Features:

1. Note for cosmological redshift z:

1. z = 0: The present in cosmic time measured from the ideal cosmic time zero of Big Bang cosmology: i.e., the Big-Bang singularity which was at lookback time equal to the age of the observable universe = 13.797(23) Gyr (Planck 2018).

2. z = 10,000: Which is a bit earlier than radiation-to-matter-dominance transition epoch (cosmic time ∼ 47,000 years, cosmic redshift z=3600) (see Wikipedia: Chronology of the universe: Matter domination; Wikipedia: Scale factor (cosmology): Matter-dominated era; Wikipedia: Age of the universe = 13.797(23) Gyr (Planck 2018)).

3. z = ∞: The ideal cosmic time zero, the time of the Big-Bang singularity which was at lookback time equal to the age of the observable universe = 13.797(23) Gyr (Planck 2018). In fact, virtually all cosmologists do NOT believe the Big-Bang singularity is real. The real observable universe probably tracks into Big Bang cosmology no earlier (and probably somewhat later) than the Planck epoch, 10**(-43) s after the ideal cosmic time zero.

2. log-log plots::
On a log-log plot, the main divisions axes are in powers of 10 rather than in equal amounts.
1. Logarithmic plots are useful for plotting quantities that vary by orders of magnitude---we need them all the time in astronomy.
2. By the by, in the jargon of logarithmic plots, a dex is a factor of 10.
3. On the log-log plots in the Image 1 and Image 2, the large tick marks are separated by factors of 10 (i.e., 1 dex). The small tick marks in each dex are at factors of 2, 5, and 8.

3. The cosmological redshift z is defined in general by the formula:

### z = (λobserved-λrest)/λrest ,

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.

4. The vertical axis represents the other cosmological distance measures in units of giga-light-years (Gly) or gigayears (Gyr). Recall giga- is the metric prefix that stands for 1 billion (10**9).

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.

5. The cosmological distance measures given here are NOT measurements.

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.

6. Note all the cosmological distance measure curves converge asymptotically as z → 0 to the same curve. This is because as z becomes small the light travel time over cosmic time (time since the Big Bang zero point) Δt/t→0, and so asymptotically the astronomical objects at those z values are all asymptotically in the cosmic present and all cosmological distance measures converge to just ordinary physical distances which by definition are distances that can be measured with a ruler at one instant in time.

7. Cosmological distance measures: explicated:

1. Luminosity = luminosity distance D_L: Luminosity distance D_L is the distance-like quantity obtained by assuming the simple inverse-square law for measured radiant flux F from an astronomical object of known luminosity, NOT in motion relative to the observer and NOT subject to extinction by cosmic dust:

### F = L/(4πD_L**2) gives D_L = sqrt{L/(4πF)}   .

Luminosity distance is a direct observable, but is only obtainable for astronomical objects of known luminosity and CAN never be obtained to the accuracy/precision of cosmological redshift z. SNe Ia (Type Ia supernovae) are circa 1998--? the best astronomical objects for obtaining luminosity distances.

2. naive Hubble = redshift distance r_red: Distance like quantity r_red is quantity obtained using the redshift velocity v_red=zc in the Hubble law formula v=H_0*r which gives r_red = zc/H_0. The Hubble law formula v=H_0*r is, in fact, an exact result for Friedmann equation (FE) models of the observable universe with r being physical distances and v being recession velocity, but neither r nor v are observables, except asymptotically as z → 0. Also asymptotically as z → 0 we have v_red→v and r_red→r. Note redshift velocity v_red=zc is a direct observable, and therefore so is r_red.

3. LOS comoving = physical distance: Physical distance is just ordinary distance which is what can be measured with a ruler at one instant in time. However, we CANNOT do that for cosmologically remote astronomical objects, except asymptotically as z → 0. So physical distance is a model-dependent result NOT an observable, except asymptotically as z → 0.

4. Lookback time is the time since a light signal started out toward us from cosmological redshift z. It is the light travel time is also model-dependent result NOT an observable, except asymptotically as z → 0 where it just physical distance r divided by vacuum light speed c = 2.99792458*10**8 m/s ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns

5. Angular diameter = angular diameter distance: Angular diameter distance D_A is given by D_A = L_ruler/θ_ruler, where L_ruler is the known size of an astronomical object and θ_ruler is angle (measured in radians) the astronomical object subtends on the sky. Angular diameter distance is a direct observable if L_ruler is known, but but the accuracy/precision is limited even for the best known cosmologically useful astronomical objects (i.e., cosmic distance indicators), the baryonic acoustic oscillations (BAOs). As z → 0, angular diameter distance asymptotically becomes physical distance insofar as the small angle approximation holds and can be corrected to physical distance when it does NOT hold.

