# Light Speed Variation with Brane/String-Inspired Space-Time Foam

Chengyi Li  Bo-Qiang Ma  School of Physics and State Key Laboratory of Nuclear
Physics and Technology, Peking University, Beijing 100871, China  Center for
High Energy Physics, Peking University, Beijing 100871, China  Collaborative
Innovation Center of Quantum Matter, Beijing, China

###### Abstract

Recently a series of studies on high energy gamma-ray burst (GRB) photons
suggest a light speed variation with linear energy dependence at the Lorentz
violation scale of $3.6\times 10^{17}~{}\mathrm{GeV}$, with subluminal
propagation of high energy photons in cosmological space. We propose stringy
space-time foam as a possible interpretation for this light speed variation.
In such a string-inspired scenario, bosonic photon open-string travels in
vacuo at an infraluminal speed with an energy dependence suppressed by a
single power of the string mass scale, due to the foamy structure of space-
time at small scales, as described by D-brane objects in string theory. We
present a derivation of this deformed propagation speed of the photon field in
the infrared (IR) regime. We show that the light speed variation, revealed in
the previous studies on GRBs time-delay data, can be well described within
such a string approach towards space-time foam. We also derive the value of
the effective quantum-gravity mass in this framework, and give a qualitative
study on the theory-dependent coefficients. We comment that stringent
constraints on Lorentz violation in the photon sector from complementary
astrophysical observations can also be explained and understood in the space-
time foam context.

###### keywords:

light speed variation, gamma ray burst, space-time foam, Lorentz invariance
violation , string theory , quantum gravity

††journal: Results in Physics, published as Results Phys. 26 (2021) 104380,
https://doi.org/10.1016/j.rinp.2021.104380

##  1 Introduction

Constant light speed in vacuo is one of the basic postulations in Einstein’s
theory of special relativity. However, it is speculated that quantum effects
of gravity may bring a tiny correction to the speed of light due to Lorentz
invariance violation (LIV) at the Planck scale $M_{\mathrm{Pl}}=\sqrt{hc/2\pi
G}\approx 1.22\times 10^{19}~{}\mathrm{GeV}/c^{2}$ [1]. Since the light speed
variation is extremely small, it is hard to detect such Planck-suppressed LIV
modification from most terrestrial experiments. Amelino-Camelia et al. [2, 3]
first proposed that distant celestial sources of high-energy photon emissions,
e.g., $\gamma$-ray bursters (GRBs), can play a crucial role in probing such
LIV effects.

Since the launching and the operation of the high-precision Fermi $\gamma$-ray
space telescope (FGST), Fermi-LAT Collaboration [4, 5] has reported that with
regard to emissions arriving from a GRB, high-energy photons have a tendency
to arrive relatively later than low-energy ones. Though such arrival-time
retardation of photons may originate from source effects (non-simultaneous
emission at the source) that we know very little, the collaboration [5] also
examined the possibilities of vacuum dispersions caused by LIV. Thanks to a
number of GRB samples cumulatively observed by the Fermi telescope, collective
analyses over the whole collection of the photon data can be performed.
Inspired by statistically robust methods developed by Ellis et al. [6, 7, 8],
a series of very recent studies on GRB photons have been carried out in Refs.
[9, 10, 11, 12, 13, 14, 15]. Therein through global analyses of time-of-flight
lags between the high-energy photons and the corresponding low-energy signals
from different GRBs, 14 Fermi-detected very-high-energy events are found to be
all compatible with the same LIV energy scale of magnitude of
$10^{17}~{}\mathrm{GeV}$, with 9 of these 14 events lining up very nicely to
form a mainline. Such regularity indicates a linearly energy-dependent
velocity of light in vacuo of the form
$c_{g}(\mathcal{E})=1-\mathcal{E}/E_{\mathrm{LIV}}$,111For brevity, natural
units where $\hslash,c\equiv 1$ are used hereafter. with the first-order
Lorentz-violation scale determined as $E_{\mathrm{LIV}}\simeq 3.60\times
10^{17}~{}\mathrm{GeV}$ [11, 12, 13]. Such finding is certainly impressive as
characterized in Refs. [12, 13, 14, 15], it is thus interesting to explore a
possible explanation for such finding with a Lorentz-violating framework.

In this paper, we interpret this previously suggested light speed variation as
a consequence of stringy interactions of energetic photons from GRBs with the
D-particle background incorporated in a Liouville-string scenario of space-
time foam [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. The basic idea is that the
photon, viewed as bosonic open-string state, scatters off the D0-brane space-
time defect to form an oscillating intermediate string which subsequently
decays to emit the outgoing photon wave with a time delay of
$\mathcal{O}\left(1/M_{\mathrm{Pl}}\right)$. The velocity of light in the
space-time vacuum is therefore deformed at infrared (IR) relative to the
string scale. We indicate that this brane/string-inspired model can serve as a
consistent framework to explain the light speed variation as observed from
GRBs, in agreement with other astrophysical results to date.

Our paper is organized as follows. In the next section, we briefly review the
D-brane realization of space-time foam and its effects on photon propagation.
In Sec. 3, we give a derivation of the deformed speed of light in this string-
inspired model. In Sec. 4 we connect observations with the theory to provide
an explanation for the finding of the speed-of-light variation as observed
from analyses of Fermi-GRB data. We then show that some complementary results,
including those severe limits from birefringence, photon decay, etc., can also
be explained in this framework. In the last section we conclude with some
comments on our proposal.

##  2 A stringy model for space-time foam

In this section we quickly review the D-brane model of space-time foam and
present its physical consequences. The intuitive picture of space-time foam
goes all the way back to Wheeler [26], who noted that there would be highly
curved quantum fluctuations with $\Delta E\sim E_{\mathrm{Pl}}$ in the space-
time continuum, at short time and small spatial scales $\Delta t\sim
1/E_{\mathrm{Pl}}$, $\Delta x\sim\ell_{\mathrm{Pl}}$, where quantum-gravity
effects become relevant.

Within a Liouville non-critical string approach towards quantum gravity (QG)
[16, 17, 18, 19], a stringy analogue of space-time foam model was developed in
Refs. [20, 21]. In this string-inspired framework, the observable
(3+1)-dimensional Universe is described as a compactified D(irichlet)3-brane
[27], roaming in a higher-dimensional bulk space-time punctured by point-like
D0-brane (“D-particle”) defects in the context of Type-IA string theory (a T
dual of Type-I superstrings [28]). The standard-model matter particles,
propagating through the brane world, are viewed as (open) string excitations
with their ends attached on the D3-brane. When the D3-brane Universe moves in
the bulk space, the D-particles traverse it and interact with the string
matter anchored on the D3-brane in a topologically non-trivial manner. For an
observer living on the brane world, the D-particles “flash” on and off, and
appear as quantum space-time foam, termed as “D-foam” [24].

