Light Cone - Hubble Radius & Time T and R Relationship

[Ledsnyder]

T (Gy) R (Gly)
0.00037338 0.00062840
0.00249614 0.00395626
0.01530893 0.02347787
0.09015807 0.13632116
0.52234170 0.78510382
2.97769059 4.37361531
13.78720586 14.39993199
32.88494318 17.18490043
47.72506282 17.29112724
62.59805320 17.29930703
77.47372152 17.29980205
92.34940681 17.29990021

http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html

is there an equation that relates T and R?

 

[marcus]

Hi Ledsnyder, great to see you using the Lightcone calculator. It can save you trouble if you just click on "scriptsize" in the "Display as…" menu. That gives the table output in Latex form ready to copy paste into a post like this:
$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&R (Gly) \\ \hline 0.00037338&0.00062840\\ \hline 0.00249614&0.00395626\\ \hline 0.01530893&0.02347787\\ \hline 0.09015807&0.13632116\\ \hline 0.52234170&0.78510382\\ \hline 2.97769059&4.37361531\\ \hline 13.78720586&14.39993199\\ \hline 32.88494318&17.18490043\\ \hline 47.72506282&17.29112724\\ \hline 62.59805320&17.29930703\\ \hline 77.47372152&17.29980205\\ \hline 92.34940681&17.29990021\\ \hline \end{array}}$$

In fact there IS a formula that works as a pretty good approximation relating T and R. Jorrie discussed it with us.

As I recall it involves sinh, or tanh , some hyperbolic trig function. And it is only approximately valid, over the matter dominated span of time. The problem is that matter and radiation behave somewhat differently under expansion. In one case the energy density falls off as the cube, in the other as the fourth power.
So if you want to allow for a MIX, a variable mix whose proportions change with expansion, then you really need to crank out a NUMERICAL solution. loops, wife calling to supper. have to go back later.

You already figured out to open the "column definition and selection" menu, I see you arranged to have 8-digit precision. So having opened that menu you can select which ever columns you want by having checks in the corner boxes instead of X's. If you click on a check it will change to an X, getting rid of a column. And viceversa, you can add a column by clicking on an X. The boxes are in the upper right corner of the menu item for each column.

I'll see if I can respond to some of your questions here, in this thread. That way, if there is any useful information that comes out, other readers can share, and they can catch my mistakes, add stuff I miss, etc.

BTW if the LaTex table in this post does not show up, click RELOAD, in the browser and reload the post. Should work then.

 

[Ledsnyder]

okay

 

[Ledsnyder]

SO its sinh or tanh?

 

[marcus]

I don't recall what it was in detail, just that it involved some hyperbolic trig function. Jorrie would probably recall.
But in any case that is just an approximate formula. To be more accurate you have to solve numerically, and that is what the Lightcone calculator does (or any of the other online calculators that implement the standard cosmic model.)

They all give the same numbers (if fed the same parameters) but Jorrie's is especially nice because it is not merely a ONESHOT. It outputs tables showing past and future history of cosmos. And it will also plot the output as curves if you choose that instead of table.

BTW The Hubble time is just a(t)/a'(t). I think you know most of what I'm saying here. At some point the question came up about relating T to R. But Hubble radius R is virtually synonymous with Hubble time a/a'.

And you know a' has units of inverse time (while a is a pure number) so a/a' has units of time. Which is right.
It shouldn't be too hard to come up with an approximate formula relating scale factor a to both those quantities you are interested in.

BTW when you look at the second friedman equation you see and expression involving rho (density) and p pressure. If you want to transcribe that, there is a greek letter rho off to the right of the box where you are typing. Click "new reply" and you will see a bunch of greek letters off to the right, that you click on.
ρ + 3p
energy density plus 3x pressure. It is confusing that ρ looks like p, but we have to keep the distinction clear. Both pressure and energy density have basically the same units. Forcexdistance = energy, so Force/area = Energy/volume.

If you are working with MASS density then you have a factor of c² to stick in there somewhere to make it possible to add the two things. Again I think you probably know this. Measuring mass by its energy equivalent. Getting quantities to be in the same terms if you want to add them.

From what I can see you are comfortable with calculus, and using google calculator etc.

I hope we can dig up that formula relating T and R. Sorry I don't recall it, it was a few years back. What I'm recalling could be two separate things relating each of them to the scale factor a, or to its reciprocal S = 1/a. That would do just as well (relating each to a third quantity.)

