X}+2X^{2}G_{4,\phi XX}\right)-\nonumber \\
 & -18H^{2}\left(1+G_{4}-7XG_{4,X}-16X^{2}G_{4,XX}-4X^{3}G_{4,XXX}\right)-\nonumber \\
 & -18XH^{2}\Bigl(6G_{5,\phi}+9XG_{5,\phi X}+2X^{2}G_{5,\phi XX}\Bigr)+\nonumber \\
 & +6\dot{\phi}XH^{3}\Bigl(15G_{5,X}+13XG_{5,XX}+2X^{2}G_{5,XXX}\Bigr)\,,\nonumber \\
w_{4}\equiv & 1+2\left(G_{4}-XG_{5,\phi}-XG_{5,X}\ddot{\phi}\right)\,.\nonumber \end{aligned}$$

All of the dynamics of linear perturbations are fully determined by the above four functions. In particular, the speed of propagation of gravitational waves, $c_{\text{T}}$, and the normalization of the kinetic term of these tensor perturbations, $Q_{\text{T}}$, is given by

$$c_{\text{T}}^{2}=\frac{w_{4}}{w_{1}}>0,\qquad Q_{\text{T}}=\frac{w_{1}}{4}>0\,,\label{eq:c_T}$$

with positivity required by stability. From the above, is can be seen that $w_{1}$ has the meaning of the normalization of the tensor perturbations, i.e. it is the effective Planck mass squared. The corresponding quantities for the scalar degree of freedom, the sound speed of dark energy, $c_{\text{s}}$, and the normalization of the kinetic energy for perturbations, $Q_{\text{S}}$, in the presence of dust with energy density $\rho_{\text{m}}$, are

$$\begin{aligned}
c_{\text{s}}^{2}= & \frac{3\left(2w_{1}^{2}w_{2}H-w_{2}^{2}w_{4}+4w_{1}w_{2}\dot{w_{1}}-2w_{1}^{2}(\dot{w}_{2}+\rho_{\text{m}})\right)}{w_{1}(4w_{1}w_{3}+9w_{2}^{2})}>0\,,\label{eq:c_s}\\
Q_{\text{S}}= & \frac{w_{1}\left(4w_{1}w_{3}+9w_{2}^{2}\right)}{3w_{2}^{2}}>0\,.\nonumber \end{aligned}$$

With above definitions in hand, we can define the five scale-independent functions $h_{1-5}$ which appeared in the result . All the observables for scalar perturbations in the *quasi-static* regime are determined by these five functions,

$$\begin{aligned}
h_{1} & \equiv\frac{w_{4}}{w_{1}^{2}}=\frac{c_{\text{T}}^{2}}{w_{1}}\,,\qquad h_{2}\equiv\frac{w_{1}}{w_{4}}=c_{\text{T}}^{-2}\,,\label{eq:h1h2}\\
h_{3} & \equiv\frac{H^{2}}{2XM^{2}}\frac{2w_{1}^{2}w_{2}H-w_{2}^{2}w_{4}+4w_{1}w_{2}\dot{w}_{1}-2w_{1}^{2}(\dot{w}_{2}+\rho_{\text{m}})}{2w_{1}^{2}}\,,\nonumber \\
h_{4} & \equiv\frac{H^{2}}{2XM^{2}}\frac{2w_{1}^{2}H^{2}-w_{2}w_{4}H+2w_{1}\dot{w}_{1}H+w_{2}\dot{w_{1}}-w_{1}(\dot{w}_{2}+\rho_{\text{m}})}{w_{1}}\,,\nonumber \\
h_{5} & \equiv\frac{H^{2}}{2XM^{2}}\frac{2w_{1}^{2}H^{2}-w_{2}w_{4}H+4w_{1}\dot{w}_{1}H+2\dot{w_{1}}^{2}-w_{4}(\dot{w}_{2}+\rho_{\text{m}})}{w_{4}}\,,\nonumber \end{aligned}$$

and where the effective mass squared, $M^{2}$, can be expressed in terms 