Historical research guide · 0963 · 2021
String cosmology backgrounds from classical string geometry
Heliudson Bernardo · Robert Brandenberger · Guilherme Franzmann
In one minute
Heliudson Bernardo, Robert Brandenberger and Guilherme Franzmann build an early-universe model out of the geometry a string actually sees, rather than the point-particle geometry of Einstein’s equations. Their starting point is ten dimensions, all nine spatial ones curled up smaller than the length of a string, filled with a hot gas of closed strings. At that size the strings prefer to wrap the compact directions, and a gas of wrapped strings pushes in a way that keeps the Einstein-frame universe still. As the box grows the wrapping unwinds, and the authors show it can unwind completely only in three directions. The result is exactly the universe we live in: three large dimensions carrying on expanding, six frozen at the string scale. What is new is the machinery — corrected equations that hold to all orders in the string length, so the static early phase that string gas cosmology used to assume is now derived. The model also predicts a short burst of accelerated expansion between stages.
Connection
Chapter 2 keeps asking what the vacuum is made of and why its energy behaves as it does; this paper is one of the few places where the question is answered with a concrete alternative geometry, in which the size of space and the inverse size of space are the same physics and six of the dimensions simply stop. For chapter 13 it is a working example of the unified move: one gas, one set of corrected equations, and the split between the three dimensions we move in and the ones we do not comes out of the dynamics rather than being put in by hand.
From the source
The model begins in ten dimensions with all nine spatial directions compactified on a torus smaller than the string length, so that winding modes dominate the string gas and the equation of state parameter is minus one ninth; that state is a de Sitter expansion in the string frame and a static phase in the Einstein frame.Section II, Summary of the Model; Section IV B 1; Section V, Conclusion
The alpha-prime-corrected equations supply, for the first time, actual dynamics for the quasi-static Einstein-frame phase that string gas cosmology previously had to postulate — and they supply it in ten dimensions rather than in the four that the original scenario assumed.Section II, closing paragraph; Section V, first paragraph
As the directions expand and the equation of state drifts toward zero, the geometry stops being isotropic and splits into two sectors: six internal directions whose pressure stays near zero because winding and momentum modes balance at the string scale, and three external directions in which winding modes can annihilate completely so that the equation of state runs on to that of radiation. That split is the string gas mechanism for producing a three-dimensional cosmology from a ten-dimensional one.Section II; Section V, Conclusion
Solutions exist with static internal directions, having vanishing pressure and a non-evolving equation of state, so the balance between winding and momentum modes at the string scale can in principle stabilise all six internal dimensions — a stabilisation that earlier work had suggested but that is derived here within the corrected framework.Section III C, Nonisotropic solutions with static internal directions
Between the later stages the external directions in the Einstein frame pass through a short phase of superexponential accelerated expansion, which would reach a finite-time singularity if it lasted; both such phases are short in this model, and the authors ask whether that acceleration alone could account for the observed spatial flatness of the universe.Section IV B, stages 2 and 3, Equations 71 and 72; Section V
The whole construction rests on T-duality — physics at characteristic radius R being equivalent to physics at the squared string length divided by R — and on the associated O(d,d) symmetry, argued in the cited literature to hold to all orders in alpha-prime; the authors propose embedding the model in double field theory, or describing its first stage in a T-dual frame where the dimensions are large and contracting instead of small and expanding.Section I, Introduction; Section V, final discussion
The way in
https://doi.org/10.1103/physrevd.103.043540How to cite it
Heliudson Bernardo, Robert Brandenberger, Guilherme Franzmann (2021) String cosmology backgrounds from classical string geometry doi:10.1103/physrevd.103.043540