Emulating the CFHTLenS Weak Lensing data: Cosmological Constraints from moments and Minkowski functionals
Andrea Petri, Jia Liu, Zoltan Haiman, Morgan May, Lam Hui, Jan M. Kratochvil
aa r X i v : . [ a s t r o - ph . C O ] M a y Emulating the CFHTLenS Weak Lensing data: Cosmological Constraints frommoments and Minkowski functionals
Andrea Petri,
1, 2, ∗ Jia Liu, Zolt´an Haiman, Morgan May, Lam Hui, and Jan M. Kratochvil Department of Physics, Columbia University, New York, NY 10027, USA Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA Department of Astronomy, Columbia University, New York, NY 10027, USA Astrophysics and Cosmology Research Unit, University of KwaZulu-Natal, Westville, Durban 4000, South Africa (Dated: September 8, 2018)Weak gravitational lensing is a powerful cosmological probe, with non–Gaussian features poten-tially containing the majority of the information. We examine constraints on the parameter triplet(Ω m , w, σ ) from non-Gaussian features of the weak lensing convergence field, including a set of mo-ments (up to 4 th order) and Minkowski functionals, using publicly available data from the 154 deg CFHTLenS survey. We utilize a suite of ray–tracing N-body simulations spanning 91 points in(Ω m , w, σ ) parameter space, replicating the galaxy sky positions, redshifts and shape noise in theCFHTLenS catalogs. We then build an emulator that interpolates the simulated descriptors as afunction of (Ω m , w, σ ), and use it to compute the likelihood function and parameter constraints.We employ a principal component analysis to reduce dimensionality and to help stabilize the con-straints with respect to the number of bins used to construct each statistic. Using the full set ofstatistics, we find Σ ≡ σ (Ω m / . . = 0 . ± .
04 (68% C.L.), in agreement with previousvalues. We find that constraints on the (Ω m , σ ) doublet from the Minkowski functionals suffera strong bias. However, high-order moments break the (Ω m , σ ) degeneracy and provide a tightconstraint on these parameters with no apparent bias. The main contribution comes from quarticmoments of derivatives. PACS numbers: 98.80.-k, 95.36.+x, 95.30.Sf, 98.62.SbKeywords: Weak Gravitational Lensing — Data analysis — Methods: analytical,numerical,statistical
I. INTRODUCTION
Weak gravitational lensing (hereafter WL) is emergingas a promising technique to constrain cosmology. Tech-niques have been developed to construct cosmic shearfields with shape measurements in large galaxy cata-logues. Although the shear two–point function (2PCF)is the most widely studied cosmological probe (see, e.g.[1]), alternative statistics have been shown to increasethe amount of cosmological information one can extractfrom weak lensing fields. Among these, high–order mo-ments ([2–6]), three–point functions ([7–9]), bispectra([10–13]), peak counts ([14–21]) and Minkowski Func-tionals ([22, 23]) have been shown to improve cosmolog-ical constraints in weak lensing analyses.In this work, we use the publicly available CFHTLenSdata, consisting of a catalog of ≈ m , σ and the darkenergy (DE) equation of state w . The statistics weconsider in this work are the Minkowski functionals(MFs) and the low–order moments (LM) of the con-vergence field. Cosmological parameter inferences fromCFHTLenS have been obtained using the 2PCF [1], anda number of authors have investigated the constrainingpower of CFHTLenS using statistics that go beyond the ∗ [email protected] usual quadratic ones. Fu et al. [24] used three–point cor-relations as an additional probe for cosmology, and foundmodest (10 − − σ constraints, utilizing the abun-dance of WL peaks. Cosmological constraints using WLpeaks in CFHTLenS Stripe82 data have also been inves-tigated by Liu et. al. [27]. Finally, closest to the presentpaper, Shirasaki & Yoshida [28] investigated constraintsfrom Minkowski Functionals, including systematic errors.Our study represents two major improvements over pre-vious work. First, constraints from the MFs in ref. [28]were obtained through the Fisher matrix formalism, as-suming linear dependence on cosmological parameters.Our study utilizes a suite of simulations sampling the cos-mological parameter space, mapping out the non-linearparameter-dependence of each descriptor. Second, we in-clude the LMs as a set of new descriptors; these yield thetightest and least biased constraints.This paper is organized as follows: we first give anoverview of the CFHTLenS catalogs, and summarize theadopted data reduction techniques. Next, we give a de-scription of our simulation pipeline, including the ray–tracing algorithm, and the procedure used to sample theparameter space. We call the statistical weak lensingobservables – the power spectrum, Minkowski function-als, and moments – ”descriptors” throughout the paper.We discuss the calculation of the descriptors, includingdimensional reduction using a principal component anal-ysis, and the statistical inference framework we used. Wethen describe our main results, i.e. the cosmological pa-rameter constraints. To conclude, we then discuss ourfindings and comment on possible future extensions ofthis analysis. II. DATA AND SIMULATIONSA. CFHTLenS data reduction
In this section, we briefly summarize our treatment ofthe public CFHTLenS data. For a more in–depth de-scription of our data reduction procedure, we refer thereader to [26].The CFHTLenS survey covers four sky patches of 64,23, 44 and 23 deg area, for a total of 154 deg . Thepublicly released data consist of a galaxy catalog createdusing SExtractor [29], and includes photometric redshiftsestimated with a Bayesian photometric redshift code [30]and galaxy shape measurements using lensfit [31, 32].We apply the following cuts to the galaxy catalog: mask < . < z < . fitclass = 0 (which requires the object tobe a galaxy) and weight w > × galaxies, 124.7 deg sky cov-erage, and average galaxy density n gal ≈ . − .The catalog is further reduced by ∼
25% when one re-jects fields with non–negligible star–galaxy correlations.These spurious correlations are likely due to imperfectPSF removal, and do not contain cosmological signal.These cuts are consistent with the ones adopted by theCFHTLenS collaboration (see [24]).The CFHTLenS galaxy catalog provides us with thesky position θθθ , redshift z ( θθθ ) and ellipticity e ( θθθ ) of eachgalaxy, as well as the individual weight factors w ( θθθ )and additive and multiplicative ellipticity corrections c ( θθθ ) , m ( θθθ ). Because the CFHTLenS fields are irregularlyshaped, we first divide them into 13 squares (subfields)to match the shape and ≈
