Light Cone - Hubble Radius & Time T and R Relationship

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    Hubble Radius Time
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Discussion Overview

The discussion revolves around the relationship between cosmic time (T) and the Hubble radius (R) in the context of cosmological models. Participants explore potential equations that could relate these two quantities, referencing approximations and numerical solutions relevant to the matter-dominated and dark energy-dominated epochs of the universe.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that there is an approximate formula relating T and R, potentially involving hyperbolic trigonometric functions like sinh or tanh, but emphasize that it is only valid under certain conditions.
  • Others argue that due to the mixed epoch of the universe, direct approximations may not be relevant, and numerical solutions are necessary for accurate modeling.
  • One participant notes that the Hubble radius R is closely related to Hubble time, which complicates the relationship between T and R.
  • Another participant provides a specific equation for the scale factor in a flat universe consisting of matter and dark energy, indicating a more complex relationship between T and R.
  • There is mention of a calculator that outputs tables and curves showing the past and future history of the universe, which could aid in understanding these relationships.
  • Participants express uncertainty about the exact form of the equation relating T and R, with references to previous discussions and equations that may need modification or clarification.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a specific equation relating T and R, and multiple competing views and uncertainties remain regarding the nature of this relationship.

Contextual Notes

The discussion highlights limitations in approximations due to the changing dynamics of matter and dark energy, as well as the need for numerical solutions in mixed epochs. There are unresolved assumptions regarding the specific forms of equations and their applicability across different cosmological scenarios.

  • #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}}
 

Attachments

  • 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.