Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 62 / 4

At linear order a coordinate displacement ${\xi }^{\mu }$ changes a metric perturbation by $q_{\mu \nu }\mapsto q_{\mu \nu }-{\mathcal{L}}_{\xi }{\bar{g}}_{\mu \nu }$. Work in conformal time with ${\bar{g}}_{\mu \nu }=a^2\mathrm{diag}(-1,1,1,1)$, write ${\xi }^0=\alpha$ and lower spatial indices with ${\delta }_{ij}$. Directly evaluating the Lie derivative of a covariant tensor field givesHere primes on $\alpha ,{\xi }_i$ mean conformal-time derivatives, while primes on $q_{0\mu }^{\prime }$ label transformed components. Setting the transformed lapse and shift perturbations to zero requiresThese are first-order time equations. Their explicit local solutions areThus four coordinate functions can impose the four synchronous conditions in any regular perturbative patch. This is synchronous gauge fixing by coordinate displacement. The arbitrary spatial functions $C,D_i$ remain as residual gauge freedom; the conditions alone do not fix the coordinate system completely.

Geometrically, synchronous coordinates follow a congruence of freely falling observers launched normally from a spatial slice, using their proper time. This gives unit lapse and zero shift in cosmic time; conversion to background conformal time gives the displayed convention. “Always possible” is local: geodesic caustics or singularities can prevent a global synchronous chart. Such global obstructions do not invalidate local linear cosmological perturbation theory.

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.

With negligible cold-dark-matter velocity, the synchronous perfect-fluid density equation gives ${\delta }_C^{\prime }=-h^{\prime }/2$. Differentiate and use the trace Einstein equation, taking the cold matter sound speed to vanish and its density to dominate:In a spatially flat matter-dominated universe, $a\propto {\tau }^2$, $\mathcal{H}=2/\tau$, and the conformal Friedmann equation gives $4\pi G{a}^2{\rho }_C=3{\mathcal{H}}^2/2=6/{\tau }^2$. HencePut ${\delta }_C\propto {\tau }^s$. The indicial equation is $s(s-1)+2s-6=(s-2)(s+3)=0$. These two independent modes span the solution space, soEquivalently the growing mode is proportional to $a$ and the decaying mode to $a^{-3/2}$. The coefficients are set by initial data. Since synchronous gauge has residual coordinate modes, interpreting the density contrast physically requires fixing its residual freedom consistently, for example by the cold-matter rest frame and metric tensor integration convention; the mode equation alone is not a gauge-invariant observable.

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.