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

Giovanni AMELINO-CAMELIA,1, 2 Antonino MARCIANO,3, 4 Marco MATASSA,5 and Giacomo ROSATI1, 2

1Dipartimento di Fisica, Universita di Roma “La Sapienza”, P.le A. Moro 2, 00185 Roma, Italy2INFN, Sez. Roma1, P.le A. Moro 2, 00185 Roma, Italy

3Department of Physics, The Koshland Integrated Natural Science Center, Haverford College, Haverford, PA 19041 USA4Department of Physics, Princeton University, New Jersey 08544, USA

5SISSA, Via Bonomea 265, I-34136 Trieste, Italy

The interest of part of the quantum-gravity community in the possibility of Planck-scale-deformedLorentz symmetry is also fueled by the opportunities for testing the relevant scenarios with analy-ses, from a signal-propagation perspective, of observations of bursts of particles from cosmologicaldistances. In this respect the fact that so far the implications of deformed Lorentz symmetry havebeen investigated only for flat (Minkowskian) spacetimes represents a very significant limitation,since for propagation over cosmological distances the curvature/expansion of spacetime is evidentlytangible. We here provide a significant step toward filling this gap by exhibiting an explicit exam-ple of Planck-scale-deformed relativistic symmetries of a spacetime with constant rate of expansion(deSitterian). Technically we obtain the first ever example of a relativistic theory of worldlines ofparticles with 3 nontrivial relativistic invariants: a large speed scale (“speed-of-light scale”), a largedistance 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 exploitingthe recent understanding of the connection between deformed Lorentz symmetry and relativity ofspacetime locality. We also offer a preliminary analysis of the differences between the scenario wehere propose and the most studied scenario for broken (rather than deformed) Lorentz symmetryin expanding spacetimes.

I. INTRODUCTION

Technically the interest attracted over the last decadeby the possibility of Planck-scale-deformed relativistickinematics, as conceived within the proposal “DSR”(doubly-special, or, for some authors, deformed-specialrelativity) introduced in Refs. [1, 2], was mainly linkedto the following observations:

• it affords us the luxury of introducing as observer-indepedent laws some of the properties for thePlanck scale that have been most popular in thequantum-gravity literature, such as a role for thePlanck length (the inverse of the Planck scale) asthe minimum allowed value for wavelengths [1, 2];

• it reflects the content of results rigorously estab-lished for 3D quantum gravity (see, e.g., Refs. [3–5]);

• it reflects the content of results rigorously estab-lished for some 4D noncommutative spacetimes(see, e.g., Refs. [6, 7]);

• it fits the indications emerging from some com-pelling semiheuristic arguments based on 4D LoopQuantum Gravity [3, 8].

A comparable “motivational list” can be claimed byother scenarios for the fate of relativistic symmetries atthe Planck scale, but in the case of these DSR scenarios acrucial additional motivation comes from the opportuni-ties in phenomenology. Some of the novel effects that canbe accommodated within a DSR-relativistic kinematicscan be tested through observations of bursts of particles

from cosmological distances [9, 10]. The key for theseanalyses are the implications of DSR deformations forthe propagation of signals, and the sought Planck-scalesensitivity is reached thanks to the huge amplificationafforded by the cosmological distances.

Our analysis takes off from the realization that thismost notable “selling point” for DSR research, basedon its phenomenological prospects, has not yet beenmade truly accessible by the DSR literature developedso far. DSR-deformed relativistic frameworks have beeninvestigated so far only for flat (Minkowskian) space-times, and this is not the case relevant for the analy-sis of signals received from sources at cosmological dis-tances, for which the curvature/expansion of spacetimeis very tangible. We here provide a significant step to-ward filling this gap by exhibiting an explicit example ofPlanck-scale-deformed relativistic symmetries of a space-time with constant rate of expansion (deSitterian). Evi-dently this will lead us to introduce the first ever exam-ple1 of a relativistic theory of worldlines of particles with3 nontrivial relativistic invariants: a large speed scale(“speed-of-light scale”), a large distance scale (inverse ofthe “expansion-rate scale’), and a large momentum scale(“Planck scale”).

There had been previous attempts of investigatingthe interplay between DSR-type deformation scales andspacetime expansion (see, e.g., Refs. [12, 13]), but with-out ever producing a fully satisfactory picture of how the

1 Studies such as those in Refs. [11–13] did contemplate the possi-bility of 3 invariants, but did not go as far as giving a consistentrelativistic picture of worldlines of particles.

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worldlines of particles should be formalized and inter-preted. In retrospect we can now see that these previousdifficulties were due to the fact that the notion of relativelocality had not yet been understood, and without thatnotion the interplay between DSR-deformation scale andexpansion-rate scale remains unintelligible.

Relative locality (on which we shall of course return inmore detail in later parts of this manuscript) is the space-time counterpart of the DSR-deformation scale ` just inthe same sense that relative simultaneity is the space-time counterpart of the special-relativistic scale c (scaleof deformation of Galilean Relativity into Special Rela-tivity). This was understood only very recently, in stud-ies such as the ones in Refs. [14–16]. Awareness of thepossibility of relative locality is already very importantin making sense of the implications of DSR-deformationsin a flat/non-expanding spacetime. And, as we shall hereshow, it plays an even more crucial role in the consistencyof the spacetime picture emerging from the interplay be-tween DSR-deformation scale and expansion-rate scale.

While our original (and still here primary) motivationwas to address these fascinating, but merely technical, is-sues concerning the interplay between DSR-deformationscale and expansion-rate scale, as shown in later parts ofthis manuscript our results appear to carry rather strongsignificance for the phenomenology. In particular, ourconstructive analysis leads to a description of the depen-dence of the novel effects on redshift that is not like any-thing imagined in the previous DSR literature (relyingthen only on heuristic arguments for the interplay be-tween DSR-deformation scale and expansion-rate scale).

We work at leading order in the DSR-deformation scale`, since (assuming it is of the order of the Planck length)that is the only realistic target for DSR phenomenologyover the next few decades. And in order to keep thingssimple, without renouncing to any of the most significantconceptual hurdles, we opt to work in a 2D spacetime(one time and one spatial dimension).

The choices of notation we adopt are relatively stan-dard and self-explanatory with the exception of the factthat occasionally we denote with X the set of spacetimecoordinates, i.e. X ≡ (t, x), and we denote with G theset of symmetry generators, i.e. G ≡ (E, p,N ).

