## Variable Hubble Constant H - part III

The cosmological principle states that the universe is homogeneous and isotropic. This principle implies that an expanding Universe conforms to the law: v = H * d (second Hubble's Law) with v being the proper speed and d the proper distance. In this document space expansion is studied for different values of H. For more information about the first and second Hubble's Law see: Comments about Hubble's law - Part I

### Example 1

The left side of Table 1 shows an expanding universe with H = 1. The initial distance = 1 and the initial speed v = 1.
 t distance v H t v delta dist 0 1.000 1.000 1.0 .0 .000 .000 1.000 1 2.718 2.718 1.0 .1 1.000 .100 1.100 2 7.389 7.388 1.0 .2 1.100 .110 1.210 3 20.085 20.083 1.0 .3 1.210 .121 1.331 4 54.598 54.589 1.0 .4 1.331 .133 1.464 5 148.413 148.387 1.0 .5 1.464 .146 1.610 6 403.428 403.348 1.0 .6 1.610 .161 1.771 7 1096.632 1096.387 1.0 .7 1.771 .177 1.948 8 2980.957 2980.215 1.0 .8 1.948 .194 2.143 9 8103.081 8100.865 1.0 .9 2.143 .214 2.357 10 22026.458 22019.900 1.0 1.0 2.357 .235 2.593

### Table 1

Line 1 at Table 1 shows that between t = 0 and 1 the distance has increased with 1.718. This increase should be at least more than 1 because the initial speed at t = 0 is 1 and starts to increase from there on. To understand (calculate) the total increase you have to subdivide the time in small increments.
The right hand side of Table 1 explains this increase between t=0 and 1 with a step size (delta) of 0.1.
• The column delta shows the value v * step size = v * 0.1
• The column dist shows the previous distance plus v * step size.
What the right hand side of Table shows is that the distance with a step of 0.1 has increased from 1 to 2.59373
If you do the same with a step size of 0.01 the distance will increase to 2.704814. With 0.001 to 2.71692 and with 0.0001 to 2.718143 and finally to the value 2.71828 which is called e.
This same increase will also happen at t=2. As such the total distance at t=2 will be e^2.

In general the distance of an expanding homogeneous universe follows the law: distance = d0 * e ^ (H * t) with d0 being the initial distance.

### Example 2

Table 2 shows 3 examples of an expanding universe.
• Test 1 shows the situation with H = Constant.
• Test 2 shows the situation with v = Constant.
• Test 3 shows the situation where v slowly decreases in time. The final value is zero.
Test 1 Test 2 Test 3
tdistancev H distancev H distancev H
0 1.069.069 1.100.100 1.199.199
1 1.071.074.069 1.100.100.090 1.189.179.151
2 1.148.079.069 1.200.100.083 1.359.160.117
3 1.231.085.069 1.300.100.076 1.509.140.092
4 1.319.091.069 1.400.100.071 1.639.120.073
5 1.414.098.069 1.500.100.066 1.749.099.057
6 1.515.105.069 1.600.100.062 1.839.080.043
7 1.624.112.069 1.700.100.058 1.909.060.031
8 1.741.120.069 1.800.100.055 1.959.040.020
9 1.866.129.069 1.900.100.052 1.989.020.010
10 2.138.069 2.100.050 1.999.000.000

### Table 2

What the middle case shows is that space expansion decreases when the Hubble Constant decreases. The right case shows that space expansion even stops.

