Searching for New Physics with Ultrahigh Energy Cosmic Rays
aa r X i v : . [ a s t r o - ph . H E ] A ug Searching for New Physics with Ultrahigh EnergyCosmic Rays
Floyd W Stecker ‡ Astrophysics Science DivisionNASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Sean T Scully
Dept. of Physics and AstronomyJames Madison University, Harrisonburg, VA 22807, USA
Abstract.
Ultrahigh energy cosmic rays that produce giant extensive showers of charged particlesand photons when they interact in the Earth’s atmosphere provide a unique tool to searchfor new physics. Of particular interest is the possibility of detecting a very small violationof Lorentz invariance such as may be related to the structure of space-time near the Planckscale of ∼ − m. We discuss here the possible signature of Lorentz invariance violation onthe spectrum of ultrahigh energy cosmic rays as compared with present observations of giantair showers. We also discuss the possibilities of using more sensitive detection techniques toimprove searches for Lorentz invariance violation in the future. Using the latest data fromthe Pierre Auger Observatory, we derive a best fit to the LIV parameter of 3 . +1 . − . × − ,corresponding to an upper limit of 4 . × − at a proton Lorentz factor of ∼ × .This result has fundamental implications for quantum gravity models.
1. Introduction
Owing to the uncertainty principle, it has long been realized that the higher the particleenergy attained, the smaller the scale of physics that can be probed. Thus, optical, UV andX-ray observations led to the understanding of the structure of the atom, γ -ray observationsled to an understanding of the structure of the atomic nucleus, and deep inelastic scatteringexperiments with high energy electrons led to an understanding of the structure of the proton.Accelerator experiments have led to an understanding of quantum chromodynamics and itis hoped that the Large Hadron Collider [1] will eventually reveal new physics at the TeVscale. This could lead to the discovery of the predicted Higgs boson and supersymmetricparticles. To go much beyond this scale of fundamental physics, to search for clues to a ‡ Corresponding author: [email protected] earching for New Physics with Ultrahigh Energy Cosmic Rays
The theory of relativity is, of course, one of the fundamental pillars of modern physics.However, because of the problems associated with merging relativity with quantum theory,it has long been felt that relativity will have to be modified in some way in order to constructa quantum theory of gravitation.The group of Lorentz transformations delineated by special relativity can be describedas a high energy modification of the unbounded group of Galilean transformations. Sincethe Lorentz group is also unbounded at the high boost (or high energy) end, in principle itmay also be subject to modifications in the high boost limit. There is also a fundamentalrelationship between the Lorentz transformation group and the assumption that space-time is scale-free, since there is no fundamental length scale associated with the Lorentzgroup. However, as noted by Planck [2], there is a potentially fundamental scale associatedwith gravity, viz. , the Planck scale. Thus, there has been a particular interest in thepossibility that a breakdown of Lorentz invariance (LI) may be associated with the Planckscale, λ P l = q G ¯ h/c ∼ − m, owing to various speculations regarding quantum gravityscenarios. This scale corresponds to an energy (mass) scale of M P l = ¯ hc/λ
P l ∼ GeV.It is at the Planck scale where quantum effects are expected to play a key role indetermining the effective nature of space-time that emerges in the classical continuumlimit. The idea that LI may indeed be only approximate has been explored within thecontext of a wide variety of suggested new Planck-scale physics scenarios. These includethe concepts of deformed relativity, loop quantum gravity, non-commutative geometry, spinfoam models, and some string theory models. Such theoretical explorations and their possibleconsequences, such as observable modifications in the energy-momentum dispersion relationsfor free particles and photons, have been discussed under the general heading of “Planck scalephenomenology”. There is an extensive literature on this subject. (See [3] for a review;some recent references are Refs. [4] – [6].)
It has been proposed that violation of LI at a high energy such as the Planck scale couldhave astrophysical consequences that might be manifested in a suppressed form at an energyscale << M
P l [7] – [9]. A surprising result of subsequent work has been the conclusion thatseveral potential effects of Lorentz invariance violation (LIV) can be explored and tested earching for New Physics with Ultrahigh Energy Cosmic Rays e.g. , Ref.[3, 10] and references therein.)Among the relevant astrophysical tests, we focus here on the ultrahigh energycosmic-ray sector. Astrophysically produced ultrahigh energy particles are the perfectvehicles to explore the potential for detection of possible violations of special relativityat ultrahigh energies. One may also search for possible evidence of Planck scalephysics and quantum gravity through photon propagation effects [9],[4]. Sucheffects may be revealed by space-based observations from the
SWIFT γ -ray burstdetector ( http://heasarc.gsfc.nasa.gov/docs/swift/swiftsc.html ), and the Fermi ( http://fermi.gsfc.nasa.gov/ ) γ -ray Space Telescope [11]. We will concentrate hereon present observations of ultrahigh energy cosmic rays. We will also discuss future satelliteprograms proposed to make observations of ultrahigh energy cosmic-ray air showers fromspace such as JEM-EUSO (Extreme Universe Space Observatory) [12] and
OWL (OrbitingWide-Angle Light Collectors) [13] (See section 7.)
