Issue 
A&A
Volume 518, JulyAugust 2010
Herschel: the first science highlights



Article Number  A38  
Number of page(s)  8  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201014147  
Published online  31 August 2010 
Triaxial collapse and virialisation of darkmatter haloes
C. Angrick  M. Bartelmann
Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik, AlbertUeberleStr. 2, 69120 Heidelberg, Germany
Received 27 January 2010 / Accepted 13 May 2010
Abstract
We reconsider the ellipsoidalcollapse model and extend it in two
ways: we modify the treatment of the external gravitational shear
field, introducing a hybrid model in between linear and nonlinear
evolution, and we introduce a virialisation criterion derived from the
tensor virial theorem to replace the ad hoc criterion employed so
far. We compute the collapse parameters
and
and find that they increase with ellipticity e and decrease with prolaticity p. We marginalise them over the appropriate distribution of e and p
and show the marginalised results as functions of halo mass and
virialisation redshift. While the hybrid model for the external shear
gives results very similar to those obtained from the nonlinear model,
ellipsoidal collapse changes the collapse parameters typically by
,
in a way increasing with decreasing halo mass and decreasing
virialisation redshift. We qualitatively confirm the dependence on mass
and virialisation redshift of a fitting formula for
,
but find noticeable quantitative differences in particular at low mass
and high redshift. The derived mass function is in good agreement with
mass functions recently proposed in the literature.
Key words: cosmology: theory  methods: analytical  dark matter  cosmological parameters  galaxies: clusters: general
1 Introduction
The sphericalcollapse model (e.g. Mota & van de Bruck 2004; Schäfer & Koyama 2008; Engineer et al. 2000; Bartelmann et al. 2006; Wang & Steinhardt 1998) is a fundamental ingredient in the theory of cosmic structure formation. Following the collapse of a slightly overdense, homogeneous sphere, it allows the derivation of two essential parameters; the overdensity of a virialised halo compared to the mean or the critical cosmic density, and the critical linear density contrast . The former is important because it allows relating sizes to masses of virialised structures, and the latter because it establishes a link between linear structure formation and the population statistics of collapsed haloes. Despite fundamental doubts as to the validity of such a simplified model for accurate cosmological predictions, the parameters derived from the sphericalcollapse model or variants thereof allow surprisingly farreaching predictions such as the halo mass function or the correlation properties of haloes, which are confirmed by numerical simulations.
The statistics of a Gaussian random field implies that
spherical collapse should not occur. In fact, the probability
distribution for the eigenvalues
of the Zel'dovich deformation tensor (Doroshkevich 1970) shows that spherical collapse has a vanishing probability,
because it requires . In this equation, , , and denotes the variance of the matter power spectrum smoothed on a scale that corresponds to a halo mass M. Modifications of the original PressSchechter mass function (Press & Schechter 1974; Bond et al. 1991) motivated by ellipsoidal collapse (Sheth & Tormen 2002) substantially improve the agreement of analytic predictions on the halo population with numerical simulations (e.g. the Millennium simulation by Springel et al. 2005).
Ellipsoidal collapse was analysed many times before (see Bond & Myers 1996; Eisenstein & Loeb 1995; Bartelmann et al. 1993, for examples). Several authors have worked with the model by Bond & Myers (1996), generalising it for different cosmologies and introducing the scale factor a as a time variable (Monaco 1998; Sheth et al. 2001; Ohta et al. 2004; Monaco 1997,1995; Sheth & Tormen 2002). We are reconsidering it here for two reasons.
First, we want to analyse how different assumptions on the treatment of the environment of a halo impact on the parameters and . Previous assumptions have been that the principalaxis system of the homogeneous ellipsoid is either identical with that of the external gravitational shear field, or that the two eigensystems do not coincide, introducing rotation and a deviation from the homogeneous mass profile. We introduce another assumption here, letting the eigensystems of the collapsing ellipsoid and its surrounding shear field follow each other until turnaround of the major principal axis and then decoupling both.
Second, we want to stop the collapse along any of the principal axes according to a physically motivated virialisation condition. Virialisation must be invoked to prevent the axes from collapsing to zero, and thus to be able to follow the entire collapse of an ellipsoid, i.e. the collapse of its three principal axes. Conventionally, the collapse of each axis is stopped when a_{i}(a)=0.177a, where a_{i} is the scale factor of the ith axis and a is the background scale factor (Sheth et al. 2001; Bond & Myers 1996). In this way, for spherical collapse in an Einsteinde Sitter universe at the time when the third axis virialises. However, the value 0.177 has no fundamental physical motivation, and there is no guarantee for it not to be different for ellipsoidal rather than spherical collapse, or whencosmologies other than EdS are to be considered. We introduce a general virialisation condition based on the tensor virial theorem that avoids introducting such an uncalibrated factor. We find substantial changes on both and from both modifications and point out several discrepancies of our results with earlier studies.
We introduce the ellipsoidalcollapse model including our modifications in Sect. 2, present our results in Sect. 3 and conclude in Sect. 4.
2 The model
in this section, we shall briefly review the ellipsoidalcollapse model of Bond & Myers (1996) for cosmologies with a cosmological constant introducing the scale factor a as time variable. Furthermore, we shall present a physically motivated virialisation condition to stop the collapse of each axis, and show how to find the proper initial ellipticity and prolaticity as a function of mass and virialisation redshift.
2.1 The evolution equations
Let
be the dimensionless principal axes of the ellipsoid, where R_{i} with
are its dimensional semimajor axes, and
