The initial distribution of spin rates of massive stars is a fingerprint of their elusive formation process. It also sets a key initial condition for stellar evolution and is thus an important ingredient in stellar population synthesis. So far, most studies have focused on single stars. Most O stars are however found in multiple systems. By establishing the spin-rate distribution of a sizeable sample of O-type spectroscopic binaries and by comparing the distributions of binary sub-populations with one another as well as with that of presumed single stars in the same region, we aim to constrain the initial spin distribution of O stars in binaries, and to identify signatures of the physical mechanisms that affect the evolution of the massive stars spin rates. We use ground-based optical spectroscopy obtained in the framework of the VLT-FLAMES Tarantula Survey (VFTS) to establish the projected equatorial rotational velocities (vrot) for components of 114 spectroscopic binaries in 30 Doradus. The vrot values are derived from the full-width at half-maximum (FWHM) of a set of spectral lines, using a FWHM vs. vrot calibration that we derive based on previous line analysis methods applied to single O-type stars in the VFTS sample. The overall vrot distribution of the primary stars resembles that of single O-type stars in the VFTS, featuring a low-velocity peak (at $vrot < 200$ kms) and a shoulder at intermediate velocities ($200 < vrot < 300$ kms). The distributions of binaries and single stars however differ in two ways. First, the main peak at $vrot sim$100 kms is broader and slightly shifted toward higher spin rates in the binary distribution compared to that of the presumed-single stars. Second, the vrot distribution of primaries lacks a significant population of stars spinning faster than 300 kms while such a population is clearly present in the single star sample.
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