Question: I am going through the text *Quantum Field Theory and the Standard Model* and have a question regarding a formula used in chapter 2 on the general solution to the differential equation: $\left(\partial\_{t}^{2}+\vec{p} \cdot \vec{p}\right) a\_{p}(t)=0$. The author states that the answer is just $\phi(x, t)=\int \frac{d^{3} p}{(2 \pi)^{3}}\left(a\_{p} e^{-i p x}+a\_{p}^{\star} e^{i p x}\right)$, a Fourier Decomposition into plane waves, however I do not have the slightest clue where this formula was obtained from. I want to add that $p x=\omega\_{p} x\_{0}-\vec{p} \cdot \vec{x} .$

Answer: 
The smoothness of the cosmic microwave background shows that there was no large scale matter-antimatter asymmetry at that time, so if the universe has separated into large separated regions of matter and antimatter it must happened since the era of recombination. The problem is that there is no conceivable mechanism for this to happen, so the conclusion has to be that the universe is not currently separated in large regions of matter and antimatter.

That the smoothness of the CMB shows no large scale separation of matter and antimatter seems so obvious as to be hardly worth questioning, but in fact the paper [Is a symmetric matter-antimatter universe excluded?](https://arxiv.org/abs/1512.08482) did exactly this and confirmed that this is the case.
If you have say two galaxies, and if one is made of matter and the other antimatter, the space in between them would show photon emissions due to contact between matter and antimatter and the resulting annihilation.

Assuming they have been separated over a large distance, for a large period of time, these photon emissions would get progressively rare since the amount of matter + antimatter annihilation would have subsided (since a lot of matter and antimatter would have been turned into energy) at the boundary between the two, until the annihilations would stop (so the regions are effectively isolated).
At this point, you are correct in that we would have no way of knowing which was the matter galaxy and which was the antimatter galaxy.

But it is true that if there were matter + antimatter regions in space, we would have observed many times, a constant glow at some boundary between these regions. The fact that we do not see this, seems to suggest that the universe is matter dominated (at least in the observable universe). In fact, we know that space is not a perfect vacuum, and there is always a certain number of particles in every volume, regardless of how small that number may be. Hence there always should be regions of matter and antimatter in between such galaxies, meaning there should always be a "photon glow" no matter how faint it is. But again, we have never observed this kind of glow.