II. PRELIMINARIES ON DEFORMEDLORENTZ SYMMETRY WITHOUT EXPANSION

For our purposes the main starting point is providedby the analysis in Ref. [14] which provided, assuming aMinkowskian spacetime (no expansion), a framework forDSR-deformed relativistic theories of worldlines of par-ticles with two nontrivial relativistic invariants: a largespeed scale c (the “speed-of-light scale”, here mute be-cause of the conventional choice of units c = 1) and alarge momentum scale `−1 (assumed to be roughly of theorder of the “Planck scale”). Ref. [14] however has alimitation that for our purposes is significant: the anal-

ysis in Ref. [14] took off from an ansatz, with a singleparameter `, for the energy(/momentum) dependence oftravel times of massless particles from a given emitterto a given detector, so it did not fully explore the pos-sible issue of a dependence of such travel times on thechoice of DSR-deformation of the on-shell relation. Tak-ing as starting point the special-relativistic on-shell rela-tion, m2 = E2−p2, one could contemplate, in particular,adding a term of form Ep2 and/or adding a term of formE3. This issue was not much of interest in Ref. [14] sinceit is easy to see that the difference between Ep2 defor-mation and E3 deformation, if analyzed in a flat/non-expanding spacetime, carries very little significance (itshould be inevitably insignificant at least for masslessparticles). But in our analysis of a first case with space-time expansion we shall find that there are some signifi-cant difference between Ep2 deformations and E3 defor-mations.

So in this section we find appropriate to offer a minorgeneralization and reformulation of the results reportedin Ref. [14], particularly suitable for our generalizationto the case of an expanding spacetime, most notably forwhat concerns the comparison between Ep2 deformationand E3 deformation.

We start by assuming that, in a Minkowskian space-time, the relativistic symmetries leave invariant the fol-lowing combination of the energy E and momentum p ofparticles:

Cα,β = E2 − p2 + `(αE3 + βEp2

), (1)

where α and β are two numerical parameters.

And we exhibit a corresponding DSR-deformed 1+1DPoincare algebra of charges compatible with the invari-ance of Cα,β , describable in terms of the following Poissonbrackets:

E, p = 0 , E,N = p− ` (α+ β) pE ,

p,N = E +1

2`(αE2 + βp2

). (2)

A conveniently intuitive picture of the deformation weare studying is obtained by giving a representation ofthese symmetry generators in terms of time and spacecoordinates t, x and variables Ω,Π canonically conjugate

3

to them2:

Ω, t = 1 , Ω, x = 0 ,

Π, t = 0 , Π, x = 1 ,

t, x = Ω,Π = 0 , (3)

We find as representation the following:

E = Ω +1

2`((1− β)Π2 − αΩ2

), (4)

p = Π , (5)

N = tp+ xE + `

(1

2αxE2 − tpE +

1

2βxp2

). (6)

As done in Ref. [14], we shall derive the worldlinesadopting a covariant formulation of classical relativisticmechanics, also introducing an auxiliary affine parameterlabeling points on the worldline of a particle of mass m.As standard in the covariant formulation of classical rela-tivistic mechanics, evolution is coded in a pure-constraintHamiltonian; specifically, the evolution in the auxiliaryparameter on the worldline is governed by a Hamilto-nian constraint: Cα,β −m2 = 0. This straightforwardlyleads [14] to the following worldlines

xm,p = x0 −(

p√p2 +m2

+ `p

)(t− t0) , (7)

which in the particularly interesting case of massless par-ticles reduces to

xm=0,p (t) = x0 −p

|p| (t− t0) (1− `|p|) . (8)

Notice that with our choices of conventions the velocityis positive when p is negative, and therefore, restrictingour focus on p < 0, one has

xm=0,p<0 (t) = x0 + (t− t0) (1− `|p|) . (9)

2 Some authors (see, e.g., Ref. [17, 18]) have studied similar de-formations of Lorentz symmetry adopting spacetime coordinateswith Poisson brackets x, t = `x. This was done in relationto the fact that there are quantum-spacetime pictures that canbe analyzed in relation to such deformed-Lorentz-symmetry sce-narios, and the most notable case is “κ-Minkowski” (see, e.g.,Ref. [19, 20]), a non-commutative spacetime with [x, t] = i`x.We opt for standard Poisson brackets since it is known that theresults of analyses of travel times give the same results both as-suming x, t = `x and assuming x, t = 0. This was shown inRefs. [21, 22], and is a simple consequence of the fact that one canobtain coordinates x, t = `x, from our coordinates (x, t = 0)by posing t = t, x = x + `Ωx. (The point is [21, 22] that xand x+ `Ωx coincide in the origin, so their difference can neveraffect the determination of when a particle reaches (or is emit-ted) from the origin of the spacetime coordinates of an observer.Ref. [23] will show explicitly that our analysis (also for the casewith spacetime expansion) redone using coordinates such thatx, t = `x leads to the same results for travel times.

And let us also stress that evidently the charges E, p,Nare conserved along the motion, in the sense that G =Cα,β , G = 0, and they generate respectively deformedtime translations, spatial translations, and boosts, byPoisson brackets. By construction we have ensured thatthese transformations all are relativistic symmetries ofthe theory, as one can explicitly verify [14] by acting withthem on the worldlines (7), finding that the worldlines arecovariant.

At this stage of the analysis the physical content ofthese worldlines is still hidden behind the relativity oflocality. A warning that this might be the case is seen inthe fact that we are analyzing a case where the on-shellrelation, in light of (1), is α and β dependent,3

m2 = E2 − p2 + `(αE3 + βEp2

),

whereas our worldlines are independent of α and β.Most importantly the coordinate velocity one infers fromthose wordlines is independent of α and β. But one ofthe main known manifestations of relative locality is amismatch [14, 21, 24] between coordinate velocity4 andphysical velocity of particles.

For what concerns specifically the analysis so far re-ported in this section, the main challenge resides in thefact that we are used to read velocities off the formulasfor worldlines, but this implicitly assumes that transla-tion transformations are trivial. Essentially we take theworldline written by a certain observer Alice to describeboth the emission of a particle “at Alice” (in Alice’s ori-gin) and the detection of the particle far away from Alice.The observer/detector Bob that actually detects the par-ticle, since he is distant from Alice, should be properlydescribed by acting with a corresponding translation onAlice’s worldline. And the determination of the “arrivaltime at Bob” (crucial for determining the physical ve-locity [21]) should be based on Bob’s description of theworldline, just as much as the “emission time at Alice”should be based on Alice’s description of the worldline.

