Introduction
- the COBE level of anisotropies at l of order 10
- the present day abundance of clusters.
- (1) the high velocity tail of the 3D velocity PDF of clusters or
- (2) the correlation length of clusters more massive than some 1014/h solar masses, corresponding to a mean intercluster separation of approximatively 60 Mph/h
We have carried out two large collisionless simulations of an isocurvature CDM model (ICDM) for structure formation which has been recently proposed by Peebles, where the one point probability function of the overdensity of the matter field is originally strongly non-gaussian. The simulations are normalized to provide a reasonable fit to simple large-scale constraints: they match both :
The goal of these simulations is twofold: to obtain reliable cluster statistics for this non-gaussian model and to build the skeleton on which we could later graft precriptions for galaxy formation and evolution to make photometric predictions at various redshifts.
Here, we first consider first and second order statistics (in contrast to more elaborate ones like the higher-order moments or the bipsectrum) of the DM and verify that second-order statistics like the power spectrum and the correlation function agree reasonably well with standard gaussian simulations on scales of interest, while the one point PDF of density or momentum show the expected strong signature of non-ganussianity. We then compare with analytical results: in particular, we find good agreement of the mass function of dark haloes with the formulae derived by Robinson and Baker (2000), hereafter RB00.
Besides the mass function of galaxy clusters at z=0, we also look for a possible signature of the strong non-gaussianity of the model in their 3D peculiar velocities and in the amplitude of their spatial clustering as a function of abundance.
We show that:
are critical points able to falsify the ICDM model, and therefore probably any strongly positively skewed initial non-gaussian DM distribution, against the currently favoured gaussian paradigm.
Note that after this project has been started, the
However, we believe this model is still interesting because of its strong
intrinsic non-gaussianity. Our final objective is therefore to check
to whether and how this pristine non-gaussianity is echoed in mock cluster
or in galaxy catalogues which we plane to simulate using
a semi-analytic algorithm, or if has been erased
by gravitational evolution and biased galaxy formation.
References
Some selected publications :
-
An Isocurvature CDM Cosmogony. I. A Worked Example of Evolution Through Inflation,
P. J. E. Peebles,
1999, ApJ, 510, 523, astro-ph/9805194
-
An Isocurvature CDM Cosmogony. II. Observational Tests,
P. J. E. Peebles,
1999, ApJ, 510, 531, astro-ph/9805212
-
Evolution of the cluster abundance in non-Gaussian models,
J. Robinson and J. E. Baker
2000, MNRAS, 311, 781, astro-ph/9905098
Normalization
-
We normalize the model by matching two constraints :
- the CMB anisotropies on large scales by means of CMBFAST (Seljak & Zaldarriaga 1996)
- the cluster abundance observed at z=0 with the help of the theoretical mass function derived in a similar way as RB00.
The precise constraints come from Bunn & White (1997) at l=10 and from the cluster data reported by Bahcall & Cen (1993) : 2 clusters more massive than 4 1014 per 1003 (Mpc/h)3.
Although the tilt proposed by Peebles enables one to fit approximately the CMB anisotropies on large scales where the Sachs-Wolfe effect dominates (see Figure 1 of Peebles 1999a), the isocurvature nature of the fluctuations in the model has been ruled out by the BOOMERANG measurements of the position and amplitude of the first acoustic peak. Refer to Hu (1998) for a description of these issues.
We have sticked to the COBE normalization on
large scales, without bothering abouth whether the peaks are consistent
with observations. However, since a larger baryon fraction yields a higher
acoustic peak in the case of isocurvature fluctuations, we have
increased Omegab to the upper bound suggested by Peebles at
a value of 0.05. The next plot gives the point of agreement in the
(ninit, sigma8) plane for three fractions of
baryons.
The Figure below shows the
mismatch of the ICDM model with the BOOMERANG
data, represented by crosses.
Parameters of the collisionless simulations
We have perfomed two large dissipationless simulations of the ICDM model normalized as above. Both have Nparts}=2563 and box sizes of 162 and 600 Mpc/h respectively.
The small-box simulation is aimed at studying galaxy formation: its size has been chosen so that the particle mass is similar to that of the \GIF simulation described in Kauffmann et al. (1999).
The large-box simulation has been carried out to get reliable statistics of massive clusters, and to probe larger scales where the power spectrum bends due to the transfer function.
