Other work in the literature

halos. It seems that the linear Kaiser eect is able to model the redshift space distortions at these redshifts on the considered k-modes. Therefore, the non-linear Kaiser eect is not required anymore on the examined k-modes. This would also explain the agreement between real and redshift space for the estimates of b₁ and b₂.

It can be concluded that the estimation of these bias parameters can be performed at high redshifts and for LAEs as tracers for the underlying matter density eld. Therefore, in a future analysis extracting b₁ and b₂ from such a data set can be carried out.

0.6 0.8 1 1.2 1.4

1 100 200 300 400 500 600 700 800 900

Ratio

configuration id 1

1.25 1.5 1.75 2 2.25 2.5 2.75

32 10 1416 18 20 22 24 26 28 30 32 34 36 38 40

Bzs(k1, k2, k3)/Brs(k1, k2, k3)

multiple of k_f

Ratio between the two curves above

z = 2.2

GIPCC theory

0.6 0.8 1 1.2 1.4

1 100 200 300 400 500 600 700 800 900

Ratio

configuration id 1

1.25 1.5 1.75 2 2.25

2.52 10 1416 18 20 22 24 26 28 30 32 34 36 38 40

Bzs(k1, k2, k3)/Brs(k1, k2, k3)

multiple of k_f

Ratio between the two curves above

z = 3.0

GIPCC theory

Figure 3.47: Ratio between the redshift space bispectrumB_zs(k₁, k₂, k₃)and the real space bispectrum B_rs(k₁, k₂, k₃) (also called boost-factor) for the LAEs at z = 2.2 and 3.0 are plotted on the left and right panel, respectively . Each panel contains two sub-panels. The upper sub-panel shows the GIPCC ratio in black and the theoretical prediction in red.

In the lower sub-panel the ratio between the two boost-factors is shown where the dotted black line indicates perfect match between the two quantities.

They extended this method from the power spectrum to the bispectrum and concluded that even at low redshifts the bispectrum can be well modeled. Redshift space was not exam-ined in their work. Therefore, it is not clear if the linear Kaiser eect can be as easily incorporated as proposed in this thesis. They also claim that their ansatz is comparable to 1-loop corrections at least at high redshifts and refer to the similarity of the reduced bispectrum. However, it was concluded in this thesis, that the reduced bispectrum cannot be modeled correctly.

In Smith et al. (2008), the bispectrum was derived for the halo model (Cooray & Sheth, 2002) in real and redshift space. Their investigation was mainly focused on the modeling for the U-shape representation itself. No parameters were tted. They found that on con-siderably large scales (k & 0.10 h Mpc⁻¹) the 2-halo term starts to contribute already a non-vanishing signal to the reduced bispectrum. For the conguration k₁ = 0.10h Mpc⁻¹ and k₂ = 0.20 h Mpc⁻¹ this can be up to 10%. It can be assumed that the same is true forB(k₁, k₂, θ₁₂) because of the calculation of the reduced bispectrum. The reduced signal from the 2-halo term due to the division by the power spectra is not present for the calcu-lation of the bispectrum and therefore this signal is not reduced. The model of Smith et al.

(2008) is in agreement with the simulation at least on the large scales in real space but in redshift space the same problems occur as discussed in this work. Their redshift modeling includes linear Kaiser eect and FoG as in this thesis but no explanation for the mismatch was given there.

Improving the theory of mode-coupling is one possibility to extend the valid range of a model. In Simpson et al. (2011), another approach was chosen. The most non-linear part, the highest density peaks, of the data will be clipped away and these high density peaks are set to a predened maximum value. The resulting bispectrum is not dominated by these extreme structures anymore and can reliably be used up to 0.7 h Mpc⁻¹ atz = 0.0 for the Millenium simulation (Springel et al., 2005) but only in real space. Their approach is comparable to the situation when only low mass halos of the L-BASICC catalogs were utilized for estimating b₁ and b₂. In this case, no information of the massive halos would be used. If the density eld is clipped, there will be still some information of the unclipped density eld present. As for the previous investigations, redshift space was not examined in Simpson et al. (2011). They know that their method extracts not the real bias parameters because of the clipping. They claim that this change of the bias is not necessarily impor-tant because the extracted matter power spectrum is close to the linear one. Therefore, an application to a real data set, where one is interested in the real bias parameters, is still not possible.

Similar results of the reduced bispectrum which were discussed in this section were also found in Pollack et al. (2012) where they claimed that the locality assumption of

Fry & Gaztañaga (1993) is violated. Of course, this would lead to a dierent biasing in the two- and three-point statistics. They smoothed the density eld and extracted the linear and the quadratic bias from the bispectrum and directly from the density eld. They stated that consistent results of these bias parameters with the original tree-level ansatz can be achieved with a smoothing scale of R ≈ 20h⁻¹ Mpc. This smoothing scale is not known a priori and set as a free t parameter. So far, their investigation was only performed in

real space. They also found that higher order corrections are required for recovering the bias parameters.

In Sefusatti et al. (2006), an attempt to extract cosmological parameters from the bispec-trum was examined. They modeled deviations between the mock catalogs generated by PTHALOS (Scoccimarro & Sheth, 2002) and the2^ndorder Lagrangian perturbation theory (Scoccimarro, 2000) simulations with the original tree-level approach. With this method they are able to use the bispectrum up to0.3hMpc⁻¹ in real and redshift space. However, this is it not a full modeling because it requires the presence of N-body simulations to which the measurement can be compared. It was concluded that the Likelihood-functions are almost the same as for the power spectrum. Only a combined analysis on the two clustering analyses will be able to give tighter constraints on cosmological parameters.

In this thesis, the modeling of the dark matter bispectrum from rst principles was not only examined in real space but also in redshift space. In contrast to the above discussed publications this is mostly not the case. An additional contribution of the analysis in this thesis is not only the extended investigation of the bispectrum in redshift space but also the wide range of examined triangle congurations (all congurations up k_max . 0.19 h Mpc⁻¹).

Chapter 4

Summary and Conclusions

This chapter is divided into three parts. In Section 4.1, the results obtained from two- and three-point statistics will be summarized. Possible improvements on the modeling of the anisotropic two-point correlation function ξ(r_p, π)and the bispectrum B(k₁, k₂, k₃) will be discussed in Section 4.2 in order to extend the range of validity of the model. This thesis will be concluded with Section 4.3 where possible future projects for real data sets will be proposed.