Extra dimensions and Lorentz invariance violation
Viktor Baukh∗ and Alexander Zhuk†
Department of Theoretical Physics and Astronomical Observatory,
Odessa National University, 2 Dvoryanskaya St., Odessa 65026, Ukraine
Tina Kahniashvili‡
CCPP, New York University, 4 Washington Place, New York, NY 10003, USA
National Abastumani Astrophysical Observatory, 2A Kazbegi Ave, Tbilisi, GE-0160 Georgia
We consider effective model where photons interact with scalar field corresponding to conformal
excitations of the internal space (geometrical moduli/gravexcitons). We demonstrate that this
interaction results in a modified dispersion relation for photons, and consequently, the photon group
velocity depends on the energy implying the propagation time delay effect. We suggest to use the
experimental bounds of the time delay of gamma ray bursts (GRBs) photons propagation as an
additional constrain for the gravexciton parameters.
PACS numbers: 04.50.+h, 11.25.Mj, 98.80.-k
Lorentz invariance (LI) of physical laws is one of the
corner stone of modern physics. There is a number of ex-
periments confirming this symmetry at energies we can
approach now. For example, on a classical level, the ro-
tation invariance has been tested in Michelson-Morley
experiments, and the boost invariance has been tested
in Kennedy-Torhndike experiments [1]. Although, up
to now, LI is well established experimentally, we can-
not say surely that at higher energies it is still valid.
Moreover, modern astrophysical and cosmological data
(e.g. UHECR, dark matter, dark energy, etc) indicate
for a possible LI violation (LV). To resolve these chal-
lenges, there are number of attempts to create new phys-
ical models, such as M/string theory, Kaluza-Klein mod-
els, brane-world models, etc. [1].
In this paper we investigate LV test related to photon
dispersion measure (PhDM). This test is based on the
LV effect of a phenomenological energy-dependent speed
of photon [2, 3, 4, 5, 6, 7, 8], for recent studies see Ref.
[9] and references therein.
The formalism that we use is based on the analogy
with electromagnetic waves propagation in a magnetized
medium, and extends previous works [8, 10, 11]. In our
model, instead of propagation in a magnetized medium,
the electromagnetic waves are propagating in vacuum
filled with a scalar field ψ. LV occurs because of an in-
teraction term f(ψ)F 2 where F is an amplitude of the
electromagnetic field. Such an interaction might have
different origins. In the string theory ψ could be a dila-
ton field [12, 13]. The field ψ could be associated with
geometrical moduli. In brane-world models the similar
term describes an interaction between the bulk dilaton
and the Standard Model fields on the brane [14]. In
Ref. [15], such an interaction was obtained in N = 4
∗Electronic address: bauch˙vGR@ukr.net
†Electronic address: zhuk@paco.net
‡Electronic address: tinatin@phys.ksu.edu
super-gravity in four dimensions. In Kaluza-Klein mod-
els the term f(ψ)F 2 has the pure geometrical origin, and
it appears in the effective, dimensionally reduced, four
dimensional action (see e.g. [16, 17]). In particular, in
reduced Einstein-Yang-Mills theories, the function f(ψ)
coincides (up to a numerical prefactor) with the volume
of the internal space. Phenomenological (exactly solv-
able) models with spherical symmetries were considered
in Refs. [18]. To be more specific, we consider the model
which is based on the reduced Einstein-Yang-Mills the-
ory [17], where the term ∝ ψF 2 describes the interaction
between the conformal excitations of the internal space
(gravexcitons) and photons. It is clear that the similar
LV effect exists for all types of interactions of the form
f(ψ)F 2 mentioned above.
Obviously, the interaction term f(ψ)F 2 modifies the
Maxwell equations, and, consequently, results in a mod-
ified dispersion relation for photons. We show that
this modification has rather specific form. For example,
we demonstrate that refractive indices for the left and
right circularly polarized waves coincide with each other.
Thus, rotational invariance is preserved. However, the
speed of the electromagnetic wave’s propagation in vac-
uum differs from the speed of light c. This difference
implies the time delay effect which can be measured via
high-energy GRB photons propagation over cosmological
distances (see e.g. Ref. [9]). It is clear that gravexcitons
should not overclose the Universe and should not result
in variations of the fine structure constant. These de-
mands lead to a certain constrains for gravexcitons (see
Refs. [17, 19]). We use the time delay effect, caused by
the interaction between photons and gravexcitons, to get
additional bounds on the parameters of gravexcitons.
The starting point of our investigation is the Abelian
part of D-dimensional action of the Einstein-Yang-Mills
theory:
SEM = −
|g|FMNFMN , (1)
http://arxiv.org/abs/0704.0314v4
mailto:bauch_vGR@ukr.net
mailto:zhuk@paco.net
mailto:tinatin@phys.ksu.edu
where the D-dimensional metric, g = gMN (X)dX
dXN = g(0)(x)µνdx
µ ⊗ dxν + a21(x)g(1), is defined on
the product manifold M = M0 × M1. Here, M0 is
the (D0 = d0 + 1)-dimensional external space. The d1-
dimensional internal space M1 has a constant curvature
with the scale factor a1(x) ≡ LPl expβ1(x). Dimensional
reduction of the action (1) results in the following effec-
tive D0-dimensional action [17]
S̄EM = −
|g̃(0)| [(1−Dκ0ψ)FµνFµν ] , (2)
which is written in the Einstein frame with the D0-
dimensional metric, g̃
µν = (exp d1β̄
1)−2/(D0−2)g
Here, κ0ψ ≡ −β̄1
(D0 − 2)/d1(D − 2) ≪ 1 and β̄1 ≡
β1 − β10 are small fluctuations of the internal space scale
factor over the stable background β10 (0 subscript de-
notes the present day value). These internal space scale-
factor small fluctuations/oscillations have the form of
a scalar field (so called gravexciton [20]) with a mass
mψ defined by the curvature of the effective potential
(see for detail [20]). Action (2) is defined under the
approximation κ0ψ < 1 that obviously holds for the
condition1 ψ < MPl. κ
0 = 8π/M
Pl is four dimen-
sional gravitational constant, MPl is the Plank mass,
D = 2
d1/[(D0 − 1)(D − 1)] is a model dependent con-
stant. The Lagrangian density for the scalar field ψ reads:
|g̃(0)|(−g̃µνψ,µψ,ν−m2ψψψ)/2. For simplicity we
assume that g̃0 is the flat Friedman-Lemaitre-Robertson-
Walker (FLRW) metric with the scale factor a(t).