12 deg size of our simulatedmaps (see below). These square-shaped subfield mapsare pixelized according to a Gaussian gridding procedure¯ e ( θθθ ) = P N s i =1 W ( | θθθ − θθθ i | )w( θθθ i )[ e obs ( θθθ i ) − c ( θθθ i )] P N s i =1 W ( | θθθ − θθθ i | )w( θθθ i )[1 + m ( θθθ )] , (1) W θ G ( θθθ ) = 12 πθ G exp (cid:18) − θθθ θ G (cid:19) , (2)where the smoothing scale θ G has been fixed at 1.0 arcmin(but varied occasionally to 1.8 and 3.5 arcmin for specifictests described below) and m, c refer to the multiplicativeand additive corrections of the galaxies in the catalog.Using the ellipticity grid ¯ e ( θθθ ) as an estimator for thecosmic shear γ , ( θθθ ), we perform a non–local Kaiser–Squires inversion [33] to recover the convergence κ ( θθθ ) from the E –mode of the shear field, κ ( l ) = (cid:18) l − l l + l (cid:19) γ ( l ) + 2 l l l + l γ ( l ) . (3)The simulated κ maps we create below are 12 deg insize and have a resolution of 512 ×
512 pixels. TheCFHTLenS catalogs contain masked regions (which in-clude the rejected fields and the regions around brightstars). We first create gridded versions of the observed κ maps matching the size and pixel resolution of oursimulated maps, with each pixel containing the numberof galaxies ( n gal ) falling within its window. We thensmooth this galaxy surface density map with the sameGaussian window function as equation (2) and removeregions where n gal < (see [28]). Regions withlow galaxy number density can induce large errors in thecosmological parameter inferences. B. Simulation design
We next give a description of our method to sam-ple the parameter space with a suite of N–body simula-tions. We wish to investigate the non–linear dependenceof the descriptors (in this work, Minkowski Functionalsand moments of the κ field) on the parameter triplet p = (Ω m , w, σ ), while keeping the other relevant pa-rameters ( h, Ω b , n s ) fixed to the values (0.7, 0.046, 0.96)(see [34]). We sampled the D –dimensional ( D = 3 in thiscase) parameter space using an irregularly spaced grid.The grid was designed with a method similar to that usedto construct an emulator for the matter power spectrumin the Coyote simulation suite [35]. Given fixed availablecomputing resources, the irregular grid design is more ef-ficient than a parameter grid with regular spacings: toachieve the same average spacing between models in thelatter approach would require a prohibitively large num-ber of simulations.We limit the parameter sampling to a box whose sidesrange over Ω m ∈ [0 . , , w ∈ [ − . , , σ ∈ [0 . , . N points x i ∈ [0 , D . Let a design D be the set of thisirregularly spaced N points. Our goal is to find an opti-mal design, in which the points are spread as uniformlyas possible inside the box. Following ref. [35], we chooseour optimal design as the minimum of the cost function C ( D ) = 2 D / N ( N − N X i 1) for d = 1 ...D .2. We shuffle the coordinates of the par-ticles in each dimension independently x di = P d (cid:16) N − , N − , ..., (cid:17) , where P , ..., P D arerandom independent permutations of (1 , , ..., N ).3. We pick a random particle pair ( i, j ) and a randomcoordinate d ∈ { , ..., D } and swap x di ↔ x dj .4. We compute the new cost function. If the valueis less than in the previous step, we keep the ex-change, otherwise we revert the coordinate swap.5. We repeat steps 3 and 4 until the relative cost func-tion change is less than a chosen accuracy parame-ter ǫ .We have found that for N = 91 grid points O(10 )iterations are sufficient to reach an accuracy of ǫ ∼ − .Once the optimal design D m has been determined, weinvert the mapping Π → [0 , to arrive at our simulationparameter sampling p s . We show the final list of gridpoints in Table I and Figure 1.For each parameter point on the grid p s we then runan N –body simulation and perform ray tracing, as de-scribed in § II C, to simulate CFHTLenS shear catalogs.Throughout the rest of this paper, we refer to this setof simulations as CFHTemu1 . Additionally, we have run50 independent N –body simulations with a fiducial pa-rameter choice p = (0 . , − . , . § III B. This additional suiteof 50 simulations will be referred to as CFHTcov . C. Ray–Tracing Simulations The goal of this section is to outline our simulationpipeline. The fluctuations in the matter density fieldbetween a source at redshift z and an observer locatedon Earth will cause small deflections in the trajectoriesof light rays traveling from the source to the observer.We estimate the dark matter gravitational potential run-ning N –body simulations with N = 512 particles, usingthe public code Gadget2 [36]. We adopted a comovingbox size of 240 h − Mpc, corresponding to a mass resolu-tion of 7 . × h − M ⊙ . The simulations include darkmatter only, and the initial conditions were generated with N-GenIC at z = 100, based on the linear mat-ter power spectrum created with the Einstein-Boltzmanncode CAMB [37]. Data cubes were output at redshift in-tervals corresponding to 80 h − (comoving) Mpc.Using a procedure similar to refs. [38, 39], the equationthat governs the light ray deflections can be written inthe form d x ( χ ) dχ = − c ∇ x ⊥ Φ( x ⊥ ( χ ) , χ ) , (5)where χ is the radial comoving distance, x ⊥ = χβββ refersto two transverse coordinates (with βββ the angular sky co-ordinates, using the flat sky approximation)a, x ( χ ) is thetrajectory of a single light ray, and Φ is the gravitationalpotential.Suppose that a light ray reaches the observer at anangular position θθθ on the sky: we want to know wherethis light ray originated, knowing it comes from a red-shift z s . To answer this question we need to integrateequation (5) with the initial condition βββ (0; θθθ ) = θθθ up toa distance χ s = χ ( z s ) to obtain the source angular po-sition βββ ( χ s ; θθθ ). Since light rays travel undeflected fromthe observer to the first lens plane, the derivative initialcondition in the Cauchy problem (5) reads ˙ βββ (0; θθθ ) = 0.We indicate the derivative of βββ with respect to χ as ˙ βββ .We use our proprietary implementation Inspector Gad-get (see, e.g., [40]) to solve for the light ray trajecto-ries based on a discretized version of equation (5) thatis based on the multi–lens–plane algorithm (see [39] forexample). Applying random periodical shifts and rota-tions to the N –body simulation data cubes, we generate R = 1000 pseudo–independent realizations of the lensplane system used to solve (5). Once we obtain the lightray trajectories, we infer the relevant weak lensing quan-tities by taking angular derivatives of the ray deflections A ( χ s ; θθθ ) = ∂βββ ( χ s ; θθθ ) /∂θθθ and performing the usual spindecomposition to infer the convergence κ and the shearcomponents ( γ , γ ), A ( χ s ; θθθ ) = (1 − κ ( χ s ; θθθ )) III − γ ( χ s ; θθθ ) σ − γ ( χ s ; θθθ ) σ (6)where III is the 2 × σ , are the first and thirdPauli matrices. We perform this procedure for each of the R realizations of the lens planes and we obtain R pseudo–independent realizations of the γγγ weak lensing field. Weuse these different random realizations to estimate themeans (from the CFHTemu1 simulations) and covariancematrices (from the CFHTcov simulations) of our descrip-tors. Since the random box rotations and translationsthat make up the CFHTcov simulations are based on 50independent N –body runs, we believe the covariance ma-trices measured from this set to be more accurate thanthe ones measured from the CFHTemu1 set.The convergence κ is related to the magnification,while the two components of the complex shear γγγ = γ + iγ are related to the apparent ellipticity of thesource. Given a source with intrinsic complex ellipticity N Ω m w σ N Ω m w σ N Ω m w σ N Ω m w σ CFHTemu1 grid points in the 3D cosmological parameter space. m −3.0−2.5−2.0−1.5−1.0−0.50.0 w m σ FIG. 1. (Ω m , w ) and (Ω m , σ ) projections of the final simulation design. The blue points correspond to the CFHTemu1 simulationset, which consists of one N –body simulation per point, while the red point corresponds to the CFHTcov simulation set, whichis based on 50 independent N –body simulations. e s = e s + ie s , its observed ellipticity will be modified to e = e s + g g ∗ e s | g | ≤ ge ∗ s e ∗ s + g ∗ | g | > g ≡ γγγ/ (1 − κ ) is the reduced shear.For each simulated galaxy, we assign an intrinsic ellip-ticity by rotating the observed ellipticity for that galaxyby a random angle on the sky, while conserving its magni-tude | e | . To be consistent with the CFHTLenS analysis,we adopt the weak lensing limit ( | γγγ | ≪ , κ ≪ g ≈ γγγ and e ≈ e s + γγγ . We also add the multiplicativeshear corrections by replacing γγγ with (1 + m ) γγγ . We notethat the observed ellipticity for a particular galaxy al-ready contains the lensing shear by large scale structure(LSS), but the random rotation makes this contributionat least second order in κ by destroying the shape spa-tial correlations induced by lensing from LSS. Consistentwith the weak lensing approximation, the lensing signalfrom the simulations is first order in κ and hence therandomly rotated observed ellipticities can be safely con-sidered as intrinsic ellipticities.We analyze the simulations in the same way as we ana-lyzed the CFHTLenS data – constructing the simulated κ maps as explained in § II A. These final simulation prod-ucts are then processed together with the κ maps ob-tained from the data to compute confidence intervals onthe parameter triplet (Ω m , w, σ ). III. STATISTICAL METHODS The goal of this section is to describe the framework tocombine the CFHT data and our simulations, and to de-rive the constraints on the cosmological parameter triplet(Ω m , w, σ ). Briefly, we measure the same set of statisti-cal descriptors from the data and from the simulations;these are then compared in a Bayesian framework in or-der to compute parameter confidence intervals. A. Descriptors The statistical descriptors we consider in this workare the Minkowski Functionals (MFs) and the low–ordermoments (LMs) of the convergence field. The threeMFs ( V , V , V ) are topological descriptors of the conver-gence field κ ( θθθ ), probing the area, perimeter and genuscharacteristic of the κ excursion sets Σ κ , defined asΣ κ = { κ > κ } . Following refs. [22, 23] we use thefollowing local estimators to measure the MFs from the κ maps: V ( κ ) = 1 A Z A Θ( κ ( θθθ ) − κ ) dθθθ,V ( κ ) = 14 A Z A δ D ( κ ( θθθ ) − κ ) q κ x + κ y dθθθ, (8) V ( κ ) = 12 πA Z A δ D ( κ ( θθθ ) − κ ) 2 κ x κ y κ xy − κ x κ yy − κ y κ xx κ x + κ y dθθθ. Here A is the total area of the field of view and κ x,y de-notes gradients of the κ field, which we evaluate usingfinite differences. In this notation Θ( x ) is the Heavisidefunction and δ D ( x ) is the Dirac delta function. The firstMinkowski functional, V , is equivalent to the cumulativeone–point PDF of the κ field, while V , V are sensitiveto the correlations between nearby pixels. The one–pointPDF of the κ field, ∂V , can be obtained by differentia-tion ∂V ( κ ) = dV ( κ ) /dκ .In addition to these topological descriptors, we con-sider a set of low–order moments of the convergence field(two quadratic, three cubic and four quartic). We choosethese moments to be the minimal set of LMs necessary tobuild a perturbative expansion of the MFs up to O ( σ )(see [41, 42]). We adopt the following definitionsLM : σ , = h κ i , h|∇ κ | i , LM : S , , = h κ i , h κ |∇ κ | i , h κ ∇ κ i , LM : K , , , = h κ i , h κ |∇ κ | i , h κ ∇ κ i , h|∇ κ | i . (9)If the κ field were Gaussian, one could express the MFsin terms of the LM moments, which are the only inde-pendent moments for a Gaussian random field. In re-ality, weak lensing convergence fields are non–Gaussianand the MF and LM descriptors are not guaranteed tobe equivalent. Refs. [41, 42] studied a perturbative ex-pansion of the MFs in powers of the standard deviation σ of the κ field. When truncated at order O ( σ ), thiscan be expressed completely in terms of the LMs up toquartic order. Such perturbative series, however, havebeen shown not to converge [23] unless the weak lensingfields are smoothed with windows of size ≥ ′ . Becauseof this, throughout this work, we treat MF and LM asseparate statistical descriptors.We note that this choice is somewhat ad-hoc. In gen-eral, the LMs that contain gradients are sensitive to dif-ferent shapes of the κ multispectra P nκ ( l , ..., l n ) becausea particular LM n has the general formLM n = Z d l ...d l n ρ ( l ...n ) P nκ ( l ...n ) (10)where ρ is a polynomial of order n in the l ’s. For exam-ple for K we have ρ ( l ) = l and this moment em-phasizes quadrilateral shapes for which one side is muchlarger than the others. On the other hand, for K wehave ρ ( l ) = ( l · l )( l · l ) and this moment is mostsensitive to trispectrum shapes that are close to rectan-gular. There are moments which include derivatives inaddition to those included in Eq. 9. In the future, wewill investigate whether there is additional constrainingpower in these additional quartic moments.In addition to the MFs and LMs, we consider the an-gular power spectrum P l ≡ P l of κ , defined as h ˜ κ ( l )˜ κ ( l ′ ) i = (2 π ) δ D ( l + l ′ ) P l , (11) Descriptor Details N b (linear spacing) V , V , V (MF) κ ∈ [ − . , . 