Cosmic velocity--gravity relation in redshift space
aa r X i v : . [ a s t r o - ph ] M a y Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 30 October 2018 (MN L A TEX style file v2.2)
Cosmic velocity–gravity relation in redshift space
St´ephane Colombi, ⋆ Michał J. Chodorowski † and Romain Teyssier ‡ Institut d’Astrophysique de Paris, CNRS, 98 bis Boulevard Arago, 75014 Paris, France Copernicus Astronomical Center, Bartycka 18, 00–716 Warsaw, Poland Commissariat `a l’Energie Atomique, Direction des Sciences de la Mati`ere, Service d’Astrophysique,Centre d’ ´Etudes de Saclay, L’orme des Merisiers, 91191 Gif-sur-Yvette Cedex, France
30 October 2018
ABSTRACT
We propose a simple way to estimate the parameter β ≃ Ω . /b from three-dimensionalgalaxy surveys, where Ω is the non-relativistic matter density parameter of the Universe and b is the bias between the galaxy distribution and the total matter distribution. Our methodconsists in measuring the relation between the cosmological velocity and gravity fields, andthus requires peculiar velocity measurements. The relation is measured directly in redshiftspace , so there is no need to reconstruct the density field in real space. In linear theory, theradial components of the gravity and velocity fields in redshift space are expected to be tightlycorrelated, with a slope given, in the distant observer approximation, by gv = p β/ β / β . We test extensively this relation using controlled numerical experiments based on a cosmolog-ical N -body simulation. To perform the measurements, we propose a new and rather simpleadaptive interpolation scheme to estimate the velocity and the gravity field on a grid.One of the most striking results is that nonlinear effects, including ‘fingers of God’, affectmainly the tails of the joint probability distribution function (PDF) of the velocity and gravityfield: the – . σ region around the maximum of the PDF is dominated by the linear theoryregime , both in real and redshift space. This is understood explicitly by using the sphericalcollapse model as a proxy of nonlinear dynamics.Applications of the method to real galaxy catalogs are discussed, including a preliminaryinvestigation on homogeneous (volume limited) “galaxy” samples extracted from the simula-tion with simple prescriptions based on halo and sub-structure identification, to quantify theeffects of the bias between the galaxy distribution and the total matter distribution, as well asthe effects of shot noise. Key words: methods: analytical – methods: numerical – cosmology: theory – dark matter –large-scale structure of Universe.
Analyzes of large-scale structure of the Universe provide estimatesof cosmological parameters that are complementary to those fromthe cosmic microwave background measurements. In particular,comparing the large-scale distribution of galaxies to their peculiarvelocities enables one to constrain the quantity β ≡ Ω . /b . Here, Ω is the cosmological non-relativistic matter density parameter and b is the linear bias of galaxies that are used to trace the underly-ing mass distribution. This is so because the peculiar velocity field, v , is induced gravitationally and is therefore tightly coupled to the ⋆ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] matter distribution. In the linear regime, this relationship takes theform (Peebles 1980) v ( r ) = Ω . Z d r ′ π δ ( r ′ ) r ′ − r | r ′ − r | , (1)where δ denotes the mass density contrast and distances have beenexpressed in km · s − . Under the assumption of linear bias δ = b − δ ( g ) , where δ ( g ) denotes the density contrast of galaxies, theamplitude of peculiar velocities depends linearly on β .Density–velocity comparisons are done by extracting the den-sity field from full-sky redshift surveys (such as the PSCz, Saunderset al. 2000; or the 2MRS, Erdo ˘g du et al. 2006), and comparing itto the observed velocity field from peculiar velocity surveys (suchas the Mark III catalog, Willick et al. 1997; ENEAR, da Costaet al. 2000; and more recently SFI++, Masters et al. 2005). The c (cid:13) Colombi, Chodorowski & Teyssier methods for doing this fall into two broad categories. One can useequation (1) to calculate the predicted velocity field from a redshiftsurvey, and compare the result with the measured peculiar veloc-ity field; this is referred to as a velocity–velocity comparison (e.g.,Kaiser et al. 1991; Willick & Strauss 1998). Alternatively, one canuse the differential form of this equation, and calculate the diver-gence of the observed velocity field to compare directly with thedensity field from a redshift survey; this is called a density–densitycomparison (e.g., Dekel et al. 1993; Sigad et al. 1998). The advan-tage of density–density comparisons is that they are purely local,but they are significantly sensitive to shot noise because the diver-gence of the observed velocity field is estimated from the sparse ve-locity sample. The integral form of velocity–velocity comparisonsmake them much less sensitive to such a noise, but their non-localnature make them affected by the tides due to the presence of fluc-tuations outside the survey volume (see, e.g., Kaiser & Stebbins1991).Still, the common problem with both types of these compar-isons is that while equation (1) involves the density field in realspace, the observed density field is given solely in redshift space.One approach to tackle with this problem is to reconstruct the real-space density field from the redshift-space one. The transformationfrom the real space coordinate, r , to the redshift space coordinate, s , is s = r + v ( r ) ˆ r , (2)where v ( r ) = v · ˆ r ; v ( r ) is the real-space velocity field, and ve-locities are measured relative to the rest frame of CMB. Therefore,one has to correct galaxy positions for their peculiar velocities. Todo so, equation (1) is used and a self-consistent solution for thedensity field in real space is usually obtained iteratively (Yahil etal. 1991). In the first iteration, to predict peculiar velocities accord-ing to equation (1), the real-space density field appearing on ther.h.s. of this equation is approximated by the redshift space-densityfield, and so on, until convergence. However, from equation (1) itis obvious that the amplitude of peculiar velocities depends on β ,the parameter to be subsequently estimated. Therefore, to performa density–velocity comparison self-consistently, one has to recon-struct the real space density field for a range of different valuesof β . For example, Branchini et al. (1999) performed such recon-structions for 10 different values of β in the range . – . . Notethat instead of this traditional algorithm, more sophisticated meth-ods now rely on Euler-Lagrange action minimization (e.g., Pee-bles 1989; Shaya, Peebles & Tully 1995; Nusser & Branchini 2000;Phelps 2002; Phelps et al. 2006) or resolution of optimal assigne-ment problems (e.g., Croft & Gazta˜naga 1997; Frisch et al. 2002;Mohayaee et a. 2003; Mohayaee & Tully 2005).Another approach, proposed by Nusser & Davis (1994; here-after ND), is to perform the comparison directly in redshift space.ND derived the density–velocity relation in redshift space in the lin-ear regime. Because velocity–velocity comparisons seem in prac-tice more robust than density–density ones, they aimed at trans-forming this relation to an integral form, i.e. at solving for the ve-locity as a functional of the redshift-space density. Due to the radialcharacter of redshift-space distortions, it turned out to be possibleonly via a modal expansion of the density and velocity fields inspherical harmonics. Using this expansion, ND were able to putconstraints on β (ND; Davis, Nusser & Willick 1996). However,the approach with the reconstruction of the real-space density fieldremained popular. Apparently, equation (1) is appealing by its sim-plicity, both in terms of reconstruction of the velocity field, and ofestimation of the parameter β . This paper is devoted to finding an equivalent of equation (1)that would hold for redshift-space quantities, but would share itssimplicity. Specifically, let us define the scaled gravity: g ( r ) ≡ Z d r ′ π δ ( g ) ( r ′ ) r ′ − r | r ′ − r | . (3)Under the assumption of linear bias, adopted here, g is proportionalto gravitational acceleration, and can be directly measured from a3-D galaxy survey. Equation (1) then implies g = β − v . (4)Now, let us assume that we measure the gravitational accelerationdirectly in redshift space: g s ( s ) = Z d s ′ π δ ( g ) s ( s ′ ) s ′ − s | s ′ − s | , (5)where δ ( g ) s denotes the density contrast of galaxies in redshiftspace. We will follow ND in a natural definition of the redshift-space velocity field: v s ( s ) ≡ v ( r ) . (6)What is the relation between g s and v s ?Equation (4) holds strictly in the linear regime. Nevertheless,numerical simulations (Cieciela¸g et al. 2003) have shown that itremains accurate to a few percent for fully nonlinear gravity andvelocity fields. These results will be fully confirmed by the presentwork and can be explained by the fact that velocity and gravityfields are dominated by long-wavelength, linear modes. Therefore,in deriving the redshift-space counterpart of Equation (4), we willapply linear theory. Unfortunately, there is no local deterministicrelation between g s and v s . However, as shown below, g s and v s are strongly correlated, so the mean relation will be a useful quan-tity. Only radial components of velocities are directly measurable,so we will be interested in the relation between g s = g s · ˆ s and v s = v s · ˆ s .This paper is thus organized as follows. In §
2, we compute theproperties of the joint probability distribution function (PDF) of thefields g s and v s in the framework of linear theory and distant ob-server limit. In the linear regime, this PDF is expected to be a Gaus-sian, entirely determined by its second order moments. The quan-tities of interest are h v s g s i / h v s i , h g s i / h v s g s i and p h g s i / h v s i :they all give an estimate of the expected ratio between g s and v s and the difference between them can be used to compute the scatteron the relation, that will be shown to be small. The validity of ourassumptions, in particular the distant observer approximation, is ex-amined a posteriori in §
3. We also justify our choice of CMB restframe redshifts, needed to avoid the so-called rocket effect (Kaiser1987; Kaiser & Lahav 1988). In §
4, linear theory is tested againstnumerical experiments, both in real and redshift space. To do thatwe use a dark matter cosmological N -body simulation with highresolution, allowing us to probe the highly nonlinear regime. Wepropose a new and simple algorithm to interpolate the velocity andgravity field on a grid from a distribution of particles. We addressextensively a number of issues, such as the validity of the distantobserver limit, edge effects, cosmic variance effects, effects arisingfrom non-linear dynamics, in particular so called ‘fingers of God’(FOG) and effects of dilution (numbers of tracers used to probethe velocity and gravity fields). We use spherical top-hat model tosupport our interpretations of the measurements (technical detailsare given in Appendix A). The effect of the bias is also examinedbriefly by extracting from the dark matter distribution two kinds of c (cid:13) , 000–000 subsamples, one where each “galaxy” is identified with a dark mat-ter halo, and the other one where each “galaxy” is identified with adark matter substructure. Finally, the main results of this work aresummarized in §
5. In this last section, we discuss observational is-sues such as discreteness effects, appropriate treatment of biasing,incompleteness, errors on peculiar velocity estimates, etc. These is-sues will be addressed in detail in a forthcoming work, where themethod will be applied to real data.
