Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 64 / 1

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.

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.