2. Ultrahigh Energy Cosmic Rays
Ultrahigh energy cosmic rays (UHECRs) produce giant air showers of charged particles whenthey impinge on the Earth’s atmosphere. Observational studies of these showers have beenundertaken using scintillator arrays and with atmospheric fluorescence detectors. In thismanner the total energies and atomic weights of the primary particles can be determinedfrom the shower characteristics. The total energy of the primary incoming particle can bededuced from the number of secondary charged particles produced at a fiducial distance fromthe shower axis or the amount of atmospheric fluorescence produced by the shower. A roughmeasurement of the atomic weight of the primary can be obtained from the determining theheight of the initial interaction in the atmosphere.The history of UHECR detection goes back almost half a century [14]. Owing to theirobserved global isotropy and ultrahigh energy that allows them to be unfettered by thegalactic magnetic field, cosmic rays above 10 EeV (1 EeV ≡ eV) are believed to be ofextragalactic origin. This fact, together with the absence of a correlation of arrival directionswith the galactic plane, indicates that if protons are the primary particles that make up theultrahigh energy cosmic radiation, these protons should be of extragalactic origin.The large air shower detector arrays and, in particular the Auger array,( ) (a.k.a. the Pierre Auger Observatory (PAO)) have opened uptwo potential new areas of research. One area is a new field of ultrahigh energy particleastronomy – the identification and exploration of powerful extragalactic sources capable ofaccelerating cosmic rays to energies above 1 joule per particle. The second area, which isthe topic of this focus paper, is the field of potential new ultrahigh energy particle physics– the search for new physical processes that may occur at energies much greater than those earching for New Physics with Ultrahigh Energy Cosmic Rays
Shortly after the discovery of the 3K cosmogenic background radiation (CBR), Greisen [15]and Zatsepin and Kuz’min [16] predicted that pion-producing interactions of such cosmicray protons with the CBR should produce a spectral cutoff at E ∼
50 EeV. The flux ofultrahigh energy cosmic rays (UHECR) is expected to be attenuated by such photomesonproducing interactions. This effect is generally known as the “GZK effect”. Owing to thiseffect, protons with energies above ∼
100 EeV should be attenuated from distances beyond ∼
100 Mpc because they interact with the CBR photons with a resonant photoproductionof pions [17].The flux and spectrum of the secondary ultrahigh energy neutrinos resulting fromthe decay of the photoproduced pions was also subsequently calculated [18, 19]. Photonswith comparable ultrahigh energy have much smaller mean-free-paths because they pair-produce electrons and positrons by interacting with radio background photons and are thusattenuated. The attenuation length for photons is somewhat uncertain, because of theuncertainties in our knowledge of the flux and spectrum of the radio background [20, 21].The GZK effect is not a true cutoff, but a suppression of the ultrahigh energy cosmicray flux owing to an energy dependent propagation time against energy losses by suchinteractions, a time which is only ∼
300 Myr for 100 EeV protons [17]. At high redshifts, z ,the target photon density increases by (1 + z ) and both the photon and initial cosmic rayenergies increase by (1 + z ). A plot of the GZK energy as a function of redshift, calculatedfor the ΛCDM cosmology, is shown in Figure 1 [22]. If the source spectrum is hard enough,there could also be a relative enhancement just below the “GZK energy” owing to a “pileup”of cosmic rays starting out at higher energies and crowding up in energy space at or belowthe predicted GZK cutoff energy [23]. At energies in the 1-10 EeV range, pair productioninteractions should take a bite out of the UHECR spectrum. Some “trans-GZK” hadronic showers with energies above the predicted GZK cutoff energyhave been reportedly observed by both scintillator and fluorescence detectors, particularlyby the scintillator array group at Akeno [24], in apparent contradiction to the expected GZKattenuation effect. While there is less evidence for such interesting events from fluorescencedetectors, the
Fly’s Eye fluorescence detector reported the detection of a 320 EeV event[25], an energy that is a factor of ∼ Auger data [26],[27] as well asthose from
HiRes , [28], have both been interpreted as indicating a GZK cutoff. This has led earching for New Physics with Ultrahigh Energy Cosmic Rays Log E G Z K ( e V ) Figure 1.
The GZK cutoff energy, defined as the energy predicted for a flux decrease of 1 /e owing to intergalactic photomeson production interactions, as a function of redshift [22]. many in the cosmic ray community to assume that there is no new physics to be discoveredat ultrahigh energies. Thus, the emphasis in the field has been on ultrahigh energy particle astronomy , i.e. , the attempt to determine which nearby extragalactic objects accelerate andemit such high energy particles. However, the subject of this paper will be the search for new physics at ultrahigh energies. In particular, we will discuss the features in the ultrahighenergy cosmic ray spectrum that would be a signal of Lorentz violation and possible Planckscale physics and would also be compatible with present observational data.
