How can we have degenerate theories? I. Gauge Redundancy: o ff -shell constraints E.g. GR or Gauge theories II. Constrained systems: on-shell constraints E.g. Non-trivial kinetic mixing between scalar and tensor [Langlois and Noui’15] III. Total Derivatives and Antisymmetry: 1st derivatives: r µ φ r µ φ ! OK 2nd derivatives: f ( φ ) g µ ν r µ r ν φ f ( φ ) g µ ν r µ r ν φ � r µ ( f ( φ ) g µ ν r ν φ ) = �r µ ( f ( φ ) g µ ν ) r ν φ 3rd derivatives: r µ r ν r α φ JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories? I. Gauge Redundancy: o ff -shell constraints E.g. GR or Gauge theories II. Constrained systems: on-shell constraints E.g. Non-trivial kinetic mixing between scalar and tensor [Langlois and Noui’15] III. Total Derivatives and Antisymmetry: 1st derivatives: r µ φ r µ φ ! OK 2nd derivatives: f ( φ ) g µ ν r µ r ν φ f ( φ ) g µ ν r µ r ν φ � r µ ( f ( φ ) g µ ν r ν φ ) = �r µ ( f ( φ ) g µ ν ) r ν φ 3rd derivatives: [ r µ , r ν ] r α φ JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories? I. Gauge Redundancy: o ff -shell constraints E.g. GR or Gauge theories II. Constrained systems: on-shell constraints E.g. Non-trivial kinetic mixing between scalar and tensor [Langlois and Noui’15] III. Total Derivatives and Antisymmetry: 1st derivatives: r µ φ r µ φ ! OK 2nd derivatives: f ( φ ) g µ ν r µ r ν φ f ( φ ) g µ ν r µ r ν φ � r µ ( f ( φ ) g µ ν r ν φ ) = �r µ ( f ( φ ) g µ ν ) r ν φ 3rd derivatives: [ r µ , r ν ] r α φ = 0 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories? I. Gauge Redundancy: o ff -shell constraints E.g. GR or Gauge theories II. Constrained systems: on-shell constraints E.g. Non-trivial kinetic mixing between scalar and tensor [Langlois and Noui’15] III. Total Derivatives and Antisymmetry: 1st derivatives: r µ φ r µ φ ! OK 2nd derivatives: f ( φ ) g µ ν r µ r ν φ f ( φ ) g µ ν r µ r ν φ � r µ ( f ( φ ) g µ ν r ν φ ) = �r µ ( f ( φ ) g µ ν ) r ν φ [ r µ , r ν ] r α φ = 1 2 R α β µ ν r β φ 3rd derivatives: JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
How can we have degenerate theories? I. Gauge Redundancy: o ff -shell constraints E.g. GR or Gauge theories II. Constrained systems: on-shell constraints E.g. Non-trivial kinetic mixing between scalar and tensor [Langlois and Noui’15] III. Total Derivatives and Antisymmetry: 1st derivatives: r µ φ r µ φ ! OK 2nd derivatives: f ( φ ) g µ ν r µ r ν φ f ( φ ) g µ ν r µ r ν φ � r µ ( f ( φ ) g µ ν r ν φ ) = �r µ ( f ( φ ) g µ ν ) r ν φ [ r µ , r ν ] r α φ = 1 2 R α β µ ν r β φ 3rd derivatives: R α β [ µ ν ; ρ ] = R α [ β µ ν ] = 0 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
2 Horndeski’s theory Horndeski’s work: [Horndeski 1974] • JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
2 Horndeski’s theory Horndeski’s work: (local+Di ff . inv. theories) [Horndeski 1974] • 1st: find most general scalar-tensor 2nd order EoM in 4D 381 S E C O N D - O R D E R S C A L A R - T E N S O R FIELD E Q U A T I O N S 2nd: find a Lagrangian that reproduces them 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 ik _ 4 ~.. 6cde cde Ih 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 -3 %/(g)(2o (4.21) + N/(g){4Jf9 - p(2,~'" + 4'~"" + p_o~ + 20/{'9)} where (4.22a) ~= f ~ Ks(O;p)dp; ~IU= - W JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting and 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). Remark. The Lagrangian which yields the tensor density mentioned in 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)
