The residual gravity acceleration effect in the Poincaré dodecahedral space

Context. In a flat space, it has been shown heuristically that the global topology of comoving space can affect the dynamics expected in the weak-field Newtonian limit, inducing a weak acceleration effect similar to dark energy. Aims. Does a similar effect occur in the case of the Poincare dodecahedral space, which is a candidate model of comoving space for solving the missing fluctuations problem observed in cosmic microwave background all-sky maps? Moreover, does the effect distinguish the Poincare space from other well-proportioned spaces? Methods. The acceleration effect in the Poincare space $S^3/I^*$ is studied, using a massive particle and a nearby test particle of negligible mass. Calculations are made in S 3 embedded in $\mathbb{R}^4$. The weak-limit gravitational attraction on a test particle at distance  r is set to be ${\propto}[R_{{\rm C}}\sin({r/R_{{\rm C}}})]^{-2}$ rather than ${\propto}r^{-2}$, where $R_{{\rm C}}$ is the curvature radius, in order to satisfy Stokes' theorem. A finite particle horizon large enough to include the adjacent topological images of the massive particle is assumed. The regular, flat, 3-torus  T 3 is re-examined, and two other well-proportioned spaces, the octahedral space $S^3/T^*$, and the truncated cube space $S^3/O^*$, are also studied. Results. The residual gravity effect occurs in all four cases. In a perfectly regular 3-torus of side length  L a , and in the octahedral and truncated cube spaces, the highest order term in the residual acceleration is the third-order term in the Taylor expansion in powers of $r/L_a$ (3-torus), or $r/R_{{\rm C}}$, respectively. However, the Poincare dodecahedral space is unique among the four spaces. The third order cancels, leaving the fifth order term ${\sim}{\pm}300 (r/R_{{\rm C}})^5$ as the most significant. Conclusions. Not only are three of the four perfectly regular well-proportioned spaces better balanced than most other multiply connected spaces in terms of the residual gravity acceleration effect by a factor of about a million (setting $r/L_a$ = $r/R_{{\rm C}}$ ~ 10 -3 ), but the fourth of these spaces is about ten thousand times better balanced than the other three. This is the Poincare dodecahedral space. Is this unique dynamical property of the Poincare space a clue towards a theory of cosmic topology?