Due to the presence of D-particle foam defects near the brane world, the
dispersion relations for certain string matters would receive ultraviolet (UV)
corrections. For massless gauge excitations, like the photon, according to the
string formation formalism [22] in this framework, each non-trivial
interaction between an open photon string and a D-particle excites an
oscillating intermediate string state which subsequently decays and re-emits
the outgoing photon wave, resulting in an infinitesimal (causal) time delay
for the incident photon propagating through the D-foam. Through modeling the
photon-D-particle interactions as stretched intermediate strings or performing
straightforward amplitude calculation, one finds that the (retarded) outgoing
waves are emitted with time delays of the order

$\Delta t\sim\ell_{s}^{2}\mathcal{E},$ (1)

where the string length $\ell_{s}$ is associated with the string mass scale
$M_{s}$ through the relation $\ell_{s}=1/M_{s}$, and $\mathcal{E}$ denotes the
incident photon energy. The overall time delay consists of each retardation
resulting from individual scattering of the photon with a D-particle defect
encountered in the foam. Therefore the time delay depends in general on the
density $n_{D}$ of foam particles in the bulk space. By taking the linear
density of defects $n_{D}/\ell_{s}$ into account, the total delay experienced
by a photon can be estimated as

$\Delta t_{\mathrm{tot}}\propto\ell_{s}^{2}{\cal
E}\frac{n_{D}}{\ell_{s}}=\frac{\mathcal{E}}{M_{s}}n_{D}.$ (2)

In addition to this leading-order delay 2, the collision of the photon with a
bulk D-particle would cause the latter to recoil at a velocity of
$\mathbf{u}$, which modifies the retarded re-emergence time of the outgoing
photon wave to become [22]

$\Delta t\sim\frac{\ell_{s}^{2}{\cal E}}{1-\lvert{\mathbf{u}}\rvert^{2}},$ (3)

which reduces to Eq. 1 in the case of $u_{i}\ll 1$. In fact the recoil of the
D-particle defect would distort the neighboring space-time [17, 19], by
inducing an off-diagonal metric element related to the recoil velocity $u_{i}$
of the D-defect. The resulting space-time fluctuations lead to an effective
Finsler-like target-space metric as

$\eta_{\alpha\beta}^{\mathrm{eff}}=\eta_{\alpha\beta}+h_{\alpha\beta},\ \
h_{0i}=u_{i}.$ (4)

Such a distortion of space-time near the recoiling D-particle then affects the
(local) canonical relationship between energy and momentum of the scattered
photon through
$k^{\alpha}k^{\beta}\eta_{\alpha\beta}^{\mathrm{eff}}=-\mathcal{E}^{2}+2\mathcal{E}k^{i}u_{i}+k^{i}k_{i}=0$,
which gives rise to the anomalous dispersion of photons in vacuo [20]

$\mathcal{E}(\mathbf{k})=\mathbf{k}\cdot\mathbf{u}+\lvert\mathbf{k}\rvert\left(1+\left(\mathbf{u}\cdot\frac{\mathbf{k}}{\lvert\mathbf{k}\rvert}\right)^{2}\right)^{1/2}.$
(5)

An important comment is that the delay 2 varies linearly with the photon
energy and presents no helicity dependence, which stems from the CPT-even
feature of the theory.

##  3 Velocity dispersion of light in the D-particle quantum-gravitational
foam medium

In order to connect the experimental finding with the above theory, in this
section we first present a derivation of the modified form of the speed of
light within this D-brane QG framework. Since we work in units of $\hslash=1$,
the photon dispersion 5 can be rewritten, in terms of the frequency $\omega$,
as

$\omega=\mathbf{k}\cdot\mathbf{u}+k\left(1+\left(\mathbf{u}\cdot\hat{e}_{\mathbf{k}}\right)^{2}\right)^{1/2},$
(6)

where $k\equiv\lvert\mathbf{k}\rvert$ is the magnitude of the wave vector, and
$\hat{e}_{\mathbf{k}}\equiv\mathbf{k}/k$ denotes the unit vector oriented
along the direction of the photon momentum $\mathbf{k}$. In the special case
of no D-particle recoil $u_{i}\rightarrow 0$, terms involving the recoil
velocity $\mathbf{u}$ in Eq. 6 vanish, and the standard relativistic relation
for photons $\omega=\lvert\mathbf{k}\rvert$ is recovered. By squaring both
sides of Eq. 6, we have

$\displaystyle\autobreak\omega^{2}=\left(\mathbf{k}\cdot\mathbf{u}\right)^{2}+\left(k\left(1+\left(\mathbf{u}\cdot\hat{e}_{\mathbf{k}}\right)^{2}\right)^{1/2}\right)^{2}+2k\left(\mathbf{k}\cdot\mathbf{u}\right)\left(1+\left(\mathbf{u}\cdot\hat{e}_{\mathbf{k}}\right)^{2}\right)^{1/2},$
(7)

where the recoil velocity of the D-particle defect is
$u_{i}=M_{s}^{-1}g_{s}\Delta k_{i}=\Delta k_{i}/M_{D}$ [29], with
$M_{D}=M_{s}/g_{s}$ being the mass for the D-particle. $\Delta k_{i}$
indicates the momentum transfer from the photon to the D-particle. Keep in
mind that the string coupling is weak, i.e., $g_{s}<1$ is assumed in the
model, and that the string scale $M_{s}$ is expected to be a large mass scale,
so these D-defects are in general massive objects with very large $M_{D}\sim
M_{\mathrm{Pl}}$ in string theory. Therefore a heavy D-particle would recoil
with a non-relativistic velocity $u_{i}\ll 1$ after the collision with a
photon open-string, as long as the incident energy $k\ll M_{s}$, which we
refer to as the IR limit (relative to the string scale). In this IR region,
one can safely expand the last term in Eq. 7 to obtain

$\displaystyle\autobreak\omega^{2}=k^{2}+2k\left(\mathbf{k}\cdot\mathbf{u}\right)+2\left(\mathbf{k}\cdot\mathbf{u}\right)^{2}+\left(\mathbf{k}\cdot\mathbf{u}\right)\left(\mathbf{u}\cdot\hat{e}_{\mathbf{k}}\right)^{2}-\frac{1}{4}k\left(\mathbf{k}\cdot\mathbf{u}\right)\left(\mathbf{u}\cdot\hat{e}_{\mathbf{k}}\right)^{4}+\mathcal{O}\left(\lvert\mathbf{u}\rvert^{7}\right).$
(8)

