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Astron. Astrophys. 337, 655-670 (1998)

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5. Non-linear regime

We showed in the previous paragraphs that a very simple model, based on the spherical dynamics, can describe the evolution of the density field up to [FORMULA]. It is very tempting to try to extend this model into the non-linear regime up to [FORMULA], which is sufficient to describe approximately the subsequent highly non-linear regime with the help of the stable-clustering ansatz (e.g. Peebles 1980 for a description of this latter approximation). However, this implies that we take into account other processes like virialization, which can only be made in a very crude way within the framework of the prescription described in the previous paragraphs, so that we cannot hope to get accurate quantitative results. In fact, this highly non-linear regime where the probability distribution of the density contrast is governed by the properties of virialized objects is probably beyond the reach of rigorous perturbative methods based on the expansion of the equations of motion given by a fluid description with an irrotational velocity field. Indeed, the fluid approximation itself breaks down after shell-crossing and cannot describe collapsing halos, so that one has to use the Liouville equation which makes the analysis more difficult. Hence it may still be worthwile to consider simple models like the one described in this article which could give some hindsight into the relevant processes.

5.1. Spherical collapse

5.1.1. Counts-in-cells

According to the spherical dynamics large overdensities decouple slowly from the general expansion of the universe, reach a maximum radius [FORMULA], turn-around and collapse to a singularity when their linear density contrast [FORMULA] is equal to a critical value [FORMULA], with [FORMULA] for [FORMULA] (we shall only consider the case of a critical universe in the following). However, one usually assumes that such an overdensity will eventually virialize (because the trajectories of particles are not purely radial) into a finite radius to form a relaxed halo. Generally, this virialization radius [FORMULA] is taken to be one half of the turn-around radius, from arguments based on the virial theorem, so we shall note [FORMULA] with [FORMULA]. If we assume that after virialization this halo remains stable, its density contrast will grow as [FORMULA], while its "linear" density contrast increases as [FORMULA]. Thus, we modify the function [FORMULA] which links [FORMULA] to [FORMULA] so that:

[EQUATION]

At the time of collapse where the linear density contrast [FORMULA] is equal to [FORMULA], the actual density contrast of the halo is [FORMULA]. Within the framework of stable clustering, a similar relation can be obtained for [FORMULA] (see VS):

[EQUATION]

where [FORMULA] is related to R in a fashion similar to [FORMULA]

[EQUATION]

which comes from the fact that these approximations are based on a Lagrangian point of view where one follows the evolution of matter elements of constant mass. One can expect [FORMULA] since all regions of space do not follow the same dynamics, and in fact the previous model only applies to overdensities, which form one half of the volume and mass in the early linear universe. Then, in a fashion similar to what was done in VS, we obtain using (16):

[EQUATION]

[EQUATION]

so that the probability distribution of the density contrast satisfies the scaling-law:

[EQUATION]

since in the regime we consider here ([FORMULA]) we have [FORMULA]. The variable x is related to the usual linear parameter [FORMULA] by:

[EQUATION]

while the scaling function [FORMULA] is simply:

[EQUATION]

In fact, this scaling is characteristic of a much wider class of models, defined by the scale-invariance of the many-body correlation functions, studied in detail by Balian & Schaeffer (1989). Thus, the model we described in this article, based on the spherical dynamics and a strong stable clustering assumption, appears as a simple way to estimate the scaling function [FORMULA] characteristic of the highly non-linear regime. Then, all the analysis developed for this general class of density fields can be applied to this peculiar model. As was noticed in VS, the approach presented in this article allows one to recover the PS mass function as it is based on the same fundamental idea: one follows the evolution of individual matter elements from the linear regime, described by gaussian probability distributions, into the non-linear regime. Indeed, if in a fashion similar to PS we identify the fraction [FORMULA] of matter embedded within just-virialized objects of mass larger than M with the mass contained in cells of scale R with a density contrast [FORMULA] (with [FORMULA] and [FORMULA]) we obtain:

[EQUATION]

