MEASURING THE GEOMETRY OF THE UNIVERSE FROM WEAK GRAVITATIONAL LENSING BEHIND GALAXY GROUPS IN THE HST COSMOS SURVEY^*

The COSMOS survey (Scoville et al. 2007a) brings together panchromatic imaging from X-ray to radio wavelengths, including the largest contiguous area observed with the Hubble Space Telescope (HST), and deep optical spectroscopic observations. The field covers an area of 1.64 deg² centered at 10:00:28.6, +02:12:21.0 (J2000) and contains identified groups, clusters, and larger structures spanning a wide range in redshift (Scoville et al. 2007b).

We consider the gravitational lensing signal behind a sample of galaxy groups selected originally via their X-ray emission (Finoguenov et al. 2007) and updated using a combined mosaic of imaging from XMM-Newton (1.5 Ms; Hasinger et al. 2007; Cappelluti et al. 2009) and the Chandra observatories (1.8 Ms; Elvis et al. 2009). Groups are detected from the combined X-ray mosaic using a wavelet filter, which can result in centering uncertainties of up to 32''. The distribution of galaxies along the line of sight to each X-ray detection is searched for a red sequence overdensity to determine the group redshift, with spectroscopic redshifts used for subsequent refinement (Finoguenov et al. 2007). Group members are selected based on their photometric redshift and proximity to the X-ray centroid, using an algorithm tested extensively on mock catalogs and spectroscopic subsamples (George et al. 2011). Stellar masses of the member galaxies are determined from multiwavelength data (see Leauthaud et al. 2012 for details). From an initial list of members, group centers are then redefined around the most massive group galaxy within the Navarro–Frenk–White (NFW) scale radius of the X-ray centroid (MMGG_scale), which optimizes the weak-lensing signal at small radii (M. George et al., in preparation). For the majority of our groups, this gives centers that agree with the X-ray centroid; a minority (approximately 20%) of groups show significant offsets between the most massive galaxy and the X-ray centroid. These offsets could be due to observational problems (such as low signal to noise (S/N) in the X-ray or optical data), or they might indicate unrelaxed, low-concentration groups with poorly defined physical centers, such as recent mergers. We will consider below both the full set of groups and a "restricted" set which excludes the systems with significant offsets. The centering algorithm will be discussed further in a forthcoming paper (M. George et al., in preparation). The full X-ray group sample, together with derived properties, is available from the NASA/IPAC Infrared Science Archive (IRSA) Web site¹³ (see George et al. 2011 for details).

We restrict the lens sample to groups at z < 1 to ensure the reliability of X-ray detections and optical associations, as well as good photometric redshifts for identifying members and centers. We further cut out of the sample poor groups, groups with centroids affected by masking, and possible mergers (this corresponds to taking only groups with FLAG_INCLUDE = 1 as defined in George et al. 2011). Our final sample consists of 129 systems (105 in the restricted set) spanning a rest-frame 0.1–2.4 keV luminosity range between 10⁴¹ and 10⁴⁴ erg s⁻¹, with estimated virial masses of 0.8 × 10¹³–2 × 10¹⁴ h⁻¹₇₀ M_☉, virial radii of 0.4–0.8 h⁻¹₇₀ Mpc, and projected angular sizes of 1'–6'. Figure 1 shows the mass, physical size, and angular size for the groups in the sample (note that units have been converted from the value H₀ = 72 km s⁻¹ Mpc⁻¹ used in the catalog to H₀ = 70 km s⁻¹ Mpc⁻¹). The virial radius is taken to be R_200c, the radius within which the mean density is equal to 200 times the critical density ρ_c(z) at the redshift of the group, and the virial mass is taken to be M_200c, the mass enclosed within R_200c. These masses and radii are estimated from the X-ray data, using X-ray scaling relations calibrated with lensing data (Finoguenov et al. 2007; Leauthaud et al. 2010; George et al. 2011).

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High-resolution imaging of the COSMOS field was obtained with the HST between 2003 October and 2005 June (Scoville et al. 2007c; Koekemoer et al. 2007). The main program consisted of 575 slightly overlapping pointings of the Advanced Camera for Surveys (ACS) Wide-Field Camera taken with the F814W (approximately I band) filter. At each pointing, four slightly dithered exposures of 507 s were obtained. Any cosmetic defects and reflection ghosts were carefully masked by hand. Using version 2.5.0 of the SExtractor photometry package (Bertin & Arnouts 1996), in a Hot–Cold configuration on the stacked images, we detected compact objects in a 015 diameter aperture down to F814W_AB = 26.6 at 5σ (Leauthaud et al. 2007).

We measure the shapes of galaxies in this catalog using the RRG method (Rhodes et al. 2000), largely following the analysis pipeline of Leauthaud et al. (2007) and calibrated against simulated ACS images generated with the simage package (Massey et al. 2004, 2007b). However, we now include two significant improvements on this earlier work.

First, we correct trailing in the ACS images due to charge transfer inefficiency via a physically motivated readout model (Massey et al. 2010) that acts at the pixel level, rather than a parametric scheme at the catalog level. This moves electrons in the raw exposures back to where they should have been read out. The method achieves a 97% level of correction and is robust to variety in galaxy morphology, local galaxy density, and sky background level. After correction, residual shears are well below statistical measurement precision.

