# CMB dipole anisotropies & Hubble's Law

Gold Member
It is known that the dipole anisotropies of the CMB radiation, can give the relative velocity of the Earth with respect to the CMB "rest" frame...
In first order, this is given by $\beta \approx 1.2 \times 10^{-3}$ or $u_{CMB} \approx 360 km/s$.
I have one question here:
Why is the velocity given as such?
In first approximation we have $T' (\theta)_{observed} \approx T_{cmb} (1 + \beta \cos \theta)$
and this depends on $\theta$, the direction in which we are looking at CMB. So I'd say that the velocity of the Earth wrt to the CMB rest frame, should also depend on $\theta$:

$1.2 \times 10^{-3} = |\frac{\Delta T}{T}| = |\frac{T - T'}{T}| = |1- T'/T| = |\beta \cos \theta|$

Also, I'm not sure how can this affect the measurements of distances with the Hubble's law:
$u = H_0 D$

Is it because the $u$ will have to get the contribution from $u_{CMB}$? but this is crazy, because the far-away galaxy we are observing, will also have to move with some velocity wrt CMB... How is Hubble's law then affected?

The velocity is in a specific direction, toward somewhere near the center of the Hydra constellation. The $\theta$ of the dipole contribution to the CMB temperature is measured as an angle from that direction.

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Dearly Missed
The velocity is in a specific direction, toward somewhere near the center of the Hydra constellation. The $\theta$ of the dipole contribution to the CMB temperature is measured as an angle from that direction.
I think you may have your directions mixed up. Solar system motion (say approx 370 km/s) is in the direction of a point in constellation Leo, in northern hemisphere.
That we can measure by taking the temperature of the CMB doppler hotspot around that point in Leo.

But we also know the solar system's ORBITAL velocity in the galaxy. With respect to the center of the galaxy. So we can do some vector arithmetic and INFER the speed and direction our galaxy is going. And by implication the Local Group. Relative to the CMB.
As I recall that is about 600 km/s in a direction in southern hemisphere. You may have heard Hydra, or Centaurus (they are large southern constellations). Actually when I looked it up the coordinates were in a smaller constellation called Krater, which neighbors on the larger better-known ones like Hydra.

We should not confuse that with solar system motion relative to CMB which we can actually measure. The Milky Way and Local Group motion relative to CMB is then inferred

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• TEFLing
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Dearly Missed
...Also, I'm not sure how can this affect the measurements of distances with the Hubble's law:
$u = H_0 D$...

they just correct for solar system motion. Subtract off the solar system velocity vector from all the data on recession velocities of all the galaxies.
that way you get estimates of recession speeds (over distance) which are symmetric.
Because they are taken from the point of view of an observer who is at rest with respect to the CMB, i.e. at rest with respect to the expansion process itself (the"Hubble flow" it used to be called and still is sometimes).
as I understand it, they used to make that correction even before they observed the CMB
i.e. correct for solar system motion w.r.t. Hubble flow. but less precisely and less confidently I imagine.

It is the same as you would want to correct Earth based observations for the Earth's orbital motion about the sun, or likewise correct for an orbital observatory's motion.
But Earth orbit speed is only 30 km/s relative to solar system.. So it does not count for much compared with the solar system motion of around 370 km/s relative to CMB.

Gold Member
So you mean that when I observe a galaxy A traveling at $V$ away from me (the earth), its real velocity is $V- u_{CMB}$ ?
It would make sense only if the galaxy A is at rest with respect to CMB rest frame (and that's why I got confused about it). If instead the galaxy is also moving, as the Earth does, this doesn't look correct to me :s

John_QPublic
The dipole may require another explanation. I say this because there are other measures of apparent velocity that conflict with it: http://arxiv.org/abs/1405.4796

"The magnitude of the peculiar velocity thus determined turns out be extremely large (9750±550km/s; ∼3% the speed of light), and is about an order of magnitude larger than the velocity determined from the dipole anisotropy in the Cosmic Microwave Background Radiation or the value determined earlier relative to the frame of distant radio sources. Even the direction of the motion is in a direction nearly opposite to the earlier determinations. The large differences in the magnitudes of inferred motion as well as their opposite signs are rather disconcerting. A genuine difference between these velocity vectors would imply highly anisotropic Universe, with anisotropy changing with epoch. This would violate the cosmological principle where the isotropy of the Universe is assumed for all epochs, and which is the basis of modern cosmological models."