Here [FORMULA] is the fraction of matter computed in the linear gaussian field, and we used the fact that in our spherical model a density contrast [FORMULA] over a scale [FORMULA] in the initial gaussian random field is associated to a density contrast [FORMULA] over a scale R in the actual non-linear density field as described in (13) and (15), as in the PS prescription. Then, the mass fraction in collapsed objects of mass M to [FORMULA] is simply:

[EQUATION]

Since we have [FORMULA] we recover as we should the PS mass function:

[EQUATION]

However, (41) is not strictly equivalent to the usual PS prescription, since in addition to the spherical model it also relies on stable clustering. One should note that we did not multiply our probability distribution by the usual factor 2, so that we only recover one half of the total matter content of the universe since the previous arguments only apply to initial overdensities. Hence the scaling function (44) verifies [FORMULA] while this integral should be normalized to unity. This implies (at least) that the approximate [FORMULA] we obtained cannot be used for any x. In fact, following the discussion of Sect. 4, we expect the spherical dynamics model we used so far to be valid only for extreme events [FORMULA]. Hence the approximate [FORMULA] and [FORMULA] we got should only apply to [FORMULA]. The normalization problem of the PS mass function is often "cured" by an overall multiplication by a mere factor 2, "justified" by the excursion set approach in the case of a top-hat in k (Cole 1989; Bond et al.1991). However, this result does not extend to other window functions (like the top-hat in real space used here) with which for large overdensities [FORMULA] the PS mass function does not suffer from the cloud-in-cloud problem in the sense that the multiplicative factor needed to correct for double counting goes to 1 (and not 2) for large masses (Bond et al.1991; Peacock & Heavens 1990; VS). Hence, it appears that one should not multiply (44) by 2, but merely restrict its application to [FORMULA]. Moreover, we can note that this normalization problem is closely related to the behaviour of underdensities, which are not well described in the usual PS approach and constitute the missing half of the matter content of the universe.

5.1.2. Density profile of virialized halos

We can notice that an overdensity with an initial density profile which is exactly given by the mean value [FORMULA] (Sect. 4) leads to a final virialized halo with a flat slope in its inner parts, since the initial density contrast [FORMULA] converges to a finite value at the center ([FORMULA] in the case [FORMULA]). This is not consistent with the results obtained from numerical simulations which find inner density profiles [FORMULA] (Navarro et al.1996; Navarro et al.1997; Tormen et al.1997) or even steeper (Moore et al.1998). However, for moderate values of [FORMULA] the correlation between the density at scale R and the density at a smaller scale [FORMULA] enclosed within the former one quickly weakens as the fluctuation [FORMULA] becomes larger than [FORMULA] ([FORMULA] diverges for small [FORMULA] while [FORMULA] remains finite). As a consequence one cannot infer the average density profile of virialized halos from [FORMULA]. In fact, it seems more reasonable to consider an initial density profile of the form:

[EQUATION]

which leads to a final density profile for the virialized halo:

[EQUATION]

where x (resp. R) is a comoving (resp. physical) coordinate. The reasoning below (48) is that if we look at a virialized halo of mass M, its characteristic density within a smaller sphere of mass [FORMULA] (centered on the peak) will be set by the maximum linear density contrast [FORMULA] realised over all spheres of mass [FORMULA] enclosed within the larger matter element M. The value of this maximum [FORMULA] will scale with [FORMULA] as [FORMULA], as soon as [FORMULA], see (37), which leads to (48). This picture also assumes that during virialization new collapsing shells which may not be centered on the density peak will roughly circularize around it. We can note that for [FORMULA] the slope (49) is [FORMULA] which is close to what is seen in simulations (Navarro et al.1997; Moore et al.1998a). The same reasoning could also be applied to underdensities, which we shall use in the next section. However, the previous arguments are quite crude and a much more detailed analysis would be required to get a good description of virialized halos. Moreover, within the approach described in the previous section the shape of the mean density profile has a rather weak meaning since the model implies that a lot of substructure is present within halos, so that a large object can be decomposed as a hierarchy of many smaller peaks with larger densities. Note that some simulations (Ghigna et al.1998) seem indeed to show that many sub-halos can survive within larger objects although not to such a large extent as in the model (this might be due to finite resolution effects).