### Example 3

In Table 3 a slightly different approach is followed. Starting point is that space increases lineair. The value of H is calculated to assure this.
 t distance v H distance v 0 10 1 .100 1 11 1 .090 2 12 1 .083 3 13 1 .076 4 14 1 .071 5 15 1 .066 5 .333 6 16 1 .062 5.333 .333 7 17 1 .058 5.666 .333 8 18 1 .055 6 .333 9 19 1 .052 6.333 .333 10 20 1 .050 6.666 .333 z+1 = 2 z+1 = 1.333333

### Table 3

However Table 3 shows more. When light from the source and is half way there is a supernova. Space expansion from that second source is indicated in the right side.
The following sketch explains that situation.
```t10           X      Y                   Z
| *         .      .                  *
|   *             .                 *
|     *    .     .                *
|       *       .               *
t5---------S2   .              *
|           * .             *
|        .   .*           *
|           .   *       *
|       .  .      *   *
t0-------X-Y---------S1--------------------
Figure 3A
```
Figure 3A shows space expansion from two sources: S1 and S2. The line S1,Z shows the space expansion from S1. The line S2,X shows the space expansion from the second source i.e. the supernova. There is also a third line Y,Y drawn which connects a point half way between S1 and Observer at t=0 and t=10.
All the three lines (X,X) (Y,Y) and (S1,Z) comply with the cosmological principle and with the law v = d * H with proper distance and speed.
What is very important that space expansion (z+1) for S1 is a factor of 2 and for S2 (half way) is approximate 1.4

The following table shows the full range of z+1 values between 1 and 10:

 t z+1 t z+1 t z+1 t z+1 t z+1 1 1.8181 2 1.6666 3 1.5384 4 1.4285 5 1.3333 6 1.2500 7 1.1764 8 1.1111 9 1.0526 10 1
Table 3B

### Example 4

The following example is the same as the previous example except that the Hubble Constant is considered constant in time.
 t distance v H distance v 0 10 4 .400 1 14.918 5.967 .400 2 22.255 8.902 .400 3 33.201 13.280 .400 4 49.530 19.812 .400 5 73.890 29.556 .400 5 2 6 110.231 44.092 .400 7.459 2.983 7 164.446 65.778 .400 11.127 4.451 8 245.325 98.130 .400 16.600 6.640 9 365.982 146.392 .400 24.765 9.906 10 545.981 218.392 .400 36.945 14.778 z+1 = 54.59814 z+1 = 7.389056

The following table shows the full range of z+1 values between 1 and 10:

 t z+1 t z+1 t z+1 t z+1 t z+1 1 1.4918 2 2.2255 3 3.3201 4 4.9530 5 7.3890 6 11.0231 7 16.4446 8 24.5325 9 36.5982 10 54.5981
Table 4A

### Hubble telescope finds 'never-seen' galaxies - Example 5

In the Usenet newsgroup sci.astro.research there is a discussion called: "Hubble telescope finds 'never-seen' galaxies". In Message 14 the first and the second Hubble's Law are discussed.
• The first Hubble Law is Z = (H/C) * d with Z being the present value and d being the distance in the past.
• The second Hubble Law is V = H * d with V, H and d being the present values.
• The second Hubble Law can also be written as V = H * d with V, H and d being the past values at moment of emission.
The question is are all those three H values the same. IMO they are not.
The following sketch explains this.

```             G2                                    G1
t10 z        *       z                             *
| .z        *       z                             *
t7 z .       *       z                             *
| z   .    *       z                             *>v7
|z      .  *      z                             *->v6
t5         *     z                             *-->v5
|           . z                            *--->v4
|          z  .                        *---->v3
t2       z       .                 *----->v2
|   z             .         *------>v1
t0-------------------G1-------------------
Figure 5
```
Figure 5 shows:
• space expansion from a Galaxy G1 between t0 and t10 with t0 being the point of emission i.e. the past and t10 being the present. What is important that space expansion stops starting from moment t7. You can argue that that is not accordingly to reality, but let us assume that that is the case.
• Each horizontal line shows the second Hubble's law. What is important that the Hubble constant H at t0 has a certain value H0 and slowly goes to zero. You can express this as: H = H0(t) = H0*(1+a*t) with a being negative.
It is clear from Figure 5 that the H values in the second Hubble's law at t0 and t10 are different.
• The left part shows the first Hubble's law for Galaxy G1. The redshift value z is identified with the letter z. This value starts from zero at t0, increases linear until t2, but this increase diminishes and becomes zero, after which z becomes constant.
• The left part also shows the first Hubble's law for Galaxy G2. Galaxy G2 is at a distance halfway between G1 and the observer.
What the Figure 5 shows is that:
• That the speeds and space expansion of G2 are approximate 50% of those of G1.
• That the z values of G2 is approximate 25% of those of G1.
• In fact Figure 5 shows that the first Hubble's Law is not true because the z value of a Galaxy at twice the distance is much larger.
• You could argue that for small distances z increases linear with distance, which implies a positive value for the Hubble constant. However this is in conflict with the second Hubble's Law at t10 where H = 0.