3 Note that the quantity we denote with m (and we loosely referto as the mass) is not the rest energy when α 6= 0. The restenergy evidently is µ such that m2 = µ2 + α`µ3. The inter-ested readers can easily verify that none of our most significantobservations (most evidently since we mainly focus on masslessparticles) depends on the difference between m2 and µ2.

4 Examples of velocity artifacts were of course known well beforethe understanding of relative locality for theories with deformedLorentz symmetry: for example in de Sitter spacetime (and anyexpanding spacetime) the coordinate velocity of a particle distantfrom the observer can be described by the observer as a veloc-ity greater than the speed of light, even though in classical deSitter spacetime the physical velocity measured by an observerclose to the particle is of course always no greater than the speedof light. Since these previously known velocity artifacts are con-nected with spacetime curvature, the recent realization that someof our currently investigated theories formulating (one form oranother of) “deformed Minkowski spacetime” are subject to rel-ative locality, with some associated velocity artifacts [14, 21, 24],was largely unexpected.

4

When translations are trivial (translation generators con-jugate to the spacetime coordinates) we can go by with-out worrying about this more careful level of discussion,since the naive argument based solely on Alice’s worldlinegives the same result as the more careful analysis usingAlice’s worldline for the emission and Bob’s descriptionof that same worldline for the detection. But when trans-lations are nontrivial, and one has associated features ofrelativity of locality, this luxury is lost.

One way to have “relative locality” is indeed the casehere of interest, with the non-trivial translation genera-tors of (4)-(5) acting on spacetime coordinates as follows:

E, t = 1− `αE , E, x = `(1− β)p ,

p, t = 0, p, x = 1 . (10)

So, following Ref. [21], let us probe the difference be-tween coordinate velocity and physical velocity throughthe simple exercise of considering the simultaneous emis-sion “at Alice” of two massless particles, one “soft” (withmomentum ps small enough that `-deformed terms in for-mulas fall below the experimental sensitivity available)and one “hard” (with momentum ph big enough that atleast the leading `-deformed terms in formulas fall withinthe experimental sensitivity available).

We of course describe the relationship between the co-ordinates of two distant observers in relative rest in termsof the Poisson-bracket action of the translation genera-tors E, p, i.e. 1 − atE, · − axp, ·, with at, ax thetranslation parameters along t and x axes. In the specificcase in which Bob detects the soft massless particle in hisorigin, which restricts us to the possibilities at = ax = L(L being the spatial distance between Alice and Bob),one finds that Bob’s coordinates are related to Alice’s asfollows:

tB0 = tA0 − L+ `LαE , (11)

xB0 = xA0 − L− `L(1− β)p . (12)

The case we are considering, with a soft and a hard mass-less particle simultaneously emitted toward Bob in Al-ice’s origin, is such that the two particles are describedby Alice in terms of the worldlines

xAm=0,ps<0(tA) = tA ,

xAm=0,ph<0(tA) = tA (1− `|ph|) .

And these worldlines, in light of (11)-(12), are describedby Bob as follows:

xBm=0,ps<0(tB) = tB ,

xBm=0,ph<0(tB) = tB (1− `|ph|)− `L (α+ β) |ph| . (13)

This allows us to conclude that Bob, who is at the detec-tor, measures the following difference of times of arrivalbetween the soft photon (detected at Bob at tB = 0) andthe hard photon

∆tB = `L|ph|(α+ β) . (14)

tA

xA

∆t = ℓL|ph|

detectorL

βtB

xB

∆t = ℓL|pBh |(α+ )

-Lsource

FIG. 1. We illustrate the results for travel times of mass-less particles derived in this section by considering the caseof two distant observers, Alice and Bob, in relative rest. Twomassless particles, one soft (dashed red) and one hard (solidblue), are emitted simultaneously at Alice and they reach Bobat different times. Because of the nontriviality of translationtransformations, in Bob’s coordinatization (bottom panel) theemission of the particles at Alice appears not to be simulta-neous. And similarly, for the difference in times of arrival atthe distant detector (where Bob is located) Alice finds in hercoordinatization (top panel) a value which is not the same asthe difference in times of arrival that Bob determines (bottompanel). The case in figure has α+ β = 4/3. For visibility weassumed here unrealistic values for the scales involved. Forrealistic values of the distance between observers and of theenergy of the hard particle, taking ` as the inverse of thePlanck scale, no effect would be visible in figure (the world-lines would coincide).

So we see that the nontriviality of the translation trans-formations does affect the difference between coordinatevelocity and physical velocity. And the relativity of lo-cality produced by the nontriviality of the translation

5

transformations, while at first appearing to be counter-intuitive, actually ensures the internal logical consistencyof the relativistic framework5. Satisfactorily the physi-cal velocity does depend on α and β just in the way oneshould expect on the basis of the role of α and β in theon-shell relation. But the dependence of the physical ve-locity on α and β is not very significant, since it comesonly in the combination α + β. It is because of this fea-ture that the difference between Ep2 deformation and E3

deformation, if analyzed in a flat/non-expanding space-time, carries very little significance. One can get thesame physical velocity of massless particles by any mix-ture of Ep2 deformation and E3 deformation (includingthe cases where one of the two is absent) as long as α+βkeeps the same value. There is nothing extraordinary orsurprising about this: we are working at leading orderin `, so in terms which already have an ` factor we canuse E = |p| (for massless particles, and using the factthat at zero-th order in ` the massless shell is E = |p|).Evidently within these approximations here of interest acorrection term of form `Ep2 is indistinguishable from acorrection term of form `E3. While this is not surprisingit was still worth devoting to it this section since one ofthe main findings of the generalization proposed in thenext sections is that essentially in presence of spacetimeexpansion correction terms of form `Ep2 are very signif-icantly different from corrections term of form `E3.

