Figure 4.12: Mass contributions to the cross-trispectrum in square configuration de-fined in Eq.(4.94)at redshiftz= 0. Shown is the effect on the one-halo term from halos in certain mass intervals as indicated by the line color. Note that the black line shows the to-tal cross-trispectrum including all halo terms which explains the deviation of the green line to the black line on large scales. Compared to the results of the galaxy power spectrum in the right panel of Fig. 4.4, the trispectrum is dominated by higher mass halos.
rises until k ' 1hMpc−1 where it slowly decreases for small scales. On large scales the reduced trispectrum goes to a constant that is different from that for the reduced bispectrum. The peak atk '1hMpc−1 is approximately one magnitude larger than for the cross-bispectrum, whereas on small scales it drops below 1. On the other hand, the dark matter trispectrum is rising for small scales in the same way as the dark matter bispectrum depicted in the upper left-hand panel of Fig. 4.7.
In Fig. 4.12 we show the different mass contributions to the one-halo term of the cross-trispectrum defined in Eq.(4.94). As expected, the dimensionless cross-trispectrum is dominated by more massive halos compared to the results of the galaxy power spectrum in Fig. 4.4 because it has a stronger mass-weighting. This can be seen for example by comparing the mass factors of the one-halo terms in Eqs. (4.82) and (4.27).
Figure 4.13: Ratio of the one-halo term of the power spectrum including a concentration parameter distribution to the one-halo term with a deterministic concentration parameter as a function of the wave-number. On the left panel we show the ratio for the dark matter power spectrum and on the right panel the ratio for the galaxy power spectrum. We display the results for four different concentration parameter dispersion, i.e.,σlnc∈ {0.1,0.2,0.3,0.4}
indicated by the different line color. Note that a ratio of 1 corresponds to perfect agreement between both spectra.
ratio
R1h(k, z;σlnc)≡ ∆1h(k, z;σlnc)
∆1h(k, z;σlnc = 0). (4.100) The result is depicted in Fig. 4.13, where we have plotted the ratio as a function of the wave-number. We show the results for four different choices of the concentration dispersion, i.e., σlnc ∈ {0.1,0.2,0.3,0.4}. Results from simulations prefer values around σlnc = 0.2–0.3 (see Jing 2000 and the recent results from the millennium run in Neto et al. 200711). These values are bracketed by the two extremes of a low and high value of the dispersion. A higher concentration dispersion leads also to an enhanced power spectrum on small scales. We see that the influence of a concentration distribution starts around k '1hMpc−1 for both panels. However, the impact of the dispersion is negligible also on small scales, especially for the galaxy power spectrum.
We define analogously the ratio of the one-halo term of the bispectrum as R1heq(k, z;σlnc)≡ ∆1heq(k, z;σlnc)
∆1heq(k, z;σlnc = 0), (4.101)
11Note that they quote best-fit values for a log-normal distribution with base 10. Therefore one needs to transform the dispersion usingσlnc= ln(10)σlog
10c.
where we defined the dimensionless form of the bispectrum for equilateral triangles
∆eq(k, z)≡ k3 2π2
pB(k, k, k, z). (4.102) The results of the ratio for the one-halo term of the bispectrum are shown in Fig. 4.14 as a function of the wave-number. On the upper left panel we depict the dark matter bispectrum and on the upper right panel the galaxy bispectrum. The dark matter bispectrum ratio grows in the extreme case ofσlnc = 0.4 to Req1h = 1.5 for the smallest considered scale of k= 102hMpc−1. For σlnc = 0.2 we find R1heq = 1.1 and for σlnc = 0.3 the ratio is R1heq = 1.26 for the smallest scale. On the other hand, we find smaller ratios for the galaxy bispectrum, i.e., R1heq = 1.05 for σlnc = 0.2 and R1heq = 1.1 for σlnc = 0.3.
Clearly, the influence of a stochastic concentration parameter is less pronounced for the galaxy bispectrum. This is expected since on small scales the galaxy bispectrum is dominated by central-satellite-satellite correlations in Eq. (4.56)which are weighted by one density profile less compared to the dark matter density profile. Hence, the galaxy bispectrum is less affected by the enhancement of the density profile due to the log-normal distribution of the concentration parameter. Thus on the smallest scales we have the approximate relation that the ratio of the dispersion for σlnc = 0.2 and σlnc = 0.3 for dark matter corresponds toσlnc = 0.3 and σlnc = 0.4 for galaxy clustering, respectively. We find similar results for the two cross-bispectra (not shown) which is due to the fact that the bispectra including galaxy correlations have on small scales the same dependence on the density profiles.
