On the streaming motions of haloes and galaxies

A simple model of how objects of different masses stream towards each other as they cluster gravitationally is described. The model shows how the mean streaming velocity of dark matter particles is related to the motions of the parent dark matter haloes. It also provides a reasonably accurate description of how the pairwise velocity dispersion of dark matter particles differs from that of the parent haloes. The analysis is then extended to describe the streaming motions of galaxies. This shows explicitly that the streaming motions measured in a given galaxy sample depend on how the sample was selected, and shows how to account for this dependence on sample selection. In addition,we show that the pairwise dispersion should also depend on sample type. Our model predicts that, on small scales, redshift space distortions should affect red galaxies more strongly than blue.


INTRODUCTION
Gravity makes objects cluster. Therefore, the motions of objects towards each other may provide information about the background cosmology. Of course, different subsets of the clustering particles may trace the underlying streaming motions differently. The scale dependence of the mean streaming v dm 12 (r) of dark matter particles has been understood for some time now (Hamilton et al. 1991;Nityananda & Padmanabhan 1994). But there has been little study of how this statistic depends on trace-particle type.
To do this, we build a model in which gravitational clustering is viewed as the combination of two processes. The first arises from the fact that gravity causes matter to stream towards local minima of the gravitational potential. This requires a model of how matter which was initially distributed rather smoothly around the centre of collapse becomes redistributed into a more centrally concentrated density profile as the collapse proceeds. The second process is that these centres around which local collapses are occurring, these clusters, are themselves moving towards each other: clusters cluster. It is the combination of these two types of motions which gives rise to the spatial distribution and streaming motions of objects today.
Section 2.1 summarizes useful results which follow from linear theory. Section 2.2 shows how the streaming motions of collapsed dark matter haloes depend on halo mass. Section 2.3 uses this to model the streaming motions of parti-cles, rather than haloes. It shows what fraction of a particle's streaming motion arises from the motion of its parent halo, and what fraction must arise from motions within the halo. These smaller scale motions are essentially a consequence of the collapse around the halo centre we referred to earlier. It then presents measurements from numerical simulations which show that the model predictions are reasonably accurate. It also shows that the model provides a reasonable description of how the second moment of the pairwise velocity distribution of the dark matter differs from that of haloes.
Section 3 shows how to extend the model to study the mean streaming motions and the pairwise velocity dispersion of galaxies and presents measurements from semianalytic galaxy formation simulations which show that the model predictions are reasonably accurate. Section 4 discusses what this model implies if one wishes to use measurements of the streaming motions of galaxies to make inferences about cosmology.

The mean streaming velocity
We will begin by reviewing the strategy which led to the derivation of how v dm 12 (r) depends on scale. The relevant starting equation is the pair conservation equation in Peebles' book (Peebles 1980), but we will start with the equa-tion in the form presented by Nityananda & Padmanabhan (1994): whereξ(r, a) is the volume averaged correlation function on proper (rather than comoving) scale r at the time when the expansion factor is a, and the Hubble constant is H. This says that if we know the correlation function for all scales r and all times a, then the assumption that the number of pairs is conserved allows us to compute how v12(r) depends on scale today. An approximate solution to this expression can be got as follows (Peebles 1980). Assume thatξ evolves according to linear theory:ξ(r, a) = [D(a)/D0] 2ξ (r, a0), where D(a) is the linear theory growth factor at a, and D0 is the growth factor at the present time when a = a0. In an Einstein de-Sitter cosmology, D(a)/D0 = a/a0. Then the left hand side is ∂ξ(r, a)/∂ ln a = 2 f (Ω)ξ(r, a), where f (Ω) ≡ ∂ ln D/∂ ln a. So, in this approximation we get . (2) On large scales, ξ ≪ 1, and so this is just the usual linear theory expression with an extra factor of (1 + ξ) in the denominator. While this approximation is fine on large scales (r ≥ 10 Mpc/h), it underestimates the exact solution by a factor of 3/2 or so on smaller scales (Juszkiewicz, Springel & Durrer 1998;. Hamilton et al. (1991) showed they could compute a good estimate of the evolution of ξ(r, a), if the initial correlation function is known (also see Nityananda & Padmanabhan 1994). Hamilton et al. also showed that by inserting their expression for the evolution of ξ(r, a) into equation (1) above, they were able to describe the shape of v dm 12 (r) well on all scales.