Before moving on to those challenges, in closing thissection let us comment briefly on boost transformations.For the purposes we had in this section the relative local-ity produced by deformed boost transformations did notcome into play. Readers interested in those features ofrelative locality can look in particular at Ref. [14]. Thepoint relevant for us is that, as inferred from Ref. [14], therelative locality produced by boosts does not affect thederivation of the physical velocity, but rather is the wayby which the relativistic theory renders the properties of

5 As discussed in greater detail in Refs. [14, 16, 21], there isan analogy between the role of relative locality in deformed-Lorentz-invariant theories and the role of relative simultaneity indeformed-Galilean invariant theories (ordinary Lorentz-invarianttheories are to be viewed, from this perspective, as deformationsof Galilean-invariant theories, which accommodate the invariantscale c and the relativity of simultaneity, one being the counter-part of the other). The introduction of the invariant momentumscale `−1 requires a deformation of Lorentz invariance since ordi-nary Lorentz transformations change the value of all momentumscales. And for the logical balance of the relativistic theory onefinds that having one more thing invariant (the scale `) requiresrendering one more thing relative, which is the role played byrelative locality. One can see that the same logics applies to thetransition from Galilean Relativity to Einstein’s Special Relativ-ity. Taking as starting point Galilean Relativity, the introduc-tion of the invariant velocity scale c requires a deformation ofGalilean boosts, since ordinary Galilean boosts change the valueof all velocity scales. And for the logical balance of the relativis-tic theory one finds that having one more thing invariant (thescale c) requires rendering one more thing relative, which is therole played by relative simultaneity.

the physical velocity compatible with boost invariance.This all comes down to the fact that `-deformed boosts donot affect the timing of events at the observer: in the sim-ple analysis reported above an observer purely boostedwith respect to Alice would also see as simultaneous theemission of the two particles Alice sees as simultaneous;and another observer, this one purely boosted with re-spect to Bob, would describe the timing of the detectionsof the two particles at Bob in a way that is completelyundeformed. Boosts do play a role in the analysis of co-incidences of events distant from the observers6, but playno role in the determination of the physical velocity.

III. DSR-DEFORMEDDE-SITTER-RELATIVISTIC SYMMETRIES AND

PHYSICAL VELOCITY

A. DSR-deformed de-Sitter-relativistic symmetriesand equations of motion

The preliminaries given in the previous section providea good starting point for the main challenge we intendto face in this manuscript. Those preliminaries summa-rize some known aspects of `-deformed Lorentz symme-try for Minkowskian (non-expanding) spacetimes, includ-ing the features of relativity of locality and the rathermarginal relevance of the differences between Ep2 defor-mations and E3 deformations. We now want to producethe first example of theory of classical-particle world-lines with `-deformed de-Sitter-relativistic symmetries.It is interesting that de Sitter relativistic symmetries canthemselves be viewed as a deformation of the special-relativistic symmetries of Minkowski spacetime such thatthe expansion-rate parameter H is an invariant. And it isknown that the invariance of the expansion rate comes atthe cost of some velocity artifacts. With the points sum-marized in the previous section we can easily contrastthe two deformations of the special-relativistic symme-tries of Minkowski spacetime which provide the startingpoints for our work. On one side we have `-deformedLorentz(/Poincare) symmetries, a deformation by a largemomentum scale `−1 which produces velocity artifactsconnected with the novel feature of relative locality. Andon the other side we have de-Sitter-relativistic symme-tries, a deformation by a large distance scale H−1 which

6 This point emerges most simply and clearly by analyzing, asdone in Ref. [14], the illustrative example of an observer Alicewhich describes two pairs of events, one pair coincident in herorigin and one pair coincident far away from her origin. Foran observer purely boosted (purely `-deformed boosted) withrespect to Alice one finds [14] that the pair of events close to theorigin are still coincident, but the pair of events distant from theorigin are not coincident. This is the boost counterpart of thefact that, as a result of the properties of deformed translations,and the associated implications for relative locality, the distantpair of events will not be observed as coincident by observerswhose origin is close to that pair of events.

6

produces velocity artifacts connected with spacetime ex-pansion. The theory we are seeking must be such thatboth of these features can be accommodated, while pre-serving the relativistic nature of the theory. So both `and H shall be relativistic invariants (for a total of 3,including the speed-of-light scale, here mute because ofour choice of units). And we shall have both expansion-induced and relative-locality-induced velocity artifacts.

The strength of these demands appears to confront uswith an unsurmountable challenge. But we shall see thatthe “preliminaries” offered in the previous section draw avery clear path toward our objective. We start by speci-fying that our `,H-deformed relativistic symmetries shallleave invariant the following combination of the energyE, momentum p and boost N charges of particles:

CH,α,β = E2 − p2 + 2HNp+ `(αE3 + βEp2

). (15)

Evidently for ` → 0 this reproduces the standard in-variant of de Sitter symmetries. Something even moregeneral than our correction term `

(αE3 + βEp2

)could

here be envisaged, but we are not seeking results of max-imum generality. On the contrary we want to make thecase as convincingly and simply as possible that thereare examples of the novel class of relativistic theorieswe are here proposing. Moreover the correction term`(αE3 + βEp2

)does have enough structure for us to in-

vestigate the interplay of Ep2 deformations and E3 de-formations with spacetime expansion, which is the oneamong our side results that we perceive as most intrigu-ing.

The path drawn in the previous section guides us toobserve that the following `-deformed (2D) de Sitter alge-bra of charges is compatible with the invariance of CH,α,β

E, p = Hp− `αHEp ,E,N = p−HN − `αE(p−HN )− `βEp ,

p,N = E +1

2`αE2 +

1

2`βp2 . (16)

One easily sees that for ` → 0 this reproduces the stan-dard properties [25] of the classical de Sitter algebra ofcharges while for H → 0 it reduces to (2).

We give a convenient representation of these symmetrygenerators in terms of “conformal coordinates”, with con-formal time η and spatial coordinate x, and variables Ω,Πcanonically conjugate to these conformal coordinates:

Ω, η = 1 , Ω, x = 0 ,

Π, η = 0 , Π, x = 1 ,

η, x = 0 . (17)

We find the following representation

E = −HxΠ + Ω−HηΩ +`

2(1− β)Π2

+`

2HηΠ2 − `

2α(Ω−HηΩ−HxΠ)2 ,

(18)

p = Π , (19)

N = x(1−Hη)Ω +

(η − H

2η2 − H

2x2)

Π

+ `

(1 +Hη

2xΠ− (1−Hη)ηΩ

)Π .

(20)

Again we intend to derive the worldlines of particlesworking within a “covariant formulation”; so we formallyintroduce an auxiliary affine parameter on the worldlineand the formal evolution in the affine parameter is gov-erned by the invariant CH,α,β .

