Radially stabilized inflating cosmic strings

In general relativity, local cosmic strings are well known to produce a static, locally flat spacetime with a wedge removed. If the tension exceeds a critical value, the deficit angle becomes larger than $2\pi$, leading to a compact exterior that ends in a conical singularity. In this paper, we investigate dynamical solutions for cosmic strings with super-critical tensions. To this end, we model the string as a cylindrical shell of finite and stabilized transverse width and show that there is a marginally super-critical regime in which the stabilization can be achieved by physically reasonable matter. We show numerically that the static deficit angle solution is unstable for super-critical string tensions. Instead, the geometry starts expanding in the axial direction at an asymptotically constant rate, and a horizon is formed in the exterior spacetime, which has the shape of a growing cigar. We are able to find the analytic form of the attractor solution describing the interior of the cosmic string. In particular, this enables us to analytically derive the relation between the string tension and the axial expansion rate. Furthermore, we show that the exterior conical singularity can be avoided for dynamical solutions. Our results might be relevant for theories with two extra dimensions, modeling our Universe as a cosmic string with a three-dimensional axis. We derive the corresponding Friedmann equation, relating the on-brane Hubble parameter to the string tension or, equivalently, brane cosmological constant.

Figure 1

The radial profile of $\alpha$ at different values of $\tau$ for a sub-critical tension. The dots indicate the shell’s position, left of which the plotted function is $\tilde{\alpha }(\tilde{r})$. After the initial perturbation is carried away in form of outgoing gravitational waves, the metric settles back to the static deficit angle geometry.

Figure 2

For sub-critical tensions, both the axial expansion rate $H$ (a) and azimuthal pressure $p_{\varphi }$ (b) oscillate and approach zero at late times, in accordance with the analytic predictions for the static solution. The numerical error estimates do not exceed the line thickness.

Figure 3

For super-critical tensions, the axial expansion rate $H$ (a) and azimuthal pressure $p_{\varphi }$ (b) both tend to constant, non-zero values at late times. This shows that the static solution is not an attractor anymore. The numerical error estimates are indicated by the gray bands. The dashed lines correspond to the analytic predictions derived in Sec. 6.

Figure 4

For super-critical string tensions, the shell’s position approaches a constant velocity in the interior and in the exterior coordinate patch. In the interior, this asymptotic velocity is $<1$ and its actual value depends on $\overline{\lambda }$. In the exterior, it is $\text{generically}=1$, implying a horizon. The coordinates in either patch only range from the axes to the shell, so the gray regions are not part of the spacetime.

Figure 5

Embedding diagrams of the radial geometry at equidistant time steps $\mathrm{\Delta }\tau$. The interior of the regularized string is drawn red (dark gray), the exterior green (light gray). In the super-critical cases, a light ray emitted from the exterior axis towards the shell is drawn as a solid dark line, visualizing the formation of a horizon.

Figure 6

Interior (left column) and exterior (right column) area of the super-critical cosmic string geometry with (upper row) and without (lower row) conical singularity at the exterior axis. The interior area approaches a constant value, confirming a successful stabilization, whereas the exterior area keeps growing.

Figure 7

There are scaling solutions only for $\overline{\lambda }<27/16\approx 1.69$. Above that value the system (51) and (57) has no real solution. Below, there are two branches, one corresponding to $\tilde{v}<4/5$ and the other to $\tilde{v}>4/5$. The red dots, showing the numerical results, single out the former branch as being the attractor solution and thus physically interesting.

Figure 8

The axial expansion rate of the super-critical string as a function of the tension $\overline{\lambda }$. The linear relation (58) corresponds to a good approximation.

Figure 9

The string can be radially stabilized by physical matter as long as $\overline{\lambda }<128/81\approx 1.58$. For larger values of the tension the equation of state drops below $-1$ indicating that a stabilized shell cannot be realized physically.

Figure 10

Region plot of parameter space, as explained in the text.

Figure 11

All admissible changes of character in vacuum regions can be read off from this diagram, by following incoming or outgoing null rays.

Figure 12

The gradient of $W$ for the Gowdy universe, $W=\sin (t^{*})\sin (r^{*})$. The corresponding spacetime character matches that of Fig. 11, with the gray regions degenerated to single lines.