# Friedmann's Equation and the path of a light ray.

27 views

### Nicolaas Vroom

Oct 10, 2011, 9:07:11 AM10/10/11
to
The Friedmann equation is the most important
laws which describes the evolution of the Universe.
In order to understand its behaviour I have written an Excel
program which allows me to study this equation for different
parameters.
The document which describes the results can be found
here:
http://users.telenet.be/nicvroom/friedmann's%20equation.htm
The idea of the document is to calculate the path of a lightray
which starts at the BigBang and which reaches
the observer now using the Friedmann equation as described
in the book "Introducing Einstein's Relativity" by d''Inverno
for three paramaters: C, labda and k.
When you have this path you can also calculate the redshift
(z) which the observer observes depending what the
origin (as a function of distance) is of the light source.
What my results show is that the results are highly non linear
The question than pops up to what extend based on actual
observation you can use z as an indication for distance.
The same problems arise if you calculate H0 which is
measured near the observer.
The question is to what extend can you use the parameter
1/H0 in order to calculate the age of the Universe.
What my results also show that the relation v = z * c
is only true near the observer. For larger distances
this relation is non linear.
Space acceleration (Global and Local) is also discussed.

What the results also show is that the more
parameters are non zero in the Friedmann
equation the more difficult it becomes to calculate
the indivudual parameters.

The source of the program is available on request.

Nicolaas Vroom
http://users.pandora.be/nicvroom/

Oct 10, 2011, 10:26:07 AM10/10/11
to
On Oct 10, 9:07�am, Nicolaas Vroom <nicolaas.vr...@telenet.be> wrote:
> What my results show is that the results are highly non linear
> The question than pops up to what extend based on actual
> observation you can use z as an indication for distance.
> The same problems arise if you calculate H0 which is
> measured near the observer.
> The question is to what extend can you use the parameter
> 1/H0 in order to calculate the age of the Universe.
> What my results also show that the relation v = z * c
> is only true near the observer. For larger distances
> this relation is non linear.

> Nicolaas Vroomhttp://users.pandora.be/nicvroom/

Isn't this exactly what would be observed if the expansion of the
Universe was
due to the introduction of "new" space rather than the "stretching" of
existing
space?
Note: I am not saying that there is a larger Universe beyond our
horizon
that is "leaking" into ours. I am asking if an unrecognized process
may be
introducing new space between gravitationally bound structures larger
than
galaxies

### eric gisse

Oct 11, 2011, 3:02:27 AM10/11/11
to
Nicolaas Vroom <nicolaa...@telenet.be> wrote in news:mt2.0-29733-
13182...@hydra.herts.ac.uk:

[...]

> What my results show is that the results are highly non linear

Yep.

> The question than pops up to what extend based on actual
> observation you can use z as an indication for distance.

It depends. Are we talking in 2011 with the cosmic distance ladder
firmly established and understood, or say 30 years ago when that wasn't
true?

In 2011 since we have the cosmological parameters well determined,
knowing the object's redshfit will give you its' distance.

> The same problems arise if you calculate H0 which is
> measured near the observer.

zuh?

The Hubble paramater is the textbook definition of something that's not
locally determined, as the Hubble flow is nonexistent for
gravitationally bound objects like everything from here until the edge
of the local group.

> The question is to what extend can you use the parameter
> 1/H0 in order to calculate the age of the Universe.

By itself? Rather minimally, as the Hubble constant alone does not
constrain the various other cosmological parameters that determine the
age of the universe.

> What my results also show that the relation v = z * c
> is only true near the observer. For larger distances
> this relation is non linear.

This is well known. Hubble's law only works out to about z=1.

> Space acceleration (Global and Local) is also discussed.
>
> What the results also show is that the more
> parameters are non zero in the Friedmann
> equation the more difficult it becomes to calculate
> the indivudual parameters.

This is well known.

Go read how the WMAP people figured it out.

http://lambda.gsfc.nasa.gov/product/map/dr4/pub_papers/sevenyear/cosmolo
gy/wmap_7yr_cosmology.pdf

### jacob navia

Oct 11, 2011, 6:11:27 PM10/11/11
to
Le 10/10/11 16:26, brad a �crit :

> Isn't this exactly what would be observed if the expansion of the
> Universe was
> due to the introduction of "new" space rather than the "stretching" of
> existing
> space?

The concept of a "space expansion" is absurd. Space can't expand since
it has no space to expand.

