Analytic study of small scale structure on cosmic strings

The properties of string networks at scales well below the horizon are poorly understood, but they enter critically into many observables. We argue that in some regimes, stretching will be the only relevant process governing the evolution. In this case, the string two-point function is determined up to normalization: the fractal dimension approaches one at short distance, but the rate of approach is characterized by an exponent that plays an essential role in network properties. The smoothness at short distance implies, for example, that cosmic string lensing images are almost undistorted. We then add in loop production as a perturbation and find that it diverges at small scales. This need not invalidate the stretching model, since the loop production occurs in localized regions, but it implies a complicated fragmentation process. Our ability to model this process is limited, but we argue that loop production peaks a few orders of magnitude below the horizon scale, without the inclusion of gravitational radiation. We find agreement with some features of simulations, and interesting discrepancies that must be resolved by future work.

Figure 1

Comparison of the model (dashed line) with the data provided by 12 (solid line) in the radiation-dominated era, for which the correlation length is $\xi \simeq 0.30t$.

Figure 2

Comparison of the model (dashed line) with the data provided by 12 (solid line) in the matter-dominated era, for which the correlation length is $\xi \simeq 0.69t$.

Figure 3

Structure on the string, $⟨h_{++}⟩$, as a function of the length $l$ at present time (solid curve). On scales larger than the critical length $l_{\mathrm{c}}\sim 3\times {10}^{-5}{d}_H$ the structure is determined by the matter era expression. On scales below $l_{\mathrm{c}}$ the transition result (2.35) gives an enhanced effect. The dashed curves show the extrapolations of the two relevant expressions: on small scales the actual structure is enhanced relative to the pure matter era result.

Figure 4

Schematic representation of the system considered, with the string lens displayed along the $x_1$-axis and the distant source and the observer located at points $S$ and $O$, respectively.