Luminosity distance
In cosmology, the luminosity distance D_L is defined as the distance to an astronomical source at which it would produce the observed bolometric flux F given its intrinsic bolometric luminosity L, according to the relation F = \frac{L}{4\pi D_L^2}.[1] This measure accounts for the effects of cosmic expansion, where photons are diluted by the increasing scale factor of the universe, leading to D_L = (1 + z)^2 D_A, with z denoting the redshift and D_A the angular diameter distance.[1] Introduced by Richard C. Tolman in 1930 as part of tests for an expanding universe, the luminosity distance concept derives from general relativity and the Friedmann-Lemaître-Robertson-Walker metric, predicting that surface brightness dims as (1 + z)^{-4} due to redshift and expansion. Tolman's framework highlighted how luminosity distance enables distance estimation in curved, dynamic spacetimes, distinguishing it from Euclidean distances and providing a tool to probe cosmic geometry. Luminosity distance plays a central role in observational cosmology, serving as a key observable for mapping the universe's expansion history through standard candles such as Type Ia supernovae, whose peak luminosities are empirically standardized.[2] Measurements of D_L as a function of redshift z reveal deviations from a decelerating expansion, with data from high-z supernovae indicating an accelerating phase driven by dark energy, as evidenced by the Supernova Cosmology Project and High-Z Supernova Search Team in the late 1990s.[2] These findings, quantified via the distance modulus m - M = 5 \log_{10}(D_L / 10 \, \mathrm{pc}), constrain cosmological parameters like the matter density \Omega_m and dark energy density \Omega_\Lambda, underpinning the \LambdaCDM model.[1]Fundamentals
Definition
The luminosity distance d_L is a fundamental measure in astronomy and cosmology that quantifies the distance to a celestial object by relating its intrinsic luminosity L—the total energy emitted per unit time across all wavelengths—to the observed flux F, which is the energy received per unit area per unit time at Earth. It represents the hypothetical distance at which an isotropic point source with luminosity L would yield the observed flux F if embedded in a flat, static, Euclidean space devoid of expansion or other relativistic effects. This concept preserves the inverse square law for light propagation in idealized conditions, providing a baseline for interpreting brightness as a distance indicator. Mathematically, the luminosity distance is given by d_L = \sqrt{\frac{L}{4\pi F}} where the factor of $4\pi accounts for the surface area of a sphere over which the luminosity is distributed uniformly. In a non-expanding universe, this d_L directly corresponds to the standard Euclidean distance, as flux diminishes solely due to geometric spreading. However, in real cosmological settings, deviations arise from effects such as flux dimming due to cosmic expansion, which stretches photon wavelengths and reduces energy density. The nomenclature "luminosity distance" stems from "luminosity," denoting the source's total radiant energy output, and "distance," signifying the effective propagation scale that links emission to observation. Distances are conventionally expressed in megaparsecs (Mpc) or gigaparsecs (Gpc), aligning with the vast scales of the universe probed by this metric.Physical basis
In an expanding universe, the luminosity distance d_L accounts for the diminished flux observed from a distant source due to the combined effects of photon dilution and redshift, which deviate from the simple inverse-square law expected in Euclidean space. Photons emitted isotropically from a source propagate along null geodesics, spreading over an expanding spherical wavefront. The number of photons reaching an observer is conserved, but their density decreases with the square of the comoving distance \chi, as the effective area over which they are distributed scales with $4\pi \chi^2. This geometric dilution already incorporates the effects of expansion during propagation via the definition of the comoving distance, with the scale factor a(t) = 1/(1+z) at emission time relating to the redshift z.[3] An additional reduction in observed flux arises from the energy loss of individual photons due to cosmological redshift. As the universe expands, the wavelength of each photon stretches proportionally to the scale factor, reducing its energy by a factor of $1/(1+z), since E \propto 1/\lambda and \lambda_{obs}/\lambda_{em} = 1+z. This effect lowers the energy flux independently of the geometric spreading.[3] Furthermore, time dilation in an expanding spacetime reduces the rate at which photons arrive at the observer. The time interval between successive photon emissions at the source is dilated by $1+z, meaning the observed photon arrival rate is decreased by another factor of $1/(1+z). Combining these effects—geometric dilution over \chi, energy redshift, and time dilation—the total flux is reduced by $1/[\chi^2 (1+z)^2], leading to a luminosity distance that scales as d_L = (1+z) \chi in simple cases without curvature or acceleration details.[3] Conceptually, this can be analogized to light propagating through an inflating balloon: as the surface expands, not only do the paths between points lengthen, but the wavefronts stretch radially and transversely, diluting the intensity more than in a static medium, with redshift akin to the "tired light" effect from the ongoing expansion.[3]Derivation in cosmology
Role in general relativity
