When does the world split in MWI I've been reading Eliezer Yudkowsky's blog post regarding decoherence and many worlds, and although he is not a physics but a strong proponent of MWI, I can basically see why he feels that MWI is a "simple" explanation. However, in all the posts there it's decoherence that's the natural (and mathematically simple) approach to QM, while the step to MWI seems to take a certain "leap of faith".
Can someone tell me the practical difference of "world splitting" in MWI, and the original "wavefunction collapse"? Even if there is no such thing as an "abrupt" split, I don't see why you couldn't also argue the same for the "collapse".
One of the reasons for why one should accept the fact that these other worlds exist is:

If a spaceship goes over the cosmological horizon relative to us, so that it can no longer communicate with us, should we believe that the spaceship instantly ceases to exist?

But, I fail to see the analogy here. The knowledge of ship's existence is there at the beginning, and there is only one ship. With many worlds, you start with a knowledge of only a single world.
Also, since decoherence seems to have a certain finite speed, is the splitting also something that propagates at a certain finite speed? In other words, if two far separated observers take measurements on two entangled photons at the same angle, whoever makes the first measurement (presuming they are sharing the same inertial frame) will decohere the mutual wave function of both photons, right?
 A: If you want to use nonrelativistic quantum mechanics you have to first start with the basics. Firstly it doesn't handle particle creation or destruction, so you need to fix how many particles you have of each type.
Then you want a function from the configuration space $\mathbb R^{3n}$ into the joint spin state $\mathbb C^{k_1}\otimes\mathbb C^{k_2}\otimes ... \otimes \mathbb C^{k_n}$. Where $k_m=2s_m+1$ when $s_m$ is the spin of the $m$th particle.
Then you need a Hamiltonian which tells you how it evolves. And the evolution includes that the region in configuration space where the wavefunction is nonzero (called the support) can change in time as determined by the present values in configuration space and the Hamiltonian.
When you model an isolated system you only need to include the particles in the isolated system.
When you want to model a measurement, you need a huge number of particles to represented the thermodynamically irreversible (so in principle could be reversed, but in practice can't) interaction whereby you get the measurement results.
The state splits into a sum of two waves, each of which could be called a world. They are called worlds because they don't meaningfully interact. And it's too hard to ever cause them to meaningfully interact in the future. It's like if you had a function whose support was one blob and then it evolves into a function whose support was two disconnected blobs. If those blobs move around in ways where the supports never overlap then each is basically following it's own wave equation as if it were the only wave.
It's like an amoeba that splits into two and then each baby amoeba gets placed into a  different rocket and sent to different planets. At some point, each amoeba could forget about the other amoeba.

With many worlds, you start with a knowledge of only a single world.

If you want to bring up knowledge, then knowledge is a fact about a world: when a world splits,  a fact is created, knowledge can be created later. When the blob that is the support of a wavefunction splits into two blobs then the wavefunction restricted to each blob could contain information about its world. But only if that blob is a world. And the location in configuration space where the blob is right now isn't the world. A world isn't a region of configuration space, it is the fully evolving independent wave that happens to equal the full wave right now when restricted to that blob in configuration space and (unlike the full wave) is zero everywhere else. And we only call it a world when it won't interact with the other worlds.
So you asked about knowledge. Knowledge has to be a fact about a world. Some facts might be the same across all worlds, such as the number of particles (since we are doing non relativistic quantum mechanics). Some of it might be localized such that a small number of particles in one world have interactions that encode a fact about the world. But this kind of knowledge can be copied and shared within the world in which it exists.
It's a very different thing. When the world splits, information is created. A kind of information that can be shared and copied within that world. And yes, the measurements you purposefully do are ultimately about distributing that information, but that happens after the world splits.
The world splitting was something that happened earlier. It happened when the support of the waves allowed distinct waves to act independently.

Also, since decoherence seems to have a certain finite speed, is the splitting also something that propagates at a certain finite speed? 

The wave has a support in configuration space. It includes a huge number of different configurations, and each one corresponds to a different configuration for all $n$ particles. And the streamlines move according to the probability current so they are constrained by the Hamiltonian, but what generally happens is that two regions of support that are already far away move even farther away and the region in between develops a gap.
It's like of you had an electrically charged slinky that was stretching but was weak in the middle. It gets longer and longer and then snaps. And then you have two slinkies that move away from each other. But the two ends of the one slinky might have been light years apart and so when it split into two, the center of mass of each slinky might instantly be super far away from each other but the smooth evolution was completely local.
The wave evolves in a continuous fashion. There isn't some place where it happens and then propagates outwards. There is one wave, and at some points you can start treating it as two waves because they are separated.

In other words, if two far separated observers take measurements on two entangled photons at the same angle, whoever makes the first measurement (presuming they are sharing the same inertial frame) will decohere the mutual wave function of both photons, right?

