Weak Lensing Probes of Modified Gravity

TL;DR

The paper investigates how three viable modified gravity theories—$f(R)$, DGP, and TeVeS—alter weak lensing observables compared to GR+DE with the same expansion history, focusing on linear scales to separate growth from geometry. It formulates generalized lensing expressions using the growth factor $D_m(k,z)$ and the Poisson factor $D_{\Phi_-}(k,z)$ within a PPF-like framework and introduces the Poisson ratio $\mathcal{P}(\ell)$ to test the matter-potential relation. Forecasts for LSST-like surveys show MG-induced deviations in galaxy-shear and shear-shear correlations up to tens of percent, with distinct scale- and redshift-dependence that differ across models, and the TeVeS case offers the strongest signature. The results demonstrate that future weak lensing data can robustly distinguish MG from smooth Dark Energy and potentially discriminate among MG theories, provided non-linearities are understood and measured in tandem with growth and Poisson tests.

Abstract

We study the effect of modifications to General Relativity on large scale weak lensing observables. In particular, we consider three modified gravity scenarios: f(R) gravity, the DGP model, and TeVeS theory. Weak lensing is sensitive to the growth of structure and the relation between matter and gravitational potentials, both of which will in general be affected by modified gravity. Restricting ourselves to linear scales, we compare the predictions for galaxy-shear and shear-shear correlations of each modified gravity cosmology to those of an effective Dark Energy cosmology with the same expansion history. In this way, the effects of modified gravity on the growth of perturbations are separated from the expansion history. We also propose a test which isolates the matter-potential relation from the growth factor and matter power spectrum. For all three modified gravity models, the predictions for galaxy and shear correlations will be discernible from those of Dark Energy with very high significance in future weak lensing surveys. Furthermore, each model predicts a measurably distinct scale dependence and redshift evolution of galaxy and shear correlations, which can be traced back to the physical foundations of each model. We show that the signal-to-noise for detecting signatures of modified gravity is much higher for weak lensing observables as compared to the ISW effect, measured via the galaxy-CMB cross-correlation.

Figures (8)

• Figure 1: Left panel: Deviation of the growth factor $D_m(k,z)$ from a GR+DE model with the same expansion history, for three different modified gravity models: $f(R)$ (red/dotted; using a value of $B_0=0.4$), DGP (blue/short-dashed), TeVeS (green/long-dashed; normalized to 1 at $z=5$ for clarity). The three lines show different scales: $k=10^{-3}\:h/{\rm Mpc}$ (thick), $k=0.01\:h/{\rm Mpc}$ (medium), and $k=0.1\:h/{\rm Mpc}$ (thin). Right panel: The Poisson factor $D_{\Phi_-}$ [equation (\ref{['eq:Pratio']})] scaled by $k^2/H_0^2/(1+z)$, which reduces to $3\Omega_m/2$ in the case of an unmodified Poisson equation, as a function of redshift for $\Lambda$CDM and modified gravity models. The lines correspond to the same scales as in the left panel.
• Figure 2: Left panel: The galaxy-shear cross power $C^{g\kappa}(\ell)/b$ for $\Lambda$CDM (black/solid) and modified gravity theories: $f(R)$ (red/dotted), DGP (blue/short-dashed), and TeVeS (green/long-dashed). In case of DGP and TeVeS, the thin lines show the corresponding predictions for a GR+DE model with the same expansion history. The foreground galaxies are from bin "F" with median redshift of $\bar{z}_f = 1.1$, and background (sheared) galaxies are from bin "B" with $\bar{z}_b=3.6$ (see text). Right panel: Relative deviation of the galaxy-shear cross power $C^{g\kappa}(\ell)$ of modified gravity models from that of a GR+DE model with identical expansion history, for the same redshift bins as in the left panel.
• Figure 3: Left panel: The shear-shear auto-correlation $C^{\kappa\kappa}(\ell)$ for $\Lambda$CDM and modified gravity theories, for high-redshift galaxies with median redshift $\bar{z}_b=3.6$. The different colors and line styles correspond to the same modified gravity and GR+DE models as in Fig. \ref{['fig:Cgk-vs-l']}. Right panel: Relative deviation of the shear power spectrum $C^{\kappa\kappa}(\ell)$ from that of GR+DE models with the same expansion history, for the same high-redshift galaxies as in the left panel.
• Figure 4: Left panel: Relative deviation of the cross power $C^{g\kappa}(\ell)$ from that of the corresponding GR+DE models, for the same redshift bins as in Fig. \ref{['fig:Cgk-vs-l']}, but in bins of $\Delta\ell = 50$ with statistical errors expected from LSST. Right panel: Same as the left panel, but for the shear-shear correlation of high-$z$ galaxies (as in Fig. \ref{['fig:Ckk-vs-l']}).
• Figure 5: Left panel: The galaxy-shear cross power $C^{g\kappa}(\ell)/b$ at $\ell=100$ for a fixed background galaxy sample with median redshift $\bar{z}_b=3.6$, as a function of the redshift $\bar{z}_f$ of foreground galaxies. The different lines correspond to the same modified gravity and GR+DE models as in Fig. \ref{['fig:Cgk-vs-l']}. Right panel: Relative deviation of the galaxy-shear power predicted by the modified gravity models from those of the corresponding GR+DE models.