Every direction we look we see radiation left over from the Big Bang, known as the Cosmic Microwave Background Radiation (CMBR). Those of us who are old enough to remember analog TV have seen some of this radiation in the “snow” we saw when we tuned to a channel between stations.
The expansion of the Universe has stretched out the wavelength of this radiation so that it is now in the microwave part of the electromagnetic spectrum, but it was once much shorter, in the optical part of the spectrum and shorter wavelengths. One remarkable aspect of this CMBR is that it nearly perfectly matches the radiation that is given off by a hot object in thermal equilibrium with its surroundings.
This means we can measure the temperature of the Universe: it is currently 2.725 degrees above absolute zero. This temperature is the same to better than one part in a thousand in every direction.
But this is a problem, because if General Relativity describes the expansion of the Universe, the Universe expanded so rapidly that different parts of the Universe could never have been in thermal contact with each other. Another way to say this is that light emitted at the beginning of the Universe has not has not had enough time to cross the visible Universe because the expansion of the Universe has been so fast. Without the ability to interact, there is no reason different places should be at thermal equilibrium with each other.
The standard explanation for having the entire Universe at the same temperature while keeping General Relativity is to postulate a period of exponential expansion, called inflation. The basic idea is that the Universe first came into thermal equilibrium, then it inflated, then the inflation stopped, replaced by the slow expansion we see today. Since the entire visible Universe came from a very small region of space, everything is at the same temperature.
Unfortunately, there is a huge problem with this explanation. Inflationary theories appear to require fine tuning to get inflation started, and more fine tuning to get it to stop. In essence, we have just replaced one problem with two different problems. If we were comfortable with fine tuning, this fine tuning could have been used to make the initial temperature of the Universe the same everywhere without a physical mechanism, so it is not clear that shifting the fine tuning to inflation has really bought us anything other than the fact that the fine tuning is now less obvious.
Antigravity for Antimatter can solve this Mystery!
Fortunately, if we have a Universe with equal parts matter and antimatter which repel each other gravitationally, then we do not need inflation to make the Universe the same temperature everywhere we look. The reason is that the initial expansion of the Universe would be much slower, and this allows all parts of the visible Universe to be in causal contact with each other, allowing them to be in thermal equilibrium. The reason for the slower expansion is easy to see: with no net gravitational charge the expansion of the Universe is essentially coasting, whereas if everything is attracting everything else (as it is in the presently accepted concordance model) the initial expansion of the Universe needs to be much faster to overcome this attraction. Furthermore, this initial expansion rate has to be incredibly fine tuned; too small and the Universe rapidly collapses back upon itself, and too fast and the Universe flys apart so fast there is no time for any structure to form. In mathematical terms, we need the Universe to be balanced at Ω=1.