the size of a spherical tophat corresponding to a mass
with the cosmological background density
.
The evolution of the three principal axes a_{i} with time t in a cosmology with a cosmological constant
is given by
(Bond & Myers 1996), where G is the gravitational constant and c the speed of light. The density contrast of the ellipsoid with respect to the background density is . The parameters b_{i} and denote the internal and external contributions to the gravitational tidal shear which occur because of the deviation from sphericity. Generally, the total tidal field is described by the tidal field tensor with the elements , where denotes the peculiar gravitational potential, and and are the internal and external contributions to the shear, respectively. After a transformation into the ellipsoid's eigensystem, which is the same as the eigensystem of in this model, the internal shear can be evaluated as
while the external shear can be approximated by
where D_{+} is the linear growth factor, and the are the eigenvalues of the Zel'dovich deformation tensor. See the first appendix of Bond & Myers (1996) for details of the calculation. Here and in the following the index ``0'' refers to initial values.
In linear approximation, the environment into which the ellipsoid is embedded evolves completely independently of it, whereas in the nonlinear approximation it is tightly coupled to the ellipsoid. In Sect. 3.1 we shall introduce the hybrid model as a third approximation for the external shear.
We can rewrite Eq. (2) by using the scale factor a as time variable using Friedmann's equation
,
where H_{0} is Hubble's constant. The expansion function of the universe, E(a), introduces the dimensionless density parameters of matter and dark energy today,
and
,
respectively. This gives
where a prime denotes differentiation with respect to a, and . Equation (5) defines a set of three coupled secondorder differential equations, for which we need six independent initial conditions compatible with the Zel'dovich approximation for early times. These are provided by
since for . Choosing , this is comfortably fulfilled. In Appendix A, we compare Eqs. (5)(7) with the results presented by Monaco (1997) and see that they differ.
In the following, we shall assume that the eigenvalues of the Zel'dovich tensor are ordered as , which implies that the ellipsoid first collapses along the direction 1, first. At that time , and the collapse of the remaining two axes can no longer be followed so that we have to add a virialisation condition for each axis preventing their collapse to zero.
2.2 The virialisation condition
Conventionally, the collapse of each axis is stopped when a_{i}(a)=0.177a, where a_{i} is the scale factor of the ith axis and a the background scale factor (e.g. Sheth et al. 2001; Bond & Myers 1996). In this way, for spherical collapse in an Einsteinde Sitter (EdS) universe at the time when the third axis is assumed to virialise. However, the value 0.177 has no fundamental motivation in the physics of the collapse, and there is no guarantee that it remain unchanged in the case of ellipsoidal instead of spherical collapse, or in other cosmologies than Einsteinde Sitter.
We shall thus follow a different approach and present a
physically wellmotivated virialisation condition in the following to
stop the collapse of each axis individually. We start from the tensor virial theorem (see Binney & Tremaine 1987, p. 213, 280),
where is the moment of inertia tensor, and are the contributions to the kinetic energy tensor coming from ordered and random motions, respectively, is the potential energy tensor, and is the external potential energy tensor. Their elements are generally defined as
and
where is the density of the fluid, and are the gravitational potentials of the ellipsoid itself and its surroundings, respectively, is the volume which is integrated over, are the velocity dispersions, and the bar indicates averaging over .
We now specialise to the case of a homogeneous ellipsoid. For a stable mass configuration, the lefthand side of Eq. (8)
has to vanish for each component of the inertial tensor individually.
Since the ellipsoid is assumed to be at rest and the ellipsoid's
eigensystem is chosen as a reference frame,
so that K_{ij}=0 and
,
with the Kronecker symbol
.
Note again that in this framework the eigensystems of the overdense
ellipsoid and the gravitational tidal field are identical. For a
homogeneous ellipsoid
,
thus
with the mass M of the ellipsoid. The sum W_{ij}+V_{ij} can be evaluated using and Eq. (2) to be
Requiring that Eq. (8) is fulfilled for each axis separately and introducing the scale factor a as time variable yields the virialisation conditions for the three axes a_{i},
When this condition is fulfilled for an axis together with , its collapse is stopped and its size is frozen in.
We emphasise that the former equation is consistent with the
virialisation condition for spherical collapse in the EdS universe,
where
.
The subscripts ``v'' and ``ta'' denote the time of virialisation and
turnaround, respectively. For the special case of EdS, Eq. (13) becomes
with . Using the parametric solution
(Engineer et al. 2000) together with the relations and R_{0}^{3}=9 GMt_{0}^{2}/2 indeed gives the expected result that Eq. (14) is satisfied when .
For bound objects in the CDM model, the virialisation condition is
where is the average kinetic energy of the halo, its average potential energy, and the average effective potential energy contributed by the cosmological constant. Using and for the homogeneous sphere as well as energy conservation between turnaround and virialisation, one arrives at a cubic equation for ,
with (Lahav et al. 1991). The relevant solution of this equation agrees precisely with the condition derived from the tensor virial theorem, Eq. (13), as Fig. 1 shows. The small deviation occurs because we choose the time of virialisation rather than collapse as a reference ( ) when using the tensor virial theorem. If we use the collapse time ( ) instead, both conditions yield identical results.
Figure 1: Comparison of the virialisation conditions from the scalar and tensor virial theorems. The curves differ slightly because they refer to different times. While the tensor virial theorem was evaluated at virialisation, the scaler virial theorem was evaluated at collapse. If we choose the collapse of the darkmatter halo as reference for z, both yield exactly the same result. 

