Stochastic backgrounds in alternative theories of gravity: Overlap reduction functions for pulsar timing arrays

In the next decade gravitational waves might be detected using a pulsar timing array. In an effort to develop optimal detection strategies for stochastic backgrounds of gravitational waves in generic metric theories of gravity, we investigate the overlap reduction functions for these theories and discuss their features. We show that the sensitivity to nontransverse gravitational waves is greater than the sensitivity to transverse gravitational waves and discuss the physical origin of this effect. We calculate the overlap reduction functions for the current NANOGrav pulsar timing array and show that the sensitivity to the vector and scalar-longitudinal modes can increase dramatically for pulsar pairs with small angular separations. For example, the $\mathrm{J}1853+1303\ensuremath{-}\mathrm{J}1857+0943$ pulsar pair, with an angular separation of about 3\ifmmode^\circ\else\textdegree\fi{}, is about ${10}^{4}$ times more sensitive to the longitudinal component of the stochastic background, if it is present, than the transverse components.