Let’s consider Eq. (2). It is worth of noting that the
D0-dimensional field strength tensor, Fµν , is gauge in-
variant.2 Secondly, action (2) is conformally invariant in
the case when D0 = 4. The transform to the Einstein
frame does not break gauge invariance of the action (2),
and the electromagnetic field is antisymmetric as usual,
Fµν = ∂µAν − ∂νAµ. Varying (2) with respect to the
electromagnetic vector potential,
−g (1−Dκ0ψ)Fµν
= 0. (3)
The second term in the round brackets Dκ0ψFµν reflects
the interaction between photons and the scalar field ψ,
and as we show below, it is responsible for LV. In par-
ticular, coupling between photons and the scalar field ψ
makes the speed of photons different from the standard
speed of light. Eq. (3) together with Bianchi identity
(which is preserved in the considered model due to gauge-
invariance of the tensor, Fµν [17]) defines a complete set
1 In the brane-world model the prefactor κ0 in the expression for
κ0ψ is replaced by the parameter proportional to M
[14].
Thus, the smallness condition holds for ψ < MEW .
2 Eq. (2) can be rewritten in the more familiar form S̄EM =
−(1/2)
|g̃(0)|F̄µν F̄
µν [17]. The field strength tensor
F̄µν is not gauge invariant here.
of the generalized Maxwell equations. As we noted, ac-
tion (2) is conformally invariant in the 4D dimensional
space-time. So, it is convenient to present the flat FLRW
metric g̃0 in the conformally flat form: g̃0µν = a
2ηµν ,
where ηµν is the Minkowski metric.
Using the standard definition of the electromagnetic
field tensor, Fµν , we obtain the complete set of the
Maxwell equations in vacuum,
∇ ·B = 0 , (4)
∇ ·E = Dκ0
1−Dκ0ψ
(∇ψ ·E) , (5)
∇×B = ∂E
− Dκ0ψ̇
1−Dκ0ψ
1−Dκ0ψ
[∇ψ ×B] , (6)
∇×E = −∂B
, (7)
where all operations are performed in the Minkowski
space-time, η denotes conformal time related to physi-
cal time t as dt = a(η)dη, and an overdot represents a
derivative with respect to conformal time η.
Eqs. (4) and (7) correspond to Bianchi identity, and
since it is preserved, Eqs. (4) and (7) keep their usual
forms. Eqs. (5) and (6) are modified due to interactions
between photons and gravexcitons (∝ κ0ψ). These mod-
ifications have simple physical meaning: the interaction
between photons and the scalar field ψ acts as an effective
electric charge eeff . This effective charge is proportional
to the scalar product of the ψ field gradient and the E
field, and it vanishes for an homogeneous ψ field. The
modification of Eq. (6) corresponds to an effective cur-
rent Jeff , which depends on both electric and magnetic
fields. This effective current is determined by variations
of the ψ field over the time (ψ̇) and space (∇ψ). For
the case of a homogeneous ψ field the effective current is
still present and LV takes place. The modified Maxwell
equations are conformally invariant. To account for the
expansion of the Universe we rescale the field components
asB,E → B,E a2 [21].
To obtain a dispersion relation for photons, we use
the Fourier transform between position and wavenumber
spaces as,
F(k, ω) =
dη d3x e−i(ωη−k·x)F(x, η) ,
F(x, η) =
(2π)4
dω d3kei(ωη−k·x)F(k, ω) . (8)
Here, F is a vector function describing either the elec-
tric or the magnetic field, ω is the angular frequency of
the electro-magnetic wave measured today, and k is the
wave-vector. We assume that the field ψ is an oscilla-
tory field with the frequency ωψ and the momentum q,
so ψ(x, η) = Cei(ωψη−q·x) , C = const . Eq. (4) implies
B ⊥ k. Without loosing of generality, and for simplic-
ity of description we assume that the wave-vector k is
oriented along the z axis. Using Eq. (7) we get E ⊥ B.
