Gravitational backreaction near cosmic string kinks and cusps

We find the leading-order effect of gravitational backreaction on cosmic strings for points near kinks and cusps. Near a kink, the effect diverges as the inverse cube root of the distance to the kink and acts in a direction transverse to the world sheet. Over time, the kink is rounded off, but only regions fairly close to the kink are significantly affected. Near cusps, the effect diverges inverse linearly with the distance to the cusp and acts against the direction of the cusp motion. This results in a fractional loss of string energy that diverges logarithmically with the distance of closest approach to the cusp.

Figure 1

A drawing of an observer point (red circle) at $(u,v)=(0,-\epsilon )$ near a kink in $A$ at $v=0$. The intersection of the light cone with the world sheet is in blue, with the relevant branch solid and the other branch dotted. The region “below” the kink is labeled by $-$, above the kink is labeled by $+$, and at the kink is labeled by $=$. As the observer approaches the kink ($\epsilon \to 0$), the distance at which the intersection crosses the kink, $\delta$, will also go to zero.

Figure 2

How a kink is modified due to backreaction. We show a segment of a world sheet function, where the region above the kink (solid blue, on the left) does not change, but the region below the kink goes from being straight (dashed light blue) to having some curvature (solid blue, on the right). Note that the curvature dies out as one goes to the right, so there is some distance after which the $A^{\prime }$ below the kink before and after backreaction are effectively identical.

Figure 3

The $w$ component of the cusp acceleration integrands for two observers with $\overline{u}=0$, one located at $\overline{v}/L=-2\times {10}^{-5}$ in the past of the cusp (left panel, purple) and the other located at the same distance in the future of the cusp (right panel, green). In both plots, the short-dashed line indicates the predicted acceleration integrand when $u\ll \overline{u}$, while the long-dashed line is for when $u\gg \overline{u}$. Note that the left plot changes sign between regimes, while the right plot maintains the same sign throughout.