Due to stochastic captures of a photon by D-defects, one should average the
above relation 8 over the D-particles in the foam, and we adopt the notation
$\langle\\!\langle\cdot\\!\cdot\\!\cdot\rangle\\!\rangle_{D}$ to denote this
type of averaged values. An anisotropic background field that explicitly
breaks Lorentz symmetry then emerges from the average recoil velocity of
D-particles $\langle\\!\langle u_{i}\rangle\\!\rangle_{D}\neq 0$ (“anisotropic
foam”), whose direction is oriented along that of the incident photon [30].
Note that string theory leads to unambiguous (minus) sign for
$\langle\\!\langle u_{i}\rangle\\!\rangle_{D}$ due to the fact that there
should be no superluminal excitations in a string spectrum. Hence,
$\langle\\!\langle\mathbf{k}\cdot\mathbf{u}\rangle\\!\rangle_{D}=-\langle\\!\langle
k\lvert\mathbf{u}\rvert\rangle\\!\rangle_{D}<0$. Denote the momentum transfer
$\Delta k$ in $u_{i}$ as a fraction of the incident momentum $k$, $\Delta
k_{i}\equiv\lambda k_{i}$, with $\lambda<1$, we can easily write down

$\displaystyle\autobreak\omega^{2}\simeq
k^{2}\Biggl{(}1-2\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}\frac{k}{M_{s}}+2\langle\\!\langle\lambda^{2}\rangle\\!\rangle_{D}g_{s}^{2}\frac{k^{2}}{M_{s}^{2}}-\langle\\!\langle\lambda^{3}\rangle\\!\rangle_{D}g_{s}^{3}\frac{k^{3}}{M_{s}^{3}}+\frac{1}{4}\langle\\!\langle\lambda^{5}\rangle\\!\rangle_{D}g_{s}^{5}\frac{k^{5}}{M_{s}^{5}}+\mathcal{O}\left(g_{s}^{7}k^{7}/M_{s}^{7}\right)\Biggr{)},$
(9)

where averaging notations for $k_{i}$ and $\mathcal{E}$ in the above equation
is omitted, also hereafter, for simplicity. The higher-order correction terms
are reserved for generality in the expansion 9. To leading order, the modified
dispersion relation is given by

$\omega^{2}\simeq
k^{2}\left(1-2\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}\frac{k}{M_{s}}\right),$
(10)

with $0<\langle\\!\langle\lambda\rangle\\!\rangle_{D}<1$. Before moving on, we
argue that on average the momentum ratio $\lambda$ is related to the bulk
density of the D-particle. The total time delay 2, mentioned in Sec. 2, may be
thought of as implying an effective quantum-gravitational phase refractive
index of the vacuum
$\eta_{\mathrm{vac}}(\omega)=1+\mathcal{O}\left(n_{D}\omega/M_{D}\right)>1$
for light with frequency $\omega$. By substituting it into
$c_{\mathrm{ph}}\equiv 1/\eta$,222Here $c_{\mathrm{ph}}\equiv c/\eta=1/\eta$
with $c=1$ being the conventional speed of light in normal vacuo of Special
Relativity (i.e., the propagation velocity of low energy photons). Recall that
in Lorentz invariant space-time, the group velocity and the phase velocity of
photons in vacuum are the same, which equal to the constant speed of light,
$c_{g}=c_{\mathrm{ph}}=1$. the phase velocity of photons in the QG D-foam
medium can be estimated as $c_{\mathrm{ph}}(k)\simeq
1-\left\lvert\mathcal{O}\left(n_{D}k/M_{D}\right)\right\rvert$. On the other
hand, from equation 10 one can derive the same phase velocity as
$c_{\mathrm{ph}}=\omega/\lvert\mathbf{k}\rvert\simeq
1-\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}k/M_{s}$. This fact yields
the expected relation

$\langle\\!\langle\lambda\rangle\\!\rangle_{D}\propto n_{D}.$ (11)

Assume that the relation $c_{g}=\partial\omega/\partial k$ still holds in QG,
the propagation velocity of light can be straightforwardly deduced, on account
of Eq. 9, as follows

$\displaystyle\autobreak
c_{g}=\frac{k}{\omega}\Biggl{(}1-3\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}\frac{k}{M_{s}}+4\langle\\!\langle\lambda^{2}\rangle\\!\rangle_{D}g_{s}^{2}\frac{k^{2}}{M_{s}^{2}}-\frac{5}{2}\langle\\!\langle\lambda^{3}\rangle\\!\rangle_{D}g_{s}^{3}\frac{k^{3}}{M_{s}^{3}}+\mathcal{O}\left((k/M_{s})^{5}\right)\Biggr{)},$
(12)

where $k/\omega$ factor is given by

$\displaystyle\frac{k}{\omega}\simeq$ $\displaystyle\
\Biggl{(}1-2\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}\frac{k}{M_{s}}+2\langle\\!\langle\lambda^{2}\rangle\\!\rangle_{D}g_{s}^{2}\frac{k^{2}}{M_{s}^{2}}-\langle\\!\langle\lambda^{3}\rangle\\!\rangle_{D}g_{s}^{3}\frac{k^{3}}{M_{s}^{3}}\Biggr{)}^{-1/2}$
$\displaystyle=$ $\displaystyle\
1+\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}\frac{k}{M_{s}}+\frac{1}{2}\langle\\!\langle\lambda^{2}\rangle\\!\rangle_{D}g_{s}^{2}\frac{k^{2}}{M_{s}^{2}}+\mathcal{O}\left((k/M_{s})^{4}\right).$
(13)

So the group velocity of photons in a D-particle foam background is modified
by

$c_{g}=1-2\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}\frac{k}{M_{s}}+\frac{3}{2}\langle\\!\langle\lambda^{2}\rangle\\!\rangle_{D}g_{s}^{2}\frac{k^{2}}{M_{s}^{2}}+\mathcal{O}\left(k^{4}/M_{s}^{4}\right).$
(14)

Note that terms proportional to odd powers of momentum, like the
$k^{3}/M_{s}^{3}$ term, in Eq. 12, cancel out in the final expression 14
except for the single-power term, which is a striking feature of this stringy
approach to QG-dispersed velocity of light, and we shall comment on that from
another viewpoint below. With the IR condition ($k\ll M_{s}$) applied, we have