While this approach is very useful for studying the statistics of dark matter particles, it is not obvious that it can be used to estimate the streaming motions of galaxies. This is because one usually assumes that galaxies form at different times. This means that the number of galaxies is not conserved, so the number of galaxy pairs is not conserved. This means, for example, that the correlation function of galaxies refers to different sets of particles at different times. Therefore, there is little reason to expect that inserting the correlation function of galaxies into the pair conservation equation should provide a good estimate of v gal 12 (r) today. We show below that, provided one makes the correct choice of what one uses for ξ gal (r, a), the pair conservation equation can be used to provide an accurate estimate of the streaming motions of galaxies.

The haloes
This subsection is concerned with the first moment of the pairwise velocity distribution of haloes identified at the present time. Every halo will be represented by one particle, say, the one at the halo centre of mass today. Imagine tracing these centre-of-mass particles back in time. By definition the number of these particles is conserved, since all we're doing is following them back to high redshift. Of course, at high redshift, few if any of the haloes would actually have collapsed around these centre-of-mass particles. Nevertheless, we will use the motions of these particles to represent the motions of the halo centre of mass. Peebles' pair conservation equation, combined with the assumption that the motion of a halo today is the same as that of its associated centre-of-mass particle, says that if we knew ξ(r, a) for these tracer particles, then we can compute v halo 12 (r) today. So, to compute v halo 12 , we are stuck with the problem of studying the spatial distribution (i.e., the bias factor) of a special marked set of particles at earlier times. The case in which the marked particles (in this case, the halo centresof-mass) are observed at a later epoch than when they were marked is familiar: e.g. this is like the Mo & White (1996) simple model for galaxies, in which galaxies formed in haloes at z = 3 but we only observe them today. Here, we are interested in the spatial distribution of the special particles at earlier epochs than when they were marked.
The halo centre-of-mass particles are biased tracers of the dark matter distribution. The large scale bias factor is the square root of the ratio of the correlation function of these particles to that of the dark matter correlation function on large scales. It depends on halo mass: and a similar equality holds forξ hh (r). Here ξ Lin dm (r) denotes the initial correlation function of the dark matter extrapolated using linear theory to the present time. In equation (3) we use the linearly extrapolated ξ Lin dm (r) rather than the present day ξ(r) as a practical way of taking into account the volume exclusion effects of haloes at small scales. See Section 2 in  for a detailed discussion.
To a good approximation, where ν(m) ≡ δc0/σ(m) is a function which increases with decreasing halo mass, and D(a) and D0 were defined earlier. At the present time, D(a) = D0 and this is the familiar Eulerian bias formula from Mo & White (1996). The Lagrangian bias factor is usually expressed as the ratio of ξ hh at the initial time to the linearly extrapolated ξ Lin dm . This means that the Lagrangian bias factor is where ai denotes the expansion factor at the initial time.
Since ai ≪ a0, bLag → (ν 2 −1)/δc0, which is another familiar expression from Mo & White (1996). So, in this approximation, It is straightforward to insert these expressions for the halo correlation function and its evolution into the pair conservation formula (equation 1) to see how different v halo 12 is from v dm 12 . If we study the streaming motions of haloes of two different masses, then we must replace b 2 (m) → b(m1) b(m2). This gives v halo 12 (r) . (6) If we insert the linear evolution approximation for the relation between the correlation function and v12 (equation 2), then this becomes v halo 12 (r) Notice that when b1 = b2 = 1, then v halo 12 (r) = v dm 12 (r). Also, in the large separation (smallξ) limit, v halo 12 (r) → (b1 +b2)/2 times v dm 12 (r). So, on average and on large separations, relative to the dark matter, massive haloes (b1 + b2) > 2 stream towards each other whereas less massive haloes (b1 + b2) < 2 stream away from each other. This makes some physical sense; clusters cluster, so they are moving towards each other, whereas smaller clumps are in or at the edges of expanding voids, so they are separating from each other. This linear bias of the streaming velocities at large separation is consistent with the linear theory analysis of Fisher et al. (1994).
