The determination of the parameters n, , , h, , and by the Levenberg-Marquardt minimization method can be realized in the following way: We vary the set of parameters n, , , h, and or some subset of them and find the minimum of . Since the process is discrete we repeat this procedure three times for =1, 2, and 3. The lowest of the three minimums is the minimum of for the complete set of free parameters. The number of degrees of freedom if all parameters are free. It increases, if some of the parameters are fixed to a certain value. (Remember that even though we have 13 power spectra points, they can be described by just 3 degrees of freedom.)
We have determined the minimum of for =1, 2, 3 in 11 different cases, where all observational data described in Sect. 2 are used.
1) n, , , , h, and are free parameters ();
2) is fixed, the remaining parameters are free ();
6) is fixed, the remaining parameters are free ();
7) is fixed, the remaining parameters are free ();
8) is fixed, the remaining parameters are free ();
9) , are fixed, the remaining parameters are free ();
10) , and are fixed, the remaining parameters are free ();
11) is fixed by the lower limit of neutrino mass inferred by the observed neutrino oscillations in the Super-Kamiokande experiment ().
For these 11 cases we find the minimum of from which we determine the parameters presented in Table 4. Note, that for all models is in the range, which is expected for a Gaussian distribution of degrees of freedom. This means that the cosmological paradigm which has been assumed is in agreement with the data. (Note here, that the reduction of the 13 not independent data points of the cluster power spectrum to three parameters is very important for our analysis. Otherwise we would obtain a which is by far too small. If we would have assumed the 13 points of the Abell cluster power spectrum as independent, resulting in , the smallness of would have indicated that something is wrong in our approach. But we might have drawn the wrong conclusion that the error bars be too large!) In Table 5 we present also the values of the different observational constraints for the best fit models found in Table 4.
Table 4. Cosmological parameters determined for the tilted MDM model with one, two and three species of massive neutrinos. In case No. 1 all parameters are free, in the other cases (No2-11) some of them are fixed, as described above.
Table 5. Theoretical predictions for the observational values of tilted MDM models found with the parameters of Table 4 (for the value of leading to the lowest ).
If all parameters are free (Table 4, case No 1), the model with one sort of massive neutrinos provides the best fit to the data, . Note, however, that there are only marginal differences in for . Therefore, with the given accuracy of the data we cannot conclude whether - if massive neutrinos are present at all - their number is one, two, or three. We summarize, that the considered observational data on LSS of the Universe can be explained by a flat MDM inflationary model with a tilted spectrum of scalar perturbations and vanishing tensor contribution. The best fit parameters are: , , , , and . The CDM density parameter is and is considerable, .
The value of the Hubble constant is close to measurements by Madore et al. (1999). The spectral index coincides with the COBE prediction. The neutrino matter density corresponds to a neutrino mass eV. The estimated cluster bias parameter fixes the amplitude of the Abell-ACO power spectrum (Fig. 6). All predictions of the measurements summarized in Table 5 are close to the experimental values and within the error bars of the data.
The predicted position of the acoustic peak () is systematically lower than the experimental value determined here from the complete data set on (). This position is nearly fixed by the requirement and is only weakly dependent of the parameters varied in this study. The acoustic peak inferred by the Boomerang experiment (Mauskopf et al. 1999) is situated at and prefers models which are very close to flat (Melchiorri et al. 1999b). The models with low (case No. 7 in Table 4) fit the observable data somewhat less good than the best model () but all predictions are still within the range. These models prefer a high Hubble parameter, and no massive neutrinos, . On the contrary, the matter dominated tilted MDM model (, models 6 in Table 4) prefers high , three sorts of massive neutrinos and a low Hubble parameter, . This can be understood by considering one of the most serious problems of standard CDM, namely that the model, when normalized to COBE, has too much power on small scales. This problem can be solved either by introducing HDM and thereby damping the spectrum on small scales or by introducing a cosmological constant which leads mainly to a `shift of the power spectrum to the left'.
Another interesting correlation can be seen in Table 4, cases No 2-4, where we have fixed h. An increasing Hubble constant is compensated by a decreasing matter density, , (i.e. increasing cosmological constant) and a decreasing baryon content due to the tight nucleosynthesis constraint on . Furthermore, increasing the number of massive neutrino species from 1 to 3 leads to an increase of from 0.06 to 0.13 and to a decrease of from 0.59 to 0.43 (case 1).
If we use the nucleosynthesis constraint by Tytler et al. 1996 (case No 5), is slightly higher than in case No 3.
Now let us discuss models with a perfectly scale invariant primordial power spectrum as predicted by the first inflationary models, fixed (cases 8-10 in Table 4). If all of the remaining parameters are free (case 8) then this data set prefers a MDM model with parameters and and a somewhat lower neutrino content than the best fit model. Models with low matter content, , prefer a high Hubble parameter, and no hot dark matter, (case No 10). The matter dominated model (case No 9) is the standard MDM model with , three sorts of massive neutrinos (eV) and .
If the HDM component is eliminated or is fixed at the small value defined by the lower limit of the neutrino mass from the Super-Kamiokande experiment , we obtain the best-fit value for the matter density parameter and Hubble constant (case No 11).
The experimental Abell-ACO power spectrum and the theoretical predictions for some best fit models are shown in Fig. 6. Recently it was shown (Novosyadlyj 1999) that due to the large error bars, the position of the peak of at h/Mpc does not influence the determination of the cosmological parameters significantly. Mainly the slope of the power spectrum on scales smaller than the scale of the peak position determines the cosmological parameters.
