the Ω dependence in equations of motion

© 1998 RAS

Mon. Not. R. Astron. Soc. 294, 457–464 (1998)

The X dependence in equations of motion

Adi Nusser and Jorg M. Colberg

Max-Planck-Institut fur Astrophysik, Karl Schwarzschild Strasse 1, D-85740 Garching bei Munchen, Germany

Accepted 1997 October 1. Received 1997 August 27; in original form 1997 May 19

A B STR ACTThe equations of motion governing the evolution of a collisionless gravitating systemof particles in an expanding universe can be cast in a form which is almostindependent of the cosmological density parameter, W, and the cosmologicalconstant, L. The new equations are expressed in terms of a time variable t=ln D,where D is the linear rate of growth of density fluctuations. The dependence on thedensity parameter is proportional to e\WÐ0.2Ð1 times the difference between thepeculiar velocity (with respect to t) of particles and the gravity field (minus thegradient of the potential); or, before shell-crossing, times the sum of the densitycontrast and the velocity divergence. In a one-dimensional collapse or expansion,the equations are fully independent of W and L before shell crossing. In the generalcase, the effect of this weak W dependence is to enhance the rate of evolution ofdensity perturbations in dense regions. In a flat universe with L80, thisenhancement is less pronounced than in an open universe with L\0 and the sameW. Using the spherical collapse model, we find that the increase of the rms densityfluctuations in a low-W universe relative to that in a flat universe with the same linearnormalization is 10.01e (W),d3., where d is the density field in the flat universe.The equations predict that the smooth average velocity field scales like W0.6, whilethe local velocity dispersion (rms value) scales, approximately, like W0.5. High-resolution N-body simulations confirm these results and show that density fields,when smoothed on scales slightly larger than clusters, are insensitive to thecosmological model. Haloes in an open model simulation are more concentratedthan haloes of the same M/W in a flat model simulation.

Key words: cosmology: theory – dark matter – large-scale structure of Universe.

1 INTRODUCTION

The cosmological background determines the growth rateof matter density fluctuations. This is the result of twoeffects. First, the initial conditions are specified in terms ofthe density contrast field d=r (x)/rbÐ1. Therefore theactual density, r (x), which dictates the dynamical evolution,as can be seen for example from the spherical top-hatmodel, depends on the mean matter density, rb. The secondeffect comes about simply because the mean matter densityvaries with time according to the assumed cosmologicalmodel. This in turn translates into a dependence of theevolution of the fluctuation field d on the parameters of thecosmological model: the density parameter, W, and thecosmological constant, L. Here we focus on the followingaspect of the dependence of dynamics on the cosmologicalbackground. Starting from an initial density fluctuation field

and a given amplitude of the evolved field, we address thequestion: how do the evolved peculiar velocity and densityfields depend on the parameters W and L? In the linear (e.g.Peebles 1980) and in the Zel’dovich quasi-linear (Zel’do-vich 1970) approximations, once an initial density fluctua-tion field is evolved to a given amplitude, it does not containany information on the parameters W and L. In theseapproximations, the peculiar velocity field is simply propor-tional to f (W, L) where f is the so-called linear growthfactor. This result is easy to understand. In the linearapproximation, the density fluctuations are merely ampli-fied by a time-dependent factor, D. In the Zel’dovichapproximation, the displacement vector is the product ofthe initial gravity field and the function D. Moreover,second-order perturbation theory calculations (e.g. Bouchetet al. 1992) have shown that moments of the density fluctua-tion field are very insensitive to W and L. Finally, in the

highly non-linear regime, N-body simulations (e.g. Davis etal. 1985) show that the final matter distribution in simula-tions with the same initial conditions changes very little asthe parameters of the cosmological background are varied.Significant differences between flat and open models arefound only in the cores of what are identified as rich clustersin these simulations. These results have proved useful inanalysing observations of the large-scale structure. Nusser& Dekel (1992), for example, used N-body simulations toargue that a recovery of the initial density fluctuations fromthe observed galaxy distribution is almost independent off (W, L), whereas a recovery from the observed peculiarvelocity field is sensitive to the assumed W. They appliedtheir reconstruction method to the POTENT compilationof the peculiar velocity data and to the 1.2-Jy IRAS surveyand concluded that Wa0.3 with high confidence. Bernar-deau et al. (1995) used second-order perturbation theory toargue that the reduced skewness of the divergence of thepeculiar velocity field is inversely proportional to f, inaccordance with the scaling implied by the Zel’dovichapproximation. They found that the measured skewness isconsistent with W of about unity.

