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Is it guaranteed that there exists a universal time on which all comoving observers will agree?

Special relativity teaches us that inertial observers in relative motion do not agree on the time they observe. Now in standard cosmology, the physical distance between the comoving observers changes with time because of the expansion of the Universe even though they sit at fixed coordinates in the comoving coordinate system.

Does this mean they do not agree on the cosmic time $t$? I'm not talking about the conformal time.

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Is it guaranteed that there exists a universal time on which all comoving observers will agree?

Suppose that comoving clocks A and B have both existed since the big bang. In a cosmological model that is homogeneous, homogeneity guarantees the environment surrounding one such clock is the same as the environment surrounding the other. Furthermore, they will agree if compared by some synchronization procedure. For example, if each clock emits a radio pulse at a time that it calls $t_1$, and this pulse is received by the other at $t_2$, then if they agree on $t_1$ they will also agree on $t_2$.

Special relativity teaches us that inertial observers in relative motion do not agree on the time they observe. Now in standard cosmology, the physical distance between the comoving observers changes with time because of the expansion of the Universe even though they sit at fixed coordinates in the comoving coordinate system.

Does this mean they do not agree on the cosmic time $t$? I'm not talking about the conformal time.

Coordinate velocities are meaningless. It is only in the limit of small distance scales that GR reduces to SR.

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