CFHTLenS: Combined probe cosmological model comparison using 2D weak gravitational lensing
We present cosmological constraints from 2D weak gravitational lensing by the large-scale structure in the Canada-France Hawaii Telescope Lensing Survey (CFHTLenS) which spans 154 square degrees in five optical bands. Using accurate photometric redshifts and measured shapes for 4.2 million galaxies between redshifts of 0.2 and 1.3, we compute the 2D cosmic shear correlation function over angular scales ranging between 0.8 and 350 arcmin. Using non-linear models of the dark-matter power spectrum, we constrain cosmological parameters by exploring the parameter space with Population Monte Carlo sampling. The best constraints from lensing alone are obtained for the small-scale density-fluctuations amplitude scaled with the total matter density . For a flat CDM model we obtain .
We combine the CFHTLenS data with WMAP7, BOSS and an HST distance-ladder prior on the Hubble constant to get joint constraints. For a flat CDM model, we find and . In the case of a curved CDM universe, we obtain , , and .
We calculate the Bayesian evidence to compare flat and curved CDM and dark-energy CDM models. From the combination of all four probes, we find models with curvature to be at moderately disfavoured with respect to the flat case. A simple dark-energy model is indistinguishable from CDM. Our results therefore do not necessitate any deviations from the standard cosmological model.
keywords:cosmological parameters – methods: statistical
Weak gravitational lensing is considered to be one of the most powerful tools of cosmology. Its ability to measure both the geometry of the Universe and the growth of structure offers great potential to obtain constraints on dark energy and modified gravity. Moreover, to first order, weak lensing does not rely on the relation between galaxies and dark matter (bias), and is therefore a key probe of the dark Universe.
Cosmic shear, denotes the distortion of images of distant galaxies due to the continuous deflection of light bundles propagating through the inhomogeneous Universe. The induced correlations between shapes of galaxies are directly related to the statistical properties of the total (dark + luminous) large-scale matter distribution. With an estimate of the redshift distribution of the lensed galaxies, theoretical predictions of weak-lensing observables can be tested to obtain constraints on cosmological parameters and models. Recent reviews which also summarize past observational results are Bartelmann & Schneider (2001); Van Waerbeke & Mellier (2003); Munshi et al. (2008); Hoekstra & Jain (2008); Bartelmann (2010).
The Canada-France Hawaii Telescope Legacy
PSF modeling and galaxy shape measurement have been performed with the forward model-fitting method lensfit, which has been thoroughly tested on simulations and improved for CFHTLenS (Miller et al. 2013).
Systematics tests have been performed in a blind way, to yield unbiased cosmological results (Heymans et al. 2012).
The cosmology-dependent covariance matrix is obtained by a mixture of numerical simulations on small scales and analytical predictions on large scales.
The reliability and accuracy of our photometric redshifts allows for three-dimensional weak lensing analyses. The measurement of the lensing correlations for different redshift combinations allows us to obtain information on the growth of structure (e.g. Hu 1999), and has a great potential to constrain dark energy and modified gravity models (e.g. Albrecht et al. 2006; Uzan 2010). In several companion papers, we perform 3D cosmic shear analyses by splitting up galaxies into redshift bins using correlation function methods presented here (lensing tomography: Simpson et al. 2013; Heymans et al. 2013; Benjamin et al. 2013).
In this paper, we perform a 2D lensing analysis using a single redshift distribution. Despite the fact that the redshift information is not used in an optimal way, our analysis has several advantages. First, it yields the highest signal-to-noise ratio () for a single measurement. This is particularly important on large angular scales, where the is too low to be used for tomography. These large scales probe the linear regime, where non-linear and baryonic effects do not play a role, and one can therefore obtain very robust constraints on cosmology (Semboloni et al. 2011). Second, we can include low-redshift galaxies without having to consider intrinsic alignments (IA; Hirata & Seljak 2004). For a broad redshift distribution, IA is expected to be a sub-dominant contribution to the cosmological shear-shear correlation with an expected bias for which is well within the statistical uncertainty (Kirk et al. 2010; Mandelbaum et al. 2011; Joachimi et al. 2011), see also a joint lensing and IA tomography analysis over the full available redshift range (Heymans et al. 2013). Therefore, despite the fact that a 2D lensing is more limited than tomography, it is less noisy and more immune to primary astrophysical systematics. It is therefore a necessary basic step and puts any further cosmological exploitation of CFHTLenS using more advanced tomographic or full 3D lensing techniques on solid grounds. Such analyses are presented in the CFHTLenS companion papers.
This paper is organized as follows. Section 2 provides the expressions for the second-order lensing observables used in this analysis, both obtained from theoretical predictions and estimated from data. The measured shear functions and covariances are presented in Sect. 3. Cosmological models and sampling methods are introduced in Sect. 4. The results on cosmological parameters and models are presented in Sect. 5, followed by consistency tests in Sect. 6. The paper is concluded with a discussion in Sect. 7.
2 Weak cosmological lensing
2.1 Theoretical background
Weak lensing by the large-scale structure measures the convergence power spectrum , which can be related to the total matter power spectrum via a projection using Limber’s equation (Kaiser 1992):
The projection integral is carried out over comoving distances , from the observer out to the limiting distance of the survey. The lens efficiency is given by
where is the Hubble constant, the speed of light, the total matter density, and the scale factor at comoving distance . The comoving angular distance is denoted with and depends on the curvature of the Universe;
The 3D power spectrum is evaluated at the wave number , where denotes the projected 2D wave mode. The function represents the weighted distribution of source galaxies.
