Gravitational waves from spin-3/2 fields Karim Benakli, Y. C, Peng - PowerPoint PPT Presentation

NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Gravitational waves from spin-3/2 fields Karim Benakli, Y. C, Peng Cheng and Ga¨ etan Hunting SUSY in the sky Lafforgue-Marmet. Motivation: Spin-3/2 Fields Yifan Chen Gravitational Waves from LPTHE, Sorbonne Universit´ e Non-Adiabatically Varying Fields arXiv:1811.11774, Spin-3/2 Fields as Karim Benakli, Y. C, Peng Cheng and Ga¨ etan Sources for Gravitational Wave Lafforgue-Marmet. Emission Gravitino from Polonyi model 24th Meeting of the New Higgs Working Group, Osaka Summary 22.12.2018

Outlines NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng and Ga¨ etan Motivation: Spin-3/2 Fields Lafforgue-Marmet. Motivation: Spin-3/2 Fields Gravitational Waves from Non-Adiabatically Varying Fields Gravitational Waves from Non-Adiabatically Varying Fields Spin-3/2 Fields as Sources for Gravitational Wave Emission Spin-3/2 Fields as Sources for Gravitational Wave Emission Gravitino from Polonyi model Gravitino from Polonyi model Summary Summary

Why Spin-3/2? NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng and Ga¨ etan ◮ In nature, we have observed Lafforgue-Marmet. Motivation: Spin − 0 , 1 / 2 , 1 , 2 . Spin-3/2 Fields Gravitational Waves from Why 3/2 is missing? Non-Adiabatically Varying Fields ◮ The only known fundamental particle with spin-3/2: Spin-3/2 Fields as Sources for Gravitational Wave Gravitino . Emission Gravitino from Polonyi model Smoking gun of SUGRA (SUSY as ultraviolet Summary symmetry).

Gravitational Waves NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng and Ga¨ etan Three kinds of sources: Lafforgue-Marmet. ◮ Astrophysical: Low redshift events from point-like Motivation: Spin-3/2 Fields sources, huge success from LIGO/VIRGO; Gravitational ◮ Stochastic GW background from Inflation: Enhanced Waves from Non-Adiabatically primordial fluctuation, to be tested from CMB B-mode, Varying Fields PTA, Space-based GW detectors; Spin-3/2 Fields as Sources for Gravitational Wave ◮ Stochastic GW background after Inflation: Emission Cosmological sources, sub-horizon k ≫ H . Gravitino from Polonyi model e.g. First-order phase transition, non-adiabatically Summary varying fields during preheating...

Gravitational Waves NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng Linear perturbation of the metric in the transverse-traceless and Ga¨ etan Lafforgue-Marmet. (TT) gauge: Motivation: Spin-3/2 Fields ds 2 = a 2 ( τ )[ − d τ 2 + ( δ ij + h ij ) dx i dx j ] . Gravitational Waves from Non-Adiabatically Einstein equation: Varying Fields ◮ Background: Friedman-Robertson-Walker (FRW) Spin-3/2 Fields as Sources for equation; Gravitational Wave Emission ◮ Equation of motion for GW: Gravitino from Polonyi model h ij + 2 H ˙ ¨ h ij − ∇ h ij = 16 π G Π TT Summary , ij

Gravitational Waves NHWG Yifan Chen LPTHE, Sorbonne Universit´ e ◮ Solution in the momentum space: convolution between arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng the Green function from free propagation and the and Ga¨ etan source term: Lafforgue-Marmet. � t Motivation: h ij ( k , t ) = 16 π G dt ′ sin [ k ( t − t ′ )] a ( t ′ )Π TT ( k , t ′ ) . Spin-3/2 Fields ij a ( t ) k Gravitational t I Waves from Non-Adiabatically ◮ Spectrum: Varying Fields Spin-3/2 Fields as � t � t Sources for 2 Gk 3 d ρ GW Gravitational Wave dt ′ dt ′′ a ( t ′ ) a ( t ′′ ) d log k ( k , t ) = Emission π a 4 ( t ) t I t I Gravitino from cos[ k ( t ′ − t ′′ )]Π 2 ( k , t ′ , t ′′ ) , Polonyi model Summary ◮ Unequal-time correlator of Π TT : ij � Π TT ( k , t )Π TTij ( k ′ , t ′ ) � ≡ (2 π ) 3 Π 2 ( k , t , t ′ ) δ (3) ( k − k ′ ) . ij

Sources: Stress Tensors under TT Projection NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng and Ga¨ etan Lafforgue-Marmet. ◮ In the momentum space, source term can be written as: Motivation: Spin-3/2 Fields Π TT ( k , t ) = Λ ij , lm (ˆ k )( T lm ( k , t ) − P g lm ) , Gravitational ij Waves from Non-Adiabatically Varying Fields where the TT projection tensor satisfies Spin-3/2 Fields as Sources for k ) k l = Λ ij , lm (ˆ k ) k m = 0 . Λ ij , lm (ˆ Gravitational Wave Emission Gravitino from ◮ k is the momentum mode of GW. Polonyi model Summary

Sources: Examples NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng ◮ Sources: kinetic terms, e.g. and Ga¨ etan Lafforgue-Marmet. � Motivation: d 3 p Λ ij , lm (ˆ ∂ l σ ( p , t ) ∂ m σ ( p ′ , t ) , k ) Spin-3/2 Fields Gravitational � Waves from d 3 p Λ ij , lm (ˆ ¯ ψ ( p , t ) γ l ∂ m ψ ( p ′ , t ) , k ) Non-Adiabatically Varying Fields Spin-3/2 Fields as � d 3 p Λ ij , lm (ˆ ∂ l A µ ( p , t ) ∂ m A µ ( p ′ , t ) Sources for k ) Gravitational Wave Emission ∂ µ A l ( p , t ) ∂ µ A m ( p ′ , t ) . + Gravitino from Polonyi model Summary ◮ Homogeneous: p ′ = p + k .

