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, Karim Benakli, Y. C, Peng Cheng and Ga¨ etan Lafforgue-Marmet. Motivation: Spin-3/2 Fields Gravitational Waves from Non-Adiabatically Varying Fields Spin-3/2 Fields as Sources for Gravitational Wave Emission Gravitino from Polonyi model Summary

Gravitational waves from spin-3/2 fields Hunting SUSY in the sky

Yifan Chen LPTHE, Sorbonne Universit´ e arXiv:1811.11774, Karim Benakli, Y. C, Peng Cheng and Ga¨ etan Lafforgue-Marmet.

24th Meeting of the New Higgs Working Group, Osaka

22.12.2018

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

Outlines

Motivation: Spin-3/2 Fields Gravitational Waves from Non-Adiabatically Varying Fields Spin-3/2 Fields as Sources for Gravitational Wave Emission Gravitino from Polonyi model Summary

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

Why Spin-3/2?

◮ In nature, we have observed

Spin − 0, 1/2, 1, 2. Why 3/2 is missing?

◮ The only known fundamental particle with spin-3/2:

Gravitino. Smoking gun of SUGRA (SUSY as ultraviolet symmetry).

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

Gravitational Waves

Three kinds of sources:

◮ Astrophysical: Low redshift events from point-like

sources, huge success from LIGO/VIRGO;

◮ Stochastic GW background from Inflation: Enhanced

primordial fluctuation, to be tested from CMB B-mode, PTA, Space-based GW detectors;

◮ Stochastic GW background after Inflation:

Cosmological sources, sub-horizon k ≫ H. e.g. First-order phase transition, non-adiabatically varying fields during preheating...

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

Gravitational Waves

Linear perturbation of the metric in the transverse-traceless (TT) gauge: ds2 = a2(τ)[−dτ 2 + (δij + hij)dxidxj]. Einstein equation:

◮ Background: Friedman-Robertson-Walker (FRW)

equation;

◮ Equation of motion for GW:

¨ hij + 2H ˙ hij − ∇hij = 16πGΠTT

ij

,

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

Gravitational Waves

◮ Solution in the momentum space: convolution between

the Green function from free propagation and the source term: hij (k, t) = 16πG a(t)k t

tI

dt′ sin [k(t − t′)]a(t′)ΠTT

ij

(k, t′).

◮ Spectrum:

dρGW dlogk (k, t) = 2Gk3 πa4(t) t

tI

dt′ t

tI

dt′′a(t′)a(t′′) cos[k(t′ − t′′)]Π2(k, t′, t′′),

◮ Unequal-time correlator of ΠTT ij

: ΠTT

ij

(k, t)ΠTTij(k′, t′) ≡ (2π)3Π2(k, t, t′)δ(3)(k − k′).

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

Sources: Stress Tensors under TT Projection

◮ In the momentum space, source term can be written as:

ΠTT

ij

(k, t) = Λij,lm(ˆ k)(T lm(k, t) − Pglm), where the TT projection tensor satisfies Λij,lm(ˆ k)kl = Λij,lm(ˆ k)km = 0.

◮ k is the momentum mode of GW.

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

Sources: Examples

◮ Sources: kinetic terms, e.g.

• d3pΛij,lm(ˆ

k) ∂lσ(p, t)∂mσ(p′, t),

• d3pΛij,lm(ˆ

k) ¯ ψ(p, t)γl∂mψ(p′, t),

• d3pΛij,lm(ˆ

k) ∂lAµ(p, t)∂mAµ(p′, t) + ∂µAl(p, t)∂µAm(p′, t).

◮ Homogeneous:

p′ = p + k.

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

Sources: Linear k dependence

◮ The linear dependence on k is projected out for spin-0

and spin-1/2: Λij,lm(ˆ k) ∂lσ(p, t)∂mσ(p′, t) = Λij,lm(ˆ k) plσ(p, t)(pm + km)σ(p′, t) = Λij,lm(ˆ k) plσ(p, t)pmσ(p′, t).

◮ The linear dependence on k is preserved for spin-1:

Λij,lm(ˆ k) ∂µAl(p, t)∂µAm(p′, t) = Λij,lm(ˆ k) pµ(pµ + kµ)Al(p, t)Am(p′, t).

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

Sources: Adiabatic Evolution

◮ If the wave-functions of the matter fields vary

adiabatically: σ(p, t) ∼ eiωt.

◮ The kinetic factor will force the four momentum

conservation:

• dt′ cos (kt′)ΠTT

ij

(k, t′) ∼

• d3p δ4(pµ ± kµ − p′µ)...

◮ Massless: mσ = 0 ◮ Helicity conservation: e.g., for spin-0

p//k, Λij,lm(ˆ k)pl = 0.

◮ Only massless spin-1 fields could emit GW in the

adiabatic limit!

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

Sources: Non-Adiabatic Evolution

◮ Non-adiabatic (NA) condition:

| ˙ ω ω2 | ≫ 1

◮ E.g., preheating, scalar field background:

V = 1 2m2φ2, φ ∼ ΦI sin (mτ). Matter coupling: g2φ2σ2, gφ ¯ ψψ, ω2

φ/ψ = k2 + g2φ2.

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

Sources: Non-Adiabatic Evolution

◮ For ω2 φ/ψ = k2 + g2φ2, NA condition leads to the

inequality: 0 ≤ k ≤ k∗ ≃ (gmΦI)

1 2 ,

in such range the matter fields are excited.

◮ For bosons, one gets large occupation number in these

modes.

◮ For fermions, the excited modes fulfill a Fermi sphere

with radius kF = k∗. One gets a model-independent spectrum parameterized by kF.

