Mordred

Might be an helpful eye opener to perhaps look into how work is defined in probability terms as well as how work is defined in thermodynamics and statistical mechanics as opposed to how work is defined in the transfer for force. You can keep the energy definition the same the change is in how work is defined. Example under QFT ( probability currents) The first three cases work is defined as the expected energy transfer over possible states. In the classical latter case its the force applied over displacement.

Unfortunately I didn't spot that till today. I knew there was something bothering me but couldn't put my finger on it till today.

Both methodologies take advantage that temperature scales as the inverse of the scale factor. Both methodologies applies the thermodynamic laws. In the paper its mentioned that its methodology is using local measurements with your use of beta function B= v/c with no further correction correlations beyond Hubble horizon as well the numbers you used in your article for density calculations you would also be local. If you were using say z=1100 none of the density numbers would match. The paper itself doesn't go into which formulas to employ but they would likely involve Maxwell Boltzmann and Bose Einstein. Both methodologies will obviously involve the Boltzmann constant. They also will both require the volume element being examined and if you think about it your employing the critical density formula relations albeit your method isnt total density. If it was then one of your formulas is wrong as its missing a (3) in its derivative. I will let you see which formula Im referring to in your paper. It may help me relate to which formulas you have derived yourself and which formulas you have used from other resources such as AI or other papers and your understanding of those formulas incorperated that were derived not using RG.

Here's a more recent method for using CMB temp to determine Hubble constant. https://arxiv.org/pdf/1910.09881 Its a quick breakdown of the methodology but has the basics of the general idea. Other methods will typically include utilizing those previous mentioned equations in particular Bose Einstein.

Forgot to add that using the above relations one can use arbitrarily use fermions to estimate Hubble constant. This is often suggested for the dominant era.

When reading a lot of papers one often misses critical details a very common setting to simplify the mathematics is the following \[ [distance] = [time] = [energy]^{-1}= [mass]^{-1}= [temperature]^{−1}\] Both the Bose Einstein and Fermi-Dirac statistics directly apply the Boltzmann constant. The above relations are still applied they are simply normalized. The difference is that from the two above relations as well as Maxwell Boltzmann for thermodynamic effects one can develop a full equation that correlates to the contributions of any number of particle species including those that do not involve the fine structure constant interactions via a particles effective degrees of freedom. The Big Bang nucleosynthesis mathematics employ the above relations to encompass the entirety of the standard model. Using the first two statistics one can further calculate the number density of any particle species and apply those relations to QM as well as QFT. In point of detail QFT has an equivalent form of all the above. One can literally use the above to throw whatever any particular particle and even atoms at a region and make reasonable predictions of what would occur. The above highlights the flexibility and the reason why the complexities develop. It is when one must get into greater details to extract different distributions and dynamics. LCDM for example is a rather HUGE set of theories and methodologies that it has incorporated into its model. There is always a good reason for doing so when you get into the nitty gritty to understand what those reasons are and how they develop via mathematical proofs. That is when one truly learns how interconnected different theories truly are. I'm sure you've seen the equations I'm referring to but just in case. Bose Einstein Statistics \[n_i = \frac {g_i} {e^{(\varepsilon_i-\mu)/kT} - 1}\] Fermi-Dirac statistics \[n_i = \frac{g_i}{e^{(\epsilon_i-\mu) / k T} + 1}\] Maxwell Boltzmann \[\frac{N_i}{N} = \frac {g_i} {e^{(\epsilon_i-\mu)/kT}} = \frac{g_i e^{-\epsilon_i/kT}}{Z}\] the nice thing about these relations is they also readily work with the Saha equations. Using the above one can determine and equation when all particles are in thermal equilibrium and through derivatives from the above derive when each species drops out of thermal equilibrium when you also include expansion rates Saha equations being applicable to atoms as opposed to particles. As mentioned previous I will often throw in counter arguments. Range of applicability in this case. When I get a chance I will dig a decent one up if I recall Zibens had a treatment sometime in late 2000's.

