Multi-scale gravity and cosmology

Abstract

The gravitational dynamics and cosmological implications of three classes of recently introduced multi-scale spacetimes (with, respectively, ordinary, weighted and qderivatives) are discussed. These spacetimes are non-Riemannian: the metric structure is accompanied by an independent measure-differential structure with the characteristics of a multi-fractal, namely, different dimensionality at different scales and, at ultra-short distances, a discrete symmetry known as discrete scale invariance. Under this minimal paradigm, five general features arise: (a) the big-bang singularity can be replaced by a finite bounce, (b) the cosmological constant problem is reinterpreted, since accelerating phases can be mimicked by the change of geometry with the time scale, without invoking a slowly rolling scalar field, (c) the discreteness of geometry at Planckian scales can leave an observable imprint of logarithmic oscillations in cosmological spectra and (d) give rise to an alternative mechanism to inflation or (e) to a fully analytic model of cyclic mild inflation, where near scale invariance of the perturbation spectrum can be produced without strong acceleration. Various properties of the models and exact dynamical solutions are discussed. In particular, the multi-scale geometry with weighted derivatives is shown to be a Weyl integrable spacetime.

Standard Model in multiscale theories and observational constraints

“…This is indeed the case for weak and strong interactions but not for quantum electrodynamics (QED); the reason of this discrepancy is that, among the gauge couplings, only the electric charge is observed directly. The theory is nontrivial also because, when gravity is switched on, the field redefinitions are associated with a change of frame which never leads to general relativity plus minimally-coupled matter [43]. Moreover, once the frame has been fixed, the measure affects the physics at mesoscopic scales, such as in thermodynamical and atomic systems [55].…”

“…In this context, we focus on theories of multiscale spacetimes [4,[30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. These have been proposed either as stand-alone models of exotic geometry [31,32,40,43] or as an effective means to study, in a controlled manner, the change of dimensionality with the probed scale (known as dimensional flow 1 of these models and of their status).…”

“…These have been proposed either as stand-alone models of exotic geometry [31,32,40,43] or as an effective means to study, in a controlled manner, the change of dimensionality with the probed scale (known as dimensional flow 1 of these models and of their status). The model with ordinary derivatives was the first to be proposed [4,46,47] but it cannot be a fundamental theory due to some issues regarding its momentum space and quantization.…”

“…The theory with fractional derivatives is most promising especially as far as renormalization is concerned but, apart from a general power-counting argument [32], its physical properties have not been studied yet; this will be done in the near future. The scenario with weighted derivatives and the one with q-derivatives, on both of which we focus here, have been analyzed in greater detail both in quantum field theory [32,38,40,41] and cosmology [43] and a wealth of new phenomenology has begun to emerge.…”

“…(B) The theories with weighted and q-derivatives have not yet reached a satisfactory level of maturity and much needs to be done to assess their relevance and viability as physical models beyond the standard lore. For instance, the Standard Model has been formulated only in the electromagnetic sector [40,42], models of cosmic inflation and late-time acceleration have been explored only preliminarily [43] and observational constraints on the fundamental scales of the geometry are either heuristic [32] (based qualitatively on toy models of dimensional regularization) or too weak [40]. Moreover, it is not even clear whether a satisfactory perturbative quantum field theory can be formulated at all in the case with weighted derivatives, due to difficulties in defining a predictive Feynman series of scattering processes [41].…”