Note on Holographic RG Flow in String Cosmology

TL;DR

The paper addresses holographic RG flow for an accelerating universe in string cosmology by proposing a string-frame C-function $C_s(t)=\frac{1}{H_s^{d-1}G_N^{(s)}}$ (with inverse $H_s^{d-1}G_N^{(s)}=G_N\left(H+\frac{\dot{\Phi}}{d-1}\right)^{d-1}$) as a generalization of Strominger's proposal to non-de Sitter futures, and compares it with the Einstein-frame central charge $C(t)=\frac{1}{H^{d-1}G}$. For exponential dilaton potentials, the analysis shows the C-theorem holds in the strong-coupling branch ($\dot{\Phi}>0$, $H_s>0$) for all equation-of-state parameters $\omega$, while in the weak-coupling branch it requires $H_s>0$; the explicit form $H_s=g^{-\frac{2}{d-1}}\left(1\pm\sqrt{\frac{d(1+\omega)}{2}}\right)H$ with $H\sim\frac{1}{t}$ is derived. In the strong-coupling (M-theory) regime, the dual description yields $H_M>0$ universally, ensuring the C-theorem in the M-frame as well, with $H_M=g^{-\frac{1}{(d-1)\sqrt{d}}}\left(H-\frac{\dot{\Phi}}{(d-1)\sqrt{d}}\right)$. These results establish a frame-consistent holographic RG flow for dilaton-driven acceleration and motivate extensions to multi-moduli and tracker-type potentials.

Abstract

We propose a new holographic C-function for the accelerating universe defined in the stringy frame motivated mainly by the fact that the number of degrees of freedom should be infinite for a physical system of infinite size. This is the generalization of Strominger's recent proposal of the holographic C-function to the asymptotically non-de Sitter universe. We find that the corresponding C-theorem holds true if the universe accelerates toward the weak coupling regime driven by the exponential dilaton potential. It also holds in other simple cases.