Fact-checked by Grok 2 weeks ago

Angular diameter distance

The angular diameter distance D_A is a key measure in cosmology defined as the ratio of an object's physical transverse size to its observed angular size (in radians), providing a way to infer proper distances from angular observations in an expanding universe.^[1] Unlike the Euclidean case where D_A increases linearly with distance, in standard cosmological models, D_A reaches a maximum around redshift z \approx 1 and decreases at higher redshifts, causing very distant objects to subtend larger angles than they would in a static universe.^[2] This distance is formally related to the transverse comoving distance D_M by the equation D_A(z) = D_M(z) / (1 + z), where the factor of $1 + z accounts for the redshift-induced dilution of angular sizes due to cosmic expansion.^[2] It also connects to the luminosity distance D_L, which governs flux observations, via D_L(z) = (1 + z)^2 D_A(z), highlighting the etherington's reciprocity relation that links these measures in general relativity-based cosmologies.^[3] For a flat universe with matter density \Omega_m and dark energy \Omega_\Lambda, D_M(z) is computed as D_M(z) = \int_0^z \frac{c \, dz'}{H(z')}, where H(z) is the Hubble parameter at redshift z'.^[3] Angular diameter distance plays a crucial role in observational cosmology, enabling the determination of physical scales for extended sources like galaxies, clusters, and cosmic microwave background features from their angular extents.^[1] It is particularly vital for applications in gravitational lensing, where the geometry of light deflection depends on D_A between lens, source, and observer, and for probing the universe's curvature and expansion history through high-redshift surveys.^[2] Measurements of D_A from baryon acoustic oscillations or supernova angular sizes help constrain cosmological parameters, such as the Hubble constant and dark energy equation of state.^[3]

Conceptual Foundations

Definition

The angular diameter distance d_A is defined as the ratio of the physical transverse diameter D of an object to its observed small angular size \delta\theta (measured in radians), such that d_A = D / \delta\theta. This measure relates the intrinsic scale of an extended source, such as a galaxy or cluster, to the angle it subtends in the observer's sky under the small-angle approximation, where \delta\theta \ll 1 radian.^[1] The formal concept of angular diameter distance in relativistic cosmology was developed by Kristian and Sachs in 1966, building upon earlier empirical observations of galaxy angular sizes by Edwin Hubble in the 1920s and 1930s to estimate nebula diameters and distances.^[4] Hubble's work, which correlated angular diameters with luminosity assumptions to infer physical extents, laid empirical groundwork for later theoretical formalizations in expanding universe models. In standard usage, d_A is expressed in units of megaparsecs (Mpc), reflecting its role as a transverse proper distance corresponding to the scale at the emission epoch for extended sources perpendicular to the line of sight. This convention facilitates comparisons across cosmological scales while accounting for the geometry of light propagation.^[1]

Physical Interpretation

The angular diameter distance d_A quantifies the relationship between an object's proper transverse size and the angle it subtends on the sky, serving as the effective distance in a hypothetical Euclidean geometry where the observed angular size matches the physical size divided by this distance. In flat, non-expanding space, d_A aligns directly with the line-of-sight distance, providing an intuitive measure of separation. However, in the curved spacetime of general relativity or the expanding universe of cosmology, d_A diverges from other distances like the luminosity distance due to factors such as metric expansion and spatial curvature, altering how physical scales map to observed angles. In the local universe, where expansion and curvature effects are minimal, d_A behaves such that for a fixed physical size, objects at greater d_A subtend smaller angular sizes, consistent with everyday Euclidean intuition—farther objects appear smaller. At cosmological scales, however, d_A as a function of redshift increases to a peak (typically around z \approx 1.5 in standard models) before declining, leading to a counterintuitive effect where high-redshift objects of fixed proper size can appear larger than expected compared to those at the turnover redshift. This angular size amplification at extreme distances arises from the integrated history of cosmic expansion compressing the effective transverse scale relative to the light-travel path. A practical illustration involves a galaxy with a physical diameter of 100 kpc subtending an angle of 1 arcminute, or \theta \approx 2.91 \times 10^{-4} radians. The angular diameter distance is then given by

d_A = \frac{D}{\theta},

yielding d_A \approx 344 Mpc (approximately 350 Mpc), which converts the observed angle into an inferred physical scale assuming the small-angle approximation holds. This calculation underscores d_A's role in bridging angular observations to intrinsic properties across cosmic distances.^[5]

