is it possible that gravitationally bound systems (clusters, galaxies, globular clusters, solar systems, planets with moons) are taken apart by the Hubble-expansion? if yes, under which circumstances and on what time scale?

bonus question: can you show that the integral over spherical harmonics almost always vanishes,

\begin{equation}

\int\mathrm{d}\Omega\:Y_{\ell m}(\theta,\phi) = 0

\end{equation}

except of course for the monopole $\ell=m=0$?

bonus question: a nice way to show that is the realisation that $Y_{00}=1/\sqrt{4\pi}$, so $\sqrt{4\pi} Y_{00}=1$, and of course $Y_{00}^*=Y_{00}$ which allows to write the integral like that \begin{equation}

ReplyDelete\int\mathrm{d}\Omega\: Y_{\ell m} =

\int\mathrm{d}\Omega\: Y_{\ell m} \times 1=

\sqrt{4\pi}\int\mathrm{d}\Omega\:Y_{\ell m} Y_{00}^* =

\sqrt{4\pi}\delta_{\ell 0}\delta_{m 0}

\end{equation}

and to use orthogonality in the last step: only for $\ell=m=0$ the integral is non-vanishing.

CQW *thinks* that it's a matter of time scales if a gravitationally bound system can be taken apart, it would need to hold that the dynamical time-scale of the system is larger than the time scale of Hubble-expansion $1/H(a)$. if that condition is not fulfilled, gravitational systems are not taken apart by the Hubble-expansion, e.g. the distance to the moon does not become larger with time. another way of looking at this would be that by no means the gravitational potentials in the solar system (or inside a globular cluster or a galaxy, or a cluster) are of the FLRW-type.

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