Light Cone - Hubble Radius & Time T and R Relationship

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    Hubble Radius Time
  • #31
N_{t}=The population size at time t
K=The carrying capacity of the population
N_{0}= The population size at time zero
r= the intrinsic rate of population increase (the rate at which the population grows when it is very small)



17.3 would be k since the value approach 17.3

[tex]{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&R (Gly) \\ \hline 0.00037338&0.00062840\\ \hline 0.00249614&0.00395626\\ \hline 0.01530893&0.02347787\\ \hline 0.09015807&0.13632116\\ \hline 0.52234170&0.78510382\\ \hline 2.97769059&4.37361531\\ \hline 13.78720586&14.39993199\\ \hline 32.88494318&17.18490043\\ \hline 47.72506282&17.29112724\\ \hline 62.59805320&17.29930703\\ \hline 77.47372152&17.29980205\\ \hline 92.34940681&17.29990021\\ \hline \end{array}}[/tex]
 
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  • #32
attachment.php?attachmentid=71068&d=1404454722.png
 
  • #33
Ledsnyder said:
Perhaps I can make a logistic equation in that form that approximates the actual curve.

The acid test would be to make a curve that approximates it for the range of say S_upper=5000, S_lower=0.5.
attachment.php?attachmentid=71081&stc=1&d=1404538430.png

I have used 50 steps for the curve and only 20 for the table below.
[tex]{\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}[/tex] [tex]{\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&R (Gly) \\ \hline 0.000025341&0.000046766\\ \hline 0.000060035&0.000107981\\ \hline 0.000138890&0.000242519\\ \hline 0.000313389&0.000530859\\ \hline 0.000690586&0.001137192\\ \hline 0.001490753&0.002395836\\ \hline 0.003164637&0.004987297\\ \hline 0.006632095&0.010296559\\ \hline 0.013767608&0.021141785\\ \hline 0.028387897&0.043256161\\ \hline 0.058260498&0.088300863\\ \hline 0.119187747&0.179988176\\ \hline 0.243304736&0.366499913\\ \hline 0.495891849&0.745461536\\ \hline 1.009044217&1.512398264\\ \hline 2.045859358&3.040370613\\ \hline 4.097971755&5.904611356\\ \hline 7.884195673&10.277245972\\ \hline 13.787205857&14.399931992\\ \hline 19.103789438&16.085296354\\ \hline 24.828656320&16.839628627\\ \hline \end{array}}[/tex]
 

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  • #34
Does anyone happen to know the value of R at time 0 or something close to time 0?
 
  • #35
Ledsnyder said:
Does anyone happen to know the value of R at time 0 or something close to time 0?
Since [itex]tanh(0)=0[/itex], R was probably near the Planck scale near time zero, which for the equation used here, was after inflation. If expansion during inflation was strictly exponential, R was constant at the Planck scale during that phase (if 'constant' means anything at Planck scales...).
 
  • #36
The closest logistic model i could calculate was
16.997769517/(1+35.79253791478e^-.6025402362228t)

Ok but certainly not satisfying.
Where t is in billions of years
 
  • #37
Ledsnyder said:
The closest logistic model i could calculate was
16.997769517/(1+35.79253791478e^-.6025402362228t)

Ok but certainly not satisfying.
Where t is in billions of years

I think George's solutions (posts 13 and 7) do much better, because for most of U's history, radiation energy density has been negligible compared to the other forms. It hence gives R_Hubble with great precision back to the epoch where stars/galaxies were formed (also as far as you wish into the future)
 
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  • #38
Jorrie,your calculator is awesome.i wish there was similar to calculate complex curves I general that don't neccesarily deal with the universe.
 
  • #39
Ledsnyder said:
i wish there was similar to calculate complex curves I general that don't neccesarily deal with the universe.

Try http://graphsketch.com/, a free graph plotter with quite a lot of flexibility.