Mordred

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

Several of the answers above have provided excellent clues into gravity vs density but let's refine that with mass density. Lets do a couple thought experiments and for simplicity we will keep the total mass constant in each case. Lets set at 1 solar mass ( mass of our sun). Case 1) spread that mass out evenly everywhere where no coordinate has greater mass than any other coordinate. No matter which location you choose you can state it's the effective center of mass. Gravity in the above case is zero everywhere. It does not matter what density of mass each coordinate has it could be as dense as one can fathom. As long as the mass density is uniform everywhere Newtons Shell theorem applies. Case 2) you have one region with higher mass density than other regions (anistropic distribution) Now you have a clear cut center of mass as the center of that region is clearly a higher density than the surrounding regions. Now you have gravity where the difference follows Newtons laws of gravity. Now Case 3 is rather special take that one solar mass above and let's assume it has the same volume as our sun. The strength of gravity one measures depends on the radius from the center of the sun. If however you collapse the radius of the sun below its Schwartzchild radius it becomes a blackhole. However the mass does not change. The radius where you can measure gravity has decreased so at the event horizon the strength is such that nothing can escape. Yet the force of gravity is still the same if you were to measure gravity from Earth. Hope that helps remember at no point did of the 3 scenarios change the total mass. It is the distribution of mass that leads to gravity and the radius from the center of mass.

Then what was the problem when I stated I had no problem with using conformal age that you felt it necessary to flame me in the manner you did since ?

Thank you so when you asked if I had a problem with conformal time as the age of the Universe. Did you specify proper age ? Instead of conformal age ? Both are valid conformal age has the side benefict of specifying what treatment your applying. Ie conformal coordinates

So age is on the rhs of the equal sign on that equation correct ? So why do you think 47 Gyrs is the age on the left hand side ?

Sounds like you already applied an age ie time zero to 13.8 haven't you

One last point if you used a(t) over the entire expansion history how did you end up with a singular value of 47 Gly the scale factor varies over time.

Thank you for admitting you are a sickpuppet account

No that formula does not today scale factor a=1 there is a difference between conformal time now and conformal time then. This will be my last post this thread be well.

Do you not understand the difference of conformal time today as opposed to conformal time during expansion history ? The equation you had during opening post only applied scale factor today not the scale factor of earlier times from BB forward hence the integral. I lost count the number of times I posted lookback time with E(z) including the evolutionary history of matter and radiation as terms. The formula you used in the opening post ignored the expansion history and you had already that the expansion history must be taken into account. Just calculating today's moment of the expansion radius is not including the expansion history. In essence the formula you used in the opening post was the equivalent of start of signal from an emitter today sending a signal to observer today how long would the signal take to arrive with no further expansion. In other words you did apply any form of look back time. Which I posted references to numerous times.

47 gly is the conformal time of the universe today correct ? As that's what you have on you opening post. That scale factor you used is the universe today. Did you apply the conformal time to age integral ? At BB n=0 Dt=a(n)dn T(n)=int_0^ n a(\prime{n})d\prime{n}

Are you sure about that ? Better supply a reference of where I am in error. Lookback time The lookback time tL to an object is the difference between the age to of the Universe now (at observation) and the age te of the Universe at the time the photons were emitted (according to the object). It is used to predict properties of high-redshift objects with evolutionary models, such as passive stellar evolution for galaxies. Recall that E(z) is the time derivative of the logarithm of the scale factor a(t); the scale factor is proportional to (1 + z), so the product (1 +z) E(z) is proportional to the derivative of z with respect to the lookback time https://ned.ipac.caltech.edu/level5/Hogg/paper.pdf

Glad to see your studying, a couple of points on conformal distance vs proper distance the latter being the actual physical distance while the conformal distance is rescaled, with the addition of the scale factor. Through this rescaling this simplifies redshift relations, angular diameter distance etc to a fundamental observer. Just a side note the calc in my signature for example initially uses conformal distance then converts to proper distance. There is a link for the tutorial on how to use it including which formulas are used This article may help understand how it simplifies some key distance relations. Though careful one detail this uses a rescaled proper distance in essence the equivalent of a commoving distance as opposed to the actual physical (proper distance). It specifies that detail and discusses it https://people.ast.cam.ac.uk/~pettini/Intro%20Cosmology/Lecture05.pdf

Yw

Not really, once you delve deep enough you start to learn how useful conformal time is with regards to measurements via luminosity distance, angular diameter distance or redshift.

Commoving time and conformal time including the formula for age of the Universe is all forms of coordinate time. When you apply a scale factor to a metric in the case of the FLRW you are specifying you are a commoving observer and the radius for that scale factor is also a form of coordinate. So the change in radius is a commoving event.. The proper time being along the worldline or null geodesic between emitter and receiver. The age of the Universe ie for example 10^-43 sec after the BB occurs prior to the formation of the CMB. For that matter it would extend beyond the Cosmic neutrino background assuming we can ever eventually measure it. The time periods prior to nucleosynthesis is needed as it provides a timeline for inflation and electroweak symmetry breaking both which occur prior to nucleosynthesis which forms the CMB. Now consider the following argument as to why conformal time is preferred ? Take a redshift value you can establish a distance as well as time the signal is emitted but that depends on coordinates between observer and emitter and not some clock following the null geodesic. As to an observer at CMB ? Well see above I already explained that the calculation for the age of the Universe is not the proper time age. The wiki link I posted yesterday specifically stated it's conformal time. The Peebles article further highlighted that detail. Look back time is the formula used for age of universe and it accounts for expansion which entails commoving coordinates

https://en.wikipedia.org/wiki/Proper_time Search Proper timeArticle Talk Language Download PDF Watch Edit In relativity, proper time (from the Latin proprius, meaning own) along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time Now given the above definition is commoving coordinate independent of its geometry ? Or is that now a coordinate time ? Any location you choose for a reference point for the emitter or observer is a coordinate dependant event location. The proper time follows the ds^2 line element aka the world line or null geodesic Here is the FLRW metric Chrisoffel not that it's needed now given the definition above of proper time.