Mordred

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 \[D_A(z)=\frac{d_L(z)}{(1+z)^2}\]

https://arxiv.org/pdf/2411.11328 Article contains proposed neutrino mass mixing matrix.

DESI constraints https://www.osti.gov/servlets/purl/3011043 Has a particular section to follow up on massive neutrinos behaving as dark matter described in above link. https://arxiv.org/abs/2507.01380

Interesting study, hadn't seen that one before and your judgement on the paper is accurate. Another factor to consider is the particle spin itself. One known problem with the particle view is say the electron in the particle view it's angular momentum exceeds c. However in the QFT field excitation view this is resolved. Im still digging into how the Pilot wave theory deals with that (assuming solutions have been presented) more for my own curiosity lol. If I do find some decent literature in that regard I will share here

On the early time measurements ie measurements using the CMB in regards to improved calibration. The process involves calibrating the Baryon acoustic oscillations (BAO) https://arxiv.org/pdf/2405.20306 This paper includes references to the Hubble tension. For reference here are the 2024 DESI constraints. https://www.osti.gov/servlets/purl/3011043 Many of the regular forums members have often seen me refer to the equations of state and how matter, radiation evolve over time and seen me post a key formula to determine Hubble rate in the past compared to Hubble rate today as a function of redshift. https://www.osti.gov/servlets/purl/3011043 See equation 2.2 of article. this post I show how equation 2.2 comes about in the key equations and includes the look back time adjustment based on the same procedure.

Ive been off and on watching for papers researching the hubble contention between early and late time measurements. Figured I would share this interesting article here. https://arxiv.org/abs/2408.06153 One of the factors involved being Leavitts law https://www.astro.gsu.edu/lab/website/4LeavittLab-1.pdf The main point however is the paper expresses no new physics is required and supports the need for tighter constraints on luminosity to distance relations including tighter calibrations. Last link above is just a quick overview of the law. Below is a more recent calibration papers. https://arxiv.org/abs/2509.16331 A simplistic way of describing the law is The longer the period of the star, the higher its absolute luminosity. Local cephieds however are used to calibrate this law however as distance increases other factors may become involved hence the last link is further studies increasing calibration constraints

I recall my high school physics, it basically covered rudimentary SR. They certainly didn't cover gravity waves let alone GR. Lol several of us had better knowledge of GR than the instructors.

This video is not too bad. I usually dont subscribe to videos other than lectures but this one isn't too bad. David Bohm’s Pilot Wave Interpretation of Quantum MechanicsScience News, Physics, Science, Philosophy, Philosophy of Science She touches on the key points between the Copenhagen interpretation ( non locality in particle physics terms specifically aa applied to interactions). The issue this causes with Lorentz invariance. The 3 principle equations of the the theory. Some details to add is that the Hamilton-Jacobi usage is nonlocal and non linear as opposed to the linearity of the Schrodinger equation. The other key point is there is no testable means of showing its more or less accurate than the Copenhagen interpretation aka standard QM. It makes no predictions that will differ from those of QM. The other issue being the non locality when it comes to QFT. There are papers available of course presenting possible solutions to this problem but it's still in the works so to speak. This should help answer sone of your questions on our cross post lol. I would like you to consider the following. A wavefunction under QM and QFT as you are aware is a probability graph. All functions are graphs but not all graphs are functions. Those wavefunctions do have relevant constraints. For example causality is a constraint with regards to time dependent wavefunctions. Example the Dirac equations or Klein Gordon, Schrodinger etc Other constraints applied include conservation laws for probability wave functions applicable to closed groups. From those constraints anything not allowed is not included in the probability wavefunction. This is a very technical article describing boundary conditions as applicable to quantum mechanics included in the article is the Borne approximation or Borne condition. "Quantum boundary conditions" https://dottorato.fisica.uniba.it/wp-content/uploads/2018/03/tesiPhD-Garnero-compressed.pdf all boundary conditions is a form of constraint. All finite groups are also constrained. What many laymen or those not mathematically versed in physics often do not realize is every statement under physics is mathematically defined or described. This includes the axioms of a physics theory, group etc. Simple example symmetry ie laws of physics must be the same regardless of reference frame in the Minkowskii group is mathematically defined via \[\mu\cdot\nu=\nu\cdot\mu\] Constraints and boundary conditions are also mathematically defined.

