Constructing general scalar-tensor theories of gravity: Are they - - PowerPoint PPT Presentation
V Postgraduate Meeting on Theoretical Physics Constructing general scalar-tensor theories of gravity: Are they viable? Jose Mara EZQUIAGA Based on: Phys. Rev. D94 , 024005 (2016) by JME , J. GARCA-BELLIDO and M. ZUMALACRREGUI arXiv
Jose María EZQUIAGA V Postgraduate Meeting on Theoretical Physics
Based on:
arXiv 1608.01982 by D. BETTONI, JME, K. HINTERBICHLER and M. ZUMALACÁRREGUI
18th of November of 2016, Oviedo V Postgraduate Meeting
GR is in very good shape…
m Laboratory 10-4 Solar System 1014 Cosmos 1026 Planck Scale 10-35
18th of November of 2016, Oviedo V Postgraduate Meeting
GR is in very good shape…
great agreement at very different scales
m Laboratory 10-4 Solar System 1014 Cosmos 1026 Planck Scale 10-35
18th of November of 2016, Oviedo V Postgraduate Meeting
… but it is not enough.
m Laboratory 10-4 Solar System 1014 Cosmos 1026 Planck Scale 10-35
18th of November of 2016, Oviedo V Postgraduate Meeting
… but it is not enough.
m Laboratory 10-4 Solar System 1014 Cosmos 1026 Planck Scale 10-35
18th of November of 2016, Oviedo V Postgraduate Meeting
1 Why scalar-tensor theories? 2 Ostrogradski’s theorem and Horndeski’s theory 3 The differential forms formalism 4 Testing Modified Gravity 5 GWs and the fate of Scalar-Tensor gravity 6 Conclusions
18th of November of 2016, Oviedo V Postgraduate Meeting
1 Why scalar-tensor theories? 2 Ostrogradski’s theorem and Horndeski’s theory 3 The differential forms formalism 4 Testing Modified Gravity 5 GWs and the fate of Scalar-Tensor gravity 6 Conclusions
[PRD94.024005 (2016)] [arXiv 1608.01982]
18th of November of 2016, Oviedo V Postgraduate Meeting
scalar
18th of November of 2016, Oviedo V Postgraduate Meeting
scalar
…in fact theories with only massless spin-2 particles are fixed to follow linearized GR in the IR
[Lectures by Arkani-Hamed at IAS]
i p2 + i✏T µνNµν,αβT αβ Nµν,αβ ∼ ηµαηνβ + ηµβηνα − 2 D − 2ηµνηαβ
αβ µν
[deWitt 1967]
*Also Unimodular Gravity (TDiff)
[Van der Bij et al. 1982]
(Diff)
18th of November of 2016, Oviedo V Postgraduate Meeting
(typically scalar fields)
Example: Kaluza-Klein tower of states
18th of November of 2016, Oviedo V Postgraduate Meeting
(typically scalar fields)
Example: Kaluza-Klein tower of states
Example: Pion in Particle Physics Example: Scalaron in Gravity
3 4 M 2 Mpl2
2 4 6 8
V()
LStarobinsky ∼ R + R2 6M 2
18th of November of 2016, Oviedo V Postgraduate Meeting
Scalar-Vector-Tensor Theories [Bekenstein 2004] Massive Gravity [de Rham, Gabadadze and Tolley 2011]
Horava-Lifshitz Gravity [Horava 2009] Bi-gravity [Hassan and Rosen 2011] Multi-gravity [Hinterbichler and Rosen 2012] Multi-scalar-tensor [Damour and Esposito-Farese 1992] Vector-Tensor Theories [Will and Nordtvedt 1972]
18th of November of 2016, Oviedo V Postgraduate Meeting
18th of November of 2016, Oviedo V Postgraduate Meeting
E a r l y U n i v e r s e L a t e U n i v e r s e
18th of November of 2016, Oviedo V Postgraduate Meeting
[Solar System Tests] [CMB data] [LSS observations] [Gravitational Waves]
18th of November of 2016, Oviedo V Postgraduate Meeting
[Ostrogradski 1850]
L = L(q, ˙ q, ¨ q, · · · )
18th of November of 2016, Oviedo V Postgraduate Meeting
[Ostrogradski 1850]
L = L(q, ˙ q, ¨ q, · · · )
18th of November of 2016, Oviedo V Postgraduate Meeting
[Ostrogradski 1850]
EoM, the system is unstable L = L(q, ˙ q, ¨ q, · · · )
18th of November of 2016, Oviedo V Postgraduate Meeting
[Ostrogradski 1850]
EoM, the system is unstable
I knew I was right… 2nd order EoM rules!