Notice that v12 scales with the sum of the bias factors. If one ignored the evolution of the bias factor when using equation (2), one would have concluded that the scaling was with the product of the bias factors-including the evolution of the bias factor is essential to getting the correct answer. Finally, notice that on smaller scales where ξ Lin dm > 1, this analysis suggests that v12 of less massive haloes should be larger than that of the dark matter, with the opposite trend being true for massive haloes. Of course, the linear theory and linear evolution approximations we used to obtain equation (7) are not accurate on small scales. Nevertheless, this provides at least some indication of the small scale behaviour of the halo streaming motions.
wheren halo ≡ dm n(m) is the average number density of haloes, b halo ≡ dm n(m)b(m) is their average bias factor, the weighting factor in the second line is the ratio of the number of m1 and m2 halo pairs at r to the total number of halo pairs at r, and the final expression follows from inserting equation (7) for v halo 12 and using equation (2) for v dm 12 . Our model, which we only expect to be accurate on large scales because our approximation for the halo correlation function, equation (3), breaks down on small scales, is reasonably accurate down to scales of order a Mpc/h or so. For comparison, the dotted curves show the Hubble velocity. The crosses show the streaming motions of the dark matter particles in the simulations, and the dashed curve shows the prediction associated with the model described in the next section.

The dark matter
The large scale net streaming motion of the dark matter can be got from our expression for the halo motions by integrating up the contribution to the streaming motion from pairs in different mass haloes, weighting by the fraction of the total number of pairs which are in such haloes, and weighting by the halo mass function: The final equality follows from inserting equation (6), noting that dm mn(m) ≡ρ, and using the fact that the bias factors are defined so that dm mn(m) b(m) ≡ρ.
We can now make two important points. The first is that, at large separations, this expression equals v dm 12 (r) = v 2halo 12 (r); in this regime the streaming motions of the dark matter particles are entirely due to the fact that the haloes which contain the particles are moving. Moreover, in this regime, v dm 12 (r) ≈ v L 12 (r), where v L 12 is got from equation (2) by using the linear theory values of ξ andξ. The second is that this expression exactly equals that in  for the contribution to v12(r) from particles which are in separate haloes (see their eq. 19). This will be important in what follows.
Notice that on smaller scales, ξ Lin dm (r) < ξ dm (r). In this regime the halo motions only account for a fraction of v dm 12 (r). The remaining contribution to v dm 12 (r) must arise from the streaming motions of pairs in which both particles are in the same halo. This means that the fact that our model for halo motions is not accurate on small scales will not matter very much for the small scale value of v dm 12 because, on small scales, the fraction of pairs which are in separate haloes, and so are affected by this inaccuracy, is small. We turn, therefore, to a discussion of the streaming motions of pairs in which both particles are in the same halo.