Sources: Linear k dependence NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng and Ga¨ etan ◮ The linear dependence on k is projected out for spin-0 Lafforgue-Marmet. and spin-1/2: Motivation: Spin-3/2 Fields Λ ij , lm (ˆ ∂ l σ ( p , t ) ∂ m σ ( p ′ , t ) k ) Gravitational p l σ ( p , t )( p m + k m ) σ ( p ′ , t ) Waves from Λ ij , lm (ˆ = k ) Non-Adiabatically Varying Fields Λ ij , lm (ˆ p l σ ( p , t ) p m σ ( p ′ , t ) . = k ) Spin-3/2 Fields as Sources for Gravitational Wave ◮ The linear dependence on k is preserved for spin-1: Emission Gravitino from Polonyi model Λ ij , lm (ˆ ∂ µ A l ( p , t ) ∂ µ A m ( p ′ , t ) k ) Summary Λ ij , lm (ˆ p µ ( p µ + k µ ) A l ( p , t ) A m ( p ′ , t ) . = k )

Sources: Adiabatic Evolution NHWG Yifan Chen LPTHE, Sorbonne Universit´ e ◮ If the wave-functions of the matter fields vary arXiv:1811.11774, Karim Benakli, Y. adiabatically: C, Peng Cheng and Ga¨ etan σ ( p , t ) ∼ e i ω t . Lafforgue-Marmet. ◮ The kinetic factor will force the four momentum Motivation: Spin-3/2 Fields conservation: Gravitational Waves from dt ′ cos ( kt ′ )Π TT Non-Adiabatically ( k , t ′ ) � Varying Fields ij d 3 p δ 4 ( p µ ± k µ − p ′ µ ) ... Spin-3/2 Fields as � ∼ Sources for Gravitational Wave Emission ◮ Massless: m σ = 0 Gravitino from Polonyi model ◮ Helicity conservation: e.g., for spin-0 Summary k ) p l = 0 . p // k , Λ ij , lm (ˆ ◮ Only massless spin-1 fields could emit GW in the adiabatic limit!

Sources: Non-Adiabatic Evolution NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng and Ga¨ etan ◮ Non-adiabatic (NA) condition: Lafforgue-Marmet. | ˙ ω Motivation: ω 2 | ≫ 1 Spin-3/2 Fields Gravitational Waves from ◮ E.g., preheating, scalar field background: Non-Adiabatically Varying Fields Spin-3/2 Fields as V = 1 2 m 2 φ 2 , Sources for φ ∼ Φ I sin ( m τ ) . Gravitational Wave Emission Gravitino from Matter coupling: Polonyi model Summary φ/ψ = k 2 + g 2 φ 2 . g 2 φ 2 σ 2 , g φ ¯ ω 2 ψψ,

Sources: Non-Adiabatic Evolution NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng and Ga¨ etan φ/ψ = k 2 + g 2 φ 2 , NA condition leads to the Lafforgue-Marmet. ◮ For ω 2 inequality: Motivation: Spin-3/2 Fields 0 ≤ k ≤ k ∗ ≃ ( gm Φ I ) 1 2 , Gravitational Waves from in such range the matter fields are excited. Non-Adiabatically Varying Fields ◮ For bosons, one gets large occupation number in these Spin-3/2 Fields as Sources for modes. Gravitational Wave Emission ◮ For fermions, the excited modes fulfill a Fermi sphere Gravitino from with radius k F = k ∗ . One gets a model-independent Polonyi model Summary spectrum parameterized by k F .

ψ µ α : Spinor and Vector NHWG Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, ◮ Rarita-Schwinger Lagrangian: Karim Benakli, Y. C, Peng Cheng and Ga¨ etan Lafforgue-Marmet. L = − 1 ψ µ γ 5 γ ν ∂ ρ ψ σ − 1 2 ǫ µνρσ ¯ 4 m 3 / 2 ¯ ψ µ [ γ µ , γ ν ] ψ ν . Motivation: Spin-3/2 Fields ◮ Equations of motion in the flat limit: Gravitational Waves from Non-Adiabatically ( i ∂ / − m 3 / 2 ) ψ µ = 0 Varying Fields Spin-3/2 Fields as γ µ ψ µ = 0 Sources for Gravitational Wave ∂ µ ψ µ = 0 Emission Gravitino from Polonyi model ◮ Wave-functions: Summary � 1 , 1 2 , l , s 2 | 3 ψ µ ˜ 2 , λ � ǫ µ p , l u ( | λ | ) � p ,λ ( t ) = 2 ( t ) , p , s s = ± 1 , l = ± 1 , 0