Of course the fact that CH,α,β is an invariant ofthe (deformed-)relativistic symmetries implies that thecharges that generate the symmetry transformations areconserved over this evolution:

E=CH,α,β , E=0 , p=CH,α,β , p=0 ,

N =CH,α,β ,N=0 .

Importantly by consistency with the chosen (deformed)form of the invariant CH,α,β we have obtained translationtransformations which are significantly deformed, specif-ically for the time direction. In fact, from (18) and (19)one finds

E, η = 1−Hη−`α(1−Hη)(−HxΠ+Ω−HηΩ) , (21)

E, x =−Hx+ `((1−β+Hη) Π

+αHx(Ω−HxΠ−HηΩ)),(22)

p, η = 0 , p, x = 1 . (23)

This prepares us to deal with relative-locality effects, ofthe sort described in the previous section but intertwinedwith the additional complexity of spacetime expansion.

But let us first note down the equations of motion thatare obtained in our framework. Following the standardprocedures for derivation of worldlines in the covariantformulation of classical mechanics one easily arrives, fora particle of mass m (i.e. CH,α,β = m2), at the followingresult for the worldlines:

xm,p = x0 +

√m2 + (1−Hη)2p2

Hp

−√m2 + (−1 +Hη0)2p2

Hp+ `(η − η0)p .

(24)

For the case of massless particles this reduces to

xm=0,p (η) = x0 −p

|p| (η − η0) (1− `|p|) , (25)

which in turn specifying p < 0 (so that the velocity ispositive along the x-axis, see related comments in theprevious section) takes the form

xm=0,p (η) = x0 + (η − η0) (1− `|p|) . (26)

By construction these worldlines (24),(26) are covariantunder the (deformed-)relativistic transformations gener-ated by the charges E, p,N , as one can also easily verifyexplicitly.

7

B. Travel time of massless particles

As already established in previous studies of the casewithout spacetime expansion (here summarized in Sec-tion II), the DSR-deformed relativistic symmetries intro-duce (as most significant among many other novel fea-tures) a dependence on energy of the travel time of amassless particle from a given source to a given detector.In the analysis we provided so far of our novel proposalof DSR-deformed relativistic symmetries of an expandingspacetime (with constant expansion rate) an indicationof dependence on energy of the travel time of a mass-less particle from a given source to a given detector isfound in Eq. (26): the equation governing the proper-ties of the worldline of a massless particle in conformalcoordinates is `-corrected and the correction term intro-duces a dependence on the energy(/momentum) of theparticle. However, as also expected on the basis of pre-vious results on the case without spacetime expansion,we are evidently working in a framework where localityis relative (evidence of which has been provided so farwithin our novel framework implicitly in Eqs. (21)-(22)).So the analysis of the equations of motion written by oneobserver is inconclusive for what concerns the notion of“travel time”, i.e. the correlation between emission timeand detection time. As illustrated for the non-expandingcase in the previous section, we must guard against thecoordinate artifacts associated to relative locality by an-alyzing the emission of the particle in terms of the de-scription of an observer near that emission point and wemust then analyze the detection of the particle in termsof the description of an observer near the detection point.

This is indeed our next task. We consider the case oftwo distant observers: Alice at the emitter and Bob atthe detector. We keep the analysis in its simplest form,without true loss of generality, by considering the case ofsimultaneous emission at Alice of only two massless parti-cles, one with “soft” momentum ps and one with “hard”momentum ph. Evidently, on the basis of the analysisreported in the previous subsection, Alice describes thetwo particles according to

xAps(ηA) = ηA , (27)

xAph(ηA) = ηA(1− `|pAh |

), (28)

where we specified xA0 = ηA0 = 0, so that the emission is

at (0, 0)A

. Since translations are a relativistic symmetryof our novel framework, we already know that the sametwo worldlines will be described by the distant observerBob in the following way

xBm=0,ps(ηB) = xB0;s + ηB − ηB0;s , (29)

xBm=0,ph(ηB) = xB0;h +

(ηB − ηB0;h

) (1− `|pBh |

), (30)

i.e. the same type of worldlines but with a differenceof parameters here codified in xB0;s, η

B0;s,x

B0;h, ηB0;h. In-

deed, Alice’s worldlines (27)-(28) and Bob’s worldlines(29)-(30) have exactly the same form, but for the ones

of Alice we had by construction (by having specifiedsimultaneous emission at Alice) that xA0;h = ηA0;h =

xA0;s = ηA0;s = 0 whereas Bob’s values of the parameters,

xB0;s, ηB0;s,x

B0;h, ηB0;h, should be determined by establishing

which (`-deformed) translation transformation connectsAlice to Bob.

This is the same task performed in some of the previ-ous studies (such as Ref. [21]) involving relative localityin DSR-deformed relativistic theories without spacetimeexpansion. But also this aspect of the analysis turnsinto a rather more challenging exercise for us, having todeal with spacetime expansion. Let us then proceed de-termining the translation transformation that connectsAlice to Bob providing enough details for the reader toappreciate the interplay between the different structureswe must handle.

As usual we shall give the action on points η, x of aworldline by a finite transformation TG;a generated bythe generic element G of the relativistic-symmetry alge-bra in terms of its exponential representation as

TG;a . X = e−aG . X ≡∞∑

n=0

(−a)n

n!G,Xn , (31)

where a is the transformation parameter and G,Xn isthe n-nested Poisson bracket defined by the relation

G,Xn =G, G,Xn−1

, G,X0 = X . (32)

And before proceeding we must also implement some-thing else that can be specified about observer Bob. Theworldline parameters xA0;h, η

A0;h, x

A0;s, η

A0;s of observer Al-

ice are fully specified (xA0;h = ηA0;h = xA0;s = ηA0;s = 0) byits being at the point of simultaneous emission of the twoparticles. We have introduced observer Bob as one thatdetects the particles (the particles worldines should crossBob spatial origin) but we have so far left completely un-specified its worldline parameters xB0;s, η

B0;s,x

B0;h, ηB0;h. We

evidently can do better than that. And actually we canalso insist, without true loss of generality, that the softparticle reaches Bob in his spacetime origin: so we arefree to enforce xB0;s = ηB0;s = 0.

