Mass function and bias of dark matter halos for nonGaussian initial conditions
P. Valageas
Institut de Physique Théorique, CEA Saclay, 91191 GifsurYvette, France
Received 4 June 2009 / Accepted 31 January 2010
Abstract
Aims. We revisit the derivation of the mass function and the bias of dark matter halos for nonGaussian initial conditions.
Methods. We use a steepestdescent approach to point out that
exact results can be obtained for the highmass tail of the halo mass
function and the twopoint correlation of massive halos. Focusing on
primordial nonGaussianity of the local type, we check that these
results agree with numerical simulations.
Results. The highmass cutoff of the halo mass function takes
the same form as the one obtained from the PressSchechter formalism,
but with a linear threshold
that depends on the definition of the halo (i.e.
for
a nonlinear density contrast of 200). We show that a simple
formula, which obeys this highmass asymptotic and uses the fit
obtained for Gaussian initial conditions, matches numerical simulations
while keeping the mass function normalized to unity. Next, by deriving
the realspace halo twopoint correlation in the spirit of Kaiser
(1984, ApJ, 284, L9) and taking a Fourier transform, we obtain good
agreement with simulations for the correction to the halo bias,
,
due to primordial nonGaussianity. Therefore, neither the halo mass function nor the bias require an adhoc parameter q (such as
), provided one uses the correct linear threshold
and pays attention to halo displacements. The nonlinear realspace
expression can be useful for checking that the ``linearized'' bias is a
valid approximation. Moreover, it clearly shows how the baryon acoustic
oscillation at
Mpc is amplified by the bias of massive halos and modified by primordial nonGaussianity. On smaller scales,
30 <x< 90 h^{1} Mpc, the correction to the realspace bias roughly scales as
.
The lowk
behavior of the halo bias does not imply a divergent realspace
correlation, so that one does not need to introduce counterterms
that depend on the survey size.
Key words: gravitation  methods: analytical  largescale structure of Universe
1 Introduction
Standard singlefield slowroll inflationary models predict a nearly
scaleinvariant and Gaussian spectrum of primordial curvature
fluctuations (e.g., Bartolo et al. 2004). This agrees with current observations of the cosmic microwave background (CMB) anisotropies (Komatsu et al. 2009) and of largescale structures (Slosar et al. 2008). Nevertheless, several inflationary models predict a
potentially observable level of nonGaussianity (e.g., Bartolo et al. 2004,
for a review), so that constraining or detecting primordial
nonGaussianity is an important task for current cosmological studies.
This would allow one to rule out some of the many inflationary models
that have already been proposed. In particular, in many
cases, the nonGaussianity is of the local type, meaning that it only
depends on the local value of Bardeen's potential .
That is, the latter can be decomposed as
where is a Gaussian random field. Simple slowroll inflation gives a parameter of 10^{2}, but this would be masked by the nonlinearities that arise from the dynamics (e.g., from the nonlinearity of Einstein's equations, see Bartolo et al. 2004) or from the physical processes involved by the observables (e.g., perturbations at recombination that affect the CMB, see Senatore et al. 2009), which give an effective close to unity. High values of can be obtained, for instance, from multifield inflation (Bartolo et al. 2002; Lyth et al. 2003), selfinteractions (Falk et al. 1993), tachyonic preheating in hybrid inflation (Barnaby & Cline 2006), or ghost inflation (ArkaniHamed et al. 2004). Current limits are from CMB (Komatsu et al. 2009) and from largescale structures (Slosar et al. 2008).
The effects of primordial nonGaussianity on largescale structures can be seen, for instance, through the mass function of virialized halos, especially in the highmass tail as the steep falloff magnifies the sensitivity to initial conditions (Lucchin & Matarrese 1988; Colafrancesco et al. 1989; Grossi et al. 2007; Maggiore & Riotto 2009). This allows using the Xray luminosity function of clusters to constrain the amount of nonGaussianity (Amara & Refregier 2004).
A second probe of nonGaussianity is provided by the clustering of
these halos, as measured through their manybody correlations.
In particular, the halo twopoint correlation can be significantly
increased if the underlying primordial density field is nonGaussian
(Grinstein & Wise 1986). More specifically, Dalal et al. (2008) have recently shown that primordial nonGaussianity of the local type (1)
gives rises to a strongly scaledependent bias on large scales, whereas
in the Gaussian case the bias is roughly constant in this range. Thus,
at linear order over
they obtain in Fourier space a correction of the form
where b_{M}(k,0) is the Gaussiancase bias (defined as the ratio of the halo and matter power spectra, b_{M}^{2}(k)=P_{M}(k)/P(k), for objects of mass M). Here is the linear matter density contrast associated with virialized objects (usually taken as ), T(k) the transfer function, and D(z) the linear growth factor, normalized as at high redshift. Then, Dalal et al. (2008) checked in numerical simulations that, in agreement with Eq. (2), the halo bias correction roughly grows as 1/k^{2} at low k. This gives rise to a significant and specific signal that has already been used to constrain (Slosar et al. 2008).
In this article, following a previous work devoted to the Gaussian case (Valageas 2009b), we revisit the derivations of the halo mass function and of the bias for primordial nonGaussianity. Although we focus on the local type (1), our approach also applies to any nonGaussian model where Bardeen's potential can be written as the sum of linear and quadratic terms over an auxiliary Gaussian field, that is, where becomes a convolution kernel. (It also extends to cases that contain higher order terms and multiple Gaussian fields.)
After introducing our notations and the quantities needed for our calculations in Sect. 2, we consider the halo mass function in Sect. 3. Here, our aim is to argue that the exponential cutoff of the highmass tail can be obtained exactly from a saddlepoint approach. This is equivalent to the saddlepoint computation of Matarrese et al. (2000), which is often used to model the nonGaussian halo mass function. However, with a different treatment, we simultaneously derive the linear density profile of this saddle point, which allows us to check that the latter is almost insensitive to primordial nonGaussianity, so that shell crossing is not amplified and exact results can be obtained provided one uses the correct linear density threshold, rather than the usual one. We also propose a simple recipe to match the dependence on of the highmass tail while keeping the mass function normalized to unity. Then, in Sect. 4 we consider the twopoint correlation of dark matter halos in real space, following the spirit of Kaiser (1984). Next, taking a Fourier transform we obtain the halo bias in Fourier space. Here, our aim is to show that one does not need to introduce free parameters to match the results of numerical simulations. Moreover, the nonlinear realspace expression is of interest by itself and it also allows one to check whether the ``linearized'' bias is valid on the range of interest. Finally, we conclude in Sect. 5.
2 NonGaussian initial conditions
We focus in this paper on nonGaussianities of the local type, where Bardeen's potential
is of the form (1), with
a Gaussian random field. On scales smaller than the Hubble
radius,
equals minus the Newtonian gravitational potential and the Poisson equation gives in Fourier space (Slosar et al. 2008)
where is the linear matter density contrast, T(k) is the transfer function and D(z) is the linear growth factor, normalized as at high redshift. Unless stated otherwise, we normalize the Fourier transform as
Note that we define by applying Eq. (1) at early times (i.e. ), which is sometimes called the ``CMB convention'', whereas some authors first linearly extrapolate at z=0 (``LSS convention''). Thus, both conventions are related by (Pillepich et al. 2010). Then, defining the timedependent Gaussian field by
we can write the linear density field at redshift z as
with
This reads in real space as
where the kernel only depends on the two vectors ,
as long as the system remains statistically homogeneous, as for the local model (1). The realspace and Fourierspace kernels are related by (note the different normalization from (4))
The relationships (6) and (8) describe any homogeneous model where the Bardeen potential can be expressed as the sum of linear and quadratic terms over some Gaussian field. Thus, our analytical results also apply to other `` type'' models than the ``local'' one (1).
For
we recover Gaussian initial conditions,
,
with a linear density power spectrum
and a twopoint linear density correlation
As usual, it is convenient to introduce the smoothed linear density contrast, , within the sphere of radius q and volume V around position ,
with a tophat window that reads in Fourier space as
Then, in the linear regime, the crosscorrelation of the smoothed linear density contrasts on scales q_{1} and q_{2} and positions and reads as
In particular, is the usual rms linear density contrast on scale q. Then, in the nonGaussian case we define the initial conditions by the same power spectrum (11) for the field and we vary the parameter (for the local type (7)). We still define the variance as in Eq. (15) from the Gaussian field .
In the following sections, where we define dark matter halos as
spherical overdensities, we shall need the average of the kernel
over spherical cells, weighted by the linear correlation (12). Thus, omitting the superscript
for simplicity, we define the quantity
where the spheres of volumes V, V_{1} and V_{2}, and radii q, q_{1} and q_{2}, are centered on the points , and . In Eq. (16) the coordinates and are integrated over all space. In terms of the Fourier kernel this reads as
Thanks to statistical homogeneity and isotropy, the kernel only depends on the lengths k_{1}, k_{2}, and on the angle between both vectors. Then, for spheres that are centered on the same point (i.e. ), Eq. (17) simplifies as
which does not depend on the position of the sphere. Here we defined , and . On the other hand, when we consider the twopoint correlation of dark matter halos, the three spheres in Eq. (16) are chosen among two possible spheres V_{a} and V_{b}, separated by a distance x. Then, we need the two quantities,
and
Their Fourier transforms with respect to the separation read as
and
3 Mass function of dark matter halos
We now extend the analysis of Valageas (2009b) to obtain the mass function of dark matter halos for nonGaussian initial conditions.
3.1 Rareevent saddle point for the density distribution
In a fashion similar to Valageas (2009b), we note that the exponential falloffs of the highmass tail of the halo mass function n(M), and of the overdensity tail of the linear density contrast distribution
,
can be exactly obtained from the constrained maximum,
Here we introduced the probability distribution of the smoothed linear density contrast within the sphere of radius q, which we can take centered on the origin,
In the Gaussian case, , we simply have , as defined in Eq. (13), and the probability distribution is a Gaussian. Then, as stressed in Valageas (2009a,b), in the limit of rare events (e.g., at fixed density contrast in the largescale or highmass limit ), the tails of the distribution are obtained from Eq. (23), where we maximize the statistical weight of the Gaussian field , under the constraint that the smoothed linear density contrast is equal to the value of interest. Equation (23) only gives the leadingorder exponential falloff. Subleading terms, such as powerlaw prefactors, may be obtained by expanding around the saddlepoint , within a steepestdescent method. This also provides the probability distribution of the nonlinear density contrast on scale r in the quasilinear regime, as well as the cumulant generating function (Valageas 2002a,b, 2009a,b)^{}. As pointed out in Valageas (2009b), this also gives the highmass tail of the halo mass function, if halos are defined as spherical overdensities with a fixed nonlinear density threshold . Then, the halo mass M and radius r are related to the Lagrangian radius q and linear density threshold through
and
where the function describes the spherical collapse dynamics. This holds as long as shellcrossing has not extended beyond radius r, so that the usual spherical collapse dynamics at constant mass is valid. This yields an upper bound for the nonlinear density threshold that can be used to define halos to take advantage of the exact asymptotic tail (23). For the preferred CDM model this gives for up to for (the dependence on mass is due to the change of slope of the matter power spectrum ).
In the Gaussian case, the constrained weight (23) simply reads as
,
and we recover the exponential tail of the usual PressSchechter mass function (Press & Schechter 1974), except that the standard threshold
must be replaced by
,
with
for
.
In the nonGaussian case, that is for
,
we can compute the weight (23) by using a Lagrange multiplier .
Thus, we define the action
,
where is the nonlinear functional that affects to the initial condition defined by the Gaussian field the linear density contrast within the sphere of radius q, obtained through Eq. (8). Then, we must look for the saddle point of the action (27), with respect to both and . Thus, differentiating the action (27) with respect to and multiplying by the operator gives
(28) 
whence, using Eq. (8),
Differentiating the action (27) with respect to gives the constraint
whence
Next, we solve the system (29)(31) as a perturbative series over the nonGaussianity kernel (i.e. over powers of the parameter ). At order zero we recover the Gaussian saddlepoint,
At first order we obtain
where the quantity f_{q;qq} is given by Eq. (18).
Figure 1: The radial profile (36, 37) of the linear density contrast of the saddle point of the action . We show the profiles obtained with a CDM cosmology for the masses M=10^{11} and . A larger mass corresponds to a lower ratio at large radii q'/q > 1. We show our results for the Gaussian case (solid line), positive (dashed line) and negative (dotted line), for the local model (7). 