If haloes are stable, then the streaming motion within a halo exactly cancels the Hubble flow: −v12(r)/Hr = 1. In this case, the contribution from pairs which are in the stable haloes equals unity times where m 2 λ(r|m) denotes the number of pairs at separation r which are in the same halo which has mass m; it depends on the density profiles of haloes. For all halo shapes of interest, this expression approaches unity at very small r, because ξ dm (r) = ξ 1halo (r) + ξ Lin dm (r) ≈ ξ 1halo (r), and ξ dm (r) ≫ 1 on scales which are smaller than a typical halo. So, if stable clustering is correct, then −v dm 12 (r)/Hr = 1 on small scales. In fact, the mean pairwise velocity on small scales depends on the low-mass behaviour of n(m) and λ(r|m)-in general, there is no guarantee that n(m) and λ(r|m) will conspire to give stable clustering (Ma & Fry 2000;. In particular, Section 4 of  shows that the small scale term is  The sum of equations (9) and (10) gives a complete description of the streaming motions of dark matter on all scales. The dashed curves in Fig. 1 show that this sum provides a good description of the dark matter streaming motions on all but the smallest scales (see  for a discussion of the discrepancy at the smallest scales). Now, the dashed curves correspond to independently identifying haloes at each epoch. On the other hand, we could have followed the approach of Section 2.2-identify or mark the haloes today, and then follow them backward in time. With this latter approach, the number density of haloes is fixed to the value today, n(m, a0), but the profile changes from, say a tophat to a more centrally concentrated shape. Equation (9) shows that the contribution to the streaming motions from particles in separate haloes is independent of the details of how this happens. However, recall that this contribution exactly equals the two-halo contribution to v dm 12 worked out by . This means that the one-halo contributions to v12 must also be the same in both approaches. In particular, this means that however the profile changes from a tophat to an NFW shape, it must change in just such a way that the final answer for the streaming motions of the dark matter particles equals equation (10). Indeed, we can use this requirement to constrain how the profile changes from the initial tophat to the final NFW cusp-the Appendix shows a worked example of how to do this.

The pairwise velocity dispersion
So far, we have shown how the mean streaming motions of the dark matter and the haloes are related.  discuss how to do this for the second moment of the pairwise velocity distribution. They argued that the dark matter particles receive substantial nonlinear kicks to their initial velocities (essentially, the virial motions within haloes), whereas the haloes do not . As a result the pairwise dispersion of the dark matter should be significantly larger than that of the haloes on all scales. Fig. 2 compares what their equation (31) predicts with the simulations (we refer the reader to their paper for details of the model). The open circles and crosses show the pairwise dispersion, σ12(r), of the haloes and the dark matter respectively, and the dashed and solid curves show the model predictions. (We do not show σ12 for the dark matter in the SCDM simulations because the simulation box is sufficiently small [85Mpc/h] that cosmic variance affects the measurement significantly.) The model is reasonably accurate on large scales, and not accurate on small scales.  discuss why this happens for the dark matter (the inaccuracy is due to the simplifing assumptions that haloes have no substructure, the pairwise dispersion from a single halo is isotropic and independent of the pair position, and the number of pairs in the infall regime around haloes is sufficiently small that the use of virial motions to model the dispersion from infalling pairs does not not lead to a large error). For the haloes, this discrepancy appears on scales which are of the order of a typical m * halo and smaller. This suggests that the discrepancy almost surely arises from using linear theory to model the spatial distribution and velocities of haloes on scales which are smaller than the smoothing scale used to make the model prediction. Despite the quantitative discrepancies on small scales, the model is in qualitative agreement with the simulation: the pairwise dispersion of the haloes is substantially smaller than that of the dark matter.

GALAXIES
The previous section showed that the first moment of the pairwise velocity distribution of haloes is different from that of the dark matter. It showed that massive haloes separated by large distances are streaming together more rapidly than less massive haloes at the same separation, and that this difference scaled with one rather than two powers of the halo bias factor. It also showed that the dark matter statistic was obtained by weighting the halo statistic by the number of dark matter particle pairs per halo. The second moments of the pairwise velocity distributions are also different. In this case, also, the dark matter statistic is got by weighting by the number of particle pairs per halo. However, the pairwise dispersion is also sensitive to the fact that virial motions within haloes can be substantially higher than the motions of the haloes themselves. As a result, the pairwise velocity dispersion of dark matter particles is substantially larger than that of haloes, on all scales. This section studies what these results imply for the pairwise motions of galaxies.
We will model galaxies as random particles in dark matter haloes. That is, the motion of the galaxies is the same as that of the dark matter particle with which they are associated. In this sense there is no velocity bias in our model; the fact that velocity statistics for the dark matter and the galaxies may, nevertheless, be different, arises solely from the fact that dark matter statistics weight each halo proportional to halo mass, whereas galaxy statistics do not. Such models for the difference between the statistics of galaxies and dark matter particles have received considerable atten-tion recently. Seljak (2000), Peacock & Smith (2000) and Scoccimarro et al. (2000) have used them to model the spatial distribution of galaxies,  describe how to model the distribution function of galaxy peculiar velocites, and  describe how to use these models to do analytically what Jing, Mo & Börner (1998) did numerically in their study of the pairwise velocity dispersion of galaxies.
