To avoid this difficulty it is better to define the biasing factor b using power spectra of matter and galaxies, which can be determined also for cases where regions of zero density of objects of interest exist. The biasing factor can be determined for different populations using varying density thresholds to define populations of different type (galaxies of different luminosity, galaxy systems of different richness etc.).
Einasto et al. (1999a) found the following relation between the power spectra of the matter, Pm(k), and that of the clustered population (galaxies or clusters of galaxies), Pc(k):
where k is the wavenumber, and Fc is the fraction of matter in the clustered population. The last equation gives for the bias factor of the clustered population bc = 1/Fc.. In other words, the biasing factor depends on the fraction of matter in the clustered population, at least in the first approximation.
These equations show that the subtraction of a homogeneous population from the whole matter population increases the amplitude of the spectrum of the remaining clustered population. In this approximation biasing is linear and does not depend on scale. These equations have a simple interpretation. The power spectrum describes the square of the amplitude of the density contrast, i.e. the amplitude of density perturbations with respect to the mean density. If we subtract from the density field a constant density background, but otherwise preserve density fluctuations, then amplitudes of absolute density fluctuations remain the same, but amplitudes of relative fluctuations with respect to the mean density increase by a factor which is determined by the ratio of mean densities, i.e. by the fraction of matter in the new density field with respect to the previous one (Einasto et al., 1999a).
To check the above relations Einasto et al. (1999a) performed numerical simulations. The relations are identical in the 2-D and 3-D cases, thus a 2-D simulation with 5122 particles and cells was made to obtain a better resolution. Simulation particles were divided into populations according to values of the local density. Particles with low density values, < 0, were called void particles; all others are called clustered particles of various density. 0 is the threshold density to separate clustered and unclustered particles. If we consider all galaxies, including the faintest dwarf galaxies, then it is natural to accept 0 = 1. For galaxies of higher luminosity and for groups/clusters 0> 1. Distributions of particles of various threshold density are shown in Fig. 7.7. For comparison also a sheet of galaxies of various absolute magnitude is shown, which crosses the Local, the Coma and the southern corner of the Hercules supercluster.
Power spectra and respective bias factors (as a function of the wavenumber k) for various samples of simulation particles are shown in Fig. 7.8. Galaxy and cluster samples are the same as used in Fig. 7.7. Calculations show that the biasing factor can be indeed calculated from the fraction of particles in the respective sample, for details see Einasto et al. (1999a).
Figure 7.8 shows that all power spectra of samples of clustered particles are similar to the power spectrum of the matter, but have higher amplitudes. This similarity of the shape of the power spectra shows that biased galaxy samples and even samples of particles in clusters contain all essential information on the amplitudes of density waves of different scales (excluding the shortest waves, which can be followed only by dwarf galaxies). From the difference in the amplitude of the power spectra of these populations with respect to the power spectrum of matter we derived the biasing parameter as a function of the wavenumber. The results are plotted in the right panel of Fig. 7.8. For most samples the biasing parameter is almost constant. As expected, the deviations are larger on small scales where the cluster sample contains almost no test objects.
Fig. 7.7 The distribution of simulated and real galaxies in a box of side-length 90 h−1 Mpc. Panel (a) gives particles in voids ( < 1); panel (b) shows the distribution of simulated galaxies in high- density regions: galaxies in the density interval 5≤ < 20 are plotted as black dots, galaxies with > 20 as filled (red) regions; panel (c) shows field galaxies in the density interval 1 ≤ < 1.5 (open blue circles), and 1.5 ≤ < 5 (dots). Densities are expressed in units of the mean density of the Universe. Panel (d) shows the distribution of galaxies in supergalactic coordinates in a sheet 0 ≤ X < 10 h−1 Mpc, horizontal and vertical axes are supergalactic Y and Z, respectively; bright galaxies (MB ≤ −20.3) are plotted as red dots, galaxies −20.3 < MB ≤ -19.7 as black dots, galaxies −19.7 < MB ≤ −18.8 as open blue circles, galaxies −18.8 < MB ≤ −18.0 as green circles (absolute magnitudes correspond to Hubble parameter h =1) (Einasto et al., 1999a).