Linear regression of v s on g s yields g s = h v s g s ih v s i v s . (7)The symbols h·i denote ensemble averaging. We can thus charac-terize the linear relation between gravity and velocity (gravity interms of velocity) by its slope, a F ≡ h v s g s ih v s i . (8)Alternatively, one can study the inverse relation (velocity in termsof gravity). Linear regression then yields v s = ( h v s g s i / h g s i ) g s ,so the inverse slope is a I ≡ h v s g s i / h g s i . To describe the linearvelocity–gravity relation, we have thus at our disposal two estima-tors of its slope, the forward slope, a F , and the reciprocal of theslope of the inverse relation, a − I = h g s ih v s g s i . (9)Due to the scatter, these two estimators are not equal: a − I − a F = a − I (1 − r ) > , (10)where r = h v s g s i σ v σ g (11)is the cross-correlation coefficient of the velocity and gravity fields; σ v = h v s i and σ g = h g s i . The linear regression, although the bestamong all linear fits to a cloud of points, visually looks biased. Fortwo correlated Gaussian variables, an unbiased slope of the isocon-tours (ellipsoids) of their joint PDF is given by the square root ofthe ratio of the variances of the two variables. Thus we have thirdestimator of the slope, a U ≡ r h g s ih v s i . (12)This is easy to show that it predicts intermediate values between a − I and a F .Second-order moments of the joint PDF for gravity and ve-locity, which appear in the above three estimators of its slope, aremuch easier to perform in Fourier space. We will adopt the distantobserver limit (DOL), in which the Jacobian of the transformationfrom real to redshift space, equation (2), simplifies to J ( r ) = 1 + ∂v∂r . (13)The Fourier transform of the redshift-space velocity field, using themapping (2), is ˜ v s ( k ) ≡ Z e i k · s v s ( s ) d s = Z e i k · r e ikv v ( r ) J ( r ) d r = Z e i k · r v ( r ) d r + O ( vδ ) ≃ ˜ v ( k ) , (14)where ˜ v ( k ) is the Fourier transform of the real-space velocity field.The linearized continuity equation yields ˜ v ( k ) = − f ( i k /k )˜ δ ( k ) ,where f ≃ Ω . and ˜ δ ( k ) is the Fourier transform of the real-spacedensity field, hence ˜ v s ( k ) = − f i k k ˜ δ ( k ) . (15)From equation (5) we have ˜ g s ( k ) = − i k k ˜ δ ( g ) s ( k ) . (16)This equation looks similar to the preceding one, but here appearsthe Fourier transform of the redshift-space galaxy density field.Moreover, unlike the preceding equation, the above equation is ex-act. Radial components of the redshift-space velocity and gravityfields are ˜ v s ( k ) = − f ( iµ/k ) ˜ δ ( k ) and ˜ g s ( k ) = − ( iµ/k ) ˜ δ ( g ) s ( k ) ,where µ = k · s / ( ks ) . From conservation of numbers of galax-ies in real and redshift space, it is straightforward to write downan equation for the Fourier transform of the redshift-space galaxydensity contrast. In the linear regime it reduces to ˜ δ ( g ) s ( k ) =˜ δ ( g ) ( k ) + fµ ˜ δ ( k ) (Kaiser 1987), or ˜ δ ( g ) s ( k ) = b (1 + βµ )˜ δ ( k ) . (17)Therefore, we obtain ˜ v s ( k ) = − f iµk ˜ δ ( k ) , (18)and ˜ g s ( k ) = − b (1 + βµ ) iµk ˜ δ ( k ) . (19)The above pair of equations enables us to calculate the aver-ages appearing in equations (8), (9), and (12). Specifically, h v s i = (2 π ) − Z e − i ( k + k ′ ) · s f iµk f iµ ′ k ′ h ˜ δ ( k )˜ δ ( k ′ ) i d k d k ′ . (20)For a homogeneous and isotropic random process, h ˜ δ ( k )˜ δ ( k ′ ) i =(2 π ) δ D ( k + k ′ ) P ( k ) , where δ D is Dirac’s delta and P ( k ) is thepower spectrum of the real-space density field. Performing the in-tegral over k ′ yields h v s i = f (2 π ) Z µ k P ( k ) d k = f π Z ∞ P ( k ) d k . (21)Similarly, h v s g s i = bf (2 π ) Z (1 + βµ ) µ k P ( k ) d k = bf [1 + (3 / β ]6 π Z ∞ P ( k ) d k . (22)Finally, h g s i = b (2 π ) Z (1 + βµ ) µ k P ( k ) d k = b π (cid:20) β β (cid:21) Z ∞ P ( k ) d k . (23)This yields a F ≡ h v s g s ih v s i = 1 + 3 β/ β , (24) c (cid:13) , 000–000 Colombi, Chodorowski & Teyssier a − I ≡ h g s ih v s g s i = 1 + 6 β/ β / β (1 + 3 β/ , (25) a U ≡ r h g s ih v s i = p β/ β / β . (26)Equations (24)–(26) provide a way to estimate β compar-ing gravitational accelerations to peculiar velocities of galaxies di-rectly in redshift space. All three estimators predict the slope that isgreater than the corresponding one for real-space quantities ( β − ),the more the bigger β . This is expected since redshift distortionsenhance linear density contrasts (eq. 17), and the amplitude of dis-tortions scales linearly with β . We can therefore consider the factor β − as the one coming from real-space dynamics, and the addi-tional factors involving β as redshift-space corrections.As stated above, in redshift space, unlike in real space, thevelocity–gravity relation is not deterministic even in the linear the-ory, so a − I > a U > a F . However, the linear gravity and veloc-ity fields are tightly correlated, so Equations (24)–(26) yield val-ues of the ratio between g s and v s very close to each other: e.g.,for β = 0 . (the value used in our numerical experiments, § a − I = 2 . , a U = 2 . , and a F = 2 . .To illustrate this point furthermore, one can examine the scat-ter on the conditional average h g i | v (mean g given v ), where wehave dropped out the subscripts ‘ s ’ for simplicity of notation. For g and v being Gaussian-distributed, h g i | v = h vg i v/ h v i , in agree-ment with linear regression, equation (7). The scatter in this re-lation, σ g | v , is the square root of the conditional variance, h ( g −h g i | v ) i | v . The conditional variance is then equal to (1 − r ) σ g (see, for example, Appendix B of Cieciela¸g et al. 2003). Thus, forGaussian velocity and gravity fields in redshift space we have σ g | v = (1 − r ) σ g , (27)where r is the cross-correlation coefficient defined in eq. (11). Notethat the scatter is just one number, i.e., it is independent of the valueof v . Equations (21)–(23) yield r = 1 − β β/ β / . (28)In particular, for β = 1 , r ≃ . , and for β = 0 . , r ≃ . :again, redshift-space velocity and gravity in the linear regime are(though not simply mutually proportional, like in real space) verytightly correlated. Inserting equation (28) into (27) yields σ g | v = 12175 β β/ β / σ g . (29)Hence, for β = 1 , σ g | v ≃ . σ g , and for β = 0 . , σ g | v ≃ . σ g . This implies that the ‘signal to noise ratio’, S/N , of theestimate of a single galaxy’s peculiar velocity from its gravitationalacceleration can be as high as ( S/N ∼ g/σ g | v ∼ σ g /σ g | v ≃ ). This is to be contrasted with the signal to noise ratio of theestimate of a galaxy’s peculiar velocity from its distance and red-shift, which is typically below unity: an expected typical % rel-ative error in distance, for a galaxy at a distance of h − Mpc, translates to the velocity error of km · s − , greater than typicalpeculiar velocities of galaxies. As a corollary, the intrinsic scatterin the redshift-space linear velocity–gravity relation is negligible See, for instance, Strauss & Willick (1995) for a review on issues relatedto peculiar velocity estimates. Where h = H / and H is the Hubble constant expressed inkm/s/Mpc. compared to that introduced by the errors of measurements of pe-culiar velocities. While equation (2) was written in the CMB rest frame, the choiceof the reference frame for computing redshifts of galaxies can bemore general, s = r + [ v ( r ) − v orig ] ˆ r (30)where v orig is the (angle dependent) radial velocity of the origin ofthe system of coordinates (in the CMB rest frame).The Jacobian of eq. (30) yields (Kaiser 1987) δ ( g ) s ( s ) = (cid:20) v ( r ) − v orig r (cid:21) − (cid:16) ∂v∂r (cid:17) − (cid:2) δ ( g ) (cid:3) . (31)This equation is valid as long as the mapping (30) does not induceany shell crossing. In general, non trivial singularities can appear in δ ( g ) s ( s ) , even if δ ( g ) is finite. However, in practice, since one alwaysperforms some additional smoothing to the data, δ ( g ) s ( s ) remainsfinite. The small r limit can be still problematic as it corresponds toa singularity in the system of coordinates. If the field v ( r ) is smoothenough, then in the neighborhood of the origin, one can write v ( r ) − v orig r ≃ v ( ) − v orig r . (32)Choosing a spherical coordinate system such that the z axis is par-allel to v ( ) − v orig , the term { v ( r ) − v orig ] /r } − in eq. (31)will create a singular surface of equation r = −| v ( ) − v orig | cos( θ ) , θ > π/ (33)that will concentrate to the origin, s = , in redshift space. Simi-larly the term (1 + ∂v/∂r ) − might become singular, but we seehere that the situation nearby the origin is not different from whathappens far away from it.The singular behavior of the form (33) is expected to occuronly in the neighborhood of the observer, but might be problem-atic when estimating the gravitational acceleration. While insert-ing this singularity in eq. (5), we notice that it should be an issueonly for s = , where it coincides with the Green function sin-gularity (this is due to the finite mass of the singular surface). Inother words, even though its small s behavior is difficult to pre-dict, the redshift space gravity should not be significantly affectedby such a singular behavior at distances sufficiently large from theobserver. In the CMB rest frame, v orig = 0 and v ( ) would cor-respond to the Local Group velocity, say, | v ( ) | ≃ km/s (seee.g., Erdo ˘g du et al. 2006), so we would need r and therefore s largeenough compared to 6 h − Mpc. More specifically, let us considerthe real space sphere of radius r = | v ( ) | . Its content is embeddedin a sphere of radius s = 2 | v ( ) | in redshift space. The mass insidesuch a volume remains finite, but its internal distribution affects thegravity field at larger s in a non trivial way. What matters is that themultipole contributions of higher order than the monopole (the sub-structures within this volume) have negligible contribution on thegravity field. Using the wisdom from treecode simulations tech-niques (e.g. Barnes & Hut 1986; Barnes & Hut 1989), this amountsto | v ( ) | s . . (34) c (cid:13) , 000–000 According to that criterion, the effect of the singular behavior nearthe origin should be of little consequence if s & | v ( ) | = 13 h − Mpc.If it is supposed now that s is indeed large enough, one canlinearly expand eq. (31) to obtain (Kaiser 1987) δ ( g ) s ( s ) = δ ( g ) − v ( r ) − v orig r − ∂v∂r . (35)(Note that, at linear order, one can assume s ≃ r .) The distant ob-server limit would consist in dropping the term v ( r ) − v orig ] /r from this equation. However, as extensively discussed in Kaiser(1987) and Kaiser & Lahav (1998), this term is in fact non neg-ligible in the redshift space gravitational acceleration as it inducesthe so-called rocket effect, resulting in a large r logarithmic diver-gence if v orig = 0 . This justifies the choice of CMB rest framecoordinate system, v orig ≡ . Still the remaining contribution of v ( r ) /r , although zero in average, might introduce some significantfluctuations on the large scale redshift space gravity field. In thelinear perturbation theory framework, v ( r ) does not correlate witheither δ ( g ) or ∂v/∂r . As a result, while computing the sum of thefluctuations of δ ( g ) s in a sphere of radius r = s ′ in eq. (5), what mat-ters is to see whether the fluctuations added by v ( r ) /r are small,or not, compared to the fluctuations added by δ ( g ) − ∂v/∂r .So let us estimate the ratio R ≡ (cid:10) [2 v ( r ) /r ] (cid:11)(cid:10) [ δ ( g ) − ∂v/∂r ] (cid:11) . (36)Calculations are similar to §