The runs were performed on 128 processors of the
CRAY T3E at the Computer Center (RZG) of
the Max-Planck Society at Garching. The
600 Mpc/h simulation was entirely completed with HYDRA, while the 162
Mpc/h simulation was started with HYDRA
(Couchmann et al. 1995) until clustering became
significant at z=1 and from then finished with
GADGET (Springel et al 2000). The force
resolution was kept fixed at comoving softening length of 30 kpc/h in
both cases, and the starting redshift was taken zinit=50.
The next table summarizes the parameters of the simulations. L,
N, M, lsoft, zinit give the
box size (units Mpc/h), the number of particles,
their mass (units 1010/h solar masses), the comoving softening lengh (units
kpc/h) and the starting redshift. The cosmological parameters are
recalled, and ninit is the power-law index of the power
spectrum of the matter density field, input to the code before it
computes the initial conditions.
| L | N | M | lsoft | zinit | Omega0 | Lambda | Omegab | h | Sigma8 | ninit |
| 600 | 2563 | 71.1 | 30 | 50 | 0.2 | 0.8 | 0.05 | 0.7 | 0.8 | -1.8 |
| 162 | 2563 | 1.40 | 30 | 50 | 0.2 | 0.8 | 0.05 | 0.7 | 0.8 | -1.8 |
Slices through the simulations
The picture below compares a slice of width 162 Mpc/h and
thickness 15 Mpc/h extracted the large numerical collisionless simulation
of the non-gaussian ICDM model to a slice of width 141 Mpc/h and thickness 15
Mpc/h cut in the GIF simulation of the gaussian adiabatic
LCDM scheme wich is currently favoured by the data.
Click the image to get the full extent (warning: 1044x2088 pixels
in gif format, some 2 MBytes). From there you will be able to
download the corresponding .gif.tar.gz file.
Dark matter statistics
-
We present here the DP of the present-day dark matter density
field, the power spectra (at starting redshift and at
present time) and finally the present-day correlation functions
of the dark matter distribution in the two simulations.
- PDF of the DM density field
- Power spectrum
- Correlation functions
The Figure below compares the present-day density PDF smoothed on 8 Mpc/h in the non-gaussian ICDM case (dashed line) to the corresponding PDF of the gaussian LCDM GIF simulation (dotted line). The solid line is the fit to the initial overdensity smoothed on 8 Mpc/h measured by RB00 in their simulations.
Due to finite box effects, note however that the exact shape of the PDF smoothed on large scales can depend on the scheme used to generate the initial conditions for the non-gaussian case, and also on the power spectrum. RB00 have not applied any transfer function to their chi-squared simulation, so their fits are given as indicative only.
We compute the power spectrum for each simulation, at zinit=50 using a TSC scheme (which we do not deconvolve when plotting) and at z=0 using NGP. The next Figure shows in solid line the power spectrum measured at z=0, in dashed line the initial power spectrum at z=50 and in dotted lines its linear extrapolation at z=0.
The data overplotted (withe error bars) have been measured in the PSCz catalogue of Sutherland et al 1999. To further guide the eye, we plot in dash-dotted line a power-law of slope n=-1.8, our theoretical input. We have checked that the measured growth of the large-scale modes is fully consistent with linear theory over this redshift range.
The left and right panels correspond to the 600 and 162 Mpc/h size
simulations.
We now discuss features of the linear power spectrum, leaving aside the analysis of the onset and development of nonlinearity, since we think it would make more sense to perform it separately or together with our mock ICDM galaxy catalogues. However, note that a theoretical basis for the analysis of the evolution of the DM power spectrum in non-gaussian models is given by Seto (2001).
Already at the initial redshift, one clearly notices a numerical artifact inducing a shallow slope of the power spectrum. As found by RB00, the actual initial slope realized in both simulations is close to -1.6 for wavenumbers k of order 0.1 h/Mpc. Furthermore, there is substantial power loss at the first modes in both simulations, again comparable to that found by RB00 and White (1999). Of course, the fact that the power index measured at the first modes in the 600 Mpc/h simulation is substantially greater than the one defined by the first modes of the 162 Mpc/h simulation is due do the transfer function.
At z=0, the somewhat low normalization of the simulations (sigma8=0.8) translates into a galaxy bias of b8=1.56, which is hardly noticeable in the power spectrum of the 162 Mpc/h simulation, due to the strong non-linear evolution.
In both boxes, one is confronted with a strong non-linear evolution at small scales, which dramatically flattens the spectrum to n=-1.1, already on scales as large as k=0.1 h/Mpc.
