More Cosmic Accounting And Some Speculations On Matter-Antimatter Asymmetry

In March of this year, I did a post on "Cosmic Accounting and Neutrino Mass". This is also a cosmic accounting post. The notion is to determine the experimentally measured particle make up of the universe to use as a reference point for evaluating theories that purport to explain this reality.

After doing that accounting with the best available data, I speculate about what this could imply in terms of dark matter and cosmology.

Baryon Number and Lepton Number in the Standard Model With Sphalerons

In the Standard Model, baryon number (the number of quarks minus the number of antiquarks, divided by three) is conserved as is lepton number (the number of charged leptons and neutrinos minus the number of charged antileptons and antineutrinos), except in sphaleron process. Per Wikipedia:

A sphaleron (Greek: σφαλερός "weak, dangerous") is a static (time-independent) solution to the electroweak field equations of the Standard Model of particle physics, and it is involved in processes that violate baryon and lepton number. Such processes cannot be represented by Feynman diagrams, and are therefore called non-perturbative. Geometrically, a sphaleron is simply a saddle point of the electroweak potential energy (in the infinite-dimensional field space), much like the saddle point of the surface z(x,y)=x2−y2 in three dimensional analytic geometry.

In the standard model, processes violating baryon number convert three baryons to three antileptons, and related processes. This violates conservation of baryon number and lepton number, but the difference B−L is conserved. In fact, a sphaleron may convert baryons to anti-leptons and anti-baryons to leptons, and hence a quark may be converted to 2 anti-quarks and an anti-lepton, and an anti-quark may be converted to 2 quarks and a lepton. A sphaleron is similar to the midpoint (\tau=0) of the instanton, so it is non-perturbative. This means that under normal conditions sphalerons are unobservably rare. However, they would have been more common at the higher temperatures of the early universe.

The trouble is that if you start with B=0 and L=0, as you would expect to in a Big Bang comprised initially of pure energy, it is hard to determine how you end up with the observed values of B and L in the universe which are so far from zero.

The mainstream view among physicists, although there are some theorists who dissent from this analysis, is that Standard Model sphaleron processes in the twenty minutes during which Big Bang Nucleosynthesis is believed to have taken place, or the preceding ten seconds between the Big Bang and the onset of Big Bang Nucleosynthesis, can't account for the massive asymmetry between baryons made of matter and baryonic anti-matter that is observed in the universe (also here) without beyond the Standard Model physics (also here).  Likewise, after that point, sphaleron processes should be so rare that they can't explain the baryon asymmetry of the universe (BAU), which is one of the great unsolved problems in physics.

What are the experimentally measured values of baryon number and lepton number in the universe?

Astronomers have been able to estimate the baryon number of the universe to one or two significant digits, but while they have a good estimate of the number of Standard Model leptons in the universe (to one significant digit) they have a less accurate estimate of the lepton number of the universe since they don't know the relative number of neutrinos and antineutrinos.

As one science education website explains:

[S]cientific estimates say that there are about 300 [neutrinos] per cubic centimeter in the universe. . . . Compare that to the density of what makes up normal matter as we know it, protons electrons and neutrons (which together are called “baryons”) – about 10^-7 per cubic centimeter. . . . the size of the observable universe is a sphere about 92 million light-years across. So the total number of neutrinos in the observable universe is about 1.2 x 10⁸⁹! That’s quite a lot – about a billion times the total number of baryons in the observable universe. . . . We don’t know the mass of a neutrino exactly, but a decent rough estimate of it is 0.3 eV (or 5.35 x 10-37 kg). Scientists have indirect proof that neutrinos do have some mass, but as far as we can tell, they’re the lightest elementary particle in the universe. So there are lots of neutrinos in the universe, but they each don’t weigh very much. So that means in a cube of volume one Astronomical Unit on a side (where one AU is the distance from the sun to the earth, or about 93 million miles), there’s only about 600 tons worth of neutrinos! So the mass of neutrinos in empty space millions of miles across contains less mass than the typical apartment building.

(In fact, a better estimate of the average neutrino mass is probably 0.02 eV or less (based on the sum of the mass differences in a normal hierarchy, see also here for cosmology based limits), so so 40 tons rather than 600 tons is probably a better guess for the one cubic AU of neutrino mass in the quoted material above.  The back of napkin estimate of the mass of all of the neutrinos in the universe on that basis is about 1/10,000th of the mass of all of the protons and neutrons in the universe combined, and about 1/5th of the mass of all of the electrons in the universe.)