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From Eqs. (32)(35) we also obtain the radial linear density profile of the saddlepoint, up to first order,
We can check that at q'=q it verifies the constraint (30), and at order zero we recover the Gaussian profile (Valageas 2009b). Equations (36, 37) give the integrated density profile, that is, is the mean linear density contrast within the Lagrangian radius q'. The local linear density contrast at radius q', , is given by
(38) 
whence
where f_{0;qq}(q') and f_{q;0q}(q') are given by Eqs. (19, 20).
We show in Fig. 1 the integrated linear density profile (36, 37) obtained for the masses M=10^{11} and , for a CDM cosmology. The dependence on mass is due to the change of slope of the linear power spectrum with scale. We plot our results for the Gaussian case ( , solid lines), large positive ( 10^{3} and 10^{4}, dashed lines) and large negative ( 10^{3} and 10^{4}, dotted lines), for the local model (7). Thus, we can see that a positive increases the relative density (i.e. with respect to the density at radius q) both at small and large radii. The very large values of required to be able to distinguish the curves in the figure imply that for realistic cases ( ) the perturbation of the density profile is very small. Therefore, the values of the upper boundary , which marks the onset of shellcrossing, obtained in Valageas (2009b) for the Gaussian case remain valid up to a very good accuracy. We can note that to obtain a similar deviation from the Gaussian profile we need a larger value of the parameter on a smaller scale. This can be understood from the expression (7), which scales as and grows as k^{2} on very large scales. The same behavior (i.e. a higher sensitivity to localtype nonGaussianity on large scales) is obtained for the bias of dark matter halos, see Eq. (2) above and Sect. 4 below.
Next, we define the constrained weight (23) as
at the relevant saddlepoint . This gives, up to first order,
Therefore, the tails of the probability distribution (23) read as
where we introduced the skewness of the linear density contrast, at first order over ,
Indeed, from Eq. (8) we have at first order
=  
(46)  
=  6 f_{q;qq}.  (47) 
Hereafter, since S_{3}^{(0)}=0, we simply note S_{3}=S_{3}^{(1)}.
3.2 Mass function
Following the PressSchechter approach (Press & Schechter 1974), the mass function that is obtained in the Gaussian case from the probability distribution
reads as
with (using the subscript ``PS'' to distinguish the PressSchechter prediction)
and
Here , where the Lagrangian scale q is related to M by Eq. (26). As stressed in Valageas (2009b), the linear threshold in Eq. (50) must be defined as , as in Eq. (25), which gives for . Then, the exponential tail of (49) is exact,
but the powerlaw prefactor and the lowmass tail of (49) have no reason to be valid (and numerical simulations indeed show that they are not exact). Then, in order to match numerical simulations, one needs to use fitting formulae. One such fit to simulations, which obeys the exact tail (51), is (Valageas 2009b)
Both mass functions (49) and (52) satisfy the normalization
which ensures that all the mass is contained in such halos:
In the nonGaussian case, that is , we may estimate the halo mass function by multiplying the Gaussian one by the corrective factor obtained in Eq. (44),
As explained above, this yields the exact highmass tail (up to first order over ) but it is not expected to hold for the lowmass tail. In particular, this gives a nonGaussian mass function that does not obey the normalization (54). In order to satisfy Eq. (54), while keeping the highmass tail of Eq. (55), a simple procedure is to make use of the scaling (48) and of the normalization (53). Thus, modifying the relationship (50) as (see Eq. (44))
we may use for the nonGaussian mass function
where we use the same scaling function f as for the Gaussian case (48). This recovers the highmass tail (55) and satisfies the normalization (54). Note that Afshordi & Tolley (2008) have recently proposed a similar rescaling as (56), which they reinterprate as a modified effective variance . However, to obtain such a rescaling they use some approximations, such as the decoupling between the local values of the fields and , in the spirit of a peakbackground split approximation, whereas the derivation presented here is asymptotically exact. Equation (57) yields for the ratio of both mass functions at fixed mass M,
Using Eqs. (50), (56), we obtain
If we use the PressSchechter mass function (49), Eqs. (57)(59) give back the result obtained in Matarrese et al. (2000), except for the fact that we use instead of 1.686 as explained above.
Note that Matarrese et al. (2000) also use a saddlepoint approach to derive the halo mass function for nonGaussian initial conditions, expanding at linear order over (or S_{3}), so that their computation is equivalent to the one described above. However, since they first compute the cumulant generating function ( in their notations, or in Valageas 2009b), the constraint (30) is expressed through a Dirac function, written as an exponential by introducing an auxiliary variable , so that they can first integrate over the Gaussian field and next expand over the expression obtained for . The somewhat simpler method described in this article has the advantage of simultaneously giving the density profile (36), (37) of the underlying saddle point. As explained above in Fig. 1, this allows us to check that realistic amounts of primordial nonGaussianity have a negligible effect on this profile, so that the onset of shellcrossing appears for almost the same nonlinear density threshold . This ensures that the rareevent and highmass tails (44) and (55) are exact (at leading order), as long as halos are defined by a nonlinear threshold below the upper bound . The simplicity of the method presented in this article also allows a straightforward application to twopoint distributions, as shown in Sect. 4 below, or to more complex primordial nonGaussianities, which may involve several fields or higher order polynomials as in Eq. (95) below.
Another approach presented in Lo Verde et al. (2008) is to use the Edgeworth expansion, which writes the probability distribution as a series over the cumulants of the nonGaussian variable . In practice, one truncates at the lowest order beyond the Gaussian, that is, at the third cumulant described by S_{3}. Thus, expanding the exponentials (44) and (55) one recovers the results of Lo Verde et al. (2008) at large (i.e. low ). However, in the rareevent limit, where computations rest on firm grounds as explained above, the Edgeworth expansion does not fare very well. In particular, although we only derived the saddlepoint and the argument of the exponential up to linear order over , see Eqs. (42), (43), it is best to keep the exponential as in Eqs. (55) or (57). Indeed, in the rareevent and small limits, the tail (55) can be a good approximation even when the corrective factor is much larger than unity (i.e. we can have the hierarchy ).
Since the lowmass tail (49) does not match numerical simulations, it is sometimes proposed to keep the ratio (58) given by the PressSchechter mass function, and to multiply the fit from simulations of the Gaussian mass function by this factor (Grossi et al. 2007, 2009; Lo Verde et al. 2008). However, this procedure clearly violates the normalization condition (54). Therefore, we suggest to use (58) with the fitting formula obtained from Gaussian simulations, that is, to use Eq. (57), which automatically satisfies the normalization (54). However, there is no reason to expect that the lowmass tail can be exactly recovered by any such procedure, even though by construction it gives the right behavior for the Gaussian case, as the lowmass slope might also depend on in some specific manner.
Figure 2: The ratio , of the mass functions obtained for over the mass function obtained for Gaussian initial conditions, as a function of M for several redshifts. The dotdashed line labeled `` '' is the multiplicative factor (55), the dashed line labeled ``PS'' is Eq. (58) with the PressSchechter mass function (44), which also corresponds to the result of Matarrese et al. (2000) (but with ), the solid line labeled ``f'' is Eq. (58) with the fitting function (52). The data points are results from the numerical simulations of Grossi et al. (2009). 

