PSF calibration requirements for dark energy from cosmic shear

The control of systematic effects when measuring galaxy shapes is one of the main challenges for cosmic shear analyses. In this context, we study the fundamental limitations on shear accuracy due to the measurement of the Point Spread Function (PSF) from the finite number of stars. In order to do that, we translate the accuracy required for cosmological parameter estimation to the minimum number of stars over which the PSF must be calibrated. We first derive our results analytically in the case of infinitely small pixels (i.e. infinitely high resolution). Then image simulations are used to validate these results and investigate the effect of finite pixel size in the case of an elliptical gaussian PSF. Our results are expressed in terms of the minimum number of stars required to calibrate the PSF in order to ensure that systematic errors are smaller than statistical errors when estimating the cosmological parameters. On scales smaller than the area containing this minimum number of stars, there is not enough information to model the PSF. In the case of an elliptical gaussian PSF and in the absence of dithering, 2 pixels per PSF Full Width at Half Maximum (FWHM) implies a 20% increase of the minimum number of stars compared to the ideal case of infinitely small pixels; 0.9 pixels per PSF FWHM implies a factor 100 increase. In the case of a good resolution and a typical Signal-to-Noise Ratio distribution of stars, we find that current surveys need the PSF to be calibrated over a few stars, which may explain residual systematics on scales smaller than a few arcmins. Future all-sky cosmic shear surveys require the PSF to be calibrated over a region containing about 50 stars.Comment: 13 pages, 4 figures, accepted by A&

Cosmic Shear Systematics: Software-Hardware Balance

Cosmic shear measurements rely on our ability to measure and correct the Point Spread Function (PSF) of the observations. This PSF is measured using stars in the field, which give a noisy measure at random points in the field. Using Wiener filtering, we show how errors in this PSF correction process propagate into shear power spectrum errors. This allows us to test future space-based missions, such as Euclid or JDEM, thereby allowing us to set clear engineering specifications on PSF variability. For ground-based surveys, where the variability of the PSF is dominated by the environment, we briefly discuss how our approach can also be used to study the potential of mitigation techniques such as correlating galaxy shapes in different exposures. To illustrate our approach we show that for a Euclid-like survey to be statistics limited, an initial pre-correction PSF ellipticity power spectrum, with a power-law slope of -3 must have an amplitude at l =1000 of less than 2 x 10^{-13}. This is 1500 times smaller than the typical lensing signal at this scale. We also find that the power spectrum of PSF size \dR^2) at this scale must be below 2 x 10^{-12}. Public code available as part of iCosmo at http://www.icosmo.orgComment: 5 pages, 3 figures. Submitted to MNRA

Optimal PSF modeling for weak lensing: complexity and sparsity

We investigate the impact of point spread function (PSF) fitting errors on cosmic shear measurements using the concepts of complexity and sparsity. Complexity, introduced in a previous paper, characterizes the number of degrees of freedom of the PSF. For instance, fitting an underlying PSF with a model with low complexity will lead to small statistical errors on the model parameters, however these parameters could suffer from large biases. Alternatively, fitting with a large number of parameters will tend to reduce biases at the expense of statistical errors. We perform an optimisation of scatters and biases by studying the mean squared error of a PSF model. We also characterize a model sparsity, which describes how efficiently the model is able to represent the underlying PSF using a limited number of free parameters. We present the general case and illustrate it for a realistic example of PSF fitted with shapelet basis sets. We derive the relation between complexity and sparsity of the PSF model, signal-to-noise ratio of stars and systematic errors on cosmological parameters. With the constraint of maintaining the systematics below the statistical uncertainties, this lead to a relation between the required number of stars to calibrate the PSF and the sparsity. We discuss the impact of our results for current and future cosmic shear surveys. In the typical case where the biases can be represented as a power law of the complexity, we show that current weak lensing surveys can calibrate the PSF with few stars, while future surveys will require hard constraints on the sparsity in order to calibrate the PSF with 50 stars.Comment: accepted by A&A, 9 pages, 6 figure

Information Gains from Cosmological Probes

In light of the growing number of cosmological observations, it is important to develop versatile tools to quantify the constraining power and consistency of cosmological probes. Originally motivated from information theory, we use the relative entropy to compute the information gained by Bayesian updates in units of bits. This measure quantifies both the improvement in precision and the 'surprise', i.e. the tension arising from shifts in central values. Our starting point is a WMAP9 prior which we update with observations of the distance ladder, supernovae (SNe), baryon acoustic oscillations (BAO), and weak lensing as well as the 2015 Planck release. We consider the parameters of the flat $\Lambda$CDM concordance model and some of its extensions which include curvature and Dark Energy equation of state parameter $w$. We find that, relative to WMAP9 and within these model spaces, the probes that have provided the greatest gains are Planck (10 bits), followed by BAO surveys (5.1 bits) and SNe experiments (3.1 bits). The other cosmological probes, including weak lensing (1.7 bits) and {$\rm H_0$} measures (1.7 bits), have contributed information but at a lower level. Furthermore, we do not find any significant surprise when updating the constraints of WMAP9 with any of the other experiments, meaning that they are consistent with WMAP9. However, when we choose Planck15 as the prior, we find that, accounting for the full multi-dimensionality of the parameter space, the weak lensing measurements of CFHTLenS produce a large surprise of 4.4 bits which is statistically significant at the 8 $\sigma$ level. We discuss how the relative entropy provides a versatile and robust framework to compare cosmological probes in the context of current and future surveys.Comment: 26 pages, 5 figure

iCosmo: an Interactive Cosmology Package

Aims: The interactive software package iCosmo, designed to perform cosmological calculations is described. Methods: iCosmo is a software package to perform interactive cosmological calculations for the low redshift universe. Computing distance measures, the matter power spectrum, and the growth factor is supported for any values of the cosmological parameters. It also computes derived observed quantities for several cosmological probes such as cosmic shear, baryon acoustic oscillations and type Ia supernovae. The associated errors for these observables can be derived for customised surveys, or for pre-set values corresponding to current or planned instruments. The code also allows for the calculation of cosmological forecasts with Fisher matrices which can be manipulated to combine different surveys and cosmological probes. The code is written in the IDL language and thus benefits from the convenient interactive features and scientific library available in this language. iCosmo can also be used as an engine to perform cosmological calculations in batch mode, and forms a convenient adaptive platform for the development of further cosmological modules. With its extensive documentation, it may also serve as a useful resource for teaching and for newcomers in the field of cosmology. Results: The iCosmo package is described with various examples and command sequences. The code is freely available with documentation at http://www.icosmo.org, along with an interactive web interface and is part of the Initiative for Cosmology, a common archive for cosmological resources.Comment: 6 pages including 2 tables and 4 figures. Accepted and published in Astronomy and Astrophysics. Public code and further resources available at http://www.icosmo.or