5.2. Underdensities

5.2.1. Density profile of underdensities

The function [FORMULA] obtained in the previous section (44) seems to show that the exponent of the small x power-law tail is [FORMULA]. However, as we argued above we do not expect this scaling function to give reasonable results for [FORMULA], so that we need to get [FORMULA] from another point of view, which considers explicitely low-density regions. Indeed, since we expect the spherical dynamics approximation to be valid mainly for rare events we shall shift from [FORMULA] to [FORMULA]: that is we now study very underdense areas. Moreover, initially non-spherical underdensities tend to become increasingly spherical as they expand (contrary to the collapse which enhances deviations from spherical symmetry), as seen in Bertschinger (1985), so that a spherical dynamics model could give reasonable results. We shall assume in this paragraph that the many-body correlation functions are scale-invariant, so that the density field is described by the scaling function [FORMULA] in the non-linear regime [FORMULA] for [FORMULA] which includes very underdense and small x regions. Then, if we consider a sphere of radius R with a density contrast [FORMULA] over R, the mean density contrast [FORMULA] on its outer shell can be shown to be:

[EQUATION]

for low density regions ([FORMULA]), where

[EQUATION]

is the slope of the two points correlation function in the highly non-linear regime and n is the slope of the initial power-spectrum [FORMULA] which we assume here to be a power-law. This means that the density profile is locally [FORMULA]. Thus, to get [FORMULA] we may consider the evolution of the density profile of a typical spherical underdensity using (50). Let us follow an underdensity with an initial profile in the early universe [FORMULA] (see (48)). Its dynamics is simply given by the spherical model:

[EQUATION]

and

[EQUATION]

Hence we have:

[EQUATION]

At late times [FORMULA], and using (3) we obtain:

[EQUATION]

which means that [FORMULA]. If we identify this exponent with [FORMULA] we obtain eventually:

[EQUATION]

which is also the value one would have inferred from (44). We can note that although the initial density is lower in the central regions of the perturbation, which expand faster, there is no shell-crossing and the density profile is given by (55) for [FORMULA] (the profile of the initial density contrast is less steep than [FORMULA]). As we could expect from the analysis we developed in the linear regime, we can note that the case [FORMULA] leads to some problems since we would have [FORMULA]. Hence it cannot be described by this simple model. We shall only consider [FORMULA], which corresponds to hierarchical clustering and cosmologically relevant power-spectra (but one may expect that [FORMULA] on very large scales [FORMULA]).

5.2.2. Contact of underdensities

We shall now develop a slightly more detailed description of the fate of underdensities, in order to follow the behaviour of the matter (one half of the total mass) which was initially embedded in these low-density areas. This means that we have to modify the function [FORMULA] for [FORMULA] too. As we noticed in Sect. 2, underdensities grow very fast according to the spherical dynamics which leads to an approximate probability distribution of the density contrast with an ever increasing normalization (instead of unity). In fact, the volume formed by any given range of underdensities will eventually outgrow the available volume of the universe within the approximation used so far. Indeed, we can write (14) and (16) as:

[EQUATION]

with

[EQUATION]

The linear variable [FORMULA] defines the underdensities as it is a constant of the dynamics. We can see from (57) that a logarithmic interval of [FORMULA] of order unity will occupy all the volume of the universe when

[EQUATION]

Naturally, we cannot keep our model unchanged for later times, since the mean volume of these underdensities should not grow faster than [FORMULA] after this date. In fact, in this picture such a range of negative density fluctuations first expands following the spherical dynamics until neighbouring underdensities come into contact and fill the entire universe. At this time the universe appears to be constituted of low density bubbles of size R and density [FORMULA] (with [FORMULA]). The matter which was "pushed" by these "voids" to form the interface between adjacent underdensities gets squeezed and reaches high densities. Thus, we shall assume the latter build virialized structures of size [FORMULA] and density [FORMULA] (for instance they may have a density [FORMULA] times higher than the density [FORMULA] of neighbouring areas). These represent the sheets and filaments one can observe in numerical simulations (e.g. Bond et al.1996) which separate low-density regions, while the spherical overdensities described in the previous sections are the nodes which form at their intersections (note that the latter, corresponding to large positive initial density fluctuations, appear first). To see more clearly what this would imply for the probability distribution of the density contrast (and for the mass functions) we shall simply consider that underdensities defined by their linear parameter [FORMULA], and their scale R, stop expanding when:

