Weak Lensing Results from the 75 Square Degree CTIO Survey
Abstract
We measure seeingcorrected ellipticities for galaxies with magnitude in 12 widely separated fields totalling 75 deg of sky. At angular scales , ellipticity correlations are detected at high significance and exhibit nearly the pure “Emode” behavior expected of weak gravitational lensing. Even when smoothed to the full field size of 25, which is Mpc at the lens distances, an rms shear variance of is detected. At smaller angular scales there is significant “Bmode” power, an indication of residual uncorrected PSF distortions. The data constrain the power spectrum of matter fluctuations on comoving scales of Mpc to have (95% CL, CDM, ), where the systematic error includes statistical and calibration uncertainties, cosmic variance, and a conservative estimate of systematic contamination based upon the detected Bmode signal. This normalization of the power spectrum is lower than, but generally consistent with, previous weaklensing results, is at the lower end of the range from various analyses of galaxy cluster abundances, and agrees with recent determinations from CMB and galaxy clustering.
The large and dispersed sky coverage of our survey reduces random errors and cosmic variance, while the relatively shallow depth allows us to use existing redshiftsurvey data to reduce systematic uncertainties in the distribution to insignificance. Reanalysis of the data with more sophisticated algorithms will hopefully reduce the systematic (Bmode) contamination, and allow more precise, multidimensional constraint of cosmological parameters.
1 Introduction
The realization that weak gravitational lensing effects could reveal the power spectrum of matter fluctuations in the Universe (Valdes, Tyson, & Jarvis, 1983; MiraldaEscudé, 1991; Kaiser, 1992) strongly motivates largescale imaging surveys of faint galaxies. Weak gravitational lensing constrains fundamental parameters under the current paradigm that the power spectrum of matter evolves from primordial fluctuations due to gravitational instability. Comparison of weak lensing power at with measurements of the cosmic background anisotropy at can ultimately test this underlying paradigm to high precision.
The coherent distortions induced by weak lensing are, however, lost in the intrinsic shape variations of the source galaxies unless a very large number of galaxy shapes can be determined to beat down this “shape noise.” This was the primary impetus behind the construction of the Big Throughput Camera (BTC; Wittman et al. (1998)) and other highefficiency CCD mosaic cameras. We report here the results of a large weaklensing survey conducted with the BTC camera and its successor, the NOAO Mosaic II imager (Muller et al., 1998).
Firm detections of weak lensing in random fields were first reported using early data from mosaic groundbased cameras (Wittman et al., 2000; Kaiser et al., 2000; Van Waerbeke et al., 2000; Bacon et al., 2000), HST (Rhodes, Refregier, & Groth, 2000), and singleCCD cameras (Bacon et al., 2000; Maoli et al., 2001). These initial efforts did not place strong constraints on the matter spectrum. This is due in part to their relatively small number of galaxy samples ( or fewer). Close inspection of these data also reveal, though, that the methods that were used to remove systematic distortions induced by PSF ellipticities have left residual signals that may contaminate the lensing observations.
More recently, Hoekstra et al. (2002)[HYG02], Van Waerbeke et al. (2002)[vW02], Bacon et al. (2002), and Refregier et al. (2002), have derived more precise constraints on and the normalization of the matter power spectrum by analyzing larger samples of galaxy shapes. In this paper we present constraints on the power spectrum from the largest weaklensing survey to date, using 75 deg of images collected with the BTC and Mosaic II imagers. HYG02, vW02, and this paper not only have large sky coverage, but also make use of the techniques presented by Crittenden et al. (2002); Schneider, van Waerbeke, & Mellier (2002a); and Pen et al. (2002) for distinguishing “Emode” distortion patterns, which should be produced by lensing, from “Bmode” power, which indicates the presence of uncorrected systematic errors. The E/B decomposition provides an important validity check as well as improving the ratio by . Our program is distinguished by relying exclusively on galaxies with magnitude , for which the redshift distribution is well measured by spectroscopic redshift surveys. The accurate calibration eliminates another source of systematic error. Our results differ as well by making use of many of the techniques developed in Bernstein & Jarvis (2002)[BJ02] for extraction of galaxy ellipticities and lensing distortion signals in the face of asymmetric PSFs. Most other results to date make use the formalism of Kaiser, Squires, & Broadhurst (1995) and its modifications. Our data, reductions methods, and survey depth are largely disjoint from other authors’. Given the subtlety of the weak lensing measurements, we seek reassurance that independent methods yield similar cosmological results.
2 Data
2.1 Observations
The data were taken using the Blanco 4 meter telescope at Cerro Tololo Interamerican Observatory (CTIO) in Chile from December, 1996 to July, 2000. The telescope changed its widefield imager from the BTC to the Mosaic II in 1999, and approximately half of the data were taken using each camera. This is beneficial to us, since some systematic effects are presumably different for the two cameras, so we can compare the results from each camera. Table 1 summarizes the observing runs.
Date  Clear Nights  Camera 

1996 Dec  3^{1}^{1}Less than half of the time in these runs was devoted to this project.  BTC 
1997 Feb  3^{1}^{1}Less than half of the time in these runs was devoted to this project.  BTC 
1998 Nov  4  BTC 
1999 Feb  5  BTC 
2000 Jan  6  Mosaic 
2000 Jul  4  Mosaic 
We observed 12 fields, each approximately 25 square. Table 2 summarizes the locations, area, and camera for each field. We also list the galactic extinction, (cf. step 4), the mean distortion for each field (cf. §3.2) and the total number of galaxies used for shape measurement after the cuts described in §2.2. Note that we were unable to finish two of the fields (K and R). These two are approximately half the full height in declination, but still give us useful statistics on scales of 25 in the RA direction.
Each sky location is observed in three distinct 5minute exposures, all in the band. Exposure pointings are taken in an interlaced pattern that places each galaxy’s image on two or three different CCDs of the mosaic. This makes it easier to detect systematic errors that depend upon chip location, as discussed further in §2.3.2. The large dithers also allow us to eliminate chip defects, scattered light features, and ghost images from our object catalogs by requiring galaxies to have multiple coincident detections. Each BTC field has 112 exposures with 4 CCDs per exposure, with a scale of 043 per pixel. Each Mosaic field has 75 exposures, 8 CCDs per exposure,^{1}^{1}1 During the July, 2000 run, two of the chips failed. One was out for two nights and the other for three nights. so more exposures were taken in some fields to compensate for the lost area per exposure. and 027 per pixel. In all, the dataset contains 1155 exposures, with CCD images. We clearly want to avoid the need to examine the images or catalogs by eye at any point in the reduction process.
The magnitude at which galaxy completeness drops to 50% is , and varies somewhat from field to field due to differences in seeing and other factors (cf. Figure 9). We use galaxies with in order to have an approximately consistent depth for all fields. The median seeing for all the exposures is 1.05 arcsec FWHM, with most of the exposures between 0.9 and 1.3 arcsec.
Label  RA  Dec  Area  Camera  Mean Distortion (%)  

(J2000)  (deg)  (mag)  ()  
K  4.2  Mixed  0.080  123  
H  6.8  BTC  0.018  160  
J  6.8  Mixed  0.022  184  
N  6.8  BTC  0.045  154  
A  6.8  BTC  0.118  168  
M  6.8  BTC  0.121  163  
Q  6.8  Mosaic  0.106  188  
L  6.8  Mixed  0.133  179  
T  6.8  Mosaic  0.111  142  
X  6.8  Mosaic  0.056  148  
R  3.4  Mosaic  0.062  91  
G  6.8  Mixed  0.031  150 
2.2 Image Reduction
Extracting useful shapes from observations of distant galaxies is a complicated procedure. The CTIO Survey data have been analyzed using a subset of the techniques described in BJ02. We list below the steps involved in our reduction process, and refer to BJ02 for a more complete description of each step. At no point in the reduction do we sum exposures: all measurements are made on individual exposures, with merging of exposures occurring at the catalog level. This is “Method 1” described in BJ02 §4.

Bias Subtraction and Flat Fielding
We do this in the normal way using the IRAF packages CCDRED and MOSRED^{2}^{2}2 IRAF is distributed by NOAO, which is operated by AURA under cooperative agreement with NSF.. Note that our observing scheme allows us to make excellent dark sky flats, since we move substantially after every exposure.

Object Detection
We use the program SExtractor (Bertin & Arnouts, 1996) for the initial object detection. As discussed in BJ §8.1, any surfacebrightness threshold detection scheme will result in a selection bias, whereby objects similar in shape to the PSF will be more likely to be detected. We set the SExtractor detection threshold very low, such that a significant fraction of the detections are noise, or unusably faint galaxies. Later (step 10), we select from our list of objects according to a significance parameter that is unbiased by the PSF shape. Requiring coincident detections on more than one exposure eliminates the vast majority of the noise detections and spurious objects. Galaxy total magnitudes and sky levels are obtained from the SExtractor catalogs.

Field Registration and Distortion Measurement
Registration of the images is done in two stages. In the first stage, we register each field by matching the bright objects () to the positions of stars given by the USNO A2.0 catalog (Monet, 1998). There are typically 100 or fewer matching stars per CCD image, which is not sufficient to adequately fit for all terms of the coordinate map.
From the first stage coordinate maps, we can match all detections of a given galaxy. Each galaxy is typically observed 3 or 4 times, and we ignore any object which is detected only once, most of which are either noise or edge objects.
In the second stage of the coordinate map determination, we fit the full polynomial telescope distortion model by minimizing the exposuretoexposure variance in the positions of multiplydetected galaxies. The USNO star positions are included in the fit to tie the solutions to the astrometric frame. Note that this stage of this procedure would be impossible if we had not dithered our exposures by large amounts, because a solution with only small shifts in galaxy positions leaves many of the distortion parameters degenerate, and there are too few USNO stars to constrain all the higherorder distortion terms.

Photometry
Precise photometry is not extremely important for this study, since we are primarily concerned with the shapes of galaxies, not their overall intensity. We do, however, need reasonable magnitude estimates for our galaxies to infer their distribution of redshifts (see §4.1). For our sample, a magnitude calibration error of leads to an erroneous factor in the inferred .
We use the SExtractor “best” magnitudes as our photometry measurement. This magnitude is then corrected for the Jacobian of the distortion map found for each image in step 3. All exposures of a given field are put on a common photometric system by minimizing the exposuretoexposure variance of multiplydetected galaxies’ magnitudes. This procedure only determines the relative magnitudes from image to image. We obtain the overall zeropoint by measuring several Landolt (1992) stars for each run. Our magnitudes are thus tied to his Cousins filter system. No significant color terms are detected. The uncertainty in the magnitudes from this zeropoint determination is mag, which is quite sufficient for our redshift calibration.

Initial Shape Measurement
The shape of a galaxy is described by two components of its ellipticity. These and quantities are determined for each galaxy on each exposure using the algorithms in §3 of BJ02, in which ellipticities are defined from weighted central second moments of each galaxy. The weights are elliptical Gaussians, with size, shape, and centroid iterated to match those of the galaxy. If the iteration does not converge, usually due to crowding by nearby objects or edge effects, then we discard the galaxy. These measurements are made in sky coordinates rather than in pixel coordinates to remove the effects of telescope distortion.

Star Identification
We need to know the pointspread function (PSF) for each exposure, so we can remove its effects. Thus we need to identify the stars in each exposure. The usual method is to use a sizemagnitude scatter plot, and look for an arm in the distribution at small sizes which separates from the main swath of galaxies. There is often an arm at smaller size, corresponding to cosmic rays, which does not extend as far to bright magnitudes as the stellar arm.
This identification is very easy to do by eye, but it is not trivial to construct an algorithm to automate the process. A full description of the algorithm we use is in a forthcoming paper (Jarvis, 2003), however it might be of interest to mention one of the particular difficulties of this analysis for large field cameras. The images from the BTC camera have significant astigmatism which enlarge the stars near the corners of the array. (The seeing on the Mosaic images is more uniform, but still has some of this effect.) Thus, the stars near the corners of the array do not have the same size as most of the stars on the sizemagnitude plot. If this is not carefully taken into account, the stars near the corners will be missed, and one will not be able to accurately describe the PSF shape across the whole chip. This will, of course, lead to significant systematic errors in the final shapes, so an accurate and complete identification of the stars is very important.

Image Convolution
Correction for the effects of the PSF on galaxy shapes is done in two steps. First we remove the ellipticity induced by the PSF by convolving with a kernel which makes all the stars look round, using the methods in §7 of BJ02. Then we correct for the dilution of the shapes analytically according to the size of the PSF at the location of each galaxy (in step 10), as described in Appendix C of BJ02.
For each star identified in step 6, we find a pixel kernel which makes the star round. In the language of the Laguerre decomposition described in BJ02 §6.3, we find a kernel for which , , and are all 0. The most important term for insuring that the PSF does not bias the galaxy shapes is . The and are higher order terms which can also cause a bias in galaxy shapes, so we make them vanish as well. The methods for finding the appropriate kernel are described in BJ02 §7.
Once we have a kernel for each star in the image, we need to interpolate this kernel across the image. The kernel is described by a set of coefficients, so we actually fit each coefficient across the image. This apparently simple interpolation task has been found by us and other practitioners (Van Waerbeke et al., 2002; Bacon et al., 2002) to require great care to avoid inducing spurious distortion power. With too little freedom, an interpolant misses real variation in some regions of the chip, but with too much freedom, the fit adds spurious wiggles in other regions or at the chip edges. We find that simple polynomial fits could not simultaneously avoid both problems, especially on the BTC PSF structure. We instead use a smoothing spline algorithm developed by Gu & Wahba (1991), which is available from NetLib as Rkpack^{3}^{3}3 http://www.netlib.org/gcv/rkpk.shar.
As noted in BJ02 §7.5, the convolution is efficient since each of the kernel components is the result of three successive kernel convolutions.
We want to avoid extrapolating the kernel into regions where the PSF is not well constrained. We hence reject all galaxies from our catalogs which are too far from any valid PSFtemplate star or which are outside the bounding convex polygon of all such stars. We also reject regions which are near extremely bright stars to avoid spurious detections due to diffraction spikes, rings or bleeding columns.

Remeasurement of Shapes
The convolved images now have stars that are round. Therefore, the intrinsic sky image has now effectively been convolved with a round PSF, rather than the elliptical PSF of the original images. When we measure the shapes of galaxies on this image, we should have shapes which are unbiased by the PSF. The measurement again uses the method of BJ02 §3.
Note, however, that the shapes are not yet the true shapes of the galaxies. A round PSF makes a galaxy appear more round than it really is. This is called PSF dilution, since the magnitude of the ellipticity is reduced by the PSF. Appendix C of BJ02 derives an approximate factor by which the shape has been diluted. Each galaxy shape must be corrected for dilution by the factor to obtain an estimate of the true shape of the galaxy. This is done in step 10.

Removal of Centroid Bias
BJ §8.2 describes the “centroid bias.” Essentially, the centroids of galaxies are more uncertain in the direction of the PSF elongation than they are perpendicular to this direction. This anisotropic error in the centroid position affects the measured shapes as well and leaves a bias in the shapes relative to the observed PSF shape.
The functional form of the bias is expected to be
(1) where is the significance of the galaxy detection (see step 10), and the are weighted second radial moments (sizes) of the PSF and the galaxy. We measure empirically simply by fitting all of the observed shapes to this form, obtaining the value for both BTC and Mosaic images. The bias is then subtracted from the shape of each galaxy.

