# [extropy-chat] WMAP Results - Cosmology Makes Sense

Amara Graps amara at amara.com
Tue Mar 21 06:03:28 UTC 2006

Damien Sullivan phoenix at ugcs.caltech.edu
>Aww, and I'd just read a SciAmer article from last year about
>deviations from the predicted curves.

If you are referring to Cooperstock's and Tieu's work, they do not
have a refereed paper yet (many are waiting for that), and none of
the cosmologists are putting weight (:-)) on it. The authors have
some assumptions in their math model that are wrong, for one thing.

I'll post what I wrote to another list four months ago on that
dark matter topic. The beginning is a repeat. I should say up
front that this is 10^{35} scales away from my own field of
expertise, all of what I know is what I read (when I have time).
I follow up with a bunch of references from ArXiV (I haven't read
them all).

The issue is a paper that claims that no exotic dark matter is needed
to fit the observed rotation curve to a reasonable ordinary matter
distribution.

In order to distinguish between kinds of dark matter:

=================================================================

CONTENTS of the UNIVERSE

Type       Likely Composition       Main Evidence     Omega Contrib
-----------------------------------------------------------------------

Visible      Ordinary matter (protons,     telescopic         0.01
Matter       neutrons) that forms          observations
stars, dust and gas

Baryonic     Ordinary matter that is       Big Bang           0.05
Dark         too dim to see (brown or      nucleosynthesis
Matter       black dwarfs, massive         and observed
compact halo objects:         deuterium abundance
MACHOS)

Nonbaryonic  Very light "exotic"           Gravity of visible  0.3
Dark Matter  particles such as             matter is insuffi-
axions, neutrinos w/mass      cient to account
or weakly interacting         for orbital speeds
massive particles:            within galaxies
WIMPS                         and galaxies within
clusters

Cosmological Cosmological Constant         Microwave back-     0.?
Dark Energy  (energy of empty space)       ground suggests
cosmos is flat but
there is not enough
baryonic/nonbaryonic
matter to make it so.

from [1]
================================================================

In 1932, Oort found evidence for extra matter within our galaxy, and
then one year later Zwicky inferred a large density of matter within
clusters of galaxies. [2]

The conceptual idea is to look at the motions of various kinds of
astronomical objects, and assess whether the visible material is
sufficient to provide the inferred gravitational force. If it is not,
the excess attraction must be due to extra invisible material.

Since the 1970s there has been a discrepancy between the observed
rotational velocities of stars in the outer regions of spiral galaxies
and the orbit velocities that one would expect according to Newton's
Laws from the distribution of visible stars in the galaxy. This
discrepancy indicates that there should be much more matter in the
outer parts of the spiral galaxies. [3]

In particular, mass is widely distributed in a galaxy, so then the
rotation rates of gas and stars should increase with distance from the
center until most of the galaxy's mass is inside their orbit, then
slow further out. From Kepler:

2
v     G M(R)
---- = ------
R        2
R

G M(R)
v  = sqrt(------)
R

At large distances, enclosing most of the visible part of the galaxy,
we expect that the rotational velocity to drop off as the square root
of R. It doesn't. Instead, galactic rotation rates never drop, the
velocity stays roughly constant. This is evidence that unseen matter
well beyond the visible disk controls the stars' velocities.  The
figure that people seem to like that clearly shows this is the
rotation curve for spiral galaxy NGC 3198, in an article published by

The Cooperstock and Tieu paper claims that the present solutions to
the General Relativity equations for this problem for determining
velocities of stars in galaxies are not correct. General Relativity is
a difficult equation to solve, so various approximations are usually
invoked. The present solutions to the distribution of matter in
galaxies are linear approximations based on perturbation theory where
the leading terms capture the main characteristics of the problem, and
the following terms contribute very little, and so the remaining terms
which would only add unnecessary complexity and are truncated.
Cooperstock and Tieu claim that a perturbation theory solution is not
valid because the remaining terms are, instead, large contributors.