 

[Jorrie]

I have already replied to the method in the acceleration thread.

Unfortunately we live in "mixed epoch" time, so the direct approximations are not very relevant. In a few billion years from now, the cosmological constant may be totally dominant and R_H will be time independent. Some 9 billion years ago, matter density was totally dominant and we could have written with good accuracy (for a spatially flat universe):

$a \approx t^{2/3}$ and $t \approx 2/(3H)$, where $H \approx H_0 \sqrt{\Omega_m/a^3}$

PS: for the far future (exponential expansion phase), $a \approx \sinh{t}$ and $H \approx H_0 \sqrt{\Omega_\Lambda}$

 

[George Jones]

For a spatially flat universe that consists of matter and dark energy (w = -1), but no radiation, the scale factor is given exactly by

$$a\left(t\right) = A \sinh^{\frac{2}{3}} \left(Bt\right),$$

where

$$A = \left( \frac{1 - \Omega_{\Lambda 0}}{\Omega_{\Lambda 0}} \right)^{\frac{1}{3}}$$

and

$$B = \frac{3}{2} H_0 \sqrt{\Omega_{\Lambda 0}} .$$

 

[Jorrie]

Thanks George, I think these were the "mixed epoch" equations we were looking for. :)

 

[marcus]

Beautiful! Thanks so much to you both. I'm glad to see Ledsnyder asking about the Lightcone numbers. It is a really interesting calculator (past and future history of universe, tables, curves) and potentially a great teaching tool.

 

[George Jones]

Ledsnyder, I am a little confused. Your thread title is "Hubble Radius and time", but in your post you write

is there an equation that relates T and R?

R is the Hubble radius, but T is not the Hubble time, T is the cosmic time at which the light left the source. Marcus has made the same point.

At some point the question came up about relating T to R. But Hubble radius R is virtually synonymous with Hubble time

 

[marcus]

George, really appreciate the help. Jorries's calculator has a lot of pedagogical potential and accompanying it with some equations that users like Ledsnyder can latch on to enriches the experience.

I remember now when back in August 2012 you contributed this:

I haven't been following this thread, so I might have missed some things. Did anyone write down the easy analytical solution to this separable differential equation?

$$\int^{Y\left(t\right)}_{Y\left(0\right)} \frac{dY}{1 - \left( \frac{Y}{Y_\infty} \right)^2} = \frac{3}{2} \int^t_0 dt'$$
gives

$$Y\left(t\right) = Y_\infty \tanh\left( \frac{3t}{2Y_\infty} \right)$$
for $Y\left(0\right)$ = $0$.

We were using Y for the Hubble time, back then. Didn't want to use subscripts as in T_H.

In this thread we are asking about an approximate equation relating age t to Hubble radius R(t)

So we could rewrite your equation here.

R(t) = R_∞ tanh (1.5c t/R_∞)

R(t) = 17.3 tanh (1.5 t/17.3 Gy) Gly.

I forget what the detailed assumptions were or whether this needed to be modified but as I recall it started us on a productive track back then.

 

[George Jones]

After a minor bit of playing around, I get the relation ship between R and T to be (with no radiation present)

$$R = \frac{3}{2B}\tanh\left(BT\right),$$

where B is given in post #7. This seems to reproduce numerically some of the relationship between T and R in the table.

[edit]Can't seem to remember my previous posts. Thanks, marcus.[/edit]

 

[George Jones]

I forget what the detailed assumptions were or whether this needed to be modified but as I recall it started us on a productive track back then.

The conditions are the same as in post #7, i.e., another way to get the result is to take the expression

$$a\left(t\right) = A \sinh^{\frac{2}{3}} \left(Bt\right)$$

and divide by its time derivative. This gives

$$R = \frac{3}{2B}\tanh\left(BT\right)$$

 

[Jorrie]

After a minor bit of playing around, I get the relation ship between R and T to be (with no radiation present)

$$R = \frac{3}{2B}\tanh\left(BT\right),$$

where B is given in post #7. This seems to reproduce numerically some of the relationship between T and R in the table.

I originally thought that the "with no radiation present" condition is sufficiently met back to T~50 thousand years (i.e. matter-radiation equality), but I now realize that this will substantially invalidate the condition. In order to stay within (say) 5% error, we should avoid using the equation at times earlier than 3 million years (S or z about 300). This includes all interesting epochs with the exception of the CMB (and earlier).

