## 3. Testing the methodIn order to test our method to determine cosmological parameters for stability, we have constructed a mock sample of observational data. We start with a set of cosmological parameters and determine for them the "observational" data which would be measured in case of faultless measurements with errors comparable to the observational errors. We then insert random sets of starting parameters into the search program and try to find the right model which corresponds to the mock data. The method is stable if we can recover our input cosmological model. Even starting very far away from the true values, our method reveals as very stable and finds the `true' model whenever possible (see Table 3).
One of the main ingredients for the solution to our search problem
is a reasonably fast and accurate determination of the transfer
function which depends on the cosmological parameters. We use the
accurate analytical approximations of the MDM transfer function
depending on the parameters
, ,
,
and The linear power spectrum of matter density fluctuations is where where . We normalize the spectra to the 4-year COBE data which determines the amplitude of density perturbation at the horizon crossing scale, (Liddle et al. 1996, Bunn and White 1997), which for a matter dominated Universe without tensor mode and cosmological constant is given by For a flat model with cosmological constant () we have (). The normalization constant
The Abell-ACO power spectrum is related to the matter power spectrum at , by the cluster biasing parameter . We assume a scale-independent, linear bias: For a given set of parameters The dependence of the position and amplitude of the first acoustic peak of the CMB power spectrum on cosmological parameters has been investigated using CMBfast by Seljak & Zaldarriaga 1996. As expected, the results are, within sensible accuracy, independent of the hot dark matter contribution (). This is illustrated in Figs. 4 and 5.
For the remaining parameters, The theoretical values of the other experimental constraints are obtained as follows: The density fluctuation is calculated according to Eq. (3) with taken from Eq. (12). We set and , where for and for , respectively. The rms peculiar velocity of galaxies in a sphere of radius Mpc is given by where is the power spectrum for the velocity field of the density-weighted matter (Eisenstein & Hu 1999), is the top-hat window function. A previous smoothing of raw data with a Gaussian filter of radius Mpc is employed here similar to the procedure which has led to the observational value. For the scales of interest . We denote the rms peculiar velocity by . The value by Gnedin 1998 from the formation of Ly- clouds constrains the rms linear density perturbation at and Mpc. In terms of the power spectrum is given by It will be denoted by . The corresponding value of the constraint by Croft et al. 1998 is at and , (where is the Hubble parameter at redshift z) will be denoted by . The slope of the power spectrum at this scale and redshift, is denoted by . For all tests except Gnedin's Ly- clouds we used the density weighted transfer function from Eisenstein & Hu 1999. For Gnedin's we use according to the prescription of (Gnedin 1998). It must be noted that even in the model with maximal () the difference between and is less than for . Finally, the values and The relative quadratic deviations of the theoretical values from their observational counterparts are given by : where and
are the experimental data and their
dispersion, respectively. The set of parameters The method was tested in the following way. Assuming a 4-year COBE normalized tilted MDM model with the parameters , , , , , and assuming further a cluster biasing parameter we have calculated mock cluster power spectrum and treated them as , . The remaining mock data , have been calculated as described above. We have assigned to these mock data the same relative `experimental' errors as in the corresponding experiments described in the previous section. We then used these mock data to search the parameters Our conclusions from the test results can be summarized as follows: 1. If all parameters are free and (the input value) the code finds the correct values of the free parameters (test 1, for in Table 3). 2. If all parameters are free and the code finds values of the free parameters which are in the range of errors (test 1, for in Table 3). 3. If some parameters are fixed and differ from the input values
(tests 2, 4, 6 in Table 3) the code finds for the remaining
search parameters values close to the correct ones. The most stable
and accurate value is . The results
for 4. If some parameters are fixed to the predetermined ones and (the input value) the code finds the correct values of the free parameters (test 3 and 5, for in Table 3), if the determined values are within the range (test 3 and 5, for in Table 3). In summary, the code determines the parameters © European Southern Observatory (ESO) 2000 Online publication: April 10, 2000 |