Open with DEXTER 
2.3 The initial ellipticity and prolaticity
Equations (6), (7) imply that one has to choose initial values for the Zel'dovich deformation tensor to define the initial deviation of the principal axes and their time derivatives from the background. We shall show how they are chosen appropriately so that they comply with the assumed Gaussian nature of the Universe's initial conditions and represent a statistical average of haloes with the same mass M and virialisation redshift but different shapes.
Figure 2: Evolution of the principal axes and the external shear for a darkmatter halo with and p=0 in the reference CDM model. Left panel: linear approximation. Central panel: nonlinear approximation. Right panel: hybrid approximation. 

Open with DEXTER 
Starting from the probability distribution for the eigenvalues of the Zel'dovich deformation tensor, Eq. (1), the conditional probability distribution for the ellipticity
and the prolaticity
,
defined as
and
,
respectively, was derived by Sheth et al. (2001) to be
To compute a statistical average of any quantity for a given halo that depends on e and p, one should marginalise over the distribution g(e,p),
However, a Taylor expansion of up to second order in e and p around their mean values and under the distribution g(e,p) gives
where and are the variances for e and p according to the distribution g. Up to first order, . Deviations occur only at second order. The expectation values for e and p as well as their variances are given by
Since the variances are , and , using the approximation introduces only a small error, which is 1% for and 3% for for CDM and OCDM. For EdS, the error is larger and 3% for and 10% for . But since the latter cosmology is scientifically only of low relevance and usually serves as a reference model only, the usage of the former approximation is well justified. Instead of sampling at several points for e and p, one only has to evaluate it once for and .
Generally, the eigenvalues
are related to e and p by (see e.g. Bardeen et al. 1986; Bond & Myers 1996)
where we have set and p=0 in the last step.
3 Results
In this Section, we show the results of the ellipsoidalcollapse model for the parameters and for three different cosmologies and discuss how they are affected by the choice of the externalshear model.
3.1 Influence of the external shear
Figure 2 shows the evolution of the three principal axes and the eigenvalues of the external shear for a reference CDM model with , , and for three different models of the external shear and , p=0. The shear in the linear approximation evolves completely smoothly over the entire collapse time. The evolution of each eigenvalue is given by D_{+}(a) so that according to Eq. (4), , , and at all times.
This is different in the nonlinear approximation: At early times, the evolution of the is the same, but soon thereafter they start evolving nonlinearly and becomes slightly negative. Noticeably there are steps in the evolution of the external shear whenever an axis virialises because , and the evolution of this volume factor changes after virialisation of each axis. In the right panel, we introduce the hybrid approximation: Initially, the evolution of is described by the nonlinear model. When one of the axes turns around, however, the corresponding eigenvalue of the external shear continues evolving linearly, i.e. its value at turnaround is then scaled by .
We believe that the hybrid model is the preferred model for the evolution of the external shear since it takes into account that the evolution of the ellipsoid itself and its vicinity should be tightly coupled in the beginning. At turnaround, however, they are definitely decoupled so that choosing this moment to switch from nonlinear to linear evolution seems appropriate. Hence, we will use the hybrid model for the evolution of the external shear in the following.
Figure 3: (left panels) and (right panels) for three different models of the external shear. Both quantities were calculated for a darkmatter halo in the reference CDM model. Top panels: as a function of virialisation redshift . Bottom panels: e=0.2 for all . In both cases, p=0. 

