1. Motivation Data Methods Results Conclusion
Constraints on Lorentz Invariance Violation from
Fermi-LAT Observations of GRBs
Camille Couturier
Vlasios Vasileiou, Agnieszka Jacholkowska,
Frédéric Piron, Julien Bolmont,
Jonathan Granot, Floyd W. Stecker,
Johann Cohen-Tanugi, Francesco Longo
C. Couturier (LPNHE Paris) Constraining Lorentz Invariance Violation July 3rd, 2013 1 / 8

2. Motivation Data Methods Results Conclusion
One manifestation of Lorentz Invariance Violation:
dispersion of light in vacuum
• Extension of the Special/General relativity
• Violation of Lorentz Invariance
→ velocity of light in vacuum could vary with energy !
• degree of dispersion τn (with n = 1 ou 2):
τn =
t2 − t1
En
2 − En
1
s±
(1 + n)
En
QGH0
kn
bla bla
with kn =
z
0
(1 + z )n dz
Ωm(1 + z )3 + ΩΛ
C. Couturier (LPNHE Paris) Constraining Lorentz Invariance Violation July 3rd, 2013 2 / 8

3. Motivation Data Methods Results Conclusion
Data of four Gamma-ray Bursts seen by Fermi-LAT
Times ti and energies Ei of photons
from 4 Gamma-Ray Bursts
observed by Fermi-LAT
Notes:
• various redshift z
• energies from 20 MeV to 30 GeV
• timescale from 3s (090510) to
25s
Energy(MeV)
2
10
3
10
4
10
5
10
Time after trigger (sec)
-2 0 2 4 6 8 10 12 14
GRB 090510
(ti, Ei) for GRB 090510
C. Couturier (LPNHE Paris) Constraining Lorentz Invariance Violation July 3rd, 2013 3 / 8

4. Motivation Data Methods Results Conclusion
Three methods to look for dispersion in the data
PairView
(PV)
1 calculation of
Li,j =
ti − tj
En
i − En
j
2 distribution of Li,j
Sharpness Maximization
(SMM)
1 For a given τn:
S(τn) =
N−ρ
i=1
log
ρ
ti+ρ − ti
2 calculation for several τn
Maximum likelihood
(ML)
1 Build a model
P(t, E|τn) =
1
Npred
Λ(E)f(t − τnEn
)
2 test it for different τn
10
TABLE II. Configuration Details
Time Range (s) ρ NE>100MeV γ Ntemplate Nfit Ecut (MeV)
All Methods SMM PV & SMM Likelihood
n=1 n=2 n=1 n=2 n=1 n=2
3.53–7.89 3.53–7.80 50 30 59 59 2.2 82 59 59 100
0.01–3.11 -0.01–4.82 50 70 157 168 1.5 148 118 125 150
.79–14.22 5.79–14.21 80 80 111 111 1.9 57 87 87 150
.92–10.77 9.3–10.76 25 30 60 58 2.2a
53 48 47 120
m for this GRB also includes an exponential cutoff with pre-set e-folding energy Ec=0.4 GeV in accordance with
t al.[44]
Photon-pair lags (s/GeV)
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
Entriesperbin(arb.norm.)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Spectral lag (s/GeV)
-0.1 -0.05 0 0.05 0.1
SharpnessMeasureS
484
486
488
490
492
16000
12000
10
TABLE II. Configuration Details
Time Range (s) ρ NE>100MeV γ Ntemplate Nfit Ecut (MeV)
All Methods SMM PV & SMM Likelihood
n=1 n=2 n=1 n=2 n=1 n=2
3.53–7.89 3.53–7.80 50 30 59 59 2.2 82 59 59 100
-0.01–3.11 -0.01–4.82 50 70 157 168 1.5 148 118 125 150
5.79–14.22 5.79–14.21 80 80 111 111 1.9 57 87 87 150
8.92–10.77 9.3–10.76 25 30 60 58 2.2a
53 48 47 120
rum for this GRB also includes an exponential cutoff with pre-set e-folding energy Ec=0.4 GeV in accordance with
nn et al.[44]
Photon-pair lags (s/GeV)
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
Entriesperbin(arb.norm.)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Spectral lag (s/GeV)
-0.1 -0.05 0 0.05 0.1
SharpnessMeasureS
484
486
488
490
492
16000
12000
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.040
2
4
6
8
10
12
14
LIV parameter (s/GeV)
-0.1 -0.05 0 0.05 0.1
SharpnessMeasureS
484
486
488
490
492
-2Δlog(L)
LIV parameter (s/GeV)
C. Couturier (LPNHE Paris) Constraining Lorentz Invariance Violation July 3rd, 2013 4 / 8