3. Violating Lorentz Invariance - A Framework
In this paper we will take the phenomenological approach to exploring the effects of LIVpioneered by Coleman and Glashow [29]. They have proposed a simple formalism via postulating a small first order perturbation in the free-particle Lagrangian. This formalismhas the advantages of (1) simplicity, (2) preserving the SU (3) ⊗ SU (2) ⊗ U (1) standardmodel of strong and electroweak interactions, (3) having the perturbative term in the earching for New Physics with Ultrahigh Energy Cosmic Rays § This formalism has provenuseful in exploring astrophysical data for testing LIV [29],[31].Coleman and Glashow start with a standard-model free-particle Lagrangian, L = ∂ µ Ψ ∗ Z ∂ µ Ψ − Ψ ∗ M Ψ (1)where Ψ is a column vector of n fields with U(1) invariance and the positive Hermitianmatrices Z and M can be transformed so that Z is the identity and M is diagonalized toproduce the standard theory of n decoupled free fields.They then add a leading order perturbative, Lorentz violating term constructed fromonly spatial derivatives with rotational symmetry so that L → L + ∂ i Ψ ǫ∂ i Ψ , (2)where ǫ is a dimensionless Hermitian matrix that commutes with M so that the fields remainseparable and the resulting single particle energy-momentum eigenstates go from eigenstatesof M at low energy to eigenstates of ǫ at high energies.To leading order, this term shifts the poles of the propagator, resulting in the freeparticle dispersion relation E = ~p + m + ǫ~p . (3)This can be put in the standard form for the dispersion relation E = ~p c MAV + m c MAV , (4)by shifting the renormalized mass by the small amount m → m/ (1 + ǫ ) and shifting thevelocity from c (=1) by the amount c MAV = q (1 + ǫ ) ≃ ǫ/ ∂E∂ | ~p | = | ~p | q | ~p | + m c MAV c MAV , (5)which goes to c MAV in the limit of large | ~p | . Thus, Coleman and Glashow identify c MAV tobe the maximum attainable velocity of the free particle. Using this formalism, it becomesapparent that, in principle, different particles can have different maximum attainablevelocities (MAVs) which can be different from c . Hereafter, we denote the MAV of a particleof type i by c i and the difference c i − c j = ǫ i − ǫ j ≡ δ ij . (6)There are other popular formalisms that are inspired by quantum gravity models orby speculations on the nature of space-time at the Planck scale. There are formidable § See Ref. [30] for a generalization to the non-isotropic case. earching for New Physics with Ultrahigh Energy Cosmic Rays ≥ M P l [10],[33]. This leads to dispersion relations having a series of smallerand smaller terms proportional to p n +2 /M nP l ≃ E n +2 /M nP l , with n ≥
1. The astrophysicalimplications of this formalism have been discussed in the literature [10], [34]-[40]. However, inrelating LIV to the observational data on UHECRs, it is useful to use the simpler formalismof Coleman and Glashow. Given the limited energy range of the UHECR data relevant tothe GZK effect, this formalism can later be related to possible Planck scale phenomena andquantum gravity models of various sorts (See Section 6.2).Let us consider the photomeson production process leading to the GZK effect. Nearthreshold, where single pion production dominates, p + γ → p + π. (7)Using the normal Lorentz invariant kinematics, the energy threshold for photomesoninteractions of UHECR protons of initial laboratory energy E with low energy photons ofthe CBR with laboratory energy ω , is determined by the relativistic invariance of the squareof the total four-momentum of the proton-photon system. This relation, together with thethreshold inelasticity relation E π = m/ ( M + m ) E for single pion production, yields thethreshold conditions for head on collisions in the laboratory frame4 ωE = m (2 M + m ) (8)for the proton, and4 ωE π = m (2 M + m ) M + m (9)in terms of the pion energy, where M is the rest mass of the proton and m is the rest massof the pion [17].If LI is broken so that c π > c p , it follows from equations (3), (6) and (9) that thethreshold energy for photomeson is altered because the square of the four-momentum isshifted from its LI form so that the threshold condition in terms of the pion energy becomes k ωE π = m (2 M + m ) M + m + 2 δ πp E π (10)Equation (10) is a quadratic equation with real roots only under the condition δ πp ≤ ω ( M + m ) m (2 M + m ) ≃ ω /m . (11)Defining ω ≡ kT CBR = 2 . × − eV with T CBR = 2 . ± .
