Recovering 21cm Monopole Signals Without Smoothness

We expect the monopole signal at the lowest frequencies below $100\,$MHz to be composed to two components: the deep Rayleigh-Jeans tail of the cosmic microwave background and two distinct features: the dark ages trough at $\sim 17\,$MHz and the cosmic dawn trough at $\sim 75\,$Mhz. These are hidden under orders of magnitude brighter foregrounds whose emission is approximately a power-law with a spectral index $\approx -2.5$. It is usually assumed that monopole signals of interest are separable from foregrounds on the basis of spectral smoothness. We argue that this is a difficult approach and likely impossible for the Dark Ages trough. Instead, we suggest that the fluctuations in the foreground emission around the sky should be used to build a model distribution of possible shapes of foregrounds, which can be used to constrain presence of a monopole signal. We implement this idea using normalizing flows and show that this technique allows for efficient unsupervised detection of the amplitude, width and center of the Dark Ages trough as well as Rayleigh-Jeans tail of the cosmic microwave background for a sufficiently sensitive experiment. We show that achromatic and smooth response significantly helps with foreground separation. We discuss the limitations of the inherent assumptions in this method and the impact on the design of future low-frequency experiments.