12] 50Power Spectrum (PS) l ∈ [300 , N b in each case. where ˜ κ ( l ) is the Fourier transform of the κ field. Previ-ous works have studied cosmological constraints from theconvergence power spectrum extensively. Here our pur-pose is to compare the constraints we obtain from theMFs and LMs to ones present in the literature, whichare based on the use of quadratic statistics (see for ex-ample [1]). The statistical descriptors used in this workare summarized in Table II.When measuring statistical descriptors on κ maps, par-ticular attention must be paid to the effect of maskedpixels. The MFs and LMs remain well–defined in thepresence of masks, since the estimators in equations (8)and (9) are defined locally, and can be computed in thenon–masked regions (with the exception of the few pix-els that are close to the mask boundaries). The situa-tion is more complicated for power spectrum measure-ments, which require the evaluations of Fourier trans-forms and hence rely on the value of every pixel in themap. Although sophisticated schemes to interpolate overthe masked regions have been studied (see for example[43]), for the sake of simplicity, we here insert the value κ = 0 in each masked pixel. Given the uniform spa-tial distribution of the masked regions in the data, weexpect that masks have a little effect on the power spec-trum at the range of multipoles in Table II, except foran overall normalization which will be the same both inthe data and the simulations. Likewise, we believe thatthe way we deal with masked sky regions – essentially ig-noring them – is robust for the MFs and LMs. Since weapply the same masks to our simulations and the data,they are unlikely to introduce biases in the resulting con-straints. Masks, of course, can still affect the sensitivityand weaken constraints. The impact of the masks andtheir treatment has been evaluated for the MFs, obtainedfrom CFHTLenS, by ref. [28], in which the authors findthat the masked regions are not a dominant source ofsystematic effects in the CFHTLenS data. B. Cosmological parameter inferences In this section, we briefly outline the statistical frame-work adopted for computing cosmological parameter con-fidence levels. We make use of the MFs and LMs, as wellas the power spectrum, as discussed in the previous sec-tion. We refer to M ri ( p ) as the descriptor measured froma realization r of one of our simulations with a choice ofcosmological parameters p (i.e. from one of the R = 1000map realizations in this cosmology), and to D i as the descriptor measured from the CFHTLenS data. In thisnotation, i is an index that refers to the particular bin onwhich the descriptor is evaluated (for example i can rangefrom 0 to 9 for the LM statistic and from 0 to N b − N b different, linearly spaced κ bins,as indicated in Table II).Once we make an assumption for the data likelihood L d ( D i | p ) and for the parameter priors Π( p ), we can useBayes’ theorem to compute the parameter likelihood L p , L p ( p | D i ) = L d ( D i | p )Π( p ) N L . (12)Here N L is a p –independent constant that ensures theproper normalization for L p . We make the usual assump-tion that the data likelihood L d ( D i | p ) is Gaussian [44] L d ( D i | p ) = [(2 π ) N b det C ] − / e − χ ( D i | p ) ,χ ( D i | p ) = [ D − M ( p )] T C − [ D − M ( p )] . (13)We assume, for simplicity, that the covariance matrix C in equation (13) is p –independent and coincides with C ( p ). The simulated descriptors M ( p ) are measuredfrom an average over the R = 1000 realizations in the CFHTemu1 ensemble M i ( p ) = 1 R R X r =1 M ri . (14)The covariance matrix C ij = 1 R − R X r =1 [ M ri ( p ) − M i ( p )][ M rj ( p ) − M j ( p )](15)is measured from the R = 1000 realizations in the CFHTcov ensemble. While eq. (15) gives an unbiased es-timator of the covariance matrix, its inverse is not anunbiased estimator of C − (e.g. ref. [39]). Given that inour case R ≫ N b , we can safely neglect the correctionfactor needed to make the estimator for C − unbiased.When computing parameter constraints from theCFHTLenS weak lensing data alone, we make a flat priorassumption for Π( p ). We postpone using different priors,incorporating external data, for future work. Parameterinferences are made estimating the location of the max-imum of the parameter likelihood in eq. (12), which wecall p ML ( D i ), as well as its confidence contours. The N σ –confidence contour of L p ( p | D i ) is defined to be thesubset of points in parameter space on which the likeli-hood has a constant value c N and Z L >c N L p ( p | D i ) d p = 1 √ π Z N − N dxe − x / . (16)Using equation (17) below, and given the low dimen-sionality of the parameter space we consider ( D = 3),we are able to directly compute the parameter likeli-hood eq. (12) for 100 different combinations of thecosmological parameters p , arranged in a finely spaced100 × × 100 mesh within the prior window Π( p ). Wedirectly compute the maximum likelihood p ML ( D i ) andthe contour levels c N without the need for more sophis-ticated MCMC methods.The data likelihood is directly available for parame-ter combinations on the simulated irregular grid p s . Weuse a Radial Basis Function (RBF) scheme to interpolate M ( p ) to arbitrary intermediate points. We approximatethe model descriptor as M ( p ) = N X s =1 λ s φ ( | p − p s | ) (17)where φ has been chosen as a multi-quadric function φ ( r ) = p r/r ) , with r chosen as the mean Eu-clidean distance between the points in the simulated grid p s . The constant coefficients λ s can be determined byimposing the N constraints M ( p = p s ) = M ( p s ), whichenforce exact results at the simulated points. The in-terpolation computations are conveniently performed us-ing the interpolate.Rbf routine contained in a Scipylibrary [45].We studied the accuracy of the emulator, built withthe CFHTemu1 simulations, by interpolating the con-vergence descriptors to the fiducial parameter setting(Ω m , w, σ ) = (0 . , − . , . 