are no modified-gravity effects at all in this quasi-static limit.

It is worth noting that one could have arrived at the form of Eqs  given just our assumptions of second-order equations of motion, the symmetries of the FRW background and quasi-staticity.

In the same quasi-static limit, from the matter conservation equation, we obtain $$\delta_{\text{m}}''+\left(2+\frac{H'}{H}\right)\delta_{\text{m}}'=-k^{2}\Psi=\frac{3}{2}\Omega_{\text{m}}\delta_{\text{m}}h_{1}\left(\frac{1+k^{2}h_{5}}{1+k^{2}h_{3}}\right)\,,$$ or $$\begin{aligned}
f'+f^{2}+\left(2+\frac{H'}{H}\right)f & =\frac{3}{2}\Omega_{\text{m}}h_{1}\left(\frac{1+k^{2}h_{5}}{1+k^{2}h_{3}}\right)\,.\label{eq:f}\end{aligned}$$ On the other hand, we can write for the weak-lensing function $\Sigma$, $$\Sigma=Y(1+\eta)=h_{6}\left(\frac{1+k^{2}h_{7}}{1+k^{2}h_{3}}\right)\,,\label{eq:sigma}$$ where we have introduced two auxiliary functions, $h_{6}=h_{1}(1+h_{2})$ and $h_{7}=(h_{5}+h_{4}h_{2})/(1+h_{2})$. As an aside, one can show that if $G_{3},G_{4}$ depend only on $\phi$ and $G_{5}=\text{const}$, i.e. we are dealing with a k-*essence* theory non-minimally coupled to gravity, then $h_{2}=1$ and $h_{7}=h_{3}$ so that $\Sigma$ becomes independent of $k$. In this limit, the gravitational potential felt by photons is not distorted (we have discussed this model in detail in Ref. [@Sawicki:2012re]).

From the observables $P_i$, $E$ we can construct a model-independent relation measuring the anisotropic stress $\eta$ as follows. From $P_{2},P_{3}$ we can obtain $f=\Omega_{\text{m}0}\Sigma/P_{2}$ and $f'=P_{3}\Omega_{\text{m}0}\Sigma/P_{2}-(\Omega_{\text{m}0}\Sigma/P_{2})^{2}$. Inserting this in Eq.  and employing Eq. , after a little algebra we obtain a simple relation. $$\frac{3P_{2}(1+z)^{3}}{2E^{2}\left(P_{3}+2+\frac{E'}{E}\right)}-1 =\eta =h_{2}\left(\frac{1+k^{2}h_{4}}{1+k^{2}h_{5}}\right)\,.\label{eq:kpol}$$ It is important to stress that the l.h.s. of Eq.  is a 