Can gravity weaken due the inflation of the universe? As I understood the gravitational force is exchanged between particles at the speed of light. If there is an increment of distance between astronomical objects is it true to think that the gravitational force between them is weakened not only due to distance, but also because the receding velocity of the distant object is somehow paragonable to the speed of gravity, so gravity does not influence the object as it would if the velocity of the object was not relativistic?
 A: With the expanding universe,  structures, or groups of them are moving away from eachother, but not every galaxy is retreating from all other galaxies. many dont have enough empty space between them for dark energy to work, sort of a scaled up version of why we arent being torn apart by expansion. For now. Anyway, many large structures, various clusters of relatively close galaxies will keep their local groups and unless something happens to change how dark energy works (we really have no clue,  but it seems to be constant, if the volume of empty space doubles, dark energy increases accordingly instead of just diluting in the now bigger void) and with dark matter all over, just hanging out and being extra invisible mass for galaxies, clusters etc. At least for the foreseeable future, a loss of gravity in any spot in the universe due to expansion wouldnt bother anything with mass, just the large, empty vacuums between them. The overall mass/gravity in any large  volume of the universe would weaken as that volume with x galaxies becomes twice the volume but the value of x galaxies doesnt change.
    For longer distances,  expansion can be faster than lightspeed, as space itself is massless, so it can do that. Theres boundaries on our current map of the universe called (among several others) the hubble radius and the particle horizon. These boundaries show where due to numerous factors, particles can no longer communicate with anything beyond, there are good wiki pages for these for more detail. So that entire section of the universe will forever be out of reach, other than observations from light/energy that was emitted before expansion forced them across the horizon. I dont believe the effects of anythings gravity at those huge distances would be noticed, but everything past that sphere around us is gone and will never be reachable. 
      I should explain that these boundaries are moved accordingly from place to place in the universe. Theres  equations  to figure out the distances, but if we were in a galaxy 2 billion LY in some direction, we would center them on us there, and have whole different part of the universe with different parts out of reach. And you asked if gravity doesnt exist between somthing at the edge of the observable universe... Each object has its own gravity still, but it cant interact. Im sure I need a lot fixs here, but hope it helps