$c_{g}\simeq
1-2\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}\frac{k}{M_{s}}.$ (15)

We comment at this point that there is another way to have the photon speed
obtained. Instead of deriving the group velocity from the squared form of the
dispersion relation 9, one can start directly from Eq. 6. Utilizing the
relation
$\langle\\!\langle\mathbf{k}\cdot\mathbf{u}\rangle\\!\rangle_{D}\simeq-
M_{s}^{-1}g_{s}\langle\\!\langle\lambda\rangle\\!\rangle_{D}k^{2}$, we have
the exact formula of the velocity

$\displaystyle\autobreak
c_{g}=1-2\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}\frac{k}{M_{s}}+\langle\\!\langle~{}\Biggl{(}1+\lambda^{2}g_{s}^{2}\frac{k^{2}}{M_{s}^{2}}\Biggr{)}^{1/2}-1\rangle\\!\rangle_{D}+\langle\\!\langle~{}\Biggl{(}1+\lambda^{2}g_{s}^{2}\frac{k^{2}}{M_{s}^{2}}\Biggr{)}^{-1/2}\lambda^{2}g_{s}^{2}\frac{k^{2}}{M_{s}^{2}}\rangle\\!\rangle_{D}.$
(16)

In the IR regime, the latter two terms in the above equation can be expressed
as the Taylor series expansion, and one can easily check that this equation
then reduces to Eq. 14. We see that the second term in 16 just corresponds to
the linear-order correction to light speed 15, and it is reasonable that odd-
power terms $\mathcal{O}\left((k/M_{s})^{2n+1}\right)~{}(n>0)$ do not exist
since the Taylor expansion would only give birth to those terms suppressed by
even powers of the string mass $M_{s}$.

We stress again that equations 14 as well as 15 are valid only if
$\langle\\!\langle\lvert\mathbf{u}\rvert\rangle\\!\rangle_{D}\ll 1$,
corresponding to $k\ll M_{s}$. Our new-developed formula 16 seemingly does not
rely on this IR limit however, so it is likely to work even when $k\sim
M_{s}$. But one should notice that Eq. 16 together with our starting point 5
actually depend on the metric deformation formalism 4 which naturally requires
the distortion of space-time not to be strong, namely $h_{0i}=u_{i}\ll 1$.
Hence in the case of relativistic recoil of D-particles $u_{i}\rightarrow 1$
that could be attained when the incident energy $k\sim M_{s}$, Eq. 14 and the
precise form 16 of the velocity are no longer valid. For this reason, in the
UV regime one should stick to the string formation formalism. At energies
$k\sim M_{s}/g_{s}$, as a matter of fact the recoiling D-particle moves at
nearly the relativistic speed of light,
$\lvert\mathbf{u}\rvert^{2}=\mathcal{O}\left(1\right)$, as a result the time
delay $\Delta t$ goes to infinity (c.f. 3), which is a clear manifestation of
the destabilization of the vacuum. This fact implies that the recoil of the
D-particle, when scattered off such ultra-high-energy photons, would induce a
very strong space-time distortion, leading to complete absorption of such
photons by the D-particle defect [31]. The defect behaves like a black hole,
capturing permanently the incident photons [32].

Before closing this section, we define, for convenience in the discussions
that follow, the (stringy) quantum-gravity “effective” scale for photons in
the D-particle foam by

$\mathcal{M}_{\mathrm{QG}\gamma}^{(\textrm{D-foam})}=\frac{M_{s}}{2\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}}\equiv
M_{s}/\varsigma_{\gamma,\mathrm{ani}}^{(\textrm{D-foam})},$ (17)

where
$\varsigma_{\gamma,\mathrm{ani}}^{(\textrm{D-foam})}=2\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}>0$
denotes a nonzero dimensionless coefficient for the anisotropic D-foam.333We
consider in this paper a uniform density of D-particles, i.e.,
$\varsigma_{\gamma,\mathrm{ani}}^{(\textrm{D-foam})}$ is a
redshift-$z$-independent coefficient. In more complicated cases, the density
$n_{D}$ could vary with the cosmological epoch [24, 33], then the effective QG
scale could depend also on the redshift:
$\widetilde{\mathcal{M}}_{\mathrm{QG}\gamma}^{(\textrm{D-foam})}(z)\sim
n_{D}(z)/M_{s}$. We emphasis here that this relevant QG scale 17 cannot be a
priori determined. To see this clearly, one should note that, though the
underlying string theory, the most promising framework for a theory of
everything, incorporating the compelling description of gravity quantization,
should have no free parameters, the scale $M_{s}$, according to modern
understanding in string theory [34], is indeed an adjustable constant to be
constrained by studies on string phenomenology. Moreover, the coefficient
$\varsigma_{\gamma,\mathrm{ani}}^{(\textrm{D-foam})}$ that appears in the QG
scale 17 relates to the microscopic setup of the model. Specifically, we see
that Eq. 11 leads to a relation
$\varsigma_{\gamma,\mathrm{ani}}^{(\textrm{D-foam})}\sim g_{s}n_{D}$, where
the D-particle density $n_{D}$ in the bulk is arbitrary in general, implying
that it is essentially a free parameter in the model [35]. Hence the scale 17
is free to vary, and thus needs to be constrained by experiments. Given the
effective QG mass defined above, the deformed velocity of light 15 can be
expressed as the following simple form

$c_{g}\simeq 1-\omega/\mathcal{M}_{\mathrm{QG}\gamma}^{(\textrm{D-foam})},$
(18)

where, to leading order, $k\simeq\omega$. In the next section this QG-modified
speed of light 18 can be used to explain the experimental finding in Refs.
[11, 12, 13] from the GRB data.

##  4 An explanation for the light speed variation in GRBs and further
discussions

Since the launching and the operation of the high-precision FGST, the Fermi-
LAT Collaboration has reported that high energy photons arrive at the Earth
later relative to low energy ones [4, 5]. Though such time-of-flight lags may
be attributed to source-intrinsic properties of GRBs, there are still
possibilities that arrival time delays originate from the speed dispersions of
photons duration the propagation, arising from LIV. In the last decade, a
number of energetic GRB samples have been cumulatively obtained by the Fermi
telescope, making it possible to perform global analyses on a range of GRBs to
disentangle LIV-induced lag from a priori unknown source-intrinsic lag and
further reveal fascinating regularities which can serve as the supports for
the light speed variation.

