1 P. Pasipoularides in Collaboration with K. Farakos (NTUA) and N. Mavromatos (Kings ‘ s College London) THESSALONIKI 2008, NEB XIII RECENT DEVELOPMENTS

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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


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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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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55 MN 4 (brane)MN

5

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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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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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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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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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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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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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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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

20

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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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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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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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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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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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

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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.

Documents

1 P. Pasipoularides in Collaboration with K. Farakos (NTUA) and N. Mavromatos (Kings ‘ s College London) THESSALONIKI 2008, NEB XIII RECENT DEVELOPMENTS