Dynamics of Cosmic Superstrings and the Overshoot Problem

TL;DR

The paper investigates the coupled cosmological evolution of a rolling volume modulus and a network of cosmic superstrings with tension that depends on the modulus. It extends a dynamical-systems analysis to a full LVS potential featuring an early runaway and a late-time minimum, considering multiple string-loop species and radiation, and examines the impact on overshoot, loop-energy density, and potential gravitational-wave signals. A key finding is that NS5-brane–induced loops (β=1/6) can stabilize the modulus even without radiation, with string loops contributing a substantial fraction of the energy density during the approach to the minimum and potentially driving detectable high-frequency gravitational waves. The study also shows that oscillating string tension does not produce resonant enhancements, while GW losses could become relevant in the late-time evolution, pointing to future work on GW spectra and multi-species loop dynamics.

Abstract

We exploit the techniques of dynamical systems to study the cosmological evolution of cosmic fundamental strings and effective strings arising from branes wrapped on internal cycles. We also include the whole potential of the volume modulus characterised by an early time run-away towards a late time minimum. We analyse the overshoot problem with and without radiation, and find that the presence of an initial population of strings arising from NS5-branes wrapped around 4-cycles is enough to ensure that the modulus stabilises in its late time minimum, even in the absence of radiation. The reason is the transfer of energy between the modulus and the effective strings caused by the fact that their tension depends on the volume modulus. Interestingly, we find that the energy density of cosmic superstrings is generically very large when the modulus is oscillating around its minimum, opening up the possibility of a detectable gravitational wave signal. We also find no evidence of an efficient resonant enhancement of cosmic superstrings due to an oscillating tension in the late time minimum.

Figures (13)

• Figure 1: Stability map of the fixed points of the dynamical system \ref{["eq:X' no fluid"]}-\ref{["eq:Y' no fluid"]} with exponential potential.
• Figure 2: Evolution of $\Phi$ towards the minimum of the potential \ref{['eq:LVS potential']} located at $\Phi_{\rm min} = 19 M_p$. For each initial value $\Phi_0$, 4 initial densities of loops of F-strings with $\beta = 1/2$ have been tested: $Z_0^2 =\{ 0.01, 0.51, 0.84, 0.99\}$. All of them overshoot the minimum and the maximum (grey and black lines respectively).
• Figure 3: Evolution of $\Phi$ from $\Phi_0 = 6 M_p$ towards the minimum of the potential \ref{['eq:LVS potential']} located at $\Phi_{\rm min} = 19 M_p$ for the 3 values of $\beta$. For each $\beta$, 4 initial densities of loops have been tested: $Z_0^2 =\{ 10^{-4}, 4 \times 10^{-4}, 1.2 \times 10^{-3}, 6.4 \times 10^{-3}\}$. For $\beta =1/2$ and $\beta =1/3$, all of them overshoot, while for $\beta= 1/6$ the trajectories with the higher values of $Z_0^2$ do not overshoot.
• Figure 4: Evolution of the fractional energy densities of the dynamical system in the case of NS5-strings with $\beta = 1/6$ and no radiation. The oscillating behaviour at late times indicates that the field settles into the minimum. The initial conditions for the field and relative energy densities are: $\Phi_0 = 6 M_p$, $X^2_0 = 0$, $Y^2_0=1-10^{-3}$, $Z_0^2 = 10^{-3}$.
• Figure 5: Final averaged fraction of energy density in NS5-loops while $\Phi$ oscillates around its minimum as a function of the initial energy density of the loop fluid $Z_0$ and for different initial values of the field $\Phi_0$. For higher values of $\Phi_0$, the peak shifts to the right. For $\Phi_0 = 6M_p$, the peak is at $Z_0^2 \simeq 8.6 \times 10^{-4}$ , close to the value chosen for Fig. \ref{['fig:NS5 no rad']}.