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Because density is a scalar field, $\tilde{\rho }(\tilde{x})=\rho (x)$. Write both sides as background plus perturbation and expand the background time shift:The coordinate shift acting on an already first-order perturbation contributes only at second order. Thus the density perturbation gauge transformation isThe uniform-density curvature perturbation in the paper's sign convention isThe derivative on ${\bar{\rho }}^{\prime }$ is essential and is present in the PDF; the TeX transcription drops it. The scalar-curvature combination transforms aswhile the density term changes by $-\mathcal{H}T$. Hence the two time-slicing changes cancel:On a uniform-density slice, $\delta \rho =0$ and this variable is the signed spatial-curvature perturbation. The construction assumes ${\bar{\rho }}^{\prime }\ne 0$; a pure constant-density cosmological constant does not define such a time slicing.

In Newtonian gauge in cosmology, $E=B=0$ and $C=-\Phi$. The background cosmological perfect-fluid continuity equation gives ${\bar{\rho }}^{\prime }=-3\mathcal{H}(\bar{\rho }+\bar{P})$, so, writing $D=\bar{\rho }+\bar{P}$,Set the cosmological adiabatic sound speed $c_a^2={\bar{P}}^{\prime }/{\bar{\rho }}^{\prime }$. Then $D^{\prime }=-3\mathcal{H}D(1+c_a^2)$. Differentiate the previous expression and insert the perturbed energy-conservation equation:Consequently the curvature evolution equation in this sign convention isThe last equality defines the total energy-frame velocity by $\mathbf{q}=D\mathbf{v}$. For adiabatic cosmological perturbations, the non-adiabatic pressure perturbation vanishes. On superhorizon scales, the stated suppression of the velocity divergence makes the remaining gradient term negligible. Thus $\zeta$ is conserved to leading order in $k/\mathcal{H}$. The positive sign of the pressure source follows from the paper's definition of $\zeta$ and must not be replaced by a formula using the opposite curvature convention.

Superhorizon conservation of uniform-density curvature lets one carry a primordial perturbation from its inflationary generation through otherwise complicated eras and predict the initial conditions for later Cosmic microwave background anisotropy and structure growth. It relies on adiabaticity: an isocurvature perturbation can source curvature evolution, so the same conclusion is not automatic in an arbitrary multifield model.

During recollapse $H<0$, so nearby comoving galaxies are increasingly blueshifted rather than redshifted. Their physical separations and angular-diameter distances shrink, making them appear larger and generally brighter; in a closed geometry sufficiently old light can also produce repeated or strongly focused images.

The cosmic microwave background temperature obeys $T_{\gamma }\propto a^{-1}$. It therefore rises without bound as $\theta \to 2\pi$, and its photons are blueshifted. Before the formal Big Crunch, the growing temperature reionizes matter, scattering makes the universe opaque, and the idealization that old galaxies and the original last-scattering surface remain directly visible eventually fails.

After subtracting the mean, the motion-induced dipole and galactic foregrounds, intrinsic Cosmic microwave background anisotropy appears as angular hot and cold patches, typically at the $10^{-5}$ fractional scale. These spatial temperature fluctuations encode early density and velocity structure. The displayed patch pattern is illustrative, with no claim to reproduce an observed sky map.

For a statistically isotropic Cosmic microwave background anisotropy, rotational invariance organizes the angular three-point function using a Wigner 3j symbol:In the usual parity-even temperature convention, one instead factors out the Gaunt integralThe coefficient $b$ is the reduced CMB bispectrum: it removes the angular geometric factor and retains primordial and transfer-function information. A Sachs-Wolfe projection of a constant bispectrum is an analytic large-angle example.

Parity matters here. Since $Y_{\ell m}(-\mathbf{n})=(-1{)}^{\ell }{Y}_{\ell m}(\mathbf{n})$, the Gaunt integral vanishes when ${\ell }_1+{\ell }_2+{\ell }_3$ is odd. Statistical isotropy alone does not impose parity invariance. A parity-odd angular three-point function therefore needs its own geometric convention and cannot generally be represented by this parity-even Gaunt factor times a finite $b$.

Ignoring velocities and evolving potentials, a thin reionization screen transmits the primary Cosmic microwave background anisotropy with amplitude $e^{-{\tau }_{\mathrm{re}}}$ and adds a direction-averaged source. A plane wave propagating a distance $\Delta \chi$ from recombination to the screen has monopole factor $j_0(k\Delta \chi )=\sin (k\Delta \chi )/(k\Delta \chi )$. This suppresses the rescattered source at $k\Delta \chi \gg 1$, while coherent large-scale modes are unchanged to leading order.

Wayne Hu developed physical descriptions of Cosmic microwave background anisotropy, including the 1997 damping-tail analysis with Martin White. Their work calibrated photon-diffusion and reionization damping while separating model-dependent changes in acoustic structure.^[ref]

Photon diffusion smooths fluctuations on scales photons traverse before decoupling. Its angular imprint therefore connects microscopic scattering and the expansion history to the measured small-scale Cosmic microwave background anisotropy.