As far I understand, the whole universe, at a redshift of 1100, became transparent to light, and the phenomenon occurred throughout the whole universe. It is in our past, not in our left or right direction, i.e, the interval is timelike, not spacelike. If this is true, then why do we call it the surface of last scattering instead of redshift/time of last scattering? What is the meaning of "distance of last scattering surface" from us if it was in our past, not at left or right direction? How is the distance calculated and how do we measure the angular diameter($2^{\circ}$) of the surface from us in that case?


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Light traveled from there to here, therefore the scattering events seen by us are at a null interval from us. Up to small corrections those events lie on a spatial surface in the past. To get the idea, throw away one spatial coordinate for a moment, and then you have a circle lying at a moment of cosmic time in our past light cone. Coming back to all three spatial dimensions, it is a spatial surface, in the past, which is spherical to good approximation.

You can compare this to the experience of being in a break in the mist on a misty day. The light arriving at you comes from a spherical region around you, in your past light cone (but of course in this example it is the very recent past).

The notion of distance to this surface can be defined in more than one way. Most large distances in the universe have this feature. One way to define it is to take those events long ago, and then extend into the cosmic present the lines at the same comoving coordinate locations, and talk about the distance in the present to those locations. Another way could be to go back down a worldline at Earth's comoving coordinate and find the distance in the past between that line and the events. You can also have luminosity distance and angular diameter distance and other such measures. You have to look carefully at any given treatment to find out how the term 'distance' is being employed.

Your question then goes on to ask how these distances are calculated. I leave that to others to answer if they like, but it might help if you did a bit of research yourself on that, by looking up some basic treatment of cosmology.


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