New Math, Old Theories

[ImplicitDemands]

The calculations used for a) just two orbital objects had to reintegrate the lagrange coordinate (gravitational equilibrium between them) in order to continuously adjust the derivative for the same reason that b) the 3 body problem or even c) the theory of expansion, were posed in the first place. 

Point a)

Quote

In Newton's law, it is the proportionality constant connecting the gravitational force between two bodies with the product of their masses and the inverse square of their distance.

As you move about in the z plane not only does gravity's effects at a distance (dz/dt) have to be recalibrated (the gravitational constant as an integral of the inverse square law), but also the curvature of space has to be accounted for, that is because space is curved and not flat.

Quote

Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space

I was exposed to the same amount of algebra and trig that Newton was when he came up with calculus but unlike Newton I was also more informed about spacetime when I came up with how to use the sine function to map the entire surface volume of the sphere in order to use perspective of how circles placed anywhere on that surface (which can be made into spheres using the same .707sin(45deg) process) as a numerical value for objects falling along the z plane. The first trick to that was knowing that only 8 spheres could fit around the center without crossing into each others space. 3^2 also confirms that it can only be 8 on the outside plus one on the inside. The next iteration is trickier because the next 18 spheres don't fall along one entire sphere but parts of eight individual spheres whose surfaces are already partially occupied. If you could find out how to expand or shrink the circumferences of even the first eight outer spheres by the exact amount if they were to be placed in the only eight points about the original sphere's surface volume you'd still only have half of the puzzle in which I've memorized for my approach to deviating from trigonometry's two dimensional world, in a similar but wholly different way than Newton did when developing his derivations. 

Point b) This mathematics has a net of spheres along the z plane that can be used to calculate that curvature without needing to reincorporate the Lagrangian. 

Point c) As for discrepancies in the redshift coming from older galaxies which are further away, being greater than they should be, there's no need for expansion if the gravitational constant isn't mathematically expressible using any form of calculus. 

Quote

Doubly special relativity considers a deformation of the special relativistic kinematics parametrized by a high-energy scale, in such a way that it preserves a relativity principle. When this deformation is assumed to be applied to any interaction between particles one faces some inconsistencies. In order to avoid them, we propose a new perspective where the deformation affects only the interactions between elementary particles. A consequence of this proposal is that the deformation cannot modify the special relativistic energy-momentum relation of a particle.

 

Edited April 18 by ImplicitDemands

[Mordred]

You might try including the math you speak of. First off you seem to have missed the detail that spacetime curvature vs flat directly describes the null geodesics of massless particles such as photons through spacetime. Hence we can test the curvature term by looking for distortions in the CMB.

 

[ImplicitDemands]

10 hours ago, Mordred said:

You might try including the math you speak of. First off you seem to have missed the detail that spacetime curvature vs flat directly describes the null geodesics of massless particles such as photons through spacetime. Hence we can test the curvature term by looking for distortions in the CMB.

 

It is not just the curvature being wrong due to a miscalculation of how the grav const is effected by range due to perspectives. The positions of the objects in question (receding galaxies) effects this, why my math differs for these positions is that it takes into account not only the position in the half sphere as it curves representing the entire volume of space observed, it also adjusts for how perspective effects the circumference of that circular slice which can be factored into the grav const without calculus.

Alas, yes light wouldn't be effected by that grav const but but the origin of that light would be as well as the space it's traversed.

Edited April 18 by ImplicitDemands

[Mordred]

As I mentioned you would need the math to show this. You keep mentioning your math so you should already have the math handy for us to examine.

I can easily show you all the mathematics behind the FLRW metric but that wouldn't help determine why you have an issue with it.

If it's an issue with not knowing how to latex the math in let me know and I can demonstrate how our format uses \[\frac{1}{2}\,] I placed a comma in the last part to prevent it from activating. Your description of using spheres for example tells me you should have a spherical coordinate system with some constant of proportionality for the scale factor  however that's based on your description. I need your math for confirmation.

Edited April 18 by Mordred

[ImplicitDemands]

The cmb is sending light through a cone but the galaxy's actual position is about a half sphere

22 minutes ago, Mordred said:

As I mentioned you would need the math to show this. You keep mentioning your math so you should already have the math handy for us to examine.

I can easily show you all the mathematics behind the FLRW metric but that wouldn't help determine why you have an issue with it.

