Cosmic structures and gravitational waves in ghost-free scalar-tensor theories of gravity Purnendu Karmakar with Nicola Bartolo, Sabino Matarrese and Mattia Scomparin (arXiv:1712.04002 [gr-qc]) Department of Physics and Astronomy “Galileo Galilei”, University of Padova, Italy February 14, 2018 GC2018, YITP, Japan Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Motivation Screening Screening in Modified Gravity Modified gravity Introduce additional degrees of freedom (often scalar): fifth force Gravity is well tested in the solar system . Screening Screening the fifth force in the solar system, and GR is recovered. e.g.: Vainshtein, Chameleon, Symmetron, disformal screening etc. • P. Brax, C. vd. Bruck, AC. Davis, J. Khoury, A. Weltman(astro-ph/0408415) • C. de Rham (1401.4173[hep-th]) • T. S. Koivisto, D. F. Mota, M. Zumalacarregui (1205.3167 [astro-ph.CO]) Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Motivation Screening Vainshtein Screening t e i n h r a s d n i i u a s V ( v c r e e n e e S d ( ) r y l a r g e ) Outside (Modified Gravity) Relativistic star (screened) Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Motivation Screening Quadratic DHOST model L = L g + L ϕ + L oth + L m , 5 � L g ≡ f R , L ϕ ≡ ζ I ( X ) L I , I =1 � � L oth ≡ A X − B Λ , (shift symmetry) where L 2 ≡ ( � ϕ ) 2 , L 1 ≡ ∇ µ ∇ ν ϕ ∇ µ ∇ ν ϕ , L 3 ≡ ( � ϕ ) ∇ µ ϕ ∇ ν ϕ ∇ µ ∇ ν ϕ , L 4 ≡ ∇ µ ϕ ∇ ν ϕ ∇ µ ∇ ρ ϕ ∇ ν ∇ ρ ϕ , L 5 ≡ ( ∇ µ ϕ ∇ µ ∇ ν ϕ ∇ ν ϕ ) 2 . DHOST Class Ia* (scalar sector alone): Ghost free ζ 3 ( X ) = − ζ 4 ( X ) = 2 X − 1 ζ 1 ( X ) , ζ 2 ( X ) = − ζ 1 ( X ) , ζ 5 ( X ) = 0 • JB. Achour, D. Langlois, K. Noui (1602.08398 [gr-qc]) • JB. Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui, G. Tasinato (1608.08135 ) Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Motivation Vainshtein screening Objective Testing Vainshtein screening mechanism in the qDHOST theory. e i n h t r a s d n i i u a s V ( v ? ? ? ) e ( r y l a r g e ) Outside (Modified Gravity) Relativistic star ( ??? ) Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Method Steps Method We study a static and spherically symmetric object embedded in de Sitter space-time for the qDHOST model. Assuming the background is spatially flat de Sitter universe. (0) = − d τ 2 + e 2 H τ � d ρ 2 + ρ 2 d Ω 2 ds 2 � 2 Introducing a static and spherically symmetric cosmic structure, ds 2 = − e ν ( r ) dt 2 + e λ ( r ) dr 2 + r 2 d Ω 2 2 Sub-Horizon, Weak-Field Limit ( Hr << 1) � 1 − H 2 r 2 � � 1 − H 2 r 2 � ν ( r ) ∼ ln + δν ( r ) λ ( r ) ∼ − ln + δλ ( r ) ϕ ( r , t ) ∼ v 0 t + v 0 1 − H 2 r 2 � � 2 H ln + δϕ ( r ) At r → ∞ , we have δν → 0, δλ → 0 and δϕ → 0 (de Sitter) Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Covariant EOM Result - 1: Covariant field Equation of HOST and qDHOST • Covariant field EOM of full quadratic Higher Order Scalar-Tensor Theory (qHOST). • It will allow you to do other cosmology analysis of all qDHOST model by setting the conditions over ζ I ( X ). Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Covariant EOM Result - 1: Covariant field Equation of HOST and qDHOST • Covariant field EOM of full quadratic Higher Order Scalar-Tensor Theory (qHOST). • It will allow you to do other cosmology analysis of all qDHOST model by setting the conditions over ζ I ( X ). Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Vainshtein screening Result - 2: Vainshtein screening breaks down Grav. force: Υ 1 G N M ′′ ( r ) d Φ( r ) = G N M ( r ) + r 2 4 dr 5 Υ 2 G N M ′ ( r ) d Ψ( r ) = G N M ( r ) − r 2 4 r 2 dr Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Vainshtein screening Result - 2: Vainshtein screening breaks down Grav. force: Υ 1 G N M ′′ ( r ) d Φ( r ) = G N M ( r ) + r 2 4 dr 5 Υ 2 G N M ′ ( r ) d Ψ( r ) = G N M ( r ) − r 2 4 r 2 dr where G N , Υ 1 , 2 parameters defined as 1 Λ σ 2 = G N = � G , 3 ζ (0) 0 ζ (0) 6 H 2 f � 1 − 2 v 2 ( B σ 2 − 1) + 1 2¯ 1 , X