Information Theoretic Bounds on Cosmic String Detection in CMB Maps with Noise

We use a convolutional neural network (CNN) to study cosmic string detection in cosmic microwave background (CMB) flat sky maps with Nambu-Goto strings. On noiseless maps we can measure string tensions down to order $10^{-9}$, however when noise is included we are unable to measure string tensions below $10^{-7}$. Motivated by this impasse, we derive an information theoretic bound on the detection of the cosmic string tension $G\mu$ from CMB maps. In particular we bound the information entropy of the posterior distribution of $G\mu$ in terms of the resolution, noise level and total survey area of the CMB map. We evaluate these bounds for the ACT, SPT-3G, Simons Observatory, Cosmic Origins Explorer, and CMB-S4 experiments. These bounds cannot be saturated by any method.