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:
2 g0i = gi0 = 0,g00 = − (1 + 2Φ ),gij = a(t) (1 + 2Ψ )δij, (116 )
where a(t) is the scale factor. Like in the static weak-field case (Eq. 73View Equation), Φ is the non-relativistic potential in units of c2, but the equality Ψ = − Φ in Eq. 73View Equation does not necessarily imply the equality in Eq. 116View Equation. 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. 73View Equation, do not harbor this equality in the perturbed FLRW case, and the quantity Φ + Ψ 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., [128Jump To The Next Citation Point] and Section 9.2).

Like in the static case, weak gravitational lensing from large-scale structure will actually depend on Φ − Ψ, whereas galaxy clustering will arise only from the non-relativistic potential Φ. 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. [498Jump To The Next Citation Point] proposed a clever method to observationally estimate Φ − Ψ. This allowed Reyes et al. [362Jump To The Next Citation Point] 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 f(X ) function of [33Jump To The Next Citation Point], 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 π∕4 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.

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