= Open dust linear growth mode
{title2=$D_+(x)=1+3/x-3\sqrt{(1+x)/x^3}\operatorname{arsinh}\sqrt x$}
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 factor>
$$
D_+(x)=D_-(x)\int_0^x\left(\frac{y}{1+y}\right)^{3/2}dy=1+\frac3x-3D_-(x)\operatorname{arsinh}\sqrt x.
$$
Its <Wronskian> with $D_-$ is $x^{-3/2}(1+x)^{-1/2}$, so the modes are independent. The limits are $D_+\sim2x/5$ during <matter domination> and $D_+\to1$ 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>.
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