Rencontres de Moriond, La Thuile, Italy
CONSTRAINTS ON LORENTZ SYMMETRY VIOLATIONS
USING LUNAR LASER RANGING OBSERVATIONS
A. Bourgoin1,2, A. Hees1, C. Le Poncin-Lafitte1,S. Bouquillon1, G. Francou1, M.-C. Angonin1,
C. Courde3, J.-M. Torre3 and J. Chaibe3
1SYRTE, Observatoire de Paris, 2University of Bologna, 3Universite Cote d’Azur, CNRS,Observatoire de la Cote d’Azur, IRD, Geoazur
March 27th, 2019
Outline
1 IntroductionTests of GRPostfit analysisLunar laser rangingDynamical modelingResults in pure GR
2 SME minimally coupledPresentationModelingSME coefficientsRealistic errors
3 SME gravity-matter couplingsPresentationModelingSME coefficientsRealistic error
Outline
1 IntroductionTests of GRPostfit analysisLunar laser rangingDynamical modelingResults in pure GR
2 SME minimally coupledPresentationModelingSME coefficientsRealistic errors
3 SME gravity-matter couplingsPresentationModelingSME coefficientsRealistic error
Alternative theories of gravity
Solar system scale frameworks
The fifth force formalisma (cf. Fig.) :
V(r) = �Gm
r(1 + ↵e
�r/�).
The PPN formalismb :
10 parameters(�, �, ⇠,↵{1,2,3}, . . .),�� 1 = (+2.1± 2.3)⇥ 10�5 c,��1 = (+0.2±2.5)⇥10�5 d.
a. Fischback et. al, 1986b. Nordtvedt, 1968 & Will, 1971c. Bertotti et. al, 2003d. Verma et. al, 2014
Konopliv et. al, 2011
Standard-model extension (SME)
Colladay and Kostelecky 1997, 1998 ; Bailey and Kostelecky 2006 ; Kostelecky andTasson 2011
Parametrizes Lorentz symmetry violations in all physics,
Contains both the standard model of particles physics and GR,
Derived from an action principle,=) not covered by PPN, fifth force, . . .
4 / 24
Alternative theories of gravity
Solar system scale frameworks
The fifth force formalisma (cf. Fig.) :
V(r) = �Gm
r(1 + ↵e
�r/�).
The PPN formalismb :
10 parameters(�, �, ⇠,↵{1,2,3}, . . .),�� 1 = (+2.1± 2.3)⇥ 10�5 c,��1 = (+0.2±2.5)⇥10�5 d.
a. Fischback et. al, 1986b. Nordtvedt, 1968 & Will, 1971c. Bertotti et. al, 2003d. Verma et. al, 2014
Konopliv et. al, 2011
Standard-model extension (SME)
Colladay and Kostelecky 1997, 1998 ; Bailey and Kostelecky 2006 ; Kostelecky andTasson 2011
Parametrizes Lorentz symmetry violations in all physics,
Contains both the standard model of particles physics and GR,
Derived from an action principle,=) not covered by PPN, fifth force, . . .
4 / 24
Data analysis
Postfit analysisLook for oscillating signatures in residuals,
1 Oscillating signatures are derived analytically,2 Oscillating signatures are fitted in residuals,
Major problems
Oscillating signatures only for short period,
Estimate only SME coefficients=) No correlations with other global parameters,=) Formal errors are over optimistic a.
a. Le Poncin-Lafitte et. al, 2016
Constraints on Lorentz symmetry violations=) Global LLR data analysis : confrontation between observations and predictions.
Simulate the LLR observable in SME framework :=) Effects on orbital motions,=) Effects on light propagation.
Short and long periodic oscillations :=) Numerical integration.
Fit and correlations with global parameters :=) Clean partial derivatives.
5 / 24
Data analysis
Postfit analysisLook for oscillating signatures in residuals,
1 Oscillating signatures are derived analytically,2 Oscillating signatures are fitted in residuals,
Major problems
Oscillating signatures only for short period,
Estimate only SME coefficients=) No correlations with other global parameters,=) Formal errors are over optimistic a.
a. Le Poncin-Lafitte et. al, 2016
Constraints on Lorentz symmetry violations=) Global LLR data analysis : confrontation between observations and predictions.
Simulate the LLR observable in SME framework :=) Effects on orbital motions,=) Effects on light propagation.
