Gravity with higher curvature terms BUM-HOON LEE SOGANG UNIVERSITY - PowerPoint PPT Presentation

고 송희성 교수님 추모 심포지엄 Gravity with higher curvature terms BUM-HOON LEE SOGANG UNIVERSITY Sogang University APRIL 12, 2017

고 송희성 교수님에 대한 기억 들 1970 년대 1975 : 관악 캠퍼스로 이전 , 학부 4 학년 시절 문리대 물리학과 + 공대 응용물리학과 + 사범대 물리교육학과 1976~ 1978 대학원 시절 , 양자역학 수강 1980 년대 1985 년 경 , 뉴욕 방문시 , 1990 년대 및 2000 년대 이론물리센터 (SRC) 물리학회 활동 , 학회장 출마 , THE 48TH WORKSHOP ON GRAVITATION AND NUMERICAL RELATIVITY FOR APCTP TOPICAL 17 May 2014 RESEARCH PROGRAM

Contents 1. Motivation 2. Black Holes in the Dilaton Gauss-Bonnet theory 3. Inflation with a Gauss-Bonnet Term 4. Summary 3

1. Motivation : Why Gauss-Bonnet Term? Low energy effective theory from string theory → Einstein Gravity + higher curvature terms Gauss-Bonnet term is the simplest leading term. Q : What is the physical effects of Gauss-Bonnet terms? 1) Effects to the Black Holes. No-Hair Theorem of Black Holes Stationary black holes (in 4-dim Einstein Gravity) are completely described by 3 parameters of the Kerr-Newman metric : mass, charge, and angular momentum (M, Q, J) Werner Israel(1967), Brandon Carter(1971,1977), Hairy black hole solution ? In the dilaton-Gauss-Bonnnet theory → Yes! David Robinson (1975) Exists the minimum mass of BH Affects the stability, etc. 2) Effects in the Early Universe.

Werner Israel(1967), No-Hair Theorem of Black Holes Brandon Carter(1971,1977), David Robinson (1975) Stationary black holes (in 4-dim Einstein Gravity) are completely described by 3 parameters of the Kerr-Newman metric : mass, charge, and angular momentum (M, Q, J) • A rotating black hole (one with angular momentum) has an ergoregion around the outside of the event horizon • In the ergoregion, space and time themselves are dragged along with the rotation of the black hole Hairy black hole solution is possible in the dilaton-Gauss- Bonnnet theory.

Colliding Galaxies: A Black Hole Merger Actual observations provide evidence and data for computer simulations. What does it look like when black holes collide? NASA / CXC / MPE / S. Komossa, et al. GW150914 Colliding Black Holes : A Black Hole Merger + Gravitational Wave Q: A Black Hole unstable ? splitting into two Black Holes ?

Holography (asymptotic) AdS Black Hole in d+1 dim ↔ Quantum System in d dim. Instability of Black Holes ↔ instability of Quantum System Hence, instability of AdS BH ↔ phase transitions in Quantum System * * Black ho hole les i in hi higher r dim imensio ions are re quite te div ivers rse !

2. Black Holes in the Dilaton Gauss-Bonnet theory W.Ahn, B. Gwak, BHL, W.Lee, Eur.Phys.J.C (2015) Shwarzschild Black Holes Action 2𝜆 2 𝑆 − 1 1 𝑇 = න 𝑒 4 𝑦 2 𝑕 𝜈𝜉 𝜖 𝜈 𝜚𝜖 𝜉 𝜚 −𝑕 where and Black Hole solution 𝐼 = 2 𝑁 𝑠 𝑠 ) 𝑒𝑢 2 + 𝑒𝑠 2 2𝑁 𝑒𝑡 2 = - (1- 𝑠 ) + 𝑠 2 𝑒Ω 2 No hair 2 𝑁 (1 − 2𝑁 Note : Horizon 𝑠 𝐼 = 2 𝑁 𝜚 𝑠 𝐼

Hairy black holes in Dilaton-Einstein-Gauss-Bonnet (DEGB) theory Guo,N.Ohta & T.Torii, Prog.Theor.Phys. Action 120,581(2008);121 ,253 (2009); N.Ohta &Torii,Prog.Theor.Phys.121,959; 122,1477(2009);124,207 (2010); K.i.Maeda,N.Ohta Y.Sasagawa, PRD80, where and 104032(2009); 83,044051 (2011) N. Ohta and T. Torii, Phys.Rev. D 88 ,064002 (2013). The Gauss-Bonnet term : Boundary term if 𝛿 = 0 Note : 1) The symmetry under allows choosing γ positive values without loss of generality. 2) The coupling α dependency could be absorbed by the r → r/ α transformation. with non-zero α coupling cases being generated by α scaling. However, the behaviors for the α = 0 case cannot be generated in this way. Hence, we keep the parameter α, to show a continuous change to α = 0.

