# Gamma-ray and Cosmic-ray Tests of Lorentz Invariance Violation and Quantum
Gravity Models and Their Implications

Floyd W. Stecker

###### Abstract

The topic of Lorentz invariance violation (LIV) is a fundamental question in
physics that has taken on particular interest in theoretical explorations of
quantum gravity scenarios. I discuss various $\gamma$-ray observations that
give limits on predicted potential effects of Lorentz invariance violation.
Among these are spectral data from ground based observations of the multi-TeV
$\gamma$-rays from nearby AGN, INTEGRAL detections of polarized soft
$\gamma$-rays from the vicinity of the Crab pulsar, Fermi Gamma Ray Space
Telescope studies of photon propagation timing from $\gamma$-ray bursts, and
Auger data on the spectrum of ultrahigh energy cosmic rays. These results can
be used to seriously constrain or rule out some models involving Planck scale
physics. Possible implications of these limits for quantum gravity and Planck
scale physics will be discussed.

###### Keywords:

quantum gravity

###### :

04.60Bc

##  1 Introduction

It has been the major goal of particle physics to discover a theoretical
framework for unifying gravity with the other three known forces, viz.,
electromagnetism, and the weak and strong nuclear forces. Such a theory must
be compatible with quantum theory at very small scales corrsponding to very
high energies. Even the possibly less ambitious goal of reconciling general
relativity with quantum theory has been elusive and may require new concepts
to accomplish.

There has been a particular interest in the possibility that a quantum gravity
theories will lead to Lorentz invariance violation (LIV) at the Planck scale,
$\lambda_{Pl}=\sqrt{G\hbar/c^{3}}\sim 1.6\times 10^{-35}$ m. This scale
corresponds to a mass (energy) scale of $M_{Pl}=\hbar/(\lambda_{Pl}c)\sim
1.2\times 10^{19}$ GeV/c2. It is at the Planck scale where quantum effects are
expected to play a key role in determining the effective nature of space-time
that emerges as general relativity in the classical continuum limit. The idea
that Lorentz invariance (LI) may indeed be only approximate has been explored
within the context of a wide variety of suggested Planck-scale physics
scenarios. These include the concepts of deformed relativity, loop quantum
gravity, non-commutative geometry, spin foam models, and some string theory (M
theory) models. Such theoretical explorations and their possible consequences,
such as observable modifications in the energy-momentum dispersion relations
for free particles and photons, have been discussed under the general heading
of “Planck scale phenomenology”. There is an extensive literature on this
subject. (See ma05  for a review; some recent references are Refs. el08  –
he09 . For a non-technical treatment of the present basic approaches to a
quantum gravity theory, see Ref. smolin ). One should keep in mind that in a
context that is separate from quantum gravity considerations, it is important
to test LI for its own sake co98 ; cg99 . LIV gratia LIV. The significance of
such an approach is evident when one considers the unexpected discoveries of
the violation of $P$ and $CP$ symmetries. In fact, it has been shown that a
violation of $CPT$ would imply LIV gr02

We will consider here some of the consequent searches for such effects using
high energy astrophysics observations, particularly observations of high
energy cosmic $\gamma$-rays and ultrahigh energy cosmic rays.

##  2 LIV Perturbations

We know that Lorentz invariance has been well validated in particle physics;
indeed, it plays an essential role in designing machines such as the new LHC
(Large Hadron Collider). Thus, any LIV extant at accelerator energies (“low
energies”) must be extremely small. This consideration is reflected by adding
small Lorentz-violating terms in the free particle Lagrangian. Such terms can
be postulated to be independent of quantum gravity theory, e.g., Refs. co98 ;
cg99 . Alternatively, it can be assumed that the terms are small because they
are suppressed by one or more powers of $p/M_{Pl}$ (with the usual convention
that $c=1$.) In the latter case, in the context of effective field theory
(EFT), such terms are assumed to approximate the effects of quantum gravity at
“low energies” when $p\ll M_{Pl}$.

One result of such assumptions is a modification of the dispersion relation
that relates the energy and momentum of a free particle or photon. This, in
turn, can lead to a maximmum attainable velocity (MAV) of a particle different
from $c$ or a variation of the velocity of a photon in vacuo with photon
energy. Both effects are clear violations of relativity theory. Such
modifications of kinematics can result in changes in threshold energies for
particle interactions, suppression of particle interactions and decays, or
allowance of particle interactions and decays that are kinematically forbidden
by Lorentz invariance cg99 .