[EQUATION]

with [FORMULA] (that is when they fill all of the universe) and keep after this date a constant radius R and density [FORMULA]. Thus, as time goes on the universe gets filled with increasingly large and underdense "bubbles" while a growing fraction of the matter progressively forms virialized structures (with a characteristic density which becomes vanishingly small as compared to the mean density). Note that within such a picture all the mass will eventually be embedded in virialized high density objects, so that the usual normalization problem is solved in a natural way. Thus, we must now use (60) to obtain a relation [FORMULA] in order to get the probability distribution of the density contrast from (57). A negative density fluctuation [FORMULA] "stops" when it reaches a density contrast [FORMULA] on a scale R such that [FORMULA], defined by (58) and (60). At late times, [FORMULA], the spherical dynamics (see (3)) leads to the approximate relation:

[EQUATION]

hence we obtain:

[EQUATION]

Note that the scale R is not the Lagrangian scale [FORMULA], and [FORMULA]. After this "stopping time" [FORMULA], the radius of the object does not evolve any longer while its density contrast increases as [FORMULA], hence initial underdensities which have already "stopped" verify:

[EQUATION]

since [FORMULA] by definition, where R is the physical radius. This is the relation [FORMULA] we needed to derive the probability distribution of the density contrast. We can note that the density contrast is now a function of both [FORMULA] and [FORMULA], contrary to the pure spherical dynamics case used previously where we had [FORMULA]. This is in fact a necessary condition to be able to get eventually all the mass of the universe within overdense virialized structures. Finally, this leads to:

[EQUATION]

where [FORMULA] is defined by (56), x by (42) and we used (39) to introduce [FORMULA]. Thus, we recover the scaling-law (42) with the same exponent [FORMULA] as previously, with logarithmic corrections. Such logarithmic terms may indeed exist, but our model is probably too crude to give a reliable estimate of their importance. As was the case for overdensities, we also obtain a relation between the linear and non-linear parameters [FORMULA] and x:

[EQUATION]

This scaling (42) of [FORMULA] applies to density contrasts larger than the one of underdense regions which are currently on the verge of filling the entire universe: [FORMULA] with:

[EQUATION]

hence

[EQUATION]

which is exactly the limit predicted by a general study of the models defined by the scale-invariance of the many-body correlation functions. Thus, the approach developed in this section "explains" in a natural way both the emergence of the scaling-law (42) for small x and its range of validity. Note that (44) derived from the behaviour of overdensities only applied to virialized objects [FORMULA], and was in fact restricted to [FORMULA] as we argued above. The previous considerations also mean that, when seen on comoving scale x at a time defined by the scale factor a, the universe appears to be covered by very underdense regions of typical density contrast [FORMULA] with (omitting logarithmic terms):

[EQUATION]

as long as one remains in the non-linear regime ([FORMULA] that is [FORMULA]). The "overdensity" [FORMULA] tends to 0 for large times or small scales as it should, while the volume occupied by most of the matter becomes increas ingly negligible.

5.2.3. Adhesion model

We shall now try to compare the approach developed above to a very different point of view: the adhesion model, in the peculiar case of 1-dimensional fluctuations (in a 3-dimensional universe). Indeed, from the picture developed in the previous paragraphs we do not expect the adhesion model to give reliable estimates for the counts-in-cells at large values of [FORMULA], since in this range we should count roughly spherical virialized halos which have a radius larger than the considered scale R: these correspond to the overdensities described by the spherical collapse seen in Sect. 5.1. Hence the finite value of the virialization radius of these objects plays a crucial role in the final probability distribution of the density contrast, which is then out of reach of the adhesion model where this radius is simply zero. However, this model could give a fairly good picture of the filaments and sheets which characterize the highly non-linear universe as the transverse thickness of these structures, smaller than their length or the radius of the neighbouring "bubbles", does not play an important role on the counts-in-cells realised at these latter scales. More precisely, we shall try to evaluate the typical density contrast seen on a given scale: this corresponds to the maximum of [FORMULA] and to the density contrast [FORMULA] of the "bubbles" which cover the universe.

