# The Radius of the Universe

1. Jul 3, 2013

### johne1618

An effort to understand the physical implications of the Hubble radius.

I start by assuming a flat radial FRW metric given by:
$$ds^2 = - c^2 dt^2 + a(t)^2 dr^2$$
Let us assume that we are holding one end of a rigid ruler with the other end out in space at a fixed proper distance $R$ away from us.
$$R = a(t) r$$
Therefore the comoving radial co-ordinate $r$ of the end of the ruler is given by
$$r = \frac{R}{a(t)}\ \ \ \ \ \ \ \ \ \ (1)$$
Now using the relation for proper time $ds = -c\ d\tau$ in the expression for the metric and dividing through by $d\tau^2$ we obtain the differential relation
$$c^2 \left(\frac{dt}{d\tau}\right)^2 - a^2\left(\frac{dr}{d\tau}\right)^2 = c^2. \ \ \ \ \ \ \ \ \ \ \ \ (2)$$
By differentiating Equation (1) by proper time $\tau$ we find
$$\frac{dr}{d\tau} = -\frac{R}{a^2}\frac{da}{dt}\frac{dt}{d\tau}$$
Substituting the above expression into Equation (2) we obtain
$$c^2\left(\frac{dt}{d\tau}\right)^2 - R^2 \left(\frac{\dot a}{a}\right)^2\left(\frac{dt}{d\tau}\right)^2 = c^2$$
Rearranging, and substituting the Hubble parameter $H=\dot{a}/a$, we obtain:
$$\frac{dt}{d\tau} = \frac{1}{\sqrt{1 - \frac{R^2 \ H^2}{c^2}}} \ \ \ \ \ \ \ \ (3)$$
Now the Hubble law, for the proper recession velocity $v$ of a galaxy at proper distance $R$, is given by
$$v = H \ R$$
Substituting into Equation (3) we finally obtain
$$\frac{dt}{d\tau} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}.$$
This implies that as one approaches the Hubble radius $R_H=c/H$, where the galaxy recession velocities approach $c$, we find that the local proper time $\tau$ slows down to a standstill compared to coordinate time $t$. (It is interesting that the time dilation formula is exactly as one would expect from a special relativity viewpoint.)

This argument seems to show that there is a cosmological event horizon at the Hubble radius. The Hubble radius really is the radius of our Universe, at the present cosmological time $t$, and the event horizon is the edge of our Universe.

I would say that the space beyond the Hubble radius event horizon is not part of our Universe just as the space beyond a black hole's event horizon is not part of our Universe.

P.S. Implication for Universal expansion rate

As the Hubble radius really is the radius of the Universe then it must expand with the Universe therefore
$$R_H = a(t) r_h$$
where $r_h$ is a constant.

Substituting into Hubble's law with $v=c$ we have
$$\frac{\dot a}{a} = \frac{c}{R_H} \\ \frac{\dot a}{a} = \frac{c}{a \ r_h}$$
Thus $\dot a$ is a constant so that $a$ must increase linearly with time.

Last edited: Jul 3, 2013
2. Jul 3, 2013

### Mordred

first off the Hubble radius is the radius at which galaxies are receeding from us at the speed of light. It is not the entire universe or the entire observable universe. Hubble's constant changes over time so it would be useful to denote H0, as Hubble's constant today.

for distances larger than the Hubble radius use the form

rhs=c/H0 where hs =Hubble's sphere. Objects receed beyond Hubbles sphere at faster than the speed of light. up to 3c at z=1090.

this forum has a handy tool that will help show that relation easiest. The link is in my signature. Cosmocalc goto the site and use the lightcone calculator. Here is a 50 step printout of that calculator. I included all the columns to show the relation between Hubble's radius, the particle horizon, the horizon distance and Vrec which is tghe recessive velocity history of the observable universe. The metrics used in the calculator is contained here

it is based on this artricle by Lineweaver and Davis.
http://arxiv.org/pdf/astro-ph/0402278v1.pdf you may find that article handy to read.

edit:there appears to be an error in the text small Jorrie may be working on it so here is the text large. its not as clean as it used to be I'll send Jorrie an email to look at it.
You can also graph the results however you have to watch the scales.

