the vast majority of licensed bands are still the age old single peak and often at insane output powers
I wouldn't call a few kW insane. Only the largest transmitters in the world go into the megawatt range; the vast majority are things like wifi, which are small and weak, at
They are looking for aliens who have set up multi-Gigawatt transmitters all over their planet, just like we have.
You obviously don't know much about transmitters. There are no multigigawatt transmitters anywhere in the world. The most powerful transmitter in the world is the Roumoules transmitter in France [wikipedia.org], which outputs 1.4MW, three orders of magnitude less than you think. It is notable that only its longwave broadcasts can be heard past 100km, because those reflect off the ionosphere. At night, the medium waves can do that too and so can be heard farther.
Notice that most of that radiated power will be reflected from the ionosphere and won't even make it as far as Earth's orbit. But for argument's sake, let's assume a full half of the signal makes it through. The antenna is somewhat directional, though the wikipedia article does not specify the beam width. Let's be generous and say it's a cone 30 degrees wide.
This cone will form a moving beam across the sky as the Earth rotates. A 30 degree beam will illuminate any particular star in its path for 2 hours each day. The study in the article we're discussing listened only for 8 hours, which is too short to always catch it.
Even if the signal is not reflected from the atmosphere, it will be significantly attenuated. Let's say 1MW makes it through. Gliese 581 is 20 light years away, ~2e17m. The base of the radiated cone is 2e17*tan30=1e17. The area of the base is pi*1e17^2=3.75e34. 1MW/3.75e34m2 = 2.67e-29W/m2.
Minimum detectable signal [wikipedia.org] with a 1kHz bandwidth is -144dBm of the 1mW reference signal. That's 4e-15W. As a dumb estimate, we can calculate that 1.5e14m2 of continuous antenna area would be necessary to receive such a signal. That's approximately equal to the cross section of the Earth.
Of course, that's if the signal can be received at all. At low levels like that electrons in the antenna are unlikely to absorb anything at all. The ground state energy, for example, is 13eV = 2e-18J, 11 orders of magnitude lower than the signal per square meter. I find it difficult to believe that any excitation can occur here.
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