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Cosmological redshift is based on the idea that the universe is expanding. When the universe doubles in size, or scale factor, the wavelength of light doubles. But the Milky Way is not expanding so my guess is that there is no cosmological redshift within the Milky Way? There is of course Doppler redshift.

Having seen the first 5 answers, which seem to confirm my guess, I am now going to add a corollary. The Doppler effect has almost nothing to do with cosmological red shift, outside our local cluster of galaxies. Is that also correct? Which makes Hubble very lucky.

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Your guess is correct. The constituents of the Milky Way, and also close dwarf galaxies like the Magellanic clouds form a gravitational bound system and have therefore, seen as a single system, decoupled from the Hubble flow. This happened because the average energy density within this system at one time in the cosmological history became much larger than the average energy density of the universe (One can check out reviews about structure formation for more details). (Of course the whole system is still part of the expansion in the sense that the distance to galaxies far away increases.) Therefore there is no cosmological redshift observed within this system, for example from Earth, for any stars within or other constituents of these galaxies.

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    $\begingroup$ "The constituents of the Milky Way, and even close dwarf galaxies..." probably even nearby galaxies (e.g. M31). AFAIK, galaxy clusters are still bound together by gravity so there shouldn't be any cosmological redshift. $\endgroup$
    – jng224
    Jul 23, 2021 at 10:42
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    $\begingroup$ @Jonas You are most probably correct, keeping in mind that M31 even gets closer to the Milky Way. I was not sure what the most distant objects are that are still gravitationally bound to the Milky Way. Could be the whole galaxy cluster; but I decided to just state what I was absolutely sure about. :-) $\endgroup$
    – Koschi
    Jul 24, 2021 at 9:44
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Even if the Milky Way is expanding with the Hubble flow (and most cosmologists believe that it isn't), the expansion would be difficult to measure.

The size of the Milky Way $d$ is approximately $10^{21} \;\text{m}$.

Using $$v=Hd$$

with Hubble's constant in SI units of about $2\times 10^{-18} \;\text{s}^{-1}$

means that even a star far away, in the Milky Way, would have a redshift corresponding to a motion of about $2000 \;\text{m}\,\text{s}^{-1}$. However peculiar velocities of stars are typically a hundred times higher than this, for example the sun is thought to be moving relative to the Milky Way at $250 \;\text{km}\,\text{s}^{-1}$.

Cosmological redshifts are used to determine distances of stars etc... that are far enough away that the peculiar velocities can be ignored.

It would be interesting to see the results of a future experiment that attempted to 'average out' the peculiar motions of millions of stars within the Milky Way, to see if there is a cosmological redshift.

It seems that at the moment the consensus amongst cosmologists is that it doesn't exist.

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  • $\begingroup$ This would in fact be easy to measure. a 2km/s expansion would be an enormous effect. Peculiar velocity is not the correct term for what is 100 times larger than this - that is an orbital speed you are quoting. An expansion could be seen perpendicular to that, where the "peculiar velocities" are more like 10 km/s for radial motion or perpendicular to the plane of the Galaxy. $\endgroup$
    – ProfRob
    Jul 24, 2021 at 13:37
  • $\begingroup$ @ProfRob There is more to the motion than the simple orbital model. The latest data from GAIA shows high radial components too arxiv.org/abs/1804.09380 quote from the abstract: "The maps show the complexity and richness of the velocity field of the galactic disc. We observe streaming motions in all the components of the velocities as well as patterns in the velocity dispersions." $\endgroup$
    – John Hunter
    Jul 25, 2021 at 7:52
  • $\begingroup$ A systematic 2 km/s expansion of stars across the galaxy would be easy to detect. Most stars in the Milky Way disk do have nearly circular orbits and have small radial velocities. $\endgroup$
    – ProfRob
    Jul 25, 2021 at 8:14
  • $\begingroup$ @ProfRob A reference would be of interest, if this easy measurement has been done, and presumably put limits on any overall expansion or contraction of the Milky Way $\endgroup$
    – John Hunter
    Jul 25, 2021 at 8:34
  • $\begingroup$ Fair point. I can't readily find anything. But given that typical line of sight velocity precisions per star from Gaia are 2 km/s, and you could have thousands and thousands of stars in a sample on say a line of sight out of the galaxy, all of which would have distances. If the dispersion sigma was 10 km/s, then a systematic gradient of 0.075 km/s per kpc should be detectable given velocities spread over 10-20 kpc. But if the distributions had various r-dependent asymmetries this could be difficult. I withdraw my objection and suggest it might be measurable. $\endgroup$
    – ProfRob
    Jul 25, 2021 at 9:01
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There is not.

