### 8.4 Weak lensing by large-scale structure

The weak-lensing method can also be applied on larger scales, i.e., mapping the shear-field induced by
large-scale structures. On these scales, the metric of the expanding-universe-forming-structure is well
represented by a Newtonianly perturbed FLRW metric:
where is the scale factor. Like in the static weak-field case (Eq. 73), is the non-relativistic
potential in units of , but the equality in Eq. 73 does not necessarily imply the equality in
Eq. 116. In GR, this equality is actually respected for both cases (apart from perturbations
around a FLRW background sourced by anisotropic stress), but the relativistic MOND theories,
which have been constructed in order to yield the equality for the static weak-field limit in
Eq. 73, do not harbor this equality in the perturbed FLRW case, and the quantity
is referred to as the gravitational slip. For instance, in the TeVeS (Sections 7.3 and 7.4) and
GEA (Section 7.7) theories, based on unit-norm vector fields, the equality is broken due to the
growth of vector perturbations in the course of cosmological evolution (see, e.g., [128] and
Section 9.2).
Like in the static case, weak gravitational lensing from large-scale structure will actually
depend on , whereas galaxy clustering will arise only from the non-relativistic potential
. By combining information on the matter overdensity at a given redshift (obtained by
measuring the peculiar velocity field) and on the weak lensing maps, Zhang et al. [498] proposed
a clever method to observationally estimate . This allowed Reyes et al. [362] to use
luminous red galaxies in the SDSS survey in to exclude one model from the original TeVeS
theory (Section 7.3) with the original function of [33], thus explicitly showing how such
measurements could be a possible future smoking-gun for all theories based on dynamical vector
fields. But note that other MOND theories such as BIMOND would not be affected by such
measurements.

However, let us finally note a caveat in the interpretation of the weak lensing shear map in the context
of relativistic MOND. While intercluster filaments negligibly contribute to the weak lensing signal in GR, a
single filament inclined by from the line of sight can cause substantial distortion of background
sources pointing toward the filament’s axis in relativistic MOND theories [148]. Since galaxies are generally
embedded in filaments or are projected on such structures, this contribution should be taken into account
when interpreting weak lensing data. This additional difficulty for interpreting weak-lensing data in MOND
is not only true for filaments, but more generally for all low-density structure such as sheets and
voids.