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We compare in Fig. 2 our results for the halo mass function with numerical simulations from Grossi et al. (2009). They use the LSS convention for , so that in terms of the CMB convention used in this article this corresponds to . We show the ratio of the nonGaussian mass function to the Gaussian one, as given by the multiplicative factor (55), the ratio (58) computed with the PressSchechter mass function (44) or with the fitting function (52). In agreement with Grossi et al. (2009), we find that the ratio (58) computed with the PressSchechter mass function, which also corresponds to the result of Matarrese et al. (2000) (but with instead of 1.686 as explained above), agrees reasonably well with simulations. Using the fitting function (52) or the simple multiplicative factor (55) yields close results in this regime and also agrees with simulations. However, using Eq. (58) with the fitting function (52) appears to agree somewhat better with simulations, especially at low masses. This could be expected from the fact that this procedure ensures that the mass function is properly normalized (in contrast, the simple multiplicative factor (55) is greater than unity and does not reproduce the crossing of both mass functions for ).
On the other hand, let us point out that, contrary to some previous works, we do not need to introduce any adhoc parameter q (e.g., through a change of the form as in Grossi et al. 2009) to obtain a good match with numerical simulations. This decrease in the linear threshold with respect to the standard value (for ) is actually obtained in our approach by using the exact linear threshold , as explained above. Therefore, the advantage of this procedure is that we do not need to run new simulations for other cosmologies to obtain a fit for such a qfactor, since the value can always be computed from the spherical collapse dynamics.
Figure 3: Same as Fig. 2, but with ( upper panel) and ( lower panel). The data points are the results of the numerical simulations of Dalal et al. (2008). 

