tokens: and the shape noise simultaneously, while Ling etal. ( 2015 ) have focused on the surface mass density at z 0.1. Furthermore, we find that the difference of the lensing MFs between the CDM and the HS model has the similar trend to a change of 8 (e.g. see fig. 2 in Shirasaki Yoshida 2014 ). Figure 4. Impact of f ( R ) gravity on the Minkowski functionals (MFs). The three panels represent the results of V 0 , V 1 and V 2 . In every panel, the error bars represent the standard error of the average (i.e. the standard deviation divided by ). Also, coloured points represent the average MF over 100 realizations for the three models, while the bottom portion shows the relative difference of MF between CDM and f ( R ) models. The dashed line shows the Gaussian prediction for CDM model. 5.2 Detectability of imprint of <inline-formula><tex-math id="TM0082" notation="LaTeX"></tex-math></inline-formula> gravity In order to quantify the detectability of f ( R ) gravity in a given statistical quantity, we start by writing a measure of a goodness of fit: where represents a theoretical model of cosmic shear statistic at the i th bin of x for a cosmological model that one wishes to test, is an observed statistic drawn from the true unknown cosmology and C is the covariance matrix of the observed data vector . In our case, corresponds to either the power spectrum, bispectrum, peak counts or MFs, while x refers to the multipole , the peak height or the normalized convergence depending on the statistics. In what follows, we also consider a data vector composed of different statistics when we examine parameter constraints from joint analyses of more than one statistic. In such cases, we properly take into account the off-diagonal components relevant to the two statistics of interest in the covariance matrix. When follows a multivariate Gaussian distribution and if we assume the correct model in , the quantity defined by equation ( 38 ) follows the 2 distribution with the degree of freedom of N bin 1 as the name suggests, where N bin represents the total number of bins for the observables . Borrowing the idea behind equation ( 38 ), which compares the levels of estimated (in the form of a covariance matrix) and measured (the actual scatter around the mean) cosmic variances, we define a similar quantity to assess the statistical power to constrain f R 0 by replacing the numerator with the difference of the expected statistics in two models: where we consider the f ( R ) cosmology characterized by f R 0 and the fiducial CDM cosmology. One can assess the discriminating power of the statistic by the (S/N) 2 defined above. Note that this quantity does not depend on the binning scheme explicitly, as long as we take a binning fine enough not to miss important features in the statistics. In the absence of degeneracy between different cosmological parameters, one can straightforwardly convert the (S/N) 2 to the expected level of constraint on f R 0 ; (S/N) 2 4 for f R 0 10 5
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