02 K, equation (11) canbe rewritten δ πp ≤ . × − ( ω/ω ) . (12) k We assume here that protons and pions are kinematically independent entities. For a treatment of theseparticles as composites of quarks and gluons, see Ref. [41]. earching for New Physics with Ultrahigh Energy Cosmic Rays
4. Kinematics
If LIV occurs and δ πp >
0, photomeson production can only take place for interactions ofCBR photons with energies large enough to satisfy equation (12). This condition, togetherwith equation (10), implies that while photomeson interactions leading to GZK suppressioncan occur for “lower energy” UHE protons interacting with higher energy CBR photonson the Wien tail of the spectrum, other interactions involving higher energy protons andphotons with smaller values of ω will be forbidden. Thus, the observed UHECR spectrummay exhibit the characteristics of GZK suppression near the normal GZK threshold, but theUHECR spectrum can “recover” at higher energies owing to the possibility that photomesoninteractions at higher proton energies may be forbidden. We now consider a more detailedquantitative treatment of this possibility, viz. , GZK coexisting with LIV.The kinematical relations governing photomeson interactions are changed in the presenceof even a small violation of Lorentz invariance. Following equations (3) and (6), we denote E = p + 2 δ a p + m a (13)where δ a is the difference between the MAV for the particle a and the speed of light in thelow momentum limit ( c = 1).The square of the cms energy of particle a is then given by √ s a = q E − p = q δ a p + m a ≥ . (14)Owing to LIV, in the cms the particle will not generally be at rest when p = 0 because v = ∂E∂p = pE . (15)The modified kinematical relations containing LIV have a strong effect on the amount ofenergy transfered from a incoming proton to the pion produced in the subsequent interaction, i.e. , the inelasticity [43, 44]. The total inelasticity, K , is an average of K θ , which dependson the angle between the proton and photon momenta, θ : K = 1 π π Z K θ dθ. (16)The primary effect of LIV on photopion production is a reduction of phase space allowedfor the interaction. This results from the limits on the allowed range of interaction anglesintegrated over in order to obtain the total inelasticity from equation (16). For real-rootsolutions for interactions involving higher energy protons, the range of kinematically allowedangles in equation (16) becomes severely restricted. The modified inelasticity that results isthe key in determining the effects of LIV on photopion production. The inelasticity rapidlydrops for higher incident proton energies.As shown in Ref. [29], in order to modify the effect of photopion production on theUHECR spectrum above the GZK energy we must have δ π > δ p , i.e. , δ πp >
0. We note that earching for New Physics with Ultrahigh Energy Cosmic Rays
9a constraint can be put on δ pγ in the case where δ pγ > E max = m p q / δ pγ . (17)above which protons traveling faster than light will emit light at all frequencies by theprocess of ‘vacuum ˇCerenkov radiation’ [29], [31], [42]. This process occurs rapidly, so thatthe energy of the superluminal protons will rapidly fall back to energy E max . Therefore,because UHECRs, assumed here to be protons, have been observed up to an upper energyof E U ≃
320 EeV [25], it follows that δ pγ ≤ m p E U ≃ × − . (18)Our requirement that δ πp > via the rapid, strong interaction, pion emission process, p → N + π . This process would beallowed by LIV in the case where δ πp is negative, producing a sharp cutoff in the UHECRproton spectrum.The empirical constraint given by equation (18) is independent of any constraint on δ πp . However, we note that if δ π ≃ δ p , no observable modification of the UHECR spectrumoccurs. Therefore, we will assume that δ π > δ p at or near threshold as a requirement forclearly observing a potential LIV signal in the UHECR spectrum. This assumption is alsomade in Ref. [43]. We will thus take δ πp ≡ δ π in the case where δ p is small and positiveas required by eq. (18). Indeed, it can be shown in this case that the dependence of theUHECR spectral shape on the δ πp parameter dominates over that on the δ p parameter [44].Figure 2 shows the calculated proton inelasticity modified by LIV for a value of δ πp = 3 × − as a function of both CBR photon energy and proton energy [44]. Otherchoices for δ πp yield similar plots. The principal result of changing the value of δ πp is tochange the energy at which LIV effects become significant. For a choice of δ πp = 3 × − ,there is no observable effect from LIV for E p less than ∼
200 EeV. Above this energy, theinelasticity precipitously drops as the LIV term in the pion rest energy approaches m π .With this modified inelasticity, the proton energy loss rate by photomeson productionis given by 1 E dEdt = − ω c π γ ¯ h c ∞ Z η dǫ ǫ σ ( ǫ ) K ( ǫ ) ln[1 − e − ǫ/ γω ] (19)where we now use ǫ to designate the energy of the photon in the cms, η is the photonthreshold energy for the interaction in the cms, and σ ( ǫ ) is the total γ -p cross section withcontributions from direct pion production, multipion production, and the ∆ resonance.The corresponding proton attenuation length is given by ℓ = cE/r ( E ), where the energyloss rate r ( E ) ≡ ( dE/dt ). This attenuation length is plotted in Figure 3 for various valuesof δ πp along with the unmodified pair production attenuation length from pair productioninteractions, p + γ CBR → e + + e − . earching for New Physics with Ultrahigh Energy Cosmic Rays Figure 2.
The calculated proton inelasticity modified by LIV for δ πp = 3 × − as afunction of CBR photon energy and proton energy [44]. @ eV D L og { @ M p c D Figure 3.
The calculated proton attenuation lengths as a function proton energy modifiedby LIV for various values of δ πp (solid lines), shown with the attenuation length for pairproduction unmodified by LIV (dashed lines). From top to bottom, the curves are for δ πp = 1 × − , × − , × − , × − , × − , earching for New Physics with Ultrahigh Energy Cosmic Rays
5. UHECR Spectra with LIV and Comparison with Present Observations
Let us now calculate the modification of the UHECR spectrum produced by a very smallamount of LIV. We perform an analytic calculation in order to determine the shape of themodified spectrum. It can be demonstrated that there is little difference between the resultsof using an analytic calculation vs. a Monte Carlo calculation ( e.g. , see Ref. [45]). In orderto take account of the probable redshift evolution of UHECR production in astronomicalsources, we take account of the following considerations:( i ) The CBR photon number density increases as (1 + z ) and the CBR photon energiesincrease linearly with (1 + z ). The corresponding energy loss for protons at any redshift z isthus given by r γp ( E, z ) = (1 + z ) r [(1 + z ) E ] . (20)( ii ) We assume that the average UHECR volume emissivity is of the energy and redshiftdependent form given by q ( E i , z ) = K ( z ) E − Γ i where E i is the initial energy of the protonat the source and Γ = 2 .
