Information theoretic bounds on cosmic string detection in CMB maps with noise

We use a convolutional neural network 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 Gμ from CMB maps. In particular, we bound the information entropy of the posterior distribution of Gμ 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.