A simple formulation for breaking LI by a small first order perturbation in
the electromagnetic Lagrangian which leads to a renormalizable treatment has
been given by Coleman and Glashow cg99 . The small perturbative noninvariant
terms are both rotationally and translationally invariant in a preferred
reference frame which one can assume to be the frame in which the cosmic
background radiation is isotropic. These terms are also taken to be invariant
under $SU(3)\otimes SU(2)\otimes U(1)$ gauge transformations in the standard
model.

Using the formalism of Ref. cg99 , we denote the MAV of a particle of type $i$
by $c_{i}$, a quantity which is not necessarily equal to $c\equiv 1$, the low
energy in vacua velocity of light. We further define the difference
$c_{i}-c_{j}\equiv\delta_{ij}$. These definitions can be generalized and can
be used to discuss the physics implications of cosmic-ray and cosmic
$\gamma$-ray observations sg01--st09.

##  3 Electroweak Interactions

In general then, $c_{e}\neq c_{\gamma}$. The physical consequences of such a
violation of LI depend on the sign of the difference between these two MAVs.
Defining

$c_{e}\equiv c_{\gamma}(1+\delta)~{},~{}~{}~{}~{}0<|\delta|\ll 1\;,$ (1)

one can consider the two cases of positive and negative values of $\delta$
separately cg99 ; sg01 .

Case I: If $c_{e}<c_{\gamma}$ ($\delta<0$), the decay of a photon into an
electron-positron pair is kinematically allowed for photons with energies
exceeding

$E_{\rm max}=m_{e}\,\sqrt{2/|\delta|}\;.$ (2)

The decay would take place rapidly, so that photons with energies exceeding
$E_{\rm max}$ could not be observed either in the laboratory or as cosmic
rays. From the fact that photons have been observed with energies
$E_{\gamma}\geq$ 50 TeV from the Crab nebula, one deduces for this case that
$E_{\rm max}\geq 50\;$TeV, or that -$\delta<2\times 10^{-16}$.

Case II: For this possibility, where $c_{e}>c_{\gamma}$ ($\delta>0$),
electrons become superluminal if their energies exceed $E_{\rm max}/2$.
Electrons traveling faster than light will emit light at all frequencies by a
process of ‘vacuum Čerenkov radiation.’ This process occurs rapidly, so that
superluminal electron energies quickly approach $E_{\rm max}/2$. However,
because electrons have been seen in the cosmic radiation with energies up to
$\sim\,$2 TeV, it follows that $E_{\rm max}\geq 2$ TeV, which leads to an
upper limit on $\delta$ for this case of $3\times 10^{-14}$. Note that this
limit is two orders of magnitude weaker than the limit obtained for Case I.
However, this limit can be considerably improved by considering constraints
obtained from studying the $\gamma$-ray spectra of active galaxies sg01 .

###  3.1 Constraints on LIV from AGN Spectra

A constraint on $\delta$ for $\delta>0$ follows from a change in the threshold
energy for the pair production process $\gamma+\gamma\rightarrow e^{+}+e^{-}$.
This follows from the fact that the square of the four-momentum is changed to
give the threshold condition

$2\epsilon
E_{\gamma}(1-cos\theta)~{}-~{}2E_{\gamma}^{2}\delta~{}\geq~{}4m_{e}^{2},$ (3)

where $\epsilon$ is the energy of the low energy photon and $\theta$ is the
angle between the two photons. The second term on the left-hand-side comes
from the fact that $c_{\gamma}=\partial E_{\gamma}/\partial p_{\gamma}$. It
follows that the condition for a significant increase in the energy threshold
for pair production is $E_{\gamma}\delta/2$ $\geq$ $m_{e}^{2}/E_{\gamma}$, or
equivalently, $\delta\geq{2m_{e}^{2}/E_{\gamma}^{2}}$. The observed
$\gamma$-ray spectrum of the active galaxies Mkn 501 and Mkn 421 while flaring
ah01  exhibited the high energy absorption expected from $\gamma$-ray
annihilation by extragalactic pair-production interactions with extragalactic
infrared photons ds02 ; ko03 . This led Stecker and Glashow sg01  to point out
that the Mkn 501 spectrum presents evidence for pair-production with no
indication of LIV up to a photon energy of $\sim\,$20 TeV and to thereby place
a quantitative constraint on LIV given by
$\delta<2m_{e}^{2}/E_{\gamma}^{2}\simeq 10^{-15}$.