8) and comparing the resultto the one expected from the CFHTcov simulations. Fig-ure 2 shows that our power spectrum emulator has arelative error smaller than 20% for the lower multipoles( l < . κ values in[ − . , . C. Dimensionality reduction The main goal of this work is constraining thecosmological parameter triplet (Ω m , w, σ ) using theCFHTLenS data. Once the N σ contours have been ob-tained, using the procedure and equations (12)–(16) out-lined above, one may ask whether the choice of binningaffects these contours. Indeed, in our previous work, wehave found that the number of bins, N b , can have a non-negligible effect on the contour sizes (see [23] for an ex-ample with simulated datasets). In order to ensure that are results are robust with re-spect to binning choices, we have implemented a Prin-cipal Component Analysis (PCA) approach. Our physi-cal motivation for this approach is that, even though weneed to specify N b numbers in order to fully characterizea binned descriptor, we suspect that the majority of theconstraining information (of a particular descriptor) iscontained in a limited number of linear combinations ofits binning. In the framework adopted by [35], for exam-ple, the authors find that the majority of the cosmologicalinformation in the matter power spectrum is containedin only 5 different linear combinations of the multipoles.Because of this, we believe that dimensionality reduc-tion techniques such as PCA can help deliver accuratecosmological constraints using only a limited number ofdescriptor degrees of freedom.In order to compute the principal components of ourdescriptor space, we use the CFHTemu1 simulations, whichsample the cosmological parameters at the N = 91 pointslisted in Table I, and allow us to compute the N × N b model matrix M pi = M i ( p ). Note that this is a rectangu-lar (non-square) matrix. Following a standard procedure(see, e.g., ref. [47]), we derive the whitened model matrix˜ M pi , defined by subtracting the mean (over the N = 91models) of each bin, and normalizing it by its variance(always over the N = 91 models). Next we proceed witha singular value decomposition (SVD) of ˜M , USV T = ˜M √ N − , (18)where S ij = S i δ ij is a diagonal matrix and V Tij is the j –thcoordinate ( j = 1 ...N b ) of the i –th principal component( i = 1 ... min[ N b , N ]) of ˜M , with the index j ranging from1 to N b . By construction, V is p –independent.To rank the Principal Components V T in order of im-portance, we note that the diagonal matrix S is simplythe diagonalization of the model covariance (not to beconfused with the descriptor covariance in eq. (15)),1 N − ˜M T ˜M = VS V T . (19)We follow the standard interpretation of PCA compo-nents, stating that the only meaningful components V Ti in the analysis (i.e. the ones that contain the relevantcosmological information) are those corresponding to thelargest eigenvalues S i , with the smallest eigenvalues cor-responding to noise in the model, due to numerical inac-curacies in the simulation pipeline. We expect our con-straints to be stable with respect to the number of com-ponents, once a sufficient number of components havebeen included. Using the fact that different principalcomponents are orthogonal, we perform a PCA projec-tion on our descriptor space by whitening the descriptorsand computing the dot product with the principal com-ponents, keeping only the first n components M ( n ) rpi = V T ( n ) ij ˜ M rpj ; D ( n ) i = V T ( n ) ij ˜ D j . (20) 300 1000 2000 3000 4000 5000l10 -3 -2 -1 ( E − M ) / (cid:0) C MM -0.04 -0.01 0.02 0.06 0.09 0.12κ PSV V V FIG. 2. Accuracy of the emulator based on the CFHTemu1 simulations. The figure shows the absolute difference between thedescriptor interpolated at the fiducial parameter setting, and the descriptor expected from the CFHTcov simulations (these arethe absoulte values of differences which oscillate around zero). The descriptors are shown in units of the standard deviation ineach bin i (determined from the diagonal elements of the CFHTcov covariance matrix). We show the accuracy results for thepower spectrum (red) and the three Minkowski functionals V (green), V (blue) and V (black). For reference, we also show,using dashed lines, the difference between the expected CFHTcov descriptors and the interpolated descriptor at the non–fiducialpoint p = (0 . , − . , . σ contour from the simulations shown in Figure 3 rightpanel, and corresponds to the target accuracy we wish to achieve Here we indicate with V T ( n ) the truncation of V T tothe first n rows (i.e. i can now range from 1 to n ). Asdescribed above, the expectation is that most of the cos-mological information is contained in a small number ofcomponents n < N b . We will describe in detail below thechoice we make for n , together with the sensitivity of ourresults to this choice.Looking at PCA from a geometrical perspective, thedimensionality reduction problem is equivalent to the ac-curate reconstruction of the coordinate chart of the de-scriptor manifold. As outlined in ref. [47], the coordinatechart constructed with the PCA projection in eq. (20) isaccurate for reasonably flat descriptor manifolds. Whencurvature becomes important, more advanced projectiontechniques (such as Locally Linear Embedding) have tobe employed. We postpone an investigation of such im-provements to future work. IV. RESULTS This section describes our main results, and is orga-nized as follows. We begin by showing the cosmologi-cal constraints from the CFHTLenS data for the triplet(Ω m , σ , w ), as well as for an alternative parameteriza-tion, (Ω m , Σ , w ), with Σ ( α ) ≡ σ (Ω m / . α . In thenext section we give a justification on why we fix α to avalue of 0 . 55. We then use our simulations to perform a robustness analysis of the parameter confidence intervalswith respect to the number of PCA components used inthe projection. We finally study whether the constraintscan be tightened by combining different descriptors. Asummary with the complete set of results, along with therelevant Figures, is shown in Table III. A. Cosmological constraints We first make use of equations (12)-(16) to computethe 1 σ constraints on cosmological parameters, usingthe triplet (Ω m , σ , w ). Figures 3 shows the constraintsin the (Ω m , σ ) plane, marginalized over w , for boththe CFHTLenS data, as well as from the mock data inour simulations. In Figure 4, we examine constraintsfrom different sets of moments, as well as using differentsmoothing scales. Figure 5 shows the confidence contoursin the ( w, Σ ) plane, marginalized over Ω m . As this fig-ure shows, and as discussed further below, no meaningfulconstraints were found on w from CFHTLenS alone.Because of the relatively small size of this survey, de-generacies among the parameters can have undesirableeffects on the constraints. The well-known strong de-generacy between Ω m and σ is evident in the long “ba-nana” shaped contours in Figures 3 and 4. To mitigatethe effect of this degeneracy, in addition to the usualtriplet (Ω m , σ , w ), we consider an alternative parame-terization, built with the triplet ( α, Σ , w ) where α is aconstant, and Σ ( α ) ≡ σ (Ω m / . α . While Ω m and σ are poorly constrained due to degeneracies, the Σ ( α )combination lies in the direction perpendicular to the er-ror “banana” at the pivot point Ω m = 0 . 