A Ubiquitous Unifying Degeneracy in 2-body Microlensing Systems

While gravitational microlensing by planetary systems [3, 4] can provide unique vistas on the properties of exoplanets [5], observations of such 2-body microlensing events can often be explained with multiple and distinct physical configurations, socalled model degeneracies. An understanding of the intrinsic and exogenous origins of different classes of degeneracy provides a foundation for phenomenological interpretation. Here, leveraging a fast machine-learning based inference framework [6], we present the discovery of a new regime of degeneracy—the offset degeneracy—which unifies the previously known close–wide [7,8] and inner–outer [9] degeneracies, generalises to resonant caustics, and upon reanalysis, is ubiquitous in previously published planetary events with 2-fold degenerate solutions. Importantly, our discovery suggests that the commonly reported close–wide degeneracy essentially never arises in actual events and should, instead, be more suitably viewed as a transition point of the offset degeneracy. While previous studies of microlensing degeneracies are largely studies of degenerate caustics, our discovery demonstrates that degenerate caustics do not necessarily result in degenerate events, which for the latter it is more relevant to study magnifications at the location of the source. This discovery fundamentally changes the way in which degeneracies in planetary microlensing events should be interpreted, suggests a deeper symmetry in the mathematics of 2-body lenses than has previously been recognised, and will increasingly manifest itself in data from new generations of microlensing surveys. Microlensing appears as the time-varying magnification and astrometric deflection of a background star when a foreground system of stellar objects passes close to the line of sight of the background star. If the foreground system contains more than one body, the resultant complex light curve can be analysed to infer some properties of the lens system, in particular the mass ratios q and projected separations s between the individual bodies [5] in units of the angular Einstein radius1 (θE), which is the characteristic scale for gravitational lensing. To date, >100 exoplanets have been discovered with microlensing2 with thousands more expected as 1Here, θE = √ κMπrel, where κ = 4G/(cAU), M is the total lens mass, and πrel =AU/Drel is the lenssource relative parallax. See [5] 2https://exoplanetarchive.ipac.caltech.edu. Accessed 08/21/2021. 1 ar X iv :2 11 1. 13 69 6v 1 [ as tr oph .E P] 2 6 N ov 2 02 1 next-generation microlensing surveys further extend the sensitivity limit from space [10, 11]. For a given 2-body lens configuration, the expected time-varying magnification signal can be accurately predicted [12] using General Relativity [13] with the gravitational lens equation [14]. However, a given set of observations is not guaranteed to map back to a unique lens configuration: there are often multiple, distinct sets of parameters that adequately explain the data. Facing such a potentially ambiguous solution space, it is vital to have prior knowledge and understanding of possible degeneracies to ensure that all viable solutions are properly recovered in the analysis of individual events. While some degeneracies arise due to insufficient time coverage or low data quality, there exists so-called “mathematical” degeneracies that arise from symmetries of the gravitational lens equation itself that appear in certain limits, and therefore persist even in the presence of the highest quality of data. Those degeneracies are best understood by considering microlensing caustics, which are the loci of points where a point source would formally be infinitely magnified by the foreground lens, and can be classified into three regimes for 2-body lenses: close, resonant, and wide [14] (Extended Data Figure 1). As significant deviations from the single-lens magnification occur near caustics, an understanding of caustic properties has been considered to provide insight into the phenomenology, and thus degeneracies, of two-body microlensing events. There are two well-studied mathematical degeneracies: the close-wide degeneracy [7,8] and the inner-outer degeneracy [9], which applies to sources passing close to the central or planetary caustics, respectively. For the close-wide degeneracy, the central caustic shape is known to be invariant under the s ↔ s−1 transformation for |1 − s| q and q 1 [21] (Extended Data Figure 1a;c). This invariance has been cited to explain degeneracies in planetary events where the source passes close to the the primary star, and thus the central caustic. For the inner-outer degeneracy, the planetary caustic(s) in the same condition of |1 − s| q and q 1 is well approximated by a Chang-Refsdal lens that is symmetrical along the star-planet axis [2]. Therefore, sources passing either side of the planetary caustic are expected to create similar magnification patterns (Figure 1a,i; Extended Data Figure 2d,h). In both cases, q is expected to be similar for the two degenerate solutions, but in the close-wide case, there is ambiguity as to whether the planet is close-in (e.g., separations similar to Earth or Mars), or farout (e.g., separations similar to Jupiter or Saturn), thus affecting studies of statistical properties of exoplanets. To probe the existence of new degeneracies, we examined the posterior parameter distributions of a large number simulated binary-lens, single-source (2L1S) events that exhibited multi-modal solutions. While such an endeavour would generally be computational prohibitive with the current status-quo microlensing data analysis approaches, our effort has been made possible by the recent application of likelihood-free-inference (LFI) [15] to 2-body microlensing [6]. The key to the LFI approach is the Neural Density Estimator (NDE), which is a particular type of neural network capable of learning conditional distributions that are complex and multi-modal. Observing that the posterior is a conditional distribution, we trained a NDE on 691,257 events simulated in the context of the next generation Roman Space Telescope microlensing survey [11]. The trained NDE can then infer posterior parameter distributions for any future event unseen during training within seconds, effectively in real time. To isolate events with multi-modal solutions, we apply a clustering algorithm [16] which classifies each of the posteriors into discrete clusters/modes (see Methods). This approach effectively allowed us to “preview” the degeneracies that would be encountered with the increased sensitivity of Roman. Visual inspection of NDE-derived, multi-modal posteriors for planetary events reveals three apparent regimes of degeneracy: the inner–outer degeneracy, the close–wide degeneracy, and degeneracies that involve the resonant caustic which have been observed (e.g. [1,49]) and stud-