Neutrino mass and neutrino hot dark matter

 

 

Introduction
Dark matter is matter in the universe that does not create light, but we detect it through the effects it has on normal (baryonic) matter, on light and, if observable, on gravitational waves. Light bending by dark matter is similar to light bending by the sun, detected by Sir Arthur Eddington in 1919 at the island of Principe, that proved the correctness of Einstein's General Theory of Relativity of 1915.

There are two types of dark matter: of galaxies and of clusters of galaxies. Galaxy dark matter consists of baryons locked up in planets of nearly earth weight. Cluster dark matter is non-baryonic and the subject of this page. We shall argue that it is just consituted by neutrinos, despite the ocean of counter arguments known in literature.

Cold dark matter (CDM)
Presently it is believed that dark matter arises from heavy particles, but their mass is not known. A variety of assumptions cover a the mass range of a few keV (kilo electron volt), or MeV, GeV, TeV, ..., depending on the considered theory and firmness of the assumptions. CDM particles should have clustered before the plasma turned in a fog of neutral gas, after which the photons were no longer scattered and could then travel to us and be observed as the Comic Microwave Background (CMB). More than two dozen experimental searches for the CDM particle have been carried out, in most of these windows, but nothing has been found so far. Still, one can often read that were are close to detecting the dark matter particle, because this new theory ... or because in the next search ... .


The galaxy cluster Abell 1689
Without any preferred candidate in mind, we aimed at describing the matter distribution of a galaxay cluster. We chose the cluster Abell 1689 (A1689), this is a relaxed cluster with a lot of dark matter, as is known from strong and weak lensing.
Galaxies have the shape pancakes (more precisely: "spiegeleieren"); when viewed under an angle they look like ellipses. The strong lensing effects of A1689 imply that galaxies, located behind the cluster, appear elongated and look like bananas (lensing arcs). See here a nice picture of A1689. In this picture, please look a while, till you notice that it looks as if there is a crystal ball: everything goes in circles! There must be much dark matter in the cluster, that bends the light from galaxies behind it, galaxies alone could not do that. The light bending by cosmic matter, dark or not, occurs just as by the glass of spectacles, that bends light to correct our eye vision.
Weak lensing occurs further out from the center (this is outside the photograph, the cluster is much larger). It is a statistical effect due to the lensing by the mostly dark matter: the randomly directed ellipse of each galaxy appears slightly rotated along circles, and the lensed galaxies are seen to have on the average a local circular direction. The amount of this circularity is determined by the total matter, dark or not, along the line-of-sight.


Virialization and violent relaxation
Clusters consist for three components: galaxies, dark matter and gas. We have described this system as follows. For the dark matter we have assumed a Fermi-Dirac distribution of degenerate masses. For the galaxies and the gas we have assumed Boltmann distributions. Next we made the assumption of virial equilibrium, that is to say, that the velocity dispersions of galaxies, dark matter particles and alpha-particles in the gas, are all the same, so that each ones temperature is proportional to its mass. The other gas constituents, protons, ions and electrons take the same speed as the alpha particles, because, being charged, they will interact a lot with them and with each other. The virial state is possible by the peculiarity of Newton's law, where the acceleration of a particle in a given gravitational field does not depend on its mass, and neither will then its typical speed do so. This has been worked out more deeply in a process known as violent relaxation. Its key point is that the energy of any individual particle, be it a neutrino, a hydrogen atom or a whole galaxy, changes in time when they move in a time dependent gravitational potential. So each particle exchanges energy with the gravitational energy of the whole cluster, which supports a virialized state. As we discuss below, this effect was strongest when the dark matter particles condensed on the cluster.
The virialized state is maintained by the X-ray radiation of the hot gas, the energy for which is provided by a slow contraction of the cluster. The situation is somewhat similar to what happens in stars like the sun, where the radiation energy is also supplied by a slow contraction.

The theory based on those three assumptions, Newton's law, quantum statistics and virial equilibrium, performs rather well. It achieves to describe the lensing data (total matter) rather well, the galaxies reasonably and the gas good near the center, though further out than 200 kpc the gas temperature decays, an effect that could also be counted for. Surprisingly, the predicted temperature of the alpha-particles coincides with the measured gas temperature. And that is super high, about 10 keV or 100 million degrees Kelvin (about the same in centegrade or Fahrenheit). Even more astonishing is that the baryons (neither the galaxies nor the gas) trace the local dark matter density well, even though they do trace the total mass enclosed inside the sphere around the origin. The latter happens by the virial assumption. This non-sequitur nullifies many conclusions in literature, e.g. that sterile neutrinos should have keV masses.

This set up fixes the mass of the dark matter particle rather precisely. Surprisingly, it is of order eV. The answer still depends on the number of modes available. The neutrinos spin can be up or down, there can be anti-neutrinos, and there are three families. That sets g=2*2*3=12 and the mass is approximately 1.45 eV (3*10^(-36) kg, three millionth of the electron mass). In the studied cluster there must be some 10^(82) of them.

