So, regarding the first part of the answer, from where does the energy come from? Gravitational potential energy? And for the second part: But even if the energy gained by the strings has an identifiable origin, and does not come out of nowhere, would the cosmic strings gain more and more energy as long as spacetime expands? And in what form would this energy manifest? Heat? Radiation? @benrg

@vengaq 1: The energy is already there in the initial conditions of ΛCDM cosmology. Inflation can (allegedly) produce those initial conditions but I don't have a good story for where it ultimately comes from. 2: The strings don't gain more and more energy. There is no totally well defined conserved energy, but there are good enough approximate conservation laws to prevent that, AFAIK. 3: The energy is the mass of the cosmic strings; it's similar to QCD flux tube energy.

So in summary are you saying that Harrison's gedanken experiment of mining the energy from spacetime expansion wouldn't work? @benrg

@vengaq It works, but you can only extract a bounded amount of energy that way (as Harrison correctly says). That would normally suggest a conservation principle of some sort at work, but Harrison nonetheless argues that there isn't one, and I think his arguments don't hold water.

But Harrison says that you can extract a limited amount in a decelerating universe. While in an accelerating one, one could (at least in principle) extract an unlimited amount. In an accelerating universe, wouldn't the strings be stretched again after the first stretch-collapse phase you described, adding more internal energy to it as long as spacetime expansion is accelerated? @benrg

@vengaq He says "The energy mined inside the Hubble sphere ... is finite" at the top of page 65... although he then goes on to say that you might get more by tethering a body past the de Sitter horizon, which is wrong (you can't fish things out from beyond event horizons). I didn't want to say too much about Harrison's paper in the answer since it seemed peripheral, but if you want me to pick it apart I can. I just realized that you may have mistaken Harrison's tethering strings (which are just rope) for cosmic strings (which he only mentioned briefly), and asked the wrong question.

No, I know the difference between the rope and the cosmic strings. I was interested in seeing his experiment applied to something "natural" (like a cosmic string) instead of an artificial situation with a man made rope... Yes, but just after that phrase says that the mined energy could be infinite (as I understand it, he talks about staying in the Hubble volume, and then he talks about going beyond it). @benrg

In the first page, he says that you can take a limited amount of energy by attaching the imaginary rope to an object, deceleraring that object by making tension in the string and then, once the object has stopped and the tension has vanished, attaching the string to another object...until everything inside the horizon is stopped. However, he says that in an accelerated universe the tension does not vanish, so the energy extracted could be (at least in principle) unlimited. He repeats this numerous times in the 2nd page of the article (page 64). @benrg

Also, why couldn't we attach the string to objects outside the cosmological horizon? Is it because we'd need to travel faster than the speed of light to catch anything beyond it? @benrg

@vengaq Page 64 is (according to the section name) a nonrelativistic argument. The nonrelativistic case ends up being very different because there's no event horizon. There are a number of old questions on this SE about why you can't lower things below an event horizon. Most are about black hole horizons, but it extends to other event horizon types. I don't mind writing more about this paper, but it seems way out of scope for this question. You could ask another. I think "Harrison says such and such, but benrg said blah blah in comments, who's right?" is a fine question.

The thing is that I don't understand a couple of things: Harrison at the end of the intro says (as a general explanation for his arguments in the paper, not focusing in any particular case) that he argues that you can extract an unlimited amount of energy in accelerating universes and a limited amount in decelerating universes. Ours is an accelerating universe and if it is so easy to explain (you've made so in a few lines) that you cannot extract unlimited energy even in an accelerating universe, why would he lie? @benrg

If that is true, couldn't we then re-attach the string to harness the energy from the receding movement? For instance Harrison, says in the 1st page that the tension in the string between the objects would vanish in decelerating universes but not necessarily in accelerating ones as a general statement (before delving into the specific cases of non-relativistic, relativistic, newtonian...). Doesn't that indicate that in an accelerated expansion, even if the object is stopped, it would start receding again, forming tension in the string? @benrg

In another question slightly related to this (physics.stackexchange.com/questions/283350/…), user pela explains that if two objects are sufficiently far apart, even if they move towards each other, they would be pulled apart due to Hubble flow. Wouldn't that be similar to the case of the object attached to the string? If it is sufficiently far apart, even if we put it at rest with the string tension, wouldn't it move again by the drag of the accelerated expansion? (More info: arxiv.org/abs/astro-ph/0104349) @benrg

@vengaq I'd recommend Christoph's answer to that question over pela's. Maybe I'll write one and try to dump some of what I've been saying here into it. In short, objects at relative rest may later gravitate toward each other, or (if $Λ>0$) away, but either way it's a force (= mass × acceleration) and is conservative. What doesn't exist is a frictional/Aristotelian effect that makes things want to go at a certain velocity. If you tether a galaxy, it doesn't "remember" its old speed and return to it, nor is it dragged back to it by galaxies you didn't tether, or by expanding-space aether.

But then would a receding galaxy by spacetime expansion (once put into rest by the rope) move away at less speed than when it was initially separating? So if we repeat the process of attaching/detaching/re-ataching the rope to the galaxy, each time we'd re-attach the string to the galaxy, the tension would be weaker and weaker, as the initial speed of the galaxy could never be recovered? I asked a similar question to another physicist, and he explained that in the case of an accelerating universe like ours, you'd get the same tension each time you reattach a galaxy at the same distance @benrg

@vengaq There is a perpetual tension, but you can't run a perpetual motion machine on perpetual tension (think of e.g. an electrically charged insulator). Even if $Λ=0$, there's a perpetual negative tension (pressure) in the tether due to gravitational attraction. You can extract work from it only until the tether's length reaches zero. If $Λ>0$, you can extract work until the tether's length reaches the Hubble length, then you hit the event horizon and the tether breaks. Reeling the mass back in before you hit the horizon costs as much energy as you extracted from letting it spool out.