If it's an issue with not knowing how to latex the math in let me know and I can demonstrate how our format uses \[\frac{1}{2}\,] I placed a comma in the last part to prevent it from activating. Your description of using spheres for example tells me you should have a spherical coordinate system with some constant of proportionality for the scale factor  however that's based on your description. I need your math for confirmation.

When I'm in the mood 

Edited April 18 by ImplicitDemands

[swansont]

25 minutes ago, ImplicitDemands said:

When I'm in the mood 

!

Moderator Note

Please be in the mood the next time you post

 

[Mordred]

4 hours ago, ImplicitDemands said:

The cmb is sending light through a cone but the galaxy's actual position is about a half sphere

When I'm in the mood 

That comment makes no sense but so far without seeing your math. Nothing you have stated makes much sense.

The CMB is literally everywhere in our observable universe you can even hear it's static on older radios that don't filter it out.

It may surprise you to know that expansion has little to do with gravity but rather it's due to thermodynamics via the equations of state for each particle species.

https://en.m.wikipedia.org/wiki/Equation_of_state_(cosmology)

If you take a uniform mass distribution and apply Newtons Shell theorem gravity is zero.

Edited April 18 by Mordred

[ImplicitDemands]

On 4/18/2024 at 5:43 PM, Mordred said:

That comment makes no sense but so far without seeing your math. Nothing you have stated makes much sense.

The CMB is literally everywhere in our observable universe you can even hear it's static on older radios that don't filter it out.

It may surprise you to know that expansion has little to do with gravity but rather it's due to thermodynamics via the equations of state for each particle species.

https://en.m.wikipedia.org/wiki/Equation_of_state_(cosmology)

If you take a uniform mass distribution and apply Newtons Shell theorem gravity is zero.

Differential Calculus

Circle slice of a cone r=3in moves forward toward the wide end of the cone by one unit of 3 inches, A = pi(r^2) & A' = 2pi(r) x dA/dt; A'/A = (6pi x dA/dt)/(9pi)) -> (6pi x (3+3))/9pi = 2. That's 2r.

New Math

Which is fine for flat space but for curved space I have a different when it moves forward that same distance but also moves up and across (x&y) by a 45 degree angle because of the curve in the surface volume of the sphere across 2 radii or one circumference away from the sphere of origin (that's 3 iterations of spheres in which you can fit a total of 27 spheres where all of the spherical surfaces make contact). The difference between discrepancies in redshift of receding galaxies could be a simple matter of the difference between the shape of a cone and the shape of a half sphere. I can map the dimensions that are parametrized by curved space, I'm just dealing with a situation right now can you please be patient until my situation changes? 

Edited April 22 by ImplicitDemands

[ImplicitDemands]

1 hour ago, ImplicitDemands said:

Differential Calculus

Circle slice of a cone r=3in moves forward toward the wide end of the cone by one unit of 3 inches, A = pi(r^2) & A' = 2pi(r) x dA/dt; A'/A = (6pi x dA/dt)/(9pi)) -> (6pi x (3+3))/9pi = 2. That's 2r.

New Math

Which is fine for flat space but for curved space I have a different number (5.121) when it moves forward that same distance but also moves up and across (x&y) by a 45 degree angle because of the curve in the surface volume of the sphere across 2 radii or one circumference away from the sphere of origin (that's 3 iterations of spheres in which you can fit a total of 27 spheres where all of the spherical surfaces make contact). The difference between discrepancies in redshift of receding galaxies could be a simple matter of the difference between the shape of a cone and the shape of a half sphere. I can map the dimensions that are parametrized by curved space, I'm just dealing with a situation right now can you please be patient until my situation changes? 

There I even gave you the new radius produced by New Math (bolded and underlined). 

[Mordred]

So how does this relate to the gravitational constant or redshift ? Why wouldn't I just use a spherical coordinate system and apply a constant of proportionality via the scale factor "a" of the FLRW metric. That metric works regardless of the curvature term. It doesn't matter if spacetime is flat, positive curved or negative curved. 

 I can calculate the proper distance to any object. I can tell you what the CMB blackbody temperature is at any given cosmological redshift value. I can even modify for different expansion rates as new data comes in for matter, radiation density. Calculate the age of the universe, as well as predict the rates of volume change of our observable universe far far into the future provided the cosmological parameters continue to  evolve as current data show.

