# Why do we see a cosmological redshift at all if space is not expanded in our solar system?

It is generally accepted that the large scale redshift of galaxies (as given by the Hubble law) is due to the expansion of the universe. According to this theory, the fabric of space itself stretches during this expansion, hence increasing the wavelength of light (as qualitatively explained at A Model of the Universe).

However, there is also a general consensus that atoms, objects or systems held together by gravitational forces like our solar system do not take part in this expansion (see If the universe is expanding, does that mean atoms are getting bigger?).

In terms of the scale factor $$a$$ of the metric, the redshift $$z$$ is given by

$$1+z=\frac{a_{\text{now}}}{a_{\text{then}}}$$

The above equation for the redshift is derived for a scale factor $$a$$ that is merely a globally identical function of time, but it should be evident from the derivation that it actually relates the scale factors at the time/location of the emission to the time/location of the observation. We can therefore generalize the equation to

$$1+z=\frac{a_{\text{now-and-here}}}{a_{\text{then-and-there}}}$$

So as the fabric of space has not expanded in our solar system since the light was emitted by some remote galaxy (i.e. $$a_{\text{now-and-here}}=a_{\text{then-and-there}}$$), does this not mean that the light should revert to its original wavelength as soon as it enters our solar system (i.e. there should not be any cosmological redshift observed at all ($$z=0$$))? If not, why not?

• By what mechanism do you propose it should "revert"? Jul 31 at 17:18
• @ProfRob Well, by the same mechanism the causes light to go red-shifted in the regions of expanded space, i.e. by adjusting to the local metric of space (which it does for instance also when it is going past a massive object) Jul 31 at 17:48
• I think this is a really good question, and I don’t understand the downvotes.
– pela
Aug 1 at 15:53
• @pela The question suggests it shouldnt happen but it does. The Milky Way might be blueshifting but the redshifting is bigger. Thats why I downvoted. Aug 1 at 18:40
• I think the problem is that, while your first equation is true when referring to the global scale factor, your second equation is not true. Taking into account the non-homogeneous expansion of space, you really have to consider the scale factor in infinitesimal steps. I tried recapping our discussion in the final section. I hope it makes sense now :)
– pela
Aug 3 at 10:25

tl;dr Because space doesn't contract inside our Solar System.

### Wavelength increase is proportional to space expansion

The prediction of general relativity — one of the most thoroughly tested and succesful theories — is that the wavelength of observed light changes in proportion to the factor by which space expands (Lemaître 1927).

If space expands by a factor of two while the light is traveling, its wavelength increases by a factor of two. If space contracts by a factor of two, its wavelength decreases by a factor of two. When and how fast this expansion or contraction takes place does not matter for the final wavelength.

### It doesn't matter how space expands

That means that, if you observe a photon that you know should have been emitted with an intrinsic wavelength of $$\lambda_0$$ to have a a wavelength of $$\lambda_1 = 2\lambda_0$$, you have no way of knowing — from this observation alone — whether space expanded by a factor of two continuously along the photon's way, in a rapid burst just after emission, in a series a small expansions, or even by a factor of a thousand followed by a contraction by a factor of 500. All you know is the difference between then and now.

Observing many sources at different distances and epochs, however, together with the seemingly fair assumption that space is homogeneous on larger scales, has convinced us that our Universe has expanded smoothly (although at a changing rate) during its history.

### The story of a photon

A photon traveling through a smoothly expanding Universe has its wavelength increasing continuously. Sometimes the photon enters an overdensity, e.g. a galaxy cluster, that expands at a slower rate (or maybe doesn't expand at all), and its wavelength increases less fast. Sometimes it enters a void, and its wavelength increases a little faster. But on average, it keeps increasing.

Then at some point, the photon encounters the Milky Way. As you say, space in the vicinity of the Milky doesn't expand, because the density is much larger than the average density of the Universe. At this point (actually already when it enters the Local Group) the photon's wavelength stops increasing. It even decreases a little bit due to gravitational blueshift, as the photon "falls" into the gravitational potential of the Milky Way (but it also decreased a little faster when it had to climb out of the potential of the galaxy from which it was emitted, so that's fair).

### And now to the answer

The total, observed redshift is the integral of infinitely many infinitesimally small redshifts. For each infinitesimal step $$dr$$, what matters for the infinitesimal increase in redshift $$dz$$ is the infinitesimal increase $$da$$ in the local scale.

When we say that "$$1+z = a_\mathrm{now}/a_\mathrm{then}$$" we are referring to the average, or global, scale factor of the Universe, but you're right that, in principle, the "local" scale factor of the Milky Way is not equal to the global scale factor $$a_\mathrm{now}$$. I think this is the source of your (very fair) confusion.

Remember, however, that you don't "see" the scale factor as you move around in space. There is no underlying coordinate system where you can check the absolute size of space. If suddenly the Universe froze in time, you would not be able to measure a difference between intergalactic space (which has followed the global expansion), and space inside our Solar System (which stopped expanding when the proto-Milky Way decoupled from the Hubble flow in the early Universe and started collapsing)$$^\dagger$$.

