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For constant $P=w\rho$, the perfect fluid stress tensor and inverse perturbed spatial metric give $T^{ij}=a^{-2}w\bar{\rho }[(1+\delta ){\delta }^{ij}-h^{ij}]$ at first order. Normalizing the velocity makes $u^0=a^{-1}$ at this order. Stress-energy conservation yields the synchronous perfect-fluid density equation and Cosmological Euler equation in synchronous gauge; the pressure projection includes $v_i{\bar{P}}^{\prime }$, which changes the velocity damping from $\mathcal{H}$ to $(1-3w)\mathcal{H}$.

// Target: cosmology.bigb

Use primes for conformal-time derivatives. The cold-dark-matter Cosmological Euler equation in synchronous gauge is $v_C^{\prime }+\mathcal{H}{v}_C=0$, so multiplying by $a$ gives $(a{v}_C{)}^{\prime }=0$. ThereforeIn an expanding universe $a$ increases, damping any initial peculiar velocity as $a^{-1}$. During radiation domination $a\propto \tau$, so the velocity scales as ${\tau }^{-1}$; during matter domination $a\propto {\tau }^2$, it scales as ${\tau }^{-2}$. This is the nonrelativistic limit of the physical-momentum redshift derived in Q1. It concerns the bulk velocity in the synchronous coordinate frame, not a growing peculiar velocity sourced in another gauge. A cold-matter rest-frame choice also removes the residual synchronous time displacement relevant to that component. The decay statement assumes expanding evolution and the pressureless, linear approximation used by the equation.

The continuity and Cosmological Euler equation in synchronous gauge combine particularly simply. Differentiate ${\delta }_C^{\prime }=-\nabla \cdot v_C-h^{\prime }/2$ and add $\mathcal{H}{\delta }_C^{\prime }$:The velocity terms cancel by the Cosmological Euler equation in synchronous gauge, so this step does not require setting the initial velocity exactly to zero.

For subhorizon modes deep in radiation domination, take the rapidly oscillating radiation perturbation to be negligible in the averaged gravitational forcing. Also neglect the subdominant matter self-gravity to leading order in ${\rho }_C/{\rho }_r$. With $a\propto \tau$ and $\mathcal{H}=1/\tau$, the resulting equation isIntegrating twice giveswhere the arbitrary reference time ${\tau }_{\ast }$ makes the logarithm dimensionless and can be absorbed into $B$. This is logarithmic growth of matter perturbations during radiation domination.

The neglect of matter self-gravity is needed for the displayed form to be exact within the leading radiation-background approximation. Uniform radiation alone leaves a nonzero term $4\pi G{a}^2{\rho }_C{\delta }_C$ if the cold-matter density is retained. Relative to ${\mathcal{H}}^2$, its coefficient is $3{\Omega }_C(a)/2$, small deep in the radiation era but not near equality. The Mészáros equation retains that effect in a matter-plus-radiation background; its solution basis tends to a constant and a logarithm at $a/a_{\mathrm{eq}}\ll 1$, recovering the present leading result. It connects smoothly to matter-era growth rather than allowing the logarithmic approximation to be extrapolated indefinitely.

Galaxy-scale modes enter the Hubble radius before matter-radiation equality. The earlier radiation forcing and horizon-entry matching ordinarily generate a nonzero logarithmic coefficient, even though subsequent oscillatory radiation forcing is small. A Fourier mode with entry time ${\tau }_h\sim k^{-1}$ accumulates $\log ({\tau }_{\mathrm{eq}}/{\tau }_h)\sim \log (k{\tau }_{\mathrm{eq}})$ before equality. Across a large scale-factor range this factor need not be small, so treating the density contrast as exactly frozen would lose an important part of the galaxy-scale seed amplitude and its scale dependence. Growth is nevertheless much slower than ${\delta }_C\propto a$ in matter domination. This is the Mészáros effect, and the logarithmic matching contributes to the small-scale matter cold-dark-matter transfer function.

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.

Write $\rho =\bar{\rho }(1+\delta )$, take a constant barotropic equation-of-state parameter $w$, and keep first-order terms in $\delta$, $v^i$ and $h_{ij}$. In synchronous gauge, the inverse metric is $g^{00}=-a^{-2}$, $g^{0i}=0$, $g^{ij}=a^{-2}({\delta }^{ij}-h^{ij})$. The normalization $u^{\mu }{u}_{\mu }=-1$ gives $u^0=a^{-1}+O(v^2)$ and $u^i=a^{-1}{v}^i$. Substitution in the perfect fluid stress tensor gives the linear perfect-fluid stress tensor in synchronous gauge:In the first component the pressure contributions cancel; in the last, the velocity product is second order and the inverse-metric perturbation supplies the minus sign.

Use projections of stress-energy conservation, ${\nabla }_{\mu }{T}^{\mu \nu }=0$, to retain all connection terms efficiently. Let $\mathcal{H}=a^{\prime }/a$ be the conformal Hubble parameter and $h={\delta }^{ij}{h}_{ij}$. Since $\sqrt{-g}=a^4(1+h/2)$, the fluid expansion isThe energy projection is $u^{\mu }{\partial }_{\mu }\rho +(\rho +P)\vartheta =0$. Its background is ${\bar{\rho }}^{\prime }+3\mathcal{H}(1+w)\bar{\rho }=0$. Expanding and subtracting that background leaves the synchronous perfect-fluid density equationFor momentum, the spatial projection is $(\rho +P)u^{\nu }{\nabla }_{\nu }{u}_i+({\delta }_i{}^{\nu }+u_i{u}^{\nu }){\partial }_{\nu }P=0$. The given connections yield $u^{\nu }{\nabla }_{\nu }{u}_i=v_i^{\prime }+\mathcal{H}{v}_i$ at first order. The pressure projection contains both ${\partial }_i\delta P$ and $v_i{\bar{P}}^{\prime }$; retaining the latter is essential. With $\delta P=w\bar{\rho }\delta$ and ${\bar{P}}^{\prime }=-3w\mathcal{H}(1+w)\bar{\rho }$, division by $\bar{\rho }(1+w)$ gives the Cosmological Euler equation in synchronous gauge,For Fourier convention $e^{i\mathbf{k}\cdot \mathbf{x}}$, these results areThe velocity equation assumes nonzero enthalpy $\bar{\rho }+\bar{P}$, so a cosmological constant with $w=-1$ is not a propagating fluid-velocity case. A time-dependent $w$ would also introduce additional terms and is not the constant-barotropic system used here.

Every term in part (d) is a spatial gradient. Extracting the longitudinal scalar coefficient, up to a spatially homogeneous function that can be absorbed into the zero mode, yields the Cosmological Euler equation in synchronous gauge: