Introduction and Purpose

This document descibes "Blad1" and "Blad2" of the Excell Program Bigbang3.xls .
The purpose of the program is to simulate Hubble's Law using different expansion scenario's.

• "Blad1" shows 3 simulation of Hubble's Law of a selected scenario with different initial values of the expansion speed v ("v in")
• "Blad2" shows the same simulations but for different distances.

The three scenarios are:

• H is constant during space expansion. This implies that the expansion speed increases exponential.
• The expansion speed v is constant.
• The expansion speed v decreases linear and becomes zero when a light pulse of distant galaxy reaches the Observer.

For a copy of the program in zip format select: BIGBANG3.XLS

Description and Operation: "Blad1"

Blad1 shows three examples of the same simulation.
Each simulation requires five input paramaters: d, v, c, corr and test.
Those five parameters can be modified by the user
The following sketch shows the meaning of those input parameters under scenario 2.

                                                                        G2  
t10             H       z                                               *---->v 
 | .             H     z                                           *---->v
 |   .           H    z                                       *---->v
t7     .          H  z                                   *---->v
 |       .        Hz                                *---->v
t5         .     z H                            *---->v
 |           . z   H                       *---->v
 |          z  .    H                 *---->v
t2       z       .  H            *---->v
 |   z        c<--- .H     *---->v
 O-------------------G1-------------------------------------------------------
t0<------------------><-------------------------------------------------> 
      distance = d                         space expansion = dt - d
                        Figure 1

The point G1 is the initial position of the Galaxy i.e. the past position.
The point G2 is the present position of the Galaxy.

• The input parameter d is the initial distance or past distance of the Galaxy.
• The input parameter v is the initial speed or past speed v of the Galaxy towards the right at t = 0.
• The input parameter c is the speed light i.e. of a light pulse towards the Observer at position O. In the above sketch v is much larger than c and approximate equal to 3 * c.
  The maximum value of of v = 6 * c.
• The input parameter test defines three types of expansion scenarios. See below.

Output values of each simulation are stored in arrays marked: Time, d, dt, vt, zt and H.

• The array Time contains the Time after light pulse is emitted from the Galaxy.
• The array d identifies the position were the light pulse is. This are the dots in Figure 1.
  The values are calculated using the following equation: d = "initial d" - c * time
• The array dt identifies the present distance were the Galaxy is. This are the ** in Figure 1.
  The values are calculated using the following equation: dt = "initial d" + v * time
• The array H shows the Hubble Constant H. This are the letter H in Figure 1.
  The values are calculated using the following equation: v = H * dt.
• The array zt shows the observed redshift i.e. space expansion of the light pulse towards the Observer.
  Suppose that during a certain period "delta t" the Galaxy distance (dt) increases with a small value ddt. Assume homogeneous space expansion than space expansion (dexp) at light distance (d) is equal to: dexp/d = ddt/ dt or dexp = ddt * d / dt .
  In order to calculate zt:
  1. Divide the time that light pulse moves from its initial distance towards the Observer in small increments dt and calculate space expansion dexp during that period. See above.
  2. In order to get the total space expansion (exp) add all those increments dexp together. Exp represents the space expansion subject to the light pulse when the light pulse travels from its position at emission (distance = d) at t0 towards the Observer.
  3. Finally divide exp by the initial Galaxy distance "d" in order to calculate zt.
• The array z represents space expansion as a function of the present Galaxy distance (dt).
  In figure 1 z = "space expansion" / "distance" = ("dt" - "d")/"d" = "dt"/"d" - 1
• The array vt contains the present speed of the galaxy i.e. speed of space expansion

In order to execute the simulation of "blad1" select Calculate.

What the simulation shows for test 2 with v = constant is that:

• For small values of v the value of H is almost constant. Initial zt (observed redshift z) increases linear when the light pulse travels towards the Observer, but this increase deminishes and goes to zero when the pulse reaches the Observer.
• For a value of v equal to c, H decreases with a factor of 2. The value of zt shows the same behaviour as before. The final value of zt is ample less than 1.
• For a value of v larger than c (i.e. v = 4 * c), the final value of zt is slightly larger than 1.

Paramater Test

The next table shows the final result of the simulation for 3 test values:

1. Test = 1: H = constant
2. Test = 2: v = constant
3. Test = 3: v = variable - final value = 0

The most important parameter is the observed redshift value zt of a light pulse emitted at time t0 at an initial distance selected by the parameter "d".

In test 2 the expansion velocity v is the same as the initial speed v of the Galaxy G1 and constant. That means that the output values v are the same as the input parameter v. The Hubble constant H is calculated by dividing vexp with the Galaxy distance. Or H = vexp / Galdist.