Short and long periodic oscillations :=) Numerical integration.
Fit and correlations with global parameters :=) Clean partial derivatives.
5 / 24
Data analysis
Postfit analysisLook for oscillating signatures in residuals,
1 Oscillating signatures are derived analytically,2 Oscillating signatures are fitted in residuals,
Major problems
Oscillating signatures only for short period,
Estimate only SME coefficients=) No correlations with other global parameters,=) Formal errors are over optimistic a.
a. Le Poncin-Lafitte et. al, 2016
Constraints on Lorentz symmetry violations=) Global LLR data analysis : confrontation between observations and predictions.
Simulate the LLR observable in SME framework :=) Effects on orbital motions,=) Effects on light propagation.
Short and long periodic oscillations :=) Numerical integration.
Fit and correlations with global parameters :=) Clean partial derivatives.
5 / 24
Lunar laser ranging
5 LLR stations :McDonald 2.7m, MLRS1, MLRS2 (Texas)Grasse Yag, Rubis, MeO (France)Haleakala (Hawaii)Matera (Italie)Apache-point (Texas)
•••••
5 retroreflectors :Apollo XIApollo XIVApollo XVLunokhod 1Lunokhod 2
•••••1 pulse contains
1018 photons
Only 1 photonover 109 is reflected
Only0.01 photonsper pulse are
detected
1 pulse ' 0.2ns
pulse separation ' 2.9ns
10 pulsesper second
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Ephemeride Lunaire Parisienne Numerique (ELPN)
Modeling
ICRF (cT, X, Y, Z), with T the TDB,
Earth, the Moon, the Sun, planets, Pluto and the 70 most massive asteroids,
Physical effects1 Newtonian point-mass interactions.
2 Figure potential of bodies :=) J2 for the Sun,=) Jn, with n = 5 for the Earth,=) Degree 2, 3, 4 and 5 for the Moon.
3 Earth orientation :=) Precession UAI 1976,=) Nutation UAI 1980 (terms in 18.6 yr).
4 Tidal and spin effects :=) Anelastic deformations (Time-lag),
5 Relativistic point-mass interactions :=) Solar system barycentre,=) TT � TDB = f (TDB),=) Geodetic precession effect,=) SME corrections.
6 Lunar librations :=) Momentums,=) Fluid lunar core (dissipation).
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Residuals ELPN in GR
⇢[c
m]
UTC [year]
McDonaldApache point
GrasseHaleakala
Matera
Stations LLR Period �ELPN [cm] �INPOP13b [cm] 1 �DE430 [cm] 1
McDonald 2.7m 1969-1985 35.0 31.9 30.2McDonald MLRS1 1983-1988 35.1 29.4 27.7McDonald MLRS2 1988-2013 9.7 5.4 4.9
Grasse Rubis 1984-1986 17.6 16.0 14.6Grasse Yag 1987-2005 4.5 6.6 5.6Grasse MeO 2009-2013 3.6 6.1 3.5
Haleakala 1984-1990 9.8 8.6 8.3Matera 2003-2013 9.1 7.1 5.8
Apache-point 2006-2013 3.8 4.9 4.21. Fienga et. al 2014 8 / 24
Outline
1 IntroductionTests of GRPostfit analysisLunar laser rangingDynamical modelingResults in pure GR
2 SME minimally coupledPresentationModelingSME coefficientsRealistic errors
3 SME gravity-matter couplingsPresentationModelingSME coefficientsRealistic error
SME minimally coupled
The pure gravitational sector (Bailey et. al, 2006)
Stot = Sg + Sm + Sf
Field action : Sg =1
2
Zd4
xp�g(R � uR + s
µ⌫R
T
µ⌫ + t↵�µ⌫
C↵�µ⌫),
Matter action : Sm = Sm[ m, gµ⌫ , u, sµ⌫ , t
↵�µ⌫ ],
Point particle action : Sf = �mc
Zds.
Functional derivativesModified field equations, m minimally coupled to gµ⌫ :=) EEP,=) geodesic equations,
Observational implications
Spontaneous Lorentz violation (u, sµ⌫ , t
↵�µ⌫),At post-Newtonian level, u and t
↵�µ⌫ play no role,=) s
µ⌫ .