2.2 Flavours of real-space second-order functions
From an observational point of view, the most direct measurement of weak cosmological lensing is in real space, by using the weak gravitational shear signal as derived from galaxy ellipticity measurements. The two-point shear correlation functions (2PCFs) and are estimated in an unbiased way by averaging over pairs of galaxies (Schneider et al. 2002),
The sum is performed over all galaxy pairs with angular distance within some bin around . With and we denote the tangential and cross-component of the galaxy ellipticity, respectively. The weights are obtained from the lensfit shape measurement pipeline (Miller et al. 2013). The 2PCFs are the Hankel transforms of the convergence power spectrum or, more precisely, of linear combinations of the E- and B-mode spectra, and , respectively. Namely,
where and are the first-kind Bessel functions of order 0 and 4, and correspond to the components and , respectively.
It is desirable to obtain observables which only depend on the E-mode and B-mode, respectively. Weak gravitational lensing, to first order, only gives rise to an E-mode power spectrum and therefore, a non-detection of the B-mode is an important sanity check of the data. To this end, we calculate the following second-order shear quantities which can be derived from the correlation functions: The aperture-mass dispersion (Schneider et al. 1998), the shear top-hat rms (Kaiser 1992), the optimized ring statistic (Fu & Kilbinger 2010), and COSEBIs (Complete Orthogonal Sets of E-/B-mode Integrals; Schneider et al. 2010). The optimized ring statistic was introduced as a generalisation of the so-called ring statistic (Schneider & Kilbinger 2007). The corresponding filter functions have been obtained to maximise the figure-of-merit of and for a CFHTLS-T0003-like survey. We use these functions for CFHTLenS which, despite the larger area, has similar survey characteristics. COSEBIs represent yet another generalisation and contain all information about the E- and B-mode weak-lensing field from the shear correlation function on a finite angular range.
Being quantities obtained from the 2PCFs by non-invertible relations, these derived functions do not contain the full information about the convergence power spectrum (Eifler et al. 2008), but separate the E- and the B-mode in a more or less pure way, as will now be described. The derived second-order functions can be written as integrals over the filtered correlation functions. They can be estimated as follows:
Here, is the bin width, which can vary with , for example in the case of logarithmic bins. With suitable filter functions and (App. A), the estimator () is sensitive to the E-mode (B-mode) only. The filter functions are defined for the various second-order observables in Table 1 and Appendix A.
All derived second-order functions are calculated for a family of filter functions. For the aperture-mass dispersion, the optimized ring statistic, and the top-hat shear root mean square (rms), these are given for a continuous parameter which can be interpreted as the smoothing scale. Here and in the following we will use the notation ’’ as the scale for the 2PCFs, and ’’ as the smoothing scale for derived functions.
For COSEBIs, the filter functions are a discrete set of functions. The latter exist in two flavours, Lin-COSEBIs and Log-COSEBIs, defined through filter functions which are polynomials on linear and logarithmic angular scales, respectively. Here we use Log-COSEBIs, for which many fewer modes are required to capture the same information as Lin-COSEBIs (Schneider et al. 2010; Asgari et al. 2012). See App. A for more details.
All of the above functions can be expressed in terms of the convergence power spectrum. The general relation is
|Name||(eq. 6)||(eq. 7)||Reference|
|Aperture-mass dispersion||(eq. 23)||eq. (24)||0||Schneider et al. (1998)|
|Top-hat shear rms||(eq. 25)||eq. (26)||0||Kaiser (1992)|
|Optimized ring statistic||(eq. 28)||n/a||Fu & Kilbinger (2010)|
|COSEBIs||(eq. 30)||n/a||Schneider et al. (2010)|
The measured shear correlation function is available only on a finite
interval . The upper limit is given
by the finite survey size. We choose arc minutes, which is
roughly the largest scale for which a sufficient number of galaxy
pairs are available on at least three of the four Wide
If the support of the filter functions and exceeds the observable range, eq. (6) leads to biased results, and a pure E- and B-mode separation is no longer guaranteed.
This is the case for the aperture-mass dispersion and the top-hat shear rms. For the former, only the lower angular limit is problematic and causes leakage of the B-mode into the E-mode signal on small smoothing scales. On scales larger than arc minutes however, this leakage is below 1.5 cent for arcsec (Kilbinger et al. 2006). We therefore choose arcmin to be the smallest smoothing scale for . Note that on scales smaller than we set the correlation function to zero and do not use a theoretical model to extrapolate the data on to this range, to avoid a cosmology-dependent bias.
The B-mode leakage for the top-hat shear rms is a function of both the lower and upper available angular scales. Over our range of scales, the predicted leaked B-mode for the WMAP7 cosmology is nearly constant with a value of .
The first pure E-/B-mode separating function for which the corresponding filter functions have finite support was introduced in Schneider & Kilbinger (2007). Following that approach, the optimized ring statistic and COSEBIs were constructed in a similar way to not suffer from an E-/B-mode leakage.
