How large are neutron-star mountains, given no continuous gravitational waves?
In plain words
A spinning neutron star with a tiny bump on its crust would emit a steady gravitational-wave hum. None has been detected, and it is not known how large a bump the crust can support or how large real bumps are.
Precise statement
Searches for continuous gravitational waves from known pulsars set upper limits on ellipticity $\epsilon$ of order $1e-8\ \text{to}\ 1e-9$ for some nearby millisecond pulsars. Determine the maximum ellipticity a realistic crust can sustain (estimates range from about $1e-9\ \text{to}\ 1e-6$ depending on how the mountain is built) and whether a floor near $\epsilon \sim 1e-9$ exists, as suggested by millisecond-pulsar spin-down.
What would settle it
A detection of continuous gravitational waves, or a crust-elasticity calculation with a self-consistent formation history giving $\epsilon_{\mathrm{max}}$ below current limits.
Status in the literature
Unverified note
Searches of LIGO-Virgo-KAGRA fourth observing run data were in progress in 2026 with no reported detection.