The errors in the best fit parameters presented in Table 4 are the square roots of the diagonal elements of the covariance matrix. More information about the accuracy of the determination of parameters and their sensitivity to the data used can be obtained from the contours of confidence levels presented in Fig. 7 for the tilted MDM model with parameters from Table 4 (case No 1, ). The same contours for cases No 6 and 7 are shown in Figs. 8 and 9, respectively. These contours show the confidence regions which contain 68.3% (solid line), 95.4% (dashed line) and 99.73% (dotted line) of the total probability distribution in the two dimensional sections of the six-dimensional parameter space, if the probability distribution is Gaussian. Since the number of degrees of freedom is 7 they correspond to 8.2, 14.3 and 21.8 respectively. The parameters not shown in a given diagram are set to their best-fit value.
As one can see in Fig. 7a the iso- surface is rather prolate from the low- - high-n corner to high- - low-n. This indicates some degeneracy in parameter plane, which can be expressed by the following equation which roughly describes the `maximum likelihood ridge' in this plane within the :
A similar degeneracy is observed in the plane in the range , (Fig. 7c). The equation for the `maximum likelihood ridge' or `degeneracy equation' has here the form:
The 3rd column of Table 4 () shows that all models except 9th with are within the 1 contour of the best fit.
The next important question is: which is the confidence limit of each parameter marginalized over the other ones. The straightforward answer is the integral of the likelihood function over the allowed range of all the other parameters. But for a 6-dimensional parameter space this is computationally time consuming. Therefore, we have estimated the 1 confidence limits for all parameters in the following way. By variation of all parameters we determine the 6-dimensional surface which contains 68.3% of the total probability distribution. We then project the surface onto each axis of parameter space. Its shadow on the parameter axes gives us the 1 confidence limits on cosmological parameters. For the best MDM model with one sort of massive neutrinos the 1 confidence limits on parameters obtained in this way are presented in Table 6.
The best fit values of all the parameters with errors obtain by maximizing the (Gaussian) 68% confidence contours over all other parameters.
It must be noted that the upper edge for h is equal to 1.08 when we marginalized over all other parameters and input observable data used here. But this contradicts the age of the oldest globular clusters (Carretta et al. 1999). Thus we have included this value into the marginalization procedure for the upper limit of h. We then have 8 degrees of freedom (24 data points) and the 6-dimensional surface which contains 68.3% of the probability is confined by the value 13.95. We did not use the age of the oldest globular cluster for searching the best fit parameters in general case because it is only a lower limit to the age of the Universe; besides, it does not change their values as one can see from the last column of Table 5.
The errors given in Table 6 represent 68% likelihood, of course, only when the probability distribution is Gaussian. As one can see from Fig. 7 (all panels without degeneracy) the ellipticity of the likelihood contours in most of the planes is close to what is expected from a Gaussian distribution. This indicates that our estimates of the confidence limits are reasonable. These errors define the range of each parameter within which the best-fit values obtained for the remaining parameters lead to . Of course, the best-fit values of the remaining parameters lay within their corresponding 68% likelihood given in the Table 6. It does however not mean that any set of parameters from these ranges satisfies the condition, .
For example, standard CDM model (, , , and best-fit value of cluster biasing parameter ()) has (!), that excludes it at very high confidence level, . When we use the baryon density inferred from nucleosynthesis ( (, )) the situation does not improve much, . Furthermore, even if we leave h as free parameter we still find () with the best-fit values and (); this variant of CDM is ruled out again by direct measurement of the Hubble constant.
The standard MDM model (, , , , , with a best value of the cluster biasing parameter ()) does significantly better: it has ( C.L.) which is out of the confidence contour but inside . With the nucleosynthesis constraint the situation does not change: ; also if we leave h as free parameter: , . But if, in addition, we let vary, we obtain with best-fit values of , , (). This means that the model is ruled out (as well as model 9 in Table 4) by the data set considered in this work at confidence level only. But also here the best-fit value for h is very low. If we fix it at the lower observational limit then (the best fit values are: , ()), which corresponds to a confidence level of 95%.
Therefore, we conclude that the observational data set used here rules out CDM models with , a scale invariant primordial power spectrum () and at very high confidence level, . MDM models with , and are ruled out at C.L.
The best-fit parameters for 31 models which are inside the range of the best model are presented in Table 4. We conclude also that the observational data set used here does not rule out any of the 32 models presented in Table 4 at high confidence level but defines the 1 range of cosmological parameters for the MDM models which match observations best.
One important question is how each point of the data influences our result. To estimate this we have excluded some data points from the searching procedure. We have determined the best-fit parameters for the cases:
The results for models with and all parameters free are presented in Table 7 (see for comparison model 1 for in Table 4). Excluding any part of observable data results only in a change of the best-fit values of n, and h within the range of their corresponding standard errors. This indicates that the data are mutually in agreement, implying the same cosmological parameters (within the still considerable error bars). The small scale constraints, the Ly- tests reduce the hot dark matter content from to . The -tests further reduce to . Including the Abell-ACO power spectrum in the search procedure, tends to enhance slightly. The most crucial test for the baryon content is of course the nucleosynthesis constraint. Its -accuracy safely keeps near its median value 0.019. The parameter in turn is only known to accuracy due to the large errors of other experimental data used here, especially for the Hubble constant. The obtained accuracy of h () is better than the one assumed from direct measurements, . Summarizing, we conclude that all data points used here are important for searching the best-fit cosmological parameters.
Table 7. Parameters determined for the tilted MDM with one sort of massive neutrinos if some of the data are excluded from the searching procedure.
© European Southern Observatory (ESO) 2000
Online publication: April 10, 2000