:-)

Space already occupies all space there is.

In a similar vein space can't "stretch" or "compress" or "condense".
This terms apply to objects in space but NOT to space itself.

Then, the question if space expands into a pre-existing space
or stretches to have more space than before is also absurd.

All this comes from a lack of understanding of basic philosophical
knowledge.

The only expanding universe is the known universe. As our scopes peer
farther and farther we can discern objects near the supposed "bang"...
and they look surprisingly old and very similar to objects we see in our
own neighborhood.

The same thing applies to "inflation" of space faster than
light...

If space "expands" faster than light, movement is impossible and all
objects have negative enrgy.

If you want to go from point A to point B you can't do it: Unless you
are a strange neutrino you can't go faster than light, and since
the expansion is separating point B from point A faster than light speed
you will never reach point B, it will have receeded faster than you can
ever run.

Conclusion: movement in such a universe is impossible. You can't go from
point A to point B no matter the distance between them.

Another unresolved paradox is the "cutout" point.

At which point does the space expansion stop objects from
moving apart?

We know objects like me or you do NOT expand, nor do the solar system,
nor our galaxy. At some point, however space expansion does separate
objects.

It is a pity that apparently ALL objects in the observable Universe
are connected to each other.

There are "rivers" of galaxies that flow between super clusters,
connecting them by thin filaments. Those filaments connect the largest
structures in the cosmos.

Those filaments should have been
destroyed by space expansion but apparently they aren't.

Why?

Maybe (just maybe) because there isn't any expansion at all.

I know that according to present day astronomy there is overwhelming
evidence for space expansion.

I am convinced that this interpretation is wrong because the arguments
above.

jacob

Oct 13, 2011, 2:25:03 AM10/13/11
to
On Oct 11, 6:11�pm, jacob navia <ja...@spamsink.net> wrote:

>
> The concept of a "space expansion" is absurd. Space can't expand since
> it has no space to expand.

> Space already occupies all space there is.

Wheeler suggested a "quantum foam" as the ultimate spacetime
configuration.
More than one theorist believes that spacetime must have a structure.
What would
be the result (if this were so) if I added 1 quanta of space to the
Universe?

>
> In a similar vein space can't "stretch" or "compress" or "condense".
> This terms apply to objects in space but NOT to space itself.

So, then; What is a gravitational field?

> Then, the question if space expands into a pre-existing space
> or stretches to have more space than before is also absurd.
>
> All this comes from a lack of understanding of basic philosophical
> knowledge.

Okay, we all have our limitations.

> The only expanding universe is the known universe. As our scopes peer
> farther and farther we can discern objects near the supposed "bang"...
> and they look surprisingly old and very similar to objects we see in our
> own neighborhood.

According to the standard Cosmological model the fit is very good.
Ancient
galaxies are deficient in heavy elements because they only contain
first
generation stars.

> The same thing applies to "inflation" of space faster than
> light...
>
> If space "expands" faster than light, movement is impossible and all
> objects have negative enrgy.
>
> If you want to go from point A to point B you can't do it: Unless you
> are a strange neutrino you can't go faster than light, and since
> the expansion is separating point B from point A faster than light speed
> you will never reach point B, it will have receeded faster than you can
> ever run.
>
> Conclusion: movement in such a universe is impossible. You can't go from
> point A to point B no matter the distance between them.

This is actually a good argument for my original post; That the
expansion is due
to new space quanta being added to the Universe!

> Another unresolved paradox is the "cutout" point.
> At which point does the space expansion stop objects from
> moving apart?

Sorry, now you've lost me. Doesn't expansion cause separation?

> We know objects like me or you do NOT expand, nor do the solar system,
> nor our galaxy. At some point, however space expansion does separate
> objects.
> It is a pity that apparently ALL objects in the observable Universe
> are connected to each other.
> There are "rivers" of galaxies that flow between super clusters,
> connecting them by thin filaments. Those filaments connect the largest
> structures in the cosmos.

Ever look at the foam atop a glass of beer? Basically bubbles! The
large scale
structure, that you refer to, looks like huge bubbles (the voids) with
matter (galaxies, et al.)
occupying the surfaces of those bubbles. In addition, the expansion is
associated
with the voids, not the matter. Seems coincidental...hmm?

>
> Those filaments should have been
> destroyed by space expansion but apparently they aren't.
> Why?

See my analogy above.

