A&A 396, 1-19 (2002)
P. Schneider 1,2 - L. van Waerbeke3,4 - M. Kilbinger1 - Y. Mellier3,5
1 - Institut f. Astrophysik u. Extr. Forschung, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
2 - Max-Planck-Institut f. Astrophysik, Postfach 1317, 85741 Garching, Germany
3 - Institute d'Astrophysique de Paris, 98bis boulevard Arago, 75014 Paris, France
4 - Canadian Institute for Theoretical Astrophysics, 60 St Georges Str., Toronto, M5S 3H8 Ontario, Canada
5 - Observatoire de Paris, DEMIRM/LERMA, 61 avenue de l'Observatoire, 75014 Paris, France
Received 13 June 2002 / Accepted 12 September 2002
Recently, cosmic shear, the weak lensing effect by the inhomogeneous matter distribution in the Universe, has not only been detected by several groups, but the observational results have been used to derive constraints on cosmological parameters. For this purpose, several cosmic shear statistics have been employed. As shown recently, all second-order statistical measures can be expressed in terms of the two-point correlation functions of the shear, which thus represents the basic quantity; also, from a practical point-of-view, the two-point correlation functions are easiest to obtain from observational data which typically have complicated geometry. We derive in this paper expressions for the covariance matrix of the cosmic shear two-point correlation functions which are readily applied to any survey geometry. Furthermore, we consider the more special case of a simple survey geometry which allows us to obtain approximations for the covariance matrix in terms of integrals which are readily evaluated numerically. These results are then used to study the covariance of the aperture mass dispersion which has been employed earlier in quantitative cosmic shear analyses. We show that the aperture mass dispersion, measured at two different angular scales, quickly decorrelates with the ratio of the scales. Inverting the relation between the shear two-point correlation functions and the power spectrum of the underlying projected matter distribution, we construct estimators for the power spectrum and for the band powers, and show that they yields accurate approximations; in particular, the correlation between band powers at different wave numbers is quite weak. The covariance matrix of the shear correlation function is then used to investigate the expected accuracy of cosmological parameter estimates from cosmic shear surveys. Depending on the use of prior information, e.g. from CMB measurements, cosmic shear can yield very accurate determinations of several cosmological parameters, in particular the normalization of the power spectrum of the matter distribution, the matter density parameter , and the shape parameter .
Key words: dark matter - gravitational lensing - large-scale structure of the Universe
Most analytical work on cosmic shear has been done on second-order statistical measures of the distortion field, such as the shear correlation functions, the shear variance in an apertures, or the aperture mass (see Sect. 2 for a definition of these quantitities). Although higher-order statistical measures, such as the skewness of the shear (Bernardeau et al. 1997), are likely to yield additional, if not even superior constraints on cosmological parameters, their theoretical predictions are more uncertain at present. Recently, an estimator for the skewness of the shear was developed (Bernardeau et al. 2002a), and applied to wide-field survey data (Bernardeau et al. 2002b), yielding a significant detection.
In this paper we consider second-order statistical measures only. All of them can be derived in terms of the correlation functions, as shown in, e.g., Crittenden et al. (2002, hereafter C02) and Schneider et al. (2002, hereafter SvWM), and since the measurement of the correlation functions is easier in practice than the other second-order statistics (e.g., gaps in the data are easily dealt with), we consider the correlation functions as the fundamental observables from a cosmic shear survey. In order to use them for determining cosmological parameters, it is important to know a practical and unbiased estimator for them, and to determine the covariance of this estimator. Two effects enter this covariance: a random part, which is due to the intrinsic ellipticity of the galaxies from which the shear is measured, together with measurement noise, and sampling (or cosmic) variance. The first of these effects is expected to dominate on small angular scales, whereas the second determines the covariance for large separations. The covariance will depend not only on the total survey area, but also on the survey geometry. As has been pointed out by Kaiser (1998), in order to decrease the sampling variance on large scales, it may be favourable to choose a survey geometry that samples those scales sparsely. In order to design an optimized survey geometry, the covariance as a function of survey geometry needs to be calculated.