The math you have shown here doesn't give me that ability. So where is the advantage of using mathematics if I cannot derive critical functions used in Cosmology ?

here is a sample

/[{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline z&T (Gy)&R (Gly)&D_{now} (Gly)&Temp(K) \\ \hline 1.09e+3&3.72e-4&6.27e-4&4.53e+1&2.97e+3\\ \hline 3.39e+2&2.49e-3&3.95e-3&4.42e+1&9.27e+2\\ \hline 1.05e+2&1.53e-2&2.34e-2&4.20e+1&2.89e+2\\ \hline 3.20e+1&9.01e-2&1.36e-1&3.81e+1&9.00e+1\\ \hline 9.29e+0&5.22e-1&7.84e-1&3.09e+1&2.81e+1\\ \hline 2.21e+0&2.98e+0&4.37e+0&1.83e+1&8.74e+0\\ \hline 0.00e+0&1.38e+1&1.44e+1&0.00e+0&2.73e+0\\ \hline -6.88e-1&3.30e+1&1.73e+1&1.12e+1&8.49e-1\\ \hline -8.68e-1&4.79e+1&1.74e+1&1.43e+1&3.59e-1\\ \hline -9.44e-1&6.28e+1&1.74e+1&1.56e+1&1.52e-1\\ \hline -9.76e-1&7.77e+1&1.74e+1&1.61e+1&6.44e-2\\ \hline -9.90e-1&9.27e+1&1.74e+1&1.64e+1&2.73e-2\\ \hline \end{array}}\]

 

this is from redshift z= 1100 to far into the future I kept the options small simply to demonstrate the calculator in my signature is far far more capable in the chart options. I can even graph each column via the same calculator as well as change my range to just after inflation forward into the future.

That is an example of the capabilities needed to to be useful for cosmologists. 

Edited April 22 by Mordred

[ImplicitDemands]

3 hours ago, Mordred said:

So how does this relate to the gravitational constant or redshift ? Why wouldn't I just use a spherical coordinate system and apply a constant of proportionality via the scale factor "a" of the FLRW metric. That metric works regardless of the curvature term. It doesn't matter if spacetime is flat, positive curved or negative curved. 

 I can calculate the proper distance to any object. I can tell you what the CMB blackbody temperature is at any given cosmological redshift value. I can even modify for different expansion rates as new data comes in for matter, radiation density. Calculate the age of the universe, as well as predict the rates of volume change of our observable universe far far into the future provided the cosmological parameters continue to  evolve as current data show.

The math you have shown here doesn't give me that ability. So where is the advantage of using mathematics if I cannot derive critical functions used in Cosmology ?

here is a sample

/[{\scriptsize

z1.09e+33.39e+21.05e+23.20e+19.29e+02.21e+00.00e+0−6.88e−1−8.68e−1−9.44e−1−9.76e−1−9.90e−1T(Gy)3.72e−42.49e−31.53e−29.01e−25.22e−12.98e+01.38e+13.30e+14.79e+16.28e+17.77e+19.27e+1R(Gly)6.27e−43.95e−32.34e−21.36e−17.84e−14.37e+01.44e+11.73e+11.74e+11.74e+11.74e+11.74e+1Dnow(Gly)4.53e+14.42e+14.20e+13.81e+13.09e+11.83e+10.00e+01.12e+11.43e+11.56e+11.61e+11.64e+1Temp(K)2.97e+39.27e+22.89e+29.00e+12.81e+18.74e+02.73e+08.49e−13.59e−11.52e−16.44e−22.73e−2

}\]

 

 

this is from redshift z= 1100 to far into the future I kept the options small simply to demonstrate the calculator in my signature is far far more capable in the chart options. I can even graph each column via the same calculator as well as change my range to just after inflation forward into the future.

That is an example of the capabilities needed to to be useful for cosmologists. 

Because the issue is that objects in proportion of space shaped like a half sphere are not constant, a cone facing upright has a perfect linear v, a half sphere facing upright is more of a u shape. Objects don't increase in scale by the same amount as they approach. If the sphere gets larger after repeating what I just did, the next time the amount in which it scales up changes. The third iteration is based off of the second iteration not the first iteration. For the redshift problem much like in the 3 body problem, Galaxy A's distance from Galaxy B depends heavily upon two different angles where this math deviates from calculus in the way I demonstrated before. On top of this, you need to know how to use that half sphere metric in order to factor in how the relative perspective of the gravitational constant is effected.  Differences in the calculus versus observation, depending upon how space is actually shaped, could produce some affects associated with the wave function.