Within the context of this model, galaxies are treated by setting v gal 12 (r) = v 1gal 12 (r) + v 2gal 12 (r), where the two terms denote the contribution to the statistic from galaxies in the same and in different haloes, respectively. The second term on the right hand side can be got by modifying equation (9): v 2gal 12 (r) wherē ρ gal = dm g(m)n(m) and Here g(m) and g2(m) denote the first and second moments of the distribution of the number of galaxies in m-haloes, and we set g2(m) = 0 if g(m) < 1. There are details associated with how one treats the central galaxy in a halo, but, for the most part, these amount to a small effect (see , so we have ignored them-they add complications but no essential change to the logic of our argument. On scales larger than a few Mpc/h, v 2gal 12 dominates over the one-halo contribution. If we assume linear theory for the evolution of the two-halo term (equation 2), then we can set 2f (Ω)ξ Lin dm /3 → −v dm 12 /Hr times 1+ξ Lin dm (r), and then equation (12) .
This shows that, on large scales, the streaming motions of galaxies can be biased relative to the dark matter. The extent to which they are biased is related to how differently they are clustered, and this, in turn, depends on the g(m) relation.
On smaller scales, the streaming motions are dominated by galaxy pairs in which both members are in the same halo. A little thought shows that this can be computed simply by settingρ →ρ gal and m 2 → g2(m) in equation (10). This is because the g(m) relation does not introduce any additional time dependence-recall that the number density of haloes in the present model is fixed to the value it has today, and Figure 3. Mean number of bright galaxies as a function of parent halo mass in the ΛCDM GIF semianalytic galaxy formation model of Kauffmann et al. (1999). Top panel shows the result of dividing the sample into two based on colour. Bottom panel shows a division based on star formation rate. Crosses, circles, squares and triangles are for objects classified as being red, blue, quiescent and star-forming galaxies respectively.
the galaxies are to be thought of simply as marked tracer particles within the haloes.
This has an interesting consequence. Suppose one wishes to use the pair conservation equation (1) to estimate v gal 12 (r). Then the model above suggests that, on small scales, simply inserting the observed galaxy correlation function into equation (1) should be reasonably accurate. However, on larger scales, doing this leads to an estimate of v gal 12 which is incorrect, for the following reason. Because there is no time dependence in g(m), the result of doing this has the same time dependence as in the case of the dark matter for which g(m) = m, or the case of the haloes for which g(m) = 1. But we know that, for the haloes, doing this results in the wrong answer, because it neglects the evolution of the bias factor. The case of galaxies is no different; if we insert the observed galaxy correlation function into the pair conservation equation, then we are incorrectly neglecting the evolution of the bias factor of the galaxies. This would lead one to conclude, incorrectly, that v12 should scale as b 2 gal rather than as b gal .