Combination of Measurements
Step 3 yields a list of multiplydetected galaxies and the exposures on which each galaxy was measured. We can now combine the set of shape measurements for each galaxy into a single maximum likelihood estimate of the shape according to the methods of BJ §4.2.
Before averaging, we correct for the dilution mentioned above by dividing each shape by the dilution (cf. step 8). Galaxy measurements with are rejected.
This step is also where we select our galaxies using an unbiased estimate of the significance . As discussed by Kaiser (2000) and in BJ02 §8.1, a surfacebrightness selection of galaxies (as for SExtractor) is inherently biased toward galaxies aligned with the PSF. Galaxies which are the same shape as the PSF are essentially matchedfiltered by the PSF, and thus are easier to detect. We estimate the significance of each galaxy according to , where is the weighted flux of the object, is the white noise density of the sky, and is the scale size of the object. If these are all measured in the convolved images, which have round PSF, then the estimate of has no shape bias. We have found empirically that SExtractor detects essentially all objects with , so galaxies more significant than this are unaffected by selection bias. We therefore cut from the galaxy sample all galaxies with .
We now have a catalog of galaxies for each field, with an estimate of the true shape of each galaxy before being affected by seeing or other PSF effects. We also keep in this catalog an estimate of the errors in the ellipticity measurements, the magnitude, and the size of the galaxy.
One may ask how our BJ02based methodology compares to the more common KSB methods. We have not tried to construct a parallel KSB pipeline for our data, so we can only speculate on the relative merits. From theoretical considerations (see BJ02), we expect our methods to reduce the impact of photon noise on the measurements of individual ellipticies by 20% or so. For circular objects the 2 methods are equivalent. There is a similar potential S/N gain from the use of our nearlyoptimal weighting when combining shapes to estimate a distortion (§3.1), since the shape noise is minimized. We expect a more significant benefit over KSB to be a reduction of systematic errors due to uncorrected PSF ellipticities (described in next section). KSB breaks down when objects aren’t Gaussian or the PSF isn’t nearly circular, and our higherorder circularization kernel is expected to do a better job with the PSFs that are present in our data.
2.3 Checks for Systematic Errors
The most daunting problem in weaklensing measurements is not the volume of the data, but rather the elimination of spurious shape correlations caused by instrumental effects, particularly asymmetric PSFs. Our decision to make a shallow survey is beneficial in that the galaxies have known and larger angular sizes than faint galaxies, but a shallow survey has a smaller lensing signal and hence is more susceptible to systematic errors. Our requirements for rejection of systematic signals are very stringent: as noted below, the lensing distortion signal at our largest scales is RMS, while a large fraction of our images contain stellar images with ellipticities of 10% or higher. This is particularly a problem for the BTC images, because significant astigmatism in the telescope combines with warped CCDs to produce larger PSF ellipticities than is typical. Galaxies measured in image regions with stellar ellipticities are discarded from the catalogs.
There are a number of checks one can make to look for systematic errors in the galaxy shapes. In particular, since the convolution should make the stars round, we check that the stars in the final image actually are round. Also, it is expected that most systematic effects will correlate with the position on the chips and/or the size and shape of the PSF. We describe here some of the tests we have done to quantify systematic errors which might remain after the processing steps described in §2.2.
2.3.1 Final Star Shapes
We make a “whisker plot” of the stars in every image to look for images where the processing may have gone wrong. For example, missed stars near the corners of the chip, a bad fit to the kernel, etc. A representative example of one of these plots along with the corresponding plot before the convolution is given in Figure 1. For reference, the horizontal whisker in the center of each corresponds to 1% ellipticity.
Clearly, the convolution does not make each star exactly round. Each whisker on the plot still has about 13% ellipticity. The important thing to check is that the convolution has removed any coherence in the whisker orientations. Measurement error on the stars’ ellipticities will leave uncorrelated random whiskers as residuals. The longer whiskers tend to correspond to fainter, noiser stars.
To search for residuals which may be coherent functions of pixel coordinates, we average the shapes of the stars from many images as a function of position on the CCD array. The whisker plots resulting from this procedure for both the BTC and Mosaic chips are shown in Figure 2.
All of the whiskers are smaller than the 1% whisker, with most of the whiskers barely visible (). The only whiskers which approach 1% in size are in the corners of the chips. This is mostly due to the fact that stars near the corners are more likely to be rejected from the kernel fits (in step 7 of §2.2). Therefore, fewer stars are being averaged, and the statistical errors tend to increase the resultant whisker size. This is not a problem for the galaxies, because galaxies in regions with no stars are also rejected.
Another way to see how well we are making the stars round is to plot the final shape of the stars against the initial shape. This plot is shown in Figure 3. There is a noticeably positive slope for both and , but the slope is of order 1/300. This means that the shape of the PSF is reduced by a factor of 300 by the convolution. The mean final ellipticity for the worst initial ellipticities is less than 0.1%, which is well below the level of our lensing signal. Moreover, the rms deviation from zero is only 0.03%.
2.3.2 Galaxy Shape vs. Chip Position
We can make whisker plots for the galaxies as well as the stars. The galaxies, however, are not each expected to be round after convolving. Even when the PSF is round, the galaxies each have a real shape variance of order 30%. To see if the convolution has affected the galaxies correctly as well as the stars, we reduce the shape noise by averaging the shapes of many galaxies. We first bin the galaxies by array position to search for systematic errors that depend upon pixel coordinates. Figure 4 shows the (null) results of this test. For neither hardware configuration do we detect any residual pattern in the galaxy shapes.
2.3.3 Galaxy Shape vs. PSF Shape
The biggest systematic effect to eliminate is the effect of the PSF. While there are other effects which vary across the chip, such as charge transfer inefficiencies, telescope distortion, and the flat field pattern, clearly the main concern is the PSF.
The most obvious test then is to bin the allegedlycorrected galaxy shapes by the “raw” PSF shape, as was done for the convolved stars. Again, each individual galaxy is not expected to be round, but the average shape of many galaxies should be independent of the PSF shapes for the galaxies if the survey is large enough to decouple the PSF variations from the true lensing signal. Figure 5 shows the results of this test.
Apparently, there is still a bias of the galaxy shapes with respect to the initial PSF shape. The plot shows a slope of approximately 0.015, which corresponds to a 0.4% effect for our worst PSFs, and a 0.2% rms effect. This is somewhat below (but of the same order as) the level of our lensing signal.
We believe the observed residual galaxy dependence upon PSF shape is primarily due to higherorder asymmetries in the PSF which have not been removed by the convolution filters. We plan to implement the full higherorder analytical PSF corrections of BJ02 §6.3.5 and expect this to have smaller residuals than the kernelconvolution method used here.
Until then, we empirically fit for the slope of this bias and subtract the bias from the galaxy shape measurements. However, it is likely that simply subtracting the bias from the galaxy shapes is not the correct thing to do. In particular the bias seems to be stronger for faint galaxies and for large galaxies (which may be due to slightly nonlinear CCD response or charge transfer efficiency). We currently use different bias slopes for and , but it is possible that we have not correctly identified the exact population of galaxies which are giving us most of the bias. Therefore, we suspect that this residual is likely the main source of the Bmode power described in §3.4.
3 Analysis
3.1 Determining Shear from Shape Averages
Once we have a catalog of galaxy shapes, we want to be able to convert these shapes into various statistics of the lensing distortion field. These statistics require finding either the average of many ellipticities (e.g. the overall shear in each field, §3.2) or the average product of pairs of ellipticities (e.g. the shear correlation functions, §3.3).
The optimal weight for averaging ellipticities to obtain the lensing distortion [BJ02 Equation (528)] requires knowledge of the intrinsic distribution of galaxy shapes. The distribution for the brighter, wellmeasured galaxies in this survey is shown in BJ02 Figure 4. BJ02 Equation (536) gives a simple weight which gives nearly optimal results:
(2) 
where is the measurement uncertainty in each component of the shape, as measured in the sheared coordinate system where the shape is circular. BJ02 Figure 5 demonstrates that this weight recovers very close to the optimal signal to noise for the estimate of the distortion.
Thus, our estimate for a distortion from a set of shapes is (from BJ02 Equations 523, 533, and 535)
(3)  
(4)  
(5)  
(6)  
(7)  
(8) 
The responsivity, , is similar to the shear polarizability of the KSB method and describes how our weighted mean ellipticity responds to an applied shear. In the simple case of an unweighted ellipticity average with unweighted shape measurements, . The shape noise, , is the variance in the intrinsic of the galaxies. For our brighter galaxies, we measure .
Figure 5 of BJ02 also demonstrates that the approximations made in the derivation of the above responsivity lead to error in the resultant distortion calibration, for the shapes and noise levels of galaxies found in this survey. Smith et al. (2001) perform a complete numerical simulation of the distortion measurement process, concluding that the overall calibration is accurate to . The BJ02 formulae improve upon those of Smith et al. (2001) so we believe the responsivity calibration is now accurate to or better.
3.2 Overall Shear in Each Field
The simplest statistic to calculate is the average distortion in each of our 12 fields. These results are listed in Table 2. The uncertainty on each of these measurements is typically . The for detection of each distortion component in each field is . Collectively, they give a strong detection of the rms fluctuation in the shear field averaged on a scale of . At the redshift at which our sensitivity to lensing matter peaks (cf. §4.1), this corresponds to a comoving smoothing scale of Mpc, the largest scale to date on which gravitational lensing effects have been detected.