Intriguing statistical analyses of photon time-of-flight over the whole
collection of Fermi-detected GRBs have been carried out in a series of very
recent studies [9, 10, 11, 12, 13, 14, 15]. In Refs. [11, 12, 13], utilizing a
general Lorentz-violation modified dispersion relation with vanishing rest
mass

$\mathcal{E}^{2}\simeq
p^{2}-s_{n}\mathcal{E}^{2}\left(\frac{p}{E_{\mathrm{LIV},n}}\right)^{n},$ (19)

the authors obtain the deformed photon propagation velocity with higher-order
terms dropped as

$v_{g}(\mathcal{E})=\frac{\partial\mathcal{E}}{\partial p}\simeq
1-s_{n}\frac{n+1}{2}\left(\frac{\mathcal{E}}{E_{\mathrm{LIV},n}}\right)^{n},$
(20)

where $n=1$ or $2$ corresponds to linear or quadratic energy dependence
respectively, $s_{n}=\pm 1$ is a sign factor of LIV correction, and
$E_{\mathrm{LIV},n}$ is the $n$th-order Lorentz-violation scale. Using the
arrival-time differences of multi-GeV photon signals from a number of Fermi-
observations, a regularity fitting well with all 14 high-energy events is
observed by the authors, indicating a first-order Lorentz-violation scale
(i.e., $n=1$) at444Compared to the quadratic ($n=2$) LIV case, as illustrated
in Ref. [11], the linear ($n=1$) case is more favored by the photon time-delay
data. In the case $n=1$ the redundant subscripts $n$ in the above formulae are
hereafter omitted.

$E_{\mathrm{LIV}}=\left(3.60\pm 0.26\right)\times 10^{17}~{}\mathrm{GeV},$
(21)

which is close to the Planck scale $E_{\mathrm{Pl}}\simeq 1.22\times
10^{19}~{}\mathrm{GeV}$. A positive sign factor $s=+1$ is suggested through
fitting the data, implying that high-energy photons travel across cosmos
slower than their low-energy counterparts. More striking regularity is that 9
out of these 14 photons falls very nicely on a same mainline, giving a strong
indication of a light speed variation suppressed by a single power of the
Planck scale.555Statistical significance of such finding has been studied in
Refs. [13, 14, 15], e.g., one finds that this light speed variation is
flavored at $3$-$5\sigma$ confidence level (C.L.) in Ref. [13], where we refer
the interested reader for further details. It is worth noting that such a LIV
scale 21 is consistent with various constraints from high-energy $\gamma$-ray
observations of pulsars [36, 37, 38, 39], active galactic nuclei (AGNs) [40,
41, 42, 43, 44] as well as GRBs [6, 7, 8, 45, 46, 47, 48, 49, 50] and it is
also compatible with the strongest robust limit to date [51] from a recent
study on 8 Fermi-LAT GRBs with bright emissions in multi-$\mathrm{GeV}$
energies [52]. We need to mention that there can be more stringent limits on
this characteristic scale of LIV from individual analyses of flaring PKS
2155-304 [53, 54],666In Ref. [53], time delays between two light curves of
different energy bands of PKS 2155-304 were used to give a lower bound of
$E_{\mathrm{LIV}}>7.2\times 10^{17}~{}\mathrm{GeV}$. However a latest work
[55] points out a novel way to understand the delays which can be considered
as a signal for the light speed variation with $E_{\mathrm{LIV}}\simeq
3.6\times 10^{17}~{}\mathrm{GeV}$, i.e., Eq. 21. We refer the reader to [55]
for details. short GRB 090510 [5, 56, 57] or striking TeV event of GRB 190114C
lately found by MAGIC [58, 59], which tend to place lower bounds
$E_{\mathrm{LIV}}\gtrsim\left(0.1-10\right)\times E_{\mathrm{Pl}}$, stronger
than that in Eq. 21. Therefore more observations are needed to further verify
the speed of light modification proposed in Refs. [11, 12, 13].

We discuss now the implications of the above phenomenological results from a
theoretical perspective. In Sec. 3 we saw that the dispersion relation and the
speed of light of the photon field are modified due to the interaction of
photon open-string with the D-particle string foam. We indicate in the
following that this fact suggests a natural way to explain the speed variation
of light mentioned above.

Note that one of the most interesting features revealed in Refs. [11, 12, 13]
is that multi-GeV events from different samples of GRBs line up surprisingly
to form a mainline, indicating a linear energy dependence in the speed of
light, $\lvert
v_{g}-1\rvert\sim\mathcal{O}\left(\mathcal{E}/E_{\mathrm{LIV}}\right)$, rather
than a quadratic (or higher-order) dependence of the photon energy, with
$E_{\mathrm{LIV}}$ characterizing such a linear suppression of LIV determined
to be a few $10^{17}~{}\mathrm{GeV}$, i.e., Eq. 21, 2-order less than the
Planck energy scale. If we consider this result in terms of the phase velocity
$v_{\mathrm{ph}}\equiv 1/\eta=\mathcal{E}/p$, taking into account the modified
dispersion relation 19, with

$\eta=1+\frac{1}{2}s_{n}\left(\frac{\mathcal{E}}{E_{\mathrm{LIV},n}}\right)^{n}.$
(22)

the light speed variation with $E_{\mathrm{LIV}}\simeq 3.60\times
10^{17}~{}\mathrm{GeV}$ just corresponds to the linear case, i.e., $n=1$ in
Eq. 22, which reduces to
$\eta=1+\frac{1}{2}\frac{\mathcal{E}}{E_{\mathrm{LIV}}}$. Thus the finding in
Refs. [11, 12, 13] implies that current data favor such a vacuum refractive
index that varies linearly with the energy of the photon. While according to
the D-brane-inspired space-time foam (c.f., Sec. 2), the non-relativistic
recoil of the foam defect when interacts with an energetic photon propagating
over our brane-world Universe can lead to a deformed background space-time
metric “felt” by the photon. As a result, the velocity of light receives a QG-
induced UV correction which grows linearly on the photon energy, $c_{g}\sim
1-\mathcal{E}/\mathcal{M}_{\mathrm{QG}\gamma}^{(\textrm{D-foam})}$, i.e., Eq.
18, with $\mathcal{M}_{\mathrm{QG}\gamma}^{(\textrm{D-foam})}$ approaching the
Planck mass. Due to the presence of the D-foam, the space-time vacuum appears
no longer transparent to photons, but behaves essentially like a dispersive
medium with non-trivial refractive indices
$\eta_{\mathrm{vac}}(\mathcal{E})=1+\mathcal{O}\left(\mathcal{E}/\mathcal{M}_{\mathrm{QG}\gamma}^{(\textrm{D-foam})}\right)$
for photon propagation. Obviously, these features of the string foam model is
exactly consistent with the linear form energy-dependence of light speed
observed by the authors of Refs. [11, 12, 13].