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Next, we compare our results with numerical simulations from Dalal et al. (2008) in Fig. 3. We again show the ratio as a function of mass, but for a greater primordial nonGaussianity, . For (upper panel), we note as in Dalal et al. (2008) that the prediction of (58) computed with the PressSchechter mass function (44) (i.e. the result of Matarrese et al. 2000, but with ) tends to overestimate the deviations from the Gaussian case (and the simple multiplicative factor (55) fares somewhat worse). However, using Eq. (58) with the correct Gaussian mass function (52) decreases this ratio somewhat (as in Fig. 2) and provides good agreement with the simulations. For (lower panel) the match is not as good. However, in that case the agreement might improve at higher masses, where there are no data points but where the predictions (55) or (58) are asymptotically exact. Moreover, the discrepancy between theoretical predictions and numerical simulations has the same order as the deviation between the different theoretical curves. Since the latter show the same exact highmass behavior (at the leading order given by the exponential cutoff (44)), these deviations show the sensitivity of the mass functions to the details of the theoretical prescriptions. These powerlaw prefactors have not been rigorously derived (and are expected not to be exact for all formulae used here). Therefore, the deviation between these theoretical predictions estimates the theoretical uncertainty for the ratio of the halo mass functions. Then, taking this theoretical uncertainty into account, we can see that the agreement with the numerical results is still reasonable. For practical purposes, when one tries to derive constraints on cosmology from observations of halo mass functions, it would be useful to consider several theoretical prescriptions in addition to the best prediction (58)(52), as in Figs. 2 and 3, so as to take the theoretical uncertainty into account in the analysis.
4 Bias of dark matter halos
4.1 Twocell saddle point and realspace bias
We now consider the bias of dark matter halos, or more precisely their twopoint correlation function. As in Valageas (2009b), following Kaiser (1984),
we identify rare massive halos with positive density fluctuations in
the linear density field. Thus, we first consider the bivariate
probability distribution,
,
of the linear density contrasts
and
within two spheres V_{1} and V_{2}, of radii q_{1} and q_{2}, and separated by the distance s. Note that we distinguish the Lagrangian distance s between the halos measured in the linear density field from their Eulerian distance x
measured in the nonlinear density field. Indeed, since halos have moved
through their mutual gravitational attraction, these two distances are
usually different. Proceeding as in Sect. 3.1 to obtain the rareevent tails,
we are led to introduce the action
which now involves the two Lagrange multipliers , and . We again obtain the minimizer of the Gaussian weight of Eq. (60) by differentiating the action with respect to , and , and we solve these equations as a perturbative series over . At order zero, we recover the Gaussian terms
where we note and from Eq. (15). At first order we obtain
and
where we note for instance f_{1;22}=f_{a;bb}(s) and f_{2;12}=f_{b;ab}(s), with V_{a}=V_{1}, V_{b}=V_{2}, from Eqs. (19), (20). Next, the Gaussian weight reads at order zero
and at first order,
Therefore, in the rareevent limit the tail of the bivariate distribution (60) reads as
with
where is given by Eqs. (68), (69) and and by Eqs. (42), (43) for each sphere V_{1} and V_{2}. Next, following Valageas (2009b), we write the halo twopoint correlation as
where the factor models the effects associated with the mapping from Lagrangian to Eulerian space. This is the local linear density contrast at radius s from a halo of mass (to keep the symmetry ), as given by Eqs. (39), (40). A sufficiently accurate approximation would be to use only the zerothorder term (39), as shown by Fig. 1, but taking the correction (40) into account brings no further difficulty. Here we approximated the nonlinear density contrast by the linear density contrast , since at a large distance where we have . Next, we must express the Lagrangian separation s in terms of the Eulerian distance x. Following Valageas (2009b), at the lowest order where we consider each halo as a test particle that falls into the potential well built by the other halo, we obtain s as the solution of the implicit equation
where is the linear density contrast within radius s of the halo of mass M_{i}, given by Eqs. (36), (37). At a large separation, this relation can be inverted as
which provides an explicit expression for s.
Finally, we define the realspace halo bias as the ratio of the halo and matter twopoint correlations,
Since at a large distance the matter correlation is within the linear regime, , we also write in this limit
which fully determines the halo bias from Eq. (72).
For equalmass halos of radius q, defined by the same threshold
,
the two Lagrange multipliers are equal,
,
and Eqs. (63, 64) and (66, 67) simplify as
and
where we note . This yields, for the Gaussian weight ,
and
Then, the difference defined in Eq. (71), which measures the correlation between rare events in the linear density field, writes as
and
We can check that vanishes for , since all mixed quantities, , f_{1;12} and f_{2;11}, go to zero.
As stressed in Politzer & Wise (1984), realspace formulae such as (72), which are obtained in the rare event (
)
and large separation (
)
limits, do not assume that the exponent
is small. In fact, as shown in Valageas (2009b),
at high redshift one can probe a regime where this exponent is large,
so that one needs to keep the nonlinear form (72), which yields a nonlinear bias. As seen from Eq. (81), this regime corresponds to very massive halos,
,
at fixed (low) ratio
.
Nevertheless, in the regime where
is small (i.e. in the large separation limit
), which covers the cases of interest encountered at low redshift, we can expand the exponential in Eq. (72). Then, at the lowest order over the terms that vanish in the large separation limit, we obtain
(83) 
for equalmass halos, whence
Hereafter we call the bias b^{2}_{M}(x) obtained from Eqs. (84) and (76) the ``linearized'' bias.
Following the approach of Kaiser (1984), Matarrese & Verde (2008) also computed the effect of localtype nonGaussianity (1) on the halo twopoint correlation. Drawing on earlier work by Matarrese et al. (1986), who computed the npoint halo correlations by expanding the relevant pathintegrals and next resumming the series within a largedistance and rareevent approximation, they obtained expressions of the form (72) without the prefactor and (84) without the terms in the first line, which arise from this prefactor, and the last term f_{1;11} in the second bracket. This term arises from the factor in the denominator of expression (80), associated with the effect of primordial nonGaussianity on the onepoint distribution (43). It can be seen as a renormalization of the Gaussian term , since it shows the same scale dependence through the function . The presence of such a term has already been noticed in Slosar et al. (2008) in Fourier space, and Desjacques et al. (2009) points out that it needs to be included to obtain good agreement with numerical simulations. This will give rise to the last term in Eq. (85) and the second term in Eqs. (91) and (93) below. On the other hand, the terms in the first line of Eq. (84), associated with the prefactor in Eq. (72), stem from the mapping from Lagrangian to Eulerian space, and the first bracket in Eq. (84) expresses the (weak) effect of primordial nonGaussianity on this mapping. Another difference between Eqs. (72) and (84) and previous works is that (within some approximation) we pay attention to the difference between Lagrangian and Eulerian distances s and x. This can play a nonnegligible role as seen in Fig. 4 below.
Figure 4: The halo bias b_{M}(x), as a function of , at fixed redshift z=0 and distance x=50 h^{1} Mpc. The solid lines ``b'' are the nonlinear theoretical prediction of Eqs. (72), (73) and (76), for and (i.e. Gaussian case, intermediate line), while the dotdashed lines ``'' are the linearized bias of Eq. (84). The upper dashed line ``s=x'' shows the result obtained in the Gaussian case by setting s=x in Eq. (72). The points are the fits to Gaussian numerical simulations, from Sheth et al. (2001) (crosses) and Pillepich (2010) (circles). 