27. This is thedirection of the lowest variance L (Ω m , σ ) for a suitablechoice of α , and hence has a much smaller relative un-certainty. We can derive the optimal value of α fromthe full three dimensional likelihood L (Ω m , w, σ ), fromwhich we can compute the expectation values E ( α ) = h Σ ( α ) i ; V ( α ) = h (Σ ( α ) − E ( α )) i (21)and minimizing the ratio √ V / E with respect to α . Theexpectation values are taken over the entire parameterbox. This procedure yields a value α ≈ . 55 for the sta-tistical descriptors that we consider, consistent with whatis found in the literature (see [1] for example). Although α can mildly depend on the type of descriptor consid-ered, we choose to keep it fixed, knowing that the widthof the Σ likelihood cannot vary significantly with differ-ent choices of α . We show the probability distributionof the best-constrained parameter Σ (marginalized overΩ m and w ) in Figure 6.We discuss the results of this section in § V below. B. Robustness The cosmological constraints should in principle be in-sensitive to N b , once a sufficient number of bins are used,but inaccuracies in the covariance (due to a limited num-ber of realizations) can introduce an N b dependence. Ourbinning choices are summarized in Table II. Here we showthat the cosmological constraints derived in this paperare numerically robust, i.e. they are reasonably stable,once we consider a large enough number n of PrincipalComponents.Figure 7 shows the PCA eigenvalues from the SVDdecomposition of our binned descriptor spaces (follow-ing the discussion in § III C), as well as the cumulativesum of these eigenvalues, normalized to unity. Figure 8shows the dependence of the (Ω m , σ ) constraints on thenumber of principal components n .These figures clearly indicate that we only need a lim-ited number of components in order to capture the cos-mological information contained in our descriptors. Theeigenvalues diminish rapidly with n , and, in particular,the confidence contours converge to good ( . n = 5 − 10 (depending on the descrip-tor). This finding also addresses the inaccuracy of theMF emulator at high thresholds, pointed out in Figure 2.By keeping a limited number of principal components, weare able to prevent the inaccurate high–threshold bins,which have a low constraining power, from contributingto the parameter confidence levels. C. Combining statistics Different statistics can include complementary cos-mological information, allowing their combinations totighten the constraints. Previous work using multiplelensing descriptors in CFHTLenS alone included combin-ing the power spectrum and peak counts [26], combiningthe power spectrum and Minkowski functionals [28] ,andcombining quadratic (2PCF) statistics with cubic statis-tics derived from the 3PCF of the CFHTLenS κ field [24].The procedure we adopt here is as follows. Con-sider two binned descriptors, d ,i , d ,j where the indices i, j correspond to bin numbers. We first compute eachsingle–descriptor constraint as a function of the numberof PCA components, as in Figure 8. We then determinethe minimum number of PCA components n , neededfor the constraints to be stable. We next construct thevector d × = { d ( n ) , d ( n ) } and consider this as thecombined ( n + n )–dimensional descriptor vector. Thisprocedure naturally allows us to account for the cross–covariance between different binned descriptors. An anal-ogous procedure can be used to combine multiple (threeor more) descriptors.We show constraints from different descriptor combi-nations in the (Ω m , σ ) and ( w, Σ ) planes in Figure 9,and on the best-constrained parameter Σ in Figure 10.We also provide a tabulated version of the Σ constraints(1 σ ) in Table IV. We discuss these findings in the nextsection. V. DISCUSSION In this section we discuss the results shown in § IVabove, with particular focus on the constraints on cos-mology.As pointed out in § III C, the choice of the numberof bins, N b , is an important issue. In order to ensurethat our results are insensitive to N b , we adopted a PCAprojection technique to reduce the dimensionality of ourdescriptor spaces. The left panel of Figure 7 shows thatthe PCA eigenvalues for all of our descriptors decrease byabout 4 orders of magnitude from n = 1 to n = 3. Theright panel of this figure shows that more than 99% ofthe descriptor variances are captured by including onlythe first n = 3 components.This does not necessarily mean, however, that the cos-mological information is captured by the first 3 PCAcomponents: in principle, one of the higher- n PCAcomponents could have an unusually strong cosmology-dependence, and could impact the confidence levels. Toaddress this possibility, we determined the 1 σ contoursizes as a function of n in Figure 8. This figure showsthat the first 3 components indeed capture essentially allthe information contained in the power spectrum. How-ever, this is not true for the other descriptors. In partic-ular, we find that n ≥ V ,and n ≥ 20 components for V and V , in order for the0 Ω m σ PS(3)V (5)V (20)V (20)Moments(9) Ω m σ PS(3)V (5)V (20)V (20)Moments(9) FIG. 3. 1 σ (68% CL) constraints on the (Ω m , σ ) parameter doublet using the power spectrum (red), the three Minkowskifunctionals ( V : green, V : blue, V : black) and the moments (orange). We show the constraints from the data (left panel) andfrom a mock observation constructed using the mean of 1000 realizations in the CFHTcov simulation suite (right panel). Thecontours are calculated from the parameter likelihood function L marginalized over w . The parentheses near the descriptorlabel refer to the number of principal components included. Ω m σ σ ,S ,K (1.0 ′ )σ ,S ,K (1.0 ′ )σ ,S ,K (1.0 ′ )σ ,S ,K (1.0 ′ )σ ,S ,K (1.0 ′ )σ ,S ,K (1.0 ′ )σ ,S ,K (1.0 ′ ) Ω m σ σ ,S ,K (1.0 ′ )σ ,S ,K (1.0 ′ × 1.8 ′ )σ ,S ,K (1.0 ′ × 1.8 ′ × 3.5 ′ ) FIG. 4. 1 σ (68% CL) constraints on the (Ω m , σ ) parameter doublet using moments, with different colors corresponding todifferent moment combinations (see eq. 9 for their definitions). We show the results from the one–point moments σ , S , K (black curves; both left and right panels). In the left panel, we also show constraints obtained adding moments of gradients tothe one–point moments. In the right panel, we combine one–point moments measured at different smoothing scales. Parameters Descriptors Short description Relevant Figures(Ω m , σ ) PS(3), V (5) , V (20) , V (20),LM(9) 1 σ constraints from CFHTLenSand mock observations 3,3b(Ω m , σ ) ( σ i , S i , K i ) 1 σ constraints from CFHTLenSusing κ momentscombined at different θ G w, Σ ) PS(3), V (10) , V (10) , V (10),LM(9) contours from CFHTLenS 5Σ PS(3), V (10) , V (10) , V (10),LM(9) L (Σ ) from CFHTLenS 6- PS, V , V , V ,LM PCA eigenvalues 7(Ω m , σ ) PS, V , V , V ,LM Stability of contours 8(Ω m , σ ) PS(3) × V (5) × V (20) × V (20) × LM(9) constraints from CFHTLenScombining statistics 9( w, Σ ) PS(3) × V (10) × V (10) × V (10) × LM(9) constraints from CFHTLenScombining statistics 9bΣ PS(3) × V (10) × V (10) × V (10) × LM(9) L (Σ ) from CFHTLenScombining statistics 10TABLE III. Summary of our results and related figures.Descriptor Σ = σ Ω . m PS(3) 0 . +0 . − . PS(3) × Moments(9) 0 . +0 . − . V (10) × V (10) × V (10) 0 . +0 . − . PS(3) × V (10) × V (10) × V (10) 0 . +0 . − . PS(3) × V (10) × V (10) × V (10) × Moments(9) 0 . +0 . − . TABLE IV. Tabulated values of 1 σ constraints on Σ corresponding to Figure 10 −1.4 −1.2 −1.0 −0.8 −0.6 −0.4 −0.2 w Σ PS(3)V (10)V (10)V (10)Moments(9) FIG. 5. 