We shall first consider 1-dimensional density fluctuations, since in this case the Zeldovich approximation is correct until shell-crossing occurs so that we expect the adhesion model to provide reliable results. We still define n as the index of the power spectrum [FORMULA] so that we obtain:

[EQUATION]

This range in n ensures that the density fluctuations increase at small scales while the fluctuations of the density potential grow at large scales, so that we are in the domain of the usual hierarchical clustering. We note x (resp. q) the comoving Eulerian (resp. initial Lagrangian) coordinate of particles, in this 1-dimensional problem (at early times [FORMULA]). The relation [FORMULA] is given by the adhesion model for the displacement field, and the density is simply obtained from:

[EQUATION]

We shall follow some of the notations used by Vergassola et al.(1994) and we introduce the reduced velocity v and velocity potential [FORMULA]:

[EQUATION]

where [FORMULA] is the peculiar velocity. We shall only consider the cases [FORMULA] where the initial velocity ([FORMULA]) has homogeneous increments and verifies the scale-invariance:

[EQUATION]

while the initial velocity potential satisfies:

[EQUATION]

with [FORMULA]. Here, [FORMULA] means "having the same statistical properties". Then, the overall density over the cell [FORMULA] is simply:

[EQUATION]

where the point [FORMULA] satisfies:

[EQUATION]

as given by the Hopf-Cole solution of the Burgers equation (Hopf 1950, Cole 1951) obtained from the adhesion model. If there is a shock at the Eulerian location x, several Lagrangian coordinates q correspond to the same x and we choose the smallest one which we note [FORMULA]. Then, we have:

[EQUATION]

(which is simply the Zeldovich dynamics). Using (73) we can check that the density [FORMULA] over a cell of size [FORMULA] satisfies:

[EQUATION]

so that one only needs to consider the time [FORMULA]. The highly non-linear regime which is of interest to us here corresponds to [FORMULA] or [FORMULA]. Numerical simulations and theoretical arguments (She et al.1992; Vergassola et al.1994) strongly suggest that the Lagrangian map [FORMULA] forms a Devil's staircase and that shock locations are dense in Eulerian space (for [FORMULA]), which can be proved rigorously for [FORMULA] (Sinai 1992). Hence for almost every Eulerian coordinate x a Lagrangian coordinate [FORMULA] exists. Using (76) we can see that for two points [FORMULA] and [FORMULA] at time [FORMULA] we have:

[EQUATION]

In the limits [FORMULA] and [FORMULA] we have from (72) the behaviour [FORMULA]. Since [FORMULA] the second term in (78) becomes negligible, so that we expect:

[EQUATION]

where [FORMULA] is the most probable value of [FORMULA] (note however that by definition the mean value of [FORMULA] is simply [FORMULA]). Using (77) we obtain:

[EQUATION]

We shall now consider the approach based on the spherical dynamics we described in the previous sections, applied to this 1-dimensional problem in order to compare its prediction to (80). The equation of motion of a 1-dimensional density fluctuation, of longitudinal size [FORMULA], centered on the origin, in an universe invariant by transverse translations, can be written:

[EQUATION]

where [FORMULA] corresponds to the Hubble expansion of the matter element: [FORMULA] and [FORMULA] as [FORMULA] (on the other hand the comoving transverse coordinates remain constant in time). Thus, [FORMULA] is the comoving coordinate of the outer front which we normalized so that [FORMULA] for [FORMULA]. We can write (81) as:

[EQUATION]

The solution of this equation is simply related to the initial conditions by:

