Velocity bias in the distribution of dark matter halos

Tobias Baldauf School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA    Vincent Desjacques Département de Physique Théorique and Center for Astroparticle Physics, Université de Genève, 24 quai Ernest Ansermet, CH-1221 Genève 4, Switzerland    Uroš Seljak Department of Physics, University of California Berkeley, CA 94720, USA Lawrence Berkeley National Laboratory, Physics Department, Berkeley, CA 94720, USA
Abstract

The standard formalism for the co-evolution of halos and dark matter predicts that any initial halo velocity bias rapidly decays to zero. We argue that, when the purpose is to compute statistics like power spectra etc., the coupling in the momentum conservation equation for the biased tracers must be modified. Our new formulation predicts the constancy in time of any statistical halo velocity bias present in the initial conditions, in agreement with peak theory. We test this prediction by studying the evolution of a conserved halo population in N-body simulations. We establish that the initial simulated halo density and velocity statistics show distinct features of the peak model and, thus, deviate from the simple local Lagrangian bias. We demonstrate, for the first time, that the time evolution of their velocity is in tension with the rapid decay expected in the standard approach.

I Introduction

The three dimensional late time matter distribution in the Universe has the potential to place stringent constraints on cosmological parameters and fundamental physics. The interpretation of the observed galaxy distribution is, however, hampered by the fact, that galaxies and their host halos are imperfect tracers of the matter distribution. The local bias model has been successful in explaining the varying clustering amplitude observed for different tracers of the large scale structure Tegmark and et al. (2006). With data from current and upcoming surveys, one would like to push the maximum wavenumber in the analysis into the weakly non-linear regime, requiring consistent bias descriptions that go beyond the simple local bias model. In this regard, it was recently demonstrated that the local bias model is not even consistent with gravitational evolution. Namely, tracers which are initially locally biased will always develop nonlocal bias contributions. Chan et al. (2012); Baldauf et al. (2012).

In this letter, we focus on a different extension of the local bias model known as the peak approach Bardeen et al. (1986), which has its roots in the initial distribution of the regions that will eventually form a dark matter halo. The peak model predicts the existence of a linear, statistical halo velocity bias which remains constant with time Desjacques and Sheth (2010); Desjacques et al. (2010). If true, this would have important consequences for our description of redshift space statistics such as the power spectrum Kaiser (1987). However, there is thus far no evidence for such an effect from -body simulations. Furthermore, the coupled-fluids approximation for the coevolution of dark matter and halos, which is widely used to compute the time evolution of bias Fry (1996); Tegmark and Peebles (1998); Hui and Parfrey (2008), predicts that any initial velocity bias rapidly decays to zero Elia et al. (2011). This stands in conflict with predictions from the peak approach.

Here, we will show that these two seemingly contradictory results can be reconciled if one recognizes that the gravitational force acting on biased tracers of the large scale structure is itself biased. Consequently, when the purpose is to compute correlators like power spectra etc., one should interpret the two-fluids approximation as describing effective/mean-field quantities and, therefore, momentum conservation must be modified accordingly. We will explain how the Euler equation should be modified, and demonstrate that the new two-fluids approximation predicts the (time) constancy of any statistical velocity bias present in the initial conditions. We will then explore the scale-dependence of Lagrangian halo bias in simulations, demonstrate the existence of an initial halo velocity bias in good agreement with peak theory and present numerical evidence that it persists until virialization.

Ii Evolution in the Peak Model

Virialized halos of mass can be traced back to the initial conditions to define proto-halos, i.e., the Lagrangian patches that will collapse into halos. In the peak approach, these are associated with peaks of the initial density field (assumed Gaussian throughout this letter) smoothed on the halo Lagrangian scale . As shown in Desjacques (2008); Desjacques and Sheth (2010), the large-scale limit of the peak 2-point correlation and mean pairwise velocity can be thought of as arising from the effective (or mean-field) relations and . Here, and are Lagrangian bias factors whereas is the characteristic scale of the peak velocity bias. Hence, we expect the Fourier modes and of the proto-halo overabundance and velocity, respectively, to scale as

