if $\mathcal{L}_{x}$ is a constant (i.e. the DE is a cosmological constant) and the matter is a perfect fluid. In the linear regime, the lensing effect is proportional to the lensing potential, which itself is driven by the density perturbations (see e.g. [@Dodelson2003]). In general, this relation can be written as $$\begin{aligned}
k^{2}\Phi_{\text{lens}} & = & k^{2}(\Psi-\Phi)=-\frac{3}{2}Y(1+\eta)\Omega_{\text{m}}\delta_{\text{m}}\\
 & = & -\frac{3}{2}Y(1+\eta)\frac{G_{\text{t}}\sigma_{\text{8,t}}\delta_{\text{t,0}}}{Z}=-\frac{3}{2}\Sigma G\Omega_{\text{m}}\sigma_{8}\delta_{\text{t,0}}\,,\nonumber \end{aligned}$$ where we have defined the “modified lensing” function $\Sigma(k,z)\equiv Y(1+\eta)$. The ellipticity correlation is an integral function of $\Phi_{\text{lens}}$ within a window function that depends on the survey geometry (see e.g. [@Zhang:2007nk]). Assuming good-enough knowledge of the galaxy distribution one can differentiate the correlation integral and obtain the quantity $$\begin{aligned}
\sigma(k,z) & \equiv & \frac{2}{3}(k^{4}P_{\Phi_{\text{lens}}})^{1/2}=\frac{1}{a^{3}E^{2}}\Omega_{\text{m}0}\Sigma G\sigma_{8}\delta_{\text{t,0}}\,.\end{aligned}$$ Then from $\delta_{\text{gal}}(k,z,\mu)$ (with e.g. $\mu=0,1$) and $\sigma(k,z)$ one can measure the three quantities $A,R,L$ defined as $$\begin{aligned}
A & =Gb\sigma_{8}\delta_{\text{t,0}}\,,\quad R=Gf\sigma_{8}\delta_{\text{t,0}}\,,\\
L & =\Omega_{\text{m0}}\Sigma G\sigma_{8}\delta_{\text{t,0}}\,.\end{aligned}$$ The quantities that connect the observations to theory (i.e. to the Lagrangian $\mathcal{L}_{x}$) are $\Omega_{x}$, $f$, $\Sigma$, so it would be optimal to estimate them directly from observations. Now, the cosmic microwave background anisotropy allows one to measure, at least in principle, the initial potential $\Psi_{\text{in}}$ through the Sachs-Wolfe effect. It is, however, impossible to 