1 σ (68% C.L.) constraints on the ( w, Σ ) parameterdoublet from the CFHTLenS data, obtained with the powerspectrum (red), the three Minkowski functionals ( V : green, V : blue, V : black) and the moments (orange). The con-tours are calculated from the parameter likelihood function L marginalized over Ω m , and the parentheses near the descrip-tor label refer to the number of principal components. (Ω m , σ ) contours to be stable at the ∼ 5% level. All ninemoments need to be included for the moments contoursto be stable to this accuracy.These results are slightly different when we study the( w, Σ ) constraints (with α fixed at α = 0 . 55 as discussedabove). In this case we find that the optimal choice forall three MFs is n = 10, while the number of componentsrequired for the PS and Moments remain at n = 3 and n = 9, respectively. (These results are not shown, butobtained analogously to the Figures above.)We now discuss the main scientific findings of thiswork. In Figure 3, we show the 1 σ constraints onthe (Ω m , σ ) doublet from the CFHTLenS data. TheMF constraints appear to be biased towards the low– σ , high–Ω m region. Here and throughout the remain-der of this paper, by ”biased” (or ”unbiased”) we re-fer to being incompatible (or compatible) with the con-cordance fiducial values at 1 σ obtained in other ex-periments. For example, the current best-fit valuesof (Ω m , σ ) = (0 . , . , (0 . , . 82) from cosmic mi-crowave background anisotropies measured respectivelyby the Planck [48] and WMAP [34] satellites lie beyondthe 99% likelihood contours obtained from the three MFs(not shown).This discrepancy may be due to uncorrected system-atics in the CFHTLenS data, amplified by the (Ω m , σ )degeneracy. As a test of our analysis pipeline, when wetry to constrain mock observations based on simulations(shown in the right panel of Figure 3), we recover the cor-rect input position of the 1 σ contours. It is important to2 Σ L ( Σ ) PS(3)V (10)V (10)V (10)Moments(9) FIG. 6. The likelihood of the best-constrained parametercombination Σ ( α ) ≡ σ (Ω m / . α from the CFHTLenSdata, obtained with the power spectrum (red), the threeMinkowski functionals ( V : green, V : blue, V : black) andthe moments (orange). The likelihood was computed with aconstant optimized α = 0 . 55, but marginalized over both Ω m and w . The parentheses near the descriptor label refer to thenumber of principal components. note, however, that the mock observations to which theright panel of Figure 3 refers, were built with the meanof R = 1000 realizations of the CFHTcov simulations. Wefound that it is possible to find some rare (10 out of the1,000) realizations for which the best fit for (Ω m , σ ) liesin the lower right corner, near the location of the best-fitfrom the data. While this could provide an alternativeexplanation of the bias from the MFs, the likelihood ofthis happening is very small ( . m , σ ). Furthermore, this constraint is un-biased, in the sense defined above: it includes the currentconcordance values for these parameters within 1 σ . Thisleads us to conclude that the bias in the constraints fromthe MFs is due to systematic errors, rather than the rarestatistical fluctuations found above. The fact that themoments are useful for deriving unbiased cosmologicalconstraints has been noted in previous work, which ex-amined the biases caused by spurious shear errors [49].In order to determine the origin of the tight bounds de-rived from moments, we studied the contribution of eachindividual moment to the constraints. Figure 4 shows theevolution of the (Ω m , σ ) constraints as we add increas-ingly higher-order moments to the descriptor set. Sincewe are constraining 3 cosmological parameters, we startby considering the set of the three traditional one–point moments which do not involve gradients, i.e. the vari-ance, skewness, and kurtosis ( σ , S , K ). We then addthe remaining six moments of derivatives one by one,starting from the quadratic moments.Figure 4 shows that the biggest improvement on theparameter bounds comes from including quartic momentsof derivatives (i.e. K i with i ≥ 1) in the descriptor set.This might explain why [24] find only relatively weakcontour tightening ( ∼ κ derivatives.Ref. [24] consider one–point, third–order moments, com-bined for multiple smoothing scales. Figure 4 explicitlyshows, however, that smoothing scale combinations arenot as effective as moments of derivatives in constrainingthe (Ω m , σ ) doublet. Our results agree with an earlyprediction [5] that the kurtosis of the shear field can helpin breaking degeneracies between Ω m and σ . Here wefound that considering quartic moments of gradients fur-ther helps in breaking this degeneracy.As noted above, the bias in the (Ω m , σ ) constraints isamplified by the cosmological degeneracy of these param-eters. To mitigate this effect, we consider the combina-tion of Ω m and σ that lies orthogonal to the most degen-erate direction, namely Σ = σ (Ω m / . . . Figure 5shows the 1 σ constraints for the ( w, Σ ) doublet, whileFigure 6 shows the marginalized Σ likelihood from theCFHTLenS data. The CFHTLenS survey constrains theΣ combination to a value of Σ = 0 . ± . σ ) usingthe full descriptor set, in agreement with values previ-ously published by the CFHTLenS collaboration [1].These figures also show that the current dataset is in-sufficient to constrain w to a reasonable precision. Thisis consistent with the previous analyses of CFHTLenS [1,24, 26, 28]. We also note that ref. [28] obtained the best-fit value of w ≈ − w = − σ ). We found a similar result when usinga Fisher matrix to compute confidence levels. Since theFisher matrix formalism is equivalent to a linear approx-imation of our emulator (in which all cosmological pa-rameter dependencies are assumed to be linear), we thusattribute this bias to the oversimplifying assumption oflinear cosmology-parameter dependence of the