### Example 6

The purpose of this example is to challenge both Hubble's Laws starting from the basics.
Consider the following example.
``` 1000km/sec <--------c
v--> 10km/sec
O--------------------X--Y
G     XY=10km
<-------------------->
distance d = 1000km
Figure 6A
```
In Figure 6A we have a galaxy G at a distance d from the origin O. The Galaxy G has a speed v to the right.
A quick glance at figure 6A will reveal that it will take 1 sec for light to reach the Observer. That means during that period Galaxy G will move a distance of 10 km to the right i.e. the distance XY. That means the redshift observed will be 10/1000 = 0.01 = z.
Using the first Hubbles Law z=(H/c)*d we can calculate H. H = z * c /d = 0.01 * 1000/1000 = 0.01
Next we get a slightly different example:
```          1000km/sec <--------c
v--> 100km/sec
O-------------------------------------------------X---------Y
G10  G9   G8   G7   G6   G5   G4   G3   G2   G1   G     XY=1000km
<------------------------------------------------->
distance d = 10000km
Figure 6B
```
It will take 10 sec for light to reach the observer. Using the same value of H of 0.01 we get z=(H/c)*d = (0.01/1000)*10000 = 0.1. This is the start value of Z. The distance XY will be z * d = 1000 km and v is XY/10=100 km/sec = H * d.
However is this calculation correct ? Let us assume that initial speed at v = 100 km/sec and constant.
• During the first second G will move a distance of 100 to the right (distance 10100 km) and light a distance 1000 towards the left to G1 (distance 9000km from O). Space expansion is approx 95 km. (in fact smaller i.e. 94.5 km)
• During the second second G will move to a distance of 10200 and light to a distance of 8000km i.e. G2. Speed v of G1 is equal to 100*9000/10100 = aprox 85km/sec. Exact space expansion = 83.8 km.
• During the third second G will move to a distance of 10300 and light to a distance of 7000km i.e. G3. Speed v of G2 is equal to 100*8000/10200 = aprox 75km/sec. Exact space expansion = 73.2 km.
• 62,8 52,7 42,7 32,9 23,3 13,8
• During the tenth second G will move to a distance of 11000 and light to a distance of 0km i.e. G10. Speed v of G9 is equal to 100*1000/10900 = aprox 5km/sec. Space expansion = 4.6 km.
• In total space expansion is only 484,2 which means that the observed value of z = 0.0484, which is much less than the initial value of 0.1.

### Hubble Law simulation program

In order to test Hubble's Law I have written a program using EXCEL.
For a copy of the program in zip format select: BIGBANG3.XLS
For a description of the program select: BIGBANG3.XLS Description and Operation

The program describes three expansion scenarios: One with H is Constant, one with expansion speed v = Constant and the final one with v going to zero.

### Reflection

Table 3B and Table 4A each show that the realation between d and z+1 is non linear. This is in conflict with the first Hubble's law which states that z = (H/c) * d i.e. that z increases linear with distance (time).
The first Hubble's law can be considered correct at small distances but not for larger values of z. In fact for larger values of z the galaxies are much closer as the first Hubble's law indicates. Figure 5 has the same message.
That is an important conclusion !

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Created: 31 January 2010
Updated: 20 Februari 2016

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