Probing Fundamental Physics with Gravitational Waves

Transcription

1 Probing Fundamental Physics with Gravitational Waves Cyril Lagger Seminar - Max-Planck-Institut für Kernphysik, Heidelberg - March 7, / 36

2 Overview Gravitational Wave (GW) detection by LIGO/Virgo is promising for theoretical physics: confirms prediction of General Relativity allows to test GR (and its modifications) in a strong and dynamical regime suggests to look for other sources of GWs in relation to particle physics: phase transitions, cosmic strings,... 2 / 36

3 Overview Gravitational Wave (GW) detection by LIGO/Virgo is promising for theoretical physics: confirms prediction of General Relativity allows to test GR (and its modifications) in a strong and dynamical regime suggests to look for other sources of GWs in relation to particle physics: phase transitions, cosmic strings,... Two topics in this talk: constraining noncommutative space-time from LIGO/Virgo waveforms (transient signal) exploring beyond the Standard Model physics with GWs from phase transitions (stochastic background) 2 / 36

4 Part I: Test of GR and noncommutative space-time 3 / 36

5 First GW signal: GW Inspiral, merger and ring-down of a binary black hole observed by LIGO. Masses of M and M. Frequency ranging from 35 to 250 Hz and velocity up to 0.5c. [LIGO/Virgo Coll., Phys. Rev. Lett. 116 (2016) ] [N. Yunes, K. Yagi, F. Pretorius, arxiv: ] 4 / 36

6 An opportunity to test GR and its modifications Einstein Field Equations (EFE) from GR predicts the waveform of such GWs : post-newtonian formalism: analytical expansion in v c for the inspiralling numerical Relativity: accurate simulations including merger and ring-down 5 / 36

7 An opportunity to test GR and its modifications Einstein Field Equations (EFE) from GR predicts the waveform of such GWs : post-newtonian formalism: analytical expansion in v c for the inspiralling numerical Relativity: accurate simulations including merger and ring-down GW data are in good agreement with GR predictions opportunity to test various models beyond GR. [LIGO/Virgo Coll., Phys. Rev. Lett. 116 (2016) ] [e.g.: N. Yunes, K. Yagi, F. Pretorius, arxiv: , N. Yunes, E. Berti, K. Yagi, arxiv: ] 5 / 36

8 An opportunity to test GR and its modifications Einstein Field Equations (EFE) from GR predicts the waveform of such GWs : post-newtonian formalism: analytical expansion in v c for the inspiralling numerical Relativity: accurate simulations including merger and ring-down GW data are in good agreement with GR predictions opportunity to test various models beyond GR. [LIGO/Virgo Coll., Phys. Rev. Lett. 116 (2016) ] [e.g.: N. Yunes, K. Yagi, F. Pretorius, arxiv: , N. Yunes, E. Berti, K. Yagi, arxiv: ] Our objective: constrain the scale of noncommutative space-time. 5 / 36

9 The post-newtonian formalism A perturbative approach to solve the EFE, h αβ = 16πG c 4 τ αβ µ h αµ = 0, as an expansion in v c. [L. Blanchet, Living Rev. Rel. 17 (2014)] 6 / 36

10 The post-newtonian formalism A perturbative approach to solve the EFE, h αβ = 16πG c 4 τ αβ µ h αµ = 0, as an expansion in v c. [L. Blanchet, Living Rev. Rel. 17 (2014)] Notation: gravitational-field amplitude: h αβ = gg αβ η αβ matter-gravitational source: τ αβ = g T αβ + ( ) O (n) O v n c n c4 16πG Λαβ 6 / 36

11 Far zone vs near zone Iterative expansions in the near and far zones and matching strategy in the overlap zone: 7 / 36

12 Far zone vs near zone Iterative expansions in the near and far zones and matching strategy in the overlap zone: Post Minkowskian (PM) - G n : h αβ = n=1 Gn h αβ n h αβ = Λ αβ h αβ n = Λ αβ n [h 1,, h n 1 ] 7 / 36

13 Far zone vs near zone Iterative expansions in the near and far zones and matching strategy in the overlap zone: ( ) Post Newtonian (PN) - 1c n: h αβ = n=2 1 c n h αβ n τ αβ = n= 2 1 c n τ αβ n 2 h αβ n = 16πG τ αβ n t hαβ n 2 Post Minkowskian (PM) - G n : h αβ = n=1 Gn h αβ n h αβ = Λ αβ h αβ n = Λ αβ n [h 1,, h n 1 ] 7 / 36

14 Matter source Consider a binary system of two black holes of masses m 1 and m 2. Usually approximated by two point-like particles: T µν (x, t) = m 1 v gg ρ 1 vσ 1 ρσ c 2 µ v1 (t)vν 1 (t) δ3 (x y 1 (t)) / 36

15 Matter source Consider a binary system of two black holes of masses m 1 and m 2. Usually approximated by two point-like particles: T µν (x, t) = m 1 v gg ρ 1 vσ 1 ρσ c 2 µ v1 (t)vν 1 (t) δ3 (x y 1 (t)) Useful parametrization: total mass: M = m 1 + m 2 reduced mass: µ = m 1m 2 M symmetric mass ratio: ν = µ M = m 1m 2 M 2 8 / 36

16 Matter source Consider a binary system of two black holes of masses m 1 and m 2. Usually approximated by two point-like particles: T µν (x, t) = m 1 v gg ρ 1 vσ 1 ρσ c 2 µ v1 (t)vν 1 (t) δ3 (x y 1 (t)) Useful parametrization: total mass: M = m 1 + m 2 reduced mass: µ = m 1m 2 M symmetric mass ratio: ν = µ M = m 1m 2 M 2 We neglect spin effects in our considerations. 8 / 36

