= Solution
The <Jeans length> is the physical wavelength at which the restoring pressure term and self-gravity balance. In the elementary static-fluid approximation,
$$
\omega^2=c_s^2k_{\rm phys}^2-4\pi G\rho,
\qquad k_J^2=\frac{4\pi G\rho}{c_s^2},\qquad
\lambda_J=\frac{2\pi}{k_J}=c_s\sqrt{\frac{\pi}{G\rho}}.
$$
Wavelengths longer than $\lambda_J$ are unstable to self-gravitating collapse, whereas shorter ones have restoring sound oscillations. Expansion and relativistic effects modify this elementary criterion. Ideal <cold dark matter> has $c_s=0$, hence no positive pressure-supported <Jeans length> in these equations. For the specified negative-pressure barotropic string fluid, $c_s^2=-1/3$; there is no real restoring-pressure Jeans threshold.
For $\eta\gg1$, the matter growth equation has leading terms
$$
D_{\eta\eta}+\frac2\eta D_\eta-\frac{3}{2\eta^3}D\simeq0.
$$
The gravitational term is one inverse power smaller than the derivative coefficients. Dropping it at leading order gives $(\eta^2D_\eta)_\eta=0$, hence a constant mode and a decaying $\eta^{-1}$ mode. A more controlled check is possible: direct substitution gives $D_-=\sqrt{1+\eta}/\eta^{3/2}$ as an exact solution. <Reduction of order> then gives the growing mode, normalized by $D_+/\eta\to1$ at early times,
$$
D_+(\eta)=\frac52\frac{\sqrt{1+\eta}}{\eta^{3/2}}
\int_0^\eta\frac{x^{3/2}}{(1+x)^{3/2}}\,dx.
$$
The integral grows as $\eta+O(\log\eta)$, while the prefactor is $\eta^{-1}[1+O(\eta^{-1})]$. Thus
$$
\boxed{D_+(\eta)\longrightarrow\frac52,\qquad
\delta_C\longrightarrow\frac52A_k,}
$$
with corrections of order $\log\eta/\eta$. The perturbation freezes because the matter fraction tends to zero while the <coasting fluid> controls the expansion.
To describe the requested <cold-dark-matter transfer function>, its normalization must be specified. The early matter-era <Hamiltonian constraint>, in the negative-expansion convention for $K$, gives $-4k^2\Psi/a^2-4H\kappa=16\pi G\delta\rho$. For the growing comoving dust mode, the <continuity equation> gives $\kappa=\dot\delta_C=H\delta_C$ in proper time. Since $16\pi G\bar\rho=6H^2$, this yields $\delta_C=-2k^2\Psi/[5(aH)^2]$. On early superhorizon scales $\Psi$ is the conserved curvature amplitude, up to the chosen overall sign, and $(aH)^2\propto a^{-1}$. This establishes the $k^2a$ initial growing behavior rather than assuming identical horizon-entry amplitudes at all epochs. For regular adiabatic modes outside the horizon during early <matter domination>, $A_k=\mathcal A k^2\mathcal R_*(k)$, with a $k$-independent constant $\mathcal A$ and primordial curvature amplitude $\mathcal R_*$. The matter equation above has no $k$, so at fixed late time
$$
\delta_C(k,\eta)=\mathcal A k^2\mathcal R_*(k)D_+(\eta).
$$
Consequently the density-per-curvature transfer is proportional to $k^2D_+$ on scales entering after equality. The conventional shape transfer, which factors out $k^2$ and the common growth factor, is \b[a plateau, $T(k)\simeq1$ on these scales]. Comparing with a matter-only universe that continues growing as $\eta$ instead gives a common amplitude suppression $D_+/\eta\simeq5/(2\eta)$, still independent of $k$. A decline caused solely by later horizon entry would contradict the supplied scale-independent matter equation: the regular synchronous growing solution already evolves outside the Hubble radius. This is the <scale-independent matter suppression by a coasting fluid>.
There is also a limiting entry scale. Since $a^2\rho_S$ is constant, define $k_*^2=(8\pi G/3)a^2\rho_S$. Then
$$
\mathcal H^2=k_*^2\left(1+\frac1\eta\right),\qquad
\eta_{\rm ent}=\frac{k_*^2}{k^2-k_*^2}.
$$
A mode enters the <Hubble radius> only if $k>k_*$; $k\gg k_*$ enters during <matter domination>, whereas $k\downarrow k_*$ enters arbitrarily late. Modes $k\leq k_*$ never enter in this ideal coasting future. The requested post-equality entrants occupy $k_*<k<k_{\rm eq}$. The following original diagram distinguishes the two transfer conventions and marks this restricted interval; it does not attempt to model modes entering during <radiation domination>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-55-transfer.png]
{title=Post-equality cold-matter transfer in the barotropic matter-string model: normalized shape versus density per primordial curvature}
The <adiabatic initial conditions> for separately conserved components have equal $\delta_N/(1+w_N)$. Therefore the radiation relation is replaced here by
$$
\boxed{\delta_S=\frac23\delta_C\quad\text{on superhorizon scales}.}
$$
To check dynamically, put $S=\delta_S-2\delta_C/3$. At $k=0$, subtracting two-thirds of the matter equation from the string equation gives $S''+2\mathcal H S'=0$. Selecting $S=0$ and no independent relative-velocity mode preserves the relation. Thus the growing adiabatic string <density contrast> follows the matter mode and is also approximately frozen at late times to leading order in gradients. This is the <adiabatic matter-string density relation>.
At late times $\mathcal H\to k_*$, while $\Omega_C$ and $\delta_C'$ vanish. The homogeneous string equation becomes
$$
\delta_S''+2k_*\delta_S'-\frac{k^2}{3}\delta_S\simeq0,
\qquad
\delta_S\propto\exp\left[\left(-k_*\pm\sqrt{k_*^2+\frac{k^2}{3}}\right)\tau\right].
$$
For $k\gg k_*$, one solution grows rapidly with exponent approximately $k/\sqrt3-k_*$, instead of undergoing radiation-like acoustic oscillations. This is a <gradient instability of a negative-pressure perfect fluid>. For $k\ll k_*$, the growing exponent is only $k^2/(6k_*)$, while the other exponent is approximately $-2k_*$. The leading zero-gradient approximation is frozen, but at fixed nonzero superhorizon $k$ a slow instability can accumulate over sufficiently long <conformal time>. The strict gradient limit and infinite-future limit therefore need not commute.
These are the apparent behaviors of the equations given. They do not establish rapid collapse of a realistic <cosmic string network>: tension, <anisotropic stress> and the constitutive sound response were excluded in replacing the network by this barotropic <perfect fluid>. Once the modeled string perturbation becomes nonlinear, the linear calculation itself ceases to apply.
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