3 World Models - An introduction

Purpose

The purpose of this document is an introduction to explain the age of the oberved galaxies for 3 different categories of world models:
  1. A Simple linear model.
    In this model the size R(t) of universe increases linear as a function of time. The speed of light towards the observer is not influenced by space expansion.
    There is a Quick Basic and a visual Basic program available to explain this model.
  2. Milne's model. This is also a linear model
    In this model the speed of light towards the observer is influenced by space expansion. There is an Excel Program available to explain this model.
  3. Models described by Friedmann's equation.
    Friedmanns equation is a function of the following cosmological parameters: Lambda, C (rho), k and age.
    The calculated cosmological parameters are: Omega(Lambda), Omega(m), Omega(k) and H0.

#1 A Simple linear model 

                                    D
                                   /|
                                  / |
                                 /  |
                                /   |
                               /    |
                              /     |
                             /      |
                            /       |
                           /        |
                          /         |
                         /          |
                        /           |
                       /            |
                      /             |
                     /              |
                    /               |
                   /                C
      e           /               . |
        .        /              .   |
          .     /             .     |
            .  /            .       |
              d           .         |
             /| .       .           |
            / |   .   .             |
           /  |     c               |
          /   |   . | .             A
         /    | .   |   .      .    |
        /     .     |     a         |
       /    . |     |.    | .       |
      /   .   | .   |     |   .     |
     /  .   . |     |     |     .   |
    / .       |     |     |       . |
   BB...............................O
   0          5     8     11        16
The sketch left demonstrates how the program operates.
  1. BB is the momement of the Big Bang at t= 0
  2. O is the position of the Observer at t = 16 Gyear.
  3. The line BB-C represents the maximum distance of an expanding Universe with an expansion factor of 1 i.e. expansion with speed c.
  4. After 8 Byear the distance reached is the point c. Light from that point will reach the Observer after 7.5 BYear. This lightpath is de line c-O.
  5. After 16 BYear the youngest galaxies you will see are 8 Gyear. You will not see any younger. You can see older galaxies. The line BB-A represents such a situation. This line intersects the line ecdO at the point a. The galaxy will be roughly 11 BYear old.
  6. The line BB-D represents the maximum distance of an expanding Universe with an expansion factor of 2 i.e. expansion with speed 2*c.
  7. After 5 BYear the distance reached is the point d. Light from that point will reach the Observer after 11 Byears. This lightpath is de line O-d.
  8. In order to see galaxies almost in their infancy the expansion factor has to be much larger. The line BB-E (not drawn) running vertical represents this situation. This line intersects the line Ocd at the point e. The galaxy will be roughly 1 BYear old.
There are also a Quick Basic, Visual Basic 5.0 and Visual Studio 2010 programs available to explain this model.
For a description, operation and a copy select: Big Bang simulation

#2 Milne's model

In World model #1 the speed of lightray c towards the observer O is not affected by space expansion.
This is not correct 
                    /               |
                   /                C
                  /               . |
        e        /              .   |
          .     /             .     |
            .  /            .       |
              d           .         |
             /  .       .           |
            /     .   .             |
           /        c               |
          /       .   .             |
         /      .       .           |
        /   x X x         .         |
       / x  .   .  x x x    .       |
      /x  .       .       x x .     |
     x  .           .           x   |
    x .               .           x |
   BB...............................O
   0          5        10           16
At point X, 5 b years after the Big Bang, there is a Supernova. Light from that point will move with the speed of light c towards the observer O. If World Model #1 is assumed this light flash will reach the Observer at O after 10 b years
However the expansion speed is also c. That means very close at point X both components cancel and the distance towards O will not change. That means initial the path is horizontal.
But now the expansion speed will decrease and become less than c. The overal consequence is that the light ray slowly starts to move towards O.
The final result is that the lightray will reach the Observer after roughly 16 billion years.
What this means in case the expansion speed is c that the youngest galaxy observed in World Model #2 is 5 Byears compared to World model 1 this is 8 Byears

However there is also a second important difference with World Model #1: The path of the light ray before point X is bended towards the Big Bang. That means in case the expansion speed is 2*c the youngest galaxy observed is roughly 1 b years old compared to World Model #1 which shows 5 b years.

This world model is called "Milne's model".
For more infomation select: Milne's model
There is also a Visual Basic 5.0 or Visual Studio 2010 program available to explain the Milne's Model. For a copy select:
Visual Basic Big Bang simulation program "VB BigBang.exe" and "VB2010 BigBang.exe

3. World Models based on Friedmann's equation.

The picture on the left shows the world model of a universe calculated using the friedmann equation with the cosmological parameters:
C (=rho) =100 Lambda=0.11551 k=0 and the age of the universe= 13.74
The derived cosmological parameters are: Omega(Lambda)=0.7332 Omega(m)=0.2668 Omega(k)=0 H0=71
Those parameters reflect the current state of art. See Precision Cosmology from WMAP

The black line shows the outer edge of the universe.
The blue line shows the path of a lightray towards an observer 13.74 b years after the Big Bang.
The brown line shows the theoretical light cone from the observer. This line makes an angle of 45 degrees with the horizontal time axis. This angle shows that the dimensions of the time axis and the x axis are the same. Both are in billion light years.
For all the three pictures this is the same. That means you can compare them. What this also shows is that the Black Line is almost identical as a linear world model with expansion factor of 3.

The most important lesson to be learned is that Milne's model with an expansion factor of three compared with the Friedmann model with the present day accepted cosmological parameters is almost identical. That means to call the Friedmann model an accelerating universe is a rather border line case because there is not much acceleration involved.
What makes this so complicated is that based on our rather local observations it is so difficult to distinquish between these two and to decide in which type of world model we live.

For more information select: Friedmann's Equation & The path of a light ray 13 Questions.

Reflection part 1

The assumption that the expansion factor is much larger to 1 explains that in principle we are still able to observe galaxies in their infancy i.e. very close after the moment of the Big Bang. However those galaxies should also look much younger and can never be very large and mature. The problem is that currently based on observations this picture is not correct. We see large grown up galaxies at large distances. To state it differently: the Universe looks much more uniform (and static) as compared to an Universe in evolution (and dynamic). What the observations also could imply that the much younger galaxies are much further away and that the total universe is much larger (as currently observed) and much older.

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Created (Partly): 29 December 2001
Modified: 5 Februari 2014

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