However, many disagree, and, according to Sean Carroll, a mathematical
physicist at the Enrico Fermi Institute, and Kavli Institute for
Cosmological Physics at the University of Chicago, approximate
solutions to Einstein's equation are close enough. He says:

"But the real reason why most astronomers and physicsts didn't
take the paper seriously is that it violates everything we know
about perturbation theory. In the galaxy, there are two
parameters that are very small - the gravitational potential is
about 10^{-6}, and the velocity of the stars (compared to the
speed of light) is about 10^{-3}. So it would be surprising
indeed if perturbation theory weren't doing a really good job in
this situation, even just including the first-order contribution.
The real reason why nobody paid much attention to Cooperstock and
Tieu is that they didn't even seem to recognize that this was a
problem, much less offer some proposed explanation as to why
perturbation theory was breaking down. Extraordinary claims
require extraordinary evidence, and we would need to be given a
compelling reason to think that our perturbative intuition was
failing before anyone would put a lot of effort into analyzing
this paper." [4]

So how were Cooperstock and Tieu solving the problem?

The essense of General Relativity is that 1) Spacetime is a curved
pseudo-Riemannian manifold with a specific metric and 2) There exists
a specific relationship between matter and the curvature of spacetime.
The manifold is four-dimensional and the metric gives a way of taking
the norm on the manifold, essentially containing all of the
information about the geometry of the manifold.  [5]

To model the galaxy, Cooperstock and Tieu considered a uniformly
rotating fluid without pressure and symmetric about its axis of
rotation in the context of General Relativity.  The exact Einstein
field solution to their galactic model contains a class of metrics
called the "van Stockum class of metrics" or the "van Sockum dust" [6]

According to some there are problems with their mathematical
approach. Sean Carroll says [5] :

"To be honest, there are a bunch of problems with this paper. For
example, equations (1) and (2) seem mutually inconsistent - they
have chosen one coordinate system in which to express the
spacetime metric, and another in which to express the spacetime
velocity of the particles in the galaxy. Ordinarily, you have to
pick one coordinate system and stick to it."

Some also find problems with this physical model. Mikolaj Korzynski of
Warsaw University says [7] that in their model, a gravitational field
is generated by a thin singular disk in addition to the galaxy matter,
so the model for this problem is unphysical to their stated goals.
This extra singular thin disk is moreover made of exotic matter,
according to another work by Vogt and Letelier [8].

So that is the status as I understand it now. After the references I
will list alot more references that I dug up on ArXiv that are
relevant, for all of you to follow up more (and me too, if I have
time). Cooperstock and Tieu's idea is certainly worthwhile, but I
don't know if it will pan out (they have not published a rebuttal
yet), and even if it does, there are other dark matter evidence than
galactic rotation curves.

Amara

REFERENCES

[1] "Cosmological Antigravity" by Lawrence M. Krauss in
Scientific American, Special Edition: The Once and Future Cosmos,
Volume 12, Number 2, 2002, pg. 33.

[2] Liddle, Andrew, _An Introduction to Modern Cosmology_, 1999,
Wiley, pg. 62-3.

[3] Hawking, Stephen, _The Universe in a Nutshell_ Bantam, 2001,
pg. 186-7.

[4]
http://cosmicvariance.com/2005/10/17/escape-from-the-clutches-of-the-dark-sector/

[5] "A No-Nonsense Introduction to General Relativity"
http://pancake.uchicago.edu/~carroll/notes/grtinypdf.pdf
Or see the the fullblown text:
http://pancake.uchicago.edu/~carroll/notes/

[6]
http://en.wikipedia.org/wiki/Fluid_solution
In general relativity, the van Stockum dust is an exact solution of
the Einstein field equation in which the gravitational field is
generated by dust particles (*) which are rotating about an axis of
cylindrical symmetry. Since the density of the dust is increasing with
distance from this axis, the solution is rather artificial, but as one
of the simplest known solutions in general relativity, it stands as a
pedagogically important example. This solution is named for Willem
Jacob van Stockum, who rediscovered it in 1937, independently of an
even earlier discovery by Cornelius Lanczos in 1924.

(* Note:  'dust particles' = fluid solution to General Relativity.)
In general relativity, a fluid solution is an exact solution of the
Einstein field equation in which the gravitational field is produced
entirely by the mass, momentum, and stress density of a fluid.

