## Chances of aliens finding us in an infinite universe

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SpitValve
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### Chances of aliens finding us in an infinite universe

This is not really a puzzle, it's just a problem I've been thinking about and would interested in some input in.

ok, current measurements make it look like the universe is (largely) Euclidean 3-space. While there's still some uncertainity, it's still a definite possibility.

One of the consequences of a universe like that is that it has infinite size.

Now the Cosmological Principle says that the universe is pretty much the same everywhere. The density of planets and life is pretty much the same everywhere, if you look on the scale of a ridiculous number of parsecs.

So (based on these assumptions) there are an infinite number of planets out there.

Now, let's assume that superluminal travel is possible. (Currently we doubt this, but for the purpose of argument let's say you're allowed to go faster than the speed of light).

Each planet has a finite chance of having life. Each alien race has a finite chance of developing faster than life technology. And each FTL race has a chance of discovering Earth.

So if there are infinite alien races out there, does that mean there would be a 100% chance that they have discovered earth by now? (i.e. hence disproving either the possibility of FTL or an infinite universe) Or does the fact that an infinite number of races have an infinite number of planets to search through make everything go screwy?

moopanda
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I usually like to think of this problem assuming FTL travel is not possible (you know, physical reasons).

This means that according any aliens further than 100 light years away, we don't even have radio technology yet. And only aliens within 50 light years could have responded. I also like to set (what I think is a very generous) upper limit on human existance of 900 years from now, meaning if we have contact with aliens *without* FTL travel is confined to a radius of 500 light years, or 1000 if they have it and we don't. What's more is that there's only a 1000 year window of opportunity (out of the 15,000,000,000 year age of the universe).

My usual conclusion is that *yes* aliens are more than likely out there, but no, we're probably never going to contact them and they certainly wouldn't make the difficult journey here to stick things up our bums. Or whatever it is that they do.

If I had more time before I head off to uni I guess we could estimate life bearing planets as a 4D Poisson distribution (which is fair I reckon), estimate the parameter... somehow... and find the distribution for the closest one in space and time. We could at least set sensible bounds on the parameter by where we know life doesn't exist...

ikefalcon
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### Re: Chances of aliens finding us in an infinite universe

SpitValve wrote:ok, current measurements make it look like the universe is (largely) Euclidean 3-space. While there's still some uncertainity, it's still a definite possibility.

Actually, the universe is shaped like a hyperbolic toroid. Imagine a horse's saddle. That's how space is curved. Imagine this type of space sitting on the surface of a balloon. I know it's difficult to conceptualize, as thats putting 4 dimensions on a 2-dimensional surface, but the Big Bang Theory describes space as shaped like this. Thus, the universe is finite in size.

Gelsamel
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### Re: Chances of aliens finding us in an infinite universe

ikefalcon wrote:
SpitValve wrote:ok, current measurements make it look like the universe is (largely) Euclidean 3-space. While there's still some uncertainity, it's still a definite possibility.

Actually, the universe is shaped like a hyperbolic toroid. Imagine a horse's saddle. That's how space is curved. Imagine this type of space sitting on the surface of a balloon. I know it's difficult to conceptualize, as thats putting 4 dimensions on a 2-dimensional surface, but the Big Bang Theory describes space as shaped like this. Thus, the universe is finite in size.

I thought they didn't know specifically what shape it was, it could be Euclidean, hyperbolic or spherical depending on the density parameter value.

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### Re: Chances of aliens finding us in an infinite universe

Gelsamel wrote:
ikefalcon wrote:
SpitValve wrote:ok, current measurements make it look like the universe is (largely) Euclidean 3-space. While there's still some uncertainity, it's still a definite possibility.

Actually, the universe is shaped like a hyperbolic toroid. Imagine a horse's saddle. That's how space is curved. Imagine this type of space sitting on the surface of a balloon. I know it's difficult to conceptualize, as thats putting 4 dimensions on a 2-dimensional surface, but the Big Bang Theory describes space as shaped like this. Thus, the universe is finite in size.

I thought they didn't know specifically what shape it was, it could be Euclidean, hyperbolic or spherical depending on the density parameter value.

I site Hawking in support of my argument.

SpitValve
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There's a parameter called Omega that tells you what the universe is shaped like.

