Review: Cosmology from cosmic shear observations

Martin Kilbinger, CEA Saclay, Service d’Astrophysique (SAp), France

Find on this page general information and updates for my recent review article (arXiv:1411.0155) on cosmic shear, Reports on Progress in Physics 78 (2015) 086901 (ads link for two-column format).

Sigma 12/17
Fig. 7 of the review article: The quantity $$\Sigma = \sigma_8 \left( \Omega_{\rm m}/0.3 \right)^\alpha$$ as function of publication year.
Updated figure!
02/2015: Added Stripe-82 and CFHTLenS peak counts
07/2015: Added DES-SV.
06/2016: Added DLS, two more CFHTLenS analyses, DES-SV peak counts, and KiDS-450.
08/2017: Added DES-Y1, and another KiDS-450 result.
12/2017: Added more KiDS-450 and DES-Y1 results (peak counts, lensing+clustering, density-split statistic).

In the video abstract of the article I talk about cosmic shear and the review for a broader audience.
Additional references, new papers
General papers, new reviews.
    • Another weak-lensing review has been published by my colleagues Liping Fu and Zu-Hui Fan (behind a pay wall, not available on the arXiv).
    • Rachel Mandelbaum’s short, pedagogical review to instrumental systematics and WL

 Sect. 2: Cosmological background

 Sect. 5: Measuring weak lensing

    • News on ensemble shape measurement methods:
      An implementation of the Bernstein & Armstrong (2014) Bayesian shape method has been published at arXiv:1403.7669. The team that participated at the great3 challenge with the Bayesian inference method “MBI” published their pipeline and results paper, see arXiv:1411.2608.
    • Okura & Futamase (arXiv:1405.1539) came up with an estimator of ellipticity that uses 0th instead of 2nd-order moments!
    • arXiv:1409.6273 discusses atmospheric chromatic effects for LSST.
    • Dust in spiral galaxies  as source of shape bias, but also astrophysical probe: arXiv:1411.6724.


Fig. 3 (b), derivatives of the convergence power spectrum with respect to various cosmological parameters.

Comments and suggestions are welcome! Please write to me at

Last updated 22 July 2015.

The galaxy-halo connection from a joint lensing, clustering and abundance analysis in the CFHTLenS/VIPERS field

Authors: J. Coupon et al.
Journal: MNRAS
Year: 2015
Download: ADS | arXiv


We present new constraints on the relationship between galaxies and their host dark matter halos, measured from the location of the peak of the stellar-to-halo mass ratio (SHMR), up to the most massive galaxy clusters at redshift


and over a volume of nearly 0.1~Gpc


. We use a unique combination of deep observations in the CFHTLenS/VIPERS field from the near-UV to the near-IR, supplemented by


secure spectroscopic redshifts, analysing galaxy clustering, galaxy-galaxy lensing and the stellar mass function. We interpret our measurements within the halo occupation distribution (HOD) framework, separating the contributions from central and satellite galaxies. We find that the SHMR for the central galaxies peaks at

Mh,peak=1.90.1+0.2×1012MM_{\rm h, peak} = 1.9^{+0.2}_{-0.1}\times10^{12} M_{\odot}

with an amplitude of


, which decreases to


for massive halos (

Mh>1014MM_{\rm h} > 10^{14} M_{\odot}

). Compared to central galaxies only, the total SHMR (including satellites) is boosted by a factor 10 in the high-mass regime (cluster-size halos), a result consistent with cluster analyses from the literature based on fully independent methods. After properly accounting for differences in modelling, we have compared our results with a large number of results from the literature up to


: we find good general agreement, independently of the method used, within the typical stellar-mass systematic errors at low to intermediate mass (

M<1011M{M}_{\star} < 10^{11} M_{\odot}

) and the statistical errors above. We have also compared our SHMR results to semi-analytic simulations and found that the SHMR is tilted compared to our measurements in such a way that they over- (under-) predict star formation efficiency in central (satellite) galaxies.

A new model to predict weak-lensing peak counts I. Comparison with N-body Simulations

Authors: C.-A. Lin, M. Kilbinger.
Journal: A&A 576, A24
Year: 2015
Download: ADS | arXiv



Weak-lensing peak counts has been shown to be a powerful tool for cosmology. It provides non-Gaussian information of large scale structures, complementary to second order statistics. We propose a new flexible method to predict weak lensing peak counts, which can be adapted to realistic scenarios, such as a real source distribution, intrinsic galaxy alignment, mask effects, photo-z errors from surveys, etc. The new model is also suitable for applying the tomography technique and non-linear filters. A probabilistic approach to model peak counts is presented. First, we sample halos from a mass function. Second, we assign them NFW profiles. Third, we place those halos randomly on the field of view. The creation of these "fast simulations" requires much less computing time compared to N-body runs. Then, we perform ray-tracing through these fast simulation boxes and select peaks from weak-lensing maps to predict peak number counts. The computation is achieved by our \textsc{Camelus} algorithm, which we make available at this http URL. We compare our results to N-body simulations to validate our model. We find that our approach is in good agreement with full N-body runs. We show that the lensing signal dominates shape noise and Poisson noise for peaks with SNR between 4 and 6. Also, counts from the same SNR range are sensitive to Ωm and σ8. We show how our model can discriminate between various combinations of those two parameters. In summary, we offer a powerful tool to study weak lensing peaks. The potential of our forward model is its high flexibility, making the use of peak counts under realistic survey conditions feasible.