Open with DEXTER 
The influence of the externalshear model on the parameters
and
is shown in Fig. 3. For a given mass and a given virialisation redshift ,
the initial overdensity is chosen such that the third axis of the ellipsoid finally virialises at .
Both parameters can then be calculated by
In this case, is defined with respect to the critical density as it can be primarily found in the literature. For the definition with respect to the background density one simply has to omit .
Using as a function of virialisation redshift and p=0 (top panels of Fig. 3), the dependence of and on is almost the same for the nonlinear and the hybrid models. However, both differ from the linear model, showing that the external shear is most important at the beginning of the ellipsoid's evolution. At that time, the nonlinear and the hybrid models agree. While is always smaller in the linear compared to the other two models, the curves for cross. This reflects the circumstance that the initial overdensity in the linear model is different from that in the two other models, leading to a different initial ellipticity and therefore to a completely different evolution history. This can be seen in the bottom panels of Fig. 3, for which we have chosen e=0.2 independently of . They also clearly show that a varying initial ellipticity drives primarily the evolution of , whereas also strongly varies for fixed e. In this case, both and are smaller in the linearshear model compared to the nonlinear and the hybrid models.
3.2 Parameters as function of mass and redshift
Before we present general results for the parameters
and
,
we should comment on a subtle but very important issue: Whenever we
want to compare our results with the ordinary sphericalcollapse model,
we have to keep in mind that we calculate all quantities at the time
when the third axis virialises.
Thus, we also have to compare these quantities those from the
sphericalcollapse model that are also calculated at the time of
virialisation and not of collapse, i.e. when
and not R=0 for EdS. This leads to slightly lower reference values of
and
since
.
Here the subscript ``col'' denotes collapse. Using the parametric solutions of Ohta et al. (2004) for the linear and the nonlinear overdensity,
respectively, and at virialisation, we find that and for the EdS universe at independent of . Recently, Lee & Ng (2009) arrived at the same values when accounting for the time of virialisation instead of collapse.
The top panels of Fig. 4 show and for three different cosmologies for e=p=0, i.e. spherical systems. The OCDM cosmology is the same as our reference CDM model except that . For the EdS model we set and . Indeed, for the EdS universe the constant values derived analytically are also reproduced by solving Eq. (5) numerically. This demonstrates again that Eqs. (5), (13) are fully consistent with the wellknown sphericalcollapse model. Note that the qualitative behaviour of is the same as (compare e.g. with Bartelmann et al. 2006). However, there is a difference for the critical linear overdensity whose shape as a function of differs substantially from the shape as a function of . This should illustrate that the time chosen in the model when virialisation actually occurs ( or ) can already have substantial impact on the qualitative behaviour of relevant quantities as a function of redshift, hence it is not necessarily a consequence of ellipsoidal collapse alone.
The bottom panels of Fig. 4 show and for a triaxial halo with e=0.2 and p=0.1. Also in this case both parameters are independent of for the EdS universe, although changes from 1.583 to 2.058, while stays almost the same, 148 instead of 147. Interestingly, the most drastic changes in the shapes of both parameters occur for the OCDM model, for which the total density is only approximately a third of the critical density, while for the CDM model their changes are small.
Figure 4: (left panels) and (right panels) for three different cosmological models and fixed initial e and p. Top panels: spherical case (e=p=0). Bottom panels: triaxial case with e=0.2 and p=0.1. 

