## 5. Non-linear regimeWe 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 . It is very tempting to try to extend this model into the non-linear regime up to , 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-cellsAccording to the spherical dynamics large overdensities decouple slowly from the general expansion of the universe, reach a maximum radius , turn-around and collapse to a singularity when their linear density contrast is equal to a critical value , with for (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 is taken to be one half of the turn-around radius, from arguments based on the virial theorem, so we shall note with . If we assume that after virialization this halo remains stable, its density contrast will grow as , while its "linear" density contrast increases as . Thus, we modify the function which links to so that: At the time of collapse where the linear density contrast is equal to , the actual density contrast of the halo is . Within the framework of stable clustering, a similar relation can be obtained for (see VS): where is related to 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 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): so that the probability distribution of the density contrast satisfies the scaling-law: since in the regime we consider here () we
have . The variable while the scaling function is simply: 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
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
of matter embedded within just-virialized objects of mass larger than
Here is the fraction of matter computed in
the linear gaussian field, and we used the fact that in our spherical
model a density contrast over a scale
in the initial gaussian random field is
associated to a density contrast over a scale
Since we have we recover as we should the PS mass function: 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 while this
integral should be normalized to unity. This implies (at least) that
the approximate we obtained cannot be used for
any ## 5.1.2. Density profile of virialized halosWe can notice that an overdensity with an initial density profile
which is exactly given by the mean value
(Sect. 4) leads to a final virialized halo with a flat slope in
its inner parts, since the initial density contrast
converges to a finite value at the center
( in the case ). This is
not consistent with the results obtained from numerical simulations
which find inner density profiles (Navarro et
al.1996; Navarro et al.1997; Tormen et al.1997) or even steeper (Moore
et al.1998). However, for moderate values of
the correlation between the density at scale which leads to a final density profile for the virialized halo: where ## 5.2. Underdensities## 5.2.1. Density profile of underdensitiesThe function obtained in the previous
section (44) seems to show that the exponent of the small for low density regions (), where is the slope of the two points correlation function in the highly
non-linear regime and Hence we have: At late times , and using (3) we obtain: which means that . If we identify this exponent with we obtain eventually: 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 (the profile of the initial density contrast is less steep than ). As we could expect from the analysis we developed in the linear regime, we can note that the case leads to some problems since we would have . Hence it cannot be described by this simple model. We shall only consider , which corresponds to hierarchical clustering and cosmologically relevant power-spectra (but one may expect that on very large scales ). ## 5.2.2. Contact of underdensitiesWe 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 for 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: The linear variable defines the underdensities as it is a constant of the dynamics. We can see from (57) that a logarithmic interval of of order unity will occupy all the volume of the universe when Naturally, we cannot keep our model unchanged for later times,
since the mean volume of these underdensities should not grow faster
than 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 with (that is when they fill all of the
universe) and keep after this date a constant radius hence we obtain: Note that the scale since by definition, where where is defined by (56), This scaling (42) of applies to density contrasts larger than the one of underdense regions which are currently on the verge of filling the entire universe: with: hence 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
as long as one remains in the non-linear regime ( that is ). The "overdensity" 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 modelWe 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 , since in this range we should
count roughly spherical virialized halos which have a radius larger
than the considered scale 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 This range in We shall follow some of the notations used by Vergassola et
al.(1994) and we introduce the reduced velocity where is the peculiar velocity. We shall only consider the cases where the initial velocity () has homogeneous increments and verifies the scale-invariance: while the initial velocity potential satisfies: with . Here, means "having the same statistical properties". Then, the overall density over the cell is simply: where the point satisfies: 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 (which is simply the Zeldovich dynamics). Using (73) we can check that the density over a cell of size satisfies: so that one only needs to consider the time
. The highly non-linear regime which is of
interest to us here corresponds to or
. Numerical simulations and theoretical
arguments (She et al.1992; Vergassola et al.1994) strongly suggest
that the Lagrangian map forms a Devil's
staircase and that shock locations are dense in Eulerian space (for
), which can be proved rigorously for
(Sinai 1992). Hence for almost every Eulerian
coordinate In the limits and we have from (72) the behaviour . Since the second term in (78) becomes negligible, so that we expect: where is the most probable value of (note however that by definition the mean value of is simply ). Using (77) we obtain: 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 , centered on the origin, in an universe invariant by transverse translations, can be written: where corresponds to the Hubble expansion of the matter element: and as (on the other hand the comoving transverse coordinates remain constant in time). Thus, is the comoving coordinate of the outer front which we normalized so that for . We can write (81) as: The solution of this equation is simply related to the initial conditions by: where is the linear theory density contrast. Indeed, we have . Thus, we obtain: , so that at late times: The linear parameter is still given by where now so that: Hence according to the model described in the previous section underdensities stop expanding and fill the entire universe when they reach the density contrast such that: where we omitted logarithmic corrections and we used (69). Here,
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 (while it occurs at for both models in the 1-dimensional case): indeed the previous arguments would give instead of (56) which leads to lower characteristic densities . 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 of a given particle evolves as: which means that for a spherical underdensity we obtain at late times while the spherical dynamics leads to , see (55). Hence the Zeldovich approximation overestimates the expansion of voids which explains the low and the high 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 correctionsAs we noticed above, we expect the spherical dynamics to describe
extreme events . Moreover, in the same way as
the "cloud-in-cloud" problem is not very important for large
overdensities , as noticed in VS, which is
necessary for the approach developed in Sect. 5.1, the
corresponding "void-in-void" problem disappears for
(since the gaussian density field is symmetric
under ) which allows one to use the
prescription presented in 5.2.2. However, we can see from (65) that
large negative values of correspond to very
small For the 1-dimensional growth, (84) implies that we have where we do not consider logarithmic corrections (i.e. ). Using and we obtain on scale (the smallest scale of the underdense regions). If we write this exponent as we get: In a similar fashion, a two-dimensional expansion leads to and: We compare on Table 3 the values obtained in numerical simulations
to these estimations. Thus, although we recover the increase of
with
## 5.3. General pictureThus, 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 and
. 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
and . The behaviour of the intermediate fluid
elements 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
around from its
exponential cutoff to its power-law tail. Note that in the regime
, low density contrast regions
are not described by linear theory, even
though 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 . It would also be of
interest to follow the evolution of initial extreme matter elements
. 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 becomes small virialized structures no longer form as the linear growth factor 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 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 . On larger scales, fluctuations will never reach the non-linear regime since tends to a constant as , which is nearly reached as soon as . 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 ) within the framework of the approximation developed in this article. © European Southern Observatory (ESO) 1998 Online publication: August 27, 1998 |