Omega > 1 means the universe is hyperbolic (the 3-surface of a 4-hyperhyperbola)
Omega = 1 means the universe is normal 3-Euclidean flat space
Omega < 1 means the universe is spherical (the 3-surface of a 4-hypersphere)

Current measurements say Omega is between 1.00 and 1.04

So the universe is either flat, or very slightly hyperbolic. Both of these result in an infinitely big universe...

Anyway, the question is easy if FTL is not possible. Then, regardless of the size of the universe, the odds of an alien finding us are really small.

That's not what I'm concerned about: this is more about probability than cosmology. If there are an infinite number of alien races with FTL spaceships, and an infinite number of planets for them to search, can we say anything about the odds of them finding Earth?

Gelsamel
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SpitValve wrote:There's a parameter called Omega that tells you what the universe is shaped like.

Omega > 1 means the universe is hyperbolic (the 3-surface of a 4-hyperhyperbola)
Omega = 1 means the universe is normal 3-Euclidean flat space
Omega < 1 means the universe is spherical (the 3-surface of a 4-hypersphere)

Current measurements say Omega is between 1.00 and 1.04

So the universe is either flat, or very slightly hyperbolic. Both of these result in an infinitely big universe...

Or the measurements are incorrect and it's Omega < 1.

So we don't have any definitave proof.

moopanda
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First of all, by "superluminal" you really have to mean "infinite speed" travel for the problem to be different from the subluminal case in an infinite universe (otherwise you just multiply all the figures by the appropriate maximum speed).

Secondly, we're going to run into problems with "infinite". For example, the probability of picking a rational number (of which there are countably infinite) from the interval [0,1] is 0, while the probability of picking out an irrational number (of which there are uncountably infinite) is 1. However, the probability of picking out a number in the interval [1/4, 3/4] is obviously 1/2, but there are an uncountably infinite number of those as well. So just claiming that two quantaties are infinite and wondering about the ratio between them is... not well defined.

SpitValve
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First of all, by "superluminal" you really have to mean "infinite speed" travel for the problem to be different from the subluminal case in an infinite universe (otherwise you just multiply all the figures by the appropriate maximum speed).

yep, I meant "no top speed".

Secondly, we're going to run into problems with "infinite".

Yep, and that's exactly what I'd like to figure out... and I think we can do better than saying it's not well defined. Whether the probability comes out to be zero or unity or something in between would conceivably make a difference in a universe with no speed limit.

As you've pointed out, even when dealing with infinities you can get good meaningful probabilities out of them.

prob of earth being found = integral over all volume of ( p(r) = prob of planet having life and FTL at r finding Earth )

due to the cosmological principle, p(r)~constant, so integral of all space of a constant is infinity... which is not good.

Maybe because that integral is not how you do probability... oh well

toms2866
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One of the consequences of a universe like that is that it has infinite size...So (based on these assumptions) there are an infinite number of planets out there...Now, let's assume that superluminal travel is possible...So if there are infinite alien races out there, does...make everything go screwy?

Does a tin foil hat of infinite density protect you against an alien mind-ray of infinite strength?

FluffyCubed
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moopanda wrote: I also like to set (what I think is a very generous) upper limit on human existance of 900 years from now

..generous? We've managed 3 million so far, and I doubt we're actually stupid enough to go ahead with something that would kill EVERYONE.

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FluffyCubed wrote:
moopanda wrote: I also like to set (what I think is a very generous) upper limit on human existance of 900 years from now

..generous? We've managed 3 million so far, and I doubt we're actually stupid enough to go ahead with something that would kill EVERYONE.

You have far, far too much faith in humanity.

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### Re: Chances of aliens finding us in an infinite universe

SpitValve wrote:ok, current measurements make it look like the universe is (largely) Euclidean 3-space. While there's still some uncertainity, it's still a definite possibility.

One of the consequences of a universe like that is that it has infinite size.

The universe may be infinite but it's not all accessible. The speed of light is finite and the universe is of finite age, making everything outside a finite subset of the universe causally disconnected from Earth. The observable universe, depending on how you define it, is a sphere of between 13 and 80 billion light years in radius. Large, but finite.
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I don't agree with any of your assumptions, so I'll treat this as a mathematical problem.