17 The balance equation 9 / 36

18 The balance equation Equations of motion - energy E: ν T µν = 0 a 1 = Gm 2 n r O(2) E = m 1v Gm 1m 2 2r 12 + O(2) / 36

19 The balance equation Equations of motion - energy E: ν T µν = 0 a 1 = Gm 2 n r O(2) E = m 1v Gm 1m 2 2r 12 + O(2) Radiated flux F: ( F = G 15 I (3) c 5 ij I (3) ij F = G c 5 ( 32G 3 M 5 ν 2 5r 5 ) + O(2) ) + O(2) 9 / 36

20 The balance equation Equations of motion - energy E: ν T µν = 0 a 1 = Gm 2 n r O(2) E = m 1v Gm 1m 2 2r 12 + O(2) Radiated flux F: ( F = G 15 I (3) c 5 ij I (3) ij F = G c 5 ( 32G 3 M 5 ν 2 5r 5 ) + O(2) ) + O(2) Conservation of energy implies the balance equation and the orbital phase: de dt = F φ = Ω(t)dt 9 / 36

21 State-of-the-art computations For data analysis, consider the waveform in frequency space: h( f ) = A( f ) e iψ( f ). 10 / 36

22 State-of-the-art computations For data analysis, consider the waveform in frequency space: h( f ) = A( f ) e iψ( f ). The phase ψ( f ) (Fourier transform of φ(t)) has been calculated to 3.5PN accuracy: ψ( f ) = 2π f t c φ c π ( ) πmg f (j 5)/3 128 ϕ j c j=0 3, 10 / 36

23 State-of-the-art computations For data analysis, consider the waveform in frequency space: h( f ) = A( f ) e iψ( f ). The phase ψ( f ) (Fourier transform of φ(t)) has been calculated to 3.5PN accuracy: ψ( f ) = 2π f t c φ c π ( ) πmg f (j 5)/3 128 ϕ j c j=0 3, where the phase coefficients are ϕ 0 = 1 ϕ 1 = 0 ϕ 2 = 3715 ϕ 3 = 16π ϕ 4 = ν ν ν2 [T. Damour, B. Iyer and B. Sathyaprakash, Phys. Rev. D 63 (2001) ] [G. Faye, S. Marsat, L. Blanchet, B. Iyer, Class. Quantum Grav. 29 (2012) ] 10 / 36

24 GR vs GW150914: bayesian analysis [LIGO/Virgo Coll., Phys. Rev. Lett. 116 (2016) ] 11 / 36

25 Noncommutative corrections to the waveform A. Kobakhidze, CL, A. Manning, PRD 94 (2016) / 36

26 Noncommutative space-time NC space-time arises in a number of contexts: Originally proposed by Heisenberg as an effective UV cutoff. Several formalisations (e.g. Snyder [Phys. Rev. 71 (1947) 38]). Noncommutative geometry [A. Connes, Inst. Hautes Etudes Sci. Publ. Math. 62 (1985) 257]. Low-energy limit of string theory [N. Seiberg and E.Witten, JHEP 9909 (1999) 032]. 13 / 36

27 Noncommutative space-time NC space-time arises in a number of contexts: Originally proposed by Heisenberg as an effective UV cutoff. Several formalisations (e.g. Snyder [Phys. Rev. 71 (1947) 38]). Noncommutative geometry [A. Connes, Inst. Hautes Etudes Sci. Publ. Math. 62 (1985) 257]. Low-energy limit of string theory [N. Seiberg and E.Witten, JHEP 9909 (1999) 032]. We focus on the canonical algebra of coordinates: [ ˆx µ, ˆx ν ] = iθ µν x µ x ν 1 2 θµν with noncommutative QFT - fields product replaced by Moyal product: + ( ) i n 1 f (x) g(x) = f (x)g(x) + 2 n! θα 1β1 θ α nβ n α1 αn f (x) β1 βn g(x) n=1 13 / 36

28 Noncommutative space-time NC space-time arises in a number of contexts: Originally proposed by Heisenberg as an effective UV cutoff. Several formalisations (e.g. Snyder [Phys. Rev. 71 (1947) 38]). Noncommutative geometry [A. Connes, Inst. Hautes Etudes Sci. Publ. Math. 62 (1985) 257]. Low-energy limit of string theory [N. Seiberg and E.Witten, JHEP 9909 (1999) 032]. We focus on the canonical algebra of coordinates: [ ˆx µ, ˆx ν ] = iθ µν x µ x ν 1 2 θµν with noncommutative QFT - fields product replaced by Moyal product: + ( ) i n 1 f (x) g(x) = f (x)g(x) + 2 n! θα 1β1 θ α nβ n α1 αn f (x) β1 βn g(x) n=1 Previous constraints on NC scale θ only at inverse TeV. [S. Carroll et al., Phys. Rev. Lett.87 (2001) ] [X. Calmet, Eur. Phys. J. C41 (2005) 269] 13 / 36

29 Noncommutative effects on GWs Expect modifications on both matter source and field equations. 14 / 36

30 Noncommutative effects on GWs Expect modifications on both matter source and field equations. Consider a Schwarzschild black hole described by a massive scalar field in noncommutative QFT[A. Kobakhidze, Phys. Rev. D79 (2009) ]: T µν NC (x) = 1 2 ( µ φ ν φ + ν φ µ φ) 1 ( ) 2 ηµν ρ φ ρ φ m 2 φ φ Similar approach as for the quantum corrections of a Schwarzschild BH. [N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, Phys. Rev. D68 (2003) ] 14 / 36