##  4 Gamma-ray Constraints on Quantum Gravity and Extra Dimension Models

As previously mentioned, LIV has been proposed to be a consequence of quantum
gravity physics at the Planck scale ga95 ; al02 . In models involving large
extra dimensions, the energy scale at which gravity becomes strong can occur
at a quantum gravity scale, $M_{QG}<<M_{Pl}$, even approaching a TeV ar98 . In
the most commonly considered case, the usual relativistic dispersion relations
between energy and momentum of the photon and the electron are modified al02 ;
ac98  by a term of order $p^{3}/M_{QG}$.

Generalizing the LIV parameter $\delta$ from equation (1) to an energy
dependent form, we find

$\delta~{}\equiv~{}{\partial E_{e}\over{\partial p_{e}}}~{}-~{}{\partial
E_{\gamma}\over{\partial
p_{\gamma}}}~{}\simeq~{}{E_{\gamma}\over{M_{QG}}}~{}-~{}{m_{e}^{2}\over{2E_{e}^{2}}}~{}-~{}{E_{e}\over{M_{QG}}}.$
(4)

It follows that the threshold condition for pair production given by equation
(3) implies that $M_{QG}~{}\geq~{}E_{\gamma}^{3}/8m_{e}^{2}.$ Since pair
production occurs for energies of at least 20 TeV, we find a constraint on the
quantum gravity scale st03  $M_{QG}\geq 0.3M_{Pl}$. This constraint
contradicts the predictions of some proposed quantum gravity models involving
large extra dimensions and smaller effective Planck masses. In a variant model
of Ref. el04 , the photon dispersion relation is changed, but not that of the
electrons. In this case, we find the even stronger constraint $M_{QG}\geq
0.6M_{Pl}$.

##  5 Energy Dependent Photon Delays from GRBs and Tests of Lorentz Invariance
Violation

One possible manifestation of Lorentz invariance violation, from Planck scale
physics produced by quantum gravity effects, is a change in the energy-
momentum dispersion relation of a free particle or a photon. If this results
from the linear Planck-supressed term as in equation (4) above, this results
in a photon velocity retardation that is of first order in $E_{\gamma}/M_{QG}$
ac98 ; el00 . In a $\Lambda CDM$ cosmology, where present observational data
indicate that $\Omega_{\Lambda}\simeq 0.7$ and $\Omega_{m}\simeq 0.3$, the
resulting difference in the propagation times of two photons having an energy
difference $\Delta E_{\gamma}$ from a $\gamma$-ray burst (GRB) at a redshift
$z$ will be

$\Delta t_{LIV}=H_{0}^{-1}{{\Delta E_{\gamma}}\over
M_{QG}}{\int_{0}^{z}}{{dz^{\prime}(1+z^{\prime})}\over{\sqrt{\Omega_{\Lambda}+\Omega_{m}(1+z^{\prime})^{3}}}}$
(5)

for a photon dispersion of the form $c_{\gamma}=c(1-E_{\gamma}/M_{QG}$), with
$c$ being the usual low energy velocity of light ja08 . In other words,
$\delta$, as defined earlier, is given by $-E_{\gamma}/M_{QG}$.

The Fermi Gamma-ray Space Telescope, (see Figure 1), with its $\gamma$-ray
Burst Monitors (GBM) covers an energy range from 8 keV to 40 MeV and its Large
Area Telescope (LAT) covers an energy range from 20 MeV to $>300$ GeV. 111See
paper the of Silvia Rainò, these proceedings. It can observe and study both
GRBs and flares from active galactic nuclei over a large range of both energy
and distance. This was the case with the GRB 090510, a short burst at a
cosmological distance corresponding to a redshift of 0.9 that produced photons
with energies extending from the X-ray range to a $\gamma$-ray of energy
$\sim$ 31 GeV. This burst was therefore a perfect subject for the application
of equation (5). Fermi observations of GRB090510 have yielded the best
constraint on any first order retardation of photon velocity with energy
$\Delta t\propto(E/M_{QG})$. This result would require a value of
$M_{QG}\mathrel{\raise 2.15277pt\hbox{$>$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}1.2M_{Pl}$ Fermi2009 222See also the paper of Francesco
de Palma, these proceedings. In large extra dimension scenarios, one can have
effective Planck masses smaller than $1.22\times 10^{19}$ GeV, whereas in most
QG scenarios, one expects that the minimum size of space-time quanta to be
$\lambda_{Pl}$. This implies a value for $M_{QG}\mathrel{\raise
1.29167pt\hbox{$<$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}M_{Pl}$ in all
cases.

