The Astrophysical Journal, 557:L7–L10, 2001 August 10 ൴ ᭧ 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A. MEASURING ANGULAR DIAMETER DISTANCES THROUGH HALO CLUSTERING Asantha Cooray,1,2 Wayne Hu,1 Dragan Huterer,3 and Michael Joffre1 Received 2001 May 7; accepted 2001 July 13; published 2001 July 20 ABSTRACT Current and upcoming wide-field surveys for weak gravitational lensing and the Sunyaev-Zeldovich effect will generate mass-selected catalogs of dark matter halos with internal or follow-up photometric redshift information. Using the shape of the linear power spectrum as a standard ruler that is calibrated by cosmic microwave background 2 14 measurements, we find that a survey of 4000 deg and a mass threshold of 10 M, can be used to determine the comoving angular diameter distance as a function of redshift. In principle, this test also allows an absolute calibration of the distance scale and measurement of the Hubble constant. This test is largely insensitive to the details of halo mass measurements, mass function, and halo bias. Determination of these quantities would further allow a measurement of the linear growth rate of fluctuations. Subject headings: cosmology: theory — large-scale structure of universe On-line material: color figures 1. INTRODUCTION to uncertainties in the mass function and the relationship be- tween the halo masses and the actual observables, e.g., the SZ A number of observational efforts are now underway or temperature decrement or lensing aperture mass. The bias of being planned to image the large-scale structure of the universe the halos is scale-dependent only on small (nonlinear) scales spanning a range of redshifts. These wide-field surveys typi- and can, in principle, be extracted to arbitrary precision from cally cover tens to thousands of square degrees of the sky: the N-body simulations. If and when these quantities are securely ongoing Sloan Digital Sky Survey, the weak gravitational lens- known, one can extract further information from the amplitude ing shear observations with instruments such as the Supernova/ and small-scale behavior of the power spectrum. In particular, Acceleration Probe (SNAP) or the Large Aperture Synoptic the linear growth rate and nonlinear scale provide extra handles Survey Telescope, and surveys of the Sunyaev-Zeldovich (SZ) on the dark energy. For illustrative purposes, we adopt the LCDM cosmology effect (Sunyaev & Zeldovich 1980). p with energy densities (relative to critical) ofQm 0.35 in mat- In addition to their primary science goals, these surveys are p p ter,Qb 0.05 in baryons,QL 0.65 in vacuum energy, the expected to produce catalogs of dark matter halos, which in p the case of lensing and SZ surveys are expected to be essentially dimensionless Hubble constant ofh 0.65 , and a scale- invariant spectrum of primordial fluctuations, normalized to the mass-selected (Wittman et al. 2001; Holder et al. 2000). Lens- p ing and other optical surveys are particularly promising in that present-day galaxy cluster abundance (j8 0.9 ; Viana & Lid- they will provide photometric redshifts on the member galaxies dle 1999). of a given halo (e.g., Hogg et al. 1998); this will render accurate determination of the halo redshift. Halo number counts as a 2. ANGULAR POWER SPECTRUM function of redshift is a well-known and powerful cosmological The angular power spectrum of halos in ith redshift bin is test. Here we consider the additional information supplied by a Limber (1954; Kaiser 1992) projection of the halo number the angular clustering of halos. density power spectrum A feature in an angular power spectrum of known physical scale and originating from a known redshift can be used to H(z) l measure the angular diameter distance between us and this C i p ͵ dz W 2 (z) P ; z ,(1) l ihh2 () redshift; this has most notably been applied to the case of the dAA(z) d cosmic microwave background (CMB) to determine the dis- 3 tance to redshiftz ∼ 10 . The angular power spectrum of halos whereWi(z) is the distribution of halos in a given redshift bin ∫ p provides a similar test based on the standard ruler defined by normalized so thatdz Wi(z) 1 ,H(z) is the Hubble param- its shape. In the adiabatic cold dark matter (CDM) model for eter, anddA is the angular diameter distance in comoving co- structure formation, this standard ruler is essentially the horizon ordinates. Note thatWi(z) comes directly from the observations at matter-radiation equality, and its absolute physical scale can of the number counts as a function of redshift and depends on be directly calibrated with CMB anisotropy data. In principle the mass function and mass sensitivity of the employed ob- then, one can