Context : 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 .

Aims : 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 .

Methods : We use ground-based optical spectroscopy obtained in the framework of the VLT-FLAMES Tarantula Survey ( VFTS ) to establish the projected equatorial rotational velocities ( \varv _ { e } \sin i ) for components of 114 spectroscopic binaries in 30 Doradus .
The \varv _ { e } \sin i values are derived from the full-width at half-maximum ( FWHM ) of a set of spectral lines , using a FWHM vs . \varv _ { e } \sin i calibration that we derive based on previous line analysis methods applied to single O-type stars in the VFTS sample .

Results : The overall \varv _ { e } \sin i distribution of the primary stars resembles that of single O-type stars in the VFTS , featuring a low-velocity peak ( at \varv _ { e } \sin i < 200 { km } \ > { s } ^ { -1 } ) and a shoulder at intermediate velocities ( 200 < \varv _ { e } \sin i < 300 { km } \ > { s } ^ { -1 } ) .
The distributions of binaries and single stars however differ in two ways .
First , the main peak at \varv _ { e } \sin i \sim 100 { km } \ > { s } ^ { -1 } is broader and slightly shifted toward higher spin rates in the binary distribution compared to that of the presumed-single stars .
This shift is mostly due to short-period binaries ( P _ { \mathrm { orb } } \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { $ \sim$ } } % } \hbox { $ < $ } } } 10 d ) .
Second , the \varv _ { e } \sin i distribution of primaries lacks a significant population of stars spinning faster than 300 { km } \ > { s } ^ { -1 } while such a population is clearly present in the single star sample .
The \varv _ { e } \sin i distribution of binaries with amplitudes of radial velocity variation in the range of 20 to 200 { km } \ > { s } ^ { -1 } ( mostly binaries with P _ { \mathrm { orb } } \sim 10 - 1000 d and/or with q < 0.5 ) is similar to that of single O stars below \varv _ { e } \sin i \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { $ \sim% $ } } } \hbox { $ < $ } } } 170 { km } \ > { s } ^ { -1 } .

Conclusions : Our results are compatible with the assumption that binary components formed with the same spin distribution as single stars and that this distribution contains few or no fast spinning stars .
The larger average spin rate of stars in short-period binaries may either be explained by spin-up through tides in such tight binary systems , or by spin-down of a fraction of the presumed-single stars and long period binaries through magnetic braking ( or by a combination of both mechanisms ) .
Most primaries and secondaries of SB2 systems with P _ { \mathrm { orb } } \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { $ \sim$ } } % } \hbox { $ < $ } } } 10 d appear to have similar rotational velocities .
This is in agreement with tidal locking in close binaries of which the components have similar radii .
The lack of very rapidly spinning stars among binary systems supports the idea that most stars with \varv _ { e } \sin i \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { $ \sim% $ } } } \hbox { $ > $ } } } 300 { km } \ > { s } ^ { -1 } in the single star sample are actually spun-up post-binary interaction products .
Finally , the overall similarities ( low-velocity peak and intermediate velocity shoulder ) of the spin distribution of binary and single stars argue for a massive star formation process in which the initial spin is set independently of whether stars are formed as single stars or as components of a binary system .