2 Horndeski’s theory Horndeski’s work: (local+Di ff . inv. theories) [Horndeski 1974] • 1st: find most general scalar-tensor 2nd order EoM in 4D 381 S E C O N D - O R D E R S C A L A R - T E N S O R FIELD E Q U A T I O N S 2nd: find a Lagrangian that reproduces them 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 ik _ 4 ~.. 6cde cde Ih 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 -3 %/(g)(2o (4.21) + N/(g){4Jf9 - p(2,~'" + 4'~"" + p_o~ + 20/{'9)} where X ⌘ � 1 There are 4 free functions of and φ 2 r µ φ r µ φ (4.22a) � ~= f ~ Ks(O;p)dp; There are non-minimal couplings (with derivatives) ~IU= - W JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting and 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). Remark. The Lagrangian which yields the tensor density mentioned in 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)
5 Full Horndeski’s theory (modern notation): L H = X • L H i i =2 L H 2 = G 2 ( φ , X ) L H 3 = G 3 ( φ , X )[ Φ ] 4 = G 4 ( φ , X ) R + G 4 ,X ([ Φ ] 2 � [ Φ 2 ]) L H 5 = G 5 ( φ , X ) G µ ν r µ r ν φ � 1 6 G 5 ,X ([ Φ ] 3 � 3[ Φ ][ Φ 2 ] + 2[ Φ 3 ]) L H ( Φ n ; ν , [ Φ n ] ≡ Φ n µ ν g µ ν ) µ ν ≡ φ µ α 1 φ ; α 1 ; α 2 · · · φ ; α n − 1 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
5 Full Horndeski’s theory (modern notation): L H = X • L H i i =2 L H 2 = G 2 ( φ , X ) L H 3 = G 3 ( φ , X )[ Φ ] 4 = G 4 ( φ , X ) R + G 4 ,X ([ Φ ] 2 � [ Φ 2 ]) L H 5 = G 5 ( φ , X ) G µ ν r µ r ν φ � 1 6 G 5 ,X ([ Φ ] 3 � 3[ Φ ][ Φ 2 ] + 2[ Φ 3 ]) L H ( Φ n ; ν , [ Φ n ] ≡ Φ n µ ν g µ ν ) µ ν ≡ φ µ α 1 φ ; α 1 ; α 2 · · · φ ; α n − 1 Incorporates most inflationary and dark energy models! • Simplest subcases: • 1 G 2 = − Λ Einstein-Hilbert+ Λ : 8 π G, G 4 = 16 π G, G 3 = G 5 = 0 1 φ Jordan-Brans-Dicke: G 2 = w ( φ ) X − V ( φ ) , G 4 = 16 π G, G 3 = G 5 = 0 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
There are codes to test the cosmology of your favorite model • [Bellini, Lesgourgues, Sawicki and Zumalacárregui] JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
There are codes to test the cosmology of your favorite model • [Bellini, Lesgourgues, Sawicki and Zumalacárregui] JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
Brief summary: � � � � � � JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
Brief summary: � i) Antisymmetry is a key ingredient to � avoid Ostrogradski’ s instabilities � � � � JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
Brief summary: � i) Antisymmetry is a key ingredient to � avoid Ostrogradski’ s instabilities � ii) Healthy second order scalar-tensor � theories (Horndeski) are constructed � using this property � JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
Brief summary: � i) Antisymmetry is a key ingredient to � avoid Ostrogradski’ s instabilities � ii) Healthy second order scalar-tensor � theories (Horndeski) are constructed � using this property � [Reminder] The mathematical objects that describes antisymmetric quantities are the differential forms JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
3 Di ff erential Forms and Gravity General Covariance (Di ff Inv.) can be reinterpreted • as the invariance under Local Lorentz Transformations (LLT) in the Tangent Space Needed to couple fermions to gravity! • In a pseudo-Riemannian manifold (usual spacetime without torsion • and metric compatible): Geometry (and Physics) is encoded in the vielbein and the 1-form connection ⇒ R a ω a θ a b b JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