Second, we model the HST's point-spread function (PSF) as a function of chip position, telescope focus offset, and velocity aberration factor. The latter two parameters reflect HST's thermal condition during each exposure: slight expansion and contraction changes the PSF. Following Rhodes et al. (2007), we measure the focus offset (the distance between the primary and secondary mirrors) with a precision of 1 μm by comparing the apparent shapes of ∼10 stars in each exposure to TinyTim models (Krist 2003). Jee et al. (2007) found that focus offset correlates with the first Principal Component of PSF variation and accounts for 97% of the power, while Schrabback et al. (2010) found that VAFOCUS correlates with the next Principal Component. We measure the shapes of all stars in the COSMOS imaging, then interpolate between them using all four measured parameters. This improves residuals compared to Massey et al. (2007a), and we retain this physically motivated approach rather than relying solely on Principal Component Analysis.

We have also revised our method for determining the variance of the tangential shear slightly. This is now determined empirically, as described in Section 3.5 of Leauthaud et al. (2012). Galaxies are binned by S/N and magnitude, and the total variance of the shear components γ₁ and γ₂ is measured directly in each bin. This empirical derivation of the shear dispersion includes both the scatter due to intrinsic variations in galaxy shape and the additional scatter due to shape measurement errors. We find that the shear dispersion varies from $\sigma _{\tilde{\gamma }} \sim 0.25$ for bright galaxies with high S/N to $\sigma _{\tilde{\gamma }} \sim 0.4$ for faint galaxies with low S/N. These measured values may be very slightly overestimated, however, as suggested by the reduced χ² of the profile fit discussed in Section 3.2 below.

Of the 129 groups in our full sample, 95% contain two or more spectroscopically confirmed members, 3% have one spectroscopically confirmed member, and the remainder have redshifts determined photometrically from the red sequence of member galaxies (as in Finoguenov et al. 2010). The average redshift error for the group ensemble is ∼0.0017, only slightly larger than their typical velocity dispersions of 300–450 km s⁻¹.

Multicolor ground-based imaging in over 30 bands (Capak et al. 2007) also provides photometric redshift information for all of the source galaxies along the same line of sight. We use photometric redshift estimates from the LePhare χ² template-fitting code, which are updated from those published in Ilbert et al. (2009) by the addition of deep H-band data, and small improvements in the template-fitting technique. We have compared photo-zs from 10801 galaxies at z ∼ 0.48, 696 at z ∼ 0.74, and 870 at z ∼ 2.2 to spectroscopic redshift measurements with the Very Large Telescope Visible Multi-Object Spectrograph (Lilly et al. 2007) and the Keck Deep Extragalactic Imaging Multi-Object Spectrograph. The rms dispersion in the offset σ_Δz between photometric and spectroscopic redshift is 0.007(1 + z) at i⁺_AB < 22.5 and 0.02(1 + z) at i⁺_AB ∼ 24 and z < 1.3 (or 0.06(1 + z) for i⁺_AB ∼ 24 and z ⩾ 1.3).

To mitigate against catastrophic failure in estimated photo-zs, for example due to confusion between the Lyman and 4000Å breaks, we reject from the sample all-source galaxies with a secondary peak in the redshift probability distribution function (i.e., galaxies where the parameter zp_sec is greater than zero in the Ilbert et al. 2009 catalog). The rejected zp_sec >0 galaxy population is expected to contain a large fraction of catastrophic errors (roughly 40%–50%; Ilbert et al. 2006, 2009). For the purposes of cosmological constraints, we further exclude from the sample objects with relative redshift uncertainties Δz/(1 + z) ⩾ 0.05, taking the average redshift error to be Δz ≡(zu68_gal–zl68_gal)/2.0, where zu68_gal and zl68_gal are the 68% confidence limits on the redshift, based on the photo-z probability distribution (Ilbert et al. 2009). Our final source sample consists of all galaxies passing these cuts that lie within 6' of a group center. Individual galaxies may enter into the final sample multiple times if they lie within 6' of more than one peak. The photo-z quality cuts reduce the number density of source galaxies to 26 galaxies arcmin⁻², for a total of 3.7 × 10⁵ galaxies (3.1 × 10⁵ in the restricted sample). The mean redshift of the final sample is 〈z〉 = 0.95 and the mean relative error in redshift is Δz/(1 + z) = 0.018, while the mean magnitude is 〈I_F814W〉 ∼ 24.

