How are time and distance measured in cosmology?
Relativity says that time isn't absolute. For example, the rate at which a clock runs depends on its state of motion. Because of this, it's not trivial to define what is meant by phrases like "the age of the universe," or "three minutes after the Big bang." Cosmologists do have a standard definition, which in this discussion we'll refer to as "universe standard time." (Standard technical terminology would be to call it the "preferred time coordinate" of a particular model, or to prefix "time" with the name of the model, as in "FRW time" for the FRW model.)
One way to think of "universe standard time" is as the age of the universe measured by observers at rest relative to the cosmic microwave background radiation (CMB). The standard models of cosmology do their book-keeping according to this time because it makes the math simple and convenient. If necessary, the model's predictions can then be translated into the frame of reference of a particular observer, such as one on earth, who is not exactly at rest relative to the CMB.
Roughly speaking the steadily declining temperature of the CMB itself provides a sort of clock. As distances increase, the ancient CMB light cools. Observers who measure the same microwave sky temperature are contemporaries---they belong to the same "time t" era.
To be more precise one would have to take account of gravity. Some observers might be near massive bodies whose intense gravity has a noticeable effect on the passage of time. But for simplicity we can picture all our observers well out in open space, away from large mass concentrations, so that gravitational effects are small and can be neglected.
Like time, distance is relative, not absolute. We can imagine a network of observers, all at rest relative to the CMB, synchronizing their clocks and agreeing to measure the distance between various objects, or between the observers themselves, as close to instantaneously as they can manage. In reality this would involve enormous amounts of planning and collaboration, but it is just an idealization. This is essentially what is meant by proper distance: the distance at a particular time t as measured by observers at CMB rest. We can only estimate proper distance because we don't have unlimited numbers of observers scattered about the universe, prepared to collaborate like that.
"Universe standard time" and the related concept of distance are used in constructing some basic tools in cosmology. They are used to define the scale factor, to formulate the Friedmann equation model (which runs on this version of time) and to formulate the Hubble law, which relates a proper distance at some moment in time to the rate that distance is expanding.
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