In particular, we note the string theory inspired model of Ref. el08 . This
model invisions space-time as a gas of D-particles in a higher dimensional
bulk where the observable universe is a D3 brane. The photon is represented as
an open string that interacts with the D-particles, resulting a retardation
$\propto E_{\gamma}/M_{QG}$. The new Fermi data appear to rule out this model
as well as other models that predict such a retardation.

The dispersion effect will be smaller if the dispersion relation has a
quadratic dependence on $E_{\gamma}/M_{QG}$ as suggested by effective field
theory considerations my03 ; ja04 . This will obviate the limits on $M_{QG}$
given above. These considerations also lead to the prediction of vacuum
birefringence (see next section).

Figure 1: Schematic of the Fermi satellite, launched in June of 2008\. The LAT
is located at the top (yellow area) and the GBM array is located directly
below.

##  6 Looking for Birefringence Effects from Quantum Gravity

A possible model for quantizing space-time which has been actively
investigated is loop quantum gravity (see the review given in Ref. pe04  and
references therein.) A signature of this model is that the quantum nature of
space-time can produce a vacuum birefringence effect. (See also the EFT
treatment in Ref. my03 .) This is because electromagnetic waves of opposite
circular polarizations will propagate with different velocities, which leads
to a rotation of linear polarization direction through the angle

$\theta(t)=\left[\omega_{+}(k)-\omega_{-}(k)\right]t/2=\xi k^{2}t/2M_{Pl}$ (6)

for a plane wave with wave-vector $k$ ga99 . Again, for simple Planck-
suppressed LIV, we would expect that $\xi\simeq 1$.

Some astrophysical sources emit highly polarized radiation. It can be seen
from equation (6) that the rotation angle is reduced by the large value of the
Planck mass. However, the small rotations given by equation (6) can add up
over astronomical or cosmological distances to erase the polarization of the
source emission. Therefore, if polarization is seen in a distant source, it
puts constraints on the parameter $\xi$. Observations of polarized radiation
from distant sources can therefore be used to place an upper bound on $\xi$.

Equation (6) indicates that the higher the wave number $|k|$, the stronger the
rotation effect will be. Thus, the depolarizing effect of space-time induced
birefringence will be most pronounced in the $\gamma$-ray energy range. It can
also be seen that the this effect grows linearly with propoagation time.

The difference in rotation angles for wave-vectors $k_{1}$ and $k_{2}$ is

$\Delta\theta=\xi(k_{2}^{2}-k_{1}^{2})d/2M_{Pl},$ (7)

replacing the time $t$ by the distance from the source to the detector,
denoted by $d$.

The best secure bound on this effect, $|\xi|\mathrel{\raise
1.29167pt\hbox{$<$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}10^{-9}$, was
obtained using the observed 10% polarized soft $\gamma$-ray emission from the
region of the Crab Nebula ma08 .

Clearly, the best tests of birefringence would be to measure the polarization
of $\gamma$-rays from GRBs. We note that linear polarization in X-ray flares
from GRBs has been predicted fa05 . Most $\gamma$-ray bursts have redshifts in
the range 1-2 corresponding to distances of greater than a Gpc. Should
polarzation be detected from a burst at distance $d$, this would place a limit
on $|\xi|$ of

$|\xi|\mathrel{\raise 1.29167pt\hbox{$<$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}5\times 10^{-15}/d_{0.5}$ (8)

where $d_{0.5}$ is the distance to the burst in units of 0.5 Gpc ja04 .
Detectors that are dedicated to polarization measurements in the X-ray and
$\gamma$-ray energy range and which can be flown in space to study the
polarization from distant astronomical sources are now being designed mi05 ;
pr05 .

##  7 LIV and the Ultrahigh Energy Cosmic Ray Spectrum

###  7.1 The “GZK Effect”

Shortly after the discovery of the 3K cosmogenic background radiation (CBR),
Greisen gr66  and Zatsepin and Kuz’min za66  predicted that pion-producing
interactions of such cosmic ray protons with the CBR should produce a spectral
cutoff at $E\sim$ 50 EeV. The flux of ultrahigh energy cosmic rays (UHECR) is
expected to be attenuated by such photomeson producing interactions. This
effect is generally known as the “GZK effect”. Owing to this effect, protons
with energies above $\sim$100 EeV should be attenuated from distances beyond
$\sim 100$ Mpc because they interact with the CBR photons with a resonant
photoproduction of pions st68 .

