= Pressure correction to a growing polytropic cosmological mode
{title2=$\delta_+(t)=C t^{2/3}[1+\Lambda^2t^{-2\nu}/(2\nu(5/3-2\nu))+\cdots]$}
Put $\delta=t^{2/3}h(t)$ in the <Bessel density modes of a polytropic expanding universe> equation. It becomes $h''+8h'/(3t)+\Lambda^2t^{-2\nu-2}h=0$. Substituting $h=1+a\Lambda^2t^{-2\nu}+\cdots$ gives $a=[2\nu(5/3-2\nu)]^{-1}$. Since $ck/R=\Lambda t^{-\nu-1}$, the fractional pressure correction is small when $(ck/R)^2t^2\ll2\nu(5/3-2\nu)$. For $0<\nu<1/3$, this is the order-of-magnitude condition $ck/R\ll\sqrt{6\pi\nu G\rho}$. It describes approach to the dust-like growing branch, not an exact universal time at which every initial perturbation starts increasing. The instantaneous <Jeans instability> balance instead compares $(ck/R)^2$ with $4\pi G\rho$.
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