Here, we calculate the covariance matrices for the shear correlation functions binned in angular separation. In Sect. 2, we introduce our notation and briefly summarize the second-order cosmic shear measures and their interrelations. Unbiased estimators of the two basic correlation functions are derived in Sect. 3, togther with the corresponding unbiased estimators of the aperture mass and the shear dispersion. The covariance matrices of these correlation function estimators are then derived in Sect. 4, expressed in terms of a set of galaxy positions. From these expressions, the covariances can be determined for an arbitrary survey geometry. In a forthcoming paper (Kilbinger et al., in preparation), we shall use the results of Sect. 4 to design an optimized geometry for a planned cosmic shear survey.
For the case of a filled survey geometry, the ensemble average of these covariances can be further analyzed; using a few approximations, we express in Sect. 5 the covariances for this case in terms of integrals. The corresponding expressions have been evaluated, for a particular cosmological model, and are illustrated in a set of figures. In Sect. 6 we derive the covariance for the aperture mass dispersion, which can be expressed simply in terms of the covariances of the correlation functions. The variance of the aperture mass dispersion, as well as the covariance, is then explicitly calculated for a survey with filled geometry, showing that indeed the aperture mass at two angular scales decorrelates quickly as the scale ratio decreases away from unity.
We then turn in Sect. 7 to a simple estimator of the power spectrum of the projected cosmic density field, which can be expresed in terms of the correlation functions. Since the correlation functions will be known only over a finite range in angular separation, the simple estimator we derive is biased. We show that, provided the angular range on which the correlation functions can be measured is as large as can be expected with the next generation of cosmic shear surveys, this bias is indeed very small over a large range of wave numbers. We derive the covariance of the power spectrum estimator and calculate it explicitly for the filled survey geometry case; the resulting error bars on the estimated power spectrum are quite a bit smaller than one might have expected, given the simplicity of the approach. In Sect. 8 we consider the accuracy with which the parameters of the cosmological model can be constrained, given a survey area. In fact, by fitting the correlation function directly to model predictions, even the currently available cosmic shear surveys can yield fairly accurate constraints on cosmological parameter. Finally, we summarize our results in Sect. 9.
In this paper we shall assume that the observable shear is due to the tidal gravitational field of the cosmological matter distribution only; in this case, the two shear components are not mutually independent. This is due to the fact that the gravitational field is a gradient field. In the language of some recent papers (e.g., C02; Pen et al. 2002; SvWM), we thus assume that the shear field is a pure E-mode field. B-modes (or the "curl component'') can in principle be generated if the intrinsic orientation of the galaxies from which the shear is measured are correlated, e.g. due to tidal interactions of dark matter halos in which these galaxies are formed (Croft & Metzler 2000; Pen et al. 2000; Heavens et al. 2001; Catelan et al. 2001; Mackey et al. 2002; Brown et al. 2002). Also, the clustering of source galaxies in redshift space generates a B-mode contribution which, however, turns out to be fairly small (SvWM). This restriction to E-modes only affects the interrelations between various two-point statistics. Inclusion of B-modes would not change the results of Sects. 3 through 5, and much of Sects. 6 and 7 will also be left unaffected in the presence of a B-mode contribution; we shall indicate this in due course.
The shear correlation functions are defined by considering pairs of
and defining the
tangential and cross-component of the shear
for this pair as
All these second-order statistics are thus linearly filtered versions of the power spectrum , where the filter functions are quite different between the various statistics. For the correlation function , the filter function is very broad, about constant for , and oscillating for large , with an amplitude decreasing as . The filter function for has the same slow decrease, but behaves as for small , and is therefore more localized than the one for . The filter function for the shear dispersion, , is a low-pass filter, i.e. constant for , and then decreasing in amplitude as for large . Finally, the filter function for the aperture mass dispersion is and thus behaves like for small , and decreases oscillatory as for . Hence, yields the most local estimate of the underlying power spectrum of the projected mass. On the other hand, because it is so localized, it contains less power in its filter, so that the value of is smaller than that of on the same filter scale .
The various second-order statistics of the shear are related to each other; in particular, they can all be expressed in terms of the two-point correlation functions, as was shown in C02, Pen et al. (2002) and SvWM. We briefly summarize the results here.