With regards to Hubble constant and the methodology of using the CMB blackbody temperature to determine Hubble constant. This isn't a new idea. There are numerous peer reviewed papers suggesting this possibility. Though in some cases the peer reviewed methodologies have a less involved first order approximation. I would recommend you straighten out your neutrino statement prior to your formulas involving the Hubble constant calculstions. The Cosmic neutrino background is a different blackbody temperature ( less than 2.73 Kelvin) but does not contribute to the photon blackbody temperature. If you think about the properties of neutrinos and the definition of Blackbody temperature this should be obvious as to the reason why this is the case. Do not confuse blackbody temperature with thermal temperature which is the average kinetic energy. Blackbody temperature is not thermal temperature. ( I mention this with regards to the distinction of the CMB with the CNB) Mathematically I dont see why your CMB temperature calculation regarding photon density will not work but you could have simplified the calculation by using baryon to photon density ratio. Avoids those clunky numbers such as the gravitational constant and Stefen Boltzmann. I should add those other papers have substantially greater weight of credibility as they also examine factors such as potential error margins when measurements are concerned and how to potentially correct for them. Those equations above should also vary accordingly as the Hubble constant varies from what I can determine

I didn't comment on this earlier. One of the most valuable lessons I ever received in my formal training was a good theorist will always try to prove their own theories wrong. It is the only path to developing a robust theory is continous examination finding flaws and then making the needed improvements to correct those flaws as well as looking for ways to improve any theory. On the quintessence case the WMAP evidence as well as other datasets were showing a non varying cosmological constant as well as incorrect e-folds. For that particular paper it was better to simply start over. Lol lets just say Ive gotten extremely good at proving any theory I develop wrong. I spend a far greater portion of time doing so than in its development.

Ah that particular degeneracy isn't something I've made much of a study of. Post Newtonian equations however are common to spectography applications as well a GW waves which was why I asked for clarification. 1PN being specifically the velocity element. Its more a topic of Astrophysics than it is of cosmology though the two branches are closely related.

I think what may help you is to provide you a more detailed understanding on the field of spectography used in Cosmology applications. Many of formulas used are not quite the same as the FLRW metric standard form for example. In particular the gravitational lensing formulas can be rather daunting when applied to spectographyy without a fundamental understanding of the particular forms of standard equations used in Spectography. When I get a chance I will post some of the more commonly applied equations and factors that deal with.

I assume 1 PN is post Newtonian coefficients if so that is not actually a bad route it commonly used in cosmographic equations for model independency particularly when you apply higher order coefficients. I would recommend in the testing stages you restrict your dataset comparisons to below 1.4 Z to avoid the corrections needed beyond that ( previously mentioned this thread.) Have you considered applying your methodology to a system where the barycenter is not central to the star ? for example the barycenter between Jupiter and the sun. That is a dynamic that is safely ignored in your mercury sun system. Another potential stage being metallicity distributions of light to heavier elements around an orbiting body such as a galaxy or plasma distributions remaining after Poynting vector removal via solar winds. The latter will relate to what types of planets form and in what orbital range is more likely. (specifically why metal heavy planets form near a star while gas giants form further away prior to migration trends inward. Another set of tests could also be a system where frame dragging becomes an issue lets for example what would occur to Mercruy's orbit if the sun were to rotate at 0.5 c. (there are datasets of systems with this dynamic). GPS data could also prov useful to you as a test as to why each satellite requires its own calibration setup for its particular orbit etc

The software I currently have access to is specifically designed to utilize a specific set of theories pertaining to Spectrographic grating. It is one of those rather specific application theories that is seldom heard of. In this particular case Kogelnik's Coupled wave theory coupled with the more commonly known Braggs law as applicable to Grotian diagrams. Though the real challenge isn't simply the grating but the noise reduction due to atmospheric conditions as well as the peculiar motion of the observatory ( dipole anistrophies etc) to filtering intervening plasma etc. I take it you didn't notice this portion of my last post. In essence every equation in mathematics is a form of mapping as they can all be graphed. So I was curious how your philosophy or ontology would address that statement. This statement from your last post isn't quite accurate. It is mathematically impossible to separate the true orbital velocity β from the inclination i using strictly 1D. Its true I wouldn't use strictly one dimensional related mathematics ( dimensionality being the number of effective degrees of freedom ). However using time elapsed spectography it is rather easy to determine the true orbital velocity. Doppler shift effects on spectograph datasets is easily identifiable. The challenge is more causation. Specifically separating gravitational, cosmological redshift from Doppler redshift. However this is where range calibrations are commonly used those type of calibrations inherently gets incorperated into the filtering software algorithms.