An additional bias arises from the removal of close galaxy pairs in the lensing analysis, as was first reported by Hartlap et al. (2011). As previously discussed, lensfit produces a shape bias for galaxies separated by less than arcsec, but this bias is random and the close pairs are therefore used in the analysis, to be correlated with other galaxies at larger distances. That said, for very close blended galaxies, where it is non-trivial to determine if there is one or two galaxies observed, galaxy shapes cannot even be attempted. These blended pairs are therefore not reliably detected and lost from our analysis. This causes a potential bias, since these galaxies are removed preferentially at low redshift, where galaxy sizes are larger, and from high-density regions compared to voids, because galaxies trace the large-scale structure. From Fig. 3 of Hartlap et al. (2011) we infer that the magnitude of this effect is at the per cent level on scales larger than arcmin for the 2PCFs.
3 CFHTLenS shear correlation data and covariance
The CFHTLenS data are described in several companion papers; for a full summary see Heymans et al. (2012). Stringent systematics tests have been performed in Heymans et al. (2012) which flag and remove any data in which significant residual systematics are detected. It is this cleaned sample, spanning per cent of the total CFHTLenS survey area, that we use in this analysis. This corresponds to 129 out of 171 MegaCam pointings. In this paper, we complement those tests by measuring the B-mode up to large scales (Sect. 3.5). Comparing this to the previous analysis of F08 shows the improved quality of the lensing analysis by CFHTLenS.
3.1 Redshift distribution
A detailed study of the reliability of our photometric redshifts, the contaminations between redshift bins, and the cosmological implications is performed in Benjamin et al. (2013). This work shows that the true redshift distribution is well approximated by the sum of the probability distribution functions (pdfs) for all galaxies. The pdfs are output by BPZ (Bayesian Photometric Redshift Estimation; Benítez 2000) as a function of photometric redshift , and have been obtained by Hildebrandt et al. (2012). The resulting is consistent with the contamination between redshift bins as estimated by an angular cross-correlation analysis (Benjamin et al. 2010). The contamination is relatively low for galaxies selected with , which is confirmed by a galaxy-galaxy-lensing redshift scaling analysis in Heymans et al. (2012). The resulting is shown in Fig. 1. The mean redshift is . In contrast, the mean redshift of the best-fitting histogram is biased low with .
3.2 Angular correlation functions
We calculate the two-point shear correlation functions by averaging
over pairs of galaxies, using the tree code
We use distances and angles on the sphere to calculate the shear correlation functions. For two galaxies at right ascension and declination , we calculate the great-circle distance with
Each galaxy’s ellipticity is measured in a local Cartesian coordinate system with the -axis going along the line of constant declination and the -axis pointing to the North pole. We project this ellipticity to the tangential and radial component with respect to the connecting great circle. For that, we calculate the angle between the great circle segments and . Then, the projection or so-called course angles are . With the sine and cosine rules on the sphere, we get
and corresponding expressions for by exchanging indices.
To estimate the smoothed second-order quantities, we compute the 2PCFs on 10,000 linear angular bins. This is large enough not to cause a significant E-/B-mode leakage due to the approximation of the integrals over the correlation functions by the direct sum (eq. 6). We verified this using CFHTLenS numerical simulations with no B-mode (see next Section) , see also Becker (2012). We choose the smallest angular distance between two galaxies to be 9 arcsec, corresponding to the first bin centre to be arcsec. We calculate the 2PCFs (eq. 4) as the weighted mean over the four Wide patches, using the number of galaxy pairs as weight for each bin.
3.3 Data covariance
To model and interpret the observed second-order shear functions, we need to estimate the data covariance and its inverse. The cosmic shear covariance C is composed of the shot-noise D, which only appears on the diagonal, a cosmic-variance contribution V, and a mixed term M (Schneider et al. 2002). The covariance of the 2PCFs is comprised of four block matrices. The diagonal consists of and which are the auto-correlation covariance matrices of and , respectively. The off-diagonal blocks are and which denote the cross-correlation covariance between and .
Since the cosmic shear field is non-Gaussian on small and medium angular scales, the cosmic variance involves four-point functions. Neglecting those can yield overly optimistic cosmological constraints (Semboloni et al. 2007; Takada & Jain 2009; Hilbert et al. 2011).
To account for non-Gaussianity, we use -body simulations from Harnois-Déraps et al. (2012). From these simulations, a ‘Clone’ of the CFHTLenS data has been produced with the same galaxy redshift distribution, galaxy clustering, masks, and noise properties. The cosmological lensing signal is added using ray-tracing through the light cones. The Clone cosmology is a flat CDM model with and . The lensing signal for each galaxy is constructed by ray-shooting through the simulated dark-matter distribution. Each simulated line of sight spans a field of view of square degrees. We fit close to MegaCam pointings on each line of sight, which is possible because of overlapping areas between pointings. A total of 184 independent lines of sight are used to calculate the field-to-field covariance matrix. The final matrix is scaled with the ratio of the effective areas (including masks) of 0.11 which corresponds to 90 per cent of the area of 16 MegaCam pointing that fit into each line of sight, divided by 129 MegaCam pointings used in this analysis. We average over three different samples of the galaxy redshift probability distribution, where galaxy redshifts were drawn from the corresponding pdf.