2, and one simply finds, in the linearregime, R = 4 β β/ β / R ∞ P ( k ) dkr R ∞ P ( k ) k dk . (37)One can furthermore assume that the linear fields are smoothed, e.g.with a Gaussian window of size ℓ . In that case one has to replace P ( k ) with P ( k ) exp[ − ( kℓ ) ] in eq. (37). For scale-free powerspectra, P ( k ) ∝ k n , one finds using eq. (4.10b) of Bardeen et al.(1986) R = 8 β n + 3) (1 + 2 β/ β / (cid:16) ℓr (cid:17) , (38)which gives R ≃ . n + 3 (cid:16) ℓr (cid:17) , for β = 0 . . (39)This means that smoothing increases the relative contribution of theterm v ( r ) /r ! In other words, smoothing makes the distant observerapproximation worse .The standard cold dark matter (CDM) cosmology consideredin the forthcoming numerical analyses assumes the non-relativisticmatter density parameter Ω = 0 . , the cosmological constant Λ =0 . and the Hubble constant H = 70 ≡ h km · s − · Mpc − .Using for instance the package of Eisenstein & Hu (1998) to com-pute P ( k ) with these cosmological parameters (and Ω baryons =0 . ), one obtains numerically, in the absence of bias, b = 1 , R ≃ . h − Mpc r , no smoothing , (40) R ≃ . h − Mpc r , ℓ = 1 h − Mpc (41) We suppose here that smoothing on v ( r ) /r is approximately equivalentto smoothing on v ( r ) , prior to dividing by r , which should be reasonable. R ≃ h − Mpc r , ℓ = 10 h − Mpc . (42)Clearly, for a catalog depth of the order of h − Mpc as consid-ered below, we expect significant deviations from our theoreticalpredictions if the smoothing scale is as large as h − Mpc . In this section we perform controlled numerical experiments to testthe velocity–gravity relation, both in real and redshift space, onthe dark matter distribution. These analyzes extend the work ofCieciela¸g et al. (2003) who performed similar work but only inreal space and on simulations using a pure hydrodynamic code ap-proximating the dynamics of dark matter, namely the Cosmolog-ical Pressureless Parabolic Advection code of Kudlicki, Plewa &R´o˙zyczka (1996, see also Kudlicki et al. 2000).This section is organized as follows. In § § N -body simulation set used in this work. In § § Ω + Λ = 1 , we adopt theapproximation f (Ω) ≃ Ω / (43)for function f (Ω) , as it is known to be slightly more accurate inthat case than the traditional Ω . fit (Bouchet et al. 1995). Finally,let us recall that ‘DOL’ means “distant observer limit”. In real observations, the catalogs used to estimate cosmic velocityand gravity are in general different. In particular, the velocity fieldtracers are not necessarily representative of the underlying densityfield. Here we assume to simplify that the same catalog is usedto probe velocity and gravity fields and that all the objects in thecatalog have the same weight, or from the dynamical point of view,the same mass. Also, since gravitational force is of long range, it isnecessary to estimate it in a domain V grav large enough comparedto the effective volume V where the velocity–gravity comparisonis actually performed. We assume here that it is indeed the case. Inthis paper, V grav will simply be the simulation cube, while V willbe a sub-volume included in this cube.The basics of our method is to perform adaptive smoothing onthe particle distribution to obtain a smooth velocity and a smoothdensity field on a regular grid encompassing V grav and of resolu-tion N g , while preserving as much as possible all the information.Then additional smoothing with a fixed window, preferably Gaus-sian, can be performed a posteriori in order, e.g., to be in the lin-ear regime, since our analytic predictions in principle apply to thatregime. We shall see however below that this additional step is notnecessarily needed and can in fact complicate the analyzes (see also § N points with three-dimensional coordinatesand a scalar velocity. This latter is either the radial velocity whenwe consider redshift space measurements, or the z coordinate of c (cid:13) , 000–000 Colombi, Chodorowski & Teyssier the velocity vector when we consider real space measurements orredshift space measurements in the DOL approximation.The main difficulty is to reconstruct a smooth velocity fieldon the sampling grid. Let us remind as well that the velocity fieldwe aim to estimate is a purely Eulerian quantity, in other words, amean flow velocity. In particular, what we aim to measure, in termsof dynamics, is something as close as possible to a moment of thedensity distribution function f ( r L , v L ) in phase-space: v ( r ) ≡ ρ Z d v L v L f ( r , v L ) (44)where ρ is density, the source term of the Poisson equation to esti-mate the gravitational potential, given by ρ ( r ) = Z d v L f ( r , v L ) . (45)Note that equation (44) applies only to real space, but we shallcome back to that below.Since we aim to estimate velocity and density on a finite res-olution grid, a more sensible way of performing the calculationsis to integrate equations (44) and (45) on a small cubic patch ∆ r corresponding to a grid element to reduce noise as much as pos-sible and to make the calculation conservative (i.e., the total massand momentum is conserved). In practice, we do not perform thiscalculation exactly, but only approximately using an approach in-spired from smooth particle hydrodynamics (SPH, see, e.g. Mon-aghan 1992), as we now describe in details.In the SPH approach, each particle is represented as a smoothcloud of finite varying size R SPH depending on local density, i.e.on the typical distance between the particle and its N SPH closestneighbors (which can be found quickly with e.g. standard KD-treealgorithm), where N SPH is usually of the order of a few tens. As aresult, smooth representation of the density and velocity fields canbe obtained at any point of space by summing up locally the contri-bution of all the clouds associated to each particle. With appropriatechoice of the SPH kernel, these functions can be easily integratedover each cell of the grid.The problem with this approach is that it does not guaran-tee that local reconstructed density is strictly positive everywhere,which can leave regions where the velocity field is undefined. Tosolve that problem, we start from the grid points, which we assumeto be virtual particles for which we find the N SPH closest neighborsto define the SPH kernel associated to this grid site. However, onehas to take into account the fact that the sought estimates shouldroughly correspond to an integral in a small cubic patch ∆ r . Inparticular, all particles belonging to a grid site should participateto such an integral. This issue can be addressed in an approximateway as follows:(i) Count and store for each grid site, the number N p of particlesit contains;(ii) If N p > N SPH , then perform SPH interpolation at the gridsite as explained below using the N p particles contributing to itinstead of the N SPH closest neighbors;(iii) If N p < N SPH , then find for the grid site the N SPH closestparticles and perform SPH interpolation at the grid site as explainedbelow.Given a choice of the SPH kernel, S ( x ) [which should be amonotonic function verifying S (0) = 1 and S ( x >
2) = 0 ], anda number N X of neighbors, with N X = N p or N SPH according tothe procedure described above, the interpolation of a quantity A onthe grid site is given by ˜ A ( i, j, k ) = 1[ R SPH ( i, j, k )] " N X − X l =1 A l W l S (cid:18) d l R SPH ( i, j, k ) (cid:19) . (46)In this equation, A l is the value of A associated to each particle l , R SPH ( i, j, k ) is half the distance to the furthest neighbor of the gridsite ( i, j, k ) among the N X , d l is the distance of the l -th particle tothe grid site, and W l is a weight given to each particle such thatthe total contribution of every particle to all the grid sites is exactlyunity: W l = 1 /S l = "X i,j,k R SPH ( i, j, k )] S (cid:18) d l R SPH ( i, j, k ) (cid:19) − . (47)In practice, the interpolated density thus reads ˜ ρ ( i, j, k ) = ˜ m ( i, j, k ) / ∆ r , (48)by taking A l = m l in equation (46), where m l is the mass of eachparticle. An interpolated velocity reads ˜ v ( i, j, k ) = ˜ p ( i, j, k ) / ˜ m ( i, j, k ) , (49)where ˜ p ( i, j, k ) is the interpolated momentum in cell ( i, j, k ) . It isobtained by taking A l = p l ≡ m l v l in equation (46), where v l isthe velocity (the radial component, or a coordinate) of particle l .Note that there can be some particles for which S l = 0 inequation (47) (in that case W l = 0 by definition), i.e. which donot contribute at all to the interpolation. In the practical measure-ments described later, this can happen at most only for a very smallfraction of the particles, typically of the order of 0.1 percent, andaffects the results insignificantly. There is, however, another notice-able defect in our method, due to the unsmooth transition betweenthe two interpolation schemes when the number of particles N p pergrid site becomes larger than N SPH , that affects in a non trivial waythe interpolated density, but again, it does not have any significantconsequences for the present work.In practice, we take N SPH = 32 for all the measurementsdescribed below and the following spline for the SPH kernel (Mon-aghan 1992) S ( x ) = 1 − . x + 0 . x , x S ( x ) = 0 .
25 (2 − x ) , x S ( x ) = 0 , x > . (50)Once the density field is interpolated on the grid, it is easyto estimate the gravitational potential from it by solving the Pois-son equation in Fourier space, keeping in mind that the edge of thevolume V , where the velocity–gravity relation is tested, should besufficiently far away from the edges of the volume V grav , whichis itself included in the grid. It would be beyond the scope of thispaper to discuss other problems related to incompleteness or edgeeffects. The main one is related to the uncertainties on the gravita-tional potential induced by the obscuring due to our own Galaxy.While this can be certainly an issue, other incompleteness prob-lems such as segregation in luminosity can be addressed by givingthe proper weight (or mass) to the galaxies in the catalog. This ofcourse needs strong assumption on the bias, and can work only forpopulations of not too bright galaxies.Note finally that, as explained before, we perform “naively”our SPH interpolation whether we are working in real space orin redshift space. The fact that we use SPH interpolation in red-shift space is sufficient for our purpose as long as we are in theDOL limit, although it does not correspond anymore exactly to asimple moment of the Vlasov equation. However, the interpolationbecomes somewhat questionable when the assumption of DOL is c (cid:13) , 000–000 dropped: the nature of the interpolation changes with distance fromthe observer. In particular, if we assume that the smoothing ker-nel was fixed, projections, e.g. passing from the velocity vector to aone dimensional quantity such as the radial velocity, and smoothingdo not commute anymore. The same problem arises for commuta-tion between the calculation of the radial part of the gravitationalforce and smoothing. It is therefore necessary to carefully checkthat the simplistic nature of our interpolation does not introduceany systematic bias in the measurements, when performing them inredshift space relaxing the approximation of a distant observer. N -body simulation set We performed a high resolution simulation using the adaptivemesh refinement (AMR) code RAMSES (Teyssier 2002). As al-ready mentioned earlier, the cosmology considered here assumes
Ω = 0 . , Λ = 0 . and H = 70 ≡ h km · s − · Mpc − . Initialconditions were set up using Zel’dovich approximation (Zel’dovich1970) to perturb a set of particles disposed on a regular grid patternto generate initial Gaussian fluctuations with a standard Λ CDMpower spectrum. To do that, we used the COSMICS package ofBerstchinger (1995). The simulation involves dark matter par-ticles on the AMR grid initially regular of size , in a periodiccube of size h − Mpc. Then, additional refinement is allowedduring runtime: if cells contain more than 40 particles they are di-vided using standard AMR technique (with a maximum of 7 levelsof refinement). Note finally for completeness that the normaliza-tion of the amplitude of initial fluctuations was chosen such that thevariance of the density fluctuations in a sphere of radius h − Mpcextrapolated linearly to the present time was given by σ = 0 . . From our RAMSES simulation, we extracted a number of dark-matter samples, as described in details in Table 1, namely:(i) A high resolution grid sample of size N g = 512 for testingthe velocity–gravity relation in real space;(ii) The analog of (i) for testing the velocity–gravity relation inredshift space, using the DOL approximation;(iii) The analog of (ii), but without using the DOL, for testing thevalidity of this approximation. In that case the radial coordinate ofthe velocity and gravity field was estimated using an observer atrest at the center of the simulation box. The comparison of veloc-ity to gravity was performed in a sphere of radius 70 h − Mpc toavoid edge effects due to the loss of periodicity while projecting inredshift space;(iv) A set of 125 low resolution grids ( N g = 128 ) in redshiftspace, for estimating (at least partly) cosmic variance effects andeffects of structures nearby the observer. These samples were gen-erated by locating the observer in the simulation box on a regulargrid of size × × . Again, in these samples, the velocity–gravityrelation was tested in a volume of radius 70 h − Mpc centered onthe observer (exploiting the periodic nature of the simulation box).Since the catalogs we consider in point (iv) represent a significantfraction of the simulation volume, we know however that effects ofcosmic variance are likely to be underestimated.We now perform a visual inspection of the fields, followed bymeasurements of the joint PDF of velocity and gravity, first in realspace, then in redshift space.