6. cosmological redshift z: As described above cosmological redshift z is a direct observable and is easily measured to accuracy/precision. Hence it is the coordinator for all the other cosmological distance measures and is the independent variable for plots of cosmological distance measures as in the images above.

For a table correlating the cosmological distance measures, see the figure below (local link / general link: cosmological_redshift_lookup_table.html).

UNDER CONSTRUCTION below, not required reading
8. The free parameters used the Λ-CDM model vary a bit between references. This is natural---different references have their own data and/or prescriptions for best determination of the free parameters.

The free parameters of Λ-CDM model used for the plots are:

1. Hubble constant H_0 = 72 (km/s/)/Mpc = (1.02857...)*h_70, where h_70 = H_0/[70 (km/s)/Mpc] is the natural unit for the Hubble constant H_0.
2. density parameter for the cosmological constant Ω_Λ = 0.732.
3. density parameter for matter Ω_matter = 0.266.
4. density parameter for radiation Ω_radiation = 0.266/3454 = 0.000077012.
5. density parameter for curvature Ω_k =1 - Ω which is chosen so that the total density parameter (AKA Ω) is 1. For Ω = 1, the spatial geometry is flat (i.e., Euclidean) for the observable universe. A flat spatial geometry is consistent with all observations (see Wikipedia: Friedmann equations: Density parameter).

9. The cosmological distance measures are:

1. Cosmological redshift z (which is the independent variable and forms the horizontal axis) given by

### z = (λ_observed-λ_rest_frame)/λ_rest_frame = a_0/a(t) - 1       ,

where λ_observed is the observed wavelength for an object participating in the mean expansion of the universe λ_rest_frame is the rest-frame wavelength (which is known for spectral lines), a_0 is the cosmic scale factor for the current cosmic time t, a(t) is the cosmic scale factor for the cosmic time when a light was emitted that is reaching us at the at the current cosmic time t = present.

A remarkably simple result in Friedmann-Lemaitre-Robertson-Walker (FLRW) models is that

### a_0/a(t) = z +1       ,

is the factor by which the observable universe has expanded since the cosmic time when a light was emitted.

The cosmological redshift is a direct observable since intrisic spectral line wavelengths are well known.

2. Luminosity distance (just "Luminosity on the plots) is the distance quantity obtained by assuming the flux from a source falls off according to the inverse-square law:

### F = L/(4*πr**2)

which inverted for r gives

### r = sqrt[L/(4*πF)]       ,

the luminosity distance. The luminosity distance would be the proper distance (distance measurable at one instant in time with a ruler) if the observable universe were static and if extinction can be neglected or corrected for.

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.

3. The naive Hubble distance (just called naive Hubble on the plots) is the distance calculated from the formula

### r = (c/H_0)*z = L_H * z       ,

where c is the vacuum light speed, H_0 is the Hubble constant taken to be a constant in cosmic time (which it isn't), L_H = c/H_0 is called the Hubble length (note the Hubble length = L_H = c/H_0 = 4.2827 Gpc/h_70 = 13.968 Gly/h_70) and z is the cosmological redshift again.

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:

### a(t) = a_0*exp(t/t_0 - 1)       ,

where a_0 = 1 is the current cosmic scale factor, exp is exponential function, and t_0 is the present cosmic time cosmic time which is also 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, but recall the expansion of the universe is NOT exponential expansion).

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.

4. LOS comoving on the plots is line-of-sight comoving distance.

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.

5. Lookback time is difference in cosmic time from the present to the cosmic time when a light signal of cosmological redshift z originated.

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.

6. Angular distance is the cosmological distance measure obtained from the formula:

### r=s/θ       ,

where is the perpendicular-to-line-of-sight size of an astronomical object and θ is the angular size of the astronomical object as measured at Earth.

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 → ∞???.

10. An obvious feature of the plots is that all the cosmological distance measures (except the cosmological redshift itself) asymptotically approach the same curve as z → 0.

This item is UNDER RECONSTRUCTION when time allows.

The reason for this is that they agree to 1st order in small 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

### f(z) = f_1*z + f_2*z**2 + f_3*z**3 + ... as long as z ≤ ∼ 1       .

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.

As z → 0, the differences between formulae for the cosmological distance measures eventually start decreasing like z**2 and the relative differences like z because the terms of order ≥ 3 become smaller at a faster rate than lower order terms.

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.

11. The coefficient f_1 by the way is given by the formulae
```  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.

12. There are many ways of defining the boundary between the local observable universe (local in both space and time) and the cosmologically remote observable universe.

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.