55. For the source evolution, we assume K ( z ) ∝ (1 + z ) . with z ≤ . K ( z ) is roughly proportional to the empirically determined z -dependenceof the star formation rate. K ( z = 0) and Γ are normalized fit the data below the GZK energy.Using these assumptions, Scully and Stecker [44] have calculated the effect of LIV on theUHECR spectrum. The results are actually insensitive to the assumed redshift dependencebecause evolution does not affect the shape of the UHECR spectrum near the GZK cutoffenergy [46, 47]. At higher energies where the attenuation length may again become largeowing to an LIV effect, the effect of evolution turns out to be less than 10%. The curvescalculated in Ref. [44] assuming various values of δ πp , are shown in Figure 4 along withthe Auger data from Ref. [26]. They show that even a very small amount of LIV that isconsistent with both a GZK effect and with the present UHECR data can lead to a “recovery”of the UHECR spectrum at higher energies.5.1. Non-Protonic UHECR
Throughout this paper, we have made the assumption that the highest energy cosmic rays, i.e. , those above 100 EeV, are protons. The composition of these primary particles ispresently unknown. The highest energy events for which composition measurements havebeen attempted are in the range between 40 and 50 EeV, and the composition of these eventsis uncertain [48]-[50].We note that in the case where the UHECRs with total energy above ∼
100 EeV are notprotons, both the photomeson threshold and the LIV effects are moved to higher energiesbecause (i) the threshold is dependent on γ ∝ E/A , where A is the atomic weight of earching for New Physics with Ultrahigh Energy Cosmic Rays @ eV D L og J H E L ´ E @ e V m - s - s r - D Figure 4.
Comparison of the latest Auger data with calculated spectra for various valuesof δ πp , taking δ p = 0 (see text). From top to bottom, the curves give the predicted spectrafor δ πp = 1 × − , × − , . × − , × − , × − , × − , × − , the UHECR [17], and (ii) it follows from equation (3) that the LIV effect depends on theindividual nucleon momentum p N → E/A. (21)In the case of photodisintegration, LIV effects can play a role. We note that for singlenucleon photodisintegration of iron nuclei, the threshold is at a higher energy than for theGZK effect [51, 21]. For He nuclei, on the other hand, the threshold energy is lower thanthe GZK energy [51].
6. Constraints on LIV δ πp It has been suggested that a small amount of Lorentz invariance violation (LIV) could turnoff photomeson interactions of ultrahigh energy cosmic rays (UHECRs) with photons of thecosmic background radiation and thereby eliminate the resulting sharp steepening in thespectrum of the highest energy CRs predicted by Greisen, Zatsepin and Kuzmin (GZK).Recent measurements of the UHECR spectrum reported by the
HiRes [28] and
Auger [26]collaborations, however, appear to indicate the presence of a GZK effect. earching for New Physics with Ultrahigh Energy Cosmic Rays @ eV D L og J H E L ´ E @ e V m - s - s r - D Figure 5.
Comparison of the latest Auger data with the stated UHECR energiesincreased by 25% (see text) shown with the calculated spectra for various values of δ πp , taking δ p = 0. From top to bottom, the curves give the predicted spectra for δ πp = 1 × − , × − , . × − , × − , × − , × − , × − , A true determination of the implications of these recent measurements for the searchfor Lorentz invariance violation at ultrahigh energies requires a detailed analysis of thespectral features produced by modifications of the kinematical relationships caused by LIVat ultrahigh energies. Scully and Stecker [44] calculated modified UHECR spectra forvarious values of the Coleman-Glashow parameter, δ πp , defined as the difference betweenthe maximum attainable velocities of the pion and the proton produced by LIV. They thencompared the results with the experimental UHECR data.We have updated these results using the very latest Auger data from the procedings ofthe 2009 International Cosmic Ray Conference [26],[27]. This update is shown in Figure 4.The amount of presently observed GZK suppression in the UHECR data is consistent withthe possible existence of a small amount of LIV. In order to quantify this, we determinethe value of δ πp that results in the smallest χ for the modeled UHECR spectral fit usingthe observational data from Auger [26] above the GZK energy. The best fit LIV parameterfound was in the range given by δ πp = 3 . +1 . − . × − , corresponding to an upper limit on δ πp of 4 . × − . ¶ This result, as it stands, is slightly more constraining than that givenin Ref. [44]. However, we note that the overall fit of the data to the theoretically expectedspectrum is somewhat imperfect, even below the GZK energy and even for the case of no ¶ The
HiRes data [28] do not reach a high enough energy to further restrict LIV. earching for New Physics with Ultrahigh Energy Cosmic Rays @ eV D L og J H E L ´ E @ e V m - s - s r - D Figure 6.