The overall amplitude shift is determined by the fraction of matter in the clustered population associated with galaxies or systems of galaxies. In other words, populations with higher values of power spectra (and correlation functions) are not more clustered, as traditionally interpreted for the cluster correlation function (Bahcall & Soneira, 1983; Klypin & Kopylov, 1983), but are more strongly normalised to take into account the smaller fraction of objects in their samples. The definition of the biasing factor using power spectra of populations is in harmony with the original definition by Kaiser, since the correlation function and the power spectrum express the same property of the distribution, one in real space, the other in Fourier space.
Fig. 7.8 Left: Power spectra of simulated galaxies. The solid bold line shows the spectrum derived for all test particles (the matter power spectrum); dashed and dotted bold lines give the power spectrum of all clustered particles (sample Gal-1) and clustered galaxies in high-density regions (sample Clust). Thin solid and dashed lines show the power spectra of samples of particles with various threshold densities and sampling rules. Right: the biasing parameter as a function of wavenumber, calculated from definition given in the text. Samples and designations are the same as in the left panel (Einasto et al., 1999a).
This explanation of the correlation function of clusters of galaxies indicates that there is no need to introduce two different power spectra for galaxies and clusters with different cut-off scales, as suggested by Dekel (1984).
Our simple biasing algorithm does not take into account physical processes of galaxy formation which influence masses and luminosities of galaxies. One of these processes is the supernova-driven wind which blows out interstellar gas from galaxies and is very important in the formation of dwarf galaxies (Dekel & Silk, 1986). A detailed discussion of related problems is, however, outside the scope of this review.
In connection with the biasing problem it is appropriate to ask the question: What is the fraction of matter in voids? This problem was studied by Einasto et al. (1994a) using numerical simulations of the ΛCDM Universe with density parameter Ωm = 0.2, for comparison also models with Ωm = 1.0 were calculated. The void matter was defined by the high-resolution density field, with local density ≤ 1, where is the density in units of the mean density of the sample. Hydrody- namical simulations confirm that in low-density regions galaxies do not form (Cen & Ostriker, 1992, 2000). By definition, half of the particles are initially located in regions with densities less than the mean density. During the evolution matter flows out from voids to systems of galaxies. How much matter is presently located in voids and in the clustered population depends on the parameters of the model. For the Ωm = 0.2 ΛCDM model Einasto et al. (1994a) found for the fraction of matter in voids Fυ = 0.15 ± 0.05.
The above formula takes into account the major factor — the absence of galaxies in voids. Differences in the distribution of dark matter and matter associated with galaxies in high-density regions influence the biasing parameter much less, as our calculations show. What is important here is the use of a high- resolution density field with smoothing scale of the order of the scale of actual structures — galaxies with dark halos and groups/clusters of galaxies. Using such a small smoothing scale we get for voids a zero density of matter associated with galaxies.
Once I discussed with Enn Saar this problem. He argued that in mathematical statistics a zero density has no meaning, since in this cas
e it is impossible to estimate errors. Thus large smoothing is needed to have everywhere non-zero density of the clustered matter. But it seems to me that biasing is not a statistical, but a hydrodynamical problem. During the early stages of the evolution of the Universe there were no galaxies, only the primordial gas, a mixture of dark matter and hydrogen-helium baryonic gas. In this gas the density of heavier chemical elements was zero. Also, remember the approach of Zeldovich to the evolution of the early Universe — he considered the problem as a hydrodynamical one, particles were used only as markers of positions, but the formula for the evolution were hydrodynamical.
In this approach the numerical value of the biasing factor depends on the density threshold 0, used to separate clustered and unclustered matter. Our numerical simulations show that the distribution of local densities (smoothed with a small kernel) is a smooth function of the density, i.e. there exists no natural value for the threshold density, which could be found from the density distribution itself. Thus the value of the threshold density must be calculated separately, taking into account processes which lead to the formation of galaxies. One such possibility is the Press & Schechter (1974) algorithm of galaxy formation (see also Bardeen et al. (1986)). According to this approximation galaxies have time to collapse in the Hubble time, if the local density of the contracting cloud has a density exceeding 1.68 times the overall mean density. I used various density thresholds in the interval from 1 to 1.68 to find the biasing parameter. Results show that the biasing parameter changes only a little when the threshold is varied within this small interval.