 δh(k)/WR(k) (1) vh(k)/WR(k) =bv(k)vm(k)=(1−R2vk2)vm(k), (2)

at leading order. The peak selection function asymptotes to unity for such that, on large scales both halo and matter velocity agree. One usually employs a Gaussian to guarantee convergence of the peak model calculation. The time evolution of and can be worked out exactly within the Zel’dovich approximation. Assuming that the motion of the proto-halo center-of-mass coincides with that of the peak position, Ref. Desjacques et al. (2010) found schematically

 cE1,pk(k,a) =bv(k)+D−1+(a)c1(k) (3) bEv,pk(k,a) =bv(k), (4)

where the superscript E labels evolved quantities and is the linear growth rate normalized to unity at the collapse redshift. The linear Lagrangian bias decays to the linear velocity bias, which remains constant throughout time.

This prediction stands in sharp contrast with that of the coupled fluid approximation to halos and dark matter. In this approach, halos and matter are modelled as pressureless fluids co-evolving in the potential determined by the matter distribution solely. Upon linearizing the continuity and momentum conservation equations for the Fourier modes of the halo and dark matter density fields, and , and velocity divergences, and , the time evolution of any initial and is

 cE1,fluid(k,a) =1+D−1+(a)(c1(k)+2bv(k)−3) +2D−3/2+(a)(1−bv(k))… (5) bEv,fluid(k,a) =1+D−3/2+(a)(bv(k)−1). (6)

In the absence of an initial velocity bias, and we recover the usual decay of the linear Lagrangian bias, Fry (1996). In general, the above equations predict the rapid decay of any non-vanishing initial velocity bias.

Can we reconcile these two apparently contradictory results? Firstly, we should bear in mind that this fluid approximation aims at predicting correlators of halos and dark matter. This is the reason why we are allowed to relate with through some (local or nonlocal) bias expansion. In other words, and should in fact be thought of as being effective or mean-field quantities given a realization of , in analogy with the interpretation of the peak bias expansion described in Desjacques (2013). Consequently, the Euler equation for halos describes the conservation of momentum of the halo mean-field density field . The halo momentum conservation equation will be the same as that of the dark matter only if the gravitational force acting on halos is statistically unbiased relative to the force acting on dark matter particles. In general, we argue that the Euler equation for halos should be changed into

 ∂θh(k)∂η+Hθh% (k)+32ba(k)H2Ωmδm(k)=m.c. (7)

Here, is the conformal time, “m.c.” designates second-order mode-coupling terms and we have omitted the dependence of and on . is the linear gravitational force or acceleration bias. It is important to note that our modification to the Euler equation does not contradict the equivalence principle since it only makes sense statistically: on an object-by-object basis, the biased tracers still satisfy the usual momentum conservation.

Since, in the linear regime, the acceleration is parallel to the initial velocity and since gravity mode-coupling cannot induce any linear contribution (by definition), must be equal to the linear, initial velocity bias . Solving the fluid equations for halos and dark matter with this new halo momentum conservation equation, we find that and evolve in accordance with the peak predictions Eqs. (3) – (4). To convince ourselves that this must be true, consider the set of points with zero initial velocities (e.g. Percival and Schäfer (2008)) as the discrete, biased tracers of the mass density field. We thus expect and the linear Eulerian bias to decay to zero. This is indeed what happens since, in the linear regime, the mass 2-point correlation grows with while the 2-point correlation of these biased tracers remains constant. We note that the mode-coupling terms may also be biased, but we leave this possibility to future studies.

Iii Initial Velocity Bias in Simulations

So far, there has not been any conclusive evidence for a statistical velocity bias in the distribution of virialized dark matter halos, except for a couple of tentative measurements from Percival and Schäfer (2008); Elia et al. (2012). These suffered from the fact that simulations are sampling the cosmic density field with a finite number of discrete tracer particles. Hence, it is difficult to define a velocity field throughout the whole simulation volume, especially for the rare, massive dark matter halos. Here, we work instead with the number-weighted (for halos) and density-weighted (for dark matter) velocity fields Okumura et al. (2012). These “momentum” fields and are well defined everywhere. To extract statistical information about them, we will measure the density-momentum correlators , where is the matter or halo momentum projected along the -axis. Since it is a cross-correlation with , it does not suffer from shot-noise, like the halo-matter cross power spectrum . We begin by assessing whether both Lagrangian and have -dependencies in agreement with that predicted by peak theory. In this regard, we consider a suite of 16 collisionless dark matter simulations. The initial conditions for the particles in the box were set up at using 2LPT Crocce et al. (2006) and subsequently evolved using GADGET2 Springel (2005). Halos are identified at with a FoF halo finder of linking length and their constituent particles were traced back to the initial conditions to define the Lagrangian halo distribution (or proto-halos). We repeat this procedure for a set of intermediate time steps. This provides us with the non-linear time evolution of a strictly conserved set of tracers of large scale structure. The mean mass of the five bins is and .