 

[Jorrie]

George, is it possible to perform something like you did in the link that Marcus provided:

Did anyone write down the easy analytical solution to this separable differential equation?

$$\int^{Y\left(t\right)}_{Y\left(0\right)} \frac{dY}{1 - \left( \frac{Y}{Y_\infty} \right)^2} = \frac{3}{2} \int^t_0 dt'$$
gives

$$Y\left(t\right) = Y_\infty \tanh\left( \frac{3t}{2Y_\infty} \right)$$
for $Y\left(0\right)$ = $0$.

for the comoving distance to a galaxy at redshift z?

The integral is given by Tamara Davis' Thesis, eq. A.10 (pp.116) as:

$$D_{now} = \int^{z}_{0} \frac{dz'}{H(z')}$$

where I have converted her comoving coordinate $\chi(z)$ to comoving distance $D_{now}$. This is exactly what LightCone 7 uses.

For a spatially flat U and again ignoring radiation:

$$H(z') = H_0 \sqrt{\Omega_\Lambda + (1+z')^3\Omega_m}$$

So the question is, is there a short-cut here as well?

 

[George Jones]

Tamara Davis' Thesis[/URL], eq. A.10 (pp.116) as:

$$D_{now} = \int^{z}_{0} \frac{dz'}{H(z')}$$

where I have converted her comoving coordinate $\chi(z)$ to comoving distance $D_{now}$. This is exactly what LightCone 7 uses.

For a spatially flat U and again ignoring radiation:

$$H(z') = H_0 \sqrt{\Omega_\Lambda + (1+z')^3\Omega_m}$$

So the question is, is there a short-cut here as well?

This depends on what "shiort-cut" means.

Yes, the integral can be done, but the result is a somewhat complicated expression involving elliptic functions.

 

[George Jones]

This depends on what "shiort-cut" means.

Yes, the integral can be done, but the result is a somewhat complicated expression involving elliptic functions.

Using $\Omega_m = 0.307$, $\Omega_\Lambda = 0.693$, $\Omega_r = 0$, Maple gives

$$
D_{now} = \frac{1}{H_0} \left( 3.2742 - 1.1973 * \mathrm{EllipticF} \left( \frac{52.821 \sqrt{709.72 + 307z}}{1407.3 + 307z}, 0.96593 \right) \right).
$$

I think that WoframAlpha can be used to evaluate ellipticF.

 

[marcus]

Impressive! this thread is showing me some math functions I didn't know about, and recalling other interesting stuff about the standard universe model. Thanks GJ and Jorrie, and also Led for having gotten started with the Lightcone cosmic table-maker and asking a very sensible question (how does the reciprocal of the growth rate change over time, how does it relate to how long expansion has been going on, is there a formula?)

 

[George Jones]

It looks like Maple will handle non-zero $\Omega_r$ for spatially flat and non-flat universes. What exact values should I use for $\Omega_m$, $\Omega_\Lambda$, and $\Omega_r$?

 

[marcus]

It looks like Maple will handle non-zero $\Omega_r$ for spatially flat and non-flat universes. What exact values should I use for $\Omega_m$, $\Omega_\Lambda$, and $\Omega_r$?

Much better to get definite word from Jorrie, but in case he is busy and does not check in here for a while let me say tentatively that the default for Lightcone is FLAT, and that Omega_Lambda is (144/173)²

and the epoch of matter radiation equality comes at z+1 = 3400

So matter and radiation have to account for 1- (144/173)²

I would assume Omega_m + Omega_m/3400 = (3401/3400)Omega_m = 1- (144/173)²

But maybe in the actual working he uses different values, rounded off etc. Let me look. It probably says.
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
YES! just click on the link and you see in the upper right corner some listing of values for Omega_Lambda and Omega_m. they look rounded off, however. I'm not sure which ones are best to use in making a comparison.

 

[Jorrie]

http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
YES! just click on the link and you see in the upper right corner some listing of values for Omega_Lambda and Omega_m. they look rounded off, however. I'm not sure which ones are best to use in making a comparison.

Given the LightCone 7 default values: $R_0=14.4,\, R_{\infty}=17.3,\, S_{eq}=3400,\, \Omega=1$, the exact conversions for output comparison are:

$z = S-1,\ \Omega_\Lambda = (R_{0}/R_{\infty})^2,\, \Omega_r = (\Omega-\Omega_\Lambda)/(1+S_{eq}),\, \Omega_m = S_{eq}\Omega_r$, with $\Omega \ge \Omega_\Lambda$.