If we consider a source galaxy (or "source" hereafter) at redshift z_S being lensed by a foreground group (or "lens" hereafter) at redshift z_L and observed at z_O, in the weak limit the tangential shear induced by the lens will be

$$\begin{equation} \gamma _t(r) = [\, \overline{\Sigma }(<r) - \overline{\Sigma }(r)]/\Sigma _c = \Delta \Sigma (r)/\Sigma _c, \end{equation} \tag{ 1 }$$

where $\overline{\Sigma }(<r)$ is the mean surface density interior to projected (physical) radius r, $\overline{\Sigma }(r)$ is the azimuthally averaged surface density at r, and Σ_c is the critical surface density defined as

$$\begin{equation} \Sigma _c \equiv {{c^2}\over {4\pi G}}{{D_S}\over {D_L D_{LS}}}. \end{equation} \tag{ 2 }$$

Here D_S denotes the angular diameter distance from the observer to the source

$$\begin{equation} D_S = f_k(\omega [z_O,z_S])a(z_S), \end{equation} \tag{ 3 }$$

where ω[z_O, z_S] is the comoving (or coordinate) distance along a radial ray between the observer and the source

$$\begin{equation} \omega [z_O,z_S] = \int _{z_O}^{z_S} d\omega = \int _{a_S}^{a_O} {{cdt}\over {a(t)}} = \int _{a_S}^{a_O} {{cda}\over {a^2H(a)}}. \end{equation} \tag{ 4 }$$

D_L and D_LS are angular diameter distances from the observer to the lens and from the lens to the source respectively, given by

$$\begin{eqnarray} D_L &=& f_k(\omega [z_O,z_L])a(z_L), \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} D_{LS} &=& f_k(\omega [z_L,z_S])a(z_S), \end{eqnarray} \tag{ 6 }$$

(e.g., Bartelmann & Schneider 2001).

If we consider the case of a flat cosmology (k = 0), f_k(ω) = ω and thus we can rewrite Σ_c more simply in terms of the comoving distances ω_L ≡ ω[z_O, z_L] and ω_S ≡ ω[z_O, z_S]

$$\begin{equation} \Sigma _c \equiv {{c^2}\over {4\pi G}}{{\omega _S}\over {\omega _L (\omega _S - \omega _L)}} (1 + z_L). \end{equation} \tag{ 7 }$$

The critical density incorporates all the geometric dependence of lensing; it is the nonlinear dependence of Σ_c on z_S and z_L that makes it hard to stack the signal from different source–lens pairs in a straightforward way. We can simplify the dependence, however, by defining the comoving distance ratio x ≡ ω_S/ω_L. Note that x > 1 for sources chosen to be behind the lens; we will consider objects with both x > 1 and x ⩽ 1 below. In terms of x

$$\begin{equation} \Sigma _c \equiv {{c^2}\over {4\pi G}}{{(1 + z_L)}\over {\omega _L}}{{1}\over {(1 - 1/x)}}. \end{equation} \tag{ 8 }$$

We can also write this in terms of Σ_{c, ∞}, the value of the critical density in the limit x → ∞, which is

$$\begin{equation} \Sigma _c = {{1}\over {(1 - 1/x)}}\Sigma _{c,\infty }, \end{equation} \tag{ 9 }$$

where

$$\begin{equation} \Sigma _{c,\infty } \equiv {{c^2}\over {4\pi G}}{{(1 + z_L)}\over {\omega _L}} \end{equation} \tag{ 10 }$$

depends only on the lens properties, not on the properties of the source galaxy.

From Equation (1), the geometric dependence of all source–lens pairs now takes on a universal form

$$\begin{equation} \gamma _t(r)\Sigma _{c,\infty }/\Delta \Sigma (r) \equiv \Gamma (x) = \left(1 - {1\over {x}}\right). \end{equation} \tag{ 11 }$$

Γ(x) corresponds, e.g., to the lensing efficiency E defined by Golse et al. (2002). In as much as the measured tangential ellipticity ε_t of each source galaxy is an estimator $\tilde{\gamma }_t$ of the true tangential shear γ_t, we can construct a weighted sum of estimates from individual source galaxies j with respect to lensing centers i to recover the universal geometric dependence:

$$\begin{eqnarray} \left(1 - {1\over {x}}\right) &=& \sum _{i,j} w_{ij}\Gamma _{ij}(x) \nonumber \\ &=& {{\sum _{i,j} w_{ij}\tilde{\gamma }_{t,ij}\Sigma _{(c,\infty) i}/\Delta \Sigma _i(r_{ij})}\over {\sum _{i,j} w_{ij}}} \end{eqnarray} \tag{ 12 }$$

with weights w_ij chosen to maximize the S/N or sensitivity to cosmological parameters, as discussed below.

Since we are just fitting the data to a fixed function, cosmology appears to have disappeared from Equation (12). In fact, it is hidden in the conversion from measured source and lens redshifts to inferred source and lens distances. For a given cosmology we convert redshifts to comoving or angular diameter distances, construct the weighted sum in Equation (12), and calculate the χ² of Γ(x) with respect to the theoretical expectation (1 − 1/x). This gives us the relative likelihood of that particular set of cosmological parameters; iterating over this process then allows parameter constraints. The only remaining problems are to determine the optimal weights w_ij and surface mass density contrast ΔΣ(r). We discuss these calculations in the next section.