Even if we allow arbitrarily fast travel, the acceleration can't be too high (much more than 9.8 m/s^2 would not be feasible for humans), and even if we could bypass that, the actual space ships should have a maximal acceleration. So if the closest alien is far enough away, it couldn't've reached us yet.

What if we allow teleportation/warping? For a single space ship searching for earth, it won't make a difference. If we define "finding earth" as being within a certain radius of it, it would be no more sensible to randomly teleport than to go in a straight line, because you won't search through a larger distance. So this is where the infinities come in.

An infinite number of space ships teleport randomly in R^3 to find earth. First off, the distribution of the points-to-teleport-to can't be uniform, because the total probability would then be infinite. So the probability to teleport to arbitrarily far away points must approach 0. So just making the density of life low enough, we can make the probability that they should've already found us arbitrarily low. In fact, even without doing so, I think (without having done any calculations) that infinitely many aliens in the universe wouldn't make the chance any greater than just having ~100 close neighbours.

I don't really know if this is the kind of answer you wanted.

SpitValve
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I don't really know if this is the kind of answer you wanted.

actually yeah, that was the kind of thing I was thinking of.

kira
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I'm curious about what you mean by:

Torn Apart By Dingos wrote:the acceleration can't be too high (much more than 9.8 m/s^2 would not be feasible for humans)

Do you mean physically? Mechanically? In space? I mean, I've ridden a roller coaster that purports to have acceleration of rates up to 19.9 m/s^2. (And, upon researching this, it seems there's another ride with acceleration of almost 27 m/s^2.)

The max sustained acceleration that can be tolerated by humans (with appropriate equipment) is around 90 m/s^2.

And, on top of that, some absolutely insane guy subjected himself to acceleration of 453 m/s^2 (and he survived).

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I wouldn't want to be on a space ship in which I constantly feel like I'm on a roller coaster. For a long journey, 9.8 m/s^2 would probably be optimal. All I'm saying is that there's a maximal force that creatures can be exposed to.

kira
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I certainly don't think the maximal force is anything near 9.8 m/s^2. Like I said, with proper equipment, humans are able to undergo acceleration of at least 10 times that amount.

Heck, astronauts withstand 3 gs (or about 29.4 m/s^2) and are still able to move around (head and arms) to control the shuttle.

With more advanced technology and better equipment (and without having to move your head and arms), it would definitely be feasible to have over 30 m/s^2 of acceleration for a prolonged period of time.

If I had the opportunity to travel into distant space just by agreeing to feel like I was on a roller coaster the entire time (provided that it wouldn't kill me), I'd jump at the chance.

Of course, the fact that you can accelerate over three times as fast as you suggested doesn't really change your premise that it would take significant time to reach any arbitrarily high velocity. As such, your point still stands.

SpitValve
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Torn Apart By Dingos wrote:I wouldn't want to be on a space ship in which I constantly feel like I'm on a roller coaster. For a long journey, 9.8 m/s^2 would probably be optimal. All I'm saying is that there's a maximal force that creatures can be exposed to.

Although they might come from a planet where surface gravity is 100 N/kg... or (as we're talking sci-fi here) they might have some sort of non-intertial gravity drive (e.g. 3001 space odyssey), or (getting very sci-fi) teleportation...

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Vaniver
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I site Hawking in support of my argument.
I like how you don't actually cite him...

If there are an infinite number of alien races with FTL spaceships, and an infinite number of planets for them to search, can we say anything about the odds of them finding Earth?
You're missing a key parameter- time it takes to search a planet. It's more appropriate to discuss the probability of how long it will take aliens to find us- if they search a planet every year the probability climbs much slower than if they search a planet every day. Given infinite planets, it will take infinite time for the probability to hit 1- but we're probably far more interested in the time it takes it to hit .99999999990. However, that depends on an infinite number of planets... which also destroys the math.

If any of these civilizations produce cultures which colonize over interstellar distances, even at a small fraction of the speed of light, the galaxy should have been completely colonized in no more than a few million years [1].
Please to note this is a few million years after the technological civilizations are formed. Do we have any reason to believe that another planet developed significantly more quickly than Earth did? It may be that three million years from now humans on every habitable planet in the galaxy look back at the Fermi paradox and laugh, but there's no reason to expect that now is several million years after the foundation of a technological civilization.