Figure 1 shows the density, the gravity field and the velocity field ina thin slice extracted from the simulation, both in real space and inthe DOL redshift space. Despite our interpolation procedure, thereare some minor discreteness artifacts left, visible on the densityfield. One can notice for example a few underdense regions wherethe initial grid pattern, distorted by large scale dynamics, is stillpresent. Such artifacts do not show up on the smoother z coordinateof the gravity field. On the other hand, this latter seems to sufferfrom a few aliasing defects in real space, as vertical spurious linesin the vicinity of deep potential wells. We did not bother tryingto understand such aliasing effects, because they have negligibleimpact on the measurements. Interesting to notice here is the niceagreement between the z coordinate of the velocity field and the z coordinate of the gravity field in underdense regions. These regionsdominate the velocity–gravity statistics, since we have a volumeweighted approach. Note the particular features in the velocity fieldassociated to the filaments in the density distribution, as well as the‘fingers of God’ (FOG) in redshift space. These FOG, which aremainly associated to the dark matter halos, are expected to inducesome particular properties on redshift space statistics: • First, and this is a straightforward consequence of passingfrom real space to redshift space, there is, on the velocity field,an ‘inversion’ effect inside FOG. In other words, inside a finger ofGod, the variations of the velocity field are opposite to what hap-pens in its nearby environment. This can be explained in the follow-ing way. Inside a halo, which can be, in real space, considered as apoint-like structure in first approximation, there are particles withpositive velocities and particles with negative velocities. Assume,to simplify, that this halo has itself a zero center of mass velocity.It will, in redshift space, look like an elongated structure. Particlesin this structure that have positive z velocity will be above the cen-ter of mass, and particles with negative velocity will be below thecenter of mass. As a result, one expects, for the z coordinate of thevelocity, the finger of God corresponding to this halo to have posi-tive velocities (towards light color on bottom right panel of Fig. 1)above the center of the halo and to have negative velocities (towardsdark color) below the center of the halo. This is indeed what we canobserve. However, in the nearby environment of the halo, the situ-ation is somewhat opposite. Indeed, the halo is expected to lie in afilament, which itself represents a local potential well. If this wellis not too strong to induce shell-crossing while passing from real toredshift space, the general trend of the velocity field is not modified,compared to real space: negative sign (towards dark color) abovethe filament, positive sign below it. If the filament corresponds to asufficiently deep potential well such that shell crossing occurs, thenthe same effect than for FOG is expected, as can be noticed on thebottom right panel of Fig. 1 for the largest filaments. • Second, due to the FOG stretching, halos are more elongatedin redshift space, and a natural consequence is that the correspond-ing potential well is less deep. That is why the z coordinate of the These aliasing effects might be related to some minor defect in the par-ticular Fourier transform algorithm we are using (Teuler 1999), but this hy-pothesis seems to be contradicted by some accuracy tests performed on thisalgorithm (Chergui 2000). A more sensible explanation is that these effectsare induced by the way we interpolate the density field, in particular by thetransition between the N p > N SPH and the N p < N SPH regimes, com-bined with the fact that our Fourier Green function for computing the grav-itational acceleration is simply proportional to k /k , without additionalfiltering.c (cid:13) , 000–000 Colombi, Chodorowski & Teyssier
Figure 1.
The density (upper panels), gravity (middle panels) and velocity field (lower panels) in a one-mesh-element-thick slice extracted from the samples(i) and (ii), which correspond to a grid resolution of . The left and right columns of panels correspond respectively to real space and distant observerredshift space. In the top panels, the color scale is logarithmic, from white to black. In the four bottom panels, the z coordinate of each field is displayed. Thecolor scale is linear and normalized in such a way that direct visual comparison between the four panels can be performed. One can notice clearly ‘fingers ofGod’ on the upper right and lower right panels. In such ‘fingers of God’, the velocity field tends to have a different signature compared to that in the nearbyenvironment, as explained in the text. Finger-of-God effects also tend to soften the gravitational potential at small scales (cf. middle panels).c (cid:13) , 000–000 Figure 2.
The same as in the right column of panels of Fig. 1, but for the sample (iii), where the assumption of a distant observer is dropped. The two lowerpanels thus show the radial component of the gravity and the velocity fields. The observer is at the center of the images, towards where ‘fingers of God’ point.c (cid:13) , 000–000 Colombi, Chodorowski & Teyssier
Table 1.
Summary of the characteristics of the dark matter samples used in this paper and the corresponding main quantitative results. The first column givesthe name of the sample as used in the text. The second one mentions whether the measurements are performed in real or redshift space, the acronym DOLmeaning that the approximation of an infinitely remote observer was used. The third column gives the number N rea of realizations considered. The fourthcolumn gives the size of the sample, which is the full periodic simulation cube of size L when the letter L is used and a sphere of radius R when the letter R is used. Scales are expressed in units of h − Mpc. The fifth column gives the resolution N g of the rectangular grid (matching the simulation box) used tosample various fields from the particle distribution. The sixth column gives the number N obj of objects that are actually contained in the sample. A star meansthat this number is only approximate, corresponding to a simple rescaling of the total number of objects in the periodic cube by taking into account the size ofthe sampled volume, a sphere of radius R . The 7th, 8th and 9th columns give the measured Ω using h g i i / h v i g i i (1st number), p h g i i / h v i i (2nd number)and h v i g i i / h v i i (3rd number) as estimators of the velocity–gravity relation, where i = s if the measurements are performed in redshift space and i = z if themeasurements are performed in real space or distant observer redshift space. In principle, all these estimators should give the same answer (nearly the sameanswer in redshift space) in the ideal case where the linear theory applies, and various sources of noise (such as shot noise, cosmic variance, systematics, etc.)are negligible. The disagreement between these various estimates can be used as a proxy to estimate error bars on actual measurements. Where possible, anerror bar obtained from the dispersion among various realizations is quoted as well. The last column explains what exactly the quoted numbers correspond to,in particular if the fields were smoothed with a Gaussian window of size ℓ = 10 h − Mpc (“smoothed”) or not (“no smoothing”), if all the PDF was used toperform the measurements (“all”) or only the most likely region defined by Equation (52) (“1.5 σ isocontour”). The symbol h· · ·i means that an average wasperformed over the number of available realizations, when relevant.Samp. Content N rea Size N g N obj Measured Ω from: Comment h g ih vg i q h g ih v i h vg ih v i (i) real space 1 L = 200 σ isocontour, no smoothing0.295 0.297 0.299 1.5 σ isocontour, smoothed(ii) DOL redshift space 1 L = 200 σ isocontour, no smoothing0.274 0.279 0.284 1.5 σ isocontour, smoothed(iii) redshift space 1 R = 70 . ∗ σ isocontour, no smoothing0.269 0.284 0.300 1.5 σ isocontour, smoothed(iv) redshift space 125 R = 70 . ∗ h all i , no smoothing0.023 0.030 0.269 error from dispersion0.177 0.200 0.226 h all i , smoothed0.022 0.021 0.023 error from dispersion0.283 0.294 0.304 h σ isocontours i , no smoothing0.039 0.040 0.041 error from dispersion0.255 0.262 0.270 h σ isocontours i , smoothed0.031 0.032 0.034 error from dispersion gravitational force is less contrasted in the middle right panel ofFig. 1 than in the middle left one.Still, we can see that both in real and in redshift space, the velocity–gravity relation is going to be in quite good agreement with ex-pectation from linear theory. Indeed and again, our measurementsare volume weighted, dominated by underdense regions. The colorscale on four bottom panels of Fig. 1 has been chosen such that iflinear theory applies, the same color pattern should be found for thegravity and velocity field, which is the case at first glimpse, exceptin the densest regions, corresponding to halos or rich filaments.Finally, similarly as the right column of panels of Fig. 1, Fig. 2displays the density, the gravity field and the velocity field in red-shift space, but without using the DOL. If one now takes into ac-count the radial nature of the projection, it is clear that the conclu-sions of the previous discussion remain unchanged, at least at thequalitative level. Figure 3 shows in grey scale the measured joint probability distri-bution function (PDF) of the z coordinate of gravity and velocityfields extracted from the real space sample (i). The striking result is that the regions of the best likelihood (larger values of the PDF,darker places) match very well the prediction given by linear the-ory (thick solid line), even in the highly non-linear regime (upperpanel). This remarkable property is mainly related to our volume-weighted approach: results are mainly influenced by underdenseregions, which are weakly evolved from the dynamical point ofview and are expected to match well linear theory predictions. This,and the ‘propeller’ shape of the bivariate PDF, can be understoodin more detail by using the spherical top-hat model as a proxy ofnonlinear dynamics. In that case, up to shell crossing, the velocity–gravity relation reads approximately g ≃ R " − (cid:18) − vβR (cid:19) / (51)(this can be easily derived from Bernardeau 1992; 1994), where R is the distance from the center of a top-hat fluctuation (in km · s − ).This equation is valid inside the fluctuation, which can be over-dense (negative v ) or underdense (positive v ). One can then imag-ine, to simplify, the density field as a patchwork of spherical fluctu-ations, which correspond to a set of curves given by Equation (51),as shown in the top panel of Fig. 3. Note here that we should,for the picture to be correct, take into account the fact that we c (cid:13) , 000–000 Figure 3.