L = L(q, ˙ q, ¨ q, · · · )
18th of November of 2016, Oviedo V Postgraduate Meeting
[Ostrogradski 1850]
EoM, the system is unstable
I knew I was right… 2nd order EoM rules! So… what about us?
L = L(q, ˙ q, ¨ q, · · · )
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
[Langlois and Noui’15]
E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
[Langlois and Noui’15]
E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
1st derivatives:
[Langlois and Noui’15]
rµφrµφ E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
1st derivatives: rµφrµφ ! OK
[Langlois and Noui’15]
E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
1st derivatives: 2nd derivatives: rµφrµφ ! OK f(φ)gµνrµrνφ
[Langlois and Noui’15]
E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
1st derivatives: 2nd derivatives: rµφrµφ ! OK f(φ)gµνrµrνφ
[Langlois and Noui’15]
f(φ)gµνrµrνφ rµ(f(φ)gµνrνφ) = rµ(f(φ)gµν)rνφ E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
1st derivatives: 2nd derivatives: 3rd derivatives: rµφrµφ ! OK f(φ)gµνrµrνφ rµrνrαφ
[Langlois and Noui’15]
f(φ)gµνrµrνφ rµ(f(φ)gµνrνφ) = rµ(f(φ)gµν)rνφ E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
1st derivatives: 2nd derivatives: 3rd derivatives: rµφrµφ ! OK f(φ)gµνrµrνφ
[Langlois and Noui’15]
f(φ)gµνrµrνφ rµ(f(φ)gµνrνφ) = rµ(f(φ)gµν)rνφ [rµ, rν]rαφ E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
1st derivatives: 2nd derivatives: 3rd derivatives: rµφrµφ ! OK f(φ)gµνrµrνφ [rµ, rν]rαφ = 0
[Langlois and Noui’15]
f(φ)gµνrµrνφ rµ(f(φ)gµνrνφ) = rµ(f(φ)gµν)rνφ E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
1st derivatives: 2nd derivatives: 3rd derivatives: rµφrµφ ! OK f(φ)gµνrµrνφ [rµ, rν]rαφ = 1 2Rα
βµνrβφ
[Langlois and Noui’15]
f(φ)gµνrµrνφ rµ(f(φ)gµνrνφ) = rµ(f(φ)gµν)rνφ E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories?
1st derivatives: 2nd derivatives: 3rd derivatives: rµφrµφ ! OK f(φ)gµνrµrνφ [rµ, rν]rαφ = 1 2Rα
βµνrβφ
Rα
β[µν;ρ] = Rα [βµν] = 0
[Langlois and Noui’15]
f(φ)gµνrµrνφ rµ(f(φ)gµνrνφ) = rµ(f(φ)gµν)rνφ E.g. Non-trivial kinetic mixing between scalar and tensor E.g. GR or Gauge theories
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[Horndeski 1974]
18th of November of 2016, Oviedo V Postgraduate Meeting
[Horndeski 1974]
1st: find most general scalar-tensor 2nd order EoM in 4D 2nd: find a Lagrangian that reproduces them
S E C O N D
R D E R S C A L A R
E N S O R FIELD E Q U A T I O N S 381 Upon comparing equation (4.18) with (3.20) we readily deduce that in a space of four-dimensions the symmetric tensor density presented in (3.20) is the Euler-Lagrange tensor corresponding to
. It~'~['7 8cde~ Ih R /to 4 " ede h l/ lk cde Ih
ik _ 4 ~.. 6cde
h / k
+ "~/(g)(~-+ 2~g/~)8~Rca fh + 2 x/(g)(2,Y,('3 - 2,3("1 + 4pX3)6~OIcI@IJ h
~'' + 4"///`' + pO~a)~lc Ic + 2x/(g)af86~le~l'f~lalh
+ N/(g){4Jf9 - p(2,~'" + 4'~"" + p_o~ + 20/{'9)} where
~= f ~ Ks(O;p)dp;
~IU= - W (4.21) and (4.22a) 5= f {,~Y"'l(q~; p) -- ff{'3(~;D)-2pX3((~;p)}d p (4.22b) To recapitulate the above work we have Theorem 4.1. In a space of four-dimensions any symmetric contravariant tensor density o frank 2 which is a concomitant of a pseudo-Riemannian metric tensor (with components gij), and its first two derivatives, together with a scalar fieM ~, and its first two derivatives, and furthermore is such that its components, A ab, satisfy
Aablb = (plaA(gij; gij, h;gij, hk; d~; O,h; (P, hk)
is the Euler-Lagrange tensor corresponding to a suitably chosen Lagrange scalar density of the form presented in equation (4.21).