A linearly polarized wave can be expressed as a super-
position of left (L, −) and right (R, +) circularly polar-
ized (LCP and RCP) waves. Using the polarization basis
of Sec. 1.1.3 of Ref. [22], we derive E± = (Ex± iEy)/
Rewriting Eqs. (4) - (7) in the components,3 for LCP
and RCP waves we get,
(1 − n2+)E+ = 0, (1− n2(−))E
− = 0 , (9)
where n+ and n− are refractive indices for RCP and LCP
electromagnetic waves
n2+ =
k2 [1−Dκ0ψ(1 + qz/k)]
ω2 [1−Dκ0ψ(1 + ωψ/ω)]
= n2− . (10)
In the case when LI is preserved the electromagnetic
waves propagating in vacuum have n+ = n− = n =
k/ω ≡ 1. For the electromagnetic waves propagating in
the magnetized plasma, k/ω 6= 1, and the difference be-
tween the LCP and RCP refractive indices describes the
Faraday rotation effect, α ∝ ω(n+ − n−) [23]. In the
considered model, since n+ = n− the rotation effect is
absent, but the speed of electromagnetic waves propaga-
tion in vacuum differs from the speed of light c (see also
Ref. [24] for LV induced by electromagnetic field cou-
pling to other generic field). This difference implies the
propagation time delay effect, ∆t = ∆l(1−∂k/∂ω) (∆l is
a propagation distance), ∆t is the difference between the
photon travel time and that for a ”photon” which travels
at the speed of light c. Here, t is physical synchronous
time. This formula does not take into account the evo-
lution of the Universe. However, it is easy to show that
the effect of the Universe expansion is negligibly small.
Solving the dispersion relation as a square equation,
we obtain
ω2ψ − q2z
(Dκ0ψ)2
, (11)
where ± signs correspond to photons forward and back-
ward directions respectively.
The modified inverse group velocity (11) shows that
the LV effect can be measured if we know the gravexciton
frequency ωψ, z-component of the momentum qz and its
amplitude ψ. For our estimates, we assume that ψ is
the oscillatory field, satisfying (in local Lorentz frame)
the dispersion relation, ω2ψ = m
ψ + q
2, where mψ is the
mass of gravexcitons4. Unfortunately, we do not have
3 We have defined the system of 6 equations with respect to 6
components of the vectors E and B. This system has non-trivial
solutions only if its determinant is nonzero. From this condition
we get the dispersion relation. The Faraday rotation effect is
absent if the matrix has a diagonal form.
4 To get physical values of the corresponding parameters we should
rescale them by the scale factor a.
any information concerning parameters of gravexcitons
(some estimates can be found in [17, 19]). Thus, we
intend to use possible LV effects (supposing it is caused
by interaction between photons and gravexcitons) to set
limits on gravexciton parameters. For example, we can
easily get the following estimate for the upper limit of
the amplitude of gravexciton oscillations:
|ψ| ≈ 1√
MPl , (12)
where for ω and mψ we can use their physical values.
In the case of GRB with ω ∼ 1021 ÷ 1022Hz ∼ 10−4 ÷
10−3GeV and ∆l ∼ 3 ÷ 5 × 109y ∼ 1017sec the typical
upper limit for the time delay is ∆t ∼ 10−4sec [9]. For
these values the upper limit on gravexciton amplitude of
oscillations is5
|κ0ψ| ≈
10−13GeV
. (13)
This estimate shows that our approximation κ0ψ < 1
works for gravexciton masses mψ > 10
−13GeV. Future
measurements of the time-delay effect for GRBs at fre-
quencies ω ∼ 1 − 10GeV would increase significantly
the limit up to mψ > 10
−9GeV. On the other hand,
Cavendish-type experiments [26, 27]) exclude fifth force
particles with masses mψ . 1/(10
−2cm) ∼ 10−12GeV
which is rather close to our lower bound for ψ field
masses. Respectively we slightly shift the considered
mass lower limit to be mψ ≥ 10−12GeV. These masses
considerably higher than the mass corresponding to the
equality between the energy densities of the matter and
radiation (matter/radiation equality), meq ∼ Heq ∼
10−37GeV, where Heq is the Hubble ”constant” at mat-
ter/radiation equality. It means that such ψ-particles
start to oscillate during the radiation dominated epoch
(see appendix). Another bound on the ψ-particles masses
comes from the condition of their stability. With re-
spect to decay ψ → γγ the life-time of ψ-particles is
τ ∼ (MPl/mψ)3tPl [17], and the stability conditions re-
quires that the decay time should be greater than the age
of the Universe. According this we consider light gravex-
citons with masses mψ ≤ 10−21MPl ∼ 10−2GeV ∼ 20me
(where me is the electron mass).
As an additional restriction arises from the condi-
tion that such cosmological gravexcitons should not
overclose the observable Universe. This reads mψ .
meq(MPl/ψin)
4 which implies the following restriction
for the amplitude of the initial oscillations: ψin .
(meq/mψ)
MPL << MPl [19]. Thus, for the range of
masses 10−12GeV ≤ mψ ≤ 10−2GeV, we obtain respec-
tively ψin . 10
−6MPl and ψin . 10
−9MPl. According to
5 We thank R. Lehnert to point that in addition of the time de-
lay effect the Cherenkov effect could be used to constrain the
electromagnetic field and ψ field coupling strength [25].
Eq. (A.3), we can also get the estimate for the amplitude
of oscillations of the considered gravexciton at the present
time. Together with the non-overcloseness condition,
we obtain from this expression that |κ0ψ| ∼ 10−43 for
mψ ∼ 10−12GeV and ψin ∼ 10−6MPl and |κ0ψ| ∼ 10−53
for mψ ∼ 10−2GeV and ψin ∼ 10−9MPl. Obviously, it is
much less than the upper limit (13). Note, as we men-
tioned above, gravexcitons with masses mψ & 10
−2GeV
can start to decay at the present epoch. However, taking
into account the estimate |κ0ψ| ∼ 10−53, we can easily
get that their energy density ρψ ∼ (|κ0ψ|2/8π)M2Plm2ψ ∼
10−55g/cm3 is much less than the present energy density
of the radiation ργ ∼ 10−34g/cm3. Thus, ρψ contributes
negligibly in ργ . Otherwise, the gravexcitons with masses
mψ & 10
−2GeV should be observed at the present time,
which, obviously, is not the case.