The velocity–gravity relation in real space as a scatter plot, compared to theoretical prediction in the linear regime (thick diagonal line), as measuredfrom sample (i) extracted from the simulation using the z coordinate of each field. The color scale is defined as follows: the darkest area is encompassed by a68 percent isocontour (eq. 52), corresponding approximately to a 1.5 σ contour in the Gaussian case. Then, lower isocontours, corresponding to lighter color,scale logarithmically, which emphasizes the tails of the PDF.The two upper panels correspond to the raw data interpolated on the grid with the SPH-like interpolation detailed in § ℓ = 10 h − Mpc was performed. The smoothing scale is indicated on three panels (in the two toppanels, it corresponds to the mesh element size). In the top panel, predictions of velocity–gravity relation according to the spherical top hat model are displayedas solid curves. As discussed in detail in the text, these curves help to understand why the linear prediction still represents the most prominent feature of therelation, even in the highly nonlinear regime, while the tails due to nonlinear dynamics induce a ‘propeller’ shape of the scatter plot, as already noticed byCieciela¸g et al. (2003). Such a propeller shape results in a bias on the measurement of the slope by using directly second-order moments of the bivariate PDF.This is demonstrated by plotting on middle (and bottom) panel the dotted line which gives the estimated slope from the ratio p h g z i / h v z i . This dotted lineshould be compared to the thick solid line. The dashed lines correspond to slopes obtained by the conditional averages h g z i / h g z v z i and h g z v z i / h v z i . In themiddle panel these slopes are significantly different, consistent with significant amount of the scatter in the relation, due to nonlinear effects. In the middlepanel, also the . σ contour expected in the Gaussian case is shown.c (cid:13) , 000–000 Colombi, Chodorowski & Teyssier are using only the z coordinate of the fields, g z = g cos θ and v z = v cos θ , where θ is the angle between the z -axis and the ra-dial vector R . In the top panel of Fig. 3, we consider the cases R cos θ = ± , ± , ± , ± km/s. Each curve has astopping point corresponding to the maximum possible value of v , v max = βR/ , which reflects the fact that there is an expected up-per bound for the expansion speed of voids. This property excludesthe upper left and lower right quadrants of the velocity–gravity di-agram to be populated too far from linear theory prediction. Onthe other hand, overdense fluctuations tend to populate the upperright and the lower left quadrants, above and below linear the-ory prediction, respectively. Furthermore, since the low- v regimeconverges to linear theory, all the curves corresponding to Equa-tion (51) superpose in that regime, creating a ‘caustic’ of best like-lihood nearby the maximum of the joint PDF, explaining the verygood agreement with linear expectation in that region. As a result,we now understand, thanks to the spherical top-hat model, both the‘propeller’ shape of the bivariate PDF, as well as the remarkableagreement with linear theory prediction nearby its maximum, evenin the highly non-linear regime (see also Cieciela¸g et al., 2003).The arguments developed here are oversimplified, but capture themain features of the dynamics of the large scale structures prior toshell crossing in real space. Beyond shell crossing, there is a mix-ing effect that tends to decorrelate velocity from gravity, implyinga widening of the bivariate PDF and even larger tails in the upperright and the lower left quadrants. However such an effect does notaffect significantly the region of best likelihood.A straightforward consequence of the above discussion is thatthe joint PDF of gravity and velocity is substantially non Gaussiandue to non-linear contributions in the dynamics, as supported bythe examination on middle panel of Fig. 3 of the 1.5 σ elliptic iso-contour of the Gaussian distribution with the same second ordermoments as the measured PDF. As a result, the direct measurementof β parameter from the moments of the joint PDF using linear the-ory predictions is biased to lower values, due the propeller shapeof the PDF. This is illustrated on middle panel of Fig. 3 by the dot-ted line, which gives the velocity–gravity relation obtained fromthe second order moments of the PDF, p h g z i / h v z i , while the twodashed lines correspond to the conditional averages h g z i / h g z v z i and h g z v z i / h v z i . In the linear regime, let us remind from § h g z i / h g z v z i > p h g z i / h v z i > h g z v z i / h v z i , and β is underesti-mated, leading to an effective bias or an underestimated value of Ω ,as shown in Table 1. Note, as expected, that additional smoothingof the fields with a Gaussian window of radius 10 h − Mpc helpsto reduce non Gaussian features as well as this effective bias, andmakes the overall relation between gravity and velocity tighter.However, Fig. 3 appeals for a more sophisticated way to mea-sure the velocity–gravity relation than simply using directly themoments of the joint PDF. Since the region around the maximumseems to agree rather well with the linear prediction, this suggeststo estimate the moments from the PDF only within that region. Toperform such an exercise, we selected the 68-percent PDF isocon-tour, P , such that Z P ( g,v ) > P dg dv P ( g, v ) = 0 . . (52)This corresponds roughly to a . σ contour in the Gaussian case.The PDF is then set to zero outside the contour of best likeli-hood. From this truncated PDF, the moments are estimated again,leading to a much better estimate of Ω , agreeing at the few per- cent level with the true value, as shown in Table 1, regardlesswhether additional Gaussian smoothing with a h − Mpc sizewindow is performed on the fields, or not. Furthermore, the estima-tors p h g z i / h v z i , h g z i / h g z v z i and h g z v z i / h v z i differ now onlyslightly from each other, reflecting the narrowness of the region ofbest likelihood around the linear expectation. Figure 4 is similar to Fig. 3, but shows the results obtained fromthe sample (ii), in the DOL redshift space. The same striking resultobtained as in real space holds: the region of maximum likelihoodagrees very well with the linear theory prediction, which gives avelocity–gravity slope larger than in real space because of the en-hancement of large scale density contrasts due to projection in red-shift space (symbol LSRD – Large Scale Redshift Distortion – onthe left panel).However, there is on Fig. 4 a noticeable new feature on thePDF visible on left panel, in addition to the propeller shape: thejoint PDF seems now to present tails in the directions orthogonalto the maximum likelihood domain. Note that these tails tend todisappear with smoothing which then makes the bivariate PDF lookvery much like the real space one.This new feature is due to the finger-of-God effects alreadydiscussed at length in § g z v z = 1 β (cid:16) β (cid:17) , (53)a value significantly smaller than what is obtained from Eqs. (24),(25) and (26): equation (53) gives g z /v z ≃ . instead of . . Thisdisagreement with the statistical expectation from linear theory isexplained in Appendix A. However, despite the limitations of thespherical top hat model, the arguments developed previously in thereal space case to explain why the linear regime dominates the mostlikely part of the joint PDF still hold.Thanks to finger-of-God effects, if no additional smoothingis applied to the interpolated fields, the measured joint PDF is nowmuch more symmetric about the linear prediction than in real space.As a consequence, the slope obtained from the direct measure-ment of p h g z i / h v z i (dotted line on left panel of Fig. 4) agreesnow well with linear theory, while we still have h g z i / h g z v z i > p h g z i / h v z i > h g z v z i / h v z i . However, when an additional Gaus-sian smoothing is performed, the fingers of God tend to be diluted.This implies on Fig. 4 a clockwise rotation of the upper left tailto the upper right and of the lower right tail to the lower left. Asa consequence, with our h − Mpc scale Gaussian smoothing,one converges to similar behavior as obtained in real space, withthe propeller effect significantly biasing the overall slope, implying c (cid:13) , 000–000 Figure 4.
The velocity–gravity relation in distant observer redshift space as measured from sample (ii) extracted from the simulation, using the z coordinateof the fields. The color scaling is the same as in Fig. 3. Again the match between the most likely part of the bivariate PDF (darkest region) and the predictionfrom linear theory (thick solid line) is close to perfect. The dotted line gives the slope obtained directly from the ratio p h g z i / h v z i . The left panel whichcorresponds to minimum amount of smoothing can be compared to the two upper panels of Fig. 3. The differences with real space results are explained bythe arrows: large scale redshift distortion (LSRD) increases the effective slope of the velocity–gravity relation, while the finger of God (FOG) effect movespoints from the upper right down to the left, and points from the lower left up to the right, creating two additional tails of the PDF. As discussed in the text,this is due to the inversion of the signature of velocity happening in fingers of God as seen in the bottom right panel of Fig. 1, and to the effective smoothingof gravitational potential due to the extended nature of the FOG. When smoothing with a Gaussian window of radius ℓ = 10 h − Mpc is performed, thesetwo tails move back to the upper right and the lower left of the panel: one recovers the propeller shape obtained in real space. However, its effect is morepronounced, because there is still some remnant of the anti-diagonal effect, depending on the level of smoothing. again that Ω is biased low. This bias is more pronounced in red-shift space than in real space, because there is still some remnantof the anti-diagonal effect, depending on the level of smoothing. Ofcourse, smoothing at larger scales would make the agreement withlinear theory prediction better again.To reduce the bias, one can play again the exercise of measur-ing the slope of the velocity–gravity relation by selecting the regionof best likelihood, as in Equation (52). The agreement with lineartheory prediction is improved as expected, and the correspondingmeasured value of Ω matches very well the real value as illustratedby Table 1. However, additional smoothing tends to mix the non-linear finger of God effects with linear features, contaminating theregion of best likelihood. As a result, the measured value of Ω isslightly biased to lower values ( . instead of . ) and changingthe likelihood contour selection does not improve significantly theresults. This mixing effect brought by smoothing suggests that infact it might be not so wise to perform additional smoothing ofthe interpolated fields. Even though smoothing at sufficiently largescales brings better overall agreement with linear theory, it makesthe measurements much more sensitive to finite volume and edgeeffects and it is furthermore not needed here since linear theoryregime always dominates the region of best likelihood. Besides, weshowed in § worse.Figure 5 is exactly the same as Fig. 4, but the measurementsare now performed in sample (iii), without assuming that the ob-server is (infinitely) remote. Qualitatively, all the conclusions de-rived from the analysis of Fig. 4 still hold. The main difference isthat the measurements are more noisy due to the size of the sam-pled volume, now 5 times smaller. Therefore, the recovered value Valid, however, by definition, for the sample currently under considera-tion, (ii). of Ω from the measurements of the moments in the region of bestlikelihood of the joint PDF of the unsmoothed fields is good only atthe ten percent level ( Ω = 0 . instead of . , see Table 1). At thislevel of accuracy, we find that the linear prediction, which was de-rived in the approximation of DOL, agrees well with the measure-ments. To improve the quality of this comparison, Figure 6 givesthe same scatter plots as in Fig. 5, but for sample (iv), i.e. after co-adding the contributions of 125 different observers. The value of Ω derived from the best likelihood region of left panel of Fig. 6 agreesnow at a few percent level with the true value of Ω (the estimatedvalue is . , see Table 1), showing that the distant observer limitis an approximation good enough for deriving the linear prediction.Note again the bias to lower values ( . ) on the measured valueof Ω brought by additional Gaussian smoothing with a 10 h − MpcGaussian window. This bias is more pronounced than in the DOLsample (ii), and this is certainly at least partly due to the fact that,as discussed in §
3, deviations from the DOL limit are not negli-gible anymore for such a smoothing scale, given the sample depth,and they add to the mixing between linear and nonlinear featuresdiscussed above. It could also come partly from the edge effect dis-cussed in caption of Fig. 6.The dispersion among the 125 different observers leads to atypical error on Ω of the order of . , suggesting that the errorsrelated to the choice of the position of the observer – which probesthe space of configurations for the s = singularity discussed in § R = 70 h − Mpc ra-dius catalog. These errors include cosmic variance effects, but theselatter are probably underestimated because our spherical redshiftsamples represent a rather significant fraction of the total simula-tion volume.Nevertheless these measurements illustrate the relative robust-ness in that respect of our velocity–gravity estimator to derive Ω from a large-scale galaxy survey. However, we reiterate that, as al-ready mentioned in beginning of § c (cid:13) , 000–000 Colombi, Chodorowski & Teyssier
Figure 5.
Same as in Fig. 4, but for the sample (iii) which does not assume that the observer is (infinitely) distant. Although the general behavior is exactly thesame as in Fig. 4, several explainable differences can be noticed. The measurement is more noisy and the apparent width of the bivariate distribution is smallerthan on Fig. 4. This is mainly related to the fact that the sampled volume covers only 18 percent of the simulation box. Still, the most likely part of the PDFfits very well the slope predicted by linear theory in the infinite remote observer approximation, lending credence to our simplified approach.
Figure 6.