Here we aim at better understanding of the dependenceof the equations of motion on the cosmological parameters.In Section 2 we write the equations of motion in a formwhich is almost independent of the background cosmology.We discuss the W dependence in toy models in Section 3. InSection 4, we use high-resolution N-body simulations toinvestigate in detail the differences in the matter distribu-tion in flat and open models. We conclude with a summaryand discussion in Section 5.

2 A LMOST X- INDEPENDENT EQUATIONS OF MOTION

We restrict our treatment to the case of a matter-dominateduniverse with a cosmological constant, i.e. we assume thatthe total mean density is rtot\rb+L/(8pG) where rb(t) isthe matter contribution and L is the cosmological constant.We use the standard notation, in which a (t) is the scale-factor, H (t)\a/a is the time-dependent Hubble factor,W\rb(t)/rc(t) and l\L/3H 2 where rc\3H 2/8pG is thecritical density. Let x and v\dx/dt be the position and pecu-liar velocity of a particle in comoving coordinates. Theequations of motion are: the continuity equation

dddt

+(1+d)H ·v\0;(1)

the Euler equation of motion,

dv

dt+2Hv\Ð

3

2 WH 2Hf; (2)

and the Poisson equation,

Df\d. (3)

Note that we have defined f=2Fg/(3WH 2) where Fg is thepeculiar gravity potential in comoving coordinates. Equa-tions (1), (2) and (3), together with the Friedman equationsfor the background quantities W and a, fully specify thedynamics of pressure-free density fluctuations. The scale-

factor a can be solved for using the Friedman equation

2da

dt32

\8p

3Grba 2+

L3

a 2Ðk, (4)

where k\+1, Ð1 and 0 correspond, respectively, toclosed, open or flat universes, and we work in units in whichc\1. Energy conservation, rba 3\constant, yields W\c0/H 2a 3 where c0\W0H 2

0 a 30 and the subscript 0 denotes quanti-

ties at the present time. Therefore, energy conservation and(4) yield

W\c0

c0+(L/3)a 3Ðka. (5)

The Hubble factor, H, can be eliminated from these equa-tions by working with a new time variable p=ln a. Wedefine a new ‘velocity’ a=dx/dp\H Ð1 v. In these new vari-ables, the Poisson equation remains unaltered while thecontinuity and Euler equations become

dddp

+(1+d)H ·a\0, (6)

da

dp+(1Ðq)a\Ð

3

2 WHf, (7)

where q (p)\W/2Ðl is the time-dependent decelerationparameter and we have used dH/dp\Ð(1+q)H to derive(7).

Attempting to eliminate W from (7), we make an addi-tional transformation from the time variable p to t definedby

t\ln D (p), (8)

where D is the linear growing density mode determined bythe equation

d2D

dt 2+2H

dD

dtÐ

3

2 WH 2D\0. (9)

Analytic solutions to (9) can be found in Heath (1977).Expressing (7) in terms of t defining the ‘velocity’ b=dx/dt,the continuity equation is

dddt

Ð(1+d)y\0, (10)

and the Euler equation is

f 2db

dt+&df

dp+(1Ðq) f'b\Ð

3

2 WHF, (11)

where y\ÐH ·b, and f\dt/dp is the linear growth factorwhich relates the density contrast, d, to the divergence of thepeculiar velocity field, v, in the linear regime. For l\0, agood approximation1 for f is f2W0.6 (Peebles 1980). For