31 Noncommutative effects on GWs Expect modifications on both matter source and field equations. Consider a Schwarzschild black hole described by a massive scalar field in noncommutative QFT[A. Kobakhidze, Phys. Rev. D79 (2009) ]: T µν NC (x) = 1 2 ( µ φ ν φ + ν φ µ φ) 1 ( ) 2 ηµν ρ φ ρ φ m 2 φ φ Similar approach as for the quantum corrections of a Schwarzschild BH. [N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, Phys. Rev. D68 (2003) ] Neglect corrections on the EFE since noncommutative gravity appears at O( θ 2 ) and is model-dependent. [X. Calmet, A. Kobakhidze, Phys. Rev. D74 (2006) ] [P. Mukherjee, A. Saha, Phys. Rev. D74 (2006) ] 14 / 36

32 Energy-momentum tensor in noncommutative space-time After quantising and keeping leading-order corrections of the Moyal product: with T µν NC (x, t) Tµν GR (x, t) + m3 G 2 8c 4 vµ v ν Θ kl k l δ 3 (x y(t)) Θ kl = θ0k θ 0l l 2 P t2 P + 2 v p c θ 0k θ pl l 3 P t P + v pv q c 2 θ kp θ lq l 4 P = θ0k θ 0l l 2 P t2 P + O(1) 15 / 36

33 Energy-momentum tensor in noncommutative space-time After quantising and keeping leading-order corrections of the Moyal product: with T µν NC (x, t) Tµν GR (x, t) + m3 G 2 8c 4 vµ v ν Θ kl k l δ 3 (x y(t)) Θ kl = θ0k θ 0l l 2 P t2 P + 2 v p c θ 0k θ pl l 3 P t P + v pv q c 2 θ kp θ lq l 4 P = θ0k θ 0l l 2 P t2 P + O(1) Binary black hole EMT with 2PN noncommutative corrections: T µν (x, t) = m 1 γ 1 v µ 1 vν 1 δ3 (x y 1 (t)) + m3 1 G2 κ 2 where κθ i = θ0i l P t P. 8c 4 v µ 1 vν 1 θk θ l k l δ 3 (x y 1 (t)) / 36

34 Noncommutative effects on gravitational waveform d(e GR + E NC ) dt = F NC F NC 16 / 36

35 Noncommutative effects on gravitational waveform d(e GR + E NC ) dt = F NC F NC Lowest-order corrections appear at 2PN: E NC = 3M3 µ(1 2ν)G 3 κ 2 8c 4 r 3 θ k θ l ˆn kl + O(5) F NC = G c 5 ( 36 5 G 5 M 7 ) c 4 r 7 ν2 (1 2ν)κ 2 + O(5) 16 / 36

36 Noncommutative effects on gravitational waveform d(e GR + E NC ) dt = F NC F NC Lowest-order corrections appear at 2PN: E NC = 3M3 µ(1 2ν)G 3 κ 2 8c 4 r 3 θ k θ l ˆn kl + O(5) F NC = G c 5 ( 36 5 G 5 M 7 ) c 4 r 7 ν2 (1 2ν)κ 2 + O(5) Lowest order modification to the waveform phase: ϕ 4 = ν ν2 + 5 (1 2ν)κ / 36

37 Noncommutativity vs GW δϕ4 NC = ϕnc 4 ϕ4 GR = (1 2ν) ν ν κ2 δϕ NC 4 20 κ / 36

38 Summary of Part I Several observations of binary system merger by LIGO/Virgo 18 / 36

39 Summary of Part I Several observations of binary system merger by LIGO/Virgo GW waveform consistent with GR 18 / 36

40 Summary of Part I Several observations of binary system merger by LIGO/Virgo GW waveform consistent with GR Explicit computation of the lowest-order (2PN) noncommutative correction to the GW waveform. 18 / 36

41 Summary of Part I Several observations of binary system merger by LIGO/Virgo GW waveform consistent with GR Explicit computation of the lowest-order (2PN) noncommutative correction to the GW waveform. Constraint on the scale of noncommutativity to around the Planck scale: θ 0i O(10) l P t P 18 / 36

42 Part II: Phase transitions and Gravitational Waves 19 / 36

43 Hot Big Bang scenario: First-order phase transition and GWs early Universe hot plasma (high T) scalar field(s) behaviour dictated by their free energy density F(ρ, T) dynamics depend on the underlying particle physics model 2nd-order transition / crossover: smooth dynamics no GWs 20 / 36

44 Hot Big Bang scenario: First-order phase transition and GWs early Universe hot plasma (high T) scalar field(s) behaviour dictated by their free energy density F(ρ, T) dynamics depend on the underlying particle physics model 2nd-order transition / crossover: smooth dynamics no GWs 20 / 36

45 Hot Big Bang scenario: First-order phase transition and GWs early Universe hot plasma (high T) scalar field(s) behaviour dictated by their free energy density F(ρ, T) dynamics depend on the underlying particle physics model 2nd-order transition / crossover: smooth dynamics no GWs 20 / 36

46 Hot Big Bang scenario: First-order phase transition and GWs early Universe hot plasma (high T) scalar field(s) behaviour dictated by their free energy density F(ρ, T) dynamics depend on the underlying particle physics model 2nd-order transition / crossover: smooth dynamics no GWs 20 / 36

47 Hot Big Bang scenario: First-order phase transition and GWs early Universe hot plasma (high T) scalar field(s) behaviour dictated by their free energy density F(ρ, T) dynamics depend on the underlying particle physics model 1st-order transition: bubble nucleation bubble collision stochastic GW background 20 / 36

48 Hot Big Bang scenario: First-order phase transition and GWs early Universe hot plasma (high T) scalar field(s) behaviour dictated by their free energy density F(ρ, T) dynamics depend on the underlying particle physics model 1st-order transition: bubble nucleation bubble collision stochastic GW background 20 / 36

49 Hot Big Bang scenario: First-order phase transition and GWs early Universe hot plasma (high T) scalar field(s) behaviour dictated by their free energy density F(ρ, T) dynamics depend on the underlying particle physics model 1st-order transition: bubble nucleation bubble collision stochastic GW background 20 / 36