Cosmological expansion (and therefore redshift) are caused by general relativity. We understand this through the Friedmann–Lemaître–Robertson–Walker metric, which describes a homogeneous, isotropic universe.

But you may notice that the Universe doesn't appear homogeneous or isotropic. Some parts are denser than others, like galaxies. And when you look around, you might even argue that the galaxies aren't exactly the same, so it cannot be isotropic.

But if you look at the Universe from the point of view of a huge Gigaparsec creature (1000x larger than Megaparsec galactic systems), the galaxies and inhomogeneities appear like sand on a beach (we are 1000x larger than millimeter grains), or atoms in air. This is a way to illustrate that at a large enough scale, the Universe does appear homogeneous and isotropic, and can be well-described by the FLRW metric.

The cosmological expansion of the Universe depends on assumptions, and those assumptions are valid at large scales, but not small scales. Hence, the prediction of general relativity is that there will be expansion on those large scales where those assumptions hold true, but not necessarily expansion on the small scales where these assumptions do not hold true. And if you had a large enough computer to simulate the equations of general relativity in the Milky Way, you would not find cosmological redshift (most cosmologists believe).

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If everything were expanding with the Big Bang expansion, stars from each other , planetary systems, atomic systems, nuclear systems there would be no possibility of measuring the expansion, because the units scientists use to measure would also be changing (without our being able to perceive it) following the expansion.

The hypothesis that allows to measure an expansion is that the bound states of the four known forces, cannot be affected by the expansion. This allows to treat galaxies, and the stars within them, as gravitational bound states ,not changing size, and allows for expansion and the infrared shifts one measures.

So it is not a matter of consensus and beliefs, it is an inevitable axiom in the model, the Big Bang, that fits the data observed. At the moment it is validated by all the known observations. Thus the hypothesis is extended to the bound states of the other three forces, which bind more strongly than gravity.

Maybe future observations and measurements might show discrepancies with this hypothesis, then a modified Big Bang would have to be used , though I think the argument of "how one could observe any change when the units change" will have to be addressed.

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    $\begingroup$ I find your answer confusing. What is this 'hypothesis' you say suggests that 'bound states of the four known forces, cannot be affected by the expansion'? The same equations, derived in the context of GR, that describe the expansion of the universe, tell us that parts of the universe with a much higher energy density will not take part in the expansion, but collapse, while gravity and electromagnetism both taken into account tell us how some of these collapsed objects can form stars. So where is the need for any hypothesis that 'is extended to the bound states of the other three forces'? $\endgroup$
    – Koschi
    Jul 23, 2021 at 17:17
  • $\begingroup$ What I mean is: Your answer seems to suggest that, while GR/cosmology might be able to explain the expansion of the universe, there is an additional hypothesis needed to explain, why the Galaxy, the solar system, the Earth etc.. do not expand. But this is NOT the case. It is NOT an 'axiom' of the Big Bang model that 'bound states are not affected by the expansion', and this is also NOT postulated to fit observations (since it is not postulated). Quite contrary: That 'bound states' like the Galaxy do not take part in the expansion is an inevitable prediction of the theory. $\endgroup$
    – Koschi
    Jul 23, 2021 at 17:27
  • $\begingroup$ @Koschi "will not take part in the expansion" note, this is not a general , but a particular way some matter coalesces. It is not a prediction of the theory that after they coalesce there is no expansion. My argument is that if everything expanded, there would be no way to know it because of the definitions of our measurements. I has to by axiomatically stated in the theory that what has coalesced into a bound state no longer expands. $\endgroup$
    – anna v
    Jul 23, 2021 at 18:24
  • $\begingroup$ I am sorry, but I have to disagree with you. It is a prediction of the theory that overdense regions, which started as small fluctuations from the homogeneous energy density, will form gravitational bound objects and therefore decouple from the expansion. Look, e.g. here: arxiv.org/pdf/astro-ph/9403011.pdf What else should we mean with 'gravitationally bound'? Of course, if EVERYTHING, i.e. even atoms, would expand evenly, we could not notice it (which would render it un-physical anyway, I guess). But I am not aware of any theory that assumes this. $\endgroup$
    – Koschi
    Jul 24, 2021 at 9:36
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For a technical discussion of this type of issue, with numerous references to other work, see "On the influence of the global cosmological expansion on the local dynamics in the Solar System" https://arxiv.org/abs/gr-qc/0602098

and

"The Influence of the Cosmological Expansion on Local Systems" https://arxiv.org/abs/astro-ph/9803097.

These papers discuss the problems of defining measurements within general relativity, and show that the overall cosmological expansion has extremely small effects (but not zero) on the scale of the solar system.

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