If you believe I was only describing the usefulness of geometry for engineering and plotting's then I obviously did not explain my stance well enough. It is also a powerful to to make predictions . Model development, calibrations of measurement equipment etc etc etc. I couldn't even do my current job of calibrating the telescope spectrographic equipment at the University of the Caribou which is roughly the size as the one used by Hubble. That obviously involves fully understanding how light behaves and how to employ gratings for frequency separation prior to the collimater Lmao I even use geometry on MRI's that I've been involved in calibrating where I also need to identify and track diffraction angles . Nor could I have done written my dissertation way back when the only decent dataset I had to work with was COBE. With regards to BAO measurements looking for signatures of quintessence at that time. My dissertation was long ago shown incorrect when WMAP findings got released ( didn't have sufficient E-folds). Not a biggie part of science. Tell me how do you think the vast majority of all physics equations got developed if not through the use of geometric relations ? lol for that matter your methodology includes geometric relations one example being orthogonality. The question I really have is why you would feel mappings is ontologically wrong. Its a very versatile tool used in every day industries not exclusive to physics. LOL every graph you have posted here is a form of mapping. Consider this then. every varying relation can be graphed. Mappings are inevitable as a result. Spacetime is a flexible tool for any mapping translations involving a volume with varying time as part of its mappings. However if you have one relationship to another that is varying over some other value that too can be graphed with or without any coordinate basis and subsequently you have a form of mapping notice the above applies to all mathematics

One of the problems of relying too much on minimalizations is you tend to overlook critical details. For example and I honestly hope the OP has considered the following and at some point made the necessary corrections is the vastness of our universe with regards to the expansion history. Take the formula which is his principle formula \[\Beta=\frac{v}{c}\] works great for near field measurements however once you hit the Hubble horizon then recessive velocity becomes greater than c. There is a means to make corrections for this but SR cannot be used to describe the distance relations of our universe not in its entirety. GR also has to account for the expansion history. Expansion itself affects any observation methodology. It affects redshift, luminosity distance, angular diameter distance, Tully Fisher relations and proper time calculations. As some key examples Angular diameter distance for example has a rather unexpected side effect caused by expansion regarding time of emission to time of signal received. The apparent diameter size of the object being measured increases the further away you go at redshift 4.9 this leads to a decrease in the distance calculations past 4.9. Without making necessary adjustments. Here's a quick list of adjustments needed for expansion. the Hubble parameter can be written as \[H=\frac{d}{dt}ln(\frac{a(t)}{a_0}=\frac{d}{dt}ln(\frac{1}{1+z})=\frac{-1}{1+z}\frac{dz}{dt}\] look back time given as \[t=\int^{t(a)}_0\frac{d\acute{a}}{\acute{\dot{a}}}\] \[\frac{dt}{dz}=H_0^{-1}\frac{-1}{1+z}\frac{1}{[\Omega_{rad}(1+z^4)+\Omega^0_m(1=z0^3+\Omega^0_k(1+z)^2+\Omega_\Lambda^0]^{1/2}}\] \[t_0-t=h_1\int^z_0\frac{\acute{dz}}{(1+\acute{z})[\Omega^0_{rad}(1+\acute{z})^4+\Omega^0_m(1+\acute{z})^3=\Omega^0_k(1+\acute{z})^2+\Omega^0_\Lambda]^{1/2}}\] second order Luminosity distance full integral \[D_L(z)=(1+z)\cdot D_M(z)\] where \(D_M(z)\) is the transverse commoving distance Universe with arbitrary curvature \[d_L(z)=\frac{c}{H_0}\frac{(1+z)}{\sqrt{|\Omega_k|}}[sinn \sqrt{|\Omega_k|}]\int^z_0\frac{\acute{z}}{E(\acute{z})}\] sinn(x) defined as sin(x) when \(\Omega_k<0\), sinh(x) when \(\Omega_k>0\), x when \(\Omega_k=0\) Expansion function (dimensionless Hubble parameter) \[E(z)=\sqrt{\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z^2)+\Omega_\Lambda}\] modern times radiation is negligible, and for k=0 simplifies to \[D_L(z)=\frac{c(1+z)}{H_0}\int^z_0 \frac{d\acute{z}}{\sqrt{\Omega_M(1+\acute{z})^3+\Omega_\Lambda}}\] angular diameter distance reprocity relation \[D_A(z)=\frac{d_L(z)}{(1+z)^2}\] Angular diameter distance integral \[d_A(z)=\frac{c}{\sqrt{|\Omega_{k,o}|H_o(1+z)}} \cdot S_k[H_o\sqrt{|\Omega_{k,o}|} \int^z_0 \frac{dz}{H(z)}\] \[S_k(x)=\begin{cases}sin(x)&k>0\\x&k=0\\sinh&k<0\end{cases}\] commoving distance \[D_c =\frac{c}{H_0} \int^z_0 \frac{d\acute{z}} {E(\acute{z}) }\] the relation to all the above is \[E_Z=[\Omega_R(1+z)^4+\Omega_m(1+z)^3+\Omega_k(1+z)^2+\Omega_\Lambda]^{1/2}\] All the second order adjustments above use this relation for measurements beyond Hubble horizon. It factors in expansion history of each equation of state to determine what measurement adjustments are needed. The real beauty is the above equations will work regardless of any spacetime curvature. Now leets try an example of a Down on Earth example. Particle accelerators. If one wishes to develop equations of motion for particles being accelerated is doesn't make sense to use the same coordinates used in Cosmology applications. This is where a new coordinate system was designed using the tools of GR as well as those of geometry. In particular applying Frenet-Serret formulas. Without going through all the derivatives one can arrive at the equations of motion Langrene curvilinear coordinate beam dynamic Langrangian \[\mathcal{L}=-mc^2\sqrt{1-\frac{1}{c^2}(\dot{x}^2+\dot{y}^2+h^2\dot{z}^2)}+e(\dot{x}A_x+\dot{y}A_y+h\dot{z}A_z)=-e\phi\] part of the derivatives used in that last expression involves Floquet theory. "In the quantum world, where the linearity of the Schrodinger equation is guaranteed from the start, the Floquet theory applies whenever the Hamiltonian governing the system is time-periodiodic. quote from article below https://www.ggi.infn.it/sft/SFT_2019/LectureNotes/Santoro.pdf The real beauty of Frenet-Seret equations is that they are well designed for Helical motion. The above are all examples of where Geometric relations are incredibly useful in problem solving so to ignore geometry is akin to throwing away one of the more versatile tools in a physicists tool pouch I for one am absolutely happy with that definition it applies at all levels of physics and has been incredibly well tested throughout the entire of physics. lol to change the definition of energy would literally mean a complete rewrite of all physics as well as engineering related equations

