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					     Cosmology with General Relativity

Foundations
Special relativity

Shortest paths between two points (geodesics) can have
various shapes

Mass alters the shapes of geodesics.


The Metric: How to specify the geometry of
space
Let dl specify an incremental distance within some
coordinate system. The metric is the mathematical
description of dl.

2-D Euclidian space

     dl 2  dr 2  r  d  .
                            2



3-D Euclidian space-time

                                          
         ds 2  c 2 dt 2  dl 2  c 2 dt 2  dr 2  r  d  .
                                                           2
                                                               
Light travels along null distances in space-time, since for it,
and it alone,
                   dl  c  dt  ds  0 .

For any particle, on the other hand,

                     dl < c dt  ds > 0.

2-D Curved space (sphere)




                dl 2  d 2  RC Sin 2  d 2 ,
                               2




where

                                
                           
                                RC .


The coordinate  measures the distance from an observer at
location O on the surface. It is the analog of the comoving
coordinate – fixed forever by the location of each point on
the surface.

Define a new coordinate x such that
                                            
                            x  RC Sin
                                      R      .
                                              
                                       C     

Eliminate coordinate increment d by noting that

                                 d               
                dx  RC Cos
                           R     
                                   R   Cos
                                            R         d ,
                                                       
                            C      C       C        

so that

                               2                  2
            dx 2  Cos 2 
                         R     d  1  Sin 2 
                                                R     d ,
                                                       
                                      
                          C                    C   

and therefore

                                 dx 2
                       d  2

                                             .
                            1  Sin 2 
                                      R       
                                               
                                       C      

Now define the curvature parameter k:

                                k 1
                                       RC .
                                        2



Clearly, for spherical surfaces k > 0. Then

                                     x2
                 1  Sin 
                        2
                         R        1  2  1  kx2 ,
                                 
                          C           RC
and the incremental distance (the metric) can thus be written
as

                             dx 2
                      dl  2
                                     x 2 d 2 .
                           1  kx 2



The proper distance dlP is the length of a line segment that
subtends angle d at O. Evidently that distance corresponds
to dx = 0, since in this case

                  dlP  dl  xd  RC Sin  d ,

which is the length of a line segment perpendicular to .

4-D Curved space-time

                        dr02                                2
   ds  c dt  R(t )           r0 d   r0 sin   d  
     2    2   2        2                  2

                       1  kr0                               

which is the Robertson-Walker metric for spherical
cosmological models.
Newtonian models: k specifies the total energy of an
arbitrary mass shell in an expanding space.

GR models: k determines the geometry of space-time:

k < 0: negatively curved space (analog of a saddle)

k = 0: ordinary, flat Euclidian space

k > 0: positively curved space (analog of a spherical
surface)
The Scale Factor in GR
It all starts with the Einstein’s Field Equation:

                   1          8G
             R   g  R   2  T , where
                   2           c

R = the Ricci Tensor, contains info about velocities,
accelerations,

g = the Metric Tensor, specifies the local curvature of
space-time,

T = the Stress-Energy Tensor, contains info about
momentum, energy.


Einstein noticed that a more general form of the Field
Equation could be written as

                   1          8G
             R   g  R   2  T  g  .
                   2           c

Here  is an arbitrary constant that introduces additional
curvature. In Newtonian models it represents a kind of
“potential energy”.
Solutions:

Assume that space is homogeneous and isotropic. The field
equation then reduces to two equations for the scale factor,
one of which is

                 2   8G 2 1
                R        R  R 2  kc2 .
                        3    3

or

                   2        
                 R  8G 1 
                               R 2  kc2 ,
                 R 
                         3  3 
                 
                               
                                

which, with  = 0, is identical to the Newtonian result. We
recover equation 7) exactly!

Set  = 0 for now.
The Size of the Observable Universe
(the horizon distance, dh)
A DISTANCE DICTIONARY

Comoving coordinate (r0): A fixed coordinate assigned to
each particle (galaxy) in the Universe. Neglecting proper
motions, the expansion has no effect on comoving
coordinate values, and the coordinate distances between
particles are forever fixed. In other words, the coordinate
system expands with space itself.