Theorem 4.1 is unique only up to the addition of scalar densities of the form (1.5) which yield identically vanishing Euler-Lagrange tensors upon varying the gii's. As an immediate consequence of Theorem 4.1 we obtain Theorem 4.2. In spaces of four-dimensions the most general Euler-Lagrange equations which are at most of second-order in the derivatives of both gq and ~, and which are derivable from a Lagrange scalar density of the form (t.5)
(local+Diff. inv. theories)
18th of November of 2016, Oviedo V Postgraduate Meeting
[Horndeski 1974]
1st: find most general scalar-tensor 2nd order EoM in 4D 2nd: find a Lagrangian that reproduces them
S E C O N D
R D E R S C A L A R
E N S O R FIELD E Q U A T I O N S 381 Upon comparing equation (4.18) with (3.20) we readily deduce that in a space of four-dimensions the symmetric tensor density presented in (3.20) is the Euler-Lagrange tensor corresponding to
. It~'~['7 8cde~ Ih R /to 4 " ede h l/ lk cde Ih
ik _ 4 ~.. 6cde
h / k
+ "~/(g)(~-+ 2~g/~)8~Rca fh + 2 x/(g)(2,Y,('3 - 2,3("1 + 4pX3)6~OIcI@IJ h
~'' + 4"///`' + pO~a)~lc Ic + 2x/(g)af86~le~l'f~lalh
+ N/(g){4Jf9 - p(2,~'" + 4'~"" + p_o~ + 20/{'9)} where
~= f ~ Ks(O;p)dp;
~IU= - W (4.21) and (4.22a) 5= f {,~Y"'l(q~; p) -- ff{'3(~;D)-2pX3((~;p)}d p (4.22b) To recapitulate the above work we have Theorem 4.1. In a space of four-dimensions any symmetric contravariant tensor density o frank 2 which is a concomitant of a pseudo-Riemannian metric tensor (with components gij), and its first two derivatives, together with a scalar fieM ~, and its first two derivatives, and furthermore is such that its components, A ab, satisfy
Aablb = (plaA(gij; gij, h;gij, hk; d~; O,h; (P, hk)
is the Euler-Lagrange tensor corresponding to a suitably chosen Lagrange scalar density of the form presented in equation (4.21).
Theorem 4.1 is unique only up to the addition of scalar densities of the form (1.5) which yield identically vanishing Euler-Lagrange tensors upon varying the gii's. As an immediate consequence of Theorem 4.1 we obtain Theorem 4.2. In spaces of four-dimensions the most general Euler-Lagrange equations which are at most of second-order in the derivatives of both gq and ~, and which are derivable from a Lagrange scalar density of the form (t.5)
(local+Diff. inv. theories)
There are 4 free functions of and
X ⌘ 1 2rµφrµφ φ
18th of November of 2016, Oviedo V Postgraduate Meeting
5
X
i=2
LH
i
LH
2 = G2(φ, X)
LH
3 = G3(φ, X)[Φ]
LH
4 = G4(φ, X)R + G4,X([Φ]2 [Φ2])
LH
5 = G5(φ, X)Gµνrµrνφ 1
6G5,X([Φ]3 3[Φ][Φ2] + 2[Φ3]) (Φn
µν ≡ φµα1φ;α1 ;α2 · · · φ;αn−1 ;ν, [Φn] ≡ Φn µνgµν)
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5
X
i=2
LH
i
G2 = −Λ 8πG, G4 = 1 16πG, G3 = G5 = 0 G2 = 1 w(φ)X − V (φ), G4 = φ 16πG, G3 = G5 = 0 Einstein-Hilbert+Λ: Jordan-Brans-Dicke: LH
2 = G2(φ, X)
LH
3 = G3(φ, X)[Φ]
LH
4 = G4(φ, X)R + G4,X([Φ]2 [Φ2])
LH
5 = G5(φ, X)Gµνrµrνφ 1
6G5,X([Φ]3 3[Φ][Φ2] + 2[Φ3]) (Φn
µν ≡ φµα1φ;α1 ;α2 · · · φ;αn−1 ;ν, [Φn] ≡ Φn µνgµν)