A new, probabilistic model for weak-lensing peak counts is being proposed in this first paper of a series of three. The model is based on drawing halos from the mass function and, via ray-tracing, generating weak-lensing maps to count peaks. These simulated maps can directly be compared to observations, making this a forward-modelling approach of the cluster mass function, in contrast to many other traditional methods using cluster probes such as X-ray, optical richness, or SZ observations.

The model prediction is in very good agreement with N-body simulations.

It is very flexible, and can potentially include astrophysical and observational effects, such as intrinsic alignment, halo triaxiality, masking, photo-z errors, etc. Moreover, the pdf of the number of peaks can be output by the model, allowing for a very general likelihood calculation, without e.g. assuming a Gaussian distribution of the observables.


Review: Cosmology from cosmic shear observations

Mean and 68% error bars for the parameter  $\sigma_8 (\Omega_{\rm m}/0.3)^\alpha$, for various cosmic shear observations, plotted as function of their publication date (first arXiv submission). Data points are second-order statistics (circles), third-order (diamonds), 3D lensing (pentagons), galaxy-galaxy lensing (+ galaxy clustering; triangle), and CMB (squares).
Mean and 68% error bars for the parameter $$\sigma_8 (\Omega_{\rm m}/0.3)^\alpha$$, for various cosmic shear observations, plotted as function of their publication date (first arXiv submission). Data points are second-order statistics (circles), third-order (diamonds), 3D lensing (pentagons), galaxy-galaxy lensing (+ galaxy clustering; triangle), and CMB (squares).

A review on cosmology from cosmic shear observations has been submitted to ROPP, and has the arXiv reference arXiv:1411.0115. Comments are very welcome! Check also the accompanying web page for more information, updates, and errata.

Reduced-shear power spectrum

Fitting formulae of the reduced-shear power spectrum for weak lensing


Martin Kilbinger, 2010, arXiv:1004.3493


We provide fitting formulae for the reduced-shear power-spectrum correction which is third-order in the lensing potential. This correction reaches up to 10% of the total lensing spectrum. Higher-order correction terms are one order of magnitude below the third-order term. The correction involves an integral over the matter bispectrum. We fit this integral with a combination of power-law functions and polynomials. We also fit the derivatives with respect to cosmological parameters. A Taylor-expansion around a fiducial (WMAP7) model provides accurate reduced-shear corrections within a region in parameter space containing the WMAP7 68% error elllipsoid.


Our fits are accurate to 1% for l<104, and to 2% for l<2·105, which reduces the bias by a factor of four compared to the case of no correction. This matches the precision lensing power spectrum predictions of recent N-body simulations.

Ratio of power spectra uncorrected (lower lines) and corrected (upper lines) for reduced-shear.


Download, install, and run the code

Download an example code which includes the fitting matrices. Use ‘make’ to compile the code. To use the code, you have to fill in Fmn(a) (eq. 10 from the paper) which involves the lensing efficiency, comoving distances and the redshift distribution(s).

The reduced-shear corrections are also implemented in the cosmology and lensing package ‘nicaea’. This code provides all necessary functions to produce lensing observables (shear power spectrum and real-space second-order functions). The cosmology and redshift distributions are set via parameter files.


Martin Kilbinger (

Optimes E-/B-mode decomposition

A new cosmic shear function:
Optimised E-/B-mode decomposition on a finite interval


Liping Fu, Martin Kilbinger, 2009, arXiv:0907.0795


We have introduced a new cosmic shear statistic which decomposes the shear correlation into E- and B-modes on a finite angular interval. The new function is calculated by integrating the shear two-point correlation function with a filter function. The filter function fulfills the E-/B-mode decomposition constraints given in Schneider & Kilbinger (2007).

Download, install, and run the code

Download the tar file decomp.tgz. Extract the archive with
tar xzf decomp.tgz

To compile and run the code:
cd Demo
make links
make decomp_eb

The package fftw3 has to be installed. If it is not in a standard directory, fftw3.h is looked for in $(FFTW)/include and libfftw3.a in $(FFTW)/lib. Change the variable `FFTW’ in the Makefile accordingly. You can download fftw3 from

The program produces two files, Tpm containing the filter functions T+ and T, and REB containing the shear functions RE and RB.


Liping Fu, Martin Kilbinger (

Weak lensing mass reconstruction using wavelets


Authors: J.-L. Starck, S. Pires, Alexandre Réfrégier
Journal: A&A
Year: 2006
Download: ADS | arXiv



This paper presents a new method for the reconstruction of weak lensing mass maps. It uses the multiscale entropy concept, which is based on wavelets, and the False Discovery Rate which allows us to derive robust detection levels in wavelet space. We show that this new restoration approach outperforms several standard techniques currently used for weak shear mass reconstruction. This method can also be used to separate E and B modes in the shear field, and thus test for the presence of residual systematic effects. We concentrate on large blind cosmic shear surveys, and illustrate our results using simulated shear maps derived from N-Body Lambda-CDM simulations with added noise corresponding to both ground-based and space-based observations.