We then measure the ratio of the initial halo-matter and matter-matter density-density and density-momentum power spectra. In the peak model, at the lowest order, these quantities are given by

 ⟨δm,i(k)δh,i(−k)⟩⟨δm,i(k)δm,i(−k)⟩ =(b10+b01k2)WR(k) (8) ⟨δm,i(k)jzh,i(−k)⟩⟨δm,i(k)jzm,i(−k)⟩ =(1−R2vk2)WR(k). (9)

In a first step we fit the linear density bias from the density correlator on large scales, then we jointly fit for the scale of the Gaussian peak selection function and the bias term in the same statistic. Subsequently we use the filter with the same scale to fit the scale-dependence of the velocity bias and find strong evidence for a non-zero initial . In App. B we reconsider the case where the scale dependence of the velocity correlator is entirely due to the window function and find a poor performance for the density statistic, thus strengthening the case for the scenario presented here. Fig. 1 shows that this parametrization is able to reproduce the scale-dependence of the proto-halo density and velocity bias reasonably well. The choice of a Gaussian for the peak selection function is motivated by the sole requirement that the spectral moments of the Gaussian field should be convergent. This would not be the case for a top-hat window, but generalized window functions might provide a better fit and still yield convergent moments. In Appendix A we show that the fitted peak density and velocity bias parameters are in agreement with theoretical predictions once a model for the mass–collapse threshold relationship has been employed. We also show that the coefficients of the corrections are fairly insensitive to the choice of the collapse threshold and thus robust to choices that go beyond the peak constraint. For this study we decide to employ the fitted parameters that describe the initial conditions and to compare the resulting predictions to the evolved halo statistics in the simulations.

Iv Time Dependence of Velocity Bias

Next, we turn to the time-dependence of the halo velocity bias. To this purpose, we consider the time evolution of the linear density-density and density-momentum correlators. These linear correlators are obtained by cross-correlating the evolved halo positions and momenta with the linear Gaussian matter density field . Considering cross-correlations with the non-linear matter field would contaminate the statistics with the poorly understood late time matter distribution and, thus, undermine a clear isolation of the scale-dependencies induced by the peak constraint. We also refrain from using halo-halo correlators, since these are likely plagued by non-trivial shot noise effects Hamaus et al. (2010).

We assume that peaks move according to their initial velocity as in Zel’dovich approximation. We calculate the resulting correlators by writing the evolved peak positions as following the steps laid out in Desjacques et al. (2010), where is the displacement field at Lagrangian position . Since the initial matter fluctuations are Gaussian, we only select the linear terms in the bias relation. We finally obtain

 ⟨δ(1)m(k)δh(−k)⟩ =D2+cE1(k,a)Gpk(k)P(k)WR(k), (10) ⟨δ(1)m(k)jzh(−k)⟩ =(bv,pk(k)−D2+σ2d,pkcE1,pk(k,a)k2) (11) ×Hf+D2+(ik⋅^zk2)Gpk(k)P(k)WR(k),

where and are defined in Eqs. (3) and (4), is the peak propagator and is the peak displacement dispersion (extrapolated to the collapse epoch), given by with . It is reduced relative to the linear matter displacement dispersion because i) is smaller for halos than for matter due to the finite smoothing scale R (which is zero for the matter), and ii) the dark matter preferentially flows onto the peaks, so statistically the peaks are more at rest than the dark matter, and the term accounts for that. The analogy with the Eulerian coevolution model in Eqs. (3) – (4) can only be seen in the low- limit due to the resummation of the displacement dispersions. In principle non-local third order bias corrections Biagetti et al. (2014); Saito et al. (2014) contribute to the density correlator at late times. As we will discuss shortly, these loop corrections are likely suppressed by the explicit smoothing scale in the peak model.

