Given the following parametric form of the Friedmann Equation for an open, dust-filled (matter-dominated) universe:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]a(x)={a_0 \Omega \over 2(1-\Omega)}(cosh(x)-1)[/tex]

[tex]t(x)={\Omega \over 2 H_0 (1- \Omega)^{3/2}}(sinh(x)-x)[/tex]

I am trying to calculate the Hubble Radius, R=c/H(t) where H(t)=(da/dt)/a and have come up with the following solution:

[tex]{da \over dt} = {da \over dx} {dx \over dt}[/tex]

[tex]{da \over dx}={a_0 \Omega \over 2(1-\Omega)}sinh(x)[/tex]

[tex]{dt \over dx}={\Omega \over 2H(1- \Omega)^{3/2}}(cosh(x)-1)[/tex]

[tex]{da \over dt}=Ha_0(1-\Omega)^{1/2}coth({x \over 2})[/tex]

Integrate to find a:

[tex]a=Ha_0(1-\Omega)^{1/2}coth({x \over 2})t[/tex]

[tex]{{da \over dt} \over a}={1 \over t}[/tex]

Therefore, the Hubble Radius:

[tex]R_H={c a \over ({da \over dt})}=ct[/tex]

I just can't seem to find any references to this solution, so I don't know if it is actually correct.

Can anyone confirm?

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# Calcluating the Hubble Radius for an open universe?

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