As shown in the upper panel of Fig. 3, the Gaussian prediction for the cosmic variance (Kilbinger & Schneider 2004) for provide a good match to the Clone covariance on intermediate scales, arcmin arcmin. On larger scales, up to arcmin, the numerical simulations under-predict the power due to the finite box size (e.g. Power & Knebe 2006). Only the last two data points show an increased variance, which is due to the finite Clone field geometry. When comparing the Clone mean correlation function to a theoretical prediction with cut-off scale Mpc, we get a rough agreement between the two, indicating that the lack of power is indeed caused by the finite box. We draw similar conclusions for the cosmic variance of , shown in the lower panel of Fig. 3. Further, we verified that a Jackknife estimate of the variance by sub-dividing the CHFLTenS data into 129 subfields gives consistent results.
Grafting the covariance matrix
We construct the total covariance out to arcmin by grafting the Clone covariance to the analytical Gaussian prediction. For the latter, we use the method developed in Kilbinger & Schneider (2004), which takes into account the discrete nature of the galaxy distribution and the field geometry. First, we add the Clone covariance to the Gaussian cosmic covariance term . The combined cosmic covariance is
where the modulation function alleviates discontinuities in the combined matrix. We choose to be a bi-level step function, with ; if both indices are smaller than the step index ; and if at least one of the indices or is larger than or equal to . The step index is chosen such that is the scale closest to 30 arcmin. Equation (11) is applied to all covariances between the two shear correlation functions, i.e. and .
The Clone covariance also contains an additional variance term, which was discovered recently (Sato et al. 2009). This so-called halo sample variance (HSV) stems from density fluctuations on scales larger than the (finite) survey size that are correlated with fluctuations on smaller scales. For example, the number of halos in the survey depend on the large-scale modes outside the survey, since halos are clustered and do not just follow a Poisson distribution. This introduces an extra variance to the measured power spectrum. The halo sample variance is proportional to the rms density fluctuations at the survey scale (Sato et al. 2009). Since our simulated light-cones are cut-outs from larger boxes of size () at redshift below (above) unity, they do contain Fourier scales outside the survey volume and their coupling to smaller scales. The halo sample variance is important on small scales, where our cosmic variance is dominated by the Clone covariance. Following Sato et al. (2009), we estimate the halo sample variance to dominate the CFHTLenS total covariance at , corresponding to 5 arc minutes which is the Clone covariance regime.
The missing large-scale Fourier modes in the simulation box cause the HSV to be underestimated. A further underestimation comes from the rescaling of the Clone lines of sight to the CFHTLenS area since, in contrast to the other covariance terms, the HSV term decreases less strongly than the inverse survey area (Sato et al. 2009). According to Kayo et al. (2012), when naively rescaling from a 25 deg-survey to 1500 deg, the signal-to-noise ratio is too optimistic by not more than 10 per cent. For a re-scaling to the smaller CFHTLenS area, this bias is expected to be much less.
Our grafted covariance of the two-point correlation function is estimated for a fiducial cosmological model, which is given by the -body simulations. In order not to bias the likelihood function of the data (Sect. 4.2) at points other than that fiducial model, we need to account for the fact that the covariance depends on cosmological parameters. We model the cosmology-dependence of the covariance matrix following Eifler et al. (2009), who suggested approximative schemes for the mixed term M and the cosmic variance term V. Accordingly, for the cosmic-variance term, we assume a quadratic scaling with the shear correlation function. This is true on large scales, where the shear field is close to Gaussian and the covariance is indeed proportional to the square of the correlation function. We calibrate the small-scale Clone covariance in the same way, as any differences in the way the non-Gaussian part might scale are likely to be small.
For the mixed term M, we use the fitting formula provided by Eifler et al. (2009). They approximate the variation with and , leaving the matrix fixed for other parameters. The shot-noise term D does not depend on cosmology. The final expression for the covariance matrix is
where the indices stand for the components ’’ and ’’, and are the angular-scale indices. Here, denotes the cosmological parameter for which the covariance is evaluated, with being the fiducial model of the Clone simulation (Sect. 3.3). The shot-noise term and the mixed term are estimated using the method of Kilbinger & Schneider (2004). The cosmic-variance term is given in eq. (11). The power-law indices and depend on the angular scales and the covariance component ’’’’, and have been obtained in Eifler et al. (2009). We note that the inverse covariance is very sensitive to two apparent outliers of and for the -part. To avoid numerical issues, we replace these two numbers by the mean of their neighbouring values. In our case the mixed term of the covariance is important and cannot be neglected (see also Kilbinger & Schneider 2004; Vafaei et al. 2010), in contrast to recent findings by Jee et al. (2012).
During the Monte-Carlo sampling (Sect. 4.2), the covariance is updated at each sample point using eq. (12). We make sure that each calculation of the covariance resulted in a numerically positive-definite matrix, and discard the (rare) sample points for which this is not the case.
In Fig. 4 we show the total covariance and compare it to the Gaussian prediction . Both cases are similar on most scales. On small scales the grafted covariance shows stronger cross-correlations between scales, indicating non-Gaussian effects. We find that the additional covariance due to the shear calibration (see Sect. 3.4) can be neglected, as can be seen in Fig. 3 and Sect. 6.2.
Inverse covariance estimator
It has been shown in Anderson (2003) and Hartlap et al. (2007) that the maximum-likelihood (ML) estimator of the inverse covariance is biased high. The field-to-field covariance from the Clone is such an ML estimate. The bias depends on the number of realisations or fields , and the number of bins . The ML estimator can be de-biased by multiplication with the Anderson-Hartlap factor (Hartlap et al. 2007).
