Cosmic string dynamics and evolution in warped spacetime

We study the dynamics and evolution of Nambu-Goto strings in a warped spacetime, where the warp factor is a function of the internal coordinates giving rise to a “throat” region. The microscopic equations of motion for strings in this background include potential and friction terms, which attract the strings towards the bottom of the warping throat. However, by considering the resulting macroscopic equations for the velocities of strings in the vicinity of the throat, we note the absence of enough classical damping to guarantee that the strings actually reach the warped minimum and stabilize there. Instead, our classical analysis supports a picture in which the strings experience mere deflections and bounces around the tip, rather than strongly damped oscillations. Indeed, 4D Hubble friction is inefficient in the internal dimensions and there is no other classical mechanism known, which could provide efficient damping. These results have potentially important implications for the intercommuting probabilities of cosmic superstrings.

Figure 1

String trajectory in two-dimensional phase space $(\ell ,v_{\ell })$, i.e. in the case of a single extra dimension $\ell$, assuming a constant 3D velocity $v_x$. The two plots correspond to different warping potentials, with warp factors $h(\ell )=A-B{tanh}^2(\ell )$ (left plot) and $h(\ell )=[A+B\ln (|\ell |)]/{\ell }^4$ (right plot). Starting at a distance away from the potential minimum (located at $\ell =0$ in the former case and $\ell =0.8$ in the latter) and with initial velocity towards it, the string oscillates around the tip, but the motion is only weakly damped by Hubble expansion.

Figure 2

Effect of Hubble damping on scalar field oscillations $\phi (t)$ in both scalar-field-dominated and radiation-dominated cosmology. In the scalar-field-dominated case (left plot), Hubble damping has an approximately constant magnitude, but in the radiation domination case (right plot) cosmological friction is much less efficient as it scales like $t^{-1}$. In the case of strings the situation is more dramatic, because the Hubble term also comes with a factor of $1-2v_x^2-v_{\ell }^2$ [see Eq. (18)] and so, as 3D velocities evolve towards their scaling value, this term becomes zero.

Figure 3

Effect of 3D velocity $v_x$ on the orbits of the string in $(\ell ,v_{\ell })$ phase space, for the case of one extra dimension $\ell$. The warping factor is taken to be of the form $h(\ell )=A-B{tanh}^2(\ell )$. Treating the 3D velocity as a constant parameter, each plot corresponds to a different value of $v_x$. From top left to bottom right: $v_x=0$, 0.3, 0.5, and 0.64. Clearly, as one increases $v_x$, Hubble damping becomes less and less important, since the coefficient of the friction term in Eq. (19) decreases. As a result, the orbit is more stable for larger values of $v_x$. To model this effect, the 3D velocity should be treated as a dynamical variable rather than a constant parameter.

Figure 4

String trajectories (left plots) near the warped minimum in two-dimensional phase space $(\ell ,v_{\ell })$ for the same warping potentials and initial conditions as in Fig. 1, but now treating $v_x$ as a dynamical variable. The 3D velocity $v_x$ oscillates due to its coupling to $v_{\ell }$ [see Eq. (20)], but this effect is small (right plots). The upper plots are for the warping factor $h(\ell )=A-B{tanh}^2(\ell )$, while the lower ones are for $h(\ell )=[A+B\ln (|\ell |)]/{\ell }^4$.

Figure 5

A sample of string trajectories in physical space, in the case of two extra dimensions ${\ell }_1$ and ${\ell }_2$. Different trajectories correspond to different initial conditions for the position $\mathit{l}=({\ell }_1,{\ell }_2)$ and velocity ${\mathbf{v}}_{\ell }=(v_{\ell 1},v_{\ell 2})$. In all cases the warping factor is $h(\mathit{l})=A-B{tanh}^2(|\mathit{l}|)$. The possibilities (depending on the initial conditions) include deflections, bounces, and bound orbits with negligible Hubble friction.