[Paper Review] Rigorous "Rich Argument" in Microlensing Parallax

Andrew Gould|arXiv (Cornell University)|Feb 3, 2020

History and Developments in Astronomy3 references4 citations

TL;DR

This paper provides a rigorous analytical derivation of the relative likelihood of discretely degenerate microlensing parallax solutions, showing that the probability ratio depends on a Galactic star count factor $ H_i $ and a Jacobian factor $ B_i = D_{L,i}/\pi_{{\rm E},i} $, which formalizes the 'Rich argument' as a geometric prior. The key contribution is a closed-form expression that validates and generalizes the heuristic $ \pi_{\rm E}^{-2} $ suppression of large-parallax solutions, offering a robust consistency check for Bayesian analyses in microlensing parallax studies.

ABSTRACT

I show that when the observables $(\vec π_{ m E},t_{ m E},θ_{ m E},π_s,\vec μ_s)$ are well measured up to a discrete degeneracy in the microlensing parallax vector $\vec π_{ m E}$, the relative likelihood of the different solutions can be written in closed form $P_i = K H_i B_i$, where $H_i$ is the number of stars (potential lenses) having the mass and kinematics of the inferred parameters of solution $i$ and $B_i$ is an additional factor that is formally derived from the Jacobian of the transformation from Galactic to microlensing parameters. The Jacobian term $B_i$ constitutes an explicit evaluation of the ``Rich Argument'', i.e., that there is an extra geometric factor disfavoring large-parallax solutions in addition to the reduced frequency of lenses given by $H_i$. Here $t_{ m E}$ is the Einstein timescale, $θ_{ m E}$ is the angular Einstein radius, and $(π_s,\vec μ_s)$ are the (parallax, proper motion) of the microlensed source. I also discuss how this analytic expression degrades in the presence of finite errors in the measured observables.

Motivation & Objective

• To rigorously formalize the 'Rich argument' that disfavors large-parallax solutions in microlensing parallax by deriving the relative likelihood of degenerate solutions.
• To derive a closed-form analytical expression for the relative probability of discrete parallax solutions, incorporating both Galactic stellar density and geometric Jacobian factors.
• To validate and generalize the heuristic $ \pi_{\rm E}^{-2} $ suppression of large-parallax solutions as a consequence of physical and geometric priors.
• To provide a consistency check for numerical Bayesian analyses in microlensing, especially for point-lens events where $ \theta_{\rm E} $ is not measured.

Proposed method

• Derives the relative likelihood $ P_i \propto K H_i B_i $, where $ H_i $ is the number of Galactic stars with physical parameters matching solution $ i $, and $ B_i = D_{L,i}/\pi_{{\rm E},i} $ is the Jacobian factor from the transformation to microlensing parameters.
• Assumes the lens-source relative proper motion distribution is approximately linear over the solution space, ensuring validity under small error bars.
• Models the Galactic prior via physical star counts and includes the Jacobian term to account for geometric distortion in parameter space.
• Applies the formalism to well-measured observables, showing that the $ \pi_{\rm E}^{-1} $ dependence in $ B_i $ generalizes the $ \pi_{\rm E}^{-2} $ heuristic of the 'Rich argument'.
• Considers cases where $ \theta_{\rm E} $ is not measured, showing that the formula still applies when $ D_L $ is estimated via Bayesian inference with a Galactic model.
• Demonstrates that the analytic result serves as a sanity check for numerical Bayesian integrations, even when full posterior distributions are computed.

Experimental results

Research questions

• RQ1 How can the relative likelihood of discretely degenerate microlensing parallax solutions be rigorously derived in closed form?
• RQ2 What is the physical and geometric origin of the 'Rich argument' that suppresses large-parallax solutions?
• RQ3 How does the Jacobian factor $ B_i = D_{L,i}/\pi_{{\rm E},i} $ relate to the heuristic $ \pi_{\rm E}^{-2} $ suppression in the literature?
• RQ4 In what cases does the analytic formula break down, and when must numerical Bayesian methods be used instead?
• RQ5 Can this formalism serve as a consistency check for numerical Bayesian analyses in microlensing, especially when $ \theta_{\rm E} $ is not measured?

Key findings

• The relative likelihood of degenerate microlensing parallax solutions is given by $ P_i = K H_i B_i $, where $ H_i $ counts Galactic stars with the physical parameters of solution $ i $, and $ B_i = D_{L,i}/\pi_{{\rm E},i} $ is a geometric Jacobian factor.
• The $ \pi_{\rm E}^{-1} $ dependence in $ B_i $ generalizes and formalizes the heuristic $ \pi_{\rm E}^{-2} $ suppression of large-parallax solutions, validating the 'Rich argument' as a geometric prior.
• The derivation holds under the assumption of linear relative proper motion distribution over the solution space, which is well-justified for small error bars.
• The analytic result provides a robust consistency check for numerical Bayesian analyses, particularly in point-lens events where $ \theta_{\rm E} $ is not measured and $ D_L $ must be inferred.
• The method remains valid even when $ \theta_{\rm E} $ is not measured, provided $ D_L $ is estimated via a Bayesian Galactic model, with the analytic formula serving as a cross-check.
• The approach is most sensitive to deviations from Gaussianity in the parallax error distribution, especially in asymmetric or arc-shaped likelihood contours, where numerical evaluation may be required.