Note : 1. All the black holes in the DEGB theory with given non- zero couplings α and γ have hairs. I.e., there does not exist black hole solutions without a hair in DEGB theory. (If we have Φ = 0, dilaton e.o.m. reduces to 𝑆 𝐻𝐶 = 0. so it cannot satisfy the dilaton e.o.m..) 2 Hair Charge Q is not zero, and is not independent charge either. 2. For the coupling α = 0, the solutions become a Schwarzschild black hole in Einstein gravity. 3. For 𝛿 = 0 , DEGB theory becomes the Einstein-Gauss-Bonnet (EGB) theory. The EGB black hole solution is the same as that of the Schwarzschild one. However, the GB term contributes to the black hole entropy and influence stability.

Coupling γ dependency of the minimum mass for fixed α 1/16. γ=√2(green), γ=1.3(cyan), γ=1. 29(blue ), γ=1/2(red), γ=1/6(black ), γ=0(purple ) Singular pt S coincides w/ pt C Singular pt S & the min. mass btwn γ=1.29(blue) & 1.30(cyan). As γ→0 , C exist for γ = √ 2. No lower branch below γ=1.29 the solution → Schw BH. Note : 1. For large γ, sing. pt S & extremal pt C (with minimum mass ෩ 𝑁 ) exist. 2. The solutions between point S and C are unstable for perturbations and end at the singular point S , I.e., there are two black holes for a given mass in which the smaller one is unstable under perturbations. 3 . As γ smaller, the singular point S gets closer to the minimum mass point C. 4 . Below γ=1.29 , the solutions are perturbatively stable and approach the Schwarzschild black hole in the limit of γ going to zero. These solutions depend on the coupling γ. 5. If DEGB BH horizon becomes larger, the scalar field goes to 0, and the BH becomes a Schwarzschild BH.

Q: How about the properties, such as Stability Implication to the cosmology etc ?

ሶ ሶ ሶ ሷ ሶ Guo,N.Ohta &Torii, Pr.Th.P.120,581(2008);121 ,253 (2009); 3.Inflation with a Gauss-Bonnet N.Ohta &Torii,Pr.Th.P.121,959;122,1477(2009);124,207 (2010); Maeda,N.Ohta Sasagawa, PRD80,104032(2009); 83,044051 (2011) An action with a Gauss-Bonnet term: • N. Ohta and T. Torii, Phys.Rev. D 88 ,064002 (2013). 2𝜆 2 𝑆 − 1 1 2 𝑕 𝜈𝜉 𝜖 𝜈 𝜚𝜖 𝜉 𝜚 − 𝑊 𝜚 − 1 Gauss-Bonnet term 2 𝑇 = න 𝑒 4 𝑦 −𝑕 2 𝜊 𝜚 𝑆 𝐻𝐶 = 𝑆 𝜈𝜉𝜍𝜏 𝑆 𝜈𝜉𝜍𝜏 − 4𝑆 𝜈𝜉 𝑆 𝜈𝜉 + 𝑆 2 2 𝑆 𝐻𝐶 𝐻 μν = 𝜆 2 𝑈 𝐻𝐶 μν + 𝑈 μν μν = 𝜖 𝜈 𝜚𝜖 𝜉 𝜚 + 𝑊 𝜚 − 1 1 𝐻 μν ≡ 𝑆 μν − 𝜆 2 = 8π𝐻 2 𝑕 μν 𝑆 2 𝑕 μν 𝑕 𝜍𝜏 𝜖 𝜍 𝜚𝜖 𝜏 𝜚 + 2𝑊 𝑈 𝜍 − 1 𝐻𝐶 = 4 𝜖 𝜍 𝜖 𝜏 𝜊𝑆 𝜈𝜍𝜉𝜏 − □𝜊𝑆 𝜈𝜉 + 2𝜖 𝜍 𝜖 (𝜈 𝜊𝑆 𝜉) − 2 2𝜖 𝜍 𝜖 𝜏 𝜊𝑆 𝜍𝜏 − □𝜊𝑆 𝑕 μν 𝑈 2 𝜖 𝜈 𝜖 𝜉 𝜊𝑆 μν ,𝜚 𝜚 − 1 2 𝑈 𝐻𝐶 = 0 𝑈 𝐻𝐶 = 𝜊 ,𝜚 𝜚 𝑆 𝐻𝐶 □𝜚 − 𝑊 2 𝑒𝑠 2 FLRW Universe metric: 𝑒𝑡 2 = - 𝑒𝑢 2 + 𝑏 2 𝑢 1−𝐿𝑠 2 + 𝑠 2 (𝑒θ 2 + 𝑡𝑗𝑜 2 θ 𝑒φ 2 ) • Einstein and Field equations yield: 𝐼 2 = 𝜆 2 1 𝜚 2 + 𝑊 − 3𝐿 𝜊𝐼 𝐼 2 + 𝐿 𝜆 2 𝑏 2 + 12 ሶ 𝑏 2 3 2 𝐼 = − 𝜆 2 𝜚 2 − 2𝐿 𝜊 𝐼 2 + 𝐿 𝐼 − 𝐼 2 − 3𝐿 𝜆 2 𝑏 2 − 4 ሷ − 4 ሶ 𝜊𝐼 2 ሶ 𝑏 2 𝑏 2 2 ,𝜚 + 12𝜊 ,𝜚 𝐼 2 + 𝐿 𝐼 + 𝐼 2 = 0 𝜚 + 3𝐼 ሶ 𝜚 + 𝑊 𝑏 2

ሶ ሶ ሶ ሶ ሷ Inflation with a Gauss-Bonnet 2𝜆 2 𝑆 − 1 1 2 𝑕 𝜈𝜉 𝜖 𝜈 𝜚𝜖 𝜉 𝜚 − 𝑊 𝜚 − 1 Action • 𝑇 = න 𝑒 4 𝑦 2 −𝑕 2 𝜊 𝜚 𝑆 𝐻𝐶 Einstein and Field equations yield: S. Koh, BHL, W. Lee, Tumurtushaa 𝐼 2 = 𝜆 2 1 𝜚 2 + 𝑊 − 3𝐿 𝜊𝐼 𝐼 2 + 𝐿 𝜆 2 𝑏 2 + 12 ሶ PRD90 (2014) ) no. no.6, 06 063527 𝑏 2 3 2 S. Koh, BHL, W. Lee, Tumurtushaa 𝐼 = − 𝜆 2 𝜚 2 − 2𝐿 𝜊 𝐼 2 + 𝐿 𝐼 − 𝐼 2 − 3𝐿 𝜆 2 𝑏 2 − 4 ሷ − 4 ሶ 𝜊𝐼 2 ሶ arX rXiv:1610.04360 𝑏 2 𝑏 2 2 ,𝜚 + 12𝜊 ,𝜚 𝐼 2 + 𝐿 𝐼 + 𝐼 2 = 0 𝜚 + 3𝐼 ሶ 𝜚 + 𝑊 𝑏 2 The duration of inflation gets shorter as the Gauss- Bonnet coupling constant increases. Increasing of the Gauss-Bonnet coupling constant makes the effective potential steeper such that the scalar field rolls faster than it does in models without Gauss-Bonnet term Hence inflation ends earlier in models with a large Gauss-Bonnet coupling. 0 = 0.5 × 10 −12 𝑊 𝜊 0 = 0 (black), 𝜊 0 = 3 × 10 6 (red), and 𝜊 0 = 3 × 10 7 (blue).

 Model-2

 Model-2

 Model-2

Reheating parameters in Gauss-Bonnet inflation Models Let us consider a mode with comoving wavenumber 𝑙 ∗ which crosses the horizon during inflation when the scale factor is 𝑏 ∗ . The comoving Hubble scale 𝑏 ∗ 𝐼 ∗ = 𝑙 ∗ at the horizon crossing time can be related to that of the present time as (6) where 𝑏 0 , 𝑏 ∗ , 𝑏 end , and 𝑏 th denote the scale factor at present, the horizon crossing, the end of inflation, and the end of reheating, respectively. By taking logarithm from both sides, we rewrite (7) where 𝑂 ∗ ≡ ln(𝑏 end /𝑏 ∗ ) is the number of e-foldings between the time of mode exits the horizon and the end of inflation, and 𝑂 th ≡ ln(𝑏 th /𝑏 end ) is the number of e-foldings between the end of inflation and the end of reheating.