$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.057&3.15&66.18\\ \hline 0.001&863.334&0.0006&0.0009&45.160&0.052&0.071&3.14&57.18\\ \hline 0.001&683.804&0.0008&0.0013&44.962&0.066&0.090&3.12&49.59\\ \hline 0.002&541.606&0.0012&0.0019&44.736&0.083&0.113&3.11&43.14\\ \hline 0.002&428.979&0.0017&0.0028&44.478&0.104&0.142&3.09&37.62\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.179&3.07&32.87\\ \hline 0.004&269.117&0.0036&0.0057&43.849&0.163&0.224&3.05&28.76\\ \hline 0.005&213.154&0.0052&0.0081&43.471&0.204&0.281&3.02&25.18\\ \hline 0.006&168.829&0.0075&0.0116&43.043&0.255&0.353&2.99&22.05\\ \hline 0.007&133.721&0.0107&0.0165&42.559&0.318&0.442&2.96&19.31\\ \hline 0.009&105.913&0.0153&0.0235&42.012&0.397&0.552&2.92&16.90\\ \hline 0.012&83.889&0.0219&0.0334&41.397&0.493&0.690&2.87&14.77\\ \hline 0.015&66.444&0.0312&0.0475&40.703&0.613&0.861&2.83&12.89\\ \hline 0.019&52.627&0.0445&0.0675&39.921&0.759&1.072&2.77&11.23\\ \hline 0.024&41.683&0.0634&0.0960&39.041&0.937&1.332&2.71&9.76\\ \hline 0.030&33.015&0.0902&0.1363&38.052&1.153&1.652&2.64&8.45\\ \hline 0.038&26.150&0.1282&0.1936&36.938&1.413&2.043&2.57&7.30\\ \hline 0.048&20.712&0.1823&0.2748&35.686&1.723&2.519&2.48&6.27\\ \hline 0.061&16.405&0.2590&0.3901&34.278&2.090&3.095&2.38&5.36\\ \hline 0.077&12.993&0.3679&0.5535&32.696&2.516&3.785&2.27&4.55\\ \hline 0.097&10.291&0.5223&0.7851&30.918&3.004&4.606&2.15&3.83\\ \hline 0.123&8.151&0.7414&1.1130&28.920&3.548&5.571&2.01&3.19\\ \hline 0.155&6.456&1.0518&1.5760&26.679&4.132&6.686&1.85&2.62\\ \hline 0.196&5.114&1.4908&2.2269&24.168&4.726&7.950&1.68&2.12\\ \hline 0.247&4.050&2.1099&3.1334&21.363&5.274&9.345&1.48&1.68\\ \hline 0.312&3.208&2.9777&4.3736&18.248&5.688&10.827&1.27&1.30\\ \hline 0.394&2.541&4.1804&6.0136&14.827&5.835&12.324&1.03&0.97\\ \hline 0.497&2.013&5.8131&8.0532&11.148&5.539&13.731&0.77&0.69\\ \hline 0.627&1.594&7.9554&10.3462&7.321&4.593&14.936&0.51&0.44\\ \hline 0.792&1.263&10.6323&12.5763&3.529&2.795&15.854&0.25&0.22\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline 1.263&0.792&17.3021&15.6609&3.145&3.971&16.845&0.22&0.25\\ \hline 1.571&0.637&20.8119&16.3885&5.640&8.860&17.041&0.39&0.54\\ \hline 1.955&0.512&24.4447&16.8085&7.716&15.084&17.146&0.54&0.90\\ \hline 2.433&0.411&28.1470&17.0395&9.417&22.906&17.199&0.65&1.34\\ \hline 3.027&0.330&31.8868&17.1633&10.797&32.683&17.222&0.75&1.90\\ \hline 3.766&0.265&35.6465&17.2287&11.913&44.870&17.229&0.83&2.60\\ \hline 4.687&0.213&39.4170&17.2627&12.812&60.048&17.263&0.89&3.48\\ \hline 5.832&0.171&43.1927&17.2806&13.536&78.940&17.281&0.94&4.57\\ \hline 7.257&0.138&46.9714&17.2899&14.118&102.451&17.290&0.98&5.93\\ \hline 9.030&0.111&50.7517&17.2946&14.586&131.709&17.295&1.01&7.62\\ \hline 11.236&0.089&54.5326&17.2972&14.962&168.116&17.297&1.04&9.72\\ \hline 13.982&0.072&58.3138&17.2985&15.264&213.419&17.299&1.06&12.34\\ \hline 17.398&0.057&62.0955&17.2991&15.507&269.791&17.299&1.08&15.60\\ \hline 21.649&0.046&65.8771&17.2995&15.702&339.938&17.300&1.09&19.65\\ \hline 26.939&0.037&69.6588&17.2997&15.859&427.224&17.300&1.10&24.70\\ \hline 33.521&0.030&73.4405&17.2999&15.985&535.837&17.300&1.11&30.97\\ \hline 41.711&0.024&77.2224&17.2998&16.087&670.988&17.300&1.12&38.79\\ \hline 51.902&0.019&81.0041&17.2999&16.168&839.161&17.300&1.12&48.51\\ \hline 64.584&0.015&84.7858&17.3000&16.234&1048.425&17.300&1.13&60.60\\ \hline 80.364&0.012&88.5677&17.2998&16.286&1308.820&17.300&1.13&75.65\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&17.300&1.13&94.38\\ \hline \end{array}}$$