###  7.2 Modification of the GZK Effect Owing to LIV

Let us consider the photomeson production process leading to the GZK effect.
Near threshold, where single pion production dominates,

$p+\gamma\rightarrow p+\pi.$ (9)

Using the normal Lorentz invariant kinematics, the energy threshold for
photomeson interactions of UHECR protons of initial laboratory energy $E$ with
low energy photons of the CBR with laboratory energy $\omega$, is determined
by the relativistic invariance of the square of the total four-momentum of the
proton-photon system. This relation, together with the threshold inelasticity
relation $E_{\pi}=m/(M+m)E$ for single pion production, yields the threshold
conditions for head on collisions in the laboratory frame

$4\omega E=m(2M+m)$ (10)

for the proton, and

$4\omega E_{\pi}={{m^{2}(2M+m)}\over{M+m}}$ (11)

in terms of the pion energy, where M is the rest mass of the proton and m is
the rest mass of the pion st68 .

If LI is broken so that $c_{\pi}~{}>~{}c_{p}$, the threshold energy for
photomeson is altered.333This requirement precludes the ‘quasi-vacuum Čerenkov
radiation’ of pions, via the rapid, strong interaction, pion emission process,
$p\rightarrow N+\pi$. This process would be allowed by LIV in the case where
$\delta_{\pi p}$ is negative, producing a sharp cutoff in the UHECR proton
spectrum. (For more details, see Refs. cg99 ; st09 ; alt07 .

Because of the small LIV perturbation term, the square of the four-momentum is
shifted from its LI form so that the threshold condition in terms of the pion
energy becomes

$4\omega E_{\pi}={{m^{2}(2M+m)}\over{M+m}}+2\delta_{\pi p}E_{\pi}^{2}$ (12)

where $\delta_{\pi p}\equiv c_{\pi}~{}-~{}c_{p}$, again in units where the low
energy velocity of light is unity.

Equation (12) is a quadratic equation with real roots only under the condition

$\delta_{\pi
p}\leq{{2\omega^{2}(M+m)}\over{m^{2}(2M+m)}}\simeq\omega^{2}/m^{2}.$ (13)

Defining $\omega_{0}\equiv kT_{CBR}=2.35\times 10^{-4}$ eV with
$T_{CBR}=2.725\pm 0.02$ K, equation (13) can be rewritten

$\delta_{\pi p}\leq 3.23\times 10^{-24}(\omega/\omega_{0})^{2}.$ (14)

###  7.3 Kinematics

If LIV occurs and $\delta_{\pi p}>0$, photomeson production can only take
place for interactions of CBR photons with energies large enough to satisfy
equation (14). This condition, together with equation (12), implies that while
photomeson interactions leading to GZK suppression can occur for “lower
energy” UHE protons interacting with higher energy CBR photons on the Wien
tail of the spectrum, other interactions involving higher energy protons and
photons with smaller values of $\omega$ will be forbidden. Thus, the observed
UHECR spectrum may exhibit the characteristics of GZK suppression near the
normal GZK threshold, but the UHECR spectrum can “recover” at higher energies
owing to the possibility that photomeson interactions at higher proton
energies may be forbidden. We now consider a more detailed quantitative
treatment of this possibility, viz., GZK coexisting with LIV.

The kinematical relations governing photomeson interactions are changed in the
presence of even a small violation of Lorentz invariance. The modified
kinematical relations containing LIV have a strong effect on the amount of
energy transfered from a incoming proton to the pion produced in the
subsequent interaction, i.e., the inelasticity st09 ; al03 ; ss08 .

The primary effect of LIV on photopion production is a reduction of phase
space allowed for the interaction. This results from the limits on the allowed
range of interaction angles integrated over in order to obtain the total
inelasticity. For real-root solutions for interactions involving higher energy
protons, the range of kinematically allowed angles becomes severely
restricted. The modified inelasticity that results is the key in determining
the effects of LIV on photopion production. The inelasticity rapidly drops for
higher incident proton energies.