Additionally, it follows from Eq. (42) in Ref. [17]
that to avoid the problem of the fine structure constant
variation, the amplitude of the initial oscillations should
satisfy the condition: ψin . 10
−5MPl which, obviously,
completely agrees with our upper bound ψin . 10
−6GeV.
Summarizing we shown that LV effects can give addi-
tional restrictions on parameters of gravexcitons. First,
we found that gravexcitons should not be lighter than
10−13GeV. It is very close to the limit following from the
fifth-force experiment. Moreover, experiments for GRB
at frequencies ω > 1GeV can result in significant shift of
this lower limit making it much stronger than the fifth-
force estimates. Together with the non-overcloseness con-
dition, this estimate leads to the upper limit on the am-
plitude of the gravexciton initial oscillations. It should
not exceed ψin . 10
−6GeV. Thus, the bound on the ini-
tial amplitude obtained from the fine structure constant
variation is one magnitude weaker than our one even for
the limiting case of the gravexciton masses. Increasing
the mass of gravexcitons makes our limit stronger. Our
estimates for the present day amplitude of the gravexci-
ton oscillations, following from the obtained above lim-
itations, show that we cannot use the LV effect for the
direct detections of the gravexcitons. Nevertheless, the
obtained bounds can be useful for astrophysical and cos-
mological applications. For example, let us suppose that
gravexcitons with masses mψ > 10
−2GeV are produced
during late stages of the Universe expansion in some re-
gions and GRB photons travel to us through these re-
gions. Then, Eq. (A.3) is not valid for such gravexcitons
having astrophysical origin and the only upper limit on
the amplitude of their oscillations (in these regions) fol-
lows from Eq. (13). In the case of TeV masses we get
|κ0ψ| ∼ 10−16. If GRB photons have frequencies up to
1 TeV, ω ∼ 1TeV, then this estimate is increased by 6
orders of magnitude.
Acknowledgments
We thank G. Dvali, G. Gabadadze, A. Gruzinov, G.
Melikidze, B. Ratra, and A. Starobinsky for stimulating
discussions. T. K. and A. Zh. acknowledge hospital-
ity of Abdus Salam International Center for Theoreti-
cal Physics (ICTP) where this work has been started.
A.Zh. would like to thank the Theory Division of CERN
for their kind hospitality during the final stage of this
work. T.K. acknowledges partial support from INTAS
061000017-9258 and Georgian NSF ST06/4-096 grants.
A. Appendix: Dynamics of Light Gravexcitons
In this appendix we briefly summarize the main prop-
erties of the light gravexcitons necessary for our inves-
tigations. The more detail description can be found in
Refs. [17, 19].
The effective equation of motion for massive cosmolog-
ical gravexciton6 is
ψ + (3H + Γ)
ψ +m2ψψ = 0 , (A.1)
where H ∼ 1/t and Γ ∼ m3ψ/M2Pl are the Hubble pa-
rameter and decay rate (ψ → γγ) correspondingly. This
equation shows that at times when the Hubble parame-
ter is less than the gravexciton mass: H . mψ the scalar
field begins to oscillate (i.e. time tin ∼ H−1in ∼ 1/mψ
roughly indicates the beginning of the oscillations):
ψ ≈ CB(t) cos(mψt+ δ) . (A.2)
We consider cosmological gravexcitons with masses
10−12GeV ≤ mψ ≤ 10−2GeV. The lower bound fol-
lows both from the fifth-force experiments and Eq. (13).
The upper bound follows from the demand that the life-
time of these particles (with respect to decay ψ → γγ)
is larger than the age of the Universe: τ = 1/Γ ∼
(MPl/mψ)
tPl ≥ 1019sec > tuniv ∼ 4 × 1017 sec. Thus,
we can neglect the decay processes for these gravexci-
tons. Additionally, it can be easily seen that these par-
ticles start to oscillate before teq ∼ H−1eq when the en-
ergy densities of the matter and radiation become equal
to each other (matter/radiation equality). According to
the present WMAP data for the ΛCDM model it holds
Heq ≡ meq ∼ 10−56MPl ∼ 10−28eV. Thus, considered
particles have masses mψ >> meq and start to oscil-
late during the radiation dominated stage. They will not
overclose the observable Universe if the following condi-
tion is satisfied: mψ . meq(MPl/ψin)
4, where ψin is the
amplitude of the initial oscillations at the moment tin
(see Eq. (18) in Ref. [19]).
Prefactors C and B(t) in Eq. (A.2) for con-
sidered light gravexcitons respectively read: C ∼
(ψin/MPl) (MPl/mψ)
and B(t) ∼ MPl (MPlt)−3s/2.
Here, s = 1/2, 2/3 for oscillations during the radiation
6 We have seen that the interaction between gravexcitons and or-
dinary matter (in our case it is 4D-photons) is suppressed by the
Planck scale. Thus, gravexcitons are weakly interacting massive
particles (WIMPs).
dominated and matter dominated stages, correspond-
ingly. We are interested in the gravexciton oscillations
at the present time t = tuniv. In this case s = 2/3 and
for B(tuniv) we obtain: B(tuniv) ∼ t−1univ ≈ 10−61MPl.
Thus, the amplitude of the light gravexciton oscillations
at the present time reads:
|κ0ψ| ∼ 10−60
. (A.3)
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http://arxiv.org/abs/gr-qc/9405021

  We consider effective model where photons interact with scalar field
corresponding to conformal excitations of the internal space (geometrical
moduli/gravexcitons). We demonstrate that this interaction results in a
modified dispersion relation for photons, and consequently, the photon group
velocity depends on the energy implying the propagation time delay effect. We
suggest to use the experimental bounds of the time delay of gamma ray bursts
(GRBs) photons propagation as an additional constrain for the gravexciton
parameters.

