Quantifying the effect of cooled initial conditions on cosmic string network evolution

Quantitative studies of the evolution and cosmological consequences of networks of cosmic strings (or other topological defects) require a combination of numerical simulations and analytic modeling with the velocity-dependent one-scale (VOS) model. In previous work, we demonstrated that a GPU-accelerated code for local Abelian-Higgs string networks enables a statistical separation of key dynamical processes affecting the evolution of the string networks and thus a precise calibration of the VOS model. Here we further exploit this code in a detailed study of two important aspects connecting the simulations with the VOS model. First, we study the sensitivity of the model calibration to the presence (or absence) of thermal oscillations due to high gradients in the initial conditions. This is relevant since in some Abelian-Higgs simulations described in the literature a period of artificial (unphysical) dissipation—usually known as cooling—is introduced with the goal of suppressing these oscillations and accelerating the convergence to scaling. We show that a small amount of cooling has no statistically significant impact on the VOS model calibration, while a longer dissipation period does have a noticeable effect. Second, in doing this analysis we also introduce an improved Markov Chain Monte Carlo based pipeline for calibrating the VOS model, Comparison to our previous bootstrap based pipeline shows that the latter accurately determined the best-fit values of the VOS model parameter, but underestimated the uncertainties in some of the parameters. Overall, our analysis shows that the calibration pipeline is robust and can be applied to future much larger field theory simulations.

Figure 1

The evolution of mean string separation according to the Lagrangian estimator (left panels) and the mean velocity squared according to equation of state estimator (right panels), averaged for sets of 12 runs at each expansion rate in the range [0.50, 0.95]. The top panel shows the results for the hot case (standard case, without cooling), while the middle and bottom panels show the warm and cold cases. Low expansion rates are at the top of the panels while high expansion rates are at the bottom of the panels. All simulations have box sizes ${512}^3$ with constant comoving width (PRS algorithm).

Figure 2

Top panels: the string network average velocity and dimensionless comoving string separation, $v=\sqrt{⟨v^2⟩}$ and $\xi /\eta$, respectively in the left and right panels, for the three cooling scenarios. Bottom panels: The momentum parameter and the energy loss function (left and right panels respectively) for the same cooling scenarios. In all cases the error bars are the statistical uncertainties from averaging over 12 simulations with different initial conditions, and the solid line is the prediction from the VOS model, with the calibrated parameters listed in Table 3. For convenience the values corresponding to simulations in the radiation and matter eras have been highlighted.

Figure 3

Same as Fig. 2, but including the uncertainty propagation described in the main text and with the solid line now being the prediction from the VOS model with the calibrated parameters listed in Table 4.

Figure 4

The corner plots for the posterior distributions in the hot (standard case). Above the 1D histogram for each variable we report the 50th quantile and use the 16th and 84th quantiles to compute and show uncertainties. These three quantiles are indicated by the dashed black lines. Contour plots between pairs of parameters are also shown. The blue lines (and dots) represent the value found via the bootstraping procedure.

Figure 5

Same as Fig. 4, for the warm case.

Figure 6

Same as Fig. 4, for the cold case.