So far, I thought that big bang scenario implies that spacetime is finite. It started at zero and now becomes bigger. In addition, at a minimum, we have a causal horizon and anything farther away shouldn't matter. Thus instead of integrating "all of space" we could equally integrate to the causal horizon.
However, now I learned that the standard argument for why spontaneous symmetry (SSB) breaking can happen in a QFT depends crucially on the assumption that the spatial volume is infinite.
This is discussed in detail, for example, in chapter 11 of A Modern Introduction to Quantum Field Theory by Michele Maggiore.
At first, he notes that the fact that SSB is possible in QFT is related to the infinite degrees of freedom of a quantum field:
SSB strictly speaking can only take place in a system with an infinite number of degrees of freedom. It is therefore a genuinely field-theoretical phenomenon, which does not appear in quantum mechanical systems with a finite number of variables. [...] SSB is a phenomenon that cannot take place in a quantum mechanical system with a finite number of degrees of freedom, since in this case, if we have a family of “vacua”, the true vacuum state is a superposition of them which respects the original symmetry.
But his punchline is:
The crucial difference is that the tunneling amplitude in this case is proportional to exp{−cV } with c a constant and V the spatial volume. [...] In an infinite volume this amplitude vanishes and there is no mixing between the two vacua. In other words, the effective height of the barrier is infinite and therefore we truly have two distinct sectors of the theory, i.e. two different Hilbert spaces H+, H− constructed above the two vacua |± with the usual rules of second quantization. There is no possibility to restore the symmetry via tunneling, and all local operators have vanishing matrix elements between a state in H+ and a state in H −.
How is this assumption of an infinite spatial volume justified? Simply by the observation that otherwise, SSB wouldn't be possible?
EDIT:
I recently read Jackiw's The Unreasonable Effectiveness of Quantum Field Theory. There, he argues that
However, in a field theory, the graph in the Figure describes spatial energy density as a function of the field, and the total energy barrier is the finite amount seen in the Figure, multiplied by the infinite spatial volume in which the field theory is defined. Therefore the total energy barrier is infinite, and tunneling is impossible. [...] But we see that this crucial ingredient of our present-day theory for fundamental processes is available to us precisely because of the infinite volume of space, which is also responsible for infrared divergences!
Before this, he discussed the infrared divergences and notes that
[I]nfrared infinity […] is a consequence of various idealizations for the physical situation: taking the region of space-time which one is studying to be infinite, and supposing that massless particles can be detected with infinitely precise energy-momentum resolution, are physically unattainable goals and lead in consequent calculations to the aforementioned infrared divergences. In quantum electrodynamics one can show that physically realizable experimental situations are described within the theory by infrared-finite quantities.
Therefore, the thing really seems to be that an infinite spacetime is just a convenient approximation. Compared to the scale of QFT processes the spacetime volume is certainly huge and instead of some large number, we use infinity. The tunneling probability is therefore maybe not zero, but small enough such that we don't get a superposition as true vacuum state. The final puzzle piece for me would be if someone could quantify when a given tunneling amplitude is small enough such that no superposition happens. (E.g. when the tunneling would have happend less than one time since the big bang or something like that.)
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