50 Hot Big Bang scenario: First-order phase transition and GWs early Universe hot plasma (high T) scalar field(s) behaviour dictated by their free energy density F(ρ, T) dynamics depend on the underlying particle physics model 1st-order transition: bubble nucleation bubble collision stochastic GW background 20 / 36

51 Example of a very recent simulation [D. Cutting, M. Hindmarsh, D. Weir, arxiv: ] 21 / 36

52 Looking for BSM physics with GWs A possible probe of new physics: no 1st-order PT in the Standard Model [K. Kajantie et al., Phys. Rev. Lett. 77 (1996) 2887] no stochastic GW background predicted in the SM various BSM models account for a 1st-order EWPT (e.g. motivated by electroweak baryogenesis) 22 / 36

53 Looking for BSM physics with GWs A possible probe of new physics: no 1st-order PT in the Standard Model [K. Kajantie et al., Phys. Rev. Lett. 77 (1996) 2887] no stochastic GW background predicted in the SM various BSM models account for a 1st-order EWPT (e.g. motivated by electroweak baryogenesis) Examples of models considered: non-linearly realised electroweak gauge group [A. Kobakhidze, A. Manning, J. Yue, arxiv: ] [A. Kobakhidze, CL, A. Manning, J. Yue, arxiv: ] Standard Model with hidden scale invariance [S. Arunasalam, A. Kobakhidze, CL, S. Liang, A. Zhou, arxiv: ] 22 / 36

54 Stochastic background from bubble collisions Stochastic background from three sources [C. Caprini et al., JCAP 1604 (2016) no ]: h 2 Ω GW ( f ) h 2 Ω col + h 2 Ω sw + h 2 Ω MHD 23 / 36

55 Stochastic background from bubble collisions Stochastic background from three sources [C. Caprini et al., JCAP 1604 (2016) no ]: h 2 Ω GW ( f ) h 2 Ω col + h 2 Ω sw + h 2 Ω MHD Ω col dominant for very strong PT (as considered here). 23 / 36

56 Stochastic background from bubble collisions Stochastic background from three sources [C. Caprini et al., JCAP 1604 (2016) no ]: h 2 Ω GW ( f ) h 2 Ω col + h 2 Ω sw + h 2 Ω MHD Ω col dominant for very strong PT (as considered here). Peak frequency and amplitude of the background mainly depend on the bubble size R at collision and kinetic energy ρ kin stored in the bubbles: f peak ( R) 1 ρ 2 kin Ω col ( RH p ) 2 (ρ kin +ρ rad ) 2 23 / 36

57 Bubble-collision simulations Going beyond dimensional analysis with numerical simulations (and redshift) [S. Huber and T. Konstandin, JCAP 0809 (2008) 022] 24 / 36

58 Bubble-collision simulations Going beyond dimensional analysis with numerical simulations (and redshift) [S. Huber and T. Konstandin, JCAP 0809 (2008) 022] Notation: α = ρ kin /ρ rad and β = v R 1 24 / 36

59 Bubble-collision simulations Going beyond dimensional analysis with numerical simulations (and redshift) [S. Huber and T. Konstandin, JCAP 0809 (2008) 022] Notation: α = ρ kin /ρ rad and β = v R 1 Amplitude: ( ) h 2 Ω col ( f ) = /3 ( ) β 2 ( ) κ 2 α 2 ( 0.11v 3 ) v g H p 1 + α v 2 S( f ) S( f ) = 3.8( f / f 0) ( f / f 0 ) / 36

60 Bubble-collision simulations Going beyond dimensional analysis with numerical simulations (and redshift) [S. Huber and T. Konstandin, JCAP 0809 (2008) 022] Notation: α = ρ kin /ρ rad and β = v R 1 Amplitude: ( ) h 2 Ω col ( f ) = /3 ( ) β 2 ( ) κ 2 α 2 ( 0.11v 3 ) v g H p 1 + α v 2 S( f ) Peak frequency: f 0 = ( S( f ) = 3.8( f / f 0) ( f / f 0 ) 3.8 Tp 1 GeV ) ( g 100 ) ( 1/6 H 1 p β v + v 2 ) Hz 24 / 36

61 Bubble-collision simulations Going beyond dimensional analysis with numerical simulations (and redshift) [S. Huber and T. Konstandin, JCAP 0809 (2008) 022] Notation: α = ρ kin /ρ rad and β = v R 1 Amplitude: ( ) h 2 Ω col ( f ) = /3 ( ) β 2 ( ) κ 2 α 2 ( 0.11v 3 ) v g H p 1 + α v 2 S( f ) Peak frequency: f 0 = ( S( f ) = 3.8( f / f 0) ( f / f 0 ) 3.8 Tp 1 GeV ) ( g 100 ) ( 1/6 H 1 p β v + v 2 ) Hz The set of parameters ( R, ρ kin, v, κ ν ) is determined by the underlying particle physics model. 24 / 36

62 Different scenarios of electroweak phase transition Typical case (quick PT): O(1) bubbles produced per Hubble volume at T n T EW they rapidly collide percolation temperature T p T n time scale of the process much shorter than Hubble time f peak millihertz range of LISA [C. Caprini et al., JCAP 1604 (2016) no ] 25 / 36

63 Different scenarios of electroweak phase transition Typical case (quick PT): O(1) bubbles produced per Hubble volume at T n T EW they rapidly collide percolation temperature T p T n time scale of the process much shorter than Hubble time f peak millihertz range of LISA [C. Caprini et al., JCAP 1604 (2016) no ] Prolonged and supercooled PT [A. Kobakhidze, CL, A. Manning, J. Yue, arxiv: ]: weaker nucleation probability less bubbles produced more time needed for them to collide T p T n T EW f peak 10 8 Hertz range of Pulsar Timing Arrays 25 / 36

