Cosmological Models with
No Big Bang
許文郁
Wun-Yi Shu
Institute of Statistics
National Tsing Hua University
Presentation at
Institute of Physics
National Chiao Tung University
2011/04/07
In the late 1990s, observations of Type Ia supernovae
led to the astounding discovery that the universe is
expanding at an accelerating rate.
.
.
The explanation of this anomalous acceleration has
been one of the greatest challenges of theoretical
physics since that discovery. The current mainstream
explanation of the accelerating expansion of the
universe is dark energy — a mysterious force so named
because researchers have never detected such thing. In
general relativity, dark energy is represented as a
cosmological constant.
.
We propose cosmological models that can explain the
cosmic acceleration via the geometric structure of
space-time, without introducing a cosmological
constant into the standard field equation, negating the
necessity for the existence of dark energy.
.
There are four distinguishing features of these models:
1) the speed of light and the gravitational “constant” are not
constant, but varies with the evolution of the universe,
2) time has no beginning and no end; i.e., there is neither a big
bang nor a big crunch singularity,
3) the spatial section of the universe is a 3-sphere, ruling out the
possibility of a flat or hyperboloid geometry, and
4) the universe experiences phases of both acceleration and
deceleration.
Outline
•
•
•
•
•
Geometry
Cosmological Models
Dynamics of the Universe
Test of the Models
Conclusion
1. Geometry
Paraboloid
2
2
2
2
2
3
(
x
,
y
,
x

y
)
:
(
x,
y
)

R
,
x

y

1

R


Geometry of Paraboloid
The distance :
On R2, (x, y) → (x+dx, y+dy)
On R3, (x, y, x2+y2) → ( x+dx, y+dy, (x+dx)2 +(y+dy)2 ) .
distance  1  4 x
2
  dx   8 xy  dxdy  1  4 y   dy  
2
2
2
2
1

4
x
4 xy 

2
  x, y   R , let  gij  x, y   
.
2
 4 xy 1  4 y 
1/2

 dx  
distance   dx, dy   g  x, y     
 dy  

ij
1/ 2
Geometry of Paraboloid
D 2    x, y   R 2 : x 2  y 2  1 
1  4 x
  x, y   D , let  gij  x, y   
 4 xy
2
2
4 xy 
.
2
1 4 y 
2
D
 ; gij ( x, y)  : Paraboloid
Geometry of sphere
D    x, y   R : x  y  1 
2
2
2
2
 1 y
2
2

1

x

y
2
  x, y   D , let gij  x, y   
 xy
 1  x 2  y 2
2
2
D
 ; gij ( x, y)  : Sphere

2
2
1 x  y 
.
2
1 x 
2
2
1  x  y 
xy
Statistical Manifold
Statistical Model:  f  x ;  :    R k 
l  x ;   ln f  x ; 
i l  x ;  

l  x ; 
i

Fisher information matrix
g    E  l  X ;   l  X ; 
ij

i
j
M   ; gij    : Statistical manifold
The geometric properties (e.g. curvature) of M play
important role in statistical inference.
Statistical Manifold of Normal Distributions
 N   , 
: - <  <  ,  > 0 
f  x ; ,  
1
2 
e
 ( x   ) 2 / 2 2
l  x ;  ,    ln f  x ;  ,  
1l  x ;  ,      l  x ;  ,  
 2l  x ;  ,     l  x ;  ,  
Fisher information matrix :
1 2
  ,    
 0
0 

2 2
 
, - < x < 
Hyperbolic Geometry
H    x, y   R : -   x   , y  0 
2
2
1/
y
0 

  x, y   H , let  gij  x, y    
.
2
 0 1/ y 
 H ; g ( x, y) 
ij
: Hyperbolic Geometry
 3,3
 0,3
 0, 2 
 2, 2 
 0,1
Equal Distance
1 2
y

 x, y   
 0
0 

1 2
y 
Hyperbolic Geometry
The Earth
The world map
Arc length in H
dt
a
t
 x t  , y t 
b
 1
2
y
t


 x t  , y t   
 0




1
2
y t  
0
 x t  dt, y t  dt 
Arc length in H
dt
a

 x t  , y t 
t
 x t  dt, y t  dt 
b
 1
2
 y t 
x  t  dt , y  t  dt 
 0


2
2








 x  t  dt
x
t
y
t




  
 
   dt 2

 y t dt   y  t    y  t   
1
2
 
 
y  t       
0
2
b
arc length  
a
2
 x(t ) 
 y (t ) 

 
 dt
 y (t ) 
 y (t ) 




