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A narrow linear object extending $\sim 60 \,{\rm kpc}$ from the centre of agalaxy at redshift $z \sim 1$ has recently been discovered and interpreted asshocked gas filament forming stars. The host galaxy presents an irregularmorphology, implying recent merger events. Supposing that each of theprogenitor galaxies has a central supermassive black hole (SMBH) and the SMBHsare accumulated at the centre of the merger remnant, a fraction of them can beejected from the galaxy with a high velocity due to interactions between SMBHs.When such a runaway SMBH (RSMBH) passes through the circumgalactic medium(CGM), converging flows are induced along the RSMBH path, and star formationcould eventually be ignited. We show that the CGM temperature prior to theRSMBH perturbation should be below the peak temperature in the cooling functionto trigger filament formation. While the gas is temporarily heated due tocompression, the cooling efficiency increases, and gas accumulation becomesallowed along the path. When the CGM density is sufficiently high, the gas cancool down and develop a dense filament by $z = 1$. The mass and velocity of theRSMBH determine the scale of filament formation. Hydrodynamical simulationsvalidate the analytical expectations. Therefore, we conclude that theperturbation by RSMBHs is a viable channel to form the observed linear object.Using the analytic model validated by simulations, we show that the CGM aroundthe linear object to be warm ($T < 2 \times 10^5 \, K$) and dense ($n > 2\times 10^{-5} (T/2 \times 10^5 \, K)^{-1} \, {\rm cm^{-3}}$).

Formation of dense filaments induced by runaway supermassive black holes