I saw that on FB as well lmao. Lot of hoaxes floating around lately. Naturally I had to comment its falsehood on the FB post I came across

Pilot wave is considered more an interpretation much like the Copenhagen interpretation under QM. Its premise is more deterministic than probabilistic however the Copenhagen is what is considered more in alignment with QM. There were numerous issues with pilot wave in so far as entanglement and hidden variables as one of the reasons as to why the Copenhagen interpretation became more accepted. Personally I dont see any means where it would tighten constraints that are not already accomplished by statistical weighted average for most likely position of a particle. Perhaps looking at how each interpretation would work with Dirac Delta functions might provide some insight.

You dont require billions of years to toy model cyclic universe models. There are numerous models avaliable that have accomplished this. Cyclic universe models is nothing new to physics. Alternately there are numerous bounce cosmology models. However toy modelling requires mathematics and for this application applicable geometric treatments. I would suggest application of the Raychaudhuri equations would be incredibly useful. ( though that methodology has already been done)

Lets start with the correct definition of mass. Mass is the resistance of inertia change or acceleration for short. It is a property of a system or state/particle. It isn't something that exists on its own. Energy is also a property,, Energy being the property of a system, state, classical objects, fields, particles etc describing the ability to perform work. Once again Energy does not exist on its own. It may surprise you but e=mc^2 is just the invariant (rest mass) it is not the full equation. The full equation is the energy momentum relation \[E^2=(pc)^2+(m_o c^2)^2\] https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation

Agreed it's not perfectly flat with regards to the Rheimann tensor as to the earlier statement I made "where (tS ) is the scale factor and k is a constant which denotes the spatial curvature of the three-space and could be normalized to the values +1, 0, –1. When k = 0 the three-space is flat and the model is called Einstein de-Sitter static model, when k = +1 and k = –1 the three-space are of positive and negative constant curvature; these incorporate the closed and open Friedmann models respectively." note the statement 3 space which also agrees with your earlier statement. https://mpra.ub.uni-muenchen.de/52402/1/MPRA_paper_52402.pdf Obviously we're both aware a static solution is considered impossible. The other use for the critical density formula as shown above ties into the fate of the universe.

I found after some digging a useful article showing some of the corrections mentioned earlier for higher redshift distances https://people.ast.cam.ac.uk/~pettini/Intro%20Cosmology/Lecture05.pdf One thing I should mention often textbooks etc on a topic gives you the first order formulas for various things like redshift, luminosity distance, angular diameter distance etc. You rarely find the more advanced formulas in common literature. Those formulas tend to be something that the instructor will have you derive yourself. The above lecture lesson is an example We have to be careful here K is a specific relationship with the critical density formula. When K=0 precisely then the energy mass density equals the critical density. If K=1 then this describes a closed universe where the energy density is greater than the critical density. If K=-1 then the mass energy density is less than critical density.( open universe). Though open closed universes are an older application of the critical density formula. With regards to inflation one of the problems inflation addresses is the fatness problem related to the above. K value remains unchanging throughout the universes expansionary history. For example if in the first case k is precisely zero the universe is static neither expanding nor contracting.

Yes there is superluminal expansion during inflation however the period during such time would be more problematic as the mean free path of photons would be too short to receive signals from emitter to observer aka the dark ages. Mean free path time estimate 10^{-32} seconds. So you wouldn't be able to recieve signals between two inertial frames of reference Good point

yes changes to the scale factor evolution is non linear over the Universes history this leads to numerous adjustments that must be made to linear relations (first order formulas) where second order relations must be incorperated example of second order being acceleration example accelerating expansion. various measurements as a result of the above non linear expansion rates that require corrections is angular diameter size, angular diameter distance, look back time (ie age of universe at a given distance usually as a function of redshift), redshift corrections, luminosity distance corrections. the above corrections must apply the equations of state for matter, radiation, Lambda and any applicable curvature term. for example if one tried to take the time dilation formula under SR once you get to recessive velocity greater than c then that equation will give wrong answers. I know your not strong in the mathematics but If your interested in the corrections for higher recessive velocities they are here corrections to look back time I would have to dig up the corrections for angular diameter distance , luminosity distance etc but the previous two examples show how the equations of state are involved. There are also other factors such as the relation between angular size and angular distance. One counter intuitive example is that above redshift 1.5 Z approximately the angular size increases rather that decreases.