Straight lines in H
Straight lines in H
Hyperbolic Geometry
1. Any two points can be joined by a straight line.
2. Any straight line segment can be extended
indefinitely in a straight line.
3. Given any straight line segment, a circle can be
drawn having the segment as radius and one
endpoint as center.
4. All right angles are congruent.
5. Through a point not on a given straight line, more than
one lines can be drawn that never meet the given line.
Through a point not on a given straight line,
More than one lines can be drawn that never
meet the given line.
2. Cosmological models
A cosmological model is defined by specifying:
•
the spacetime geometry determined by a
metric g ab ,
•
the mass-energy distribution of the universe described in
terms of a stress-energy-momentum tensor Tab , and
•
the interaction of the geometry and the mass-energy, which
is depicted through a field equation.
2.1 Geometry of Space-time
Henri-Emile-Benoit Matisse
1869/12/31~1954/11/03
Woman with a hat. 1905 (81×60cm). Private collection.
Blue Nude IV. 1952. The Matisse Museum, Nice.
Le Bateau. 1953. The Museum of Modern Art, NY.
The Museum of Modern Art hung the print upside-down for 47 days in 1961.
In this period of time 11,600 people has passed through the gallery.
The longest period of time for which a modern
painting has hung upside down in a public gallery
unnoticed is 47 days. This occurred to Le Bateau
by Matisse in the Museum of Modern Art, New
York City. In this time 11,600 people has passed
through the gallery.
.
— The Guinness Book of Records,
Guinness Superlatives, Ltd..
A simple to ask,
but hard to answer question:
What are Space and Time ?
The progress of science can be measured by
revolutions that produce new answers to it.
Newton’s View of Space and Time
Newton’s View of Space and Time
Absolute space, in its own nature, without
relation to anything external, remains always
similar and immovable,
,
Absolute, true, and mathematical time, of
itself, and from its own nature, flows equably
without relation to anything external.
.
[From the scholium in the Principia]
Einstein’s View of Space and Time
Space is nothing apart from the things that
exist; it is only an aspect of the relationships that
hold between things.
.
Time has no absolute meaning. There is no
time apart from change. Time is described only in
terms of change in the network of relationships
that describes space. Time is nothing but a
measure of change – it has no other meaning.
.
The spacetime in the neighborhood
of the sun
ct
Geodesic curves → Geometry
Einstein’s Field Equation
1
R ab  R
2
g ab  8
G
c
4
Tab
Rab : Ricci curvature
R : scalar curvature
Space-time tells matter how to move,
matter tells space-time how to curve.
— John A. Wheeler
Spacetime geometry in the
neighborhood of the sun
g  (t , , , )  
  2M 
  1 







0


0


0

0
 2M 
1 




0
0
0
-1
0
2
0

0




0


0

 2 sin 2  
Spacetime geometry in the
neighborhood of the sun
2.1.1 Geometry of the Universe
Cosmological Principle:
On the large scales, the universe is assumed to be homogeneous and isotropic.
Expressed in the synchronous time coordinate and co-moving spatial Spherical/
hyperbolic coordinates (t,  ,  ,  ) , the line element of the spacetime metric takes
the form:


d 2  sin 2  d 2  sin 2  d 2

2
2
2
2
2
2
2
2
2
ds  c dt  a (t ) 
d   d  sin  d

2
2
2
2
2
d


sinh

d


sin

d






,

where the three options listed to the right of the left bracket correspond to the
three possible spatial geometries: a 3-sphere, 3-dimensional flat space, and a
3-dimensional hyperboloid, respectively.
.
The spherical coordinate
S3    x, y, z , w   R 4 : x 2  y 2  z 2  w2  1 
Spherical coordinate : ( ,  ,  )
d
2

 d 2  sin 2  d 2  sin 2  d 2

2.1.1 Geometry of the Universe with
Spatial Geometry S3
g  (t , , , )  
  c2

 0
 0

 0
0
a 2 (t )
0
0


0
0


a 2 (t ) sin 2 ( )
0

0
a 2 (t ) sin 2 ( ) sin 2  
0
0
2.1.1 Geometry of the Universe with
Spatial Geometry R3
g  (t , , , )  
  c2