Same as in Fig. 5, but we now consider sample (iv), where 125 different observer positions are located on a regular grid in the simulation box andwhere the interpolated field is computed only on a grid. As expected, the resulting bivariate PDF is smoother than in Fig. 5, since the simulation volumeis now fully sampled by all these observer positions.Note the significant asymmetry of the tails of the bivariate PDF around the major axis of the maximum likelihood region. This is an expected consequenceof edge effects. Indeed, while performing redshift space projection to construct a spherical catalog of finite radius R max , the galaxies near the edge of thecatalog with positive radial velocity and radial position r < R max tend to get outside the sample. On the other hand, galaxies near the edge of the catalogwith negative radial velocity and radial position r > R max tend to get inside the catalog. As a result one expects more galaxies with negative radial velocitiesthan with positive ones, hence the asymmetry of the PDF. This effect mainly affects fingers of God, so the anti-diagonal tails of the bivariate PDF. Importantly,this effect does not affect too much the region of best likelihood. pute gravity field should be significantly larger than the actual vol-ume used to perform the velocity–gravity comparison. The presentanalyses suppose it is the case as indeed achieved by most recentthree dimensional galaxy catalogs such as the 2MRS, for whichthe median redshift corresponds to a half-depth of ≃ h − Mpc(Erdo ˘g du et al. 2006). Up to now, we have measured the velocity–gravity relation in veryrich catalogs, where the number of objects was so large that dis-creteness effects could be considered as negligible. In real galaxy catalogs, the number of objects is much smaller, particularly whentracers of the velocity field are taken at concern. Before addressingthe issue of biasing between the galaxy distribution and the darkmatter distribution, we therefore examine pure discreteness effects,by diluting our dark matter samples. This dilution will not onlybring a shot noise contribution, it will also increase the overall levelof smoothing which is performed by our interpolation procedure.To be able to quantify as accurately as possible the biases in-duced by discreteness in the mock “galaxy” catalogs considered innext section, we dilute randomly our N -body sample in two kindof subsamples (see Table 2):(D1) 125 realizations of dark matter catalogs involving 50000 c (cid:13) , 000–000 Table 2.
Same as Table 1, but for the dilute dark matter sub-samples, D1 and D2.Samp. Content N rea Size N g N obj Measured Ω from: Comment h g ih vg i q h g ih v i h vg ih v i (D1) 50000, real space 125 L = 200
128 50000 0.163 0.213 0.278 h all i , no smoothing0.004 0.006 0.008 error from dispersion0.237 0.252 0.268 h all i , smoothed0.007 0.008 0.009 error from dispersion0.283 0.300 0.319 h σ isocontour i , no smoothing0.011 0.012 0.013 error from dispersion0.277 0.288 0.299 h σ isocontour i , smoothed0.012 0.013 0.013 error from dispersion50000, redshift space 125 R = 70 ∗ h all i , no smoothing0.025 0.027 16.11 error from dispersion0.100 0.184 0.378 h all i , smoothed0.027 0.022 0.118 error from dispersion0.227 0.264 0.309 h σ isocontours i , no smoothing0.030 0.034 0.050 error from dispersion0.209 0.237 0.269 h σ isocontours i , smoothed0.026 0.028 0.039 error from dispersion(D2) 11000, real space 125 L = 200
128 11000 0.192 0.250 0.326 h all i , no smoothing0.011 0.017 0.026 error from dispersion0.216 0.255 0.300 h all i , smoothed0.015 0.020 0.026 error from dispersion0.251 0.292 0.340 h σ isocontour i , no smoothing0.023 0.027 0.033 error from dispersion0.249 0.281 0.318 h σ isocontour i , smoothed0.024 0.028 0.033 error from dispersion11000, redshift space 125 R = 70 ∗ h all i , no smoothing0.028 0.035 1.687 error from dispersion0.096 0.197 0.492 h all i , smoothed0.031 0.034 0.238 error from dispersion0.190 0.253 0.347 h σ isocontours i , no smoothing0.040 0.048 0.099 error from dispersion0.182 0.233 0.308 h σ isocontours i , smoothed0.040 0.044 0.089 error from dispersion points, similarly as for the mock with mass thresholding M th =5 . M ⊙ used in § M th = 4 . M ⊙ used in § gridsfor interpolating the fields and we have 125 different realizations)and in samples (iv) (in redshift space). In the latter case, note thatthe observer position in the simulation cube is different for eachrealization, chosen exactly to be on a regular patter covering thefull simulation volume as in samples (iv).The results are summarized in Table 2 and illustrated byFigs. 7 and 8. We first discuss real space measurements using thesecond order moments of the full PDF and compare the values ob-tained for Ω to those from the undiluted sample (i), given in Table 1.Note that this sample uses a grid for the interpolation insteadof the one for samples D1 and D2. Using a resolutiongrid increases the measured value of Ω in first line of Table 1 from . to . . Taking that fact into account, we notice that, due tothe dilute nature of the samples, our interpolation procedure usesan adaptive kernel of larger size: as shown by Fig. 7, this makes thefields more linear, closer to the Gaussian limit, and decreases thelevel of effective bias brought by the “propeller” shape of the bivari-ate PDF. As a result, prior to additional smoothing with a Gaussian window of size ℓ = 10 h − Mpc, the measured Ω is larger for thedilute samples than for the full one, and the convergence with lineartheory prediction improves with level of dilution. The mean inter-particle distance in the sparser sample is of the order of λ = 9 h − Mpc, slightly lower than the size of the post-processing Gaussiansmoothing kernel. One thus expects rough agreement between fullsample and samples D1 and D2 after smoothing with such a win-dow, which is approximately the case. Furthermore, the measuredvalue of Ω in D2 is not very sensitive to whether additional smooth-ing is performed or not, since λ ≃ ℓ .Still examining real space measurements, we now consider thevalues of Ω measured from second order moments of the PDF, butusing only the region of best likelihood, Eq. (52), similarly as inprevious paragraph. As expected, the effective bias due to nonlinearcontributions (or propeller effects) is tremendously reduced, andone recovers a value of Ω compatible with the true value, giventhe errorbars. These latter, which estimate pure shot noise, are ofthe order of 4 and 10% for D1 and D2 respectively. As for the fullsample, this result stands also when additional smoothing is appliedto the data, although one can notice a slight bias to lower values of Ω . We now turn to redshift space measurements and first considerthe measured value of Ω using the second order moments of the fullPDF. The results obtained in previous section still hold (compareTable 1 to Table 2), except that the measurements performed be- c (cid:13) , 000–000 Colombi, Chodorowski & Teyssier
Figure 7.
Similarly as in Fig. 3, but for the 125 realizations of diluted samples D1 (upper panels) and D2 (lower panels). the left panels correspond to theraw interpolated fields on a grid, while for the right panels, a smoothing with a Gaussian window of size ℓ = 10 h − Mpc was performed prior to themeasurements. Since the samples are now more diluted, the adaptive interpolating kernel is of much larger size, which decreases considerably nonlinear tails:the propeller shape of the PDF is less apparent but it still affects the measurements: the slope of the dotted line is larger than the linear expectation given bythe thick solid line. fore additional Gaussian smoothing give a lower value of Ω , . for D1 and . for D2: the low bias effect on Ω gets worse withdilution, since the larger size of the adaptive kernel tends to reducethe effect of fingers of God: as noticed previously, finger-of-God ef-fects help to reduce the asymmetry brought by the “propeller”, themain source of the low bias on Ω . Additional Gaussian smoothingwith a window of size ℓ = 10 h − Mpc improves the convergencebetween D1, D2 and the full sample, as expected, but induces ahighly underestimated value of Ω ∼ . , because of the propellereffects which are then prominent. These arguments are supportedby examination of Fig. 8. Note that the asymmetry of the tails ofthe bivariate PDF around the major axis of the maximum likeli-hood region, already observed and explained in Fig. 6, is now morepronounced, at least from the visual point of view.Measurements are improved while selecting the region of bestlikelihood, but not as well as in the real space case or in the red-shift space case with full sampling: at best, Ω is underestimatedby about 12 percent. Additional smoothing or passing from D1 toD2 expectingly increases the bias. This underestimate comes againfrom the fact that the adaptive interpolating kernel is now of muchlarger extension than for the full particles sample, which induces biases comparable to what was observed for the full samples withadditional smoothing ( § Ω by narrowing the region of best likelihood at the price ofan increase of the errors. For instance, taking a 38 percent confi-dence region enclosed by a ≃ σ isocontour gives Ω = 0 . and Ω = 0 . for D1 and D2, respectively. Our procedure formeasuring Ω can thus certainly be improved with a more sophis-ticated treatment of the region of best likelihood. For instance, away to extract in an unbiased way the parameter Ω from the datacould consist in measuring the local slope of the skeleton of thesurface representing the bivariate PDF (see Novikov, Colombi &Dor´e, 2006) after appropriate (adaptive) smoothing of the velocity–gravity scatter plot. This is left for future work; in what follows, weshall still use for the sake of simplicity our . σ likelihood contourtechnique, while staying aware of the bias brought by dilution. To estimate in a sufficiently realistic way how biasing affects theresults on the velocity–gravity relation, we extracted from the sim-ulation four “galaxy” catalogs, corresponding to two methods of c (cid:13) , 000–000 Figure 8.
Similarly as in Fig. 6, but for the diluted samples D1 (upper panels) and D2 (lower panels). treating dark matter halos. In the first method, we consider eachdark matter halo as a galaxy. In the second method, we considereach substructure present in dark matter halos as a galaxy. The de-tails are given in § M > . M ⊙ and M > . M ⊙ respectively. In all cases, each galaxy isgiven the same weight while density and velocity are interpolatedas explained in § § To extract halos and substructures from the simulation, we usethe publically available software adaptaHOP (Aubert, Pichon & Colombi, 2004). AdaptaHOP builds an ensemble of trees. Eachtree corresponds to a halo which is a connected ensemble of par-ticles with SPH density ρ > . The branches of the trees arecomposite structures of which the connectivity is controlled by thesaddle points in the particle distribution. The leaves of the treescorrespond to the smallest possible substructures one can find inthe simulation. From this ensemble of trees, we extract two kindsof catalogs, one where each galaxy is identified to a tree, the otherone where each galaxy is identified to the leaves of the trees (if atree has only one leaf, it means that the halo is its own single sub-structure). Note that velocities of these galaxies are computed as theaverage velocity of all the particles belonging to the corresponding(sub-)structure.Additional mass thresholding is used to control the number of“galaxies” in the catalogs,. In final, 4 mock catalogs are obtained(see also Table 3):(v) A “halos” catalog, involving 43482 dark matter halos withmasses larger than M th = 5 . M ⊙ ;(vi) A “massive halos” catalog, involving 11934 dark matter ha-los with masses larger than M th = 4 . M ⊙ ; The parameters used in adaptaHOP are the same as in Aubert et al.(2004, Appendix B), namely N SPH = 64 , N HOP = 16 , ρ TH = 81 and f Poisson = 4 .c (cid:13) , 000–000 Colombi, Chodorowski & Teyssier
Table 3.