Last edited: Jul 3, 2013
3. Jul 3, 2013

### Mordred

apparently Jorrie has a new version out,
that looks better lol
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html

in particular formula 2
http://arxiv.org/pdf/1304.3823v1.pdf

$${\small\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.057&3.15&66.18\\ \hline 0.001&863.334&0.0006&0.0009&45.160&0.052&0.071&3.14&57.18\\ \hline 0.001&683.804&0.0008&0.0013&44.962&0.066&0.090&3.12&49.59\\ \hline 0.002&541.606&0.0012&0.0019&44.736&0.083&0.113&3.11&43.14\\ \hline 0.002&428.979&0.0017&0.0028&44.478&0.104&0.142&3.09&37.62\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.179&3.07&32.87\\ \hline 0.004&269.117&0.0036&0.0057&43.849&0.163&0.224&3.05&28.76\\ \hline 0.005&213.154&0.0052&0.0081&43.471&0.204&0.281&3.02&25.18\\ \hline 0.006&168.829&0.0075&0.0116&43.043&0.255&0.353&2.99&22.05\\ \hline 0.007&133.721&0.0107&0.0165&42.559&0.318&0.442&2.96&19.31\\ \hline 0.009&105.913&0.0153&0.0235&42.012&0.397&0.552&2.92&16.90\\ \hline 0.012&83.889&0.0219&0.0334&41.397&0.493&0.690&2.87&14.77\\ \hline 0.015&66.444&0.0312&0.0475&40.703&0.613&0.861&2.83&12.89\\ \hline 0.019&52.627&0.0445&0.0675&39.921&0.759&1.072&2.77&11.23\\ \hline 0.024&41.683&0.0634&0.0960&39.041&0.937&1.332&2.71&9.76\\ \hline 0.030&33.015&0.0902&0.1363&38.052&1.153&1.652&2.64&8.45\\ \hline 0.038&26.150&0.1282&0.1936&36.938&1.413&2.043&2.57&7.30\\ \hline 0.048&20.712&0.1823&0.2748&35.686&1.723&2.519&2.48&6.27\\ \hline 0.061&16.405&0.2590&0.3901&34.278&2.090&3.095&2.38&5.36\\ \hline 0.077&12.993&0.3679&0.5535&32.696&2.516&3.785&2.27&4.55\\ \hline 0.097&10.291&0.5223&0.7851&30.918&3.004&4.606&2.15&3.83\\ \hline 0.123&8.151&0.7414&1.1130&28.920&3.548&5.571&2.01&3.19\\ \hline 0.155&6.456&1.0518&1.5760&26.679&4.132&6.686&1.85&2.62\\ \hline 0.196&5.114&1.4908&2.2269&24.168&4.726&7.950&1.68&2.12\\ \hline 0.247&4.050&2.1099&3.1334&21.363&5.274&9.345&1.48&1.68\\ \hline 0.312&3.208&2.9777&4.3736&18.248&5.688&10.827&1.27&1.30\\ \hline 0.394&2.541&4.1804&6.0136&14.827&5.835&12.324&1.03&0.97\\ \hline 0.497&2.013&5.8131&8.0532&11.148&5.539&13.731&0.77&0.69\\ \hline 0.627&1.594&7.9554&10.3462&7.321&4.593&14.936&0.51&0.44\\ \hline 0.792&1.263&10.6323&12.5763&3.529&2.795&15.854&0.25&0.22\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline 1.263&0.792&17.3021&15.6609&3.145&3.971&16.845&0.22&0.25\\ \hline 1.571&0.637&20.8119&16.3885&5.640&8.860&17.041&0.39&0.54\\ \hline 1.955&0.512&24.4447&16.8085&7.716&15.084&17.146&0.54&0.90\\ \hline 2.433&0.411&28.1470&17.0395&9.417&22.906&17.199&0.65&1.34\\ \hline 3.027&0.330&31.8868&17.1633&10.797&32.683&17.222&0.75&1.90\\ \hline 3.766&0.265&35.6465&17.2287&11.913&44.870&17.229&0.83&2.60\\ \hline 4.687&0.213&39.4170&17.2627&12.812&60.048&17.263&0.89&3.48\\ \hline 5.832&0.171&43.1927&17.2806&13.536&78.940&17.281&0.94&4.57\\ \hline 7.257&0.138&46.9714&17.2899&14.118&102.451&17.290&0.98&5.93\\ \hline 9.030&0.111&50.7517&17.2946&14.586&131.709&17.295&1.01&7.62\\ \hline 11.236&0.089&54.5326&17.2972&14.962&168.116&17.297&1.04&9.72\\ \hline 13.982&0.072&58.3138&17.2985&15.264&213.419&17.299&1.06&12.34\\ \hline 17.398&0.057&62.0955&17.2991&15.507&269.791&17.299&1.08&15.60\\ \hline 21.649&0.046&65.8771&17.2995&15.702&339.938&17.300&1.09&19.65\\ \hline 26.939&0.037&69.6588&17.2997&15.859&427.224&17.300&1.10&24.70\\ \hline 33.521&0.030&73.4405&17.2999&15.985&535.837&17.300&1.11&30.97\\ \hline 41.711&0.024&77.2224&17.2998&16.087&670.988&17.300&1.12&38.79\\ \hline 51.902&0.019&81.0041&17.2999&16.168&839.161&17.300&1.12&48.51\\ \hline 64.584&0.015&84.7858&17.3000&16.234&1048.425&17.300&1.13&60.60\\ \hline 80.364&0.012&88.5677&17.2998&16.286&1308.820&17.300&1.13&75.65\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&17.300&1.13&94.38\\ \hline \end{array}}$$

