Quasar Clustering and Spacetime Geometry

The non-Euclidean geometry of spacetime induces an anisotropy in the apparent correlation function of high-redshift objects, such as quasars, if redshifts and angles are converted to distances in "naive" Euclidean fashion. The degree of angular distortion depends on cosmological parameters, especially on the cosmological constant Λ, so this effect can constrain Λ independent of any assumptions about the evolution of luminosities, sizes, or clustering. We examine the prospects for distinguishing between low-density (Ω₀ = 0.1-0.4) cosmological models with flat and open space geometry using the large quasar samples anticipated from the Two Degree Field Survey (2dF) and the Sloan Digital Sky Survey (SDSS). Along the way, we derive a number of results that are useful for studies of the quasar correlation function. In particular, we show that even these large quasar surveys are likely to reside in the "sparse sampling" regime for correlation function measurements, so that the statistical fluctuations in measurements are simply the Poisson fluctuations in the observed numbers of pairs. As a result, (1) one can devise a simple maximum likelihood scheme for estimating clustering parameters, (2) one can generate Monte Carlo realizations of correlation function measurements without specifying high-order correlation functions or creating artificial quasar distributions, and (3) for a fixed number of quasars, a deeper survey over a smaller area has greater statistical power than a shallow, large-area survey. If the quasar correlation length is equal to the value implied by recent (quite uncertain) estimates, then the 2dF and SDSS samples can provide clear discrimination between flat and open geometries for Ω₀ ≤ 0.2 but only marginal discrimination for Ω₀ = 0.4. Clear discrimination is possible for Ω₀ = 0.4 if the true quasar correlation length is a factor of 2 larger, and a high-density survey of 30,000 quasars in 200 deg² would provide clear discrimination even for the lower correlation length. Detection of quasar clustering anisotropy would confirm the cosmological spacetime curvature that is a fundamental prediction of general relativity.