Gravitational Wave in Modified Gravities Shinichi Nojiri Department - PowerPoint PPT Presentation

Gravitational Wave in Modified Gravities Shin’ichi Nojiri Department of Physics & Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya Univ. Aug. 9, 2018 S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 1 / 29

Mainly based on S. Capozziello, M. De Laurentis, S. Nojiri and S. D. Odintsov, “Evolution of gravitons in accelerating cosmologies: The case of extended gravity,” Phys. Rev. D 95 (2017) no.8, 083524 doi:10.1103/PhysRevD.95.083524 arXiv:1702.05517 [gr-qc] S. Nojiri and S. D. Odintsov, “Cosmological Bound from the Neutron Star Merger GW170817 in scalar-tensor and F ( R ) gravity theories,” Phys. Lett. B 779 (2018) 425 doi:10.1016/j.physletb.2018.01.078 arXiv:1711.00492 [astro-ph.CO]. K. Bamba, S. Nojiri and S. D. Odintsov, “Propagation of gravitational waves in strong magnetic fields,” Phys. Rev. D 98 (2018) no.2, 024002 doi:10.1103/PhysRevD.98.024002 arXiv:1804.02275 [gr-qc]. S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 2 / 29

Introduction Gravitational Waves ⇐ linearizing ( g µν → g µν + h µν ) the Einstein equation, R µν − 1 2 g µν R = κ 2 T µν by choosing the transverse and traceless gauge, ∇ µ h µν = g µν h µν = 0 ⇒ [ ] 1 −∇ 2 h µν − 2 R λ ρ ν µ h λρ + R ρ µ h ρν + R ρ ν h ρµ − h µν R + g µν R ρλ h ρλ 2 = κ 2 δ T µν . T µν depends on the metric. The dependence carries the informations on the mechanism of the expansion of the universe. δ T µν can be different in models even if the expansion history of the universe is identical. S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 3 / 29

Introduction 1 Example: Scalar Tensor Theory 2 Example: Quantum Thermodynamical Scalar Field 3 Speed of Propagation 4 Propagation of Light 5 Progagation in Scalar-Tensor Theory by GW170817 6 Propagation in F ( R ) Gravity by GW170817 7 Gravitational Wave from Early Universe or in Future? 8 Summary 9 S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 4 / 29

Example: Scalar Tensor Theory ∫ d 4 x √− g L φ , L φ = − 1 2 ω ( φ ) g µν ∂ µ φ∂ ν φ − V ( φ ) , S φ = ⇒ T µν = − ω ( φ ) ∂ µ φ∂ ν φ + g µν L φ , ⇒ δ T µν = h µν L φ + 1 2 g µν ω ( φ ) ∂ ρ φ∂ λ φ h ρλ , S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 5 / 29

Assuming a FRW spatially flat metric ds 2 = − dt 2 + a ( t ) 2 ∑ ( dx i ) 2 , i =1 , 2 , 3 and φ = φ ( t ), we may choose φ = t ( ) 2 ) ( ) ⇒ ω ( φ ) ∂ µ φ∂ µ = ˜ ( ) φ∂ µ ˜ ( ) ( ( )) φ ′ ( ˜ ˜ ∂ µ ˜ ˜ ˜ ˜ , φ = φ φ ω φ φ , ˜ ω φ ≡ ω φ φ φ ( ) H ≡ ˙ a the FRW equations a ( H + 3 H 2 ) κ 2 H 2 = ω 3 − 1 = ω 2 ˙ 2 + V , 2 − V , κ 2 ( H + 3 H 2 ) ⇒ ω = − 2 V = 1 κ 2 ˙ ˙ H , . κ 2 Then ( t ) α V ( φ ) = 3 α 2 − α 2 α a ( t ) = ⇔ ω ( φ ) = 0 φ 2 , 0 φ 2 . κ 2 t 2 κ 2 t 2 t 0 t 0 , α : real constants S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 6 / 29

