3 Constraints from a single ${\vec z} {_2'}$ measurement

To investigate the potential of z₂' measurements for constraining $\Omega _{\rm M}$ and $\Omega _\Lambda$ I now use $(\Omega _{\rm M}, \Omega _\Lambda ) = (0.3, 0.7)$ as an input or reference cosmology in defining the quantity

 

$$\chi^2(\Omega_{\rm M}, \Omega_\Lambda) = \sum_{i=1}^{n_p} \left[ ... ...ga_{\rm M}, \Omega_\Lambda; z_1^i, z_2^i, \alpha^i)}{\sigma^i_{z_2'}} \right]^2$$ (10)

(not to be confused with the radial comoving coordinate). The sum goes over $n_{\rm p}$ pairs of objects, where the ith pair is defined by z₁ⁱ, z₂ⁱ and $\alpha^i$. $\sigma _{z_2'}$ is the uncertainty in the measurement of z₂'. In the absence of a specific notion of how a measurement of z₂' might be achieved, I consider $\sigma _{z_2'}$ as an arbitrary scaling parameter for the moment.

Figure 1: Examples of the constraints placed on $\Omega _{\rm M}$ and $\Omega _\Lambda$ by a single measurement of z2'. Each panel shows a different choice of z1, z2 and $\alpha$. This figure demonstrates that, in principle, almost any given linear combination of $\Omega _{\rm M}$ and $\Omega _\Lambda$ can be constrained by a suitable choice of the cosmological triangle. The solid contours show the 68, 90 and 99% "confidence levels'' (cf. Eq. (10)). In each panel, $\chi ^2$ reaches its minimum everywhere along the dashed line, and $\sigma _{z_2'}$, which scales the width of the contours, was chosen to give approximately similar widths in all panels. The cross marks the input cosmology of $(\Omega _{\rm M}, \Omega _\Lambda ) = (0.3, 0.7)$. The shaded region in the upper left corner of each panel represents "bouncing universe'' cosmologies with no big bang in the past ( $E(z) \le 0$ at a finite z > 0, see e.g. Carroll et al. 1992).

First, let us consider $n_{\rm p} = 1$, in which case $\chi ^2$ simply reflects the variation of z₂' as a function of $\Omega _{\rm M}$ and $\Omega _\Lambda$. In Fig. 1 I plot $\chi^2(\Omega_{\rm M}, \Omega_\Lambda)$ for a number of different choices of z₁, z₂ and $\alpha$. Since one "data'' point cannot constrain two free parameters, $\chi ^2$ attains its minimum value not at a point but everywhere along a line in the $\Omega _{\rm M}$- $\Omega _\Lambda$plane. I find that a single measurement of z₂' generally constrains a linear combination of the form $C = \Omega _\Lambda + m \Omega _{\rm M}$, although there are clearly exceptions to this rule, in particular at z > 2(cf. the last two panels of Fig. 1). The examples in Fig. 1 further demonstrate that almost any value of mis possible, including the cases $m = -\frac{1}{2}$, $m \rightarrow \infty$, m = 1 and 0, where the linear combination constrained is C = -q₀ (the deceleration parameter), $\Omega _{\rm M}$, $\Omega_{\rm total}$and $\Omega _\Lambda$ respectively.

In order to show the low-redshift characteristics of this constraint in more detail I have parameterised it in terms of the angle of the $\chi^2_{\rm min}$-line with the $\Omega _{\rm M}$-axis, $\theta = \arctan(-m)$, and the formal error on C, $\sigma_C$, which is equivalent to the width of the contours in Fig. 1 (measured perpendicular to the $\chi^2_{\rm min}$-line). In Fig. 2 I plot $\theta$ and $\sigma_C$ for 284 different triangles with $0.1 \le z_1, z_2 \le 1.5$ and $0\hbox{$^\circ$ }\le \alpha \le 80\hbox{$^\circ$ }$. Since $\sigma_C$ scales approximately linearly with the arbitrary value of $\sigma _{z_2'}$ I plot the ratio of the two.

Figure 2: Detailed behaviour of the constraint placed on the linear combination $C = \Omega _\Lambda + m \Omega _{\rm M}$ by a single z2' measurement at $z \le 1.5$. Each position along the x-axis corresponds to a different combination of z1, z2 and $\alpha$. Each change in background shading (grey and white) marks a change in the value of z1 as indicated along the bottom axis. Similarly, for each value of z1, hashed and non-hashed backgrounds mark the various values of z2, which are also indicated along the bottom axis. Finally, for each combination of z1 and z2, $\alpha$ varies from $0\hbox {$^\circ $ }$ to $80\hbox {$^\circ $ }$. Constellations which do not constrain the cosmological model, i.e. $z_1 \le z_2$ and $\alpha = 0\hbox {$^\circ $ }$, have been excluded. There are no numerical errors or noise in this plot, the small-scale "spiky'' behaviour is solely caused by the variation of $\alpha$. The top panel shows z2'. The middle panel shows $\theta = \arctan(-m)$. When $\theta = 27\hbox {$^\circ $ }$, z2' constrains C = -q0; $\theta = 90\hbox {$^\circ $ }$ corresponds to a constraint on $C = \Omega _\Lambda$. The bottom panel shows the formal error on C scaled by $\sigma _{z_2'}$. This combination is approximately independent of the (arbitrary) value of $\sigma _{z_2'}$.

In the low-redshift limit, where all distances are small, the triangle cannot constrain the curvature but only q₀. This is a well-known property of many cosmological tests and due to the fact that, to second order, the expansion of the integral in Eq. (3) involves only q₀. For the redshift range shown, $\theta$ takes on all values from this low-redshift limit to > $90\hbox{$^\circ$ }$ where $C = \Omega _\Lambda$. In fact, the same value of $\theta$ can be produced by several, quite different configurations. For a given z₁, or if the values of z₁ and z₂ are exchanged, $\theta$ is generally larger for z₂ < z₁, with the largest values achieved for $\alpha = 0\hbox {$^\circ $ }$.

Over the redshift range considered, $\sigma_C$ decreases by nearly four orders of magnitude. The increasing sensitivity to $\alpha$ with z₁ is due to a strong anti-correlation of $\sigma_C$ with z₂'. The longer the third side of the triangle, the better the constraints on the cosmological parameters. This also explains why for a given z₁ the tightest constraints are achieved for the largest z₂ and $\alpha$.