64 Different scenarios of electroweak phase transition [From rhcole.com/apps/gwplotter/] 25 / 36

65 Prolonged electroweak phase transition A. Kobakhidze, CL, A. Manning, J. Yue [Eur.Phys.J. C77 (2017), arxiv: ] 26 / 36

66 Realisation of SU(2) L U(1) Y Main idea: G coset = SU(2) L U(1) Y /U(1) Q is gauged with broken generators T i = σ i δ i3 I and Goldstone bosons π i (x) physical Higgs as a singlet ρ(x) (1, 1) 0 27 / 36

67 Realisation of SU(2) L U(1) Y Main idea: G coset = SU(2) L U(1) Y /U(1) Q is gauged with broken generators T i = σ i δ i3 I and Goldstone bosons π i (x) physical Higgs as a singlet ρ(x) (1, 1) 0 ( ) SM Higgs doublet identified as H(x) = ρ(x) e i 2 πi (x)t i 0, i {1, 2, 3} / 36

68 Realisation of SU(2) L U(1) Y Main idea: G coset = SU(2) L U(1) Y /U(1) Q is gauged with broken generators T i = σ i δ i3 I and Goldstone bosons π i (x) physical Higgs as a singlet ρ(x) (1, 1) 0 ( ) SM Higgs doublet identified as H(x) = ρ(x) e i 2 πi (x)t i 0, i {1, 2, 3} 2 1 SM particle content but BSM interactions 27 / 36

69 Realisation of SU(2) L U(1) Y Main idea: G coset = SU(2) L U(1) Y /U(1) Q is gauged with broken generators T i = σ i δ i3 I and Goldstone bosons π i (x) physical Higgs as a singlet ρ(x) (1, 1) 0 ( ) SM Higgs doublet identified as H(x) = ρ(x) e i 2 πi (x)t i 0, i {1, 2, 3} 2 1 SM particle content but BSM interactions Minimal setup (usual SM configurations except Higgs potential): V (0) (ρ) = µ2 2 ρ2 + κ 3 ρ3 + λ 4 ρ4. 27 / 36

70 Realisation of SU(2) L U(1) Y Main idea: G coset = SU(2) L U(1) Y /U(1) Q is gauged with broken generators T i = σ i δ i3 I and Goldstone bosons π i (x) physical Higgs as a singlet ρ(x) (1, 1) 0 ( ) SM Higgs doublet identified as H(x) = ρ(x) e i 2 πi (x)t i 0, i {1, 2, 3} 2 1 SM particle content but BSM interactions Minimal setup (usual SM configurations except Higgs potential): V (0) (ρ) = µ2 2 ρ2 + κ 3 ρ3 + λ 4 ρ4. For additional details, see e.g.: [M. Gonzalez-Alonso et al., Eur. Phys. J. C 75 (2015) 3, 128] [D. Binosi and A. Quadri, JHEP 1302 (2013) 020] [A. Kobakhidze, arxiv: ] [R. Contino et al., JHEP 1005 (2010) 089] 27 / 36

71 Tree-level potential Model specified by one parameter: κ = κ m2 h v 63.5 κ GeV. Barrier in the Higgs potential at tree level likely to allow a strong 1st-order EWPT. 28 / 36

72 Tree-level potential Model specified by one parameter: κ = κ m2 h v 63.5 κ GeV. Barrier in the Higgs potential at tree level likely to allow a strong 1st-order EWPT. 28 / 36

73 Bubble nucleation probability Decay probability per unit volume per unit time: Γ(T) A(T)e S(T) [A. Linde, Nucl. Phys. B216 (1983) 421] 29 / 36

74 Bubble nucleation probability Decay probability per unit volume per unit time: Γ(T) A(T)e S(T) [A. Linde, Nucl. Phys. B216 (1983) 421] Computation of the Euclidean action: β S[ρ, T] = 4π dτ 0 0 dr r 2 [ 1 2 ( ) dρ dτ 2 ( ) dρ 2 + F(ρ, T)] dr 2 ρ τ ρ r ρ r r F ρ (ρ, T) = 0 + boundary conditions S 4 [ρ, T] = 2π 2 0 S[ρ, T] 1 T S 3[ρ, T] = 4π T 0 d r r 3 [ 1 2 dr r 2 [ 1 2 ( ) dρ 2 + F(ρ, T)] d r ( ) dρ 2 + F(ρ, T)] dr, T R 1 0, T R / 36

75 Bubble nucleation probability Decay probability per unit volume per unit time: Γ(T) A(T)e S(T) [A. Linde, Nucl. Phys. B216 (1983) 421] Some numerical results: Standard scenario: number of bubbles O(1) requires min S / 36

76 Phase transition dynamics General formalism in expanding universe: [M. Turner et al., Phys. Rev. D46 (1992) 2384]. 30 / 36

77 Phase transition dynamics General formalism in expanding universe: [M. Turner et al., Phys. Rev. D46 (1992) 2384]. Probability for a point of space-time to remain in the false-vacuum: [ p(t) = exp 4π t ] t dt Γ(t )a 3 (t )r 3 (t, t ) r(t, t ) = dt v(t ) 3 t t a(t ) Completion of the PT requires p(t) 0 Percolation temperature ( collision) [L. Leitao et al., JCAP 1210 (2012) 024]: p(t p ) / 36