### Nicolaas Vroom

Oct 14, 2011, 2:14:36 PM10/14/11
to
"eric gisse" <jowr.pi...@gmail.com> schreef in bericht
news:mt2.0-26635...@hydra.herts.ac.uk...

> Nicolaas Vroom <nicolaa...@telenet.be> wrote in news:mt2.0-29733-
> 13182...@hydra.herts.ac.uk:
>
>> The question than pops up to what extend based on actual
>> observation you can use z as an indication for distance.
>
> It depends. Are we talking in 2011 with the cosmic distance ladder
> firmly established and understood, or say 30 years ago when that wasn't
> true?
>
> In 2011 since we have the cosmological parameters well determined,
> knowing the object's redshfit will give you its' distance.

That is the whole point.
If you look at document (2011) : Table1
http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.4717v1.pdf
Only for 2 SN the Tully Fisher relation is used to establish distance.
The distance involved 22 Mpc and that is very short.
For 2 SN the redshift, z is used to calulate distance.
This distance is roughly 100 Mpc and that is not what we want.

What you want is to establish the relation between z versus distance
and both should be measured indepently.

The document Hubble Constant (2010) page 33
http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.1856v1.pdf
shows that there are only 6 high quality SN measurements
available to calculate Hubble constant.

Figure 10 (page 77) shows distances based on TF relation
for distances until 100 pc versus v. (using v=c*z you can
back calculate z, which is observed)

My point is that those distances are too short to demonstrate
the non linearities involved.
To be more specific: To calculate the parameters C, labda and k.

In general you cannot calculate distance as a function of
using v=c*z (1) and v=H*d because (1) is only valid for small
distances.

>> The question is to what extend can you use the parameter
>> 1/H0 in order to calculate the age of the Universe.
>
> By itself? Rather minimally, as the Hubble constant alone does not
> constrain the various other cosmological parameters that determine the
> age of the universe.

That is my whole point of the document in question.
For an assumed age of 14 billion years you get for almost
any combination of the cosmological parameters C, labda and k
a different value of H0.
Does that mean that only those combinations are valid that result in
a 1/H0 value of 14 billion years ?

>> What my results also show that the relation v = z * c
>> is only true near the observer. For larger distances
>> this relation is non linear.
>
> This is well known. Hubble's law only works out to about z=1.

What exactly do you mean with: works out.
Does this mean to calculate distances ?
Does this mean to calculate age of Universe ?
specific IMO this is not possible.

Nicolaas Vroom
http://users.pandora.be/nicvroom/

[Mod. note: please DO NOT link to the arXiv cache, but direct to the
article of interest -- mjh]

Oct 15, 2011, 2:49:26 AM10/15/11
to
In article <mt2.0-21457...@hydra.herts.ac.uk>, Nicolaas Vroom
<nicolaa...@telenet.be> writes:

> > In 2011 since we have the cosmological parameters well determined,
> > knowing the object's redshfit will give you its' distance.
>
> That is the whole point.
> If you look at document (2011) : Table1
> http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.4717v1.pdf
> Only for 2 SN the Tully Fisher relation is used to establish distance.
> The distance involved 22 Mpc and that is very short.
> For 2 SN the redshift, z is used to calulate distance.
> This distance is roughly 100 Mpc and that is not what we want.
>
> What you want is to establish the relation between z versus distance
> and both should be measured indepently.

The whole idea of standard candles is that the apparent magnitude
measures the distance and z is easily observed.

> The document Hubble Constant (2010) page 33
> http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.1856v1.pdf
> shows that there are only 6 high quality SN measurements
> available to calculate Hubble constant.
>
> Figure 10 (page 77) shows distances based on TF relation
> for distances until 100 pc versus v. (using v=c*z you can
> back calculate z, which is observed)
>
> My point is that those distances are too short to demonstrate
> the non linearities involved.
> To be more specific: To calculate the parameters C, labda and k.

The idea is this: the Hubble constant is calculated at a redshift which
is large enough so that the cosmological redshift dominates over
peculiar motion, but small enough so that the relationship is still
linear. Objects at larger redshift are used for the non-linearity. But
the non-linearity can be observed even if the Hubble constant is
completely unknown.

> That is my whole point of the document in question.
> For an assumed age of 14 billion years you get for almost
> any combination of the cosmological parameters C, labda and k
> a different value of H0.
> Does that mean that only those combinations are valid that result in
> a 1/H0 value of 14 billion years ?