458 A. Nusser and J. M. Colberg

© 1998 RAS, MNRAS 294, 457–464

1For l\0, the function f satisfies (1ÐW)df/d ln WÐ(1ÐW/2) f+3/2WÐf 2\0, which to first order in 1ÐW yields f2W4/7 for W21 (seealso Lightman & Schechter 1990). However, the general solutionto this equation is better fitted by f2W0.6 for Ws0.7. A fit whichworks well for 0.05sWs1 is f\W4/7+(1ÐW)3/20.

l80, Lahav et al. (1991) found that f2W0.6+l (1+W/2)/70.Therefore, for reasonable values of the cosmological con-stant we neglect the dependence of f on l in the approxi-mate forms for f (Lahav et al. 1991). Note that the velocityb\v/(H ). In the Zel’dovich approximation, this is the dis-placement vector of a particle from its initial to presentposition. Using (9), we find

d f

dp+(1Ðq) f\

3

2 WÐf 2, (12)

so the Euler equation (11) is

db

dtÐbÐ

3

2 [1+e ·W)] (gÐb)\0, (13)

where

e (W)=W

f 2Ð12WÐ0.2Ð1, (14)

and we have defined g\ÐHf. Similar equations werederived by Gramann (1993) and Mancinelli & Yahil (1995).The weak dependence on W in (13) through e (W) couples tothe difference between the velocity, b, and the gravity field,g. Since W11 at early times, initially the function e almostvanishes, thus any changes in the dynamics as a result of thisweak dependence on W occur at later times.

In virialized regions, the acceleration of a particle isdominated by the gravity field g. It is easy to see that, byneglecting the terms involving the velocity in (13) and work-ing with a new time variable with respect to which thevelocity is (1+e)Ð1/2b2W0.1b, we obtain an Euler equationwhich is independent of W. This velocity is approximatelyequal to the comoving peculiar velocity divided by HW0.5.This scaling with W is not surprising, since the virial theoremimplies that the rms value of the physical velocities in viri-alized regions with a given density contrast has similarscaling with W. Note, however, that in our derivation wehave not taken into account the dependence of the shapesof virialized regions on W. Therefore we expect this scalingto be of limited validity (see Fig. 9 below).

3 X DEPENDENCE IN TOY MODELS

It is clear from the form of (13) that the source term whichdrives the evolution is larger for lower W. Therefore weexpect to see more evolved clustering in a low-W universethan in an W\1 universe with the same initial conditionsand linear normalization. It is instructive to investigate theeffect of the term e in cases of special symmetry. Considerfirst the spherical expansion or collapse before the occur-rence of shell crossing. In cases of special symmetry, we findit easier to solve directly for d and y rather than for g and b.Therefore we take the divergence of (13) and use the Pois-son equation to obtain

dydt

ÐyÐP2Ð3

2 [1+e (W)] (dÐy)\0, (15)

where P2\∑ i, j (qxibj)

2\y 2/N with N\1, 2 and 3 at thecentres of configurations with planar, cylindrical and spheri-

cal symmetry, respectively. The term P2 appears as a resultof taking the divergence of the non-linear term in the con-vective derivative db/dt\qb/qt+b ·Hb. Therefore, in thespherical top-hat model, the equations (15) and (10)together with the equations relating a, t, W and f are suffi-cient to determine the evolution of the quantities y and d.For L\0, the spherical collapse model can be solved analyt-ically (e.g. Peebles 1980) if the initial peculiar velocity isneglected. Here we numerically integrate equations (14)and (15) under the initial conditions di\yi with !di!ss1,where the subscript i refers to quantities at the initial time.These initial conditions are realistic as they arise naturallyin linear theory (cf. Peebles 1980).

