Probing cosmic strings by reconstructing polarization rotation of the cosmic microwave background

Cosmic birefringence -- the rotation of the polarization of the cosmic microwave background (CMB) photons as they travel to the Earth -- is a smoking gun for axion-like particles (ALPs) that interact with the photon. It has recently been suggested that topological defects in the ALP field called cosmic strings can cause polarization rotation in quantized amounts that are proportional to the electromagnetic chiral anomaly coefficient $\mathcal A$, which holds direct information about physics at very high energies. In this work, we study the detectability of a random network of cosmic strings through estimating rotation using quadratic estimators (QEs). We show that the QE derived from the maximum likelihood estimator is equivalent to the recently proposed global-minimum-variance QE; the classic Hu-Okamoto QE equals the global-minimum-variance QE under special conditions, but is otherwise still nearly globally optimal. We calculate the sensitivity of QEs to cosmic birefringence from string networks, for the Planck satellite mission, as well as for third- and fourth-generation ground-based CMB experiments. Using published binned rotation power spectrum derived from the Planck 2015 polarization data, we obtain a constraint $\mathcal A^2\,\xi_0 < 0.93$ at the 95\% confidence level, where $\xi_0$ is the total length of strings in units of the Hubble scale per Hubble volume, for a phenomenological but reasonable string network model describing a continuous distribution of string sizes. Looking forward, experiments such as the Simons Observatory and CMB-S4 will either discover or falsify the existence of an ALP string network for the theoretically plausible parameter space $\mathcal A^2\,\xi_0 \gtrsim 0.01$.