I imagine that most science fiction fans will be familiar with the so called Fermi Paradox. From Wikipedia that quotes the Hart-Tipler argument:
1. There are billions of stars in the galaxy that are similar to the Sun, and many of these stars are billions of years older than the Solar system.
2. With high probability*, some of these stars have Earth-like planets, and if the Earth is typical, some may have already developed intelligent life. And some of these civilizations may have developed interstellar travel, a step the Earth is investigating now (but we would only see these aliens only if they have faster than light travel).
3. Even at the slow pace of currently envisioned interstellar travel, the Milky Way galaxy could be completely traversed in a few million years.**I don't think there is an actual paradox, and I'm not the first to come to this conclusion. I'd be shocked if I was.
But here's the thing I realized.
A first order approximation for going around our galaxy at light speed, assuming you don't stop, would take 314 thousand years. The minimum if stopping once (as in go halfway around the galaxy and colonize one planet), and you would have to quadruple that to allow for acceleration and deceleration, which means 1,236,000 million years have passed.
Okay, that doesn't sound so bad, but assuming you want to stop at all the planets that can support life (minimum approximation 100,000***), then one has to make a decision how many expeditions can be sent at the same time.
This leaves a range that falls between a minimum of 1,236,000 million years up to 123,600,000,000 one hundred and twenty-three billion years (1 to 100,000 expeditions) to colonize the galaxy. Not a very useful range, but hopefully you can where I'm going with this?
But I seriously doubt that one can colonize the galaxy at a one-to-one equivalent of light-speed.
Therefore, starting again with the assumption that the galaxy can be colonized, then it will have to be done at sub-light speeds. The best theoretical drive would be a black hole (the interesting stuff in the video start at around 8 minutes). For a more detailed explanation of a black hole drive aka Torch engine.
Even so, my best guess for the maximum speed would be 0.5c or 0.25 average for a journey to allow for deceleration again. But that's a highball number, 0.1c or 0.05c average might be more realistic (for definitions of realistic when applied to engineering that's way beyond our current level).
So a minimum of 4,944,000 million years up to 494,400,000,000 four hundred and ninety-four billion years and change.
At 0.1c then the quickest the galaxy can be colonized (100,000 worlds) is 12,360,000 million years up to a staggering 1,236,000,000,000 one point two trillion years.
If we take an average of the highest and lowest times then it would roughly take 31.4 billion years to colonize the galaxy.
So, the universe hasn't been around long enough for us to meet aliens.
NB: When I finished this I realized that there was yet another way to calculate the time to explore/colonize the galaxy and that was work out the average distance between habitable star systems, which a ball park figures comes out to 3,126 light years.****
So running the numbers will depends on how many expeditions you send out. The minimum would again be quite modest 3,142 years, but I don't think it's feasible to travel that fast.
Assuming 0.1c, then the minimum would rise to 31,400,000 and obviously rise higher since the assumption is sending 100,000 expeditions out at once. Your guess is as good as mine about how many expeditions would be sent and how much time would occur between them.
So again I'm left with the conclusion that we haven't been around long enough to have met or discovered aliens.
Addendum: There is another explanation for lack of contact too. It's by Geoffrey A. Landis and is called, The Fermi Paradox: An Approach Based on Percolation Theory. It comes to the conclusion that given how large space is, not all possible planets will be colonized (physic theory of percolation).
In conclusion, this is what I research when writing. Enjoy.
* The Drake Equation.
** Not feasible unless you have faster than light travel.
*** No one really knows (see The Drake Equation for the assumptions) as there could be one or two trillion stars in our galaxy (1,000,000,000,0000 to 2,000,000,000,0000 stars).
**** Figure derived from a first order approximation of the volume of the galaxy divided by the number of habital star systems to arrive at an average distance between them.