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 [66
]
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 [214
]
From Eqs. (13.2) and (13.8
) we obtain the matter perturbation
for
:
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]:
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
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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