Download 2. Velocity dispersions of galaxies

Finsler Geometry
vs.
Dark Matter and Dark Energy Hypothesis
CHANG Zhe
Institute of High Energy Physics
Chinese Academy of Sciences
17/11/2009
at ITP
I. Experimental observations
1. Galactic rotation curves
2. Velocity dispersions of galaxies
3. Missing matter in clusters of galaxies
4. Large scale structure formation
5. Accelerated expanding Universe
6. Pioneer anomaly
7. Anomalous increase of the Astronomical
Unit
8. Anomalous orbital-energy changes during
spacecraft flyby of earth
9 ……
1. Galactic rotation curves
In the late 1960s and early 1970s
V. Rubin
from Carnegie Institution of Washington
presented that most stars in spiral galaxies orbit
at roughly the same speed.
Rotation curve of a typical spiral galaxy:
predicted (A) and observed (B).
2. Velocity dispersions of galaxies
Rubin's pioneering work has stood the test
of time.
Measurements of velocity curves in spiral
galaxies were soon followed up with
velocity dispersions of elliptical galaxies.
While sometimes appearing with lower
mass-to-light ratios, measurements of
ellipticals still indicate a relatively high dark
matter content.
3. Missing matter in clusters of
galaxies
X-ray measurements of hot intracluster gas
correspond closely to Zwicky's observations
of mass-to-light ratios for large clusters of
nearly 10 to 1. Many of the experiments of
the Chandra X-ray Observatory use this
technique to independently determine the
mass of clusters.
Strong gravitational lensing as observed by the Hubble Space Telescope
in Abell 1689 indicates the presence of dark matter - Enlarge the image
to see the lensing arcs.
4. Large scale structure formation
Observations suggest that structure formation
in the universe proceeds hierarchically, with
the smallest structures collapsing first and
followed by galaxies and then clusters of
galaxies. As the structures collapse in the
evolving universe, they begin to "light up"
as the baryonic matter heats up through
gravitational contraction and the object
approaches hydrostatic pressure balance.
5. Pioneer anomaly
Pioneer 10 (also called Pioneer F) was the first
spacecraft to travel through the asteroid belt,
which it entered on July 15, 1972, and to
make direct observations of Jupiter, which it
passed by on December 3, 1973. It was
launched from Cape Canaveral Air Force
Station's Launch Complex 36A on March 3,
1972 at 01:49:00 UTC. Pioneer 10 is heading
in the direction of Aldebaran, located in
Taurus. By some definitions, Pioneer 10 has
become the first artificial object to leave the
solar system. It is the first human-built object
to have been set upon a trajectory leading out
of the solar system.
– The Pioneer anomaly is the observed deviation from
predicted trajectories and velocities of various
unmanned spacecraft visiting the outer solar system,
most notably Pioneer 10 and Pioneer 11.
– Both Pioneer spacecraft are escaping from the solar
system, and are slowing down under the influence of
the Sun's gravity. Upon very close examination,
however, they are slowing down slightly more than
expected.
6. Anomalous increase of the Astronomical unit
Transits of Venus across the face of the Sun were for long the
best method of measuring the astronomical unit
The latest planetary ephemerides gave the
accurate value of AU
Reports from Krasingky and Brumberg
Celest. Mech. Dyn. Astrn. 90, 267 (2004).
Standish
Proc. IAU Colloq. 196, 163 (2005)
7. Accelerated expanding Universe
• Cosmology is the scientific study of the large scale
properties of the Universe as a whole.
• It endeavors to use the scientific method to
understand the origin, evolution and ultimate fate
of the entire Universe.
• Cosmology involves the formation of theories or
hypotheses about the universe which make
specific predictions for phenomena that can be
tested with observations.
• Depending on the outcome of the observations, the
theories will need to be abandoned, revised or
extended to accommodate the data.
Ricci tensor
By making use of the energy-momentum tensor, the Einstein
equation reads
Time-time component:
Space-space components:
The space-time components give 0=0
Friedmann equation
II.Finsler geometry
In 1854 Riemann saw the difference between the
quadratic differential form--Riemannian
geometry and the general case.
The study of the metric which is the Fourth root of
a quartic differential form is quite time-consuming and does not throw new light to the
problem." Happily, interest in the general case
was revived in 1918 by Paul Finsler's thesis,
written under the direction of Caratheodory.