Our final 2PCF covariance, however, is the mixture of an ML estimate and analytical expressions. The ML estimator is modulated with the Gaussian cosmic variance via eq. (11), to which we add the shot-noise and mixed terms, eq. (12). We quantify a possible bias of the inverse covariance by varying for a fixed . For a step index , corresponding to arcmin, Fig. 5 shows that the trace of does not depend on the ratio for our grafted cosmic covariance matrix. Multiplication with for results in an overcorrection, causing a strong decrease of with . A similar albeit less strong decrease is seen when naively taking into account the fact that the Clone covariance only contributes to an effective number of scales , according to eq. (11) with . The three curves for the inverse seem to converge for . Therefore, we have reason to be confident that any bias of the unaltered inverse of eq. (12) is small, and hence we do not need to apply the scalar correction factor . The addition of a deterministic component to the ML covariance seems to be sufficient to render the estimate of the inverse to be unbiased.
Covariance of derived second-order functions
Expressions for the covariance of the derived second-order statistics (eq. 6) are straight-forward to obtain, and can be calculated by integrating the covariance of the 2PCFs (Schneider et al. 2002). However, the necessary precision for the numerical integration requires a large number of angular bins for which the 2PCF covariance has to be calculated, which is very time-consuming. Consequently, for all derived second-order functions we choose not to graft the Clone covariance to the Gaussian covariance, but instead only use the Clone to calculate the total covariance of the derived functions. To include shot noise, we add to each galaxys’ shear an intrinsic ellipticity as a Gaussian random variable with zero mean and dispersion = 0.38. The latter is calculated as , where the sum goes over all CFHTLenS galaxies in our redshift range. Therefore, the covariance between the 184 Clone lines of sight gives us the total covariance . Contrary to the case of the 2PCFs (previous section), this covariance stems from a pure ML estimate, and therefore the inverse needs to be de-biased by the Anderson-Hartlap factor . With a typical number of angular scales of to the corresponding is of order . We show that our cosmological results are independent of the number of realisations in Sect. 6.2. Note that for the all derived estimators, the cosmology-dependence of the covariance is neglected.
For upcoming and future tomographic surveys such as
3.4 Ellipticity calibration corrections
We apply the shear calibration as described in Heymans et al. (2012), which accounts for a potential additive shear bias and multiplicative bias ,
The additive bias is found to be consistent with zero for . The second ellipticity component shows a signal-to-noise ratio () and size-dependent bias which we subtract for each galaxy. This represents a correction which is on average at the level of . The multiplicative bias is modelled as a function of the galaxy and size . It is fit simultaneously in 20 bins of and , see Miller et al. (2013). We use the best-fitting function and perform the global correction to the shear 2PCFs, see eqs. (19) and (20) of Miller et al. (2013). Accordingly, we calculate the calibration factor as the weighted correlation function of ,
The final calibrated 2PCFs are obtained by dividing and by . The amplitude of is around 0.91 on all scales. The errors on the correlation function from the fit uncertainty are negligible compared to our statistical errors. Furthermore, we calculate the covariance matrix for the correlation function from this uncertainty, and show in Sect. 6.2 that the cosmological results remain unchanged by adding this term to the analysis.
Figure 6 shows the combined and corrected 2PCFs, which are the weighted averages over the four Wide patches with the number of pairs as weights. Note that the data points are strongly correlated, in particular on scales larger than about 10 arcmin. Cosmological results using this data will be presented in Sect. 5. The correlation signal split up into the contributions from the four Wide patches is plotted in Fig. 7. There is no apparent outlier field. The scatter is larger than suggested by the Poisson noise on large scales, in agreement with the expected cosmic variance.
3.5 E- and B-modes
The aperture-mass dispersion is shown in the upper panel of Fig. 8. The B-mode is consistent with zero on all scales. We quantify this by performing a null test, taking into account the B-mode Poisson covariance as measured on the Clone,
Since here the covariance is entirely estimated from the Clone line-of-sight, the inverse has to be de-biased using the Anderson-Hartlap factor. We consider the B-mode over the angular range arcmin. As discussed before, the lower scale is where the B-mode due to leakage is down to a few per cent. The upper limit is given by the largest scale accessible to the Clone, which is much smaller than the largest CFHTLenS scale: It is 280 arcmin, resulting in an upper limit of of half that scale. The resulting of , corresponding to a non-null B-mode probability of 46 per cent. Even if we only take the highest six (positive) data points, we find the per degree of freedom (dof) to be , which is less than significance. The non-zero B-mode signal at around 50 - 120 arcmin from F08 is not detected here.
The top-hat shear rms B-mode is consistent with zero on all measured scales, as shown in the middle panel of Fig. 8. Note, however, that of all second-order functions discussed in this work, is the one with the highest correlation between data points. The predicted leakage from the B- to the E-mode is smaller than the measured E-mode, but becomes comparable to the latter for arcmin, where the leakage reaches up to 50 per cent of the E-mode.
The optimized ring statistic for is plotted in the lower panel of Fig. 8. Each data point shows the E- and B-mode on the angular range between and , the latter of which is labelled on the -axis. The B-mode is found to be consistent with zero, a null test yields a 35 per cent probability of a non-zero B-mode.