[7] Astrophysics, abstract: http://arxiv.org/abs/astro-ph/0508377
From: Mikolaj Korzynski
Date: Wed, 17 Aug 2005 17:27:48 GMT   (4kb)

Singular disk of matter in the Cooperstock and Tieu galaxy model
Authors: Mikolaj Korzynski

Recently a new model of galactic gravitational field, based on
ordinary General Relativity, has been proposed by Cooperstock and
Tieu in which no exotic dark matter is needed to fit the observed
rotation curve to a reasonable ordinary matter distribution. We
argue that in this model the gravitational field is generated not
only by the galaxy matter, but by a thin, singular disk as well.
The model should therefore be considered unphysical.

[8] Astrophysics, abstract: http://arxiv.org/abs/astro-ph/0510750
From: Patricio S. Letelier
Date: Wed, 26 Oct 2005 19:50:07 GMT   (3kb)

Presence of exotic matter in the Cooperstock and Tieu galaxy model
Authors: D. Vogt, P. S. Letelier

We analyze the presence of an additional singular thin disk
in the recent General Relativistic model of galactic
gravitational field proposed by Cooperstock and Tieu. The
physical variables of the disk's energy-momentum tensor are
calculated. We show that the disk is made of exotic matter,
either cosmic strings or struts with negative energy density.

======================================================================

Search:
http://xxx.lanl.gov/find/astro-ph/1/abs:+AND+dark+matter/0/1/0/past/0/1

"What is the Evidence for Dark Matter?" by J.A. SellwoodSellwood
http://arxiv.org/abs/ astro-ph/0401398 January 22, 2004

Abstract. Newtonian mechanics indicates that galaxies and galaxy clusters
are much more massive than we would have guessed from their luminosities,
with the discrepancy being generally attributed to dark matter
halos. An alternative hypothesis is that accelerations in very weak
gravitational
fields are larger than predicted by Newton's laws, and there is
no need for dark matter. Even though we do not currently have a satisfactory
theory associated with this rival hypothesis, we can ask whether
any observational tests could rule it out or prefer it over the dark matter
hypothesis. Current evidence suggests that neither hypothesis enjoys a
decisive advantage over the other. If dark matter turns out to be the correct
interpretation however, then theories of galaxy formation face some
quite severe fine-tuning problems.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0510123

From: Paolo Salucci [view email]
Date: Wed, 5 Oct 2005 14:00:34 GMT   (259kb)

The Distribution of Dark Matter in Spirals
Authors: Paolo Salucci
Comments: 6 pages, 5 Fig. Invited Talk at 21st IAP Colloquium." Mass
Profiles and Shapes of Cosmological Structures", 4-9 July 2005

In the past years a wealth of observations allowed to unravel the
structural properties of the Dark Matter Halos around spirals.
First, their rotation curves follow an Universal profile (URC)
that can be described in terms of an exponential thin stellar disk
and a dark halo with a constant density core, whose relative
importance increases with galaxy luminosity. Careful studies of
individual objects reveal that dark halos have a core, whose size
$r_0$ correlates with the central density $\rho_0$.
These properties are in serious discrepancy with the cuspy density
distribution predicted by N-body simulations in collisionless
$\Lambda$CDM Cosmology.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0506676

From: Paul Frampton [view email]
Date: Tue, 28 Jun 2005 17:59:05 GMT   (11kb)

Introduction to Dark Energy and Dark Matter
Authors: Paul H. Frampton
Comments: 9 pages. Talk at 40th Rencontre de Moriond, La Thuile,
Italy. March 5-12, 2005

In an introductory manner, the nature of dark energy is addressed,
how it is observed and what further tests are needed to
reconstruct its properties. Several theoretical approaches to dark
energy will be discussed. Finally, the dark matter, especially
WIMPs, is introduced.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0505266

From: Philip D. Mannheim [view email]
Date (v1): Thu, 12 May 2005 19:15:30 GMT   (150kb)
Date (revised v2): Mon, 1 Aug 2005 14:53:47 GMT   (155kb)

Alternatives to Dark Matter and Dark Energy
Authors: Philip D. Mannheim (University of Connecticut)
Comments: LaTeX, 87 pages, 3 figures. To appear in Progress in
Particle and Nuclear Physics, 2005. Final version, contains expanded
references and footnotes

