Mordred

Well I can understand that, if that's the goal then I would recommend including a planar wave examination and a comparison between the two. here is an example which should help. Though primarily it simply explains that there is a difference between the two but the article is lacking in what those differences entail in acoustics. https://ccrma.stanford.edu/~jay/subpages/Lectures/Lecture1-Acoustics.pdf I didn't spot any mistakes in the first article of equations you applied however still looking it over as I have time

I would like a little clarity on the goal in terms of acoustics. I fully agree that there is distinctions in the behavior of plane waves and spherical waves in terms of the specific acoustic impedence. This is known, the specific acoustic impedence is the relations the calculator that Swansont linked above applies. For plane waves the Z as it's commonly denoted stays relatively constant. However in the spherical wave you will have a 1/r relation. This affects the sound intensity. Is our goal specifically reverberation time as per Sabines equation ?

The initial roll depends on the model which has some variations but roughly 10^{-32} seconds after the big bang. Don't think of it as a particle rolling downhill that would be incorrect. Instead think of it as the potential energy density of the early universe decreases. Today we may or may not be in a metastable state but at lower potential. or we could be close to the minimum. There are competing models on this.

range of force is the particle momentum and mean lifetime of the mediator. So a graviton would need to be stable as per what Migl stated above. The HUP does also apply in the decay rates of particles

I'll look at it in more detail later on but on a quick glance I'm impressed good job so far

@Joigus something to keep in mind in topological spaces you will have connected and disconnected sets. For any topological space dealing with the Feymann path integrals. You will be dealing with connected graphs (sets) where momentum conservation is applied and the graph is oriented. Just an FYI to help better understand path integrals. A way to help is think of any point on a graph or if you prefer manifold/geometry etc. In Feymann path integrals that point is a vertex. In GR its a reference frame, In Feymann Integrals that vertex can be connected to other vertexes or disconnected. If its disconnected its assigned a valency of zero. If its connected to another vertex it will have a valency of 1 or greater depending on the number of connections. so your path integral has a valency of 1 it has 2 vertexes joined by an edge.(the two vertexes are the initial state/event and the final state/event. The edge being the path defined by the Euler Langrangian for the extremum of the action integral which will be the geodesic. Which will also apply the affine connections through the Christoffels with the use of parallel transport. This then gets into the metric connection which that parallel transport of two vectors must stay parallel along a curve in metric space. (covariant derivative of the above gauge groups). Hopefully the correlation for 4d spacetime will help with regards to the various topological space connections. You can readily see how they may apply in degrees of freedom

Well trust me I can relate when I first started studying gauge groups I didn't realize the above and kept getting lost and mislead. Once I realized the reasoning of gauge group axioms life got a whole lot simpler in understanding numerous relations described by the group. In particular the (U(1), SU(2) and SU(3) groups including the corresponding SO(N) groups. As SO(N) groups are a double cover of the SU(N) groups.

You really don't have much here to show a shrinking matter theory. Trying to show shrinking matter and how it supposedly accounts for universe expansion measurements isn't at all viable. For starters if matter shrinks then the fine structure constant itself would vary along with numerous other coupling constants. There is zero evidence this ever occurs. You moon conjecture has absolutely zero relation as the orbitals mathematics as it applies to escape velocity and the conservation laws is sufficient to explain the moons orbit. The paper you posted examines cosmological redshift of quasars vs super nova and shows the two have different redshift relations due to various factors related to luminosity distance relations etc. Unfortunately the paper didn't really do a good job as it doesn't take into account the non linearity past the Hubble horizon of z. However that's just my opinion. However regardless. The paper does nothing to support a shrinking matter case. (your first paper You already admit your second paper doesn't relate Lastly the very scales of change were talking about is like seeing a tiny change at one end to a massive change in the measurements. The two do not equate.... You mentioned in your opening posts concerning constraints . 1) the effect on coupling constants 2) doesn't account for BB nucleosythesis 3) cosmological redhift is a logarithic exponential change it isn't linear 4) the temperature history corresponds to that redshift as being proportional to the scale factor. 5) the cosmic microwave background we can measure infalling and out flows of matter in terms of the corresponding sound wave modes E and B modes why do we not see evidence of an expanded matter field in terms of c ? That's a start

The reference frame is the commoving frame however the FLRW metric uses this to apply commoving time. The commoving frame is set at mean average mass density of the the metric.

