2. The experimental data set
2.1. The Abell-ACO cluster power spectrum
One might expect that the most favorable data for the determination of cosmological parameters are power spectra constructed from the observed distribution of galaxies. But the power spectra of galaxies obtained from the two-dimensional APM survey (e.g. Maddox et al. 1996 , Tadros & Estathiou 1996, and references therein), the CfA redshift survey (Vogeley et al. 1992, Park et al. 1994), the IRAS survey (Saunders et al. 1992) and/or from the Las Campanas Redshift Survey (Da Costa et al. 1994, Landy et al. 1996) differ both in the amplitude and in the behavior near the maximum. Moreover, nonlinear effects on small scale must be taken into account in their analysis. For these reasons we do not include galaxy power spectra for the determination of parameters in this work. Here, we use the power spectrum of Abell-ACO clusters (Einasto et al. 1997, Retzlaff et al. 1998) as observational input. This power spectrum is measured in the range MpcMpc. The cluster power spectrum is biased with respect to the dark matter distribution. We assume that the bias is linear and scale independent in the range of scales considered. The position of the maximum (Mpc) and the slope at lower and larger scales are sensitive to the baryon content , the Hubble constant h, the neutrino mass and the number of species of massive neutrinos (Novosyadlyj 1999). The Abell-ACO cluster power spectrum (here and in the following a tilde denotes observed quantities) has been taken from Retzlaff et al. 1998. We present 13 values of and the errors in Table 1 and in Fig. 1. As a first step, we have assumed that the 13 points in this power spectrum given below are independent measurements. The value of obtained under this assumption is much smaller than the number of degrees of freedom (see below). We interpret this as a hint that the 13 points of given in Table I cannot be considered as independent measurements. We therefore describe the power spectrum by three parameters A, and to be of the form
A fit of the parameters to the observed power spectrum gives
Table 1. The Abell-ACO power spectrum by Retzlaff et al. 1998
In Fig. 1 we show the observed power spectrum together with the fit. The cosmological model parameters obtained using the full power spectrum information or the three parameter fit are in good agreement, but the latter prescription leads to a more reasonable value of .
This point is quite important since it illustrates that a small need not mean that the error bars of the data are too large but it can be due to data points depending only on a few parameters and therefore not being independent. If a power spectrum, like the one above can be modeled by 3 parameters, then, by varying three cosmological parameters, like e.g. the cluster bias , the HDM contribution and the Hubble parameter, h, we can in general (if there is no degeneracy) fit all three parameters A, and and thereby the entire power spectrum. The number of degrees of freedom in such a fit is 0 and not 10 as one would infer from the number of points of the power spectrum.
To make best use of the observational information, we nevertheless use the full 13 points of the power spectrum to fit the data, but we assign it for the number of degrees of freedom.
2.2. CMB data
We normalize the power spectrum using the COBE 4-year data of CMB temperature fluctuations (Bennett et al. 1996, Liddle et al. 1996, Bunn and White 1997). We believe that using all available experimental data on on angular scales smaller than the COBE measurement is not an optimal way for searching best-fit parameters because some data points in CMB spectrum contradict each other. Therefore, we use only the position and amplitude of the first acoustic peak derived from observational data as integral characteristics of CMB power spectrum, which are sensitive to some of the model parameters.
To determine the position, , and amplitude, , of the first acoustic peak we use the set of observational data on CMB temperature anisotropy given in Table 2 (altogether 51 observational points). For each experiment we include the effective harmonic, the amplitude of the temperature fluctuation at this harmonic, the upper and lower error in the temperature, and the effective range of the window in -space. In those cases when original papers do not contain effective harmonics and band width we have taken them from Max Tegmark's CMB data analysis center (Tegmark 1999b) dated Nov 25 1999. We fit the experimental data points by a polynomial of 6-th order using the Levenberg-Marquardt method to determine the position and amplitude of the first peak: . The best-fit values of the coefficients are: , , , , , , (). The amplitude and position of first acoustic peak determined from data fitting curve are and 253 correspondingly. Our result differs only slightly from the numbers obtained by Lineweaver & Barbosa 1998 who found 260 and . Fig. 2 shows the observational data used together with the polynomial best fit (solid line).
Table 2. Observational data on CMB temperature fluctuations (in )
We estimated the error of and in the following way.