Furthermore, we saw that the result in these previous works leads also to a
positive sign factor (i.e., $s=+1$) for the linear LIV correction, indicating
the fact that photons with higher energies propagate more slowly than lower
ones, corresponding to a subluminal photon propagation.777It might be argued
that the non-observation of superluminal photon propagation in Refs. [11, 12,
13] may be attributed to $\gamma$-decays allowed by a modified superluminal
dispersion effect. However such superluminal vacuum dispersion has been
severely constrained by the absence of $\gamma$-decay [60, 61], which, as
demonstrated below, can serve as an evidence to support the D-particle foam
interpretation of light speed variation presented in this paper. From the
viewpoint of the phase velocity, this result implies $\eta>1$ in Eq. 22.
Within the D-brane foam context, the capture and re-emission processes of the
photon by D-particle defects induce a causal time delay 2, resulting in a
deceleration of high-energy photons by the D-particle foam. This effect
manifests itself through a subluminal velocity $c_{g}(\mathcal{E})\simeq
1-\varsigma_{\gamma,\mathrm{ani}}^{(\textrm{D-foam})}\mathcal{E}/M_{s}$ with
the quantity
$\varsigma_{\gamma,\mathrm{ani}}^{(\textrm{D-foam})}/M_{s}\sim\langle\\!\langle\lambda\rangle\\!\rangle_{D}/M_{D}$
(denoted by $\mathrm{LIV}_{\textrm{D-foam}}$) being positive by construction
(c.f., Sec. 3). Meanwhile, such a time delay 2 may be thought of as implying
an increasing refractive index of light,
$\eta_{\mathrm{vac}}=1+\mathcal{O}\left(\ell_{s}\mathcal{E}\right)>1$, with
photon energy. As a result, the more energetic one photon is, the slower it
travels through space. It becomes clear that such prediction of the space-time
foam model is consistent with the subluminal feature of photons, as exposed in
Refs. [11, 12, 13] from GRB observations.

With the established consistency between the propagation velocity from the
generalized LIV-modified dispersion relation in Eq. 20 with the pair
$(n,s_{n})$ setting to $(1,+1)$ and the velocity from the stringy space-time
foam scenario in Eqs. 15 and 18, we can relate
$\mathcal{M}_{\mathrm{QG}\gamma}^{(\textrm{D-foam})}$ to the Lorentz-violation
scale $E_{\mathrm{LIV}}$. On account of Eq. 21, we arrive at a relation

$\mathcal{M}_{\mathrm{QG}\gamma}^{(\textrm{D-foam})}\simeq
E_{\mathrm{LIV}}\simeq 3.60\times 10^{17}~{}\mathrm{GeV},$ (23)

or, equivalently we have

$\frac{\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}}{M_{s}}\simeq\frac{1}{2E_{\mathrm{LIV}}}\simeq
1.39\times 10^{-18}~{}\mathrm{GeV}^{-1}.$ (24)

The result 23 leads to a claim of an effective quantum-gravity scale $\sim
10^{17}~{}\mathrm{GeV}$ for the string/D-particle foam model. This is very
close to the expectation that this scale is taken to be very large, typically
an order of magnitude or so below the Planck scale,
$\mathcal{M}_{\mathrm{QG}\gamma}^{(\textrm{D-foam})}\lesssim\mathcal{M}_{\mathrm{Pl}}=\mathcal{O}\left(10^{18}\right)~{}\mathrm{GeV}$,
where $\mathcal{M}_{\mathrm{Pl}}=\sqrt{1/8\pi G}\approx 2\times
10^{18}~{}\mathrm{GeV}$ is the reduced Planck mass. If we take this result as
the lower bound on the scale of stringy quantum gravity in the space-time foam
scenario, then our result 23 is very similar to that obtained recently in Ref.
[51]. Given that
$\langle\\!\langle\lambda\rangle\\!\rangle_{D}\lesssim\mathcal{O}\left(1\right)$,
we can further deduce the constraint

$\frac{M_{s}}{g_{s}}<7.20\times 10^{17}~{}\mathrm{GeV},$ (25)

which is compatible with the natural assumption [33] for the value
$M_{s}/g_{s}\sim 10^{19}~{}\mathrm{GeV}$ of the mass of the foam defect.
Meanwhile, we find that our result leads to a rough estimate about the scale
$M_{s}\sim 3.60\times 10^{17}~{}\mathrm{GeV}$, which gives a string mass scale
nearly $0.15\times\mathcal{M}_{\mathrm{Pl}}$ if
$\varsigma_{\gamma,\mathrm{ani}}^{(\textrm{D-foam})}$ is of order $1$. While
in general, taking Eq. 24 into account, we get

$M_{s}\simeq 7.20\times
10^{17}\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}~{}\mathrm{GeV}.$
(26)

If
$\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}\sim\mathcal{O}\left(1\right)$
and then an intriguing estimation of $M_{s}$ is at the scale of
$\mathrm{a~{}few}\times 10^{17}~{}\mathrm{GeV}$, which is consistent with the
expectation in modern string theory that $M_{s}\neq M_{\mathrm{Pl}}$ in
general. While usually it is assumed in the string foam model that the
combination $\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}$ is (much)
lower than unity, then the string scale $M_{s}$ is roughly not higher than
$10^{16}~{}\mathrm{GeV}$ and it can be as low as a few $\mathrm{TeV}$, which
could be attained if $\langle\\!\langle\lambda\rangle\\!\rangle_{D}\ll 1$ or
$g_{s}\ll 1$. In this case, the result may be compatible with several schemes
of low-scale string [62, 63] where the mass scale $M_{s}$ can be decreased
arbitrarily low, though it cannot be lower than $\sim 10^{4}~{}\mathrm{GeV}$
since if this were the case we should have already observed fundamental
strings in the experiments at the Large Hadron Collider (LHC). In addition, we
also obtain analogous estimations of the string length scale $\ell_{s}$ and
the universal Regge slope $\alpha^{\prime}$ as

$\displaystyle\ell_{s}\simeq$ $\displaystyle\ 1.39\times
10^{-18}\frac{1}{\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}}~{}\mathrm{GeV}^{-1},$
(27) $\displaystyle\alpha^{\prime}\simeq$ $\displaystyle\ 1.93\times
10^{-36}\frac{1}{\langle\\!\langle\lambda\rangle\\!\rangle_{D}^{2}g_{s}^{2}}~{}\mathrm{GeV}^{-2}.$
(28)

For fundamental string length $\ell_{s}$, our estimate 27 further implies
$\ell_{s}\simeq 1.69\times
10^{-19}\frac{1}{\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}}~{}\mathrm{GeV}^{-1}\left(\frac{\ell_{\mathrm{Pl}}}{10^{-28}~{}\mathrm{eV}^{-1}}\right)\approx
1.7\frac{\ell_{\mathrm{Pl}}}{\langle\\!\langle\lambda\rangle\\!\rangle_{D}g_{s}}\sim\ell_{\mathrm{Pl}}/n_{D}g_{s}$,
with the Planck length $\ell_{\mathrm{Pl}}\simeq 1.6\times
10^{-35}~{}\mathrm{m}$.