18th of November of 2016, Oviedo V Postgraduate Meeting
[Bellini, Lesgourgues, Sawicki and Zumalacárregui]
18th of November of 2016, Oviedo V Postgraduate Meeting
[Bellini, Lesgourgues, Sawicki and Zumalacárregui]
18th of November of 2016, Oviedo V Postgraduate Meeting
Brief summary:
18th of November of 2016, Oviedo V Postgraduate Meeting
Brief summary:
avoid Ostrogradski’ s instabilities
18th of November of 2016, Oviedo V Postgraduate Meeting
Brief summary:
avoid Ostrogradski’ s instabilities ii) Healthy second order scalar-tensor theories (Horndeski) are constructed using this property
18th of November of 2016, Oviedo V Postgraduate Meeting
[Reminder] The mathematical objects that describes antisymmetric quantities are the differential forms
Brief summary:
avoid Ostrogradski’ s instabilities ii) Healthy second order scalar-tensor theories (Horndeski) are constructed using this property
18th of November of 2016, Oviedo V Postgraduate Meeting
as the invariance under Local Lorentz Transformations (LLT) in the Tangent Space
and metric compatible): Geometry (and Physics) is encoded in the vielbein and the 1-form connection
θa ωa
b
⇒ Ra
b
18th of November of 2016, Oviedo V Postgraduate Meeting
as the invariance under Local Lorentz Transformations (LLT) in the Tangent Space
[Lovelock 1971, Zumino 1986]
L(l) =
l
^
i=0
Raibi ∧ θ?
a1b1···albl
✓?
a1···ak =
1 (D − k)!✏a1···akak+1···aD✓ak+1 ∧ · · · ∧ ✓aD and
where 2l ≤ D
and metric compatible): Geometry (and Physics) is encoded in the vielbein and the 1-form connection
θa ωa
b
⇒ Ra
b
18th of November of 2016, Oviedo V Postgraduate Meeting
Differential Forms Dictionary
Metric Formalism Vielbein Formalism
g = gµνdxµ ⊗ dxν = ηabθa ⊗ θb gµν θa = ea
µdxµ
Γλ
µν
ωa
b
Rλ
µνρ
Ra
b
rµgαβ = 0 ωab = ωba Γλ
µν = Γλ νµ
T a = Dθa = 0
ηab ✏a1a2···aD
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Ψa ⌘ raφrbφ θb Φa ⌘ rarbφ θb
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
Ψa ⌘ raφrbφ θb Φa ⌘ rarbφ θb …construct a basis of Lagrangians invariant under LLT in a pseudo- Riemannian manifold
L(lmn) =
l
^
i=1
Raibi ^
m
^
j=1
Φcj ^
n
^
k=1
Ψdk ^ θ?
a1b1···alblc1···cmd1···dn
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
Ψa ⌘ raφrbφ θb Φa ⌘ rarbφ θb …construct a basis of Lagrangians invariant under LLT in a pseudo- Riemannian manifold
L(lmn) =
l
^
i=1
Raibi ^
m
^
j=1
Φcj ^
n
^
k=1
Ψdk ^ θ?
a1b1···alblc1···cmd1···dn
Clear structure in terms of the number of fields: Finite basis due to antisymmetry Contains well-known theories, e.g. Horndeski and Beyond Horndeski p ≡ 2l + m + n ≤ D
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
S =
p≤D
X
l,m,n
Z
M
αlmnL(lmn)
18th of November of 2016, Oviedo V Postgraduate Meeting
S =
p≤D
X
l,m,n
Z
M
αlmnL(lmn)
(η ≡ √−gd4x, Φn
µν = φ;µα1φ;α1 ;α2 · · · φ;αn−1 ;ν, [Φn] ≡ Φn µν gµν)
L(001) = Ψa ^ θ?
a = rµφrµφη ⌘ 2Xη
L(010) = Φa ^ θ?
a = [Φ]η
L(110) = Rab ^ Φc ^ θ?
abc = (2Rµ⌫ + Rgµ⌫)Φµ⌫η = 2(Gµ⌫Φµ⌫)η
L(030) = Φa ^ Φb ^ Φc ^ θ?
