Horizon problem

Remarks

The size of the particle horizon follows from the scale factor. In a decelerating model with $a \propto t^{\alpha}$ and $0 < \alpha < 1$, the particle horizon has proper distance $d_p(t) = t/(1 - \alpha)$. This is comparable to the Hubble radius $1/H = t/\alpha$. Regions that are at the edge of today's observable universe were, at the time of emission, much farther apart than this. The standard proposed solution is cosmic inflation. An early epoch of accelerated expansion makes the integral $\int dt/a$ from the beginning large. Then the whole observable universe comes from a single causally connected region. Other proposals, such as variable speed of light theories, exist, but inflation is the most widely accepted.