The reason for their haste is that the energy needs of their civilization would have reached the point where entire stars would only serve as temporary fuel sources for them. And there are only so many stars in a galaxy that are suitable for harvesting.
Fermi National Accelerator Laboratory (FermiLab) researcher Dan Hooper wrote in his paper that galaxies that are not tethered by gravity to our own galaxy will go past the cosmic horizon of Earth in a hundred billion years. Once that happens, they will no longer be visible from here and cannot be reached even by starships that can travel faster than the speed of light.
Given that humans are aware of universal expansion, Hooper reasoned, advanced civilizations would arguably know as well. They would go to those galaxies, capture stars, and haul them to their home galaxies for exploitation before dark matter can send those resources out of their grasp forever. (Related: Scientists are actually losing ground on their understanding of dark matter, as new research contradicts previous findings.)
Not every interstellar civilization will be able to harvest stars. They need to have reached Level III on the so-called Kardashev scale, which measures the technological advancement of a civilization based on its energy use.
For comparison, humankind is stuck at Level I, the lowest part of the scale. Level III civilizations, on the other hand, are able to harness the energy of an entire star in much the same way humans use fossil fuels for energy.
Level III civilizations are expected to be capable of building a Dyson sphere, a swarm of orbital structures that completely surround a star. These satellites will collect energy directly from the star and send it to a planet in the same star system for maximum efficiency.
Hooper's paper suggested that a Dyson sphere could haul the star to another region of the galaxy where its power is urgently required. The star will provide more than enough power for the move.
This massive movement would produce large amounts of electromagnetic energy that can be detected by civilizations on the lower end of the Kardashev scale, such as humankind.
In addition to looking for stolen stars that are in transit, Hooper also suggested that human astronomers investigate galaxies that are missing stars of a certain size. That is because star-harvesting aliens need a star to be the right size.
The red dwarf stars that make up most of the stellar population in the universe are not going to be bright enough for their purposes. On the other hand, very big stars are probably on the verge of exploding into supernova.
The best candidates for harvesting are stars that are around 20 to 100 times the mass of our sun. Stars of that size happen to generate higher levels of specific wavelengths of light.
"The spectrum of starlight from a galaxy that has had its useful stars harvested by an advanced civilization would be dominated by massive stars and thus peak at longer wavelengths than otherwise would have been the case," Hooper remarked in his paper, which he published in the journal arXiv.
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Sources include:
ArXiv.org [PDF]