[EQUATION]

where [FORMULA] is the linear theory density contrast. Indeed, we have [FORMULA]. Thus, we obtain: [FORMULA], so that at late times:

[EQUATION]

The linear parameter [FORMULA] is still given by [FORMULA] where now [FORMULA] so that:

[EQUATION]

Hence according to the model described in the previous section underdensities stop expanding and fill the entire universe when they reach the density contrast [FORMULA] such that:

[EQUATION]

where we omitted logarithmic corrections and we used (69). Here, x is the considered comoving scale we noted [FORMULA] in (80). Thus we recover exactly the behaviour seen above within the framework of the adhesion model (80). This is due to two effects: i) in such a 1-dimensional problem the Zeldovich approximation gives the exact dynamics until a shock appears, ii) the formation of these large underdensities corresponds to peaks of the initial velocity potential which are global maxima over a large scale but all particles located in the final broad low-density areas come from a small Lagrangian region (the peak of [FORMULA]) so that inner properties are given by local characteristics and the spherical model is reasonable (the stop of the expansion at [FORMULA] models the global constraints, related to the fact that these peaks are only maxima over a finite scale). In fact, both models lead to very similar pictures: most of the universe is filled by the expansion of initial low-density peaks while most of the mass is squeezed between this underdensities in virialized objects (for the spherical model) or infinitesimally thin shocks (in the adhesion model).

For the usual 3-dimensional case, the adhesion model does not lead to the same results as the spherical prescription and there would be a break at [FORMULA] (while it occurs at [FORMULA] for both models in the 1-dimensional case): indeed the previous arguments would give [FORMULA] instead of (56) which leads to lower characteristic densities [FORMULA]. We think this discrepancy could be due to the fact that the Zeldovich approximation does not give the exact dynamics any more, even before a shock forms, so that the shape and size of structures built at late times is not correctly described. Indeed, within this approximation the physical coordinate [FORMULA] of a given particle evolves as:

[EQUATION]

which means that for a spherical underdensity we obtain at late times [FORMULA] while the spherical dynamics leads to [FORMULA], see (55). Hence the Zeldovich approximation overestimates the expansion of voids which explains the low [FORMULA] and the high [FORMULA] it gives. Thus, the adhesion model appears to confirm our spherical prescription for 1-dimensional fluctuations, where the former has a rather firm foundation, which builds confidence in our model which predicts moreover scaling-laws which are actually seen in numerical simulations. For the generic 3-dimensional case, this latter result suggests that our prescription is still valid, while the adhesion model should worsen.

5.2.4. Non-spherical corrections

As we noticed above, we expect the spherical dynamics to describe extreme events [FORMULA]. Moreover, in the same way as the "cloud-in-cloud" problem is not very important for large overdensities [FORMULA], as noticed in VS, which is necessary for the approach developed in Sect. 5.1, the corresponding "void-in-void" problem disappears for [FORMULA] (since the gaussian density field is symmetric under [FORMULA]) which allows one to use the prescription presented in 5.2.2. However, we can see from (65) that large negative values of [FORMULA] correspond to very small x. Indeed, with [FORMULA] we can check that [FORMULA] leads to [FORMULA], [FORMULA] and [FORMULA]. Thus, the range of x which can be studied in current numerical simulations [FORMULA] may be too small to recover the power-law tail with exponent [FORMULA] and the exponential cutoff predicted by our approach. Hence, non-spherical corrections may change the value of [FORMULA] obtained in numerical simulations. To get an idea of the magnitude and direction of such effects, we can study the case where underdensities only expand along 1 or 2 directions (planar or cylindrical symmetry) while the other axis remain(s) constant in comoving coordinates. Thus the 1-dimensional problem considered in the previous section corresponds to the expansion along only one direction, while the spherical model represents a growth along all three axis.