3 Di ff erential Forms and Gravity General Covariance (Di ff Inv.) can be reinterpreted • as the invariance under Local Lorentz Transformations (LLT) in the Tangent Space Needed to couple fermions to gravity! • In a pseudo-Riemannian manifold (usual spacetime without torsion • and metric compatible): Geometry (and Physics) is encoded in the vielbein and the 1-form connection ⇒ R a ω a θ a b b Example: Lovelock’s Theory • [Lovelock 1971, Zumino 1986] l ^ R a i b i ∧ θ ? and L ( l ) = where 2 l ≤ D a 1 b 1 ··· a l b l i =0 1 ( D − k )! ✏ a 1 ··· a k a k +1 ··· a D ✓ a k +1 ∧ · · · ∧ ✓ a D ✓ ? a 1 ··· a k = JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
Di ff erential Forms Dictionary g = g µ ν dx µ ⊗ dx ν = η ab θ a ⊗ θ b Metric Formalism Vielbein Formalism θ a = e a µ dx µ g µ ν Γ λ ω a µ ν b R λ R a µ νρ b r µ g αβ = 0 ω ab = � ω ba T a = D θ a = 0 Γ λ µ ν = Γ λ ν µ Invariant objects: and • ✏ a 1 a 2 ··· a D η ab Basic operations: wedge product, exterior di ff erential, integration… • Basic identities: Cartan’s structure equations and Bianchi’s identities • JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
3 The di ff erential forms formalism [PRD94.024005 (2016)] Define 1-forms with derivatives of the scalar field (at lowest order) • Ψ a ⌘ r a φ r b φ θ b Φ a ⌘ r a r b φ θ b JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
3 The di ff erential forms formalism [PRD94.024005 (2016)] Define 1-forms with derivatives of the scalar field (at lowest order) • Ψ a ⌘ r a φ r b φ θ b Φ a ⌘ r a r b φ θ b …construct a basis of Lagrangians invariant under LLT in a pseudo- Riemannian manifold l m n ^ R a i b i ^ ^ Φ c j ^ ^ Ψ d k ^ θ ? L ( lmn ) = a 1 b 1 ··· a l b l c 1 ··· c m d 1 ··· d n i =1 j =1 k =1 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
3 The di ff erential forms formalism [PRD94.024005 (2016)] Define 1-forms with derivatives of the scalar field (at lowest order) • Ψ a ⌘ r a φ r b φ θ b Φ a ⌘ r a r b φ θ b …construct a basis of Lagrangians invariant under LLT in a pseudo- Riemannian manifold l m n ^ R a i b i ^ ^ Φ c j ^ ^ Ψ d k ^ θ ? L ( lmn ) = a 1 b 1 ··· a l b l c 1 ··· c m d 1 ··· d n i =1 j =1 k =1 Clear structure in terms of the number of fields: p ≡ 2 l + m + n ≤ D Finite basis due to antisymmetry Contains well-known theories, e.g. Horndeski and Beyond Horndeski JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
p ≤ D Z • Action of a general scalar-tensor theory: X S = α lmn L ( lmn ) M l,m,n JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
p ≤ D Z • Action of a general scalar-tensor theory: X S = α lmn L ( lmn ) M l,m,n • Examples: some 4D Lagrangians ( η ≡ √− gd 4 x, Φ n µ ν = φ ; µ α 1 φ ; α 1 ; α 2 · · · φ ; α n − 1 ; ν , [ Φ n ] ≡ Φ n µ ν g µ ν ) L (001) = Ψ a ^ θ ? a = r µ φ r µ φη ⌘ � 2 X η L (010) = Φ a ^ θ ? a = [ Φ ] η abc = ( � 2 R µ ⌫ + Rg µ ⌫ ) Φ µ ⌫ η = � 2( G µ ⌫ Φ µ ⌫ ) η L (110) = R ab ^ Φ c ^ θ ? L (030) = Φ a ^ Φ b ^ Φ c ^ θ ? abc = ([ Φ ] 3 � 3[ Φ ][ Φ 2 ] + 2[ Φ 3 ]) η abcd = ( R µ ⌫⇢� R µ ⌫⇢� � 4 R ↵� R ↵� + R 2 ) η L (200) = R ab ^ R cd ^ θ ? JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
p ≤ D Z • Action of a general scalar-tensor theory: X S = α lmn L ( lmn ) M l,m,n • Examples: some 4D Lagrangians ( η ≡ √− gd 4 x, Φ n µ ν = φ ; µ α 1 φ ; α 1 ; α 2 · · · φ ; α n − 1 ; ν , [ Φ n ] ≡ Φ n µ ν g µ ν ) L (001) = Ψ a ^ θ ? a = r µ φ r µ φη ⌘ � 2 X η L (010) = Φ a ^ θ ? a = [ Φ ] η abc = ( � 2 R µ ⌫ + Rg µ ⌫ ) Φ µ ⌫ η = � 2( G µ ⌫ Φ µ ⌫ ) η L (110) = R ab ^ Φ c ^ θ ? L (030) = Φ a ^ Φ b ^ Φ c ^ θ ? abc = ([ Φ ] 3 � 3[ Φ ][ Φ 2 ] + 2[ Φ 3 ]) η abcd = ( R µ ⌫⇢� R µ ⌫⇢� � 4 R ↵� R ↵� + R 2 ) η L (200) = R ab ^ R cd ^ θ ? • The basis is closed under exterior derivatives if contractions with the gradient field are included • Notation: over bar indicates contractions with r a φ 10) = r a φ Φ a ^ θ ? e.g. b r b φ L (0¯ • Additional Lagrangians: and L (¯ L ( l ¯ mn ) lmn ) JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