5. Motivation Data Methods Results Conclusion
Constraints on the total degree of dispersion τn
Linear case (n = 1) k1 =
z
0
(1+z ) dz
√
Ωm(1+z )3+ΩΛ
1k
1 1.5 2 2.5 3 3.5 4 4.5
)-1
(sGeV1
τ
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
080916C
090510
090902B090926A
080916C
090510
090902B
1k
0.96 0.98 1 1.02 1.04 1.06 1.08 1.1
)-1
(msGeV1τ
-80
-60
-40
-20
0
20
40
2
)-2
(sGeV2
τ
-1
-0.5
0
0.5
1
1.5
2
090510
090510
1
)-2
(msGeV2τ
-4
-3
-2
-1
0
1
C. Couturier (LPNHE Paris) Constraining Lorentz Invariance Violation July 3rd, 2013 5 / 8

6. Motivation Data Methods Results Conclusion
Constraints on the total degree of dispersion τn
Quadratic case (n = 2) k2 =
z
0
(1+z )2 dz
√
Ωm(1+z )3+ΩΛ
16
4 4.5
080916C
080916C
2
k
2 4 6 8 10 12 14
)-2
(sGeV2
τ
-1
-0.5
0
0.5
1
1.5
2
080916C
090510
090902B090926A
080916C
090510
090902B2k
1 1.2 1.4 1.6 1.8 2
)-2
(msGeV2τ
-4
-3
-2
-1
0
1
C. Couturier (LPNHE Paris) Constraining Lorentz Invariance Violation July 3rd, 2013 5 / 8

7. Motivation Data Methods Results Conclusion
Accounting for GRB-intrinsic dispersions
τn = τLIV + τintrinsic
(data) (LIV effect) (source)
Modeling of τint, assuming observations are dominated by source effects:
• width(τint) = width(τn) and τint = τn = 0
• PDF of τint set to match distribution of possibilities for τn
→ symmetric confidence intervals on τLIV
• worst case scenario
→ most conservative limits
C. Couturier (LPNHE Paris) Constraining Lorentz Invariance Violation July 3rd, 2013 6 / 8

8. Motivation Data Methods Results Conclusion
95% lower limits on EQG (subluminal case)
Linear case (n = 1) τ1 = ∆t
∆E
2
EQGH0
k1(z)
Redshift
1 1.5 2 2.5 3 3.5 4 4.5
=+1)
±
(n=1,sPl
/EQG
LowerLimitonE
-1
10
1
10
080916C
090510
090902B
090926A
PV
SMM
ML
GRB09051095%
0.5 1
=+1)
±
GeV)(n=2,s10
(10QG
LowerLimitonE
-1
10
1
10
090510
C. Couturier (LPNHE Paris) Constraining Lorentz Invariance Violation July 3rd, 2013 7 / 8

9. Motivation Data Methods Results Conclusion
95% lower limits on EQG (subluminal case)
Quadratic case (n = 2) τ2 = ∆t
∆E2
3
E2
QGH0
k2(z)
4.5
080916C
PV
SMM
ML
GRB09051095%
Redshift
0.5 1 1.5 2 2.5 3 3.5 4 4.5
=+1)
±
GeV)(n=2,s10
(10QG
LowerLimitonE
-1
10
1
10
080916C
090510
090902B
090926A
PV
SMM
ML
PKS2155-30495%
GRB090510High confi dence
C. Couturier (LPNHE Paris) Constraining Lorentz Invariance Violation July 3rd, 2013 7 / 8

10. Motivation Data Methods Results Conclusion
Conclusion: QG models predicting EQG < EPlanck disfavored
• 4 Fermi-LAT Gamma-Ray Bursts;
• 3 different methods give consistent results;
• Limits on the Quantum Gravity energy scale are improved;
→ over the Planck mass for n = 1 even w/ GRB-intrinsic effect !
→ QG models predicting EQG < EPlanck disfavored
• Phys. Rev. D 87, 122001 (2013) [arXiv:1305.3463]
Obrigado pela atenção !
C. Couturier (LPNHE Paris) Constraining Lorentz Invariance Violation July 3rd, 2013 8 / 8