We first test our calculation of COSEBIs on the CFHTLenS Clone with noise, where we measure a B-mode of at most a few for and arcmin. Even though this is a few orders of magnitudes larger than the B-mode due to numerical errors from the estimation from theory, it is insignificant compared to the E-mode signal. When including the largest available scales for the Clone however, arcmin, the B-mode increases to be of the order of the E-mode. This is true independent of the binning or whether noise is added. We presume that this is due to insufficient accuracy with which the shear correlation function is estimated from the simulation on these very large scales, from only a small number of galaxy pairs. Further, for a similarly large B-mode is found for some cases of . Again, the accuracy of the simulations is not sufficient to allow for precise numerical integration over the rapidly-oscillating filter functions of Log-COSEBIs for higher modes (Becker 2012). We will therefore restrict ourselves to for the subsequent cosmological analysis.
The measured COSEBIs modes are shown in Fig. 9. We use as smallest scale arcsec, and two cases of of 100 and 250 arcmin. In both cases we do not see a significant B-mode. The signal-to-noise ratio of the high mode points decreases when the angular range is increased: For arcmin only the first two modes are significant. This is not unexpected, since the filter functions for arcmin sample larger angular scales and put less weight on small scales where the signal-to-noise ratio in the 2PCFs is larger.
A further derived second-order quantity are the shear E-/B-mode correlation functions (Crittenden et al. 2002; Pen et al. 2002), which have been used in F08. Whereas they share the inconvenience with the top-hat shear rms of a formal upper infinite integration limit, they offer no advantage over the latter, and will therefore not be used in this work.
3.6 Conclusion on estimators
We compared various second-order real-space shear functions, starting with the fundamental two-point correlation functions . From the 2PCFs we calculated a number of E-/B-mode separating functions. The top-hat shear rms is of limited use for cosmological analysis because of the cosmology-dependent E-/B-mode leakage. For the aperture-mass dispersion this leakage is confined to small scales, whereas the optimized ring statistic and COSEBIs were introduced to avoid any leakage. The drawback of the 2PCFs is that they are sensitive to large scales outside the survey area and thus may contain an undetectable B-mode signal (Schneider et al. 2010). COSEBIs capture the E-/B-mode signals in an optimal way on a finite angular-scale interval . The interpretation of COSEBIs and the matching of modes to angular scales are not straightforward since the corresponding filter functions are strongly oscillating.
For lensing alone, we obtain cosmological parameter constraints on and for the different estimators discussed in this section. The results and comparisons are presented in Sect. 5.1.
We decided to use the 2PCFs to compute cosmological constraints in combination with the other probes, for the following reasons. The goal of this paper is to explore the largest scales available for lensing in CFHTLenS. This is only possible with a sufficiently large signal-to-noise ratio when using the 2PCFs. We note that on these large scales our systematics tests, the star-galaxy shape correlation (Heymans et al. 2012) and the E-/B-mode decomposition (this work), were not possible. However, since both tests have revealed no systematics on smaller scales, we are confident that the shear signal up to very large scales is not significantly contaminated. Moreover, the implementation of a cosmology-dependent covariance is currently only feasible for the 2PCFs.
4 Cosmology set-up
4.1 Data sets
We use the following data sets and priors:
CFHTLenS two-point shear correlation functions and covariance as described in Sect. 3. We choose the smallest and largest angular bins to be 0.9 and 300 arcmin, respectively. This includes galaxy pairs between 0.8 and 350 arcmin.
Cosmic microwave background (CMB) anisotropies: WMAP7 (Larson et al. 2011; Komatsu et al. 2011). The released WMAP code
12is employed to calculate the likelihood, see also Dunkley et al. (2009). We use CAMB 13(Lewis et al. 2000) to get the theoretical predictions of CMB temperature and polarisation power- and cross-spectra.
Baryonic acoustic oscillations (BAO): SDSS-III (BOSS). We use the ratio of the apparent BAO at to the sound horizon distance, as a Gaussian random variable, from Anderson et al. (2012).
Hubble constant. We add a Gaussian prior for the Hubble constant of from Cepheids and nearby type-Ia supernovae distances from HST (Riess et al. 2009, hereafter R09).
In contrast to Kilbinger et al. (2009) we do not include supernovae of type (SNIa) Ia. BOSS puts a tight constraint on the expansion history of the Universe, which is in excellent agreement with corresponding constraints using the luminosity distance from the most recent compilation of SNIa (Conley et al. 2011; Suzuki et al. 2012). Both BOSS and SNIa are geometrical probes, and adding SNLS to WMAP7+BOSS yields little improvement on cosmological parameter constraints with the exception of (Sánchez et al. 2012).
All data sets are treated as independent, neglecting any covariance between those probes. Experiments which observe the same area on the sky are certainly correlated since they probe the same cosmological volume. However, this is a second-order effect, like CMB lensing, the integrated Sachs-Wolfe effect (ISW), or the lensing of the baryonic peak (Vallinotto et al. 2007). Compared to the statistical errors of current probes, these correlations can safely be ignored at present, but have to be taken into account for future surveys (Giannantonio et al. 2012).
