A&A 386, 784–795 (2002) Astronomy DOI: 10.1051/0004-6361:20020274 & c ESO 2002 Astrophysics Cosmological information from quasar-galaxy correlations induced by weak lensing B. M´enard1,2 and M. Bartelmann1 1 Max-Planck-Institut f¨ur Astrophysik, PO Box 1317, 85741 Garching, Germany 2 Institut d’Astrophysique de Paris, 98bis Bld. Arago, 75014 Paris, France Received 4 December 2001 / Accepted 14 February 2002 Abstract. The magnification bias of large-scale structures, combined with galaxy biasing, leads to a cross- correlation of distant quasars with foreground galaxies on angular scales of the order of arcminutes and larger. The amplitude and angular shape of the cross-correlation function wQG contain information on cosmological parame- ters and the galaxy bias factor. While the existence of this cross-correlation has firmly been established, existing data did not allow an accurate measurement of wQG yet, but wide area surveys like the Sloan Digital Sky Survey now provide an ideal database for measuring it. However, wQG depends on several cosmological parameters and the galaxy bias factor. We study in detail the sensitivity of wQG to these parameters and develop a strategy for using the data. We show that the parameter space can be reduced to the bias factor ¯b,Ω0 and σ8, and compute the accuracy with which these parameters can be deduced from SDSS data. Under reasonable assumptions, it should be possible to reach relative accuracies of the order of 5%–15% for ¯b,Ω0,andσ8. This method is complementary to other weak-lensing analyses based on cosmic shear. Key words. cosmology – large-scale structure of Universe – gravitational lensing 1. Introduction distribution only. Measurements of the lensing-induced QSO-galaxy correlation function therefore have the poten- Weak gravitational lensing by large-scale structures gives tial to constrain the dark-matter power spectrum, several rise to a magnification bias on distant quasar samples. cosmological parameters, and the galaxy bias. Quasars behind matter overdensities are magnified and Many such measurements have been undertaken therefore preferentially included into flux-limited samples. (e.g. Ben´ıtez & Mart´ınez-Gonz´alez 1995; Norman & Impey Area magnification dilutes quasars on the sky and counter- 1999; Ben´ıtez et al. 2001; Norman & Impey 2001). The ex- acts the magnification to some degree, but the number istence of highly significant large-scale QSO-galaxy corre- count function of bright quasars is sufficiently steep to lations has been convincingly demonstrated (cf. Fugmann provide a huge reservoir of faint quasars to be magnified 1990; Bartelmann & Schneider 1993, 1994; Bartelmann above the flux limit. Bright quasars in flux-limited samples et al. 1994; Rodrigues-Williams & Hogan 1994; Ben´ıtez thus occur preferentially behind matter overdensities. & Mart´ınez-Gonz´alez 1995; Norman & Impey 1999), but Galaxies are biased tracers of dark matter. If the bias is the measurement of the amplitude and shape of the positive, their number density is higher in matter overden- cross-correlation function is still highly uncertain (see sities. Since these overdensities act as gravitational lenses Bartelmann & Schneider 2001 for a review). This is mainly for background sources, they give rise to purely lensing- because homogeneous, sufficiently deep galaxy surveys of induced cross-correlations between background quasars large portions of the sky have so far been unavailable. and foreground galaxies. Upcoming wide-field surveys, above all the Sloan Typical angular scales for this effect range from a Digital Sky Survey (SDSS; York et al. 2000), will pro- few arcminutes to one degree. The lensing effect of in- vide huge, homogeneous samples of quasars and galaxies dividual galaxies does not matter on such scales. Group- covering a substantial fraction of the sky. The detection or cluster-sized haloes contribute to the correlation only and analyses of QSO-galaxy cross-correlations with high on scales below about one arcminute. On larger scales, signal-to-noise ratio is thus coming within reach. We in- the expected signal is caused by the large-scale matter vestigate in this paper which parameters can best be con- Send offprint requests to:B.M´enard, strained with such a measurement, and what accuracy we e-mail: [email protected] can expect to achieve. Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20020274 B. M´enard and M. Bartelmann: QSO-Galaxy correlations 785 The main route to extract cosmological information provided that the magnification µ is weak, |µ − 1|1. from weak lensing has so far been the exploitation of Here, α is the logarithmic slope of the cumulative quasar cosmic shear. This approach measures the gravitational number-count function near the flux limit and wκδ(θ)is tidal field of the dark matter between background galax- the cross-correlation function between the lensing conver- ies and the observer. Despite the many difficulties in gence κ and a suitably weighted projected density contrast measuring exact shape parameters of faint galaxies, im- δ¯. The convergence itself is a weighted projection of the pressive results have been obtained in the past year, density contrast, so that the correlation function wκδ(θ) demonstrating the power of the method (Bacon et al. 2000; is straightforwardly related to the projected dark-matter Hoekstra et al. 2002a; Kaiser et al. 2000; van Waerbeke power spectrum. et al. 2001; Wittman et al. 2000). However, it depends Equation (2) provides an operational definition of a upon the crucial assumption that galaxy ellipticities are mean galaxy bias factor ¯b(θ) on a given angular scale θ. intrinsically uncorrelated (see Heavens et al. 2000 and ref- As defined, it averages over all galaxies considered in the erences therein). cross correlation at an angular separation θ from the near- Quasar-galaxy cross-correlations induced by weak est quasar, and thus involves averaging over redshift and lensing provide a complementary method for analysing possibly also morphological type, unless galaxies are se- weak lensing by large-scale structures which does not de- lected according to their morphology. Being linear in the pend on this assumption, and does not need any shape galaxy number density, the QSO-galaxy cross-correlation measurement. It therefore allows an independent and wel- is insensitive to a possible stochasticity of the bias. In come cross-check of the cosmic shear results and, in addi- practice, measurements will determine the wQG within an tion, it allows the determination of the galaxy bias factor. angular range R around θ. We will abbreviate the average ¯ The ratio Ω0/b can be constrained using the ratio bias factor within that interval as bR. ¯ wQG/wGG,wherewGG is the angular auto-correlation Of course, the relation of the bias factor bR to com- function of the foreground galaxies (Ben´ıtez & Sanz 2000). monly used definitions of the galaxy bias factor depends However, this method neglects all information contained on how the bias depends on physical scale, redshift, and in the amplitude and angular shape of wQG. Motivated by galaxy type. We will further discuss this issue in Sect. 2.2 recent cosmic shear results, we investigate here how well below. cosmological parameters can be constrained if the full in- formation provided by quasar-galaxy cross-correlations is 2.1. The κ-δ cross correlation used. In this paper, we briefly outline the lensing theory of We now describe briefly how the cross-correlation func- QSO-galaxy cross-correlations and investigate on which tion wκδ between lensing convergence and projected den- parameters the cross-correlation function depends. We sity contrast can be evaluated. More detail can be found show that the dimensionality of the parameter space can in Bartelmann (1995), Dolag & Bartelmann (1997), Sanz be substantially reduced. We then use simulated measure- et al. (1997) and Bartelmann & Schneider (2001). ments to investigate how accurately the remaining param- Since both κ and δ¯ are weighted line-of-sight projec- eters can be constrained. tions of the density contrast, we aim at an expression re- lating wκδ to the matter power spectrum. Specifically, we can write κ as the projection Z 2. Theory wH κ(θ)= dwpκ(w) δ[fK(w)θ,w], (3) The theory of the QSO-galaxy cross-correlation function 0 caused by weak lensing was introduced and developed in where w is the radial comoving distance along the line- several earlier studies. We can therefore be brief here and of-sight, f (w) is the comoving angular diameter distance limit the discussion to the issues required later on. K and p (w) is the projector Let n and n be the number densities on the sky of κ Q G quasars and galaxies, respectively. The QSO-galaxy cross 3 H 2 p (w)= Ω 0 correlation function is then κ 2 0 c Z h − − i wH 0 − 0 [nQ(φ) n¯Q][nG(θ + φ) n¯G] × dw 0 fK(w) fK(w w )· wQG(θ)= , (1) WQ(w ) 0 (4) n¯Q n¯G w a(w) fK(w ) where the average extends over all positions φ and all di- The ratio between the angular diameter distances fK rep- rections of θ. The average number densities of quasars and resents the usual effective lensing distance, WQ(w)isthe normalised distance distribution of the sources, in our case galaxies aren ¯Q andn ¯G, respectively. If there is no overlap in redshift between the quasar and galaxy populations, the quasars, and a(w) is the cosmological scale factor. ¯ this correlation is exclusively due to lensing, and can be Similarly, the projected density contrast δ can be writ- written (Bartelmann 1995): ten as Z ∞ ¯ w (θ)=2(α − 1)¯b(θ) w (θ), (2) δ(θ)= dwpδ(w) δ[fK(w)θ,w], (5) QG κδ 0 786 B.