Figure 6 shows a scatter plot of the 12 distortion values, along with their error bars for each component. The mean distortions appear randomly distributed as expected for a real signal. A tendency to align on the axis would indicate a systematic error aligned with the CCD axes, such as charge transfer nonlinearities—no such effect is seen.
We assume that the average of each distortion component or in a 25 square box has a Gaussian probability distribution with width . The expected distribution is then broadened by the measurement error in each field (Equation (4)). A maximum likelihood analysis of the 24 values then yields . Circles at 1 and 2 are also drawn on Figure 6 for reference. For comparison to other results which use the shear, , rather than the distortion, .
3.3 Shear Correlation Functions
MiraldaEscudé (1991) introduced the shear correlation functions of the ellipticities of pairs of galaxies measured with respect to the line separating them. We will not make direct use of the correlation functions to constrain cosmology, but rather follow the prescriptions of Crittenden et al. (2002) (as clarified in Pen et al. (2002) and Schneider, van Waerbeke, & Mellier (2002a)) for constructing the and statistics (described below) from the correlation function data.
The simplest way to calculate these functions is to treat the lens distortion as a complex number, . To conform with other authors, we express the correlations in terms of the shear , related in the weak limit to the distortion by . Three correlation function are defined by:
(9)  
(10) 
Note that if the galaxies in a pair are swapped, the first equation turns into its conjugate. Thus, this quantity can be made manifestly real by double counting every pair of galaxies. With only single counting, the imaginary term gives an estimate of the statistical error of the other three quantities. Further, taking a mirror image of the entire field will turn the second equation into its conjugate. Thus, the imaginary part of this quantity is also expected to go to zero in the absence of systematic effects.
Nearoptimal estimators for these twopoint functions are constructed using the weight function and responsivity defined in Equations (2–8):
(11)  
(12) 
where the sum is taken over all pairs of galaxies and with separation within some bin. The variance of due to shape noise and shot noise is also simply estimated, but cosmic variance and bin covariances are more difficult to estimate. We bypass these complications by constructing fieldtofield covariance matrices, as described below.
As our primary use of the correlation functions will be to calculate other quantities by integrating over a range of , we calculate the correlation functions in fairly small bins with .
3.4 Aperture Mass Statistic
The aperture mass statistic () is useful for estimating cosmological parameters. The idea is fairly straightforward: a mass concentration at the center of a given aperture will tend to produce a tangential pattern to the galaxy ellipticities around the center of the aperture. For cosmic shear measurements, we do not detect the mass concentrations individually [see Wittman et al. (2001) for a rare (so far) example], but rather the magnitude of the mass fluctuations, so we really want the rms variation of the value as the aperture is swept across the sky.
The statistic for a single aperture of radius is (Schneider et al., 1998)
(13)  
(14) 
where is the tangential component of the shear (as estimated from the galaxy ellipticities). For many independent apertures, and the variance, , probes the power spectrum of the effective convergence. The window function for this statistic is narrow in space and centered at (Schneider et al., 1998).
One advantage of the statistic is that there is a natural test for systematics. If each galaxy is rotated in place by 45 degrees, the integral should vanish if due purely to lensing. This test is essentially measuring the curl of the shear field, and is therefore often called the “B mode,” while measures “Emode” power. We designate this Bmode version of the statistic as . Most systematics are expected to add equal power to the E and B modes, hence the data are a sensitive test for contaminating systematics.
Another advantage to using the statistic is that is very weakly correlated with when differs from by a factor of (Schneider et al., 2002). Thus, for our range of , we have essentially 7 independent points with which to constrain cosmology.
The problem with calculating in the obvious way (scanning the aperture across the images and calculating variance) is that each aperture is not uniformly filled with galaxies. There are holes due to foreground bright stars, edge effects, bad columns, etc. These holes can then bias the resulting estimates and produce spurious power. Specifying a mask for our entire survey would require a painfully long time, as would the development of software to automate the task. Fortunately, Crittenden et al. (2002) (detailed also in Pen et al., 2002; Schneider, van Waerbeke, & Mellier, 2002a) express and as integrals over the shear correlation functions, which do not require knowledge of the survey geometry. The relevant formulae are:
(15)  
(16) 
where equations for and are given in Schneider, van Waerbeke, & Mellier (2002a). The result of this calculation for our data are shown in Figure 7 and discussed in the following sections.
Another common statistic of the shear field is the windowed variance, , which is the variance of the shear when smoothed with a circular window of radius . We have presented in §3.2 the results for the windowed shear in 25 squares. For quantitative comparison to cosmological models, the naive summation would require knowledge of the survey mask geometry. Again the above references show how to express as an integral over and , thereby removing the need to know the mask. If correlation function data are available out to separation , then all of , and are unambiguously determined for .
The statistic is inferior to in several respects: its window function in space is much broader, and hence measurements at different are more highly correlated. Also does not separate Emode from Bmode power, so the noise is higher and there is no systematicpower null test. The information in this statistic is mostly degenerate with the statistic, with the important exception that the variance probes the power spectrum for , whereas probes . Therefore for near half our 25 field size probes a portion of the power spectrum at larger physical scales than is accessible to . We will therefore make use of for . The statistic is plotted above the statistic in Figure 7.
Pen et al. (2002) and Schneider, van Waerbeke, & Mellier (2002a) do give prescriptions for producing versions of the and statistics which separate the E and B contributions. However, these expressions have indeterminate constants of integrations which render the E and B modes degenerate on the scale of the field of view. Shear that is nearly constant over the field obviously cannot be classified as having either E or B properties. Since we only use for values of near the field size, we will not use this decomposition.
3.5 Covariance Matrix
For each of our twelve fields, we have a vector of observations, , including the values from 1′ to 100′, and the values from 50′ to 100′. Each of these vectors has many data points, since we calculate each statistic at rather small intervals in aperture radius . However, these data are highly degenerate in their information content. values become essentially independent at a factor of 2 in , giving only about 7 independent data points. And values are highly correlated with each other, with the effect that these values only add one more independent data point. So the mean of the twelve vectors, , gives us essentially 8 independent points with which to constrain cosmology.
To quantify this degeneracy more exactly, we construct the full covariance matrix, , for the vectors by equally weighting each of the twelve fields:
(18) 
where ranges over the fields, and are the indices of the data values.
This construction of the covariance matrix means that we ignore the (approximately identical) nominal error bars for each point, using the actual fieldtofield variation as our estimate of the error. This has the advantage that it automatically includes cosmic variance in the uncertainty as well as measurement noise both on and off the diagonal. The disadvantage is that the halfsize fields, which have slightly larger statistical error bars, are given equal weight to the fullsize fields.
3.6 Bmode Power
It is evident in Figure 7 that there is significant power in the statistic, indicating that we do have some Bmode power, and hence some systematic contamination in our data. The effect becomes much weaker at scales , suggesting that the large scale data are probably free of this contamination. Since the Bmode drops to essentially zero at the largest scales of , we also expect that the statistic will be free of the contamination for , since it probes power at even larger angular scales.
Note that the observed Bmode signals are much larger than those to be expected from intrinsic galaxyshape correlations (Crittenden et al., 2002) or secondorder gravitational lensing effects (Schneider, van Waerbeke, & Mellier, 2002a; Cooray, 2002). We believe that they are more likely due to uncorrected highorder PSF effects or inexact kernel fitting, both of which become more important at smaller scales.
Most systematics effects, including uncorrected PSF variation, can increase and much more easily than decrease. However, one can also conceive of systematic effects which simply mix power from the Emode into Bmode rather than add power to either one. For example, if each ellipticity vector’s orientation is rotated slightly but the magnitude is unchanged, then no power is added, but the power is mixed somewhat into the Bmode.
Therefore, a conservative estimate is that the values can be in error in either direction by the amount of the values. Further, it is not sufficient to simply increase the error bars on individual data points by this amount, since the effect is presumably in the same direction at all (or many) values of . Instead, one should consider the two cases of adding the Bmode to all points and of subtracting it from all points. The range of these two cases will then give an estimate of the full potential effect of the systematic error. If the spurious systematic signal has more E power than B power, we could still have overestimated the lensing E signal. Such behavior is expected only, however, for intrinsic correlations, which we believe to be a small constituent of our systematic contamination.
In any case, for the purpose of constraining cosmology, we want to limit our consideration to the range of that has the least amount of Bmode contamination. We will use the values only for , and still use from . Thus, our data vectors (cf. §3.5) now only have essentially 4 independent data points. The increase in the statistical error from this reduced range is however much more than compensated by the decrease in the systematic error from the Bmode; in this range the systematic error is smaller than the statistical error.
EMode  BMode  Reduced Covariance Matrix  