Edited April 23 by ImplicitDemands

[Mordred]

  The standard model knows how to deal with the geometry of spheres, cones, or any other volume determined by a shape.

Spacetime curvature doesn't describe its volumetric shape. It describes  its affect on the geodesic equation for photon paths in regards to redshift or any signals we recieve due to particle paths. Hence it describes spacetime in geometric terms. With invariance under the metric choice the coordinate choice doesn't particularly matter. 

 The use of differentials a huge part of GR as a conformal metric, so your using differentials is nothing new. However one can easily also choose to use integrals as per the QFT related theories such as loop quantum gravity. So that choice doesn't matter either.  The FLRW metric already includes the radius for spheres in its metric. That is how the scale factor "a" is determined for the volume element but equally important is that the formula includes the velocity and the acceleration terms to describe expansion rates.  However the volume element is the easiest thing to describe mathematically speaking with regards to the Observable universe. The FLRW metric further employs thermodynamics to determine what causes the expansion rates.

The math you have posted here simply doesn't have that capability. We already know how to handle spheres, we already know to to ray cast spheres which would have equivalency with cones or even just use cone segments.

That stuff is covered in differential calculus which is already employed. Differential geometry is one of the most used tools used in physics right along side with integrals The choice doesn't matter as its trivial to convert between them

How does relations you showed here

11 hours ago, ImplicitDemands said:

Differential Calculus

Circle slice of a cone r=3in moves forward toward the wide end of the cone by one unit of 3 inches, A = pi(r^2) & A' = 2pi(r) x dA/dt; A'/A = (6pi x dA/dt)/(9pi)) -> (6pi x (3+3))/9pi = 2. That's 2r.

 

add anything at all we don't already employ where appropriate ? 

1 hour ago, ImplicitDemands said:

 gravitational constant is effected.  Differences in the calculus versus observation, depending upon how space is actually shaped, could produce some affects associated with the wave function.

As mentioned we already take into consideration optical physics via differential calculus. It's a huge part of gravitational lensing for example. We even treat under the entire EM spectrum. The techniques involved in that is Huge part of spectrum analysis. Used all the time to for example to determine  how much hydrogen is in a region via the 21 cm line in spectrography.  Or determining or geometry (null geodesic paths) by looking for distortions caused by any non flat spacetime in the CMB.

Once again it's one of the commonly used tools in observation. Nothing in your relations adds anything we don't already know how to do. That obviously includes wavefunctions. Which is a huge part of luminosity  which is a useful tool in and of itself.

For that we can even check for different expansion rates in a region or difference in gravitational potential of different  regions via the Sache Wolfe effect. All that involves optics 

 

Edited April 23 by Mordred

[ImplicitDemands]

17 hours ago, Mordred said:

The standard model knows how to deal with the geometry of spheres, cones, or any other volume determined by a shape.

I know I know. If you are curious what exactly it is I am doing to place a second circle atop the surface of a sphere with r=3 at the top right to get the r=5.121 value, or at what point the two given volumetric surfaces make contact, I assure you it is not a trigometric function that has been discovered yet or it would have been covered in my education which has been extensively mathematical oriented. Derivations were unknown until discovered. I'm sorry, I must correct something I said earlier about how many spheres can make contact about the surface volume of the first iteration. constant at any iteration at any 3-sided pyramidal angle gravitational interaction though I said 9 but but if you look at the values you will find that you can only wrap two in the front of the first sphere and one in the back. Much like the up and down quarks in a nucleus. 

It is the dimensions being used in the mathematics of motion other than calculus which also does this but it misses how relative perspective factors into the parameters governing change over time. One needs to know how to factor (not integrate as if we were doing calculus) in the gravitational variance. As all shapes have a triangle at their core, trigonometry is really the basis of all geometry, and therefore change over time. If I were speaking to Newton I would to stress the importance taking a second or a third look at trigonometry, considering it as the primary basis for the geometry of motion. Then, he might have had a more complete representation of the third dimension, so as to understand why things became so murky when orbital bodies exceeded two, using his process. 

As an engineer, it is easier when looking at gravity in this sense, to affect my thinking about how time may be why some atomic mediums can react to light passing through, and other do not. And by react I mean change energy state, for one. Maybe not a wave fluctuation but a velocity governing time in a relativistic sense. Note, this is not what is written here about observational variance in the gravitational interaction due to relative perspective, there is a distinction between perspective and the interactions. I'm trying to clarify that this math with no name IS the former. 