Formulation in Different Metrics

In Euclidean Space

In Euclidean space, the angular diameter distance serves as a baseline measure for relating the observed angular size of an object to its physical extent, assuming a flat, static geometry. This framework posits an isotropic and homogeneous space without cosmic expansion, where light travels in straight lines, enabling the use of classical Euclidean geometry for distance calculations.^[6] The derivation relies on the small-angle approximation for objects where the subtended angle is much less than 1 radian. Consider an object of transverse physical size D located at a radial distance d from the observer, with the line of sight perpendicular to the object's plane. The angular size \delta\theta (in radians) is then \delta\theta \approx D / d. The angular diameter distance d_A is defined such that d_A = D / \delta\theta, yielding d_A = d in this geometry.^[7] This relation finds practical use in astronomy for nearby objects, where curvature and expansion effects are negligible, such as on solar system scales. For instance, the Moon's average angular diameter of approximately 0.5 degrees, combined with its known average distance of about 384,000 km, allows direct estimation of its physical diameter using D \approx d \cdot \delta\theta (with \delta\theta converted to radians).^[8]

In Friedmann–Lemaître–Robertson–Walker Metric

In the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, which describes a homogeneous and isotropic expanding universe, the angular diameter distance d_A to a source at redshift z is derived by considering the propagation of light along null radial geodesics.^[9] The line element of the FLRW metric is

ds^2 = -c^2 \, dt^2 + a(t)^2 \left[ \frac{dr^2}{1 - k r^2} + r^2 \, d\Omega^2 \right],

where a(t) is the scale factor with a(t_0) = 1 at present time t_0, k is the curvature parameter (with units of inverse length squared), r is the comoving radial coordinate, and d\Omega^2 = d\theta^2 + \sin^2\theta \, d\phi^2.^[9] For null geodesics (ds = 0, d\Omega = 0), the comoving distance \chi to the source is obtained by integrating along the light path:

\chi = \int_{t_e}^{t_0} \frac{c \, dt}{a(t)} = \int_0^z \frac{c \, dz'}{H(z')},

where t_e is the emission time related to redshift by $1 + z = 1/a(t_e), and H(z) = \dot{a}(t)/a(t) is the Hubble parameter at redshift z'.^[9] The comoving angular diameter distance is then f_k(\chi), where f_k is the curvature-dependent function satisfying df_k / d\chi = \sqrt{1 - k f_k^2} with boundary condition f_k(0) = 0, leading to explicit forms: f_k(\chi) = [\chi](/page/Chi) for k = 0, f_k(\chi) = (1/\sqrt{k}) \sin(\sqrt{k} [\chi](/page/Chi)) for k > 0, and f_k(\chi) = (1/\sqrt{-k}) \sinh(\sqrt{-k} [\chi](/page/Chi)) for k < 0. The observed angular size \theta of a source with proper transverse size l at emission relates to d_A by \theta = l / d_A, yielding

d_A(z) = \frac{1}{1 + z} f_k\left( \int_0^z \frac{c \, dz'}{H(z')} \right),

with the factor (1 + z)^{-1} accounting for the angular dilution due to expansion since emission.^[9] For a flat universe (k = 0), this simplifies to

d_A(z) = \frac{1}{1 + z} \int_0^z \frac{c \, dz'}{H(z')}.

^[9] In general, for non-flat universes, the expression is

d_A(z) = \frac{1}{1 + z} \frac{1}{\sqrt{|k|}} \, S_k \left( \sqrt{|k|} \int_0^z \frac{c \, dz'}{H(z')} \right),

where S_k(x) = x for k = 0, \sin x for k > 0, and [\sinh x](/page/Sinh) for k < 0.^[9]

Cosmological Applications

Model Dependence

In the standard \LambdaCDM model, the angular diameter distance d_A(z) depends sensitively on cosmological parameters, reflecting the expansion history and geometry of the universe. The matter density parameter \Omega_m influences d_A(z) differently across redshift ranges: at low z, higher \Omega_m decreases d_A(z) due to greater deceleration in the recent universe, while at high z, it reduces d_A(z) by suppressing the distance peak through enhanced matter domination.^[9] The dark energy density parameter \Omega_\Lambda promotes accelerated expansion, thereby increasing d_A(z) particularly at higher z.^[9] Overall, d_A(z) scales inversely with the Hubble constant H_0, as distances are proportional to c/H_0.^[9] Curvature further modulates d_A(z) in non-flat models. In open universes (k < 0, \Omega_k > 0), d_A(z) is larger than in flat models because diverging light rays increase the effective transverse scale. Conversely, in closed universes (k > 0, \Omega_k < 0), d_A(z) is smaller due to focusing effects that converge light rays, reducing the inferred distance for a given angular size.^[9] As an illustrative example, adopting the Planck 2018 cosmological parameters (\Omega_m = 0.315, \Omega_\Lambda = 0.685, H_0 = 67.4 km/s/Mpc) in a flat \LambdaCDM model yields d_A(z=1) \approx 1700 Mpc.