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SME minimally coupled
The pure gravitational sector (Bailey et. al, 2006)
Stot = Sg + Sm + Sf
Field action : Sg =1
2
Zd4
xp�g(R � uR + s
µ⌫R
T
µ⌫ + t↵�µ⌫
C↵�µ⌫),
Matter action : Sm = Sm[ m, gµ⌫ , u, sµ⌫ , t
↵�µ⌫ ],
Point particle action : Sf = �mc
Zds.
Functional derivativesModified field equations, m minimally coupled to gµ⌫ :=) EEP,=) geodesic equations,
Observational implications
Spontaneous Lorentz violation (u, sµ⌫ , t
↵�µ⌫),At post-Newtonian level, u and t
↵�µ⌫ play no role,=) s
µ⌫ .
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Orbital and time delay effects
Orbital motion
Equations of motion (cT, X, Y, Z) :
aJ =
GNM
r3
s
JK
tr
K �32
sKL
tr
Kr
Lr
J + 3sTK
VK
rJ � s
TJV
Kr
K � sTK
VJr
K
+ 3sTL
VK
rK
rLr
J + 2�m
M
⇣s
TKv
Kr
J � sTJ
vK
rK
⌘�.
Rescaled Newtonian constant : GN = G(1 + 53 s
TT).
3D Traceless tensor sJKt = s
JK � 13 s
TT�JK .
Gravitational light time delay
Time delay (cT, X, Y, Z) :
�⌧g =X
b=S,T
(2GNmb
c3
1 �
23
sTT � s
TJd
J
�ln
"rbO + rbR + d
rbO + rbR � d
#
�GNmb
c3
23
sTT � s
TJd
J � sJK
th
J
bh
K
b
�⇣r
K
bRd
K � rK
bOd
K
⌘
�GNmb
c3
13
sTT
dK
hK
b� s
TJh
J
b+ s
JK
td
Jh
K
b
�hb
�rbR � rbO
�
rbOrbR
),
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Correlations
Data analysis
sµ⌫ =) 10 parameters : (sTT , s
TX , sTY , s
TZ , sXXt
, sXYt
, sXZt
, sYYt
, sYZt
, sZZt
),
Brutal fit of sµ⌫ with 46 other global physical parameters,
=) Some high correlations,=) SME coefficient correlations (cf. Tab.).
sTT
sTX
sTY
sTZ
sXX
ts
XY
ts
XZ
ts
YY
ts
YZ
ts
ZZ
t
sTT
1.00
sTX -0.11 1.00
sTY -0.03 -0.01 1.00
sTZ 0.03 0.02 -0.99 1.00
sXX
t-0.01 0.08 -0.02 0.02 1.00
sXY
t0.01 -0.03 -0.09 0.09 -0.14 1.00
sXZ
t-0.04 -0.01 -0.05 0.05 -0.16 0.30 1.00
sYY
t0.02 -0.11 -0.02 0.01 0.95 -0.12 -0.12 1.00
sYZ
t-0.08 0.73 -0.05 0.07 0.14 -0.07 -0.09 -0.12 1.00
sZZ
t0.07 -0.62 0.03 -0.05 0.40 -0.02 -0.01 0.64 -0.84 1.00
TABLE – Table of correlations between SME coefficients of the minimal SME.
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LLR data sensitivity
SME coefficients
4 SME coefficients : (sTT , sTX , s
XYt
, sXZt
),
3 linear combinations : (sA, sC , s
D) where
sA = s
XX
t� s
YY
t,
sC = s
TY + 0.37sTZ ,
sD = s
XX
t+ s
YY
t� 2s
ZZ
t� 1.89s
YZ
t.
sTT
sTX
sXYt
sXZt
sA
sC
sD
sTT
1.00
sTX -0.07 1.00
sXYt
0.00 0.04 1.00
sXZt
-0.04 0.09 0.28 1.00
sA 0.00 -0.01 -0.02 -0.07 1.00
sC 0.03 -0.04 0.03 -0.05 0.04 1.00
sD -0.02 0.04 -0.08 -0.12 -0.31 0.00 1.00
TABLE – Table of correlations between SME coefficients of the minimal SME.
Maximum sensitivity
With LLR : |sTT | < 10�4,
With VLBI : |sTT | < 10�5 (Le Poncin-Lafitte et. al, 2016).