Before proceeding we should note that our simple stacking analysis ignores several complications. First, it ignores the distinction between true and reduced shear (e.g., Shapiro 2009). In the weak shear limit the two are identical, and for the groups considered here the surface mass density is low enough that the contribution from non-weak shear corrections is unimportant outside ∼50–100 h⁻¹₇₀ kpc. The effect of the second-order correction term is illustrated in Figure 4 of Leauthaud et al. (2010). Its contribution is roughly comparable to that of the stellar mass in the central group galaxy (see Figure 2 below) at large projected radii and always less than the stellar contribution at radii less than ∼50–100 h⁻¹₇₀ kpc. Given only that ∼2% of our sources lie at such small projected radii, the effect of these contributions on our fits should be negligible (excluding from the sample all sources within 15' of group centers,¹⁴ for instance, has no effect on the final results).

Second, we have also assumed flatness in separating the dependence on the lens distance and the source distance. While current cosmological constraints indicate an almost completely flat universe (e.g., Larson et al. 2011 find |Ω_k| ≲ 0.01 from various sets of constraints), it would be nice to be able to relax this assumption. Unfortunately, while it is still possible to fit shear ratios between individual pairs in the general case, there is no simple way of stacking all measurements together into a single functional form since the dependence on the two redshifts can no longer be factored out of f_k(ω[z_L, z_S]) in a simple way. We can estimate the effect of curvature by considering the series expansion for f_k in the limit |Ω_k| ≪ 1. For typical values of ω ∼ c/H₀, the next term in the series is smaller by a factor 1/6(ω/R₀)² ∼ 1.6 × 10⁻³, where $R_0 \equiv c/(H_0 \sqrt{|\Omega _k|})$. Thus in realistic non-flat cosmologies, we expect a correction of the order of 1.6 × 10⁻³ to our values of Γ(x). Compared to our cosmological sensitivity ΔΓ ∼ 0.05 (see Section 4.2 below), this represents a 3% correction to our derived parameters. This correction is smaller than errors on Ω_X we obtain below, although it would quickly become important in larger surveys. In what follows we will ignore the complication of non-zero spatial curvature.

Third and lastly, our analysis assumes a specific functional form for the surface mass density contrast ΔΣ(r), namely a projected NFW profile. We will show below that this functional form is in fact an excellent fit to the stacked data. We could use instead an empirical profile based directly on the data itself, but given the agreement between the NFW model and the data, this would not affect our results significantly.

Our goal is to measure the redshift dependence of the group lensing signal. The radial dependence of the surface mass density contrast ΔΣ(r) around groups, although intrinsically very interesting, is essentially a nuisance parameter in this calculation. We need to determine ΔΣ(r), however, in order to weight measurements of individual source galaxy shapes optimally when estimating Γ(x). The density contrast around groups was studied in detail in Leauthaud et al. (2010); we reproduce the same calculation here, stacking with respect to physical radius the signal from all groups with well-determined redshifts and centers.

The density contrast profile includes contributions from four main terms¹⁵: the weak shear contribution of the main halo (the "one-halo" term), the average weak shear contribution from nearby halos (the "two-halo" term), a weak shear contribution from the stellar mass of the central group galaxy, and the second-order corrections to the shear in the center of the main halo. Of these, only the one-halo term is important here; the two-halo term only becomes significant at large projected radii (r ∼ 4 h⁻¹₇₀ Mpc), while the stellar and second-order terms are only significant at small projected radii (r ≲ 50–100 h⁻¹₇₀ kpc) where we have very few galaxies in the source sample, as discussed in Section 3.1.

We expect that the one-halo term for a single group should follow a projected NFW profile Σ(r), whose form f_NFW(r/r_s) is given, e.g., in Wright & Brainerd (2000). The profile has two free parameters, a scale radius r_s and a normalization Σ₀, or alternately it can be defined in terms of a virial radius r_vir and a concentration c ≡ r_vir/r_s. The expected values of these parameters can be inferred from X-ray fluxes, X-ray scaling relations, and theoretical concentration–mass relations. Using the concentration relations of Zhao et al. (2009), for instance, Leauthaud et al. (2010) predicted concentrations in the range 3.6–4.6 for the COSMOS groups (these values assume the definition r_vir = R_200c; thus they correspond to the values c_200c from Zhao et al. 2009). Simulations suggest that individual halos will have significant (∼50%) scatter around these mean values (Zhao et al. 2009). Finally, we note that here we are considering an average profile for all groups in the sample, where the averaging is weighted by surface mass density contrast ΔΣ. At a fixed redshift and fixed concentration, r_s and r_vir for each group should scale as M^1/3 and our stacked profile would be similar to a mass-weighted average. Variations in concentration and redshift complicate this behavior, but we can still use mass-weighted averaging to guide our expectations as to the final values for the concentration or scale radius. Using the concentration relations of Zhao et al. (2009) and assuming a WMAP7 cosmology, for instance, we predicted a mass-weighted average scale radius of r_s = 154 h⁻¹₇₀ kpc for our groups.

