2.7 Correlation functions
2.7.3 Redshift errors for the three-point statistic?
Before this chapter will be concluded, a qualitative discussion will be given, why the ex- amination of redshift errors on the three-point statistics is excluded from this thesis. In Figure 2.13, a cuboid with a side length of200h−1 Mpc for the x- and z- direction and500
h−1 Mpc for the y-direction at z = 0.5 of the L-BASICC simulation number 48, in which
ξ48(r) behaves mostly like the mean ξ(r), is projected along the z-axis. On the top panel,
the blue dots represent the position of the halos in real space and the red ones the halos in redshift space. Their positions are distorted along the y-axis of the simulation box, which was dened to be the line-of-sight direction. The shift can easily be identied.
In redshift space, the structures look squashed, as it is expected. These structures are still visible and it is possible to extract a clear clustering signal from this simulation. On the bottom panel, Gaussian redshift errors with a rms of σz = 0.03 were applied to the same
L-BASICC simulation in redshift space with periodic boundary conditions (of course along the y-axis). Almost no structures can be detected anymore. As shown for the BAO peak inξ(rp, π), prominent features are washed out.
The signal of the three-point clustering statistics will also be reduced by redshift errors. The signal-to-noise is anyway lower the higher the clustering statistics is and with redshift errors the situation gets worse. The triangles which are counted by the three-point corre- lation function will look elongated in redshift error space. The spherical average for the calculation of the three-point statistics would not make any sense because homogeneity and isotropy are destroyed by the redshift errors anyway. Therefore, such an analysis is excluded from this thesis.
As mentioned earlier, the three-point statistics is not as thoroughly investigated as the two-point statistics. The basic eects of non-linear structure growth, biasing and pecu- liar velocities must be examined rst, before further complication are incorporated in the model.
0 100 200 300 400 500 0 50 100 150 200 y [Mpc/h] x [Mpc/h] 0 100 200 300 400 500 0 50 100 150 200 y [Mpc/h] x [Mpc/h]
Figure 2.13: A cuboid projected along the z-axis with a side lengths (lx, ly, lz) = (200
h−1 Mpc, 500 h−1 Mpc, 200 h−1 Mpc) was taken from the dark matter halo L-BASICC
simulation box number 48 at z = 0.5. Real and redshift space are combined in the upper panel and are represented by the blue and the red dots, respectively. The lower panel shows the eect of Gaussian distributed redshift errors with a rms of σz = 0.03.
Chapter 3
Investigation of
ξ(r
p
, π)
and
B(k
1
, k
2
, k
3
):
Extracting the parameter of the
equation of state of dark energy and the
bias parameters
This chapter is divided into two smaller main sections. The rst section will be focused on the two-point statistics whereas the second is mainly interested in the bispectrum. Section 3.1 will describe how to extract the equation of state parameter of dark energy and the linear bias from ξ(rp, π)as well as from w(rp). A huge part of this section is from
Schlagenhaufer et al. (2012). In Section 3.2, the extraction of the linear and the quadratic bias from the bispectrum will be investigated in detail.
3.1 Two-point statistics: Determination of
w
DEand
b
This rst big section will present the results of the estimation of the equation of state parameter of dark energywDE and the linear biasb by means of ξ(rp, π)andw(rp). In thediscussion of the two-point statistics, the linear bias is called b and is dened by Equation (2.75), while for the three-point statistics the linear biasb1 is associated with the rst term
of the series expansion of Equation (2.77).
It is clear from this denitions that the b-variable is an eective quantity which contains contributions from all bn-parameters of the series expansion. The anisotropic two-point
correlation functionξ(rp, π), see Section 3.1.2, and the projected correlation functionw(rp),
see Section 3.1.3, will be utilized for the determination of wDE andb. These two quantities
were selected for this task because they have special properties.
ξ(rp, π) is a two-dimensional function and its information content is not combined into a
few bins like it is the case forξ(r). More information can be used to constrain cosmological
parameters, in this thesiswDE andb. This underlines the importance of a careful modeling
w(rp)is the projected correlation function and the result when ξ(rp, π)is integrated along
the line-of-sight component (theπ-direction). Ideally, this integration should be performed to innity in order to recover the three-dimensional real space correlation function ξ(r).
Then, w(rp) would be better suited for extracting wDE and b. Therefore, w(rp) will also
be examined.
The exact procedure for performing the analysis will be discussed in Section 3.1.1. The investigation is carried out at z = 0.5 only. This is the expected mean redshift of Pan- STARRS (Cai et al., 2009), on which the model will be applied to in a future analysis. In the Sections 3.1.2 and 3.1.3,wDE andbare extracted from the L-BASICC dark matter halo
catalogs by means ofξ(rp, π)andw(rp), respectively. This whole section will be concluded
with Section 3.1.4 in which the introduced model ofξ(rp, π)will be qualitatively compared
to similar approaches.