We quantify the impact of galaxy formation on dark matter halo shapes using cosmological simulations at redshift $z=0$. The haloes are drawn from the IllustrisTNG project, a suite of magneto-hydrodynamic simulations of galaxies. We focus on haloes of mass $10^{10-14} M_odot$ from the 50-Mpc (TNG50) and 100-Mpc (TNG100) boxes, and compare them to dark matter-only (DMO) analogues and other simulations e.g. NIHAO and Eagle. We further quantify the prediction uncertainty by varying the baryonic feedback models in a series of smaller 25 Mpc $h^{-1}$ boxes. We find that: (i) galaxy formation results in rounder haloes compared to the DMO simulations, in qualitative agreement with past hydrodynamic models. Haloes of mass $approx 2times 10^{12} M_odot$ are most spherical, with an average minor-to-major axis ratio of $left< s right> approx 0.75$ in the inner halo, an increase of 40 per cent compared to their DMO counterparts. No significant change in halo shape is found for low-mass $10^{10} M_odot$ haloes; (ii) stronger feedback, e.g. increasing galactic wind speed, reduces the impact of baryons; (iii) the inner halo shape correlates with the stellar mass fraction, which can explain the dependence of halo shapes on different feedback models; (iv) the fiducial and weaker feedback models are most consistent with observational estimates of the Milky Way halo shape. Yet, at fixed halo mass, very diverse and possibly unrealistic feedback models all predict inner halo shapes that are closer to one another than to the DMO results. This implies that a larger observational sample would be required to statistically distinguish between different baryonic prescriptions due to large halo-to-halo variation in halo shapes.
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