More on the Spectrum of Perturbations in String Gas Cosmology

TL;DR

The paper recasts string gas cosmology in the Einstein frame to better understand background dynamics and cosmological perturbations. It shows that a strong coupling Hagedorn phase with a frozen dilaton can resolve strong-coupling issues and enlarge the horizon, enabling thermal equilibrium over large scales. It demonstrates that perturbation spectra depend sensitively on the dilaton dynamics: a negligible dilaton velocity yields a scale-invariant metric spectrum, whereas significant dilaton motion produces Poisson spectra, with frame transformations linking the two viewpoints. The work also discusses matching conditions across the Hagedorn-to-radiation transition and outlines avenues for modeling the strong-coupling phase and exploring alternative early-universe scenarios.

Abstract

String gas cosmology is rewritten in the Einstein frame. In an effective theory in which a gas of closed strings is coupled to a dilaton gravity background without any potential for the dilaton, the Hagedorn phase which is quasi-static in the string frame corresponds to an expanding, non-accelerating phase from the point of view of the Einstein frame. The Einstein frame curvature singularity which appears in this toy model is related to the blowing up of the dilaton in the string frame. However, for large values of the dilaton, the toy model clearly is inapplicable. Thus, there must be a new string phase which is likely to be static with frozen dilaton. With such a phase, the horizon problem can be successfully addressed in string gas cosmology. The generation of cosmological perturbations in the Hagedorn phase seeded by a gas of long strings in thermal equilibrium is reconsidered, both from the point of view of the string frame (in which it is easier to understand the generation of fluctuations) and the Einstein frame (in which the evolution equations are well known). It is shown that fixing the dilaton at some early stage is important in order to obtain a scale-invariant spectrum of cosmological fluctuations in string gas cosmology.

Figures (4)

• Figure 1: Sketch (based on the analysis of BV of the evolution of temperature $T$ as a function of the radius $R$ of space of a gas of strings in thermal equilibrium. The top curve is characterized by an entropy higher than the bottom curve, and leads to a longer region of Hagedorn behaviour.
• Figure 2: Space-time diagram (sketch) showing the evolution of fixed comoving scales in string gas cosmology. The vertical axis is string frame time, the horizontal axis is comoving distance. The Hagedorn phase ends at the time $t_R$ and is followed by the radiation-dominated phase of standard cosmology. The solid curve represents the Hubble radius $H^{-1}$ which is cosmological during the quasi-static Hagedorn phase, shrinks abruptly to a micro-physical scale at $t_R$ and then increases linearly in time for $t > t_R$. Fixed comoving scales (the dotted lines labeled by $k_1$ and $k_2$) which are currently probed in cosmological observations have wavelengths which are smaller than the Hubble radius during the Hagedorn phase. They exit the Hubble radius at times $t_i(k)$ just prior to $t_R$, and propagate with a wavelength larger than the Hubble radius until they reenter the Hubble radius at times $t_f(k)$. Blindly extrapolating the solutions (\ref{['E7']}, \ref{['E8']}) into the past would yield a dilaton singularity at a finite string time distance in the past of $t_R$, at a time denoted $t_s$. However, before this time is reached (namely at time $t_c$) a transition to a strong coupling Hagedorn phase with static dilaton is reached. Taking the initial time in the Hagedorn phase to be $t_0$, the forward light cone from that time on is shown as a dashed line. The shaded region corresponds to the strong coupling Hagedorn phase.
• Figure 3: Space-time diagram (sketch) showing the evolution of fixed comoving scales in string gas cosmology. The vertical axis is Einstein frame time, the horizontal axis is comoving distance. The solid curve represents the Einstein frame Hubble radius ${\tilde{H}}^{-1}$ which is linearly increasing after ${\tilde{t}}_c$. Fixed comoving scales (the dotted lines labeled by $k_1$ and $k_2$) which are currently probed in cosmological observations have wavelengths which are larger than the Einstein frame Hubble radius during the part of the Hagedorn phase in which the dilaton is rolling. However, due to the presence of the initial strong coupling Hagedorn phase, the horizon becomes much larger than the Hubble radius. The shaded region corresponds to the strong coupling Hagedorn phase.
• Figure 4: Space-time diagram (sketch) showing the evolution of scales in inflationary cosmology. The vertical axis is time, and the period of inflation lasts between $t_i$ and $t_R$, and is followed by the radiation-dominated phase of standard big bang cosmology. During exponential inflation, the Hubble radius $H^{-1}$ is constant in physical spatial coordinates (the horizontal axis), whereas it increases linearly in time after $t_R$. The physical length corresponding to a fixed comoving length scale labelled by its wavenumber $k$ increases exponentially during inflation but increases less fast than the Hubble radius (namely as $t^{1/2}$), after inflation.