7.2. The inflationary energy scale

The most solid observational result is the interpretation of the cosmic microwave anisotropies seen by COBE as giving the amplitude of the initial power spectrum. COBE is a particularly powerful probe because its large beam size makes it sensitive only to scales much larger than the horizon size when the microwave background formed. The perturbations are therefore seen in their primordial form, and depend only on the initial perturbations and not all the other parameters. ⁽⁹⁾

The COBE normalization requires the perturbation at the present Hubble scale, _{H H}(k = a₀H₀), to be given by [25]

(66)

Since

(67)

then unless proves to be tiny (say much less than a hundredth) this will give

(68)

at the time when observable scales crossed outside the horizon, pretty much the scale that particle physicists associate with Grand Unified Theories.

⁹ There is a residual dependence on ₀ and which determine the relation between the metric perturbations and the matter perturbations, and also the evolution of perturbations, but that is easily dealt with. I will assume critical density for simplicity. Back.