= Cold-matter growth in a matter-coasting-fluid universe
{title2=$D''+\frac{3+4\eta}{2\eta(1+\eta)}D'-\frac{3D}{2\eta^2(1+\eta)}=0$}
In a spatially flat universe containing separately conserved <pressureless matter> and a <coasting fluid> with $w=-1/3$, the ratio $\eta=\rho_s/\rho_m$ is proportional to the <scale factor>. If the coasting component is modeled by the barotropic fluid perturbation equations with no <anisotropic stress>, its contribution to the trace gravitational source vanishes because $1+3w=0$. The matter growing mode is
$$
D(\eta)=\frac52\frac{\sqrt{1+\eta}}{\eta^{3/2}}\int_0^\eta\frac{x^{3/2}}{(1+x)^{3/2}}\,dx.
$$
A decaying solution is $D_-=\sqrt{1+\eta}/\eta^{3/2}$. The <reduction of order> formula with integrating factor $\eta^{3/2}\sqrt{1+\eta}$ produces the integral solution. It has $D=\eta[1-4\eta/7+O(\eta^2)]$ during <matter domination> and tends to $5/2$ during coasting-fluid domination. Thus growth freezes rather than continuing in proportion to the <scale factor>. The background geometry is flat despite the same background expansion law as an open dust universe.
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