A weak acceleration effect due to residual gravity in a multiply connected universe

¹ Toruń Centre for Astronomy, N. Copernicus University, ul. Gagarina 11, 87-100 Toruń, Poland e-mail: boud@astro.uni.torun.pl
² Copernicus Astronomy Centre, ul. Bartycka 18, 00-716 Warsaw, Poland
³ Astronomy & Cosmology Department, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland

Received: 7 February 2006
Accepted: 29 November 2006

Abstract

Context.Understanding dark energy and measuring the topology of the Universe are two of the biggest open questions in physical cosmology. It was previously shown that multiple connectedness, via the twin paradox of special relativity, provides a novel physical justification for an assumption of the standard FLRW model: it implies a favoured space-time splitting (comoving coordinates).

Aims.Could cosmic topology also imply dark energy?

Methods.We use a weak field (Newtonian) approximation of gravity and consider the gravitational effect from distant, multiple copies of a large, collapsed (virialised) object today (i.e. a massive galaxy cluster), taking into account the finite propagation speed of gravity, in a flat, multiply connected universe, and assume that due to a prior epoch of fast expansion (e.g. inflation), the gravitational effect of the distant copies is felt locally, from beyond the naïvely calculated horizon.

Results.We find that for a universe with a  spatial section, the residual Newtonian gravitational force (to first order) provides an anisotropic effect that repels test particles from the cluster in the compact direction, in a way algebraically similar to that of dark energy. For a typical test object at comoving distance χ from the nearest dense nodes of the cosmic web of density perturbations, the pressure-to-density ratio w of the equation of state in an FLRW universe, is , where L is the size of the fundamental domain, i.e. of the Universe. Clearly, . For a  spatial section of exactly equal fundamental lengths, the effect cancels to zero. For a  spatial section of unequal fundamental lengths, the acceleration effect is anisotropic in the sense that it will tend to equalise the three fundamental lengths.

Conclusions.Provided that at least a modest amount of inflation occurred in the early Universe, and given some other conditions, multiple connectedness does generate an effect similar to that of dark energy, but the amplitude of the effect at the present epoch is too small to explain the observed dark energy density and its anisotropy makes it an unrealistic candidate for the observed dark energy.