Mordred

Pointless continuing particularly with your attitude bye

You already have your answer one last time yes that's the correct row

bottom row is todays universe an observer today uses this row

That last post has an error in it I was about to correct it higher Z >1 in past lower Z in future from the observer past/future events when the universe is expanding. The reason for this directly involves the CMB blackbody temperature. An expanding universe cools down as it expands

No in that case its blueshift and not redshift. The sign changes

if a is greater than 1 then the universe radius is larger than the radius of the observers universe if expansion continues ( a future event in that case). Quite frankly that should be obvious by now and if you didn't understand that relation then its no wonder you thought there was something wrong.

Or possibly in this case an honest misunderstanding. Particularly if one legitimately didn't know how the scale factor works

no only a_0 will be one forever that's the person doing the observing. Just like a drafting scale ruler a 1 to 1 scale is always 1. its no different the length or scale on top is either greater, lesser of equal to the scale on bottom denominator.

Lets put it simply the one doing the Observing the scale factor and his observable universe radius is always =1 so the denominator is always set at 1 event being observed is always the numerator and your comparing the radius of the universe of the observer to the radius of the event. If you had some past observer his value will be 1 as he is the one observing and still the denominator the event remains as the numerator on top that observer is measuring

of course but if I already provide the answer can you understand my response now do you confirm

look at my top row \[a= 9.08\times 10^-4\] its right there on top

wow just wow 1/2=0.5=50% radius now=1 radius at previous or time yet to come is the numerator same with the scale factor. If radius of the universe is half the radius as it is now a=0.5 from observer now \[{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline z&Scale (a)&S&T (Gyr)&D_{then}(Gly) \\ \hline 1.10e+3&9.08e-4&1.10e+3&3.66e-4&4.11e-2\\ \hline 5.46e+2&1.83e-3&5.47e+2&1.16e-3&8.18e-2\\ \hline 2.70e+2&3.69e-3&2.71e+2&3.55e-3&1.61e-1\\ \hline 1.34e+2&7.43e-3&1.35e+2&1.06e-2&3.16e-1\\ \hline 6.58e+1&1.50e-2&6.68e+1&3.08e-2&6.09e-1\\ \hline 3.22e+1&3.01e-2&3.32e+1&8.92e-2&1.15e+0\\ \hline 1.55e+1&6.07e-2&1.65e+1&2.57e-1&2.08e+0\\ \hline 7.18e+0&1.22e-1&8.18e+0&7.36e-1&3.54e+0\\ \hline 3.06e+0&2.46e-1&4.06e+0&2.10e+0&5.27e+0\\ \hline 1.01e+0&4.96e-1&2.01e+0&5.79e+0&5.56e+0\\ \hline 0.00e+0&1.00e+0&1.00e+0&1.38e+1&0.00e+0\\ \hline \end{array}}\] universe now on this graph is a=1 bottom row value of a at z=1100 top row is top row

your comparing two radius relations I already provided the answer. \(R_0=1\)\ and \(a_0=1\) when the universe is half our size you will get the answer 0.5

\[a=\frac{R}{R_o}\] set radius now as 1 (\(R_0=1\) if the radius in the numerator is the same the by simple division the answer is 1 1/1=1. Value of Z is zero...

If your discussing cosmological redshift YOU MUST include the scale factor a and understand how it works. Especially when it comes to how cosmological redshift defines Z mathematically Once again I can prove that using GR but that would likely be a waste of time 1 give you a clue the radius of a describes today is the size of the observable universe today if the size of the universe then you have no cosmological redshift

how many times have I stated above including the mathematics that its not constant The term constant of proportionality simply describes a dimensionless scalar value. look directly at the following relation \[\frac{a}{a_0}\] where \(a_0\) is scale factor today now answer your own question what would happen if you place a zero in the denominator ?

You do understand that the scale factor is a dimensionless constant do you not understand how that constant is defined ? ask yourself what calculation allows the scale factor to become dimensionless and ask yourself once again the question you just posed.

yes but you can get that answer directly from the calculator in my link including any associated graphs https://light-cone-calc.github.io/ its quite versatile but doesn't port the graphs well even getting the latex here on this site requires altering the latex commands. You can arbitrarily set any cosmological parameters you choose to use.

If you cannot get a specific value at a specific point on any graph does it matter ?

Already provided a dataset use that it incorporates the full FLRW metric with the relations I described previously and used the Lineweaver and Davies paper for its benchmark tests.

would you like me to prove you wrong under full blown GR treatment ? I can readily provide the Full GR treatment for each case if you like. of course that will barrage you with a large list of equations as I would also have to include the mathematical proof under 4 momentum along the null geodesics involved.

Ok I'm done with attitude in a post you have my answer it's your choice to understand the answer or ignore it. The lineweaver paper your comparing to uses factors you choose to ignore. I provided the relations of those factors to the redshift relation Z. You cannot inverse the Doppler shift relation including the relativistic Doppler and expect to get the same results as the Cosmological redshift because the Cosmological redshift relation involves factor not included in relativistic Doppler the very terms you choose to ignore. I'm tired of repeating myself as well Ever stop to wonder why your receiving negative rep points ? For the record the Moderator you just talked back to also has a PhD in physics

Still waiting for it but I already identified where your error is and you just admitted you did not account for the evolution of the scale factor. You didn't build the first top graph I already knew that however I know how that graph was derived and the originator used the details I mentioned but just on the GR line though as per the comment on the paper provided (Lineweaver and Davies) The equations specifically describing the difference between SR and what they termed the GR line is described at the beginning of the article To quote "conversion from cosmological redshift to recession velocity is different from the conversion from an SR Doppler shift to peculiar velocity"

Great mathematically include it in your graphs by simply knowing it but not knowing how to incorporate it into your the graphs. Not going to work you must know how to include those variations mathematically

How will that integral work if you do not include how the scale factor and the goemetry changes over time? It won't period in order for the integral to work you must include and account for all time dependent variations. That has been precisely what I have been trying to get through to you but you keep missing that very point.