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

and the effective density field where the subscript “0” represents present values with . Using the relation with Eqs. (13.1) and (13.2), it follows thatWriting 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 [66]

where is a comoving radial distance ( is a redshift). The convergence and the shear can be derived from the components of the matrix , as and . For a redshift distribution of the source, the convergence can be expressed as . Using Eqs. (13.2) and (13.4) it follows that where is the maximum distance to the source and .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 assume that the sources are located at the distance (corresponding to the redshift ), giving the relations and . From Eq. (13.3) can be expressed as , where is the matter power spectrum. Hence the convergence spectrum (13.6) readsWe 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 [214]

where is the initial potential generated during inflation, is a transfer function describing the epochs of horizon crossing and radiation/matter transition (), and is the growth function at late times defined by ( corresponds to the scale factor at a redshift ). Our interest is the case in which the transition redshift is smaller than 50, so that we can use the standard transfer function of Bardeen–Bond–Kaiser–Szalay [58]: where and . In the CDM model the growth function during the matter dominance is scale-independent (), but in BD theory with the action (10.10) the growth of perturbations is generally scale-dependent.From Eqs. (13.2) and (13.8) we obtain the matter perturbation for :

The initial power spectrum generated during inflation is , where is the spectral index and is the amplitude of [71, 214]. Therefore we obtain the power spectrum of matter perturbations, as Plugging Eq. (13.11) into Eq. (13.7), we find that the convergence spectrum is given by where Note that satisfies the differential equation .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 [595]:

This estimation is reliable for the transition redshift much larger than 1. In the simulation of Figure 14 the numerical value of for at is 0.056 (with ), which is slightly smaller than the analytic value estimated by Eq. (13.14). The deviation of the spectral index of from the CDM model will be useful to probe modified gravity in future high-precision observations. Note that the galaxy-shear correlation spectrum will be also useful to constrain modified gravity models [528].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 [73]. For BD theory with the action (10.10) the quasi-static results (10.38) and (10.39) of the gravitational potentials give

Since in the scalar-tensor regime (), one can realize in BD theory. Of course we need to wait for further observational data to reach the conclusion that modified gravity is favored over the CDM model.To conclude this session we would like to point out the possibility of using the method of gravitational lensing tomography [574]. 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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