HL in the linear quasi-static limit. This is the second main result of this paper. Needless to say, a cosmological constant satisfies this consistency relation.

On the other hand, if there are consistent solutions then we obtain an indication in favor of the HL and also direct constraints on it. For instance, if observationally we find that $P_{2},P_{3}$ do not depend on $k$, then from Eq.  we see that the condition $h_{4}=h_{5}$ must be satisfied.

If the consistency relation is not satisfied, the only possible way out of our conclusion is that the conditions for the linear quasi-static limit that we employed to derive Eq.  are not satisfied. This can occur if the rate of change of the functions $h_{1-5}$ is very large, e.g. if $h_{j}'/h_{j}\approx c_{\text{s}}^{2}k^{2}\gg1$ for some $j$. However if the field $\phi$ drives the current accelerated expansion (this is indeed an additional assumption) we expect it to be slow rolling on time scales of the order of $\dot{h}_{j}/h_{j}\sim H$, i.e. $h_{j}'/h_{j}\approx1$. If the sound speed squared $c_{\text{s}}^{2}$ is of order unity, then on typical astrophysical scales of $100$ Mpc$/h$ one has $k^{2}\approx10^{3}$, so the quasi-static limit should be very well satisfied. However if $c_{\text{s}}^{2}$ is less than 