The surface mass density contrast ΔΣ(r) is related to the tangential shear by

$$\begin{equation} \Delta \Sigma\, (r) = \Sigma _c \times \gamma _t(r). \end{equation} \tag{ 13 }$$

Thus, ΔΣ(r) can be estimated as in Equation (8) of Leauthaud et al. (2010)

$$\begin{equation} \Delta \Sigma\, (r) = {{\sum _{ij}w_{ij}{\tilde{\gamma }}_{t,ij}\Sigma _{c,ij}}\over {\sum _{ij}w_{ij}}}. \end{equation} \tag{ 14 }$$

Figure 2 shows the surface mass density contrast for our stacked sample of groups, assuming a WMAP7 cosmology. Points with error bars indicate the mean value inferred from Equation (14), binned logarithmically in radius. The thin dotted lines indicate contributions from a projected NFW profile (middle blue line), the stellar mass of the central galaxy in the group (lower red line), and the sum of these two components (upper green line). The normalization of the stellar contribution is based on the mean stellar mass of the central group galaxy, as inferred from photometry (see Leauthaud et al. 2012 for details).

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The main part of the profile is well fit by a projected NFW profile Σ(r) with a scale radius r_s ∼ 160–180 h⁻¹₇₀ kpc and a normalization Σ₀ ∼ 200 h₇₀ M_☉ pc⁻² at r_s. The best-fit values of these two parameters are strongly correlated, so we choose instead to normalize the profile at a radius r_fix = 250 h⁻¹₇₀ kpc where the measurement errors are small and the amplitude is less dependent on r_s. Thus the profile can be written

$$\begin{equation} \Sigma (r) = \Sigma _0 {{f_{{\rm NFW}}(r/r_s)}\over {f_{{\rm NFW}}(r_{\rm fix}/r_s)}}. \end{equation} \tag{ 15 }$$

This gives us constraints on Σ₀ = Σ(r_fix) and r_s which are more independent of each other.

We determine values for Σ₀ and r_s by calculating χ² with respect to a projected NFW model. We do not bin the data in radius, since the bin boundaries would change with cosmology or scaling, but sum the contribution from each galaxy individually. For the full group sample and assuming WMAP7 cosmology, we obtain the lowest reduced χ² by considering all sources at x > 1.1, with no other cut on redshift errors. This gives a slightly high value for the scale radius r_s = 187⁺⁵⁴ _{− 29} h⁻¹₇₀ kpc, however. Considering only the restricted sample and/or sources with Δz/(1 + z) ⩽ 0.05, we obtain values of r_s closer to the expected value. This suggests that the remaining 20% of the group sample may be affected by centering problems or that it may include many unrelaxed or disturbed groups with lower mean concentrations. The dependence on photo-z cuts could indicate that the lensing signal in all groups is diluted by group member contamination in the source population when the limits on photo-z errors are relaxed. For the restricted group sample with the cut on source redshift errors, we obtain best-fit values Σ₀ = 98.8 ± 11 h₇₀ M_☉ pc⁻² and r_s = 158⁺⁵⁵ _{− 28} h⁻¹₇₀ kpc. The reduced χ² is marginally higher for this sample than for the much larger uncut sample, but the best-fit value of r_s closer to the expected value, so we will take this as our fiducial profile, and marginalize over values of r_s and Σ₀ in this range for our cosmological calculations. The best-fit value of r_s also places some constraints on possible centering errors for the groups. We have tested the effect of centering errors by adding random offsets to the individual group centers, with rms values of 6'', 12'', and 24'' in each coordinate. The resulting profiles are still well fit by our model, but the best-fit value of r_s increases to 220, 260, and 340 h⁻¹₇₀ kpc for the three cases, respectively. This suggests average centering errors are ≲ 6'' ∼ 25–50 h⁻¹₇₀ kpc in each coordinate, consistent with other estimates of the centering uncertainty (M. George et al., in preparation).

We have also investigated other forms of stacking. In principle we could correct for the predicted variations in concentration, for instance, stacking in r/r_s, or we could stack in comoving rather than physical coordinates. Testing scaling the radius by r_s, r_vir, or (1 + z_L), we find little or no significant improvement in the χ² of the fit to the radial profile. In particular, we find only a marginal indication of any trend in concentration with mass or redshift. Given that the halo-to-halo scatter measured in simulations is comparable or larger than to the average trends over the mass and redshift range spanned by our group sample, this is perhaps unsurprising. Furthermore, since the concentration relations are themselves dependent on cosmology, we would have to account for this dependence in our marginalization over cosmological parameters, so we will not attempt to correct for variations in concentration. We can also consider other analytic fits to the profile. We find that NFW is preferred over a cored isothermal profile at the 95% confidence level and preferred over a singular isothermal (Σ(r)∝r⁻¹) profile at 97%–98% confidence. Thus, our stacked profile provides empirical confirmation of the NFW model in agreement with other recent high-precision measurements of cluster density profiles (Umetsu et al. 2011; Okabe et al., in preparation).

Finally, we note that the fit to the radial profile gives us an independent check of our empirical shear variance estimates. Because we fit the profile without binning, we have very large number of degrees of freedom and thus a narrow range of expected scatter in the reduced χ². The best-fit NFW profile has a reduced χ² of 0.931 with a very small (0.0027) expected scatter, so we conclude that our empirical variance is probably overestimated by ∼7%, corresponding to error bars which are 3.6% too big. We correct for this in all our subsequent analysis, multiplying the empirical shear variance by a factor of 0.931.

