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I know that cosmic time is the time that "moves with the expanding universe". More precisely, in cosmology we study the Friedmann–Lemaître–Robertson–Walker metric: $$ d \tau^2 = dt^2 - a(t)^2d\Sigma^2 $$ where $\tau$ is proper time, $t$ is ``cosmic time'', $a(t)$ is the scale factor, and $d\Sigma^2$ is a metric with constant scalar curvature on a 3d-space manifold. If we fix a point in the manifold, then $$ d\tau^2 = dt^2, $$ so proper time is the same as time at a point in the space manifold of the expanding universe. When it's said the universe is 13.7 billion years old, they are referring exactly to this time, the time in the reference frame moving with the expansion (the "comoving reference frame").

My Question: This "cosmic time" is not the same as "time on earth", since the earth is technically not "moving with the expanding universe". We are moving around the sun, which is rotating around the galaxy, our time is influenced by gravity of the earth, sun, etc. Due to these factors, "cosmic time" is not "our time" (the time measured on earth). Does anyone know the difference (a formula would be great if there is one) between "cosmic years" and "our earth years"? Books on cosmology don't seem to ask (nor answer) this question.

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If you check Plank's latest results, earth is moving wrt to the CMB rest frame at $\sim 370 km/s << c$. Furthermore, gravitational pull due to the sun and the center of the milky way is very weak by GR standards. Hence the gravitational and relativistic redshift in time will be very low. So for most practical effects you may approximate "earth's time" by cosmological time.

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    $\begingroup$ I guess my next question is why the CMB rest frame is identical to the comoving reference frame. This is probably obvious! Is there an easy explanation of this somewhere? $\endgroup$
    – Curiosity
    Oct 11, 2021 at 19:34
  • $\begingroup$ Well, the CMB was produced in electron-positron annihilation that had average velocity 0 in the comoving reference frame, hence you expect the CMB rest frame to be the comoving reference frame. $\endgroup$ Oct 11, 2021 at 19:47

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