derive from this information the present power spectrum $\delta_{\text{t,0}}$ since it also depends on a scale- and time-dependent transfer function. Absent a model for DE, this transfer function is unknown and since it acts to process the total perturbation spectrum, changing its $k$-dependence, it also makes $\delta_{\text{t,0}}$ an unknown without further assumptions. This argument shows that the only $\delta_{\text{t,0}}$-independent quantities directly measurable from linear cosmological observations are ratios of $A,R,L$, and their $N$-derivatives, i.e. $$\begin{aligned}
P_{1} & \equiv & R/A=f/b,\\
P_{2} & \equiv & L/R=\Omega_{\text{m}0}\Sigma/f,\\
P_{3} & \equiv & R'/R=f+f'/f.\end{aligned}$$ All other possible $\delta_{\text{t,0}}$-independent ratios, such as $A'/A$, $L'/L$ or $R'/L$ or higher-order $N$-derivatives, can be obtained as combinations of $P_{1-3}$ and their derivatives, for instance $L'/L=P_{2}'/P_{2}+P_{3}$. Other linear-perturbation probes, such as integrated Sachs-Wolfe, cross-correlations or 21-cm flux measurements, add statistics and might extend the observational range but do not break the fundamental degeneracy.

The quantity $P_{1}$ is a well-known observable quantity, often denoted $\beta$ [@Dodelson2003]; since it involves the bias function $b$, related in an unknown way to the model of dark energy, we will not consider it any longer in this paper. The quantity $P_{2}$ has already been introduced in [@Zhang:2007nk] as $E_{G}$ as a test of modified 