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With ${\beta }^2=\Omega R^2/D$ and $\epsilon =2V_{s}R/(3B\Omega D)$, the first angular harmonic of the steady cell conservation equation satisfies $a^{\prime \prime }+a^{\prime }/r-a/r^2-i{\beta }^{2}a=0$ and $a^{\prime }(1)=-1$. The regular solution is $a=-I_1(qr)/[q{I}_1^{\prime }(q)]$, $q=e^{i\pi /4}\beta$. At large $\beta$ it gives the displayed layer with width $1/\beta$. The exact Modified Bessel function of the first kind expression remains regular at the axis, unlike extrapolation of the boundary-layer formula there.

For fixed real $\nu$ as $z\to +\infty$, the Modified Bessel function of the first kind and Modified Bessel function of the second kind have asymptotic expansionsThe expansion and its error bounds establish actual growing and decaying solutions, beyond formal matching of a differential equation. These formulas are documented in NIST DLMF, equations 10.40.1–2.

The endpoint Laplace method for a Legendre polynomial ceases to localize when $n\sqrt{x-1}=O(1)$. On the scale $x=1+\nu /n^2$, its real integral representation givesThe Modified Bessel function of the first kind equals one at $\nu =0$ and has large-$\nu$ behavior $e^{\sqrt{2\nu }}/\sqrt{2\pi \sqrt{2\nu }}$, matching the Hyperbolic Debye asymptotic for Legendre polynomials. This boundary layer resolves an increasingly steep function at an endpoint where its value remains fixed.

Write total pressure and gravitational potential asHere $\pi$ is the Eulerian perturbation of a fluid variable applied to pressure, while $\psi$ is the corresponding gravitational potential perturbation. They have units $\mathrm{Pa}$ and ${\mathrm{m}}^2{\mathrm{s}}^{-2}$, respectively. The constant interior mass density is $\rho$.

The Euler equations areSubtract the hydrostatic equilibrium relation $\nabla p_0/\rho +\nabla {\Phi }_0=0$ and discard terms quadratic in the perturbation. Mass conservation gives incompressible flow, and the gravitational Poisson equation has no bulk density perturbation. Thus, strictly inside the fluid,The last equation is Laplace equation, not a statement that self-gravity has disappeared. The displaced surface carries a first-order surface mass density $\rho \eta$, where $\eta$ is its radial displacement, and this supplies the source and matching conditions for $\psi$ at the moving interface. The axisymmetric gravitational potential of a displaced cylindrical interface, expressed using modified Bessel functions, incorporates that boundary source.

It is useful to define $W=\psi +\pi /\rho$. Taking the divergence of the momentum equation gives ${\nabla }^{2}W=0$. For an axisymmetric Fourier mode with axial wavenumber $k>0$, regularity at $r=0$ selects radial dependence $I_0(kr)$, the Modified Bessel function of the first kind, since $K_0(kr)$ is singular there. In what follows $k=|k_z|$ can always be chosen nonnegative.

Use complex amplitudes and take their real parts at the end. Write the radial surface displacement as $\eta =ϵe^{i(kz-\omega t)}$ and the combined potential asThe linear kinematic boundary condition is $u_r(R)={\partial }_t\eta$. Differentiate it once more in time and use the radial momentum equation, ${\partial }_t{u}_r=-{\partial }_{r}W$, evaluated at the unperturbed radius $R$:With $\kappa =kR$, the mode amplitudes therefore satisfyThe prime differentiates the Modified Bessel function of the first kind with respect to its dimensionless argument, rather than with respect to $r$. The result has units $\mathrm{m}{\mathrm{s}}^{-2}$, as required because $ϵ{\Pi }_0$ is a potential. It applies to $k>0$; the uniform $k=0$ limit requires separate treatment.

Let $C=N_{0}n$ be the cell number density. The steady cell conservation equation with isotropic translational diffusion isUsing the rapid-rotation mean gyrotactic orientation, put $a_s=2V_s/(3B\Omega )$, so the swimming drift is $-a_s{\mathbf{e}}_1$. In the transverse plane, $x_1=Rr\cos {\theta }^{\prime }$ and $x_3=Rr\sin {\theta }^{\prime }$. Thus $\mathbf{u}\cdot \nabla =\Omega {\partial }_{{\theta }^{\prime }}$ andMultiplying the conservation equation by $-R^2/D$ giveswith ${\beta }^2=\Omega R^2/D$ and $\epsilon =a_{s}R/D=2V_{s}R/(3B\Omega D)$. The base fluid velocity is tangent to the wall. Zero normal cell flux there is $-a_{s}n\cos {\theta }^{\prime }-D{n}_r/R=0$, orNormalization requires the cross-sectional mean of $n$ to be one.

For $\epsilon \ll 1$, set $n=1+\epsilon \mathrm{Re}[a(r)e^{i{\theta }^{\prime }}]+O({\epsilon }^2)$. The drift term acting on the leading constant vanishes, and the first angular harmonic satisfiesWriting $q=e^{i\pi /4}\beta$ turns this into the Modified Bessel differential equation of order one. Its regular solution iswhere $I_1$ is the Modified Bessel function of the first kind. The singular second solution is excluded at the axis, and the displayed coefficient enforces the wall boundary condition. The angular harmonic has zero area average, so the normalization is satisfied at this order.