()  
9.52  1.38  3.44  1.25  1  0.80  0.20  0.13  
7.24  2.00  1.23  1.19  0.80  1  0.14  0.02  
3.84  1.09  0.21  1.00  0.20  0.14  1  0.11  
26.4  6.20      0.13  0.02  0.11  1 
For reference, an abridged listing of the data and covariance matrix is given in Table 3. It includes both the shear variance and the aperture mass statistics (both E and B mode in the latter case) for selected values within this range. It also gives the reduced covariance matrix as described in §3.5 for these values. It is evident that the four statistics are relatively uncorrelated with only and being somewhat correlated.^{4}^{4}4 Takada & Jain (2002) predict that the kurtosis of the shear field is beginning to be significant on these scales and could be the cause of this correlation. This is expected, since they sample the power spectrum at relatively disjoint ranges in frequency. Each statistic can be written:
(19) 
and Figure 8 shows for each of these four statistics.
4 Cosmological Implications
4.1 Redshift Distribution
In BJ02 it is shown that the distortion estimators in §3.1 will converge to the lensing distortion. In reality the expected distortion is a function of the source redshift , and we determine some mean distortion . We do not know the redshift of every source galaxy, but we can have some knowledge of its distribution vs. magnitude from spectroscopic redshift surveys. If we divide our source galaxies into redshift bins then the measured signal should converge to
(20)  
(21)  
(22) 
Here we have defined to be the total weight in the redshift bin, and is the mean responsivity, as per Equation (5), of the galaxies in the bin. We will make the assumption that is independent of redshift. For galaxies with magnitude , we do see a significant dependence of the effective upon surface brightness, with as we select subsets of extremely high or low surface brightness and high or low measurement noise. On the other hand, each redshift bin contains galaxies with a wide range of surface brightness and measurement noise, so the responsivity variation with should be much smaller than this extreme range.
With the taken as constant, the measured distortion (22) becomes
(23)  
(24) 
where is the probability of a given galaxy having redshift , weight , and apparent magnitude . Ideally we would determine the function by conducting a redshift survey over a statistically significant sample of galaxies with known and for our survey conditions. Existing surveys, however, can only give us the conditional distribution of redshift for a given magnitude. We will therefore make the further assumption that, within a given magnitude bin, the redshift [or more precisely ] is statistically independent of the weight , so that , in which case
(25) 
Since the redshift data are sparse, the integrals are calculated in 0.5magnitude bins. Let the lensing survey field contain a total weight of galaxies in the magnitude bin, which we apportion among the galaxies in the redshift survey that lie within this bin. If redshiftsurvey galaxy has redshift and lies in magnitude bin , then the expected signal becomes
(26)  
(27) 
Figure 9 plots the distribution of weight vs. magnitude for galaxies which pass the selection criteria in step 10. Each of the 12 fields are plotted separately. We truncate the galaxy sample at to minimize fieldtofield variations in effective depth, and to keep the galaxy sample to the magnitude range for which sizable redshift surveys are available. We also truncate at , since bright galaxies are negligibly lensed.
The Caltech Redshift Survey (Cohen et al., 2000)[CRS] is almost complete for for a 437 galaxy sample surrounding the Hubble Deep Field. The Steidel & Hamilton (1992) photometric system (not to be confused with our symbol for responsivity) differs slightly in both zeropoint and bandpass from the system defined by Landolt (1992). Convolution of synthetic galaxy spectra at a variety of redshifts (R. Somerville, private communication) suggests that mag for or galaxies, rising to mag at as the 400 nm break in galaxy spectra moves between the two filters. We apply this correction to the CRS magnitudes to define an sample that is 97% complete. The CRS magnitudes are then corrected for Galactic extinction in the same way as our program fields.
We apply Equation (26) to this CRS sample, assuming that either (a) all the galaxies with unknown redshift are at , or (b) galaxies with unknown redshift have the same distribution as other galaxies of similar magnitude. For the underlying weight distribution, we take that of the field of median depth, field T. We also examine the effect of taking the shallowest (A) and deepest (N) fields. Choosing case (a) or (b) for incompleteness, or fields T, A, or N, changes the expected signal by at most 4% from the canonical case. The implied uncertainty in is, at 2%, insignificant compared to the measurement errors.
A calibration of our signal can also be done using the CanadaFrance Redshift Survey (Lilly et al., 1995)[CFRS], which is complete to . While band magnitudes are not measured in the CFRS, Smith et al. (2001) collect band images in the CFRS fields so that a sample of 783 galaxies can be defined, after correcting for Galactic extinction in the CFRS fields. This subset of the CFRS sample is not quite representative of our source galaxies because the bluer galaxies at do not make the cut of the CFRS sample. The CFRS sample is also only 88% complete in redshift. The incompleteness in redshift and the depth/color mismatch make the CFRS data less reliable for our purposes than the CRS. Nonetheless, treating the incompleteness by method (a) above () gives a calibration within 5% of the nominal CRS case. Using method (b), however, gives an expected signal that is 15% below that of the nominal CRS value. Given the known shortcomings of the CFRS sample, we will take this as a very conservative 95% CL bound on the possible error of the CRS depth calibration. Since at our larger scales, the 95% CL error on is 7%, to be added in quadrature with the statistical errors and Bmode corrections.
There is an additional uncertainty due to our assumptions that and are independent of redshift. Note that this assumption is implicit in all previous cosmicshear measurements. A detailed test of this assumption requires larger redshift surveys, and we will for the time being ignore the effect as we believe it is weak.
4.2 Prediction of Signals
Using the notation of Schneider et al. (1998), and approximating the source distribution as a set of weighted delta functions at the observed redshiftsurvey ’s, the angular power spectrum of the lensing convergence is expected to be
(28) 
Here is the conformal distance from , is the horizon, are the distances to the redshiftsurvey galaxies, is the comoving angular diameter distance, and is the mass power spectrum at a given comoving scale and epoch. Figure 10 plots the bracketed quantity in Equation (28) [with an additional factor of ] as a function of the lens redshift , for several of the assumptions about sourcegalaxy redshift distributions detailed above—they are all seen to give very similar results. The sensitivity function has a broad peak at .
There are two important issues in making model predictions for the and statistics. First, for the angular scales of interest, nonlinear clustering and its evolution must be taken into account. Jain & Seljak (1997) show that the effect of nonlinear enhancement exceeds a factor of 4 in the variance on arcminute scales, and is about 50% at 10′ (for close to unity). Thus essentially all existing shearcorrelation measurements in the literature probe the nonlinear regime. Since our survey measurements extend to beyond 1 degree (albeit at shallower depth), we span an interesting dynamic range that includes the nonlinear regime (below 5 arcminutes), the quasilinear (1020 arcminutes) and the nearly linear regime on larger scales. In any case, to compare our full set of measurements to model predictions with some degree of accuracy requires that we use a well calibrated model for the nonlinear mass power spectrum. The fitting formulae developed by Hamilton et al. (1991); Peacock & Dodds (1994); Jain, Mo & White (1995); Peacock & Doods (1996) provide empirical but fairly accurate predictions for the nonlinear power spectrum. We will use the Peacock & Doods (1996) formulae to compute the shear variances for different models. Recently vW02 have discussed the accuracy achieved with these formulae and found that on arcminute scales there is some uncertainty in the theoretical predictions which precludes parameter estimation at much better than the 10% percent level. The large angular scale coverage of our measurements makes this uncertainty on small scales less of an issue.
The second issue in making model predictions is the choice of cosmological parameters to vary. One approach is to choose a physical model and vary parameters that have specific meaning within such a model. Our approach however will be closer to an empirical one in which we will choose the parameters that lensing is most sensitive to and take other parameters to be unknown or fixed. Based on earlier theoretical work (Kaiser, 1992; Bernardeau et al., 1997; Jain & Seljak, 1997), we choose the primary parameter space to be that of , such that the amplitude of the linear power spectrum is , and , the present mean mass density parameter of the universe. We will parameterize the shape of the power spectrum by the standard parameter, which has a specific physical meaning for cold dark matter models. We will fix and the primordial spectral index in most of the analysis, but as discussed below, we will explore the sensitivity of our constraints to the value of . We have above explored the sensitivity to the uncertainty in our redshift distribution. Our analysis is similar in spirit to that of vW02, but our choice of survey depth and angular scale makes the interpretation insensitive to unmeasured parameters of and , so marginalization over these quantities is not required.
The theoretical predictions for and are given by the following equations:
(29)  
(30) 
where and are the first and fourthorder Bessel functions, and is given by Equation (28). In the linear regime, for an Einsteinde Sitter universe the 3dimensional mass power spectrum grows in time as , so the dependence on the redshift coordinate is contained in the term in square brackets in Equation (28). However for other cosmologies and in the nonlinear regime the growth rate is different, so the dependence on is more complicated and can be scale dependent. Thus the dependence on cosmological parameters enters in rather complicated ways through the distance factors as well as the power spectrum. For a reasonable class of models that are at all consistent with other cosmological probes, the dependence of the second moment of lensing statistics on the cosmological constant is weak. The main dependences then are on and , and as shown in previous work, the combination that enters is close to the one in cluster abundances and is given roughly by . There are further dependences on the shape of the power spectrum and the redshift distribution which we will explore below.
4.3 Fit of Models to the Data
As described in §3.5 above, our data are essentially reduced to a mean vector, , of and values, along with the covariance matrix of these values, . We calculate the corresponding vector for each of our cosmological models and compute a value for each model:
(31) 
One important consideration in the above calculation is that our covariance matrix is largely degenerate. Our vector has over 100 elements, since we calculate the statistics at fairly small separations in aperture radius. However, as discussed in §3.5 and §3.6, there are really only 3–4 independent data points among all of these. Rather than selecting 4 values of arbitrarily, we use a singular value decomposition to calculate , and take only the largest 4 eigenvalues of the decomposition. This automatically uses only the nondegenerate components of the matrix, and in some sense finds the best combination of the data to use for our 4 values. The 4 values selected for Table 3 are therefore merely representative of the data used for the constraints.
Figure 11 shows a contour plot of as a function of and . The nominal overall best fit is at . However, there is clearly a strong degeneracy in this plot indicating that we really only constrain the combination , which is found to have a value of (95% confidence interval).
This error bar includes only the statistical and cosmological variations which went into the calculation of the covariance matrix. However, we need to account for the systematic uncertainty due to the Bmode power. As discussed above in §3.6, we allow for the two possibilities: that the Emode should be either increased or decreased by the amount of the Bmode. For these two extremes (also plotted in Figure 11), we find that ranges from 0.263 to 0.377 at the 95% confidence level. That is, these are the minimum and maximum values at 95% confidence for any of the three cases: E, E+B or EB. These values then span the full range allowed by our data taking into account both the systematic and the statistical errors.
Our calibration and redshift uncertainties of 5% and 7% respectively are smaller than the above errors but need to be included. We also examined the dependence of our result on , and found that for the range [0.15,0.50], our estimate of varied as , which even for the extremes of this range is only a 2% result, and is therefore negligible compared to our other uncertainties.
We add all of these uncertainties in quadrature to get a final estimate of:
(32) 
which includes all systematic, statistical, and calibration uncertainties.
Models for the high and low extremes of this range along with the best fit model are the dotted curves plotted in Figure 7.
4.4 Potential Causes of the BMode
It is particularly unfortunate that we cannot use the data at , since one of the expected benefits from our survey was the large dynamic range over which we are able to measure shear. We make a significant lensing detection from up to , over 2 decades of powerspectrum range. In the absence of contaminating power, our uncertainty in would be approximately half of our present error bar. The benefit of having a larger scale range is that it can break the degeneracy seen in Figure 11 between and . Larger values of tip the predicted curve up at small scales, whereas smaller values tip it down. So with the full range of data and no Bmode contamination, we would start to gain some constraint on .
If we repeat the above calculation using the entire range of 1′–100′ for the statistic and the same 50′–100′ for the statistic, we find that the statistical error bars drop from about 17% to about 7%. However, the systematic errors dominate in this case, so that the final 95% CL estimate is . Thus, while the statistical precision is nominally improved, the Bmode degrades the expected accuracy, and therefore we believe the above estimate (Equation (32)) is more appropriate.
Given the obvious benefits to removing whatever is causing the Bmode contamination, we have spent considerable time trying to determine the cause and which steps have brought the B mode down to its current level. A detailed discussion is in Jarvis (2002), which we summarize here.
One step which we believe is a source of spurious power is the fit of the kernel across the image (§2.2, step 7). We found the Bmode to drop somewhat when we switched from a polynomial fit to a smoothing spline. There are unfortunately only stars per image, which means that even without noise, a fit can only probe variations on scales larger than of the chip size (). Smoothing the PSF fit to average measurement noise means that real variations in the PSF on scales of several arcminutes will be missed.^{5}^{5}5 Indeed the sparseness of PSF test points, a.k.a. stars, may be the ultimate limitation for groundbased weak lensing measurements on aperture scales of a few arcminutes or below (A. Refregier, private communication). Orbiting observatories, with PSFs that are stable over time, can accumulate a PSF map from the stars of many exposures.
Another source of spurious power could be higherorder asymmetries left in the PSF after application of our circularization kernel. We currently use a kernel which removes the lowest 3 orders of the PSF bias. Stars in our fields do have significant power in higher orders. We tried to test this hypothesis by increasing our kernel size from to (which we still use here), but the Bmode did not change significantly.
Nonlinearities in the CCD response could produce spurious shear power, because we use relatively bright stars to measure the size and shape of the PSF. If this were the case, however, the effect would likely be very different for the two cameras used in our observations. The measured E and B powers for BTC, Mosaic, and mixed fields are all consistent with each other, as illustrated in Figure 12.
While we do not know which of the above effects is causing our Bmode, shortcomings in the PSFcircularization kernel seem the most likely cause. We may be able to do better by implementing the analytic deconvolution technique described in BJ02.
To be complete, there is another possible cause of the Bmode contamination—intrinsic correlations of the galaxy ellipticities. Crittenden et al. (2001) calculate the correlation function that may exist due to spin correlations between galaxies. The predicted power, scaled to our median redshift of 0.5, is not much lower than the Bmode power seen in our data. They indicate in their conclusions that there are a number of reasons to suspect that the predictions are an overestimate of the true power from intrinsic correlations; however, it is possible that a large portion of our Bmode power is due to this effect. Note that if this is the case, then Crittenden et al. (2002) calculate that the power should be approximately evenly split between the Emode and Bmode, in which case, the EB fit above is the appropriate one for weak lensing constraints.
4.5 Consumer’s Guide to Current Weak Lensing Results
Table 4 lists some of the most recent weaklensing results, quoting the value of at and in order to make the comparisons more obvious. Note, however, that all authors actually constrain a parameter , where is usually 0.4–0.6, and ranges from 0.02 for our survey (cf. §4.3) to 0.15 for Refregier et al. (2002). Our result is clearly somewhat lower than the other recent values, but the relatively large error bars for most of these are such that we are nominally consistent with all of them (although we are only barely consistent with vW02 at 95% confidence).
^{1}^{1} Assuming CDM model, normalized to , .  Mag Limit  Max scale  Survey Area  B Mode?  