Higgs cross sections partial width's \[\Gamma(H\rightarrow f\bar{f})=\frac{G_Fm_f^2m_HN_c}{4\pi \sqrt{2}}(1-4m^2_f/m^2_H)^{3/2}\] \[\Gamma(H\rightarrow W^+ W^-)=\frac{GF M^3_H\beta_W}{32\pi\sqrt{2}}(4-4a_w+3a_W^2)\] \[\Gamma(H\rightarrow ZZ)=\frac{GF M^3_H\beta_z}{64\pi\sqrt{2}}(4-4a_Z+3a_Z^2)\]

as this subject is on Dm being the sterile neutrino here is some useful tidbits to chew on. In the standard model model SU(3) gauge used to describe neutrinos. The left hand (particle) is a doublet. However the right hand neutrino is a singlet. This has consequence which takes quite a bit of math to explain However let me know if anyone would like to see the related formulas. I will gladly post them. Anyways this has consequences in the weak mixing angles of the CKM matrix via the Higgs seesaw mechanism will have a higher mass term than the left hand neutrino ( a massive partner.) hence cold dark matter (vs warm or hot). so as matter doesn't generate pressure p=0 equation of state. While neutrinos do w=1/3

right DM can readily pass by other particles in the galaxy without interaction while baryonic matter will interact and those baryonic particles are more likely to join the galaxy as you described

Another theorem that's a good study is density wave theorem this will cover the flattening effect due to mass rotation. it also covers metallicity distributions as it applies to star formations. as well as how the spiral arms come about. The theory also applies to the rings of Saturn, as the momentum terms apply the same in both cases.

Its an apparent horizon where the radius can alter depending on observer. Another term commonly used is a coordinate horizon or coordinate singularity. Mathius Blau if I recall also discusses this in his lecture notes on General relativity on arxiv. It is more often described in literature as an "artifact of coordinate choice". If I recall the Kruskal-Szekeres coordinates will eliminate the R=0 singularity condition at EH. Page 182 covers it in Lecture Notes on General Relativity Sean M. Carroll https://arxiv.org/pdf/gr-qc/9712019.pdf LMAO I guess one could say the frozen can be thawed with a different choice of coordinates

Just so your aware I'm still looking into this as I get the time, part of the problem is the separating topological rigidity problem that is apparently more common than I initially realized lol. The local conditions often fit the criteria to establish invariance. Or in the case common in particle physics the c.m. (center of mass frame) or lab frame. Gauge theory by criteria must be Lorentz invariant, however not all groups are Lorentz invariant particularly with the first order QM operators. The Schrodinger equation for example isn't Lorentz invariant which is one of the reasons QFT uses the second order Klein Gordon equation. Granted another reason directly applies to the conservation laws which are symmetric relations in closed systems. You have via the continuum equation \[\frac{\partial\rho}{\partial t}+\nabla\cdot j=0\] for a conserved quantity. QM,QFT,QED, Navier Stokes and fluid dynamics have variations of this equation but in each instance can be shown the equivalent via the related mathematical proofs. Noether's theorem shows the symmetry relations to a conserved quantity being symmetric. from wiki "In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups)." does that help address the reason for local vs global. https://en.wikipedia.org/wiki/Gauge_theory edit: @joigus Have you considered the above in terms if the rigidity like properties in regards to the topological spaces you have studied ? just food for thought but there is a good likelihood it will relate.

Why would that make any difference the moons angular momentum is far too low to even make any difference however even then those same lasers can still test for it. You still seem to not understand that inertial mass is identical to gravitational mass for all practical tests. If you ever plan on ever getting any form of funding you will require explicit mathematics to present your case. No one invests in haphazard guesses an conjecture.

As Markus mentioned the question is rather meaningless. We can only measure differences in potential from one geometric location to another. There is technically no limit to the amount of difference involved. Even if you took the entire mass of the universe and calculated that mass within a single infinitesimal and compared that to a region of zero mass density you still would not have a determination of an upper limit.