By varying all coefficients we determine the -hyper-surface in 7-dimension parameter space which contains deviations lower than . If the probability distribution obeys Gaussian statistics, this corresponds to a 68.3% confidence level. It is well known, that present CMB anisotropy data even on small scales do not obey Gaussian statistics and thus this procedure is somewhat arbitrary. However, it can be assumed that this gives us a good indication for the error bars in position and amplitude of the first acoustic peak. For 44 degrees of freedom (51 data points minus 7 parameters) this hyper-surface corresponds to . For values of parameters which have a we calculate the 's and find the peak amplitude and the position . They are in the contour line in the plane shown in Fig. 3. The upper-lower and right-left extremal points indicate statistical errors: and . Uncertainties of effective harmonics of each experiment do not influence the error of the amplitude of the first acoustic peak but must be taken into account for the full error in the peak position, so that , where the last term is the mean band width around . We estimate it as the mean width of all the experiments weighted by acoustic peak amplitude
where the weighting factor is calculated using polynomial fit. This finally leads to . (Without weighting the value is ). Therefore, the errors of determination of first acoustic peak amplitude and position are and respectively. We use these errors below in our search of cosmological parameters.
It is interesting to note that no 6th order polynomial fits the data really well. For our best fit polynomial we obtain for 51 data points and 7 parameters. The probability for this polynomial leading to the observed data is about 1%. This big can have two origins. First, the probability distribution is non-Gaussian and, therefore, the probability to obtain this value of is higher than 1%. Second, some data seem to be contradictory. For example, if we ignore all the Python V points we obtain a best fit polynomial with which is even slightly too low. (Removing these points does not change essentially the result amplitude and position of acoustic peak, , without them). But of course we are not allowed without any good reason, to leave away some experimental results. It may well be that Python V is correct and some other experiments are wrong. Therefore, we adopted this somewhat hand waving way to extract information from these data. Clearly, a more thorough analysis with true, non-Gaussian likelihood functions would be in order, which we leave for the future (see Bartlett et al. 1999).
For comparison of the models with the CMB data we use, apart from the COBE normalization, only the two parameters obtained by the fitting procedure described above: the effective harmonic of the peak position and the amplitude of the peak . Clearly, this position and height of the first acoustic peak is not strictly implied by the present data and can therefore be criticized. In this sense it has to be considered primarily as a working hypothesis which will be confirmed or contradicted in the future by more accurate data.
2.3. Other experimental constraints
A constraint of the amplitude of the fluctuation power spectrum at cluster scale can be derived from the cluster mass and the X-ray temperature functions. It is usually formulated as a constraint for the density fluctuation in a top-hat sphere of 8 Mpc radius, , which can be calculated for a given initial power spectrum :
where is the Fourier transform of a top-hat window function. A recent optical determination of the mass function of nearby galaxy clusters (Girardi et al. 1998) gives . Several groups have found similar results using different methods and different data sets (for a comprehensive list of references see Borgani et al. 1999). To take into account the results from other authors we have decided to use more conservative error bars:
From the existence of three very massive clusters of galaxies observed so far at a further constraint has been established by Bahcall & Fan 1998
where if and if with The relation of this value to other tests will be analyzed too.
Another constraint on the amplitude of the linear power spectrum of density fluctuations in our vicinity comes from the study of galaxy bulk flows in spheres of large enough radius around our position. Since these data may be influenced by the local super-cluster (cosmic variance), we will use only the value of bulk motion - the mean peculiar velocity of galaxies in the sphere of radius Mpc given by Kolatt & Dekel 1997,
An essential constraint on the linear power spectrum of matter clustering on small scales (Mpc comes from the Ly- forest of absorption lines seen in quasar spectra (Gnedin 1998, Croft et al. 1998 and references therein). Assuming that the Ly- forest is formed by discrete clouds with a physical size close to the Jeans scale in the reionized inter-galactic medium at , Gnedin 1998 has obtained a constraint on the value of the rms linear density fluctuations
Taking into account the new data on quasar absorption lines, the effective equation of state and the temperature of the inter-galactic medium at high redshift were re-estimated recently by Ricotti et al. 2000. As a result, the value of Jeans scale at has moved to Mpc (Gnedin 1999).
The procedure of recovering the linear power spectrum from the Ly- forest has been elaborated by Croft et al. 1998. Analyzing the absorption lines in a sample of 19 QSO spectra they have obtained the following constraint on the amplitude and slope of the linear power spectrum at and Mpc,
(95% CL). In addition to the power spectrum measurements we will use the constraints on the value of the Hubble constant
© European Southern Observatory (ESO) 2000
Online publication: April 10, 2000