Extra dimensions and Lorentz invariance violation
Viktor Baukh∗ and Alexander Zhuk†
Department of Theoretical Physics and Astronomical Observatory,
Odessa National University, 2 Dvoryanskaya St., Odessa 65026, Ukraine
Tina Kahniashvili‡
CCPP, New York University, 4 Washington Place, New York, NY 10003, USA
National Abastumani Astrophysical Observatory, 2A Kazbegi Ave, Tbilisi, GE-0160 Georgia
We consider effective model where photons interact with scalar field corresponding to conformal
excitations of the internal space (geometrical moduli/gravexcitons). We demonstrate that this
interaction results in a modified dispersion relation for photons, and consequently, the photon group
velocity depends on the energy implying the propagation time delay effect. We suggest to use the
experimental bounds of the time delay of gamma ray bursts (GRBs) photons propagation as an
additional constrain for the gravexciton parameters.
PACS numbers: 04.50.+h, 11.25.Mj, 98.80.-k
Lorentz invariance (LI) of physical laws is one of the
corner stone of modern physics. There is a number of ex-
periments confirming this symmetry at energies we can
approach now. For example, on a classical level, the ro-
tation invariance has been tested in Michelson-Morley
experiments, and the boost invariance has been tested
in Kennedy-Torhndike experiments [1]. Although, up
to now, LI is well established experimentally, we can-
not say surely that at higher energies it is still valid.
Moreover, modern astrophysical and cosmological data
(e.g. UHECR, dark matter, dark energy, etc) indicate
for a possible LI violation (LV). To resolve these chal-
lenges, there are number of attempts to create new phys-
ical models, such as M/string theory, Kaluza-Klein mod-
els, brane-world models, etc. [1].
In this paper we investigate LV test related to photon
dispersion measure (PhDM). This test is based on the
LV effect of a phenomenological energy-dependent speed
of photon [2, 3, 4, 5, 6, 7, 8], for recent studies see Ref.
[9] and references therein.
The formalism that we use is based on the analogy
with electromagnetic waves propagation in a magnetized
medium, and extends previous works [8, 10, 11]. In our
model, instead of propagation in a magnetized medium,
the electromagnetic waves are propagating in vacuum
filled with a scalar field ψ. LV occurs because of an in-
teraction term f(ψ)F 2 where F is an amplitude of the
electromagnetic field. Such an interaction might have
different origins. In the string theory ψ could be a dila-
ton field [12, 13]. The field ψ could be associated with
geometrical moduli. In brane-world models the similar
term describes an interaction between the bulk dilaton
and the Standard Model fields on the brane [14]. In
Ref. [15], such an interaction was obtained in N = 4
∗Electronic address: bauch˙vGR@ukr.net
†Electronic address: zhuk@paco.net
‡Electronic address: tinatin@phys.ksu.edu
super-gravity in four dimensions. In Kaluza-Klein mod-
els the term f(ψ)F 2 has the pure geometrical origin, and
it appears in the effective, dimensionally reduced, four
dimensional action (see e.g. [16, 17]). In particular, in
reduced Einstein-Yang-Mills theories, the function f(ψ)
coincides (up to a numerical prefactor) with the volume
of the internal space. Phenomenological (exactly solv-
able) models with spherical symmetries were considered
in Refs. [18]. To be more specific, we consider the model
which is based on the reduced Einstein-Yang-Mills the-
ory [17], where the term ∝ ψF 2 describes the interaction
between the conformal excitations of the internal space
(gravexcitons) and photons. It is clear that the similar
LV effect exists for all types of interactions of the form
f(ψ)F 2 mentioned above.
Obviously, the interaction term f(ψ)F 2 modifies the
Maxwell equations, and, consequently, results in a mod-
ified dispersion relation for photons. We show that
this modification has rather specific form. For example,
we demonstrate that refractive indices for the left and
right circularly polarized waves coincide with each other.
Thus, rotational invariance is preserved. However, the
speed of the electromagnetic wave’s propagation in vac-
uum differs from the speed of light c. This difference
implies the time delay effect which can be measured via
high-energy GRB photons propagation over cosmological
distances (see e.g. Ref. [9]). It is clear that gravexcitons
should not overclose the Universe and should not result
in variations of the fine structure constant. These de-
mands lead to a certain constrains for gravexcitons (see
Refs. [17, 19]). We use the time delay effect, caused by
the interaction between photons and gravexcitons, to get
additional bounds on the parameters of gravexcitons.
