Numerical simulations of Peebles's Isocurvature Cold Dark Matter model |
H. Mathis and S. D. M. White (MPA Garching)
Introduction/ References/ Normalization/ Parameters of the collisionless simulations/ Slices through the simulations/ Dark matter statistics/ Cluster mass function/ Cluster peculiar velocities/ Cluster correlation lengths/ Links/
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 :
Note that after this project has been started, the BOOMERANG measurement of the height
and the position of the first "acoustic peak" in the CMB fluctuations
has ruled out the ICDM model as it was originally proposed.
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.
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 Omega_{b} to the upper bound suggested by Peebles at
a value of 0.05. The next plot gives the point of agreement in the
(n_{init}, sigma_{8}) plane for three fractions of
baryons.
The Figure below shows the
mismatch of the ICDM model with the BOOMERANG
data, represented by crosses.
We have perfomed two large dissipationless simulations of the ICDM model normalized as above. Both have N_{parts}}=256^{3} 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 z_{init}=50.
The next table summarizes the parameters of the simulations. L,
N, M, l_{soft}, z_{init} give the
box size (units Mpc/h), the number of particles,
their mass (units 10^{10}/h solar masses), the comoving softening lengh (units
kpc/h) and the starting redshift. The cosmological parameters are
recalled, and n_{init} 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 | l_{soft} | z_{init} | Omega_{0} | Lambda | Omega_{b} | h | Sigma_{8} | n_{init} |
600 | 256^{3} | 71.1 | 30 | 50 | 0.2 | 0.8 | 0.05 | 0.7 | 0.8 | -1.8 |
162 | 256^{3} | 1.40 | 30 | 50 | 0.2 | 0.8 | 0.05 | 0.7 | 0.8 | -1.8 |
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.
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 z_{init}=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.
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
r_{0}=4.3 Mpc/h.
Last modified: March 16, 2002. For Questions / Comments / Remarks : hmathis@mpa-garching.mpg.de