Proper distance: The instantaneous (dt = 0) distance
between any two points. The distance between any two
events which occur simultaneously.



As the Universe ages, observers have time to receive light
from increasingly large distances. In other words, the size
of the causally connected part of the Universe grows.

The general expression for an increment of proper distance
is given by the Robertson-Walker metric with dt = 0:

                    ds 2  c 2  02  R 2 dl 2 .

For a radial path, the angular parts of dl2 vanish, and so
                                  dr02
                         dl  R
                           2       2

                                1  kr0 .


We therefore evaluate the integral


                    d                    r0    dr0 '
20)          d         (ds)  R  
                               2
                  0                    0
                                              1  kr02 .


Recall that r0 tracks the same particle as the Universe
expands.

NOTES: The proper distance itself is a function of time,
through R(t). As the Universe expands, galaxies really do
become farther apart.

At any time, the distance between galaxies depends on both
the “size” and “shape” of the Universe. The shape is given
by the value of k.

k = 0:                    d  R(t )  r0


Now return to the task of finding the horizon distance.
Let a photon be emitted at time t = 0 from a source at
comoving coordinate r0 = r0,e. At some later time t the
photon is received by an observer at comoving coordinate
r0 = 0 (just for convenience). Since photons travel along
null world lines, i.e. paths for which ds = 0, the Robertson-
Walker metric which describes the path is

                                    dr02
                     0  c dt  R
                              2   2   2

                                  1  kr0 .

The integral of this is

                         dt'
                          t         r0 ,e  dr0 '
21)                 c           
                      0 R (t ' )   0
                                          1  kr02 .

Note that the limits on the RHS are reversed because the
photon is moving towards smaller r0.

The proper distance of the source follows from using 20) to
eliminate the integral on the RHS of 21). The result is

                                               dt'
                                               t
22)                   d  d h  R (t )  c 
                                            0 R (t ' ) .



Choose a value for k, and hence for R(t), and evaluate the
RHS.
k = 0:

Use equation 11), which is

                                                    2/3
                                           t   
                                    2/3
                         3
                      R                
                                          t    
                                                         .
                        2                H   

Plug in for R(t ) in order to evaluate the integral in 22).
Then plug in again for R(t). The result is

23)                        d h  3ct .

Express the RHS in terms of the corresponding redshift
using 17), still for k = 0:

                                        3 / 2
                      t  t H  1  z  .
                               2
                               3

Plugging into 23) gives the result

                                         2c
                      d h ( z) 
                                   H 0 1  z  .
                                               3/ 2




For us, z = 0. Taking H0 = 71 gives

                    2c 2  3  105 kms1
          d h (0)                       8450Mpc .
                    H0   71kms1 Mpc1
FOOTNOTE: For k = 0 we just found that dh  t, whereas
R(t)  t2/3. Thus, the horizon distance expands faster than
the Universe, and more of the latter becomes visible.
Observational Cosmology
THE COSMOLOGICAL REDSHIFT

What happens to light emitted “long ago in a galaxy far, far
away…”?

Let a wave packet of duration t1 be emitted from a source
at r0 at time t1. An observer at r0 = 0 at the present epoch,
time t0, receives the packet over a duration t0.
Remembering that light travels along null world paths
(ds = 0), the Robertson-Walker metric is:

                                   dr02
                    0  c dt  R
                              2   2   2

                                 1  kr0 .

The path taken by the leading edge of the wave packet is
described by

                        t0    dt'      r0 dr0 '
                   c               
                     t1      R(t ' ) 0 1  kr02

As before, choosing the coordinate origin at the observer
introduces an extra minus sign, reversing the order of
integration on the RHS. The trailing edge of the wave
packet, which follows the same path, moves according to
                            t0  t0    dt'      r0 dr0 '
                       c                     
                        t1  t1       R(t ' ) 0 1  kr02 .

Because the duration of the packet at each end of its trip is
exceedingly small, the LHS can be well approximated by

                t0    dt'     ct0 ct1      r0 dr0 '
           c                           
             t1      R(t ' ) R(t0 ) R(t1 ) 0 1  kr02 .