The starting point of our investigation is the Abelian
part of D-dimensional action of the Einstein-Yang-Mills
theory:
SEM = −
|g|FMNFMN , (1)
http://arxiv.org/abs/0704.0314v4
mailto:bauch_vGR@ukr.net
mailto:zhuk@paco.net
mailto:tinatin@phys.ksu.edu
where the D-dimensional metric, g = gMN (X)dX
dXN = g(0)(x)µνdx
µ ⊗ dxν + a21(x)g(1), is defined on
the product manifold M = M0 × M1. Here, M0 is
the (D0 = d0 + 1)-dimensional external space. The d1-
dimensional internal space M1 has a constant curvature
with the scale factor a1(x) ≡ LPl expβ1(x). Dimensional
reduction of the action (1) results in the following effec-
tive D0-dimensional action [17]
S̄EM = −
|g̃(0)| [(1−Dκ0ψ)FµνFµν ] , (2)
which is written in the Einstein frame with the D0-
dimensional metric, g̃
µν = (exp d1β̄
1)−2/(D0−2)g
Here, κ0ψ ≡ −β̄1
(D0 − 2)/d1(D − 2) ≪ 1 and β̄1 ≡
β1 − β10 are small fluctuations of the internal space scale
factor over the stable background β10 (0 subscript de-
notes the present day value). These internal space scale-
factor small fluctuations/oscillations have the form of
a scalar field (so called gravexciton [20]) with a mass
mψ defined by the curvature of the effective potential
(see for detail [20]). Action (2) is defined under the
approximation κ0ψ < 1 that obviously holds for the
condition1 ψ < MPl. κ
0 = 8π/M
Pl is four dimen-
sional gravitational constant, MPl is the Plank mass,
D = 2
d1/[(D0 − 1)(D − 1)] is a model dependent con-
stant. The Lagrangian density for the scalar field ψ reads:
|g̃(0)|(−g̃µνψ,µψ,ν−m2ψψψ)/2. For simplicity we
assume that g̃0 is the flat Friedman-Lemaitre-Robertson-
Walker (FLRW) metric with the scale factor a(t).
Let’s consider Eq. (2). It is worth of noting that the
D0-dimensional field strength tensor, Fµν , is gauge in-
variant.2 Secondly, action (2) is conformally invariant in
the case when D0 = 4. The transform to the Einstein
frame does not break gauge invariance of the action (2),
and the electromagnetic field is antisymmetric as usual,
Fµν = ∂µAν − ∂νAµ. Varying (2) with respect to the
electromagnetic vector potential,
−g (1−Dκ0ψ)Fµν
= 0. (3)
The second term in the round brackets Dκ0ψFµν reflects
the interaction between photons and the scalar field ψ,
and as we show below, it is responsible for LV. In par-
ticular, coupling between photons and the scalar field ψ
makes the speed of photons different from the standard
speed of light. Eq. (3) together with Bianchi identity
(which is preserved in the considered model due to gauge-
invariance of the tensor, Fµν [17]) defines a complete set
1 In the brane-world model the prefactor κ0 in the expression for
κ0ψ is replaced by the parameter proportional to M
[14].
Thus, the smallness condition holds for ψ < MEW .
2 Eq. (2) can be rewritten in the more familiar form S̄EM =
−(1/2)
|g̃(0)|F̄µν F̄
µν [17]. The field strength tensor
F̄µν is not gauge invariant here.
of the generalized Maxwell equations. As we noted, ac-
tion (2) is conformally invariant in the 4D dimensional
space-time. So, it is convenient to present the flat FLRW
metric g̃0 in the conformally flat form: g̃0µν = a
2ηµν ,
where ηµν is the Minkowski metric.
Using the standard definition of the electromagnetic
field tensor, Fµν , we obtain the complete set of the
Maxwell equations in vacuum,
∇ ·B = 0 , (4)
∇ ·E = Dκ0
1−Dκ0ψ
(∇ψ ·E) , (5)
∇×B = ∂E
− Dκ0ψ̇
1−Dκ0ψ
1−Dκ0ψ
[∇ψ ×B] , (6)
∇×E = −∂B
, (7)
where all operations are performed in the Minkowski
space-time, η denotes conformal time related to physi-
cal time t as dt = a(η)dη, and an overdot represents a
derivative with respect to conformal time η.
Eqs. (4) and (7) correspond to Bianchi identity, and
since it is preserved, Eqs. (4) and (7) keep their usual
forms. Eqs. (5) and (6) are modified due to interactions
between photons and gravexcitons (∝ κ0ψ). These mod-
ifications have simple physical meaning: the interaction
between photons and the scalar field ψ acts as an effective
electric charge eeff . This effective charge is proportional
to the scalar product of the ψ field gradient and the E
field, and it vanishes for an homogeneous ψ field. The
modification of Eq. (6) corresponds to an effective cur-
rent Jeff , which depends on both electric and magnetic
fields. This effective current is determined by variations
of the ψ field over the time (ψ̇) and space (∇ψ). For
the case of a homogeneous ψ field the effective current is
still present and LV takes place. The modified Maxwell
equations are conformally invariant. To account for the
expansion of the Universe we rescale the field components
asB,E → B,E a2 [21].
To obtain a dispersion relation for photons, we use
the Fourier transform between position and wavenumber
spaces as,
F(k, ω) =
dη d3x e−i(ωη−k·x)F(x, η) ,
F(x, η) =
(2π)4
dω d3kei(ωη−k·x)F(k, ω) . (8)
Here, F is a vector function describing either the elec-
tric or the magnetic field, ω is the angular frequency of
the electro-magnetic wave measured today, and k is the
wave-vector. We assume that the field ψ is an oscilla-
tory field with the frequency ωψ and the momentum q,
so ψ(x, η) = Cei(ωψη−q·x) , C = const . Eq. (4) implies
B ⊥ k. Without loosing of generality, and for simplic-
ity of description we assume that the wave-vector k is
oriented along the z axis. Using Eq. (7) we get E ⊥ B.
