Mordred

Evidently you have no clue how the metric actually works. Observer is us on Earth, the CMB surrounds Earth and exists everywhere in the Universe today. Its current blackbody temperature is 2.73 Kelvin. It did not exist at every moment in the past ie previous to 380 million years after BB. Resorting to try and insult me doesn't cut it. Particularly since I do have credentials in the field of Cosmology. However as you cannot counter my points I made with anything related to actual physics then its pointless for this thread to continue. For the third time conformal time is not proper time. I provided you with the proper time corrections as well as the reason why they are required.

Why ? What about a geometry without a CMB the metric is perfectly capable of accurate conformal time simply using observer ie the commoving observer now and emitter to any past object including those objects prior to the CMB. You dont require some special period in our universe history for the equations as is to work. Why would you want to restrict the flexibility it has and apply unnecessary limits ? You can literally take any object at a given redshift and use that as well as the expansion history with the equations above and get an approximate age of when the signal was sent. See the calculator in my signature( it has that very feature). After all the only two requirements is a geometry and a change in volume over a given time period. The rest of the formulas allows you to determine the volume at a given redshift to determine the scale factor. You dont require a CMB for that Thanks merry Xmas and happy new year to you as well

Its because if your following what Im describing instead of delivering straight answers you might think about it. It seems your trying to find preferred frame. You dont want that. Keep it emitter/ observer. As observer you already have a frame of reference which is already assigned by the usage of the scale factor to some other past moment provided by the redshift function. The CMB is only 1 possible past moment and even the surface of last scatterring spans a number of years. Which arbitrary point will you choose ? Its isn't some discrete point in time. I doubt you've worked with the Saha equations with regards to hydrogen dropping out of thermal equilibrium which traps the free electrons. For example the aforementioned 3000 kelvin mentioned this thread represents the temp where 75% of the hydrogen formation. At 4000 kelvin it's 50% etc. Its not some one point in time. After all the static on your radio is the noise from the CMB today and not the CMB at say Z=1050 or 1100 Which time during the CMBs presence will you choose? As it's still present today as well as 13+ billion years ago ? Keep in mind I could have stated on my first post that how the age of the Universe was determined is already using conformal time from my first post. However it was clear you were not aware of that nor aware of the distinction between conformal time vs proper time as it pertains to GR four momentum.

were discussing proper age vs cosmological age. Conformal time does not describe proper time and its age is in cosmological time as per its usage in the FLRW metric equations which uses the commoving observer on a commoving coordinate system. Proper time is coordinate independent. Conformal time is not, it relies on those previously mentioned criteria. That should answer your last question as I did mention commoving observer and conformal time uses commoving coordinates. Proper time however does not and that distinction is extremely important when it comes to how GR or SR applies to the FLRW metric.

if you were to apply the Lorentz transformations under SR once you exceed the Hubble Horizon then the recessive velocities given by Hubble's law will exceed c. At Z=1100 for example the recessive velocity is 3.2 c. To get the corrections you have to apply the evolution densities of matter, radiation and the cosmological constant to account for beyond the Hubble horizon the the cosmological event horizon or particle horizon. The Hubble horizon is z=1.46 here is the methodology for the corrections a couple of posts up in that thread. I also have the more accurate or modern used look back time corrections its compatible with Peeple's equation 14

No it's not do you want the proper time corrections beyond Hubble Horizon for when the recessive velocity exceeds c ? SR without those corrections will give you the wrong answer. Same as GR without accounting for those higher recessive velocities. Hint proper time uses proper distance not commoving distance. Cosmological time uses commoving distance to a commoving observer. Age of the Universe is determined by the latter not the former

cosmology like to allow for all 3 possibilities hence its versatility and you still missed that we already use conformal distance to calculate the age of the universe. equation 14 and 30.... hence cosmic time is not the same as proper time

only in a spatially flat universe not a curved. see above you missed my edit

when you realize that the radius of the universe given by Google is the proper distance and not the commoving distance which is required by conformal time which would not be 47 Gyrs edit sorry other way around the point you should be seeing is commoving time is distinct from proper time when it comes to SR and GR treatments. Conformal distance is a rescaling in that regard. Your opening post argued that proper distance is preferred over commoving distances but obviously you didn't look at those 2 formulas and determined which is being applied did you ?