No. But it's also strange that you want to use non relativistic quantum mechanics and then your go to example was a massless photon. If you have two electrons with an entangled spin to have identical spins upon measurement and send them through a Stern-Gerlach device then it starts out with support in a square in configuration space (representing the spread of positions of each particle) and if you measured just one particle the square gets wider and splits and the spin state continuously changes so that on the right branch it becomes up-up and on the left branch it becomes down-down. A later measurement of the spin of the other particle just deflects the left square down or deflects the right square up.  If you measure the other particle the initial square support gets longer and splits and the spin state continuously changes so that on the top branch it's up-up and on the bottom branch it is down-down. A later measurement of the spin of the first particle just deflects the top square right or deflects the bottom square left.
If both happen at the same time, parts of the original support have streamlines that head to the top right corner and the spin state for that portion ends up becoming up-up. And other parts of the original support have streamlines that head to the bottom left corner and the spin state for that portion becomes down-down.
And every other possibility unfolds exactly as the Hamiltonian says it will. A frame is irrelevant. The Hamiltonian is a function of the entire configuration space. And the configuration space includes the configuration of the measurement device, so points in the configuration space that correspond to one device being used at one event have the wave start to start to split at that event. And another configuration that corresponds to the one device being used at a different event have the wave start to split at that different event.
But a frame is irrelevant. There is a wave defined on configuration space, and it evolves according to a completely local PDE and it doesn't care which frame you picked. The Hamiltonian just evolves the wave. Nothing else.
If you watch the mathematical evolution, you'll see regions of support that are currently connected stretch and disconnect and then move around independently and you can call those independently moving things worlds.
And if you use Occam's razor then at some point you can say that each world when modelling itself can ignore the other ones since they now move independently. That's many worlds.
Or if you like magic or solipsism you can pretend that one world is somehow somewhen magically selected and that the others then cease to exist. That's an untestable claim, but it isn't falsifiable so you won't really make errors in your predictions. It's straight up solipsism, but it doesn't make predictions. And that's Copenhagen. The magic part is that you never saw any evidence for anything other than evolution according the Hamiltonian, yet you proposed something else anyway, and you hide the different dynamics specifically to be in a place that can't be tested.
A: 
Can someone tell me the practical difference of "world splitting" in MWI, and the original "wavefunction collapse"? Even if there is no such thing as an "abrupt" split, I don't see why you couldn't also argue the same for the "collapse".

There is no world splitting. A state that starts out as having a single value for some relevant observable gradually changes to have a non-zero amplitude for various states. As a result of information spreading between systems, different versions of the same system gradually become less able to interfere with one another. Worlds are a large scale, approximate and emergent feature of the multiverse.

Also, since decoherence seems to have a certain finite speed, is the splitting also something that propagates at a certain finite speed? In other words, if two far separated observers take measurements on two entangled photons at the same angle, whoever makes the first measurement (presuming they are sharing the same inertial frame) will decohere the mutual wave function of both photons, right?

Wave functions are not terribly useful for thinking about the flow of information, and not terribly useful in relativistic theories. Measuring the electromagnetic field in some region will decohere the field in that region. It will also decohere quantum information that can be obtained by measurements on the field in that region alone. However, entanglement involves locally inaccessible quantum information. It is possible for a system to instantiate quantum information that can't be accessed by measuring that system alone. Rather, the information can only be obtained by comparing the results of measurements on your system with those on another system with which it is entangled - the information is locally inaccessible. For detailed discussions, see
http://arxiv.org/abs/quant-ph/9906007
http://arxiv.org/abs/1109.6223.
As a result that information does not decohere until some system can contain information about correlations between measurement outcomes on both systems. In that situation the multiverse is not divided into universes with respect to the joint observable until that comparison can take place.

One of the reasons for why one should accept the fact that these other worlds exist is:

If a spaceship goes over the cosmological horizon relative to us, so that it can no longer communicate with us, should we believe that the spaceship instantly ceases to exist?

But, I fail to see the analogy here. The knowledge of ship's existence is there at the beginning, and there is only one ship. With many worlds, you start with a knowledge of only a single world.

You do start with knowledge of only one world, but single world explanations of how reality works are ruled out by experiment and are incompatible with quantum theory. To take one example, the explanation of the EPR experiments given in the papers above require that decoherent systems are represented by quantum observables. Those observables do not represent a single version of the systems involved: they represent a complex structure that includes multiple versions of the systems involved. And there is no other explanation. The standard account of that experiment runs as follows: the quantum state represents the system until it is measured and then somehow the systems are each represented by a single number, and those numbers somehow end up correlated as predicted by quantum theory. This is not an explanation.
I haven't read the essay you linked to, but the logic of a sensible argument involving that analogy would run as follows. Our only existing explanation of cosmology requires the existence of stuff outside the horizon. For example, the rocket doesn't just vanish because that would violate conservation of energy. If you're going to deny that stuff outside the horizon exists because you can't see it, then you have no explanation of where the energy in the rocket went. More generally, the stuff that exists is the stuff entailed by explanations that solve problems and haven't been ruled out by other problems such as contradicting experiments. To understand the multiverse and the theory of knowledge better see "The Fabric of Reality" and "The Beginning of Infinity" by David Deutsch.
One last note. One commentator on this question claimed that quantum field theory somehow makes the MWI unnecessary. This is false. QFT still has unsharp observables, and interference and entanglement. QFT adds more structure on top of that, like the fact that observables in space like separated regions commute, identical particles and other stuff, but none of that gets rid of the multiverse.