Equating the two expressions to eliminate the RHS, and
remembering that R(t0) = 1, we find that

                                       ct0   ct1
                                            
                                        1     R(t1 ) ,

or just

                                                  t1
                                        t 0 
                                                 R (t1 ) .

RESULT: The expansion of space during the packet’s trip
has increased its duration ( because R(t1) < 1). The same
argument applies to all time periods, including those defined
by photon frequencies. In particular, photon frequencies are
lowered, and wavelengths lengthened.
THE LUMINOSITY DISTANCE

How can we find values for 0, H0 ??


                                    L
                           dL 
                            2

                                  4  f ,

where L is the luminosity of some remote source whose
measured flux is f.

Prof. Einstein tells us that the observed flux is written

                              L         1
                     f            
                           4  r02 1  z 2 .

The factors of (1+z) are courtesy of the Cosmological
Redshift:

a) As we have seen, time dilation expands the intervals
between successive photon oscillations, i.e. lowers their
frequencies. That lowers the energy per photon (E = h).
Furthermore,

b) the same time dilation increases the time interval
between the arrival of successive photons. This further
reduces the received flux.
From the above we see that

                               4r02 1  z  ,
                       1 L  1
               dL       
                2                             2

                      4 f 4

or in other words

24)                           d L  r0  1  z  .

Now we need to find out how r0 depends on the choice of
world model.

Start with equation 21):

                         dt'
                         t          r0 ,e  dr0 '
                    c           
21)                   0 R (t ' )   0
                                          1  kr02 .

We’d rather integrate the LHS directly over R, so use the
fact that

                                     dt  dR  ,
                                            R

which allows us to write

                         R0   dR           r0 ,e    dr0 '
                    c           
                                      
                      Re
                              RR
                                         0
                                                   1  kr02 .
Equations 6) and 7), and the one for the critical density, can
be used to show that

                        2
                                            
                       R  H 0   0   0  1 .
                             2

                                  R          

We also know that R0 = 1, and that

                                  Re  1  z  ,
                                              1




so that, for any GR cosmology we have

           1                  dR                         r0 ,e    dr0 '
         c 1                                       
25)        1 z   H 0 0 R  0  1R 2              0
                                                                 1  kr02 .


k = 0:

This is equivalent to 0 = 1. Therefore 25) looks simpler!

                       c       1      dR     r0 ,e

                       H0      1
                               1 z    R
                                           dr0'
                                            0      .

This is easy to integrate and solve for r0,e  r0. The result is


26)                          r0 
                                    2c
                                    H0
                                        
                                       1  1  z 
                                                   1 / 2
                                                          .
k  0:

The starting point is the same, but the derivation more
complex. C&O present the process on pages 1267,68. The
bottom line is

27)         r0 
                   2c
                      
                            1
                   H 0 1  z  0
                                 2
                                                        
                                    0 z  2   0  1   0 z  1    .

which turns out to hold for both open and closed universes.

CHECK: Do we recover equation 26) if 0 = 1? Yes!

The luminosity distance of a source at redshift z can now be
determined by using 26) or 27) to replace r0 in equation 24).

Collect the pieces for the general case:

24)                      d L  r0  1  z  ,



27)      r0 
                2c
                   
                         1
                H 0 1  z  0
                              2
                                                              .
                                 0 z  2   0  1   0 z  1


Doing the substitution gives


28)             dL 
                       H 0 0
                                                    
                           2 0 z  2  0  1  0 z  1 .
                       2c 1
                                                                   
Check to see if this makes sense in the familiar limits.

0 = 1:

Then 28) becomes

      dL 
             2c
             H0
                         
                 z  1 1  z  1  
                                     2c
                                     H0
                                               
                                         z 1 1 z .         

z << 1:

Use the binomial series expansion for the square root,
keeping just first-order terms in z:

          dL 
                 2c
                    z  1  1  1 z   2c  1 z   cz ,
                                  2
                 H0                        H0 2         H0

which looks familiar!
THE REDSHIFT-MAGNITUDE RELATION

If we can find nice standard candles at large distances,
equation 28), together with the distance modulus relation,
will predict the run of apparent magnitude with redshift for
various world models.