abc = ([Φ]3 3[Φ][Φ2] + 2[Φ3])η
L(200) = Rab ^ Rcd ^ θ?
abcd = (Rµ⌫⇢Rµ⌫⇢ 4R↵R↵ + R2)η
18th of November of 2016, Oviedo V Postgraduate Meeting
S =
p≤D
X
l,m,n
Z
M
αlmnL(lmn)
gradient field are included
(η ≡ √−gd4x, Φn
µν = φ;µα1φ;α1 ;α2 · · · φ;αn−1 ;ν, [Φn] ≡ Φn µν gµν)
L(001) = Ψa ^ θ?
a = rµφrµφη ⌘ 2Xη
L(010) = Φa ^ θ?
a = [Φ]η
L(110) = Rab ^ Φc ^ θ?
abc = (2Rµ⌫ + Rgµ⌫)Φµ⌫η = 2(Gµ⌫Φµ⌫)η
L(030) = Φa ^ Φb ^ Φc ^ θ?
abc = ([Φ]3 3[Φ][Φ2] + 2[Φ3])η
L(200) = Rab ^ Rcd ^ θ?
abcd = (Rµ⌫⇢Rµ⌫⇢ 4R↵R↵ + R2)η
L(¯
lmn)
L(l ¯
mn)
raφ L(0¯
10) = raφΦa ^ θ? brbφ
e.g.
18th of November of 2016, Oviedo V Postgraduate Meeting
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
arbitrary dimensions
identities relating different theories
θa φ
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
arbitrary dimensions
identities relating different theories
The calculations greatly simplifies We find the possible Lagrangians with 2nd order EoM We determine the number of independent Lagrangians θa φ
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
arbitrary dimensions
identities relating different theories
The calculations greatly simplifies We find the possible Lagrangians with 2nd order EoM We determine the number of independent Lagrangians There are 10 independent elements in the basis of Lagrangians Only 4 independent linear combinations give rise to 2nd order EoM
θa φ (4D)
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
p = 0 : L(000) p = 1 : L(001) L(010) L(0¯
10)
p = 2 : L(100) L(011) L(020) L(¯
100)
L(0¯
20)
p = 3 : L(101) L(110) L(021) L(030) L(1¯
10)
L(¯
110)
L(0¯
30)
p = 4 : L(200) L(111) L(120) L(031) L(040) L(¯
200)
L(1¯
20)
L(¯
120)
L(0¯
40)
(81) (82) (83) (84) (85) (86) (87) (88) (89) (73) (74) (75) (76) (77) (78)
18th of November of 2016, Oviedo V Postgraduate Meeting
p = 0 : L(000) p = 1 : L(001) L(010) L(0¯
10)
p = 2 : L(100) L(011) L(020) L(¯
100)
L(0¯
20)
p = 3 : L(101) L(110) L(021) L(030) L(1¯
10)
L(¯
110)
L(0¯
30)
p = 4 : L(200) L(111) L(120) L(031) L(040) L(¯
200)
L(1¯
20)
L(¯
120)
L(0¯
40)
(81) (82) (83) (84) (85) (86) (87) (88) (89) (73) (74) (75) (76) (77) (78)
DLD−1
(lmn)[Gi] = Gi,φL(lm(n+1))−Gi,XL(l(m+1)n)+Gi
⇣ L(l(m+1)n) − mL((l+1)(m−1)n) − nL(l(m+1)n) ⌘
L(lm1) = −2lL(¯
lm0) − mL(l ¯ m0) − 2XL(lm0)
LH
2 [G2] =G2L(000)
LH
3 [G3] =G3L(010)
LH
4 [G4] =G4L(100) + G4,XL(020)
LH
5 [G5] =G5L(110) + 1
3G5,XL(030)
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Summary and Outlook:
18th of November of 2016, Oviedo V Postgraduate Meeting
Summary and Outlook:
i) Differential Forms Formalism can be used to construct general ST theories, simplifying the calculations and clarifying the structure
18th of November of 2016, Oviedo V Postgraduate Meeting
Summary and Outlook:
i) Differential Forms Formalism can be used to construct general ST theories, simplifying the calculations and clarifying the structure ii) It also allows further generalizations and naturally incorporates the description of field redefinitions
[to appear soon]
18th of November of 2016, Oviedo V Postgraduate Meeting
[Question] How can we test these general models?