Comparison of the latest Auger data with calculated spectra for various negativevalues of δ p , taking δ π = 0. From top to bottom, the curves give the predicted spectra for δ p = − × − , − × − , − × − , − × − . We also show by a dashed curve thecase where δ p = − × − and δ π = − × − . LIV. It appears that the
Auger spectrum seems to steepen even below the GZK energy. Asa conjecture, we have taken the liberty of assuming that the derived energy may be too lowby about 25%, within the uncertainty of both systematic-plus statistical error given for theenergy determination. By increasing the derived UHECR energies by 25%, we arrive at theplot shown in Figure 5, again shown with the theoretical curves. In Figure 5 one sees betteragreement between the theoretical curves and the shifted data. The constraint on LIV wouldbe only slightly reduced if this shift is assumed.The results for LIV modified spectra given in Ref. [44] were calculated under theassumption that δ p ≡ δ pγ = 0. It follows from equation (14) that if δ p is slightly negativethen the spectra are additionally modified because of the reduced cms energy of the protonfor a given lab momentum. This affects the photomeson interaction rate in a different andstronger way than for the δ pγ = 0 case shown in Fig. 4. Here, the reduced cms protonenergy results in a reduction of the phase space allowed for the interactions when √ s p givenby equation (14) is near zero.The results for negative δ p are shown in Figure 6. They lie within the range of constraintson δ πp given above. However, it is clear that even a relatively small negative value for δ p hasa stronger LIV effect on the UHECR spectrum than a positive value of δ π . We also show acase where δ p and δ π are both negative (dashed line in the figure). The dashed curve showsthat the same δ p produces an almost identical effect on the spectrum in both cases, again earching for New Physics with Ultrahigh Energy Cosmic Rays @ eV D L og J H E L ´ E @ e V m - s - s r - D Figure 7.
Comparison of the latest Auger data with calculated spectra for δ πp = 4 . × − and for δ p = 0 and 0 . × − as discussed in the text. In the later case the cutoff fromthe vacuum ˇCerenkov effect is apparent. demonstrating that the negative δ p parameter gives the dominant LIV effect.We also present here, for comparison, the spectrum for a slightly positive δ pγ . Figure7 shows two curves for δ πp = 5 × − . The spectrum with a vacuum ˇCerenkov radiationcutoff at 300 EeV is for δ pγ = 0 . × − (see equation (18)). The other curve assumes δ pγ = 0 as in Figure 4. An effective field theory approximation for possible LIV effects induced by Planck-scalesuppressed quantum gravity for E ≪ M P l was considered in Ref. [40]. These authorsexplored the case where a perturbation to the energy-momentum dispersion relation for freeparticles would be produced by a CPT-even dimension six operator suppressed by a termproportional to M − P l . The resulting dispersion relation for a particle of type a is E a = p a + m a + η a p M P l ! (22)In order to explore the implications of our constraints for quantum gravity, we willequate the perturbative terms in the dispersion relation given by our equation (13), for bothprotons and pions, with the equivalent dimension six dispersion relations given by equation(22). We note that the perturbative term in equation (22) has an energy dependence, whereasour dimension four case does not. However, since we are only comparing with UHECR data earching for New Physics with Ultrahigh Energy Cosmic Rays E f ∼
100 EeV, we will make theidentification at that energy.Using this identification, we find that in most cases an LIV constraint of δ πp < . × − at a proton fiducial energy of E f ∼
100 EeV indirectly implies a powerful limit on therepresentation of quantum gravity effects in an effective field theory formalism with Plancksuppressed dimension six operators. Equating the perturbative terms in both the protonand pion dispersion relations2 δ πp ≃ ( η π − η p ) (cid:18) . E f M P l (cid:19) , (23)where we have adopted the terminology of Ref. [40] and we have taken the pion fiducialenergy to be ∼ . E f , as at the ∆ resonance [17]. Since we require δ πp > δ p > δ π > δ p , which is the assumption made in Refs. [43] and [44]. Equation(23) also indicates that LIV by dimension six operators is suppressed by a factor of at least O (10 − M − P l ), except in the unlikely case that η π − η p ≃
0. This suppression is over andabove that of any dimension four terms in the dispersion relation as we have considered here.These results are in agreement with the conclusions of Ref. [40] who also find a suppressionof O (10 − M − P l ), except with the equivalent loophole. We note that in Ref. [40] a series ofMonte Carlo runs are used in order to obtain their results. It can thus be concluded that aneffective field theory representation of quantum gravity with dimension six operators thatsuppresses LIV by only a factor of M P l is effectively ruled out by the UHECR observations,as concluded in Ref. [40].
