# Why do we say the universe is isotropic when we are clearly moving w.r.t the CMB?

Modern cosmology is built on the Friedmann equations, which in turn rely on isotropy — the idea that the universe looks the same in every direction — as a fundamental assumption.

However, there's a very noticeable dipole in the CMB, the standard interpretation of which is that we are moving with respect to the frame in which the CMB is at rest.

But if the CMB looks different in different directions, then the universe is clearly not isotropic. Why do we still believe that the universe is isotropic then?

• The anisotropy corresponds to the peculiar velocity of Earth (and the Solar System, and the Milky Way) with respect to the CMB. Although it is an interesting question whether we have independent verification of the Milky Way velocity apart from that. I hope someone can provide that answer. Jul 6, 2022 at 13:42
• The dipole can easily be flattened out by considering the observer's velocity, if the universe was truly inhomogenic and anisotropic that would not work. Local inhomogenities aside, since that assumption is only for large scales, on which it holds. Jul 6, 2022 at 13:44
• If I as a layman can interpret the above two comments, the answer is essentially that there is no wind, but it seems like there is because we are moving through the atmosphere, which gives us the perception of relative wind even though there is no absolute wind (relative to the ground, that is)? Jul 7, 2022 at 7:30

The best way to understand this I found is in terms of the possible symmetries of spacetime. Minkowski spacetime has 10 independent ones:

• 3 independent spatial translations;
• 3 independent spatial rotations;
• 3 independent boosts;
• time translation.

FLRW spacetime (which is the base of our cosmological models) has 6 of these: translations (homogeneity) and rotations (isotropy).

The fact that boost symmetry is missing means that there is a preferential velocity for observers, and moving with any velocity except for it will mean we see something different from the isotropic, homogeneous FLRW universe. However, the point is that there exists a frame with all these symmetries. It's just not the Earth's one, since as you note we see a dipole modulation in the CMB.

• Is this related to what is called the "kinematic interpretation of the CMB dipole" in this article? arxiv.org/abs/2009.14826 Jul 10, 2022 at 11:50
• @TorkelBjørnson-Langen yes, that's exactly what I'm referring to! Jul 10, 2022 at 13:22

The key property of the CMB is statistical isotropy. This does not mean that different directions are the same, but that on average (over very large scales, $$\gtrsim 100\ {\rm Mpc}$$) there is no preferred direction.

In more detail, the spherical harmonic coefficients of the temperature fluctuations, $$a_{\ell m}$$, obey the conditions (Eq 30 of https://arxiv.org/abs/0802.3688) $$\begin{eqnarray} \langle a_{\ell m} \rangle &=& 0 \ \ (\ell > 0) \\ \langle a_{\ell m}^\star a_{\ell' m'} \rangle &\propto& \delta_{\ell \ell'} \delta_{mm'} \end{eqnarray}$$ We can remove the dipole $$\langle a_{2m}\rangle$$ with a trivial change of coordinates (corresponding to a boost counteracting our local motion, as you said). Once we do this, the above conditions are satisfied, as far as we can tell observationally (eg: https://arxiv.org/abs/0908.0963). The fact that the two point correlation function is diagonal in $$m-m'$$ space is a particularly stringent constraint that would not be satisfied if the Universe were (noticeably) statistically anisotropic.

To state this in a pithy but flippant way: saying the Universe is anisotropic because we observe a dipole, would be like telling a police officer that you weren't speeding in your frame of reference (in your frame of reference the cop was speeding!).

• Relativity says that the frame of reference does NOT matter for the question who is speeding. (I.e. I believe this answer is giving the wrong reasons, but unfortunately I don't know the correct answer myself; I think it should be either an effect of all of the universe's mass, which is responsible for generating centripetal forces if you use a rotating frame of reference, or maybe it's space curvature due to nearby masses.) Jul 7, 2022 at 11:32
• @user132372 What you are referring to in the second half of your comment is Mach's principle, which inspired Einstein but it is not clear today whether it is fully compatible with GR. Anyway, you are right that there is no preferred frame in special relativity. However, in the context of general relativity, there is a preferred frame in cosmology: the frame at which the matter in the Universe is at rest on large scales (statistically). This is the frame in which the conditions in my answer apply. Jul 7, 2022 at 11:52
• @Andrew: Your last comment is incorrect. Einstein's special relativity does not assert that there is no preferred frame. It simply does not assert anything at all about preferred frames. Having a special frame is 100% consistent with special relativity. Jul 7, 2022 at 12:41
• @user132372: See my above comment. Jul 7, 2022 at 12:41
• @user21820 With respect, you are making an extremely strong claim that contradicts the standard interpretation of special relativity, while providing no evidence. I don't know what point you are trying to make, but I doubt the comment section to this post is the right place to make it. Jul 7, 2022 at 13:09

I do not understand the math, but as I understand, yes this is an unresolved problem of cosmology. The Wikipedia article on the "cosmological principle" starts like this:

Unsolved problem in physics: Is the universe homogeneous and isotropic at large enough scales, as claimed by the cosmological principle and assumed by all models that use the Friedmann–Lemaître–Robertson–Walker metric, including the current version of the ΛCDM model, or is the universe inhomogeneous or anisotropic?

It is all very well covered on Wikipedia. Here are some selective quotes from the same article:

Karl Popler wrote: the “cosmological principles” were, I fear, dogmas that should not have been proposed.

Although the universe is inhomogeneous at smaller scales, according to the ΛCDM model it ought to be isotropic and statistically homogeneous on scales larger than 250 million light years. However, recent findings have suggested that violations of the cosmological principle exist in the universe and thus have called the ΛCDM model into question, with some authors suggesting that the cosmological principle is now obsolete and the Friedmann–Lemaître–Robertson–Walker metric breaks down in the late universe.

European Space Agency says:

Planck's new image of the CMB suggests that some aspects of the standard model of cosmology may need a rethink, raising the possibility that the fabric of the cosmos, on the largest scales of the observable Universe, might be more complex than we think.

https://sci.esa.int/web/planck/-/51551-simple-but-challenging-the-universe-according-to-planck

To sum up:

"Why do we still believe that the universe is isotropic then?"

I do not know. Probably because it is the best explanatory model we have. More data supports it than not.

As to the more general question: why do people believe the things they do? I do not think physics can explain that. Try metaphysics and/or philosophy :)