Summary and Outlook:
i) Differential Forms Formalism can be used to construct general ST theories, simplifying the calculations and clarifying the structure ii) It also allows further generalizations and naturally incorporates the description of field redefinitions
[to appear soon]
18th of November of 2016, Oviedo V Postgraduate Meeting
[Review by C. Will 2014]
m Laboratory 10-4 Solar System 1014 Cosmos 1026 Planck Scale 10-35
18th of November of 2016, Oviedo V Postgraduate Meeting
[Review by C. Will 2014]
m Laboratory 10-4 Solar System 1014 Cosmos 1026 Planck Scale 10-35
18th of November of 2016, Oviedo V Postgraduate Meeting
[Review by C. Will 2014]
m Laboratory 10-4 Solar System 1014 Cosmos 1026 Planck Scale 10-35
[Review by P. Brax 2013]
18th of November of 2016, Oviedo V Postgraduate Meeting
1 Neutron Star 0.1 White Dwarf 10-5 Sun 10-7 Moon 10-10
[Review by D. Psaltis 2008]
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1 Neutron Star 0.1 White Dwarf 10-5 Sun 10-7 Moon 10-10
[Review by D. Psaltis 2008]
✏ ≡ GM 2rc2
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1 Neutron Star 0.1 White Dwarf 10-5 Sun 10-7 Moon 10-10
[Review by D. Psaltis 2008]
✏ ≡ GM 2rc2
[Eardley 1974]
18th of November of 2016, Oviedo V Postgraduate Meeting
m Laboratory 10-4 Solar System 1014 Cosmos 1026 Planck Scale 10-35
[Review by K. Koyama 2015]
18th of November of 2016, Oviedo V Postgraduate Meeting
m Laboratory 10-4 Solar System 1014 Cosmos 1026 Planck Scale 10-35
[Review by K. Koyama 2015]
18th of November of 2016, Oviedo V Postgraduate Meeting
[Bellini et al. 2016] [Alonso et al. 2016]
m Laboratory 10-4 Solar System 1014 Cosmos 1026 Planck Scale 10-35
[Review by K. Koyama 2015]
O(0.1 − 0.01) O(1 − 0.5)
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[GW Group at Cambridge]
18th of November of 2016, Oviedo V Postgraduate Meeting
[GW Group at Cambridge]
18th of November of 2016, Oviedo V Postgraduate Meeting
[GW Group at Cambridge]
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
L ∝ hµνGαβ∂α∂βhµν
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
L ∝ hµνGαβ∂α∂βhµν = hµν(C⇤ + Wαβ∂α∂β)hµν
i) Disformal effective gravitational metric
ii) Vacuum expectation value for the scalar
Gµν 6= Ω(x)gµν φ(x) W ∂φ, rrφ · · ·
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
L ∝ hµνGαβ∂α∂βhµν = hµν(C⇤ + Wαβ∂α∂β)hµν L = G(X)R + G0(X)([Φ]2 − [Φ2])
i) Disformal effective gravitational metric
ii) Vacuum expectation value for the scalar
Gµν 6= Ω(x)gµν φ(x) W ∂φ, rrφ · · ·
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
L ∝ hµνGαβ∂α∂βhµν = hµν(C⇤ + Wαβ∂α∂β)hµν L = G(X)R + G0(X)([Φ]2 − [Φ2]) c2
g =
G G − G0 ˙ φ2
i) Disformal effective gravitational metric
ii) Vacuum expectation value for the scalar
Gµν = G(X)gµν + G0(X)∂µφ∂νφ
Gµν 6= Ω(x)gµν φ(x) W ∂φ, rrφ · · ·
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
A) : GW-EM (or neutrino) counterpart cg ' c cg c − 1 = 5 × 10−17 ✓200Mpc D ◆ ✓∆t 1s ◆
∆t = ∆tarrive − (1 + z)∆temit
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
A) : GW-EM (or neutrino) counterpart cg ' c cg c − 1 = 5 × 10−17 ✓200Mpc D ◆ ✓∆t 1s ◆
∆t = ∆tarrive − (1 + z)∆temit Kill any theory with anomalous speed! cg = c
cg 6= c
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
A) : GW-EM (or neutrino) counterpart cg ' c B) : No possible counterpart at cosmological scales cg 6= c cg c − 1 = 5 × 10−17 ✓200Mpc D ◆ ✓∆t 1s ◆
∆t = ∆tarrive − (1 + z)∆temit Kill any theory with anomalous speed! cg = c
cg 6= c Difference in the time of arrival becomes cosmological!
