How does one integrate the mass density over a closed Universe (a 3-sphere?) to obtain the total mass of that Universe?(adsbygoogle = window.adsbygoogle || []).push({});

Is this the correct integral?

[tex]

M = R(t)^3 \rho\int_0^1 4 \pi r^2 \frac{dr}{\sqrt{1-r^2}}

[/tex]

where [itex]R(t)[/itex] is the radius of the Universe at cosmological time [itex]t[/itex].

By making the substitution [itex]r=\sin \chi[/itex] one finds that the above integral gives:

[tex]

M = \pi^2 R(t)^3 \rho.

[/tex]

According to wikipedia the hyperarea of a 3-sphere is [itex]2\pi^2 R^3[/itex] so I'm out by a factor of two.

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# How to calculate mass of closed Universe?

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