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Register a new userHigher order moments of lensing convergence - I. Estimate from simulations
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Large area lensing surveys are expected to make it possible to use cosmic shear tomography as a tool to severely constrain cosmological parameters. To this end, one typically relies on second order statistics such as the two - point correlation fucntion and its Fourier counterpart, the power spectrum. Moving a step forward, we wonder whether and to which extent higher order stastistics can improve the lensing Figure of Merit (FoM). In this first paper of a series, we investigate how second, third and fourth order lensing convergence moments can be measured and use as probe of the underlying cosmological model. We use simulated data and investigate the impact on moments estimate of the map reconstruction procedure, the cosmic variance, and the intrinsic ellipticity noise. We demonstrate that, under realistic assumptions, it is indeed possible to use higher order moments as a further lensing probe.
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The unprecedented quality, the increased dataset, and the wide area of ongoing and near future weak lensing surveys allows to move beyond the standard two points statistics thus making worthwhile to investigate higher order probes. As an interesting step towards this direction, we expolore the use of higher order moments (HOM) of the convergence field as a way to increase the lensing Figure of Merit (FoM). To this end, we rely on simulated convergence to first show that HOM can be measured and calibrated so that it is indeed possible to predict them for a given cosmological model provided suitable nuisance parameters are introduced and then marginalized over. We then forecast the accuracy on cosmological parameters from the use of HOM alone and in combination with standard shear power spectra tomography. It turns out that HOM allow to break some common degeneracies thus significantly boosting the overall FoM. We also qualitatively discuss possible systematics and how they can be dealt with.
We compare the efficiency of moments and Minkowski functionals (MFs) in constraining the subset of cosmological parameters (Omega_m,w,sigma_8) using simulated weak lensing convergence maps. We study an analytic perturbative expansion of the MFs in terms of the moments of the convergence field and of its spatial derivatives. We show that this perturbation series breaks down on smoothing scales below 5, while it shows a good degree of convergence on larger scales (15). Most of the cosmological distinguishing power is lost when the maps are smoothed on these larger scales. We also show that, on scales comparable to 1, where the perturbation series does not converge, cosmological constraints obtained from the MFs are approximately 1.5-2 times better than the ones obtained from the first few moments of the convergence distribution --- provided that the latter include spatial information, either from moments of gradients, or by combining multiple smoothing scales. Including either a set of these moments or the MFs can significantly tighten constraints on cosmological parameters, compared to the conventional method of using the power spectrum alone.
Weak gravitational lensing is becoming a mature technique for constraining cosmological parameters, and future surveys will be able to constrain the dark energy equation of state $w$. When analyzing galaxy surveys, redshift information has proven to be a valuable addition to angular shear correlations. We forecast parameter constraints on the triplet $(Omega_m,w,sigma_8)$ for an LSST-like photometric galaxy survey, using tomography of the shear-shear power spectrum, convergence peak counts and higher convergence moments. We find that redshift tomography with the power spectrum reduces the area of the $1sigma$ confidence interval in $(Omega_m,w)$ space by a factor of 8 with respect to the case of the single highest redshift bin. We also find that adding non-Gaussian information from the peak counts and higher-order moments of the convergence field and its spatial derivatives further reduces the constrained area in $(Omega_m,w)$ by a factor of 3 and 4, respectively. When we add cosmic microwave background parameter priors from Planck to our analysis, tomography improves power spectrum constraints by a factor of 3. Adding moments yields an improvement by an additional factor of 2, and adding both moments and peaks improves by almost a factor of 3, over power spectrum tomography alone. We evaluate the effect of uncorrected systematic photometric redshift errors on the parameter constraints. We find that different statistics lead to different bias directions in parameter space, suggesting the possibility of eliminating this bias via self-calibration.
A calculation method for higher-order moments of physical quantities, including magnetization and energy, based on the higher-order tensor renormalization group is proposed. The physical observables are represented by impurity tensors. A systematic summation scheme provides coarse-grained tensors including multiple impurities. Our method is compared with the Monte Carlo method on the two-dimensional Potts model. While the nature of the transition of the $q$-state Potts model has been known for a long time owing to the analytical arguments, a clear numerical confirmation has been difficult due to extremely long correlation length in the weakly first-order transitions, e.g., for $q=5$. A jump of the Binder ratio precisely determines the transition temperature. The finite-size scaling analysis provides critical exponents and distinguishes the weakly first-order and the continuous transitions.
Gravitational lensing surveys have now become large and precise enough that the interpretation of the lensing signal has to take into account an increasing number of theoretical limitations and observational biases. Since the lensing signal is the strongest at small angular scales, only numerical simulations can reproduce faithfully the non-linear dynamics and secondary effects at play. This work is the first of a series in which all gravitational lensing corrections known so far will be implemented in the same set of simulations, using realistic mock catalogues and non-Gaussian statistics. In this first paper, we present the TCS simulation suite and compute basic statistics such as the second and third order convergence and shear correlation functions. These simple tests set the range of validity of our simulations, which are resolving most of the signals at the sub-arc minute level (or $ell sim 10^4$). We also compute the non-Gaussian covariance matrix of several statistical estimators, including many that are used in the Canada France Hawaii Telescope Lensing Survey (CFHTLenS). From the same realizations, we construct halo catalogues, computing a series of properties that are required by most galaxy population algorithms. These simulation products are publicly available for download.
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