When $\beta \gg 1$, the gyrotactic concentration layer at a rotating-cylinder wall has stretched coordinate $s=\beta (1-r)$. The leading radial equation is $a_{ss}-ia=0$. The solution decaying into the interior is proportional to $e^{-e^{i\pi /4}s}$; the wall condition fixes its coefficient to $-1/q$. HenceThis also follows from the large-argument asymptotic expansion of $I_1$, which gives $a\sim -r^{-1/2}{e}^{-q(1-r)}/q$ near the wall; $r^{-1/2}=1+O({\beta }^{-1})$ within the layer. Its thickness is $R/\beta =\sqrt{D/\Omega }$. The quoted exponential is a wall-layer approximation, not an exact solution all the way to $r=0$; the Modified Bessel function of the first kind formula supplies the regular continuation through the axis.

Change independent variable from proper time to the scale factor. Then $\dot{\delta }=\dot{a}{\delta }_a$ and $\ddot{\delta }={\dot{a}}^2{\delta }_{aa}+\ddot{a}{\delta }_a$. Under radiation domination, $\ddot{a}=-{\dot{a}}^2/a$ and $H^2\simeq 8\pi G{\bar{\rho }}_r/3$. Since ${\bar{\rho }}_m/{\bar{\rho }}_r=a/a_{\mathrm{eq}}$, the linear matter perturbation growth equation becomesPutting $\delta =au$ givesThis keeps the matter self-gravity term in a radiation-dominated background. It is not an exact background equation through radiation-matter equality.

At $a/a_{\mathrm{eq}}\ll 1$, neglecting that small term gives ${\delta }_{aa}+{\delta }_a/a=0$. Integration yieldsThus matter has at most logarithmic growth during this leading radiation approximation, together with a constant independent mode. The constant is non-growing, not a mode that literally falls as $a^{-1}$; that power belongs to $u$ rather than ${\delta }_m$.

For an increasing/decreasing basis of the displayed equation with matter self-gravity retained, put $x=\sqrt{6a/a_{\mathrm{eq}}}$. Its equation becomes $x^2{\delta }_{xx}+x{\delta }_x-x^2\delta =0$. HenceThe Modified Bessel function of the first kind gives $I_0(x)=1+3a/(2a_{\mathrm{eq}})+O((a/a_{\mathrm{eq}}{)}^2)$, which increases slowly. The Modified Bessel function of the second kind gives $K_0(x)=-\frac{1}{2}\ln (a/a_{\mathrm{eq}})+\frac{1}{2}\ln (2/3)-{\gamma }_E+O((a/a_{\mathrm{eq}})|\ln (a/a_{\mathrm{eq}})|)$, where ${\gamma }_E$ is Euler's constant; this mode decreases as $a$ increases. Their leading span is precisely the constant/logarithmic pair above. Mode labels depend on the chosen basis and normalization; no rapid matter-era growth $\delta \propto a$ occurs here. Neglected background corrections can change subleading terms, so the Bessel basis should not be extrapolated through equality.

Put $q={\mu }^2-1/4$ and $S=\log y$. The differential equation becomes the Riccati equation $S^{\prime \prime }+(S^{\prime }{)}^2=1/4+q/x^2$. The inverse-power exponential hierarchy begins withso choose $S_0=\epsilon x/2$, $S_1=0$ with $\epsilon =\pm 1$; additive constants specify the overall normalization. At order $x^{-2}$, $2S_0^{\prime }{S}_2^{\prime }=q/x^2$, and at subsequent ordersThese equations are formally equivalent coefficient by coefficient; the order-zero transport term happens to be constant, so its chosen zero value is not interpreted as a literal denominator in a successive-ratio hypothesis. The ansatz $S_j=c_j{x}^{-(j-1)}$ is consistent for $j\ge 2$, withEmpty sums are zero. In particular $c_3=-q$, which agrees with direct expansion of the Riccati equation.

Exponentiating the inverse-power series gives $y_{\epsilon }\sim e^{\epsilon x/2}{\sum }_{j\ge 0}{A}_j^{\epsilon }{x}^{-j}$ with $A_0=1$. Substitute this amplitude series into $u^{\prime \prime }+\epsilon u^{\prime }=q{x}^{-2}u$ to obtainFormal matching alone does not prove existence of actual solutions with these asymptotics. Here actual solutions are obtained from the Modified Bessel differential equation and the large-argument asymptotic expansion of a modified Bessel function: substituting $y=\sqrt{x}w(x/2)$ yields solutions $\sqrt{\pi }\sqrt{x}{I}_{\mu }(x/2)$ and $\sqrt{x/\pi }{K}_{\mu }(x/2)$, using the Modified Bessel function of the first kind and Modified Bessel function of the second kind, whose positive-real-axis asymptotic expansions have precisely these normalized growing and decaying series. They are linearly independent. The series need not converge; it is an asymptotic expansion with a remainder after each fixed truncation. When $q=j(j+1)$ the amplitude recurrence terminates.

In a radiation-dominated background with matter self-gravity retained, ${\delta }_{aa}+{\delta }_a/a-3\delta /(2a{a}_{\mathrm{eq}})=0$. A change to $x=\sqrt{6a/a_{\mathrm{eq}}}$ gives the Modified Bessel differential equation of order zero. The Modified Bessel function of the first kind increases as $1+3a/(2a_{\mathrm{eq}})+\cdots$; the Modified Bessel function of the second kind decreases with a leading negative logarithm. The early solution space has the constant/logarithmic behaviors, not the $a$ growth of matter domination. This background approximation must not be extrapolated through equality.