Future dark energy constraints from measurements of quasar parallax: Gaia, SIM and beyond

Fiona Ding 0 1 Rupert A. C. Croft 0 1 0 Department of Physics, Carnegie Mellon University , Pittsburgh, PA 15213 , USA 1 Bruce and Astrid McWilliams Center for Cosmology, Carnegie Mellon University , Pittsburgh, PA 15213 , USA A B S T R A C T A consequence of the Earth's well-measured motion with respect to the cosmic microwave background is that over a 10-yr period it will travel a distance of 800 au. As first pointed out by Kardashev in 1986, this distance can be used as a baseline to carry out astrometric measurements of quasar parallaxes, so that only microarcsecond precision is necessary to detect parallax shifts of objects at gigaparsec distances. Such precision will soon be approached with the launch of the astrometric satellites Gaia and Space Interferometry Mission (SIM). We use a Fisher matrix formalism to investigate the constraints that these and future, even more ambitious, missions may be able to place on the cosmological distance scale and the parameters describing dark energy. We find that by observing around a million quasars as planned, an extended 10 yr Gaia mission should have the capability to detect quasar parallax shifts at the 2.8 level and so measure the Hubble constant to within 25 km s1. For the interferometer SIM (in its currently proposed SIMLite configuration) a Key Project using 2.4 per cent of the total mission time to observe 750 quasars could detect the effect at the 2 level and dedicated use of the instrument at the 3.3 level. In a concordance cosmological model, Gaia and dedicated SIMLite only weakly constrain the presence of a cosmological constant at the 1 levels. We also investigate a range of future mission concepts, such as an interferometer similar in scope and design to NASA's Terrestrial Planet Finder. This could in principle measure the dark energy parameters w0 and wa with precision w0 = 0.02 and wa = 0.05, respectively, yielding a Figure of Merit larger than the stage IV experiments considered in the report of the Dark Energy Task Force. Unlike perhaps all other probes of dark energy there appear to be no obvious astrophysical sources of systematic error on these measurements. There is however uncertainty regarding the statistical errors. As well as measurement error, there will be small additional contributions from image centroiding of variable sources, quasar peculiar motions and weak microlensing by stars along the line of sight. - The quest to measure the cosmological distance scale has been continued over the decades since the discovery of the expansion of the Universe in many different contexts, from studies of the deceleration parameter, to more recently dark energy parameters and modified gravity (see e.g. Frieman, Turner & Huterer 2008; Jain & Zhang 2008). The success of supernova standard candles in revealing the acceleration of the Universe (Riess et al. 1998; Perlmutter et al. 1999) has shown the power of classical tests, while at the same time much effort has been and will be spent in dealing with the many possible systematic errors in the measurements. The simplest and most direct classical test, using pure geometry to measure distances of objects from their parallax shift over time is arguably the most free of systematic uncertainty and easiest to interpret. More importantly it seems as though carrying out parallax measurements on cosmological scales should be feasible, through the combination of astrometric satellites, statistical averaging over many objects and the long baseline afforded by the Earths motion with respect to the cosmic microwave background (CMB). In this paper we investigate how well this combination can be expected to lead to cosmic distance scale and dark energy constraints in the future. The parallax distance to an object in an expanding Universe was first calculated theoretically and published by McCrea (1935), although he noted that it was unlikely to be measurable. We give the result in the context of dark energy cosmological models in Section 2. We note that the calculation was also performed by Kardashev, Parijskij & Umarbaeva (1973) who explored the possibility of measurements using radio interferometry. It appears also in the textbook by Weinberg (1972), and the case of inhomogeneous universes was treated by Novikov (1977) (and also Kasai 1988). Kardashev (1986) was the first to propose that the Earths motion with respect to the CMB would provide a much longer usable baseline for parallax measurements and make measurements much less technically challenging than using the Earths annual parallax. This effect is a variant of the secular parallax (see e.g. Binney & Merrifield 1998, section 2.2.3), and is in principle easier to measure because the signal increases linearly with time, while the annual (also known as trigonometric) parallax repeats at a constant (small) value. Rosquist (1988) pointed out that parallax distances of distant objects can be used to determine number densities of conserved classes of objects such as galaxies even if no dynamical model is assumed. Pierce & Cash (2004) explored the possibility that future X-ray interferometers may be able to measure the differential parallax between quasar pairs, and hence characterize dark energy. Most recently, Quercellini, Quartin & Amendola (2009) showed how alternative anisotropic models such as LemaitreTolmanBondi cosmologies with off-centre observers would produce a secular parallax effect in distant quasars even for a stationary observer and how upcoming astrometric satellites may be able to put competitive constraints on those models. Our plan for this paper is as follows. In Section 2 we summarize prior results for the parallax of extragalactic objects, and generalize them to the case of time varying dark energy models. In Section 3 we give details of planned future surveys of quasars with astronometric satellites, as well as outlining some hypothetical, more futuristic surveys. In Section 4 we deal with how the quasar data sets should be analysed, and what the systematic and statistical uncertainties are likely to be. We describe how well these future surveys can be expected to constrain dark energy parameters in Section 5, and in Section 6 we summarize and discuss our results. 2 Q UA S A R PA R A L L A X I N D A R K E N E R G Y M O D E L S The Solar system is moving with respect to the CMB frame at a velocity of 369.5 3.0 km h1 towards an apex with galactic latitude and longitude l = 264.4 0.3, b = 48.4 0.5 (Kogut et al. 1993). As a result, all extragalactic objects will experience a parallax shift, increasing linearly with time, towards the antapex with amplitude proportional to sin , where is the angle between the object and the direction of the apex. Over a 10-yr period a baseline of l = 3800 pc is therefore available for measures of parallax. We summarize below the expressions for the parallax shift of a distant extragalactic source (first computed by McCrea 1935), using the notation due to Kardas (...truncated)