The evolution of the density and velocity bias in our simulations for mass bin II is shown in Fig. 2 and in Fig. 3 for the other bins. We divide the cross-correlators of halo density/momentum with the linear matter density by those of the linear matter density/momentum with the linear matter density. On large scales, the density bias in the upper panels evolves from the Lagrangian to the Eulerian bias, i.e. it changes by unity. On smaller scales, most of the initial enhancement in the density bias is washed out by the linear evolution of the peak density bias and the additional damping due to the peak displacement dispersion in the propagator. We show the full prediction of Eq. (10) (solid lines) and the case in which the velocity bias part of decays as as predicted by coevolution (dashed lines). At all redshifts, the model with decaying velocity bias provides a damping that is too small, thus overpredicting the clustering. The improved agreement in the case of conserved velocity bias is due to a partial cancellation of the coefficient .

The velocity bias in the lower panels of Figs. 2 and 3 is unity on large scales as required by mass and momentum conservation. On smaller scales, with evolving time we see a sharpening of the dip until redshift when it starts being damped by the exponential propagator arising from the displacement dispersion. We show the full prediction of the peak model with conserved velocity bias Eq. (11) as the solid line. We also overplot the prediction of the simple linear coevolution model Eq. (6) as dashed lines, where we have kept the exponential part of Eq. (11) that is not part of the linear evolution but clearly necessary for unequal time statistics. The model with decaying velocity bias is clearly less compatible with the non-linear evolution measured in the data than the model with constant velocity bias. This is particularly evident at intermediate times, where the propagator is not dominant yet. Comparing the various mass bins, we see that the magnitude of the initial and final velocity biases increases with increasing mass.

In Fig. 4 we show the cross correlation between the final halo density and the linearly evolved initial Gaussian matter density field. This quantity asymptotes to the Eulerian bias on large scales. On smaller scales, the data show a distinct upturn that is very well described by the peak model in presence of undamped velocity bias. The inclusion of non-local third order bias does not deteriorate this agreement. The third order non-local bias would contribute through a loop correction, i.e. an integral over the power spectrum weighted by a kernel

 Pb3,nl(k)∝P(k)WR(k)∫d3q(2π)3P(q)K3,nl(q,−q,k)W2R(q). (12)

In contrast to standard implementations Saito et al. (2014); Biagetti et al. (2014), we explicitly account for the smoothing scale in the relation between halo overdensity and matter, which significantly suppresses the loop corrections. In practice, we have adopted the third order non-local bias amplitude as predicted by Eulerian coevolution , and set the smoothing scale to the Lagrangian scale.

V Conclusions

We have focused on modeling correlators with the linear, Gaussian density field in order to isolate the effects of the scale-dependent, linear peak bias. These statistics are, admittedly, not those that will be directly measured in observations. However, they allow us to understand key physical properties of the joint evolution of halos and matter under gravity and, therefore, they will help predicting the true observables more accurately and reliably. Our result also demonstrates that, at the level of linear bias, the Zel’dovich evolved peaks provide a very good description of the time evolution of the halo-matter correlators. While the interior structure of a dark matter halo is highly non-linear, the effective dynamical evolution of its center of mass is more amenable to perturbative treatments than the dark matter itself. Finally, we insist on the interpretation of , in the halo-matter fluid approximation as mean-field/effective quantities (and not counts-in-cells), as long as the purpose is to compute ensemble averages.

Our measurement of a halo velocity bias constant throughout time may have important consequences for the modeling of halo clustering in redshift space, see e.g. Desjacques and Sheth (2010). However, further developments are required until peak theory can provide an accurate template for the halo-halo correlation function. Since the full, non-perturbative initial peak-peak correlation function correctly accounts for exclusion effects Baldauf et al. (2013), it should lead to a comprehensive description of all the scale-dependent bias effects. On a final note, the effective field theory descriptions of halo statistics McDonald and Roy (2009); Assassi et al. (2014) may lead to a functional form similar to that predicted by the peak model, but the model presented in this work complements it by a dynamical perspective and provides physical arguments for the values of the operators.