3 Results [PRD94.024005 (2016)] JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
3 Results [PRD94.024005 (2016)] • We compute the EoM both for the scalar field and the vielbein for φ θ a arbitrary dimensions • We obtain all the exact forms (total derivatives) and antisymmetric algebraic identities relating di ff erent theories JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
3 Results [PRD94.024005 (2016)] • We compute the EoM both for the scalar field and the vielbein for φ θ a arbitrary dimensions The calculations greatly simplifies We find the possible Lagrangians with 2nd order EoM • We obtain all the exact forms (total derivatives) and antisymmetric algebraic identities relating di ff erent theories We determine the number of independent Lagrangians JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
3 Results [PRD94.024005 (2016)] • We compute the EoM both for the scalar field and the vielbein for φ θ a arbitrary dimensions The calculations greatly simplifies We find the possible Lagrangians with 2nd order EoM • We obtain all the exact forms (total derivatives) and antisymmetric algebraic identities relating di ff erent theories We determine the number of independent Lagrangians There are 10 independent elements in the basis of Lagrangians • Results: (4D) Only 4 independent linear combinations give rise to 2nd order EoM -This set can be associated with Horndeski’s theory JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
How scalar-tensor theories are related? • p = 0 : L (000) (81) p = 1 : L (001) L (010) (73) L (0¯ (82) 10) p = 2 : L (100) L (011) L (020) (74) L (¯ L (0¯ (83) (84) 100) 20) p = 3 : L (101) L (110) L (021) L (030) (75) (76) L (1¯ L (¯ L (0¯ (85) (86) 10) 110) 30) p = 4 : L (200) L (111) L (120) L (031) L (040) (77) (78) (87) (88) (89) L (¯ L (1¯ L (¯ L (0¯ 200) 20) 120) 40) JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
How scalar-tensor theories are related? • L H p = 0 : 2 [ G 2 ] = G 2 L (000) L (000) L H 3 [ G 3 ] = G 3 L (010) (81) L H p = 1 : 4 [ G 4 ] = G 4 L (100) + G 4 ,X L (020) L (001) L (010) (73) 5 [ G 5 ] = G 5 L (110) + 1 L H 3 G 5 ,X L (030) L (0¯ (82) 10) p = 2 : L (100) L (011) L (020) (74) L (¯ L (0¯ (83) (84) 100) 20) p = 3 : L (101) L (110) L (021) L (030) (75) (76) L (1¯ L (¯ L (0¯ (85) (86) 10) 110) 30) p = 4 : L (200) L (111) L (120) L (031) L (040) (77) (78) (87) (88) (89) L (¯ L (1¯ L (¯ L (0¯ 200) 20) 120) 40) ⇣ ⌘ DL D − 1 ( lmn ) [ G i ] = G i, φ L ( lm ( n +1)) − G i,X L ( l ( m +1) n ) + G i L ( l ( m +1) n ) − m L (( l +1)( m − 1) n ) − n L ( l ( m +1) n ) L ( lm 1) = − 2 l L (¯ lm 0) − m L ( l ¯ m 0) − 2 X L ( lm 0) JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
Summary and Outlook: � � � � � � � � � JM. Ezquiaga 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 � [PRD94.024005 (2016)] � � � � � JM. Ezquiaga 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 � [PRD94.024005 (2016)] � � ii) It also allows further generalizations and � naturally incorporates the description of field � redefinitions [to appear soon] � JM. Ezquiaga 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 � [PRD94.024005 (2016)] � � ii) It also allows further generalizations and � naturally incorporates the description of field � redefinitions [to appear soon] � [Question] How can we test these general models? JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
4 Testing Modified Gravity • Gravity can be tested at very di ff erent scales [Review by C. Will 2014] Planck Scale Laboratory Solar System Cosmos m 10 - 35 10 - 4 10 14 10 26 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
