4.3. The horizon problem and homogeneity

The inflationary expansion also solves the horizon problem. The basic strategy is to ensure that

(32)

so that light can travel much further before decoupling than it can afterwards. This cannot be done with standard evolution, but can be achieved by inflation.

An alternative way to view this is to remember that inflation corresponds to a decreasing comoving Hubble length. The Hubble length is ordinarily a good measure of how far things can travel in the Universe; what this is telling us is that the region of the Universe we can see after (even long after) inflation is much smaller than the region which would have been visible before inflation started. Hence causal physics was perfectly capable of producing a large smooth thermalized region, encompassing a volume greatly in excess of our presently observable Universe. In Figure 2, the outer circle indicates the initial Hubble length, encompassing the shaded smooth patch. Inflation shrinks this dramatically inwards towards the dot indicating our position, and then after inflation it increases while staying within the initial smooth patch. ⁽³⁾

Figure 2. Solving the horizon problem. Initially the Hubble length is large, and a smooth patch forms by causal interactions. Inflation then shrinks the Hubble length, and even the subsequent expansion again after inflation leaves the observable Universe within the smoothed patch.

³ Although this is a standard description, it isn't totally accurate. A more accurate argument is as follows. [2] At the beginning of inflation particles are distributed in a set of modes. This may be a thermal distribution or something else; whatever, since the energy density is finite there will be a shortest wavelength occupied mode, e.g. for a thermal distribution _max ~ 1/T. Expressed in physical coordinates, once inflation has stretched all modes including this one to be much larger than the Hubble length, the Universe becomes homogeneous. In comoving coordinates, the equivalent picture is that the Hubble length shrinks in until it's much smaller than the shortest wavelength, and the Universe, as before, appears homogeneous. Back.