Lmao @StringJunky beat me to the punchline

No worries one detail when dealing with probability distributions or multi measurements over an ensemble of measurements. The area of the distribution ie highest distribution is what becomes relevant. For example if you take 100 samples and 20 of those samples are in close proximity to one another while the rest are scattered in without a discernible pattern. The area of those 20 samples is your higher probability region Here is a simple example of gaussian distribution. https://introcs.cs.princeton.edu/python/appendix_gaussian/

You run into ppl like that. Its one of the reasons I try to supply reference papers for statements I make. However some ppl fail to even look at those reference papers or fail to understand them. However I always consider adding them useful for other readers of the thread as well. Glad to hear you learned something from that thread. You asked earlier on this normal distribution. As it is a probability density function you won't have a negative curve. All probability functions regardless of type are positive norm. However I should note some terminology is a little loose. For example the Dirac Delta function used to describe point mass isn't a true function but a measurement distribution. As such it's handled a little differently via Lebesque integration. Example here https://arxiv.org/pdf/2508.11639 Edit forgot to note a simple function has a finite range this isn't the case with Dirac Delta functions

Not quite accurate any solution to the Dirac equation is a solution of the Klein Gordon equation. It is treated as a foundation equation of QFT. Though today it's main use is bosons such as Higgs.(scalar) Once you involve spin (under Dirac use of spinors) then the Dirac equations are used. For other readers

I Agree excellent video one of the better ones Ive seen. I love how he went from classical wave theory, included SR to QM and then QFT in a very well laid out format. Lol literally covered several chapters of most textbooks in a short video. One added detail however the Schrodinger equation isn't lorentz invariant it doesn't work well with SR however the Klein Gordon equation used by QFT is. It does so by factoring in the mentioned energy momentum relation into its equations (in essence employs the 4 momentum.) Its an important distinction between QM and QFT. Some of you may have heard me mention the term canonical ( this is a quantized field theory) a conformal theory however isn't quantized. ( string theory as one example). Just some side tid bits

Assuming all particles reach thermal equilibrium the entropy can be safely described strictly via temp. As the mediator for temp is the photon entropy will end up being S=2 same as the entropy at 10^-43 seconds. However the problem at the low temperature end is that only massless particles travel at c and all massive particles will likely remain massive so wouldn't be in thermal equilibrium such as our universe beginning. Too many variables with regards to how particles would remain coupled for the mass terms to give any good guess. Will the coupling constants operate the same is anyone's guess. According to QM zero point energy you will always have quantum fluctuations hence absolute zero is impossible via current understanding of QM. Then there is still BH evaporation times to consider lol which is far greater than the time frame I mentioned above for a one solar mass BH. One could consider we understand electroweak symmetry breaking processes at the hot end better than we understand thermal equilibrium states on the cold end.

Keep in mind heat death is only one possibility. One that relies on the cosmological constant remaining constant. Its still viable at some point that this may no longer hold true and the universe could start to collapse. The key equation being the critical density relation which was originally used to determine the inflection point from and expanding universe to a collapsing universe. Aka cyclic bounce models Using Planck 2018+BAO dataset values roughly 45 B years into the future the Hubble constant will hit 55.7 km/Mpc/sec. It will remain roughly this value up to universe age 93 B years old. Thats as far as the cosmological calc in my signature goes. At that time the CMB balckbody temp will be roughly 0.0273 Kelvin. It will never hit absolute zero but that temp is still too warm for Bose-Einstein and Fermi-Dirac condensates so you will still have particles not in thermal equilibrium as per the standard model today. That of course is under the assumption the cosmological constant remains constant.

Wilsonian renormalization group with regards to Higgs https://www.physics.mcgill.ca/~keshav/675/wilsonianaturalness.pdf https://arxiv.org/abs/2310.10004 https://scoap3-prod-backend.s3.cern.ch/media/files/84579/10.1103/PhysRevD.109.076008.pdf https://www.db-thueringen.de/servlets/MCRFileNodeServlet/dbt_derivate_00035352/Sondenheimer_PhD-thesis.pdf

I recall that video always enjoyed Guths lectures as well as articles. Static vs inertial in terms of different observers can often give surprising results. Guth does an excellent job demonstrating some of the effects in that video