where λ_observed is the observed wavelength and λ_rest is the wavelength in the rest frame of emission. Note z is dimensionless number (i.e., it has no units).
Cosmological redshift is itself the most easily obtained high accuracy/precision direct-observable cosmological distance measure.
Thus, it the most basic direct-observable cosmological distance measure and astronomers customarily use it in preference to all others which harder to obtain and usually much less accurate/precise or are model-dependent which means NOT direct NOR indirect observables.
The upshot is that cosmological redshift is the natural independent variable for plots displaying the other cosmological distance measures as dependent variables which is why it is used for that purpose in Image 1 and Image 2.
A giga-light-year is the distance light travels in a gigayear moving at the vacuum light speed relative to an inertial frame of reference.
The vacuum light speed c = 2.99792458*10**8 m/s ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns is exactly 1 Gly/Gyr.
They are calculated for the Λ-CDM model (AKA concordance model) which is fitted to observations.
The Λ-CDM model currently fits all observations within uncertainty and is considered the most trustworthy cosmological model currently available.
Further improved observations may require it to be revised or abandoned, but for now it the best we have.
But even if the Λ-CDM model is revised or abandoned, it still fits the observable universe so well, the cosmological distance measures it predicts must still be correct to good accuracy/precision.
For a table correlating the cosmological distance measures, see the figure below (local link / general link: cosmological_redshift_lookup_table.html).
The free parameters of Λ-CDM model used for the plots are:
A remarkably simple result in Friedmann-Lemaitre-Robertson-Walker (FLRW) models is that
The cosmological redshift is a direct observable since intrisic spectral line wavelengths are well known.
Well extinction can be neglected or corrected for, but the observable universe is NOT static---we live in an expanding universe.
So luminosity distance is NOT in general true physical distance. Note the true physical distance is called proper distance Relativityspeak.
However, luminosity distance is a direct observable given that you know intrinsic luminosity (which sometimes you do), you can measure flux F (which is usually easy) and extinction can be neglected or corrected for (which sometimes you do.)
In fact, observed luminosity distances are key observations in determining the free parameters of the Λ-CDM model.
However, z → 0: luminosity distance becomes proper distance to 1st order in small z: i.e., it becomes proper distance asymptotically as cosmological redshift → 0.
The naive Hubble distance would be the proper distance if the universal expansion were exponential: i.e., if the cosmic scale factor a(t) obeyed the formula:
The naive Hubble distance is NOT a direct observable.
It is plotted on the diagram just a simple reference case.
A cosmological model with exponential expansion is the De Sitter universe.
Comoving distance is a time-independent measure of separation between astronomical objects participating in the mean expansion of the universe.
Comoving distance r_c is related to proper distance r_p (which is true physical distance) by the formula
r_p = r_c * a(t)where a(t) is the cosmic scale factor again.
Since a(t=present) = 1 by convention, comoving distance and proper distance are equal at present---and so both are shown on the plot.
Neither comoving distance nor proper distance are direct observables.
They can only be calculated from a cosmological model---in the shown plots from the Λ-CDM model as aforesaid.
We note that a z → ∞, the comoving distance asymptotically goes to a constant.
This constant is the comoving radius of the observable universe.
The value for this comoving radius from the Λ-CDM model is ∼ 46.6 Gly = 14.3 Gpc (see Wikipedia: Observable universe).
The edge of the observable universe is called the particle horizon.
A light signal that started at the particle horizon at the mythical time zero of the FLRW models (which include the Λ-CDM model) would be reaching us just now.
No such light signal from the particle horizon can actually be reaching us since: (a) the FLRW models CANNOT actually be extrapolated back to time zero, (b) the light signal would be redshifted to zero energy if it came from z = ∞.
Actually, the observable universe was NOT transparent to light until the recombination epoch about 380,000 years after time zero which corresponds to z ≅ 1100 (see Wikipedia: Recombination).
At present we can only observe to to z ≅ 1100. In the future, it may be possible to observe to higher z using neutrinos or gravitational waves.
We note that a z → ∞, the comoving distance asymptotically goes to a constant which is just the age of the observable universe = 13.797(23) Gyr (Planck 2018) (see Planck 2018: Age of the observable universe = 13.797(23) Gyr; Wikipedia: Λ-CDM model: Parameters).
Neither lookback time in general nor cosmic time are direct observables.
They can only be calculated from a cosmological model---in the shown plots from the Λ-CDM model as aforesaid.
Angular distance is proper distance if spatial geometry is Euclidean (which it is within uncertainty) and the observable universe is STATIC (which it is NOT).
So angular distance is NOT proper distance in general.
However, it is a direct observable in principle. But you have to know s and θ. Now θ can be often measured for astronomical objects, but s can usually be only estimated with poor accuracy for most astronomical objects.
However s can be determined to some accuracy/precision for baryonic acoustic oscillations (BAO), and thus these are very useful standard rules for cosmology. Angular distances can be used to fit cosmological models.
You may wonder why angular distance in the log-log plot reaches a maximum as z increases and then declines.
Well as you look out you look back in cosmic time and eventually you look back far enough in cosmic time that an astronomical object is seen when it was much closer than now and its angular distance goes down.
In fact, angular distance → 0 as z → ∞???.
To explain 1st order agreement in this context, we note that the formulae the cosmological distance measures can be written in the power series expansion form
The exponent z is raised to defines the order of the term. The coefficients of the terms are subscripted by the order of the term.
To agree to 1st order means that the f_1 coefficient is the same in for the formulae, but the higher order coefficients do NOT agree in general.
However, as z → 0, lookback time differs from the other cosmological distance measures only by a factor of the vacuum light speed c.
The choice the units gigayear (Gyr) for time and giga-light-years (Gly) for distance cause c = 1 Gly/Gyr which lookback time agree with the distance-like cosmological distance measures to 1st order in small z.
Yours truly has failed to find a compact general reference for the 1st-order for the cosmological distance measures.
There is NO analytic formulae for the cosmological distance measures, except for the naive Hubble distance formula which we gave above.
f_1 = L_H = c/H_0 = (13.968 ... Gly)/h_70 for the distance-like quantities and f_1 = t_H = 1/H_0 = (13.968 ... Gyr)/h_70 for lookback time,where t_H=1/H_0 is called the Hubble time.
People choose whatever definition suits their needs.
One convenient choice of boundary is z = 0.5.
For z ≤ 0.5, the cosmological distance measures all agree to within ∼ 50, whereas for z > 0.5 they diverge more and more.
So z = 0.5 marks a qualitative line in the behavior of the cosmological distance measures.
Another good feature is that the lookback time for z = 0.5 is ∼ 5 Gyr.
The Sun and Solar System is 4.6 Gyr (see Wikipedia: Formation and evolution of the Solar System: Timeline of Solar System evolution).
So if we want to consider the Sun itself as a measure of the observable universe---and we often do since it is our vantage point in time and space---then a lookback time of 5 Gyr and z =0.5 are convenient markers between the old and young observable universe.