The Ricci flow and the early universe

• Published in: Annals of physics Vol. 458; p. 169452
• Main Author: Luo, M.J.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.11.2023
• Subjects: Inflationary universe; Quantum gravity; Ricci flow; Ricci flow; Inflationary universe; Quantum gravity
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2023.169452

A framework of quantum spacetime reference frame is proposed and reviewed, in which the quantum spacetime at the Gaussian approximation is deformed by the Ricci flow. At sufficient large scale, the Ricci flow not only smooths out local small irregularities making the universe a homogeneous and isotropic Friedmann–Robertson–Walker metric, but also develops a local singularity at the physical-time origin. Due to the phenomenological suppression of the non-Gaussian primordial perturbations, we assume the validity of the Ricci flow applying to the high curvature region near the local singularity of the early universe. The no-local-collapsing theorem of Perelman ensures the existence of a canonical neighborhood around the large curvature pinching point, which resembles a gradient shrinking Ricci soliton (GSRS) solution of the Ricci flow. Without any inflaton field, the GSRS naturally reproduces an exact inflationary deSitter universe near the singularity at the leading order. Without any rolling-down behavior of inflaton, the deviation from exact deSitter described by the “slow roll parameters” can be calculated by a small deviation from the singular flow-time via the Ricci flow, and the primordial perturbations can also be studied on the GSRS background, the power spectrum of the scalar perturbation agrees with present observations, and the one of the tensor perturbation is predicted too small to be detectable than the standard inflation. The previous treatment of the cosmological constant and the effective gravity are also briefly reviewed in the framework. So we argue that the Ricci flow provides us a possible unified view and treatment of the late epoch accelerating expansion and early epoch inflation of the universe without introducing dark energy or inflaton (dark energy of the second kind). •In section 2, we mainly discuss the background of the paper, they are the quantum reference frame and the Ricci flow of the quantum spacetime.•We introduce the quantum clock time model as a preliminary simple model for quantum reference frame in section 2.1. The non-linear sigma model as a full model of quantum spacetime reference frame, and the consequences including the Ricci flow and gravity are introduced in section 2.2-2.6.•In section 3, we discuss the early universe based on the Ricci flow limit, i.e. the gradient shrinking Ricci soliton (GSRS).•In section 3.1, we discuss that the no-local-collapsing theorem of the Ricci flow allows a “canonical neighborhood” to model the spacetime region near the singularity of the early universe.•The inflation, its ”slow roll”, its ending and the primordial perturbations are studied within the framework of GSRS in section 3.2-3.6.