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

ψµ

α: Spinor and Vector

◮ Rarita-Schwinger Lagrangian:

L = −1 2ǫµνρσ ¯ ψµγ5γν∂ρψσ − 1 4m3/2 ¯ ψµ[γµ, γν]ψν .

◮ Equations of motion in the flat limit:

(i∂ / − m3/2)ψµ = γµψµ = ∂µψµ =

◮ Wave-functions:

˜ ψµ

p,λ(t) =

• s=±1,l=±1,0

1, 1 2, l, s 2|3 2, λǫµ

p,lu(|λ|) p, s

2 (t),

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

Sources: Spin-3/2 Fields

◮ Stress tensor:

Tαβ = ecα 2e δ(eL) δeβ

c

+ (α ↔ β) = i 4 ¯ ψµγ(α∂β)ψµ − i 4 ¯ ψµγ(α∂µψβ) + h.c.

◮ Linear dependence on k is preserved under projection:

Λij,lm(ˆ k)ǫµ

p(pµ + kµ)γlǫm p′. ◮ Equivalence theorem: in the relativistic limit p ≫ m3/2,

ǫµ

p,0 =

1 m3/2 (p,

• p2 + m2

3/2ˆ

p) = pµ m3/2 + O(m3/2 p ).

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

Sources: Helicity-1/2 Component

◮ Wave-function of Helicity-1/2 component:

˜ ψµ

p,± 1

2 (t) =

• 2

3ǫµ

p,0u ( 1

2 )

p,± 1

2 (t) +

• 1

3ǫµ

p,±1u ( 1

2 )

p,∓ 1

2 (t).

◮ One expects the dominant contribution to

ΠTT

ij

(k, t)ΠTTij(k′, t′) with the most ǫµ

p,0 due to the

equivalence theorem.

◮ However, terms with ǫµ 0ǫµ 0ǫµ 0ǫµ 0 and ǫµ 0ǫµ 0ǫµ 0ǫµ ±1 vanish by

simply replacing ǫµ

0 with pµ m3/2 due to the Levi-Civita

symbol in − 1

2ǫµνρσ ¯

ψµγ5γν∂ρψσ.

◮ The dominant result comes from term with ǫµ 0ǫµ 0ǫµ ±1ǫµ ±1.

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

Result: GW from fermions

dρGW dlogk (k, t) =

Gk3 π3a4(t)

• dp dθ K(p, k, θ, m3/2)

{|Ic(k, p, θ, t)|2 + |Is(k, p, θ, t)|2}.

◮ The second line contains wave-functions, which are

universal for fermions with the same Fermi-radius.

◮ Scaling factor K:

Helicity-3/2 : p2k2f3/2(θ, θ′), Helicity-1/2 :

p4k2 m2

3/2 f1/2(θ, θ′),

Spin-1/2 : p4 sin3 θ.

◮ For spin-1/2, spectrum scales k3 and decreases

exponentially after kF. For spin-3/2, spectrum scales k5 near the peak and the amplitude gets enhanced by

k2

F

m2

3/2 .

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

Gravitino

◮ The only known fundamental particle with spin-3/2. ◮ First candidate to be proposed as a dark matter, before

the neutralino, by Fayet in 1981 and Pagels & Primack in 1982.

◮ Problematic issue of its detectability: only R-parity

violating decay.

◮ Super-partner of graviton. ◮ After spontaneous breaking of SUSY, the helicity 1/2

component swallows the goldstone fermion, the super-partner of sGoldstino through super-Higgs mechanism: m3/2 = F √ 3MPl , MSUSY ∼ F Λmess .

◮ Naturally mass splitting from other super-particles.

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

Polonyi Model

K = |z|2 − |z|4 Λ2 , W = µ2z + W0,

◮ Spontaneous SUSY broken minimum with positive

cosmological constant.

◮ Near the minimum, m3/2 ≫ H. We can use the

quantization of gravitino in the flat limit.

◮ Large coupling to helicity-1/2 component, e.g,

non-linear SUSY: z = GG

2Fz . ◮ Equivalence theorem can be used for m3/2 ≪ p ≪ √Fz.

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

Polonyi Model: Gravitino Production

◮ The equation of motion for the helicity-1/2 component

has the same form as spin-1/2 fermions: ¨ f± + [p2 + (θ + m3/2a)2 ± i( ˙ θ + ˙ m3/2a)]f± = 0, with the source term θ(t) = −

am2

zδz

2 √ 3m3/2MPl . ◮ Effective Yukawa coupling:

˜ g = m2

z

2 √ 3m3/2MPl = m2

z

2F ≪ 1.

◮ Peak frequency now:

fp ≃ 6 · 1010˜ g

1 2 Hz.

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

Polonyi Model: GW Spectrum

◮ Current spectrum:

h2ΩGW (fp) ≃ 3 · 10−10( fp 6 · 1010Hz)12( zI mz )2.

1×108 5×108 1×109 5×109 1×1010 fp(Hz) 10-29 10-24 10-19 10-14 10-9 h2ΩGW zI mz

1014 1011 108 105 102

◮ Typically at ultra-high frequency, needs new design of

GW detectors in the future.

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

Conclusion

◮ Spin-3/2 is the only missing piece in the nature with

spin from 0 to 2. Its existence could also be the smoking gun of SUSY.

◮ Fields with non-adiabatically varying wave-functions can

emit gravitational wave.

◮ Non-adiabatic production of fermions fulfills a

Fermi-sphere whose radius depends on the preheating model.

◮ The helicity-1/2 component dominates the GW

• production. Near the peak, the scaling goes k5 instead
• f k3 like spin-1/2 fermions.

◮ In the simplest model describing SUSY broken in

de-Sitter space, the corresponding peak frequency is too high, which calls for the future design of ultra-high frequency GW detectors.

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

Thank you!