 

[Ledsnyder]

Thank you everyone! I will try and dissect these equations.

 

[Ledsnyder]

the graph of t and r looks like a logistic equation.

$N_{t} = \frac{K}{1 + \frac{K-N_0}{N_0} e^{-rt}}$ (equation 1.1)

 

[Ledsnyder]

\[/attach]

 

[Jorrie]

The graph of t vs r looks like a logistic equation.
$$N_{t} = \frac{K}{1 + \frac{K-N_0}{N_0} e^{-rt}}$$ (equation 1.1)

Your equation did not show because you did not put

[tex] ... [/tex]

tags around it.

What you said is true if we look coarsely over a large time range, especially into the future. For most of your curve, Lambda is totally dominant, giving exponential expansion. If you would limit the time to say 25 Gy or so (try setting S_lower to 0.4 or so), you may find the correlation is not so good, because matter en radiation then prevent an exponential law being followed.

 

[George Jones]

$$R = \frac{3}{2B}\tanh\left(BT\right),$$

the graph of t and r looks like a logistic equation.

$N_{t} = \frac{K}{1 + \frac{K-N_0}{N_0} e^{-rt}}$ (equation 1.1)

Yes, for large arguments, these two expressions have the same functional form.

$$\begin{align}
\tanh T &= \frac{\sinh T}{\cosh T} \\
&= \frac{e^T - e^{-T}}{e^T + e^{-T}} \\
&= \frac{e^{-T}\left(e^T - e^{-T}\right)}{e^{-T}\left(e^T + e^{-T}\right)} \\
&= \frac{1 - e^{-2T}}{1 + e^{-2T}} \\
&= \frac{1}{1 + e^{-2T}} - \frac{e^{-2T}}{1 + e^{-2T}}.
\end{align}$$

For large $T$, the second term is negligible.

 

[George Jones]

I wasn't sure if the total density was exactly unity. Maple didn't much like the fractions, so I used $\Omega_m = 0.30716$, $\Omega_\Lambda = 0.69284 - 9.0315\times10^{-5}$, and $\Omega_r = 9.0315\times10^{-5}$. Maple gives

$$
D_{now} = \frac{1}{H_0} \left( 3.27430 - 1.19705 * \mathrm{EllipticF} \left( -\frac{6.35122\sqrt{(2.31157 + z)(3401.99 + z)}}{563.240 + 122.970 z}, 0.965983 \right) \right).
$$

I hope that I have transcribed this correctly.

Using this, I have used Maple to reproduce the $D_{now}$ column on the calculator webpage. I am lucky in that my employer has a Maple site license, so I get full Maple on my computers legally for free.

 

[Jorrie]

This is very interesting, George.

Using this, I have used Maple to reproduce the $D_{now}$ column on the calculator webpage. I am lucky in that my employer has a Maple site license, so I get full Maple on my computers legally for free.

Do you mean that your result agreed with the $D_{now}$ column of the calculator (within round-off errors)? If so, it would be an interesting 'quality check' of the numerical integration results of LightCone 7.

PS: There may be a sign error in the first argument inside the EllipticF bracket of either reply 17 or 27, because with the minor changes in the other parameters, a change of sign in that argument seems unlikely(?)

 

[Ledsnyder]

what is the eliptic part for?

 

[Ledsnyder]

Perhaps I can make a logistic equation in that form that approximates the actual curve.

 

[Ledsnyder]

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

$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&R (Gly) \\ \hline 0.00037338&0.00062840\\ \hline 0.00249614&0.00395626\\ \hline 0.01530893&0.02347787\\ \hline 0.09015807&0.13632116\\ \hline 0.52234170&0.78510382\\ \hline 2.97769059&4.37361531\\ \hline 13.78720586&14.39993199\\ \hline 32.88494318&17.18490043\\ \hline 47.72506282&17.29112724\\ \hline 62.59805320&17.29930703\\ \hline 77.47372152&17.29980205\\ \hline 92.34940681&17.29990021\\ \hline \end{array}}$$

 

[Ledsnyder]

 

[Jorrie]

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.

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

 

[Ledsnyder]

Does anyone happen to know the value of R at time 0 or something close to time 0?

 

[Jorrie]

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...).