determine the angular diameter distance as a servable. function of redshift and test the properties of the dark energy. If the halos trace the linear density field, As a purely geometric test, this method is largely insensitive p 22lin Phh(k; z) Ab MS (z)D (z)P (k; 0), (2) 1 Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637; [email protected], whu@ where ASbM is the mass-averaged halo bias parameter, oddjob.uchicago.edu, [email protected]. lin 2 P (k;0)is the present-day matter power spectrum computed Sherman Fairchild Senior Research Fellow, Department of Physics, Cal- in linear theory, andD(z) is the linear growth function ifornia Institute of Technology, MS 130-33, Pasadena, CA 91125. lin lin 3 Department of Physics, University of Chicago, 5640 South Ellis Avenue, d (k; z) p D(z)d (k;0). A scale-independent halo bias is Chicago, IL 60637; [email protected]. commonly assumed in the so-called halo model (e.g., Seljak L7 L8 MEASURING ANGULAR DIAMETER DISTANCES Vol. 557 2000) and should be valid at least in the linear regime. Equa- tion (1) then becomes l C i p ͵ dz W 2(z)F(z)P lin ;0, (3) l i ()i dA H(z) p 22 F(z) 2 D(z) AbMS (z). (4) dA (z) The underlying linear power spectrum contains two physical scales: the horizon at matter-radiation equality, p ͱ 22ϩ ∝ keq 2Qm H0eq(1 z ) Qm h ,(5) which controls the overall shape of the power spectrum, and the sound horizon at the end of the Compton drag epoch, 22 ksm(Q h ,Q bh ), which controls the small wiggles in the power spectrum. The angular or multipole locations of these features p shift in redshift asleq, s kdeq, sA(z i) . We propose the following i 14 Fig. 1.—Angular power spectrum of halos withM 1 10 M, in a wide- test: measureCl in several redshift bins and, using the fact that ldscales with(z ) , constrain the angular diameter distance field survey in bins ofz p 0.3–0.4 and 1.0–1.2. The binned errors are 1 j eq Ai and assume a survey of 4000 deg2, within reach of upcoming weak lensing as a function of redshift. To be conservative, we ignore the and SZ surveys. The angular power spectrum at high redshifts is shifted toward additional information supplied byls . the right proportional to the increase in the comoving angular diameter distance. In Figure 1, we illustrate the proposed test. The two curves The oscillations are due to baryons, but we ignore the additional information show the halo power spectra in two redshift bins: 0.3 ! z ! they contain. [See the electronic edition of the Journal for a color version of 0.4and 1.0 ! z ! 1.2 . The angular power spectrum correspond- this figure.] ing to the higher redshift bin is shifted to the right in accordance ∼ Assuming Gaussian statistics, we can express the uncertainty to the ratio of angular diameter distances(dl/l ddAA/d ) . Much ≈ in the measurements of the angular power spectrum as of this shift simply reflects the Hubble law,dA z/H0 . Since the physical scale of the two features—the overall shape of the (C iiϩ N ) spectrum and the baryon oscillations—can be calibrated from DC i p ll,(6) l ͱ ϩ the morphology of the CMB peaks, these measurements can, (l 1/2)fsky in principle, be used to determine the Hubble constant inde- p 2 pendently of the distance ladder and distance to last scattering wherefskyVp deg /129,600 is the fraction of the sky covered i surface. by a survey of dimension Vdeg in degrees andNl is the noise In addition to the horizontal shift due to the change in angular power spectrum. We assume that the dominant source of noise i { ¯¯ diameter distance, the power spectra in Figure 1 are shifted is the shot noise so thatNli1/NN , where i is the surface vertically due to the change inF(z) (eq. [4]). By ignoring the density of the halos in the ith redshift bin. We use the PS mass ¯ information contained inF(z) , this purely geometric test is function to predictNi . In Figure 1, the two bins contain roughly robust against uncertainties in the mass selection, mass func- 4 and 6 halos degϪ2 above our minimum mass. In the same tion, and linear bias. Of course, if these uncertainties are pinned figure, we show band power measurement errors following down independently, bothF(z) and the halo abundance in equation (6). Wi(z) will help measure the growth rate of structure. To estimate how well halo clustering can recover cosmo- logical information, we construct the Fisher matrix, 3. PARAMETER ESTIMATION i Nbinsl max (l ϩ 1/2)f ѨC iiѨC Even though the angular diameter distance test is robust p ͸͸ sky ll Fab ii2 ,(7) p p ϩ Ѩ Ѩ against uncertainties in the halo selection function, number i 1 l lmin (CllN ) pabp density, and bias, these quantities enter into the consideration of the signal-to-noise ratio for a realistic survey.