One of the most pressing open problems in theoretical physics is the incompatibility of Einstein's classical theory of general relativity and the quantum field theories of the standard model of particle physics. Both theories are well developed and have been tested, and been confirmed, to great precision. Yet, they have continued to stand apart. In regions in which space-time curvature becomes large (compared to the Planck scale), as towards the center of black holes and in the early universe, we can no longer treat both theories as disparate entities. We need a theory for quantum gravity. Physicists search for a theory of quantum gravity not only because it is unaesthetic that two areas of physics stand apart, but because we know that with the current theories our description of nature is inconsistent and incomplete.
The problem is not new. In fact, it's been more than 50 years that physicists have been after a theory of quantum gravity. And while we have during that time better understood the problems, and some promising approaches, like string theory, have been developed to great sophistication, we arrived in the 21st century still lacking a theoretically satisfactory and experimentally confirmed theory of quantum gravity. Given this slow progress, during the last decade an increasing amount of effort has been invested into the development of phenomenological models for quantum gravity. Such models do not in and by themselves constitute candidates for a fundamental theory. Instead, they are extensions of established theories that allow to describe and - ideally - test the consequences of specific features that the fundamental theory might have. Compared to the electromagnetic, strong and weak interaction, gravity is an extremely weak force. Quantum gravitational effects are therefore hard to come by. Nevertheless, there are some areas where experiment is sensitive even to smallest deviations from the standard model.
The best known example are departures from Lorentz-invariance. Many approaches to quantum gravity that rely on discretizing space-time or that describe gravity as an emergent phenomenon violate Lorentz-invariance. With the exception of Causal Sets, all discrete approaches with a finite spacing either explicitly break Lorentz-invariance and introduce a preferred frame, or they exhibit so-called 'deformations' of special relativity: modifications of the usual Lorentz-invariance at high energies, but without a preferred frame. Models that break Lorentz-invariance can be parametrized by higher order operators added to the standard model Lagrangian. These terms have many consequences for collider and astro-particle physics, for example by modifying reaction thresholds or enabling interactions normally forbidden.
These consequences for particle interactions allow to constrain Lorentz-invariance violation very tightly. Deformations of Lorentz-invariance can cause a modification of the dispersion relation that is potentially observable in light reaching us from distant gamma ray bursts. The models with deformations are not (yet) as tightly constrained as those with a breaking of Lorentz-invariance, but with data from the Fermi satellite, constraints are getting tighter every couple of months.
Another very general expectation that we have of quantum gravity is that space-time ceases being a smooth background on shortest distance scales and instead is more like a space-time "foam;" dynamic and possibly topologically non-trivial. There have been several models proposed how to deal with that, from small perturbations of the light-cone to random walks. Typical effects from this space-time foam include blurring of spectral lines, vanishing of Airy rings of distant stars, and noise in gravitational wave interferometers. More recently, also a violation of the equivalence principle has been shown to be a possible consequence. The magnitude of these effects depends on the model, and some (for example the random walk model) are ruled out already.
Finally one has to mention here the exciting possibility that quantum gravitational effects are not as feeble as we naively expect them to be. Our expectation relies on an extrapolation of the strength of gravity. But we do not actually have experimental evidence this extrapolation holds further than to the energy scales we have tested so far, some TeV with the start of the LHC, and the naive extrapolation might just be wrong. It is theoretically possible that quantum gravity becomes relevant not far past the scale we have tested, and models with additional spatial dimensions have been one way to investigate this possibility.
Here at Nordita, we study models with extra dimensions and modifications of Lorentz-invariance with the hope to tighten the connections both with experiment and with approaches to a fundamental theory of quantum gravity.
A more extensive survey of the topic can be found in http://arxiv.org/abs/1010.3420