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Dearly Missed
So you mean that when I observe a galaxy A traveling at $V$ away from me (the earth), its real velocity is $V- u_{CMB}$ ?...
I wouldn't say that. I think that subtracting off the solar system's peculiar velocity is a way of removing statistical bias.
The distant galaxies are presumably all moving relative to CMB---they have their own peculiar velocities acquired during structure-formation as clumps of matter fall towards other clumps and clouds condense.
These individual velocities are typically small compared with distance expansion. A few hundred km/s and seemingly average out over large scale. It is even possible to study radial components of individual galaxy motion (separate from distance expansion) for example in clusters of galaxies. If a cluster looks stable and it is at a certain distance, and that distance is expanding at a certain rate, then let's ignore that and look at the (random) motions within the cluster. what must the mass of the cluster be in order to hold together, with members moving around at the observed speeds?
One can study the rotation curves of distant galaxies the same way.

Merely because one subtracts off the solar system velocity to remove a dipole bias from the data does not imply that one CLAIMS that distant objects do not also have individual motions themselves!

they can also have individual motions of a few hundred km/s that seem to be more or less random over very large scale (although people look for patterns and

so when one measures the Hubble expansion rate it improves the precision to subtract off the solar system motion, but that does not constitute a claim as to the 'real velocity' of any particular galaxy.

Expansion of distances is not like ordinary motion we are used to---nobody gets anywhere by it, everybody just becomes farther away at a certain percentage rate per unit of universe time. Nobody approaches a destination. Relative positions don't change. that's all part of the Friedmann equation cosmic model.
Galaxies have small individual motions on top of that (not part of the change in geometry itself)

Gold Member
so when one measures the Hubble expansion rate it improves the precision to subtract off the solar system motion, but that does not constitute a claim as to the 'real velocity' of any particular galaxy.

Yes, but it can give you a "correction" to when measuring distances... For example finding the distance that gives a deviation from Hubble's law of ~ X% ... did I understand that right?

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Dearly Missed
I suppose so, Chris. I'm not sure I fully understand the issue you have in mind. We may need someone more expert than myself to give a really satisfactory answer. The way I see it there are two contexts in which one wants to use that correction (subtract off the CMB dipole that corresponds to solar system motion of circa 370 km/s in direction of Leo).

In the first context, one naturally wants to subtract out the dipole from the CMB temperature map itself. Because the minute temperature fluctuations that seem to have originated long ago before there was a galaxy falling thru space and a solar system orbiting in it----these tiny temperature fluctuations which are on the order of 10-5 are incredibly interesting! And the doppler dipole is really gross! It is on the order of 1/1000!
It is comparatively recent, 100 times bigger, comes from a different cause, and you want to take it out of the date right away, in order to study the delicate statistics of primordial fluctuations that the solar system motion dipole would totally overwhelm.

The second context is where you are measuring the expansion rate H(t) and how it evolves over time. In this context, 370 km/s is a small speed. It is only on the order of 1/1000 of speed of light. So it corresponds to a difference in measured redshift of only 0.001. So you would think that it should not effect the estimates of H(t).
So why bother to take it out of the data? The error bounds for measuring H0 are, as I recall, not that tight! Why not be a little sloppy? No one will notice if there is a bit of extra redshift, like 0.001 in one part of the sky and a little bit less in the opposite direction. Well maybe some people don't think it worth correcting for. I'm not an expert and I don't really know the prevailing practice. Maybe someone with direct hands-on experience will jump in here. I'm going on what I've been told and just never bothered to question---that one corrects for solar system motion (even though in a redshift survey it is admittedly a very small correction).

Gold Member
Yes but hubble's law gives velocity wrt to distances...
So if the distances are small, the velocities will also have to be small, and so the CMB velocity will become "important" at some point(and so you will have deviations from the Hubble's law). At large distances, this is not a problem because as you say, we have very small speed (370 km/s) ...

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