(95% CL)  (source)  (arcmin)  (sq. deg.)  
This Work  0.66  100  75  yes  spectroscopic  irrelevant  
Hamana et al. (2002)^{8}^{8} The results of Brown et al. (2002) and Hamana et al. (2002) were published as preprints while this paper was being reviewed. We have added a summary of their results here, although they are not discussed in the text of §4.5.  40  2.1  yes  marginalized^{4}^{4} Same as (3), but with a flat prior for .  marginalized^{7}^{7} Assuming a flat prior [0.1,0.4].  
Brown et al. (2002)^{8}^{8} The results of Brown et al. (2002) and Hamana et al. (2002) were published as preprints while this paper was being reviewed. We have added a summary of their results here, although they are not discussed in the text of §4.5.  0.740.18  0.85  15  1.25  yes  photometric  
Hoekstra et al. (2002)^{2}^{2} See §4.5 for his preliminary results using the Smith et al. (2002) formulation of the nonlinear power spectrum.  0.58  50  53  yes  marginalized^{3}^{3} A functional form for N(z) is taken from spectroscopic surveys. The fit is marginalized over a free parameter, of this function, with a Gaussian prior for based on photometric redshifts.  marginalized^{6}^{6} Assuming a Gaussian prior based on the 2dF galaxy redshift survey. (With a flat prior, the estimate of increases to .)  
Van Waerbeke et al. (2002)  0.960.23  0.84  30  8.5  yes  marginalized^{4}^{4} Same as (3), but with a flat prior for .  marginalized^{7}^{7} Assuming a flat prior [0.1,0.4].  
Bacon et al. (2002)  0.970.26  0.9  12  1.6  unknown  photometric^{5}^{5} The median redshift is estimated from a photometric extrapolation of spectroscopic surveys.  
Refregier et al. (2002)  0.940.28  0.95  1.4  0.04  unknown  photometric^{5}^{5} The median redshift is estimated from a photometric extrapolation of spectroscopic surveys. 
Direct comparison of various results is complicated, however, because the uncertainties included in the error bar vary, even though all the error bars are 95% confidence.^{6}^{6}6 When quoted error bars were 1sigma, we doubled the uncertainty to get the 95% confidence limits. We note here some of the important distinctions between current results which must be considered when making detailed comparisons. Table 4 serves as a “scorecard” for recent cosmicshear results.