4 Testing Modified Gravity • Gravity can be tested at very di ff erent scales [Review by C. Will 2014] • Classical tests: Eötvös experiment, deflection of light, Shapiro time delay… Planck Scale Laboratory Solar System Cosmos m 10 - 35 10 - 4 10 14 10 26 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
4 Testing Modified Gravity • Gravity can be tested at very di ff erent scales [Review by C. Will 2014] • Classical tests: Eötvös experiment, deflection of light, Shapiro time delay… • Modified Gravity: Screening Mechanism [Review by P. Brax 2013] Planck Scale Laboratory Solar System Cosmos m 10 - 35 10 - 4 10 14 10 26 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Gravity can be tested at very di ff erent regimes [Review by D. Psaltis 2008] Moon Sun White Dwarf Neutron Star BH horizon � 10 - 10 10 - 7 10 - 5 0.1 1 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Gravity can be tested at very di ff erent regimes [Review by D. Psaltis 2008] ✏ ≡ GM • Strong Gravity Regime: Compact Objects, AGNs, Binary Systems… 2 rc 2 Moon Sun White Dwarf Neutron Star BH horizon � 10 - 10 10 - 7 10 - 5 0.1 1 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Gravity can be tested at very di ff erent regimes [Review by D. Psaltis 2008] ✏ ≡ GM • Strong Gravity Regime: Compact Objects, AGNs, Binary Systems… 2 rc 2 • Specific signatures in alternatives to GR, e.g. scalar radiation [Eardley 1974] Moon Sun White Dwarf Neutron Star BH horizon � 10 - 10 10 - 7 10 - 5 0.1 1 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Cosmological tests: CMB (T, B-modes), LSS (Lensing, Clustering), 21-cm… [Review by K. Koyama 2015] Planck Scale Laboratory Solar System Cosmos m 10 - 35 10 - 4 10 14 10 26 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Cosmological tests: CMB (T, B-modes), LSS (Lensing, Clustering), 21-cm… [Review by K. Koyama 2015] Planck Scale Laboratory Solar System Cosmos m 10 - 35 10 - 4 10 14 10 26 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Cosmological tests: CMB (T, B-modes), LSS (Lensing, Clustering), 21-cm… [Review by K. Koyama 2015] • Constraints on Horndeski: Present and Future O (1 − 0 . 5) O (0 . 1 − 0 . 01) [Bellini et al. 2016] [Alonso et al. 2016] Planck Scale Laboratory Solar System Cosmos m 10 - 35 10 - 4 10 14 10 26 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• New window to the Universe with Gravitational Wave Astronomy [GW Group at Cambridge] JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• New window to the Universe with Gravitational Wave Astronomy [GW Group at Cambridge] JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• New window to the Universe with Gravitational Wave Astronomy [GW Group at Cambridge] JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
5 The fate of Scalar-Tensor gravity [arXiv 1608.01982] • Fundamental analysis: Test speed of gravity JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
5 The fate of Scalar-Tensor gravity [arXiv 1608.01982] • Fundamental analysis: Test speed of gravity • Some general Scalar-Tensor gravity predicts anomalous propagation speed • At small scales for arbitrary backgrounds L ∝ h µ ν G αβ ∂ α ∂ β h µ ν JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
5 The fate of Scalar-Tensor gravity [arXiv 1608.01982] • Fundamental analysis: Test speed of gravity • Some general Scalar-Tensor gravity predicts anomalous propagation speed • At small scales for arbitrary backgrounds L ∝ h µ ν G αβ ∂ α ∂ β h µ ν = h µ ν ( C ⇤ + W αβ ∂ α ∂ β ) h µ ν i) Disformal e ff ective gravitational metric G µ ν 6 = Ω ( x ) g µ ν -Captured by a Weyl tensor in the EoM ii) Vacuum expectation value for the scalar φ ( x ) -Derivative coupling to the Weyl W � ∂φ , rr φ · · · JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