To conclude the discussion of the biasing problem I can say that our results have two consequences:
(1) the biasing parameter b is not a free parameter which can be chosen to bring models into agreement with observations, as done in early simulations of CDM models of critical cosmological density (Davis et al., 1985; White et al., 1987); actually it depends on the fraction of matter in the clustered population;
(2) there is no possibility of hiding large amounts of dark unclustered matter in voids; most of the low-density matter has been ‘eaten’ by galaxies and systems of galaxies, increasing the fraction of the clustered matter. This excludes CDM models with Ω = 1, since the observed value of the density of matter associated with galaxies is Ω ≈ 0.2 (Einasto et al., 1974d; Ostriker et al., 1974).
7.1.5 Power spectra of galaxies
In numerical simulations of the evolution of the structure in the universe for every time step both the mass density and the gravitational potential fields are calculated; the potential field is needed to find the velocities of particles. Using these data it is easy to calculate also the power spectrum of the density field for every evolution step. During our visits to the Cambridge Institute of Astronomy with Mirt Gramann we compared results of simulations with real observational data. Soon we understood that the same technique can be used to calculate the power spectra for galaxies and clusters of galaxies. We adapted the simulation program to apply it to volume limited samples of galaxies and clusters of galaxies. We presented the principal results of our analysis in October 1990 at the Ringberg workshop and at a conference in Rome, and in March 1991 at the 2nd DAEC workshop in Paris. More detailed versions were published by Gramann & Einasto (1992) and Einasto etal. (1993).
Our analysis had two goals. The first goal was to compare the power spectra for galaxies of various luminosity and for clusters of galaxies to detect the possible variations of the power spectra with sample location, luminosity limit, richness etc. The second goal was to compare the real power spectra with spectra for various cosmological models. For an ideal sample of infinite volume the power spectrum is directly related to the correlation function, as they form a Fourier transform pair. But in reality the situation is more complicated. The correlation function characterises the structure of galaxy systems well on small scales, whereas the structure on large scales can be better described in terms of the power spectrum.
Fig. 7.9 Comparison of the observed power spectra with the predictions in different CDM models. Triangles and open circles show the observed spectra in the Perseus and Virgo-Coma regions, respectively. The linear predictions of models are plotted as follow: CDM model with Ω = 1, h = 0.5 — dotted line; CDM Ω = 0.2 model for h = 0.5 and h = 1 with a dotted and dashed line, respectively. Deviations of models from observations on small scales almost disappear when non-linear evolution of models is taken into account. On large scales the evolution is linear, the absence of large-scale power in the Ω = 1 model is clearly seen (Gramann & Einasto, 1992).
Gramann & Einasto (1992) and Einasto et al. (1993) calculated power spectra for various volume limited galaxy samples in the Virgo-Coma and the Perseus supercluster regions, and for Abell clusters of galaxies. The analysis showed that the observational power spectra for galaxies of various luminosity limits, and for clusters of galaxies have a similar form in the log–log presentation, but have different amplitudes. This effect is easily explained by different fractions of matter in the clustered populations, i.e. by different biasing levels as discussed in the previous section.
We compared the observational power spectra with the spectra of the standard CDM model (Ω = 1), and of the open CDM model with Ω = 0.2 and for various Hubble parameter values. The results of the comparison are shown in Fig. 7.9. In this Figure we showed linearly evolved models, but in a more detailed comparison we used ΛCDM models with Ωm = 0.2 instead of open CDM models with a similar density parameter. The Figure shows that the standard CDM model with Ω = 1 has much less power on large scales than the real data and open CDM (or ΛCDM) models with Ωm = 0.2. Our finding of the absence of large-scale power in the standard CDM model confirmed independent analyses by Efstathiou et al.(1990) and Peacock (1991). Efstathiou et al. calculated angular correlations for the deep APM galaxy survey and for the large-scale clustering of IRAS galaxies. Peacock derived power spectra for the CfA galaxy survey and for the IRAS QDOT infrared galaxy survey. In all these studies the deficit of the large-scale power in the standard CDM model was clearly seen.
