Cosmological Phase Transitions and Radius Stabilization in Higher Dimensions

TL;DR

This paper analyzes how large extra dimensions reshape early-universe cosmology by examining cosmological phase transitions and radius stabilization. Using a field-theoretic KK framework, it shows that for $T\gg R^{-1}$ the high-temperature effective potential acquires KK-enhanced terms proportional to $RT$ and, crucially, lacks cubic mass terms, delaying symmetry restoration and often precluding first-order transitions in $D>4$; multi-loop resummation via the CJT formalism reveals plasma-mass screening that further alters the dynamics. Extending to string theory, the authors discuss the Hagedorn phenomenon, showing how the density of states with parameters $b,c$ yields a true Hagedorn temperature $T_H=c^{-1}$ whose interpretation as a limiting temperature or phase transition depends on $b$, and they distinguish open versus closed string thermodynamics. They then propose a thermal mechanism to generate and stabilize large radii of compactification, particularly in Type I strings, where finite-temperature effects tend to push radii to large values and entropy considerations can stabilize them. Overall, the work highlights substantial qualitative and quantitative differences in cosmological evolution when extra dimensions are present and offers mechanisms for both generating and stabilizing large extra dimensions with potential implications for inflation, baryogenesis, and relic production.

Abstract

Recently there has been considerable interest in field theories and string theories with large extra spacetime dimensions. In this paper, we explore the role of such extra dimensions for cosmology, focusing on cosmological phase transitions in field theory and the Hagedorn transition and radius stabilization in string theory. In each case, we find that significant distinctions emerge from the usual case in which such large extra dimensions are absent. For example, for temperatures larger than the scale of the compactification radii, we show that the critical temperature above which symmetry restoration occurs is reduced relative to the usual four-dimensional case, and consequently cosmological phase transitions in extra dimensions are delayed. Furthermore, we argue that if phase transitions do occur at temperatures larger than the compactification scale, then they cannot be of first-order type. Extending our analysis to string theories with large internal dimensions, we focus on the Hagedorn transition and the new features that arise due to the presence of large internal dimensions. We also consider the role of thermal effects in establishing a potential for the radius of the compactified dimension, and we use this to propose a thermal mechanism for generating and stabilizing a large radius of compactification.

Figures (5)

• Figure 1: The ratio $x/x_{{\rm 1-loop}}$ as a function of $\lambda RT$.
• Figure 2: Individual contributions to the effective potential as a function of the radius of compactification, evaluated at the temperature $a_T=1/2$ or $T/M_{\rm string}=1/4\pi$. The total effective potential (lower right) shows a clear tendency to push the radius out to large values.
• Figure 3: The effective potential as a function of the radius of compactification, for temperatures ranging from $a_T=0.5$ (or $T/M_{\rm string}=1/4\pi$) to $a_T=0.7$ (or $T/M_{\rm string}=7/20\pi$). In all cases, the radius is pushed out to large values. Note that the potential diverges to negative values at the Hagedorn temperature $a_T=1/\sqrt{2}\approx 0.707$.
• Figure 4: Solid lines: Entropy as a function of the string compactification radius, for temperatures (a) $a_T=0.333$; (b) $a_T=0.4$; (c) $a_T=0.5$; (d) $a_T=0.6$; and (e) $a_T=0.667$. Dashed lines: Values of the entropy for temperatures (f) $a_T=0.7$; (g) $a_T=0.69$; (h) $a_T=0.68$; and (i) $a_T=0.667$, all calculated at $R/\sqrt{\alpha'}=1$ and held constant as a function of radius. If we assume that a cooling phase of the universe becomes adiabatic (entropy-conserving) at a given initial temperature [(f) through (i)] when the compactification radius is at the string scale, then hierarchically large compactification radii are generated at lower temperatures [(a) through (e)]. Note that it is the Hagedorn phenomenon at $a^\ast_T=1/\sqrt{2}\approx 0.707$ that leads to the dramatic rise in the initial entropy as a function of temperature which in turn generates such hierarchically large radii of compactification.
• Figure 5: Solid line: Entropy as a function of the temperature, evaluated for fixed radius $R/\sqrt{\alpha'}=1$. For small temperatures, the entropy behaves as expected in field theory, while the stringy Hagedorn behavior becomes dominant at higher temperatures. Dashed line: the Hagedorn limiting temperature.