Last time I argued
that Kardashev civilizations should quickly make the jump from type II's to type III's, and in the comments I argued that this means that the Drake equation is actually incorrect. That is, it assumes a steady-state of short-lived civilizations (technically, those that start and stop communicating after some period). I would argue that there will be one galactic super-civilization per galaxy, eventually, so any galaxies that don't have a K3 yet don't have any space-faring aliens (it's very unlikely you would catch one "in the act" of colonizing its galaxy).
where R* is the rate of star formation in the Galaxy, fi is the probability of the ith step towards spacefaring (or interstellar communication in Drake's original formulation), Πi is the product over those probabilities (probability that the star has a planets, times probability the planets are habitable, times probability they have life, times... etc.), and L is the lifetime of a typical civilization. N is then the number of civilizations in the Galaxy. If the star formation rate, probabilities, or lifetimes are actually evolving with time (and they should be, I think) then you'll have to do an integral over time of the log of these quantities instead of a strict product, but this formulation is elegant and gets the point across.
I am arguing that once a civilization gets going, it's going to take over the whole galaxy quickly, and that L is actually longer than the current age of the Universe. If this means that subsequent civilizations are unlikely to arise, then N= 0 or 1 for most galaxies (0, in fact, since most galaxies don't look like they're full of Dyson spheres). The proper formulation for the expectation value of N (the average value over many galaxies) is then
where t is the time since the galaxy's stars started forming (roughly the age of the Universe), and the product of the probabilities is much less than 1/(number of stars in a galaxy). That is, N is small, or space-faring aliens are rare.
So what would they look like in our telescopes? We can continue to parameterize with a simple toy model. It is an interesting fact that the light from most galaxies is dominated by cool, giant stars (the pretty spirals and "train wrecks" are filled with bright blue stars, but they are the photogenic exceptions, not the rules; most stars are in red ellipticals). We can approximate their spectrum as that of a K giant, which is roughly a blackbody emitter at 4,500 Kelvins (about 5,000 degrees Celcius, or 7,500 degrees Fahrenheit), somewhat cooler than the Sun.
Some fraction of the stars will have Dyson spheres around them. We don't really know what that fraction is, though I argue it should approach 1 quickly. At any rate, let's call this fraction of absorption of starlight α (which may remind astronomers of an absorption coefficient). If aliens have some sort of alternative energy supply (dark energy engines, Romulan warp core singularities, black hole mass-to-energy plants) then we'll see waste heat from that, too. We'll call that parameter ε (reminding stellar structure students of the energy generation parameter) and normalize it by the starlight available in the galaxy (just as we did for α). I've already argued that most of the available energy in the galaxy is starlight, so I don't expect ε to be large (much less than 0), although in principle it could be greater than 1.
Conservation of energy says that all of the starlight plus all of the non-starlight energy must escape eventually as waste-heat. Let's parameterize all of the thermal photons that escape this way with the symbol γ (reminding physicists of the symbol for a photon, though presumably these photons will be in the mid- or far-infrared, not the γ rays that symbol usually represents) and assume they are emitted at a characteristic temperature Twaste roughly corresponding to the typical operating temperature of the civilization (that would be a mid-infrared color temperature, to be precise) . Finally, if there is some other way to get rid of waste heat (kinetic energy, gravity waves, non-thermal jets, dark energy sinks, neutrinos) we have to account for that, so we'll parameterize that component of energy loss as ν (for neutrino, though it could be any non-thermal-photon form). We'll scale both of these numbers by the total starlight in the galaxy, so all 4 parameters are on the same scale.
For reference, if we look at humanity this way (normalizing by the incident sunlight on the earth) we get
it's not really fair not to count sunlight we use for passive heating and agriculture, but that's hard to estimate in a logically coherent way, so this parameterization works best for K2 and K3's, not K1's like us.
So, if we further approximate that
that is, that there is pure photonic waste heat and all of a civilization's energy comes from starlight, then we have a simple calculation for the spectrum of a galaxy containing a K3:
where T* = 4500 K, B(T) refers to the blackbody (thermal) spectrum, and the other parameters are defined above. This makes α=1 civilizations very faint in the optical but very, very bright in the mid-infrared if the waste heat temperature is low.
This is the basis for using IRAS and other mid-infrared surveys to look for Dyson spheres.
A major difficulty in looking for Dyson spheres is that dust, like solar collectors and radiators, also blocks starlight (high α) and reemits the light at a lower radiation temperature (α=γ). So dusty stars will look like Dyson spheres, and dusty galaxies will look like K3's. Blue compact dwarf galaxies in the local universe, ultra-luminous infrared galaxies (ULIRGS), asymptotic giant branch stars (AGB stars), and even ordinary red giant stars with huge amounts of dust in the way will all have a spectrum similar to this.
Similar, but not identical: dust has its own spectral characteristics, and is easy to trace in the Galaxy, so we know where it will be a problem. Dusty galaxies are also typically star-forming galaxies, so they will also have other signatures, like lots of blue stars and emission lines characteristic of stars being born.
Is all of that enough to distinguish these confounders from massive alien civilivations? I don't know. But we're going to find out.