Open with DEXTER 
Figure 5: (left panels) and (right panels) as a function of halo mass and virialisation redshift for three different cosmologies marginalised over e and p. Top panels: dependence on virialisation redshift for a halo with . Bottom panels: dependence on mass M for a halo with . Thin dashed lines show the reference values from spherical collapse in the EdS universe, and . 

Open with DEXTER 
The influence of the halo mass M and the virialisation redshift on both and are illustrated in Fig. 5. For both decreasing mass and decreasing redshift, is a monotonically decreasing function, approaching the reference values from the sphericalcollapse model in the EdS universe for large masses and high virialisation redshifts, the situation for is much more complicated. For all three models, it has a minimum at redshifts 45 and at a mass of . For smaller values of mass and redshift, it is a monotonically decreasing function of both M and for the OCDM and EdS models. It is also monotonically decreasing as a function of M in the CDM model, but reaches a maximum at . This is a direct result of the definition of with respect to the critical density. If it was defined with respect to the background density, the factor would not appear in Eq. (26), and all three curves would increase with decreasing .
In Fig. 5 one can clearly see that the intervals that are covered for both and are the largest for the EdS model, indicating that there is a stronger dependence of the ellipsoid's evolution on the total amount of matter in the Universe compared to the size of the cosmological constant if varying initial ellipticities are taken into account.
For either or , both and must reach the reference values for spherical collapse in the EdS universe since the initial ellipticity and both decrease with increasing mass, and has to be higher the earlier the structure is required to collapse. This expected behaviour can be clearly seen for , whereas for this happens finally for very large M and . We should stress again in this context that a crucial portion of the dependence on mass and virialisation redshift is driven by the change in the initial ellipticity, comparing Figs. 4 and 5.
Figure 6: Comparison of the fitting formula of Sheth et al. (2001) for with the results of this work for the reference CDM model. Left panel: as a function of mass and z=0. Right panel: as a function of redshift and . 

Open with DEXTER 
We compare in Fig. 6 the results of this work for
as a function of mass and redshift with the fitting formula of Sheth et al. (2001) given by
where is the redshiftdependent linear overdensity of the usual sphericalcollapse model. The left panel shows that the dependence on mass is similar for both the fitting formula and the result of this work. However, the fit by Sheth et al. (2001) is less steep as a function of mass resulting from the differences of their underlying work to ours: first, they use the linearshear model instead of the hybrid model, leading to a larger for all redshifts as shown in Fig. 3. Second, the collapse of each axis is stopped using the artificial condition a_{i}=0.177a, leading to as a reference value at high mass. Thus, the fitting formula provides a larger value for large masses, while our approach leads to a for due to the virialisation condition that we apply. Third, for a given initial overdensity , Sheth et al. (2001) use the most probable value instead of the expectation value given by Eq. (21), which leads to initial ellipticities that are slightly too low so that the asymptotic limit for high redshifts is reached earlier than in our case.
The right panel of Fig. 6 shows a similar behaviour for as a function of redshift. The differences between the fitting formula and the result of our work again occur due to the different virialisation time and condition, and the difference between the most probable and the expectation value of e.
For applications, fitting formulae for both
and
may be useful. We provide expressions here which are inspired by Eq. (28) of Sheth et al. (2001). For
we suggest
where b=0.6536, c=0.6387, and both and are cosmologydependent quantities. Note again that the redshift of virialisation, , has to be chosen as reference for z for both and . For a spatiallyflat CDM model with matter density parameter in the range , , and , the maximal error is 1.8% with a mean error of 0.4%.
A similar functional dependence can be found for
.
Only a small correction term has to be added to arrive at a satisfactory accuracy. We find
with a=0.3819, b=0.5379, c=0.7589, and . In the same range as above, the maximal error is 5% with a mean error of 1%.
Figure 7: Influence of the initial ellipticity e and prolaticity p on the parameters (top panels) and (bottom panels) for three different cosmological models. Left panels: CDM. Central panels: OCDM. Right panels: EdS. The large circle in each figure indicates the value of the respective parameter for and p=0. The small circles indicate values for a combination of and (see Eqs. (21), (22)). For a fixed e both and grow for decreasing p. 

