Open dust linear growth mode ($D_{+}(x)=1+3/x-3\sqrt{(1+x)/x^3}\mathrm{arsinh}\sqrt{x}$)

For pressureless matter in a Friedmann-Lemaître-Robertson-Walker metric with negative spatial curvature of an FLRW universe and no cosmological constant, set $x=(1-{\Omega }_{m0})a/{\Omega }_{m0}>0$. A decaying linear growth factor is $D_{-}(x)=\sqrt{(1+x)/x^3}$. Reduction of order gives the regular growing linear growth factorIts Wronskian with $D_{-}$ is $x^{-3/2}(1+x{)}^{-1/2}$, so the modes are independent. The limits are $D_{+}\sim 2x/5$ during matter domination and $D_{+}\to 1$ in a curvature-dominated universe. Multiplying by $5/2$ normalizes the early mode to $x$. Negative spatial curvature of an FLRW universe thus stops late linear amplification even though the model contains no dark energy.