Flow-geometry microstates

We construct geometric microstates for a class of two-dimensional flow geometries$-$spacetimes that interpolate from an asymptotic AdS$_2$ boundary to a dS$_2$ static patch in the interior$-$by inserting particles behind the horizon. We show that this mechanism produces dS microstates with an Einstein-Rosen bridge of infinite length behind the horizon. The state-counting of these microstates, including wormhole contributions, reproduces the Gibbons-Hawking entropy, $S_{\rm dS}=A^{\rm dS}_{\rm horizon}/4G$. Furthermore, we extend the microstate-counting method to the case of a finite-length Einstein-Rosen bridge. As a result, the Hilbert space of the dS horizon in the flow geometry can be spanned by states with a purely dS Einstein-Rosen bridge, containing no AdS portion on the time-symmetric slice. This provides a concrete realization of dS microstates within a controlled holographic framework.