1 p. pasipoularides in collaboration with k. farakos (ntua) and n. mavromatos (kings ‘ s college...
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P. Pasipoularides in Collaboration with
K. Farakos (NTUA) andN. Mavromatos (Kings ‘s College London)
THESSALONIKI 2008, NEB XIII RECENT DEVELOPMENTS IN GRAVITY
Title: BULK PHOTONS IN Title: BULK PHOTONS IN ASYMMETRICALLY WARPED ASYMMETRICALLY WARPED
SPACETIMESSPACETIMES
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BRANE WORLD MODELS
Theorists, in an attempt to solve the hierarchy problem, invented new phenomenological models with extra dimensions, which are known as brane world models (non stringy brane models).
Brane models give new physics and predictions in: Astrophysics and cosmology Modifications to Newtons Law (r~160μm) High energy particle physics (1TeV-10TeV LHC)
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Brane
Bulk Bulk
z (extra dimension)
Our world is trapped in a three dimensional brane which is embedded in a multidimensional space (BULK)
Only gravitons can propagate in the bulk , hence only gravitons possess KK excitations.
However, beyond the standard brane world scenario, there are other models where standard model particles, or some of standard
model particles, can live in the bulk.
BRANE WORLD MODELS
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I. LARGE EXTRA DIMENSIONS (R≤160μm) : ADD Model (I. Antoniadis, Arkani-Hammed, Dimopoulos and Dvali). Flat space-time.
II. INFINITE EXTRA DIMENSIONS: SECOND RANDALL-SUNDRUM MODEL. (but there is an effective size for the extra dimension due to the curvature of the extra space)
III. SMALL EXTRA DIMENSIONS (r~10^(-33)cm): FIRST RANDALL-SUNDRUM MODEL. (Bulk Gauge fields and fermions, Higgs field Localized on the negative tension brane)
BRANE WORLD MODELS
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FIRST RANDALL-SUNDRUM MODELSMSM or VisibleVisible brane
Planck Planck or Hidden Hidden brane
5 2 2 2 2 2RSds =α z (-dt +dx )+dz
r
-k zRSα (z)=e
5D 19*
c
1M k 10 GeV
r: : :
c-kπrEW PL cM =e M , k r 12
19EW PLM 1TeV, M 10 GeV
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FIRST RANDALL-SUNDRUM MODEL
i. The RS metric is a solution of the Einstein equations only when the following fine-tuning is satisfied.
ii. The induced metric on the brane is Minkowski.
iii. The RS metric preserves 4D Lorentz invariance in the bulk.
21Λ=- σ
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2 2 2 2 2RSds =α z (-dt +dx )+dz
r
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ASYMMETRYCALLY WARPED
SPACETIMES Beyond the Randall-Sundrum metric we can
assume a more general answatz:
If the space and time warp factors are different the 5D spacetime is called asymmetrically warped
2 2 2 2 2 2 2ds =-α z dt + z dx + z dz r
z z
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ASYMMETRICALLY WARPED
SPACETIMES The induced metric on the brane (z=0) is Minkowski
if we assume that
However, 4D Lorentz invariance is violated in the Bulk, due to the difference between the space and time warp factors.
0 0 1
2 2 2 2 2indds =-α 0 dt + 0 dx
r
2 22 2 2 2 2α z zds =- dt + dx + z dzr
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ASYMMETRICALLY WARPED SPACETIMES
Asymmetrically warped Static solutions:
1) M. Visser, Physics Letters B159, 22-25 (1985).2) Csaba Csaki, Joshua Erlich and Christophe
Grojean, Nucl.Phys.B604:312-342,2001.3) S.L. Dubovsky, JHEP 0201:012,2002.4) Peter Bowcock, Christos Charmousis and Ruth
Gregory, Class.Quant.Grav.17:4745-4764,2000. Extra matter in the bulk is necessary
otherwise the Einstein equation is not satisfied by the assymetrically warped static solutions
2 2 2 2 2 2 2ds =-α z dt + z dx + z dz r
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Static solutions like that of RS-Model can be used as background approximately only for a short period of time around our epoch t=t0. For larger periods of time the complete cosmological evolution must be taken into account.
The corresponding static solution is obtained if we set t=t0 In general we expect different warp factors in our epoch
Daniel J.H. Chung, Edward W. Kolb and Antonio Riotto, Phys.Rev.D65:083516, 2002
ASYMMETRICALLY WARPED SPACETIMES
2 2 2 2 2 2 2ds =-α z,t dt + z,t dx + z,t dz r
0 0α z,t z,t
Cosmological evolution reasons for asymmetrically warped brane models
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BRANE AND BULK FIELDS IN ASYMMETRICALLY
WARPED SPACETIMES Brane fields (completely pinned on the brane) can not
“see” the difference between the warp factors. The space-time for these field is Minkowski.