Tell you what take equation 14 and apply it to equation 30 https://arxiv.org/pdf/astro-ph/9905116 this is your graph here https://ned.ipac.caltech.edu/level5/Hogg/Hogg10.html then look at the difference using proper distance vs commoving distance each has its use but look back time is how you determine the age of the universe see first link for equation and relations with an multicomponent universe like our own. That should answer why you need all components contributing to expansion >particularly when it comes to angular diameter distance etc.

edit forgot to add why not just use radiation also pertains to angular diameter distance. You need to account for all influences when it comes to distance measurements.

Radiation is not the driving force to expansion today that is the cosmological constant. Our universe underwent 3 distinct era... radiation dominant, matter dominant and today matter dominant. I hope you realize that conformal time specifies the use of commoving distance and not proper distance if not see the first 2 graphs here https://arxiv.org/abs/astro-ph/0310808

sigh obviously were going to use the evolutionary history of our universe over time which I already know your using for your conformal distance to time relations. ie the radius of the observable universe in this case. Which is preciseley my point to my previous statements. The temperature will vary with the scale factor. Your formula you provided specified that above. Now back to my question which you evidently don't want to answer. If you removed matter your evolutionary history of the rate the scale factor will change..... so why would you use that as a benchmark for age of the universe with a non linear curvature to the scale factors evolution through time given varying rates of expansion.

oh gee all physical processes such as temperature change can be related to age. Age is not strictly a factor of matter only. it is a duration of events. the evolutionary history of our universe can be segmented using age. Nuclear decay is simply one type of aging

do yourself a favor calculate the look back time for a radiation only universe. You will not get 47 Gyrs. Nor will you get 13.8 Gyrs I've given you a source material to see what I mean

you do understand that a radiation only universe would not expand at the same rate as a multi component universe do you not ? That would effect your scale factor \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{r}(1+z)^4+\Omega_{\Lambda}}\]

either way my question remains even if you apply a radiation only solution to the FLRW metric radiation still causes expansion. Barbera Ryden's introductory to cosmology will give you the look back times for single component universe. If you apply a single component universe your expansion rates will differ significatly and you would not get 47 Gy for your conformal time as matter contributes to expansion. The expansion contributors are matter, radiation lambda and curvature terms though for our close to flat universe curvature can be safely ignored. So the conformal time you have would be incorrect as you did not calculate the expansion rate for a universe with no matter.

what is the new latex command tags for this site its been awhile and secondly why does copy paste give me a box for font when I am copy pasting a formula from a previous post ? and how can I remove said box I tried selecting plain text but that box still remained

matter isnt the only factor when it comes to expansion as matter is strictly fermionic particles. You also have the cosmological constant as well as radiation such as photons that contribute to expansion. unless you universe is completely empty if so then can you consider it as a universe ?

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Im curious given that conformal time is the total comoving distance light could have travelled. Why you would feel this better defines the age of the Universe? If you look at the curvature of the accelerating rate of expansion from our location to the radius of the observable universe as well as the detail that conformal time is unit less. This would mean your seconds will change in duration from one second to the next. Whereas proper time this isn't the case except for time dilation effects however that's accounted for in the look back time to determine the age of the universe. Try this thought experiment let's say light has already traveled one light year. Now add expansion. The region light has already traveled is expanding as well as the distance to the observer ahead of the beam. This is occurring every second by the Hubble constant value So ask yourself this why would this be a better representation for age ? Particularly given that the Hubble constant varies over time

Bump going to get back to this project now that RL jobs has relaxed

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Might help if you look at the time aspect imsteas of writing \[a=\frac{dv}{dt}\] Write \[a(t)=\frac{dv}{dt}\] Secondly momentum includes mass via \[p=m*v\] So where is the issue ? Others have tried explaining this to you in the older thread

No worries hopefully you resolve your PC issues.