We review the underpinnings of the standard Newton-Einstein theory
of gravity, and identify where it could possibly go wrong. In
particular, we discuss the logical independence from each other of
the general covariance principle, the equivalence principle and
the Einstein equations, and discuss how to constrain the matter
energy-momentum tensor which serves as the source of gravity. We
identify the a priori assumption of the validity of standard
gravity on all distance scales as the root cause of the dark
matter and dark energy problems, and discuss how the freedom
currently present in gravitational theory can enable us to
construct candidate alternatives to the standard theory in which
the dark matter and dark energy problems could then be resolved.
We identify three generic aspects of these alternate approaches:
that it is a universal acceleration scale which determines when a
luminous Newtonian expectation is to fail to fit data, that there
is a global cosmological effect on local galactic motions which
can replace galactic dark matter, and that to solve the
cosmological constant problem it is not necessary to quench the
cosmological constant itself, but only the amount by which it
gravitates.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0504512

From: Jerome Drexler [view email]
Date: Fri, 22 Apr 2005 22:00:17 GMT   (231kb)

Identifying Dark Matter Through the Constraints Imposed by Fourteen
Astronomically Based 'Cosmic Constituents'
Authors: Jerome Drexler (New Jersey Institute of Technology)

Mankind has not yet explained dark matter, the accelerating
expansion of the Universe, the 'knee' and 'ankle' of the cosmic
ray energy spectrum graph, the low star formation rates of low
surface brightness (LSB) dwarf galaxies, the ignition of hydrogen
fusion reactions in the first generation stars or the departing
locations of earthbound high-energy cosmic ray protons. A new
research hypothesis has been developed by the author based upon
finding astronomically based 'cosmic constituents' of the Universe
that may be created or influenced by or have a special
relationship with possible dark matter candidates. A list of 14
relevant and plausible 'cosmic constituents' of the Universe was
developed by the author, which was then used to establish a list
of constraints regarding the nature and characteristics of the
long-sought dark matter particles. A dark matter candidate was
then found that best conformed to the 14 constraints established
by the 'cosmic constituents.'

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0504422

From: David Merritt [view email]
Date: Tue, 19 Apr 2005 19:47:40 GMT   (42kb)

Dark Matter Dynamics and Indirect Detection
Authors: Gianfranco Bertone, David Merritt
Comments: 17 pages. Invited review for Modern Physics Letters A
Journal-ref: Mod.Phys.Lett. A20 (2005) 1021

Non-baryonic, or "dark," matter is believed to be a major
component of the total mass budget of the universe. We review the
candidates for particle dark matter and discuss the prospects for
direct detection (via interaction of dark matter particles with
laboratory detectors) and indirect detection (via observations of
the products of dark matter self-annihilations), focusing in
particular on the Galactic center, which is among the most
promising targets for indirect detection studies. The
gravitational potential at the Galactic center is dominated by
stars and by the supermassive black hole, and the dark matter
distribution is expected to evolve on sub-parsec scales due to
interaction with these components. We discuss the dominant
interaction mechanisms and show how they can be used to rule out
certain extreme models for the dark matter distribution, thus
increasing the information that can be gleaned from indirect
detection searches.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0504422

From: David Merritt [view email]
Date: Tue, 19 Apr 2005 19:47:40 GMT   (42kb)

Dark Matter Dynamics and Indirect Detection
Authors: Gianfranco Bertone, David Merritt
Comments: 17 pages. Invited review for Modern Physics Letters A
Journal-ref: Mod.Phys.Lett. A20 (2005) 1021

Non-baryonic, or "dark," matter is believed to be a major
component of the total mass budget of the universe. We review the
candidates for particle dark matter and discuss the prospects for
direct detection (via interaction of dark matter particles with
laboratory detectors) and indirect detection (via observations of
the products of dark matter self-annihilations), focusing in
particular on the Galactic center, which is among the most
promising targets for indirect detection studies. The
gravitational potential at the Galactic center is dominated by
stars and by the supermassive black hole, and the dark matter
distribution is expected to evolve on sub-parsec scales due to
interaction with these components. We discuss the dominant
interaction mechanisms and show how they can be used to rule out
certain extreme models for the dark matter distribution, thus
increasing the information that can be gleaned from indirect
detection searches.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0502118

From: Andrew R. Zentner [view email]
Date: Sun, 6 Feb 2005 15:24:46 GMT   (46kb)