This is particulary convenient because it involves the“soft particle”, i.e. the one whose momentum ps hasbeen chosen to be small enough to render the `-deformedeffects inappreciable within the experimental sensitivitiesavailable to Alice and Bob. The fact that both xA0;s =

ηA0;s = 0 and xB0;s = ηB0;s = 0 for a soft particle leads us tofocus on the case of the observer Bob connected to Aliceby the following transformation

Bob = e−axp . e−aηE .Alice , (33)

with

ax =1− e−Haη

H. (34)

Through this we are essentially exploiting the fact thatthe deformation is ineffective on the soft particle as a way

8

for us to focus on a distant observer Bob whose relation-ship to Alice (translation parameters connecting Alice toBob) can be specified using only known results on theundeformed/standard relativistic properties7. Indeed inclassical (relativistically undeformed) de Sitter spacetimeone easily finds that a massless particle emitted in theorigin of some observer Alice will cross the origin of allobservers connected to Alice by a conformal-time trans-lation of parameter aη followed by a spatial translation ofparameter ax with the request that ax = H−1[1−e−Haη ]

Using our translation generators (18)-(19) one easilyfinds that for points on the worldline of the hard particlethe map from Alice to Bob is such that

ηBh =1−eHaηH

+eHaηηAh

+` α aηeHaη (1−HηAh )

(EAh +Haxp

Ah

),

xBh = eHaη (xAh − ax) + `βsinh (Haη)

HpAh

−`αaηeHaηH(ax − xAh )(EAh +Haxp

Ah

)

−`(1− e−Haη

) (1 + eHaηHηAh

)

HpAh ,

pBh = e−Haη(pAh + `αHaηp

Ah

(EAh +Haxp

Ah

)).

Crucial for us is the fact that, in light of this result for thelaws of transformation from Alice’s ηAh , x

Ah , p

Ah to Bob’s

ηBh , xBh , p

Bh , we can deduce that the worldline of the hard

particle emitted at Alice is described by Bob as follows

xBm=0,ph(ηB)=ηB− `|pBh |

(ηB+αaη+β

e2Haη−1

2H

), (35)

where we made use of all the specifications discussed

above, including ax = 1−e−HaηH .

In turn this allows us to obtain the sought result for thedependence on energy(/momentum) of the travel timesof massless particles: by construction of the worldlinesand of the Alice→Bob transformation the soft masslessparticle emitted in Alice’s spacetime origin reaches Bob’sspacetime origin, whereas from (35) we see that the hardmassless particle also emitted in Alice’s spacetime originreaches Bob at a nonzero conformal time. Specificallythe difference in conformal travel times derivable from(35) is

∆ηB = ηBh

∣∣∣xBh=0

= `|pB |(αaη + β

e2Haη − 1

2H

). (36)

We summarize the relativistic properties of this travel-time analysis, inconformal coordinates, in Fig. 2. Bycomparison with Fig. 1 one sees that in conformal coor-dinates the qualitative picture of the energy dependenceof travel times is very similar to the one of the case

7 This also implicitly requires [21] that the clocks at Alice and Bobare synchronized by exchanging soft massless particles.

ηA

xA

∆η = ℓL|pAh |

detectorL

βηB

xB

∆η = ℓL|pBh |(α 1

H ln( 11−HL ) +

L−HL2

2

(1−HL)2

)

− L1−HLsource

FIG. 2. We illustrate the results for travel times of masslessparticles derived in this section, adopting conformal coordi-nates. We consider the case of two distant observers, Aliceand Bob, connected by a pure translation, and two masslessparticles, one soft (dashed red) and one hard (solid blue andsolid violet), emitted simultaneously at Alice. And we con-sider two combinations of values of α and β producing thesame α+ β: the case α = 2/3, β = 2/3 (Bob’s hard worldlinein blue) and the case α = 1/3, β = 1 (Bob’s hard worldlinein violet). As in the case without expansion (Fig. 1) we havehere that in Bob’s coordinatization (bottom panel) the emis-sion of the particles at Alice appears not to be simultaneous.Similarly for the difference in times of arrival at the distantdetector Bob Alice finds in her coordinatization (top panel)not the same value as the difference in times of arrival thatBob determines. Comparison of the blue and violet world-lines shows that in the case with spacetime expansion thetravel time does depend individually on α and β (not just onα + β as in the case without spacetime expansion). Againfor visibility we assumed here unrealistic values for the scalesinvolved.

without spacetime expansion. But here, with spacetimeexpansion, there are tangible differences between Ep2 de-formations and E3 deformations.

9

In Fig. 3 we characterize the results of our travel-timeanalysis in comoving coordinates, which for most studiesof spacetime expansion are the most intuitive choice ofcoordinates. [The differences between Fig. 2 and Fig. 3all are a straightforward manifestation of the relationshipη = H−1(1 − e−Ht) between the conformal time η andthe comoving time t.]

tA

xA

∆t = ℓL|pAh |

detectorL

βtB

xB

∆t = ℓL|pBh |(α 1

H ln( 11−HL ) +

L−HL2

2

(1−HL)2

)

− L1−HLsource

FIG. 3. We here illustrate the results for travel times of mass-less particles derived in this section, adopting comoving co-ordinates. The situation here shown is the same situationalready shown (there in conformal coordinates) in Fig. 2.

IV. IMPLICATIONS FOR PHENOMENOLOGY

Our main motivation for this analysis was conceptual.There is at this point rather robust evidence that theo-ries in non-expanding spacetime with DSR-deformed rel-ativistic symmetries do have a solid internal logical con-sistency and are therefore viable candidates for quantum-gravity researchers. Some of their properties, like the rel-ativity of spacetime locality, are “extremely novel” andthis will understandably affect how each researcher sub-

jectively gauges the likelihood (or lack thereof) of DSR-deformations of relativistic symmetries for the quantum-gravity realm. But the objective fact is that, if we couldconfine our considerations to Minkowskian spacetimes,the DSR option is at this point clearly viable. However,it is of course a well established experimental fact that weare not in a Minkowskian spacetime. Even postponing,as we did here, the much reacher collection of experi-mental facts on gravitational phenomena and the Ein-steinian description of spacetime, we felt DSR researchshould at least start facing the fact that spacetime ex-pansion is very tangible in our data. So we here soughtto make the first step away from the extremely confiningarena of Minkowskian spacetime, by showing that DSR-deformations of relativistic symmetries are also viable indeSitterian spacetime, expanding at a constant rate.