Gramann & Einasto (1992) analysed also correlation functions of various galaxy samples. All samples had a characteristic shoulder on a scale r ≈ 3 h−1 Mpc, visible already in our previous data, see Fig. 7.2. This shoulder corresponds to the transition from almost round clusters/groups of galaxies to elongated filaments of galaxies.A similar change is observed in the power spectrum at a scale λ ≈ 10 h−1 Mpc; this scale is usually interpreted as the scale of transition from the linear evolution on larger scales to the non-linear evolution on smaller scales.
An independent analysis of power spectra was made by Vogeley et al. (1992) for galaxies in the second CfA redshift survey, and by Park et al. (1992a) for galaxies in the Southern Sky Redshift Survey. Changbom Park developed a very accurate method to calculate power spectra for observed galaxy samples. Both galaxy samples show that at small scales (λ ≤ 30 h−1 Mpc) the observed power spectrum is consistent with the standard CDM model, but on larger scales data indicate an excess of power over the standard model. The open CDM model with Ωh = 0.2 is consistent with the observed spectrum over all scales.
The characteristic diameter of supervoids between rich superclusters is about 100 h−1 Mpc (Jõeveer et al., 1978), the same scale was found by Broadhurst et al. (1990) between peaks in the galaxy distribution towards Galactic poles. This rises the question: What is the scale of a homogeneous Universe? This problem was investigated by Einasto & Gramann (1993) by studying the behaviour of the power spectrum for galaxies on large scales. The transition scale to homogeneity was defined as the scale where the spectral index of the matter power spectrum reaches the value n = 1, predicted by the Harrison-Zeldovich model. We used various observational samples to calculate the power spectra (Abell clusters, CfAII galaxies, QDOT galaxies), and compared the observed power spectra with models. Our result was that at scale λ ≈ 175 h−1 Mpc the power spectrum reaches the Harrison-Zeldovich form.
To see the in
fluence of large-scale waves to the density fields of matter and of gravitational potential Einasto & Gramann (1993) calculated the evolution of two-dimensional CDM models of size L = 512 h−1 Mpc for a 5122 mesh with 5122 particles. Two cases were studied, one with the full CDM-type power spectrum, and the other with the power spectrum cut on a scale λ = L/4, so that the amplitude of all waves of scales larger than the cutoff scale was put to zero. Both models had the same realisation of initial density fluctuations. The final distribution of particles in clustered regions, and the shape of potential wells corresponding to high-density regions is shown in Fig. 7.10. The Figure shows that the distributions of clustered particles in both simulations are very similar, however the shape of potential wells is very different. In the model with the full power spectrum potential wells are deeper, and their mutual distance from each other is larger than in the truncated model. This exercise shows that large waves play an important role in the formation of the structure of the Universe.
Fig. 7.10 The upper panels show the distribution of simulated galaxies in two-dimensional models (particles in voids are removed), the lower panels show the respective potential wells — regions of negative gravitational potential. In the left panels we plot data for the model with a sharp truncation of the spectrum, in the right panels for the CDM model with a smooth transition to the Harrison- Zeldovich spectrum on large scales. Equipotential lines have been drawn in both panels at the same levels (in dimensionless units) (Einasto & Gramann, 1993).
In late 1990’s I turned to the study of the power spectrum again, motivated by the BEKS effect (Broadhurst et al., 1990; Einasto et al., 1997a). Together with my collaborators I tried to find the present power spectrum for galaxies (Einasto et al., 1999c), to calculate more accurately the biasing correction (Einasto et al., 1999a), and to find the primordial power spectrum of matter (Einasto et al., 1999b). However, the data available at this time were insufficient to find a much better power spectrum than had been found earlier. The modern SDSS data yield much better power spectra, but the detailed description of these new results is beyond the scope of this book.
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