# Reference frame for the Cosmic Neutrino Background

It is well known that there exists a reference frame where the total momentum of the Cosmic Microwave Background is zero (a basic fact of special relativity applied to a collection of massless particles) and the radiation is homogeneous and isotropic. This can be observed in the dipole anisotropy.

Neutrinos also decoupled from matter shortly after the Big Bang, and therefore relic neutrinos must form a Cosmic Neutrino Background ($\mathrm{C}\nu\mathrm{B}$). These neutrinos are extremely hard, if not impossible, to detect with current technology.

The $\mathrm{C}\nu\mathrm{B}$ also defines a reference frame of zero total momentum. Is it expected that this reference frame is coincident with that of CMB?

• can you please give a link for the claim "there exists a reference frame where th total momentum of the cmb zero" – anna v Jul 2 '16 at 11:47
• @annav For example, arxiv.org/abs/1008.1183 (and lots of posts in this very own site...) – Bosoneando Jul 2 '16 at 11:57
• Yes we expect, in FRW cosmology generally, that the two frames are the same and equal to the "comoving frame". – Ihle Jul 2 '16 at 13:32
• I would suspect not, since AFAIK the neutrinos are not strictly mass-less whence their velocity vector must be timelike. – Erik Jörgenfelt Jul 2 '16 at 14:14
• Note that the CMB is amazingly uniform, and that the fluctuations you can see in the WMAP depiction are greatly exaggerated. Ethan Siegel referred to that here. Ergo I expect the CνB reference frame is coincident with that of the CMB. – John Duffield Jul 5 '16 at 7:16

As is discussed in this answer, the "rest frame" of the cosmic neutrino background would be very similar to that defined by the cosmic microwave background if neutrinos were very light (say $<0.1$ eV). The Sun would be moving with respect to this frame at around 370 km/s.

But if neutrinos were more massive(say getting on for 1-2 eV) then they are expected to be very non-relativistic with speeds comparable to the escape speed of the Milky Way. In those circumstances they will cluster around the Milky Way and would have a rest frame that was more similar to that of the Milky Way galaxy itself - i.e. they would be orbiting around the Galaxy and the Sun would move on average at about 220 km/s with respect to the average neutrino.

At present neutrino rest masses are probably somewhere between these extremes and so the cosmic neutrino background is thought to have a small dipole anisotropy that will be somewhat offset with respect to that of the cosmic microwave background. By that I mean that if you subtracted off the dipole anisotropy implied by the CMB you would still be left with an anisotropy in the C$\nu$B.

The basic assumption in the Big Bang cosmolocigal model is that the CMB departures from a uniform black body radiation observed in the map

All-sky map of the CMB, created from 9 years of WMAP data.

are a relic of the density of matter at the time of the decoupling of photons, about 380000 years after the BB. In contrast, neutrino decoupling, if the neutrino cosmic background could be measured would be about one second after the BB.

The topology of the density of the stress energy tensor describing the universe 1 second after the BB cannot be the same as that of 380.000 years as electromagnetic and gravitational forces would be very active in the interval . Neutrinos would give a snapshot at 1 second from the BB.

Thus any presumed rest frame of CMB has small probability of being the same as for ($\mathrm{C}\nu\mathrm{B}$) . Think of shapes in fluid flow. They are different downstream.

• I do not understand this answer. What do you mean by "small probability of being the same"? Clearly, they have probability zero of being EXACTLY the same. Still they should be very close to being the same, and in the limit where the universe is homogeneous and isotropic they will be exactly the same. Do you agree? – Ihle Jul 2 '16 at 22:06
• Forgive me, but would this not also require the limit of zero mass neutrinos? My thinking is that since CMB velocity is null and non-zero neutrino mass velocity is timelike, the comoving frames must differ. Is that wrong? – Erik Jörgenfelt Jul 2 '16 at 22:32
• @Ihle Think of blobs of paint in a flowing river. Their shapes are distorted . A snapshot of the density of blobs at the beginning of the flow will be different than a snapshot ten minutes downstream. So they cannot make up the same momentum vector. – anna v Jul 3 '16 at 4:14
• @ErikJörgenfelt I believe you are correct that the fact that neutrinos have mass will enter the problem. But rest frames depend on the invariant mass of the whole conglomerate, the more than one photon define an invariant mass even though they are massless ( think of the ) – anna v Jul 3 '16 at 4:17
• that should be "think of the pi0" – anna v Jul 3 '16 at 10:58