Acknowledgements.
The authors would like to thank Roman Scoccimarro, Ravi Sheth and Zvonimir Vlah for useful discussions. T.B. gratefully acknowledges support from the Institute for Advanced Study through the W. M. Keck Foundation Fund. V.D. acknowledges support by the Swiss National Science Foundation. U.S. is supported in part by the NASA ATP grant NNX12AG71G.

Appendix A Peak Bias Parameters

Predicting the peak bias parameters from the model requires one to choose a relation between halo mass and collapse threshold. While the spherical collapse model suggests a unique collapse threshold of regardless of the halo mass, measurements in simulations by Robertson et al. (2009) indicate that the top hat, linear overdensity around proto-haloes is increasing with decreasing mass. The value of is attained only for very high mass objects. The top hat window can not be used for calculations in the peak model since it does not yield convergent higher derivative moments . A Gaussian window or a mixture of top hat and Gaussian filters can be employed to bypass this problem. A measurement of the initial overdensity smoothed with a Gaussian window was performed by Elia et al. (2012) for FoF halos. They provided a fitting function, and also found that the collapse threshold increases with decreasing mass. In contrast to Elia et al. (2012) however, we decided not to tighten the filter scale to the halo mass scale, but rather adjust it so that we reproduce the cutoff in the initial condition density and velocity power spectra. The disagreement between the filter scale and the halo mass likely originates from the fact that the Gaussian filter does not reproduce the proto-halo mass density profile very well. One may want to consider mixed filtering like Paranjape et al. (2013) or a more general window that interpolates between a Gaussian and a top hat. This is clearly beyond the scope of this paper.
We use the following collapse threshold as a function of radius

 δc(R)=2.17−0.32log(R/1 h−1Mpc). (13)

Note that, on rescaling our radii to coincide with the ones predicted by the mass scale, our approximation would be close to the average collapse threshold measured by Elia et al. (2012). For the mean peak curvature we rely on Bardeen et al. (1986)

 ¯u=γν+3(1−γ2)+(1.216−0.9γ4)e−(γ/2)(γν/2)2√3(1−γ2)+0.45+(γν/2)2+γν/2 (14)

With these ingredients the peak bias parameters can be calculated according to Desjacques (2013)

 b10= 1σ0ν−¯uγ1−γ2, (15) b01= 1σ2¯u−νγ1−γ2, (16) R2v= σ20σ21. (17)

Note that the velocity bias is independent of the collapse threshold and solely depends on the size of the proto-halo patch. We show the bias parameters in Fig. 5 as a function of filter scale. The density bias and the velocity bias are not sensitive to our choice of collapse threshold. We thus consider their amplitude to be fairly model independent. By contrast, the density bias strongly depends on the collapse threshold, which is rather unsurprising since it encodes the response to long wavelength density fluctuations.

To summarize this section, let us emphasize that the model parameters can be predicted from first principles with reasonable accuracy so long as one adds a few ingredients in addition to the peak constraint. We still consider the evolution predictions based on initial condition measurements in the simulations to be a decisive test of the persistence of velocity bias that is independent of these additional assumptions.

Appendix B No velocity bias

In the main text we have fitted the initial density and momentum correlators assuming the functional form of the peak model. In this Appendix, we would like to consider the case, in which the scale dependence of the momentum correlator is entirely due to the window, i.e., that there is no velocity bias beyond the averaging over the halo patch. In this case the momentum correlator defines the “velocity” window

 Wv(k)≡⟨δm,i(k)jzh,i(−k)⟩⟨δm,i(k)jzm,i(−k)⟩ (18)

In real space, this window has some odd properties, in particular it is not positive definite and oscillatory for most of the mass bins (see right panel of Fig. 6). We can now employ this window to describe the protohalo-matter correlator as , where the change in the window function induces a change in the part of the bias. The result is shown in the left panel of Fig. 6. Dividing by the window, the scale dependence can initially be fitted by a term, but this approach fails once the velocity window crosses zero at . This is a considerably smaller range than the range over which the Gaussian window together with the scale dependent density and velocity bias presented in the main text can describe the measurements.
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