2 ⇔ α = 3 (1 + w ) . w : equation of state (EoS) parameter (when Universe is filled with perfect fluid). w = 0 ⇔ dust ∼ cold dark matter (CDM) 4 2 ω ( φ ) = 0 φ 2 , V ( φ ) = 0 φ 2 . 3 κ 2 t 2 3 κ 2 t 2 w = 1 3 ⇔ radiation 1 1 ω ( φ ) = 0 φ 2 , V ( φ ) = 0 φ 2 . κ 2 t 2 4 κ 2 t 2 S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 7 / 29

Example: Quantum Thermodynamical Scalar Field Free real scalar field φ with mass M ( ) − 1 2 g ρσ ∂ ρ φ∂ σ φ − 1 2 M 2 φ 2 T µν = ∂ µ φ∂ ν φ + g µν . Estimation in finite temperature T and chemical potential µ in the flat background, ⟨ ⟩ 1 ∫ ∞ e − β ( k 2 + M 2 ) 2 − i µ k 4 : ∂ T ij 1 12 π 2 δ ij δ kl : = √ . dk k 2 + M 2 1 ∂ g kl 1 − e − β ( k 2 + M 2 ) 2 − i µ 0 T S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 8 / 29

Tensor structures: ⟨ ⟩ � ( ) � : ∂ T ij ∂ T ij ∝ 1 ∝ δ ij δ kl ⇔ � δ k i δ l j + δ l i δ k : � j ∂ g kl ∂ g kl 2 T Scalar Tensor Theory ⇐   ( ) ∑ ∂ T ij = 1  + 1  π 2 − ( ∂ n φ ) 2 − M 2 φ 2 δ k i δ l j + δ l i δ k 2 δ ij ∂ k φ∂ l φ . j ∂ g kl 4 n =1 , 2 , 3 In case of thermal quanta, ( ) E 2 − k 2 − M 2 = 0 1st term= 0 by on-shell condition , ⟨ k k k l ⟩ ∝ δ kl . 2nd term ∼ ( ) M 2 φ 2 ⇒ V ( φ ) In case of scalar tensor theory , φ = φ ( t ) ⇒ 2nd term= 0. S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 9 / 29

When the number N of the particles is fixed N = N 0 and T → 0 ( ⇔ Cold Dark Matter (CDM)) � ⟨ ⟩ � : ∂ T ij = ∂ T ij � : = 0 , � ∂ g kl ∂ g kl T =0 , N = N 0 Scalar Tensor Theory but in general, � ⟨ ⟩ � : ∂ T ij ̸ = ∂ T ij � : , � ∂ g kl ∂ g kl T Scalar Tensor Theory for example, w = 1 3 (radiation). S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 10 / 29

Speed of Propagation B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett. 119 (2017) no.16, 161101 arXiv:1710.05832 [gr-qc] Gravitational Wave from Neutron Star Merger � � � c 2 � � < 6 × 10 − 15 . � � GW − 1 � c 2 c : propagating speed of the light c GW : the propagating speed of the gravitational wave S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 11 / 29

In case of covariant Galileon model C. Deffayet, G. Esposito-Farese and A. Vikman, “Covariant Galileon,”, Phys. Rev. D 79 (2009) 084003, doi:10.1103/PhysRevD.79.084003, [arXiv:0901.1314 [hep-th]] (( ) ) 2 − ∇ µ ∇ ν φ ∇ µ ∇ ν φ ∇ 2 φ L = X + G 4 ( X ) R + G 4 , X , X = − 1 2 ∂ µ φ∂ µ φ , G 4 ( X ) = M 2 + 2 c 0 φ + 2 c 4 X 2 , Pl Λ 6 2 M Pl 4 c 4 term induces the modification of the effective metric for the gravitational wave, g µν → g µν + C ∂ µ φ∂ ν φ , � � � � ˙ � c 2 � � 4 c 4 x 2 � φ � GW � � � ⇒ − 1 � = � , x = . � � c 2 1 − 3 c 4 x 2 HM Pl J. Sakstein and B. Jain, “Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories,” arXiv:1710.05893. S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 12 / 29