Same as Table 1, but for the “galaxy” catalogs, (v)–(viii). In the second column, it is specified if they correspond to a halo (“HAL.”) or a substructurecatalog (“SUBS.”) and what mass threshold was used to select the dark matter (sub-)structures. The values of Ω displayed in columns seven to nine are obtainedfrom the measured β assuming linear theory predictions with no bias. As a guidance to understand the results, we show in the second column of the table themeasured value of the bias b at the simulation box size scale (using the power-spectrum of the interpolated density), as well as Ω eff = Ω /b / , the expectedvalue of Ω from the measured β if unity bias is assumed.Samp. Content N rea Size N g N obj Measured Ω from: Comment h g ih vg i q h g ih v i h vg ih v i (v) HAL. > M ⊙ , real space 1 L = 200
128 43482 0.304 0.375 0.464 all, no smoothing b = 0 . , Ω eff = 0 . σ isocontour, no smoothing0.329 0.370 0.415 1.5 σ isocontour, smoothedHAL. > M ⊙ , red. space 125 R = 70 ∗ h all i , no smoothing b = 0 . , Ω eff = 0 . h all i , smoothed0.035 0.039 0.055 error from dispersion0.311 0.360 0.419 h σ isocontours i , no smoothing0.054 0.063 0.082 error from dispersion0.310 0.345 0.385 h σ isocontours i , smoothed0.058 0.064 0.078 error from dispersion(vi) HAL. > M ⊙ , real space 1 L = 200
128 11934 0.206 0.254 0.313 all, no smoothing b = 0 . , Ω eff = 0 . σ isocontour, no smoothing0.231 0.258 0.288 1.5 σ isocontour, smoothedHAL. > M ⊙ , red. space 125 R = 70 ∗ h all i , no smoothing b = 0 . , Ω eff = 0 . h all i , smoothed0.030 0.032 0.045 error from dispersion0.215 0.255 0.305 h σ isocontours i , no smoothing0.044 0.049 0.065 error from dispersion0.215 0.245 0.282 h σ isocontours i , smoothed0.045 0.049 0.061 error from dispersion(vii) SUBS. > M ⊙ , real space 1 L = 200
128 55044 0.215 0.253 0.298 all, no smoothing b = 0 . , Ω eff = 0 . σ isocontour, no smoothing0.264 0.277 0.292 1.5 σ isocontour, smoothedSUBS. > M ⊙ , red. space 125 R = 70 ∗ h all i , no smoothing b = 0 . , Ω eff = 0 . h all i , smoothed0.023 0.019 0.041 error from dispersion0.234 0.264 0.299 h σ isocontours i , no smoothing0.030 0.035 0.045 error from dispersion0.224 0.244 0.267 h σ isocontours i , smoothed0.028 0.031 0.038 error from dispersion(viii) SUBS. > M ⊙ , real space 1 L = 200
128 11221 0.189 0.226 0.269 all, no smoothing b = 1 . , Ω eff = 0 . σ isocontour, no smoothing0.218 0.236 0.255 1.5 σ isocontour, smoothedSUBS. > M ⊙ , red. space 125 R = 70 ∗ h all i , no smoothing b = 1 . , Ω eff = 0 . h all i , smoothed0.023 0.021 0.040 error from dispersion0.178 0.211 0.251 h σ isocontours i , no smoothing0.030 0.033 0.046 error from dispersion0.177 0.201 0.230 h σ isocontours i , smoothed0.030 0.032 0.041 error from dispersion (vii) A “sub-structures” catalog, involving 55044 dark mattersub-halos with masses larger than M th = 5 . M ⊙ ;(viii) A “massive sub-structures” catalog, involving 11221 darkmatter sub-halos with masses larger than M th = 4 . M ⊙ .From each of these catalogs, we compute the velocity and gravityfield on a grid as explained in § h − Mpc Mpc centered onthe observer exactly as was done to generate the pure dark matterrealizations (iv) in § c (cid:13) , 000–000 can consider the sub-structure catalogs as the most realistic. Thisis known to be true only to a limited extent. Indeed, sub-halos tendto be tidally disrupted while they spiral in their host halo, thereforeone expects less sub-structures than galaxies in the core of darkmatter halos (e.g. Diemand et al. 2004; Nagai & Kravtsov 2005).This is all the more true since we apply a mass thresholding tocontrol the number of objects in our catalogs. However, the effectof such a depletion should be noticeable only at the smallest scalesand should not affect significantly the results of the analyses at thelevel of accuracy reached in this paper.(b) dark matter halos are representative of the galaxy distribu-tion in terms of an ensemble of structures composed of clusters,groups of galaxies and field galaxies. Therefore, if one considersthe galaxy distribution from a slightly different perspective, thesestructures can be used as well to study the velocity–gravity relation,with the appropriate weighting. The potential advantage of such anapproach is to reduce considerably the fingers of God effects dis-cussed previously, since all the galaxies belonging to one cluster orgroup of galaxies are collapsed to a single point. Note that we over-simplify the analyses here by purposely giving the same weight toall the dark matter halos: this rather extreme procedure is expectedto introduce quite significant (anti-)biasing effects on the gravityfield determination. The results of our analyses are summarized in Table 3 and illus-trated by Figs. 9 and 10. As shown in § β to lower values, both in real and redshift space.To reduce this systematic effect, we propose to consider only theregion of best likelihood. Strictly speaking, this approach makessense only if the bias between the galaxy distribution and the darkmatter distribution is either inexistent or linear. The concept of anon local, scale dependent bias as experienced here complicatesconsiderably the analyses. However, the aim is to capture the lin-ear contribution of the dynamics, which in terms of our volume-weighted measurements is dominated by underdense regions. Inthese regions, the size of the adaptive kernel is large: inside thecontour of best likelihood of the joint PDF, the measurements areexpected to be dominated by the large scale bias, b ( k ) , with smallwavenumber k . In that regime, the bias is in general roughly linearor close to linear. The need to actually consider the region of bestlikelihood of the PDF to perform the measurements is thereforefurthermore justified. As a result, in what follows, only results ofTable 3 obtained from selecting the region inside the . σ contourof the PDF are considered. Since additional smoothing tends to in-troduce additional bias in the redshift space measurements, it willnot be discussed either, although the results are shown in Table 3for completeness.We first discuss real space measurements and try to under-stand the effects of the bias. Sample (v) is the halo catalog withlow mass threshold. This catalog is expected to present signifi-cant antibias. Indeed, if small halos, which mainly lie in moder-ately dense and underdense regions, trace rather well the underly-ing dark matter distribution, the largest ones, which are collapsedto a single point, induce a significant underestimate of the over-all strength of the dark matter gravity field, resulting in the antib-ias effect. As shown in Table 3, the value of the bias measuredat the simulation box size (using the Fourier modes at the largestscales) gives b ( k = 2 π/L ) = 0 . . If one uses, as argued above, this value of b as a reference, the effective value of Ω obtainedfrom linear theory, ignoring the bias, is larger than the true one, Ω eff = Ω /b / = 0 . . This is to be compared to the value of . obtained from p h g i / h v i . Although this latter result is signifi-cantly lower than the “expected” value, two facts have to be takeninto account:(a) As an effect of non local, scale dependent biasing, the cor-relation between gravity and velocity field is not as tight as for thedark matter diluted sample, D1, albeit this latter contains the samenumber of objects. Indeed, one measures Ω = 0 . and . from h g i / h vg i and h vg i / h v i respectively in sample (v), while Ω wasranging between . and . for D1. This loss of tightness in thevelocity-gravity relation obviously affects in a non trivial way theregion of best likelihood.(b) Our approach for estimating the effects of the bias is rathercrude and does not give account of the additional subtleties relatedto our adaptive smoothing procedure.Taking these two points into account, the measurements are ratherconsistent with the expectations, at least at the qualitative level.In the halo catalog (vi), small mass “galaxies” are removed,so the underdense regions are not populated anymore, leaving uswith a set of more clustered objects than in sample (v), since thehigher is their mass, the more significantly the halos are clustered.The consequence is that the antibias found in catalog (v) is re-duced to b ( k = 2 π/L ) = 0 . , resulting in an effective ex-pected value of Ω eff = 0 . , to be compared with the value of . obtained from the measurement of p h g i / h v i . Again thearguments (a) and (b) developed just above apply, leading us toconclude that this result is consistent with expectations, and sim-ilarly for the substructures catalogs: sample (vii) is the closest tothe underlying dark matter distribution, with a very slight antib-ias of b ( k = 2 π/L ) = 0 . and Ω eff = 0 . to be comparedto . from p h g i / h v i ; sample (viii) presents a small positivelarge scale bias, with b ( k = 2 π/L ) = 1 . and Ω eff = 0 . , tobe compared to . from p h g i / h v i .We can conclude here that we understand what are the var-ious effects influencing the measurements in the mock catalogs.The general trend is that the “true” value of parameter β (i.e. thevery large scale one) seems to be underestimated by about 10 to20 percent. We give here quotes, because such a value is not welldefined. Indeed, the interpretation of the measurements is compli-cated by adaptive filtering, combined with a non local, scale de-pendent bias. But as discussed above, we really want to capture thevalue of β in the linear regime limit, where the bias is expectedto be roughly linear, scale independent and in fact rather close tounity, as observed in our “galaxy” catalogs. Note interestingly thatthe conditional moment h vg i / h v i gives a rather good estimate ofthe expected value of β . Without going too far exploiting such aproperty, which may be specific to our catalogs, it is clear that theslopes given by the two conditional moments could be used as prox-ies to estimate a range of possible values for β , or in other words,errorbars. Indeed, the difference between these slopes is related tothe width of the likelihood region, i.e. it estimates the tightnessof the correlation between gravity and velocity as measured in thecatalog. However, note that this difference does not properly giveaccount of systematic effects discussed at length in this paper, aswell as cosmic variance effects, although it is indirectly related tothem.We now turn to redshift space measurements. What matters isthat they should be self-consistent with the real space ones. This isnearly the case, except again that they tend to underestimate real c (cid:13) , 000–000 Colombi, Chodorowski & Teyssier
Figure 9.
The velocity–gravity relation in real space as measured in our mock “galaxy” catalogs (v)–(viii). The left column of panels corresponds to the directSPH alike interpolation on a grid, while an additional Gaussian smoothing with a window of size h − Mpc was performed in the right column ofpanels. The informations on the catalogs are given as titles on each panel, namely, from top to bottom, halo catalog with mass thresholding M th = 5 . M ⊙ ,sub-structure catalog with M th = 5 . M ⊙ , halo catalog with M th = 4 . M ⊙ and sub-structure catalog with M th = 4 . M ⊙ . The predictionfrom linear theory in absence of bias is given as the solid line while the slope obtained directly from the ratio p h g z i / h v z i is represented as a dotted line. space measurements by up to ten percent, an effect of sparse sam-pling discussed in § Ω in redshift space andthe “expected’, effective one, Ω eff , is about 25 percent, with a sys-tematic bias to lower values, as always. Obviously our mock cata- logs are very peculiar in a sense that they represent quite an extremecase in the range of possibilities. And still, without any assumptionon the bias, or in other words, assuming that b is of the order ofunity, we can determine Ω with an accuracy of the order of 30 per-cents without significantly strong prior. Furthermore, the quality of c (cid:13) , 000–000 Figure 10.
Same as in Fig. 9, but in redshift space. The scatter plots are obtained from the average of 125 observer positions disposed on a regular patternspanning the simulation volume. As expected, finger of God effects are more prominent on the sub-structures catalogs than on the halos catalogs. this estimate can certainly be improved with appropriate weightingof the galaxies. However, as discussed in the conclusion that fol-lows, the sources of systematic errors related to the instrumentalnoise or to the method used to provide velocities were not consid-ered here.