Making use of the orthogonality of Bessel functions,
the power spectrum can be expressed in terms of the correlation
by multiplying Eq. (2) by
respectively, and then integrating over
The two Eqs. (2) and (5) allow us to express
in terms of ,
and reversely (see SvWM for a derivation),
Next we express the shear dispersion (3) in terms of the
correlation function, by inserting (5) into
(3); this yields (van Waerbeke 2000; SvWM)
Similarly, one can express the aperture mass dispersion in terms of
the correlation functions, by inserting (5) into
The fact that we can express the shear dispersion and the aperture mass dispersion directly in terms of the correlation function over a finite interval is expected, given that the estimators of both statistics include products of ellipticities of pairs of objects separated by no more than the diameter of the aperture. However, what could not have been guessed a priori is that can be expressed by a finite integral over and separately. The determination of these statistics in terms of the correlation function is in practice easier than laying down apertures, owing to the holes and gaps in a data set; in addition, a comparison of the directly determined shear and aperture mass dispersion with those obtained from the correlation functions yields a useful check on the integrity of the data.
Next we calculate the covariance of the various estimators, starting
with the correlation functions. Hence, we define
To calculate the covariance matrix for the
function, we first write
Next we shall evaluate the second term of (23) which is of
the form (28), with
inserting these expressions into
(29), the expectation value of the second term in
Several issues are worth mentioning: (1) only the first term containing the "delta function'' depends on the bin width ; thus, the bin width only affects the autovariance. (2) All terms are proportional to A-1; hence, the relative contribution of the terms is independent of the survey area, at least for separations well below the "diameter'' of the survey area for which the foregoing procedure of the ensemble averaging is valid. (3) The terms denoted by "r'' are independent of the intrinsic ellipticity dispersion and of the number density of galaxies. Hence, these terms describe the cosmic variance and thus provide a limit on the accuracy of the determination of the correlation function for a given survey geometry, independent of the observing conditions which determine n.
The expectation values of the other covariance matrices are calculated
in a similar manner. Consider the "-'' covariance (24)
next; the first term agrees with that of (23). For the
second term, we can apply (29), after expanding the cosine;
can be applied to the third term of (24), after expanding
the cosine; using (30) term by term, and combining the
results afterwards, one obtains
|Figure 1: The correlation functions. In the left panel, we have plotted the covariance matrices (thick solid curves) and , i.e. the covariance matrices with the shot-noise term removed. For , the contours are linearly spaced, with the lowest value at 10-9 (outer-most contour) and highest value for small , . For , contours are logarithmically spaced, with consecutive contours differing by a factor 1.5. The solid contours display positive values of , starting from 10-14, with the maximum value of in the upper right corner, and dotted contours show negative values of , starting at -10-15. In the right panel, is shown, again with logarithmically spaced contours differing by a factor of 1.5. Solid contours are for positive values of , starting at 10-14, negative values are shown by dotted contours, starting at -10-13.|
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We have obtained numerical estimates for the ensemble-averaged covariance matrices derived above (see Fig. 1). In the numerical estimates given in this paper (except for Sect. 8.2), we have used a standard set of parameters. The cosmological parameters are those of a by-now standard -dominated universe, , . The power spectrum of the density fluctuations is described by its primordial slope of n=1, a shape parameter , and a normalization of . We used the fit formula of Bardeen et al. (1986) for the linear power spectrum, and the prescription of Hamilton et al. (1991) in the form given in Peacock & Dodds (1996) to describe the non-linear evolution of the power spectrum. Furthermore, we fix the survey properties to be described by a fiducial area of , a number density of source galaxies, and an intrinsic ellipticity dispersion of . The source galaxies were assumed to have a redshift distribution , so that the mean redshift is . For the examples shown in Sects. 5 through 7, we take z0=1. Note that all covariances simply scale with A-1, so that the results displayed here are easily translated to other survey sizes. This scaling is also implicitly implied when considering scales of order a degree or more - all the numerical estimates are for the ensemble averaged covariances, and their validity as given here depends on the assumption .