Propagation of Light ( √− gg µρ g νσ F ρσ ) 1 = ∇ 2 A ν − ∇ ν ∇ µ A µ + R µν A µ , 0 = ∇ µ F ν µ = √− g ∂ µ ⇒ ∑ 0 = ∂ i ( ∂ i A t − ∂ t A i ) , i =1 , 2 , 3   ∑ 0 = ( ∂ t + H ) ( ∂ i A t − ∂ t A i ) + a − 2  △ A i − ∂ i  , ∂ j A j j =1 , 2 , 3 by assuming a FRW spatially flat metric ds 2 = − dt 2 + a ( t ) 2 ∑ ( dx i ) 2 , i =1 , 2 , 3 S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 13 / 29

Landau gauge: √− g ∂ µ ( √− gg µν A ν ) = − ∂ t A t + 3 HA t + a − 2 ∑ 1 0 = ∇ µ A µ = i =1 , 2 , 3 ∂ i A i ⇒ 0 = ∇ 2 A ν + R µν A µ . Assume 0 = A t = ∑ i =1 , 2 , 3 ∂ i A i ( ) A i + a − 2 △ A i . ∂ 2 0 = − t + H ∂ t S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 14 / 29

de Sitter space-time H = H 0 , a = e H 0 t . Assume A i ∝ e i k × (the part only depending on t ) △ by − k 2 ≡ − k · k . s ≡ e − H 0 t ( d 2 ) ds 2 + k 2 ⇒ 0 = A i , H 2 0 ( k ) ⇒ A i = A i 0 cos s + θ 0 . H 0 S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 15 / 29

Propagation in Scalar-Tensor Theory by GW170817 Gravitational wave in de Sitter space-time by cosmological constant, u = k h ij = s − 1 2 l ij . s , H 0 ⇒ ( ) ( 5 ) 2 d 2 du 2 + 1 2 0 = u + 1 − l ij , u 2 Bessel’s differential equation ⇒ Bessel functions J ± 5 2 ( u ). Black hole/neutron star merger s ≡ e − H 0 t ∼ 1. k H 0 ≫ 1. ( k ) h ij ∼ 1 s + ± 5 + 1 s cos π . H 0 4 ⇒ c = c GW . S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 16 / 29

( ) α t Power-law expansion a ( t ) = in Scalar-Tensor Theory. t 0 V ( φ ) = 3 α 2 − α 2 α ω ( φ ) = 0 φ 2 , 0 φ 2 . κ 2 t 2 κ 2 t 2 2 ∼ perfect fluid with a constant equation of state prameter w , α = 3(1+ w ) . H = α H = − α ˙ t , t 2 . Black hole/neutron star merger ⇒ H ∼ a constant, H ∼ H 0 . H 2 ∼ ˙ H ⇒ ˙ H ∼ a constant, ˙ H = H 1 ( ) t + △ H + 6 H 2 + H ∂ t − ∂ 2 2 ˙ 0 = h ij a 2 ⇒   ( 5 ) 2 − 2 H 1  d 2 du 2 + 1 H 2 2  l ij , 0 = u + 1 − 0 u 2 S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 17 / 29

Solution J ( u ). √ 1 − 4 H 1 ± 5 2 25 H 2 0 ( k s + ± 5 √ 1+ β + 1 ) h ij ∼ 1 β ≡ − 4 H 1 s cos π , , 25 H 2 H 0 4 0 The propagation of the light is not changed. The propagation of the gravitational wave is not changed, either. The difference is in phase, β = − 4 25 α = − 6(1 + w ) , 25 S. Nojiri (Nagoya U. & KMI) Gravitational Wave in Modified Gravities Aug. 9, 2018 18 / 29