Figure 2 shows the calculated proton inelasticity modified by LIV for a value
of $\delta_{\pi p}=3\times 10^{-23}$ as a function of both CBR photon energy
and proton energy ss08 . Other choices for $\delta_{\pi p}$ yield similar
plots. The principal result of changing the value of $\delta_{\pi p}$ is to
change the energy at which LIV effects become significant. For a choice of
$\delta_{\pi p}=3\times 10^{-23}$, there is no observable effect from LIV for
$E_{p}$ less than $\sim 200$ EeV. Above this energy, the inelasticity
precipitously drops as the LIV term in the pion rest energy approaches
$m_{\pi}$.

Figure 2: The calculated proton inelasticity modified by LIV for $\delta_{\pi
p}=3\times 10^{-23}$ as a function of CBR photon energy and proton energy ss08
.

With this modified inelasticity, the proton energy loss rate by photomeson
production is given by

${{1}\over{E}}{{dE}\over{dt}}=-{{\omega_{0}c}\over{2\pi^{2}\gamma^{2}}\hbar^{3}c^{3}}\int\limits_{\eta}^{\infty}d\epsilon~{}\epsilon~{}\sigma(\epsilon)K(\epsilon)\ln[1-e^{-\epsilon/2\gamma\omega_{0}}]$
(15)

where we now use $\epsilon$ to designate the energy of the photon in the cms,
$\eta$ is the photon threshold energy for the interaction in the cms,
$K(\epsilon)$ denotes the inelasticity, and $\sigma(\epsilon)$ is the total
$\gamma$-p cross section with contributions from direct pion production,
multipion production, and the $\Delta$ resonance.

The corresponding proton attenuation length is given by $\ell=cE/r(E)$, where
the energy loss rate $r(E)\equiv(dE/dt)$. This attenuation length is plotted
in Figure 3 for various values of $\delta_{\pi p}$ along with the unmodified
pair production attenuation length from pair production interactions,
$p+\gamma_{CBR}\rightarrow e^{+}+e^{-}$.

Figure 3: The calculated proton attenuation lengths as a function proton
energy modified by LIV for various values of $\delta_{\pi p}$ (solid lines),
shown with the attenuation length for pair production unmodified by LIV
(dashed lines). From top to bottom, the curves are for $\delta_{\pi p}=1\times
10^{-22},3\times 10^{-23},2\times 10^{-23},1\times 10^{-23},3\times
10^{-24},0$ (no Lorentz violation) ss08 .

##  8 UHECR Spectra with LIV and Comparison with Present Observations

The effect of by a very small amount of LIV on the UHECR spectrum was
analytically calculated in Ref. ss08  in order to determine the resulting
spectral modifications. It can be demonstrated that there is little difference
between the results of using an analytic calculation vs. a Monte Carlo
calculation (e.g., see Ref. ta09 ). In order to take account of the probable
redshift evolution of UHECR production in astronomical sources, they took
account of the following considerations:

(i) The CBR photon number density increases as $(1+z)^{3}$ and the CBR photon
energies increase linearly with $(1+z)$. The corresponding energy loss for
protons at any redshift $z$ is thus given by

$\displaystyle r_{\gamma p}(E,z)=(1+z)^{3}r[(1+z)E].$ (16)

(ii) They assumed that the average UHECR volume emissivity is of the energy
and redshift dependent form given by $q(E_{i},z)=K(z)E_{i}^{-\Gamma}$ where
$E_{i}$ is the initial energy of the proton at the source and $\Gamma=2.55$.
For the source evolution, we assume $K(z)\propto(1+z)^{3.6}$ with $z\leq 2.5$
so that $K(z)$ is roughly proportional to the empirically determined
$z$-dependence of the star formation rate. $K(z=0)$ and $\Gamma$ are
normalized fit the data below the GZK energy.

Using these assumptions, one can calculate the effect of LIV on the UHECR
spectrum. The results are actually insensitive to the assumed redshift
dependence because evolution does not affect the shape of the UHECR spectrum
near the GZK cutoff energy be88 ; st05 . At higher energies where the
attenuation length may again become large owing to an LIV effect, the effect
of evolution turns out to be less than 10%. The curves calculated in Ref. st09
assuming various values of $\delta_{\pi p}$, are shown in Figure 4 along with
the latest Auger data from Ref. sch09 . They show that even a very small
amount of LIV that is consistent with both a GZK effect and with the present
UHECR data can lead to a “recovery” of the UHECR spectrum at higher energies.