descrip-tors. Although the right panel in Figure 9 shows thatthe moments confine w to isolated regions in parameterspace, we note that w = − 1, the value favored by otherexisting experiments, is excluded at the 1 σ level. The2 σ contours (not shown in the figure) join, and include w = − (Figure 6).Finally, we studied whether the combination of dif-ferent statistical descriptors can help in tightening the3 i -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 S i PSV V V Moments n Σ n i = S i / S t o t FIG. 7. Results from the Principal Components Analysis (PCA) of the binned power spectrum (red), the three Minkowskifunctionals ( V : green, V : blue, V : black) and the moments (orange). The left panel shows the magnitudes of the PCAeigenvalues S i and the right panel shows their cumulative sum, normalized to unity. A dashed vertical (black) line has marks n = 3 for reference. cosmological constraints. We show the effects of some ofthese combinations in Figures 9 and 10. The left panel ofFigure 9 shows that, although combining the power spec-trum and the moments with the Minkowski functionalshelps tighten the (Ω m , σ ) constraints, it does not helpin reducing the inherent parameter bias of the MFs. Theright panel of Figure 9 shows that even with these statis-tics combined, w remains essentially unconstrained. [50]found that even weak lensing tomography alone is un-able to constrain w sensibly. Figure 10 shows that theΣ combination is already well constrained by any of thedescriptors alone, without the need of combining differentdescriptors. This further clarifies that the non–quadraticdescriptors mainly help to break degeneracies, tighteningcontours along the degenerate direction. VI. CONCLUSIONS In this final section we summarize the main conclusionsof this work: • We find that the power spectrum, combined withthe moments of the κ field provides the tight-est constraint on the (Ω m , σ ) doublet from theCFHTLenS survey data. The tightness of theseconstraints comes mainly from the moments. Evi-dence of the unbiased nature of constraints from themoments has been found in [49]. We further findthat the largest improvement on parameter boundsis achieved when we include the quartic moments ofderivatives in the descriptor set. This level of im- provement cannot be achieved by combining one-point moments at different smoothing scales. • Although weak lensing surveys are a promisingtechnique to constrain the DE equation of stateparameter w , reasonable constraints cannot be ob-tained with the CFHTLenS survey alone, evenwhen using additional sets of descriptors that gobeyond the standard quadratic statistics. • When studying the cosmological information con-tained in the CFHTLenS data, special attentionmust be paid to the effect of residual systematicbiases. While these residual systematics are foundto be unimportant when constraining cosmologywith the power spectrum alone, we find that thesesystematics need to be corrected to obtain unbi-ased constraints on the (Ω m , σ ) doublet using theMinkowski functionals. We are aware that, whentrying to explain the discrepancy between weaklensing and CMB constraints using the Minkowskifunctionals, there might be other effects to be con-sidered, namely non–Gaussian error correlations inthe descriptors and inaccuracies of the simulationson small scales. These inaccuracies could in princi-ple affect the excursion set reconstruction at high κ thresholds. We will investigate these additionalsources of error in future work. • For the CFHTLenS data set, Minkowski functionalscan effectively constrain the non-degenerate direc-tion in parameter space, Σ , where the amplifyingeffects of degeneracy are mitigated.The Minkowski4 Ω m σ V n =3n =4n =5n =10n =20n =30n =40n =50 Ω m σ ∂V (PDF) n =3n =4n =5n =10n =20n =30n =40n =49 Ω m σ V n =3n =4n =5n =10n =20n =30n =40n =50 Ω m σ V n =3n =4n =5n =10n =20n =30n =40n =50 Ω m σ PS n =3n =4n =5n =10n =20n =30n =40n =50 Ω m σ Moments n =3n =4n =5n =6n =8n =9 FIG. 8. The dependence of the 1 σ contours in the (Ω m , σ ) plane on the number of PCA components, obtained from a mockobservation constructed with the CFHTcov simulations. The different panels refer to the different descriptors (from left to right,top to bottom) V , ∂V (PDF), V , V , power spectrum and moments. The labels in each panel show the number of PCAcomponents included to obtain contour with different colors. Ω m σ PS(3)PS(3) × Moments(9)V (5) × V (20) × V (20)PS(3) × V (5) × V (20) × V (20)PS(3) × V (5) × V (20) × V (20) × Moments(9) −1.4 −1.2 −1.0 −0.8 −0.6 −0.4 −0.2 w Σ PS(3)PS(3) × Moments(9)V (10) × V (10) × V (10)PS(3) × V (10) × V (10) × V (10)PS(3) × V (10) × V (10) × V (10) × Moments(9) FIG. 9. 1 σ constraints on the (Ω m , σ ) (left panel) and ( w, Σ ) (right panel) doublets, using the power spectrum (PS) alone(red), the MFs alone (blue), as well as using different combinations of descriptors: PS × Moments (green), PS × MFs (black) andPS × MFs × Moments (orange). The likelihood function has been marginalized over w (left panel) and Ω m (right panel). Theparentheses next to each descriptor label refers to the number of PCA components included. Σ L ( Σ ) PS(3)PS(3) × Moments(9)V (10) × V (10) × V (10)PS(3) × V (10) × V (10) × V (10)PS(3) × V (10) × V (10) × V (10) × Moments(9) FIG. 10. The probability distribution of the best-constrainedparameter Σ from the CFHTLenS data, using the powerspectrum (PS) alone (red), the MFs alone (blue), as wellas using different combinations of descriptors: PS × Moments(green), PS × MFs (black) and PS × MFs × Moments (orange).The likelihood function has been marginalized over Ω m and w . The parentheses next to each descriptor label refers to thenumber of PCA components. functionals alone are sufficient to constrain the Σ combination to a value of Σ = 0 . ± . 04 at 1 σ significance level. This agrees with the value pre-viously published by the CFHTLenS collaborationwithin 1 σ . Some tensions with Planck [48] still re-main.Possible future extensions of this work include simu-lating higher–dimensional parameter spaces (including,for example, the Hubble constant H , and allowing atilt in the power spectrum or a time-dependence of theDE equation of state w ), and combining the CFHTLenSconstraints with different cosmological probes from large-scales structures and the CMB. The latter can help inbreaking the Ω m , σ degeneracy, and allow improvementsin the the constraints on w . The techniques developedhere can be applied to larger, soon-forthcoming surveydata sets, such as the Dark Energy Survey (DES) [51],Subaru [52], WFIRST [53] and LSST [54]. 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