|Figure 2: Left panel: the square root of the variances and shown as dotted and long-dashed curves, together with the correlation functions and as solid and short-dashed curves, respectively. The model parameters are as described in the text; in particular, a fiducial value of the survey area of has been taken. For the diagonal part of the covariance matrix, we have assumed a relative bin size of . For small , the variance behaves as , as it is dominated by the noise from the intrinsic ellipticity of the source galaxies, i.e. the term D (27), whereas for larger values of , the main contribution comes from cosmic variance. Right panel: the correlation coefficient , as defined in (39), as a function of , for various values of . Solid curves show , dashed curves show . The value of corresponding to each curve can be read off from the point where a curve attains the value .|
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Figure 2 displays in the left panel the square root of the (auto)variance of and , with . To calculate the value of D (27) which enters the diagonal part of the covariance matrix, we have assumed a bin width of . The square root of the variance - or noise per bin - for is smaller than for , and larger for larger angles (for the assumed value of ), whereas the noise for is smaller than in an interval of . The determination of on small angular scales is much more difficult than for , owing to the smallness of for small . Note that the noise scales like A-1/2, so that from a survey of , like the DESCART survey (see van Waerbeke et al. 2001) one should be able to obtain reliable measurements of for , and of for in bins of relative width of 0.1. Of course, the covariance of the shear will not only depend on the survey area, but also on its geometry; one might therefore design survey geometries which yield the desired noise behaviour as a function of angular scale (see, e.g., Kaiser 1998).
In order to show how strongly the correlation estimators at two
angular scales are correlated, we define the correlation coefficient
The first term in (42) yields the covariance in the absence
of cosmic shear correlations, i.e. the covariance of the estimator
due to the intrinsic ellipticity of the source galaxies. This
term can be written as
|Figure 3: Left panel: the square-root of the autovariance of as a function of angular scale. The long-dashed and dash-dotted curves show the minimum variance, i.e. in the absence of a shear correlation; this variance is due solely to the intrinsic ellipticity of source galaxies. Curves are shown for K+=0, 1/2, 1, where the minimum variance is the same for K+=0 and 1. The solid, dashed and dotted curves show the variance in the presence of a cosmic shear; also here, the cases K+=0 and K-=1 are nearly the same, and the variance is smallest for K+=1/2. For comparison, is plotted as thick solid curve. As for the other figures shown before, our standard set of parameters , and has been used; the variance scales as A-1. Right panel: the correlation coefficient of the covariance of the estimator is plotted as a function of , for various values of ; the values of can be localized as those points where the correlation function attains the value unity. The solid curves are for K+=0, i.e. when only the correlation function is used in the estimate of , the dotted curves are for K+=1/2, and the dashed curves for K+=1. The width of all three families of curves is very similar and (in logarithmic terms) basically independent of . The K+=0 curves do not develop a tail of anticorrelation, as is the case for K+=1 (and therefore also for K+=1/2). Hence, whereas K+=1/2 yields the smallest variance of the estimator , it leads to a small but long-range correlation between different angular scales.|
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We have plotted in Fig. 3 the square root of the variance of , both for the case of no correlations, and for our standard model for the cosmic shear. For small angular scales, the variance is completely dominated by the intrinsic source ellipticities, whereas the cosmic variance is the important noise error for larger angular scales. For the parameters used here, the transition between these two regimes occurs at a few arcminutes. It must be noted that the shape of the variance curves are independent of the survey area A. The results shown in Fig. 3 can be seen as an a posteriori justification of using the Gaussian assumption for the calculation of the shear four-point function in Sect. 4. In Fig. 4 of van Waerbeke et al. (2002), the influence of the non-linear density evolution on the kurtosis of was studied, using ray-tracing simulations through numerically generated cosmological matter distributions. The non-Gaussian effects start to become non-negligible for angular scales below . As can be seen in Fig. 3, this is about the scale where the transition occurs between the variance being dominated by the intrinsic ellipticity distribution and the cosmic variance. Hence, we can expect that a more advanced treatment of the shear four-point function would yield a slightly larger variance in this transition region around : for significantly smaller scales, it is dominated by the intrinsic ellipticity noise, and for larger scales, the shear four-point function is basically Gaussian.
