page_title: Distance measures (cosmology)
section_title: Overview
hierarchical_section_title: Distance measures (cosmology) / Overview
caption_attribution_description: English: A comparison of cosmological distance measures, from redshift zero to redshift of 0.5. The background cosmology is Hubble parameter 72 km/s/Mpc, Omega_lambda = 0.732, Omega_matter = 0.266, Omega_radiation = 0.266/3454, and Omega_k chosen so that the sum of Omega parameters is one. Wesino 13:25, 3 December 2006 (UTC)
context_page_description: Distance measures are used in physical cosmology to give a natural notion of the distance between two objects or events in the universe. They are often used to tie some observable quantity to another quantity that is not directly observable, but is more convenient for calculations. The distance measures discussed here all reduce to the common notion of Euclidean distance at low redshift.
  In accord with our present understanding of cosmology, these measures are calculated within the context of general relativity, where the Friedmann–Lemaître–Robertson–Walker solution is used to describe the universe.
context_section_description: There are a few different definitions of "distance" in cosmology which all coincide for sufficiently small redshifts. The expressions for these distances are most practical when written as functions of redshift , since redshift is always the observable. They can easily be written as functions of scale factor , cosmic time  or conformal time  as well by performing a simple transformation of variables. By defining the dimensionless Hubble parameter and the Hubble distance , the relation between the different distances becomes apparent.
  Here,  is the total radiation energy density,  is the total matter density,  is the dark energy density,  represents the curvature,  is the Hubble parameter today and  is the speed of light. The Hubble parameter at a given redshift is then .
  To compute the distance to an object from its redshift, we must integrate the above equation. Although for some limited choices of parameters (e.g. matter-only: ) the comoving distance integral defined below has a closed analytic form, in general—and specifically for the parameters of our Universe—we can only find a solution numerically. Cosmologists commonly use the following measures for distances from the observer to an object at redshift  along the line of sight:
  Comoving distance:
  Where here  is the Hubble distance, defined as the speed of light times the Hubble time; .  
  Transverse comoving distance:
  Angular diameter distance:
  Luminosity distance:
  Light-travel distance:
  Note that the comoving distance is recovered from the transverse comoving distance by taking the limit , such that the two distance measures are equivalent in a flat universe.
  Age of the universe is , and the time elapsed since redshift  until now is
caption: A comparison of cosmological distance measures, from redshift zero to redshift of 0.5. The background cosmology is Hubble parameter 72 km/s/Mpc, 
    
      
        
          
            Ω
            
              Λ
            
          
          =
          0.732
        
      
      {\displaystyle \Omega _{\Lambda }=0.732}
    
  , 
    
      
        
          
            Ω
            
              
                m
                a
                t
                t
                e
                r
              
            
          
          =
          0.266
        
      
      {\displaystyle \Omega _{\rm {matter}}=0.266}
    
  , 
    
      
        
          
            Ω
            
              
                r
                a
                d
                i
                a
                t
                i
                o
                n
              
            
          
          =
          0.266
          
            /
          
          3454
        
      
      {\displaystyle \Omega _{\rm {radiation}}=0.266/3454}
    
  , and 
    
      
        
          
            Ω
            
              k
            
          
        
      
      {\displaystyle \Omega _{k}}
    
   chosen so that the sum of Omega parameters is 1.
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