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Figure 5: The halo bias b_{M}(x) as a function of distance x, at redshifts z=0 ( upper panel) and z=1 ( lower panel) for several masses. We show the cases (solid lines), (dashed lines), and (dotted lines). The divergences at Mpc come from the halo and matter correlations not changing sign at the same distance. 

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Figure 6: The realspace ratio of the correction to the Gaussian bias b_{M}(x,0), divided by the factor x_{50}^{2} with , as a function of distance x. We show the cases (upper lines) and (lower lines) for several masses at redshifts z=0 ( upper panel) and z=1 ( lower panel), from Eq. (72). 

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We show in Fig. 4 the halo bias b_{M}(x) as a function of at fixed distance x=50 h^{1} Mpc and redshift z=0, for and . We display both the nonlinear result of Eq. (72) and the linear result of Eq. (84). In addition, for the Gaussian case we also display the bias obtained by setting s=x in Eq. (72). As in Valageas (2009b), for the Gaussian case ( ) we obtain good agreement with the fits to numerical simulations of Sheth et al. (2001) and Pillepich (2010). Moreover, we can see that it is important to take the displacement of the halos into account through Eq. (73), as the approximation s=x significantly overestimates the bias. We checked that using the simpler Eq. (74) gives a close result to the one obtained with Eq. (73) and also agrees with the simulations. Thus, for practical purposes it is sufficient to use Eq. (74). We can see that for all cases shown in Fig. 4 the linear bias from Eq. (84) gives results that are very close to the fully nonlinear expression (72). This justifies the use of such linearized expressions in this regime. This will be especially useful in Sect. 4.2, where we consider the bias of dark matter halos in Fourier space. Indeed, it is easier to take the Fourier transform of Eq. (84), which allows us to recover the results obtained in previous works. Thus, one interest of the realspace results (72)(84) is to provide a check on whether linearized predictions (i.e. where the halo correlation only involves the matter power spectrum at linear order) are valid. Then, in agreement with those studies (which were mostly performed in Fourier space and led to Eq. (2), see Dalal et al. 2008; Slosar et al. 2008), and with Eq. (84), we can see that the deviation from the Gaussian bias, , grows linearly with b_{M}(x,0) at large bias and has the same sign as .
Figure 7: The halo ( ) and matter () twopoint correlations at redshifts z=0 ( upper panel) and z=1 ( lower panel). We show the curves obtained for the masses M=10^{13} and . 