Statistical and Cosmic Variance: Uncertainties in shear power estimates of all the papers in Table 4 include the contributions of random (shape) noise and cosmic variance. For vW02 and RRG02 these are estimated analytically. The other authors use measured fieldtofield covariance matrices, which will automatically include shape noise and cosmic variance.

Estimates of Spurious Power: “Cosmic shear” measurements are difficult, since spurious sources of shear variance are easily mistaken for lensing signals. The E/B decomposition is one test for the presence of spurious power. HYG02 and vW02 are the only other authors to look for (and find) Bmode in their data; the RRG02 and BMRE02 surveys have field sizes too small for a useful E/B decomposition. Figure 13 compares the Bmode power detected in the three relevant efforts, which is significant in all cases, so must be included in the error analyses.
Both vW02 and HYG02 add the Bmode signal to the error bar of the Emode signal (in quadrature). This implicitly assumes that the effect of the systematic error is independent (or weakly correlated) at different angular scales. It seems more likely to us that the direction of the error is highly correlated at all scales, in which case the HYG02/vW02 procedure could underestimate the systematic uncertainty—hence our approach of subtracting (or adding) the B mode from the signal of the Emode at all angular scales in order to bound systematic errors. The analysis technique of vW02 and HYG02 could underestimate the systematic uncertainty. See Jarvis (2002) for further discussion of the Bmode signals present in our, HYG02, and vW02 data.

Source Redshift Distribution: Inaccuracies in the assumed for source galaxies will cause scale errors in the derived . In our shallow survey, and its uncertainties can be taken directly from nearlycomplete spectroscopic galaxy surveys. The other papers in Table 4 assume a parametric form for , with parameters fit to the measured median photometric redshifts in the Hubble Deep Field (and other data). HYG02 and vW02 then marginalize the resultant over an estimated prior distribution for the parameter. RRG02 and BMRE02 fix the median redshift based on their limiting magnitudes, but propagate the resultant uncertainty into their uncertainty in .

Power Spectrum Shape: Cosmic shear studies currently provide useful constraint only on the overall normalization of the mass power spectrum, not its shape. If the shear measurement is at scales far from the Mpc where the spectrum normalization is defined, then will depend upon the assumed shape of the power spectrum. The (linear) CDM power spectrum is specified by the primordial index and the parameter . All papers to date assume ; vW02 and HYG02 marginalize over a prior for . For our results, the measurement is on larger physical scales, near the window for itself; this means that our result is very weakly dependent upon and .
Note that in the current results listed in Table 4, the trend is that surveys using smaller scale tend to have larger measured values for . If taken at face value, this would imply that is larger than the usually assumed value of 0.21. However, this is at best suggestive, and shouldn’t be taken too seriously yet.

NonLinear Mass Evolution: Constraint of cosmological parameters with weak lensing data requires a model for nonlinear evolution of the power spectrum. Various authors have put forth heuristic methods for estimating nonlinear evolution, but these have not been verified (at the level) for the range of cosmologies considered here, and the baryonic contribution to the smallscale power spectrum is also significant when accuracies of a few percent are desired. Peacock & Doods (1996) claim that their prescription for nonlinear predictions has an accuracy of 15%, which is roughly half the discrepency between the studies using small angular scales and our results.
Further, Hoekstra has redone the analysis for the RCS survey using the new Smith et al. (2002) formulation for the nonlinear evolution. He obtained a value of 0.80 for which is about 8% lower than the published value which uses the formulation of Peacock & Doods (1996) (preliminary results; Hoekstra, private communication), indicating that the other results relying on the nonlinear regime may also be biased high. This potential source of systematic error is reduced for largerscale surveys such as this work.
There seems to be a trend in Table 4 wherein the lower values are obtained by the shallower, largerscale surveys (ours and HYG02). This may indicate that misestimation of and nonlinear evolution are causing biases on for the deeper, smallerscale surveys. We also note that it is easier for uncorrected PSF variations to falsely inflate than to decrease it. We would not, however, choose to overinterpret these apparent discrepancies until such time as larger datasets that are free of Bmode become available.
A close comparison between our result and HYG02 is illustrative given the similarities of the datasets. The weight of the HYG02 data peak near mag, only mag deeper than our data, so we expect the signals to agree within , but their higher result implies that they have measured a more than 1.5 times ours. At large scales, , the HYG02 signal is in fact lower than, but consistent with ours, whereas at the intermediate scales that we ignore, the HYG02 signal is higher. Hence the differential between the results is partly due to different scales that we have fit. As noted above, a change in the nonlinear growth model reduces the difference between the HYG02 and ours.^{7}^{7}7 We also note that the assumed by HYG02 for their sample has a lower mean than we have assumed, despite their deeper sample. This discrepancy in models accounts for roughly half of the difference in resultant .
4.6 Comparison to XRay Cluster Measurements
^{1}^{1}Assuming CDM model, normalized to , .  