Dark Matter Halos: Shapes, The Substructure Crisis, and Indirect Detection
Authors: Andrew R. Zentner (KICP, UChicago), Savvas M. Koushiappas
(ETH), Stelios Kazantzidis (Zurich, KICP, UChicago)
Comments: 7 Pages, 3 Figures, Review to appear in The Proceedings of
the Fifth International Workshop on the Identification of Dark Matter

In this proceeding, we briefly review three recent results. First,
we show that halos formed in simulations with gas cooling are
significantly rounder than halos formed in dissipationless
$N$-body simulations. The increase in principle axis ratios is
$\delta (c/a) ~ 0.2 - 0.4$ in the inner halo and remains
significant at large radii. Second, we discuss the CDM
substructure crisis and demonstrate the sensitivity of the crisis
to the spectrum of primordial density fluctuations on small
scales. Third, we assess the ability of experiments like VERITAS
and GLAST to detect $\gamma$-rays from neutralino dark matter
annihilation in dark subhalos about the MW.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0501231

From: Dirk Puetzfeld [view email]
Date: Wed, 12 Jan 2005 23:22:02 GMT   (276kb)

Prospects of Non-Riemannian Cosmology
Authors: Dirk Puetzfeld
Comments: 5 pages, 3 figures, 22nd Texas Symposium on Rel.
Astrophysics, Stanford University, December 2004

In this work we provide the motivation for considering
non-Riemannian models in cosmology. Non-Riemannian extensions of
general relativity theory have been studied for a long time. In
such theories the spacetime continuum is no longer described by
the metric alone but endowed with additional geometric quantities.
These new quantities can be coupled to the intrinsic properties of
matter in a very natural way and therefore provide a richer
gravitational theory, which might be necessary in view of the
recent cosmological evidence for dark matter and dark energy. In
this work we mainly focus on the concepts in metric-affine gravity
and point out their possible significance in the process of
cosmological model building.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0412297

From: Joseph Silk [view email]
Date: Mon, 13 Dec 2004 14:14:25 GMT   (23kb)

Dark Matter and Galaxy Formation: Challenges for the Next Decade
Authors: Joseph Silk
Comments: To be published in joint proceedings for Mitchell Symposium
on Observational Cosmology and Strings and Cosmology Conference,
College Station, April 2004, eds. R. Allen and C. Pope, AIP, New York,
and in proceedings for PASCOS04/NathFest, Boston, August 2004, eds. G.
Alverson and M. Vaughan, World Scientific, Singapore

The origin of the galaxies represents an important focus of
current cosmological research, both observational and theoretical.
Its resolution involves a comprehensive understanding of star
formation, galaxy dynamics, the cosmology of the very early
universe, and the nature of the dark matter. In this review, I
will focus on those aspects of dark matter that are relevant for
understanding galaxy formation, and describe the outlook for
detecting the most elusive component, non-baryonic dark matter.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0412195

From: John W. Moffat [view email]
Date (v1): Wed, 8 Dec 2004 22:20:21 GMT   (105kb)
Date (revised v2): Mon, 20 Dec 2004 22:02:55 GMT   (106kb)
Date (revised v3): Thu, 5 May 2005 16:01:46 GMT   (107kb)

Gravitational Theory, Galaxy Rotation Curves and Cosmology without Dark Matter
Authors: J. W. Moffat
and references. Corrections. To be published in Journal of Cosmology
and Astroparticle Physics (JCAP)
Journal-ref: JCAP 0505 (2005) 003

Einstein gravity coupled to a massive skew symmetric field
F_{\mu\nu\lambda} leads to an acceleration law that modifies the
Newtonian law of attraction between particles. We use a framework
of non-perturbative renormalization group equations as well as
observational input to characterize special renormalization group
trajectories to allow for the running of the effective
gravitational coupling G and the coupling of the skew field to
matter. The latter lead to an increase of Newton's constant at
large galactic and cosmological distances. For weak fields a fit
to the flat rotation curves of galaxies is obtained in terms of
the mass (mass-to-light ratio M/L) of galaxies. The fits assume
that the galaxies are not dominated by exotic dark matter and that
the effective gravitational constant G runs with distance scale.
The equations of motion for test particles yield predictions for
the solar system and the binary pulsar PSR 1913+16 that agree with
the observations. The gravitational lensing of clusters of
galaxies can be explained without exotic dark matter. An FLRW
cosmological model with an effective G=G(t) running with time can
lead to consistent fits to cosmological data without assuming the
existence of exotic cold dark matter.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0412059

