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Astron. Astrophys. 356, 418-434 (2000)

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3. Testing the method

In 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 [FORMULA] 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).


Table 3. Test of the method: results of parameter search from mock data for the tilted [FORMULA]MDM model ([FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA]). In test 1, all parameters are determined; in the 2nd to 6th tests, some parameters are fixed. For each test the first row corresponds to the case when the number of species of massive neutrinos is equal to the input value (2) and the second - when [FORMULA].
*) - fixed parameters.

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 [FORMULA] depending on the parameters [FORMULA], [FORMULA], [FORMULA], [FORMULA] and h by (Eisenstein & Hu 1999 and Novosyadlyj et al. 1999).

The linear power spectrum of matter density fluctuations is


where A is the normalization constant and [FORMULA] is the linear growth factor, which can be approximated by (Carroll, Press & Turner 1992)


where [FORMULA].

We normalize the spectra to the 4-year COBE data which determines the amplitude of density perturbation at the horizon crossing scale, [FORMULA] (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 ([FORMULA]) we have


([FORMULA]). The normalization constant A is then given by


The Abell-ACO power spectrum is related to the matter power spectrum at [FORMULA], [FORMULA] by the cluster biasing parameter [FORMULA]. We assume a scale-independent, linear bias:


For a given set of parameters n, [FORMULA], [FORMULA], h, [FORMULA], [FORMULA] and [FORMULA] theoretical values of [FORMULA] can now be obtained for the values [FORMULA] of Table 1. We denote them by [FORMULA] ([FORMULA]).

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 ([FORMULA]). This is illustrated in Figs. 4 and 5.

[FIGURE] Fig. 4. The dependence of the acoustic peak amplitude [FORMULA] on neutrino content [FORMULA]

[FIGURE] Fig. 5. The dependence of the acoustic peak position [FORMULA] on neutrino content [FORMULA]

For the remaining parameters, n, h, [FORMULA] and [FORMULA], we have determined the resulting values [FORMULA] and [FORMULA] with CMBfast for a network of model parameters. The values [FORMULA] in-between grid points are then obtained by 4-dimensional interpolation. This allows a fast and sufficiently accurate calculation of the peak position and amplitude for a given set of parameters in the range [FORMULA], [FORMULA], [FORMULA] and [FORMULA] considered in this work. The accuracy of this interpolation is estimated to be within 2%. We denote [FORMULA] and [FORMULA] by [FORMULA] and [FORMULA] respectively.

The theoretical values of the other experimental constraints are obtained as follows: The density fluctuation [FORMULA] is calculated according to Eq. (3) with [FORMULA] taken from Eq. (12). We set [FORMULA] and [FORMULA], where [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA], respectively.

The rms peculiar velocity of galaxies in a sphere of radius [FORMULA]Mpc is given by


where [FORMULA] is the power spectrum for the velocity field of the density-weighted matter (Eisenstein & Hu 1999), [FORMULA] is the top-hat window function. A previous smoothing of raw data with a Gaussian filter of radius [FORMULA]Mpc is employed here similar to the procedure which has led to the observational value. For the scales of interest [FORMULA]. We denote the rms peculiar velocity by [FORMULA].

The value by Gnedin 1998 from the formation of Ly-[FORMULA] clouds constrains the rms linear density perturbation at [FORMULA] and [FORMULA]Mpc. In terms of the power spectrum [FORMULA] is given by


It will be denoted by [FORMULA]. The corresponding value of the constraint by Croft et al. 1998 is


at [FORMULA] and [FORMULA], (where [FORMULA] is the Hubble parameter at redshift z) will be denoted by [FORMULA]. The slope of the power spectrum at this scale and redshift,


is denoted by [FORMULA].

For all tests except Gnedin's Ly-[FORMULA] clouds we used the density weighted transfer function [FORMULA] from Eisenstein & Hu 1999. For Gnedin's [FORMULA] we use [FORMULA] according to the prescription of (Gnedin 1998). It must be noted that even in the model with maximal [FORMULA] ([FORMULA]) the difference between [FORMULA] and [FORMULA] is less than [FORMULA] for [FORMULA].

Finally, the values [FORMULA] and h are denoted by [FORMULA] and [FORMULA] respectively.

The relative quadratic deviations of the theoretical values from their observational counterparts are given by [FORMULA]:


where [FORMULA] and [FORMULA] are the experimental data and their dispersion, respectively. The set of parameters n, [FORMULA], [FORMULA], h, [FORMULA], [FORMULA] and [FORMULA] or some subset of them can be determined by minimizing [FORMULA] using the Levenberg-Marquardt method (Press et al. 1992). The derivatives of the predicted values with respect to the search parameters which are required by this method are calculated numerically using a relative step size of [FORMULA] with respect to the given parameter.

The method was tested in the following way. Assuming a 4-year COBE normalized tilted [FORMULA]MDM model with the parameters [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and assuming further a cluster biasing parameter [FORMULA] we have calculated mock cluster power spectrum [FORMULA] and treated them as [FORMULA], [FORMULA]. The remaining mock data [FORMULA], [FORMULA] 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 n, [FORMULA], [FORMULA], h, [FORMULA], and [FORMULA] ([FORMULA] was fixed). As starting parameters for the search program we assumed random values within the allowed range. We have searched for the parameters assuming the "true" value of two species of massive neutrinos as well as assuming three species of massive neutrinos. The parameters obtained for different cases are presented in Table 3. The errors in the determined parameters are calculated as root square from diagonal elements of the standard error covariance matrix. In all cases the code found all the previous known parameters with high accuracy. This means that the code finds the global minimum of [FORMULA] independent of the initial values for the parameters.

Our conclusions from the test results can be summarized as follows:

1. If all parameters are free and [FORMULA] (the input value) the code finds the correct values of the free parameters (test 1, for [FORMULA] in Table 3).

2. If all parameters are free and [FORMULA] the code finds values of the free parameters which are in the [FORMULA] range of errors (test 1, for [FORMULA] 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 [FORMULA]. The results for n, [FORMULA] and [FORMULA] are in the [FORMULA] range of the correct values. The most uncertain solutions are found for n and [FORMULA] if an incorrect value for [FORMULA] has been assumed (test 6 in Table 3).

4. If some parameters are fixed to the predetermined ones and [FORMULA] (the input value) the code finds the correct values of the free parameters (test 3 and 5, for [FORMULA] in Table 3), if [FORMULA] the determined values are within the [FORMULA] range (test 3 and 5, for [FORMULA] in Table 3).

In summary, the code determines the parameters n, [FORMULA], [FORMULA], h, [FORMULA] and [FORMULA] correctly, if the observational data are correctly measured and the cosmological model assumed is correct; i.e. no curvature, a negligible amount of tensor perturbations and a primordial spectrum of scalar perturbations which is scale free from the present horizon size down to the scale of the Ly-[FORMULA] clouds.

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© European Southern Observatory (ESO) 2000

Online publication: April 10, 2000