Given a functional form for the radial dependence of the surface mass density contrast ΔΣ(r), we can proceed to estimate Γ(x) via Equation (12). The weights in the sum can be calculated as the inverse variance of the Γ_ij

$$\begin{equation} w_{ij} = ({\rm var}[\Gamma _{ij}])^{-1} = \left({\Delta \Sigma _i(r_{ij})}\over {\Sigma _{(c,\infty) i}}\right)^2{\rm var}[{\tilde{\gamma }}_{t,ij}]^{-1}, \end{equation} \tag{ 16 }$$

where the variance of the tangential shear is determined empirically, as described in Section 2.2.

The "model" here, the geometric sensitivity function (1 − 1/x), is fixed, while the data vary as we change Σ₀ and r_s, which both change ΔΣ(r), and the cosmological parameters, which map the redshifts (z_S, z_L) onto x-values and also determine Σ_{c, ∞}. If we restrict ourselves to flat cosmologies with two components, matter and dark energy, then the goodness of fit depends on Σ₀, r_s, Ω_X, and w.

Figure 3 shows Γ(x) over the range x = [0, 5] for a WMAP7 cosmology with Ω_M = 0.27, $\Omega _\Lambda = 0.73$. The density contrast profile parameters are fixed to the best-fit values r_s = 158 h⁻¹₇₀ kpc and Σ₀ = 98.8 h₇₀ M_☉ pc⁻². The points are weighted averages in bins of 0.3 in x, while the solid (red) curve is the theoretical expectation:

$$\begin{eqnarray} \Gamma (x) &=& 0 {\rm \ \ \ \ \ \ \ \ \ \ \ for\ \ } x < 1\,; \\ \nonumber &=& 1 - 1/x {\rm \ \ for\ \ } x \ge 1. \end{eqnarray} \tag{ 17 }$$

Weights here are inverse variance, as in Equation (16). The error bars on the data points are calculated as usual for an inverse-variance-weighted average:

$$\begin{equation} \sigma _{\Gamma, {\rm bin}} = \big(\sum _{\rm bin} w_{ij}\big)^{-1/2}, \end{equation} \tag{ 18 }$$

where the sum is over all pairs (i, j) with values of x in the bin.

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Clearly the geometric signal is present in the COSMOS data and measured to reasonable significance over a broad range of distance ratio x. Given the possible systematics in the measurement discussed below, the excellent agreement between theory and data illustrates the potential of the method. On the other hand, "χ²-by-eye" is somewhat misleading for this figure, as the binning in x may hide systematics at particular distance ratios. (There is a 2.1σ indication of positive signal in one bin at x < 1, for instance. This could indicate that photo-z errors are scattering sources to lower redshifts, but given the number of bins there is a 26% chance that this is simply a random statistical fluctuation.) As discussed below, in a small field structures at a few redshifts can dominate the lensing signal, introducing excesses or deficits of mass along the line of sight that dominate the signal at particular values of x. The bins chosen here are broad enough to smooth out many of these features, but clearly a goodness-of-fit measurement over the whole data set is required to determine the statistical significance of the apparent agreement in Figure 3.

We can estimate goodness of fit by calculating

$$\begin{equation} \chi ^2 = \sum _{ij}w_{ij}[(1 - 1/x_{ij}) - \Gamma _{ij}]^2. \end{equation} \tag{ 19 }$$

This sum depends on Σ₀, r_s, and the cosmological parameters, so marginalizing over the first two parameters gives constraints on the dark energy density Ω_X and the equation-of-state parameter w. While the resulting value of χ² will tell us whether the data are a good fit to the model, this is not necessarily the most sensitive way of determining cosmological parameters. In particular, for flat (k = 0) cosmological models with a cosmological constant with density parameter $\Omega _X = \Omega _\Lambda$, as the value of $\Omega _\Lambda$ increases all distances will increase, and therefore so will x and (1 − 1/x). Thus, the signal from a given source–lens redshift pair will be compared to Γ(x) at a value of x which is larger by some factor. The most sensitive probes of this re-scaling will be points at large x. To constrain Ω_X more precisely, individual measurements should be weighted by this sensitivity.

The exact sensitivity to cosmology itself depends on the cosmological parameters. We can estimate a sensitivity factor and thus a weighting that will be close to optimal over the whole range of $\Omega _\Lambda$, however, by calculating

$$\begin{eqnarray} \Delta \Gamma (z_S, z_L) &=& \Gamma _{01}[x(z_S,z_L)] - \Gamma _{09}[x(z_S,z_L)]\nonumber \\ &=& [x_{09}(z_S,z_L)]^{-1} - [x_{01}(z_S,z_L)]^{-1}, \end{eqnarray} \tag{ 20 }$$

where Γ₀₁ Γ₀₉ are the model Γ(x) evaluated for cosmologies with $\Omega _\Lambda = 0.1$ and 0.9, respectively (with w = −1 in each case). Figure 4 shows this sensitivity function versus (z_S, z_L). (Note that this sensitivity function has been calculated previously, e.g., in Figure 3 of Golse et al. 2002). For z_S ⩽ z_L, the sensitivity function is zero since the model value Γ(x) = 0 independent of cosmology. For z_S ⩾ z_L, sensitivity generally increases with source or lens redshift.

