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The angular diameter distance is defined by $d_A=x/\theta$ for a small transverse physical size $x$ subtending angle $\theta$. In a spatially flat Friedmann-Lemaitre-Robertson-Walker metric,where $\chi$ is the comoving radial distance.

For an expanding Friedmann-Lemaitre-Robertson-Walker metric with positive energy density, put $\Omega =8\pi G\rho /(3H^2)$ and $w=P/\rho$, in units $c=1$. The Friedmann equation, Friedmann acceleration equation and cosmological perfect-fluid continuity equation giveFor constant $w$, linearizing at $\Omega =1$ gives $\Omega -1\propto a^{1+3w}$. For variable $w$, replace the power by $\exp \int (1+3w(a))d\log a$. This describes the Flatness problem for decelerating expansion, away from epochs at which $H=0$.

The Lambda-CDM model combines a cosmological constant with cold dark matter and the standard baryonic and radiation components in an expanding Friedmann-Lemaitre-Robertson-Walker metric.

Linear cosmological perturbation theory expands the metric and matter fields to first order about a homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker metric.

For a spatially flat Friedmann-Lemaitre-Robertson-Walker metric, negative extrinsic curvature, and $K=-3H+\kappa$, the Hamiltonian constraint gives the displayed scalar perturbation equation. Indeed $K^2=9H^2-6H\kappa$ and $K^i{}_j{K}^j{}_i=3H^2-2H\kappa$ to first order. The spatial Ricci scalar is $4\Delta \Phi$; subtract the background equation $6H^2=16\pi G\bar{\rho }$ and divide by four.

Use units $c=1$ throughout. Set $\chi =|\mathbf{x}|$, so the spatial Euclidean metric is $d{\mathbf{x}}^2=d{\chi }^2+{\chi }^{2}d{\Omega }^2$. On either hemisphere of the embedded three-sphere, the constraint gives $u^2=R^2-{\chi }^2$ and henceIntroduce the dimensionless comoving coordinate $r=\chi /R$. ThenThis is the positively curved Friedmann-Lemaitre-Robertson-Walker metric, with $a(t)$ the physical curvature radius. An equally valid convention retains $r=\chi$, takes $a=b$, and sets $K=R^{-2}$; only the coordinate normalization differs. The chart with $r\le 1$ describes a hemisphere, not the entire closed slice. Writing $r=\sin \psi$ gives $d{\ell }^2=R^2(d{\psi }^2+{\sin }^2\psi d{\Omega }^2)$, with $0\le \psi \le \pi$, displaying the full spherical spatial slices of a closed FLRW universe.

In cosmic time, all three Friedmann-Lemaitre-Robertson-Walker metrics can be writtenwhereThe closed spatial slice is a three-sphere of radius $a(t)$, soThe complete flat and open spatial slices are noncompact and have

For a homogeneous fluid in the Spatially flat FLRW metric, the equations in part (a) reduce to the cosmological perfect-fluid continuity equationwhile homogeneity makes the spatial relativistic Euler equation automatic. The Einstein field equations give the flat Friedmann equation and Friedmann acceleration equation,equivalently $\ddot{a}/a=-4\pi G(\rho +3P)/3$.

For the Friedmann-Lemaitre-Robertson-Walker metric in Cartesian spatial coordinates, the only time-dependent components are $g_{ij}=a^2{\delta }_{ij}$. The Christoffel symbol formula therefore givesSubstituting these into the definition of the Ricci tensor givesThus, with no summation over $i$, the requested components are ${\Gamma }^0{}_{ii}=a\dot{a}$, ${\Gamma }^i{}_{0i}={\Gamma }^i{}_{i0}=\dot{a}/a$, and $R_{ii}=a\ddot{a}+2{\dot{a}}^2$.

For $k=-1$, the Friedmann-Lemaitre-Robertson-Walker metric isThe empty first Friedmann equation, with $\Lambda =0$, givesChoosing the expanding branch and shifting the origin of proper time yields

A tensor cosmological perturbation is the transverse, trace-free part of a metric perturbation about a Friedmann-Lemaitre-Robertson-Walker metric. In conformal coordinates, its spatial metric has the form $g_{ij}=a^2({\delta }_{ij}+h_{ij})$, with ${\partial }_i{h}_{ij}=0$ and $h_{ii}=0$. Its two independent polarizations describe a gravitational wave. Neutrino tensor anisotropic stress can damp its evolution through neutrino free streaming.