Echo: DDR as a Test Model

This is a copy of my post at the blog to our discussion group 'Quantum Gravity in the Lab?!' about our meeting on March 6th. The discussion was lead by Michele Arzano, and this week's topic was 'Planck-scale departures from relativistic symmetries: test theories and status of predictions'.

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Michele Arzano arrived as postdoc at PI around the same time as I did, in fall 2006. Before I start explaining what he told us in the discussion this week, let me point out that he's DJed for some while, and had the idea to organize PI's first 'Decoherence Dance' that will take place on Saturday in the old PI building. In case you're around, drop in... Michele has a quite impressive number of works on Hopf-algebras, black hole thermodynamics, and modified dispersion relations. Though he prefers to call these 'deformed dispersion relations' (DDR).

He started with explaining the general idea to confront a (not further specified) 'candidate theory of quantum gravity' (QG) with the real world by use of a test model. Such a test model, he explained, would generally have a 'main feature', and possible 'additional features'. I think of this as parameters which might not necessarily all vanish, but could do so under certain circumstances. An example that he gave later might be a DDR that could but doesn't have to come along with a violation of conservation laws, and the resulting threshold modifications.

Michele then briefly summarized evidence for DDRs from QG.

First, dispersion relations get modified in theories with non-commutative space-times. Such could arise in certain string scenarios, where Lorentz invariance is broken (for a review see e.g. Review of the Phenomenology of Noncommutative Geometry), or the non-commutativity arises through a modification of the Lie-Algebra from standard flat-space symmetries a to a κ-Minkowski Lie-Algebra (for references see e.g. Hopf-algebra description of noncommutative-spacetime symmetries).

Second, evidence for DDRs has been found in approaches from LQG (see e.g. Loop quantum gravity and light propagation, and Quantum symmetry, the cosmological constant and Planck scale phenomenology)

He then defined the test model by generally parameterizing an expansion of the DDR. The coefficient for the first non-vanishing term in energy over Planck energy E_p is the most important one:


Since the expansion parameter is typically less than 10^-16, it is of utmost important whether the first power n is 1 or >1.

This deformation is what he referred to as the 'main feature'. Since this is not a theory, but only a single equation, this might come with additional features like a modification of energy-momentum conservation, or an energy dependent speed of light. Generally, he said, it is an open question whether the dynamics can be described by an effective field theory. The above expansion includes DSR approaches as well as an explicit breaking of Lorentz invariance with a preferred frame.

[At this point Michele had already talked one hour instead of half an hour.]

He summarized two prediction that arise from this approach.

1. In the general case of a DDR the speed of a photon depends on its energy. This means that a signal composed of different frequencies shows an unusual dispersion. Roughly spoken, higher energetic photons are faster than one would think they are. The problem is that this difference in the time of flight is hard to detect, since the ratio of the photon's energy over the Planck energy tiny is for typical photons that we observe. However, a difference in time of flight can add up given that the signal composed of different frequencies travels over a long distance.
   
   If one inserts the typical scales it turns out that γ-ray bursts provide a source that would make such an effect - tiny as it is - observable with the GLAST satellite. The bursts have a high energetic contribution that can reach up to 1 GeV, and a typical distance of a Gpc. In the case of n=1, the accumulated difference in time of flight between the higher and lower energies becomes comparable to the typical duration of the burst itself (of the order milliseconds), and thus potentially detectable. (I mentioned that the energies inserted in the equation were taken in a specific restframe, that of the cosmic microwave background.)
   


2. In case energy-momentum conservation is modified, one obtains a modification of thresholds for particle production. This effect has been used to explain the to-be-confirmed absence of the GZK cutoff for cosmic rays. To briefly recall the issue: cosmic rays are commonly believed to be created from incoming high energetic protons that are not produced in nearby sources. If the protons move fast enough relative to the cosmic microwave background however, they will eventually scatter on the background radiation and produce pions (pions being the lightest mesons). If the threshold for this reaction is crossed, the typical travel distance (mean free path) of the protons drops considerably, and they can not reach us any more. One thus expects a sharp cut-off in the spectrum that should occur for proton energies around 10¹⁹eV (in the earth rest frame. In the center of mass frame this is roughly a GeV).
   
   Whether the threshold is raised or lowered depends on the sign of η. I forgot whether positive or negative would raise the threshold as necessary, sorry. (As Joy pointed out, the experimental situation on the GZK cutoff is far from clear, and there are many issues that have to be taken into account. I mentioned that the energies inserted in the equation were taken in a specific restframe, that of the cosmic microwave background. Yes, I know, I insist on that point. )
   

[Somewhere around here the discussion exceeded 90 minutes. I think I must be the worst discussion leader that the world has ever seen.]

Michele didn't have the time to elaborate on the question whether or not n>1 effects would be observable as well.

In my opinion another test for the test model is whether or not it can be put into a consistent framework. It is unfortunate that this test model still only consists of a couple of equations, and can not be understood as a theory. I think the requirement of having the main- and/or additional features arise from a theory would significantly improve the reliability of the predictions, and help to clarify ambiguities of the model. Overall seen I find the approach interesting, though direct connections to quantum gravity seem to be weak, and more motivations than actual derivations. It is hard for me to judge on whether the discussed features must necessarily arise from certain approaches, or are just a general possibility that one can't (and doesn't want to) exclude. In this regard, I hope that Lee will enlighten us next week.

Suggested literature:

• Introduction to Quantum-Gravity Phenomenology
• Probing The Structure of Space-Time with Cosmic Rays
• Gamma Ray Bursts as Probes of Quantum Gravity
• TeV Astrophysics Constraints on Planck Scale Lorentz Violation
• GRBs Neutrinos as a Tool to Explore Quantum Gravity induced Lorentz Violation