Comparison with simulations
To illustrate how our model works, we will use the g(m) relations we obtained from the semianalytic GIF ΛCDM model of Kauffmann et al. (1999) which are now publically available.  provide a fitting formula for the g(m) relation of a GIF galaxy catalog which was constructed by choosing all galaxies brighter than MV = −17.7 + 5 log h, after accounting for the effects of dust. We divided that catalog up into two subsamples based on colour (galaxies labelled as being redder or bluer than B − I = 1.8) and on star-formation rate (rates greater or less than 2M⊙/yr). Fig. 3 shows these relations. The dashed lines show the following fits: N All (m) = (m11/700) 0.9 + 0.5 e −4[log 10 (m 11 /5.6)] 2 Figure 4. Correlation functions of different tracers of the dark matter density field in the ΛCDM GIF semianalytic galaxy formation model. Filled circles are for the dark matter, crosses are for red galaxies, squares for galaxies which have low star formation rates, triangles for galaxies with high star formation rates, and open circles for blue galaxies. The two solid curves show our model predictions for the red and blue galaxies, and the dashed curves show what happens if we use the second factorial moment of the galaxy counts, rather than the second moment when making our model predicition. For comparison, the dotted curve shows the predicted dark matter correlation function. +(m11/30) 0.75 e −(2/m 11 ) 2 N Blue (m) = (m11/500) 0.8 + 0.6 e −4[log 10 (m 11 /6.2)] 2 N Red (m) = N All (m) − N Blue (m) N hSFR (m) = 0.015 + (m11/7000) 0.9 + e −8[log 10 (m 11 /56. where m11 is the halo mass in units of 10 11 M⊙/h. The solid lines are the same in both panels; they show N All (m), and they equal the sum of the two dashed lines. These relations can be used to compute the statistics of one of these galaxy samples, rather than dark matter particles, by setting g(m) equal to the appropriate N gal (m) relation. In addition to these mean N gal (m) relations, our models also require the second moment of the number of galaxies per halo distribution. We have approximated it by setting g2(m) = 0 if g(m) < 1, and g2(m) = µ 2 (m) g 2 (m) + µ g(m) ifg(m) ≥ 1, where µ(m) = log 10 [20 g(m)] 1/2 if g(m) ≤ 5, and µ = 1 when g(m) is larger. Because µ = 1 for a Poisson distribution, this approximately accounts for the fact that the scatter in galaxy counts is sub-Poisson in low mass haloes. If we Figure 5. The mean streaming velocity (left) and pairwise velocity dispersion (right) in the ΛCDM GIF semianalytic galaxy formation model. Filled circles are for the dark matter, crosses are for red galaxies, squares for galaxies which have low star formation rates, triangles for galaxies with high star formation rates, and open circles for blue galaxies. The solid and dotted curves show our predictions for the red and blue galaxies, and the dark matter, respectively.
use the same galaxy sample that Scoccimarro et al. (2000) did, then our model for the scatter is similar to theirs.
The correlation functions for these four subsamples are shown in Fig. 4: crosses, open circles, triangles and squares show ξ gal (r) for red, blue, star-forming and quiescent galaxies in the GIF simulation. Filled circles show the correlation function of the dark matter particles. Notice how similar the correlation function of the blue sample is to that of the starforming sample, how similar the red and quiescent samples are, and how different the blue and star-forming samples are from the red and quiescent samples.
The two solid curves show the result of using our model to compute the correlation functions of the red and blue samples, because these differ the most from each other. We did this by setting g(m) in equation (13) equal to the appropriate N gal (m) relation. For comparison, the dashed curves show the result of using the second factorial moment, rather than the second moment, when computing the galaxy correlation functions; the difference only matters on small scales. The dotted curve shows our calculation of ξ dm (r) which has g(m) = m. The bottom panel shows how b ≡ ξ gal /ξ dm (the ratio of the solid and dotted curves) depends on scale. Both panels show that our model provides a good description of the simulation results. In computing our model predictions, we assumed that the two samples both trace their parent dark matter haloes similarly. That is, we used the same function λ(r|m) for both the red and the blue samples. If the red galaxies were more centrally concentrated than the blue, we could have incorporated it into our anal-ysis by adjusting λ(r|m). In fact, Fig. 2 of Diaferio et al. (1999) shows that, in the semianalytic model, the red galaxies in massive clusters are concentrated more towards the centres of their parent halos than the blue ones are. The agreement between the simulation and our model curves in which we made no such adjustment for this suggests that it must amount to only a weak effect.
We turn, therefore, to the first and second moments of the pairwise velocity distribution for these four subsamples. Fig. 5 shows results for the same semianalytic galaxy samples shown in Fig. 4. As before, crosses, squares, triangles and open circles show galaxies classified as being red, quiescent, star-forming, or blue. Filled circles show the corresponding statistics of the dark matter particles. As for the correlation functions, the blue and star-forming samples are quite different from the red and quiescent samples. Our model shows that this arises simply from the fact that these samples have rather different N gal (m) relations-there are only a few blue, star forming galaxies in clusters.