7. Beyond Constraints: Seeking LIV
As we have seen (see Figure 4), even a very small amount of LIV that is consistent with botha GZK effect and with the present UHECR data can lead to a “recovery” of the primaryUHECR spectrum at higher energies. This is the clearest and the most sensitive evidence ofan LIV signature. The “recovery” effect has also been deduced in Refs. [40] and [52] + . Inorder to find it (if it exists) three conditions must exist: ( i ) sensitive enough detectors needto be built, ( ii ) a primary UHECR spectrum that extends to high enough energies ( ∼ iii ) one much be able to distinguish the LIV signature from otherpossible effects. In order to meet our second condition, we require the existence of powerful cosmic rayaccelerators, the so-called zevatrons (1000 EeV = 1 ZeV) . In this “bottom up” scenario + In Ref. [52], a recovery effect is also claimed for high proton energies in the case when δ πp <
0. However,we have noted that the ‘quasi-vacuum ˇCerenkov radiation’ of pions by protons in this case will cut off theproton spectrum and no “recovery” effect will occur. earching for New Physics with Ultrahigh Energy Cosmic Rays ∼
300 EeV [25]. The most widelyconsidered acceleration mechanism is shock acceleration, particularly in the lobes of powerfulradio galaxies ( e.g. , Refs. [53] – [55].) Blanford [54] discusses the problems associated withthis “conventional” mechanism of accelerating particles to the highest observed energies.Other acceleration mechanisms have been proposed. In particular, it has recently beenargued that the plasma wakefield acceleration mechanism, operating within relativistic AGNjets, is capable of accelerating particles to energies ∼ Our signature signal of LIV is a “recovery” of theprimary UHECR spectrum at higher energies (see Figure 4). Such an LIV signal must bedistinguished from the presence of a higher energy component in the UHECR spectrumpredicted to be produced by so-called “top-down” models. The top-down scenarios invokethe decay or annihilation of supermassive particles or topological or quantum remnants ofthe very early universe usually associated with some grand unification energy scale. Suchprocesses result in the production of a high ratio of pions to nucleons from the resultingQCD fragmentation process (see , e.g. , [21] for a review). Owing to QCD fragmentation,top-down scenarios predict relatively large fluxes of UHE photons and neutrinos as comparedto nucleons, as well as a significant diffuse GeV background flux that could be searched forby the
Fermi γ -ray space telescope.A higher energy UHECR component arising from top-down models can indeed bedistinguished from the LIV effect. Contrary to the predictions of relatively copious pionproduction in the top-down scenario, the LIV effect cuts off UHE pion production at thehigher energies and consequent UHE neutrino and photon production from UHE pion decay.We also note that LIV would therefore not produce a GeV photon flux.In this regard, we note that the Pierre Auger Observatory collaboration has providedobservational upper limits on the UHE photon flux [57, 58] that have already disfavoredtop-down models. The upper limits from the Auger array indicate that UHE photons makeup at best only a small percentage of the total UHE flux. This contradicts predictions oftop-down models that the flux of UHE photons should be larger than that of UHE protons.Upper limits on the UHE neutrino flux from
ANITA also strongly disfavor top-down models[59], [60].
It is possible that the apparent modified GZKsuppression in the data may be related to an overdensity of nearby sources related to alocal supergalactic enhancement [17]. However, at this point in time, no clear correlation of earching for New Physics with Ultrahigh Energy Cosmic Rays ∗ More and better data will berequired in order to resolve this question. An LIV effect can be distinguished from a posssiblelocal source enhancement by looking for UHECRs at energies above ∼
200 EeV, as can beseen from Figure 4. This is because the small amount of LIV that fits the observationalUHECR spectra can lead to the signature recovery of the cosmic ray flux at higher energiesthan presently observed. Such a recovery is not expected in the case of a local overdensity.Searching for such a recovery effect will require obtaining a future data set containing amuch higher number of UHECR air shower events.
8. Obtaining UHECR Data at Higher Energies
We now turn to examining the various techniques that can be used in the future in orderto look for a signal of LIV using UHECR observations. As can be seen from the precedingdiscussion, observations of higher energy UHECRs with much better statistics than presentlyobtained are needed in order to search for the effects of miniscule Lorentz invariance violationon the UHECR spectrum.
In the future, such an increased number of events may be obtained. The
Auger collaborationhas proposed to build an “
Auger North” array that would be seven times larger than thepresent southern hemisphere Auger array ( ). Further into the future, space-based telescopes designed to look downward at large areasof the Earth’s atmosphere as a sensitive detector system for giant air-showers caused bytrans-GZK cosmic rays. We look forward to these developments that may have importantimplications for fundamental high energy physics.Two future potential spaced-based missions have been proposed to extend our knowledgeof UHECRs to higher energies. One is
JEM-EUSO (the Extreme Universe SpaceObservatory) [12], a one-satellite telescope mission proposed to be placed on the JapaneseExperiment Module (JEM) on the International Space Station. The other is
OWL (OrbitingWide-angle Light Collectors) [13], a two satellite mission for stereo viewing, proposed fora future free-flyer mission. Such orbiting space-based telescopes with UV sensitive cameraswill have wide fields-of-view (FOVs) in order to observe and use large volumes of the Earth’satmosphere as a detecting medium. They will thus trace the atmospheric fluorescence trailsof numbers of giant air showers produced by ultrahigh energy cosmic rays and neutrinos.Their large FOVs will allow the detection of the rare giant air showers with energies higher ∗ A correlation with nearby AGN was hinted at in earlier Auger data [61]. However, the HiRes group hasfound no significant correlation [62] and no correlation has now been found in the more recent Auger datawith an increased number of events (Westerhoff, private communication). earching for New Physics with Ultrahigh Energy Cosmic Rays
Auger . Such missions willthus potentially open up a new window on physics at the highest possible observed energies. has beenselected as one of two candidate missions for the second utilization of the JapaneseExperiment Module on the International Space Station. It may be launched in 2013 bya Japanese heavy lift rocket. It will employ a double plastic Fresnel lens system telescopewith a 30 ◦ FOV and will help to advance the technology of such missions. Further into thefuture, a proposed “Super EUSO” mission is in the preliminary planning stage.
The
OWL (Orbiting Wide-field Light-collectors), aproposed dual satellite mission to have a larger total aperture than
JEM-EUSO , hasbeen designed to be sensitive enough to obtain data on higher energy UHECRs and onultrahigh energy neutrinos. Its detecting area and FOV will be large enough to provide theevent statistics and extended energy range that are crucial to addressing these issues. Toaccomplish this,
OWL will also make use of the Earth’s atmosphere as a huge “calorimeter”to make stereoscopic measurements of the atmospheric UV fluorescence produced by airshower particles.
