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Relativity and gravitational theory Definition

Distance measures in cosmology

Builds on Friedmann–Lemaître–Robertson–Walker metricScale factorCosmological redshiftFriedmann equations

Definition

Let g=−dt2+a(t)2(dχ2+Sk(χ)2 dΩ2)g = -dt^2 + a(t)^2 \left( d\chi^2 + S_k(\chi)^2 \, d\Omega^2 \right) be a Friedmann–Lemaître–Robertson–Walker metric in units with c=1c = 1. Here aa is the scale factor, normalized by a(t0)=1a(t_0) = 1 at the time t0t_0 of observation. The form dΩ2d\Omega^2 is the round metric on the unit sphere. The function Sk(χ)S_k(\chi) is sin⁡(kχ)/k\sin(\sqrt{k}\chi)/\sqrt{k}, χ\chi or sinh⁡(−kχ)/−k\sinh(\sqrt{-k}\chi)/\sqrt{-k} according as k>0k > 0, k=0k = 0 or k<0k < 0. Let a comoving source emit light at time te<t0t_e < t_0 that reaches a comoving observer at χ=0\chi = 0 at time t0t_0. Let zz be its cosmological redshift.

  1. The comoving distance is DC=∫tet0dta(t)D_C = \int_{t_e}^{t_0} \frac{dt}{a(t)}, which is the comoving coordinate χ\chi of the source.
  2. The transverse comoving distance is DM=Sk(DC)D_M = S_k(D_C).
  3. The angular diameter distance is DA=DM/(1+z)D_A = D_M / (1 + z).
  4. The luminosity distance is DL=(1+z) DMD_L = (1 + z)\, D_M.

Remarks

When the Hubble parameter is known as a function of redshift, DC=∫0zdz′H(z′)D_C = \int_0^z \frac{dz'}{H(z')}, with H(z)H(z) given by the Friedmann equations. An object of proper size ℓ\ell at the source, perpendicular to the line of sight, subtends the small angle ℓ/DA\ell / D_A; a source of power LL gives the observed flux L/(4πDL2)L / (4\pi D_L^2). These two facts are the reasons for the definitions of DAD_A and DLD_L, and together they give the reciprocity relation DL=(1+z)2DAD_L = (1 + z)^2 D_A. Example: in the Einstein–de Sitter model, k=0k = 0 and H(z)=H0(1+z)3/2H(z) = H_0 (1 + z)^{3/2}, so DM=DC=2H0(1−(1+z)−1/2)D_M = D_C = \frac{2}{H_0}\left( 1 - (1 + z)^{-1/2} \right). At small zz all four distances agree to first order with z/H0z / H_0. The angular diameter distance is not monotone in zz in general: in the Einstein–de Sitter model it is largest at z=5/4z = 5/4.

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Cosmological constant, concept, no evidence levelCosmological constantconceptCosmological principle, concept, no evidence levelCosmological principleconceptCosmological redshift, concept, no evidence levelCosmological redshiftconceptEinstein field equations, concept, no evidence levelEinstein field equati…conceptPerfect fluid, concept, no evidence levelPerfect fluidconceptScale factor, concept, no evidence levelScale factorconceptSpace form, concept, no evidence levelSpace formconceptSpacetime, concept, no evidence levelSpacetimeconceptFriedmann–Lemaître–Robertson–Walker metric, example, no evidence levelFriedmann–Lemaître–Ro…exampleFriedmann equations, statement, no evidence levelFriedmann equationsstatementDistance measures in cosmology, definition, no evidence levelDistance measures in …definition
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  1. Cosmological constant concept
  2. Cosmological principle concept
  3. Cosmological redshift concept
  4. Einstein field equations concept
  5. Perfect fluid concept
  6. Scale factor concept
  7. Space form concept
  8. Spacetime concept
  9. Friedmann–Lemaître–Robertson–Walker metric example
  10. Friedmann equations statement
  11. Distance measures in cosmology definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Distance_measure
About this page
TagW767YTWAJE
URIm2://W767YTWAJE
Kinddefinition
WritingW0 stub
MSC83F05
DefinesDistance measures in cosmology
UsesFriedmann–Lemaître–Robertson–Walker metric, Scale factor, Cosmological redshift, Friedmann equations
Changed3 November 2026

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