In Figs 1 and 2 we show the density contrast versus thetime t for positive (sia0) and negative (dis0) top-hatperturbations for three background cosmologies: (W0,l0)\(1, 0); (0.2, 0.8); and (0.2, 0). Fig. 1 shows curves fortwo values of the initial density contrast corresponding tobound and unbound perturbations for the W0\0.2 cosmolo-gies. Although the equations were integrated from t\Ð5.4to 0, for the sake of clarity Fig. 1 shows results only fortaÐ1. For bound perturbations, the growth of d and y isfastest in the open universe case, (W0\0.2, l0\0). How-ever, significant deviations appear only when d is larger than10 or so. This is consistent with the work of Peacock &Dodds (1996) who found that non-linear effects in the evo-lution of power spectra in N-body simulations are strongerin an open universe than in a flat W+l universe of the sameW. Although the W+l\1 case shows more rapid evolutionthan the W\1 case, the corresponding curves are very simi-lar even when the densities are larger than their values atturnaround. The reason for this is clear from Fig. 3 which

The W dependence in equations of motion 459

© 1998 RAS, MNRAS 294, 457–464

Figure 1. The quantities d and y versus the time t for a positiveperturbation for various values of W0 and l0 as indicated in thefigure. The upper (steeply rising) and the lower curves, respec-tively, correspond to bound and unbound perturbations in theW0\0.2 cases. The values of d at the turnaround radii of the boundperturbations are 4.5, 6.7 and 11.5 for (W0, l0)\(1, 0), (0.2, 0.8)and (0.2, 0) respectively.

shows W versus t. For l080 we see that W is almost unityuntil relatively late times. Therefore, until late times, theevolution of d and y is very similar to the W0\1 case. Forpositive unbound and negative perturbations, the effect ofthe cosmology on the evolution of d is almost negligible.The y curves show some differences. In voids, y grows moreslowly in the low W0 models once the density contrastapproaches Ð1. Since perturbations in flat universes with acosmological constant evolve similarly to those with l\0,we do not discuss the case L80 further.

The top-hat model can be used to evaluate the way inwhich the variance of an evolved generic density fluctuationfield depends on W. We require here that the density field issmoothed on large enough scales such that shell crossing isremoved. By inspecting curves of the numerical solution fordW=d (Ws1) and d1=d (W\1) we find that, for any given t,the following (empirical) relation is satisfied:

dW\d1 exp & d1

D (W)', (16)

where

D\85

e (1+e). (17)

This relation works remarkably well for 0.1sWs1 andd1s400. For generic configurations we assume that therelation (17) is still valid. However, we should take intoaccount the fact that the ‘dimensionality’ of the collapseaffects the amplitude of the W dependence; for example, inthe one-dimensional collapse, before shell crossing, theequations are free of W. Therefore we replace D, in (16),with aND where aN\4/(NÐ1)2 and N is the ‘dimension-ality’ of the collapse at each point in space. Other than forpurely symmetric configurations, the quantity N is some-what ambiguous. One possibility is to define it at any pointin space in terms of the eigenvalues of the initial velocitydeformation tensor, say, as the ratio of the square of thesum of the eigenvalues to the sum of their squares. Never-theless, for our purposes it is not crucial to specify the formof N and we simply treat it as a factor which depends on thelocal topology of the density field. Note that because (16) isnot exact, the spatial average of dW as estimated from (16)given d1 does not vanish in general. This can be remedied byadding a constant to the right-hand side of (16). However,for simplicity we assume that the relation (16) ensures amean value of zero for dW . It is useful to think of the changein the density field with W as being caused by a one-to-onemapping from the Eulerian space of a flat to that of an W81universe. Assuming that such a mapping indeed exists, thevariance of the field dW in a region of size V,

d 2W\

1

V h (1+dW)2 dVÐ1, (18)

can be written in terms of the field d1 as

s 2W\

1

V h [1+dW(d1)]2dV

dV1

dV1. (19)

The factor dV/dV1, in the last equation, accounts for thechange in the volume element occupied by a patch of matterin an W81 relative to a flat universe and we have assumedthat the change of the total volume V as a result of thismapping is negligible. Given the assumed one-to-one map-ping, it is easy to see that dV/dV1\(1+d1)/(1+dW). If P isthe probability distribution function of the field d1, then, byvirtue of the ergodic theorem, we find that

s 2(W)\h P (d1)(1+d1)(1+dW) dd1Ð1. (20)


© 1998 RAS, MNRAS 294, 457–464

Figure 2. The same as Fig. 1 but for negative perturbations. The dcurves are almost indistinguishable.

