Light Cone - Hubble Radius & Time T and R Relationship

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The discussion centers around the relationship between cosmic time (T) and Hubble radius (R), with a focus on an approximate formula involving hyperbolic functions like sinh or tanh. It highlights that while an analytical solution exists, numerical methods are preferred for accuracy, especially in a mixed epoch where matter and radiation densities vary. The Lightcone calculator is noted for its utility in visualizing cosmic history and generating relevant tables and curves. Participants also explore the implications of using approximations at different epochs, emphasizing the need for caution in applying formulas before certain time thresholds. Overall, the conversation underscores the complexity of cosmic expansion and the importance of precise calculations in cosmology.
  • #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

{\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}}
 
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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.
{\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}} {\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}}
 

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  • R_H(T) curve.png
    R_H(T) curve.png
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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 tanh(0)=0, 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.