EDP Sciences
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Volume 502, Number 1, July IV 2009
Page(s) 27 - 35
Section Cosmology (including clusters of galaxies)
DOI http://dx.doi.org/10.1051/0004-6361/200911881
Published online 19 May 2009
A&A 502, 27-35 (2009)
DOI: 10.1051/0004-6361/200911881

The residual gravity acceleration effect in the Poincaré dodecahedral space

B. F. Roukema and P. T. Różański

Toruń Centre for Astronomy, Nicolaus Copernicus University, ul. Gagarina 11, 87-100 Toruń, Poland

Received 19 February 2009 / Accepted 17 March 2009

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 Poincaré 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 Poincaré space from other well-proportioned spaces?
Methods. The acceleration effect in the Poincaré space S3/I* is studied, using a massive particle and a nearby test particle of negligible mass. Calculations are made in S3 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 T3 is re-examined, and two other well-proportioned spaces, the octahedral space S3/T*, and the truncated cube space S3/O*, are also studied.
Results. The residual gravity effect occurs in all four cases. In a perfectly regular 3-torus of side length La, 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/La (3-torus), or  $r/R_{{\rm C}}$, respectively. However, the Poincaré 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/La = $r/R_{{\rm C}}$ ~ 10-3), but the fourth of these spaces is about ten thousand times better balanced than the other three. This is the Poincaré dodecahedral space. Is this unique dynamical property of the Poincaré space a clue towards a theory of cosmic topology?

Key words: cosmology: theory -- cosmological parameters -- large-scale structure of Universe -- early Universe

© ESO 2009