In other words, the number of baryons in the universe is about 4*10⁷⁹, and the number of neutrinos in the universe is about 1.2*10⁸⁹. We know that the ratio of baryon antimatter to baryon matter (and the ratio of charged leptons to charged antileptons) is on the order of 10^-11. And, we know that to considerable precision there are 2 neutrons for every 14 protons in the universe (this is a confirmed prediction of Big Bang Nucleosynthesis), and that the number of charged leptons is almost identical to the number of protons in the universe. (The number of exotic mesons and baryons other than neutrons and protons is negligible at any given time in nature since they are so short lived and generated only at high energies.)

What Is The Experimentally Measured Value of B-L?

What is the value of the conserved quantity B-L (which is conserved not only in the Standard Model even in sphaleron processes, but also in the vast majority of beyond the Standard Model theories)?

The number of protons and number of charged leptons cancel, leaving the number of neutrons (about 5*10⁷⁸) minus the number of neutrinos plus the number of antineutrinos. The combined number of neutrinos is 1.2 x 10⁸⁹, but we don't have nearly as good of an estimate of the relative number of neutrinos and antineutrinos.

The number of neutrinos in the universe outnumber the number of neutrons in the universe by about 2.4*10¹⁰ (i.e. about 24 billion to 1), so the conserved quantity B-L in the universe (considering only Standard Model fermions) is almost exactly equal to the number of antineutrinos in the universe minus the number of neutrinos in the universe.

As of March 2013, the best available observational evidence suggests that antineutrinos overwhelmingly outnumber neutrinos in the universe. As the abstract of a paper published in the March 15, 2013 edition of the New Journal of Physics by Dominik J Schwarz and Maik Stuke, entitled "Does the CMB prefer a leptonic Universe?" (open access preprint here), explains:

Recent observations of the cosmic microwave background at smallest angular scales and updated abundances of primordial elements indicate an increase of the energy density and the helium-4 abundance with respect to standard big bang nucleosynthesis with three neutrino flavour. This calls for a reanalysis of the observational bounds on neutrino chemical potentials, which encode the number asymmetry between cosmic neutrinos and anti-neutrinos and thus measures the lepton asymmetry of the Universe. We compare recent data with a big bang nucleosynthesis code, assuming neutrino flavour equilibration via neutrino oscillations before the onset of big bang nucleosynthesis. We find a preference for negative neutrino chemical potentials, which would imply an excess of anti-neutrinos and thus a negative lepton number of the Universe. This lepton asymmetry could exceed the baryon asymmetry by orders of magnitude.

Specifically, they found that the neutrino-antineutrino asymmetry supported by each of the several kinds of CMB data was in the range of 38 extra-antineutrinos per 100 neutrinos to 2 extra neutrinos per 100 neutrinos, a scenario that prefers an excess of antineutrinos, but is not inconsistent with zero at the one standard deviation level.

A 2011 paper considering newly measured PMNS matrix mixing angles (especially theta13), and WMAP data had predicted a quite modest relative neutrino-antineutrino asymmetry (if any), but it doesn't take much of an asymmetry at all to make B-L positive and for the neutrino contribution to this conserved quantity to swamp the baryon number contribution.

Intuitively, an antineutrino excess over ordinary neutrinos make sense because beta decay, a common process in nature, generates antineutrinos, and there is no other process which obviously creates an equivalent number of neutrinos.  Beta decay conserves lepton number (indeed, the antineutrino was proposed theoretically in order to conserve lepton number in beta decay since the electron produced has L=1 and the antineutrino has L=-1, an argument in my mind also against Majorana neutrinos in which the particle and antiparticle are one and the same), but beta decay does tend to unbalance the relative number of neutrinos and antineutrinos in the universe, and there is no particularly obvious reason why this would be exactly cancelled out in other processes at approximately the same time.

Thus, B-L is about 85%-90% likely to be greater than zero in the universe, at least to the extent that Standard Model particles are considered, although this is not a sure thing.

Beyond The Standard Model Conjectures

In addition to the missing baryonic matter, something must account for the 23% of mass energy ascribed to dark matter by the six parameter lambda CDM model used to explain the CMB from first principles.