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Next, we display in Fig. 5 the dependence on the distance x of the bias obtained for several masses at redshifts z=0 and z=1. More precisely, we show the ratio , since the halo and matter correlations do not change sign at the same point. We plot the cases , as well as the Gaussian case . While the Gaussian bias is roughly constant on large scales, up to Mpc (in agreement with previous studies, Mo & White 1996; Mo et al. 1997), the nonGaussian bias shows a strong scale dependence, with a deviation from the Gaussian bias that roughly grows as x^{2} up to Mpc. This agrees with the k^{2} behavior observed in Fourier space, see Eq. (2) above (Dalal et al. 2008; Slosar et al. 2008) and Eq. (93) below.
To see the scaling of the realspace correction to the Gaussian bias more clearly, we show in Fig. 6 the ratio , with . We display the results obtained from Eq. (72) for several masses at z=0 (upper panel) and z=1 (lower panel) for . We can see that over the range 30<x<90 h^{1} Mpc all curves roughly collapse onto one another. This means that the realspace correction roughly scales as over this range, which roughly agrees with the Fourierspace scaling (2) (here we neglected any constant offset, such as the factor 1 in Eq. (2), see the discussion of Eq. (94) below). It appears that our predictions scale more closely as x^{2}, as shown in Fig. 6, than as , which would be suggested by Eqs. (2), (93) (at these scales the transfer function already deviates from unity). This agrees with the behavior observed in Fourier space in Fig. 10 below. The masses shown in Fig. 6 span the range 1.2<b_{M}(x,0)<5.9 at x=50 h^{1} Mpc and z=0, and 2.4<b_{M}(x,0)<15.4 at z=1, so that the linear scaling with b_{M}(x,0) of the correction appears to be a good approximation.
Below 30 h^{1} Mpc higher masses show steeper scale dependence for . At very large distance, x > 100 h^{1} Mpc, the oscillations seen in Figs. 5 and 6 are caused by the baryon acoustic oscillation. Indeed, the baryon oscillations seen in the halo and matter twopoint correlations are not exactly proportional, since the halo correlation is not exactly proportional to , even in the Gaussian case and in the linear regime (for instance it involves the smoothing scale q, see Eq. (84)). This yields the nonmonotonic behavior seen in Figs. 5 and 6 around 100 h^{1} Mpc. For the same reason, the halo and matter correlations do not exactly vanish at the same distance, which gives rise to the divergent spike at Mpc. These features simply mean that it is no longer useful to work with the bias b_{M} on these scales, which only makes sense if the halo and matter correlations are roughly proportional. In this range, where the correlations show some oscillations and change sign, it is no longer a good approximation to write the halo correlation in terms of the matter correlation multiplied by some slowly varying bias factor. Then, one instead needs to directly study the halo and matter correlations themselves.
Thus, we compare in Fig. 7 the halo and matter twopoint correlations. We focus on large scales to see how the baryon acoustic oscillation is modified when one uses massive halos as a tracer of the initial matter power spectrum. In agreement with previous works (Desjacques 2008), we can see that the oscillation is strongly amplified for massive halos that have a strong bias. This amplification still holds for significant primordial nonGaussianity ( ), although it appears to be slightly lower for positive . Moreover, the peak of the oscillation shows no significant shift, so that a measure of its position appears to be a robust ruler for constraining cosmology, independently of the halo bias and of the primordial nonGaussianity. In contrast, the distance at which the twopoint correlation changes sign is not significantly modified as one goes from the matter to the halo correlation in the Gaussian case, but it is fairly sensitive to the primordial nonGaussianity. In particular, a positive shifts this point to a greater distance. However, theoretical and observational error bars may be too large to use this effect to constrain in a competitive manner compared to other probes.
4.2 Fourierspace bias
Rather than the realspace twopoint correlation, recent works have mostly studied the effect of primordial nonGaussianity on the halo power spectrum, where at lowest order the Poisson Eq. (3) directly gives an estimate of the form (2) for the deviation from the bias obtained with Gaussian initial conditions (Dalal et al. 2008; Slosar et al. 2008).
It is not convenient to take the Fourier transform of the nonlinear correlation (72), but at moderate redshifts, the linearized form (84) provides a very good approximation. Then, if we also make the approximation
,
which is valid at the lowest order, the Fourier transform of Eq. (84) readily gives the halo power spectrum as
where the quantities are obtained from Eqs. (21), (22). As discussed below Eq. (84), the terms and in Eq. (85) have already been obtained by Matarrese & Verde (2008), following Kaiser (1984) by identifying massive halos with rare fluctuations in the linear density field. Defining the Fourierspace bias as (note that this is not the Fourier transform of the realspace bias (75))
where we used at low k for the matter power spectrum, we obtain b_{M}^{2}(k) from Eq. (85). We can also obtain the bias of differentmass halos in a similar fashion, first expanding Eq. (72) and next taking the Fourier transform. To consider the displacement of the halos (i.e. ), we can also make the approximation
where we use the explicit expression (74) for s(x). This expresses that Lagrangianspace wavelengths (i.e. measured in the linear density field) correspond to smaller Eulerianspace wavelengths (i.e. measured in the nonlinear density field) because of the displacement of massive halos, which usually have come closer because of their mutual attraction (x<s). This also follows the spirit of the Hamilton et al. (1991) ansatz, translated in Fourier space in Peacock & Dodds (1996).
For large negative
the halo power spectrum (85)
can become negative, because we looked for an expression of the halo
correlation, or of the halo power spectrum, at linear order over
as in Eq. (85). However, the power spectrum P_{M}(k) must be positive by definition. Then, in cases where expression (85) turns negative one should consider higherorder terms over
,
which would ensure that the power spectrum remains positive.
Nevertheless, since such highorder terms are beyond the scope of this
article, we consider below the following simple procedure that ensures
that P_{M}(k), or the squared bias b_{M}^{2}(k), remain positive. Up to linear order over
,
the bias (86) reads as
where is the deviation from the Gaussian halo power spectrum, and we expanded the squareroot. Then, we may use the last expression (88) as the prediction of the halo bias. This amounts to making the transformation
where the two sides only differ by higherorder terms over (as up to linear order over ) and the right side is always positive.
Figure 8: The correction to the halo power spectrum due to primordial nonGaussianity. We show the ratio for (upper curves) and (lower curves). The solid and dashed curves that are almost indistinguishable are Eqs. (85) and (87). The dotdashed curves that are above the solid curves at low k correspond to Eq. (89) which ensures that the halo power spectrum is always positive. The theoretical predictions are for M=2 and the data points are the numerical simulations of Desjacques et al. (2009), for M>2 . 

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Figure 9: Same as Fig. 8, but for M=1.5 . The data points are the numerical simulations of Desjacques et al. (2009), for 10^{13}<M<2 . 

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If we take the limit of very rare events, which is
in Eq. (85), we can only keep the last two terms (note that
,
see Eq. (92) below),
At low k, with , and at linear order over , the square root of Eq. (86) gives with Eq. (90) the bias
The first term, , is the result obtained by Kaiser (1984) for rare massive halos (i.e. with a strong bias) for Gaussian initial conditions. It is interesting to note that the second term gives a scaleindependent correction to the Gaussian bias. This term has already been noticed in Slosar et al. (2008) and Afshordi & Tolley (2008). As stressed in Desjacques et al. (2009), taking this term into account is required to obtain a good match to numerical simulations. We checked that this is indeed the case to match the numerical results in Figs. 8 and 9 below using Eq. (85). The last two terms depend on the wavenumber k. There is an additional term that we neglect in this paper, which arises from the dependence of the matter power spectrum on (i.e. the denominator in Eq. (86)). However, since this term is much smaller than the other ones, we disregard it here (see Desjacques et al. 2009). For the local nonGaussianity (7), we find from Eqs. (21), (22)
which yields from Eq. (91)
Thus, we recover the k^{2} dependence at low k brought by the localtype nonGaussianity (1), through the factor in the last term. The second term in Eq. (93) is the constant shift due to the nonGaussianity noticed above.
In spite of the k^{2} dependence at low k obtained in Eq. (93) for the halo bias, the realspace halo twopoint correlation is welldefined and finite, as seen in Sect. 4.1. Indeed, Eq. (93) only applies to a limited range, and one cannot write the realspace twopoint correlation as a Fourier transform of the form , which would diverge at low k. Thus, the advantage of the realspace approaches, such as the one described in this paper, is that we obtain welldefined results in both real space and Fourier space, and we do not need to regularize integrals by introducing a counterterm associated with a surveysize window, as in Wands & Slosar (2009). This is reassuring, since one does not expect the halo correlation on a given scale to depend on the size of the survey. Mathematically, the lack of divergence in our approach comes from the fact that it is the halo power spectrum itself which contains a term of the form , see Eqs. (85) and (92), and it is only by expanding the squareroot as in (88), , that b can be written as in Eq. (93). As we shall see below, in Figs. 8 and 9, the expression (85) is sufficient to explain the behavior observed in numerical simulations, without introducing worrying divergences.
We compare in Figs. 8 and 9 our results for the Fourierspace bias b_{M}^{2}(k) with numerical simulations from Desjacques et al. (2009). Since the dependence on mass is rather weak, we show our results in Fig. 8 for M=2 , whereas the data points are for M>2 , and we show our results in Fig. 9 for M=1.5 , whereas the data points are for 10^{13}<M<2 . The predictions (85) and (87) are almost indistinguishable in this regime, and they agree reasonably well with the simulations, except at low k for where they give a negative halo power spectrum. Equation (89), gives a much better fit to simulations at low k for negative , as could be expected from the fact that it always gives a positive halo power spectrum. However, a priori one should not give too much weight to this improved accuracy in this regime. Indeed, as is clear from Eqs. (88), (89), the solid and dotdashed curves in Figs. 8, 9 only differ by terms of order and beyond. Since all our results have been derived at linear order over , one can expect that Eq. (89) does not include all terms of order . Then, although for practical purposes it is better to use Eq. (89) in this regime (i.e. negative at low k), it is still useful to also consider Eq. (85), as the deviation between both predictions should give an estimate of the theoretical uncertainty.
In contrast to some previous approaches (e.g., Grossi et al. 2009; Desjacques et al. 2009), the good agreement with numerical simulations shown in Figs. 8 and 9 is obtained from Eq. (85) without any fitting parameter (such as the rescaling parameter q in Grossi et al. 2009; or the mass function parameters in Desjacques et al. 2009). As for the halo mass function studied in Sect. 3.2, the role of this parameter is partly played by the use of the exact linear threshold , which is predicted by the spherical dynamics of the rareevent saddle points. This makes formulae such as Eq. (85) fully predictive for any values of cosmological parameters.
Figure 10: The Fourierspace ratio of the correction to the Gaussian bias b_{M}(k,0), divided by the scaling factor defined in Eq. (94), as a function of wavenumber k. We show the case for several masses, M=10^{13} (blue), 10^{14} (red), and (green) at redshifts z=0 ( left panel) and z=1 ( right panel). The curves labeled ``s=x'', ``s>x'', and ``P_{M}>0'', correspond to Eqs. (85), (87), and (89), respectively. 