4.2 Sampling the posterior
To obtain constraints on cosmological parameters, we estimate the posterior density of a set of parameters , given the data and a model . Bayes’ theorem links the posterior to the likelihood , the prior and the evidence ,
To estimate the true, unknown likelihood distribution , a suite of -body simulations would be necessary (e.g. Hartlap et al. 2009; Pires et al. 2009). This is not feasible in a high-dimensional parameter space, and for the number of cosmological models probed in this work. Instead, to make progress, we use a Gaussian likelihood function , despite the fact that neither the shear field nor the second-order shear functions are Gaussian random fields. Nevertheless, this is a reasonable approximation, in particular when CMB is added to lensing (Sato et al. 2010).
The construction towards the true likelihood function can be informed by further features of the estimators, for example constrained correlation functions (Keitel & Schneider 2011). These constraints are equivalent to the fact that the power spectrum is positive. We do however not attempt to make use of these constaints. The expected deviations are minor compared to the statistical uncertainty of the data. The likelihood function is thus given as
where denotes the theoretical prediction for the data for a given -dimensional parameter vector and model .
We sample the posterior with Population Monte Carlo (PMC; Wraith et al. 2009; Kilbinger et al. 2010), using the publicly available code
The main difficulty for importance sampling is to find a suitable importance function. PMC remedies this problem by creating an iterative series of functions . In each subsequent iteration, the importance function is a better representation of the posterior, so the distribution of importance weights gets progressively narrower. A measure for this quality of the importance sample is the normalised Shannon information criterion,
As a stopping criterion for the PMC iterations, we use the related perplexity ,
which lies between 0 and 1, where 1 corresponds to maximum agreement between importance function and posterior.
Most PMC runs reach values of after 10 or 15 iterations. To obtain a larger final sample, we either perform a last importance run with five times the number of points, sampled under the final importance function, or we combine the PMC samples with the five highest values of . In each iteration we created 10k sample points; the final sample therefore has 50k points.
An estimate of the Bayesian evidence
is obtained at no further computing cost from a PMC simulation Kilbinger et al. (2010),
4.3 Theoretical models
We compare the measured second-order shear functions to non-linear
models of the large-scale structure, with a prediction of the density
power spectrum from the halofit fitting formulae of
Smith et al. (2003). For dark-energy models, we adopt the
scheme of the icosmo
We also see a good agreement with the CDM simulations of
Harnois-Déraps et al. (2012). A more accurate non-linear power spectrum on
a wider range of cosmological parameters could be obtained from the Coyote
We individually run PMC for CFHTLenS and WMAP7, respectively. For the combined posterior results, we importance sample the WMAP7 final PMC sample, multiplying each sample point with the CFHTLenS posterior probability.
For weak lensing only, the base parameter vector for the flat CDM model is . It is complemented by and for dark-energy and non-flat models, respectively. With CMB, we add the reionisation optical depth and the Sunyaev-Zel’dovich (SZ) template amplitude to the parameter vector. Moreover, we use as the primary normalisation parameter, and calculate as a derived parameter. We use flat priors throughout which, when WMAP7 is added to CFHTLenS, cover the high-density regions and the tails of the posterior distribution well.
For model comparison, we limit the parameter ranges to physically well-motivated priors for those parameters which vary between models. This is important for any interpretation of the Bayesian evidence, since the evidence directly depends on the prior. The prior is an inherent part of the model, and we want to compare physically well-defined models.
Thus, we limit the total matter and dark-energy densities and , setting a lower physical limit, and creating a symmetrical prior for the curvature of , which is bound from below by the physical limit of an empty universe. Note that by sampling both and , the curvature prior is no longer uniform but has triangular shape.
For the model comparison cases we limit to , therefore excluding phantom energy and dark-energy models which are non-accelerating at the present time. These priors are the same as for the models that were compared using the Bayesian evidence in Kilbinger et al. (2010). The prior ranges for the other parameters are , , , and . For the dark-energy model runs for parameter estimation, which are not used for model comparison, we use a wide prior on which runs between and .
5 Cosmological results
The most interesting constraints from 2D weak lensing alone are obtained for and , which we discuss below for the four cosmologies considered here. Table 2 shows constraints from lensing alone on the combination , which is the direction orthogonal to the - degeneracy ‘banana’ To obtain , we fit a power law to the log-posterior values using histograms with optimal bin numbers for estimating the posterior density (Scott 1979). (Fig. 10). We also discuss constraints on (for cases with free curvature) and (for CDM models). Table 3 shows the combined constraints from CFHTLenS+WMAP7 and CFHTLenS+WMAP7+BOSS+R09. The comparison between cosmological models is shown in Table 4 and described in Sect. 5.4.
|Parameter||flat CDM||flat CDM||curved CDM||curved CDM|
|Parameter||flat CDM||flat CDM||curved CDM||curved CDM|
Flat Cdm For a flat CDM universe, the constraints in the plane (left panel of Fig. 10) from CFHTLenS are nearly orthogonal to the ones for WMAP7. CFHTLenS improves the joint constraints for these parameters by a factor of two. Lensing plus CMB constrain and to better than 5 per cent and 2 per cent, respectively. Adding BOSS and R09 decreases the error on to per cent, but does not improve the constraint on .
Flat Cdm If the dark-energy equation-of-state parameter is kept free, CMB and lensing display the same degeneracy direction between and (left panel of Fig. 11). Combining both probes only partially lifts this degeneracy, the uncertainty on remains at the 25 per cent level. This uncertainty decreases to 10 percent with the addition of the BOSS BAO distance measure.