Mathematical Problems
Lecture delivered before the International
Congress of Mathematicians at Paris in 1900
By Professor David Hilbert
• 4. Problem of the straight line as the
shortest distance between two points
• 23. Further development of the methods
of the calculus of variations
1926, L. Berwald: Berwald connection Torsion
free: yes g-compatibility: no
1934, E. Cartan: Cartan connection
Torsion free: no g-compatibility: yes
1948, S. S. Chern: Chern connection
Torsion free: yes g-compatibility: no
Chern connection differs from that of Berwald's
by an À term
Finsler structure of M
.
with the following properties:
(i) Regularity: F is C on the entire slit tangent bundle TM\ 0
(ii) Positive homogeneity : F(x,  y)=  F(x,y), for all  >0
(iii) Strong convexity: the Hessian matrix
Is positive-definite at every point of TM\0
The symmetric Cartan tensor
Cartan tensor Aijk=0 if and only if gij has no ydependence
A measurement of deviation from Riemannian
Manifold
Euler's theorem on homogenous function gives
Where li=yi/F
1. Chern connection
transform like
The nonlinear connection Nij on TM\0
where ijk is the formal Christoffel symbols of the second kind
Chern Theorem guarantees the uniqueness of
Chern connection.
S. S. Chern, Sci. Rep. Nat. Tsing Hua Univ. Ser. A 5, 95 (1948);
or Selected Papers, vol. II, 194, Springer 1989.
Torsion freeness
Almost g-compatibility
Torsion freeness is equivalent to the absence of
dyi terms in ij
together with the symmetry
Almost g-compatibility implies that
where
2.Curvature
The curvature 2-forms of Chern connection are
The expressionof ijin terms of the natural basis
is of the form
where R, P and Q are the hh-, hv-, vv-curvature
tensors of the Chern connection, respectively.
III.Gravity and large scale structure
The tangent spaces (TxM, Fx) of an arbitrary Finsler
manifolds typically not isometric to each other.
Given a Berwald space, all its tangent spaces are
linearly isometric to a common Minkowski space
A Finsler structure F is said to be of Berwald type if
the Chern connection coefficients ijk in natural
coordinates have no y dependence. A direct
proposition on Berwald space is that hv--part of
the Chern curvature vanishes identically
X. Li and Z. Chang, Toward a Gravitation Theory in Berwald-Finsler Space ,gr-qc/0711.1934.
Gravitational field equation on Berwald space
Z. Chang and X. Li, Modified Newton’s gravity in Finsler space as a
possible alternative to dark matter hypothesis, Phys. Lett. B668,
453(2008).
To get a modified Newton's gravity, we consider a particle moving
slowly in a week stationary gravitational field. Suppose that the metric is
close to the locally Minkowskian metric
A modified Newton's gravity is obtained as the weak field
approximation of the Einstein's equation
Limit the metric to be the form
a0is the deformation parameter of Finsler geometry
The deformation of Finsler space should have cosmological
significance.
One wishes naturally the deformation parameter relates
with the cosmological constant ,
The geometrical factor of the density of baryons
In the zero limit of the deformation parameter, familiar results
on Riemann geometry are recovered
The acceleration a of a particle in spiral galaxiesis
M. Milgrom, Astrophys. J. 270, 365 (1983).
M. Milgrom, The MOND paradigm, astro-ph/0801.3133.
G. Gentile, MOND and the universal rotation curve: similar
phenomenologies, astro-ph/0805.1731
The MOND
Universal Rotation Curves
IV.Possible model of accelerated
expanding Universe
Z. Chang and X. Li,
Modified Friedmann model in Randers-Finsler space of
approximate Berwald type as a possible alternative to dark
energy hypothesis, Phys. Lett. B676 (2009) 173,
arXiv 0901.1023
Robertson-Walker metric
satisfies the requirements of
homogeneity, isotropy and closure
• Ricci tensor
By making use of the energy-momentum tensor, the Einstein equation reads
Time-time component:
Space-space components:
The space-time components give 0=0
Friedmann equation
The Randers-Finsler metric
Let
to be the Robertson-Walker type
The Friedmann equation
Omitting the O(b2) term and combining with the
the space-space component of the field
equations, we obtain
One can see clearly that the accelerated
expanding universe is guaranteed by the
constraint
So that the complete constraint on Randers-Finsler
structure to support accelerated expanding
universe is
It means that a negative provides an
effective repulsive force in the course of
universe expanding.
V. Conlusions
A possible unified scenario for large structure
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Secular increase of astronomical unit
Pioneer anomaly
Galactic rotation curves
Missing matter in clusters of galaxies
Large scale structure formation
Accelerated expanding Universe
Finsler geometry may really supply a
framework of astronomy and cosmology
without invoking dark
matter and dark energy hypothesis.
Thanks for your attention!