Cosmological evolution of cosmic string loops

Abstract

The existence of a scaling evolution for cosmic string loops in an expanding universe is demonstrated for the first time by means of numerical simulations. In contrast with what is usually assumed, this result does not rely on any gravitational back reaction effect and has been observed for loops as small as a few thousandths the size of the horizon. We give the energy and number densities of expected cosmic string loops in both the radiation and matter eras. Moreover, we quantify previous claims on the influence of the network initial conditions and the formation of numerically unresolved loops by showing that they only concern a transient relaxation regime. Some cosmological consequences are discussed.

Figures (4)

• Figure 1: The $(100\ell_\mathrm{c})^3$ comoving volume in the matter era when the observable universe occupies one eighth of the box.
• Figure 2: Evolution in the matter and radiation eras of the energy density associated with long strings and the energy density of loops of physical size $l_\mathrm{phys} = \alpha d_\mathrm{h}$. The time variable is the rescaled conformal time $\eta/\ell_\mathrm{c}$. The energy density of long strings rapidly reaches a scaling regime where $\rho_\infty \propto 1/d_\mathrm{h}^2$, up to some damped relaxation oscillations. These plots show that the energy density of $l_\mathrm{phys}$-sized loops also reaches a long string-like scaling regime where $\mathrm{d} \rho_{\circ} \propto 1/d_\mathrm{h}^2$. This loop scaling regime appears after a relaxation period during which the energy density increases and decreases (the bumps). This transient energy excess signs the relaxation of the initial string network towards its cosmological stable configuration. Note that the transient regime is longer for the smaller loops, as expected from the hierarchical process of loop formation.
• Figure 3: The rescaled loop number density distributions with respect to $\alpha$ (top panels) and $l_\mathrm{phys}=\alpha d_\mathrm{h}$ (bottom panels). They are plotted for different times starting at $t=1.1$ and $t=0.8$ (short dashed curves), with a physical time sampling equal to $1.1$ and $0.8$ for the matter and radiation runs, respectively. The scaling regime propagates from the large scales toward the small scales, while the relaxation bump around the initial correlation length $\ell_\mathrm{c}$ is progressively damped. The $\alpha$ intervals used to determine the scaling function in Eq. (\ref{['eq:powerlaw']}) are represented, as well as the best power law fit (long dashed line) and its estimated systematic errors (dotted lines). Note that the overall maxima of the loop distributions in the top panels correspond to the knee centered around the initial resolution length $\ell_\mathrm{r}$ in the bottom panels.
• Figure 4: Influence of the initial resolution length $\ell_\mathrm{r}$ on the rescaled loop distributions at the end of three small $(40 \ell_\mathrm{c})^3$ radiation era runs having an initial sampling of $10$, $20$ and $40$ ppcl, and a dynamic range of $45$ in physical time (top panels). As in the previous figures, $\alpha = l_\mathrm{phys}/d_\mathrm{h}$ is the loop length in units of the horizon size. In the bottom panels, the influence of the $3$ points cutoff is quantified for the $20$ ppcl run by checking for stress energy conservation.