Figure 3. The density parameter versus the time t for an openuniverse (dashed line) and a flat universe with a cosmologicalconstant (solid line).

By expanding (16) to third order in d1 and substituting theresult in (20) we find

s 2(W)

s 21

\1+1

aND+S 212+aND3 s 2

1

a 2N D2

, (21)

where s 21\,d 2

1. and S\,d 31./,d 2

1.2 are, respectively, thevariance and the reduced skewness of the field d1, and0sNs3 is some number describing the dimensionality ofa typical collapse configuration. Note that, in deriving thelast relation, we have neglected any local correlationbetween N and d1. The factor should depend on the powerspectrum of the initial fluctuations and it can be determinedempirically from N-body simulations.

Consider now the effect of changing W0 on the dynamicsof shell-crossing regions in the case of one-dimensional col-lapse. This case is particularly instructive since solutions(b\g) to the one-dimensional equations of motion are fullyindependent of W until the occurrence of shell crossing.Unfortunately, even in the simple one-dimensional col-lapse, we have no analytic solutions in shell-crossingregions. Therefore we first use the Zel’dovich solution untilthe formation of the first singularity, then we switch on to aone-dimensional N-body code to move particles accordingto (13) in the shell-crossing phase. The initial density fieldwe choose is di;cos(x). Results of simulations with W0\1and W0\0.2 without a cosmological constant are shown inFig. 4. The density profile (upper panel) is more concen-trated in the open than in the flat case. This is similar to thebehaviour of density perturbations in the top-hat modeldiscussed above. The distribution and velocities of particlesin the open model seem to be more evolved in time thanthose in the flat model.

4 X DEPENDENCE IN N -B ODY SIMUL ATIONS

We use N-body simulations to study the W dependenceunder general initial conditions. These simulations areespecially useful in orbit mixing regions where, according to(13), the effect is most important.

We ran two simulations having W0\1 and W0\0.29respectively. Both simulations were started from the sameinitial conditions. The initial conditions were generatedfrom the power spectrum for a standard cold dark matter(CDM) universe with H0\50 km sÐ1 MpcÐ1. Each simula-tion contained 1283 particles in a cubic box of length 60 hÐ1

Mpc. The simulations were evolved until the linear rmsdensity fluctuations in a sphere of 800 km sÐ1 was 0.5. Bothmodels have roughly the right small-scale power as mea-sured by the galaxy pairwise velocity, but produce fewer richclusters than observed. However, given the scale of oursimulations, our choice of the power spectrum and the nor-malization is appropriate for our purposes. A model with ahigher normalization would result in too much merging ofsmaller objects in a few larger objects. Choosing a steeperpower spectrum leads to a similar effect.

The simulations were run using a modified parallel ver-sion (MacFarland et al., in preparation) of Couchman’sP3M code (Couchman, Thomas & Pearce 1995) which usesexplicit message passing. The simulations had a softeningparameter of 13.2 per cent of the mean particle separationand a mesh of 512 in one dimension. They were run using 64processors on the CRAY T3E supercomputer at the Com-puter Center of the Max Planck Society (RZG), Garching.

Fig. 5 shows the particle distribution in a slice of thickness1 hÐ1 Mpc in the two simulations. The left and right panelscorrespond to the flat and open models respectively. Thelower panels zoom in on the ‘clusters’ seen near the centresof the upper panels. On large scales (upper panels), the twosimulations are remarkably similar. Some differences can bespotted in the lower panels. Clusters in the low-W simulationappear to be more concentrated and evolved. The differ-ences between the two simulations seem to be negligible onscales larger than 1 hÐ1 Mpc. Indeed, the rms value of thedifference between the positions of the same particles in thetwo simulations is 0.25 hÐ1 Mpc and the largest difference isless than 1.5 hÐ1 Mpc. The correlation functions for the twosimulations, plotted in Fig. 6, confirm the visual impressionfrom Fig. 5. The correlation functions differ only on scalessmaller than 1.6 hÐ1 Mpc. On ‘cluster’ scales x0.6 hÐ1 Mpc,the low-W correlation function is larger, and on scales0.6–1.6 hÐ1 Mpc (roughly corresponding to the scale ofinfall regions around clusters in the simulations) it is smallerthan the correlation function in the flat model.