The variances of for the cases K+=0 and K+=1 are basically identical, and larger by a factor than the variance of the estimator for K+=1/2. Hence, to minimize the variance of the estimator , K+=1/2 should be chosen. With this choice, the results are unchanged even in the presence of a B-mode contribution (see SvWM). As was already mentioned by C02, using cosmic shear estimators which use and with equal weight reduces the resulting noise by a factor 2-1/2. One notes that the variance for large rises, but very slowly. We can compare the behavior of the variance of with that derived in SvWJK for a more direct estimator for the aperture mass dispersion; using Eqs. (5.12) and (5.16) of that paper, one finds in the limit of large angles (and, to make the estimate comparable to the one obtained here, zero kurtosis) that , where is the estimator used in SvWJK. The functional behavior with is similar to that seen in Fig. 3, but the amplitude is lower by a factor of about 2 for K+=0, 1, and about 3 for K+=1/2; this again shows the superiority of the estimator considered here in comparison to laying down independent apertures on the data field.
To investigate the correlation of the estimator
angular scales, we define the correlation coefficient
|Figure 4: The thick solid line displays the dimensionless projected power spectrum , whereas the other two curves show the "observed'' power spectrum, as defined in (46). The dotted curve is for K+=0, i.e. only enters the determination of the observed power spectrum in this case; the dashed curve is for K+=1. In this plot is was assumed that the correlation functions are known between and .|
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We have plotted the "observed'' power spectrum as in Fig. 4, for K+=0 and K+=1, assuming that and . A comparison with the underlying power spectrum (shown as heavy solid curve) shows that traces the true power spectrum over a wide range of -values, though in an oscillatory way. If is determined solely from (i.e. K+=0), it substantially underestimates the power for (that is, approximately for ), but traces the true power spectrum out to the largest values of plotted. Conversely, the observed power determined from yields good estimates of the true power for small values of , but deviates from it strongly for , that is for values of much less than . The different behavior of the two estimates again is due to the different filter function through which correlation function and power spectrum are related. Figure 4 suggests that the best estimate for the power spectrum is obtained by choosing K+=1 for small values of , and K+=0 for large .
The covariance matrix of
reads, for K+=1,
To estimate the power spectrum from cosmic shear data, it is useful to
define the power in a band with upper and lower -values
|Figure 5: The large panel shows the estimates of the band power, shown as horizontal bars whose length indicates the bins used. The error bar on each bin shows the square root of the autovariance of the band power, and the solid curve is the underlying power spectrum, . For this figure, we have assumed that the correlation functions are measured for , from a survey of . The inset figure shows the correlation coefficient between the 13 different bins, where the triangles indicate the center of each bin. One sees that the bands are very little correlated, except for the three bins with smallest ; in fact, the first three band power estimates are fully correlated. This explains why the band-power estimator yields reasonable results even for - this is just a coincidence.|
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In Fig. 5 we have plotted the band power for our reference model parameters, in 13 bins of width , between and . The band power is shown as crosses, and vertical error bars show the range of the bins. For comparison, the solid curve shows ; as expected, with this new choice of , the band power traces the underlying power spectrum over a very wide range of wavenumbers. Only in the bins with the smallest and largest value of is there a significant bias; over the range , the band power estimator is practically unbiased. Next we calculated the error bars on the band power, by taking the square root of the diagonal part of (51). For this calculation, we have assumed to have a total area of , for which the condition for the validity of the treatment of the ensemble average in Sect. 5 is approximately satisfied. The square root of this autovariance is plotted as errorbars on the band power in Fig. 5; as can be seen from this figure, the signal-to-noise ratio is larger than unity in all bins shown, and in fact very large for intermediate values of . Hence, the power spectrum can be measured over a broad range of for the parameters chosen here.
Of course, in order to interpret the error bars correctly, it is important to see the degree of correlated noise between different bands. The correlation matrix for the bins (defined in full analogy to (44)) was calculated and its values are plotted in the inset of Fig. 5. One sees that errors of the bins for intermediate and high values of are essentially uncorrelated (the correlation coefficient for neighboring bins is 10% for ); however, for the bins become strongly correlated. In fact, the agreement of the band powers with the underlying power spectrum is forticious for : the three band powers at lowest are nearly fully correlated, so that these three points contain practically the same information of the underlying power spectrum.