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To maximize S/N in our cosmological constraints, we apply this weighting quadratically in our previous expression for χ²:

$$\begin{equation} \chi _w^2 = \sum _{ij}\Delta \Gamma ^2_{ij}w_{ij}[(1 - 1/x_{ij}) - \Gamma _{ij}]^2/\sum _{ij}\Delta \Gamma ^2_{ij}. \end{equation} \tag{ 21 }$$

This χ² can be converted to a likelihood by assuming the error distribution is Gaussian. This is only approximately true in our case, but determining more accurate error distributions would require a significantly more complex error analysis, so we will leave this to future work. Figure 5 shows the likelihood function for $\Omega _\Lambda$, calculated using the restricted group sample and normalized so the area under the curve is 1. The equation of state is fixed to w = −1 and we have marginalized over Σ₀ and r_s. The dashed vertical line indicates the value where the likelihood peaks. Dotted vertical lines indicate 68.2%, 95.4%, and 99.7% (1σ, 2σ, and 3σ) confidence regions. The solid vertical line and shading indicate the mean WMAP seven-year value $\Omega _\Lambda = 0.727$ and 68% confidence range from Larson et al. (2011).

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The results are insensitive to the priors in Σ₀ provided they are reasonably broad, but they depend strongly on the smallest values of r_s considered. This is because of the dependence of inferred physical distance on cosmology. For low values of Ω_X, the transverse distance inferred from a given angular separation on the sky is smaller. Conversely, if we fix r_s to a small value, small values of Ω_X are preferred. Because the radial variation of the profile over the range of our data is much stronger than the redshift variation, the radial fit drives the χ² values and thus biases our cosmological results if unphysically small values of r_s are allowed. The average scale radius is predicted to be r_s ∼ 160 h⁻¹₇₀ kpc for our sample, based on theoretical concentration relations. Simulations show ∼50% scatter in concentration from halo to halo (e.g., Zhao et al. 2009; Reed et al. 2011 and references therein), but most of these variations should average out in the set of ∼100 objects considered here, provided they represent a reasonably unbiased sample. On the other hand the conversion from angular to physical radius will vary by ∼10%–15% for the range of cosmologies considered, and the range of uncertainty on our fitted value of r_s is ∼130–210 h⁻¹₇₀ kpc. Thus, we take a uniform prior on r_s over the range 120–200 h⁻¹₇₀ kpc, allowing for a variation of ±25% around the fiducial value r_s = 160 h⁻¹₇₀ kpc. Extending the range of our priors to higher values of r_s has little effect on the results, while extending it to lower values of r_s decreases the lower limit on Ω_X.

Overall, we obtain the estimate $\Omega _X (w = -1) = \Omega _\Lambda = 0.848^{+0.0435}_{-0.187}$, corresponding to a detection of dark energy at more than 99% confidence. This value is consistent with the most recent WMAP analysis of CMB anisotropies, which finds a mean value 0.727^+0.030 _{− 0.029} (Larson et al. 2011). Our 68% confidence range is approximately four times wider than that of WMAP; given the small size of the field considered here (1.64 deg²), however, this level of precision demonstrates the power of the geometric test. We note, however, that our error estimates do not include systematic effects. We estimate the magnitude of some of the possible systematics in the next section.

In a cosmology with multiple components with equation-of-state parameters w and density parameters Ω_w, cosmic dynamics can be characterized by the deceleration parameter:

$$\begin{equation} q = {1\over 2}\sum _w \Omega _{w} (1+3w). \end{equation} \tag{ 22 }$$

For a flat universe with components Ω_m and $\Omega _\Lambda$, we find a value q₀ = −0.77^+0.28 _{− 0.066} at the present day. Present-day acceleration, which corresponds to q₀ < 0 or $\Omega _\Lambda > 1/3$ if w = −1, is detected at greater than the 98% CL (solid [green] vertical line in Figure 5).

We can also extend the constraints to more general dark energy models with w ≠ −1. Figure 6 shows the likelihood surface for models with a dark energy component Ω_X with an equation-of-state parameter w. Our current results provide an upper bound of w < −0.4 (68.2% CL). They do not provide a lower bound, although there is some information in the constraints on w as a function of $\Omega _\Lambda$. The shape of the confidence regions is similar to those derived by geometric tests using strong lensing (e.g., Jullo et al. 2010), although our contours are shifted upward to less negative values of w, perhaps because of the redshift distribution of our lenses. The shape of the confidence regions also differs from those derived from observations of the CMB (Larson et al. 2011), supernovae (Sullivan et al. 2011), or BAOs (Percival et al. 2010; Beutler et al. 2011), providing interesting complementarity with these other methods.

