A dynamical systems approach to studying the equivalence principle in dilaton gravity

TL;DR

The paper addresses how cosmological evolution in a string-inspired dilaton model within the Damour--Polyakov regime governs deviations from the equivalence principle. It develops a closed autonomous dynamical-system description, expanded around the least-coupling point, and identifies a stable attractor at $(x,y)=(0,0)$ whose approach is governed by Jacobian eigenvalues $\lambda_\pm$. Crucially, at any finite epoch the ambient coupling $\alpha_{\rm env}$ is set by the residual displacement $x_{\rm env}$ and determines the strength of the scalar fifth force, with the relation $(m_g^{(p)}/m_{\rm in}-1) \propto x_{\rm env}^2$. This framework provides a direct dynamical bridge between cosmological relaxation and weak-field departures from General Relativity, offering a route to connect cosmic history with terrestrial and Solar-System tests of gravity.

Abstract

We study a string-inspired dilaton cosmology in the Damour--Polyakov (DP) regime using dynamical-systems methods, aiming to make explicit how cosmological relaxation controls deviations from the equivalence principle. Working in the Einstein frame, we consider a spatially flat FLRW universe filled with pressureless matter and a universally coupled dilaton. Expanding the conformal coupling function and the scalar potential around the least-coupling point, we obtain a closed and self-consistent autonomous system governing the late-time evolution of the scalar-matter sector. The resulting phase space contains a stable fixed point associated with least coupling, approached only asymptotically along cosmological trajectories. Therefore, at any finite epoch the solution typically retains a small displacement from the fixed point. In the DP regime this finite-epoch displacement sets the ambient coupling, and thus determines the magnitude of fifth-force effects and deviations from the equivalence principle in the nonrelativistic limit. By linearising the system around a finite-epoch reference state, we show that the damping of the displacement is controlled by the Jacobian eigenvalues of the DP fixed point. This yields a direct dynamical estimate of how rapidly deviations from the equivalence principle are reduced during cosmological evolution. The mechanism is global and cosmological in origin, and is conceptually distinct from local environmental screening as in chameleon or symmetron scenarios. Overall, our results illustrate how phase-space techniques provide a clear bridge between cosmological dynamics and weak-field departures from General Relativity.

Figures (2)

• Figure 1: Quasi-static phase portraits of the linearised Damour--Polyakov subsystem in the $(x,y)$ plane for two background matter fractions: (a) $\Omega_{m0}=0.25$ and (b) $\Omega_{m0}=0.32$, both with $h_0=1.0$. The fixed point at $(0,0)$ corresponds to the least-coupling configuration.
• Figure 2: Evolution of the dimensionless scalar displacement $x(N)=(\phi-\phi_\ast)/M_{\rm Pl}$ as a function of $N=\ln a$ for representative initial conditions $(\Omega_0,h_0)$. In all cases $x$ is damped and relaxes towards $x\to 0$, approaching the least--coupling configuration. Since the ambient matter coupling in the Damour--Polyakov regime scales as $\alpha_{\rm env}\propto x$, the decay of $x(N)$ implies a progressive suppression of composition-dependent fifth forces and deviations from the equivalence principle at late times.