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I had always imagined the universe as a sort of 4-dimensional sphere. Like Pac-Man. No matter which way you go you'll always (eventually) end up where you started. Yeah, the cosmos is pretty f-ed up. But i think the chances of finding intelligent life are crippled by:
1) The possibility that they function differently that humans, creating an impenetrable barrier (think humans and buggers in Ender's game)

and 2) This has no scientific merit at all, but what if the aliens aren't even made out of the same stuff we are? (heavy elements, radioactive materials, maybe dark matter)

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What is interesting is aliens wouldn't necessarily be based on DNA, seeing what they were based on would be sweet.

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toms2866 wrote:Does a tin foil hat of infinite density protect you against an alien mind-ray of infinite strength?

1. No: Tin-foil INcreases mind-ray conductivity, use lead

2. Yes: It will crush and thereby kill you, rendering you invulnerable to ray-based mind-affectors
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moopanda wrote:I also like to set (what I think is a very generous) upper limit on human existance of 900 years from now...

My feeling is that if human civilization manages to survive for another 900 years, it will live a LOT longer too, and that if we kill ourselves off, it will happen relatively soon.

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EvanED wrote:
moopanda wrote:I also like to set (what I think is a very generous) upper limit on human existance of 900 years from now...

My feeling is that if human civilization manages to survive for another 900 years, it will live a LOT longer too, and that if we kill ourselves off, it will happen relatively soon.

Believe what you want to.

So lets see... in an infinite universe, there are all possibilities and therefore infinite amounts of life forms. Now we would be just some random planet in an infinity of space in this situation.

So...
Lifeforms: infinity
Chances of any finding us: 1/infinity

We get the possibility as infinity/infinity or 0*infinity

And... we have a problem.
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Even if there are an infinite number of aliens out there with infinitely fast travel and so on, they still might never find us - what if they're all coincidentally heading away from Earth?

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Like the rest of the universe!

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torne wrote:Even if there are an infinite number of aliens out there with infinitely fast travel and so on, they still might never find us - what if they're all coincidentally heading away from Earth?

Infinite aliens in infinite space? would they not ALL have to be infinitely small?

...

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Well I guess it all depends how you define alien.

But you can easily have infinite, massive, aliens in inifinite space. Consider each alien being a 1m cube. Then arbitrarily assign an origin. Now position each alien on an integer point in space (measured in metres). Viola. Infinite aliens in infinite space.

Even better. Position them on every point whose coordinates are a multiple of 9.4605284 Ã— 10^15 meters (thanks google). Now you have inifinite aliens in infinite space, with a minimum distance between them of 1 light year.

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The infinite universe assumption is highly suspect to me. Spacetime may be infinite, but that's not equivalent to the universe being infinitely large spatially.

moopanda
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And inifinite speed travel isn't suspect? Not just a lil bit?

SpitValve
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moopanda wrote:And inifinite speed travel isn't suspect? Not just a lil bit?

oh absolutely I doubt it'll ever be possible to go faster than the speed of light. I was posing a hypothetical question.

mnemophobe wrote:The infinite universe assumption is highly suspect to me. Spacetime may be infinite, but that's not equivalent to the universe being infinitely large spatially.

hmm... not sure about that... I've only done a little bit of general rel & differential geometry though, enough to know not to be sure about anything

ohthecommotion
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I believe the view set forth by Men In Black. Maybe our whole universe is just one marble in an alien's marble sack!

I guess that has nothing to do with whether or not aliens find us.

...or DOES IT?

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BigMcStrongmuscle
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If this gets long-winded, its because I'm making this up as I go along. Please bear with me...

Assumptions:
1) The Universe is infinite and has similar characteristics (like, say, randomly distributed planets) throughout.
2) The long-run odds of any given section of the universe (of arbitrary but constant size) housing a planet at any particular moment in time is a constant k(0). The long-run odds of any given plant housing sentient life at any particular moment is a constant k(1). The long-run odds of any given sentient race having interstellar technology at any particular moment is a constant k(2). Therefore, the probability that in any given section of the universe, there exists a race capable of looking for alien life is a constant = k(0)*k(1)*k(2) = P.
3) Sentient races capable of interstellar flight will each scour an average volume of space V over time t. Call that rate R.
4)The values of all k, of V, and of t (and therefore of P and of R) are arbitrary, but constant and finite, taking a long enough view of things.

