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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.