Isochronous cosmologies

Abstract

The possibility has been recently demonstrated to manufacture (nonrelativistic, Hamiltonian) many-body problems which feature an isochronous time evolution with an arbitrarily assigned period T yet mimic with good approximation, or even exactly, any given many-body problem (within a large, physically relevant, class) over timesT which may also be arbitrarily large (but of course such thatT < T ). Purpose and scope of this paper is to explore the possibility to extend this finding to a general relativity context. For simplicity we restrict our consideration to the case of homogeneous and isotropic metrics and show that, via an approach analogous to that used for the nonrelativistic many-body problem, a class of homogeneous and isotropic cyclic solutions of Einstein's equations may be obtained. For these solutions the duration of the cycles does not depend on the initial conditions, so we call these models isochronous cosmologies. We give a physical interpretation of such metrics and in particular we show that they may behave arbitrarily closely, or even identically, to the Friedman-Robertson-Walker solutions of Einstein's equations for an arbitrarily long time (of course shorter than their period, which can also be assigned arbitrarily), so that they may reproduce all the satisfactory phenomenological features of the standard cosmological Λ-CDM model in a portion of their cycle; while these isochronous cosmologies may be geodesically complete and therefore singularity-free.

Isochronous Spacetimes

“…non invertible) metrics. In particular we have shown [1] that for any homogeneous, isotropic and spatially flat metric g µν satisfying Einstein's equations and providing a model of the universe, it is possible to find a different (also homogeneous, isotropic and spatially flat) metric solutiong µν which is locally (in time) diffeomorphic to g µν -hence yields the same cosmology as g µν for any observation over an arbitrary time intervalT -and is cyclic 1 (in fact periodic with an arbitrary period T >T in the time coordinate t ); but it is degenerate at an infinite, discrete sequence of times t n = t 0 ± nT /2, n = 0, 1, 2, ... . We interpreted these metrics as corresponding to isochronous cosmologies [1].…”

“…In this paper we revisit the findings reported in [1] and report some additional considerations relevant for a better understanding of the validity-in the context of theoretical and mathematical physics-of those results.…”

“…Let us also recall [1] that this result is not restricted to homogeneous, isotropic and spatially flat metrics: it can be easily extended to any synchronous metric, therefore it is quite general since most metrics can be written in synchronous form by a diffeomorphic change of coordinates.…”

“…For this reason in our previous paper [1] we investigated whether and how this result could be extended to that context. We concluded that this is possible provided one extends general relativity by allowing degenerate (i.e.…”

“…In particular, the metricg µν -while being cyclic on time scales larger thanT -may yield over the time intervalT just the standard sequence of domination firstly by radiation, then by matter and dark energy, as well as the cosmological perturbations consistent with all observational tests [5,6,7,8,9,10,11,12,13,14,15], which characterize the metric g µν of the standard Λ-CDM cosmological model (see for instance [16] for a review). Furthermore, it has been shown that such isochronous metricsg µν can be manufactured to be geodesically complete [1] and therefore singularity-free 2 , so that the geodesic motion as well as all physical quantities described by scalar invariants are always well defined [1] and the Big Bang singularity may be avoided-even when some of the phenomenological observations can nevertheless be interpreted as remnants of a past Big Bang, which however may never be actually attained by the metricg µν , neither in the past nor in the future. This implies that these isochronous metricsg µν may describe a singularity-free universe which-while featuring a time evolution which reproduces identically (up to diffeomorphic time reparameterization; locally in time, except at a discrete set of instants t n ) the standard cosmological model characterized by the metric g µν -features an expansion which stops at some instant, to be followed by a period of contraction, until this phase of evolution stops and is again followed by an expanding phase, this pattern being repeated ad infinitum.…”