Math
1) The average percentage of the universe scoured by aliens for sentient life in time t ends up being P*R. For the moment, assume the areas of overlap are more or less negligible, and that P*R < 1. If our aliens are so closely packed that that's false, then Earth's discovery is pretty likely anyway.
2) Assume that these infinite races of aliens, taken en masse, don't share results. For all intents and purposes, they have no memory of where they've already looked. This will produce a conservatively low chance of discovery, but it means that the random portions of space searched in each interval t become statistically independent, which makes sense if these alien civilizations are constantly rising and falling.
3) Therefore, the odds that any singular point is *not* searched in an unrelated interval of time T are: (P*F)^(t/T). Because P is universally constant, infinity doesn't actually enter the picture at all. And since P*F is less than one (which follows from assuming negligible overlap), that gives us a nice, tractable function. As time T (measured from when we start watching) increases to infinity, the likelihood of evading discovery approaches 0, and the probability of alien contact approaches 1.

Notes and Problems:
1) This doesn't take into account interstellar expansion by humanity, humans doing things like broadcasting radio spikes to draw attention, or aliens being smart enough not to look at the same planet twice, but those would be a jerk to model and only increase our chance of discovery anywho.
2) The gory, horrific demise of the human race in a nuclear disaster, plague, supernova, velociraptor attack, or Zombiepocalypse puts an upper bound on T, in whatever imminent or remote place strikes your fancy.
3) The Big Bang kinda throws a wrench in the works by invalidating the first assumption.
4) You'd also need a metric boatload of data to compute accurate values of F and k(0, 1, and 2). But at least it comes down to constants, which tells us something.
5) I signed up just to post this. I bet this would tell a psychiatrist something.
Last edited by BigMcStrongmuscle on Sun Nov 12, 2006 3:10 am UTC, edited 1 time in total.

SpitValve
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BigMcStrongmuscle wrote:3) Therefore, the odds that any singular point is *not* searched in an unrelated interval of time T are: (P*F)^(t/T). Because P is universally constant, infinity doesn't actually enter the picture at all. And since P*F is less than one (which follows from assuming negligible overlap), that gives us a nice, tractable function. As time T (measured from when we start watching) increases to infinity, the likelihood of evading discovery approaches 0, and the probability of alien contact approaches 1.

Is that supposed to be (P*F)^(T/t)? Because otherwise, at T->infinity, t/T->0, and (P*F)^(t/T)->1...

BigMcStrongmuscle
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Yeah. I swapped the variables midway through to comply with convention and forgot to change it.

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I dunno, you all seem to be assuming that there is an infinite amount of life because the universe is infinite. But surely that only works with an infinite amount of matter? What we have is some probability of life forming that tends towards 0 as the amount of matter runs out.

However, if life does happen, I imagine it will be non-random- it will surely depend on the positioning of planets into a nice formation, which means that planets that are probable to create life might well clump together.

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mister k wrote:I dunno, you all seem to be assuming that there is an infinite amount of life because the universe is infinite.

With an infinite universe, probability states that no matter how improbable an event is, it will occur somewhere in the universe. Also, because of this, these events will happen an infinite amount of times. It is similar to rolling a die an infinite amount of times. Eventually you will see "2543", "35142", or even "425312322" rolled, after some very long time. An infinite universe means that all things are possible, no matter how improbable it may seem. This also means that these possibilities happen an infinite amount of times.

Besides, infinite universe means infinite matter. A finite amount of matter in an infinite universe means virtually no matter density at all. A simple proof by contradiction:

Let's assume that matter the infinite universe in some finite number. If life is present at all in the universe, the density of the universe must be more than 0.

Density is defined as matter/volume. In an infinite universe, there is infinite volume. This means density must be zero for any matter amount less than infinity or more than 0. This contradicts our first original statement.

We must conclude that either mass is infinite in an infinite universe or an infinite universe doesn't contain any matter at all and can't contain life.

Choose your pick. If our universe in infinite, then it obviously isn't the second statement.
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