© 1998 RAS, MNRAS 294, 457–464

Figure 4. The density profile (upper panel) and the ‘velocities’(lower panel) in the shell-crossing region in one dimension forW0\1 (solid) and W0\0.2 (dotted) with no cosmological constant.The initial density perturbation is a cosine wave symmetric aboutx\0.

We now quantify the differences between the density andvelocity fields. We first use the cloud-in-cell (CIC) inter-polation scheme to evaluate the density and velocity fieldson a cubic grid of cell length equal to the mean particleseparation. This produces a mass weighted average velocityon the grid points. We then further smooth the resultantdensity and velocity maps with a top-hat filter. In Fig. 7 weplot the densities in the open versus those in the flat fieldmodel for 4096 randomly chosen grid points. Even with onlyCIC smoothing on the grid scale, the correlation betweenthe two density fields is very tight. For densities larger than


© 1998 RAS, MNRAS 294, 457–464

10 or so, the densities in the open model are larger. Thescatter almost vanishes when the density fields aresmoothed with a top-hat window of width a200 km sÐ1.Note, however, that for moderate densities (0sds5), thedensities in the flat model are slightly larger than in theopen model. This is not surprising since the general ten-dency is that matter flows out of regions with moderatedensities into higher density regions. Since the open modelis slightly more evolved, these moderate density regions aresomewhat less dense in the open than in the flat model. InTable 1 we list the values of the rms, sd , and the reducedskewness, S=,d 3./s 4

d of the density fields as a function ofthe smoothing scale. It appears, from the table, that S varieswith W more strongly than (12/14)W2/63 as predicted fromsecond-order perturbation theory (Bouchet et al. 1992).That theory is, however, valid only for ss1. We can useTable 1 to determine Neff in (21), which relates sd in anopen universe to that in an W\1 universe. A comparison of(21) with Table 1 suggests that Neff22. This value isreasonable since non-linear collapse configurations arelikely to have pancake-like shapes. Recall that the simula-tions were stopped when the linear value of sd , smoothedwith a top-hat filter of width Rs\800 km sÐ1, was 0.5. Theactual value computed from the simulation is very close to

Figure 5. The particle distribution in the low W (right) and W\1 (left) simulations. Slice thickness is 1 hÐ1 Mpc. The lower panels focus onthe group of ‘clusters’ appearing near the centres of the upper panels.

Figure 6. The two-point correlation functions for the particle dis-tributions in the two simulations.

0.44 in the two simulations. Thus, even though non-lineareffects are clearly important, the difference between the sd

in the open and flat simulations is negligible. We now con-sider the evolved velocity fields. Fig. 8 compares one of thecomponents of the velocity fields in the two simulations.The velocity fields in the open model are scaled by the factorf (W0). Even for large velocities and small smoothing widths,the velocity fields in the two simulations seem to be relatedby the factor f. A close inspection of the scatter plot forRs\400 km sÐ1 reveals that the slope of the regression of v/fon v1 is slightly less than unity. This is because large veloci-ties are generally associated with strong non-linear effects,which tend to spoil the scaling by f.

We mentioned at the end of Section 2 that the motion ofparticles in bound objects is independent of W in terms of atime variable which corresponds to a velocity which is the

peculiar comoving velocity divided by W0.5. Therefore weexpect the velocity dispersion (rms velocity) in groups ofparticles identified in the simulation to scale, approxima-tely, like W0.5. To test this conjecture, we have used a friends-of-friends (FOF) algorithm (kindly supplied by A. Diagerio)to identify groups in the simulations. We then computed theone-dimensional velocity dispersion of particles in eachgroup. Fig. 9 (upper panel) shows the mean velocity disper-sion in groups as a function of the number of particles theycontain. Velocities in the plot are scaled by W0.5