OWL is thus proposed to consist of a pair of satellites placed in tandemin a low inclination, medium altitude orbit. The
OWL telescopes will point down at theEarth and will together point at a section of atmosphere about the size of the state of Texas( ∼ × km ).The baseline OWL instrument, shown in Figure 8 (left), is a large f/1 Schmidt camerawith a 45 ◦ full FOV and a 3 meter entrance aperture. The entrance aperture will contain aSchmidt corrector. The deployable primary mirror has a 7 meter diameter. OWL would benormally operated in stereo mode and the two “OWL eye” instruments will view a commonvolume of atmosphere.The satellites can be launched together on a Delta rocket into a proposed 1000 kmcircular orbit with an inclination of 10 ◦ . Figure 8 (right) shows both satellites stowed forlaunch as well as a depiction of one of the Schmidt telescopes. Stereoscopic observationresolves spatial ambiguities and allows determination of corrections for the effects of clouds.In stereo, fast timing provides supplementary information to reduce systematics and improvethe resolution of the arrival direction of the UHECR. By using stereo, differences inatmospheric absorption or scattering of the UV light can be determined. Detector missionssuch as the proposed OWL mission can provide the statistics of UHECR events that wouldbe needed in the 100 to 1000 EeV energy range to search for the effects of a very smallamount of Lorentz invariance violation at the highest energies.We look forward to such future detector developments. As we have seen, they may haveimportant implications for fundamental high energy physics as well as the astrophysics ofpowerful extragalactic “zevatrons”. earching for New Physics with Ultrahigh Energy Cosmic Rays Figure 8.
Left: Schematic of the Schmidt optics that form an
OWL “eye” in the deployedconfiguration. The spacecraft bus, light shield, and shutter are not shown. Right: Schematicof the stowed
OWL satellites in the launch vehicle.
Acknowledgments
We would like to thank John F. Krizmanic for helpful discussions. We would also like tothank Alan Watson for bringing the new
Auger data to our attention.
References [1] Evans L 2007
New J. Phys. Mitt. Thermodynamics , Folg. 5[3] Mattingly D 2005
Living Rev. Relativity Phys. Lett. B , 412[5] Smolin L e-print arXiv:0808.3765.[6] Henson J 2009 e-print arXiv:0901.4009[7] Sato H and Tati T 1998
Prog. Theor. Phys. Sov. J. Nucl. Phys. et al. Nature
Phys. Rev. Letters et al. (the Fermi Collaboration) 2009, Science , 1688[12] Ebusaki T et al.
Proc. 30th Intl. Cosmic Ray Conf. (Merida,Mexico) (ed. R. Caballero et al. ) earching for New Physics with Ultrahigh Energy Cosmic Rays Figure 9.
Two OWL satellites in low-Earth orbit observing the fluorescent track of a giantair shower. The shaded cones illustrate the field-of-view for each satellite.[13] Stecker F W et al.
Nucl. Phys. B
433 e-print astro-ph/0408162[14] Linsley, J 1963
Phys. Rev. Letters Phys. Rev. Letters Zh. Eks. Teor. Fiz., Pis’ma Red. Phys. Rev. Letters Astrophys. and Space Sci. Phys. Rev. D Astropart. Phys. J. Phys. G R47[22] Scully S T and Stecker F W 2002
Astropart. Phys. Nature et al.
Astropart. Phys. (2003) 447[25] Bird D J et al. Astrophys. J.
Proc. 31st Intl. Cosmic Ray Conf.
L´od´z e-printarXiv:0906.2189[27] [28] Abbasi R U et al.
Phys. Rev. Letters
Phys. Rev.
D59
Astropart. Phys. Astropart. Phys. Phys. Rev. Letters Phys. Rev. D Astropart. Phys. Phys. Rev. D Phys. Rev. Letters Phys.Rev.Lett. , 021102.[39] Maccione L, and Liberati S 2008,
J. Cosmology and Astropart. Phys. earching for New Physics with Ultrahigh Energy Cosmic Rays [40] Maccione L A, Taylor A M, Mattingly D M and Liberati S 2009, J. Cosmology and Astropart. Phys.
Phys. Rev. D Phys. Rev. Letters Phys. Rev.
D67
Astropart. Phys. Phys. Rev. D in press, e-print arXiv:0811.0396[46] Berezinsky V I and Grigor’eva S I 1988
Astron. Astrophys.
Astropart. Phys. et al. (the Auger Collaboration) 2009, e-print arXiv:0902.3787[49] Belz J (for the HiRes Collaboration) 2009, Nucl. Phys. B
Proc. Suppl. , 5[50] Ulrich R et al. (the Auger Collaboration) 2009, in
Proc. 31st Intl. Cosmic Ray Conf.
L´od´z e-printarXiv:0906.0418[51] Stecker F W 1969
Phys. Rev. et al.
Phys. Rev.
D79
Astron. Astrophys. Phys. Scr. bf T85 191; e-print arXiv:astro-ph/9906026[55] Dermer C D et al. et al.
Phys. Rev. Letters et al. (the Auger Collaboration) 2008
Astropart. Phys. et al. Astropart. Phys. et al. Phys. Rev. Letters et al. et al. Astropart. Phys. et al.et al.