78 Phase transition dynamics General formalism in expanding universe: [M. Turner et al., Phys. Rev. D46 (1992) 2384]. Probability for a point of space-time to remain in the false-vacuum: [ p(t) = exp 4π t ] t dt Γ(t )a 3 (t )r 3 (t, t ) r(t, t ) = dt v(t ) 3 t t a(t ) Completion of the PT requires p(t) 0 Percolation temperature ( collision) [L. Leitao et al., JCAP 1210 (2012) 024]: p(t p ) 0.7 Number density of produced bubbles: ( ) dn dr (t, t a(tr ) 4 p(tr ) R) = Γ(t R ) a(t) v(t R ) 30 / 36

79 Phase transition dynamics General formalism in expanding universe: [M. Turner et al., Phys. Rev. D46 (1992) 2384]. Probability for a point of space-time to remain in the false-vacuum: [ p(t) = exp 4π t ] t dt Γ(t )a 3 (t )r 3 (t, t ) r(t, t ) = dt v(t ) 3 t t a(t ) Completion of the PT requires p(t) 0 Percolation temperature ( collision) [L. Leitao et al., JCAP 1210 (2012) 024]: p(t p ) 0.7 Number density of produced bubbles: ( ) dn dr (t, t a(tr ) 4 p(tr ) R) = Γ(t R ) a(t) v(t R ) Nucleation temperature T n : maximum of dn dr (t p, t R ) 30 / 36

80 Bubbles properties at collision By definition: most bubbles collide at t p majority of them produced at t n bubble physical radius: R = a(t p )r(t p, t n ) 31 / 36

81 Bubbles properties at collision By definition: most bubbles collide at t p majority of them produced at t n bubble physical radius: R = a(t p )r(t p, t n ) Kinetic energy stored in bubble-walls: tp E kin = κ ν 4π dt dr t n dt (t, t n)r 2 (t, t n )ɛ(t) ɛ(t): latent heat ( vacuum energy) κ ν : fraction of energy going into the wall motion (vs. heating the plasma) 31 / 36

82 Bubbles properties at collision By definition: most bubbles collide at t p majority of them produced at t n bubble physical radius: R = a(t p )r(t p, t n ) Kinetic energy stored in bubble-walls: tp E kin = κ ν 4π dt dr t n dt (t, t n)r 2 (t, t n )ɛ(t) ɛ(t): latent heat ( vacuum energy) κ ν : fraction of energy going into the wall motion (vs. heating the plasma) R and E kin : key parameters to deduce the GW spectrum 31 / 36

83 Some assumptions Entire dynamics specified by Γ(t), ɛ(t), κ ν, v(t) and a(t). 32 / 36

84 Some assumptions Entire dynamics specified by Γ(t), ɛ(t), κ ν, v(t) and a(t). Very strong PT: large amount of vacuum energy released κ ν 1 [A. Kobakhidze et al, arxiv: ] v 1 (runaway bubbles) [C. Caprini et al., JCAP 1604 (2016) no ] 32 / 36

85 Some assumptions Entire dynamics specified by Γ(t), ɛ(t), κ ν, v(t) and a(t). Very strong PT: large amount of vacuum energy released κ ν 1 [A. Kobakhidze et al, arxiv: ] v 1 (runaway bubbles) [C. Caprini et al., JCAP 1604 (2016) no ] Consider a radiation-dominated Universe: a(t) t 1/2 t = ( 45M 2 p 16π 3 g ) 1/2 1 T 2 32 / 36

86 Numerical results Probability p(t): 33 / 36

87 Numerical results Number density distribution for κ = 1.9: T n 49 GeV 33 / 36

88 Numerical results κ [m 2 h / v ] T GeV T n GeV T p GeV ( RH p ) 1 ρ kin /ρ rad Observations: new feature: T p T n Hubble-size bubbles at collision ρ rad ρ kin : confirm very strong scenario 33 / 36

89 GW spectra: results Current constraints: EPTA, PPTA, NANOGrav Possible detection: Square Kilometre Array [Moore et al., Class. Quant. Grav. 32 (2015) ] 34 / 36

90 Summary of Part II Stochastic background of GWs as a signature of new physics Different possible scenarios of 1st-order transitions: standard electroweak transition at T 100 GeV signal in LISA prolonged electroweak transition signal in PTA Not limited to the model discussed here 35 / 36

91 General Conclusion The detection of Gravitational Waves represents a milestone by itself. 36 / 36

92 General Conclusion The detection of Gravitational Waves represents a milestone by itself. It also provides new opportunities to probe various area of fundamental physics from General Relativity to Particle Physics. 36 / 36

93 General Conclusion The detection of Gravitational Waves represents a milestone by itself. It also provides new opportunities to probe various area of fundamental physics from General Relativity to Particle Physics. There are lot of expectations regarding the future experiments like KAGRA, LISA, SKA, etc 36 / 36

94 Backup slides 37 / 36

95 Standard Model with hidden scale invariance Scale invariant models are attractive to address the hierarchy problem e.g.: [K. Meissner, H. Nicolai, PLB 648 (2007) 312] [R. Foot et al., PRD 77 (2008) ] [S. Iso et al., PLB 676 (2009) 81] 38 / 36

96 Standard Model with hidden scale invariance Scale invariant models are attractive to address the hierarchy problem e.g.: [K. Meissner, H. Nicolai, PLB 648 (2007) 312] [R. Foot et al., PRD 77 (2008) ] [S. Iso et al., PLB 676 (2009) 81] Assume existence of UV complete scale invariant model (string theory,...) 38 / 36

97 Standard Model with hidden scale invariance Scale invariant models are attractive to address the hierarchy problem e.g.: [K. Meissner, H. Nicolai, PLB 648 (2007) 312] [R. Foot et al., PRD 77 (2008) ] [S. Iso et al., PLB 676 (2009) 81] Assume existence of UV complete scale invariant model (string theory,...) Focus on low-energy effective field theory: Standard Model Higgs potential at UV scale Λ [ 2 V(Φ Φ) = V 0 (Λ) + λ(λ) Φ Φ vew(λ)] spontaneously broken scale invariance manifests through dilaton field χ Λ Λ χ f χ αχ v 2 ew(λ) v2 ew(αχ) χ f 2 ξ(αχ) χ 2 2 χ 2 V 0 (Λ) V 0(αχ) χ 4 ρ(αχ) fχ 4 4 χ 4 38 / 36