From: Michael R. Merrifield [view email]
Date: Thu, 2 Dec 2004 14:43:09 GMT   (36kb)

Dark Matter on Galactic Scales (or the Lack Thereof)
Authors: M.R. Merrifield (University of Nottingham)
Comments: 10 pages, 5 figures. Invited review talk presented at
IDM2004 5th International Workshop on the Identification of Dark
Matter, Edinburgh, Scotland, September 2004

This paper presents a brief review of the evidence for dark matter
in the Universe on the scales of galaxies. In the interests of
critically and objectively testing the dark matter paradigm on
these scales, this evidence is weighed against that from the only
other game in town, modified Newtonian dynamics. The verdict is
not as clear cut as one might have hoped.

------------------------------

Astrophysics, abstract
http://arxiv.org/abs/astro-ph/0411503

From: Angele Sene [view email]
Date (v1): Wed, 17 Nov 2004 16:37:49 GMT   (719kb)
Date (revised v2): Tue, 18 Jan 2005 18:23:04 GMT   (721kb)
Date (revised v3): Mon, 28 Feb 2005 16:27:22 GMT   (721kb)

Dark Matter Direct Detection
Authors: Gabriel Chardin
Comments: 46 pages, 17 figures, to appear in "Cryogenic Particle
Detection", edited by Christian Enss, (Springer, Heidelberg, 2005) ;
one figure and two references modified ; typographical corrections

Solving the Dark Matter enigma represents one of the key
objectives of contemporary physics. Recent astrophysical and
cosmological measurements have unambiguously demonstrated that
ordinary matter contributes to less than 5 % of the energy budget
of our Universe, and that the nature of the remaining 95 % is
unknown. Weakly Interacting Massive Particles (WIMPs) represent
the best motivated candidate to fill the Dark Matter gap, and
direct detection Dark Matter experiments have recently reached
sensitivities allowing them to sample a first part of
supersymmetric models compatible with accelerator constraints.
Three cryogenic experiments currently provide the best
sensitivity, by nearly one order of magnitude, to WIMP
interactions. With systematic uncertainties far less severe than
other present techniques, the next generation of cryogenic
experiments promises two orders of magnitude increase in
sensitivity over the next few years. The present results,
perspectives and experimental strategies of the main direct
detection experiments are presented. Challenges met by future
ton-scale cryogenic experiments in deep underground sites, aiming
at testing most of the SUSY parameter space, are critically
discussed.

------------------------------

General Relativity and Quantum Cosmology, abstract
http://arxiv.org/abs/gr-qc/0411104

From: Jean-Paul Mbelek [view email]
Date: Mon, 22 Nov 2004 10:37:08 GMT   (67kb)

Modelling the rotational curves of spiral galaxies with a scalar field
Authors: J.P. Mbelek
Comments: Latex, 5 pages with 3 Postscript figures
Journal-ref: Astron.Astrophys. 424 (2004) 761-764
DOI: 10.1051/0004-6361:20040192

In a previous work (Mbelek 2001), we modelled the rotation curves
(RC) of spiral galaxies by including in the equation of motion of
the stars the dynamical terms from an external real
self-interacting scalar field, $\psi$, minimally coupled to
gravity and which respects the equivalence principle in the weak
fields and low velocity approximation. This model appeared to have
three free parameters : the turnover radius, $r_{0}$, the maximum
tangential velocity, $v_{\theta max} = v_{\theta}(r_{0})$, plus a
strictly positive integer, $n$. Here, we propose a new improved
version where the coupling of the $\psi$-field to dark matter is
emphasized at the expense of its self-interaction. This
reformulation presents the very advantageous possibility that the
same potential is used for all galaxies. Using at the same time a
quasi-isothermal dark matter density and the scalar field helps to
better fit the RC of spiral galaxies. In addition, new
correlations are established.

------------------------------
--

Amara Graps, PhD      www.amara.com
Istituto di Fisica dello Spazio Interplanetario (IFSI)
Istituto Nazionale di Astrofisica (INAF),
Roma, ITALIA    Amara.Graps at ifsi-roma.inaf.it