0 for the openmodel. It seems that this scaling works well. Given theuncertainties in identifying group members by the FOFalgorithm, the deviations from this scaling for large groupsare not significant. We conclude that while the (smoothed)average velocity of particles scales like f (W), the rms velo-city, roughly, scales like W0.5. It is interesting to compare theabundance of groups in the two simulations. The lowerpanel of Fig. 9 is a plot of the abundance as a function of thenumber of particles for the two simulations. The abundanceof groups is slightly higher in the open model. This is easy tounderstand; because groups in the open model are tighterthan groups in the flat model, the FOF algorithm naturallyassigns more particles to them. Note that observations natu-rally provide the abundance of groups as a function of themass. Since the mass of groups with the same number ofparticles is proportional to W0, abundance curves, whenplotted versus the mass, look significantly different in thetwo simulations.

5 SUMM A RY

We have shown that gravitational dynamics of a pressure-less fluid in an expanding universe is almost independent ofthe cosmological parameters. According to the equations ofmotion, expressed in terms of the linear growing mode, thefinal structure in a low-W model, with or without a cosmo-


© 1998 RAS, MNRAS 294, 457–464

Figure 7. Densities in the open model versus densities in the flatmodel. The lower-left panel shows densities after the CIC inter-polation on a cubic grid of mean particle separation cell sizes. Theother panels show CIC densities smoothed with a top-hat windowof width Rs as indicated in the plot.

Figure 8. The same as Fig. 7 but for peculiar velocities. The veloci-ties in the open model were scaled by the factor f (W0)\0.477.

Table 1. Moments of the density field in thetwo simulations after CIC and top-hatsmoothing of width, Rs , expressed in kmsÐ1.

logical constant, is more evolved than in a flat universe. Weused toy models and N-body simulations to investigate theeffect of changing the cosmological background on the evo-lution of fluctuations and, in particular, on the final velocityand density fields. The present density, when smoothed onscales slightly larger than cluster scale, is almost insensitiveto the cosmological background. The background can affectthe structure of bound objects (or haloes). Haloes, charac-terized by the same ratio of mass to background density, aremore centrally concentrated in an open than in a flatuniverse. However, this effect is weak and is likely todepend on the initial power spectrum. On the other hand,peculiar velocities have a strong dependency on W. We findit remarkable that the smoothed non-linear velocity fieldscales with the growth factor, f, just as it does in lineartheory.

Since f depends very weakly on the cosmological con-stant, the final velocity field is mainly sensitive to W. There-fore the observed peculiar velocity and density fields in thenearby universe contain information mainly on W (Lahav etal. 1991). Constraints on both W and L can, in principle, beobtained by measuring the clustering amplitude at differentredshifts, for example via the correlation function (Lahav etal. 1991). As we have shown, it is rather difficult to detectsignatures of the cosmological background in the structureof density fields. Fortunately, observations provide the dis-tribution of galaxies in redshift space. Thanks to the strongdependence on the velocity field on W, the anisotropy ofclustering in redshift space can provide a measure of W.However, such estimates of W involve an assumptionregarding the relationship between the distribution ofgalaxies to that of the dark matter. Estimates of W indepen-dently of the galaxy distribution can be obtained from theobserved peculiar velocity field alone (cf. Dekel 1994). Thismakes peculiar velocity catalogues a very powerful tool withwhich to constrain the cosmological model. It is especiallyimportant that future peculiar velocity measurements aimat greater sky coverage and denser sampling rate.

ACKNOWLEDGMENTS

We especially thank Ravi Sheth for many useful comments,and Tom MacFarland for his valuable contributions to run-ning and improving the parallel P3M N-body code. We alsowish to thank the referee, Bob Mann, for a thorough read-ing of the manuscript, Simon White for useful discussions,and Antonaldo Diagerio for allowing the use of his groupfinding code.


© 1998 RAS, MNRAS 294, 457–464

Figure 9. Top: the mean one-dimensional velocity dispersion (rmsvelocity deviations) divided by W0.5

0 in groups versus the number ofparticles they contain for the two simulations. The lengths of theerror bars give the 1-s scatter about the mean for the flat model.The scatter in the open model is similar. Bottom: the abundance ofgroups versus the number of particles they contain. Groups areidentified using a FOF algorithm.

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