I am having some problems with understanding what it means to be in the horizon of something else. In particular I'm looking into axion cosmology model.
In this paper, and in particular in the strings and wall decay sections: http://arxiv.org/abs/astro-ph/0610440
Let me try to elaborate some of the ideas. When PQ symmetry is broken and after QCD phase transition it gets broken: [itex]U(1) \rightarrow Z(N)[/itex], I think that this means that in the mexican hat potential, the flat direction becomes like this:
/\/\/\/\/\/\/\/\
with N sharp points. Between these points there are created the strings, correct?
If not, what does he mean by:

?

Also in the same paragraph he takes [itex]N \ge 2[/itex] so at least the flat direction gets at least two spikes along its way (/\/\):

So I am wondering, why is that "hence" true? the domain wall appears in the field's configuration space, and not in the general spacetime...any help?

I think it helps to visualize how these phase transitions occur, and why domain walls form. They don't occur instantaneously: you get a nucleation in one specific location that spreads. All phase transitions work this way. So a specific domain starts when one tiny region of space-time settles into the new semi-stable vacuum state, and that specific state is random. This new vacuum state then spreads outward at nearly the speed of light, until it collides with another vacuum state where it isn't energetically favorable to continue spreading.

This is where the horizon comes in: even at the speed of light, the vacuum state cannot spread beyond the future horizon of that original nucleation. So you're guaranteed a domain wall somewhere within its future horizon (because eventually another nucleation will occur close enough to collide with this one).

So suppose I get two points, causally non-connected, A and B.
In A,B I will have this breaking (transition) and so let's take the case [itex]N=2[/itex] there will be created two vacua (1 /\ 2/\)... So let's say that A settles in 1 (randomly).
This will start spreading with the speed of light. If B also settles in 1, then they will both be in the same vacuum once they get causally connected. If it settles in 2 I will have two different vacua "colliding" and so 1 wall between them...
Is that interpretation correct?

Yes. But the expectation is that the number of possible vacuum states is quite vast (infinite in some models), so it's highly unlikely for two causally-disconnected points to fall into the same vacuum state.

true, but then you shouldn't expect of the order of one domain walls per horizon, but it should be like ...
Taking eg [itex]N=3[/itex] (3 degenerate vacua), we can choose 3 points in the same logic, that fall in vacuum 1, 2 and 3.
Where they meet each other, you will have 3 walls (1-2 , 2-3, and 1-3).
And doing the same for a general N , and N distinct points, you get way more walls.

It depends a bit upon the model. If a stable domain wall is rare, then you generally won't get more than one domain wall. For example, if a domain wall will always spread in the direction of the lower-energy vacuum state at nearly the speed of light, and vacuum states with the same energy don't exist (or are very rare), then every domain will rapidly expand to encompass its entire horizon, or else be overcome by a different domain with lower vacuum energy.

But either way, it's easy enough to take N=1 as a reasonable lower-bound on the expected frequency of domain walls.

Are the domain walls in this case different to the ones you get in the magnetic moments examples? http://en.wikipedia.org/wiki/Domain_wall_(magnetism)
Because then I don't understand the "stability" or "spreading"...

The main difference there is that each domain has equal energy, so that domain walls collide but remain stationary. Think of it more like the domain walls between liquid and gas in a vat of boiling water.

Hmmm... it sounds logical to me... because all points within the domain will fall in the same vacuum.
If all points of domain 1 are in vacuum 1, and all points of domain 2 are in vacuum 2, but vacuum1=vacuum2 (coming from Z(N) SSB), then also the domains should be degenerate.
I think I am missing something important or doing something wrong...

Looked into the specific model they're using in a bit more detail, and they're saying that while the domains have identical energies, the domain walls repel one another gravitationally. This repulsion is likely what gives rise to the ~1 domain wall per horizon: as long as the repulsion is large enough compared to the decay rate, so that the wall expands to fill its horizon before there's a chance for another domain to form, then you'll only get approximately one domain wall per horizon on average.