(95% CL)  
Pierpaoli et al. (2002)  0.770.10 
Bahcall et al. (2002)  0.720.14 
Viana et al. (2002)  0.610.09 
Seljak (2002)  0.750.12 
Reiprich & Böringer (2002)  0.680.11 
Borgani et al. (2001)  0.720.12 
Proty Wu (2001)  0.870.13 
Silberman et al. (2001)  1.010.18 
Pierpaoli et al. (2001)  1.010.14 
It is also interesting to compare these weak lensing results with the recent results from Xray cluster measurements, summarized in Table 5. The precision of the two types of measurement are comparable, but the cluster method suffers from very different systematics than do weak lensing measurements. In particular, there is significant uncertainty in the MassTemperature relation which relates to the parameter of the assumed density profiles. This systematic error is seen to dominate the uncertainty in the measurements, since the scatter of the measurements is significantly larger than the quoted 95% confidence intervals. (This systematic effect is discussed in more detail by Pierpaoli et al. (2002).)
Huterer & White (2002) characterize the MassTemperature relation as:
(33) 
where is the mean overdensity inside the virial radius in units of the critical density, . In their investigation of how cluster mass measurements constrain cosmology, they conclude that the parameter, is directly related to the combination . Since this is essentially the same as what we measure, our measurement implies (according to their Figure 1) a value of keV, which is at the high end of their “favored range”.
5 Conclusions
We have presented the results of a 75 square degree survey of galaxy shapes involving 12 wellseparated fields and totaling approximately 2 million galaxies. We have applied the analysis techniques of BJ02 to our data, and show many tests for residual systematic effects. Most of these tests show no residual effect, and for the one exception (cf. Figure 5), the bias is small compared to our lensing signal and we correct our shapes for this bias as well.
We calculate two lensing statistics for our data, the shear variance and the aperture mass, the results of which are plotted in Figure 7. The presence of a socalled Bmode in the aperture mass statistic indicates that we probably have some residual systematic effect which is causing spurious correlations in our galaxy ellipticities. While we do not believe that it is due to intrinsic correlations between the galaxies, we cannot rule out this possibility.
In any case, the presence of Bmode in the smallerscale portion of our data precludes us from using this part of the range for constraining cosmology. Using the data for angular scales , where the Bmode is small, we are able to jointly constrain the parameters and . They are found to be largely degenerate to the extent that we effectively only constrain the combination:
(34) 
where the error bars are 95% confidence and include the statistical, calibration and systematic uncertainties. There is no dependence upon the Hubble Parameter , and dependence upon the power spectrum parameters and are insignificantly small over the range of reasonable values.
Our value for is lower than all other cosmic shear results to date, but formally consistent with all but one at the 2 level. While it is possible that the discrepency may be due to systematic errors from the Bmode contamination discussed above, it may also be related to the treatment of source redshifts and/or nonlinear mass evolution. The other weak lensing result with which we most closely agree is HYG02. This study is the most similar to ours in both source redshifts and angular scale.
The fairly large scales to which we have confined our analysis are less sensitive to any systematic errors which may exist in the nonlinear predictions. The relatively bright source galaxies in our survey also allow us to minimize the potential systematic error due to the redshift distribution. We are able to use existing spectroscopic redshift surveys to calibrate our redshift distribution, as described in §4.1.
Also, our results are consistent with the latest several results using cluster abundances, but are inconsistent with the “older” paradigm. There are clearly some systematic effects in this field which must be worked out, but once these are worked out, our value of could have significant consequences for the physics of clusters. For example, as we pointed out above, our results may imply a relatively large value for (as defined by Huterer & White (2002)).
Finally, as also pointed out by vW02, our likelihood contours are roughly orthogonal to those of Lahav (2002) from CMB+2dF constraints. (See also Melchiorri & Silk (2002) for a similar investigation.) The intersection of our contours with theirs (cf. their Figure 2) falls roughly at , . This result, along with the CMB result that , thus supports the currently popular model with . In particular, our results (and in fact almost all of the weak lensing results to date) are inconsistent with an flat model.
In a relatively short time, weak gravitational lensing measurements have advanced to “precision cosmology” status, constraining at least one parameter combination to the 10% level. Unfortunately the accuracy of the method is currently limited by residual PSF contamination, as indicated by Bmode power. Improvements in image quality and analysis techniques should lead to further rapid improvements in the power of weak lensing cosmological constraints.
References
 Bacon et al. (2000) Bacon, D., Refregier, A., Clowe, D., & Ellis, R., 2000, MNRAS, 318, 625
 Bacon et al. (2002) Bacon, D., Massey, R., Refregier, A., & Ellis, R., 2002, MNRAS (in press, astroph/0203134)
 Bahcall et al. (2002) Bahcall, N., et al. 2002, ApJ (in press, astroph/0205490)
 Bernardeau et al. (1997) Bernardeau, F., van Waerbeke, L., & Mellier, Y., 1997, A&A, 322, 1
 Bernstein & Jarvis (2002) Bernstein, G. & Jarvis, M., 2002, AJ, 123, 583
 Bertin & Arnouts (1996) Bertin, E. & Arnouts, S., 1996, A&AS, 117, 393
 Borgani et al. (2001) Borgani, S., Rosati, P., Tozzi, P., Stanford, S., Eisenhardt, P., Lidman, C., Holden, B., Della Ceca, R., Norman, C., & Squires, G., 2001, ApJ, 561, 13
 Brown et al. (2002) Brown, M., Taylor, A., Bacon, D., Gray, M., Dye, S., Meisenheimer, K., & Wolf, C., 2002, MNRAS (submitted, astroph/0210213)
 Cohen et al. (2000) Cohen, J., Hogg, D., Blandford, R., Cowie, L., Hu, E., Songaila, A., Shopbell, P., & Richberg, K., 2000, ApJ, 538 29
 Cooray (2002) Cooray, A., 2002, Physical Review D, 65, 063512
 Crittenden et al. (2001) Crittenden, R., Natarajan, P, Pen, U., & Theuns, T., 2002, ApJ, 559, 552
 Crittenden et al. (2002) Crittenden, R., Natarajan, P, Pen, U., & Theuns, T., 2002, ApJ, 568, 20
 Gu & Wahba (1991) Gu, C. & Wahba, G., 1991, SIAM J. Sci. Statist. Comput., 12, 383
 Hamana et al. (2002) Hamana, T., Miyazaki, S., Shimasaku, K., Furusawa, H., Doi, M., Hamabe, M., Imi, K., Kimura, M., Komiyama, Y., Nakata, F., Okada, N., Okamura, S., Ouchi, M., Sekiguchi, M., Yagi, M., & Yasuda, N., 2002, ApJ (submitted, astroph/0210450)
 Hamilton et al. (1991) Hamilton, A. J. S., Kumar, P., Lu, E., & Matthews, A., 1991, ApJ, 374, L1
 Hoekstra et al. (2002) Hoekstra, H., Yee, H. K. C., & Gladders, M. D., 2002, ApJ (in press, astroph/0204295) [HYG02]
 Huterer & White (2002) Huterer, D. & White, S., 2002, ApJ (in press, astroph/0206292)
 Jain, Mo & White (1995) Jain, B., Mo, H. J., & White, S. D. M., 1995, MNRAS, 276, L25
 Jain & Seljak (1997) Jain, B. & Seljak, U., 1997, ApJ, 484, 560
 Jarvis (2003) Jarvis, M., 2003, (in preparation).
 Jarvis (2002) Jarvis, M., 2002, PhD dissertation, University of Michigan.
 Kaiser (1992) Kaiser, N., 1992, ApJ, 388, 272
 Kaiser (2000) Kaiser, N., 2000, ApJ, 537, 555
 Kaiser et al. (2000) Kaiser, N., Wilson, G., & Luppino, G., 2000, astroph/0003338
 Kaiser, Squires, & Broadhurst (1995) Kaiser, N., Squires, G., & Broadhurst, T., 1995, ApJ, 449, 460
 Lahav (2002) Lahav, O., Bridle, S., Percival, W., Pecock, J., Efstathiou, G., et al., 2002, MNRAS, 333, 961
 Landolt (1992) Landolt, A., 1992, AJ, 104, 340
 Lilly et al. (1995) Lilly, S., Le Fèvre, O., Crampton, D., Hammer, F., & Tresse, L., 1995, ApJ, 455 50
 Maoli et al. (2001) Maoli, R., van Waerbeke, L., Mellier, Y., Schneider, P., Jain, B., Bernardeau, F., Erben, T., & Fort, B., 2001, A&A, 368, 766
 Melchiorri & Silk (2002) Melchiorri, A. & Silk, J., 2002, PhysRevD, 66, 041301
 MiraldaEscudé (1991) MiraldaEscudé, J., 1991, ApJ, 380, 1
 Monet (1998) Monet, D., 1998, AAS, 19312003
 Muller et al. (1998) Muller, G., Reed, R., Armandroff, T., Boroson, T., & Jacoby, G., 1998, Proc. SPIE 3355 577
 Peacock & Dodds (1994) Peacock, J. & Dodds, S., 1994, MNRAS, 267, 1020
 Peacock & Doods (1996) Peacock, J. A. & Dodds, S. J., 1996, MNRAS, 280, L19
 Pen et al. (2002) Pen, U., van Waerbeke, L., & Mellier, Y., 2002, ApJ, 567, 31
 Pierpaoli et al. (2001) Pierpaoli, E., Scott, D., & White, M., 2001, MNRAS, 325, 77
 Pierpaoli et al. (2002) Pierpaoli, E., Borgani, S., Scott, D., & White, M., 2002, MNRAS (submitted, astroph/0210567)
 Proty Wu (2001) Proty Wu, J., 2001, MNRAS, 327, 629
 Refregier et al. (2002) Refregier, A., Rhodes, J., & Groth, E. J., 2002, ApJ, 572, L131
 Reiprich & Böringer (2002) Reiprich, T. & Böringer, H., 2002, ApJ, 567, 716
 Rhodes, Refregier, & Groth (2000) Rhodes, J., Refregier, A., & Groth, E. J., 2000, ApJ, 536, 79
 Schlegel et al. (1998) Schlegel, D., Finkbeiner, D., & Davis, M., 1998, ApJ, 500, 525
 Schneider et al. (1998) Schneider, P., van Waerbeke, L., Jain, B., & Kruse, G., 1998, MNRAS, 296, 873
 Schneider, van Waerbeke, & Mellier (2002a) Schneider, P., van Waerbeke, L., & Mellier, Y., 2002, A&A (in press, astroph/0112441)
 Schneider et al. (2002) Schneider, P., van Waerbeke, L., Kilbinger, M., & Mellier, T., 2002, A&A (in press, astroph/0206182)
 Seljak (2002) Seljak, U., 2002, MNRAS (in press, astroph/0111362)
 Silberman et al. (2001) Silberman, L., Dekel, A., Eldar, A., & Zehavi, I., 2001, ApJ, 557, 1025
 Smith et al. (2001) Smith, D. R., Bernstein, G. M., Fischer, P., & Jarvis, R. M., 2001, ApJ, 551, 643
 Smith et al. (2002) Smith, R. E., Peacock, J. E., Jenkins, A., White, S. D. M., Frenk, C. S., Pearce, F. R., Thomas, P. A., Efstathiou, G., & Couchman, H. M. P., 2002, MNRAS (submitted, astroph/0207664)
 Steidel & Hamilton (1992) Steidel, C. & Hamilton, D., 1992, AJ, 104, 941
 Takada & Jain (2002) Takada, M. & Jain, B., 2002, MNRAS (in press, astroph/0205055)
 Valdes, Tyson, & Jarvis (1983) Valdes, F., Tyson, J. A., & Jarvis, J. F., 1983, ApJ, 271, 431
 Van Waerbeke et al. (2000) Van Waerbeke, L., Mellier, Y., Erben, T., Cuillandre, J., Bernardeau, F., Maoli, R., Bertin, E., McCracken, H., Le Fèvre, O., Fort, B., DantelFort, M., Jain, B., & Schneider, P., 2000, A&A, 358, 30
 Van Waerbeke et al. (2002) Van Waerbeke, L., Mellier, Y., Pelló, R., Pen, U.L., McKracken, H. J., & Jain, B., 2002, A&A, 358, 30 [vW02]
 Viana et al. (2002) Viana, P., Nichol, R., & Liddle, A., 2002, ApJ, 569, L75
 Wittman et al. (1998) Wittman, D., Tyson, J. A., Bernstein, G., Lee, R., Dell’Antonio, I., Fischer, P., Smith, D., & Blouke, M., 1998, Proc. SPIE, 3355, 626
 Wittman et al. (2000) Wittman, D., Tyson, J. A., Kirkman, D., Dell’Antonio, I., & Bernstein, G., 2000, Nature, 405, 143
 Wittman et al. (2001) Wittman, D., Tyson, J. A., Margoniner, V., Cohen, J., & Dell’Antonio, I., 2001, ApJ, 557, L89