Pressure correction to a growing polytropic cosmological mode (${\delta }_{+}(t)=C{t}^{2/3}[1+{\Lambda }^2{t}^{-2\nu }/(2\nu (5/3-2\nu ))+\cdots ]$)

Put $\delta =t^{2/3}h(t)$ in the Bessel density modes of a polytropic expanding universe equation. It becomes $h^{\prime \prime }+8h^{\prime }/(3t)+{\Lambda }^2{t}^{-2\nu -2}h=0$. Substituting $h=1+a{\Lambda }^2{t}^{-2\nu }+\cdots$ gives $a=[2\nu (5/3-2\nu ){]}^{-1}$. Since $ck/R=\Lambda t^{-\nu -1}$, the fractional pressure correction is small when $(ck/R{)}^2{t}^2\ll 2\nu (5/3-2\nu )$. 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 $4\pi G\rho$.