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The cosmic microwave background radiation should provide kind of a global reference frame, because you can determine your speed relative to it using the redshift.

Is it known how fast we are moving in relation to the CMB? If you subtract the various orbital motions (Earth around the Sun, Sun around the Galaxy), are we standing still in the expanding universe, or traveling in a certain direction?

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Yes, our (i.e. the Sun's) motion in the "global", or comoving, reference frame can be measured accurately from the dipole of the cosmic microwave background. The latest results from the Planck Collaboration et al. (2018) yielded a velocity of $$369.82\pm0.11\,\mathrm{km}\,\mathrm{s}^{-1} $$ in the direction $$ \begin{array}{rcl} \ell & = & 264.021º\pm0.011º\\ b & = & 48.253º\pm0.005º \end{array} $$ (in Galactic coordinates).

Since Earth orbits the Sun with some $30\,\mathrm{km}\,\mathrm{s}^{-1}$, there's a small, biannual correction to this result. On much larger timescales ($\sim100\,\mathrm{Myr}$) our motion round the Milky Way alters our comoving velocity with the order of $\sim100\,\mathrm{km}\,\mathrm{s}^{-1}$.

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    $\begingroup$ How can the Planck results be precise to 0.11 km/s if there's a 60 km/s correction depending on the time of the year? $\endgroup$
    – Allure
    Sep 27, 2019 at 22:27
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    $\begingroup$ @Allure: The Earth's orbital velocity around the Sun is quite well known already. And even if it wasn't, they could just average it out over a whole year (or several). $\endgroup$
    – Ilmari Karonen
    Sep 27, 2019 at 22:30
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    $\begingroup$ @Allure Yes, exactly; that's why I wrote "our (i.e. the Sun's)" :) $\endgroup$
    – pela
    Sep 28, 2019 at 6:36
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    $\begingroup$ @nick012000 Because what you've said isn't true. The small departure from this isotropy is what leads to the measurement quoted in the answer. $\endgroup$
    – ProfRob
    Sep 28, 2019 at 11:46
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    $\begingroup$ @nick I'm sure you've seen CMB images like this, which show that the CMB is isotropic to better than one part in 10,000. Such images have been corrected for our peculiar motion; otherwise, the CMB details would be swamped by the Doppler effect of the peculiar motion. A raw map of the CMB (i.e., without that correction), looks like this. $\endgroup$
    – PM 2Ring
    Sep 29, 2019 at 9:41

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