From the above discussion we see that the light speed variation proposed
previously from GRB photons [11, 12, 13] can be described in a D-brane/string-
inspired model of space-time foam with
$\mathcal{M}_{\mathrm{QG}\gamma}^{(\textrm{D-foam})}\simeq
10^{17}~{}\mathrm{GeV}$. While there are still several classes of
complementary (astrophysical) constraints on LIV in the photon sector, which
need to be considered in our attempt to interpret the finding in Refs. [11,
12, 13]. These constraints could come from birefringence effects as well as
photon decay processes. We indicate below that these results can also be
understood in the framework of D-particle space-time foam, as shown in Ref.
[25].

Table 1: Some constraints on LIV-induced vacuum birefringence from
polarimetric observations of astrophysical sources. Lower bounds on LIV scale
for birefringence propagation
$E_{\mathrm{LIV}}^{(\mathrm{bire})}\gtrsim{\bm{\varepsilon}}=\left(4\ell_{\mathrm{Pl}}\lvert\chi\rvert_{\mathrm{upper}}\right)^{-1}$
are calculated.

Source | Distance |  $\lvert\chi\rvert_{\mathrm{upper}}$tn:a | ${\bm{\varepsilon}}~{}(\mathrm{GeV})$
---|---|---|---
3C 256 | $z\simeq 1.82$ |  $5\times 10^{-5}$ [64] | $6.1\times 10^{22}$
GRB 020813 | $z\simeq 1.26$ |  $3\times 10^{-7}$ [65] | $1.0\times 10^{25}$
Crab pulsar | $\sim 1.9~{}\mathrm{kpc}$ |  $2.3\times 10^{-10}$ [66] | $1.3\times 10^{28}$
Cygnus X-1 | $\sim 2.2~{}\mathrm{kpc}$ |  $8.7\times 10^{-12}$ [67] | $3.5\times 10^{29}$
GRB 041219A | $z\sim 0.31$ |  $5.5\times 10^{-15}$ [68] | $5.5\times 10^{32}$
GRB 110721A | $z\simeq 0.38$ |  $2.6\times 10^{-16}$ [69] | $1.2\times 10^{34}$
GRB 160509A | $z\simeq 1.17$ |  $4\times 10^{-17}$ [69]tn:b | $7.6\times 10^{34}$
GRB 061122 | $z\simeq 1.33$ |  $2.5\times 10^{-17}$ [70] | $1.2\times 10^{35}$

  * a

The numerical factors between our notation from those of quoted references are
accounted.

  * b

In Ref. [69], 12 latest polarized GRBs were assembled to place strict limits
on $\chi$ in the order of $\mathcal{O}\left(10^{-15}-10^{-17}\right)$. We
refer the reader to [69] for more details on these impressive results.

We focus first on strong constraints based on vacuum birefringence.
Birefringence is predicted in certain models of QG, with both Lorentz
invariance and reflexion symmetry (parity/CPT) violation, where the velocity
variation of two polarization modes (denoted by $h_{\pm}$) can be parametrized
by

$\delta v_{g}(\mathcal{E},h_{\pm})=\mp 4\chi\mathcal{E}/M_{\mathrm{Pl}}.$ (29)

In Eq. 29 $\chi$ is a parameter following from the underlying theory, which
characterizes the helicity dependence of photon velocity and governs CPT-odd
LIV effects. Limits on the parameter $\chi$ can be interpreted in terms of an
analogous LIV energy scale for birefringence propagation

$E_{\mathrm{LIV}}^{(\mathrm{bire})}=\left\lvert
4\chi\ell_{\mathrm{Pl}}\right\rvert^{-1}.$ (30)

Strong constraints on $\chi$ (or $E_{\mathrm{LIV}}^{(\mathrm{bire})}$) have
been set in astro-particle physics by wide studies on polarization
measurements from various sources. In Table 1, we list some results obtained
in the last two decades. For example, the observation of polarized lights from
the Crab Nebula constrains
$\mathcal{O}\left({\mathcal{E}}/M_{\mathrm{Pl}}\right)$ LIV in Eq. 29 to the
level $\lvert\chi\rvert<2\times 10^{-10}$ at $95\%$ C.L. [66], which
corresponds to $E_{\mathrm{LIV}}^{(\mathrm{bire})}\gtrsim 1\times
10^{28}~{}\mathrm{GeV}$. The most stringent constraint comes from GRB 061122
and the birefringence parameter $\chi$ is determined to be
$\lvert\chi\rvert\lesssim 10^{-17}$ [70], tightening by about seven orders of
magnitude the Crab Nebula constraint. Hence one finds from the table that
polarimetric observations of light from remote astronomical sources put very
strong limits on a modified birefringent speed of light 29 and almost rule out
LIV-motivated birefringence effects. While in the stringy space-time foam
scenario, the coupling of the photons to the D-foam medium is independent of
photon polarization. Specifically, we saw in Eqs. 1, 2 and 10 (or Eq. 15) that
string calculations do not yield a dependence of the speed of light on its
polarization mode, thereby the coefficients
($\varsigma_{\gamma,\mathrm{ani}}^{(\textrm{D-foam})}$) for left- and right-
handed circularly polarized states are necessarily the same,
$\varsigma_{\gamma,\mathrm{ani},h_{+}}^{(\textrm{D-foam})}=\varsigma_{\gamma,\mathrm{ani},h_{-}}^{(\textrm{D-foam})}$.
Photons with opposite “helicities” but the same frequency have the same
amounts of speed variation