f 2 ζ (0) 0 ζ (0) 1 − v 2 1 , X 2 ζ (0) 2 � ( B σ 2 − 1) 1 Υ 1 = � � � ζ (0) 0 − ζ (0) ζ (0) 0 − 2 ζ (0) 1 , X v 2 1 , X v 2 1 1 � � 2 ζ (0) 2 ζ (0) 0 − 3 ζ (0) 1 , X v 2 1 1 � ( B σ 2 − 1) Υ 2 = − � � � ζ (0) 0 − ζ (0) ζ (0) 0 − 2 ζ (0) 1 , X v 2 1 , X v 2 5 1 1 Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Vainshtein screening Υ 1 G N M ′′ ( r ) d Φ( r ) = G N M ( r ) + r 2 dr 4 5Υ 2 G N M ′ ( r ) d Ψ( r ) = G N M ( r ) − r 2 4 r 2 dr n r a d i i u e t s h s ( v n e i a r r e e n e d y V c ) S ( l a r g M ′′ = M ′ = 0 e ) Outside (Modified Gravity) Relativistic star, NOT screened M ′′ � = 0 � = M ′ Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Vainshtein screening Result-3: Condition to recover Vainstein Screening 2 ζ (0) 2 1 � ( B σ 2 − 1) Υ 1 = � � � ζ (0) 0 − ζ (0) ζ (0) 0 − 2 ζ (0) 1 , X v 2 1 , X v 2 1 1 2 ζ (0) � 2 ζ (0) 0 − 3 ζ (0) � 1 , X v 2 1 1 � ( B σ 2 − 1) Υ 2 = − � � � ζ (0) 0 − ζ (0) ζ (0) 0 − 2 ζ (0) 1 , X v 2 1 , X v 2 5 1 1 Fully Vainshtein screening, Υ 1 = Υ 2 = 0 Condition on the free functions of the qDHOST, ζ (0) = 0 1 1 G N = � G � 2¯ 2 B σ 2 − 1 f May help in imposing the constraints on the qDHOST functions. Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Vainshtein screening GLPV Beyond Horndeski ( L 4 , bH ) Υ 1 = Υ 2 = − 1 1 − B σ 2 � � . 3 Condition: ζ (0) 0 ζ (0) + v 2 1 , X = 0. 1 3 � G G N = 2¯ 5 B σ 2 − 2 � f • E. Babichev, K. Koyama, D. Langlois, R. Saito, J. Sakstein (1606.06627) • T. Kobayashi, Y. Watanabe, D. Yamauchi (1411.4130 [gr-qc]) Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Vainshtein screening Propagation of Gravitational Waves T / c 2 − 1 | ≤ 5 × 10 − 16 GW170817/GRB170817A constraint: | c 2 ζ (0) ϕ 2 c 2 ˙ 1 (0) T = 1 − , c 2 f + ζ (0) ϕ 2 ˙ 1 (0) = 1 + α T , T / c 2 = 1 can be obtained in principle by setting ζ (0) c 2 = 0, 1 without setting ζ 1 ( X ) = 0 . A + Λ B ζ (0) ζ (0) 2 1 ϕ ′ 1 (0) f However, = . 2 A − Λ B ζ 1 , X (0) f ζ (0) 1 A small deviation of ζ (0) from 0 ⇒ a large amount to α T 1 (huge fine-tuning of the parameters) ζ 1 ( X ) = 0 Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Vainshtein screening Status after GW170817 constraint and screening test DHOST: 5 � L = f ( ϕ, X ) R + ζ I ( ϕ, X ) L I Horndeski I =1 + A ( ϕ, X ) + B ( ϕ, X ) � ϕ + cubic . . . Screening GW (same ref.) Independent of background: qDHOST: ζ 1 = ζ 2 = 0, Horndeski All cubic & higher: ruled out Ezquiaga, Zumalacrregui, GW 1710.05901. Langlois, Saito, Yamauchi, screening Creminelli, Vernizzi 1710.05877. K. Noui, 1711.07403. Baker et. al. 1710.06394 Crisostomi, Koyama, 1711.06661. Bartolo, Arai, Nishizawa, 1711.03776 makar, Matarrese, Scomparin 1712.04002 G 5 = 0 , G 4 = G 4 ( ϕ ); All ζ I ( ϕ, X ) = 0, L = f ( ϕ ) R + A ( ϕ, X ) + B ( ϕ, X ) � ϕ Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Summary Take Home Message h t e i n r a n s d i i u a s V ( v S c r e e n e d e ( ) r y l a r g e ) Outside (Modified Gravity) Relativistic star (NOT screened) • Studied a static, spherically symmetric object embedded in de Sitter space-time for the qDHOST model. • The Vainshtein mechanism breaks down inside matter. • Found the possible conditions of healthy Vainshtein screening within the qDHOST scenario. • Remaining DHOST after GW170817 and screening. Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
Result Summary Take Home Message h t e i n r a n s d i i u a s V ( v S c r e e n e d e ( ) r y l a r g e ) Outside (Modified Gravity) Relativistic star (NOT screened) • Studied a static, spherically symmetric object embedded in de Sitter space-time for the qDHOST model. • The Vainshtein mechanism breaks down inside matter. • Found the possible conditions of healthy Vainshtein screening within the qDHOST scenario. • Remaining DHOST after GW170817 and screening. Thank You Purnendu Karmakar, University of Padova, Italy Cosmic structures in DHOST
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