$k\equiv k_{\text{phys}}/(aH)\gg1$, and inside the Jeans length of the scalar, $c_{\text{s}}k\gg1$, such that the terms containing $k$ dominate over the time-derivative terms. The sound speed $c_{\text{s}}$ is a particular function of the HL functions $K,G_{3-5}$ evaluated at the *background* level and we have presented it in Eq.  [@DeFelice:2011bh]. In this quasi-static limit for a model belonging to the HL, one obtains [@DeFelice:2011hq] $$\begin{aligned}
\eta & =h_{2}\left(\frac{1+k^{2}h_{4}}{1+k^{2}h_{5}}\right)\,,\, Y=h_{1}\left(\frac{1+k^{2}h_{5}}{1+k^{2}h_{3}}\right)\,.\label{eq:etay}\end{aligned}$$ for suitably defined functions $h_{1-5}.$ In this limit, one also has $Z=Y\eta$.

The functions $h_{1-5}$ express the modification of gravity induced by the HL. In real space they induce a time-dependent Yukawa correction to the Newtonian potential. They are all combinations of HL functions $K,G_{3,4,5}$ and their derivatives with respect to $\phi$ and $X$, all evaluated on the background and are therefore time- but not $k$-dependent, $$h_{i}\equiv h_{i}(z)\equiv h_{i}(\phi,X).$$ The explicit expressions for the functions $h_{i}$ are very complicated and not particularly illuminating; we have nonetheless presented them in appendix \[sec:App\]. For $\Lambda$CDM one has simply $h_{1,2}=1$ and $h_{3,4,5}=0$. If the two gravity-coupling functions in the HL, $G_{4},G_{5}$, are constant, i.e. the effective Planck mass is constant, then $\eta=1$; if moreover $G_{3}$ depends only on $\phi$ (i.e. k-*essence*), then also $Y=1$ and there 