Yes, IF you know both the age and the Hubble constant. In practice, it
is almost the other way around: we now know the Hubble constant and the
other parameters quite well and the age is calculated, not assumed nor
measured "directly".

### eric gisse

Oct 15, 2011, 2:52:33 AM10/15/11
to
Nicolaas Vroom <nicolaa...@telenet.be> wrote in news:mt2.0-21457-
13186...@hydra.herts.ac.uk:

> "eric gisse" <jowr.pi...@gmail.com> schreef in bericht
> news:mt2.0-26635...@hydra.herts.ac.uk...
>> Nicolaas Vroom <nicolaa...@telenet.be> wrote in news:mt2.0-29733-
>> 13182...@hydra.herts.ac.uk:
>>
>>> The question than pops up to what extend based on actual
>>> observation you can use z as an indication for distance.
>>
>> It depends. Are we talking in 2011 with the cosmic distance ladder
>> firmly established and understood, or say 30 years ago when that
wasn't
>> true?
>>
>> In 2011 since we have the cosmological parameters well determined,
>> knowing the object's redshfit will give you its' distance.
>
> That is the whole point.
> If you look at document (2011) : Table1
> http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.4717v1.pdf
> Only for 2 SN the Tully Fisher relation is used to establish distance.
> The distance involved 22 Mpc and that is very short.
> For 2 SN the redshift, z is used to calulate distance.
> This distance is roughly 100 Mpc and that is not what we want.
>
> What you want is to establish the relation between z versus distance
> and both should be measured indepently.

I'll gladly discuss the subject but I'm not going to teach it.

>
> The document Hubble Constant (2010) page 33
> http://arxiv.org/PS_cache/arxiv/pdf/1004/1004.1856v1.pdf

Why do people keep posting links to the pdf?

Regardless, I've read this before. Its' a decent paper on the subject.

> shows that there are only 6 high quality SN measurements
> available to calculate Hubble constant.

And where does it say that? I wonder how you imagine we can get such a
small systematic error in H_0 with a sample size of six.

>
> Figure 10 (page 77) shows distances based on TF relation
> for distances until 100 pc versus v. (using v=c*z you can
> back calculate z, which is observed)
>
> My point is that those distances are too short to demonstrate
> the non linearities involved.
> To be more specific: To calculate the parameters C, labda and k.
>
> In general you cannot calculate distance as a function of
> using v=c*z (1) and v=H*d because (1) is only valid for small
> distances.

Welcome to the obvious problem of the past 40 years of cosmology.

The Hubble constant can be discerned through a few ways but currently
the best non-model dependent answer is only known to about 5% and that's
a factor of 3 better than we were able to say 15 years ago.

If you allow model dependent answers ala concordance cosmology via WMAP
or twenty) then the answer is known even better.

At the moment the answers match within their error margins so I'm not
abundantly concerned.

You just cited an 80 page paper, which I doubt you've fully much less
partially read, which discusses this in intricate detail.

>
>>> The question is to what extend can you use the parameter
>>> 1/H0 in order to calculate the age of the Universe.
>>
>> By itself? Rather minimally, as the Hubble constant alone does not
>> constrain the various other cosmological parameters that determine
the
>> age of the universe.
>
> That is my whole point of the document in question.

As far as I'm able to determine you are looking at an individual data
point in the vacuum while ignoring how it fits in with a far, far larger
body of work.

The hubble constant only makes sense in 2011 when you look at the big
picture. Look at the big picture. Stop screwing around with the small
picture.

> For an assumed age of 14 billion years you get for almost
> any combination of the cosmological parameters C, labda and k
> a different value of H0.
> Does that mean that only those combinations are valid that result in
> a 1/H0 value of 14 billion years ?

http://lambda.gsfc.nasa.gov/product/map/dr4/pub_papers/sevenyear/cosmolo
gy/wmap_7yr_cosmology.pdf

>
>>> What my results also show that the relation v = z * c
>>> is only true near the observer. For larger distances
>>> this relation is non linear.
>>
>> This is well known. Hubble's law only works out to about z=1.
> What exactly do you mean with: works out.

Hubble's law is an approximation. This is well known.

> Does this mean to calculate distances ?
> Does this mean to calculate age of Universe ?
> specific IMO this is not possible.

If you say so.