For the other two tests a different method is followed. Starting point in all cases is that the final distance fd is the same. Or: fd = d + tmax * v

• For test 1 (H constant) the final distance (fd) of a Galaxy with initial distance d with constant H over a period tmax is also equal to: fd = d * exp (tmax * H)
  As such we get: d + tmax * v = d * exp (tmax * H)
  or tmax * H = log (d + tmax * V)/d
  or H = (log (d + tmax * V)/d) / tmax
  Multiplying H with the Galaxy distance gives the speed of Galaxy. Or vout = Galdist * H
• For test 3 (v goes to 0) v goes from an initial value lineair to zero.
  That means 0.5 * alpha * tmax * tmax = final distance - initial distance = tmax * v
  or alpha (acceleration) = 2 * v / tmax The expansion speed v at any moment t is than calculated as: vexp = alpha * (tmax - t)
  The Hubble constant H is calculated by dividing vexp with the Galaxy distance. Or: H = vexp / Galdist.

Test = 1: H = constant

v	vt	z	zt	H
100	111	0,11	0,05	0,01
1000	2718	1,72	0,5	0,1
4000	218218	53,55	2	0,4

Test = 2: v = constant

v	vt	z	zt	H
100	100	0,1	0,0484	0,01 - 0,009
1000	1000	1	0,3863	0,1 - 0,05
4000	4000	4	1,0120	0,4 - 0,08

Test = 3: v = variable - final value = 0

v	vt	z	zt	H
100	0	0,05	0,0327	0,01 - 0
1000	0	0,5	0,2810	0,1 - 0
4000	0	2	0,8078	0,4 - 0

Starting point of each test is the second Hubble's Law: V = H * d.
The initial value of H is : v / d. That means that for "v" of resp. 100, 1000 and 4000 and with d = 10000 the starting value of H is resp: 0.01, 0.1 and 0.4
In the case of "test 2" the expansion velocity is constant. That means with a time of t = 10 the final distances are for "v" of resp. 100, 1000 and 4000 are 11000, 20000 and 50000.
Using the equation z = (final distance - initial distance)/initial distance we get the following theoretical values of z: 0.1, 1 and 4. You get the same theoretical values using the Doppler shift equation v = c * z

What the above simultions for "test 2" show is that the simulated (observed) values of z (i.e. zt) are lower than the theoretical values. The same is also true for the theoretical values of z versus the simulated values of z (i.e. zt) for the other two tests. The highest values of zt are obtained in "test 1".

Description and Operation: "Blad2"

"Blad2" Shows the same simualations as on "Blad1". Standard execution of the simulation of "Blad2" is performed by selecting Calculate on "Blad1". When you do that the parameter values from "Blad1" are copied to "Blad2"

"Blad1" shows the evolution of z for one light pulse emitted at distance "d" for three different expansion scenarios at time t0. The idea behind "Blad2" is to shows a range of observed z values when there are additional supernovae during the time (in increments of 10%) that the original light pulse travels towards the observer at the same distance of the light pulse. The idea behind this is to test if there exists a lineair relation between z and distance under the same expansion scenario.

Column

• "d" contains the distance of the Supernova at the moment of the explosion.
• "dt" contains the present distance of the Supernova.
• "v" contains the velocity of Supernova at the moment of the explosion.
• "vt" contains the present velocity of Supernova.
• "z" contains the theoretical redshift value i.e. ("dt"- "d") / "d"
• "zt" contains the simulated observed redshift value.

Test = 1: H = constant: v in = 1000

time	d	dt	v	vt	z	zt	H
0	10000	27181	1000	2718	1,7181	0,5	0,1
5	5000	8244	500	824	0,6488	0,2502	0,1
10	0	0	0	0	0	0	0,1

Test = 2: V in = 1000 and constant

time	d	dt	v	vt	z	zt	H
0	10000	20000	1000	1000	1	0,3863	0,1
5	5000	6667	333	333	0,3334	0,1508	0,06666
10	0	0	0	0	0	0	0,05

Test = 3: V in = 1000 and variable

time	d	dt	v	vt	z	zt	H
0	10000	15000	1000	0	0,5	0,2810	0,1
5	5000	5455	182	0	0,0909	0,0585	0,03636
10	0	0	0	0	0	0	0

First Hubble's Law z = H/c * d

The First Hubble's Law describes a linear relation between the past distance of an event and the observed red shift value. In this simulation this are the parameters "d" and "zt"

What the above tests show (v = 1000) is that only in the case of "test 1" there exits a linear relation between the past distance "d" (0, 5000 and 10000) and the observed (simulated) value "zt" (0, 0,25 and 0,5). The same is true for all values of "v" i.e. "v" = 100 in "test 1" and "v" = 4000 in "test 3".

The explanation is that only in "test 1" space expansion is constant in time and the Hubble constant is constant in time. In "test 2" and "test 3" this is not the case because H is variable and decreases in time.

Second Hubble's Law v = H * d

The First Hubble's Law describes a linear relation between the past distance of an event and the velocity of that event. In this simulation this are the parameters "d" and "v"

What the above tests show (v = 1000) is that only in the case of "test 1" there exits a linear relation between the past distance "d" (0, 5000 and 10000) and the past speed "v" (0, 500 and 1000). The same is true for all values of "v" i.e. "v" = 100 in "test 1" and "v" = 4000 in "test 3".

