Projected two-point correlation function

For the analysis of the two-point correlation function on small scales often the projected correlation function has been calculated, which is in theory independent of any radial distortions (Peebles, 1980; Davis & Peebles, 1983). For small angles r2 = r_p2+π2. Thus the projected correlation function is dened as

w(rp) =

Z _∞

−∞

ξ(rp, π) dπ . (3.1)

Note that w(rp) has dimensions of length. If it was possible in practice to integrate to

innity, it would in principle be possible to recover the three-dimensional real space corre- lation functionξ(r), andw(rp)would be far better suited to infer cosmological parameters

from the spatial distribution of galaxies than ξ(rp, π), which suers from redshift space

distortions. However, since integrating out even to very large distances without signi- cantly increasing the noise is not feasible (in particular if the signal is smeared out and the amplitude diminished by large redshift errors), the integration limits have to be nite and even rather small, see also Norberg et al. (2009) for an illustration of this. This means that a part of the clustering signal, which depends on the pairwise redshift probability dis- tribution function as well as on the real and the assumed cosmology, can not be recovered. This is illustrated in Figure 3.10, where w(rp)is shown for dierent widths of the assumed

redshift errors (σz =0.015, 0.03, 0.06, and 0.12, respectively) and two dierent choices of

the integration limits, ∆π= 163.5 h−1Mpc and ∆π= 298.5 h−1Mpc.

The resulting shape of w(rp) depends strongly on both the width of redshift errors and

the size of the integration limits: if the integration limits are very large, most of the signal can be recovered and the dierence between real and redshift space and the correlation function aected by errors is, although small, still visible. It is not advisable to choose

∆π= 298.5h−1Mpc, as the measurement will be dominated by noise. On the other hand,

Figure 3.10: The projected correlation function w(rp) for real and redshift space (black

and red lines, repsectively), and for dierent widths of the assumed redshift errors, left: for integration limits of ∆π= 298.5 h−1Mpc; right: ∆π = 163.5h−1Mpc.

if the BAO ring is supposed to be fully included in the integration, the limits cannot be much smaller than ∆π ≈ 150 h−1Mpc in which case the resulting w(rp) is extremely

dependent on the size of the redshift errors (i.e. the fraction of the signal which can be recovered).

Figures 3.11 and 3.12 show the projected correlation functionw(rp)of the L-BASICC dark

matter halos integrated up to πmax = 298.5 h−1Mpc and πmax = 163.5 h−1Mpc, respec-

tively, for redshift errors of σz = 0.015, σz = 0.03, σz = 0.06, and σz = 0.12, as well as the

best-tting model for each case. The amplitude has to be taken into account, otherwise the t fails: there is not enough information in the shape alone. The corresponding tted values of the dark energy equation of state parameter wDE and the bias b and their errors

are listed in Table 3.3, and shown in Figure 3.13 as a function of σz.

Although the t is not biased for σz . 0.06, the errors are, as expected, much larger and

more quickly increasing with increasing redshift errors than for the corresponding ts of ξ(rp, π). Hence, it is concluded that for the analysis of the large scale two-point correlation

function as a means to constrain cosmological parameters from photometric data, ξ(rp, π)

is better suited than the projected correlation function w(rp).

σz wDE(πmax = 298.5 h−1Mpc) b(πmax = 298.5 h−1Mpc) 0.0 −1.018±0.326 2.477±0.717 0.015 −1.018±0.419 2.683±0.551 0.03 −0.980±0.433 2.741±0.551 0.06 −1.017±0.437 2.725±0.608 0.12 −1.197±0.463 2.835±0.862 σz wDE(πmax = 163.5 h−1Mpc) b(πmax = 163.5 h−1Mpc) 0.0 −1.078±0.386 2.683±0.470 0.015 −0.962±0.363 2.702±0.446 0.03 −0.979±0.372 2.706±0.456 0.06 −1.034±0.420 2.626±0.550 0.12 −1.143±0.576 2.764±0.804

Table 3.3: The best-tting values of wDE and b, as deduced from the projected correlation

function w(rp), with integration limits of πmax = 298.5 h−1Mpc (top rows) and πmax =

163.5 h−1Mpc (bottom rows), respectively, for Gaussian redshift errors with σz = 0.015,

Figure 3.11: The projected correlation function w(rp)of the L-BASICC dark matter halos

integrated up to πmax = 298.5 h−1Mpc for redshift errors of σz = 0.015 (top right),

σz = 0.03 (top left), σz = 0.06 (bottom right) and σz = 0.12 (bottom left), black solid

lines: mean, error bars: 1σ-deviation calculated from the variance of the 50 boxes, red solid line: best-tting wCDM model, blue dot-dot-dashed line: ΛCDM case, green dotted

Figure 3.12: The projected correlation function w(rp)of the L-BASICC dark matter halos

integrated up to πmax = 163.5 h−1Mpc for redshift errors of σz = 0.015 (top right),

σz = 0.03 (top left), σz = 0.06 (bottom right) and σz = 0.12 (bottom left), black solid

lines: mean, error bars: 1σ-deviation calculated from the variance of the 50 boxes, red solid line: best-tting wCDM model, blue dot-dot-dashed line: ΛCDM case, green dotted

line: redshift space (σz = 0.00).

Figure 3.13: Fitted values of the dark energy equation of state parameter wDE (left) and

the bias b (right) against the width of the redshift errors applied to the L-BASICC halos and the model. In black w(rp) was integrated up to πmax = 298.5 h−1Mpc for the t,