Due to the shallow slope of the power spectrum, we expect a correlation function steeper than predicted. For a pure power-law spectrum of the dark matter overdensity with index n, the correlation function is proportional to r-(n+3), yielding a theoretical r-1.22. Since the scales probed by the correlation function are non-linear at z=0, we use of course the measured, non-linear power spectrum to obtain the right slope.
The index of the DM correlation function measured in both simulations is closer to -1.8, as predicted from a n=-1.1 non-linear power spectrum on scales of interest (say, 5 to 10 Mpc/h). The correlation function matches surprisingly well the gaussian results obtained in the LCDM model, for instance, both in slope and in amplitude. The next plot shows the mass correlation function for the two small and large ICDM simulations, in dashed and dash-dotted lines, compared to the LCDM \GIF mass correlation function, in solid line.
A power-law fit to
the 600 Mpc/h needed to compute the correlation
length of haloes has a slope of -1.86 and
r0=4.3 Mpc/h.
Mass functions
-
The next Figure compares the z=0 mass function of haloes measured
in the smal- and large-box simulations (dashed and dash-dotted lines)
with the predictions of the
extended Press-Schechter theory of RB00. The haloes have been identified using a
friend-of-friend FOF algorithm (Davis et al. 1985)
with a linking length of b=0.2 times the mean interparticle
separation, as is usually done when dealing with gaussian fluctuations
(Mo et al. 1996). The agreement between simulations and analytical
predictions is excellent.
Peculiar velocities of clusters
-
The large-scale velocity field as probed by the peculiar motions of
clusters of galaxies identified in simulations
has been studied by, e.g., Bahcall et al. 1994 and Moscardini et al. 1996. They
found tha for gaussian, CDM-like, initial fluctuation fields, the
modulus of the present-day three-dimensional velocity field of the simulated
clusters is well fitted by the expected maxwellian.
- (1) this model has an evolved, large-scale velocity PDF which is still non-gaussian at z=0 and of course if
- (2) the observations of cluster peculiar velocities are consistent with gaussianity, an agreed proposition.
When comparing to observations however, they found that the observed long tail in the cumulative probability P(>v) of the clusters having a 1D velocity greater than some 1500 km/s was not reproduced in numerical simulations. Bahcall and Oh (1996) later showed how the observed tail was due to large velocity uncertainties, and that the new data of Giovanelli et al. 1997 were well accomodated by the gaussian LCDM model.
As noted by Kofman et al. 1994 and Bahcall et al. 1994, the velocity distribution of clusters of galaxies can provide a means of falsifying a given non-gaussian model of structure formation, obviously if it can be shown that:
In the case of the ICDM model indeed, we find the shape of the three-dimensional velocity distribution of the clusters to depart from a Maxwellian distribution at both small (50 km/s) and very large (1000 km/s) velocities : the 1D velocity field has a positive reduced fourth moment. This is expected to some degree given the similar shape of the PDF of the matter velocity field when smoothed on large scales, but recall that the peculiar velocities of clusters are not straightforwardly related nin theory to the smoothed peculiar velocities of the matter distribution. As recently stressed by Colberg et al. (2000), one has to use instead the velocities of the initial peaks of the density field : the results have been derived by BBKS (Bardeen et al. 1986) in the gaussian case. Even there, however, Colberg et al. 2000 show that non-linear effects like mergers of clusters result in a measured cluster 3D peculiar velocity dispersion substantially higher than both "peak" and "smooth" predictions.
If the shape of the velocity PDF of clusters can already discriminate between ICDM and LCDM for instance, its dispersion can as well, even if it is only a two-point statistics. For a given power spectrum and measuring scale, the velocity dispersion according to linear theory depends on f=Omega00.6sigma8, modulo the caution of the last alinea about the statistics of peaks. ICDM and LCDM differ in both this quantity f and in the shape of the power spectrum.
To illustrate these points, we have selected clusters with masses beyond Mtot >1014/h solar masses, to get both a reasonable level of noise in the curves and to stay at a scale as linear as possible. The following Figure compares the results of the clusters of the 600 Mpc/h ICDM simulation in solid histogram to the results measured for the clusters of a 480 Mpc/h LCDM simulation of N. Yoshida (private communication) in dotted histogram. Maxwellian distributions with the measured dispersions v3D,rms=301 km/s and v3D,rms=530 km/s in ICDM and in LCDM respectively are plotted as solid and dotted lines. Maxwellian distributions with the peak formalism prediction, although it was derived for a gaussian density field are shown for both models as dashed and dash-dotted lines : the predicted 3D velocity dispersions are 171 and 376 km/s for ICDM and LCDM respectively.