using energy to describe changes of angles does not conform to the above definition for energy not without causation due to some application of force.

Judging from your posts Its obvious to me your not recognizing the true power and versatility of geometry. Anytime I see any new theory that a member is developing I always point out the need for a geometry the second most common recommendation I mention is " Comparison against existing models and theories or methodologies. Comparisons isn't strictly accuracy but also includes flexibility of application and ease of use. ( Your previous mentioned Occam's razor ). Now in order help others develop these skills I often have to take the opposite stance in discussions. So for this post I am going to use this technique by detailing the power of geometry and the usefulness of mappings. I will endeavor to keep this as simple as possible. I mentioned before the main goal isn't just making accurate predictions that is fundamentally just a confirmation that the mathematics your employing has some measure of validity. I will be employing the devils advocate in future posts just so you are forewarned and are now aware of what the reasons is. Hopefully nothing I state gets misconstrued on any personal level (All too common in discussions I've had with other members banned or otherwise.) In my professional real world experience the single most important tool to the all the work I've ever done is graphs. If you can't interpret a graph of results from measurements you will never get work as a physicist plain and simple, no exceptions in my experience. Doesn't make any difference what field of physics your applying. A graph however is not restricted to (x, y ,z etc) any parameter can be used as a replacement. Manifolds in particular employ the fact that x, y and z are convenient labels nothing more. ( coordinate basis). One can arbitrarily graph the number of apples compared to the number of oranges grown on a time dependent graph where coordinates serve zero practical application. Yes I have noted some of your articles included a few graph comparisions. Were comparing flexibility of methodology specifically spacetime vs strictly energy for the mathematics below I will restrict myself to a simple Newtonian 3D geometry to demonstrate the flexibility. for the start I will use basic Euclid geometry \(ds^2=x^2+y^2+z^2\) Nothing fancy about that, as you stated its simply describing our container. Yet that container has incredible usage as an aid to visualize, and in many ways simplify an incredible range of mathematical relations. However you don't need to stick to those geometric relations. I could very well be comparing apples, oranges, grapes.(apples, oranges, grapes). LOL we all know of graphs that have nothing to do with geometry... or at least should. That being said lets play with the above geometry. We all know you can assign any scalar value to any coordinate that's trivial. One overlooked flexibility is the " visual aid ". Any graph whether or not the graph is coordinate basis or not is irrelevant. The true power comes in when you look for specific relations of :distribution. From those distributions one can find patterns and one can map those patterns upon the same graph. After all that is the whole point of a graph in the first place. A solid good mapping of any graph is how well whether or not its coordinate basis or not, is how one can develop mathematical relations of change and rate of change. This obviously is where differential geometry comes into play, It doesn't matter if your differential geometry uses calculus of variations, scholastic ( probability), or differentials. These mathematics are obviously not restricted to just physics. They apply to any engineering trade as well and if you think about it to any programmer. Now lets take that geometry above along with a scalar distribution of values. They could very well be just those energy quantities you have in your model development. Lets say you see a pattern that all quantities are increasing in distance or any other value not involving distance from each other. It could very well be they all have an identical increase in value over some time (rate of change). As they are all identical they are all symmetric in rate of change. So I can readily describe this by one parameter. A scale factor or more accurately a constant of proportionality. Common symbol "a" but I also want a time dependency for rate \(ds^2=a(\tau)^2(dx^2+dy^2+dz^2)\) if however I notice the pattern of rotation on any principle axis (a reference) one can simply add a new term of proportionality to represent that rotation. say for example in the above \(ds^2=a(\tau)^2(dx^2+\omega(dy^2)+dz^2)\) or alternately apply a vector field using \(z=f(x,y)\). if I wish to embed some state with determined boundaries such as a hyperbolic paraboloid \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{z}{c}\) I have already established a means of point by point translations to embed any number of geometric objects at any specific location on the original mappings. We should all recognize this makes symmetry relations far more flexible in so far as spatial and rotation translations of those embedded systems or states or even when it comes to symmetry rotations and translations of the original mappings. The above describes the flexibility of mappings via coordinates or any other parameter assigned manifold. With spacetime this also makes parallel transport operations much more flexible to mathematically describe. Perfect example is developing affine connections pertaining to path integrals such as the geodesic equations. I simply do not see that flexibility in any of your work. I would honestly hate to try any curve fitting operation or finding other mathematical relations describing embedded systems or states at a given location with your methodology let alone performing any symmetry operations. Yes I know you have a graphical simulation of an orbiting body. However in order to program that simulation you would have eventually relied on some form of coordinate system simply to employ the instruction set of the software you used. For that matter many ppl are not aware that AI uses a weighted sum of patterns in answers to a question and employs tensors of the weighted averages to determine the most common answer to a question asked.

