How to calculate mass of closed Universe?

[johne1618]

How does one integrate the mass density over a closed Universe (a 3-sphere?) to obtain the total mass of that Universe?

Is this the correct integral?

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

where $R(t)$ is the radius of the Universe at cosmological time $t$.

By making the substitution $r=\sin \chi$ one finds that the above integral gives:

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

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

 

[Simon Bridge]

You need to tell us what each thing is.
Note - if R is a function of time, then the "cube" is in the wrong place. $R(t)^3\neq R^3(t)$

However, you appear to have received an answer already:
http://physics.stackexchange.com/questions/72602/how-to-calculate-mass-of-a-closed-universe

 

[johne1618]

You need to tell us what each thing is.
Note - if R is a function of time, then the "cube" is in the wrong place. $R(t)^3\neq R^3(t)$

However, you appear to have received an answer already:
http://physics.stackexchange.com/questions/72602/how-to-calculate-mass-of-a-closed-universe

Thanks.

 

[dchartier]

You need to tell us what each thing is.
Note - if R is a function of time, then the "cube" is in the wrong place. $R(t)^3\neq R^3(t)$

However, you appear to have received an answer already:
http://physics.stackexchange.com/questions/72602/how-to-calculate-mass-of-a-closed-universe

This may be a very stupid question, but what's the difference between $R(t)^3$ and $R^3(t)$?

 

[Simon Bridge]

The former applies the cube to the brackets, the latter applies the cube to the function.
$R(t)^3=Rt^3$
$R^3(t)=\big [ R(t)\big ]^3$

compare: $\cos(kx)^3$ with $\cos^3(kx)$
Since it is also valid to write $\cos kx$ the first version does not make it clear what is intended.
Is it the cosine of the cube of a product or the cube of the cosine of a product? OP only had one variable in the brackets, which gives notice to the reader.

... its really just a notation foible.
I don't think anyone would get confused in such a small equation but in big long multiline equations it gets easier to be misread.

 

[dchartier]

The former applies the cube to the brackets, the latter applies the cube to the function.
$R(t)^3=Rt^3$
$R^3(t)=\big [ R(t)\big ]^3$

compare: $\cos(kx)^3$ with $\cos^3(kx)$
Since it is also valid to write $\cos kx$ the first version does not make it clear what is intended.
Is it the cosine of the cube of a product or the cube of the cosine of a product? OP only had one variable in the brackets, which gives notice to the reader.

... its really just a notation foible.
I don't think anyone would get confused in such a small equation but in big long multiline equations it gets easier to be misread.

I understand, thanks!