Ok I do have tchnically 2 threads ongoing with the related mathematics of Higgs inflation. https://www.scienceforums.net/topic/128412-musings-of-a-mad-scientist-inflation-as-cosmological-constant/ the first part is just the FLRW metric as I needed to get the equations of state for comparison for a single scalar Higgs field. The second part details the standard Higgs inflation mathematics. I'm still working on this part in greater detail in this thread I have in speculation but its rather random as I research each portion. https://www.scienceforums.net/topic/128332-early-universe-nucleosynthesis/ anyways if the Higgs inflation is correct and for the record the equations of state are identical to the standard inflaton field. A quick run down is as follows, the initial hot dense state had an extremely high kinetic energy term that when applied to the equations of state exceeded the critical density at that time. As the kinetic energy exceeded the potential energy by such an extreme expansion is a consequence to begin with. So your maximal false state is roughly \[10^{16\rightarrow..19^2}.. GeV\] As the expansion is already underway, you get the subsequent cooling (the slow roll stage) quark gluon plasma state where all particles are still in thermal equilibrium. At the volume curvature is still meaningless and in that thermal equilibrium state the effective degrees of freedom is a mere 2... hence the low entropy beginning. The two degrees of freedom is the polarities of the photon. As the effective degrees of freedom re the result of temperature with the photon as the mediator. At this stage you apply the Goldstone bosons correlations to the Higg's field. (invariant massless field). The consequence of the expansion allowed a sufficient drop in temperature (you an apply the photon redshift relations to the cosmological redshift formula) for a rough estimate Symmetry breaking occurs, this correlates with the rapid descent of the Sombrero hat. (number of e-folds in excess of 60 to fit observational data. The result of the rapid expansion leads to supercooling. Now this stage is important to understand prior to inflation the universe curvature didn't particularly matter due to the small volume in essence. However inflation of 60 e-folds later left a very close to critical density so k=0, approximate. the effective equation of state is the ultra relativistic radiation. The supercooling and critical density value both apply to the slow roll stage. Inclusive in this is the addition of a mass term to the particle species that have dropped out of thermal equilibrium. The "Friction is a correlation to that additional mass term". Another important detail is the transition from the false vacuum state to the true vacuum state need not be smooth you can have numerous other semi states and still match observational data. This detail likely be the result of various particle species decoupling, the result of each decoupling alters the effective expansion rate. quote from above Further, if you make a real world model of the sombrero potential and use a marble as the universe's potential, you find that the marble oscillates across the brim, before coming to a complete stop at the lowest point. Could the same oscillations be occurring to the universe's potential, and account for periods of increased and decreased expansion rates ? Or, am I reading too much into the model ? Not really due to the equations of state for radiation relation to the Hubble volume which relates to how the E-folds are calculated. If the kinetic energy term can exceed to Hubble radius in a given short time frame as defined by the E-fold logarithmic function quickly decreases but as it stabilizes it essentially merges with the radiation equation of state rather it becomes dominates by the radiation equation of state we do not know if were currently at minimal there is some conjecture we may also be in a semi stable stage.

Sterile neutrinos are still in consideration as they are predicted by the standard model. I have been looking into the literature to find examples of what the projected cross section would like. However as we have never observed sterile neutrinos it's all naturally conjective

I will have to look closer at the affine connections however I understand where your coming from will closer as to how the operators are handled

The common feeling is that DM doesn't interact with the strong or EM field. It may interact with itself or other weakly interactive particles. All particles obviously interact with gravity

you appear to have solved it. LOl yeah doesn't everyone find 800 pages an article ??? lol

Early Universe Cross section list Breit Wigner cross section \[\sigma(E)=\frac{2J+1}{2s_1+1)(2S_2+1)}\frac{4\pi}{k^2}[\frac{\Gamma^2/4}{(E-E_0)^2+\Gamma/4)}]B_{in}B_{out}\] E=c.m energy, J is spin of resonance, (2S_1+1)(2s_2+1) is the #of polarization states of the two incident particles, the c.m., initial momentum k E_0 is the energy c.m. at resonance, \Gamma is full width at half max amplitude, B_[in} B_{out] are the initial and final state for narrow resonance the [] can be replaced by \[\pi\Gamma\delta(E-E_0)^2/2\] The production of point-like, spin-1/2 fermions in e+e− annihilation through a virtual photon at c.m. \[e^+,e^-\longrightarrow\gamma^\ast\longrightarrow f\bar{f}\] \[\frac{d\sigma}{d\Omega}=N_c{\alpha^2}{4S}\beta[1+\cos^2\theta+(1-\beta^2)\sin^2\theta]Q^2_f\] where \[\beta=v/c\] c/m frame scattering angle \[\theta\] fermion charge \[Q_f\] if factor [N_c=1=charged leptons if N_c=3 for quarks. if v=c then (ultrarelativistic particles) \[\sigma=N_cQ^2_f\frac{4\pi\alpha^2}{3s}=N_cQ^2_f\frac{86.8 nb}{s (GeV^2)}\] 2 pair quark to 2 pair quark \[\frac{d\sigma}{d\Omega}(q\bar{q}\rightarrow \acute{q}\acute{\bar{q}})=\frac{\alpha^2_s}{9s}\frac{t^2+u^2}{s^2}\] cross pair symmetry gives \[\frac{d\sigma}{d\Omega}(q\bar{q}\rightarrow \acute{q}\acute{\bar{q}})=\frac{\alpha^2_s}{9s}\frac{t^2+u^2}{t^2}\]

I've studied a lot of literature on Feymann integrals and have usually found them lacking or simply don't describe the steps to solving them in great detail. I recently came across a reference that I am thoroughly enjoying the scope of how it details the integrals in a wide range of related theories. (warning extremely math intense). If anyone wants a good solid reference I highly recommend this article. Feynman Integrals by Stefan Weinzierl https://arxiv.org/pdf/2201.03593.pdf

Good examination +1