### Nicolaas Vroom

Oct 24, 2011, 9:53:00 AM10/24/11
to
schreef in bericht news:mt2.0-23482...@hydra.herts.ac.uk...
> In article <mt2.0-21457...@hydra.herts.ac.uk>, Nicolaas Vroom
> <nicolaa...@telenet.be> writes:
>
>> What you want is to establish the relation between z versus distance
>> and both should be measured indepently.
>
> The whole idea of standard candles is that the apparent magnitude
> measures the distance and z is easily observed.

I agree

>> My point is that those distances are too short to demonstrate
>> the non linearities involved.
>> To be more specific: To calculate the parameters C, labda and k.
>
> The idea is this: (1) the Hubble constant is calculated at a redshift
> which
> is large enough so that the cosmological redshift dominates over
> peculiar motion, but small enough so that the relationship is still
> linear. (2) Objects at larger redshift are used for the non-linearity.
> (3) But the non-linearity can be observed even if the Hubble
> constant is completely unknown.

I agree with (1)
The issues (2) and (3) are discussed in the following document 1
from 2003:
http://arxiv.org/PS_cache/astro-ph/pdf/0303/0303428v1.pdf
Fig 4 of that document demonstrates this non-linearity.

The same and even more data is shown the document 2 from 2011:
http://arxiv.org/PS_cache/arxiv/pdf/1104/1104.1443v1.pdf
at Fig 5 page 32.

Document 1 shows the relation between z and distance
when you assume z is linear scale.
Document 2 shows the relation between z and magtitude

Using the Friedmann equation I have tried to calculate
the parameters C, labda and k based on those two documents.
The results are at
http://users.telenet.be/nicvroom/friedmann's%20equation.htm#Q6
I have added a new question number 6.
The section shows two figures: 11 and 12.
Figure 11 shows the relation between z and magtitude.
The black line is the observed curve and is a copy of document 2.
Document 2 is the source document/data of also figure 12.
Figure 12 shows the relation between z and distance.
The black line is in good agreement with document 1.
Specific what Figure 12 shows is that the shape of the observed
curve is different as the shape of all calculated values for different
combinations of the parameters labda.
The parameter L has no direct influence on this shape.

What this means is that it is not possible to calculate the parameters
C, Labda and k based on the observations.

One possible explanation could be that for higher values of z
the ralation between magtitude and distance is different then
for low values of z

Nicolaas Vroom
http://users.telenet.be/nicvroom/

Oct 25, 2011, 2:39:43 AM10/25/11
to
In article <mt2.0-18379...@hydra.herts.ac.uk>, Nicolaas Vroom
<nicolaa...@telenet.be> writes:

> > The idea is this: (1) the Hubble constant is calculated at a redshift
> > which
> > is large enough so that the cosmological redshift dominates over
> > peculiar motion, but small enough so that the relationship is still
> > linear. (2) Objects at larger redshift are used for the non-linearity.
> > (3) But the non-linearity can be observed even if the Hubble
> > constant is completely unknown.
>
> I agree with (1)
> The issues (2) and (3) are discussed in the following document 1
> from 2003:
> http://arxiv.org/PS_cache/astro-ph/pdf/0303/0303428v1.pdf
> Fig 4 of that document demonstrates this non-linearity.

This non-linearity has been known for almost 100 years. There is no
"issue" with it; it's standard textbook stuff.

@ARTICLE {SRefsdalSdL67a,
AUTHOR = "Sjur Refsdal and Rolf Stabell and
F. G. de Lange",
TITLE = "Numerical calculations on relativistic
cosmological models",
JOURNAL = MRAS,
YEAR = "1967",
VOLUME = "71",
PAGES = "143"}

They use sigma and q instead of lambda and Omega, but the relationship
is simple (sigma = Omega/2 and lambda = sigma - q) and show how these
parameters can be used to calculate various observable quantities as a
function of redshift over a large redshift range into the very
non-linear regime. If one has good observational, it is easy to
determine the parameters via curve-fitting. All of the difficulties,
debate, discussion, problems and current work is involved in getting
good observational data, not interpreting them.