98 Standard Model with hidden scale invariance Scale invariant models are attractive to address the hierarchy problem e.g.: [K. Meissner, H. Nicolai, PLB 648 (2007) 312] [R. Foot et al., PRD 77 (2008) ] [S. Iso et al., PLB 676 (2009) 81] Assume existence of UV complete scale invariant model (string theory,...) Focus on low-energy effective field theory: Standard Model Higgs potential at UV scale Λ [ 2 V(Φ Φ) = V 0 (Λ) + λ(λ) Φ Φ vew(λ)] spontaneously broken scale invariance manifests through dilaton field χ Λ Λ χ f χ αχ v 2 ew(λ) v2 ew(αχ) χ f 2 ξ(αχ) χ 2 2 χ 2 V 0 (Λ) V 0(αχ) χ 4 ρ(αχ) fχ 4 4 χ 4 We get an effective scale invariant potential: [ V(Φ Φ, χ) = λ(αχ) Φ Φ ξ(αχ) 2 ] 2 χ 2 + ρ(αχ) χ / 36

99 Hierarchy and light dilaton Scale invariance is broken by quantum effects: λ (i) (αχ) = λ (i) (µ) + β λ (i)(µ) ln (αχ/µ) + β λ (i) (µ) ln 2 (αχ/µ) / 36

100 Hierarchy and light dilaton Scale invariance is broken by quantum effects: λ (i) (αχ) = λ (i) (µ) + β λ (i)(µ) ln (αχ/µ) + β λ (i) (µ) ln 2 (αχ/µ) +... Minimisation conditions and small vacuum energy density: V V χ = 0, Φ=vew,χ=v χ Φ = 0, V(v ew, v χ ) = 0 Φ=vew,χ=v χ 39 / 36

101 Hierarchy and light dilaton Scale invariance is broken by quantum effects: λ (i) (αχ) = λ (i) (µ) + β λ (i)(µ) ln (αχ/µ) + β λ (i) (µ) ln 2 (αχ/µ) +... Minimisation conditions and small vacuum energy density: V V χ = 0, Φ=vew,χ=v χ Φ = 0, V(v ew, v χ ) = 0 Φ=vew,χ=v χ We obtain dimensional transmutation and hierarchy of VEVs (Λ v χ ): ρ(v χ ) = 0, β ρ (v χ ) = 0, ξ(v χ ) = v2 ew v 2 χ 39 / 36

102 Hierarchy and light dilaton Scale invariance is broken by quantum effects: λ (i) (αχ) = λ (i) (µ) + β λ (i)(µ) ln (αχ/µ) + β λ (i) (µ) ln 2 (αχ/µ) +... Minimisation conditions and small vacuum energy density: V V χ = 0, Φ=vew,χ=v χ Φ = 0, V(v ew, v χ ) = 0 Φ=vew,χ=v χ We obtain dimensional transmutation and hierarchy of VEVs (Λ v χ ): ρ(v χ ) = 0, β ρ (v χ ) = 0, ξ(v χ ) = v2 ew v 2 χ ξ(v χ ) can be hierarchically small (technical naturalness) 39 / 36

103 Hierarchy and light dilaton Scale invariance is broken by quantum effects: λ (i) (αχ) = λ (i) (µ) + β λ (i)(µ) ln (αχ/µ) + β λ (i) (µ) ln 2 (αχ/µ) +... Minimisation conditions and small vacuum energy density: V V χ = 0, Φ=vew,χ=v χ Φ = 0, V(v ew, v χ ) = 0 Φ=vew,χ=v χ We obtain dimensional transmutation and hierarchy of VEVs (Λ v χ ): ρ(v χ ) = 0, β ρ (v χ ) = 0, ξ(v χ ) = v2 ew v 2 χ ξ(v χ ) can be hierarchically small (technical naturalness) Prediction of a light dilaton: m 2 χ β ρ(v χ ) 4ξ(v χ ) v2 ew m χ m h ξ 39 / 36

104 Recovering the Standard Model at µ = v ew Consider the running of parameters between v ew and v χ Λ 40 / 36

105 Recovering the Standard Model at µ = v ew Consider the running of parameters between v ew and v χ Λ Require that m 2 χ(v ew ) > Λ/GeV m t /GeV 40 / 36

106 Recovering the Standard Model at µ = v ew Consider the running of parameters between v ew and v χ Λ Require that m 2 χ(v ew ) > Λ/GeV m t /GeV Dilaton mass at v χ Λ M P : m χ 10 8 ev 40 / 36

107 Recovering the Standard Model at µ = v ew Consider the running of parameters between v ew and v χ Λ Require that m 2 χ(v ew ) > Λ/GeV m t /GeV Dilaton mass at v χ Λ M P : m χ 10 8 ev Indicative only and requires higher-loop corrections 40 / 36

108 Electroweak and QCD phase transitions In the Standard Model, both electroweak and QCD PTs are crossover [K. Kajantie et al., Phys. Rev. Lett. 77 (1996) 2887] [Y. Aoki et al, Nature 443 (2006) 675] no stochastic GW background predicted in the SM 41 / 36

109 Electroweak and QCD phase transitions In the Standard Model, both electroweak and QCD PTs are crossover [K. Kajantie et al., Phys. Rev. Lett. 77 (1996) 2887] [Y. Aoki et al, Nature 443 (2006) 675] no stochastic GW background predicted in the SM In the model with hidden scale invariance: flat direction in the Higgs-dilaton potential at tree level vacua are almost degenerate no EWPT until T T EW 41 / 36