Figure 4: Comparison of the latest Auger data with calculated spectra for
various values of $\delta_{\pi p}$, taking $\delta_{p}=0$ (see text). From top
to bottom, the curves give the predicted spectra for $\delta_{\pi p}=1\times
10^{-22},6\times 10^{-23},4.5\times 10^{-23},3\times 10^{-23},2\times
10^{-23},1\times 10^{-23},3\times 10^{-24},0$ (no Lorentz violation) st09 .

###  8.1 Allowed Range for the LIV Parameter $\delta_{\pi p}$

Stecker and Scully st09  have updated compared the theoretically predicted
UHECR spectra with various amounts of LIV to the latest Auger data from the
procedings of the 2009 International Cosmic Ray Conference sch09 , data . This
update is shown in Figure 4. The amount of presently observed GZK suppression
in the UHECR data is consistent with the possible existence of a small amount
of LIV. The value of $\delta_{\pi p}$ that results in the smallest $\chi^{2}$
for the modeled UHECR spectral fit using the observational data from Auger
sch09  above the GZK energy. The best fit LIV parameter found was in the range
given by $\delta_{\pi p}$ = $3.0^{+1.5}_{-3.0}\times 10^{-23}$, corresponding
to an upper limit on $\delta_{\pi p}$ of $4.5\times 10^{-23}$. 444The HiRes
data ab08  do not reach a high enough energy to further restrict LIV. 555We
note that the overall fit of the data to the theoretically expected spectrum
is somewhat imperfect, even below the GZK energy and even for the case of no
LIV. It appears that the Auger spectrum seems to steepen even below the GZK
energy. As a conjecture, one can assume that the derived energy may be too low
by about 25%, within the uncertainty of both systematic-plus statistical error
given for the energy determination. This gives better agreement between the
theoretical curves and the shifted data st09 . The constraint on LIV would be
only slightly reduced if this shift is assumed.

###  8.2 Implications for Quantum Gravity Models

An effective field theory approximation for possible LIV effects induced by
Planck-scale suppressed quantum gravity for $E\ll M_{Pl}$ was considered in
Ref. ma09 . These authors explored the case where a perturbation to the
energy-momentum dispersion relation for free particles would be produced by a
CPT-even dimension six operator suppressed by a term proportional to
$M_{Pl}^{-2}$. The resulting dispersion relation for a particle of type $a$ is

$E_{a}^{2}=p_{a}^{2}+m_{a}^{2}+\eta_{a}\left({{p^{4}}\over{M_{Pl}^{2}}}\right)$
(17)

In order to explore the implications of our constraints for quantum gravity,
one can take the perturbative terms in the dispersion relations for both
protons and pions, to be given by the dimension six dispersion terms in
equation (17) above. Making this identification, the LIV constraint of
$\delta_{\pi p}<4.5\times 10^{-23}$ in the fiducial energy range around
$E_{f}=100$ EeV indirectly implies a powerful limit on the representation of
quantum gravity effects in an effective field theory formalism with Planck
suppressed dimension six operators. Equating the perturbative terms in both
the proton and pion dispersion relations, one obtains the relation st09

$2\delta_{\pi
p}\simeq(\eta_{\pi}-25\eta_{p})\left({{0.2E_{f}}\over{M_{Pl}}}\right)^{2},$
(18)

where the pion fiducial energy is taken to to be $\sim 0.2E_{f}$, as at the
$\Delta$ resonance that dominates photopion production and the GZK effect st68
. Equation (18), together with the constraint $\delta_{\pi p}<4.5\times
10^{-23}$, indicates that any LIV from dimension six operators is suppressed
by a factor of at least ${\cal{O}}(10^{-6}M_{Pl}^{-2})$, except in the
unlikely case that $\eta_{\pi}-25\eta_{p}\simeq 0$. These results are in
agreement with those obtained independently by Maccione et al. from the Monte
Carlo runs ma09 . It can thus be concluded that an effective field theory
representation of quantum gravity with dimension six operators that suppresses
LIV by only a factor of $M_{Pl}^{2}$ i.e. $\eta_{p},\eta_{\pi}\sim 1$, is
effectively ruled out by the UHECR observations.