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
∆x h+ ∆t ∆t ∆t ∆t t
r x y ∆x + +
c ✓ c cg − 1 ◆
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
∆x h+ ∆t ∆t ∆t ∆t t
r x y ∆x + +
c ✓ c cg − 1 ◆
E.g. WDS J0651+2844
[Brown etal. 2012]
18th of November of 2016, Oviedo V Postgraduate Meeting
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
language of differential forms.
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
language of differential forms.
systematic classification of general scalar-tensor theories and the relations among them.
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
language of differential forms.
systematic classification of general scalar-tensor theories and the relations among them.
at the practical and conceptual level:
[PRD94.024005 (2016)]
18th of November of 2016, Oviedo V Postgraduate Meeting
language of differential forms.
systematic classification of general scalar-tensor theories and the relations among them.
at the practical and conceptual level:
[PRD94.024005 (2016)]
redefinitions or systematically study ST theories with higher derivative EoM.
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
18th of November of 2016, Oviedo V Postgraduate Meeting
[arXiv 1608.01982]
18th of November of 2016, Oviedo V Postgraduate Meeting
to measure the speed of gravity.
[arXiv 1608.01982]
18th of November of 2016, Oviedo V Postgraduate Meeting
to measure the speed of gravity.
that it is sourced by a non-conformal effective metric with spontaneous breaking of LI by the scalar.
[arXiv 1608.01982]
18th of November of 2016, Oviedo V Postgraduate Meeting
to measure the speed of gravity.
that it is sourced by a non-conformal effective metric with spontaneous breaking of LI by the scalar.
[arXiv 1608.01982]
cg = c cg 6= c
Find more details at
jose.ezquiaga@uam.es
arXiv 1608.01982 by D. BETTONI, JME, K. HINTERBICHLER and M. ZUMALACÁRREGUI
18th of November of 2016, Oviedo V Postgraduate Meeting
[Ostrogradski 1850] Non-degeneracy: L = L(q, ˙ q) ⇒ ∂L ∂q − d dt ∂L ∂ ˙ q = 0 = ⇒ ¨ q = F(q, ˙ q) ∂2L ∂ ˙ q2 6= 0 Q ≡ q P ≡ ∂L ∂ ˙ q 2 initial value data
H(Q, P) ≡ P ˙ q − L
18th of November of 2016, Oviedo V Postgraduate Meeting
L = L(q, ˙ q, ¨ q) ⇒ ∂L ∂q − d dt ∂L ∂ ˙ q + d2 dt2 ∂L ∂¨ q = 0 = ⇒ .... q = F(q, ˙ q, ¨ q, ... q ) ∂2L ∂¨ q2 6= 0 Non-degeneracy: [See review by Woodard 2015] P2 ≡ ∂L ∂¨ q P1 ≡ ∂L ∂ ˙ q − d dt ∂L ∂¨ q Q1 ≡ q Q2 ≡ ˙ q 4 initial value data
H(Q1, Q2, P1, P2) ≡
2
X
i=1
Piq(i) − L = P1Q2 + P2A(Q1, Q2, P2) − L(Q1, Q2, A)
18th of November of 2016, Oviedo V Postgraduate Meeting
Disformal transformations: ADM and Unitary Gauge: Generalized Generalized Galileons
˜ gµν = C(φ, X)gµν + D(φ, X)rµφrνφ
[Zumalacárregui and García-Bellido 2013] [Gleyzes, Langlois, Piazza and Vernizzi 2014]
Degenerate Theories: Extended Scalar-Tensor Theories
[Langlois and Noui 2015]
(Horndeski’s theory was derived covariantly)
There can be higher order spatial derivatives
[de Rham and Matas 2016]
(inducing Lorentz breaking)
[Crisostomi, Hull, Koyama, and Tasinato 2016]
Hidden Constraints Two new Lagrangians Not healthy with full Horndeski
[Crisostomi, Koyama, and Tasinato 2016]
Hamiltonian Analysis Full Classification
18th of November of 2016, Oviedo V Postgraduate Meeting
gµ⌫ = ηabea
µeb ⌫
θa = ea
µdxµ
Γ
µ⌫ = 1
2g↵(∂µg↵⌫ + ∂⌫gµ↵ ∂↵gµ⌫) ωab = 1 2
R
µ⌫⇢ = ∂⌫Γ µ⇢ ∂⇢Γ µ⌫ + Γ µ⇢Γ ⌫ Γ µ⌫Γ ⇢
Ra
b = Dωa b = dωa b + ωa c ^ ωc b
rµg↵ = 0 ωab = ωba Γ
µ⌫ = Γ ⌫µ
T a = Dθa = 0
Differential Forms Dictionary (extended)
Metric Formalism Vielbein Formalism
g = gµνdxµ ⊗ dxν = ηabθa ⊗ θb
18th of November of 2016, Oviedo V Postgraduate Meeting
(i) Take perturbations: (ii) Use (simple) relations between building blocks: (iii) Identify higher order terms: (iv) Choose appropriate coefficients: δΦa = Draδφ, δRab = 0, δGi(φ, X) = Gi,φδφ Gi,Xrzφrzδφ DΨa = Φa ^ Dφ, DΦa = Ra
zrzφ, DRab = 0
rzΦa rzRab rzΦa = raΦz + irRaz rzRab ^ θ?
