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The flatness problem in a nutshell One of the Friedman equation is given by $$ H^2\equiv\Big(\frac{\dot{a}}{a}\Big)^2=\frac{8\pi G}{3}\sum\limits_{i}\rho_i-\frac{k}{a^2}.\tag{1} $$ In terms of the density ratio $\Omega_{\rm tot}=\sum\limits_{i}\frac{\rho_i}{\rho_c}$ where $\rho_c=3H^2/8\pi G,$ equation (1) can be re-written as $$1-\Omega_{\rm tot}(t)=-\frac{k}{(aH)^2}.\tag{2}$$ If the universe started with a curved geometry in the past i.e., $k\neq 0$, then $\Omega(t)$ deviates from unity. Note that, $$\frac{1}{aH}=\frac{1}{H_0}a^{(3w+1)/2}$$ Thus if, $w>-1/3$, the the LHS is growing with time which implies $|1-\Omega|$ too is an ever-increasing function of time. Then the question is why is it so small today?

My concern The constant $k$ in the FRW metric can be rescaled to have values $k=0,\pm 1$. How is this a problem when $k$ can always be rescaled and not a measurable parameter?

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The physical quantity is $k/a^2$. Rescaling $k$ must be accompanied by a corresponding redefinition of $a$. If we take $k = -1,0,1$ as is standard, then $a$ gives the radius of curvature.

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