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Flatness Problem

The surprising observation that the universe is spatially flat, requiring an extremely fine-tuned initial condition without inflationary theory.
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The statement of the theorem

Consider the spatial curvature parameter kk in the Friedmann equation. The density parameter Ω\Omega is defined as Ω=ρρc\Omega = \frac{\rho}{\rho_c}. For the universe to be spatially flat, Ω\Omega must equal 1, implying k=0k=0. The evolution of the deviation from flatness, Ω−1\Omega - 1, is governed by the equation: dΩdt=−2a˙a(Ω−1)\frac{d\Omega}{dt} = -2\frac{\dot{a}}{a} \left( \Omega - 1 \right) For Ω\Omega to remain close to 1 over cosmological timescales, the initial value Ω(tinitial)\Omega(t_{initial}) must be fine-tuned such that ∣Ω(tinitial)−1∣≪1|\Omega(t_{initial}) - 1| \ll 1, a condition naturally satisfied by inflationary dynamics.
Source: Wikipedia