Notice that blue galaxies (circles) have the smallest streaming motions, and red galaxies (crosses) have the largest v12 values. Our model predicts that larger streaming motions at large separations indicate a higher amplitude of clustering on those scales. Comparison with Fig. 4 shows that the correlation functions of the red galaxies are biased high relative to the dark matter, whereas the blue galaxies are biased low. This is in qualitative agreement with our model.
The solid lines in Fig. 5 provide a more quantitative comparison between our model predictions for the red and blue galaxies, and the values of the galaxy velocities measured in the GIF simulation. The model predictions are in reasonable agreement with the simulation, although the agreement is certainly not as good as it was for the correlation functions.  discuss the reason for the overestimate in v12(r) on large scales (e.g., these models do not satisfy the integral constraint). The bottom panels show the ratio of the galaxy velocities to those of the dark matter; i.e., bv 12 ≡ v gal 12 /v dm 12 , and similarly for bσ 12 . This ratio is scale dependent on smaller scales. Our model describes the scale dependence reasonably well.

DISCUSSION
We presented a simple model of how the streaming motions of haloes and galaxies depends on separation. We tested the model using the publically available simulations of Kauffmann et al. (1999). In the semi-analytic galaxy formation model, the mean streaming motions depend rather strongly on how the galaxy sample was selected. For example, blue galaxies have smaller streaming motions than red galaxies. We showed that our model was able to describe the differences between a wide range of simulated galaxy catalogues rather well (Fig. 5).
Our model predicts a very close relationship between the streaming motions of the galaxies and their spatial distribution. Optical and IRAS galaxies cluster differently (e.g. Marzke et al. 1995;Fisher et al. 1994). Therefore, they must be biased differently relative to the dark matter. If the streaming motions of optical galaxies are the same as the dark matter, then our model predicts that the streaming motions of IRAS galaxies must be different from that of the dark matter. In other words, whether or not the streaming motions of a given galaxy sample trace the motions of the underlying dark matter depends on how the sample was selected. Thus, in the absence of strong arguments for why a given galaxy sample is expected to be unbiased, one should be cautious when interpretting measurements of the streaming motions of galaxies.
If the correlation function and the streaming motions of two different galaxy samples have been measured, then the model described here (equations 11 and 14) says that the square root of the ratio of the two correlation functions (at, say, 20 Mpc/h) should equal the ratio of the streaming motions on the same scale. This can be used to test the validity of the model. Again, however, caution is required because this relationship is only true on large scales. For the semi-analytic galaxy samples we presented, this simple linear biasing was a good approximation on scales larger than about 10 Mpc/h (although our model is able to describe the scale dependence of this ratio even on smaller scales). In this context, we think it worth noting that the semi-analytic galaxy samples we presented here cannot explain the results of Juszkiewicz et al. (2000) who found that ellipticals and spirals have the same value of v12 on separations of about 10 Mpc/h, even though they estimate that the spatial distributions have bias factors which differ by a factor of two: bE/bS ≈ 2. It will be interesting if future data sets confirm this.
In addition to studying how the mean streaming velocity depends on galaxy type, our Fig. 5 also shows that the second moment of the pairwise velocity distribution depends strongly on galaxy type. For example, our model of the pairwise dispersion suggests that the dispersion of blue galaxies should be substantially smaller than that of red ones (although this difference depends on the colour cut), especially on scales of 1 Mpc/h or so. Thus, it is not surprising that σ12 for optical galaxies (Marzke et al. 1995) is almost a factor of two larger than for IRAS galaxies (Fisher et al. 1994). The effects of redshift space distortions are larger if the pairwise dispersion is larger. This means that, on small scales, the amplitude of the redshift space correlation function of red galaxies should be substantially smaller than the real space correlation function, but the difference between real and redshift space correlation functions of blue galaxies should not be as dramatic. This is a generic prediction of these sorts of galaxy formation models. This suggests that galaxy redshift samples cut by colour should provide a useful and direct test of these models.