29)                     m  M  5 log d L  5 .

So one has

                      m  M  5  5 log d L ,

with


28)          dL                                     
                        2 0 z  2  0  1  0 z  1 .
                    2c 1
                    H 0 0
The entire derivation can (and has!) been repeated including
a cosmological constant, .

Supernova Ia hunters at Berkeley and CTIO are trying to
push beyond z = 1 in order to measure both 0 and .

Recent results, in conjunction with studies of the CMB
fluctuations, are pointing towards


                         TOTAL  1 ,

with

                    0.7 , and  B  0.3
The Cosmological Constant
(An idea whose time has come…and gone…and come…and
?…)


The most general solution of the Field Equation is


                   2   8G 2 1
30)               R        R  R 2  kc2 .
                          3    3

Newtonian interpretation:

The third term on the LHS represents a new “potential
energy” added to the original equation for energy
conservation in the expanding shell. The original equation
would look like

           1 2 GM r m 1             1
             mv      mc 2 r 2   kmc2 r02 .
           2      r   6             2

The new potential energy term corresponds to a force given
by
                        ~
                        F   U  r ,
                                    ˆ
                         1     2 2
                          mc r r ,
                                      ˆ
                       r  6       

                           1
                            mc 2 rr
                                    ˆ
                           3
Result: If  > 0, then an outward force exists which
counteracts gravity, as Einstein originally intended.

Furthermore, the force grows as the Universe expands!
A candidate for  - vacuum energy
Classical Physics: Vacuum energy is irrelevant!

                          E  K U ,

and the force on a particle can be written as

                 F  U   U  const. ,
                 ~


so the motions of particles are not affected by choosing
another zero point for potential energy. E(minimum) need
not be zero!

Modern Physics: Vacuum energy is required!

Particle Physics View: “Vacuum” is just the lowest energy
state of any physical theory. If the lowest energy state looks
the same for all observers, then it must be invariant under
the Lorentz transformation (“lowest energy” must represent
the same thing in all inertial reference frames).

Consequence: the “vacuum” is a fluid whose equation of
state (pressure-density relation) must be

                        PVAC   VAC c 2 .


Quantum Mechanics View: Uncertainty Principle allows
spontaneous appearance and disappearance of
virtual particles. Particle Physics views these as just the
“vacuum “ states of real particles. These particles, then,
become the source of mass-energy.

Uncertainty Principle also forbids any particle from being
isolated in a state with zero total energy. The vacuum must
                                            correspond to a
                                            non-zero energy
                                            state.
Incorporating  in Cosmological Theory
Equation 30) can be rewritten as

                2   8G 2 8G  2
               R        R       R  kc2 ,
                       3      3

where  is the mass density associated with . In the same
way as the customary density parameter is defined, one can
define

                                 8G
                                    ,
                                 3H 0

which means that

                           3H 0    .
                                2




Following the spirit of the Newtonian formalism, the
generalized form of equation 10) is

                     H 0  0     1  kc2 .
                       2




So the condition for a flat universe, k = 0, is that

                          0    1 .

As we have seen, there are powerful reasons for wanting
k = 0 (i.e. for believing in Inflation).
The Age of the Universe
Making  > 0 increases the age of the Universe for k = 0
cosmologies. The current age is given by

                 2 1  1            1  1 / 2  
            t0         1 / 2 ln         
                                                   
                                    1   1 / 2  .
                 3 H 0  
                                                

With H0 = 71, the age is

                         1         1  1 / 2 
                                    1   1 / 2  Gyr.
             t0  9.2   1 / 2 ln         
                                                   
                        
                                              

Studies of CMB fluctuations and remote supernova
currently point toward

                               0.7 ,

which means that the age is

                                           1  0.71 / 2 
           t0  9.2  0.7
                            1 / 2
                                           0.31/ 2  Gyr,
                                      ln               
                                                        


                     9.2  1.45  13.3 Gyr.

This greatly relieves, and perhaps solves, the “age problem”.

				
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