After learning a bit from the thread "The Dark Sky Ahead", I have tried to think through the dynamics of gravitational interactions in a simplified scenario. Unfortunately the GR math of the problem is over my head. My attempt below, using a simpler approach, leaves me quite confused. If my analysis below is correct, this would seem to suggest that the approach that is commonly used, comparing escape velocity with Hubble velocity to determine if a system is bound with respect to the universe expansion, is wrong. Consider an (almost) empty universe expanding exponentially with a single point mass and a test particle. Let p be the test particle. Let P be the point M. Let M be the mass of P. Let R be the distance between P and p. Let h be the fixed constant value of the Hubble constant. Let G be the Gravitational constant. The questions I want to investigate are about stable configurations. Scenario 1. p is stationary, that is, not moving with respect to P Case (a): escape velocity = Hubble velocity The square of the escape velocity Vesc2 = 2 G M / RThe square of the radial velocity of p relative to P due to the expanding universe Vexp2 = h2 R2The value of R for which these two velocities squared values are equal is R1a = (2 G M / h2)1/3 Question 1a: If p is undisturbed by any outside force, is this static configuration with R = R1a possible. (I understand that if it is, any disturbance would cause p to eventually move to infinity.) Case (b): gravitational acceleration = Hubble acceleration The gravitational acceleration is Ag = G M / R2The Hubble acceleration is what p would experience relative to P if M = 0. It is Ah = h2 R The value of R for which these two accelerations are equal is R1b = (G M / h2)1/3 Question 1b: Same as question (1a) except that R = R1b. Note: Clearly Q1a and Q1b cannot both be answered "yes". Scenario 2. p is in an orbit about P. Question 2a: Is a circular orbit possible? Question 2b: If so, is the orbital velocity affected by the universe expansion? Question 2c: If so, how? The following are my guesses at answers. Guess (2a): yes. Guess (2b): yes. Guess (2c): The square of the orbital velocity with h = 0 is Vorb2 = G M / RThe square of the radial velocity of p relative to P due to the expanding universe is Vexp2 = h2 R2The total velocity squared is the sum Vtot2 = G M / R + h2 R2The square of the escape velocity is Vesc2 = 2 G M / RVtot = Vesc implies G M / R = h2 R2,and the value of R is R2c = (G M / h2)1/3 Note: R2c = R1b My guess is that the inward gravitational acceleration must equal the sum of the outward accelerations: Hubble plus centrifugal. G M / R2 = h2 R + Vorb2 / R Vorb2 = G M / R - h2 R2When Vorb = 0, G M / R = h2 R2This is the same result as in Scenario 1 Case (b). The analysis in Scenario 2 seem to suggest the conclusion I included at the beginning of this post.