Does our past light-cone have some extension in three-dimensional space? On spacetime diagrams cosmologists represent our past light-cone as a two-dimensional surface extending back in time, on which are situated all light-emitting events which we can observe today. This is especially clearly shown when cosmic comoving coordinates are used. But shouldn't the past light-cone also have some (causal) extension in the 3rd spatial dimension and how large could this "thickness" be?
 A: You're absolutely right, but how would you propose drawing that on a piece of paper? Spacetime is four-dimensional with lorentzian signature. Paper is two-dimensional with euclidean signature.
Remember the equivalence principle: in any sufficiently small neighborhood, spacetime is approximately flat. Your intuition about light-cones in flat spacetime is a good approximation in any region that is small compared to the cosmological scale (ignoring tiny blemishes like galaxies, as cosmologists usually do). Large-scale curvature distorts the light-cones, but it doesn't flatten them. To give a more specific answer may require knowing exactly what pictures you're looking at, but the bottom line is that you're right. It's just hard to draw.
If a picture is worth a thousand words, then an equation is worth a thousand pictures. Math communicates concepts that pictures can't even begin to touch. A good picture can help us ease into the math, and it can help us check our understanding of the math, but only in very simple cases can a picture actually communicate everything that matters. Cosmology is not one of those cases.
A: The past light cone is a concept in the theory of special relativity that represents the set of all events that could have caused an event in the present. It is a 4-dimensional structure, with three spatial dimensions and one time dimension, and it has a finite extension in 3D space. The past light cone is determined by the speed of light, which is the maximum speed at which information can travel, and it is used to define the causality of events in spacetime.
A: The past light cone is a concept of the SRT as well as a cosmological concept. Strictly, the past light cone at any defined time on the time axis (eg at t = present time) has 2 spatial dimensions (the cone's shell or "skin") with zero spatial "thickness", extending back in the time dimension; so it is a 3D structure relating causal events. If we evolve the past light cone in time it would acquire a small 3D thickness which would contain sources that could be inferred from the immediate environment. Do you agree with this description? - See also related: What is the volume of spacetime we can survey?
