Caption: Our light cone for the observable universe according to the Λ-CDM model (AKA concordance model).
Explanation of the graph:
The Λ-CDM model may NOT be the final expanding universe model but it probably approximates that final expanding universe model for most of cosmic time.
The Λ-CDM model is an accelerating universe model with a non-zero cosmological constant (AKA Lambda or Λ) chosen to fit the acceleration of the universal expansion.
Time zero of cosmic time is formally the big bang singularity---a time of infinite mass-energy density within expanding universe models.
Virtually, everyone believes that the expanding universe models can apply to sometime later than time zero, but earlier than the time of Big Bang nucleosynthesis---which started at about 10 s after time zero.
The Λ-CDM model gives the cosmic time of our NOW as age of the observable universe = 13.797(23) Gyr (Planck 2018) for one determination of Λ-CDM model parameters. Other determinations give slightly different values, but they are all about 13.8 Gyr.
t_0 is the commonly called the age of the universe, but it is only that within expanding universe models and NOT be probably in an ultimate sense.
Another way of putting is that lookback time is cosmic time with our NOW as time zero.
Physical distance is just what you would measure with a ruler at one instant in time---i.e., exactly what you would ordinarily call distance.
Physical distance scales to past and future cosmic time by the cosmic scale factor a(t). At our NOW, we cosmic scale factor a(t_0)=1---this the convenient conventional choice.
Objects participating in the expansion of the universe have the same comoving distance at all times.
We don't move in comoving distance to good approximation for the purposes of this graph.
Where we are in spacetime is marked Here & Now on the graph.
Within the expanding universe models, observable universe is a sphere in comoving distance centered on us.
The expanding universe models assume the cosmological principle---i.e., they assume that the observable universe is homogenuous (same in all places) and isotropic (same in all directions) for average quantities when analyzed on a large enough scale at one instant in cosmic time.
Observationally, the cosmological principle is verified from scales larger than about 250 Mly (ie., 80 Mpc) according to one reference (see Wikipedia: Cosmological principle: Justification), but there is evidence that you may need to analyze over a scale of about 2 Gly (see Moskowitz, C. 2015, SciAm, Space Supervoid Sucks Energy from Light).
Given the cosmological principle, expanding universe models necessarily give the comoving distance to the edge of the observable universe as the same in all directions. Hence, the observable universe is a sphere centered on us (i.e., on HERE).
In other words, the red lines world lines of ideal beams light (AKA electromagnetic radiation)
The red lines define the light cone of Here & Now for the observable universe.
Anything inside the light cone can be casually linked to Here & Now (i.e., affect Here & Now) and anything outside CANNOT.
Higher speeds/velocities exist in various senses.
Note that they increase going back in cosmic time until they reach a maximum of 5.67 Gly at cosmic time about 4 Gyr and then they decrease back to time zero.
The reason for this behavior is that the signals started out in the past when the observable universe was smaller.
A signal starts its flight through spacetime from a point at a cosmological physical distance smaller than the present-day cosmological physical distance (which is also the comoving distance) for the start point.
The space grows under the signal while it travels.
As the start point approaches the zero point of cosmic time, the cosmological physical distance goes to zero.
The expansion of the universe dragged those signals along way before they reached Here & Now.
The comoving distance of a start point always increases as you go back in cosmic time.
The comoving distance of a signal that reaches Here & Now starting at time zero defines the edge of the observable universe.
This edge is called the particle horizon.
The comoving distance to the particle horizon is about 46 Gly according to the Λ-CDM model.
This distance is also the cosmological physical distance to the edge of the observable universe at the present time.
The asymptotic behavior is because as lookback time goes to zero, the expansion of space during the lookback time goes to zero and it is as if the observable universe we static, NOT counting the peculiar velocities of astronomical objects which are superimposed on the expansion of the universe.
How do we then know comoving distance, cosmological physical distance, and lookback time for an astronomical object?
For a given expanding universe model (e.g., the Λ-CDM model) and a direct observable proxy for the cosmological distance measures comoving distance, cosmological physical distance, and lookback time, we can calculate those cosmological distance measures.
Say we observe λ_obs for the wavelength of a spectral line in the spectrum of a cosmologically remote astronomical object for which the peculiar velocity is negiligble compared the recession velocity (the velocity of expansion relative to the observer).
We recognize the spectral line from the pattern of spectral lines in the spectrum and then know its rest-frame wavelength λ_rest from laboratory measurements.
The cosmological redshift z is calculated from the formula
z = (λ_obs-λ_rest) / λ_rest ,and so z is a relative shift in wavelength
The cause of the cosmological redshift is the expansion of space as the light propagates through space.
The wavelength of the light cosmic scale factor a(t):
λ_obs/a(t_0)=λ_rest/a(t) ,where t_0 is present cosmic time and t is the cosmic time of emission of the light.
A little algebra shows that
a(t_0)/a(t) = 1 + z(see Wikipedia: Cosmic scale factor).
Note that a(t=0)=0 in expanding universe models.
So z = ∞ for light signal that starts at t=0.
A signal that starts at time zero and reaches us started at the edge of the observable universe (i.e., at the particle horizon) and is redshifted to infinite wavelength at Here & Now---this means they have gone to zero energy and are unobservable.
Light signals from beyond the particle horizon have NOT had time to reach us since time zero.
Cosmological redshift is direct observable---it is NOT cosmological-model-dependent---and it is usually easily observed.
We note that the comoving horizon has increased with cosmic time As cosmic time increases more the universe in comoving distance size becomes the observable universe.
Light signals can reach us from farther away.
But this will NOT go on forever if the acceleration of the of the expansion of the universe continues at its present rate into the far future.
Eventually, everything beyond the Local Group of galaxies will disappear from sight (see Abrabram Loeb, p. 3).
The expansion of space between us and that realm will become so large that light traveling from there will be dragged away from us with space faster than it can cross space.
The observable universe will then have shrunk to just the Local Group (in whatever form it has then).
The Local Group as a gravitationally bound system does NOT participate in the expansion of the universe.
Never fear, we won't be confined to the Local Group for tens of gigayears (Abrabram Loeb, p. 3).
And it might never happen. Extrapolating the acceleration of the expansion of the universe in the far future is highly speculative.
At the moment, the positive contribution to the relative rate of acceleration is well fitted by a constant and the negative contribution is decreasing in time (see Wikipedia: Friedmann equations: Equations). However, we have no idea if this is right.
It may well be that the positive contribution varies in time, perhaps strongly.