Flatness problem

In a decelerating expanding universe, $|\Omega -1|$ grows, so its small present value requires extremely fine-tuned early initial data unless an earlier mechanism drives $\Omega$ toward one.

The continuity equation givesDuring expansion with $\rho +3P<0$, the product $\rho a^2$ grows and $|{\Omega }^{-1}-1|\propto (\rho a^2{)}^{-1}$ decreases.

For positive slope parameter $A$, the exponential-potential field increases during cosmic inflation. The quantity $aH$ grows by $e^{(1-{ϵ}_H)N}$ with constant ${ϵ}_H=A^2/(16\pi )$ and $N=8\pi ({\varphi }_R-{\varphi }_i)/(A{m}_{\mathrm{pl}})$. Starting from order-one curvature and demanding order-one or smaller present curvature gives the displayed upper bound on the initial field. A specified tighter final curvature tolerance adds one-half its logarithm to the numerator. Thus sufficient cosmic inflation sets an upper, not a lower, initial-field bound for this sign of the potential slope. Instantaneous reheating and subsequent entropy conservation determine $a_R{H}_R/a_0{H}_0$.