Influence of cosmological expansion on local systems

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Does the expansion of space mean that everything in it is stretched? Galaxies? Atoms? A shallow answer to this question is: "bounded" systems do not take part in the expansion. However, if space is stretched, then how can these systems not experience some, at least minimal, extension? Should bounded systems be stretched less intensively? The following several problems attempt to clarify the question by the example of a simple model: a classical atom, which consists of a negatively charged electron with negligible mass, rotating around a positively charged massive nucleus.

Let us place this atom in a homogeneous Universe which expands with scale factor $a(t)$. We will use two sets of spatial coordinates for its description, both spherical with the atom at the origin. The first set consists of physical coordinates $R,\theta,\varphi$, with $R$ being the distance between the electron and the nucleus at given time. The second set $r,\theta,\varphi$ is the comoving coordinates, the fixed points that partake in the cosmological expansion. The two sets are related through \[R = a(t)r.\] The angular coordinates are the same, as we assume that the cosmological expansion is radial.

Problem 1.

How can we understand in terms of the physical and comoving coordinates whether the atom partakes in the cosmological (Hubble) expansion or not?



Problem 2.

Derive the equation of motion for the atom's electron accounting for the cosmological expansion.


Problem 3.

Write the effective potential for the electron for the case of exponential expansion $a(t) = e^{\beta t}$ and use it to analyze the dynamics for the case $L^2=C$, where $C$ is the constant of electrostatic interaction.


Problem 4.

Why does the Solar system not expand despite of expansion of all the Universe? Give quantitative arguments.