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 $10^{-8}\ \text{to}\ 10^{-9}$ for some nearby millisecond pulsars. Determine the maximum ellipticity a realistic crust can sustain (estimates range from about $10^{-9}\ \text{to}\ 10^{-6}$ depending on how the mountain is built) and whether a floor near $\epsilon \sim 10^{-9}$ 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 $\epsilon_{\max}$ below current limits.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
Searches of LIGO-Virgo-KAGRA fourth observing run data were in progress in 2026 with no reported detection.
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