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As in Fig. 6, in order to see the scaling of the correction
to the Fourierspace Gaussian bias more clearly, we show the ratio
in Fig. 10, where we now define
The correction obtained from a simple peakbackground split argument instead gives , as in Eq. (2), see Dalal et al. (2008); Slosar et al. (2008). However, in our formalism, both the leading term (third term in Eq. (85)) and the subleading term (first term in Eq. (85)) are modified by primordial nonGaussianity (the terms in the brackets that follow). The term arises from the Lagrangian to Eulerian space mapping (the prefactor in Eq. (72)), and it corresponds to the factor 1 in the more usual Eq. (2), which is not modified in the simplest model. However, in general we can expect nonGaussianities to also affect this Lagrangian to Eulerian space mapping, and our model gives an estimate of this effect through Eq. (40). Therefore, we scale with b_{M}(k,0) rather than with [b_{M}(k,0)1] in Eq. (94). We show the results obtained at redshifts z=0 and z=1 in Fig. 6 for several masses. The curves labeled ``s=x'', ``s>x'' and ``P_{M}>0'' correspond to Eqs. (85), (87), and (89) respectively. We can see that the collapse of the curves obtained for different masses is not exact, as could be expected since Eq. (85) is more complex than Eq. (94), but the scaling (94) still captures most of the dependence on the halo mass M. In fact, most of the dispersion seen in Fig. 6 does not arise from the different masses but from the various approximations (85), (87), and (89). This should provide an estimate of the theoretical uncertainty. As in Figs. 8 and 9, Eq. (89), which involves higherorder terms over to ensure that P_{M}(k) is always positive, rises above Eq. (85) at low k (k<0.01 h Mpc^{1}) and is indistinguishable at higher k, whereas Eq. (87) rises above (85) at high k (k>0.01 h Mpc^{1}), where the correction is quite small, and is indistinguishable at lower k. Approximations (89) and (87) are likely to be most accurate at low and high k, respectively, see also Figs. 8 and 9. In any case, we can see that the scaling with wavenumber, , is slightly broken, although this still provides a good approximation. As in the numerical simulations of Desjacques et al. (2009), the ratio is suppressed at high k, which means that the correction decreases slightly faster than over the range 0.010.1 h Mpc^{1}. In agreement with the realspace behavior seen in Fig. 6, it appears that the correction scales slightly more closely as k^{2} than as over this range, although neither of these two behaviors is exact.
5 Conclusion
We have shown in this article how to extend to nonGaussian initial
conditions the computation of the mass function and of the bias of dark
matter halos presented in Valageas (2009b)
for the Gaussian case. This relies on a saddlepoint approach that
allows to derive the highmass asymptotic tails of the quantities of
interest from the statistical weight of the initial conditions,
supplemented by additional nonlinear constraints. Then, focusing on the
case of ``
type''
primordial nonGaussianity, where the linear gravitational potential
can be written as the sum of linear and quadratic terms over an
auxiliary Gaussian field ,
we explained how to obtain the relevant saddlepoints as a perturbative series over the nonlinear parameter
.
This method is very general, and it applies to any case of small
primordial nonGaussianity, where Bardeen's potential
(or equivalently, below the Hubble radius, the gravitational
potential or the density field) can be written as a polynomial over a
Gaussian field as
where the nonlinear parameters f_{i} are small. Then, following the method presented in Sects. 3, 4, the onecell and twocell saddle points (associated with the onepoint and twopoint density distributions, whence the halo mass function and bias) can be computed as a perturbative series over the coefficients f_{i}, which need not be of the same order. This includes the case of a cubic term in particular. Of course, it also extends to the cases where the coefficients f_{i} are not mere numbers but convolution kernels and to several Gaussian fields (which can have a nonzero crosscorrelation). In all such cases, the highmass asymptotics are set by the statistical weight of these Gaussian fields, taken at the saddle point associated with the maximization of this weight under appropriate nonlinear constraints that express the mapping from to the relevant quantity, such as the nonlinear density within a spherical cell.
Focusing on the case of localtype primordial nonGaussianity, we described how to obtain, up to linear order over , the onecell saddle point associated with the probability distributions and of the linear and nonlinear density contrasts within spherical cells. This gives the quasilinear limit of these distributions, as well as the highmass exponential falloff of the halo mass function. One advantage of our method is that it allows us to explicitly check that realistic amounts of primordial nonGaussianity have no significant effect on the density profile of this saddle point. This ensures that shell crossing appears for (almost) the same nonlinear density (see Valageas 2009b), so that the highmass tail of the halo mass function can be derived provided halos are defined by a nonlinear density threshold that is below this upper bound (which is indeed the case). Although this procedure only gives the highmass tail, we proposed a simple change of variable, applied to the mass function fitted to Gaussian numerical simulations, that obeys this highmass asymptotic while keeping the mass function normalized to unity. If one uses the PressSchechter mass function, this gives back the result of Matarrese et al. (2000), but we argue that this procedure is somewhat more natural if one wishes to recover a more accurate mass function in the Gaussian case. We also checked that this agrees with results from nonGaussian numerical simulations.
Next, we applied this method to the twopoint correlation of massive halos, following the approach of Kaiser (1984). As in Valageas (2009b) we take the displacement of halo pairs under their mutual gravitational attraction into account. This gives the realspace halo correlation , whence the realspace bias b_{M}(x). Since this approach does not assume that the halo correlation is weak, the nonlinear formula it yields can be used to check whether the ``linearized'' form (where one only keeps the linear term over the matter correlation ) is a good approximation in the regime of interest. As expected, we find that the correction to the Gaussian bias grows with b_{M}(x,0) and with scale, roughly as , up to Mpc. Beyond this scale, the baryon acoustic oscillation and the fact that the halo and matter correlations do not change sign at the same point lead to strong oscillations and divergent spikes for b_{M}(x). This means that, for x > 100 h^{1} Mpc, the bias is no longer a useful quantity, and one should directly work with the halo and matter correlations. In agreement with Desjacques (2008), we find that the twopoint correlation of massive halos, which have a large bias, strongly amplifies the baryon acoustic oscillation. In addition we also obtain the modifications associated with primordial nonGaussianity. The baryon oscillation remains strongly amplified, with a small shift, but somewhat less so for positive .
Finally, we used the ``linearized'' form of the halo twopoint correlation to derive the halo power spectrum and the halo bias in Fourier space. We also give a simple recipe that ensures that the halo power spectrum always remains positive. (This only differs from the direct prediction by terms of order and higher.) We obtain good agreement with numerical simulations without introducing any free parameter. Moreover, the two formulae described above allow one to estimate the range over which linear approximations over are sufficient. Thus, we find that terms of order start playing a role at low k ( k < 0.01 h Mpc^{1}) for large negative ( ), where the direct formula would give a negative power spectrum.
We also pointed out that the k^{2} behavior observed at low kfor the halo bias does not imply any divergence for the realspace twopoint correlation. Indeed, this behavior is only obtained within a certain limit, and it is the halo power spectrum itself (i.e. rather than ) that shows this k^{2} factor. We showed that this is sufficient to explain the behavior observed in numerical simulations. Moreover, it avoids the need to introduce counterterms, that depend on the size of the survey, so as to obtain finite realspace correlations. This is an advantage of realspace approaches, such as the one presented in this paper, which are better suited to describing the nonlinear effects associated with the bias of massive halos.
These results, which do not involve free parameters except for the mass function (if one requires its full shape, where one needs the fit to numerical simulations for Gaussian initial conditions) should be useful for constraining primordial nonGaussianities from observations of largescale structures. Thus, neither the highmass tail of the halo mass function nor the bias require rescaling parameters (such as ), because such a correction to the linear threshold is achieved through the use of the exact linear threshold predicted by the spherical dynamics of rareevent saddle points. This makes this approach more predictive than some of the previous works, since one does not need to run new simulations to fit for such qfactors in order to investigate other cosmologies. In particular, as discussed above for Eq. (95), the method presented in this article is quite general and can be applied to a large class of models. Moreover, since it provides results in both real space and Fourier space (i.e. the halo twopoint correlation and power spectrum), it gives a complete and consistent description of halo clustering. As for previous approaches, the most reliable use of these models to constrain cosmology is to take advantage of the specific shape of the dependence on mass (for the mass function) or scale (for the bias) brought by primordial nonGaussianity to constrain , rather than the change in the amplitude at a given mass or scale.
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Footnotes
 ...2009a,b)^{}
 As shown in Valageas (2009a), for the closely related adhesion model, where the same procedure can be applied, one can explicitly check that the asymptotic results obtained by this approach agree with the complete distribution that is exactly known for two cases (Brownian and whitenoise linear velocity in 1D, corresponding to a powerlaw linear density power spectrum with n=2 and n=0).
All Figures
Figure 1: The radial profile (36, 37) of the linear density contrast of the saddle point of the action . We show the profiles obtained with a CDM cosmology for the masses M=10^{11} and . A larger mass corresponds to a lower ratio at large radii q'/q > 1. We show our results for the Gaussian case (solid line), positive (dashed line) and negative (dotted line), for the local model (7). 

