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/



Introduction






References

Some selected publications :





Normalization

    We normalize the model by matching two constraints :


    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






Slices through the simulations






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

  • 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.



  • Power spectrum

  • 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.

  • Correlation functions

  • 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