1 Introduction

Measuring cosmological parameters is a major industry and it has been argued persuasively that the era of quantitative (Efstathiou et al. 2002) or even precision (Schramm & Turner 1998) cosmology has arrived. In particular, much attention has been devoted to the cosmological densities of mass, $\Omega _{\rm M}$, and vacuum energy, $\Omega _\Lambda$, because they determine the expansion history (and future) as well as the curvature of the universe. Over the decades, innumerable ways of measuring $\Omega _{\rm M}$and/or  $\Omega _\Lambda$ have been devised, ranging from the classical tests such as the magnitude-redshift (e.g. Riess et al. 1998; Perlmutter et al. 1999), the angular size-redshift (e.g. Kellermann 1993; Buchalter et al. 1998) or the number-magnitude (e.g. Phillipps et al. 2000) relations to their refinements such as the Alcock-Paczynski test (Alcock & Paczynski 1979) and the modern tests involving, e.g., the power spectrum of CMB fluctuations (e.g. de Bernardis et al. 2002), the clustering of galaxies (e.g. Peacock et al. 2001) or quasars (e.g. Hoyle et al. 2002) or the statistics of gravitational lensing (e.g. Fukugita et al. 1992).

Here, I discuss the constraints that could be placed on $\Omega _{\rm M}$ and $\Omega _\Lambda$ if it were possible to measure (independently of cosmological parameters) the redshift of an object at cosmological distance as observed by a second distant object. In Liske (2000) (hereafter L00) I showed how to calculate this redshift from the redshifts and angular separation of the two objects as observed by us (see also Roukema 2001). The result depends only on $\Omega _{\rm M}$ and $\Omega _\Lambda$ and hence a comparison with measurements constrains these parameters.

The general idea is the following: if the curvature of the underlying manifold is known then two sides and an angle of a triangle fully determine the rest of the triangle. Conversely, three sides and an angle fully determine both the triangle and the underlying curvature. As usual, by expressing distances in terms of redshifts we make the connection to the expansion history of the manifold.

This idea combines many of the positive features of other cosmological tests while avoiding some of their problems. As pointed out above, $\Omega _{\rm M}$ and $\Omega _\Lambda$ are the only cosmological parameters involved. Any possible degeneracies with other parameters, such as the Hubble constant, H₀, are hence avoided. In principle, the test makes no assumption about the properties, the physics or the evolution of the objects involved. In particular, it does not require a standard candle or standard ruler. Hence there is also no restriction on the type of object. Any object observable over cosmological distances (e.g. galaxy, QSO, absorption line system) is, in principle, a potential candidate.

The main drawback of this test is that the measurement of a cosmological triangle's third side appears quite impossible! Just as we measure distances or redshifts between us and other objects exclusively from the photons received by them, we require as a prerequisite for the measurement of a triangle's third side the appearance of an object (in the widest possible sense) to depend on the radiation received from a another, specific object at cosmological distance. This is a rare scenario. For example, although the physics of galaxies is generally thought to depend on the cumulative radiation from other galaxies and quasars, i.e. the diffuse background, it is difficult to imagine any discernible difference in the appearance of, say, the Milky Way caused by the photons received from any other specific galaxy in the universe.

However, an effect may be observed in the vicinity of particularly luminous sources. For example, it is well known that the UV radiation of a QSO alters the ionization fraction of its surrounding intergalactic medium (the proximity effect, see e.g. Liske & Williger 2001). Phillipps et al. (2002) pointed out that, potentially, this effect could be used to infer the luminosity distance between a QSO and a nearby absorber. In effect, this would be a measurement of the third side of a triangle. However, this measurement involves the intrinsic luminosity of the QSO which must be inferred from its apparent brightness and hence it is not independent of the metric.

Although no less useful, such tests, where the measurement of the third triangle side depends on the cosmological model, are less generic and conceptually different from the test proposed here. They depend on the details of the measurement of the third side and must be studied on a case-by-case basis, as was done by Phillipps et al. (2002) for the above example.

It appears then that the measurement of a cosmological triangle's third side is quite removed from observational reality. Hence the test must be considered as purely hypothetical, a thought experiment. Indeed, it is not even clear exactly which quantity might be "easiest'' to measure (redshift, luminosity distance, etc.). Here, I have chosen to express the third side in terms of redshift, thus treating all three sides equally.