Dividing a galaxy sample by colour allows another simple test of these models. At large separations, where most pairs are in separate haloes, the model described above predicts that the cross-correlation function of the two colour samples should simply be the geometric mean of the two individual samples. If the two galaxy samples both trace their parent dark matter haloes in the same way (our Fig. 4 shows that this assumption describes the semianalytic model well), then this will be approximately true even on smaller scales. (It will not be exactly true because the scatter in the N gal (m) relations are, typically, sub-Poisson.) Data sets currently available should be able to test this prediction. Figure A1. The evolution of the mass profile if we require that the density profile have the NFW form at all times. For most of the comoving volume, the mass in a given comoving shell increases with time. Only within the core of the object does the mass decrease with time. We have truncated the profiles at the point where they all enclose the same mass m.
where the subscript '0' denotes the values of quantities at the present time. This shows that the number of pairs on comoving scale x at the present time is the same as the number of pairs which, at the earlier time when the linear growth factor was D = D(a)/D0 were on the larger comoving scale x/D 2/(3+n) .
The correlation function decreases monotonically with scale, so the expression above implies that ξ 1halo dm was smaller at early times than it is today. At very early times, therefore, the correlation function might plausibly be dominated by the two-halo term (equation A2). The evolution of this term can be got from inserting the evolution of the bias factor (equation 4) into it. The integrals over m can be done analytically, with the result that ξ 2halo dm (x, a) = D 2 ξ Lin dm (x, a0). At sufficiently early times, this two-halo term dominates on all scales, so our model for the correlation function reduces correctly to the linear theory expression.
Our requirement that the same NFW form hold at all times means that the profile shape evolves as ρa(s) ρ = φ(ca)/3 s(s + (Xa/ca)/Xi) 2 = ρa 0 (S) ρ0 D 3/(3+n * ) S + 1/c0 S + D 2/(3+n) /c0 2 , (A10) where s is the comoving distance from the centre in units of the initial comoving scale Xi, whereas S is in units of the virial radius at a0. The second equality follows from the scalings above for the evolution of φ(ca) and Xa/ca, and setting D ≡ D(a)/D0. The profile evolves in such a way that the density on scales s > (Xa/ca)/Xi grows as a increases, as we expect. On much smaller scales, s ≪ (Xa/ca)/Xi, and the profile shape is more like (caXi/Xa) 2 φ(ca)/S ∝ D −1/(3+n) /S: on very small scales, the density decreases with time! Because this small scale behaviour of the density seems contrary to our intuition, we thought it worth studying the evolution of the mass as a function of comoving scale. The scalings above imply that the fraction of the total mass m = M (s)/M which is in the range ds around s from the halo centre when the growth factor is D ≡ D(a)/D0 is dm(s) ds ∝ sφ(c0)D 3/(3+n * ) s + D 2/(3+n * ) /[c0(∆ nl /Ω) 1/3 ] 2 .
(A11) Fig. A1 shows an example of how the mass gets redistributed as the profile evolves. It was constructed by setting n * = −1.5, c0 = 9 and D nl /Ω = 180 in equation (A11) (the first two values approximate those of an m * halo in a ΛCDM simulation). The solid, dashed, and dot-dashed curves show equation (A11) at D = 1, 0.75 and 0.5 respectively. For most of the volume of the halo, the mass in a given comoving shell increases as D increases. Only well within the core of the object does it decrease with time. We only show the shape of the profile out to the radius Xa which contains the mass m. At D = 1, Xa is the virial radius, which is at s = 1/5.6 ≈ 0.18; Xa was larger earlier, so that the total mass contained in the profile remains constant. By D = 0.6 or so Xa > 1, indicating that the model profile must extend beyond Xi if it is to enclose mass m; at this point the model has really broken down. A similar analysis of the Hernquist profile shows the same qualitative features: ρa(S) ρ = ρa 0 (S) ρ0 D 5/(3+n * ) S + b0 S + b0D 2/(3+n * ) 3 . (A12) Presumably, this apparently unphysical behaviour is a consequence of our unphysical requirement that the profile have the same functional form at all times.