Cosmological constraints with weak lensing peak counts and second-order
  statistics in a large-field survey
  Peak statistics in weak lensing maps access the non-Gaussian information
contained in the large-scale distribution of matter in the Universe. They are
therefore a promising complement to two-point and higher-order statistics to
constrain our cosmological models. To prepare for the high-precision data of
next-generation surveys, we assess the constraining power of peak counts in a
simulated Euclid-like survey on the cosmological parameters
$\Omega_\mathrm{m}$, $\sigma_8$, and $w_0^\mathrm{de}$. In particular, we study
how the Camelus model--a fast stochastic algorithm for predicting peaks--can be
applied to such large surveys. We measure the peak count abundance in a mock
shear catalogue of ~5,000 sq. deg. using a multiscale mass map filtering
technique. We then constrain the parameters of the mock survey using Camelus
combined with approximate Bayesian computation (ABC). We find that peak
statistics yield a tight but significantly biased constraint in the
$\sigma_8$-$\Omega_\mathrm{m}$ plane, indicating the need to better understand
and control the model's systematics. We calibrate the model to remove the bias
and compare results to those from the two-point correlation functions (2PCF)
measured on the same field. In this case, we find the derived parameter
$\Sigma_8=\sigma_8(\Omega_\mathrm{m}/0.27)^\alpha=0.76_{-0.03}^{+0.02}$ with
$\alpha=0.65$ for peaks, while for 2PCF the value is
$\Sigma_8=0.76_{-0.01}^{+0.02}$ with $\alpha=0.70$. We therefore see comparable
constraining power between the two probes, and the offset of their
$\sigma_8$-$\Omega_\mathrm{m}$ degeneracy directions suggests that a combined
analysis would yield tighter constraints than either measure alone. As
expected, $w_0^\mathrm{de}$ cannot be well constrained without a tomographic
analysis, but its degeneracy directions with the other two varied parameters
are still clear for both peaks and 2PCF. (abridged)