Angular Size–Redshift Relation

In cosmology, the angular diameter distance d_A(z) follows a characteristic relation with redshift z that encodes the universe's expansion history. At low redshifts, where z \ll 1, this relation simplifies to a linear form approximating the Hubble law: d_A(z) \approx cz / H_0, with c the speed of light and H_0 the present-day Hubble constant. This approximation arises because, for nearby sources, the cumulative effects of expansion on transverse measurements are minimal, allowing the angular size to scale inversely with a Euclidean-like distance.^[9] As redshift increases, the functional form of d_A(z) deviates from linearity, reflecting the integrated influence of cosmic expansion. In decelerating universes, such as matter-dominated models, d_A(z) initially rises with z, reaches a plateau, and eventually decreases, driven by the accelerating dilution of proper transverse sizes relative to the angular subtend due to the ongoing expansion. This non-monotonic behavior highlights how the geometry of light paths in an evolving spacetime alters perceived object sizes at greater cosmic depths.^[9] The theoretical basis for this redshift dependence originates from the propagation of light in an expanding universe, where the scale factor evolves as a(t) = 1/(1+z) from emission to observation. The angular diameter distance emerges from integrating this scale factor over the light travel time along null geodesics, which quantifies the ratio of physical transverse size to observed angular size and incorporates the redshift-induced stretching of wavelengths and transverse dimensions.^[9] Graphical representations of d_A(z) versus z vividly illustrate this relation, typically showing a steep initial rise at low z, a broad peak around z \sim 1, and a decline at higher z in decelerating scenarios. These plots, often normalized by the Hubble distance c/H_0, are essential for visualizing the evolution of apparent angular sizes and interpreting data from standard rulers like galaxies or radio sources across cosmic time.^[9]

Angular Diameter Turnover

The angular diameter turnover refers to the redshift z_t at which the angular diameter distance d_A(z) attains its maximum value, beyond which d_A(z) diminishes as redshift increases further. This phenomenon implies that objects at redshifts greater than z_t subtend larger angular sizes on the sky compared to those at intermediate redshifts, counterintuitively reversing the expected trend in an expanding universe.^[10] The turnover redshift z_t is calculated by solving for the point where the derivative \frac{d}{dz} d_A(z) = 0, which requires evaluating the integral form of d_A(z) in the Friedmann–Lemaître–Robertson–Walker metric and finding its extremum. In the standard flat \LambdaCDM cosmology with matter density parameter \Omega_m \approx 0.3 and dark energy density parameter \Omega_\Lambda \approx 0.7, this yields z_t \approx 1.6; observational estimates place it at z_t \approx 1.70 \pm 0.20.^[10]^[11] In the Einstein–de Sitter model, a matter-only universe with \Omega_m = 1 and no dark energy, the peak occurs at a lower value of z_t \approx 1.25.^[10] Physically, the turnover arises from the evolving expansion history of the universe, particularly the transition from a matter-dominated phase characterized by deceleration to a dark energy-dominated phase driving acceleration, which alters the focusing of light rays from distant sources.^[11] This shift modifies the balance between gravitational lensing effects during deceleration and the defocusing influence of accelerated expansion, positioning the maximum d_A at a redshift preceding the onset of net acceleration around z \sim 0.7.