 0
 0

 0
0
0
a 2 (t )
0
0
a 2 (t )  2
0
0


0


0

a 2 (t )  2 sin 2  
0
2.1.1 Geometry of the Universe with
Spatial Geometry Hyperboloid
g  (t , , , )  
  c2

 0
 0

 0
0
a 2 (t )
0
0


0
0


a 2 (t ) sinh 2 ( )
0

0
a 2 (t ) sinh 2 ( ) sin 2  
0
0
2.1.2 Geometry of the Universe with
Varying Speed of Light
We view the speed of light as simply a conversion factor between
time and space in spacetime. It is a Nature’s manifestation of the
structure of spacetime geometry. Since the universe is expanding, we
speculate that the conversion factor somehow varies in accordance
with the evolution of the universe, hence the speed of light varies with
cosmic time. Denoting the speed of light as a function of cosmic time
by c(t), we modify the metric as:
.


d 2  sin 2  d 2  sin 2  d 2

2
2
2
2
2
ds 2  c 2 (t )dt 2  a 2 (t ) 
d   d  sin  d

2
2
2
2
2
d


sinh

d


sin

d







2.1.2 Geometry of the Universe with
Varying Speed of Light ( S3 )
g  (t , , , )  
 c 2 (t )

 0
 0

 0
0
a 2 (t )
0
0


0
0


a 2 (t ) sin 2 ( )
0

0
a 2 (t ) sin 2 ( ) sin 2  
0
0
2.1.2 Geometry of the Universe with
Varying Speed of Light ( R3 )
g  (t , , , )  
 c 2 (t )

 0
 0

 0
0
0
a 2 (t )
0
0
a 2 (t )  2
0
0


0


0

a 2 (t )  2 sin 2  
0
2.1.2 Geometry of the Universe with
Varying Speed of Light (Hyperboloid)
g  (t , , , )  
 c 2 (t )

 0
 0

 0
0
a 2 (t )
0
0


0
0


a 2 (t )sinh 2 ( )
0

0
a 2 (t )sinh 2 ( ) sin 2  
0
0
2.2 The Stress-energy-momentum Tensor
The content of the universe is described in terms of a stressenergy-momentum tensor Tab . We shall take Tab to be the
general perfect fluid form:
T
ab
P a b

    2  u u  Pg ab
c 

ua : the 4-velocity of the cosmological fluid.
.
ρ : the proper average mass density.
.
P : the pressure as measured in the instantaneous rest .
frame.
.
2.2 The Stress-energy-momentum Tensor
The components of Tab :
  (t )c4 (t )

0

T   
0

 0


2
P(t )a (t )
0
0


0
P(t )a2 (t )sin 2 
0

0
0
P(t )a2 (t )sin 2  sin 2  
0
0
0
2.3 Einstein’s Field Equation
1
R ab  R
2
g ab  8
G
c
4
Tab
Rab : Ricci curvature
R : scalar curvature
Space-time tells matter how to move,
matter tells space-time how to curve.
— John A. Wheeler
2.3 Einstein’s Field Equation with
Cosmological Constant
R ab 
1
R ab 
1
R
2
2
R
g ab  g ab  8
g ab  8
G
c
4
G
c
4
Tab
Tab  g ab
Cosmological constant can be viewed physically as the
vacuum energy.
.
. .
Much later, when I was discussing cosmological problems
with Einstein, he remarked that the introduction of the
cosmological term was the biggest blunder of his life.
.
—
George Gamow, My World Line, 1970.
2.3.1 New Field Equation
R ab 
1
2
R
G
g ab  8
c
4
Tab ,
G (t )
 constant
2
c (t )
R ab 
1
2
Rg ab  8
1
c2
Tab  8 T*ab
Geometry of Spcetime
 c 2 (t )

0

 0

 0
0


0
0


a2 (t ) sin 2 ( )
0

2
2
2
0
a (t ) sin ( ) sin  
0
a 2 (t )
0
0
0
The Stress-energy-momentum Tensor
  (t )c4 (t )