The value of the Hubble constant from both CFHTLenS+WMAP7 () and BOSS+WMAP7 () are slightly lower when compared to the R09 result, , although it is within the 1- error bar. Since is degenerate with all other parameters except , those parameter means change with the inclusion of the R09 prior. This causes the relatively large and and low if R09 is not added. The joint Hubble constant with all four probes is .
Curved Cdm With curvature left free and no additional priors, CMB anisotropies cannot determine anymore, since there is a degeneracy between matter density, curvature and the Hubble constant. Lensing however shows a similar dependency on and to the flat model case. Therefore, the improvement on from CFHTLenS + WMAP7 with respect to WMAP7 alone is an order of magnitude, to yield a 8 per cent error. The joint error on is 3.5 per cent.
Curved Cdm The - degeneracy holds nearly the same as in the previous cases of models with fewer parameters, as displayed in the left panel of Fig. 12. The value of is slightly increased but well within the error bars. The joint CFHTLenS+WMAP7 results on and are similar to the flat CDM case.
The BOSS+R09+WMAP7 results indicate a slightly smaller and larger . The joint CFHTLenS+WMAP7+BOSS+R09 allowed region is therefore on the upper end of the CFHTLenS+WMAP7 banana. The reason for this is, as in the flat CDM case, the degeneracy of and with the Hubble constant. WMAP7 alone prefers a low value, , which increases to when BOSS+R09 is added. As a consequence, decreases and increases. On the other hand, adding CFHTLenS to WMAP7 leaves the Hubble constant at the relatively low value of .
5.2 Dark energy
For the following results on the dark-energy equation-of-state parameter , we use the flat prior .
Flat Cdm 2D weak gravitational lensing alone is not able to tightly constrain dark energy, in contrast with 3D tomographic weak lensing. The 68 per cent confidence limits for (flat CDM) are of order unity, . In combination with WMAP7 only, these errors decrease by a factor of four, and gets constrained to about 30 per cent. The CFHTLenS+WMAP7+BOSS constraints on dark energy are . We discuss this deviation from CDM in Sect. 7. Adding the R09 prior on does not reduce the error but shifts the mean to the CDM value, .
Curved Cdm The case of dark energy is similar in the curved case. CFHTLenS alone results in . Adding WMAP7 reduced this uncertainty to 30 per cent. CFHTLenS+WMAP7+BOSS yield . Adding the R09 prior on , we find the CDM-consistent value of .
CFHTLenS helps to improve the constraint on the curvature density . For CDM, the uncertainty decreases by a factor of 10 from around (WMAP7 alone) to (CFHTLenS+WMAP7). Adding BOSS+R09 decreases the error by another factor of two to around . The combined constraints are thus consistent with a flat universe within . For a CDM model, this uncertainty is of the same order.
5.4 Model comparison
In Table 4, the evidence and the logarithms of the evidence ratios, between the baseline flat model and the other three models are shown. Here, is the evidence for flat CDM and the evidence for one of the three models curved CDM, flat CDM, and curved CDM, respectively. is called the Bayes factor between model ‘0’ and model ‘1’.
An empirical scale to interpret such evidence values was suggested by Jeffreys (1961), see also Trotta (2008). Accordingly, two models are not distinguishable when . If the log-Bayes factor is between 1 and 2.5, the evidence is called weak. Moderate evidence is assumed for , and strong for values larger than that.
We compute the evidence for the two cases of probes CFHTLenS+WMAP7 and CFHTLenS+WMAP7+BOSS+R09. Although the evidence for the flat dark energy model is slightly larger than the one for the cosmological constant model, the models are indistinguishable: Their respective evidence values, or posterior odds, are within a factor of two. The evidence against curved models is moderate, with log-Bayes factor ratios between and , or posterior odd ratios between 25 and 130 in favor of flat CDM. The significance increases when adding BOSS and R09, but stays in the moderate range. We remind the reader that the dark-energy models considered here have the flat prior for , which corresponds to an accelerating non-phantom dark energy component.
6 Comparison of weak-lensing statistics, systematics and consistency tests
In this section we obtain cosmological constraints from the derived second-order estimators which were discussed in Sect. 2.2. The following tests are all performed under a flat CDM model. The results are listed in Table 5.
|2PCF, constant covariance|
6.1 Derived second-order functions
As expected, the constraints from the derived second-order estimators are less tight than from the 2PCFs, since they always involve information loss. Moreover, we use a smaller range of angular scales, cutting off both on the lower and higher end, as discussed before. All estimators give consistent results.
Aperture-mass dispersion and top-hat shear rms give very similar constraints compared to the 2PCFs. The position and slope of the banana are nearly identical, although the width is larger by a factor two (see Table 2). For , we analyse two approaches of dealing with the finite survey-size E-/B-mode leakage:
We add a constant offset of to the measured E-mode points. This corresponds to the theoretical leakage for the WMAP7 best-fitting CDM model with . On scales arcmin, the assumption of a constant offset is clearly wrong; however, the constant is two orders of magnitudes smaller than the measured signal and does ânot influence the result much.
The difference between both cases is about half of the statistical uncertainty (Table 2). More sophisticated ways to deal with this leakage, e.g. going beyond a constant offset, or marginalising over a parametrized offset, are expected to yield similar results. Since they all have the disadvantage of depending on prior information about a theoretical model which might bias the result towards that model, we do not consider this second-order estimator further.
The function on scales between and arcmin, for (implying