tion of motion for test particles can be accounted for in $T_{\mu\nu}$, and we still have $R=4\Lambda_{DE}G/c^2+8\pi
  GT/c^4$ in Eq. $(\ref{extendL})$. For massless particles, $v^\nu\nabla_\nu \left(\mathfrak{R}[4+8\pi
    T/\Lambda_{DE}c^2]v^\mu\right)=0$ instead. With the reparametization $dt \to \mathfrak{R} dt$, the extended GEOM for massless test particles reduces to the GEOM. Our extended GEOM does not affect the motion of photons.

Because the geodesic Lagrangian is extended covariantly, Eq. $(\ref{extendL})$ explicitly satisfies the strong equivalence principal. For $T_{\mu\nu}$, we may still take $T_{\mu\nu} = (\rho+p/c^2)v_\mu v_\nu - p g_{\mu\nu}$ for an inviscid fluid with density $\rho$ and pressure $p$ [@ADS]. While for the GEOM $T^{\hbox{\scriptsize{geo-Dust}}}_{\mu\nu}=\rho v_\mu v_\nu$ for dust, for the extended GEOM the pressure does not vanish [@ADS]; it is a functional of $\rho$ and $\mathfrak{R}$. Nevertheless, in the nonrelativistic limit $p<<\rho c^2$, and $T_{\mu\nu}^{\hbox{\scriptsize{Ext-Dust}}}\approx\rho v_\mu 
v_\nu$ still [@ADS]. Moreover, because $v^\mu v_\mu =c^2$ for the extended GEOM, the first law of thermodynamics still holds for the fluid, and *the standard thermodynamical analysis of the evolution of the universe under the extended GEOM follows much in the same way as before.*

All dynamical effects of extension can be interpreted as the rest energy gained or lost by the test particle due to variations in the local curvature. For these effects not to have already been seen, $\mathfrak{D}(4+8\pi T/\Lambda_{DE}c^2)$ must change very slowly at current experimental limits. As such, we take $
\mathfrak{D}(x) =
\chi(\alpha_\Lambda)\int_x^\infty(1+s^{1+\alpha_\Lambda})^{-1}ds$, where $\alpha_\Lambda \ge 1$ and $\chi(\alpha_\Lambda)$ is set by $\mathfrak{D}(0)=1$. This $\mathfrak{D}(x)$ was chosen for three reasons. First, there is only one free parameter, $\alpha_\Lambda$, to determine. Second, it ensures that the effects of the additional terms in the extended GEOM will not alre