4. Jul 3, 2013

5. Jul 3, 2013

### Mordred

try this notation may help

${dl^2}=d{r^2}+{R^2}{sin^2}\frac{r}{R}S{\gamma^2}$

where
$d\gamma^2=d\theta^2+sin^2\theta d\phi^2$

here R is the curvature radius of the 3 dimensional sphere and r is the distance from the origin.
so
$ds^2=c^2dt^2+dl^2$

a generalization of the Eienstein model is to allow the curvature radius R(t) to be a function of time

gives the FLRW metric in the form

$ds^2 = -c^2dt^2 +a^2(t)[dr^2 + S\kappa^2(r)d\gamma^2]$

$$S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sinH(r/R) &(k=-1) \end {cases}$$

in this notation a(t) is dimensionless it is defined so that a(t0=1 at time t0
when the curvature radius is R0. At other times the curvature radius is a(t)R0

a(t) is replaced by R(t) where R(t) is units of length.

edit had to correct should have been sinH for k=-1

Last edited: Jul 3, 2013
6. Jul 3, 2013

### Jorrie

The calculator has two font sizes for TEX output: 'small' and 'script'. Below is a comparison. It seems that the 'script' size fits the font size of PF well. There are other forums where 'small' fits best.

With 'Tex small' selected:

$${\small\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.057&3.15&66.18\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.179&3.07&32.87\\ \hline \end{array}}$$

With 'Tex script' selected:

$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.057&3.15&66.18\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.179&3.07&32.87\\ \hline \end{array}}$$

I recommend changing any older sig. links to LightCone7.

7. Jul 4, 2013

### johne1618

My result for the time dilation $dt/d\tau$ at a fixed proper distance from the origin in an expanding Universe is general and does not depend on a flat metric.

Following Mordred's suggestion I will use a general FRW metric with the following notation used by Davies and Lineweaver in http://arxiv.org/pdf/astro-ph/0310808:
$$ds^2 = - c^2 dt^2 + R(t)^2\left[d\chi^2 + S_k^2(\chi)d\psi^2\right]$$
where $dt$ is the time coordinate separation, $d\chi$ is the dimensionless comoving coordinate separation and $d\psi^2=d\theta^2+\sin^2\theta\ d\phi^2$, where $\theta$ and $\phi$ are the polar and azimuthal angles in spherical coordinates. The scalefactor, $R$, has dimensions of distance. The function $S_k(\chi)=\sin \chi$, $\chi$ or $\sinh \chi$ for closed ($k=+1$), flat ($k=0$) or open ($k=-1$) universes respectively.

The proper distance $D$, at time $t$, in an expanding universe, between an observer at the origin and a distant galaxy is defined to be a surface of constant time ($dt=0$). We are interested in the radial distance so $d\psi=0$. The FRW metric then reduces to $ds = R \ d\chi$ which, upon integration, gives the proper distance $D(t)$ as
$$D(t) = R(t) \ \chi.$$
Now I wish to define a rigid proper ruler. Let us assume that we are holding one end at the origin with the other end out in space at a fixed proper distance $D$ away from us.

Therefore the radial coordinate $\chi$ of the end of the ruler is given by
$$\chi = \frac{D}{R(t)}\ \ \ \ \ \ \ \ \ \ \ (1)$$
where $D$ is constant.

I now wish to derive an expression for the proper time $\tau$ that elapses at the far end of the ruler. Using the relation for an interval of proper time $ds = -c \ d\tau$ in the metric and dividing through by $d\tau^2$ we obtain the differential relation
$$c^2 \left(\frac{dt}{d\tau}\right)^2 - R(t)^2\left(\frac{d\chi}{d\tau}\right)^2 = c^2. \ \ \ \ \ \ \ \ \ \ \ \ (2)$$
By differentiating Equation (1) by proper time $\tau$ we find
$$\frac{d\chi}{d\tau} = -\frac{D}{R^2}\frac{dR}{dt}\frac{dt}{d\tau}$$
Substituting the above expression into Equation (2) we obtain
$$c^2\left(\frac{dt}{d\tau}\right)^2 - D^2 \left(\frac{\dot R}{R}\right)^2\left(\frac{dt}{d\tau}\right)^2 = c^2$$
Rearranging, and substituting the Hubble parameter $H=\dot{R}/R$, we obtain:
$$\frac{dt}{d\tau} = \frac{1}{\sqrt{1 - \frac{D^2 \ H^2}{c^2}}} \ \ \ \ \ \ \ \ (3)$$
It is interesting to express the above relationship in terms of the proper velocities of galaxies at proper distance $D$.