In general relativity, light rays from a distant source propagate along null geodesics, which are the shortest paths in spacetime satisfying the condition ds^2 = 0, where ds^2 is the line element defined by the metric tensor g_{\mu\nu}. These geodesics are parameterized by an affine parameter \lambda, providing a coordinate-independent measure of proper distance along the path, determined by the geodesic equation \frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\lambda} \frac{dx^\beta}{d\lambda} = 0, with Christoffel symbols \Gamma derived from the metric. The observed flux from the source depends on the geometry of a narrow bundle of such null geodesics connecting the emitter to the observer. The flux F is the energy received per unit proper time per unit proper area transverse to the direction of propagation. In curved spacetime, this area is governed by the geodesic deviation equation, which quantifies the relative acceleration of neighboring geodesics within the bundle: \frac{D^2 \xi^\mu}{d\lambda^2} = -[R^\mu_{\nu\rho\sigma}](/page/Riemann_curvature_tensor) \frac{dx^\nu}{d\lambda} \xi^\rho \frac{dx^\sigma}{d\lambda}, where \xi^\mu is the deviation vector, R^\mu_{\nu\rho\sigma} is the Riemann curvature tensor, and D/d\lambda denotes covariant differentiation along the geodesic. The cross-sectional area A of the bundle evolves according to \frac{dA}{A d\lambda} = \theta, where \theta is the expansion scalar of the null congruence, measuring the fractional rate of change of the bundle's cross-sectional area and incorporating the metric's influence on photon spreading or focusing. Integrating this over the bundle yields the effective area. The luminosity distance d_L emerges as a coordinate-independent observable, defined such that the flux relates to the source's intrinsic luminosity L (energy emitted per unit proper time at the source) by F = \frac{L}{4\pi d_L^2}. This definition accounts for the dilution of photon number density across the bundle's area and the redshift-induced energy reduction per photon, both arising from the null geodesic structure without reliance on specific coordinate systems. In relativistic cosmology, d_L thus serves as a fundamental quantity bridging local source properties to global spacetime geometry.Formula in expanding universe
In the Friedmann–Lemaître–Robertson–Walker (FLRW) model of cosmology, the luminosity distance d_L(z) is derived under the assumptions of a homogeneous and isotropic universe on large scales, as dictated by the cosmological principle, with the expansion history governed by the energy content including matter, radiation, and dark energy components.[4][3] In a flat FLRW universe (k = 0, \Omega_k = 0), the luminosity distance to an object at redshift z is given by d_L(z) = (1 + z) \int_0^z \frac{c \, dz'}{H(z')}, where c is the speed of light and H(z) is the Hubble parameter at redshift z', defined as H(z) = H_0 E(z) with H_0 the present-day Hubble constant and E(z) = \sqrt{\Omega_m (1 + z)^3 + \Omega_r (1 + z)^4 + \Omega_\Lambda + \Omega_k (1 + z)^2}, incorporating the density parameters for matter (\Omega_m), radiation (\Omega_r), dark energy (\Omega_\Lambda), and curvature (\Omega_k = 1 - \Omega_m - \Omega_r - \Omega_\Lambda = 0 for flatness).[3] For non-flat universes, the formula generalizes to account for spatial curvature through the density parameter \Omega_k \neq 0, using the transverse comoving distance: d_L(z) = (1 + z) \frac{c}{H_0 \sqrt{|\Omega_k|}} \, \mathrm{sinn} \left( \sqrt{|\Omega_k|} \int_0^z \frac{dz'}{E(z')} \right), where \mathrm{sinn}(x) = \sin x if \Omega_k < 0 (closed universe), x if \Omega_k = 0 (flat), and \sinh x if \Omega_k > 0 (open universe). The argument of \mathrm{sinn} is dimensionless, ensuring consistency.[3] The Hubble parameter H(z) encapsulates the expansion history, with matter dominating at intermediate redshifts (z \sim 0.5), radiation at high redshifts (z \gtrsim 1000), and dark energy driving late-time acceleration (z < 1); these components are parameterized in the standard \LambdaCDM model.[3] In practice, d_L(z) is evaluated numerically due to the lack of closed-form solutions for general cosmologies, typically via direct integration of the defining integral using quadrature methods (e.g., Simpson's rule or Gaussian quadrature) or by solving the coupled differential equations from the Friedmann equations with high-precision codes like CLASS or CAMB, achieving sub-percent accuracy for redshifts up to z \approx 10.[5]Interconnections with other distances
Comparison to angular diameter distance
The angular diameter distance d_A is defined as the ratio of an object's physical transverse size l (measured at the time of emission) to its observed angular size \theta (in radians), such that d_A = l / \theta. This distance measure accounts for the geometry of the universe and the effects of expansion on the apparent size of distant objects. Etherington's reciprocity theorem establishes a fundamental relationship between the luminosity distance d_L and the angular diameter distance d_A, given by d_L = (1 + z)^2 d_A, where z is the redshift. This relation holds in any metric theory of gravity where photons follow null geodesics and their number is conserved, independent of the specific matter content or curvature of the universe.[6] Originally derived by Etherington in 1933, the theorem arises from the conservation of surface brightness and the photon phase space density, linking how flux diminishes with distance to how angles subtend sizes. In a static, flat spacetime (corresponding to z = 0), the reciprocity simplifies to d_L = d_A, reflecting the equivalence of distance measures without expansion. However, in an expanding universe, the factor (1 + z)^2—arising from redshift-induced energy loss and time dilation—ensures d_L > d_A for z > 0, emphasizing the distinct physical effects on luminosity versus angular measurements. In the Friedmann–Lemaître–Robertson–Walker (FLRW) metric for a flat universe, both distances derive from the comoving distance \chi = \int_0^z c \, dz' / H(z'), with d_A = \chi / (1 + z) and d_L = (1 + z) \chi. The expansion rate H(z) varies by epoch, altering the functional forms:| Epoch | H(z)/H_0 | Comoving Distance \chi | d_A Relation | d_L Relation |
|---|---|---|---|---|
| Matter-dominated | (1 + z)^{3/2} | \frac{2c}{H_0} \left[1 - (1 + z)^{-1/2}\right] | \chi / (1 + z) | (1 + z) \chi |
| Dark energy-dominated (pure \Lambda) | $1 (constant) | \frac{c z}{H_0} | \chi / (1 + z) | (1 + z) \chi |