For the 1-dimensional growth, (84) implies that we have [FORMULA] where we do not consider logarithmic corrections (i.e. [FORMULA]). Using [FORMULA] and [FORMULA] we obtain [FORMULA] on scale [FORMULA] (the smallest scale of the underdense regions). If we write this exponent as [FORMULA] we get:

[EQUATION]

In a similar fashion, a two-dimensional expansion leads to [FORMULA] and:

[EQUATION]

We compare on Table 3 the values obtained in numerical simulations to these estimations. Thus, although we recover the increase of [FORMULA] with n, non-spherical corrections appear to be non-negligible. Hence we expect the value of [FORMULA] measured in numerical simulations, which is presently close to the 2-D result, to increase somewhat for smaller x at smaller scales where [FORMULA] is larger, and to get closer to the value (56) obtained from the spherical dynamics model.


[TABLE]

Table 1. Exponent [FORMULA] for various indexes n of the power-spectrum. The lines 1D, 2D and 3D corresponds to (88), (89) and (56). The last two lines present the results of numerical simulations: Colombi et al.(1997) for A and Munshi et al.(1998) for B.


5.3. General picture

Thus, in the non-linear regime virialized objects should form through two different processes according to our model. First, large overdensities with a roughly spherical shape collapse as in the PS approach to build high-density virialized halos. This corresponds to matter elements described by [FORMULA] and [FORMULA]. Second, large underdensities expand until they fill the entire universe and see their dynamics influenced by their interaction with neigbhouring "voids". The matter "pushed" by these regions forms high-density filaments and sheets at their interface, which builds virialized structures of increasingly large scale and low density (increasingly lower than the mean density of the universe). Obviously the dynamics of the filaments and walls cannot be described by a spherical model, but our approach takes advantage of the fact that the low-density "bubbles" they surround may still be described by a spherical dynamics, with the addition of other processes to take into account the global constraints which stop their expansion. This models matter elements with [FORMULA] and [FORMULA]. The behaviour of the intermediate fluid elements [FORMULA] is certainly quite complex and depends on non-local properties through the influence of neigbhouring peaks and voids. It is not described by our model and corresponds to the transition interval of the scaling function [FORMULA] around [FORMULA] from its exponential cutoff to its power-law tail. Note that in the regime [FORMULA], low density contrast regions [FORMULA] are not described by linear theory, even though [FORMULA] is small, because of shell-crossing. Thus, our model does not provide as accurate predictions as the PS mass function, but it can be tested in numerical simulations by studying the density profiles of large halos or voids, as well as the asymptotic behaviour of [FORMULA]. It would also be of interest to follow the evolution of initial extreme matter elements [FORMULA]. Moreover, we think the explicit description of underdensities is a necessary step, which was not considered in detail in the PS formulation. It ensures in a natural way that all the mass will eventually be embedded within high-density virialized structures (which occupy a negligible volume), so that there is no normalization problem, and it allows to describe the complex structure formed by low-density bubbles, filaments and walls, which is seen in numerical simulations (Cole et al.1997; Weinberg et al.1996; Bond et al.1996). One can note that according to our model, substructures should exist within large overdensities, which is not always seen in numerical simulations. However, as Klypin et al.(1998) argue this may be due to a lack of numerical resolution, moreover substructures do appear through counts-in-cells numerical studies.

In the non-linear regime, we have only considered the case of a critical universe so far. However, in a low-density universe when [FORMULA] becomes small virialized structures no longer form as the linear growth factor [FORMULA] tends to a constant: the density perturbations freeze in comoving coordinate. Thus, on small scales where the density fluctuations are large, structures formed early when [FORMULA] so that the analysis developed above for a critical universe can be applied. To get the characteristics of virialized structures today at these scales one simply needs to consider these early times and then extrapolate until today using the approximation (which was used throughout above) that virialized structures keep a constant scale and density while the mean density of the universe decreases as [FORMULA]. On larger scales, fluctuations will never reach the non-linear regime since [FORMULA] tends to a constant as [FORMULA], which is nearly reached as soon as [FORMULA]. Hence, on these scales one can simply use the quasi-linear description developed in Sect. 2.2. Thus, one gets a complete picture of the density field in a low-density universe (except for a transitory range where [FORMULA]) within the framework of the approximation developed in this article.

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Online publication: August 27, 1998
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