There are, in fact! However, these tend to be more guidelines on what to do than actual rules, and aren’t binding in an international court of law. There is a branch of law for dealing with outer space, called International Space Law, which are a set of rules that the UN has laid out and all UN member countries have agreed to abide by. It is a string of very sensible rules for behavior in space, like “do not put nuclear weapons in space”, “your country cannot claim a piece of space”, “you cannot build a military base on the moon”, and “space is for all countries”.
Most of the guidelines for what to do in case we find or are found by aliens have a similar set of instructions, whether it be found in the SETI Institution’s version, entitled “Protocols for an ETI Signal Detection” (ETI standing for Extra Terrestrial Intelligence), the “Declaration of Principles for Activities Following the Detection of Extraterrestrial Intelligence”, which has been signed off on by a whole slew of administrative bodies. But really, they’ve signed off on agreeing to be guided by the guidelines.
The guidelines go roughly like this: if you think you have found a signal from extraterrestrial life, please make extra, seriously, super-duper sure that what you think is an ET signal doesn’t have any other more plausible explanation. Given how much press the alien megastructures got as a possible explanation for the star KIC 8462852's bizarre dimming (which is certainly a weird signal from that particular star, but almost certainly not alien megastructures), I can only imagine the sort of media explosion and cultural whiplash we would have on our hands if we announced that we had discovered intelligent life. This is a “be careful” initial guideline.
Secondly, pass what you think to be a detection to other scientists, and let them check your work. This checking of your work may mean getting more data, more observations, or just someone else verifying that they can reproduce your numbers. (The recent LIGO press conferences was a great example of following this guideline - their paper was peer-reviewed and accepted before they made their announcement.) At this stage, please do not leak this to the media. It may well be that what you have found is super interesting, but not aliens. If a scientist immediately announced, “I found aliens”, and then it turned out that what they had actually found was something astrophysically bizarre, but not aliens, there is an instant loss of trust in that person. This second stage is to make sure that at the very least, reliable numbers and information are announced.
Next: if the signal has not gone away and still looks like aliens after some serious checking, you should tell the Secretary General of the UN, and a whole string of scientific bodies, which will let the rest of the scientific community know. You should also swiftly tell the rest of the world. But hey- if it was your data that did the discovering, you get the honor of hosting a very high-profile press conference, and will probably spend the next year and a half answering questions.
Publish your data. However you detected intelligent aliens, please tell other scientists so we can do more of it. This means explaining what you did at conferences and probably making your data public. Also, please put your data in as many places as possible. The last thing we would want at this stage is to lose it. Put it in 10 places. Put it everywhere. Please do not lose the evidence of aliens to a hard drive failure.
If the aliens were detected on a particular frequency (say we heard a radio transmission of theirs), that frequency should probably be protected so we can keep listening without too much interference. Radio quiet areas are pretty rare in the US, and a lot of things can cause radio waves (see for instance the peryton paper, which discovered that a bizarre chirp in the radio could be produced by one particularly well-placed microwave door being opened to stop the microwave).
And last in the guidelines, you do not get to write back to the aliens. Not without getting the majority of the planet on board, anyhow- at that point, contacting aliens has an impact on the whole human race, not just science, and not just the country the scientist who discovered it lives in. The decision of what to do next should be discussed by everyone.
But because these are only guidelines, if some private company happened to do the discovering, and disregarded all of the above rules, the way things sit currently, there would be no legal ramifications whatsoever. They are, after all, only guidelines. At the moment, it seems that any situation that might require them is still at least several decades away, though I would love to be wrong.
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