CFHTLenS: Cosmological constraints from a combination of cosmic shear two-point and three-point correlations

Higher-order, non-Gaussian aspects of the large-scale structure carry valuable information on structure formation and cosmology, which is complementary to second-order statistics. In this work we measure second- and third-order weak-lensing aperture-mass moments from CFHTLenS and combine those with CMB anisotropy probes. The third moment is measured with a significance of $2\sigma$. The combined constraint on $\Sigma_8 = \sigma_8 (\Omega_{\rm m}/0.27)^\alpha$ is improved by 10%, in comparison to the second-order only, and the allowed ranges for $\Omega_{\rm m}$ and $\sigma_8$ are substantially reduced. Including general triangles of the lensing bispectrum yields tighter constraints compared to probing mainly equilateral triangles. Second- and third-order CFHTLenS lensing measurements improve Planck CMB constraints on $\Omega_{\rm m}$ and $\sigma_8$ by 26% for flat $\Lambda$CDM. For a model with free curvature, the joint CFHTLenS-Planck result is $\Omega_{\rm m} = 0.28 \pm 0.02$ (68% confidence), which is an improvement of 43% compared to Planck alone. We test how our results are potentially subject to three astrophysical sources of contamination: source-lens clustering, the intrinsic alignment of galaxy shapes, and baryonic effects. We explore future limitations of the cosmological use of third-order weak lensing, such as the nonlinear model and the Gaussianity of the likelihood function.