In cosmic density–velocity comparisons, since the density field isgiven solely in redshift space, redshift-space effects have to be ac-counted for. One method is to perform the analysis directly in red-shift space, as proposed by ND (Nusser & Davis 1994). Anotherone is to reconstruct the real-space density field, but this method iscomplicated by the fact that redshift-space corrections require as-suming a value for β . As β is not known a priori, one is forced to c (cid:13) , 000–000 Colombi, Chodorowski & Teyssier reconstruct the density field for a range of values of β , which makesthe real-space comparison involved, despite its apparent simplicity.In the present paper we have proposed a method that works in red-shift space, but is simpler than the formalism of ND, which is basedon a spherical harmonics expansion. It is essentially as simple as inreal-space; the only difference is that here a directly estimated pa-rameter is not just β , but its simple function.This method relies on a tight correlation between cosmic grav-ity and cosmic velocity in redshift space. We have derived thevelocity–gravity relation in redshift space and its scatter analyti-cally, under the assumptions of the linear regime, distant observerlimit (DOL) and CMB rest frame for redshift measurements. Thisrelation has turned out to be a simple modification of the corre-sponding one in real space and in the linear regime. Then we haveperformed a dark matter N -body experiment to test the velocity–gravity relation both in real space and in redshift space. We alsoextracted mock galaxy catalogs from this data set, with rather ex-treme prescription for selecting “galaxies” from the dark matter dis-tribution in order to analyze the effects of non-trivial biasing on thevelocity–gravity relation. The main results are the following: • To perform the measurements, we propose to interpolate thegravity and the velocity fields on a regular grid using a new adap-tive smoothing procedure described in § sample analyzed in this paper. • In real space, simple measurements using moments of the fulljoint probability distribution function (PDF) bias Ω to low values.This is due to the fact that these moments are contaminated by thetails of the PDF. These latter are influenced by nonlinear dynamics,which induces a global ‘propeller’ shape for the PDF. • In redshift space, fingers of God (FOG) induce additional“anti-diagonal” tails to the PDF. This effect seems to somewhatcompensate the bias due to the propeller shape. However this resultcould be coincidental and is too sensitive to the level of smoothingapplied to the data. Note that in actual peculiar velocity catalogs,clusters of galaxies are in general collapsed into one point to im-prove the signal to noise ratio of the distance estimates. AlthoughFOG are thus less of an issue in real data, rich filaments can stillcontribute to an effect similar to FOG as discussed in § • Selecting about a . σ (68% confidence) region around themaximum of the joint PDF of gravity and velocity fields makes thedata to match the linear theory predictions very well, both in realand in redshift space. This result stands even in the highly nonlinearregime (no additional smoothing except the adaptive interpolationon the grid). It is easily explained by the fact that our measurementsare volume weighted, giving most of the statistical weight to under-dense regions. Therefore, it is sufficient to exclude only the tails ofthe distribution to obtain unbiased estimate of Ω . • Additional smoothing does not improve the agreement withlinear theory predictions: first, it is expected to increase deviationsfrom the DOL limit; second, it makes the situation worse in theregion of best likelihood, where it mixes linear contributions with Using CMB rest frame is crucial to avoid the so-called rocket effect onthe redshift space gravitational acceleration, as discussed in § nonlinear ones and biases β to lower values, an effect which is min-imized as much as possible with our adaptive smoothing procedure. • The measurements obtained in our mock catalogs provide avalue of β in agreement with intuition, namely that large scale biasdominates if the region of best likelihood is selected. This is in-herent to our volume weighted approach which gives more weightto underdense regions (see also Berlind, Narayanan & Weinberg2001). However, due to the dilute nature of real catalogs, we ex-pect an effective bias on β to lower values. This bias was foundto be of the order of 10–20 % for the mock samples consideredin this paper. In addition, the velocity–gravity relation looses sometightness due to effects of non-local biasing between luminous anddark matter distributions. Nevertheless, our analyses show that di-rect measurement of Ω from real catalogs assuming no bias shouldgive an answer accurate to about
30 % at worse. Moreover, this er-ror can probably be significantly improved to values as low as 10 % , with the appropriate weighting of the data (purposely not per-formed here), as discussed below.There are several points we did not address in our numericalanalyses, and we leave them for future work: - Noise: through the analysis of dilute samples and mock cata-logs, we addressed to some extent the effects of shot noise for real-istic modern catalogs. However, we used the same catalog for ve-locity and gravity fields, implying that the same adaptive kernel canbe used for both fields, which is not realistic. Furthermore we didnot take into account the fact that there is in practice a significantrelative error on galaxy distance estimates, which can be as large as20 percent. Thus the velocity–gravity relation is certainly not ex-pected to be as tight in real catalogs as in the samples considered inthis paper. Note interestingly, though, that since the measurementsare performed in redshift space, issues related to Malmquist bias(e.g., Strauss & Willick 1995) should be irrelevant, as long as thedistance estimates are unbiased (just noisy). - Incompleteness:
Another closely related issue is that the cata-logs used to estimate the gravity field are incomplete. First, edgeeffects are expected to be significant because gravity force is oflong range. The galaxy catalog used to estimate the gravity fieldhas to be significantly deeper than the region used to perform thevelocity–gravity comparison. We did not estimate in this paper howdeep it has to be, not to mention the problem of obscuring by ourown Galaxy. Second, the apparent density of galaxies decreaseswith increasing distance from the observer. In order to estimate theinterpolated density field, one can, under the assumption of no biasbetween the galaxy distribution and the mass distribution, weightthe galaxies by the inverse of the selection function, φ . In the linearregime, in the absence of bias and for redshifts measured relative tothe CMB rest frame, the redshift space density contrast measuredthat way reads (Kaiser 1987) δ s = δ − (cid:16) d ln φd ln r (cid:17) vr − ∂v∂r . (54)Under the DOL, the second term on the right hand side of this equa-tion drops, so the effect of the selection function disappears. How-ever, since remote parts of the catalogs are sparser, they might begiven lesser statistical weight, hence augmenting the relative con-tribution of this deviation from the distant observer limit. - Better handling of the bias: our analyses in the simple mockcatalogs show that biasing affects the velocity–gravity relation in If the catalogs are different, a common smoothing kernel must be deter-mine to find a compromise between the velocity and the gravity samples.c (cid:13) , 000–000 a non trivial way. However, we used the most naive prescriptionto compute the gravity field, giving equal weight to all galaxies.Clearly, a better description would be to assign to each galaxy aweight proportional to its supposed host halo mass, for exampleassuming constant mass to light ratio, M/L , or a more sophisti-cated weighting using a function
M/L = f ( L ) derived from ob-servations or obtained from theoretical models of galaxy formation.That would allow one to correct to a large extent for the effects ofbiasing, at the cost of additional priors. Still, unless the galaxy cat-alog used to compute the gravity field is very deep, i.e. includesfaint galaxies, even giving the proper mass to light ratio to eachgalaxy does not correct for the fact that underdense regions can beartificially underpopulated, especially far away from the observer.To tackle with that, one could add a background population of lowmass objects that would give account of the missing mass in the cat-alog and that would act as a shielding effect on the gravity field (see,e.g., Phelps et al. 2006). Again, some strong priors are required todeal with such a background, in particular on its clustering proper-ties and how it correlates with the population already present in thereal catalog.Note finally that while our approach presents the advantageof simplicity compared to ND (and most other methods), we donot expect it to be as accurate as of ND, since it relies on the dis-tant observer approximation and uses in its current form adaptivesmoothing, which complicates the interpretation of the results. (Onthe other hand, we clearly showed that the region of best likelihoodof the joint velocity–gravity distribution is dominated by the lin-ear regime prediction, even in the highly nonlinear regime, whichpresents a noticeable advantage.) Similarly, our method is not ex-pected to perform as well as sophisticated reconstructions of La-grangian nature, which try to minimize the Euler-Lagrange action(e.g., Peebles 1989; Shaya, Peebles & Tully 1995; Nusser & Bran-chini 2000; Phelps 2002; Phelps et al. 2006) or to solve optimalassignment problem (e.g., Croft & Gazta˜naga 1997; Frisch et al.2002; Mohayaee et al. 2003; Mohayaee & Tully 2005), althoughthis remains to be verified. ACKNOWLEDGMENTS
This work was carried out within the framework of the EuropeanAssociated Laboratory “Astronomy Poland–France” and was per-formed within the Numerical Investigations in Cosmology (NIC)group as a task of the HORIZON project. This research has beenalso supported in part by the Polish State Committee for ScientificResearch grant No. 1 P03D 012 26, allocated for the period 2004–2007. The computational resources (HP cluster) for the present nu-merical simulation were made available to us by Centre de Calculen Recherche et Technologie (CCRT, CEA).
REFERENCES ˘g (cid:13) , 000–000 Colombi, Chodorowski & Teyssier
APPENDIX A: SPHERICAL TOP-HAT IN REDSHIFTSPACE
The general result in real space is g v = β − δθ , (A1)where δ and θ are the real-space density contrast and velocity diver-gence, respectively. In redshift space, in the distant observer limit,the general result is g v = 32 C β − (cid:16) δθ + β (cid:17) . (A2)Here, C = 2(1 − e ) e (cid:20) (1 − e ) − / − sin − ee (cid:21) , (A3)eccentricity e = (1 − ̺ ) / , (A4)and ̺ = 1 − βθ/ . (A5)In the linear regime, C = 2 / and θ = δ . Hence, g v = β − (cid:16) β (cid:17) . (A6)We see thus that the spherical top-hat model does not give the sta-tistical result obtained from linear theory. The reason for this is thefollowing. In the spherical model, the velocity–gravity relation isfully deterministic, but we could also derive it calculating the rele-vant statistical averages. While in the linear regime, in general ˜ δ ( g ) s ( k ) = b (1 + βµ )˜ δ ( k ) (A7)[where µ = k · s / ( ks ) ], in the spherical top-hat we have δ ( g ) s , TH = b (1 + β/ δ , (A8)that immediately yields ˜ δ ( g ) s , TH ( k ) = b (1 + β h µ i )˜ δ ( k ) , (A9)since h µ i = 1 / . In other words, the expression for the redshiftspace density contrast in the top-hat model is an average of gener-ally valid expression (A7) over possible orientations of the vector k . In the case of the estimator h v s g s i / h v s i , the average over anglesyields an expression proportional to h µ (1 + βµ ) ih µ i = 1 + β h µ ih µ i = 1 + 35 β . (A10)In the top-hat model, this is modified to (cid:10) µ (cid:0) β h µ i (cid:1)(cid:11) h µ i = 1 + β (cid:10) µ (cid:11) h µ i = 1 + 13 β . (A11)Loosely speaking, the top-hat model performs a part of this average‘too early’. So the top-hat model yields quantitatively different re-sults from the results of a rigorous statistical calculation. However,it is sufficient for qualitative purposes, needed here. In particular, itcorrectly predicts that the slope of the velocity–gravity relation inredshift space gets steeper.Turn-around in real space corresponds to extreme flatness ofthe pancake in redshift space. When θ → θ t ≡ β − , then ̺ → and C tends to its maximal value, . Hence, g v → β − (cid:16) δ t θ t + β (cid:17) ≃ . . (A12) Here, δ t is related to θ t by the formula of Bernardeau (1992). Infact, Bernardeau et al. (1999) invented its more accurate modifica-tion: δ ≃ (1 + θ/α ) α , (A13)where α is slightly greater than / . However, for our purposes α = 3 / should be accurate enough; moreover, it better describesthe evolution of voids [for this value of α , δ ( θ = − .
5) = − ].This formula works for mildly non-linear densities.For θ > β − , the structure in redshift space is inverted: ve-locity, and so the ratio of g to v , changes sign. Then ̺ = βθ/ − , (A14)and g v = − C β − (cid:16) δθ + β (cid:17) . (A15)For θ = θ i ≡ β − we have ̺ = 1 , so the structure in redshiftspace is momentarily an (inverted) sphere, hence without furthereffort, C = 2 / . Therefore, then g v = − β − (cid:16) δ i θ i + β (cid:17) ≃ − . . (A16)For θ ≫ β − (highly nonlinear infall), the (inverted) structure inredshift space becomes very elongated (finger of God). Then ̺ = βθ/ − ≃ βθ/ ≫ and e ≃ (1 / βθ i . After some algebra,one can show that then C → βθ/ βθ/ . (A17)Moreover, then one cannot use formula (A13), but δ → ( θ/ (Bilicki & Chodorowski, in preparation), hence δ/θ ∝ θ . Thisyields − g v ∝ θ − ln θ. (A18)In other words, g becomes very small compared to | v | . c (cid:13)000