Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 62 / 4

Write $\delta ={\delta }_{C,\mathbf{k}}$ and $\mathcal{H}=a^{\prime }/a$, the conformal Hubble parameter. The flat Friedmann equation givesOutside the horizon, the stated adiabatic cosmological perturbation relation is ${\delta }_R=(4/3)\delta$. It expresses equality of ${\delta }_i/(1+w_i)$ for dust and radiation in the density convention of the supplied equations. During radiation domination, $a\propto \tau$, so $\mathcal{H}=1/\tau$ and ${\rho }_C$ is negligible. Consequently $8\pi G{a}^2{\rho }_R=3/{\tau }^2$, and the linear cosmological density perturbation equation becomesThe trial solution $\delta \propto {\tau }^q$ gives $q(q-1)+q=4$, hence $q=2,-2$.

During matter domination, $a\propto {\tau }^2$, so $\mathcal{H}=2/\tau$. Now the radiation source is negligible and $4\pi G{a}^2{\rho }_C=(3/2){\mathcal{H}}^2=6/{\tau }^2$. The equation becomeswhose exponents satisfy $q(q-1)+2q=6$, or $(q-2)(q+3)=0$. Thus the growing solutions in both eras areThe other modes are ${\tau }^{-2}$ in the radiation limit and ${\tau }^{-3}$ in the matter limit. These are statements in the density and gauge convention of the supplied evolution equation; a density contrast outside the horizon is not a gauge-independent observable by itself.

After cosmological horizon crossing, the rapidly oscillating radiation source can be averaged away as specified. Well inside radiation domination, the remaining cold dark matter self-gravity is also subleading because ${\rho }_C/{\rho }_R\ll 1$. To leading order the linear cosmological density perturbation equation is thereforeTwo integrations giveThe reference time makes the logarithm dimensionless. A growing superhorizon mode reaches entry with a nonzero derivative, so matching generally excites $B\ne 0$. For example, an abrupt leading-order match from $\delta \propto {\tau }^2$ gives $B={\tau }_H{\delta }_H^{\prime }\simeq 2{\delta }_H$, whereas its exact numerical coefficient depends on the transition and the radiation forcing.

This logarithmic growth is the Mészáros effect. It is an asymptotic radiation-era result: after dropping the oscillating radiation perturbation, the original equation still contains $4\pi G{a}^2{\rho }_C\delta$. Retaining that term and the changing background leads to the Mészáros equation, which provides the interpolation near matter-radiation equality. It would be incorrect to claim that a pure logarithm solves the complete mixed-background equation exactly.

During matter domination, the radiation source can be neglected irrespective of whether the mode is inside or outside the horizon in the supplied linear cosmological density perturbation equation. The Friedmann equation gives $\mathcal{H}=2/\tau$ and $4\pi G{a}^2{\rho }_C=6/{\tau }^2$, just as in part (a). ThereforeSubstituting ${\tau }^q$ gives $(q-2)(q+3)=0$, so for every Fourier modeThus the matter-era growing and decaying density modes have growing behaviour ${\delta }_{C,\mathbf{k}}\propto {\tau }^2\propto a$ for all $k$. There is no pressure-gradient term introducing a $k$-dependent growth rate for cold dark matter. The coefficients retain their different earlier histories; common growth does not erase the spectral shape. A specially chosen pure decaying solution is of course possible, so the growing-mode statement does not assert that every initial condition grows.

Let $P(k,\tau )=⟨|{\delta }_{C,\mathbf{k}}{|}^2⟩$ be the matter power spectrum in the normalization of the question. The Harrison-Peebles-Zeldovich spectrum gives $P\sim \mathcal{C}{\tau }^{4}k$ before entry. Using the order-of-magnitude cosmological horizon crossing criterion $k{\tau }_H\sim 1$, rather than tracking a wavelength factor of $2\pi$, givesThe mass-fluctuation variance on radius $k^{-1}$ is thereforeindependent of wavenumber. Equal logarithmic ranges of scales enter with comparable dimensionless fluctuation strength. This is the scale invariance encoded by the HPZ spectrum; it is not the claim that the dimensional matter power spectrum is independent of $k$ at a fixed time.

First consider $k{\tau }_{\mathrm{eq}}<1$. Such a mode remains outside the horizon through matter-radiation equality, grows as ${\tau }^2$ on both sides, and keeps the same matter-era growth after entry. Squaring this growth preserves the shape of the HPZ spectrum:up to a scale-independent matching coefficient.

For $k{\tau }_{\mathrm{eq}}\gg 1$, entry occurs in radiation domination at ${\tau }_H\sim k^{-1}$, with $P_H\sim \mathcal{C}{k}^{-3}$. The density amplitude then grows only logarithmically until equality. ThusDuring matter domination, the matter-era linear growth factor of the density is $(\tau /{\tau }_{\mathrm{eq}}{)}^2$, so its square multiplies the matter power spectrum:The turnover at $k\sim {\tau }_{\mathrm{eq}}^{-1}$ records whether a mode entered during radiation domination. Earlier-entering small-scale modes miss the rapid ${\tau }^2$ growth and receive only the logarithmic Mészáros effect. This is the origin of the logarithmic small-scale matter transfer. The formula is a large-$k$ asymptote: the additive constant in the logarithmic solution and smooth equality matching must be retained near $k{\tau }_{\mathrm{eq}}=1$, where the spectrum does not vanish.

Multiplying by $k^3$ gives the small-scale mass-fluctuation varianceThis gives the requested estimate for modes entering before equality, with the logarithm replaced by a smooth order-one factor in the transition region. On comoving scale $L\sim k^{-1}$, unit variance today requiresFor a numerical illustration faithful to this question's flat, matter-plus-radiation background, take $h=0.7$ and ${\Omega }_r=4.2\times 10^{-5}{h}^{-2}$, hence ${\Omega }_C=1-{\Omega }_r\simeq 1$ rather than the $0.3$ appropriate to a vacuum-dominated model. Part 1(b), now with ${\Omega }_m={\Omega }_C$, gives ${\tau }_{\mathrm{eq}}\simeq 33\mathrm{Mpc}$. In this two-component model the same integral is valid all the way to today, givingWith $L=3\mathrm{Mpc}$, ${\tau }_0/{\tau }_{\mathrm{eq}}\simeq 258$ and $\ln ({\tau }_{\mathrm{eq}}/L)\simeq 2.39$. Therefore $\mathcal{C}\sim [258^4(2.39{)}^2{]}^{-1}\simeq 4\times 10^{-11}$, corresponding to a horizon-entry root-mean-square fluctuation of roughly $6\times 10^{-6}$. A very small primordial amplitude, of order $10^{-10}$ in this schematic normalization, can reach unit small-scale variance by today. Factors from the Fourier convention, smoothing window and equality matching preclude a precise normalization here. The calculation marks the onset of nonlinear structure formation, where extrapolating the linear solution ceases to be valid. For a universe with late vacuum domination, one must replace the matter-only growth law by its actual growth factor before making a numerical inference.