It is possible that the "Cold Dark Matter" signal isn't really "Cold Dark Matter". As many prior posts at this blog have explored, evidence of "Warm Dark Matter" and "Cold Dark Matter" are almost indistinguishable in CMB observations, and Warm Dark Matter is a better fit to the other data than Cold Dark Matter.

But, given that all experimental evidence related to "dark energy" can be entirely explained with just a single constant in the equations of general relativity (the measured value of the cosmological constant) which is part of the equations for a force rather than a physical thing made up of particles, it could very well be the case that another tweak to the equations of general relativity (perhaps in a quantum gravity formulation) could also explain the CDM figure.

If there is a lot of dark matter out there, this could "balance the books" and bring the value of B-L to zero, the a priori expectation of cosmologists who assume that the Big Bang started with pure energy with B=0 and L=0.

If dark matter particles have a positive lepton number, or a negative baryon number (or both, asymmetrically, with a net value that was a positive lepton number or a negative baryon number), then B-L could equal zero.
This would also suggest, if true, that some sort of mechanism that links the magnitude neutrino-antineutrino asymmetry to the creation of dark matter is involved in the production of dark matter.

The example of a single kind of stable 2 keV sterile neutrino warm dark matter particle.

Consider a simple model in which dark matter consists entirely of sterile neutrinos (with virtually no sterile anti-neutrinos) with lepton number 1 for each particle, in a number exactly counterbalancing the B-L total from other methods. If this sterile neutrino had a mass on the order of 2 keV (the empirically preferred warm dark matter mass),one could deduce the anticipated number of sterile neutrinos that make up dark matter. This could then be used to predict the extent of the neutrino-antineutrino asymmetry in the universe and compare it against future measurements of that quantity which is currently not known very accurately.

Given that a typical nucleon has a mass of about 1 GeV and that there are about 4*10⁷⁹ baryons in the universe, and that the mass of leptons relative to baryons in the universe is negligible, and that the ratio of dark matter to baryonic matter in the universe is about 23 to 4.6, it follows that the total mass of dark matter in the universe is about 2*10⁸⁶ keV and so you would expect that there are 10⁸⁶ warm dark matter particles in the universe.

Now, given that there are 1.2*10⁸⁹ neutrinos in the universe, these warm dark matter particles would balance the B-L imbalance in the universe to zero, if the number of antineutrinos in the universe exceeds the number of neutrinos in the universe by about one part per 1,000, a number that would still cause the lepton asymmetry in the universe in favor of antimatter to exceed the baryon asymmetry in the universe in favor of matter, in terms of raw numbers of particles (since 2*10⁸⁶ is much greater than 4*10⁷⁹).

A heavier dark matter particle would imply a neutrino-antineutrino asymmetry of one part per more than 1,000, and a lighter one would imply a neutrino-antineutrino asymmetry of one part per less than 1,000 down to the counter-factual limit of hot dark matter where dark matter has the same mass as a neutrino.

Given the current range of possible neutrino-antineutrino asymmetry proportions, this value is still consistent with the CMB data at the one sigma level, but just barely, as that data tends to favor a greater asymmetry which would take more (and hence lighter) dark matter particles to counterbalance.  Given the limitations posed by Neff, the lightest possible dark matter particle in a B-L balancing hypothesis would be about 10 eV (otherwise Neff would be about 4.1, rather than about 3.4), which would allow a 10 extra-antineutrino per 100 neutrinos asymmetry, much closer to the center of the CMB data preferred range.  But, a variety of experimental observations used to place lower bounds on the mass of a warm dark matter particle would seem to rule out this possibility.

Note also, that if a keV dark matter particle interacted with the Higg field with a Yukawa proportional to its mass, that this would be well within the experimental measurement error in the calculation that the sum of the Yukawas of all the Standard Model particles with a suitable adjustment for the Higgs boson self-interaction equals one, which the current data favor to a high degree of precision, because the Yukawa of a keV particle is so tiny.

An alternative conjecture with a non-zero B-L immediately after the Big Bang.

Suppose, however, that any non-Standard Model particles that exist (including dark matter particles), have B=0 and L=0 (or B=1 or L=-1, for example), or that there simply aren't enough of them to balance the cosmic accounting ledgers, and that the strong indication of existing experimental and theoretical knowledge which support the conclusions that B-L is always conserved in all interactions no matter how high their energy.