ab = 2raRbz ^ θ? ab
δ(GiL(lmn)) δ(GiL(lmn)) ! higher order ⇠ rzRab δ(FiL(l0m0n0)) ! higher order ⇠ raRbz δ(FiL(l0m0n0)) ⇒ Fi(Gi)
18th of November of 2016, Oviedo V Postgraduate Meeting
δ(G4L(100)) =δG4 ^ Rab ^ θ?
ab
=δφ ^
^ θ?
ab
δ(F4L(020)) =δF4 ^ Φa ^ Φb ^ θ?
ab + 2F4 ^ δΦa ^ Φb ^ θ? ab
=δφ ^ (F4, ^ Φa + rz (F4,Xrzφ) ^ Φa + 2F4,Xrzφ ^ rzΦa) ^ Φb ^ θ?
ab
+2δφ ^ (ra (F4,Dφ) ra (F4,Xrzφ) ^ Φz) ^ Φb ^ θ?
ab
+2δφ ^
ab + D (F4) ^ raΦb ^ θ? ab
ab + F4rzφ ^ raRbz ^ θ? ab
ab + 2F4 ∧ δΦa ∧ Φb ∧ θ? ab
δ(G4L(100)) =δG4 ∧ Rab ∧ θ?
ab
G4(φ, X)L(100) F4(φ, X)L(020)
4 =G4 ∧ Rab ∧ θ? ab + G4,X ∧ Φa ∧ Φb ∧ θ? ab
=
LH
4 =G4 ∧ Rab ∧ θ? ab + G4,X ∧ Φa ∧ Φb ∧ θ? ab
18th of November of 2016, Oviedo V Postgraduate Meeting
αlmn(φ, X)L(lmn)
L2nd(αlmn) = αlmnL(lmn) +
l
X
j=1
α(l−j)(m+2j)nL((l−j)(m+2j)n) +
m/2
X
k=1
α(l+k)(m−2k)nL((l+k)(m−2k)n)
α(l−j)(m+2j)n = 2(l − (j + 1)) (m + 2j)(m + 2j − 1) ∂(α(l−(j−1))(m+2(j−1))n) ∂X , α(l+k)(m−2k)n = (m − 2(k − 1))(m − 1 − 2(k − 1)) 2(l + k) Z α(l+(k−1))(m−2(k−1))ndX
δ(αlmnL(lmn)) =δαlmn ^ L(lmn) + αlmn ^ δL(lmn) =δαlmn ^ L(lmn) + mαlmn ^ δΦa ^ [L(l(m−1)n)]a + nαlmn ^ δΨa ^ [L(lm(n−1))]a =δφ ^ ⇣ αlmn,φ + rz(αlmn,Xrzφ)
+mrzΦa ^ [L(l(m−1)n)]a + nrzΨa ^ [L(lm(n−1))]a
αlmn,XrzφraΦz ^ [L(l(m−1)n)]a Φz ^ ra(αlmn,Xrzφ[L(l(m−1)n)]a)
⌘ δ(αlmnL(lmn)) =δαlmn ∧ L(lmn) + αlmn ∧ δL(lmn) =δαlmn ∧ L(lmn) + mαlmn ∧ δΦa ∧ [L(l(m−1)n)]a + nαlmn ∧ δΨa ∧ [L(lm(n−1))]a