say $10^{-2}$ then the simple form of Eq.  is no longer valid. One might then expect oscillating terms in the $Y,\eta$ equations; it is possible that this behavior could be probed, and possibly rejected, by a similar method we are discussing here but a full analysis of this “cold dark energy” scenario would be required. Another potential difficulty is the fact that these more general scalar field theories contain non-linearities in principle independent of those in the matter perturbations. It may prove difficult to determine on which scales the linear approximation for the dark energy is valid, if at all.

From $P_{2}$ and $P_{3}$ one can build other consistency equations, e.g. by differentiating $P_{2}$ or the combination $P_{2}'/P_{2}+P_{3}=\Sigma'/\Sigma+f$ with respect to $N$ and again employing Eqs , . These relations however require derivatives of the observables $P_{2},P_{3}$ and will introduce derivatives of the $h_{1-5}$ functions, so appear to be less useful than Eq. .

We observe also that the propagation speed $c_{\text{T}}$ of gravitational waves is a function of the HL coupling functions, see Eq.  [@Kobayashi:2011nu]. A detection of a source both in gravitational and electromagnetic waves could allow for a measurement of $c_{\text{T}}$ and therefore new independent constraints on 