The method presented here for the determination of the power spectrum has the virtue of its simplicity. Other methods for determining the power spectrum from shear data have been investigated, e.g. by Kaiser (1998), Seljak (1998) and Hu & White (2001). Our approach has the property that it makes use only of the shear correlation functions, not on the spatial distribution of the shear. Since the shear correlation function contains all two-point statistical information of the shear field, no information loss occurs. Comparing the results of Fig. 5 with those of Hu & White (2001) it seems that both methods yield very similar error bars of the power spectrum, and that in the respective -range of applicability, the decorrelation between neighboring bins is equally quick. Since our method does not require the "pixelization'' of shear data, it can estimate the power spectrum to larger values of .
|Figure 6: 1-, 2- and 3- confidence contours in the --plane, where is a scaled version of the power spectrum normalization parameter , as indicated. Dotted, dashed and solid contours correspond to , and . In the left panel, the shape parameter of the power spectrum , whereas in the right panel, . The reference model is the one used before, i.e. , , . The contours are obtained by assuming a survey area of , and that the correlation functions were measured in the range .|
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In Fig. 6 we have plotted contours of constant in the - plane, corresponding to 1-, 2- and 3- confidence regions. In the left panel, we kept the shape parameter of the power spectrum fixed, , whereas in the right panel we used , adequate for a (dimensionless) Hubble constant of h=0.7. Shown are the confidence regions for all three functions , as indicated. The first point to note is that, for a given value of , is very well constrained, to within a few percent. This implies that the normalization of the power spectrum is very well determined from cosmic shear observations. Secondly, the 1- uncertainty on is about 0.1 for an assumed survey size of ; indeed, the left panel of Fig. 6 can be compared directly with similar figures in van Waerbeke et al., and the constraints on are quite similar. Third, if the shape parameter changes with , the confidence regions are narrower than when setting as constant, which implies that the shear correlation functions are sensitive measures for .
|Figure 7: 1-, 2- and 3- confidence contours for the maximum likelihood analysis on the four parameters , , and the source redshift parameter (see text). The six possible pairs of parameters are displayed. On each figure, the two hidden parameters are marginalized such that , , and , and the cosmological constant is fixed to . The reference model is , , and . The survey area is , the galaxy ellipticity rms is 0.3, and the correlation functions are measured in the range .|
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|Figure 8: Same as Fig. 7 with strong priors: in each figure, the two hidden parameters as assumed to be known perfectly. These plots show the degeneracy directions among all the possible pairs of parameters obtained from , , and . Note that the wiggles at the edge of the contours are not real features of the probability constraints. Their are inherent to the sampling limitation of the 4-dimensional cube of models given the memory limit of our machines. Also note that the upper left panel is the analogue of Fig. 6, but without the scaling employed there.|
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Next, considering a survey geometry of a single compact region of solid angle A, we have calculated the ensemble average of the covariances, using approximations which a valid for separations . The ensemble average of the covariances can then be reduced to integrals which are readily evaluated numerically. The estimate for the correlation function decorrelates quickly, i.e. estimates of for two angular scales which differ by more than a factor 2 are essentially decorrelated. On the other hand, the estimates of are correlated over much larger angular scales. The cross-correlation between and is significant for , which is due to the properties of the different filters with which these correlation functions are related to the power spectrum .
Using these ensemble-averaged covariances for the correlation functions, we have obtained the covariances for other second-order measures of the cosmic shear, primarily the aperture mass dispersion and the power spectrum. Of particular interest is the reconstruction of the power spectrum from the correlation functions; we have constructed a simple estimator for and the band powers of it in terms of the 's and found that the band power can be obtained with surprisingly large accuracy from even a moderately-sized cosmic shear survey. Finally, we have investigated the confidence regions for the most relevant cosmological parameters ( , , and ) with a maximum likelihood approach. We studied our ability to constrain simultaneously these parameters from a measurement of the shear correlation function, as well as the effect of some level of lack of knowledge using the marginalization technique.
In a future paper, we shall investigate strategies for conducting cosmic shear surveys by optimizing the survey geometry.
We thank the referee Dipak Munshi for his constructive comments on the manuscript. This work was supported by the TMR Network "Gravitational Lensing: New Constraints on Cosmology and the Distribution of Dark Matter'' of the EC under contract No. ERBFMRX-CT97-0172, by the German Ministry for Science and Education (BMBF) through the DLR under the project 50 OR 0106, and by the Deutsche Forschungsgemeinschaft under the project SCHN 342/3-1.