Though the analysis of the A1689 cluster does not tell us what particle constitutes its dark matter, we can infer this indirectly. Indeed, the cluster is so large, that we may assume that its dark matter is representative for the whole cosmos. But then we know that particles that decouple early in the thermal history of the Universe have a low number density, because, being decoupled, they were not heated and their number was not enhanced by the freeze out of e.g. quarks/anti-quarks and electrons/positrons. So early decouplers are ruled out as the dark matter particle. This applies to e.g. all supersymmetric and stringy particles. Bosons, such as the axion are ruled out because, having a different spatial profile, they do not fit the A1689 data.

Identification of the dark matter particle
The only late decoupler is the neutrino. The mass of 1.45 eV is enough to explain 9.5 percent of all the matter being dark. That is much more than currently allowed for neutrinos, so the underlying theory, the cold dark matter model is ruled out if neutrinos indeed have that mass. This is not completely unexpected, because with many new data coming in from e.g. the Hubble Space Telescope, every month there appears another problem for the CDM, not to mention the ones that always existed, but were ignored, such as: how does CDM explain the old globular star clusters?

Occupancy of sterile modes
In the cluster all left- and righthanded states can be occupied. It is believed that in the very early Universe, only lefthanded neutrinos were created and righthanded antineutrinos, the so-called "active neutrinos". The other ones, e.g. righthanded neutrinos, are "sterile". If they have also been created early on, there may be twice as much neutrino dark matter, so 19%, which is about what is estimated by most current approaches. This filling seems possible provided there is also a Majorana mass matrix. That approach is known from the see-saw mechnism for sterile neutrinos, but we want a small Majorana term for having nearly degenarate masses. The Majorana coupling allows neutrionoless double beta-decay. Double beta-decay occurs when in a nucleus two neutrons decay simultaneously in two protons, two electrons and two antineutrinos; in neutrionoless double beta-decay the emitted anti-neutrinos annihilate each other. (Normally it is believed that only a particle and its anti-particle can annihilate each other, here the new aspect is that two anti-particles annihilate eeach other; clearly, the lepton number is then not conserved.) With the protons remaining in the nucleus, there are only two outgoing electrons, that must go in opposite directions, a clear experimental signal. There are ongoing searches for this effect; quite better ones may be needed to observe it.

Hot dark matter
There have been many reasons in the past to conclude that neutrinos could not be of much relevance for dark matter. Their type of dark matter is called hot dark matter (HDM). The present consensus is that HDM cannot explain the observed large scale structures and CMB peaks. But people have overlooked the gravitational hydrodynamics of the protoplasma. In that plasma there are many photons and they scatter wildly from the free electrons. This creates a big viscosity. So the plasma is expanding (with space) and becoming more and more viscous. Before the decoupling the viscous length becomes smaller than the horizon scale, so the plasma will actually fragment. The free streaming neutrinos, that are responsible for a very homogeneous plasma till then, will not have enough time to repair the inhomogenities, so they are the basis of the structures we observe.

Reionization
There is much gas in clusters and it appears to be ionized, having a large temperature of several keV's, dozens of millions degrees Kelvin. This effect is called reionization, because the primeordial plasma was ionized already. Reionization is nowadays attributed to presumed early heavy stars, that in their decay should have created many gamma-rays which then should have ionized the gas.
Neutrinos are notoriously weak interacters. Basically, they don't, they just move freely, this is called free streaming. When the primordial plasma turned into a neutral gas, with fragmented structures, they just kept on streaming freely. After they become non-relativistic the cross section is only 10^(-59) m^2. But the expansion of the Universe slowed them down. At some moment they were so slow that they condensed on the clusters, such as A1689 cluster. This was when the Universe had redshift of about z=28, and age of some 100 million years. Then the loosely bound gas had to go to its virial equilibrium with its very high temperature of 10 keV. On its route to there, it became so hot that it ionized.
We thus see that the neutrinos caused the reionization, without the need for heavy stars, by bringing the gas to its high temperature virialized state.

[C45] Theo M. Nieuwenhuizen,
The case of 1.5 eV neutrino hot dark matter
arXiv:1003.0459 , (3 pp). Proceedings Marcel Grossmann XII, Eds. Thibault Damour, Robert T Jantzen and Remo Ruffini, World Scientific, Singapore, 2010

Europhysics Letters: Highlights from previous volumes
Th. M. N.: Do non-relativistic neutrinos constitute the dark matter?
Europhysics Letters 89, 00000 (2010) (2pp)

[L53] Theo M. Nieuwenhuizen, Do non-relativistic neutrinos constitute the dark matter? Europhysics Letters 86, 59001 (2009)

See also: The editorial by R. A. Treumann, Europhysics Letters 86, 50000 (2009)