### Nicolaas Vroom

Nov 12, 2011, 12:18:40 PM11/12/11
to
schreef in bericht news:mt2.0-15665...@hydra.herts.ac.uk...
>
> This non-linearity has been known for almost 100 years. There is no
> "issue" with it; it's standard textbook stuff.
>
> Please check out this paper:
>
> @ARTICLE {SRefsdalSdL67a,
> AUTHOR = "Sjur Refsdal and Rolf Stabell and
> F. G. de Lange",
> TITLE = "Numerical calculations on relativistic
> cosmological models",
> JOURNAL = MRAS,
> YEAR = "1967",
> VOLUME = "71",
> PAGES = "143"}
>
> They use sigma and q instead of lambda and Omega, but the relationship
> is simple (sigma = Omega/2 and lambda = sigma - q) and show how these
> parameters can be used to calculate various observable quantities as a
> function of redshift over a large redshift range into the very
> non-linear regime. If one has good observational, it is easy to
> determine the parameters via curve-fitting. All of the difficulties,
> debate, discussion, problems and current work is involved in getting
> good observational data, not interpreting them.
>

The above mentioned document is online available at:
I have a problem with the parameter q.
The problem is that parameter is relative easy to calculate
using the Friedmann equation but difficult to calculate
using observations. Because the parameter is a function of:
distance, speed and acceleration.
Specific acceleration of a galaxy is difficult to observe.
http://users.telenet.be/nicvroom/friedmann's%20equation.htm#Q9

The most accurate observed function between magtitude
and z is available at:
http://arxiv.org/PS_cache/arxiv/pdf/1104/1104.1443v1.pdf
See figure 5 at page 32.
My simulations show the relation between x and z.
The issue is the relation between magtitude and x.
In order to calculate the magtitude i used the relation:
F = L / 4 pi d^2.
When you do that there is a mismatch between simulation
and observation for the larger z values.
http://users.telenet.be/nicvroom/friedmann's%20equation.htm#Q7
In order to try to get a better fit I tried: F = L / 4 pi d^3
The result is a better match for higher values of z.
Physical this means that galaxies further away are weaker as
expected.
I also tried: F = L / 4 pi d^2*(1+d).
This gives the best match for Labda = 0.05 and k=0

I also tried different ages for the Universe (28 and 42)
In fact that has no influence. See question 8
That raises the question if it is very difficult to calculate
the age of the Universe based observations of
the parameters z and m (or d)

Nicolaas Vroom
http://users.telenet.be/nicvroom/

Nov 14, 2011, 4:37:50 PM11/14/11
to
In article <mt2.0-18809...@hydra.herts.ac.uk>, Nicolaas Vroom
<nicolaa...@telenet.be> writes:

> The above mentioned document is online available at:
> I have a problem with the parameter q.
> The problem is that parameter is relative easy to calculate
> using the Friedmann equation but difficult to calculate
> using observations. Because the parameter is a function of:
> distance, speed and acceleration.

No-one measures q "directly". For that matter, no-one measures velocity
directly. If once assumes that the Friedmann-Lemaitre equation holds,
then it doesn't matter which parameters one uses since the conversions
are clear. If not, then you still have essentially measurements of
redshift, brightness and angular size---you still haven't measured
anything of interest "directly" but need a(nother) framework in which to
interpret the observations.

In other words, since q is Omega/2 - lambda, it doesn't matter if I
measure q or measure Omega and lambda and calculate q. In particular,
one can't say that the former is worse since one can't "directly"
measure q; one can't "directly" measure Omega and lambda either.

> Specific acceleration of a galaxy is difficult to observe.

See above.

> The most accurate observed function between magtitude
> and z is available at:
> http://arxiv.org/PS_cache/arxiv/pdf/1104/1104.1443v1.pdf
> See figure 5 at page 32.
> My simulations show the relation between x and z.
> The issue is the relation between magtitude and x.
> In order to calculate the magtitude i used the relation:
> F = L / 4 pi d^2.

What is d? It has to be the luminosity distance. Also, the equation
applies only for bolometric luminosity---not much use in practice. Look
into the K-correction, and also evolutionary corrections if the
luminosity of your objects changes with time.

> In order to try to get a better fit I tried: F = L / 4 pi d^3
> The result is a better match for higher values of z.

A wrong equation coincidentally cancels another known effect.

> Physical this means that galaxies further away are weaker as
> expected.
> I also tried: F = L / 4 pi d^2*(1+d).
> This gives the best match for Labda = 0.05 and k=0

Ditto.

> I also tried different ages for the Universe (28 and 42)
> In fact that has no influence. See question 8
> That raises the question if it is very difficult to calculate
> the age of the Universe based observations of
> the parameters z and m (or d)

No, it is trivial and has been known since the 1920s. Again, the
problem is getting good observations and interpreting them directly.
The theoretical background is textbook stuff.