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Let $w=c_s^2$ be constant and small, and write $\theta =i\mathbf{k}\cdot \mathbf{v}$. The supplied synchronous gauge equations implyDifferentiating the first and eliminating $\theta$ givesFor the dominant single component, the Einstein equation gives $h^{\prime \prime }+\mathcal{H}{h}^{\prime }=-3{\mathcal{H}}^2(1+3w)\delta$. Therefore the exact algebraic intermediate relation isThe requested nonrelativistic density equation in synchronous gauge follows to leading nonrelativistic order: discard pressure corrections to the background, Hubble damping, and metric tensor terms while retaining $c_s^2{k}^2\delta$, which can be comparable to gravity at the Jeans wavenumber. Since $w\ll 1$, that scale lies well inside the Hubble radius, so keeping its enhanced gradient term is consistent. The resulting equation iswhere the Friedmann equation gave $3{\mathcal{H}}^2/2=4\pi G{a}^2{\bar{\rho }}_m$. Now ${\delta }^{\prime }=a\dot{\delta }$ and ${\delta }^{\prime \prime }=a^2(\ddot{\delta }+H\dot{\delta })$, soThis is the specified approximation, not an exact finite-$w$ relativistic identity. The pressure-gradient term stabilizes short wavelengths, while gravity drives the long-wavelength Jeans instability.

Now apply the stipulated equation to $P=K{\rho }^{4/3}$ with $K>0$. In the pressure-negligible matter background,Thus $c_s^2{k}^2/a^2$ is proportional to $t^{-2}$, exactly like the gravitational term. DefineThe comoving Jeans wavenumber and $q$ are constant in this era. The equation becomes the Euler-Cauchy equationSetting $\delta \propto t^p$ gives $p^2+p/3+2(q-1)/3=0$. For distinct real roots the full pair of modes isThese are the Jeans modes of a four-thirds polytropic cosmological fluid. The physical Jeans length isHere ${\lambda }_J\propto a$, so its comoving value is constant. For $\lambda >{\lambda }_J$, $q<1$ and $p_{+}>0$, giving genuine growth; the other solution decays. In the long-wavelength pressure-free limit, $p_{+}=2/3$ and $p_{-}=-1$, the familiar matter-era modes. At $\lambda ={\lambda }_J$, the two powers are $0$ and $-1/3$, so the leading mode is constant rather than growing.

For $1<q<25/24$ both powers are negative: pressure prevents growth, although the modes still decay without oscillation. At $q=25/24$ they coincide and the complete solution isFor $q>25/24$ they become a complex-conjugate pair; a real basis isThese short-wavelength acoustic oscillations have a decaying envelope. The onset of instability is $q=1$, while the threshold for oscillatory time dependence is slightly larger, $q=25/24$; expansion damping explains why these two thresholds are not identical.

Put $\theta =i\mathbf{k}\cdot \mathbf{v}$. The synchronous perfect-fluid density equation and Cosmological Euler equation in synchronous gauge giveDifferentiate the first equation and use the second and the scalar trace metric equation. Before making a nonrelativistic approximation, the result isFor $w=w_m=c_s^2\ll 1$, discard relative pressure corrections to the background gravitational and damping terms, while retaining $c_s^2{k}^2$ because a large wavenumber can compensate a small sound speed. Matter domination and the flat Friedmann equation give $\frac{3}{2}{\mathcal{H}}^2=4\pi G{\bar{\rho }}_m{a}^2$. Thus the leading nonrelativistic density equation in synchronous gauge isThe approximation sign records the small finite-$w$ terms just displayed; the printed Newtonian-form equation is not an exact relativistic equation at nonzero $w$.

The comoving Jeans wavenumber balances pressure and gravity: $c_s^2{k}_J^2=4\pi G{\bar{\rho }}_m{a}^2$. Defining physical wavelength as $2\pi a/k$, the physical Jeans length isFor wavelengths longer than this scale, gravity can overcome pressure and density modes grow; shorter modes are pressure-supported acoustic oscillations, with expansion modifying both growth and damping. Before cosmological recombination, the tightly coupled photon-baryon fluid has substantial photon pressure, $c_{s,\gamma b}^2=1/[3(1+R_b)]$, where $R_b=3{\bar{\rho }}_b/(4{\bar{\rho }}_{\gamma })$. The baryonic Jeans scale is correspondingly large, so baryons oscillate instead of freely collapsing on small scales. Cold dark matter has a much smaller pressure scale and can supply gravitational potential wells. After photon decoupling, baryons lose photon pressure support and their sound speed falls to its thermal-gas value; the baryon Jeans length across recombination drops, allowing baryons to fall into those wells. The earlier photon-coupled fluid is not itself a constant-small-$w$ fluid; its role is a physical comparison of pressure support, not an extension of the preceding approximation through recombination.

Define a normalized smoothing window on physical radius $R$, with comoving radius $r=R/a$. If $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}^{\ast }⟩=(2\pi {)}^3{\delta }_D(\mathbf{k}-{\mathbf{k}}^{\prime })P(k,\tau )$, the smoothed matter density variance isThus ${\sigma }_R$ is the root-mean-square amplitude. For a Gaussian window, $W(u)=e^{-u^2/2}$; this choice makes the scale-free integral ultraviolet convergent.

On scales where pressure is negligible, a flat matter-dominated universe has $a\propto {\tau }^2$, $\mathcal{H}=2/\tau$, and $4\pi G{\bar{\rho }}_m{a}^2=6/{\tau }^2$. The density equation has powers satisfying $s(s-1)+2s-6=0$, soSelect the growing mode and exclude residual synchronous gauge modes. Starting with $P(k,{\tau }_{\mathrm{eq}})=Ak$ gives $P(k,\tau )=Ak(\tau /{\tau }_{\mathrm{eq}}{)}^4$. The dimensionless cosmological power spectrum is thenAt cosmological horizon crossing, $k\sim aH=\mathcal{H}$, hence $k\tau \sim 2$, andFor the smoothed variance, change variables to $u=kr$:The Gaussian integral equals $1/2$. At a physical Hubble-radius window, $R=H^{-1}$ and $r=\tau /2$, giving the Gaussian horizon-crossing variance for a Harrison-Zeldovich spectrumThe window-dependent numerical constant is secondary; cancellation of the scale dependence is the Harrison-Zeldovich spectrum property. This demonstration uses the growing pressureless matter-era idealization and therefore applies to modes crossing in that era. Modes which crossed before equality require radiation-era transfer and cannot be evolved through that period with the matter equation. A pure $P\propto k$ spectrum with a real-space top-hat window also has a logarithmically divergent ultraviolet variance unless a physical high-$k$ cutoff or transfer function is supplied; the Gaussian choice makes the stated idealized variance well defined.