Finally, we define the ratio for the one-halo term of the dark matter and cross-trispectrum
R1hsq(k, z;σlnc)≡ ∆1hsq(k, z;σlnc)
∆1hsq(k, z;σlnc = 0), (4.103) where we used the definition of the dimensionless trispectrum in Eq. (4.94). The results are shown in Fig. 4.14 in dependence of the Fourier mode, where the lower left panel shows the predictions of the dark matter trispectrum and the lower right panel the results for the galaxy-dark matter cross-trispectrum. Compared to the predictions in Fig. 4.14 the influence of a stochastic concentration is enhanced. This is due to the fact that the trispectrum is weighted by four density profiles in contrast to the bispectrum which is only weighted by three density profiles. We find R1hsq = 1.16 for dark matter and Rsq1h = 1.08 for the cross-trispectrum for σlnc = 0.2 at the smallest scale. Furthermore, we find for σlnc = 0.3, R1hsq = 1.4 for the dark matter and Rsq1h = 1.2 the cross-trispectrum, respectively.
In summary, we find that the inclusion of a concentration parameter distribution is less important for galaxy and galaxy-dark matter cross-spectra compared to dark matter spectra. We showed this effect for the power spectrum, bispectrum and trispectrum.
The effect is most important for the trispectrum.
Figure 4.14: Ratio of the one-halo term of the equilateral bispectra including a concentration parameter distribution to the one-halo term with a deterministic concentration parameter as defined in Eq. (4.101) as a function of the Fourier mode k and at redshift z= 0 (upper panels). Additionally, we give the corresponding ratio for the one-halo terms of the trispectra in square configuration as a function of the wave-number kas defined in Eq.(4.103) (lower panels). All left-hand panels give the results for the dark matter spectra for reference. In the right-hand panels we plot the ratio of the galaxy bispectrum (upper right panel) and the cross-trispectrum (lower right panel). The four models of the concentration dispersion have different line colors as indicated in the figure. On large scales we see no deviations, whereas on small scales the ratio is larger than one.
Chapter 5
Weak Gravitational Lensing
Traditional probes of the large-scale structure like angular and redshift galaxy surveys basically measure the distribution of luminous matter in the Universe. However, as we already mentioned, the matter distribution of the Universe is dominated by non-luminous dark matter. A recent promising tool is weak gravitational lensing which is sensitive to the total mass distribution. The method makes use of the fact that inhomogeneities of the matter distribution induce multiple distortions of the shape of distant galaxies. Analyzing the statistics of this distortion pattern of galaxies is directly proportional to the dark matter power spectrum when we consider Gaussian fluctuations. This is a clear advantage over the analyses of galaxy surveys since it is then not necessary to consider the bias which involves the modeling of complex baryonic effects.
The deflection of light rays by mass concentrations is a prediction of general relativity.
Already in 1919, Eddington performed empirical tests of this theory by observing the light of stars passing close to the Sun during a solar eclipse. He found that the light of the stars was slightly displaced and that the measured deflection could be explained by general relativity predictions1. Soon afterwards, it was suspected that in ideal source-lens configurations massive astronomical objects could lead to multiple images of high-redshift sources. However, only in 1979 the first double imaged quasar was observed (Walsh et al. 1979). Today the most prominent observed lensing features are giant arcs around the central region of massive galaxy clusters (discovered by Lynds &
Petrosian 1986). They can be used to estimate the mass of the lensing galaxy cluster and the results can be compared to dynamical X-ray estimates. These strong effects are rare, however, less distorted images of background galaxies, so-called arclets, can be identified in many galaxy clusters. A perfectly alignment of source, lens and observer allows for a ring-like image around the lens, the so-called Einstein ring (e.g., Impey et al. 1998). All these effects occur in the regime of strong gravitational lensing.
Another method which uses the gravitational lensing effect is microlensing. It utilizes the fact that a moving object shows an enhanced magnification when it enters the line which connects a background source with our line-of-sight. This can be utilized for the search of non-luminous massive compact halo objects (MACHOs) in our Galaxy.
1In fact, a Newtonian consideration yields a deflection angle which is half as large as the general relativity prediction.
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Intensive observations revealed that stellar-mass MACHOs cannot explain the amount of dark matter in our Galaxy (Alcock et al. 2000). Furthermore, the method can be used for the detection of extrasolar planets (e.g., Beaulieu et al. 2006). In summary, gravitational lensing applications provide us with a wealth of information of the matter distribution of the Universe at different length scales.
Although this chapter deals with the weak gravitational lensing effect, we first need to introduce the basic concepts of gravitational lensing which is done in Sect. 5.1.
Subsequently, Sect. 5.2 provides a discussion of weak gravitational lensing induced by the large-scale structure of the Universe which is commonly referred to as cosmic shear.