Edited April 23 by ImplicitDemands

[Mordred]

2 hours ago, ImplicitDemands said:

I know I know. If you are curious what exactly it is I am doing to place a second circle atop the surface of a sphere with r=3 at the top right to get the r=5.121 value, or at what point the two given volumetric surfaces make contact, I assure you it is not a trigometric function that has been discovered yet or it would have been covered in my education

It is the dimensions being used in the mathematics of motion other than calculus which also does this but it misses how relative perspective factors into the parameters governing change over time. One needs to know how to factor (not integrate as if we were doing calculus) in the gravitational variance. As all shapes have a triangle at their core, trigonometry is really the basis of all geometry, a

Well as an Engineer you certainly know that your field requires mathematical rigor. It's no different for physicist or mathematical theory.

So if it's your goal to present some new trig function and have it gain weight in the Professional circles then you will need a mathematical proof. One that doesn't rely on words /pictures or descriptives. Anything less simply wouldn't cut it.

I'm sure you can recognize the need fir that  no forum has any particular influence in the Professional circles. Forums are useful but mainly to help others learn . Nothing discussed in a forum will ever alter how the scientific or mathematical community does things. That requires a professional peer review paper examined by other experts.

The mathematical proof would be needed for that.

For example every single physics formula has a corresponding mathematical proof no formula ever gets accepted without one.

Edited April 23 by Mordred

[ImplicitDemands]

4 hours ago, Mordred said:

The mathematical proof would be needed for that.

You understand that this math is designed to be factored into change the gravitational variance along a scale that must be adjusted for f(n) each time (n)sin(any deg) changes the radius of a circle along the z plane. With both the degree angle changing and the radius changing so there can be no constant which is why it is variant. A forum I feel is as good a place as any. Accredited or not, if it works to predict the orbital behavior of planetary objects in relation to one another it works. However, it is getting late. 

Edited April 24 by ImplicitDemands

[Mordred]

yes I did understand that but I'm trying to ascertain your eventual goals with this to provide direction for improvement. If you think about we do much the same with the use of the scale factor under the FLRW However a key point is that G is a constant under the FLRW so your going to have to explain why you feel G would change as a result of change in radius ?

[ImplicitDemands]

11 hours ago, Mordred said:

yes I did understand that but I'm trying to ascertain your eventual goals with this to provide direction for improvement. If you think about we do much the same with the use of the scale factor under the FLRW However a key point is that G is a constant under the FLRW so your going to have to explain why you feel G would change as a result of change in radius ?

Because the gravitational constant appears smaller at a distance. This dynamic is literally missing dimensions when you use calculus, it is the difference between how the cylindrical portion of the cone constantly decreases from base to dip to how that same cylinder constantly curve from top to bottom on a half sphere. 

Look, if the universe has said volume, is that the volume of a cube or of a sphere? 

Edited April 24 by ImplicitDemands

[Mordred]

How does a coupling constant appear smaller ?

If you apply \(F=G\frac{m_1m_2}{r^2}\) the  coupling constant remains constant what changes is the force exerted by the coupling between two masses as a function of radius. Not the coupling constant itself.

We describe our observable universe itself in the FLRW metric we know the universe extends beyond that it could be finite or infinite  as we can never measure beyond that we deal with what we can Observe and measure. (Region of shared causality)

Edited April 24 by Mordred

[ImplicitDemands]

3 hours ago, Mordred said:

How does a coupling constant appear smaller ?

If you apply F=Gm1m2r2 the  coupling constant remains constant what changes is the force exerted by the coupling between two masses as a function of radius. Not the coupling constant itself.

We describe our observable universe itself in the FLRW metric we know the universe extends beyond that it could be finite or infinite  as we can never measure beyond that we deal with what we can Observe and measure. (Region of shared causality)

You don't know whether the redshift is higher than it should be or not if the problem is a lack of some sinusoidal application to factor in the proximity values of galaxy A and B relative to the observer as I explained earlier. So you don't even know whether the cosmic event horizon or CMBR is the oldest light that's had time to reach the lens or whether it is just a blending that makes objects invisible as the tip of the cone becomes infinitesimal. 

Edited April 24 by ImplicitDemands

[Mordred]

The redshift has little to do with gravitational constant and we have means of testing redshift by understanding the processes involved. We can for example examine hydrogen which is one of most common elements in our universe and using spectrography. There is nothing random that isn't cross checked by numerous means involving redshift. We don't even rely on it as our only means of distance calculation. Quite frankly no one method works for every distance range. A huge portion of papers can be found studying the accuracy of redshift at different ranges and those cross checks using other means such as interstellar parallax.

