how far away could we detect that Earth has life?
My guess is that life bearing planets are too far apart to be detected. I think we can only find the ones within a sphere around our planet that is 100s of light years in diameter but I suspect life bearing planets may be much farther away than that.
I would like to estimate the diameter of the sphere within which we could detect life on another planet and then estimate the probability that there is life within that sphere.
For example, give our current technology what is the furthest distance would be able to detect life on Earth?
How many stars like our sun are in that sphere?
How long would it take for SETI to rule out each of those stars?
There are a lot of ways we could figure out that there's life somewhere, such as radio transmissions. Is there a specific method you have in mind, or is this more of a general overview?
I'm not sure which is the best detection technology but we should pick the one that allows us to have the biggest detection sphere. If that detection sphere contains too many stars to search, we could select a smaller sphere based on our estimates that there should be a least one life bearing planet within the smaller sphere.
Depends on what you mean by detecting life. As is explained in this what-if post by Randall Munroe, the algae on earth will tell the aliens about us before we can tell them about us.
If you consider the presence of liquid water or the presence of $O_2$ as detecting life, then such detection can be made by studying the spectrum of extra-solar planets, measurements that we can currently make. The furthest extra-solar planet discovered so far is at a distance of 27,700 light years. So, a partial answer to your questions would be to study the spectrum of every extra-solar planet found within the circumstellar habitable zone to look for signatures of tell-tale signs of life. We currently do have the technology to measure the optical reflection spectrum of an extra-solar planet, for example ESO's VLT, the Gemini Observatory and the OSIRIS instrument on GTC but I don't know if SETI has that capability. You could further look up the work of Dr. Sara Seager.
Could you elaborate on what current technology we have that enables the detection of oxygen in an exoplanetary spectrum? There have been claims of water (steam) in some hot Jupiters, but that's not oxygen and there can be no liquid water on a hot Jupiter.
@RobJeffries: Why claims? The data is out there. Hubble WFC3 and Spitzer in it's warm mission were successful in giving us the first ~20 transmission spectra of Hot Jupiters during their transits. And in those planets that are not dominated by Rayleigh-scattering, usually water is seen. There are even survey papers published. If you're interested in them I can check my notes.
@AtmosphericPrisonEscape I work in a department with exoplanet experts. They say "claims" - implying that they believe the evidence is less than conclusive. But perhaps things have moved on since my comment 20 months ago.
@RobJeffries: Hmm I think this would best be resolved by looking at the data. The spectra reviewed in Sing+2015, doi:10.1038/nature16068 looks conclusive to me, that there is water. But I'm not a spectroscopist, so I cannot say how degenerate those features are with other possible molecules. Maybe you have some time/motivation to skim over the article.
@AtmosphericPrisonEscape It hardly changes my point, which is that the only chance of detecting something at present is *steam* in the atmospheres of transiting hot Jupiters, which is not liquid water on an Earth analogue. JWST will improve matters a lot but it isn't here yet.