Bulk fields are described by a wave function. Due to the extension of the wave function in the bulk the bulk field “sees” the difference between the warp factors and 4D Lorenzt symmetry is violated.
i - t+p xx,z e z z z
z
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ASYMMETRICALLY WARPED SPACETIMES
Photons pinned on the brane
Gravitons traveling
in the bulk
extra dimension z
Gravitational violation of Lorentz symmetry. Direct signal from superluminous propagation of gravitational waves.
BULK BRANE
α z α 0
β zC C
β 0z
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ASYMMETRICALLY WARPED SPACETIMES
55 MN 4 (brane)MN
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1 1= d x g R -2Λ B B d x g
16 G 4 matterS L
AdS-Reissner Nordstrom Black Hole Solution
2 2 -2 2 2 -1 2ds =-h r dt + r d +h r drl
2 2 2 2d dσ d
2 2
2 2 4
μrh r =
r r
Q
l
BMN 5 MNG 8 T T
matterMN M Ng G z
μ and Q are the mass and charge of the five dimensional AdS Black Hole
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ASYMMETRICALLY WARPED SPACETIMES
2 2 -2 2 2 -1 2kds =-h r dt + r d +h r drl
0
1r=r z 0, k=k ze
l
AdS-Reissner Nordstrom Black Hole Solutionas a linearized perturbation around the RS-metric.
2 2 2 2 2 -1 2ds =-α z h z dt + z dx +h z dzr
2 k ze
2 2 2
4k z 6k z4 60 0
μh z 1 e e
r r Ql l
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ASYMMETRICALLY WARPED SPACETIMES
Ads RN Black Hole metric as a linearized perturbationaround the RS-metric.
-2k z -2k z2 2 2 2ds =-e 1- dt +e dx + 1+ dh z zz h r
2 2 2
4k z 6k z4 6 c0 0
c
μh z e e
r r1, 0 z z =πr
Q=
l l
We assume that δh(z) is small everywhere in the bulk
We consider Z2 symmetry and a second brane at the position z=zc, in order to achieve the structure (S1/Z2) of the first RS-model.
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BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES
We aim to study bulk photons in an asymmetrically warped spacetime which is a linear perturbation around the RS metric.
2 2 2 2 2 2RS RSds =- 1- dt + dx + 1+h z h dzz z z
r
NS
MN RSM NS F,
1gg g F 0
gN S S NA A
0 ZA=0, A 0, 0, A Coulomb Gauge Condition
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BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES
xj
μipj , p ω,A χ px,z
z e
2 2 2z RS z j j 0 j- α z 1-δh z A - A + 1+δh z A =0
2 2 2α z 1- h z χ p 1+ h z ω χ 0
z RS z
Equation of motion in the case of 5D U(1) Gauge fields
Plane wave Answatz
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BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES
20 , n=0,1,2,..., χ χ χ π 0 χ 0n n n cn nH H z rm z
Boundary Condition on the
brane 20 Unperturbed Hamiltoni n a α z , z RS zH
2 2 erturbation HamiltonianPα z δh z -δh z ω , z RS zH
If we use the formulation of time independent perturbation theory we obtain:
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BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES
20 0 0-2k
0
zn
2
n
0n0
0 n
e χ χ
χ χ
z z
n
nz m z
H z m z
c0 -kπrn n 0 n, J xm = 0, n=1,2,..x k e . 1TeV
Zero Mode plus Discrete Spectrum
00
0
1χ z = =consta Zero nt,
NMode
k z 0 nn 1
n
1 mχ z = e J , n 0
N kk ze
Unperturbed Equation
Photons in the case of the RS1-model
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BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES
Time independent perturbation theory:
2 2 20 1 22
0 0 0 0 ...m m m m
21 0
0 020
0 ω= χ χm H
Zero mode 00 0m
20 0
2 0 n20 2 2
0 000
4χ χ
ωn
n
Hm
m m
First order correction
Second order correction
Zero order
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BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES
ph 2G G
ω 1υ
p 1+α +β ω
2G G
g 2G G
1+α +β ωdωυ
dp 1+α +2β ω
Energy dependent phase and group velocity of light
cπr
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G 0
0
α = dz χ z δh z
c2πr
0 0G 0 n2
0n 0 0n
1b = dz χ z χ z δh z
m0
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BULK PHOTONS IN ASYMMETRICALLY WARPED SPACETIMES
2G
ph 3
2GG
βω 1υ ω
p 1+α 2 1+α
2G
g 3
2GG
3βdω 1υ ω
dp 1+α 2 1+α
Final Formulas for phase and group velocity of light when βG<<1
G
1
1+αlightc
Quadratic dependence
on the energy
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Our main result is a subluminous effective refractive index for vacuum
light 2G
ph G
c βω 1 ω
υ 2 1+αeffn
We see that, photons with different energies propagate with different velocities. Hence, we will observe a time lag of the arrival times of photons, which were emitted simultaneously by a remote astrophysical source. In particular photons with smaller energies will arrive first, and photons with larger energies will follow.