Open with DEXTER 
3.3 Influence of initial ellipticity and prolaticity
In Fig. 7 we plot both and as a function of the initial ellipticity e and prolaticity p centered around their expectation values given by Eq. (21) for three different cosmologies. For increasing e and decreasing p, both parameters grow qualitatively in the same way as already reported by Sheth et al. (2001, cf. their Fig. 1). Quantitative deviations arise from the differences in the applied algorithm as discussed in Sect. 3.2. For a given mass and virialisation redshift, the initial overdensity for the EdS universe is larger compared to both CDM and OCDM due to a shorter physical time interval that corresponds to the same redshift interval, resulting in a larger and , but also in larger curvatures of and with respect to e and p. These are the sources of the larger error in the approximation discussed in Sect. 2.3. Since the redshifttime relation is not very different between CDM and OCDM, the dependences of and on e and p are comparable.
3.4 Mass function
Using Eq. (29), we are able to construct the mass function of darkmatter haloes using the extended PressSchechter formalism developed by Bond et al. (1991) and Lacey & Cole (1993), which is based on the firstupcrossing distribution of the density contrast as a function of the ``time variable'' . We shall proceed similarly as Sheth & Tormen (1999,2002) and define the scaled variable to derive the mass function for our standard CDM cosmology.
As Sheth & Tormen (2002) pointed out, expressing the firstupcrossing distribution f as a function of
has the advantage that it is only necessary to calculate
for a barrier of height
at one arbitrary redshift to infer the mass function n(M) at any other redshift by a simple rescaling. For a given firstupcrossing distribution ,
the differential mass function can be calculated using the relation
where is the background density of the Universe.
First, we want to find an accurate fit to the firstupcrossing
distribution of a moving barrier which is given by the massdependent
linear overdensity parameter of the ellipsoidal collapse,
(see Eq. (29)). The parameter is evaluated at . We ran one million random walks and recorded the firstupcrossing values for in 100 equidistant bins in logarithmic space. The resulting distribution is nicely expressed by the function
thus, our suggested fitting formula is a mixture of the functional forms proposed by Sheth & Tormen (1999) and Sheth & Tormen (2002). The remaining bestfit parameters are A=0.357, p=0.212 and a=1.171. The result is shown in Fig. 8.
Figure 8: Comparison between the firstupcrossing distribution of the moving barrier, Eq. (32), inferred from an ensemble of one million random walks, and the fitting formula, Eq. (33). 

Open with DEXTER 
Second, to find a viable mass function from the firstupcrossing distribution, we proceed as Sheth & Tormen (1999) and Sheth et al. (2001), normalise
to unity and rescale the variable a such that we are in agreement with the standard Sheth & Tormen mass function and a mass function based on Nbody simulations proposed by Courtin et al. (2010). The latter is based on a firstupcrossing distribution that has the same functional form as that proposed by Sheth & Tormen (1999), but slightly different bestfit parameters,
with , , and . Note that in their definition of , the linear density contrast has to be taken at collapse. Normalising the firstupcrossing distribution based on the moving barrier of our ellipsoidalcollapse model to unity yields a rescaled parameter . We compare the resulting mass function with those by Sheth & Tormen (1999) and Courtin et al. (2010) for three different redshifts in Fig. 9. The parameter a was rescaled by . Deviations from the Sheth & Tormen mass function at high masses occur at large redshifts which is compatible with the Courtin et al. mass function. Overall, our proposed mass function lies in between these two, suggesting that differences between Nbody simulations and the Sheth & Tormen mass function might be due to an imprecise treatment of the ellipsoidalcollapse dynamics.
Figure 9: Comparison of the Sheth & Tormen and the Courtin et al. mass functions with the mass function derived from the rescaled upcrossing distribution, Eq. (33), based on our treatment of the ellipsoidalcollapse dynamics. 