In the plot below, we show the corresponding cumulative probability distributions. Again the ICDM model is drawn in solid line, the LCDM model in dotted line, and the data of Giovanelli et al. 1997 in dashed line. The standard LCDM model fits the data much better than the ICDM :
Cluster correlation lengths
-
The study of the correlation length of clusters dates back to
Hauser & Peebles (1973) who noted that the correlation function of rich clusters
of galaxies is higher than that of single galaxies. Bahcall and
Soneira (1983) and
Klypin & Kopylov (1983) found that the cluster correlation function is well
described by a power-law. Bahcall & Cen (1992), hereafter
B92, obtained a linear, much discussed, relation
between the correlation length r0 and the mean cluster separation
rcl of the sample: r0=0.4 rcl, which
they checked against numerical simulations.
However, Croft & Efstathiou (1994) measured systematically smaller correlation lengths in the APM survey. They suggested that the high correlation lengths found by B92 might be due to incompleteness in the Abell samples they used. Croft et al. (1997) and Governato et al. (1999) also performed numerical simulations in agreement with lower values for the correlation lengths. Colberg et al. (2000) used the Hubble Volume simulations of the VIRGO consortium to obtain very good statistics even beyond a mean cluster separation of 100 Mpc/h, and agreed with a weak dependence of r0 on rcl, consistent with the recent theoretical predictions of Sheth et al. (2001), hereafter SMT. There now seems to be agreement in the gaussian LCDM case that the correlation lengths obtained in the simulations are well below the values of B92, and well predicted by SMT.
We have selected haloes in the 600 Mpc/h simulation above a series of lower virial mass thresholds, and computed the mean intercluster separation, and correlation function for each of these sets. We have measured the correlation length by fitting a power-law to each of the correlation functions, and evaluating the distance where if reaches unity. We could repeat this process up to the point where the correlation function becomes too noisy because of the small number of haloes remaining above the mass threshold: we have limited ourselves to M200=4 1014/h solar masses, corresponding to a mean separation of about 75 Mpc/h in the ICDM model. Recall that the power-law fit to the DM correlation funtion used to compute the theoretical expectations has a slope of -1.86 and r0=4.3 Mpc/h.
We have also checked the SMT scheme in the case of the gaussian LCDM case by comparing their predictions to the correlation lengths obtained by Colberg et al. (2000) from one of the Hubble Volume simulations, again taking a fit to the measured DM correlation function as input to the theory.
In the next Figure, we plot the measured correlation lengths of the haloes in the 600 Mpc/h ICDM model with diamond symbols, together with the theoretical expectation (solid line) that we have worked out combing the newest analytical results of (1) Sheth et al. (2001) who predict halo bias in a family of gaussian CDM models, and of (2) Koyama et al. (1999) who derive an expression for the difference between gaussian and non-gaussian halo bias given the known shape of the PDF of the initial dark matter distribution. The square symbols show the results from the Hubble Volume simulation of the ``standard'' gaussian LCDM model, and the dashed line is the associated analytical prediction.
In both non-gaussian and gaussian cases, we find remarkably good agreement, although the theoretical predictions are highly sensitive to the linear power spectrum and to the z=0 non-linear correlation function.
These curves provide an obvious tool to discriminate between ICDM and LCDM if as noted by Robinson (2000), one sorts in decreasing order the observed, volume-limited clusters along any propriety which is monotonic with the cluster mass (e.g. core X-ray temperature). One then fixes the threshold when the number of selected clusters in the observed region gives a mean separation of about 60 Mpc/h, and where the correlation function can be accurately measured, There, the cluster correlation length can discriminate between the competing models as it is expected to reach some 25 Mpc/h in the non-gaussian ICDM model and 19 Mpc/h in gaussian LCDM with ``favoured'' cosmological parameters.
The next plot gives the two same data sets as in Figure 2 of Robinson (2000), obtained by two groups analyzing the APM sample : Croft et al. (1997) and Lee & Park (1999), in squares and triangles respectively> We show on top and with diamonds our results for the ICDM 600 Mpc/h simulation. The straight line is the linear relation of B92. At rcl=55 Mpc/h there is substantial difference in the estimation of the correlation lengths, for the same observed clusters. Note that Lee & Park (1999) are in agreement with the linear scaling of B92, but it seems that their results should be handled with caution. Sticking to the results of Croft et al. (1997), the ICDM model is obviously falsified.
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Last modified: March 16, 2002. For Questions / Comments / Remarks : hmathis@mpa-garching.mpg.de