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Doing some crude calculations (using the value of $H_0$ at this point of time only, since it is time dependent but not distance dependent thanks to Johannes answer) what is the radius of cosmological event horizon at this point of time? (not looking for the changes of CEH through time)

From here we have for $H_0 $:

$$H_0 = 73.8 \pm 2.4 (\frac{km}{s})\frac{1}{Mpc}\tag{I}$$

We are seeking the distance $L$ s.t. $H_0L = c = 3\times 10^6 \frac{km}{s}$

$$L=\frac {c}{H_0} = \frac {3\times 10^6}{73.8 \pm 2.4} Mpc \tag{II}$$

Where 1 pc = 3.26 light years ($ly$),

$$ L=\frac {c}{H_0} = \frac {3\times 10^6\times10^6\times3.26}{73.8 \pm 2.4} ly \tag{III}$$

$$ L= \frac {9.78\times 10^{12}}{73.8 \pm 2.4} ly \tag{IV}$$

$$L=1.3\pm0.1\times 10^{11} ly \tag{V}$$

Is this calculation correct? Would the correct calculation make sense? (By making sense I mean it would seem in accordance with some observation and not in contradiction to some other observations? Or results like this are unconfirmable, just mere flights of fancy were they do not relate to anything physical?

The only thing I could use to see it is not invalid (yes double negative, I cannot say it was valid) is the fact that observable universe is $45.7×10^9 ly$ but then again by that account $L=10^{123}ly$ would seem just as valid.

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2 Answers 2

The Hubble length $c/H_0$ does not coincide with the radius of the observable universe.

Your calculation assumes a Hubble parameter that doesn't change over time. This is not correct: the Hubble parameter $H$ changes over time, and $H_0$ (the Hubble constant) indicates the current value of $H$. To refer to $H_0$ as a 'constant' is a bit of a misnomer, it is effectively a constant in space, not in time.

Also note that if $H$ would have been constant over time, the Hubble time $1/H$ would be the time taken for the universe to increase in size by a factor of $e$. It is a coincidence that the current value for $H$ leads to a Hubble time very close to the current age of the universe.

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Is The Hubble length suppose to coincide with the radius of the observable universe? or not? Sorry I am confused, I don't know why Hubble length and radius of the observable universe relate to each other. –  Arjang Dec 26 '12 at 3:35
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No, there is no reason to expect the Hubble length to coincide with the radius of the observable universe. –  Johannes Dec 26 '12 at 3:40
    
Dear Arjang, Johannes' (right) answer was all about these two things' not being equal, so why did you ask again whether they were equal? Do you know derivatives? The Hubble constant is $(da/dt)/a=d\ln a / dt$ where $a$ is the distance between two galaxies or other two points. We're interested how this distance increases right now. But only if $a=Kt$ for all $t$, a linear function, $(da/dt)/a$ would be the same thing as $1/t$. In our Universe, the dependence of $a$ on $t$ wasn't linear/proportional. –  Luboš Motl Dec 26 '12 at 6:41
    
The current radius of the visible Universe is wolframalpha.com/input/?i=radius+of+visible+universe 46 billion light years. It's much greater than 13.7 billion years because the places close to the cosmic horizon were recently expanding to huge distances even though they correspond to very short periods of time right after the Big Bang - all the metric distances/times have expanded since the Big Bang. –  Luboš Motl Dec 26 '12 at 6:43
    
@LubošMotl : because of the first line of the answer , I wasn't sure if my question was implying the contrary, I wasn't questioning Johannes but confirming that my question is not implying the contrary. the result I ended up with was $13\times10^{10}$ it is 3 times the answer of from alpha. My question is at this point of time, how far is the cosmological event horizon? If it was possible to take a snap shot of universe, with given $H_0$ at this point of time, since it is not dependent on time. –  Arjang Dec 26 '12 at 8:20

The answer by Johannes is correct - the proper horizon distance in the concordance cosmology is ~46 billion light years. The reason that the answer in (1) was three times larger than that, when it should have been three times smaller, is that the value of c used was incorrect: The speed of light is $3 \times 10^5\ \mathrm{km}/\mathrm{s}$.

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