CGM cloud sizes from refractive FRB scattering

We explore constraints on the size of cool gas clouds in the circumgalactic medium (CGM) obtainable from the presence, or lack thereof, of refractive scattering in fast radio bursts (FRBs). Our refractive analysis sets the most conservative bounds on parsec-scale CGM clumpiness as it does not make assumptions about the turbulent density cascade. We find that the bulk of low-redshift cool CGM gas, constrained to have densities of $n_{\rm e} \lesssim 10^{-2}\,{\rm cm^{-3}}$, likely cannot produce two refractive images and, hence, scattering. It is only for extremely small cloud sizes $\lesssim 0.1$ pc (about a hundred times smaller than the so-called shattering scale) that such densities could result in detectable scattering. Dense $n_{\rm e} \gtrsim 0.1\,{\rm cm^{-3}}$ gas with shattering-scale cloud sizes is more likely to inhabit the inner several kiloparsecs of the low-redshift CGM: such clouds would result in multiple refractive images and large scattering times $\gtrsim 1 - 10$ ms, but a small fraction FRB sightlines are likely to be affected. We argue that such large scattering times from an intervening CGM would be a signature of sub-parsec clouds, even if diffractive scattering from turbulence contributes to the overall scattering. At redshift $z\sim 3$, we estimate $\sim 0.1\%$ of FRBs to intersect massive proto-clusters, which may be the most likely place to see scattering owing to their ubiquitous $n_{\rm e} \approx 1\,{\rm cm^{-3}}$ cold gas. While much of our discussion assumes a single cloud size, we show similar results hold for a CGM cloud-size distribution motivated by hydrodynamic simulations.