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I searched Physics Stack Exchange and google and could only find wordy articles on this, but what I am after is the actual mathematical calculation. I took General Relativity in Physics, and I tried calculating the radius of the visible universe myself, but my calculation is not quite right. What am I missing? My calculation is as follows:

Take a photon emitted 13.8 billion years go. The distance it must have been from us, at the moment it should be

$R_{0}=c\times t_{now}=3\times 10^{8}m/s\times13.8 \mbox{ billion years}=13.6 \mbox{ billion lightyears}$

We of course need to account for the expansion of space over the intercal of 13.8 billion year. The Hubble constant is roughly equal to

$H_{0}=73.8 \mbox{km/s/Mpc}=2.3917462 \times10^{-18} \mbox{/s}$

The metric for the expansion of the universe is:

$ds^{2}=-c^{2}dt^{2}+a(t)^{2}dr^2$

with approximately

$a(t)=a_{0}e^{\frac{t}{t_{H}}}$

where $t_{H}=\frac{1}{H_{0}}=4.181046 \times 10^{17}s$

For convenience I choose $a_{0}=1$ so that the coordinate $r$ of the location where the photon was emitted is given by:

$r=R_{0}=13.6 \mbox{ billion lightyears}$

we next have that:

$a_{now}=1 \times e^{\frac{13.8 \mbox{billion years}}{13.249217 \mbox{billion years}}}=2.834$

Now consider, again the the point in space that the photon was emitted from. It will be at the same coordinate $r=13.6 \mbox{ billion lightyears}$, but because of the expansion of space, its distance from us now should be

$R=a_{now} \times r = 2.834 \times 13.8 \mbox{ billion lightyears} = 39.1 \mbox{ billion lightyears} $

Obviously though, the actual radius of the visible universe is believed to be about 46.5 billion lightyears, whereas I am calculating 39.1 billion lightyears, so I am under-calculating by 6.4 billion lightyears.

My questions are therefore: Where am I going wrong? Is there a paper that presents the actual detailed calculation? I tried googling this like crazy, but could not find the actual calculation.

Rory Cornish
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1 Answers1

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The Hubble parameter is not a constant. It was larger at earlier epochs; therefore $\tau_H$ was smaller in the past.

You are also confused about definitions of $a$. In particular, $a_{\rm now} = a_0 = 1$.

The scale factor does not get bigger as $\exp(Ht)$ in a universe where matter is a significant component of the energy density.

Some details about how to calculate it correctly are given in https://physics.stackexchange.com/a/374164/43351 and https://physics.stackexchange.com/a/57538/43351

ProfRob
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  • Thanks. One of the things, in particular I wasn't sure of was whether or not it was true that the time for a photon emitted a distance r away from us was simple r/c, or not. I see not now is not. You do need to account for the fact that space is stretching as the photon is heading your way; I wasn't sure before. It will still go at c, but as time passes space is getting further apart, giving it more distance to cover due to the stretching, so in the symbols I used above, I think the correct equation is $a(t) \frac{dr}{dt}=c$; taken from your second link. – Rory Cornish Jul 23 '20 at 15:57