Multi-messenger constraints on Abelian-Higgs cosmic string networks

Nielsen-Olesen vortices in the Abelian-Higgs (AH) model are the simplest realisations of cosmic strings in a gauge field theory. Large-scale numerical solutions show that the dominant decay channel of a network of AH strings produced from random initial conditions is classical field radiation. However, they also show that with special initial conditions, loops of string can be created for which classical field radiation is suppressed, and which behave like Nambu-Goto (NG) strings with a dominant decay channel into gravitational radiation. This indicates that cosmic strings are generically sources of both high-energy particles and gravitational waves. Here we adopt a simple parametrisation of the AH string network allowing for both particle and gravitational wave production. With a reference to a specific model for NG-like loop distribution, this sets the basis for a ``multi-messenger'' investigation of this model. We find that, in order to explain the NANOGrav detection of a possible gravitational wave background, while satisfying the constraint on NG-like loop production from simulations and bounds from the cosmic microwave background, the tension of the AH string in Planck units $G\mu$ and the fraction of the NG-like loops $f_{\rm NG}$ should satisfy $G\mu f_{\rm NG}^{2.6} \gtrsim 3.2\times 10^{-13}$ at 95$\%$ confidence. On the other hand, for such string tensions, constraints from the diffuse gamma-ray background (DGRB) indicate that more than 97$\%$ of the total network energy should be converted to dark matter (DM) or dark radiation. We also consider joint constraints on the annihilation cross-section, the mass, and the relic abundance of DM produced by decays of strings. For example, for a DM mass of 500 GeV, the observed relic abundance can be explained by decaying AH strings that also account for the NANOGrav signal.