A linearly polarized wave can be expressed as a super-
position of left (L, −) and right (R, +) circularly polar-
ized (LCP and RCP) waves. Using the polarization basis
of Sec. 1.1.3 of Ref. [22], we derive E± = (Ex± iEy)/
Rewriting Eqs. (4) - (7) in the components,3 for LCP
and RCP waves we get,
(1 − n2+)E+ = 0, (1− n2(−))E
− = 0 , (9)
where n+ and n− are refractive indices for RCP and LCP
electromagnetic waves
n2+ =
k2 [1−Dκ0ψ(1 + qz/k)]
ω2 [1−Dκ0ψ(1 + ωψ/ω)]
= n2− . (10)
In the case when LI is preserved the electromagnetic
waves propagating in vacuum have n+ = n− = n =
k/ω ≡ 1. For the electromagnetic waves propagating in
the magnetized plasma, k/ω 6= 1, and the difference be-
tween the LCP and RCP refractive indices describes the
Faraday rotation effect, α ∝ ω(n+ − n−) [23]. In the
considered model, since n+ = n− the rotation effect is
absent, but the speed of electromagnetic waves propaga-
tion in vacuum differs from the speed of light c (see also
Ref. [24] for LV induced by electromagnetic field cou-
pling to other generic field). This difference implies the
propagation time delay effect, ∆t = ∆l(1−∂k/∂ω) (∆l is
a propagation distance), ∆t is the difference between the
photon travel time and that for a ”photon” which travels
at the speed of light c. Here, t is physical synchronous
time. This formula does not take into account the evo-
lution of the Universe. However, it is easy to show that
the effect of the Universe expansion is negligibly small.
Solving the dispersion relation as a square equation,
we obtain
ω2ψ − q2z
(Dκ0ψ)2
, (11)
where ± signs correspond to photons forward and back-
ward directions respectively.
The modified inverse group velocity (11) shows that
the LV effect can be measured if we know the gravexciton
frequency ωψ, z-component of the momentum qz and its
amplitude ψ. For our estimates, we assume that ψ is
the oscillatory field, satisfying (in local Lorentz frame)
the dispersion relation, ω2ψ = m
ψ + q
2, where mψ is the
mass of gravexcitons4. Unfortunately, we do not have
3 We have defined the system of 6 equations with respect to 6
components of the vectors E and B. This system has non-trivial
solutions only if its determinant is nonzero. From this condition
we get the dispersion relation. The Faraday rotation effect is
absent if the matrix has a diagonal form.
4 To get physical values of the corresponding parameters we should
rescale them by the scale factor a.
any information concerning parameters of gravexcitons
(some estimates can be found in [17, 19]). Thus, we
intend to use possible LV effects (supposing it is caused
by interaction between photons and gravexcitons) to set
limits on gravexciton parameters. For example, we can
easily get the following estimate for the upper limit of
the amplitude of gravexciton oscillations:
|ψ| ≈ 1√
MPl , (12)
where for ω and mψ we can use their physical values.
In the case of GRB with ω ∼ 1021 ÷ 1022Hz ∼ 10−4 ÷
10−3GeV and ∆l ∼ 3 ÷ 5 × 109y ∼ 1017sec the typical
upper limit for the time delay is ∆t ∼ 10−4sec [9]. For
these values the upper limit on gravexciton amplitude of
oscillations is5
|κ0ψ| ≈
10−13GeV
. (13)
This estimate shows that our approximation κ0ψ < 1
works for gravexciton masses mψ > 10
−13GeV. Future
measurements of the time-delay effect for GRBs at fre-
quencies ω ∼ 1 − 10GeV would increase significantly
the limit up to mψ > 10
−9GeV. On the other hand,
Cavendish-type experiments [26, 27]) exclude fifth force
particles with masses mψ . 1/(10
−2cm) ∼ 10−12GeV
which is rather close to our lower bound for ψ field
masses. Respectively we slightly shift the considered
mass lower limit to be mψ ≥ 10−12GeV. These masses
considerably higher than the mass corresponding to the
equality between the energy densities of the matter and
radiation (matter/radiation equality), meq ∼ Heq ∼
10−37GeV, where Heq is the Hubble ”constant” at mat-
ter/radiation equality. It means that such ψ-particles
start to oscillate during the radiation dominated epoch
(see appendix). Another bound on the ψ-particles masses
comes from the condition of their stability. With re-
spect to decay ψ → γγ the life-time of ψ-particles is
τ ∼ (MPl/mψ)3tPl [17], and the stability conditions re-
quires that the decay time should be greater than the age
of the Universe. According this we consider light gravex-
citons with masses mψ ≤ 10−21MPl ∼ 10−2GeV ∼ 20me
(where me is the electron mass).
As an additional restriction arises from the condi-
tion that such cosmological gravexcitons should not
overclose the observable Universe. This reads mψ .
meq(MPl/ψin)
4 which implies the following restriction
for the amplitude of the initial oscillations: ψin .
(meq/mψ)
MPL << MPl [19]. Thus, for the range of
masses 10−12GeV ≤ mψ ≤ 10−2GeV, we obtain respec-
tively ψin . 10
−6MPl and ψin . 10
−9MPl. According to
5 We thank R. Lehnert to point that in addition of the time de-
lay effect the Cherenkov effect could be used to constrain the
electromagnetic field and ψ field coupling strength [25].
Eq. (A.3), we can also get the estimate for the amplitude
of oscillations of the considered gravexciton at the present
time. Together with the non-overcloseness condition,
we obtain from this expression that |κ0ψ| ∼ 10−43 for
mψ ∼ 10−12GeV and ψin ∼ 10−6MPl and |κ0ψ| ∼ 10−53
for mψ ∼ 10−2GeV and ψin ∼ 10−9MPl. Obviously, it is
much less than the upper limit (13). Note, as we men-
tioned above, gravexcitons with masses mψ & 10
−2GeV
can start to decay at the present epoch. However, taking
into account the estimate |κ0ψ| ∼ 10−53, we can easily
get that their energy density ρψ ∼ (|κ0ψ|2/8π)M2Plm2ψ ∼
10−55g/cm3 is much less than the present energy density
of the radiation ργ ∼ 10−34g/cm3. Thus, ρψ contributes
negligibly in ργ . Otherwise, the gravexcitons with masses
mψ & 10
−2GeV should be observed at the present time,
which, obviously, is not the case.