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LLR data sensitivity
SME coefficients
4 SME coefficients : (sTT , sTX , s
XYt
, sXZt
),
3 linear combinations : (sA, sC , s
D) where
sA = s
XX
t� s
YY
t,
sC = s
TY + 0.37sTZ ,
sD = s
XX
t+ s
YY
t� 2s
ZZ
t� 1.89s
YZ
t.
sTT
sTX
sXYt
sXZt
sA
sC
sD
sTT
1.00
sTX -0.07 1.00
sXYt
0.00 0.04 1.00
sXZt
-0.04 0.09 0.28 1.00
sA 0.00 -0.01 -0.02 -0.07 1.00
sC 0.03 -0.04 0.03 -0.05 0.04 1.00
sD -0.02 0.04 -0.08 -0.12 -0.31 0.00 1.00
TABLE – Table of correlations between SME coefficients of the minimal SME.
Maximum sensitivity
With LLR : |sTT | < 10�4,
With VLBI : |sTT | < 10�5 (Le Poncin-Lafitte et. al, 2016).
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Systematic errors
Constraints
Realistic error,
Chi-square fit may underestimates formal errors,=) Look for neglected systematics.
Indice (l) N{l} Station Instrument Period0 - - - -1 1590 McDonald 2.7m 1969-19752 1739 McDonald 2.7m 1975-19833 187 McDonald 2.7m 1983-19874 573 McDonald MLRS1 1983-19895 469 McDonald MLRS2 1988-19946 2048 McDonald MLRS2 1994-20007 767 McDonald MLRS2 2000-20158 1182 Grasse Rubis 1984-19879 1494 Grasse Yag 1987-1991
10 1942 Grasse Yag 1991-199511 3479 Grasse Yag 1995-200112 1398 Grasse Yag 2001-200613 1661 Grasse MeO 2009-201514 757 Haleakala - 1984-199115 2378 Apache-point - 2006-201516 107 Matera - 2003-2015
TABLE – Table of definition of LLR data subsamples by LLR stations.14 / 24
Systematic errors
Constraints
Realistic error,
Chi-square fit may underestimates formal errors,=) Look for neglected systematics.
Indice (l) N{l} Station Instrument Period0 - - - -1 1590 McDonald 2.7m 1969-19752 1739 McDonald 2.7m 1975-19833 187 McDonald 2.7m 1983-19874 573 McDonald MLRS1 1983-19895 469 McDonald MLRS2 1988-19946 2048 McDonald MLRS2 1994-20007 767 McDonald MLRS2 2000-20158 1182 Grasse Rubis 1984-19879 1494 Grasse Yag 1987-1991
10 1942 Grasse Yag 1991-199511 3479 Grasse Yag 1995-200112 1398 Grasse Yag 2001-200613 1661 Grasse MeO 2009-201514 757 Haleakala - 1984-199115 2378 Apache-point - 2006-201516 107 Matera - 2003-2015
TABLE – Table of definition of LLR data subsamples by LLR stations.14 / 24
Data subsamples by LLR station
FIGURE – Evolution of formal error of with subsamples by LLR stations.
15 / 24
Data subsamples by LLR retroreflectors
FIGURE – Evolution of formal error with subsamples by LLR retroreflectors.
16 / 24
Jackknife resampling methode
Jackknife recipe
Let �j
n(D) be an estimator of the jth SME coefficient, defined for sample D = (D1, · · · ,Dn).
We construct n independent estimation of the jth SME coefficient with mean �j
n(D), labeled asps
j
i(D) and called pseudovalues
psj
i(D) = �j
n(D) + (n�1)
h�j
n(D)� �j
n�1(D[i])i
.
Where D[i] means the sample D with the ith value Di deleted from the sample.
We compute the mean and the sample variance of pseudo-values with the central limit theorem
psj = 1
n
nX
i=1
psj
i(D) and Vps(D) = 1
n�1
nX
i=1
hps
j
i(D)� ps
j
i2,
The Jackknife estimator is given by, psj(D)± �jackknife with �jackknife =
q1n
Vps(D).
Application to LLR
Jackknife applied to LLR stations (�sta),
Jackknife applied to LLR retroreflectors (�ref),
Realistic final error : �r =q
�2 + �2sta + �2
ref .
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Constraints on SME coefficients
Analysis Postfit Global
SMETechnique LLR & Atom.