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COMPARING WITH THE EXPERIMENTAL DATA OF MAGIC
Magic is an imaging atmospheric Cherenkov telescope which can detect very high energy electromagnetic particles (VHE), in particular gamma rays.
VHE photons have energies between 0.1TeV-30TeV. They are photons from conversion of gravitational Energy, when a very massive rotating star is collapsing to a supermassive black hole (Blazar or AGN).
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J. Albert et al, astro-ph/070008
Magic observations during a flare (which lasts twenty minutes) of the nearby blazar (AGN) Markarian 501 ( Z=0.034), in July 9 (2005), indicates a 4±1 min time delay between the peaks of the time profile envelops for photons with energies smaller than ω<0.25 TeV and photons with ω>1.2 TeV
150-250 GeV
250-600 GeV
600-1200 GeV
1.2-10 TeV
4 min
s-b
ins
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COMPARING WITH THE EXPERIMENTAL DATA OF MAGIC
It is an energy dependent effect in the source (SSC mechanism).
New physics induces an effective refractive index for vacuum.
Most of Quantum Gravity Models seem to predict a dispersion relation for vacuum.
Or may be brane models with Bulk photons and an assymetrically warped metric (especially in our model we have quadratic prediction for the refractive index).
Possible interpretations of delays of more energetic photons
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COMPARING WITH THE EXPERIMENTAL DATA OF MAGIC
n
n=ω
ω 1 , 2,...1, effQGn
nM
A numerical analysis based on Magic results, which aim at the reconstruction of the original electromagnetic pulse maximizing its energy, assuming a refractive index for vacuum (two cases linear and quadratic):
MAGIC Coll. & Ellis, Mavromatos, Nanopoulos, Sakharov, MAGIC Coll. & Ellis, Mavromatos, Nanopoulos, Sakharov, Sarkisyan, arXive: 07082889 [astro-ph]Sarkisyan, arXive: 07082889 [astro-ph]
1 111 2
8 0.6 10 GeV0.4 10 GeV, QGQG MM
predicts the following values for the two mass scales
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COMPARING WITH THE EXPERIMENTAL DATA OF MAGIC
2G
2
2 G
βωω ω 1 ω
2 1+,1
αe ffffQG
ennM
Numerical analysis:Magic results fitting
Theoretical analysis:Brane Models
2 G
2G
β
2 1+αQGM
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COMPARING WITH THE EXPERIMENTAL DATA OF MAGIC
2 2Gβ 10 δh TeV
We compute numerically the parameter βG
AdS-Schwarzchild Black Hole Solution
AdS-Reissner Nordstrom Black Hole Solution
2 2Gβ 2.95 δh TeV
cπr
0c
1δh = δh z dz
πr
-2k z -2k z2 2 2 2ds =-e 1- dt +e dx + 1+ dh z zz h r
Average deviation around the RS1-Metric
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COMPARING WITH THE MAGIC EXPERIMENT
112
2 -2G
2 -2G
2G 2
0.6 10 GeV
β 10 δh TeV
β 2.95
Schwazchild
Reissner Nordstδh TeV rom
β 2 ,
QG
QG
M
M
-8δh 0.745 10 Schwarzchild
-8δh 1.37 10 Reissner Nordstrom
The small values we obtain for <δh> are consistent with the weak nature of the perturbation
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CONCLUSIONS We study asymmetrically warped brane models with bulk
photons. We show that the standard Lorentz invariant dispersion relation for 4D photons, possesses nonlinear corrections which lead to an Energy-dependent speed of light.
We compared with the experimental data of Magic and we set concrete restrictions to the specific brane models we examined.
We propose further investigation for other types of particles such as, gravitons and fermions. Similar dispersion relations are expected.