Open with DEXTER 
4 Conclusions
We have reconsidered the collapse of a homogeneous, triaxial ellipsoid in an expanding background universe and extended the treatment of Bond & Myers (1996) in two ways:
 We have introduced a physically motivated criterion for the onset of virialisation along each principal axis of a collapsing ellipsoid. We derive this criterion from the tensor virial theorem, demanding that the inertial tensor stabilises and its second time derivative vanishes. This approach is a generalisation of the usual virialisation condition for the sphericalcollapse scenario and thus fully consistent with it. It replaces the conventional requirement that virialisation is assumed along an axis i when the dimensionless semimajor axis a_{i} of the ellipsoid along that axis reaches a_{i}(a)=0.177a.
 We have introduced a hybrid model for the influence of the external gravitational shear field acting on the ellipsoid, supplementing the two different models of Bond & Myers (1996). The principal axes of the shear field evolve with those of the halo before it turns around and decouples from the background expansion, and then continues to evolve linearly while the halo evolves nonlinearly and collapses. We have shown that the differences between the hybrid and the nonlinear model are relatively small.
Our main results are as follows:
 The collapse parameters and depend only weakly on the model for the external gravitational shear. The hybrid model and the nonlinear model by Bond & Myers (1996) give approximately the same results.
 When supplied with our virialisation condition derived from the tensor virial theorem, the ellipsoidalcollapse model returns values and that differ substantially from those obtained with the sphericalcollapse model. Depending on halo mass and redshift, deviations of order are common. After marginalisation over e and p, and increase with decreasing halo mass and with decreasing virialisation redshift.
 Both parameters increase with increasing initial ellipticity e and decrease with increasing prolaticity p, as already suggested by Sheth et al. (2001).
 Our results for qualitatively confirm the dependence on halo mass and virialisation redshift given by a fitting formula by Sheth et al. (2001). Deviations in particular at low mass and high redshift occur due to the differences in the virialisation condition, the model for the external shear, and the marginalisation over e.
 The mass function based on our refined treatment of the ellipsoidalcollapse model is in good agreement with those proposed by Sheth & Tormen (1999) for low redshifts and Courtin et al. (2010) for high redshifts. This suggests that differences between the Sheth & Tormen mass function and results from Nbody simulations at large redshifts may occur due to an imprecise treatment of the ellipsoidalcollapse dynamics.
C.A. wants to thank Francesco Pace for fruitful discussions, and the Deutsche Forschungsgemeinschaft for financial support under grant number BA 1369/121, the Heidelberg Graduate School of Fundamental Physics, and the IMPRS for Astronomy & Cosmic Physics at the University of Heidelberg.
Appendix A: Comparison to a previous study
Here we compare our results for the evolution equations of the collapsing ellipsoid with those presented by Monaco (1997) for a flat CDM and an OCDM model.
Starting from Eq. (5), we can replace E(a) and E'(a) for a flat CDM model using
since and the curvature parameter . This gives
This equation differs from Eq. (B11) of Monaco (1997) in the second and third term: a factor a^{2} in the denominator of both terms was omitted. Additionally, the vacuum term as well as a factor 3 were not included in the nominator of the third term.
For an OCDM model, we have
since and . Inserting this again into Eq. (5) yields
Equation (B12) of Monaco (1997) is again slightly different: the factor 3 in the last term was omitted.
There is one last difference concerning the initial conditions: Comparing Eq. (B17) of Monaco (1997) with Eq. (6) of this paper, one can find an additional factor a_{0} in front of which should be dropped.
References
 Angrick, C., & Bartelmann, M. 2009, A&A, 494, 461 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Bardeen, J. M., Bond, J. R., Kaiser, N., & Szalay, A. S. 1986, ApJ, 304, 15 [NASA ADS] [CrossRef] [Google Scholar]
 Bartelmann, M., Ehlers, J., & Schneider, P. 1993, A&A, 280, 351 [NASA ADS] [Google Scholar]
 Bartelmann, M., Doran, M., & Wetterich, C. 2006, A&A, 454, 27 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Binney, J., & Tremaine, S. 1987, Galactic dynamics, ed. J. P. Ostriker, Princeton Series in Astrophysics (Princeton, NJ: Princeton University Press) [Google Scholar]
 Bond, J. R., & Myers, S. T. 1996, ApJS, 103, 1 [NASA ADS] [CrossRef] [Google Scholar]
 Bond, J. R., Cole, S., Efstathiou, G., & Kaiser, N. 1991, ApJ, 379, 440 [NASA ADS] [CrossRef] [Google Scholar]
 Courtin, J., Rasera, Y., Alimi, J., et al. 2010, MNRAS, submitted,[arXiv:1001.3425] [Google Scholar]
 Doroshkevich, A. G. 1970, Astrophysics, 6, 320 [NASA ADS] [CrossRef] [Google Scholar]
 Eisenstein, D. J., & Loeb, A. 1995, ApJ, 439, 520 [NASA ADS] [CrossRef] [Google Scholar]
 Engineer, S., Kanekar, N., & Padmanabhan, T. 2000, MNRAS, 314, 279 [NASA ADS] [CrossRef] [Google Scholar]
 Lacey, C., & Cole, S. 1993, MNRAS, 262, 627 [NASA ADS] [CrossRef] [Google Scholar]
 Lahav, O., Lilje, P. B., Primack, J. R., & Rees, M. J. 1991, MNRAS, 251, 128 [NASA ADS] [CrossRef] [Google Scholar]
 Lee, S., & Ng, K. 2009, [arXiv:0910.0126] [Google Scholar]
 Monaco, P. 1995, ApJ, 447, 23 [NASA ADS] [CrossRef] [Google Scholar]
 Monaco, P. 1997, MNRAS, 287, 753 [NASA ADS] [Google Scholar]
 Monaco, P. 1998, Fundamentals of Cosmic Physics, 19, 157 [NASA ADS] [Google Scholar]
 Mota, D. F., & van de Bruck, C. 2004, A&A, 421, 71 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Ohta, Y., Kayo, I., & Taruya, A. 2004, ApJ, 608, 647 [NASA ADS] [CrossRef] [Google Scholar]
 Press, W. H., & Schechter, P. 1974, ApJ, 187, 425 [NASA ADS] [CrossRef] [Google Scholar]
 Schäfer, B. M., & Koyama, K. 2008, MNRAS, 385, 411 [NASA ADS] [CrossRef] [Google Scholar]
 Sheth, R. K., & Tormen, G. 1999, MNRAS, 308, 119 [NASA ADS] [CrossRef] [Google Scholar]
 Sheth, R. K., & Tormen, G. 2002, MNRAS, 329, 61 [NASA ADS] [CrossRef] [Google Scholar]
 Sheth, R. K., Mo, H. J., & Tormen, G. 2001, MNRAS, 323, 1 [NASA ADS] [CrossRef] [Google Scholar]
 Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Wang, L., & Steinhardt, P. J. 1998, ApJ, 508, 483 [NASA ADS] [CrossRef] [Google Scholar]
All Figures
Figure 1: Comparison of the virialisation conditions from the scalar and tensor virial theorems. The curves differ slightly because they refer to different times. While the tensor virial theorem was evaluated at virialisation, the scaler virial theorem was evaluated at collapse. If we choose the collapse of the darkmatter halo as reference for z, both yield exactly the same result. 