Int. Binary pulsars LLR & Plan.Eph. LLR
sTX [10�9 ] +50 ± 620 �5.2 ± 5.3 +4.3 ± 2.5 �5.5 ± 7.9
sXYt
[10�12] �600 ± 1500 �35 ± 36 +65 ± 32 �3.4 ± 5.7s
XZt
[10�12] �2700 ± 1400 �20 ± 20 +20 ± 10 +0.2 ± 8.6s
A [10�12] �1200 ± 1600 �100 ± 100 +96 ± 56 +1.6 ± 9.3s
C [10�9 ] �100 ± 2900 �10 ± 9 �5 ± 24 +6.3 ± 7.3s
D [10�11] +200 ± 3800 �12 ± 12 +16 ± 8 +1.0 ± 2.3
TABLE – Table of pseudo-constraints (postfit analysis) and constraints (global analysis) on SMEcoefficients of the minimal SME. Results from LLR & Atom. Int. are taken from Battat et. al,2007 and Chung et. al, 2009. Results from binary pulsars are taken from Shao 2014 and resultsLLR & Plan. Eph. are taken from Hees et. al, 2015. Results determined by the global analysisare taken from Bourgoin et. al, 2016 and are given at 1�r .
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Outline
1 IntroductionTests of GRPostfit analysisLunar laser rangingDynamical modelingResults in pure GR
2 SME minimally coupledPresentationModelingSME coefficientsRealistic errors
3 SME gravity-matter couplingsPresentationModelingSME coefficientsRealistic error
SME gravity matter couplings
General presentation (Kostelecky et. al, 2011)
Stot = Sg + Sm + Sf
Field action : gravitational minimal SME,Matter action : Sm = Sm[ m, gµ⌫ , u, s
µ⌫ , t↵�µ⌫ , cµ⌫ , (aeff)µ],
Point particle action : Sf = �Z
cd�hm
p(gµ⌫ + cµ⌫)uµu⌫ + (aeff)µu
µi
Functional derivativesModified field equations, m is not minimally coupled to gµ⌫ :=) violation EEP,=) 6= geodesic equations.
Observational implications
Spontaneous Lorentz violation (cµ⌫ , (aeff)µ),cµ⌫ = 0,=) (aeff)µ.
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Orbital and time delay effect
Orbital motion
Equations of motion (cT, X, Y, Z) :
aJ =
GN M
r3
(s
JK
tr
K �32
sKL
tr
Kr
Lr
J � sTJ
VK
rK � s
TKV
Jr
K + 3sTL
VK
rK
rLr
J
+ 3
sTK �
23
X
w
nw
3
M↵(a
w
eff)K
�V
Kr
J + 2�m
M
s
TK +X
w
nw
2
�m↵(a
w
eff)K
�v
Kr
J
� 2�m
M
s
TJ +X
w
nw
2
�m↵(a
w
eff)J
�v
Kr
K
).
Depends on composition w = {e, p, n},
Gravitational light time delay
Time delay (cT, X, Y, Z) :
�⌧g =X
b=S,T
(2GN mb
c3
1 �
23
sTT +
X
w
Nw
b
mb
↵(aw
eff)T
�
sTJ �
X
w
Nw
b
mb
↵(aw
eff)J
�d
J � 2X
w
nw
3
M↵(a
w
eff)T
!ln
"rbO + rbR + d
rbO + rbR � d
#
�GN mb
c3
23
sTT +
X
w
Nw
b
mb
↵(aw
eff)T �
s
TJ �X
w
Nw
b
mb
↵(aw
eff)J
�d
J � sJK
th
J
bh
K
b
!⇣r
K
bRd
K � rK
bOd
K
⌘
�GN mb
c3
13
sTT
dK
hK
b�
sTJ �
X
w
Nw
b
mb
↵(aw
eff)J
�h
J
b+ s
JK
td
Jh
K
b
!hb
�rbR � rbO
�
rbOrbR
).
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LLR data sensitivity
Analysis
12 parameters ↵(aw
eff)µ [GeV/c2] for w = {e, p, n}.