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4.3.1. Basic Noise Sources

4.3.2. Cosmic Sample Variance—Empirical Estimates

A more complicated source of systematic uncertainty in the measurement comes from structure along the line of sight. The COSMOS survey probes a relatively small field and several large structures are clearly visible in the galaxy redshift distribution below redshift z = 1. We can look for evidence of cosmic sample variance effects in various ways. One simple estimate of the importance of cosmic variance can be obtained by boot-strapping, splitting the sample into two or more disjoint sets. We can test for systematics by measuring how Δ₆₈, the width of the 68% confidence region for $\Omega _\Lambda$, changes relative to the Poissonian expectation $\Delta _{68} \propto \sqrt{n_c/n_0}$ when we cut the sample from n₀ sources down to n_c. Splitting the field into four quadrants with roughly equal numbers of source galaxies, we find Δ₆₈ increases by a factor of 1.9–2.5, so this seems consistent with the factor $\sqrt{n_c/n_0} = 2$ expected from Poisson scaling. We note, however, that one quadrant gives a best-fit value of $\Omega _\Lambda = 0$ (albeit with a 68% uncertainty of +0.6) whereas the others give values of ∼0.9. Examining the lenses and source distribution in this quadrant, it is not immediately obvious whether specific structures produce this shift. This is a sobering lesson about the possible effects of cosmic sample variance.

Splitting our lenses into two groups by redshift, each with n_c ∼ n₀/2, we find that Δ₆₈ increases by a factor of 1.33 for z_L > 0.4 and 2.4 for z_L < 0.4. Thus the noise increases at a roughly Poissonian rate in the high-redshift sample, while in the low-redshift sample it increases much more quickly. Similarly, in a sample cut at z_S > 1, Δ₆₈ increases by 1.33, while for z_S < 1, it increases by 1.73. These results suggest the signal at low redshift is more prone to systematics. We have tested alternative weighting schemes that attempt to correct for the trend with source redshift. Down-weighting sources with z_S < 1.5 by a factor of 0.5 or 0.33 reduces Δ₆₈ by 30% or 40%, respectively, but moves the peak of the likelihood 1.3σ or 1.7σ away from the WMAP7 value. Thus, it seems there is some trade-off between precision and accuracy in the redshift weighting. Clearly this subject requires further theoretical work, using realistic simulations of large-scale structure. For the moment, in the absence of an optimal weighting scheme motivated by theory, we choose not to apply either weighting to our final results.

One final concern is that the average line of sight to all our groups could be slightly over- or underdense. This would introduce a baseline shift in our model of Γ(x), modifying it to

$$\begin{eqnarray} \Gamma (x) &=& \ \Gamma _0 \phantom{(1 - 1/x) + } {\rm \ \ for\ \ } x < 1\,; \nonumber \\ &=& (1 - 1/x) + \Gamma _0 {\rm \ \ for\ \ } x \ge 1. \end{eqnarray} \tag{ 23 }$$

We can test for a constant offset Γ₀ ≠ 0 most easily in the range x = [0, 1] where the expected signal is zero. We find Γ₀ = 0.02 ± 0.03, so there is no significant evidence for an offset. Furthermore, adding Γ₀ = ±0.02 to our model gives essentially identical constraints on $\Omega _\Lambda$ (the peak value shifts by less than 0.1σ), so the method appears to be robust to any small offset of this kind.

4.3.3. Cosmic Sample Variance—Theoretical Prediction

We can also use the error description of Taylor et al. (2007) to estimate the effect on our measurements of structure along the line of sight. They calculate that the tangential shear induced by large-scale structure between the observer and two background galaxies at redshifts z_i and z_j introduces a covariance in shear measurements given by

$$\begin{equation} {\rm var}[\gamma _{t,ij}]=\int _0^{\infty }\frac{\ell {\rm {d}} \ell }{2\pi } C^{\gamma \gamma }_{ij}(\ell)\left(\frac{2[1-J_0(\ell \theta)]}{(\ell \theta)^2}-\frac{J_1(\ell \theta)}{(\ell \theta)}\right)^2, \nonumber \\ \vspace*{4pt} \end{equation} \tag{ 24 }$$

where we have integrated over a circular aperture of radius θ by multiplying by the Fourier transform of the aperture (the term in brackets), and C^γγ_ij(ℓ) is the tomographic cosmic shear power spectrum (Hu 1999).

In our case, we are only concerned with the auto-correlations between redshift bins with i = j. These will give an estimate of the excess variance added to our shear measurements by cosmic structure, as a function of source redshift. Figure 7 shows this error term calculated for an aperture of 6' in a WMAP7 cosmology and assuming the COSMOS redshift distribution. We have calculated the shear error using 20 discrete bins in redshift between 0 < z ⩽ 2. Ideally we would use a continuous cosmic shear in this measurement, as described in Kitching et al. (2011), but since the error contribution we find here is small and smoothly varying, this approximation seems adequate.

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We see that the extra error term is always less than our empirical shear dispersion $\sigma _{\tilde{\gamma }} \ge 0.25$, and that it reaches a maximum of ∼6% of the empirical dispersion. This suggests that the contribution from cosmic shear is much smaller than the excess variance seen in the previous section, which may then be due to individual halos or to other systematics. Clearly more detailed simulations are needed to determine realistic cosmic sample variance errors for our particular technique, but these estimates reassure us that systematics do not completely dominate our current results.