Same applies to luminosity distance.

By the way the redshift formula you find in textbooks is only useful at short distances cosmological scale. 

 

Edited April 24 by Mordred

[Mordred]

This is the FLRW metric

\[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2\]

\[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\]

This is the redshift equation(cosmological) that gets used at all ranges as it takes the evolution of matter, radiation and Lambda.

\[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\]

 

Edited April 24 by Mordred

[ImplicitDemands]

On 4/22/2024 at 11:06 AM, ImplicitDemands said:

Circle slice of a cone r=3in moves forward toward the wide end of the cone by one unit of 3 inches, A = pi(r^2) & A' = 2pi(r) x dA/dt; A'/A = (6pi x dA/dt)/(9pi)) -> (6pi x (3))/9pi = 2. That's 2r

A=pi*3^2=9pi; A'=(2*3)^(2-1)pi the integral is (6pi/(1+1))^(1+1)=(6/2)^2 * pi=9pi

If you have a radius of R around an inner circle with a radius of r and wanted to maximize the amount of space in that outer ring R and minimize what is in the center circle r you would say lim  x->infinity f(R)=r+r/x ; x=r, meaning that R=r+1 I suppose Newton is some shadow program working on maths adjacent to me 

Also:

Having to make some corrections here, the 5.121 number extruded another radius from the original radius of 3 inches. So let's so how close we are with 2.121 (I just went back in and realized I did my own math wrong it was 2.74 something that was the point). And yes I realized you can still fit 9 spheres inside the second iteration without all of their surfaces touching so it is like squaring the volume of a sphere to get a hypersphere. Shouldn't second guess myself. 

 

Edited April 25 by ImplicitDemands

[Mordred]

fair enough, something to keep in mind if your looking at cosmological redshift is that the expansion rates are not linear. The equation above shows this as the resultant is to determine the Hubble value at a given Z compared to the value today. The relations under the square root is the evolution of the energy density for matter, radiation and Lambda. You can learn these here. 

https://en.wikipedia.org/wiki/Equation_of_state_(cosmology)

this related to the FLRW acecleration equations. described here

https://en.wikipedia.org/wiki/Friedmann_equations

that link supplies some very useful integrals with regards to the scale factor 

the evolution of the scale factor "a" using the above relations gives

\[\frac{\ddot{a}}{a}=-\frac{4G}{3}(\rho+3P)+\frac{\Lambda}{3}\]

however to get the FLRW metric cosmological redshift equation you will also need the Newton weak field limit treatments as per GR. Particularly for curvature K=0

if your interested in that let me know and I'll provide more details

 

Edited April 25 by Mordred

[ImplicitDemands]

28 minutes ago, Mordred said:

fair enough, something to keep in mind if your looking at cosmological redshift is that the expansion rates are not linear. The equation above shows this as the resultant is to determine the Hubble value at a given Z compared to the value today. The relations under the square root is the evolution of the energy density for matter, radiation and Lambda. You can learn these here. 

https://en.wikipedia.org/wiki/Equation_of_state_(cosmology)

this related to the FLRW acecleration equations. described here

https://en.wikipedia.org/wiki/Friedmann_equations

that link supplies some very useful integrals with regards to the scale factor 

 

His problem when you consider that a 9pi area over the integral of its radius is just 4 which is exactly what he's factoring in for the gravitational constant. And you should really use my optimization of the surface volume about an origin sphere instead because it isn't factoring in 4 it is factoring in 3.87553041018 that's only a 1.3% difference which adds up when you consider all of the angular momenta involved in the observation of gravitational bodies. 

Edited April 25 by ImplicitDemands

[Mordred]

If your certain of your equations and it's validity I'm sure your going to want to test them. If you think about it I provided the essential equations to do just that with a given dataset such as Planck. 

I certainly do when I model build or simply test and cross check any new relations/interactions. Those equations apply LCDM.  to the cosmological redshift. As far as a new value of G well all I can say to that is good freaking luck on that score with what you have shown so far. 

this is a listing of the various types of studies and results form them for variations of G tests for spatial dependence is page 200 onward

http://www2.fisica.unlp.edu.ar/materias/FisGral2semestre2/Gillies.pdf

 

Edited April 25 by Mordred