Let us consider BD theory with the action (10.10), which includes f (R) gravity as a specific case. Note that the method explained below can be applied to other modified gravity models as well. The equations of matter perturbations and gravitational potentials in BD theory have been already derived under the quasi-static approximation on sub-horizon scales (), see Eqs. (10.38), (10.39), and (10.40). In order to discuss weak lensing observables, we define the lensing deflecting potential
Writing the angular position of a source and the direction of weak lensing observation to be and , respectively, the deformation of the shape of galaxies can be quantified by the amplification matrix . The components of the matrix are given by 
The convergence is a function on the 2-sphere and hence it can be expanded in the form , where with and integers. We define the power spectrum of the shear to be . Then the convergence has the same power spectrum as , which is given by [66, 601]
We recall that, during the matter era, the transition from the GR regime ( and ) to the scalar-tensor regime ( and ) occurs at the redshift characterized by the condition (10.45). Since the early evolution of perturbations is similar to that in the CDM model, the weak lensing potential at late times is given by the formula :
From Eqs. (13.2) and (13.8) we obtain the matter perturbation for :[71, 214]. Therefore we obtain the power spectrum of matter perturbations, as
In Figure 14 we plot the convergence spectrum in f (R) gravity with the potential (10.23) for two different values of together with the CDM spectrum. Recall that this model corresponds to the f (R) model with the correspondence . Figure 14 shows the convergence spectrum in the linear regime characterized by . The CDM model corresponds to the limit , i.e., . The deviation from the CDM model becomes more significant for smaller away from 1. Since the evolution of changes from to at the transition time characterized by the condition , this leads to a difference of the spectral index of the convergence spectrum compared to that of the CDM model :.
Recent data analysis of the weak lensing shear field from the Hubble Space Telescope’s COSMOS survey along with the ISW effect of CMB and the cross-correlation between the ISW and galaxy distributions from 2MASS and SDSS surveys shows that the anisotropic parameter is constrained to be at the 98% confidence level . For BD theory with the action (10.10) the quasi-static results (10.38) and (10.39) of the gravitational potentials give
To conclude this session we would like to point out the possibility of using the method of gravitational lensing tomography . This method consists of considering lensing on different redshift data-bins. In order to use this method, one needs to know the evolution of both the linear growth rate and the non-linear one (typically found through a standard linear-to-non-linear mapping). Afterward, from observational data, one can separate different bins in order to make fits to the models. Having good data sets, this procedure is strong enough to further constrain the models, especially together with other probes such as CMB [322, 320, 632, 292].
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