MOGRA2018, August 8-10 ,2018 Shin’ichi Hirano Rikkyo U. MATTER BISPECTRUM BEYOND HORNDESKI ( 進一 平野 ) based on 1801. 07885 SH , T. Kobayashi, S. Yokoyama (Nagoya U.), T. Hiroyuki (Nagoya U.) SH , T. Kobayashi, D. Yamauchi (Kanagawa U.), S. Yokoyama, in preparation
Introduction ■ Modified gravity: alternative to cosmological constant ・ Cosmological scale: late-time acceleration ・ Small scale: recovering the result of gravitational test ⇒ screening mechanism ■ Horndeski theory Horndeski (1972), Kobayashi+ (2011), Deffaiyet+ (2011) ・ Most general scalar-tensor theory with 2nd-order EoMs ・ Vainshtein screening thanks to 2nd-order derivative non-linear ints. ・ GW170817, GRB170817A: | c T − 1 | < 10 − 15 Abbott+ (2017) √− g = f ( φ ) L R + G 2 ( φ , X ) − G 3 ( φ , X ) ⇤ φ 2 1/18
<latexit sha1_base64="wKpvRKEKZEjv2fRd/KVFUjzD7CI=">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</latexit> <latexit sha1_base64="wKpvRKEKZEjv2fRd/KVFUjzD7CI=">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</latexit> <latexit sha1_base64="wKpvRKEKZEjv2fRd/KVFUjzD7CI=">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</latexit> <latexit sha1_base64="wKpvRKEKZEjv2fRd/KVFUjzD7CI=">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</latexit> Recent progress & our work ■ Beyond Horndeski (higher-order EoMs, no Ostrogradski ghost) GLPV theory Gleyzes+ (2014), Gleyzes+ (2015) DHOST theory Langlois & Noui (2015), Achour+ (2016), Achour+ (2016) ✓ Models with and c T = 1 ( ∂∂φ ) 2 ✓ Partial breaking of Vainshtein screening inside matter ( ) δ � 1 non-linear int. Kobayashi+ (2015), Langlois+ (2017), … Our aim How much is the effect of non-linear ints. at “ cosmological scale” ? δ ⌧ 1 Matter bispectrum beyond Horndeski 2/18
Plan of talk ■ Our setup ■ Cosmological perturbations ■ Matter bispectrum beyond Horndeski ■ Summary 3/18
OUR SETUP
quadratic DHOST Langlois, Noui (2015,2016), Koyama+ (2016), de Rham, Matas (2016) L qD √− g = G 2 ( φ , X ) − G 3 ( φ , X ) ⇤ φ + G 4 ( φ , X ) R + C µ νρσ φ µ ν φ ρσ (2) φ µ ν φ ρσ = a 1 ( φ , X ) φ 2 µ ν + a 2 ( φ , X )( ⇤ φ ) 2 + a 3 ( φ , X ) ⇤ φ ( φ µ φ µ ν φ ν ) C µ νρσ (2) + a 4 ( φ , X ) φ µ φ µ ν φ νρ φ ρ + a 5 ( φ , X )( φ µ φ µ ν φ ν ) 2 non-linear ints. X = � 1 ( ) 2( r φ ) 2 , φ µ = r µ φ , ⇤ φ = r 2 φ , φ µ ν = r ν r µ φ ■ includes Horndeski and GLPV at Lagrangian level Horndeski: a 1 = − a 2 = − G 4 X , a 3 = a 4 = a 5 = 0 , GLPV: … ■ The non-trivial relation between arbitrary func. ( G 4 , C (2) ) in order to evade Ostragradski ghost ← “degeneracy conditions” 4/18
Viable DHOST after GW170817 Degenerate Scalar-Tensor theory DHOST Horndeski (2nd-order EoMs) class I class II, III quintessence, f(R), KGB (higher-order EoMs) Covariant Galileon, … GLPV stable cosmological sol. de Rham & Matas (2016) mimetic gravity, extended mimetic gravity 5/18
Viable DHOST after GW170817 <latexit sha1_base64="wslD7Z6uGROVXiT/Qme0KjwS0FQ=">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</latexit> <latexit sha1_base64="wslD7Z6uGROVXiT/Qme0KjwS0FQ=">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</latexit> <latexit sha1_base64="wslD7Z6uGROVXiT/Qme0KjwS0FQ=">ACg3ichVHLSsNAFD2N1kd9tOpGcFMsiCUGxEUQSi60KVqwWVksRpDU2TmKSFWty58gdcuFIQUZf6B278ARd+grhUcOPCmzQgWtQ7THLmzD13ztxRbUN3PaKniNTWHu3o7OqO9fT29cTA4MbrlV1NJHTLMNy8qriCkM3Rc7TPUPkbUcoFdUQm2p50d/frAnH1S1z3avbYqeilEy9qGuKx5SViOMCS1BQ4aGgDySzJgQ2GdEhUSK0hREshXIUghjBUrcYlt7MKChiqXFzKY2xwcZfHFmQbOZ20GDOYaQH+wKHiLG2ylmCMxRmy/wt8WorZM3AqGDsqzU+xeDpsDKJMXqkK3qlB7qhZ/r4tVYjqOF7qfNfbWqFXYgfD6+9/6vym+Rh70v1p2cPRcwGXnX2bgeMfwutqa8dnLyuza2ONcbpnF7Y/xk90T3fwKy9aRdZsXqKGD+A/LPdrWBjKi1TWs5OpzIL4VN0YQSjmOB+zyCDZawgx+ce4Rq3uJOi0qQ0JU03U6VIqBnCt5DmPwEvyo9t</latexit> <latexit sha1_base64="LoN+so8WU7mEePEt6DB6+aSkNMs=">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</latexit> Degenerate Scalar-Tensor theory DHOST Horndeski (2nd-order EoMs) class I class II, III quintessence, f(R), KGB (higher-order EoMs) Covariant Galileon, … conformal &disformal trans. g µ ν = Ω ( φ , X ) g µ ν + Γ ( φ , X ) φ µ φ ν ˜ GLPV disformal Γ X 6 = 0 stable cosmological sol. de Rham & Matas (2016) mimetic gravity, extended mimetic gravity 5/18
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