Flow-geometry microstates

A bstract 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, $$ {S}_{\mathrm{dS}}={A}_{\mathrm{horizon}}^{\mathrm{dS}}/4G $$ S dS = A horizon dS / 4 G . 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.