In that case, it necessarily follows that B-L was not equal to zero immediately following the Big Bang.

My personal speculation to resolve this situation, which just a handful of experimental results could make a reality in the near future, is that B-L immediately after the Big Bang is positive and counterbalanced by a B-L which is negative immediately before the Big Bang, on the theory that matter wants to move forward in time, while antimatter is simply identical stuff moving backward in time and hence would tend to move to a location before the Big Bang in time, rather than to a location after it. If matter and antimatter were created in equal amounts, most of the matter would end up after the Big Bang, most of the antimatter would end up before the Big Bang, and the Big Bang itself could be viewed no just as a source of pure energy, but as an explosion caused by the collision of all the antimatter in an antimatter dominated universe and all of the matter in a matter dominated universe.

Footnote Regarding The Neff Mystery

CMB observations predict an effective number of neutrino species.  The minimum is 3 which have been observed experimentally, for technical reasons this would be expected to produce an Neff of 3.046 (supported by what we know about the process and the measured magnitude of theta13 in the PMNS matrix, see, e.g. this 2011 paper).  If there were four neutrino species, Neff ought to be something on the order of 4.05.

But, in the 9 year data report, WMAP observed Neff to be 3.26 +/- 0.35 (later corrected upward to 3.84 +/- 0.4), which was a great downward reduction from the 4.34+/-0.87 number estimated based on the 7 year WMAP data.

Paper sixteen from the Planck experiment's data dump stated that Neff was 3.30 +/- 0.27 (within one sigma of the 3 species value and about three sigma from the four species value).  But, within a few days the Planck number was revised upward to about 3.62 +/- 0.5 which slightly favors N=4 over N=3 (with a fourth neutrino species having a mass of less than 0.34 eV at the 95% confidence interval).  Paper sixteen itself say 3.62 +/- 0.25 in the conclusion, and states that the adjustment is based on resolving tensions with the Hubble constant measurement revised by the Planck experiment as well, but the paper has a variety of estimates based upon the assumptions invoked, none of which is truly definitive, and two of which have best fit values as low as 3.08.

Polarization data from Planck that is supposed to be released in early 2014 is supposed to reduce the uncertainty from about +/- 0.27 to +/- 0.02 providing powerful limits on additional species of light particles or strongly favoring a fourth neutrino species if the updated values with polarization data are 3.0-3.1 or 4.0-4.1, but it will leave us with a real conundrum if the final result turns out to be in the range 3.35-3.75 or so.

The mystery is whether this estimate truly is an experimental measurement uncertainty, or if the true value, if measured with arbitrary precision, would be between 3.046 and 4.05, implying some phenomena that creates a fractional effective neutrino species, and if so, what that could be.

Could this be some unknown form of radiation?  Could it be a particle with a mass on the boundary between that of neutrinos and of particles that are not captured by the operationalized Neff measurement (e.g. a particle of 10 eV)?  Could it be a force that partially mimics a neutrino, perhaps transmitted by a massive, but light boson?  I don't know and have seen little literature review of the subject, but I would love to know more about the possibilities that could explain the result (both in terms of sources of experimental error and in terms of BSM or Standard Model physics).

Rereading Paper Sixteen at length, I find it much more likely that the conclusion at the end of the day will be that there are three rather than three fertile and one sterile neutrino species (a fourth fertile neutrino species with a mass of a fraction of 1 eV is clearly ruled out by W and Z boson decays).

Footnote Regard The Missing Baryonic Matter

Studies of the comic microwave background radiation (CMB) in the universe, studies by experiments like WMAP and the Planck satellite have estimated that the universe has about 4.6% ordinary "baryonic" matter, about 23% "dark matter" and about 72.4% "dark energy".

 About 10% of the ordinary baryonic matter is in galaxies that we can observe with telescopes.

 The Hubble space telescope has identified another 40% of it or so in the form of interstellar gas and other "barely" luminous ordinary matter between galaxies, and hope to find another 20% of it or so with the Cosmic Origins Spectrograph and other UV range searches around "filaments" of dark matter believed to stretch between galaxies. The composition of the missing baryonic matter (which I often call "dim matter" to distinguish it from the technical meaning of "dark matter" in physics, is believed to be similar to that of observed baryonic matter, but less compact.

But, about 30% of the baryonic matter in the universe is not only dim, but is so dim that we don't even have good ideas about where to look for it.