Deformed Lorentz symmetry and relative locality in a curved/expanding spacetime

The interest of part of the quantum-gravity community in the possibility of Planck-scale-deformed Lorentz symmetry is also fueled by the opportunities for testing the relevant scenarios with analyses, from a signal-propagation perspective, of observations of bursts of particles from cosmological distances. In this respect the fact that so far the implications of deformed Lorentz symmetry have been investigated only for flat (Minkowskian) spacetimes represents a very significant limitation, since for propagation over cosmological distances the curvature/expansion of spacetime is evidently tangible. We here provide a significant step toward filling this gap by exhibiting an explicit example of Planck-scale-deformed relativistic symmetries of a spacetime with constant rate of expansion (de Sitterian). Technically, we obtain the first ever example of a relativistic theory of worldlines of particles with 3 nontrivial relativistic invariants: a large speed scale (“speed-of-light scale”), a large distance scale (inverse of the “expansion-rate scale”), and a large momentum scale (“Planck scale”). We address some of the challenges that had obstructed success for previous attempts by exploiting the recent understanding of the connection between deformed Lorentz symmetry and relativity of spacetime locality. We also offer a preliminary analysis of the differences between the scenario we here propose and the most studied scenario for broken (rather than deformed) Lorentz symmetry in expanding spacetimes.

Figure 1

We illustrate the results for travel times of massless particles derived in this section by considering the case of two distant observers, Alice and Bob, in relative rest. Two massless particles, one soft (dashed red) and one hard (solid blue), are emitted simultaneously at Alice and they reach Bob at different times. Because of the nontriviality of translation transformations, in Bob’s coordinatization (bottom panel) the emission of the particles at Alice appears not to be simultaneous. And similarly, for the difference in times of arrival at the distant detector (where Bob is located) Alice finds in her coordinatization (top panel) a value which is not the same as the difference in times of arrival that Bob determines (bottom panel). The case in figure has $\alpha +\beta =4/3$. For visibility, we assumed here unrealistic values for the scales involved. For realistic values of the distance between observers and of the energy of the hard particle, taking $\ell$ as the inverse of the Planck scale, no effect would be visible in figure (the worldlines would coincide).

Figure 2

We illustrate the results for travel times of massless particles derived in this section, adopting conformal coordinates. We consider the case of two distant observers, Alice and Bob, connected by a pure translation, and two massless particles, one soft (dashed line) and one hard (solid line), emitted simultaneously at Alice. And we consider two combinations of values of $\alpha$ and $\beta$ producing the same $\alpha +\beta$: the case $\alpha =2/3$, $\beta =2/3$ and the case $\alpha =1/3$, $\beta =1$ These two cases correspond to the same hard worldline for Alice (solid line, top panel) but give two distinct worldlines according to Bob (two solid lines in bottom panel). This shows that in the case with spacetime expansion the travel time does depend individually on $\alpha$ and $\beta$, not just on $\alpha +\beta$ (in the bottom panel the distance between the hard/solid worldline with $\alpha =2/3$, $\beta =2/3$ and the soft/dashed worldline is slightly smaller than the distance between the hard/solid worldline with $\alpha =1/3$, $\beta =1$ and the soft/dashed worldline). As in the case without expansion we have here that in Bob’s coordinatization (bottom panel) the emission of the particles at Alice appears not to be simultaneous. Similarly for the difference in times of arrival at the distant detector Bob, Alice finds in her coordinatization (top panel) not the same value as the difference in times of arrival that Bob determines. For visibility we assumed here unrealistic values for the scales involved.

Figure 3

We here illustrate the results for travel times of massless particles derived in this section, adopting comoving coordinates. The situation here shown is the same situation already shown (there in conformal coordinates) in Fig. 2.

Figure 4

Here we show the dependence on redshift of the expected time-of-arrival difference divided by the difference of energy of the two massless particles. Two such functions are shown, one for the case $\alpha =0$, $\beta =0.65$ (violet) and one for the case $\alpha =1.3$, $\beta =0$ (blue). We also show the upper limits that can be derived from data reported in Refs. 31, 32, 33, setting momentarily aside the fact that our analysis adopted the simplifying assumption of a constant rate of expansion (whereas a rigorous analysis of the data reported in Refs. 31, 32, 33 should take into account the nonconstancy of the expansion rate). The values $\alpha =0$, $\beta =0.65$ and $\alpha =1.3$, $\beta =0$ have been chosen so that we have consistency with the tightest upper bound, the one established in Ref. 31. The main message is coded in the fact that at small values of redshift the blue and the violet lines are rather close, but at large values of redshift they are significantly different (this is a log-log plot). In turn, this implies that at high redshift the difference between adding correction terms of form $E{p}^2$ and adding correction terms of form $E^3$ can be very tangible.

Figure 5

Here in the left panel we show the constraint on the $\alpha$, $\beta$ parameter space that can be obtained from Ref. 33, concerning a source at the relatively small redshift of $z=0.116$, where we can confidently apply our results for constant rate of expansion as a reliable first approximation. Through this we show that even within the confines of our analysis Planck-scale sensitivity (values of $|\alpha |$ and $|\beta |$ smaller or comparable to 1) is not far. In the right panel we show the much tighter (indeed “Planckian”) constraint on the $\alpha$, $\beta$ parameter space which would be within our reach if we could assume our analysis to apply also to redshifts close to 1, as for GRB090510 observed by the Fermi telescope [31]. By comparing the left and the right panel one also finds additional evidence of how the difference between adding correction terms of form $E{p}^2$ and adding correction terms of form $E^3$ becomes more significant at higher redshifts: in the left panel (data on source at small redshift of $z=0.116$) the bound on the $\alpha$ parameter is nearly as strong as on the $\beta$ parameter, whereas in the right panel (data on source at redshift of $z\simeq 0.9$) the constraint on the alpha parameter is significantly weaker than the constraint on the $\beta$ parameter.