Shadowy Man

You are correct that this is a simple version of MOND, and as you point out is has problems.

Sanders & McGaugh (ARAA 40:263, 2002) acknowledge the problem of modifying Newtonian dynamics by applying a length scale. They then suggest that the modification is at an acceleration level, not a length scale.

It's a good article. I'm not an expert in this stuff, but it was pretty understandable.

Friar Bellows

You can read that article online at arXiv.org:

http://arxiv.org/abs/astro-ph/0204521

lpetrich

This "softened acceleration" idea would get around the expanding-Universe problem, but it has some problems of its own.

Acceleration is a vector, and while

g_MOND = sqrt(g_Newt*g0)

may look OK as a scalar expression, as a vector expression, it would be

g_MOND = g_Newt * sqrt(g0/|g_Newt|)

or something like

g_MOND = g_Newt * sqrt(1 + g0/|g_Newt|)

for a smooth transition.

(g0 is the MOND acceleration constant).

The earlier-mentioned modified Poisson law can be turned into

D.(g*mu(g)) = 4*pi*G*rho

For a radially-symmetric force, it becomes

d(g*mu(g))/r + (2/r)*(g*mu(g)) = 4*pi*G*rho

or

g*mu(g) = g_Newt = GM(r)/r^2

To get something like MOND requires for mu(g) ~ g/g0 for small g, suggesting a form like

mu(g) = |g|/(g0 + |g|)

for g being a vector.

lpetrich

Alternatively, mu(g) = |g|/sqrt(g0^2 + g^2)

Which still has an awkward factor of |g| (length of vector g), with its singularity at g = 0 (try taking derivatives there).

So I still maintain that "dark matter" and "dark energy" are more reasonable hypotheses. I will now explain what I think is a very likely "dark matter" particle.

First, a rundown of the "Standard Model" of elementary particle physics.

Spin-0 particles: none

Spin-1/2 particles (the elementary fermions):

Quarks (they interact by QCD or color force):

Charge: 2/3
up - 4 MeV
charm - 1.5 GeV
top - 176 MeV

Charge: -1/3
down - 8 MeV
strange - 0.15 GeV
bottom - 4.7 GeV

Leptons (no QCD interactions)

Charge: 0
electron neutrino < 7 eV
muon neutrino < 0.3 MeV
tau neutrino < 30 MeV

Charge: -1
electron 0.511 MeV
mu 0.106 GeV
tau 1.777 GeV

Note that these particles come in threes, meaning that they can be grouped into three "generations".

Spin-1 particles (gauge particles):

Carrier of electromagnetic interaction; electric charge is strength of interaction with this particle
Photon (charge 0) < 10^(-15) eV (Jupiter's magnetosphere size)

Weak-interaction carrier; does electronlike <-> neutrinos and up-like <-> down-like quarks
W (charge 1) 80 GeV

Weak-interaction carrier; does not change particles
Z (charge 0) 91 GeV

QCD-force carrier; also interacts with itself
Gluon (charge 0) < a few MeV

Spin-2 particles:

Carrier of gravitational interaction (space-time curvature):
Graviton (charge 0) < 10^(-31) eV (expansion of Universe)

If this seems to you like some horrid mess, you are not alone. Like the multiplying chemical elements, nuclei, and hadrons in decades past, these multiplying elementary particles have provoked a search for some Grand Unified Theory in which they all look like versions of a few different particles -- or only one.

A step on the way has been the development of "electroweak theory", the unification of the electromagnetic and weak interactions. It posits that the "bare" W and Z are massless, and that they acquire their masses by interacting with a spin-0 particle called a "Higgs particle" that has a nonzero field strength in its ground state -- a nonzero "vacuum expectation value". This Higgs particle also interacts with the elementary fermions, giving them their masses in the same fashion.

A Higgs particle has never been observed, though the Large Hadron Collider and similar accelerators should be able to produce such particles if they exist.

So now we have spin-0, spin-1/2, spin-1, and spin-2 particles. What could relate these particles? The favorite hypothesis for doing so is "supersymmetry", which states that every particle has a superpartner with a spin different by 1/2. Though it has theoretically attractive features, it does have a certain problem: no known particle is the superpartner of some other known particle.

