The best-fit parameters obtained in this paper are within the allowed range of parameters found by other authors using different constraints. For example, for the MDM model with scale-invariant primordial power spectrum and one sort of massive neutrinos which contributes 10-20% of matter density, Valdarnini et al. (1998) found for , and for . Similar constraints have been given by Primack & Gross (1999) for MDM models with two species of massive neutrinos. Our values of are within these ranges. But at the boundary of this parameter range , which is outside of the confidence contour.
Recently Bahcall et al. (1999) have shown that the CMB anisotropy data, the cluster evolution and the SNIa magnitude-redshift relation indicate a flat Universe with accelerated expansion, compatible with a and if CDM () is assumed. As we can see from Table 4 (case No 11, 1), our analysis leads to the same conclusion if we set density of hot dark matter to the minimum value compatible with the Super-Kamiokande experiment (less than 1% of the total density). However, if is a free parameter, the observational data considered in this work lead to a MDM with a slightly blue spectrum of primordial fluctuations (case No 1 in Table 4).
In our preferred tilted MDM models (case No 1) the masses of neutrinos are eV for a model with , eV when and eV for a model with . The accuracy of neutrino mass or density determination is modest because the observational constraints depend more strongly on and n than on and . In models with fixed low matter density (case No 7 and 10) the best-fit values of the neutrino density are , i.e. even below the lower limit of the massive neutrino contribution to the cosmological density indicated by the Super-Kamiokande experiment. However, the contours of the low models include the Super-Kamiokande limit (see Fig. 9b).
In the last column of Table 5 we also indicate the age of the Universe,
for each model as well as the age of the oldest globular clusters (Carretta et al. 1999). All models with parameters taken from Table 4 have ages which are in agreement with the oldest objects of our galaxy.
We have used a scale independent, linear bias factor as free parameter in order to fit the Abell-ACO power spectrum amplitude.
Let us discuss in more detail how the model predictions presented in Table 5 match each observable constraint separately. The predicted position of the acoustic peak for all models is lower than the one determined from the observational data set presented in Table 2 (). Tilted MDM models prefer . This is due to the fact that the peak position depends only very weakly on the parameters discussed in this work. It is determined mainly by the spatial curvature which we have set to zero here (together with the initial conditions which we have assumed to be adiabatic). However, our result is in good agreement with the most recent and so far most accurate determination of the peak position from one single experiment, the North American test flight of Boomerang (Mauskopf et al. 1999, Melchiorri et al. 1999b), which led to with maximum likelihood near 1 for adiabatic CDM models. The prediction of our best model for position of the first acoustic peak () matches the value given by Boomerang experiment very well. The central value from the combination of all available experiments, , may very well be contaminated by mutual calibration inconsistencies.
Finally we want to discuss the possibility of using the averaged power spectrum of galaxies obtained by Einasto et al. 1999 to determine the parameters. This averaged spectrum of galaxies is determined in a wide range of scales (MpcMpc) and has substantially lower errors than the Abell-ACO power spectrum used here. Its errors are on small scales and at large scales versus and respectively for the Abell-ACO power spectrum. It is interesting to compare the predictions obtained from the power spectrum of galaxies with our analysis, because, as already mentioned in the introduction, the correction of the linear power spectrum for nonlinear evolution must be included into the algorithm. We use the fitting function by Smith et al. 1997, which transfers the linear into the nonlinear power spectrum, and the observational constraint for the Hubble constant (Saha et al. 1999,Tammann et al. 1999) as well as the nucleosynthesis constraint for the baryon content by Burles et al. 1999. Under these assumptions we find the following best fit parameters: , , , , , and galaxy biasing parameter .
If we add the remaining observations described in Sects. 2.2 and 2.3 the best-fit parameters remain practically unchanged due to the large number of (probably not independent) data points in the galaxy power spectrum. A model with these parameters has serious problems reproducing the experimental data set used here. Indeed, with these parameters we obtain , for the data set used in the rest of this work, far outside 3 contour. The model predictions and are lower than the corresponding observational values by Girardi et al. 1998 and Bahcall & Fan 1998. Moreover, the peculiar velocity is lower than the observed value, is and are and lower than the corresponding values inferred from the Ly- measurements. Therefore, we conclude that a model with parameters determined by the galaxy power spectrum is ruled out by the observations discussed in this work.
This result is not completely unexpected, because the galaxy power spectrum on small scales is probably influenced by a scale dependent bias (see for example Kravtsov & Klypin 1999, Fig. 3) which is not taken into account here. Moreover, the fitting formula for nonlinear evolution at 1 h/Mpc may be incorrect. If we disregard the short wavelength part of galaxy power spectrum we find parameters close to those presented in Table 4.
© European Southern Observatory (ESO) 2000
Online publication: April 10, 2000