In order to be observed with zero redshift, space inside the Milky Way (or our Solar System) would — during the time that the photon was finishing the last part of its journey — have to contract by the same factor by which the Universe had expanded during all the time from the photon left its mother galaxy and until it approached the Milky Way. But it doesn't! Space in sufficiently dense regions of the Universe is static.

$$^\dagger$$ You would of course find more particles per cm3 inside the Solar System than in the intergalactic medium, which would give you a hint, but space itself would be the same.

• Comments are not for extended discussion; this conversation has been moved to chat. Aug 2 at 2:20

Even though the fabric of space is expanding throughout the universe, our solar system is not. Our atoms remain the same size. So do the planets, moons, and stars, as well as the distances separating them. Even the galaxies in our Local Group aren't expanding away from one another.

The math tells about the possible solutions, but it is necessary to look to the physical universe to find which one of these describes the process. The farther away an object is from another, the more "stretching" occurs. The superclusters of the universe, populated with galaxies and stretching for over a billion light years are being stretched and pulled apart by the universe's expansion.

The Milky Way and all the local group galaxies will stay bound together, eventually merging together under their own gravity. Earth will revolve around the Sun at the same orbital distance, Earth itself will remain the same size, and the atoms making up everything on it will not expand. Milky Way's nearest large galaxy cluster, the Virgo cluster, at 50 million light years away, will never pull us into it. Despite a gravitational pull that's more than a thousand times as powerful as our own.

Because the expansion of the universe only has any effect where another force (gravitational, electromagnetic or nuclear) hasn't yet overcome it. The reason for this is subtle, and is related to the fact that the expansion itself isn't a force, but rather a rate. The fabric of space itself may still be expanding everywhere, but it doesn't have a measurable effect on every object.

Unfortunately, the answers I received are rather unsatisfactory. I don't feel they give a convincing argument why the photon would be red-shifted when the fabric of space expands explicitly with time, but not blue-shifted when the fabric of space contracts implicitly with time (when it enters a region of un-expanded space).

Howwever, I came now a across a paper by Fulvio Melia (published in Monthly Notices of the Royal Astronomical Society) which shows that the cosmological redshift can be interpreted as a Doppler shift through a suitable choice of coordinate system (he shows this explicitly for the case of constant spacetime curvature, but argues that it may hold generally as well).

And if one sees the redshift as a Doppler shift, it obvious that a photon should not reverse its redshift again when it enters our own galaxy/ solar system. All parts of the latter have more or less the same relative speed with regard to the remote galaxy, so the redshift would just stay the same.

• While this explanation is equally valid, your first sentence actually supports my explanation as well: You ask "why light is not blueshifted when space contracts". The point is, light is blueshifted in contracting space. But the MW is not contracting, hence light is not blueshifted. That's it.
– pela
Aug 26 at 15:21
• @pela Read again what I said: it is contracting as an implicit function of time (by the photon moving from an expanded region of space into a non-expanded region) not as an explicit function of time. As an analogy, consider a thermometer that you move from a region where the temperature is constant (or even increasing) with time into some sub-region with a much colder temperature constant with time. The temperature the thermometer shows will be falling not stay constant, Aug 26 at 22:14
• I think you're still assuming that there is some difference between a region of space outside the MW and a region inside, because the former has expanded while the latter has not. But there isn't, so that analogy doesn't hold. But given our previous discussion I also think that we will never agree on this, so I'll let it be :)
– pela
Aug 27 at 13:24
• @pela You agreed previously that a change of the metric of space would affect the photon. For instance, a photon approaching a mass from infinity will be blue shifted due to metric becoming more contracted, And this is in principle not different to a photon going from one region of space to another that has a different metric due to a different history of metric expansion. So there should be a frequency shift in this case as well, unless you are saying that the expansion of the universe is not related to a change of metric at all, in which case we could interpret it as a Doppler shift only. Aug 31 at 20:35

Why is the wavelength of a photon stretching? Because the parts of the excited field accompanying it (a photon is just an excitation of a field) are moving away from each other if the photon travels towards us. This stretching doesn't happen in the galaxies because of the positive warping of gravity. Space does not expand in the Milky Way. Not yet. In the future it will, when the positive curvature induced by mass/energy is overcome by the negative one of dark energy. Expanding space is a misleading way though of saying that particles are just accelerating away from each other but never mind, I'll stick to the mainstream language. When a photon arrives in the Milky way it is no longer stretched and it is seen as stretched. It is squeezed back again (blue-shifte) but not enough not to be seen as stretched. Note that if the particles we are made of would be expanding along with space (if we place ourselves in intergalactic space) then the shift would not be noticable. But because the particles stay put during expansion, because of the three basic forces, the expansion and growing wavelength is seen. This means that the increase in wavelength is a relative effect. There is no energy lost in reality.

• I was specifically asking about an issue with the mainstream theory of the cosmological redshift, and how mainstream theory would resolve this Aug 1 at 20:49
• @Thomas I gave you an answer within the mainstream view: "I'll stick to the mainstream language." By the way, you are right that space contracts in the milky way. But just not enough to counter the redshift. There is blueshift but not enough to neutralize the redshift. Aug 1 at 20:51
• @Thomas What do you think if all particles (or their associated excited quantum fields) would all expand along, without forces between them? Aug 1 at 20:59
• Suggestions for different redshift mechanism is not something I would like to discuss here Aug 2 at 18:05