Revisiting induced gravity in scalar-tensor thermodynamics

TL;DR

This work revisits induced gravity, a globally scale-invariant scalar-tensor theory, within the thermodynamics of scalar-tensor gravity. It derives the gravity temperature $\mathcal{K}\mathcal{T}$ and its evolution equation, clarifying how spontaneous scale symmetry breaking generates the Planck scale and leads to a de Sitter vacuum that acts as a stable GR equilibrium with a cosmological constant. For the induced gravity case with $G_4 = \xi \phi^2$, the authors provide the explicit form $\mathcal{K}\mathcal{T} = 2 \epsilon \, \frac{|\dot{\phi}|}{\phi}$ and a reduced evolution equation, which reduces to the Brans-Dicke attractor analysis in the conformal-matter limit $T^{(\mathrm{m})}=0$. They identify a necessary condition for a late-time GR attractor, $\lim_{\tau\to\infty} (\lambda \phi^2 - \xi R) = 0$, and show that the attractor mechanism aligns with previous BD results, offering a thermodynamic perspective on the emergence of GR with a cosmological constant from induced gravity.

Abstract

Induced gravity, defined as a globally scale-invariant ``first-generation'' scalar-tensor theory, is investigated within the framework of the thermodynamics of modified gravity theories. The ``temperature of gravity'' and its evolution equation are derived for this model, and the resulting expressions are used to analyse General-Relativity equilibrium states and to investigate the possible existence of an attractor mechanism toward Einstein's theory with a cosmological constant.