we will explore other velocity assignment methods whichmay alleviate the problem of the sampling artifact. Wewill also extend to galaxy velocity bias with mock galaxycatalogues, where the sampling artifact should also becorrected.Finally, we address that the RSD determines velocityindirectly through the galaxy number density distribu-tion and, therefore, the RSD inferred velocity statisticscan be free of the sampling artifact. This is another ad-vantage of RSD over conventional velocity measurementmethods. It is only when comparing the RSD determinedvelocity power spectrum with that measured in simula-tions that we must worry about the sampling artifact inthe simulation part.
Acknowledgement .— We thank Cris Sabiu’s forhis help in proofreading this paper. This workwas supported by the National Science Foundation ofChina (Grants No. 11025316, No. 11121062, No.11033006, No. 11320101002, and No. 11433001),the National Basic Research Program of China (973Program 2015CB857001), the Strategic Priority Re-search Program “The Emergence of Cosmological Struc-tures” of the Chinese Academy of Sciences (Grant No.XDB09000000), and the key laboratory grant from theOffice of Science and Technology, Shanghai MunicipalGovernment (No. 11DZ2260700). [1] J. A. Peacock, S. Cole, P. Norberg, C. M. Baugh,J. Bland-Hawthorn, T. Bridges, R. D. Cannon, M. Col-less, C. Collins, W. Couch, et al., Nature (London) ,169 (2001), arXiv:astro-ph/0103143.[2] P. Zhang, M. Liguori, R. Bean, and S. Dodelson, PhysicalReview Letters , 141302 (2007), 0704.1932.[3] L. Guzzo, M. Pierleoni, B. Meneux, E. Branchini, O. LeF`evre, C. Marinoni, B. Garilli, J. Blaizot, G. De Lu-cia, A. Pollo, et al., Nature (London) , 541 (2008),0802.1944.[4] B. Jain and P. Zhang, Phys. Rev. D , 063503 (2008),0709.2375.[5] B. Li, W. A. Hellwing, K. Koyama, G.-B. Zhao, E. Jen-nings, and C. M. Baugh, MNRAS , 743 (2013),1206.4317.[6] A. Johnson, C. Blake, J. Koda, Y.-Z. Ma, M. Col-less, M. Crocce, T. M. Davis, H. Jones, J. R. Lucey,C. Magoulas, et al., ArXiv e-prints (2014), 1404.3799.[7] R. Watkins and H. A. Feldman, ArXiv e-prints (2014),1407.6940.[8] D. Schlegel, F. Abdalla, T. Abraham, C. Ahn, C. AllendePrieto, J. Annis, E. Aubourg, M. Azzaro, S. B. C. Baltay,C. Baugh, et al., ArXiv e-prints (2011), 1106.1706.[9] J. M. Bardeen, J. R. Bond, N. Kaiser, and A. S. Szalay,Astrophys. J. , 15 (1986).[10] V. Desjacques, Phys. Rev. D , 103503 (2008),0806.0007.[11] V. Desjacques and R. K. Sheth, Phys. Rev. D , 023526(2010), 0909.4544.[12] A. Elia, A. D. Ludlow, and C. Porciani, MNRAS ,3472 (2012), 1111.4211.[13] T. Baldauf, V. Desjacques, and U. Seljak, ArXiv e-prints(2014), 1405.5885.[14] M. Biagetti, V. Desjacques, A. Kehagias, and A. Riotto,Phys. Rev. D , 103529 (2014), 1408.0293. [15] J. M. Colberg, S. D. M. White, T. J. MacFarland,A. Jenkins, F. R. Pearce, C. S. Frenk, P. A. Thomas,and H. M. P. Couchman, MNRAS , 229 (2000).[16] C.-H. Chuang, F. Prada, F. Beutler, D. J. Eisenstein,S. Escoffier, S. Ho, J.-P. Kneib, M. Manera, S. E. Nuza,D. J. Schlegel, et al., ArXiv e-prints (2013), 1312.4889.[17] F. Bernardeau and R. van de Weygaert, MNRAS ,693 (1996).[18] S. Pueblas and R. Scoccimarro, Phys. Rev. D , 043504(2009), 0809.4606.[19] Y. Zheng, P. Zhang, Y. Jing, W. Lin, and J. Pan, Phys.Rev. D , 103510 (2013), 1308.0886.[20] P. Zhang, Y. Zheng, and Y. Jing, Phys. Rev. D ,043522 (2015), 1405.7125.[21] Y. Zheng, P. Zhang, and Y. Jing, Phys. Rev. D ,043523 (2015), 1409.6809.[22] E. Jennings, C. M. Baugh, and D. Hatt, ArXiv e-prints(2014), 1407.7296.[23] P. Col´ın, A. A. Klypin, and A. V. Kravtsov, Astrophys.J. , 561 (2000), astro-ph/9907337.[24] Y. P. Jing, Y. Suto, and H. J. Mo, Astrophys. J. ,664 (2007), arXiv:astro-ph/0610099.[25] Y. Yu, J. Zhang, Y. Jing, and P. Zhang, ArXiv e-prints(2015), 1505.06827.[26] P. Zhang, J. Pan, and Y. Zheng, Phys. Rev. D , 063526(2013), 1207.2722.[27] Galaxies in a halo have extra velocities relative to thehost halo. The correlation length of this velocity fieldis < ∼ /h , so it does not contribute to velocity at k ∼ . h/ Mpc of interest. Therefore the large scale halovelocity and galaxy velocity are identical, statisticallyspeaking.[28] For example, the velocity power spectrum determinedfrom RSD is volume weighted in the framework of [26].[29] For this requirement, DMCs are more sparse than thehalo samples.[30] For all the investigated halo bins, except A z = 2),¯ n h > × − (Mpc /h ) − . Based on [21], we expect ∼
1% accuracy in the step one correction (Eq. 4). For A z = 2), the number density is lower (0 . × − ).However Eq. 4 is more accurate at z = 2 than z = 0[21]. Hence we also expect Eq. 5 to be accurate at ∼ n h → ∞→ ∞