5.2 Flatness problem

Observations of the CMB and Type Ia supernovae tell us that the Universe appears very close to spatially flat, that is, the curvature parameter of the Universe is essentially zero, . The flatness of the Universe is described by the deviation of (the total matter and energy content) from 1:

Here is the speed of light, is the scale factor and is the Hubble parameter. The observations constrain this deviation from flatness to be at the present time, but although remains constant, and are evolving with time. It can be shown that over the period when first radiation and then matter dominated in the early Universe, the deviation of from 1 should have evolved according to:

where and are the current overall density parameter, current radiation density parameter and current matter density parameter, respectively. This leads to a prediction that the deviation of from 1 must decrease substantially with time, so that at the Planck time (about seconds after the big bang), is predicted to be .

Although everyday experience of the world around us leads us to think that a flat spatial geometry is perhaps most ‘natural’, this flatness has historically been understood to present a fine-tuning problem for cosmological theory, known as the flatness problem. It is unclear what physics contrives to ensure the matter and energy content of the Universe at the Planck time was precisely that needed to exactly match the critical density .

Inflation removes this worry, because during an inflationary era Equation 6 no longer applies, and instead the denominator on the right-hand side in Equation 5 grows exponentially. Therefore instead of growing, the deviation from flatness rapidly drops very close to zero in this scenario, making it possible to start the radiation-dominated era from a point close enough to to avoid the flatness problem.