Good write up I think I may pick up that textbook. I like his writing style.

Think of the above this way if Im handed a distribution. Specifically involving sum of forces. I am also handed a geometry pertaining to that distribution ( energy mass density ) k=0 in this case. I dont need any mathematics to automatically recognize the sum of forces acting upon the planet or star is zero. ( assuming the nearest object to that region is substantially far enough away to automatically know its too far for any gravitational influence. If you were to hand me your energy distribution where I have to sit down and perform a bunch of conversions before hand and seperate all the different contributors to your energy totals. Which way would you say is easier ?

Lol after I spent an hour teaching him how to read fractions on a tape measure and teaching him how to add, multiply and divide fractions lmao

Pray tell if the only piece of information I have is a table of energy values with "Absolutely no other information " or any reference to any mainstream identification terms including any orientation involving geometry. Using nothing but the values generated by your methodology. How can you possibly have uniform energy distribution. Distance is energy Mass is energy Angles is energy Direction is energy Force is energy The only terms you are allowed is velocity and time. By the above using meters is not a unit of measure of energy. How many meters is in a joule? How would you even identify when a change in energy is due to distance, direction , orientation, binding energy, kinetic or potential energy if the only piece of information I am handed is nothing more than intervals and energy relations. How would you go about identifying any uniform distribution which by your mathematical method you shouldn't even get ? Distance is interval How would that be uniform ?

Had a chance to look into it the Laplace operator is positive semi definite while the D'alambertian is not hence its usage with hyperbolic geometry

Lol today I encountered a tradesman that didn't even know how to read a tape measure. Any time he had to determine say for example 3/8" of an inch. He would google the answer Guess the world is becoming reliant on media and software resources.

When you get a chance here's another test for you. According to Newtons Shell theorem if you energy mass distribution is uniform throughout your system/ state. G=0 Regardless of what the energy mass density is.

Ok this is something I can't shake. If you have a static event at x=10, y=10 and z=10. How do you intuitively identify their orientation from (0,0,0) including angles between principle axis ? In essence Occams razor Recognizing you also want to avoid any mappings pertaining to geometry If you were to hand me an E value how would I identify and seperate the components that make up the total E. ( in essence working backwards from total E in reverse and correctly identifying each contributor including quantity of each contribution.

I would have to double check this after work but if I recall one of the distinctions between the Laplace operator and the D'Albertian operator is that the former is positive norm. This in GR applications relates to the signature with regards to one of its usage. However as I mentioned I'm at work so will have to check that later