Spectrum of pure $R^2$ gravity: full Hamiltonian analysis

TL;DR

This work provides a definitive Hamiltonian constraint analysis of pure $R^2$ gravity, confirming that the full nonlinear theory propagates $3$ degrees of freedom, while the linearized theory around Minkowski space (and more generally around traceless-Ricci $R=0$ backgrounds) shows no propagating modes. The disappearance of DOF arises from a discontinuous change in the constraint structure upon linearization, with ten second-class constraints turning first-class and the momentum constraint losing its transverse component; this strong-coupling feature is shown to extend to Schwarzschild, Kerr, and radiation-dominated cosmologies. Higher-order perturbations around these backgrounds fail to recover any DOF, underscoring that perturbation theory around $R=0$ surfaces is nonperturbative, even though the full theory remains well-defined away from these surfaces. A cosmological phase-space analysis demonstrates that the evolving universe can cross the $R=0$ surface, highlighting potential physical relevance for cosmology and for understanding $f(R)$ theories near points where $f'(R)=0$ and the frame relation becomes singular.

Abstract

We perform a full Hamiltonian constraint analysis of pure Ricci-scalar-squared ($R^2$) gravity to clarify recent controversies regarding its particle spectrum. While it is well established that the full theory consistently propagates three degrees of freedom, we confirm that its linearised spectrum around Minkowski spacetime is empty. moreover, we show that this is not a feature unique to Minkowski spacetime, but a generic property of all traceless-Ricci spacetimes that have a vanishing Ricci scalar, such as the Schwarzschild and Kerr black hole spacetimes. The mechanism for this phenomenon is a change in the nature of the constraints upon linearisation: ten second-class constraints of the full theory become first-class, while the three momentum constraints degenerate into a single constraint. Furthermore, we show that higher order perturbation theory around these singular backgrounds reveals no degrees of freedom at any order. This is in conflict with the general analysis and points to the fact that such backgrounds are surfaces of strong coupling in field space, where the dynamics of perturbations becomes nonperturbative. We further show via a cosmological phase-space analysis that the evolving universe is able to penetrate through the singular $R=0$ surface.

Figures (4)

• Figure 1: Compactified phase space flow for the system of autonomous equations in (\ref{['DynSys1']}). The four sectors of evolution are indicated in different pastel colors defined in Table \ref{['Sectors']}. The stream lines in green and blue partitions originate in respective sources indicated by green dots in the corners of the plot, and terminate at the red curve representing an expanding de Sitter attractor; the stream lines in red and yellow partitions originate at the green curve representing a contracting de Sitter repulsor, and terminate at respective sinks indicated by red dots in corners. Sources and sinks appear confined at the corners, as a result of our hyperbolic compactification. The lines at $X\!=\!0$ and $Y\!=\!0$ partition the flow at all points except for the magenta point at the origin--- at this point we claim that the blue and green evolutions penetrate the strongly coupled surface. The upper-right quadrant matches that given in Chakraborty:2021mcf.
• Figure 2: Compactified phase space flow for the system of autonomous equations in (\ref{['DynSys2']}), producing an alternative perspective to that shown in Fig. \ref{['DSplot1']}. In these coordinates, the blue and yellow evolutions are split over the two branches, which must be glued along the matching cuts indicated by black dashed lines. The flow lines bounce tangentially from these dashed lines when transitioning from one branch to another. Both blue and yellow evolutions clearly pass through the strongly coupled surface at $Y\!=\!0$, indicated by the dashed magenta lines, which degenerate into the single magenta point at the origin in Fig. \ref{['DSplot1']}. The de Sitter repulsor (green) and attractor (red) lie along the line $Z\!=\!0$ for $Y\!>\!0$. The sources and sinks in the corners are indicated by green and red dots, respectively.
• Figure 3: Compactified phase space flow for the system of autonomous equations in (\ref{['DynSys3']}), producing an alternative perspective to that shown in Figs. \ref{['DSplot1']} and \ref{['DSplot2']}. In these coordinates, all evolutions are split over the two branches, which must be glued along the matching cuts indicated by dashed lines. The flow lines bounce tangetially from the black dashed lines, and continue orthogonally through the magenta dashed lines, that represent the singular surface of the vanishing Ricci scalar. Sources and sinks in corners on the left plot are indicated by green and red dots, respectively, while de Sitter repulsor and attractor on the right plot are indicated by the green and red lines, respectively.
• Figure 4: Compactified flow trajectory for the system in (\ref{['DynSys1D']}). The direction of the flow is with respect to the dimensionless time defined in (\ref{['TauTime']}). However, note that the dimensionful time follows this flow if the Ricci scalar velocity is positive, $\dot{R}\!>\!0$, and actually flows in reverse direction when $\dot{R}\!<\!0$. This is the reason why the right curve captures both the blue and the yellow sectors defined in Table \ref{['Sectors']}, and the left curve captures the green and red sectors. For sectors with positive Ricci velocity the evolution in dimensionful time follows the clockwise direction indicated, while for sectors with negative Ricci velocity this flow is anti-clockwise. The penetration through the strongly coupled surface at $y\!=\!0$ is evident for the right curve, that captures both blue and yellow partitions from Figs. \ref{['DSplot1']}--\ref{['DSplot3']}.