5 The fate of Scalar-Tensor gravity [arXiv 1608.01982] • Fundamental analysis: Test speed of gravity • Some general Scalar-Tensor gravity predicts anomalous propagation speed • At small scales for arbitrary backgrounds L ∝ h µ ν G αβ ∂ α ∂ β h µ ν = h µ ν ( C ⇤ + W αβ ∂ α ∂ β ) h µ ν i) Disformal e ff ective gravitational metric G µ ν 6 = Ω ( x ) g µ ν -Captured by a Weyl tensor in the EoM ii) Vacuum expectation value for the scalar φ ( x ) -Derivative coupling to the Weyl W � ∂φ , rr φ · · · • E.g. Shift symmetric, quartic Horndeski theory L = G ( X ) R + G 0 ( X )([ Φ ] 2 − [ Φ 2 ]) JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
5 The fate of Scalar-Tensor gravity [arXiv 1608.01982] • Fundamental analysis: Test speed of gravity • Some general Scalar-Tensor gravity predicts anomalous propagation speed • At small scales for arbitrary backgrounds L ∝ h µ ν G αβ ∂ α ∂ β h µ ν = h µ ν ( C ⇤ + W αβ ∂ α ∂ β ) h µ ν i) Disformal e ff ective gravitational metric G µ ν 6 = Ω ( x ) g µ ν -Captured by a Weyl tensor in the EoM ii) Vacuum expectation value for the scalar φ ( x ) -Derivative coupling to the Weyl W � ∂φ , rr φ · · · • E.g. Shift symmetric, quartic Horndeski theory L = G ( X ) R + G 0 ( X )([ Φ ] 2 − [ Φ 2 ]) G c 2 g = G µ ν = G ( X ) g µ ν + G 0 ( X ) ∂ µ φ∂ ν φ G − G 0 ˙ φ 2 JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Two scenarios: [arXiv 1608.01982] JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Two scenarios: [arXiv 1608.01982] A) : GW-EM (or neutrino) counterpart c g ' c ✓ 200Mpc ◆ ✓ ∆ t ◆ c g c − 1 = 5 × 10 − 17 D 1 s ∆ t = ∆ t arrive − (1 + z ) ∆ t emit JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Two scenarios: [arXiv 1608.01982] A) : GW-EM (or neutrino) counterpart c g ' c ✓ 200Mpc ◆ ✓ ∆ t ◆ c g c − 1 = 5 × 10 − 17 D 1 s ∆ t = ∆ t arrive − (1 + z ) ∆ t emit Kill any theory with anomalous speed! • : GR, BD, cubic Horndeski, Kinetic Conf. c g = c • : quartic and quintic Horndeski, BH c g 6 = c JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Two scenarios: [arXiv 1608.01982] A) : GW-EM (or neutrino) counterpart c g ' c ✓ 200Mpc ◆ ✓ ∆ t ◆ c g c − 1 = 5 × 10 − 17 D 1 s ∆ t = ∆ t arrive − (1 + z ) ∆ t emit Kill any theory with anomalous speed! • : GR, BD, cubic Horndeski, Kinetic Conf. c g = c • : quartic and quintic Horndeski, BH c g 6 = c B) : No possible counterpart at cosmological scales c g 6 = c Di ff erence in the time of arrival becomes cosmological! JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Test speed of gravity with periodic sources [arXiv 1608.01982] JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Test speed of gravity with periodic sources [arXiv 1608.01982] • Phase Lag Test: measure di ff erence in phase of GWs and EM radiation ✓ c ∆ t = r ( t ) ◆ − 1 c c g y x ∆ x ∆ x - + - + h + - r t ∆ t ∆ t ∆ t ∆ t JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
• Test speed of gravity with periodic sources [arXiv 1608.01982] • Phase Lag Test: measure di ff erence in phase of GWs and EM radiation • There are sources already identified: eLISA verification binaries E.g. WDS J0651+2844 [Brown etal. 2012] ✓ c ∆ t = r ( t ) ◆ − 1 c c g y x ∆ x ∆ x - + - + h + - r t ∆ t ∆ t ∆ t ∆ t JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [PRD94.024005 (2016)] JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [PRD94.024005 (2016)] • There is a great variety of Modified Gravity theories (imply extra DoF) JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [PRD94.024005 (2016)] • There is a great