On the radial velocity detection of additional planets in transiting, slowly rotating M-dwarf systems: the case of GJ 1132

M-dwarfs are known to commonly host high-multiplicity planetary systems. Therefore M-dwarf planetary systems with a known transiting planet are expected to contain additional small planets ($r_p \le 4$ R$_{\oplus}$, $m_p \lesssim 20$ M$_{\oplus}$) that are not seen in transit. In this study we investigate the effort required to detect such planets using precision velocimetry around the sizable subset of M-dwarfs which are slowly rotating ($P_{\mathrm{rot}} \gtrsim 40$ days) and hence more likely to be inactive. We focus on the test case of GJ 1132. Specifically, we perform a suite of Monte-Carlo simulations of the star's radial velocity signal featuring astrophysical contributions from stellar jitter due to rotationally modulated active regions and keplarian signals from the known transiting planet and hypothetical additional planets not seen in transit. We then compute the detection completeness of non-transiting planets around GJ 1132 and consequently estimate the number of RV measurements required to detect those planets. We show that with 1 m s$^{-1}$ RV precision per measurement, only $\sim 50$ measurements are required to achieve a 50\% detection completeness to all non-transiting planets in the system and to planets which are potentially habitable. Throughout we advocate the use of Gaussian process regression as an effective tool for mitigating the effects of stellar jitter including stars with high activity. Given that GJ 1132 is representative of a large population of slowly rotating M-dwarfs, we conclude with a discussion of how our results may be extended to other systems with known transiting planets such as those which will be discovered with TESS.