Observational Relevance

Measurement Techniques

One of the earliest attempts to measure angular diameter distances involved the Tolman surface brightness test, proposed in the 1930s, which used the observed angular sizes and surface brightnesses of galaxies at various redshifts to probe the geometry of the universe.^[12] This method compared the expected dimming of surface brightness with increasing redshift in an expanding universe, where the angular diameter distance d_A(z) influences the apparent size and flux of extended sources, providing an initial empirical constraint on cosmic expansion.^[13] A primary modern technique employs standard rulers, such as baryon acoustic oscillations (BAO), which imprint a characteristic comoving scale of approximately 150 Mpc in the large-scale structure of the universe from early universe physics.^[14] By measuring the angular scale of this BAO feature in galaxy clustering surveys at a given redshift z, the angular diameter distance d_A(z) is derived as d_A(z) = \frac{r_d}{(1 + z) \theta_{\rm obs}}, where r_d is the sound horizon at the drag epoch (a known scale from CMB data) and \theta_{\rm obs} is the observed angular scale of the BAO feature (in practice, analyses use scaling parameters relative to a fiducial cosmology).^[15] This approach has been applied in surveys like the Dark Energy Survey (DES), achieving a 2.1% precision measurement of d_A(z) at effective redshifts around z \approx 0.8 from galaxy clustering data.^[16] The cosmic microwave background (CMB) temperature and polarization anisotropies provide another fundamental technique, serving as a standard ruler through the angular scale of acoustic peaks. This yields precise measurements of D_A(z_*) at recombination (z_* \approx 1090) with ~0.1% precision from Planck 2018 data, which also calibrates the BAO sound horizon r_d.^[17] Another key method utilizes galaxy clusters, where the angular diameter distance is determined by combining X-ray observations of the intracluster medium with measurements of the Sunyaev–Zel'dovich (SZ) effect, which causes a distortion in the cosmic microwave background due to inverse Compton scattering by hot cluster gas.^[18] The physical size of the cluster is inferred independently from X-ray surface brightness profiles assuming spherical symmetry and isothermality, allowing d_A(z) to be calculated as the ratio of the physical radius to the observed angular extent, with the SZ effect providing an additional constraint on the electron pressure along the line of sight. This technique has yielded distance estimates to clusters at redshifts up to z \approx 1, with systematic uncertainties dominated by assumptions about cluster geometry, typically at the 10–20% level in early applications.^[19] Supernovae, particularly Type Ia events in clusters, contribute indirectly by providing luminosity distances that can be cross-checked against angular size measurements, though direct angular sizing of supernova remnants or host galaxies at known physical scales offers supplementary constraints on d_A(z). Recent advancements from large-scale surveys have significantly improved precision; for instance, the Dark Energy Spectroscopic Instrument (DESI) Year-1 data release (2024), incorporating BAO from over 5.7 million galaxies and quasars, constrains d_A(z) to ~0.5–1.5% precision in key redshift bins for z < 2, with DR2 (2025) further improving combined BAO precision to ~0.3% on relevant scales.^[20] Similarly, the Euclid mission, launched in 2023, uses weak lensing and galaxy clustering to forecast sub-percent level constraints on d_A(z) through its wide-field imaging and spectroscopy, with first science data releases in 2025 expected to contribute initial measurements in combination with ground-based surveys.^[21]

Relation to Other Cosmological Distances

In the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, the angular diameter distance d_A relates to the luminosity distance d_L through Etherington's duality, expressed as d_L = (1 + z)^2 d_A, where z is the redshift. This relation, first derived in 1933, connects the observed angular size of extended sources to the flux from point-like sources, assuming conservation of photon number and the validity of general relativity in metric theories of gravity. It holds universally in FLRW cosmologies without additional assumptions about the energy content, provided there is no scattering or absorption of photons along the line of sight.^[22] The angular diameter distance d_A quantifies transverse scales, determining how the proper size of an object at redshift z corresponds to its observed angular diameter \theta via \theta = l / d_A, where l is the object's physical length.^[23] In contrast, the luminosity distance d_L probes radial distances through the observed flux f of standard candles, with f = L / (4\pi d_L^2), where L is the intrinsic luminosity.^[23] For the comoving distance \chi in a flat universe, the relation simplifies to \chi = (1 + z) d_A, where \chi represents the invariant coordinate separation that expands with the scale factor.^[23] Violations of Etherington's duality would imply departures from standard general relativity, such as models with varying speed of light or non-conservation of photons due to interactions with exotic fields.^[24] These deviations are tested observationally to probe new physics and have implications for resolving tensions in cosmological parameters, including the Hubble constant (H_0) discrepancy between local and early-universe measurements, where breaking the relation could reconcile differing distance ladders.^[22]