The First Hubble's Law can also be described a linear relation between the present or proper distance of this event and the present velocity of the event. In this simulation this are the parameters "dt" and "vt"

What the above tests show (v = 1000) is that:

• In the case of "test 1" there exits this linear relation between the present distance "dt" (0, 8244 and 27181) and the present speed "vt" (0, 824 and 2718).
• But also in the case of "test 2" there exits this linear relation between the present distance "dt" (0, 6667 and 20000) and the present speed "vt" (0, 333 and 1000).

In "test 1" both definitions the value of H is the same and 0.1
In "test 2" the value of H is 0.2 and different from the values in the table.

Doppler shift law v = c * z

What the above tests show (v = 1000) is that

• In the case of "test 1" there exits a linear relation between the observed (simulated) value "zt" (0, 0,25 and 0,5) and past speed ("v" of the supernova. (0, 500 and 1000). C = 2000
• In the case of "test 2" there exits a linear relation between the doppler shift "z" (0, 0,3334 and 1) and past speed "v" and the present speed "vt" of the supernova. (0, 333 and 1000). C = 1000.

Paramater Corr

The parameter Corr can have two values: 0 and 1.

• In the above simulation the speed of light is a global concept. That means that the speed of light is everywhere the same and constant. This is the case when the parameter corr=0.
• It is also possible to perform the simulation using the assumption that the speed of light is a global concept. That means that the speed of light is a constant relative to the expanding universe. This is the case when the parameter corr=1

There is not much difference between the two cases when the speed of the galaxy is small relatif to the speed of light.
There is a difference when the speed of the galaxy approaches the speed of light and when the speed is much larger. In general the difference implies that it takes much longer for a light pulse to reach the Observer.
Assume "test 2" and that space expansion is constant.

• When the speed of the galaxy is c the speed of light (at the moment of emission) relatif to the Observer is zero. That means the distance between the light pulse and the Observer stays constant. However the distance between the Galaxy and the Observer will increase.
  The next moment, because space expansion (the speed of the Galaxy) is constant, local space expansion of the light pulse will decrease and becomes less than c, implying that the light pulse starts moving towards the Observer.
  This effect becomes stronger the further away the pulse becomes from the Galaxy.
• When the speed of the galaxy is larger than c, initially the pulse moves away from the Observer and the distance becomes larger. The pulse also moves away from the Galaxy meaning that the Local space expansion becomes less. As soon when space expansion becomes equal to c the light pulse starts to approach the Observer.

Test = 1: H = constant : corr = 1

v	vt	z	zt	H
100	112	0,11	0,0535	0,01
500	1007	1,01	0,3863	0,05
960	25104	25,15	2,3521	0,096

Test = 2: v = constant : corr = 1

v	vt	z	zt	H
100	100	0,11	0,0516	0,01 - 0,009
1000	1000	1,8	0,7171	0,1 - 0,0357
4000	4000	53,6	12,3865	0,4 - 0,00733

Test = 3: v = variable - final value = 0 : corr = 1

v	vt	z	zt	H
100	0	0,0517	0,0342	0,01 - 0
1000	0	0,7234	0,4471	0,1 - 0
4000	0	15,0398	6,5234	0,4 - 0

What the simulations show is that the theoretical value z and the simulated observed value z0 are much larger.

Paramater Corr and "Blad2"

Column

• "d" contains the distance of the Supernova at the moment of the explosion.
• "dt" contains the present distance of the Supernova.
• "v" contains the velocity of Supernova at the moment of the explosion.
• "vt" contains the present velocity of Supernova.
• "z" contains the theoretical redshift value i.e. ("dt"- "d") / "d"
• "zt" contains the simulated observed redshift value.

Test = 1: H = constant: v in = 990

time	d	dt	v	vt	z	zt	H
0	10000	1000082	990	99008	99,0082	3,6523	0,09900
30,66	8000	38467	792	3808	3,8084	0,9829	0,09900
41,43	4000	6623	396	656	0,6560	0,2733	0,09900
46,52	0	0	0	0	0	0	0,09900

Test = 2: V in = 1000 and constant

time	d	dt	v	vt	z	zt	H
0,00	10000	27988	1000	1000	1,7998	0,7171	0,1
6,96	8000	13207	472	472	0,6509	0,2763	0,5896
12,81	4000	4910	175	175	0,2277	0,0900	0,04384
17,18	0	0	0	0	0	0	0,03679

Test = 3: V in = 1000 and variable

time	d	dt	v	vt	z	zt	H
0,00	10000	17234	1000	0	0,7234	0,4471	0,1
5,46	8000	9556	345	0	0,1946	0,1266	0,04317
10,38	4000	4139	68	0	0,0349	0,0231	0,01698
14,47	0	0	0	0	0	0	0

First Hubble's Law z = H/c * d

What the above tests show is that in general there exists no linear relation between the past distance (Light distance) (0, 5000 and 10000) and the observed (simulated) value zt (0, 0,25 and 0,5). The only exception is for small values of "v" in "test 1" but that is no surprise because small means relative to the speed of light.

Second Hubble's Law v = H * d

Same remarks.

Doppler shift law v = c * z

Same remarks.

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