Finite Energy Geodesic Rays in Big Cohomology Classes

For a big class represented by $\theta$, we show that the metric space $(\mathcal{E}^{p}(X,\theta),d_{p})$ for $p \geq 1$ is Buseman convex. This allows us to construct a chordal metric $d_{p}^{c}$ on the space of geodesic rays in $\mathcal{E}^{p}(X,\theta)$. We also prove that the space of finite $p$-energy geodesic rays with the chordal metric $d_{p}^{c}$ is a complete geodesic metric space. With the help of the metric $d_{p}$, we find a characterization of geodesic rays lying in $\mathcal{E}^{p}(X,\theta)$ in terms of the corresponding test curves via the Ross-Witt Nystr\"om correspondence. This result is new even in the K\"ahler setting.