Cosmology from the two-dimensional renormalization group acting as the Ricci flow

Daniel Friedan

Paper · Sep 3, 2019

Cosmology from the two-dimensional renormalization group acting as the Ricci flow

Daniel Friedan

astro-ph.COarXiv:1909.01374v2

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Abstract

The two-dimensional renormalization group acting as the Ricci flow $Λ \frac{\partial}{\partial Λ} g_{μν} = R_{μν}$ produces a specific 1+3 dimensional space-time metric which describes an expanding universe that starts with a big bang $a \sim t^{1/ 3}$ then decelerates until $z = 0.2$ then accelerates until ending at $t_{m a x} = 1.6 t_{H}$ with a big blowup $a \sim (t_{m a x} - t)^{- 1/ 3}$ . The only free parameters are the overall time scale and the value of the present time $t_{0}$ . These are fixed by the Hubble constant $H_{0} = t_{H}^{- 1}$ and the present deceleration parameter $q (t_{0})$ . This crude calculation of cosmology omits all but the gravitational field. The only energy-momentum is purely gravitational dark matter and energy. This is a preliminary exploration towards a specific, comprehensive, testable calculation of cosmology from a fundamental theory in which physics is produced by a quantum version of the two-dimensional renormalization group.

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Submitted: Sep 3, 2019
Last revised: May 11, 2020 (v2)
Subjects: astro-ph.CO, gr-qc, hep-th, math.DG

Comments: v2 corrects one typo; 6 pages, 2 figures, supplementary material (a pdf file showing calculations and three SageMath notebooks performing numerical calculations)

Ricci flow Singularities 3-manifolds Surfaces Physics

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
11A.12
Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles
Uniformization by Ricci flow, and Hamilton’s entropy.
11B.4
Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

[1] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[2] 11A.12
Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles
Uniformization by Ricci flow, and Hamilton’s entropy.

[3] 11B.4
Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

Cosmology from the two-dimensional renormalization group acting as the Ricci flow

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