But that has not kept particle physicists from working out a Minimal Supersymmetric Standard Model (MSSM).

For every elementary fermion, it has a corresponding spin-0 particle -- scalar electrons (selectrons), scalar neutrinos (sneutrinos), and scalar quarks (squarks). For every gauge particle, it has a corresponding spin-1/2 "gaugino" -- photinos, winos, zinos, and gluinos. And for every Higgs particle, it has a correspinding spin-1/2 "higgsino". Furthermore, the photino/wino/zino states are mixed with the higgsino states.

All these particles are expected to be observable in upcoming accelerators like the Large Hadron Collider -- if they exist, of course.

Supersymmetry applied to gravity yields "supergravity", according to which the graviton has a counterpart with spin-3/2 -- the gravitino.

Worse and worse, you must be thinking. But it is expected that there is a Lightest Supersymmetric Particle, which is a mixture of the photino, zino, and neutral higgsinos. It is spin-1/2 and electrically neutral -- and stable.

And it's expected that it will be produced in about the "right" abundance for it to be "dark matter".

So I'm still holding out for the elementary-particle solution.

Shadowy Man

lpetrich:

Have you read the Sanders & McGaugh paper to which I referred (and to which Friar Bellows has so graciously provided a link)?

If so, to which specific points of theirs do you take exception?

MOND would require new physics, but so does supersymmetry. I don't know which solution will eventually win out, but I'm glad that someone is studying MOND.

Hopefully it doesn't hurt their careers too much to be advocating an unpopular view.

lpetrich

I'm not impressed. It still looks very contrived -- and difficult to fit into GR.

Also, supersymmetry is motivated by more than cosmology -- it is mainly motivated by the quest for a Grand Unified Theory. The relationships between particles of different spins suggest that the ultimate GUT is one where different-spin particles become versions of one particle. Which is a consequence of superstring theory.

And the quantum-gravity problem exists for MOND as well as for GR.

So it's not SUSY vs. MOND, it's SUSY vs. SUSY + MOND. And not QG vs. MOND, it's QG vs. QG + MOND.

Finally, am I supposed to embrace this idea just because it's heretical?

Shadowy Man

Dark matter is contrived as well. And has more free parameters.

No. Do you embrace any idea for that reason? Did I ask you to embrace the idea? Is the idea that heretical?

Whatever... I guess we're done arguing.

lpetrich

I'm sorry if I seemed excessively peeved when I wrote my comment that "am I supposed to embrace this idea just because it's heretical?".

And now that I think about it, one could avoid the awkward gravity-acceleration singularity at value zero with

mu(g) = sqrt(g^2 + g1^2)/sqrt(g^2 + g0^2)

where g1 << g0. But MOND still seems too contrived for my taste.

I will concede that if "dark matter" was the main motive for exploring supersymmetry (SUSY), it would be more of a toss-up between SUSY and MOND. However, SUSY is motivated by more than "dark matter" -- in particular, it is motivated by the hope of finding a GUT in which particles with different spins can be related.

So it is SUSY + MOND vs. SUSY alone.

And as to a multiplicty of parameters, the Standard Model has plenty of them, and I will concede that even the Minimally Supersymmetric Standard Model (MSSM) would add even more.

The Standard Model has:

3 gauge-field coupling constants
1 electroweak-breaking mass scale (makes W and Z massive)
9 elementary-fermion masses
4 quark-state mixing angles, including a CP-violating phase (in weak interactions, the up-like quark mass states do not exactly map onto the down-like ones)

A grand total of 17 constants.

Massive neutrinos would yield 3 more masses and 4 more mixing angles, increasing the total to 24 constants.

And, of course, the MSSM adds more, like multiple Higgs mass terms.

Including gravity would yield yet another constant, the Planck mass, but the other masses could be assumed to be relative to it.

Shadowy Man

Well, if you get the chance, I do recommed reading the Sanders & McGaugh paper. Though it may be filled with a bit of Astronomy jargon, you should be able to get the idea of most of it.

It's not just that it explains flat rotation curves, it also naturally explains the Tully-Fisher and Faber-Jackson relations, and a couple of other interesting observed relationships.

And like I said, even if MOND is wrong, the fact that it works as well as it does is telling you something very interesting about the distribution of dark matter in galaxies.