$\delta c_{g}(\mathcal{E})\simeq-n_{D}g_{s}\mathcal{E}/M_{s},~{}\text{for
states}~{}~{}h_{+},h_{-},$ (31)

and as a result, the usual degeneracy of $c_{g}$ amongst polarizations is
preserved when Lorentz symmetry is broken. Birefringence is therefore not
expected to present in this parity-even theory. This fact leads to a natural
way to account for the above-mentioned lower limits for
$E_{\mathrm{LIV}}^{(\mathrm{bire})}$. Since the D-particle string foam does
not predict vacuum birefringence, the polarimetric observation cannot be used
to bound $\mathrm{LIV}_{\textrm{D-foam}}$. Thus, those tight restrictions on
$E_{\mathrm{LIV}}^{(\mathrm{bire})}$ calculated in Table 1 cannot translate to
upper limits for the quantity $n_{D}g_{s}/M_{s}$ in Eq. 31, and in such a way
that the above stringent constraints are no longer valid, thereby producing no
incompatibility with that given in Eqs. 23 and 24. On the other hand,
polarized light measurements do result in severe limits on linearly suppressed
LIV, as seen from Table 1 (i.e., for linear models 29 the lower bound on
$E_{\mathrm{LIV}}^{(\mathrm{bire})}$ exceeds the Planck scale by as much as
sixteen orders of magnitude). However, with the stringy foam model, one can
easily understand the reason for that. Indeed, if the vacuum birefringence
effect does not exist at all, as theoretically predicted, then it would be
natural to expect that extremely tight constraints on any effects induced by a
helicity-$h_{\pm}$-dependent speed variation 29 should be placed by
experiments. This is exactly the situation for current observational results.
Therefore, it is clear that the D-particle foam model is capable of explaining
well the constraints from the absence of detectable birefringence.

Moreover, following that same logic, we can interpret the results from another
LIV phenomenon, namely the photon decay, $\gamma\rightarrow e^{+}e^{-}$,
allowed by superluminal LIV in Eq. 29. Since photon decay in flight from the
source would lead to a hard cutoff in astrophysical $\gamma$-ray spectra [71],
a lower bound for
$E_{\mathrm{LIV}}^{(\gamma\textrm{-decay})}=M_{\mathrm{Pl}}/4\lvert\chi\rvert$
then emerges directly from any observed ultra-high-energy cosmic photon [72,
73]. Actually, the observations of multi-TeV events from the Crab Nebula [74]
have already cast stringent limits on photon decay, with the LIV parameter
constrained to the level $\lvert\chi\rvert\lesssim 10^{-2}$, thanks to HEGRA
[75, 76] and H.E.S.S. experiments [77, 78]. The strongest result to date comes
from recent observations of $\gamma$-rays above $100~{}\mathrm{TeV}$ with
HAWC. In Ref. [79] the HAWC $2\sigma$ ($95\%$ C.L.) LIV constraint turns out
to be $E_{\mathrm{LIV}}^{(\gamma\textrm{-decay})}\gtrsim 2.22\times
10^{22}~{}\mathrm{GeV}$, over 1800 times the Planck scale
($E_{\mathrm{Pl}}\approx 10^{19}~{}\mathrm{GeV}$), in such a way that HAWC
observations could exclude the linear-order LIV scale of new physics. However,
this is not the case, and we point out that these results do not constrain the
linear speed of light modification of the type we considered here (c.f., 18).
The reason is that photons only possess subluminal in-vacuo dispersion 10 in
the quantum-gravitational D-brane foam medium, as characterized by the
quantity $\mathrm{LIV}_{\textrm{D-foam}}>0$, without any superluminal
propagation predicted. This implies that photons are stable (i.e., no photon
decay is kinematically allowed), and therefore, using the results based on
superluminal dominant reactions, e.g., $\gamma$-decays, one cannot derive the
related constraints for the QG coefficients
$M_{s}/\varsigma_{\gamma,\mathrm{ani}}^{(\textrm{D-foam})}$ within such a
model. There is, again, no incompatibility between these strong limits with
our main result 24 presented above. From another point of view, since the
photon decay does not even exist in this framework, it is straightforward to
expect that no observation would favor such a process to take place for
photons at high energies, and therefore any attempts to search for a hard
cutoff compatible with the superluminal part of 29 in the observed photon
spectra would necessarily result in very strong limits on relevant parameter
$\chi$ or related energy scale $E_{\mathrm{LIV}}^{(\gamma\textrm{-decay})}$.
In this way, we finally get a consistent description with the $\gamma$-decay
limits in the stringy space-time foam scenario.

In summary, given the positive indication of Lorentz violation when analyzing
the GRB time delays detected by FGST, we observe that the finding in Refs.
[11, 12, 13] can be described in a stringy model of space-time foam, while
being compatible with other astrophysical constraints available to date. We
indicate that the strong constraints on LIV imposed from quantum gravitational
birefringence effects and superluminal dominant phenomena, e.g., LIV photon
decays, can be fully understood in this framework as well. This fact in turn
serves as an evidence to support the D-particle string foam interpretation of
light speed variation we present here. Worthy to mention that, with the
improvement of thirteen orders of magnitude over the past two decades, we
could expect to have extra improvements on constraining possible vacuum
birefringence, as well as those effects due to superluminal LIV. If more
stringent limit can be cast in the future, then this would be a strong support
of our D-particle foam interpretation, for reasons explained above. Of course,
the validity of the work in Refs. [11, 12, 13] still needs confirmation
through future observations of gamma-rays from the Fermi telescope or next
generation of satellites (e.g., the incoming GrailQuest [80, 81]), and the
interpretation of such light speed variance also needs to be continuously
subject to further scrutiny.

##  5 Closing remarks

In this paper, we present an inspiring way to consistently describe the light
speed variation suggested previously from analyses of GRB photons [9, 10, 11,
12, 13, 14] in the framework of D-particle space-time foam. The fact that this
theory coincides well with the observations from Fermi data to date is rather
intriguing and important, not only because we provide a novel perspective on
accounting for the finding in Refs. [11, 12, 13], while in agreement with
other constraints on LIV, but also because these high-energy astrophysical
measurements could further validate hopefully the theoretical details of this
type of theories. We conclude that the anomalies of the cosmic photon
propagation might have a realistic string-theory origin. More importantly,
such an explanation for the light speed variation can also provide hints for a
very interesting and long-lasting conjecture that space-time may not be
smooth, but “foamy” at tiny scales, thus could provide an insightful
understanding of the nature of our Universe.

## Acknowledgements

We are grateful to Tianjun Li and Hao Li for valuable discussions during the
developments of this work. This work is supported by National Natural Science
Foundation of China (Grant No. 12075003).

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