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According to the definition I have encountered for the concept of cosmological time, it is defined in the following way:

  • The cosmological principle states that, at each location in the universe, it is possible to define a hypothetical observer to whom the universe appears homogeneous and isotropic. These observers are called comoving observers, and the comoving coordinate system is the system for which the comoving observers are at rest.

  • The time measured by the comoving observers is called cosmological time, and as a reference we set it to zero at the time of the Big Bang.

But, according to special relativity, the time interval that an observer measures as the duration of a certain event depends on the velocity of the observer. So, why is cosmological time unique, that is, independent of the point in space where the comoving observer is located? Wouldn't this mean that, in an expanding universe, all comoving observers have the same physical velocity? Am I misunderstanding some concept here?

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One perspective is symmetry. If the universe is homogeneous, every comoving observer is just as good, so how could some of them measure longer times than others?

Another is that relative velocities of distant objects are not meaningful in curved spacetimes. It doesn't even make sense to say that distant observers are moving at the same or different physical velocities.

But maybe it will help to think about an expanding universe with a flat spacetime. This is a universe with negligible energy density, and it's simple to show that the scale factor grows linearly with time. Since spacetime is flat, we can meaningfully compare velocities of distant objects. With respect to the (special relativistic) reference frame of a comoving observer, comoving spatial surfaces (the surfaces that we normally think of as "space" in cosmology) look like the gray curves in this spacetime diagram (time is vertical, space is horizontal): coordinates in the Milne universe

(image source)

The worldlines of comoving observers are the black lines; our reference observer is in the middle. Notice that the spatial surfaces "bend into the future" at long distances. This effect exactly compensates the time dilation of distant observers, so that any comoving observer on the same spatial surface will say that the universe has the same age.

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