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Integrating the given expansion equation gives $R^{3/2}=t/t_0$, with the integration constant fixed by $R(t_0)=1$. Thus the scale factor and Hubble parameter are $R=(t/t_0{)}^{2/3}$ and $H=2/(3t)$. The Newtonian derivation of the flat Friedmann equation gives $H^2=8\pi G\rho /3$, soUse the comoving coordinate $\mathbf{q}=\mathbf{x}/R$ and the density contrast $\delta ={\rho }_1/\rho$. Linearized mass conservation, the peculiar-velocity Euler equation for fluid motion and the Poisson equation for the peculiar gravitational potential areDifferentiate the first equation, substitute the second and use the third. A Fourier component of comoving wavenumber $k$ obeysSince $c^2{k}^2/R^2=k^2{c}^2(t_0)t_0^{2\gamma -2/3}{t}^{-2\gamma +2/3}$, the normalization is $\Lambda =kc(t_0)t_0^{\gamma -1/3}$, as printed in the PDF. The resulting linear cosmological density perturbation equation isTo reduce it to a Bessel equation, put $\nu =\gamma -4/3$, $z=\Lambda /(\nu t^{\nu })$ and $\epsilon =t^{a}y(z)$. As $\dot{z}=-\nu z/t$, multiplying the substituted equation by $t^{2-a}$ givesTaking $a=-1/6$ makes the first two coefficients proportional to those in the Bessel equation, and the remaining constant is $-25/36$. HenceFor noninteger $\lambda$, the Bessel density modes of a polytropic expanding universe areThere is an exceptional case within the allowed interval: if $\lambda =n$ is an integer, $J_{-n}=(-1{)}^n{J}_n$, so these are not two independent solutions. The full solution then uses $t^{-1/6}[A{Y}_n(z)+B{J}_n(z)]$, with the Bessel function of the second kind providing the missing branch. For example $\gamma =29/18$ gives $\lambda =3$. A basis using $J_{\lambda },Y_{\lambda }$ is valid for every allowed exponent.

For early times, $z\to \infty$. The large-argument asymptotic expansion of the Bessel function of the first kind has oscillatory phase $z-\pi a/2-\pi /4$ and envelope $(2/(\pi z){)}^{1/2}$ for order $a$; the Bessel function of the second kind has the same envelope. ThusThis decreases because $3\gamma <5$. At late times $z\to 0$, the leading powers are $J_{\pm \lambda }(z)\propto z^{\pm \lambda }$ for noninteger order, giving $t^{-1}$ and $t^{2/3}$. The $Y_n$ branch gives the same growing power in the integer case. Therefore a generic perturbation grows as $t^{2/3}$; an initial condition with zero growing-mode coefficient instead has the decaying $t^{-1}$ mode.

To find the growth scale including its dependence on $\nu$, set $\epsilon =t^{2/3}h(t)$. The equation becomesSubstitute $h=1+a{\Lambda }^2{t}^{-2\nu }+\cdots$. The leading coefficient equation is $-2\nu (5/3-2\nu )a+1=0$. The pressure correction to a growing polytropic cosmological mode is consequentlySince $ck/R=\Lambda t^{-\nu -1}$ and $t^{-2}=6\pi G\rho$, the correction becomes small when $(ck/R{)}^2{t}^2$ is small compared with $2\nu (5/3-2\nu )$. The factor $2(5/3-2\nu )$ is between two and $10/3$, giving the stated order-of-magnitude onset conditionFor a well-developed asymptotic growing mode, replace $\lesssim$ by $\ll$. This is a Jeans instability interpretation: the physical acoustic frequency must be sufficiently low compared with the gravitational expansion scale, or equivalently the physical wavelength must be sufficiently long for self-gravity to overcome pressure support. It is an approximate condition for the dust-like growing branch, not an exact initial-condition-independent instant at which any perturbation begins increasing. The instantaneous sign change of the coefficient in the original equation instead occurs at $c^2{k}^2/R^2=4\pi G\rho$.

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 $4\pi G\rho$.