A&A 382, 431449 (2002)
DOI: 10.1051/00046361:20011675
Dynamics of gravitational clustering
III. The quasilinear regime for some nonGaussian initial conditions
P. Valageas
Service de Physique Théorique, CEN Saclay, 91191 GifsurYvette, France
Received 11 July 2001 / Accepted 23 November 2001
Abstract
Using a nonperturbative method developed in a previous work (Paper II), we derive the probability distribution
of the density contrast within spherical cells in the quasilinear regime for some nonGaussian initial conditions. We describe three such models. The first one is a straightforward generalization of the Gaussian scenario. It can be seen as a phenomenological description of a density field where the tails of the linear density contrast distribution would be of the form
,
where
is no longer restricted to 2 (as in the Gaussian case). We derive exact results for
in the quasilinear limit. The second model is a physically motivated isocurvature CDM scenario. Our approach needs to be adapted to this specific case and in order to get convenient analytical results we introduce a simple approximation (which is not related to the gravitational dynamics but to the initial conditions). Then, we find a good agreement with the available results from numerical simulations for the pdf of the linear density contrast for
.
We can expect a similar accuracy for the nonlinear pdf
.
Finally, the third model corresponds to the small deviations from Gaussianity which arise in standard slowroll inflation. We obtain exact results for the pdf of the density field in the quasilinear limit, to firstorder over the primordial deviations from Gaussianity.
Key words: cosmology: theory  largescale structure of Universe
In usual cosmological scenarios, largescale structures in the universe are generated by the growth of small primordial density perturbations, through gravitational instability. At large scales or at early times one can use a perturbative approach to obtain the first few moments of the evolved density field. Moreover, as described in Bernardeau (1992) and Bernardeau (1994a) one can actually sum up the perturbative series at leading order in the limit
(where
is the rms density fluctuation) to obtain all order cumulants. This yields the probability distribution function (pdf)
of the density contrast within spherical cells. However, this method only applies to Gaussian initial conditions. Moreover, it may miss some nonperturbative effects.
In a previous paper (Paper II) we developed a nonperturbative method to derive the pdf
in this quasilinear regime. It is based on a steepestdescent approximation which yields asymptotically exact results in this limit. This allowed us to justify the results obtained by perturbative means and to correct some misconceptions related to nonperturbative effects. Another advantage of our approach is that in principle it can also be applied to nonGaussian primordial density fluctuations. However, the determination of the relevant saddlepoints needed in order to get simple analytic results may then be more difficult. In particular, the method may need to be adapted to specific cases.
Thus, in this article we show in details how to obtain the pdf
for three specific nonGaussian models, in the quasilinear regime. Indeed, although observations are consistent with Gaussian initial conditions so far, it is of interest to investigate a broader class of models until any nonGaussianity is definitely ruled out. Moreover, there exist some reasonable physical scenarios (though somewhat more contrived than the standard CDM model) which give rise to nonGaussian primordial fluctuations. Finally, even standard inflationary scenarios lead to small deviations from Gaussianity for the primordial density fluctuations.
This article is organized as follows. First, in Sect. 2 we recall the pathintegral formulation which allows one to write an explicit expression for the pdf
in terms of initial conditions. Then, in Sect. 3 we apply our method to a nonGaussian model which is a straightforward generalization of the Gaussian case and could be used as a phenomenological tool to reproduce observations in case some departure from Gaussianity would be measured (e.g., in the CMB data). Then, in Sect. 4 we investigate a physically motivated model which describes an isocurvature cold dark matter scenario (Peebles 1999a). In particular, we obtain a good agreement with the available results from numerical simulations for the pdf of the linearly evolved density field. Finally, in Sect. 5 we consider the small primordial deviations from Gaussianity which arise in standard slowroll inflation.
2 Generating functions. Gaussian initial conditions
In this article, we investigate the probability distribution function (pdf)
of the density contrast
within a spherical cell of comoving radius R, volume V:

(1) 
Here
is the nonlinear density contrast at the comoving coordinate ,
at the time of interest. In this section, following the method developed in Paper II, we recall how one can express the pdf
as a function of the initial conditions through a pathintegral formalism. In order to introduce our approach we first briefly consider the case of Gaussian initial conditions which is most familiar. In the next section, we extend our formalism to a nonGaussian model which can be seen as the simplest generalization of the usual Gaussian case.
Rather than trying to directly evaluate the pdf itself, it is actually more convenient to study its Laplace transform
given by:

(2) 
Here, the symbol
expresses the average over the initial conditions. Then, the last term in Eq. (2) can also be seen as the definition of the pdf
.
Indeed, the pdf can be recovered from
through the standard inverse Laplace transform:

(3) 
Moreover, the generating function
also yields the moments
through:

(4) 
Then, we need to compute
as an average over the initial conditions, using the first equality in Eq. (2). To do so, one simply needs two pieces of information. First, we must know the functional
which yields the exact nonlinear density contrast
over the cell V which arises from the gravitational dynamics of the linear density field
.
Indeed, as shown for instance in Paper I the initial conditions can be defined by the linear density contrast
where we only keep the linear growing mode. This does not assume that the exact nonlinear density field
can be written as a series expansion over powers of the linear field
.
In fact, such a series is only asymptotic (e.g., Papers I, V). Hereafter, we note
the growing mode of the linear density contrast at the time of interest, where we compute
(i.e., to simplify notations we do not write explicitly the time dependence). Second, in order to perform the average in Eq. (2) we need the weight which is associated with all possible fields
.
In other words, we must specify the probability distribution of the random field
.
Thus, the first point expresses the physics of gravitational interactions while the second point describes the initial conditions of the system.
In the case of Gaussian initial conditions, the statistics of the random field
are fully defined by the twopoint correlation:

(5) 
The kernel
is symmetric, homogeneous and isotropic:
.
It is convenient to express
in terms of the powerspectrum P(k) of the linear density fluctuations. To do so, we define the Fourier transform of the density field as:

(6) 
Next, we define the Fourier transform of the kernel
by the property:

(7) 
for any real fields f_{1} and f_{2}, where we introduced the shorthand notation:

(8) 
Using Eq. (6) this implies:

(9) 
which gives:

(10) 
where we defined the powerspectrum P(k) of the linear density contrast by:

(11) 
Finally, the inverse
of the kernel
is (see Paper II):

(12) 
which implies that
is positive definite since we have:

(13) 
where we used
for real fields
,
see Eq. (6).
Thus, for Gaussian initial conditions the statistical properties of the random field
are determined by the kernel
given in Eq. (10) (or equivalently by
given in Eq. (12)). In particular, the average over the initial conditions in Eq. (2) can be written as the pathintegral:

(14) 
This expression merely means that in order to compute the average
we simply need to sum up the contributions of all possible linear density fields
to which we associate a Gaussian weight which is proportional to
.
The normalization factor
ensures that ,
as implied by the definition (2). Here
is the determinant of the kernel
.
Thus, Eq. (14) yields an explicit expression for the Laplace transform
(though it may be difficult to obtain numerical results from such a pathintegral). This provides in turn the pdf
through the inverse transform (3). In Paper II we showed how to get the generating function
from Eq. (14) in the limit
(i.e. for small rms density fluctuations) using a steepestdescent method. In the next section, we apply this method to a closely related nonGaussian model.
3 Generalization of the Gaussian weight
In this section, we describe a nonGaussian model which can be seen as the simplest generalization of the Gaussian scenario. This model is not derived from physical principles. It simply provides a phenomenological tool which can describe initial conditions such that the tails of the linear pdf
of the density field are of the form
,
where
is no longer restricted to 2 (as in the Gaussian case). In addition, it allows us to show on a simple example the power of the method we developed in Paper II and how it can be applied to nonGaussian initial conditions.
3.1 A nonGaussian model
In the case of Gaussian primordial density fluctuations, the average involved in the definition (2) of the generating function
(i.e. the Laplace transform of the pdf
)
is simply given by the Gaussian weight
in the pathintegral (14) which takes the average of
over all possible initial states, as we recalled in the previous section. A straightforward generalization of the Gaussian case (14) is to modify this weight. Thus, one can consider the case where the average over the field
is given by:

(15) 
with:

(16) 
where
and
.
Here
is a normalization constant, so that ,
and we still take the kernel
to be of the form (12) hence
is positive definite. We also define a quantity
in terms of
by:

(17) 
However, contrary to the Gaussian case we now have:

(18) 
The form (15) corresponds to a large density contrast tail of the form
for the linear density field.
In order to compute the Laplace transform
we simply follow the steps of the calculation developed in Paper II. Thus, we first define a new generating function
through:

(19) 
in order to factorize the amplitude of the kernel
.
This yields from Eq. (15):

(20) 
where the "action''
is given by:

(21) 
The quasilinear regime corresponds to the limit
at fixed y. Then, we see that all terms with
in the action
in Eq. (21) vanish as
since
,
see Eq. (17). Therefore, in the limit
we are left with the action
:

(22) 
with
and we normalized
such that
.
The fact that the terms with
disappear in the quasilinear limit expresses the fact that this limit is actually a "rareevent limit''. Indeed, as seen in Paper II for the Gaussian case (and this remains valid here) a finite y corresponds to a finite density contrast. Then, in the limit
these finite density contrasts (even though small) become very rare events. On the other hand, the tails of the functional distribution
defined by the weight
are governed by the highest power in Eq. (16). This implies that in the quasilinear limit the only relevant term is the highest power
.
Note that we can expect the validity of the quasilinear limit to extend to larger values of
in the case N=1 than for N>1. Indeed, in this latter case we can expect the terms
to make a significant contribution for
which is not taken into account at all by the quasilinear limit.
3.2 Steepestdescent method
The action
is independent of the normalization of the kernel
.
Thus, as in Paper II we can apply the steepestdescent method in the limit
.
Indeed, it is clear than in this limit the pathintegral (20) is dominated by the global minimum of the action
while the contributions from other points
are exponentially damped. Then, the steepestdescent method yields asymptotically exact results in this limit. We briefly describe below the main steps of this steepestdescent method. Since the derivation is very close to the one performed in Paper II for the Gaussian case we refer the reader to that paper for the details of the calculation. Note that the Gaussian case actually corresponds to .
In order to apply the steepestdescent approximation we first need to find the global minimum of the action
.
The condition which expresses that the point
is an extremum (or a saddlepoint) is:

(23) 
where
is the functional derivative with respect to
at the point .
This constraint also writes:



(24) 
Thus, we obtain exactly the same profile for the saddlepoint as in the Gaussian case. Indeed, we can see that
only modifies a numerical multiplicative factor in the r.h.s. of Eq. (24) (i.e. independent of
and
)
which could formally be absorbed into y through
.
Then, as in Paper II we obtain a spherically symmetric saddlepoint. Note that the existence of a spherical saddlepoint could be expected a priori, since the initial conditions are homogeneous and isotropic and we study the density contrast within spherical cells. Therefore, the very problem we investigate is spherically symmetric. However, a priori this spherical saddlepoint is not necessarily the global minimum of the action (in full generality it might as well be a maximum). This will need to be checked a posteriori.
The fact that we obtain a spherical saddlepoint greatly simplifies the problem since it means that this point
is described by the wellknown spherical collapse solution of the gravitational dynamics. This reads:

(25) 
where the function
is given by the usual spherical collapse solution of the equations of motion (e.g., Peebles 1980; Paper II). The second equation in (25) merely expresses the conservation of mass and the fact that the matter enclosed within the radius R in the actual nonlinear density field actually comes from a Lagrangian comoving radius ,
for a spherical initial condition. The Eq. (25) provides the functional
for spherical states
.
As shown in Paper II this is sufficient to derive the spherical saddlepoint which obeys the condition (24). Using the results of Paper II with the substitution
we eventually obtain:



(26) 
together with:

(27) 
These two equations fully define the spherically symmetric saddlepoint
.
The implicit Eq. (26) determines
while the radial profile of this initial state is given by Eq. (27). In Eq. (27) we introduced the spherically averaged kernel
defined by:

(28) 
As in Paper II these equations can be simplified by introducing the functions
and
defined by:

(29) 
and:

(30) 
where
is defined by Eq. (29). Using Eq. (29), we note that for a powerlaw "powerspectrum''
we have
so that Eq. (30) simplifies to:

(31) 
Note however that in this nonGaussian case the "powerspectrum'' P(k) does not obey Eq. (11). Nevertheless, it still provides a measure of the power associated with the wavenumber k.
Finally, the limiting generating function
defined by
is given by the value of the action
at this minimum which yields the implicit system:

(32) 
Of course, this result is similar to the Gaussian case which can be recovered by taking .
Note that Eq. (32) has only been derived for real y (hence
is real), so that
is the absolute value of .
Thus, the steepestdescent method allows us to compute the Laplace transform
in the quasilinear regime through Eq. (32) which yields
as in Eq. (19). This will give the pdf
through Eq. (3). However, we first need to check that the spherical saddlepoint defined by Eq. (26) and Eq. (27) is the global minimum of the action
.
For small positive y this can be rigorously proved as for the Gaussian case, using the fact that the kernel
is positive definite. We shall not repeat this discussion here (see Sect. 3.4 in Paper II). For negative y this cannot be proved as it may actually happen that the spherical saddlepoint is only a local minimum. This also occurs in the Gaussian case for a powerspectrum with n<0. This case is discussed in great details in Paper II where we show that the steepestdescent method remains useful but requires a careful justification. Since the same discussion can be applied to the nonGaussian model we investigate here we refer the reader to Paper II for a description of such cases and we shall only give a brief comment on this point in the next section.
3.3 Geometrical construction
As for the Gaussian case studied in Paper II we can give a geometrical construction of the generating function
defined by Eq. (32). Indeed, this latter expression can also be written:

(33) 
The minimum which appears in this expression merely expresses the fact that by sheer definition of the steepestdescent method the generating function
is governed by the global minimum of the action
.
Then, we see that the geometrical construction displayed in Fig. 4 in Paper II still applies to this nonGaussian case, with the modification that the parabola
are replaced by the curves
.
That is, the minimum point
which yields
in Eq. (33) is given by the first contact of the curve
with the curves
of varying height h. For y>0 we start from below at
while for y<0 we start from above at .
As in the Gaussian case, we find that for some values of n the action
has no global minimum. This can be seen from the geometrical construction shown in Fig. 4 or simply from Eq. (33). Indeed, as noticed in Paper II from Eq. (31) we have the following asymptotic behaviour for
:

(34) 
Using this expression we see that for y<0 the term in the brackets in the r.h.s. of Eq. (33) is not bounded from below if
,
since in this case it goes to
for
.
For the Gaussian case this occurs for n<0. However, as explained in Paper II the steepestdescent method developed in Sect. 3.2 remains useful and the pdf
is still governed by the spherical saddlepoint defined by Eqs. (26) and (27). This can be seen by the following remark. If we consider the quantity
rather than
,
where q is an odd integer, we can again apply the steepestdescent method which yields the associated generating function
as in Eq. (33). However, the function
is now given by
.
Then, if we choose a sufficiently large value for q the behaviour of the r.h.s. bracket in Eq. (33) is dominated by the term
at large
so that the action now exhibits a global minimum. Of course, this point is still the spherical saddlepoint we obtained in Sect. 3.2 (since the initial conditions
which give rise to a fixed value of
do not depend on whether we study
itself or
!). Then the pdf
could be recovered from
through a mere change of variable. This implies in turn that the pdf
is still governed by the standard spherical saddlepoint, even though the latter is no longer the global minimum of the action. However, as described in Paper II the treatment of this case requires some additional care and it is actually associated with a breakup of perturbative theory.
3.4 Calculation of the pdf
Finally, we describe how to compute the pdf
itself from the results obtained in Sect. 3.2. We consider the case n=1 and
where the action
always shows a global minimum. Then, the Laplace transform
is fully defined by the generating function
given by Eq. (32).
In order to compute the pdf from Eq. (32) we simply use the inverse relation (3) which yields:

(35) 
Then, to perform the integration over y we need the analytic continuation of
over the complex plane. Indeed, note that Eqs. (32) and (33) were actually derived for real y (hence
was real). In particular, we must continue the absolute value
over the complex plane. Moreover, we need to specify the integration path over y (since
will usually be singular at the origin). First, the integration path starts from the real axis at the saddlepoint
given by:

(36) 
From Eq. (32) we have:

(37) 
Thus, as for the Gaussian case, we see from Eq. (30) that the triplet
is also the triplet
we obtained in Sect. 3.2 to get
.
This simply means that
at the point
is governed by the neighbourhood of the saddlepoint
obtained in Sect. 3.2, which obeys
.
This is actually quite natural. Then, we need to continue the function
over the complex plane from the neighbourhood of the point
on the real axis. Thus, if
(i.e.
)
we replace
in Eq. (32) by ,
while if
(i.e.
)
we replace
by .
Then, we have the usual analytic continuation of the powerlaws
or
.
The argument of these powerlaws is real positive at the starting point
.
This provides the analytic continuation of the generating function
.
The function
is also analytic around the real axis because the spherical collapse solution
can be expressed in terms of analytic functions (e.g., trigonometric or hyperbolic functions for a critical density universe).
Next, the integration path in the complex plane follows the steepestdescent contour which runs through the saddlepoint .
It is given by the constraint
and it is orthogonal to the real axis at the point .
This path is also symmetric about the real axis which implies that the result for
is real. Note that in the quasilinear limit
the contributions to the integral (35) only come from an infinitesimal neighbourhood of the saddlepoint
around the real axis.
Usually the function
is not regular at the origin since we have
for small real y. Then, the pdf
is not analytic at
and the moments of the pdf cannot be recovered from the expansion (4). This is not the case for Gaussian initial conditions where both
and
are regular at the origin.

Figure 1:
The pdf
for n=1,
,
and .
This corresponds to
.
The solid line shows the theoretical prediction from (32). The dashedline displays the linear pdf
of the linearly evolved density field. The dottedlines show the linear and nonlinear pdfs obtained for Gaussian initial conditions with the same linear rms density fluctuation
(the nonlinear pdf is the asymmetric curve with the extended highdensity tail). 
Open with DEXTER 
As an example, we show in Fig. 1 the pdf
for the case
and n=1 (solid line). We also have
where
is the rms density fluctuation. Since
we numerically compute
from the second moment of the linear pdf
.
Indeed, the steepestdescent method developed in Sect. 3.2 also yields the linear pdf
of the linearlyevolved density field. In this case we have
and
,
hence we simply need to take
in Eq. (32) (of course, for
this gives back the Gaussian). The nonlinear evolution of the density field increases the value of
at large positive
(it actually spreads the linear pdf towards larger value of the density) as in the Gaussian case but the cutoff remains much sharper. It also induces a cutoff at low densities
which expresses the fact that the actual density
is positive.
For comparison, we also display in Fig. 1 the results we obtain for Gaussian initial conditions with the same linear variance
(dotted lines). These curves were derived in Paper II. As could be anticipated from Eq. (15) the large density contrast cutoff of the linear pdf is much sharper than for the Gaussian case: it goes as
since here we take .
Moreover, we can check that for the actual nonlinear pdfs this trend remains valid. More precisely, as shown in Paper II the highdensity tail of the nonlinear pdf for Gaussian initial conditions is of the form:

(38) 
while for nonGaussian initial conditions we obtain:

(39) 
These behaviours are obtained from Eqs. (32), (35) and (37) which yield:

(40) 
while the asymptotic behaviour of
for large
is given by Eqs. (34) and (37).
As noticed above, the nonGaussian model (16) is not derived from a physical scenario of the primordial universe. It should be viewed as a simple model to describe initial density fluctuations which exhibit a nonGaussian tail. Then, the analysis performed in the previous sections shows how one may derive in a rigorous manner the gravitational dynamics of density fluctuations in the quasilinear regime. Moreover, we can expect the form (39) of the highdensity tail to apply to any model which obeys
at large densities, whatever the details of the model. In fact, we have shown that in the quasilinear regime the tail (39) (actually the generating function
itself !) is common to all models of the class described by Eq. (16) with
.
In practice, since observations have not shown any deviations from Gaussianity so far, the parameter
is constrained to be close to 2 (the Gaussian value). Note that this family of models presents the advantage of containing the usual Gaussian scenario as a particular case.
4 An isocurvature CDM model
In addition to generalized nonGaussian models as the one studied in Sect. 3 which are rather adhoc tools, there exist some specific nonGaussian models which arise from a physically motivated description of the primordial universe. These scenarios are of great interest as they could provide a reasonable alternative to the standard simple inflationary model. Moreover, it is prudent not to disregard a priori sensible models of structure formation in view of the rather indirect character of the probes of the early universe which are available to us.
4.1 Initial conditions
Thus, in this section we consider a specific isocurvature cold dark matter model which was presented in Peebles (1999a). This is an inflationary scenario which involves three scalar fields and it gives birth to nonGaussian isocurvature CDM fluctuations. We refer the reader to Peebles (1999a) for a description of the physical processes which give rise to this scenario. Then, at the end of inflation (at time )
the CDM mass distribution is (Peebles 1999a):

(41) 
where
is a Gaussian random field with zero mean and powerspectrum
:

(42) 
On the scales of interest for structure formation we have:

(43) 
while below the coherence length
(i.e. for
)
the powerspectrum decreases as
with s<3. This coherence length ensures that
is finite, as required by Eq. (41). In the case of the model described in Peebles (1999a) we have
pc. In the linear regime below the horizon the CDM density perturbations grow as
(we only keep the growing mode) hence we write:

(44) 
where we normalized
by
and the growing mode by
D_{+}(t_{i})=1. Note that in Eq. (44) and in the following we only consider "small'' comoving scales
Mpc, where the isocurvature CDM transfer function is close to unity (Peebles 1999b). On larger scales the density fluctuations are damped with respect to Eq. (44) and we should take into account the decrease of the transfer function. Since we are interested in the statistics of the density field at an arbitrary time t it is convenient to rescale the Gaussian field .
Thus, we define
so that Eq. (44) now reads:

(45) 
at the redshift of interest. The powerspectrum
is still of the form (43). Note that the relation between the fields
and
depends explicitly on time through the last term in Eq. (45) (to simplify notations we usually do not write explicitly the time dependence of the various random fields). Indeed, we must recall here that
is not the spatial average of
but the mean over all realizations of the Gaussian random field .
Therefore, it is independent of the peculiar realization
which gives rise to a given linear density field through Eq. (45) (even though by ergodicity both averages happen to be equal). We define the twopoint correlation of the field
by:

(46) 
and the linear variance over the scale R by:

(47) 
Note that Eq. (46) also implies that we can define the average
by:

(48) 
which does not depend on
since the initial conditions are invariant through translations. From Eq. (45) and the fact that
is Gaussian one obtains the variance
of the linearlyevolved density contrast as:

(49) 
since the twopoint correlation
of the linear density field is given by:

(50) 
From Eqs. (47) and (49) we see that for
we get
and the slope n of the linear powerspectrum P(k) of the density fluctuations is related to
by:
.
This implies
as in Eq. (43) in order to match observational constraints (Peebles 1999b). For numerical calculations we shall use:

(51) 
4.2 Laplace transform
We again define the generating function
as in Eq. (2) but the Gaussian average over
which appeared in Eq. (14) must now be replaced by a Gaussian average over
:

(52) 
Thus, our goal is now to estimate the Laplace transform .
To do so, we shall not introduce a rescaled generating function
as in Eq. (19) and we work directly with .
First, using Eq. (45) it is convenient to introduce the linear density field
into the expression (52) through the Dirac functional
:



(53) 
Here and in the following we do not write the normalization constant of the pathintegral (it will be recovered in the final expressions). Then, using the integral representation of the Dirac functional we obtain:



(54) 
where
is an auxiliary real field. Note that the exponent is actually quadratic over
and we can write Eq. (54) as:



(55) 
where we introduced the kernel:

(56) 
Then, the Gaussian integration over
is straightforward and it yields:



(57) 
Note indeed that the factor
in Eq. (55) does not depend on the random field
over which we integrate, as shown by Eq. (48). Next, multiplying the r.h.s. of Eq. (57) by
(i.e. we change the implicit normalization constant of the pathintegral) and using the relation
we get:



(58) 
At this point, it is instructive to consider the linear regime when the density field is linearly evolved:
.
In this case, we have:

(59) 
where
is a tophat with obvious notations. Then, the integration over
is straightforward and it yields the Dirac functional
.
The integration over
is now obvious and we obtain for the generating function
which describes the linear density field (again, the subscript "L'' refers to "linear''):

(60) 
where the matrix
is given by:

(61) 
The normalization of the result derived from the pathintegral in Eq. (60) is obtained from the condition
,
as implied by the definition of
(since
). As shown in Eq. (4) the generating function
gives the moments
of the pdf
of the linear density contrast. However, it is usually convenient to introduce the generating function
of the cumulants
which is given by (see any textbook on statistical theory):

(62) 
Thus, from Eq. (60) we obtain the exact expression:

(63) 
One can easily check by expanding the logarithm over y that
is actually quadratic over y (see also Eq. (75) below). Hence
as it should. Moreover, we can check that we recover Eq. (49) for the variance of the linear density contrast.
4.3 Saddlepoint contributions
In the nonlinear case it is not so easy to compute the expression (58) since the integration over
no longer yields a Dirac functional. Besides, even for the linear density field the expression (63) is not very convenient since it is not obvious to compute the logarithm. Thus, for practical purposes it only provides the first few cumulants of
,
which can be obtained by expanding the logarithm as a series over y and computing the trace of each power of
.
Hence we can try a steepestdescent approximation to the pathintegral (58), in the spirit of the calculations presented in Paper II for the Gaussian case or in Sect. 3 for a simple nonGaussian model. More precisely, we may try to evaluate the contributions of all possible saddlepoints to the pathintegral (58).
First, we can look for the saddlepoints of the exponent with respect to
.
This yields:

(64) 
Since the Gaussian random field
is homogeneous and isotropic we still have a spherical symmetry and we can again look for a spherical saddlepoint. Then, taking the mean of Eq. (64) over a spherical cell V' we get:

(65) 
From the very definition of derivatives we can also write Eq. (65) as:



(66) 
where R'' is a dummy variable and we introduced the kernel:
We again define the linear density contrast
and the Lagrangian mass scale
as in Eq. (25) and we obtain:

(68) 
in a fashion similar to the calculation presented in Paper II for the Gaussian case. The only term in the r.h.s. of Eq. (68) which depends on R' is the factor
hence we have:

(69) 
which yields the profile:

(70) 
which also defines the normalization .
Note that this profile for the field
is of the same form as the one obtained for the linear regime which involved the Dirac functional
,
see the derivation of Eq. (60).
Then, for such a state
we can estimate the trace which appeared in Eq. (58) as follows. Let us note this trace as
.
Then, we have:

(71) 
with:
using the definition of the trace:
.
Next, from Eq. (56) we see that for the spherical state (70) the matrix
is given by:

(73) 
which yields:

(74) 
The first term T_{1} (i.e. q=1) is simply:

(75) 
where we used Eq. (46). Note that it is set by the shortdistance behaviour (
)
of the twopoint correlation
,
since we actually have x=0. On the other hand, for
the other terms T_{q} with
are set by the longdistance behaviour of the correlation
,
i.e. by the scale R one is looking at by studying
.
This is the case for the powerspectra
of cosmological interest, see Eq. (43). Moreover, for large q the different factors
in Eq. (74) are almost uncorrelated and it makes sense to use the approximation:

(76) 
Besides, for a powerlaw powerspectrum
as in Eq. (43) we have (at scales larger than the coherence length ):

(77) 
Then, since
is close to 3 the correlation
is almost constant over most of the integration volume so that the approximation (76) should be quite reasonable. This yields for the trace
defined in Eq. (71):

(78) 
Next, we could try to consider a saddlepoint with respect to both
and
.
However, one can check that this procedure does not give meaningful results. In fact, as described above in the derivation of the exact Eq. (60) the generating function
of the linear density field itself is not governed by a unique saddlepoint. Indeed, we had to take into account the contributions from all real fields
in the pathintegral to get the Dirac functional which eventually provides the correct result. Then, in order to keep this structure while using the spherical saddlepoints (70) it is natural to try to approximate the pathintegral (58) by:



(79) 
Here
is given by the spherical collapse solution of the equation of motions, while the matrix
is given by Eq. (73). Thus, we have replaced the integration over the real fields
and
in Eq. (58) by an integration over the real numbers
and .
In other words, we have replaced the average over
and
by an average over spherical saddlepoints of the form (70). In the case of the linear regime when
we simply have
and
.
Then we can perform the integration over
which yields the Dirac function
.
The integration over
is now straightforward and we recover exactly the rigorous result (60) (this also justifies the normalization factor
we introduced in Eq. (79)). Note in addition that the structure of the calculation is also the same (the Dirac functional is simply replaced by a Dirac function). Thus, the expression (79) is actually exact in the linear regime. We show in Appendix A how to rigorously derive the expression (79) from Eq. (58) in the linear regime. We also explain how it can be derived for the fully nonlinear density field in the quasilinear limit. Thus, Eq. (79) is actually exact to leading order in the limit
.
More precisely, for the nonlinear density field it yields the exact exponential cutoff of the pdf
which dominates the dependence on
but the normalization is only correct up to a multiplicative factor of order unity which may depend on
but not on .
This factor arises from the Jacobians J_{a} and J_{b} in Eq. (A.15). We shall come back to this point below in Eq. (92).
As explained above, the factor
is not very convenient for practical calculations hence it is natural to use the approximation (78) which was derived for such spherical fields
.
This yields:



(80) 
where we removed the subscripts of the integration variables
and .
Note that in this approximation the term
has disappeared. In the linear regime, where
and
,
we can again perform the integrations over
and
which yields:

(81) 
Then, expanding the logarithm around y=0 we obtain the cumulants
from Eq. (62). This leads to
and:

(82) 
In particular, we obtain for the variance:
,
which must be compared with Eqs. (47) and (49). Of course, the deviation from the exact result (49) is entirely due to the approximation (76). For a powerlaw powerspectrum
as in Eq. (43) with
we obtain for this deviation :

(83) 
where we used Eq. (77) and the result (see Peebles & Groth 1976):

(84) 
Thus, we see that the approximation (76) is quite reasonable. Note that it becomes exact in the limit
.
Then, from Eq. (82) we get for the skewness and the kurtosis:

(85) 
since we have:

(86) 
On the other hand, for the same powerspectrum (
)
Peebles (1999b) obtains by a direct numerical calculation (i.e. without the approximation (76)):

(87) 
and:

(88) 
Thus, we see that the approximation (76) is quite satisfactory. In particular, the behaviour of the highorder cumulants
is very well reproduced as soon as
.
Then, we can expect the expression (80) to provide a good approximation to ,
both for the linear and the quasilinear regimes, since we do not add any approximation in order to describe the nonlinear effects encoded in the relations
and
.
Going back to Eq. (80), we can make the change of variable
which yields:



(89) 
Then, after a change of variable and using the integral representation of the Euler Gamma function:

(90) 
we obtain:



(91) 
where the factor
is the Heaviside function with obvious notations. Let us recall that in Eq. (91) we have:
and
.
4.4 The pdf
and
In fact, since the generating function is written in an integral form we can directly derive the pdf
using the inverse Laplace transform (3). Indeed, the integration over y simply gives the Dirac function
.
After a trivial integration over
we obtain:



(92) 
where the quantity
is given by the condition:
and
.
In particular, the pdf
of the linearly evolved density field is given by:



(93) 
It is obtained from Eq. (92) by setting
and
.
For Gaussian initial conditions the pdf
exhibits a specific scaling over the variable
,
which contains all the time and scale dependence of the pdf. As described above, within the approximation (76) we have
hence we get:

(94) 
Then, we see that the linear pdf (93) shows a scaling over
as in the Gaussian case:

(95) 
but the pdf
is now given by:

(96) 
As noticed in Paper II, in the Gaussian case the nonlinear highdensity tail is well described by a simple spherical model. In fact, this model gives the exact exponential dependence of the pdf on the variance ,
at leading order in the limit
,
as was also shown in Valageas (1998). Let us recall here how to build this simple model. It is based on the approximation:

(97) 
which merely states that the fraction of matter within spherical cells of radius R with a nonlinear density contrast larger than
is approximately equal to the fraction of matter which was originally enclosed within spherical cells of radius
with a linear density contrast larger than
.
Here
and
are related to the nonlinear variables R and
by the usual relation (25), as in Eq. (92). Note that this is similar to the usual PressSchechter prescription (Press & Schechter 1974), without the factor 2. Then, substituting the scaling (95) into Eq. (97) and differentiating with respect to
we obtain:

(98) 
Here the subscript "s'' refers to the "spherical'' model. In order to recover the variables used in the Gaussian case (Paper II) it is convenient to introduce the variable
given by:

(99) 
which removes the dependence on the amplitude of the rms linear density fluctuation, and to define the function
by:

(100) 
which obeys again Eq. (31). Then, the relation (98) writes:

(101) 
which yields:



(102) 
We can see that we recover the expression (92), except for the multiplicative factor where
has been replaced by
.
Thus, as in the Gaussian case, we find that the simple spherical model (97) yields the exact exponential dependent term of the pdf. However, the multiplicative factor may not be exact. Here, we note that the multiplicative factor which appears in Eq. (92) is not exact either, as stated above, below Eq. (79), see also Appendix A. In the Gaussian case, we checked by a comparison with numerical simulations (e.g., Valageas 1998; Paper II) that the spherical model (97) provides good results up to
.
We can expect a similar accuracy in the nonGaussian case studied here. Therefore, the expression (102) should give a good description of the nonlinear pdf. This means that the multiplicative factor which appears in Eq. (102) gives a reasonable approximation of the term induced by the Jacobians which arise in the nonlinear case, see Eq. (A.15).

Figure 2:
The pdf
for the isocurvature CDM scenario, with
and
.
This corresponds to
.
The solid line shows the theoretical prediction from (102) for
.
This corresponds to the "spherical'' model. The dotted line is given by Eq. (102) where no attempt is made to take into account the Jacobians which arise in the nonlinear case. The solid line (spherical model) should give better results. The dashedline displays the pdf
of the linearly evolved density field, from Eq. (93). The histogram shows the results of numerical simulations by Robinson & Baker (2000) for the linear pdf
. 
Open with DEXTER 
We show in Fig. 2 our results for
and
.
The dashedline shows the pdf
of the linearly evolved density field, from Eq. (93). As could be seen from Eq. (93) it is strongly nonGaussian, whatever the value of
.
In particular, for large overdensities
the pdf exhibits a simple exponential cutoff (multiplied by a power
)
which is much smoother than a Gaussian cutoff. Hence the number of extreme events (i.e.
)
is much larger than for the Gaussian case, as can be checked by comparison with the lower dotted curve in Fig. 1. We can see that for
our predictions agree very well with the results of numerical simulations from Robinson & Baker (2000) for the linear density field (shown by the histogram). Note that for the theoretical curve we used the exact value of
obtained from Eq. (83) in order to get a meaningful comparison. Hence the variance of the approximate pdf (93) underestimates the exact result by a factor
.
However, we can check that the agreement is quite good for
so that for practical purposes one can directly use Eq. (93). For instance, Robinson & Baker (2000) actually use the linear pdf to apply the usual PressSchechter recipe (Press & Schechter 1974) in order to estimate the mass function of justcollapsed objects.
On the other hand, we note that our estimate (93) fails for
.
In particular, we see that the pdf vanishes for
,
that is the exact lower bound
has been replaced by
.
This is due to the approximation (76). Indeed, the exact linear pdf obtained from Eqs. (3) and (60) is:



(103) 
This expression clearly shows that the linear density contrast obeys the lower bound
.
Indeed, for
we can push the integration path over y to the right (i.e. Re
)
so that the integral vanishes (we have
). Note that this behaviour is rather different from the Gaussian case where the linear density contrast can take negative values of arbitrarily large amplitude. In fact, the lower bound for
can be directly seen from Eq. (45). Note that this bound is directly related to the time t_{i} at the end of inflation where Eq. (41) holds. Indeed, from Eq. (44) we see that
where the growing mode has been normalized by
D_{+}(t_{i})=1. This minimum value of
is also overestimated by the numerical simulations which only start at a redshift
.
Nevertheless, it is clear that the approximate bound
gives the value below which underdensities become very rare (i.e. the pdf should decline for
). Besides, for practical purposes one is often mainly interested in the behaviour of overdensities
so that the expressions (92) and (93) should remain useful.
As noticed in Eq. (95), the linear pdf given by Eq. (93) exhibits a scaling over the one variable
defined in Eq. (94), as in the usual Gaussian case. This scaling property was actually used in order to obtain
from the numerical simulations. However, this property is not exact and it fails for underdensities since as noticed above the linear density contrast also satisfies the lower bound
which does not scale as
(e.g., contrary to
it is independent of scale).
Next, we show the theoretical prediction
from Eq. (92) (dotted line) and from Eq. (102) (solid line) from the spherical model (97) for the actual nonlinear density field. In Fig. 2 we used the approximation:

(104) 
which has been shown to provide a very good fit to the exact spherical collapse solution for all values of
and
,
see Fig. 2 in Bernardeau (1994b). This means that the dependence on
and
is negligible (in the quasilinear regime where we do not consider virialized objects) and the pdf only depends on the rms fluctuation
(and the slope
of the powerspectrum). Unfortunately, there are no available results from numerical simulations to compare with our prediction. However, in view of the reasonable accuracy of our prediction for the linear pdf
and the good results obtained for the Gaussian case (see Paper II) we can expect the expression (102) to provide a good estimate of the exact pdf
.
It should perform better than Eq. (92) where no attempt was made to take into account the Jacobians which arise in the nonlinear case, see Appendix A. Of course, we recover the usual features of the nonlinear evolution which increases the highdensity tail of the pdf. Note again the much larger number of high density events with respect to the Gaussian case with the same variance (see Fig. 1).
5 Standard slowroll inflation
5.1 Initial conditions
The two classes of models we investigated in Sects. 3 and 4 are strongly nonGaussian (except for
in the first case) and they apply to "nonstandard'' scenarios for the generation of primordial density fluctuations. For instance, the second model is based on a multifield inflationary scenario. By contrast, in standard slowroll inflation the primordial density perturbations should be very close to Gaussian. Nevertheless, they may still show some small deviations from Gaussianity. In particular, one is led to consider models where the perturbed primordial gravitational potential
is given by:

(105) 
where
is a Gaussian random field with zero mean and powerspectrum
,
as in Eq. (42). Such models arise in standard slowroll inflation if we keep track of the perturbations of the inflaton up to the secondorder (e.g., Gangui et al. 1994; Falk et al. 1993). In fact, Eq. (105) can be seen as the first two terms of a Taylor expansion so that it applies to a large class of slightly nonGaussian models. Thus, the parameter
is taken to be small so that the primordial density fluctuations are close to Gaussian. Note that
has the dimensions of .
We shall define below what we mean by "
being small'' (see Eq. (138)). Then, our goal here is to obtain the pdf
of the density fluctuations in the quasilinear regime up to first order in
(to go beyond this order we should first take into account the possible higherorder terms in Eq. (105)). Note that for
we must recover the case of Gaussian primordial density fluctuations studied in Paper II. The gravitational potential
is related to the primordial linear density fluctuations
by the Poisson equation (in comoving coordinates):

(106) 
where a(t) is the scalefactor and
the mean comoving density. Taking the Fourier transform of Eq. (106) one can eventually write the linear density field
at the time of interest as:

(107) 
where the kernel
is given by:



(108) 
Here we introduced the Hubble constant H_{0} and the density parameter
today, while D_{+}(t) is the usual linear growing mode at the time of interest, normalized by
D_{+}(t_{0})=1 today. The function T(k) is simply the adiabatic CDM transfer function (normalized to unity for
). Indeed, contrary to the isocurvature model described in Sect. 4 we need to take into account the deviations of T(k) from unity on the scales of interest for largescale structure formation. The factor k^{2} arises from the Laplacian in the l.h.s. in Eq. (106). Note that the kernel
is homogeneous, isotropic and symmetric since it only depends on
.
Hence for any real fields f_{1} and f_{2} we have:



(109) 
Thus, Eq. (107) defines the initial conditions of our system.
We can note that the pdf
of the linearly evolved density field was already studied in Matarrese et al. (2000) using pathintegral methods like those described in Sect. 5.2 below. However, they did not investigate the effects of the nonlinear dynamics. On the other hand, Verde et al. (2000) studied the observational tests which may constrain such deviations from Gaussian initial conditions.
5.2 Generating function
In order to derive the pdf
we again introduce the Laplace transform
as in Eq. (2), which yields again Eq. (52). Next, we introduce the linear density field through the Dirac functional
.
This Dirac functional can again be expressed through an auxiliary real field
so that the analog of Eq. (54) now reads:



(110) 
where we defined the inverse
of the twopoint correlation
of the Gaussian random field ,
which is again given by Eq. (46). The mean
is again given by Eq. (48) and it does not depend on the field
over which we integrate. As in Sect. 4 we do not write explicitly the normalization factor of the pathintegrals. Next, we introduce the kernel
defined by:

(111) 
so that Eq. (110) reads:



(112) 
As explained in Sect. 5.1 we only consider the firstorder term in .
Therefore, we expand the exponential in Eq. (112) up to firstorder in .
This yields:



(113) 
This pathintegral is Gaussian over the random field .
Thus, we first make the change of variable
so that Eq. (113) writes:
Here we used the fact that all kernels W,
and
are symmetric. Then, using Wick's theorem we can perform the Gaussian integration over
which yields:



(115) 
where we used
.
Next, we define the twopoint correlation
of the linear density field in the case
by:
where we used Eq. (107). Here the subscript "0'' refers to "
''. The relation (116) also writes:

(117) 
The pathintegral (115) is Gaussian over the field
.
Thus, making the change of variable
and integrating over
using Wick's theorem we finally get:



(118) 
where we used
.
Here we introduced the shorthand notation
for the vector:

(119) 
Note that for
we recover the expression (14) derived for Gaussian initial conditions. This was to be expected since for
the linear density field is actually Gaussian. Indeed, in this case we have
and
is a Gaussian random field. Of course, the procedure we described above can be extended up to any order in
(provided we know the initial conditions up to the required order). Indeed, by expanding the exponential which appears in Eq. (110) up to the needed order in
the integrations over the fields
and
are Gaussian and can be easily performed using Wick's theorem. Then, one eventually obtains an expression of the form (118), where the term in brackets is an expansion over
up to the required order. Note that this is similar to the standard Edgeworth expansion where one directly expands the Laplace transform
around the Gaussian value
(see any textbook on probability theory). Therefore, the expansion over
does not necessarily converge: it may only provide an asymptotic series (e.g., Cramer 1946).
5.3 Steepestdescent method
We now need to evaluate the pathintegral (118). As for the Gaussian case studied in Paper II or the nonGaussian generalization described in Sect. 3 we can use a steepestdescent method for the quasilinear regime. We refer the reader to Paper II for a detailed presentation of this method. First, we define the rescaled generating function
by:

(120) 
where
is the variance of the linear density field for
:

(121) 
Using Eq. (118) we get:



(122) 
where we introduced the "action''
given by:

(123) 
In the quasilinear limit
the pathintegral (122) is governed by the minimum of the action S. As shown in Paper II this spherically symmetric saddlepoint is given by:
where
is a dummy variable. The variable
is given by the implicit equation:

(125) 
Here the function
is the usual spherical collapse solution while
is the Lagrangian mass scale, see Eq. (25). These results were also used in Sect. 3.2 where we investigated a simple generalization of the Gaussian weight. Next, we introduce the functions
and
by:

(126) 
and:

(127) 
as in Eqs. (29) and (30). Then, we define the function
as the value of the action S at this spherically symmetric saddlepoint (i.e. the minimum of the action). As shown in Paper II it is given by the implicit system:

(128) 
in a fashion similar to Eq. (32). Then, at leading order in
we can write the pathintegral (122) in the quasilinear limit as:

(129) 
where we defined:

(130) 
Here
and
are dummy variables. The normalization of
in Eq. (129) is set by the constraint
.
Next, we define the dimensionless quantity
by:

(131) 
It is convenient to express the quantities
and J_{3} in terms of the powerspectrum
of the Gaussian random field
.
This yields:

(132) 
and:



(133) 
Here
is the Fourier transform of the window function
defined in Eq. (108) (i.e. it is related to the kernel W by
)
while F(kR) is the Fourier transform of the tophat of radius R:

(134) 
Note that in Eq. (134) we did not introduce the factor
used in the definition (6) of the Fourier transform, in order to obtain the usual tophat window F(kR). As in the previous sections we approximate the powerspectrum P_{0}(k) of the linear density fluctuations (when
)
by a powerlaw. Since we have
(i.e. a scaleinvariant primordial powerspectrum, with small and large scale cutoffs) for the standard inflationary scenarios we investigate here this means that
on the scales of interest. Indeed, the powerspectrum P_{0}(k) is related to
by:

(135) 
as can be seen for instance from Eq. (132). Then, from Eqs. (132) and (133) we get:

(136) 
which implies that the quantity
defined in Eq. (131) does not depend on ,
within this approximation. Moreover, from Eqs. (132) and (133) we see that we have the order of magnitude estimate:

(137) 
Going back to the definition (105) of the initial conditions we see that we can write the primordial perturbations of the gravitational potential as:

(138) 
where
stands for a numerical factor of order unity. It is clear from this expression that for a scaleinvariant primordial powerspectrum the initial conditions are close to Gaussian if
.
Hence
is the relevant dimensionless parameter which describes the amplitude of the deviations from Gaussianity and the usual inflationary scenarios lead to
.
Here it is interesting to consider the linear regime, that is the statistics of the linear density field. Then, we have
and
,
so that
.
This yields from Eq. (129):

(139) 
where the subscript "L'' refers to the "linearly evolved'' density field. Then, from the expansion (4) we obtain for the thirdorder moment of the linear density field
and for the skewness D_{3}:

(140) 
at firstorder in
.
Thus, we see that the parameter
is directly related to the skewness of the linear density field on the scale of interest, due to the slight nonGaussianity of the primordial density field.
5.4 The pdf of the density contrast
From the rescaled Laplace transform
obtained in Eq. (129) we can derive the pdf
,
using the inverse Laplace transform (3). This yields:

(141) 
where the function
is given by Eq. (128). The pdf
of the linearly evolved density field is obtained from Eq. (141) by using
and
which gives:

(142) 
Then we can perform the Gaussian integration over y in Eq. (142) which yields (see Gradshteyn & Ryzhik 1965, Sect. 3.462.4):

(143) 
where H_{3}(x) is the Hermite polynomial of order 3 defined by:

(144) 

Figure 3:
The pdf
for n=1 and
,
for slightly nonGaussian initial conditions:
.
The solid line shows the prediction of Eq. (141) for the nonlinear pdf
while the dashedcurve shows the pdf
of the linearly evolved density field from Eq. (143). The dotted lines display the results obtained with
,
i.e. for Gaussian initial conditions. 
Open with DEXTER 
Finally, we display in Fig. 3 our results for slightly nonGaussian initial conditions:
.
In order to compare the pdf with the case of Gaussian initial conditions we also show the results obtained for
(dotted curves). Let us recall that Eq. (142) for the pdf of the linearly evolved density field is exact to firstorder in
.
In agreement with Eq. (140) we can check that
leads to a small positive skewness for the linear density field. In particular, the highdensity tail of the pdf is slightly enhanced. We can see that this feature remains valid for the nonlinear pdf
.
However, as gravitational clustering proceeds the deviations from Gaussianity become dominated by the nonlinear dynamics over the slight primordial nonGaussianity. More precisely, we can obtain the loworder moments of the density field through the expansion (4). At the lowest order in y we have
and
where the parameter S_{3} is given by the standard result (e.g., Bernardeau 1994a):

(145) 
Then, we obtain from Eq. (129) the expansion:

(146) 
This yields for the lowest order moments:

(147) 
and for the skewness D_{3}:

(148) 
This relation clearly shows that the relative importance of the primordial deviation from Gaussianity is greater at earlier times before gravitational clustering builds up, as could be expected. Note that in Fig. 3 we used the generating function
defined by the branch of Eq. (128) which is regular at the origin y=0. Since here we considered the case n=1 this function actually shows a branch cut along the real negative axis for
(with
). Then, for large density contrasts one needs to take into account the second branch of
defined over the range
which is singular at the origin. However, as shown in Fig. 6 in Paper II this is irrelevant for the case displayed in Fig. 3 for
.
We refer the reader to Paper II for a detailed discussion of this point. Of course, for the nonGaussian model we study here one simply needs to follow the procedure outlined in Paper II for Gaussian initial conditions and to add to the relevant expressions the term in the brackets in Eq. (141) which describes the effects due to the small primordial deviation from Gaussianity (to first order in
).
Thus, using a nonperturbative method developed in a previous work (Paper II), we have described in details how to obtain the pdf
of the density contrast within spherical cells in the quasilinear regime for three specific nonGaussian models.
The first case is a straightforward generalization of the Gaussian scenario and it can be seen as a phenomenological description of a density field where the tails of the linear pdf are of the form
,
where
is no longer required to be equal to two (as in the Gaussian case). Then, we have shown that the derivation of the pdf presented for the Gaussian case can be directly extended to this model. This provides again exact results for the pdf in the quasilinear limit.
The second scenario is a physically motivated model of isocurvature cold dark matter presented in Peebles (1999a). It arises from an inflationary scenario with three scalar fields. This case is slightly more difficult as one needs to adapt the method to this specific model. Moreover, in order to get simple analytical results one must introduce a simple approximation (which is not related to the gravitational dynamics but to the nonGaussian properties of the initial conditions). However, we have shown that even with this approximation we get good results for the linear pdf
for
by comparison with numerical simulations. For lack of data we could not check our prediction for the nonlinear pdf
but we can expect a good agreement of the same accuracy.
Finally, the third scenario corresponds to the small primordial deviations from Gaussianity which arise in standard slowroll inflation. We obtained exact results for the pdf of the density field in the quasilinear limit, to firstorder over the primordial deviations from Gaussianity.
This study shows that our approach is powerful enough to be applied to a large variety of initial conditions. Note that our predictions for the linear pdf
for such nonGaussian scenarios may be used to estimate the mass function of justcollapsed objects, using a straightforward extension of the PressSchechter recipe. Note that we shall discuss the PressSchechter prescription in the light of the formalism developed in Paper II and used in this work in a companion article (Paper IV).
To conclude, we note that our approach being nonperturbative it can in principle be applied to the nonlinear regime. Indeed, it does not rely either on the hydrodynamical description. We shall present a study of this nonlinear regime in a future work, see Paper IV, for Gaussian initial conditions. However, it is clear that this can be extended to the nonGaussian models described here.
Appendix A: Isocurvature scenario. Reduction to ordinary integrals
In this appendix we show how to derive the expression (79) from Eq. (58). We first consider the linear regime where Eq. (79) is exact. The pathintegral (58) involves the functional measure
.
The latter can be defined from a discretization procedure (i.e. a spatial grid for the coordinate
with an infinitesimal spacing) but an equivalent formulation (e.g., ZinnJustin 1989) is to expand the function
on a complete set of real orthonormal functions f_{q} (in the Hilbert space
):

(A.1) 
The vectors f_{q} obey the relation:

(A.2) 
which expresses the fact that they form an orthonormal basis. Here
is the usual Kronecker symbol. We introduced a factor 1/V in the scalar product (A.2) so that the functions f_{q} are dimensionless, but this is not essential (here V is a constant). Then, the functional measure
can be defined as (e.g., ZinnJustin 1989):

(A.3) 
where
is a normalization constant. In order to get a discrete basis in Eq. (A.1) we restricted the density field to a large finite volume .
However, this is not essential: we choose
to be much larger than any relevant length scale and we can eventually take the limit
.
Next, we can choose for the first basis vector f_{0} the tophat of radius R:

(A.4) 
where
is the usual tophat with obvious notations. Then, from Eqs. (A.1) and (A.2) we have:

(A.5) 
We can expand the auxiliary field
on the same basis:

(A.6) 
so that the pathintegral (58) now writes:



(A.7) 
where the subscript "L'' refers to the "linearly evolved'' density field (so that
)
and we did not write the normalization constant of the integrals. Then, the integration over the variables a_{q} with
yields the Dirac functions
.
The integration over b_{r} is now straightforward for
and we obtain:



(A.8) 
where the matrix
is obtained from
.
Then, using
from Eq. (A.5) and making the change of variable
we obtain Eq. (79) in the linear regime (i.e.
and
).
Now, we investigate how the previous derivation is modified when we study the nonlinear density field
.
We can again define the functional measure
(and
)
by Eq. (A.3). However, we now introduce the new variable
,
which also defines
and
.
Then, we define the functions
by:

(A.9) 
so that we still have the orthonormalization property:

(A.10) 
Then, we write the linear density field
as:

(A.11) 
which also implies:

(A.12) 
This yields:

(A.13) 
while we write the auxiliary field
as:

(A.14) 
Then, the pathintegral (58) now writes:



(A.15) 
where we made the changes of variables
and
.
In Eq. (A.15) the Jacobians J_{a} and J_{b} of these transformations are given by the determinants:

(A.16) 
These Jacobians do not depend on the amplitude
of the density fluctuations. On the other hand, the quasilinear regime corresponds to the limit
for a fixed finite y (which leads to finite
and
,
as shown in the Gaussian case in Paper II). In this limit, the generating function
is governed by the exponent in Eq. (A.15) which depends on
and sets the cutoffs of order
for the density contrast
.
Then, we can neglect the Jacobians in Eq. (A.15): they do not contribute to
at leading order for
(however, they give a independent prefactor which appears in ,
see the discussion below). Moreover, if we make the approximation
we can again integrate over a_{q}' for ,
which gives the Dirac functions
,
and we eventually recover Eq. (79).
Let us now discuss the approximation
used above. The point is that in the limit
the pathintegral (58) and the ordinary integrals (A.15) are dominated by the saddlepoints of the exponent. As discussed in Sect. 4.3, because of the spherical symmetry of the physics we investigate there exist some spherical saddlepoints, of the form (70). Then, if there are no other saddlepoints (or if some other saddlepoints exist but they yield a lower contribution to integral, which will vanish in the limit
)
the pathintegral is given at leadingorder by its value for these spherical saddlepoints. Note that the case of Gaussian initial conditions studied in Paper II exhibits the same behaviour in a simpler manner. Indeed, in this case there exists only one spherical saddlepoint for both the linear and nonlinear generating functions
and .
Then, in order to keep only the leading order for
one defines the rescaled generating function
by:

(A.17) 
as in Eq. (19), which can be expressed through the pathintegral:

(A.18) 
in a fashion similar to Eq. (20). Next, in the quasilinear limit
the pathintegral in the r.h.s. in Eq. (A.18) is dominated by one saddlepoint and taking the logarithm of both sides in Eq. (A.18) one obtains in this limit:

(A.19) 
as described in Paper II (see also Sect. 3.2). Going back to
this means that in the quasilinear limit the generating function
is given by the maximum of the exponent which appears in the relevant pathintegral. This corresponds to the procedure we described above for the isocurvature CDM scenario in order to get Eq. (79). However, the difference is that in this nonGaussian case there exists an infinite number of spherical saddlepoints with respect to
.
Therefore, at leading order the generating function
is not given by a unique contribution but rather by the sum of all contributions due to these saddlepoints which are parameterized by the variables
and .
Finally, we note that the minimum of the action
,
or the value of the exponent in Eq. (A.15), only provides the leading order contribution to
in the limit
.
This means that it yields the exact exponential dependent term of the pdf
which describes the cutoffs of the pdf for density contrasts which are large relative to .
In our case, this is the factor
which appears in Eq. (92). However, it does not give the exact multiplicative prefactor of the generating function
and of the pdf
.
Indeed, the Jacobians J_{a} and J_{b} in Eq. (A.15) (or the Gaussian integration around the saddlepoint in the Gaussian case) contribute to this term. In the case we study here, this means that Eq. (79) differs from the exact result by a multiplicative factor which may depend on
and .
Therefore, the expression (92) obtained for the pdf is correct up to a multiplicative factor which may depend on
but not on .
This point is also discussed in Sect. 4.4 where we compare Eq. (92) with the result (102) of a simple spherical model. The latter can be seen as an attempt to model this prefactor. Note that for the pdf of the linear density field we do not encounter this difficulty since, as shown above, Eq. (A.8) is actually exact.
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Copyright ESO 2002