In facing successfully this challenge, of primarily “aca-demic” interest, we stumbled upon results which are ofrather sizable significance for phenomenology, for reasonsthat were hinted at in earlier parts of this manuscript andwe shall now discuss in greater detail.

Our first step toward the phenomenological issues ofinterest in this section is to reformulate the results fortravel times of massless particles derived in the previoussection in a way that is in closer correspondence with thetype of facts that are established experimentally, whenour telescopes observe distant sources of gamma rays.Many of the analyses performed at our telescopes amountto timing the detection of photons emitted from a sourceat a known redshift z. Postponing for a moment the factthat the relevant contexts are not such that we couldassume a constant expansion rate, let us observe thatthe results we derived in the previous section are eas-ily reformulated as the following prediction for the dif-ferences in detection times of photons of different ener-gies(/momenta) emitted simultaneously by a source atredshift z

∆t = `|p|(α

ln (1 + z)

H+ β

z + z2

2

H

). (37)

Here again ∆t is the difference in detection times be-tween a hard gamma-ray photon of momentum p and areference ultrasoft photon emitted simultaneously to thehard photon at the distant source. And notice that with∆t we denote the comoving time but in the relevant tim-ing sequences of detections at telescopes the differencebetween comoving and conformal time is intangible: thetelescopes are operated for a range of times within thet = 0 of their resident clock which is relatively small,much smaller than the time scale H−1, so that confor-mal time and comoving time essentially coincide (one hasη ≡ H−1(1 − e−Ht) ' t for t H−1). For what con-cerns redshift we simply relied on the standard formula[23, 26] for the case of constant expansion rate which, inour notation, gives z = eHaη − 1.

A first aspect of phenomenological relevance whichmust be noticed in our result (37) is the dependence onthe parameters α and β, i.e. the difference between Ep2

10

deformations and E3 deformations. The fact that for thenon-expanding spacetime case this dependence only goeswith α+β is of course recovered in our result (37) in thelimit of small redshift z. This is expected since for smallredshift the expansion does not manage to have appre-ciable consequences. What is noteworthy is that alreadyat next-to-leading order in redshift the dependence on αand β is no longer fully of the form α + β, as seen byexpanding our result (37) to second order in z:

∆t∣∣∣z1' `|p|

H

[(α+ β)z + (β − α)

z2

2

]. (38)

In a non-expanding spacetime the difference between Ep2

deformations and E3 deformations would not have signif-icant implications on travel-time determinations (since itdepends on α + β one gets the same result reducing theamount of Ep2 deformation in favor of an equally sizableincrease of the amount of E3 deformation). The situ-ation in expanding spacetimes is evidently qualitativelydifferent in this respect since Ep2 deformations and E3

deformations produce corrections to the travel times thathave different functional dependence on redshift. So weare learning that for determinations of travel times fromdistant astrophysical sources the difference between Ep2

deformations and E3 deformations is a phenomenologi-cally viable (phenomenologically determinable) issue.

Related to this observation is also the other point wewant to make on the phenomenology side, which con-cerns the comparison with analogous studies of scenar-ios where relativistic symmetries are actually “broken”(allowing for a preferred/“aether” frame), rather thanDSR-deformed as in the case which was of interest forus in this manuscript. Broken-Lorentz-symmetry theo-ries with a preferred frame are far simpler conceptuallythan DSR-deformed relativistic theories, and indeed theissue of the interplay between scale of Lorentz-symmetrybreakdown and spacetime expansion has been usefully in-vestigated already for several years [27–29], even produc-ing a rather universal consensus on the proper formaliza-tion that should be adopted [29, 30, 33]. Again thanks tothe simplicity of broken-Lorentz-symmetry theories theseresults apply to the general case of varying expansionrate, and for the massless particles they take the form

∆t = λLIV |p|∫ z

0

dz

a(z)H(z), (39)

where a is the scale factor, H is the expansion rate, andλLIV (“LIV” standing for Lorentz Invariance Violation)is the counterpart of our scale `: just like ` for us isthe inverse-momentum scale characteristic of the onset ofthe DSR-deformation of relativistic symmetries, λLIV isthe inverse-momentum scale characteristic of the onset ofthe LIV-breakdown of relativistic symmetries. Of course,there is a crucial difference between the properties of `and the properties of λLIV : `, as characteristic scale of adeformed-symmetry picture, takes the same value for allobservers, while λLIV , as characteristic scale of a broken-symmetry picture, takes a certain value in the preferred

frame and different values in frames boosted with respectto the preferred frame.

Setting momentarily these differences aside we cancompare our results for DSR-deformed symmetries withconstant rate of expansion with the special case of thebroken-symmetry formula (39) obtained for constant rateof expansion:

∆t = λLIV |p|z + z2

2

H. (40)

Comparing this to our result (37) we find that fixingα = 0 in the deformed-symmetry case one gets a for-mula (valid in all reference frames) which is the same asthe formula of the broken-symmetry case in the preferredframe. So if α = 0 the differences between deformed-symmetry and broken-symmetry cases would be tangibleonly by comparing studies of travel times of massless par-ticles between two telescopes with a relative boost: thedifference there would be indeed that ` takes the samevalue for studies conducted by the two telescopes whereasfor λLIV the two telescopes should give different values.

We must stress however that within our deformed-symmetry analysis we found no reason to focus specif-ically on the choice α = 0. And if α 6= 0 in the deformedsymmetry case even studies conducted by a single tele-scope could distinguish between the case of symmetrydeformation and the case of symmetry breakdown.

MKN501PKS2155-304

GRB090510Α=1.32,Β=0

Α=0,Β=

0.65

0.05 0.10 0.50 1.00 5.00 10.00

0.001

0.01

0.1

1

z

Dt

Dp

@sG

eV-

1 D

FIG. 4. Here we show the dependence on redshift of the ex-pected time-of-arrival difference divided by the difference ofenergy of the two massless particles. Two such functions areshown, one for the case α = 0, β = 0.65 (violet) and one forthe case α = 1.3, β = 0 (blue). We also show the upper lim-its that can be derived from data reported in Refs. [31–33],setting momentarily aside the fact that our analysis adoptedthe simplifying assumption of a constant rate of expansion(whereas a rigorous analysis of the data reported in Refs. [31–33] should take into account the non-constancy of the expan-sion rate). The values α = 0, β = 0.65 and α = 1.3, β = 0have been chosen so that we have consistency with the tight-est upper bound, the one established in Ref. [33]. The mainmessage is coded in the fact that at small values of redshiftthe blue and the violet lines are rather close, but at large val-ues of redshift they are significantly different (this is a log-logplot). In turn this implies that at high redshift the differencebetween adding correction terms of form Ep2 and adding cor-rection terms of form E3 can be very tangible.