##  9 Beyond Constraints: Seeking LIV

As we have seen (see Figure 4), even a very small amount of LIV that is
consistent with both a GZK effect and with the present UHECR data can lead to
a “recovery” of the primary UHECR spectrum at higher energies. This is the
clearest and the most sensitive evidence of an LIV signature. The “recovery”
effect has also been deduced in Refs. ma09  and bi09  666In Ref. bi09 , a
recovery effect is also claimed for high proton energies in the case when
$\delta_{\pi p}<0$. However, we have noted that the ‘quasi-vacuum Čerenkov
radiation’ of pions by protons in this case will cut off the proton spectrum
and no “recovery” effect will occur.. In order to find it (if it exists) three
conditions must exist: (i) sensitive enough detectors need to be built, (ii) a
primary UHECR spectrum that extends to high enough energies ($\sim$ 1000 EeV)
must exist, and (iii) one much be able to distinguish the LIV signature from
other possible effects.

###  9.1 Obtaining UHECR Data at Higher Energies

We now turn to examining the various techniques that can be used in the future
in order to look for a signal of LIV using UHECR observations. As can be seen
from the preceding discussion, observations of higher energy UHECRs with much
better statistics than presently obtained are needed in order to search for
the effects of miniscule Lorentz invariance violation on the UHECR spectrum.

####  9.1.1 Auger North

Such an increased number of events may be obtained using much larger ground-
based detector arrays. The Auger collaboration has proposed to build an “Auger
North” array that would be seven times larger than the present southern
hemisphere Auger array (http://www.augernorth.org).

####  9.1.2 Space Based Detectors

Further into the future, space-based telescopes designed to look downward at
large areas of the Earth’s atmosphere as a sensitive detector system for giant
air-showers caused by trans-GZK cosmic rays. We look forward to these
developments that may have important implications for fundamental high energy
physics.

Two potential spaced-based missions have been proposed to extend our knowledge
of UHECRs to higher energies. One is JEM-EUSO (the Extreme Universe Space
Observatory) EUSO , a one-satellite telescope mission proposed to be placed on
the Japanese Experiment Module (JEM) on the International Space Station. The
other is OWL (Orbiting Wide-angle Light Collectors) OWL , a two satellite
mission for stereo viewing, proposed for a future free-flyer mission. Such
orbiting space-based telescopes with UV sensitive cameras will have wide
fields-of-view (FOVs) in order to observe and use large volumes of the Earth’s
atmosphere as a detecting medium. They will thus trace the atmospheric
fluorescence trails of 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 than those presently observed
by ground-based detectors such as Auger. Such missions will thus potentially
open up a new window on physics at the highest possible observed energies.

##  10 Conclusions

The Fermi timing results for GRB090510 rule out and string-inspired D-brane
model predictions as well as other quantum gravity predictions of a
retardation of photon velocity that is simply proportional to $E/M_{QG}$
because they would require $M_{QG}>M_{Pl}$. More indirect results from
$\gamma$-ray birefringence limits, the non-decay of 50 TeV $\gamma$-rays from
the Crab Nebula, and the TeV spectra of nearby AGNs also place severe limits
on violations of special relativity (LIV). Limits on Lorentz invariance
violation from observations of ultrahigh energy cosmic-rays provide severe
constraints for other quantum gravity models, appearing to rule out
retardation that is simply proportional to $(E/M_{QG})^{2}$. Various effective
field theory frameworks lead to such energy dependences.

New theoretical models of Planck scale physics and quantum gravity need to
meet all of the present observational constraints. One scenario that may be
considered is that gravity, i.e. $G$, becomes weaker at high energies. We know
that the strong, weak and electromagnetic interactions all have energy
dependences, given by the running of the coupling constants. If $G$ decreases,
then the effective $\lambda_{Pl}=\sqrt{G\hbar/c^{3}}$ would decrease and the
effective $M_{Pl}=\hbar/(\lambda_{Pl}c)$ would increase. In that case, the
space-time quantum scale would be less than the usual definition of
$\lambda_{Pl}$. Such speculation is presently cogitare ex arcis, but might be
plausible if a transition to a phase where the various forces are unified
occurs at very high energies st80 .

At the time of the present writing, high energy astrophysics observations have
led to strong constraints on LIV. Currently, we have no positive evidence for
LIV. This fact, in itself, should help guide theoretical research on quantum
gravity, already ruling out some models. Will this lead to a new null result
comparable to Michelson-Morley? Will a totally new concept be needed to
describe physics at the Planck scale? If all of the known forces are unified
at the Planck scale, this would not be surprising. One thing is clear: a
consideration of all empirical data will be necessary in order to finally
arrive at a true theory of physics at the Planck scale.

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