variety of Modified Gravity theories (imply extra DoF) • We have presented a new formulation for scalar-tensor theories in the language of di ff erential forms. JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [PRD94.024005 (2016)] • There is a great variety of Modified Gravity theories (imply extra DoF) • We have presented a new formulation for scalar-tensor theories in the language of di ff erential forms. • This novel approach simplifies the computations and allows for a systematic classification of general scalar-tensor theories and the relations among them. JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [PRD94.024005 (2016)] • There is a great variety of Modified Gravity theories (imply extra DoF) • We have presented a new formulation for scalar-tensor theories in the language of di ff erential forms. • This novel approach simplifies the computations and allows for a systematic classification of general scalar-tensor theories and the relations among them. • There are interesting potential applications of this new formalism both at the practical and conceptual level: JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [PRD94.024005 (2016)] • There is a great variety of Modified Gravity theories (imply extra DoF) • We have presented a new formulation for scalar-tensor theories in the language of di ff erential forms. • This novel approach simplifies the computations and allows for a systematic classification of general scalar-tensor theories and the relations among them. • There are interesting potential applications of this new formalism both at the practical and conceptual level: • E.g. fermions in ST theories of gravity, explore general field redefinitions or systematically study ST theories with higher derivative EoM. JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [arXiv 1608.01982] JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [arXiv 1608.01982] • Modified Gravity theories can be tested at very di ff erent scales and regimes. JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [arXiv 1608.01982] • Modified Gravity theories can be tested at very di ff erent scales and regimes. • GWs astronomy opens a new window to the Universe. A fundamental test is to measure the speed of gravity. JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [arXiv 1608.01982] • Modified Gravity theories can be tested at very di ff erent scales and regimes. • GWs astronomy opens a new window to the Universe. A fundamental test is to measure the speed of gravity. • General ST theories can have anomalous propagation speed. We have shown that it is sourced by a non-conformal e ff ective metric with spontaneous breaking of LI by the scalar. JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
6 Conclusions [arXiv 1608.01982] • Modified Gravity theories can be tested at very di ff erent scales and regimes. • GWs astronomy opens a new window to the Universe. A fundamental test is to measure the speed of gravity. • General ST theories can have anomalous propagation speed. We have shown that it is sourced by a non-conformal e ff ective metric with spontaneous breaking of LI by the scalar. • There are two possible scenarios: • If : a GW-EM measurement will kill many ST theories c g = c • If : need periodic sources (phase lag test) c g 6 = c JM. Ezquiaga 18th of November of 2016, Oviedo V Postgraduate Meeting
Thank you Find more details at Phys. Rev. D94 , 024005 (2016) by JME, J. GARCÍA-BELLIDO, M. ZUMALACÁRREGUI arXiv 1608.01982 by D. BETTONI, JME , K. HINTERBICHLER and M. ZUMALACÁRREGUI or by e-mail jose.ezquiaga@uam.es
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