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Synthesizing Exoplanet Demographics: A Single Population of Long-Period Planetary Companions to M Dwarfs Consistent with Microlensing, Radial Velocity, and Direct Imaging Surveys

We present the first study to synthesize results from five different exoplanet surveys using three independent detection methods: microlensing, radial velocity, and direct imaging. The constraints derived herein represent the most comprehensive picture of the demographics of large-separation (>~ 2 AU) planets orbiting the most common stars in our Galaxy that has been constructed to date. We assume a simple, joint power-law planet distribution function of the form d^2N_{pl}/[dlog(m_p)dlog(a)] = A(m_p/M_{Sat})^{alpha}(a/2.5 AU)^{beta} with an outer cutoff radius of the separation distribution function of a_{out}. Generating populations of planets from these models and mapping them into the relevant observables for each survey, we use actual or estimated detection sensitivities to determine the expected observations for each survey. Comparing with the reported results, we derive constraints on the parameters {alpha, beta, A, a_{out}} that describe a single population of planets that is simultaneously consistent with the results of microlensing, RV, and direct imaging surveys. We find median and 68% confindence intervals of alpha = -0.86^{+0.21}_{-0.19} (-0.85^{+0.21}_{-0.19}), beta = 1.1^{+1.9}_{-1.4} (1.1^{+1.9}_{-1.3}), A = 0.21^{+0.20}_{-0.15} dex^{-2} (0.21^{+0.20}_{-0.15} dex^{-2}), and a_{out} = 10^{+26}_{-4.7} AU (12^{+50}_{-6.2} AU) assuming "hot-start" ("cold-start") planet evolutionary models. These values are consistent with all current knowledge of planets on orbits beyond ~2 AU around M dwarfs.