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There is a question here Energy extraction from universe expansion? where answer claims that it's possible to extract energy from expansion of universe. Original question proposes to attach a rope to an object, which doesn't sound practical at cosmic scales, but there's another way. Let's assume there's hyper-massive object A and another, smaller object B. Orbit of B  is a stretched ellipsoid, i.e. difference in distances from A to B in between closest and farthest apsisdes of B is very big. Due to expansion of the universe, distance from A to apsisdes should increase over time. There's also a mechanism which harnesses tidal energy from B, by slowing it down, which making its apsides closer to A. Mechanism located either on object A or, if that’s impossible (e.g., if A is a black hole), on third object C, which also orbits A. Now, I do not know how to calculate complete power of such system (i.e. total energy produced per orbital cycle of B divided by time it takes to complete it's orbital cycle), but I've tried to estimate increase in potential energy of B relative to A. Let's say, that B is planet with mass equal mass of earth, A is hyper-massive black hole with mass of one million sun (black hole in center of our galaxy has mass four times greater than that), and farthest distance from A to B is equal to one parsec. The potential energy formula is mgh. Mass and height are known. Value of g is a little trickier, since g are not constant and we actually need definite integral of g(h), with lowest and highest border of integral (sorry, i don't know correct mathematical terms, i hope you understand what i mean) is difference in heights. For simplicity I'll use value of g at a distance of 1 pc. Since g will increase as distance decreases, true value will be bigger than that. Since $g=G×M/R^2$, where G is gravitational constant $6.67×10^{-11}$, M of object A is equal a million mass of suns or $(10^6)×1.98×10^{30} kg$ and R is distance, which is 1 pc or $3.08×10^{16}$ meters, so $$g=\frac{G×M}{R^2}=\frac{6.67×(10^{-11})×(10^6)×1.98×(10^{30})}{(3.08×10^{16})^2}=0.84×10^{-7} m/s^2$$ which is approximally $0.84×10^{-7} m/s^2$. Hubble constant $H_0$ is not accurately measured yet as far as I can tell, so I approximate it to 70 km/s/Mpc (value used in wikipedia). Thus, increase in potential energy U is equal to M×g×H0×h×t, where M is mass of object B equal to earth mass of $5.97×10^{24}$ kg, g is gravitational acceleration which we just calculated, $H_0$ is hubble value (using meters) and h is distance between A and B in Megaparsecs, which at distance of 1 pc equal $10^-6$, and t is time omitted in calculation bellow, since turning it from energy to power cancels it out. Which means that at farthest apsise object B will generate $$M×g×H_0×h=5.97×(10^{24})×0.84×(10^{-7})×70×(10^3)×10^{-6}=351,036×10^{14}W$$ or 35,1033.6 thousands of Gigawatts. Total power of system, of course, will be lower depending on closest distance between A and B, since the universe expansion is lower at close distances. Now, question is — is there wrong assumptions or other errors in my reasoning?

Red Eagle
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Due to expansion of the universe, distance from A to apsisdes should increase over time

This doesn’t happen. The AB system is bound. Over time, the expansion of the universe increases the distance between non-bound systems, not the size of bound systems.

The expansion increases the size of a bound system compared to a non-expanding universe. But does not increase it over time. A bound system in this universe already has that effect included, and it is not going to increase more.

Dale
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