References

[1]
angular diameter distance
ANGULAR DIAMETER DISTANCE. The angular diameter distance DA is defined as the ratio of an object's physical transverse size to its angular size (in radians).
[2]
[PDF] Distance Measures In Cosmology
6 Angular diameter distance. The angular diameter distance DA is defined as the ratio of an object's physical transverse size to its angular size (in radians).
[3]
[PDF] Lecture 11: Observational Cosmology - MIT OpenCourseWare
Angular diameter distance: The angular diameter distance DA is defined such that. ϕ = l. DA. (316) for an object that has angular size ϕ. Then the endpoints of ...<|control11|><|separator|>
[4]
Republication of: Relativistic cosmology | General Relativity and ...
This is a republication of a paper by GFR Ellis first published in Proceedings of the International School of Physics: General Relativity and Cosmology, 1971.Missing: 1970s | Show results with:1970s
[5]
Small Angle Formula | Imaging the Universe - Physics and Astronomy
The Small Angle Formula is 'Theta equals big D (physical size) over little d (distance)', where θ is angular size in radians. It can be used to find D or d.
[6]
[PDF] ASTR 610 Theory of Galaxy Formation - Yale University
In a static, Euclidean space the angular extent and flux f of an object are related ... and the angular diameter distance is equal to the luminosity distance. dA.
[7]
[PDF] Calculating angular diameter distances
Figure 1 shows an observer in Euclidean space, looking at an object at distance D, perpendicular to the ... angular diameter distance D is defined in such a way ...
[8]
Chandra :: Photo Album :: Scales and Angular Measurement
Jul 5, 2011 · The Sun and the moon have angular diameters of about half a degree, as would a 4-inch diameter orange at a distance of 38 feet.
[9]
[astro-ph/9905116] Distance measures in cosmology - arXiv
May 11, 1999 · Abstract: Formulae for the line-of-sight and transverse comoving distances, proper motion distance, angular diameter distance, luminosity ...Missing: earliest | Show results with:earliest
[10]
Planck 2018 results - VI. Cosmological parameters
12 The quantity DM is (1+z)DA, where DA is the usual angular diameter distance. 13 Doppler aberration due to the Earth's motion means that θ∗ is expected ...
[11]
angular-diameter distance maximum and its redshift as constraints ...
As in Hellaby (2006), we also use units such that G=c= 1. Our thanks to George Ellis and Charles Hellaby for discussions and comments, and to an anonymous ...
[12]
maximum angular-diameter distance in cosmology - Oxford Academic
Jul 23, 2018 · Below, we will trace the history that has brought us to this point where we can meaningfully measure the redshift zmax at which dA(z) ...
[13]
THE TOLMAN SURFACE BRIGHTNESS TEST FOR THE REALITY ...
The Tolman SB test is particularly interesting because its principle is so clear as to give a major predicted difference in observational data between an ...
[14]
Observations contradict galaxy size and surface brightness ...
Mar 21, 2018 · As Tolman (Tolman 1930,1934) demonstrated, in any expanding cosmology, the surface brightness (SB) of any given object is expected to decrease ...
[15]
[2409.08759] Dark Energy Survey: 2.1% measurement of the Baryon ...
Sep 13, 2024 · Here, we present the angular diameter distance measurement obtained from the measurement of the Baryonic Acoustic Oscillation (BAO) feature ...
[16]
Dark Energy Survey Year 1 results: measurement of the baryon ...
ABSTRACT. We present angular diameter distance measurements obtained by locating the baryon acoustic oscillations (BAO) scale in the distribution of galaxi.
[17]
Dark Energy Survey: A 2.1% measurement of the angular Baryonic ...
Feb 16, 2024 · We present the angular diameter distance measurement obtained with the Baryonic Acoustic Oscillation feature from galaxy clustering in the completed Dark ...
[18]
Angular diameter distance estimates from the Sunyaev-Zel'dovich ...
Oct 28, 2005 · We discuss the implications of our results for an accurate determination of the Hubble constant H_0 and of the density parameter Omega_m. We ...
[19]
Probing the cosmic distance-duality relation with the Sunyaev-Zel ...
The angular diameter distances toward galaxy clusters can be determined with measurements of Sunyaev-Zel'dovich effect and X-ray surface brightness.
[20]
New tests of cosmic distance duality relation with DESI 2024 BAO ...
Jun 15, 2025 · In this paper, we test the cosmic distance duality relation (CDDR), as required by the Etherington reciprocity theorem, which connects the angular diameter ...
[21]
Euclid: Forecast constraints on consistency tests of the ΛCDM model
We find that in combination with external probes, Euclid can improve current constraints on null tests of the ΛCDM by approximately a factor of three.
[22]
The resilience of the Etherington–Hubble relation - Oxford Academic
The Etherington reciprocity theorem, or distance duality relation (DDR), relates the mutual scaling of cosmic distances in any metric theory of gravity.
[23]
[PDF] Distance measures in cosmology
The angular diameter distance is plotted in Figure 2. At high redshift, the angular diameter distance is such that 1 arcsec is on the order of 5 kpc. There ...Missing: earliest | Show results with:earliest
[24]
Distance-duality in theories with a nonminimal coupling to gravity
Oct 21, 2021 · Etherington's relation, also known as the distance-duality relation (DDR), directly relates luminosity and angular-diameter distances in GR, ...