@Lorenz:2010sm]. The former [@Blanco-Pillado:2013qja] gives the distribution of string loops of given size at fixed time, under the assumption that the momentum dependence of the loop production function is weak. The latter [@Lorenz:2010sm] is based on a different numerical simulation [@Ringeval:2005kr], and gives the distribution of non-self intersecting loops at a given time [^3].

The corresponding limits found by combining O1 and O2 data are $G\mu/c^2\leq 1.1 \times 10^{-6}$ for the model of [@Blanco-Pillado:2013qja] and $G\mu/c^2\leq 2.1 \times 10^{-14}$ for the model of [@Lorenz:2010sm]. The Advanced LIGO constraints are stronger for the model of [@Lorenz:2010sm] because the predicted spectrum is larger at 100 Hz for that model. This can be compared with the pulsar timing limits, $G\mu/c^2 \leq 1.6 \times 10^{-11}$ and $G\mu/c^2 \leq 6.2 \times 10^{-12}$, respectively [@Lasky:2015lej].

*Test of General Relativity*— Alternative theories of gravity generically predict the presence of vector or scalar gravitational-wave polarizations in addition to the standard tensor polarizations allowed in general relativity. Detection of the  would allow for direct measurement of its polarization content, enabling new tests of general relativity [@TestingGR_stoch; @stoch_nongr_O1].

When allowing for the presence of alternative gravitational-wave polarizations, the expectation value of the cross-correlation statistic becomes $$\langle \hat C(f) \rangle = 