0

2
c (t ) 
0

 0
0
0
P(t )a 2 (t )
0
0
P(t )a 2 (t )sin 2 
0
0


0


0

2
2
2
P(t )a (t )sin  sin  
0
2.4 The Equation of Motion
R ab 
1
2
Rg ab  8 T*ab
a T* ab  0

P(t )  a(t )
 (t )  3   (t )  2 
0
c (t )  a(t )


d
  (t )a 3 (t )   0,
dt
 (t )a3 (t )  constant ,
3. Dynamic of the Universe
R ab 


1
2
Rg ab  8 T*ab

a(t )  a(t ) c(t )   4   (t )  3 P(t )  c2 (t ) .
a(t ) a(t ) c(t ) 3 
c2 (t ) 
2


a
(
t
)

  2M  k ,
 c(t ) 
a(t )



M  4 (t )a3 (t ) / 3
3.1 The Conversion between
Time and Space
3.1.1 The speed of light
We view the speed of light as simply a
conversion factor between time and space in
spacetime. As gravitation, it is a Nature’s
manifestation of the structure of the spacetime
geometry.
dt
.
c(t )dt .
3.1.2 The cosmic density
When converting the magnitude of increment in time, dt,
into that in space, Nature needs a universal standard to
refer to. Noting that the concept of time arises from the
observation that the distribution of matter contained in the

universe is dynamic and the rate of change,  (t ) , of the
cosmic density is the very quantity that manifests the
dynamicity of a homogeneous universe. We postulate that
it is the standard taken by Nature. The cosmic density
plays the role of ultimate clock in a homogeneous
universe.
.
3.1.2 The cosmological density (conti.)
Accordingly, when being converted into that in space,
the magnitude of increment in time, dt, is normalized

with  (t ) . The conversion between time and space
can then be expressed as:
.
dt



|

(
t
)
|
 0



dt
3.2 The relationship between
the speed of light and space
dt



|

(
t
)
|
 0



dt , dt

c(t )dt .
 (t )a3 (t )  constant ,
c(t )  1 |  (t ) | ,

 (t )  a4(t )
a (t )

c(t )  
.


4
a
(
t
)

,



|
a
(
t
)
|


3.3 Dynamic of the Universe
2
 

a
(
t
)

  2M  k ,
 c(t ) 
a(t )


c(t )  


a
(
t
)


2
 a (t ) 




2


4
a
(
t
)

,



|
a
(
t
)
|


4
2M

 k.
a(t )
3.4 Dynamic of the Universe with
spatially 3-sphere geometry (k=1)
a(t ) 
2M
1  t /  
,   t   ,
4/ 3
where   2 3 1/ 2 M .
c (t ) 
(8 M / 3 )
1  (t /  )
4/3

2
 t / 
1/ 3
 1

 4 M  (t ) 
 1
  (t ) 

1/ 2
2

2
1/ 4
,  (t)  a(t ) / 2M
Figure 1 ︳The evolution of the 3-sphere universe.
The hyper-radius of the universe, a , can never reach zero. The universe
is accelerating in the epoch when γ < 7/8 and is decelerating when when
γ > 7/8 .
Figure 2｜The evolution of the flat/ hyperboloid universe.
The dynamics of two versions of the universe composed of pressure-free
dust: with spatially flat geometry, and with spatially hyperboloid geometry.
Figure 3 ︳The evolution of the universe. Time development of
the universe according to the Friedmann model.
3.5 The Planck’s “Constant”
c, wavelength λ , and frequency ν are
related by c =λν, a varying c could be interpreted in different ways.
We assume that a varying c arises from a varying λ with ν kept
Since the speed of light
constant. We further assume that the relation between the energy E
of a photon and the wavelength λ of its associated electromagnetic
wave is given by equation E(t) =η/λ(t) , where η is a constant that
does not vary over cosmic time. Consequently, from relation
λ(t)=c(t)/ν, it follows that E(t)=[η/c(t)]ν≡ h(t)ν . Therefore, the so
called Planck’s constant h actually varies with the evolution of the
universe.
.
4. Test of the Model
Suppose that a photon of frequency (wavelength)
ν e (λ e ) is emitted at cosmic time t e by an
isotropic observer E with fixed spatial coordinates
(ΨE , θE , ψE) .
Suppose this photon is observed at time to by
another isotropic observer O at fixed co-moving
coordinates. We may take O to be at the origin of
our spatial coordinate system. Let νo (λo) be the
frequency (wavelength) measured by this second
observer.
.
4.1 The Cosmological Redshift
o c(to ) / o c(to )a(to )
1 z 


.
e c(te ) / e
c(te )a(te )

1  (te /  ) 
 te /  


1 z 
.
3
1/
3
4/3


1

(
t
/

)
o

  to /  
4/3
3
1/ 3
 (t )  a(t ) 2M ,  e   (te ), and  o   (to ) ,
1/  e  1/  e  1
3
1/ 4
 1  z 1/  o  1/  o  1
3
1/ 4
.
4.2 The Theoretical Predictions
ds 2  c 2 (t )dt 2  a 2 (t )d 2  sin 2   d 2  sin 2  d 2  .
From the fact that ds = dθ = dψ= 0 along the photon
path, we have:
.
a
2
(t )d  c (t )dt .
2
 E 
2
to
2
c (t )
 a (t ) dt
te
.
4.2 The Theoretical Predictions (conti.)
E 
to
c (t )
 a (t ) dt
.
te
1/ 2
1/ 2
1
1

 2 tan 1/  e 1  tan 1/  o 1 


dP  z   a(to ) E
dL  z   a(to ) 1  z  sin E
 2M (to ) 1 z  sin E
4.2 The Theoretical Predictions (conti.)
mBeffective
 MB  5log dL ( z)  25 ,
dL ( z)  2M  (to ) 1  z  sin E
mBeffective ( z)    5log  o 1+z  sin  z,  o 
 m o , ( z ) ,
  MB  5log 2M  25
1/ 2
1/ 2
1
1

 z,  o   2 tan 1  e ( z) 1  tan 1  o 1  .
4.3 Data Fitting
Observations of Type Ia supernova :


z*i , mi* , i  1, 2, ... , n .
zi*  zi   z ,i , mi*  m 0 ,  ( zi )   m,i ;
i  1, 2, ... , n .
 * 2  m*  m ( z ) 2 
zi  z
i
 0 ,
2
*
*



d  zi , mi , m 0 ,  ( )   min 
 
  .
 
 
z  0  

z
m
i
 
 i  


n


2
*
*

,
d
z
,
m
,
m
(
)

i
i

,

0


0 0 1, 0
i 1
( o* ,  * )  arg min
Hubble
Diagram
Figure 4 ｜Hubble diagram Hubble diagram for: (a) 62 low-redshift Type Ia supernovae,
18 from the Calán/Tololo Supernova Survey (Data Set 1) and 44 assembled by the
Supernova Legacy Survey (Data Set 3), and (b) 115 high-redshift Type Ia supernovae, 42
discovered by the Supernova Cosmology Project (Data Set 2) and 73 by the Supernova
Legacy Survey (Data Set 4). The solid curve is the theoretical value m  o ,  ( z) as predicted
by our model with parameters  o  0.001 and   49.321 .
.
5. Conclusion
Essentially, this work is a theory about how the
magnitudes of the three basic physical dimensions:
time, length, and mass are converted into each
other, or equivalently, a theory about how the
distribution of mass-energy and the geometry of
spacetime interact. The theory resolves problems
in cosmology, such as those of the big bang, dark
energy, and flatness, in one fell stroke by
postulating that.
.
 4 
a
(t ) 

c(t )   
and


|
a
(
t
)
|



G (t )
 2  constant
c (t )
Physical Constants
In Einstein’s theory of relativity, there are two
constants, namely, c and G , while in ours, the two
constants are κ, the factor relating to the conversion
between time and space, and τ , the conversion
factor between mass and space. These two constants,
κ and τ , together with η , the constant relating the
energy of a photon and the wavelength of its
associated electromagnetic wave, can be used to
define the natural units of measurement for the three
basic physical dimensions. Using dimensional
analysis, we obtain:
.
The Natural Units
 4

a
(
t
)
 and
c(t )    


|
a
(
t
)
|



G (t )
 constant
2
c (t )
the natural unit of mass 
4
The natural unit of time  4
The natural unit of length 

 3
1

4


It seems that I opened Pandora’s Box,
the debates have only just begun!