Undergrad Matter density in Weinberg's Cosmology book

[jouvelot]

Hi everyone,

On Page 72 of S. Weinberg's Cosmology book, it's mentioned, just after Equation 1.9.16, that, for the universe matter density ρ(r) to be an analytic function near the origin (spherical symmetry), it has to be given near r = 0 by a power series of r². I'm not a math wizard, so can anyone explain this little detail to me, please (why no odd powers of r)?

Thanks in advance.

Pierre

 

[Orodruin]

If it contained powers of $r$, then it would not be analytic. Consider the function $r$ itself.

 

[jouvelot]

Hi,

Thanks a lot for taking the time to answer my question, but I still don't get it. Analytic means, to me, for a function to be locally identical to its (convergent) Taylor expansion. Why wouldn't this work for r near 0, using spherical coordinates?

Thanks.

Bye,

Pierre

 

[Orodruin]

You are looking at it as a one-dimensional function. It is a three-dimensional function. Note that the coordinate system is singular at r=0.

 

[jouvelot]

Hi,

Indeed, but since ρ(r) is supposed to only depend on r and the coordinate divergence on the gradient only comes from θ and φ, I assumed this wouldn't be a factor. But I guess a 0 doesn't remove the 1/r divergence ;)

Thanks.

Bye,

Pierre

 

[jouvelot]

Indeed, but since ρ(r) is supposed to only depend on r and the coordinate divergence on the gradient only comes from θ and φ, I assumed this wouldn't be a factor. But I guess a 0 doesn't remove the 1/r divergence ;)

Answering my own question, if the gradient has only a 1/r divergence for r=0, the vector Laplacian diverges in 1/r², thus mandating ρ(r) to be a function of r².