Horizon problem and relation to cosmological principle I am taking an introductory course in cosmology and we have just introduced inflation as a way to solve the horizon problem (why the CMB is so uniform if not all parts of it have been in causal contact).
My question is why is this a problem? The cosmological principle states that the universe is isotropic and homogeneous, so shouldn't we expect the CMB to look uniform (if it didn't it would violate the cosmological principle)? What does it mean for something to be too uniform even for a homogeneous universe? How do we quantify this?
 A: If the early Universe had been precisely homogenous, I suspect your point of view would be more widely accepted. However, while the Universe is statistically homogenous and isotropic, it is not literally homogenous and isotropic -- we see more structure in some directions than others. This structure is widely believed to be seeded by random fluctuations in the early Universe.
The temperature fluctuations in the CMB are of order $\delta T/T\sim 10^{-5}$ in every direction on the sky. Since there clearly were differences in temperature from place to place in the early Universe, why should those differences be so small, and of the same order of magnitude, in regions of the Universe that were not in causal contact?
Said differently, the cosmological principle is a nice philosophical idea, but there's no dynamical reason it should be true. From generic initial conditions, you would expect regions of the Universe not in causal contact initially to behave differently. The idea that different regions that have had no causal contact would have exactly the same temperature to five decimal places makes many people uneasy; inflation resolves this tension by effectively making all patches of the Universe we observe in causal contact.
It should be noted that the horizon problem is not a logical problem -- it is logically possible that the Universe simply started with initial conditions that lead to what we observe today. The horizon problem is rather a fine-tuning problem -- it arises from a belief that what we observe should be derivable from generic initial conditions plus a dynamical mechanism that drives the Universe toward what we observe, for some definition of generic.
