matter growth rate and $G(k,z)$ is the growth function for matter perturbations, both of which are scale-dependent for general DE models. This velocity field generates redshift distortions as a function of the direction cosine $\mu=(\vec{k}\cdot\vec{\ell})/k$ where $\vec{\ell}$ is the unit line-of-sight vector. The observable $\delta_{\text{gal}}$ can therefore be expressed as [@Kaiser:1987qv] $$\delta_{\text{gal}}(k,z,\mu)=G_{\text{t}}B\sigma_{8\text{,t}}\left(1+\frac{f}{b}\mu^{2}\right)\delta_{\text{t,0}}(k)\,,$$ where $\sigma_{\text{8,t}}$ is the present normalization of the total density spectrum. Now we can write $G_{\text{t}}B=Z\Omega_{\text{m}}\delta_{\text{m}}B/(Z\Omega_{\text{m}}\delta_{\text{m}})_{0}=Gb/(Z\Omega_{\text{m}})_{0}$, so we have the almost-standard expression $$\delta_{\text{gal}}(k,z,\mu)=Gb\sigma_{8}\left(1+\frac{f}{b}\mu^{2}\right)\delta_{\text{t,}0}(k)\,.$$ In this expression we set, using the 8 Mpc$/h$ spherical Fourier space window function $W(k)$, $$\begin{aligned}
\sigma_{8}=\frac{\sigma_{8,\text{t}}}{(Z\Omega_{\text{m}})_{0}} & = & Z_{0}^{-1}\left(\int Z_{0}^{2}P_{\text{m}0}W^{2}(k,R_{8})\mathrm{d}k\right)^{1/2},\end{aligned}$$ which is equivalent to the usual normalization $\sigma_{8}$ if $Z$ depends weakly on $k$.

To discuss weak lensing, we introduce the standard perturbed metric in longitudinal gauge $\mathrm{d}s^{2}=-(1+2\Psi)\mathrm{d}t^{2}+a^{2}(1+2\Phi)\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}$. Since later we will confine ourselves to a scalar dark energy, as part of assumption (*a*), we assume that the DE does not excite vector and tensor modes so that only scalar modes need to be included. It is helpful to introduce the function $Y$, the effective gravitational constant for matter, and the anisotropic stress $\eta$ defined as (see e.g. [@Amendola:2007rr; @DeFelice:2011hq]) $$\begin{aligned}
Y(k,a) & =-\frac{2k^{2}\Psi}{3\Omega_{\text{m}}\delta_{\text{m}}}\,,\quad\eta(k,a)=-\frac{\Phi}{\Psi}\,.\label{eq:poisson-1}\end{aligned}$$ Both $\eta$ and $Y$ are unity on sub-Hubble scales 