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A fundamental observer follows the mean cosmic matter flow and is at rest with respect to the homogeneous cosmological frame. Cosmological isotropy permits no preferred spatial acceleration, rotation vector or anisotropic expansion for this flow: the congruence is geodesic, irrotational and shear-free, with isotropic expansion. Its orthogonal homogeneous spatial slices define simultaneity. Choosing fixed spatial labels along these worldlines and their common proper time as $t$ gives synchronous coordinates, $g_{tt}=c^2$ and $g_{ti}=0$.

In the Friedmann-Lemaître-Robertson-Walker metric, $t$ is this cosmic proper time and $\chi$ is radial comoving coordinate. On a constant-$t$ radial slice, the proper distance in cosmology from $0$ to a fixed ${\chi }_0$ is $D(t)=a(t){\chi }_0$. ThereforeThis is the Hubble parameter. The metric describes expansion when $\dot{a}>0$; its form alone also permits static or contracting histories, so expansion additionally requires that sign of the scale factor derivative.

For a radial photon, the metric components relevant to its motion are $g_{tt}=c^2$ and $g_{\chi \chi }=-a^2(t)$, both independent of $\chi$. The angular momentum components vanish. The covariant-momentum form of the geodesic equation therefore givesOnly this radial contraction is needed: angular metric coefficients can depend on $\chi$, but multiply zero angular components. The conserved quantity is the lower-index $p_1=p_{\chi }$, rather than the contravariant component $p^1$.

For outward travel put $P=-p_{\chi }>0$. The null condition $c^2(p^t{)}^2-a^2(p^{\chi }{)}^2=0$ givesHere the time coordinate is $t$, rather than $ct$. A comoving observer has $u^t=1$ and measures photon energy $E=p_{\mu }{u}^{\mu }=p_t$. Thus energy and frequency decrease as $a^{-1}$, giving the cosmological redshiftThe reciprocal relation for wavelengths follows from their common local propagation speed $c$.

Finally center the spatial coordinates on the receiving comoving observer. At the source, the transverse spatial line element is $dl=a(t_E)S_K(\chi )d\theta$. For a short transverse rod and a small observed angle $\Delta \theta$,Thus the angular diameter distance is $d_A=a(t_E)S_K(\chi )$. Writing the reception epoch as $t_0=t_R$ gives the displayed formula in the question. The length is the proper transverse length at emission; $t_0$ cannot be replaced by an unrelated epoch while retaining that expression.

For the estimate implied by the supplied Hubble constant, use the low-redshift Hubble law $D\simeq cz/H_0$. ThenWith ${\theta }_E=\pi /10800$ radians, inversion of the Einstein angular radius relation givesThe corresponding projected Einstein radius is $R_E=D_L{\theta }_E\simeq 233\mathrm{kpc}$.

This is the projected enclosed mass of the galaxy cluster, including all matter, not just its stars. Strictly, the gravitational lensing formula uses angular diameter distances, and at $z=0.5$ the low-redshift approximation is only an estimate. An exact value requires a specified cosmological expansion history and a consistent lens-to-source angular diameter distance; it cannot be determined from $H_0$ alone. The later estimates use the same distance approximation consistently.

Normalize the scale factor to $a(t_0)=1$ and set $a_d=10^{-3}$. In the same matter-only model $a=(t/t_0{)}^{2/3}$, soIts present proper size after expansion with the scale factor isThe present comoving distance to decoupling isThe angular diameter distance is $D_A=a_d{\chi }_d$, so the angular size of a transverse length equal to one particle horizon radius isIf “the horizon” denotes the diameter of the whole causally connected ball, its angular diameter is twice this, approximately $3.74^{\circ }$. The degree-scale conclusion is the same. Neglecting early radiation is an explicit approximation inherited from the question.

A point-mass gravitational lens is the ideal gravitational lens obtained by concentrating a positive mass $M$ at one location. Use geometrical optics, the weak-field approximation, small angles in radians, and a stationary lens whose deflecting region is short compared with the observer-lens and lens-source distances. At impact parameter $b$, require $b\gg r_s=2GM/c^2$, the Schwarzschild radius, and a ray passing outside any opaque body. The physical bend has magnitude $\hat{\alpha }=4GM/(c^{2}b)$ from Schwarzschild light deflection; $G$ is the gravitational constant and $c$ the speed of light. Multiple images can occur within this weak-field approximation: strong gravitational lensing does not require a ray to approach a black hole closely.^[ref]

Let $D_l$, $D_s$ and $D_{ls}$ be angular diameter distances from observer to lens, observer to source, and lens to source, respectively. They refer to the background geometry used in the thin gravitational lens equation. In nearby Euclidean geometry, $D_{ls}=D_s-D_l$; in cosmology, do not make that subtraction. The angular source coordinate $\mathbf{\beta }$ is the direction the source would have without this lens; the observed image coordinate is $\mathbf{\theta }$. Both are measured relative to the observer-lens optical axis.^[ref]

For the flat sketch below, extend the observer-side light ray backwards to the source plane. Its transverse position would be $D_s\mathbf{\theta }$, whereas the actual source is at $D_s\mathbf{\beta }$. Their separation is $D_{ls}\hat{\mathbf{\alpha }}$. Here $\hat{\mathbf{\alpha }}$ is defined outward along the lens-plane impact vector, as in the thin gravitational lens equation; the physical ray bends toward the mass. Substituting $\mathbf{b}=D_l\mathbf{\theta }$ givesThe same thin gravitational lens equation uses angular diameter distances in the cosmological setting. The factor $D_{ls}/D_s$ converts the physical bend into the reduced angular deflection; these are different angles.

A source-frame transverse physical speed $v$ produces an observed angular rate $\dot{\theta }$. cosmological time dilation converts source time to observer time, so this operational distance equals $(1+z)$ times the angular diameter distance. In an FLRW geometry it is the transverse comoving distance $R_{0}S(\chi )$. It should be derived with the time convention stated, rather than interpreting the source speed and angular rate as measurements made by the same clock.

Hot electrons in galaxy clusters scatter photons of the cosmic microwave background, producing a characteristic spectral distortion. The thermal contribution measures a line-of-sight integral of electron pressure, whereas X-ray emission depends approximately on the integral of electron density squared times a temperature-dependent emissivity. Combining the two with a geometrical model can constrain a cluster's physical size and hence its angular diameter distance. Temperature structure and departures from the assumed geometry introduce systematic uncertainty.