11

In Fig. 4 we compare the dependence on redshift ofour deformed-symmetry effect among two limiting casesof balance between α and β, and we also compare theseresults to bounds on travel-time anomalies [31–33] ob-tained in studies of sources at redshift smaller than 1(where the assumption of a constant rate of expansion itnot completely misleading).

In Fig. 5 we illustrate the “constraining power” of ourresults and of foreseeable generalizations of our analysisgaining access to cases with non-constant rate of expan-sion. For better visibility and as a way to offer more in-telligible visual messages we restrict our focus to the casein which both α and β are positive. [There is no reasonfor making this assumption, but on the other hand it iseasy to see how the constraints established within this as-sumption can be adapted to the more general situationwith α and β allowed to also be negative.]

In the left panel of Fig. 5 we show how the bound ob-tained from Ref. [32], at the relatively small redshift ofz = 0.116 (where we should really be able to apply ourresults for constant rate of expansion as a reliable firstapproximation), provides a constraint on the α, β param-eter space. For these purposes we can of course fix thevalue of ` to be exactly the Planck length (the inverseof the Planck scale) since any rescaling of ` can be re-absorbed into an overall rescaling of α and β. Withinthis choice of conventions the target “Planck-scale sen-sitivity” would manifest itself as the ability to constrainvalues of α and β of order 1. As shown in the left panelof Fig. 5, even restricting our focus on cases with redshiftmuch smaller than 1 (as needed because of the presentlimitation of applicability of our approach to constant, orapproximately constant, rate of expansion) this Planck-scale sensitivity is not far. In the right panel of Fig. 5 weshow the much tighter constraint on the α, β parameterspace which would be within reach if we could assumeour analysis to apply also to redshifts close to 1, as forGRB090510 observed by the Fermi telescope [33]. Ana-lyzing data from sources at redshift of ' 1 assuming apicture with constant rate of expansion cannot produceconservative experimental bounds, but the content of theright panel of Fig. 5 serves the purpose of providing evi-dence of the fact that full Planck-scale sensitivity will bewithin reach of improved versions of our analysis, extend-ing our results to the case of expansion at non-constantrate.

V. SUMMARY AND OUTLOOK

We here addressed a long-standing issue for the studyof quantum-gravity-inspired deformations of relativisticsymmetries. It led us to propose the first ever exam-ple of a relativistic theory of worldlines of particles with3 nontrivial relativistic invariants: a large speed scale(“speed-of-light scale”), a large distance scale (inverse ofthe “expansion-rate scale’), and a large momentum scale(“Planck scale”). And on the basis of the observations

PKS2155-304

z=0.116

0 5 10 15 200

5

10

15

20

Α

Β

GRB090510

z=0.9

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

1.2

Α

Β

FIG. 5. Here in the left panel we show the constraint on theα, β parameter space that can be obtained from Ref. [32], con-cerning a source at the relatively small redshift of z = 0.116,where we can confidently apply our results for constant rateof expansion as a reliable first approximation. Through thiswe show that even within the confines of our analysis Planck-scale sensitivity (values of |α| and |β| smaller or comparableto 1) is not far. In the right panel we show the much tighter(indeed “Planckian”) constraint on the α, β parameter spacewhich would be within our reach if we could assume our anal-ysis to apply also to redshifts close to 1, as for GRB090510observed by the Fermi telescope [33]. By comparing the leftand the right panel one also finds additional evidence of howthe difference between adding correction terms of form Ep2

and adding correction terms of form E3 becomes more sig-nificant at higher redshifts: in the left panel (data on sourceat small redshift of z = 0.116) the bound on the α parame-ter is nearly as strong as on the β parameter, whereas in theright panel (data on source at redshift of z ' 0.9) the con-straint on the alpha parameter is significantly weaker thanthe constraint on the β parameter.

reported in the previous section it is clear that this is notmerely an exercise providing a more rigorous derivation ofprevious heuristic proposals: our constructive approachproduces results that reshape the targets of the relevantphenomenology.

In order to fully empower these phenomenological ap-plications the next step would be to generalize our ap-proach in such a way to render it applicable to caseswith varying expansion rate. For what concerns appli-cations in astrophysics this will open the way to exploit-ing data from the farthest sources, with the associatedbenefit of even larger distances amplifying the minutePlanck-scale effects. And with the ability of handlingvarying expansion rates this research programme couldalso gain access to data in cosmology. The main oppor-tunity from this perspective comes from the analyses ofcosmology data which were at first inspired by varying-speed-of-light theories (see, e.g., Refs. [34–36] and refer-ences therein). In our deformed-relativistic theories thespeed of light is not varying in the sense originally in-tended: enforcing (however deformed) relativistic invari-ance we did not find room for the speed-of-light scaleto have different values between now and earlier epochsof the Universe, but we did find a scenario for havingthat (with equal strength at all stages of the evolution of

12

the Universe) the speed of photons depends on their en-ergy. This energy-dependence of the speed of photons wefound for scenarios with deformed relativistic symmetriesmay affect cosmology in ways that are similar to (and yetpotentially interestingly different from [13]) the varying-speed-of-light scenarios, simply because of the fact thatthe Universe was hotter in its earlier stages of evolution,so that the typical energy of particles was higher.

Of course, besides possibly leaping toward these oppor-tunities available when generalizing our results to caseswith non-constant expansion rate, a lot more could bedone with the constant-expansion-rate case. We heremanaged, and only at the cost of facing some significantcomplexity, to obtain a consistent theory of classical-

particle worldlines, but of course much more is neededbefore having established a consistent scheme of relativis-tic deformation of our current theories. In this respect, atop priority would be to formulate quantum field theorieswith these deformed relativistic theories.

ACKNOWLEDGEMENTS

We gratefully acknowledge conversations with GiuliaGubitosi, Salvatore Mignemi and Tsvi Piran. This workwas supported in part by a grant from the John Temple-ton Foundation. AM also acknowledges support from aNSF CAREER grant.

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