I've hear several theories stating that the universe is expanding faster than the speed of light, others claim that the universe expands faster the further away you measure it. Which of this is correct and how do you prove it (mathematically)? Furthermore, does this correctly imply, then, that eventually galaxies will be so far away, and moving so fast, that we will never see them again?
Thank you in advance!
Both are correct, although the first can be further explained a bit. I won't give you a mathematical proof, though; Instead I'll play with characters.
Let's assume, for the sake of gedankenexperiment, that:
This is our current universe, and we launch a photon from body A aiming at body E.
(Space generated each second is marked with a # symbol.)
T=0s A----B----C----D----E Bodies
* Photon - 19 chars to E
T=1s A--#--B--#--C--#--D--#--E
--#--* 17 chars to E
T=2s A--#---B-#----C#-----#D----#-E
--#-----#-* 17 chars to E
T=3s A--#----B#-----#-C---#----D#-----#-E
--#-----#-----#* 18 chars to E
T=4s A--#-----#B----#----C#-----#---D-#-----#--E
--#-----#-----#-----#* 19 chars to E
I know, the graph isn't too granular, and the space generation isn't evenly distribute. I apologize for that, but it's for the sake of demonstration.
Notice that at T=2 some space is already generated between A and the photon. But that's irrelevant: E is sitting at the event horizon, and will never be reached by the photon, because the amount of space being generated between photon * and body E is equal, or superior, to the speed of light.
Given any positive expansion rate, there will be an event horizon - a point where the accumulated dilation of space is more than the amount of space a particle moving at the speed of light can travel.
A galaxy sitting initially at say, 1000 characters from A at T=0, will be at staggering 1200C at T=1 - that's 40 times our speed of light.
At T=16s, B (that was passed by the original photon at T=1) will be sitting exactly where E was relative to A, and at T=17 will fall out of our event horizon. A new photon emitted from A will never reach it.
This was supposed to be more a comment than an answer, but since I can not comment due to reputation lack I will spend few words here. First of all, the "theories" you mentioned are not inconsistent each other.
We know the simple Hubble law:
$v = H D$
where $v$ is the receding velocity of a galaxy, $H$ is the Hubble constant, $D$ is the distance of the considered galaxy. This means that the further is the galaxy you observe, the faster this galaxy is receding. At some point it will become faster than light (or superluminal). At some point, the space between us and the light emitter will grow so fast that the light can never reach us, and this will make those objects invisible. Indeed, all we can observe is by definition our observable universe. This is growing with time, but still some objects will stay invisible forever. The very first thing you mention, I suppose you should put it more correctly, since it should be better to talk about expansion rate of the universe (instead of velocity), and this is given itself by the Hubble constant, around $70 km/s/Mpc$. Take care of the units of this "constant", and you will grasp why this argument is not so intuitive. Please, wait for more experienced people, since this was just a very rough summary of cosmology concepts.
The universe expands with about 70 km/s per Mega parsec, due to Hubble's law. This means, that the velocity two objects move away from each other is proportional to their distance. At one Mega parsec it's 70 km/s. That's an average value, which doesn't need to hold for each single object.
By dividing the speed of light of about 300,000 km/s by the Hubble constant, you get, that objects further away than about 4300 Mega parsecs move faster away from each other than the speed of light.
The expansion is measured by the redshift of spectra, meaning absorption and emission lines are shifted. Together with distance estimates, based on several methods, the expansion per distance i.e. the Hubble constant, can be estimated.
The observable universe is 879,873,000,000,000,000,000,000 kilometers across. Using the mean measured Hubble Constant of 75 kilometers per second per megaparsec (30,800,000,000,000,000,000 km) for the expansion of space, you can then calculate the rate of expansion for the entire universe and that number is 2,113,636 kilometers per second. That says the expansion of the universe across the entire diameter is expanding at 7.05 (+/-2.33) times the speed of light.
Locally this works out to ...
The Milky way should experience 3.46 kilometers per second expansion or 108,988,052 kilometers per year. (using 75 kps - within margin of error)
There are a few papers now reporting an indirect but measurable increase in space locally. One peer reviewed paper published in 2015 in Gravitation and Cosmology periodical states the measured effect on Earths orbit is about 5 meters per year (about 1/2 of the hubble constant) Manifestations of dark energy in the solar system Manifestations of dark energy in the solar system
