4 Constraints from extragalactic redshift surveys

The 2dF Galaxy and QSO Redshift Surveys (2dFGRS, Colless et al. 2001; 2QZ, Croom et al. 2001) are two of the largest redshift databases for the low and high redshift universes currently available. As an "application'' of our thought experiment, consider the possibility that similar surveys have been carried out from the vantage point of another distant galaxy and that the results are sent to us. What would we be able to learn from these extragalactic redshift surveys?

In the following I consider two different scenarios with respect to the location of our "extragalactic collaborators'' (object 1 in the terminology of Sect. 2). Both the 2dFGRS and 2QZ cover two distinct survey regions, near the North and South Galactic Poles (NGP and SGP), which lie approximately in opposite directions on the sky. First I will consider the case where object 1 lies in a direction roughly perpendicular to the NGP-Earth-SGP line and at a distance much smaller than the median redshifts of the 2dFGRS and 2QZ ($\approx$0.1 and 1.5 respectively). Specifically I choose $\alpha = 06^{\rm h}$, $\delta = +80\hbox{$^\circ$ }$ and z₁ = 0.01. In this case (which I will label case A) combining our own redshift surveys with those from object 1 affords us a stereoscopic view of these regions of the universe.

Figure 3: Constraints on $\Omega _{\rm M}$ and $\Omega _\Lambda$ derived from combining the 2dFGRS and 2QZ with their extragalactic counterparts obtained from a distant galaxy (object 1). Each panel shows a different combination of case A/B (referring to the supposed location of object 1, see text) and redshift survey used. Each set of contours marks the 68, 90 and 99% confidence levels. Solid contours show the constraints derived from the NGP alone, dashed contours refer to the SGP (omitted in panel d for clarity) and the greyscale contours show the joint NGP+SGP constraint. The "+'' in each panel marks the input cosmology of $(\Omega _{\rm M}, \Omega _\Lambda ) = (0.3, 0.7)$ and the "$\times$'' in panels c), e) and f) shows the best fit. In panels a), b) and d) $\Omega _{\rm M}$ and $\Omega _\Lambda$ are not well constrained individually but only in a linear combination which is shown as the dotted lines.

In the second scenario (case B) I take object 1 to be itself a representative member of the 2dFGRS or 2QZ. Specifically, in each case I choose a survey object near the centre of the NGP region and at the median redshift of the survey. For the 2dFGRS I choose object TGN182Z064 at z₁ = 0.1 and for the 2QZ I choose object J115952.8-012236 at z₁ = 1.5.

The datasets used in the following consist of the 100 K and 10 K public data releases of the 2dFGRS and 2QZ respectively. From the 2dFGRS I have selected all objects with quality $Q \ge 3$ and 0.01 < z₂ < 0.5 which resulted in 34 489 and 53 861 galaxies in the NGP and SGP respectively. From the 2QZ I have selected all objects classified as QSOs and with quality q₁ = 11, yielding 3965 and 6488objects.

For case A I assume that all of the objects selected above have also been observed by object 1, which implies that object 1's survey parameters are very similar to those of the actual 2dFGRS and 2QZ. For the case B/2dFGRS combination I make the same assumption, now implying that object 1's survey must be both wider and deeper. For the B/2QZ combination most of the SGP QSOs lie at very large redshifts as seen from object 1 and so I impose an upper limit of z₂'< 5, which leaves only 167 SGP QSOs in common to the 2QZ and its case B counterpart from object 1.

In Fig. 3 I show $\chi^2(\Omega_{\rm M}, \Omega_\Lambda)$ (see Eq. (10)) for each of the four combinations A,B/2dFGRS,2QZ as well as the joint constraints from combining the two surveys. Here, I have added Gaussian noise both to $z_2'(\Omega_{\rm M}=0.3, \Omega_\Lambda=0.7)$ and z₂, assuming the same noise characteristics for z₂' as for z₂, i.e. for the 2dFGRS we have $\sigma_{z_2'} = 64, 89, 123$ km s^-1 for Q=5, 4, 3 (Colless et al. 2001) and for the 2QZ we have $\sigma_{z_2'} = 0.0035$(Croom et al. 2001).

 

 

Table 1: Summary of the constraints derived in Fig. 3.

Case	Quantity constrained	Error (99%)
A 2dFGRS	$\Omega_\Lambda- 0.62 \Omega_{\rm M}$	$1.6 \times 10^{-3}$
A 2QZ	$\Omega_\Lambda- 2.08 \Omega_{\rm M}$	$3.6 \times 10^{-3}$
A 2dFGRS+2QZ	$\Omega_{\rm M}, \Omega_\Lambda$	$5.3,4.0 \times 10^{-3}$
B 2dFGRS	$\Omega_\Lambda- 0.74 \Omega_{\rm M}$	$1.2 \times 10^{-4}$
B 2QZ	$\Omega_{\rm M}, \Omega_\Lambda$	$3.4,7.9 \times 10^{-4}$
B 2dFGRS+2QZ	$\Omega_{\rm M}, \Omega_\Lambda$	$3.3,2.4 \times 10^{-4}$

In case A neither the 2dFGRS nor the 2QZ by themselves can constrain both $\Omega _{\rm M}$ and $\Omega _\Lambda$ individually (panels a and b). Both surveys only constrain a linear combination as in Sect. 3, shown as the dotted lines. However, since the linear coefficient m is significantly different for the two surveys (cf. Table 1) using them jointly breaks the degeneracy and constrains $\Omega _{\rm M}$ and $\Omega _\Lambda$ to within $\sim$1% (panel c). Remarkably, it so happens that the 88 350 low-z objects of the 2dFGRS contribute almost equally to these constraints as the 10 453 high-z objects from the 2QZ. In this sense the two surveys are fortuitously well matched. The difference between the constraints from the NGP and SGP is purely due to the different numbers of objects in these regions.

For the B/2dFGRS combination (panel d) this difference is much larger. The reason is that in this case object 1 lies among the NGP galaxies which hence appear at much smaller z₂' than the SGP galaxies and thus give much weaker constraints. Again, only a linear combination of $\Omega _{\rm M}$ and $\Omega _\Lambda$ is constrained. In contrast, the combination B/2QZ does break the degeneracy (panel e). This is almost entirely due to the NGP QSOs, while the 167 SGP QSOs at z₂'< 5contribute additional constraints on $\Omega _{\rm M}$ (but not $\Omega _\Lambda$). Jointly the two surveys constrain  $\Omega _{\rm M}$ and  $\Omega _\Lambda$ to within $\sim$0.1%.