Cosmological expansion around ropes between galaxies The FLRW cosmology is a fair approximation of cosmological expansion for space between galaxies and clusters, however within galaxies themselves, is known to not hold, and in fact observational constraints limit expansion within galaxies to being negligible.
It is interesting to ponder other edge cases between expanding and non-expanding cosmology dynamics. For example, consider the asymptotic expansion of taking a cluster of galaxies and connect them with matter tubes or tendrils of certain thickness and density. It seems that such construction can be made so it would shift a negligible percentage of mass from the galaxies into the tubes. In summary, I want to consider the geometric evolution of a connected network of galaxies.
Qualitatively, the long term cosmological expansion of such clusters would seem to have a few distinct possibilities:

*

*Either the tendrils cannot be made thick/dense enough to be stable enough over cosmological ages, and still be considered essentially unidimensional, relative to the bidimensional scales of disc-like galaxies themselves. Hence they will expand and dillute over time.

*tendrils can be made stable enough, they will remain of constant density just like galaxies, but this disrupts topologically the FLRW approximation, in such a way as inhibiting cosmological expansion on the surrounding relatively empty intergalactic space between the galaxy nodes, and its tendril edges.

*tendrils can be made stable enough, they will remain of constant density just like galaxies, but they cannot however disrupt the FLRW approximation in the surrounding relatively empty intergalactic space, which means that this space keeps expanding like bubbles, deviating over time from being approximately flat, and becoming an Asymptotically Hyperbolic Void around galaxies and their connecting ropes.

Question: what is the right qualitative behavior that we should expect in this type of cosmological matter distributions?
Further clarification: In order to make the definition of an hyperbolic void clear within this context, consider a graph ABCD of four galaxies representing the sequential edges of a square. Light travels between these 4 galaxies in finite time across the tubes, but light traversing directly from A to C or B to D in the line-of-sight across intergalactic void will never reach the other side (It might expand Faster-Than-Light at some point, depending on the evolution of dark energy).
 A: You're mistaken. Mass doesn't mysteriously turn off expansion. There's just a distance regime (for a given mass) for which gravity dominates the combined effect of gravity and expansion.
Looking at the problem in one dimension:
Given $x'' = H^2x - GM/x^2$, for constants H, GM, there's some regime of small $x$ in which latter term dominates and the former can be approximated as zero. In SI units, on the scale of galaxies, $H^2$ is a tiny number (the Hubble factor squared has an order of $10^{-36} s^{-2}$) and  $GM$ is a huge number ($GM_{galaxy}$ has an order of $10^{32} m^3s^{-2}$), so the regime for which $x$ counts as "small" is also a huge number: order of $10^{22} m$ to get the expansion term to be 1/100 as large as the gravity term; an order of magnitude more distance and expansion starts to overtake gravity. So: we should predict that clusters of mass on the order of the mass of galaxies should retain satellites within distances of about $10^{22}m$ to $10^{23} m$. Outside of that distance, expansion should dominate, resulting in large and usually-increasing separations between groups of masses.
This prediction matches reality. The Local Group of galaxies (with a mass on the order of 10 times that of the Milky Way) is slightly less than $10^{23}m$ in diameter and its size is typical.
It's unlikely that if mass had a mysterious unknown interaction that turned off expansion within a certain radius, reality would work out to be exactly what we would predict just by adding Newtonian coordinate acceleration due to gravity to coordinate acceleration due to Hubble expansion and comparing which terms dominate over what domains.

Notes
In using only one term for gravity, I'm assuming that the second mass is much smaller than the first. However, even supposing we used two equal masses, it would be of little impact to an order of magnitude estimate.
Any galaxy can be used as the origin for the one-dimensional problem. Expanding the problem to three dimensions, any inertial trajectory can be used as the origin, although picking one that isn't comoving with a galaxy's center will make the math a pain.