Neutral macroscopic bodies : Ne
b = Np
b,
=) ↵(ae+p
eff )µ = ↵(ae
eff)µ + ↵(ap
eff)µ=) 8 parameters,With LLR : |↵(ae+p
eff )T | ' |↵(an
eff)T | < 10�4,With free fall (WEP) a : |↵(ae+p
eff )T | ' |↵(an
eff)T | < 10�10
=) 6 parameters.
a. Kostelecky and Tasson, 2010
SME coefficients
2 SME coefficients : (sXY , sXZ),
4 linear combinations : (sA, sD, s
E, sF) where
sA = s
XX
t� s
YY
t,
sD = s
XX
t+ s
YY
t� 2s
ZZ
t� 1.89s
YZ
t,
sE = s
TX � 1.22↵(ae+p
eff )X � 1.23↵(an
eff)X ,
sF = s
TY + 0.37sTZ � 0.12↵(ae+p
eff )Y � 0.12↵(an
eff)Y
� 0.052↵(ae+p
eff )Z � 0.052↵(an
eff)Z .
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LLR data sensitivity
Analysis
12 parameters ↵(aw
eff)µ [GeV/c2] for w = {e, p, n}.
Neutral macroscopic bodies : Ne
b = Np
b,
=) ↵(ae+p
eff )µ = ↵(ae
eff)µ + ↵(ap
eff)µ=) 8 parameters,With LLR : |↵(ae+p
eff )T | ' |↵(an
eff)T | < 10�4,With free fall (WEP) a : |↵(ae+p
eff )T | ' |↵(an
eff)T | < 10�10
=) 6 parameters.
a. Kostelecky and Tasson, 2010
SME coefficients
2 SME coefficients : (sXY , sXZ),
4 linear combinations : (sA, sD, s
E, sF) where
sA = s
XX
t� s
YY
t,
sD = s
XX
t+ s
YY
t� 2s
ZZ
t� 1.89s
YZ
t,
sE = s
TX � 1.22↵(ae+p
eff )X � 1.23↵(an
eff)X ,
sF = s
TY + 0.37sTZ � 0.12↵(ae+p
eff )Y � 0.12↵(an
eff)Y
� 0.052↵(ae+p
eff )Z � 0.052↵(an
eff)Z .
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Constraints on SME coefficients
Linear combinations
sA = s
XX
t � sYY
t ,
sD = s
XX
t + sYY
t � 2sZZ
t � 1.89sYZ
t ,
sE = s
TX � 1.22↵(ae+p
eff )X � 1.23↵(an
eff)X ,
sF = s
TY + 0.37sTZ � 0.12↵(ae+p
eff )Y � 0.12↵(an
eff)Y
� 0.052↵(ae+p
eff )Z � 0.052↵(an
eff)Z .
SME Valuess
A (+0.4 ± 1.7)⇥ 10�11
sXY
t (�3.9 ± 5.5)⇥ 10�12
sXZ
t (+0.7 ± 7.8)⇥ 10�12
sD (+3.9 ± 6.6)⇥ 10�12
sE (�1.1 ± 1.7)⇥ 10�8
sF (+6.1 ± 7.4)⇥ 10�9
TABLE – Constraints on minimal SME and gravity matter couplings with LLR. Results aredetermined from a global analysis and are given at 1�r . Bourgoin et. al 2017.
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Conclusion
Work realizedNew numerical lunar ephemeris in the SME framework,Found independent linear combinations of SME coefficients,=) Some fundamental SME coefficients.Estimate realistic errors with a resampling method,=) First real constraints on SME coefficients from LLR observations,=) No evidence for Lorentz symmetry violations.
Other results : correlationSME coefficients and the rotational motion of the Moon,SME coefficients and the Earth potential.
Perspectives
Coupled analysis with LLR and GRAIL space mission or SLR.
24 / 24
Conclusion
Work realizedNew numerical lunar ephemeris in the SME framework,Found independent linear combinations of SME coefficients,=) Some fundamental SME coefficients.Estimate realistic errors with a resampling method,=) First real constraints on SME coefficients from LLR observations,=) No evidence for Lorentz symmetry violations.
Other results : correlationSME coefficients and the rotational motion of the Moon,SME coefficients and the Earth potential.
Perspectives
Coupled analysis with LLR and GRAIL space mission or SLR.
24 / 24
Conclusion
Work realizedNew numerical lunar ephemeris in the SME framework,Found independent linear combinations of SME coefficients,=) Some fundamental SME coefficients.Estimate realistic errors with a resampling method,=) First real constraints on SME coefficients from LLR observations,=) No evidence for Lorentz symmetry violations.
Other results : correlationSME coefficients and the rotational motion of the Moon,SME coefficients and the Earth potential.
Perspectives
Coupled analysis with LLR and GRAIL space mission or SLR.
24 / 24