Additionally, it follows from Eq. (42) in Ref. [17]
that to avoid the problem of the fine structure constant
variation, the amplitude of the initial oscillations should
satisfy the condition: ψin . 10
−5MPl which, obviously,
completely agrees with our upper bound ψin . 10
−6GeV.
Summarizing we shown that LV effects can give addi-
tional restrictions on parameters of gravexcitons. First,
we found that gravexcitons should not be lighter than
10−13GeV. It is very close to the limit following from the
fifth-force experiment. Moreover, experiments for GRB
at frequencies ω > 1GeV can result in significant shift of
this lower limit making it much stronger than the fifth-
force estimates. Together with the non-overcloseness con-
dition, this estimate leads to the upper limit on the am-
plitude of the gravexciton initial oscillations. It should
not exceed ψin . 10
−6GeV. Thus, the bound on the ini-
tial amplitude obtained from the fine structure constant
variation is one magnitude weaker than our one even for
the limiting case of the gravexciton masses. Increasing
the mass of gravexcitons makes our limit stronger. Our
estimates for the present day amplitude of the gravexci-
ton oscillations, following from the obtained above lim-
itations, show that we cannot use the LV effect for the
direct detections of the gravexcitons. Nevertheless, the
obtained bounds can be useful for astrophysical and cos-
mological applications. For example, let us suppose that
gravexcitons with masses mψ > 10
−2GeV are produced
during late stages of the Universe expansion in some re-
gions and GRB photons travel to us through these re-
gions. Then, Eq. (A.3) is not valid for such gravexcitons
having astrophysical origin and the only upper limit on
the amplitude of their oscillations (in these regions) fol-
lows from Eq. (13). In the case of TeV masses we get
|κ0ψ| ∼ 10−16. If GRB photons have frequencies up to
1 TeV, ω ∼ 1TeV, then this estimate is increased by 6
orders of magnitude.
Acknowledgments
We thank G. Dvali, G. Gabadadze, A. Gruzinov, G.
Melikidze, B. Ratra, and A. Starobinsky for stimulating
discussions. T. K. and A. Zh. acknowledge hospital-
ity of Abdus Salam International Center for Theoreti-
cal Physics (ICTP) where this work has been started.
A.Zh. would like to thank the Theory Division of CERN
for their kind hospitality during the final stage of this
work. T.K. acknowledges partial support from INTAS
061000017-9258 and Georgian NSF ST06/4-096 grants.
A. Appendix: Dynamics of Light Gravexcitons
In this appendix we briefly summarize the main prop-
erties of the light gravexcitons necessary for our inves-
tigations. The more detail description can be found in
Refs. [17, 19].
The effective equation of motion for massive cosmolog-
ical gravexciton6 is
ψ + (3H + Γ)
ψ +m2ψψ = 0 , (A.1)
where H ∼ 1/t and Γ ∼ m3ψ/M2Pl are the Hubble pa-
rameter and decay rate (ψ → γγ) correspondingly. This
equation shows that at times when the Hubble parame-
ter is less than the gravexciton mass: H . mψ the scalar
field begins to oscillate (i.e. time tin ∼ H−1in ∼ 1/mψ
roughly indicates the beginning of the oscillations):
ψ ≈ CB(t) cos(mψt+ δ) . (A.2)
We consider cosmological gravexcitons with masses
10−12GeV ≤ mψ ≤ 10−2GeV. The lower bound fol-
lows both from the fifth-force experiments and Eq. (13).
The upper bound follows from the demand that the life-
time of these particles (with respect to decay ψ → γγ)
is larger than the age of the Universe: τ = 1/Γ ∼
(MPl/mψ)
tPl ≥ 1019sec > tuniv ∼ 4 × 1017 sec. Thus,
we can neglect the decay processes for these gravexci-
tons. Additionally, it can be easily seen that these par-
ticles start to oscillate before teq ∼ H−1eq when the en-
ergy densities of the matter and radiation become equal
to each other (matter/radiation equality). According to
the present WMAP data for the ΛCDM model it holds
Heq ≡ meq ∼ 10−56MPl ∼ 10−28eV. Thus, considered
particles have masses mψ >> meq and start to oscil-
late during the radiation dominated stage. They will not
overclose the observable Universe if the following condi-
tion is satisfied: mψ . meq(MPl/ψin)
4, where ψin is the
amplitude of the initial oscillations at the moment tin
(see Eq. (18) in Ref. [19]).
Prefactors C and B(t) in Eq. (A.2) for con-
sidered light gravexcitons respectively read: C ∼
(ψin/MPl) (MPl/mψ)
and B(t) ∼ MPl (MPlt)−3s/2.
Here, s = 1/2, 2/3 for oscillations during the radiation
6 We have seen that the interaction between gravexcitons and or-
dinary matter (in our case it is 4D-photons) is suppressed by the
Planck scale. Thus, gravexcitons are weakly interacting massive
particles (WIMPs).
dominated and matter dominated stages, correspond-
ingly. We are interested in the gravexciton oscillations
at the present time t = tuniv. In this case s = 2/3 and
for B(tuniv) we obtain: B(tuniv) ∼ t−1univ ≈ 10−61MPl.
Thus, the amplitude of the light gravexciton oscillations
at the present time reads:
|κ0ψ| ∼ 10−60
. (A.3)
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