Open with DEXTER  
In the text 
Figure 2: The ratio , of the mass functions obtained for over the mass function obtained for Gaussian initial conditions, as a function of M for several redshifts. The dotdashed line labeled `` '' is the multiplicative factor (55), the dashed line labeled ``PS'' is Eq. (58) with the PressSchechter mass function (44), which also corresponds to the result of Matarrese et al. (2000) (but with ), the solid line labeled ``f'' is Eq. (58) with the fitting function (52). The data points are results from the numerical simulations of Grossi et al. (2009). 

Open with DEXTER  
In the text 
Figure 3: Same as Fig. 2, but with ( upper panel) and ( lower panel). The data points are the results of the numerical simulations of Dalal et al. (2008). 

Open with DEXTER  
In the text 
Figure 4: The halo bias b_{M}(x), as a function of , at fixed redshift z=0 and distance x=50 h^{1} Mpc. The solid lines ``b'' are the nonlinear theoretical prediction of Eqs. (72), (73) and (76), for and (i.e. Gaussian case, intermediate line), while the dotdashed lines ``'' are the linearized bias of Eq. (84). The upper dashed line ``s=x'' shows the result obtained in the Gaussian case by setting s=x in Eq. (72). The points are the fits to Gaussian numerical simulations, from Sheth et al. (2001) (crosses) and Pillepich (2010) (circles). 

Open with DEXTER  
In the text 
Figure 5: The halo bias b_{M}(x) as a function of distance x, at redshifts z=0 ( upper panel) and z=1 ( lower panel) for several masses. We show the cases (solid lines), (dashed lines), and (dotted lines). The divergences at Mpc come from the halo and matter correlations not changing sign at the same distance. 

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In the text 
Figure 6: The realspace ratio of the correction to the Gaussian bias b_{M}(x,0), divided by the factor x_{50}^{2} with , as a function of distance x. We show the cases (upper lines) and (lower lines) for several masses at redshifts z=0 ( upper panel) and z=1 ( lower panel), from Eq. (72). 

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In the text 
Figure 7: The halo ( ) and matter () twopoint correlations at redshifts z=0 ( upper panel) and z=1 ( lower panel). We show the curves obtained for the masses M=10^{13} and . 

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In the text 
Figure 8: The correction to the halo power spectrum due to primordial nonGaussianity. We show the ratio for (upper curves) and (lower curves). The solid and dashed curves that are almost indistinguishable are Eqs. (85) and (87). The dotdashed curves that are above the solid curves at low k correspond to Eq. (89) which ensures that the halo power spectrum is always positive. The theoretical predictions are for M=2 and the data points are the numerical simulations of Desjacques et al. (2009), for M>2 . 

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In the text 
Figure 9: Same as Fig. 8, but for M=1.5 . The data points are the numerical simulations of Desjacques et al. (2009), for 10^{13}<M<2 . 

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In the text 
Figure 10: The Fourierspace ratio of the correction to the Gaussian bias b_{M}(k,0), divided by the scaling factor defined in Eq. (94), as a function of wavenumber k. We show the case for several masses, M=10^{13} (blue), 10^{14} (red), and (green) at redshifts z=0 ( left panel) and z=1 ( right panel). The curves labeled ``s=x'', ``s>x'', and ``P_{M}>0'', correspond to Eqs. (85), (87), and (89), respectively. 

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In the text 
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