Now the Hubble law, for the proper recession velocity $v$ of a galaxy at proper distance $D$, is given by
$$v = H \ D$$
Substituting into Equation (3) we finally obtain
$$\frac{dt}{d\tau} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}.$$
This expression is exactly the same as the time dilation at a moving galaxy assuming a simple special relativistic model of receeding galaxies.

Last edited: Jul 4, 2013
8. Jul 4, 2013

### jackmell

"Not part of our Universe?" What do you mean by that? Well it matters that's what. I mean if it's part of say something else beyond the Hubble radius but we can never, ever communicate with it, run tests on it, measure it, quantify it, then even though we are part of that bigger something, then I too believe beyond the Hubble radius is "not part of our Universe".

Last edited: Jul 4, 2013
9. Jul 4, 2013

### Chalnoth

I haven't read the whole thing, but this isn't correct. Gravity determines how the expansion rate changes over time, and we write its effects in the Friedmann equations, the first of which is usually sufficient:

$$H^2(a) = {8\pi G \over 3} \rho(a) - {kc^2 \over a^2}$$

Here $\rho$ is the energy density of the universe, in this case including the cosmological constant, and $k$ is the spatial curvature. What this means is that how the universe expands depends upon how the various components of the universe change in density as the universe expands. For example, radiation dilutes as $1/a^4$, normal and dark matter dilute as $1/a^3$, and the cosmological constant doesn't dilute at all.

Right now, for example, we are approaching a universe where nearly all of the energy density is likely to be in the cosmological constant, in which case the Hubble parameter is a constant, which means:

$${1 \over a}{da \over dt} = H_0$$
$${da \over dt} = H_0 a$$

The solution to this differential equation is simply:
$$a(t) = e^{H_0 t}$$

(Generally this differential equation has an overall multiplicative integration constant out in front, but this isn't necessary here as we define $a(t=0) = 1$.)

10. Jul 4, 2013

### George Jones

Staff Emeritus
No, not for a flat universe that includes dark energy. For the scale factor in this case, look at the attachment to

11. Jul 4, 2013

### Mordred

lol I was busy typing and struggling with the latex, a full commoving to proper distance example. I happened to notice your post and article. Covered everything I was typing lol. I had forgotten that redshift article you had previously posted. I also could have saved time by simply posting this wiki page. lol

http://en.wikipedia.org/wiki/Comoving_distance

coincidentally it has the metrics I was struggling with.(by the way how do you latex the integral in the first formula?)

http://albrecht.ucdavis.edu/Courses/P262/FRW-262.pdf

by the way if you can, one of the best text books covering distance measurements in the FLRW metrics that I have read is

Barbera Ryden's "Introductory to cosmology" I found her format far better than say Donaldson's "Modern cosmology"

http://www.amazon.ca/dp/0805389121

if you can afford it, its worth every penny

Last edited by a moderator: May 6, 2017
12. Jul 4, 2013

### Chalnoth

Note that in this equation $t=0$ is the current time.

13. Jul 5, 2013

### Mordred

This is a form of the metric I had in my notes from some Peeble's article though I cannot recall which one.

$ds^2=dt^2-a(t)^2[\frac{dr^2}{1\pm r^2/R^2}+r^2(d\theta^2+sin^2\theta d\phi^2]$

expansion can thus be defined as

$H^2=(\frac{\dot{a}}{a})^2=\frac{8}{3} \phi Gp\pm \frac{1}{a^2R^2}+\frac {\Lambda}{3}$

which can be approximated as

$H^2=H_0^2[\Omega(hz)^2+k(hz)^2+λ]$

the above includes matter,curvature and lambda or the stress energy tenser which is constant

Last edited: Jul 5, 2013