Light-Cone (Space-Time) Diagrams
Concordance Model

Light cone diagram using proper distance and normal time.

World lines (blue dashes on left) show cosmic expansion for objects currently at 8, 16 & 46 Gly (the latter is the particle horizon). Black dotted lines on the right show where the expansion speed is 2c, c, c/2 (expansion speed c is the Hubble distance -- currently c/Ho = 4.167 Gpc = 13.58 Gly). Black dot-dashed line on the left is the expanding particle horizon (approx 2c during the matter dominated phase), which joins with the world line of our current particle horizon at time=now (off the diagram).

Solid red line shows, on both sides, the trajectory of a light ray which starts at the big bang from our current particle horizon. It is also the event-line which we witness now as light comes to us, ie we see only those events lying on the red line (light rays starting at other space-time events have either already arrived or will arrive in the future) -- the light cone is for light arriving now. Notice that for the first 4 Gyr, the light is actually getting further from us, carried by cosmic expansion. At 4 Gly it crosses the Hubble distance at that time (ie expanding at c) and finally begins to get closer. Other lines of constant expansion speed are also shown (c/2 and 2c). Notice that the light cone finally arrives at 45 degrees, since locally the expansion is essentially zero.

Redshift, z, and comoving (current) distance, ro, scales are also indicated on the right and left halves of the light cone (note that the ro values match the world line values at their intersection).


  Light cone diagram using comoving distance and normal time.

The equivalent space-time diagram, but with space now in comoving (current) distance (note that the scales are no longer 1:1, so the apex doesn't seem to be 45 degrees, though in fact it is). World lines are now simple vertical lines. Notice also that the Hubble distance (ie where v=c) reaches a maximum and begins to approach us after acceleration has begun. The scales marked along the light cone are redshift (right) and emission distance, re (left).

As before, the black dot-dashed line shows the expanding particle horizon, meeting the 46 Gly world line of our current particle horizon at time t=now. This distance, in comoving coordinates, is also equal to c conformal time, which will be used as the time axis in the third and final space-time diagram.


Line cone diagram using comoving distance and conformal time.

The final space-time diagram recovers the familiar 45 degree form, as long as we use comoving distance and conformal time ((t)   =   dt/a   0 to t), and plot these at the same scale. As before, redshift and emission distance are marked along the cone. Notice how rapid early expansion stretches conformal time near the big bang, so that even the CMB is visibly offset from the y=0 axis (and would be impossible to see with a normal time axis).

Notice that the Hubble distance is rapidly decreasing during acceleration, and in fact becomes zero at conformal time 65 Gly, which corresponds to the infinite future (we see a smaller and smaller comoving part of the Universe as exponential expansion proceeds).


Figures made for this website.