Big-Bounce in Quantum f(R)-Cosmology: Polymer Dynamics with Internal Time

TL;DR

This work addresses the classical singularities of general relativity by combining metric $f(R)$ gravity in the Jordan frame with polymer quantum mechanics applied to the Universe volume, using the non-minimally coupled scalar field as an internal clock. By performing ADM reduction and quantizing in the momentum representation, the authors show a non-singular bouncing cosmology: an asymmetric bounce in internal time and a symmetric-like evolution in the reduced clock, depending on the representation and initial conditions. Gaussian wave-packet analysis confirms quasi-classical trajectories for the volume and its conjugate momentum, resembling loop quantum cosmology/polymer results and indicating a well-posed quantum evolution with a minimal volume $V_{ ext{min}} eq 0$. The study advances the synthesis of modified gravity and quantum discreteness, suggesting robust nonsingular behavior in more general settings and inviting further exploration of connections to fundamental quantum gravity mechanisms.

Abstract

We construct the Hamiltonian formulation of the isotropic Universe in a generic metric f(R)-theory in the Jordan frame. We canonically quantize the Universe volume via a polymer formulation, and we adopt the scalar field naturally arising from this scenario as a physical clock. Being within the limit of cut-off values of the space volume, we are legitimized to neglect, at first approximation level, the self-interacting potential term associated with the scalar field. We first study the semi-classical polymer dynamics, outlining the emergence of a bouncing cosmology, both in the internal as well as in the synchronous time. In this latter time variable, we are also able to compare the obtained picture with that of a standard polymer Big-Bounce. We see that in the studied case, the collapsing and expanding branches are no longer symmetric with respect to the minimum volume configuration. Then, we fully quantize the system dynamics in the momentum representation, constructing a suitable dynamical Hilbert space and setting up the dynamics of localized wave packets. The mean value dynamics, both for the momentum and volume spaces, is characterized by a bouncing dynamics as described via the internal time, which closely resembles that one obtained in Loop Quantum Cosmology and Polymerization, respectively.

Figures (9)

• Figure 1: Plot of semiclassical evolution laws: $V(\xi)$ and $p_V(\xi)$. For $V(\xi)$, we considered different values of the constant $\tilde{p}_\xi$, setting $\mu_0=1$. For $p_V(\xi)$ we considered different values of $\mu_0$.
• Figure 2: Semiclassical evolution of the volume with respect to the coordinate time in the synchronous reference frame. Different values of $\mu_0$ are considered while fixing $\tilde{p}_\xi=1$.
• Figure 3: Semiclassical polymer evolution law $V[\xi_{cl}(t)]$ (solid line) compared with the classical $V_{cl}[\xi_{cl}(t)]$ (dashed line) in the synchronous reference frame. In both cases, we fixed $\tilde{p}_\xi=-1$.
• Figure 4: Plot of semiclassical evolution laws $V(\phi)$ and $p_V(\phi)$. For $V(\phi)$, we considered different values of the constant $p_\phi$, setting $\mu_0=1$. For $p_V(\phi)$, we considered different values of $\mu_0$.
• Figure 5: Comparison between semiclassical bounces in the synchronous reference system from $f(R)$ theory (solid line) and standard GR with scalar external field (dashed and dotted lines). Here $\mu_0=1$.