Open with DEXTER  
In the text 
Figure 2: Evolution of the principal axes and the external shear for a darkmatter halo with and p=0 in the reference CDM model. Left panel: linear approximation. Central panel: nonlinear approximation. Right panel: hybrid approximation. 

Open with DEXTER  
In the text 
Figure 3: (left panels) and (right panels) for three different models of the external shear. Both quantities were calculated for a darkmatter halo in the reference CDM model. Top panels: as a function of virialisation redshift . Bottom panels: e=0.2 for all . In both cases, p=0. 

Open with DEXTER  
In the text 
Figure 4: (left panels) and (right panels) for three different cosmological models and fixed initial e and p. Top panels: spherical case (e=p=0). Bottom panels: triaxial case with e=0.2 and p=0.1. 

Open with DEXTER  
In the text 
Figure 5: (left panels) and (right panels) as a function of halo mass and virialisation redshift for three different cosmologies marginalised over e and p. Top panels: dependence on virialisation redshift for a halo with . Bottom panels: dependence on mass M for a halo with . Thin dashed lines show the reference values from spherical collapse in the EdS universe, and . 

Open with DEXTER  
In the text 
Figure 6: Comparison of the fitting formula of Sheth et al. (2001) for with the results of this work for the reference CDM model. Left panel: as a function of mass and z=0. Right panel: as a function of redshift and . 

Open with DEXTER  
In the text 
Figure 7: Influence of the initial ellipticity e and prolaticity p on the parameters (top panels) and (bottom panels) for three different cosmological models. Left panels: CDM. Central panels: OCDM. Right panels: EdS. The large circle in each figure indicates the value of the respective parameter for and p=0. The small circles indicate values for a combination of and (see Eqs. (21), (22)). For a fixed e both and grow for decreasing p. 

Open with DEXTER  
In the text 
Figure 8: Comparison between the firstupcrossing distribution of the moving barrier, Eq. (32), inferred from an ensemble of one million random walks, and the fitting formula, Eq. (33). 

Open with DEXTER  
In the text 
Figure 9: Comparison of the Sheth & Tormen and the Courtin et al. mass functions with the mass function derived from the rescaled upcrossing distribution, Eq. (33), based on our treatment of the ellipsoidalcollapse dynamics. 

Open with DEXTER  
In the text 
Copyright ESO 2010
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.