Consistency relations and degeneracies in (non)commutative patch inflation

The consistency equations of patch inflation are considered in a next-to-leading-order slow-roll (SR) expansion. Some general aspects of braneworld degeneracy are pointed out, both with an ordinary scalar field and a Born-Infeld tachyon. The discussion is then extended to the maximally symmetric noncommutative case.Comment: 6 pages; v3: version to appear in Phys. Lett.

Observational effects from quantum cosmology

The status of quantum cosmologies as testable models of the early universe is assessed in the context of inflation. While traditional Wheeler-DeWitt quantization is unable to produce sizable effects in the cosmic microwave background, the more recent loop quantum cosmology can generate potentially detectable departures from the standard cosmic spectrum. Thus, present observations constrain the parameter space of the model, which could be made falsifiable by near-future experiments.Comment: 16 pages, 3 figures. Invited review article also containing original material. v2: new section on holonomy corrections, introduction extended, references added, typos corrected; v3: includes corrections of eqs. (3), (62) and (64c) of the erratu

Introduction to multifractional spacetimes

We informally review the construction of spacetime geometries with multifractal and, more generally, multiscale properties. Based on fractional calculus, these continuous spacetimes have their dimension changing with the scale; they display discrete symmetries in the ultraviolet and ordinary Poincar\'e symmetries in the infrared. Under certain reasonable assumptions, field theories (including gravity) on multifractional geometries are generally argued to be perturbatively renormalizable. We also sketch the relation with other field theories of quantum gravity based on the renormalization group.Comment: 26 pages, 6 figures; v2: references added. Lectures given at Sixth International School on Field Theory and Gravitation 2012 (Petropolis, Brazil

Detecting quantum gravity in the sky

Getting signatures of quantum gravity is one of the topical lines of research in modern theoretical physics and cosmology. This short review faces this challenge under a novel perspective. Instead of separating quantum-gravity effects of a specific model between UV and IR regimes, we consider a general feature, possibly common to many frameworks, where all scales are affected and spacetime geometry is characterized by a complex critical exponent. This leaves a log-oscillating modulation pattern in the cosmic microwave background spectrum and gives a unique opportunity, illustrated with the example of a multi-fractional theory, to test quantum gravities at cosmological scales.Comment: 4 pages, 1 figure. In Proceedings of EPS-HEP 2017, European Physical Society conference on High Energy Physics, July 5-12, 2017, Venice, Italy. v2: minor corrections to match the published version (https://pos.sissa.it/314/033/

ABC of multi-fractal spacetimes and fractional sea turtles

We clarify what it means to have a spacetime fractal geometry in quantum gravity and show that its properties differ from those of usual fractals. A weak and a strong definition of multi-scale and multi-fractal spacetimes are given together with a sketch of the landscape of multi-scale theories of gravitation. Then, in the context of the fractional theory with $q$-derivatives, we explore the consequences of living in a multi-fractal spacetime. To illustrate the behavior of a non-relativistic body, we take the entertaining example of a sea turtle. We show that, when only the time direction is fractal, sea turtles swim at a faster speed than in an ordinary world, while they swim at a slower speed if only the spatial directions are fractal. The latter type of geometry is the one most commonly found in quantum gravity. For time-like fractals, relativistic objects can exceed the speed of light, but strongly so only if their size is smaller than the range of particle-physics interactions. We also find new results about log-oscillating measures, the measure presentation and their role in physical observations and in future extensions to nowhere-differentiable stochastic spacetimes.Comment: 20 pages, 1 figure. v2: typos corrected, minor improvements of the tex

Lorentz violations in multifractal spacetimes

Using the recent observation of gravitational waves (GW) produced by a black-hole merger, we place a lower bound on the energy above which a multifractal spacetime would display an anomalous geometry and, in particular, violations of Lorentz invariance. In the so-called multifractional theory with $q$-derivatives, we show that the deformation of dispersion relations is much stronger than in generic quantum-gravity approaches (including loop quantum gravity) and, contrary to the latter, present observations on GWs can place very strong bounds on the characteristic scales at which spacetime deviates from standard Minkowski. The energy at which multifractal effects should become apparent is $E_*>10^{14}\,{\rm GeV}$ (thus improving previous bounds by 12 orders of magnitude) when the exponents in the measure are fixed to their central value $1/2$. We also estimate, for the first time, the effect of logarithmic oscillations in the measure (corresponding to a discrete spacetime structure) and find that they do not change much the bounds obtained in their absence, unless the amplitude of the oscillations is fine tuned. This feature, unavailable in known quantum-gravity scenarios, may help the theory to avoid being ruled out by gamma-ray burst (GRB) observations, for which $E_*> 10^{17}\,{\rm GeV}$ or greater.Comment: 12 pages, 1 figure. v2: discussion expanded at several points, comparison with the Lorentz-violating Standard-Model extension added, references adde