On phase-space singular surfaces in $f(R)$ gravity

Abstract

We perform a Hamiltonian constraint analysis of metric $f(R)$ gravity in the Jordan frame and show that the regular constraint classification degenerates on singular phase-space surfaces located at $f^{\prime }(R)=0$ and $f^{\prime \prime }(R)=0$. We then study the perturbative implications of these surfaces. For exact backgrounds satisfying $f(R)=0$ and $f^{\prime }(R)=0$, the linearized spectrum is empty; the known pure $R^2$ result is therefore a special case of a more general degeneracy in $f(R)$ gravity. We also show that FLRW trajectories in the Starobinsky model can cross the surface $f^{\prime }(R)=0$, but that inhomogeneous perturbations develop a degenerate constraint structure at the crossing. The resulting crossing condition is better interpreted as a regularity condition for perturbative evolution than as an ordinary constraint within the Dirac--Bergmann algorithm. Together, these results distinguish backgrounds that lie entirely on a singular surface from backgrounds that cross one dynamically, and show that the two situations lead to different perturbative degeneracies.

27 pages, 3 figures, 7 tables