110 Electroweak and QCD phase transitions In the Standard Model, both electroweak and QCD PTs are crossover [K. Kajantie et al., Phys. Rev. Lett. 77 (1996) 2887] [Y. Aoki et al, Nature 443 (2006) 675] no stochastic GW background predicted in the SM In the model with hidden scale invariance: flat direction in the Higgs-dilaton potential at tree level vacua are almost degenerate no EWPT until T T EW QCD-induced electroweak phase transition: supercooling until T T QCD at T QCD : chiral phase transition with 6 massless quarks quark condensates reduce the barrier in the Higgs potential EWPT 41 / 36

111 Electroweak and QCD phase transitions In the Standard Model, both electroweak and QCD PTs are crossover [K. Kajantie et al., Phys. Rev. Lett. 77 (1996) 2887] [Y. Aoki et al, Nature 443 (2006) 675] no stochastic GW background predicted in the SM In the model with hidden scale invariance: flat direction in the Higgs-dilaton potential at tree level vacua are almost degenerate no EWPT until T T EW QCD-induced electroweak phase transition: supercooling until T T QCD at T QCD : chiral phase transition with 6 massless quarks quark condensates reduce the barrier in the Higgs potential EWPT See also: [E. Witten Nucl.Pys.B177 (1981) 477] [W. Buchmuller, D. Wyler, PLB 249 (1990) 281 ] [S. Iso et al., PRL 119 (2017) ] [B. von Harling, G. Servant, JHEP 1801 (2018) 159] 41 / 36

112 Thermal Higgs-dilaton potential + quark condensates Thermal contributions to the Higgs-dilaton potential barrier along the flat direction: V T (h, χ(h)) AT [ 4λ(Λ) + 6y 2 t (Λ) g2 (Λ) + 3 ] 2 g 2 (Λ) h 2 T / 36

113 Thermal Higgs-dilaton potential + quark condensates Thermal contributions to the Higgs-dilaton potential barrier along the flat direction: V T (h, χ(h)) AT [ 4λ(Λ) + 6y 2 t (Λ) g2 (Λ) + 3 ] 2 g 2 (Λ) h 2 T Quark-antiquark condensate with N massless quarks [J. Gasser, H. Leutwyler, PLB 184 (1987) 83] : qq T = qq [ T 2 1 (N 2 1) 12N fπ 2 1 ( T 2 ) 2 2 (N2 1) 12N fπ ] 42 / 36

114 Thermal Higgs-dilaton potential + quark condensates Thermal contributions to the Higgs-dilaton potential barrier along the flat direction: V T (h, χ(h)) AT [ 4λ(Λ) + 6y 2 t (Λ) g2 (Λ) + 3 ] 2 g 2 (Λ) h 2 T Quark-antiquark condensate with N massless quarks [J. Gasser, H. Leutwyler, PLB 184 (1987) 83] : qq T = qq [ T 2 1 (N 2 1) 12N fπ 2 1 ( T 2 ) 2 2 (N2 1) 12N fπ ] Quark-Higgs Yukawa interactions induce a linear term in the potential: V T (h) V T (h) + y q 2 qq T h 42 / 36

115 Thermal Higgs-dilaton potential + quark condensates Thermal contributions to the Higgs-dilaton potential barrier along the flat direction: V T (h, χ(h)) AT [ 4λ(Λ) + 6y 2 t (Λ) g2 (Λ) + 3 ] 2 g 2 (Λ) h 2 T Quark-antiquark condensate with N massless quarks [J. Gasser, H. Leutwyler, PLB 184 (1987) 83] : qq T = qq [ T 2 1 (N 2 1) 12N fπ 2 1 ( T 2 ) 2 2 (N2 1) 12N fπ ] Quark-Higgs Yukawa interactions induce a linear term in the potential: V T (h) V T (h) + y q 2 qq T h This linear term dominates over the barrier for small enough T 42 / 36

116 Results and dynamics of the transitions For N = 6 and f π 93 MeV, qq Tc = 0 at T c 132 MeV 43 / 36

117 Results and dynamics of the transitions For N = 6 and f π 93 MeV, hq qitc = 0 at Tc 132 MeV For T 127 MeV the barrier disappears and the EWPT completes V GeV T=132 MeV T=130 MeV T=128 MeV T=126 MeV h GeV 43 / 36

118 Results and dynamics of the transitions For N = 6 and f π 93 MeV, qq Tc = 0 at T c 132 MeV For T 127 MeV the barrier disappears and the EWPT completes The Higgs-dilaton rolls down the potential (smooth transition) 43 / 36

119 Results and dynamics of the transitions For N = 6 and f π 93 MeV, qq Tc = 0 at T c 132 MeV For T 127 MeV the barrier disappears and the EWPT completes The Higgs-dilaton rolls down the potential (smooth transition) However, SU(6) R SU(6) L chiral symmetry breaking is 1st-order for massless quarks [D. Pisarski, F. Wilczek, PRD 29 (1984) 338] 43 / 36

120 Results and dynamics of the transitions For N = 6 and f π 93 MeV, qq Tc = 0 at T c 132 MeV For T 127 MeV the barrier disappears and the EWPT completes The Higgs-dilaton rolls down the potential (smooth transition) However, SU(6) R SU(6) L chiral symmetry breaking is 1st-order for massless quarks [D. Pisarski, F. Wilczek, PRD 29 (1984) 338] Implicit assumption: chiral transition completes quickly 43 / 36

121 Results and dynamics of the transitions For N = 6 and f π 93 MeV, qq Tc = 0 at T c 132 MeV For T 127 MeV the barrier disappears and the EWPT completes The Higgs-dilaton rolls down the potential (smooth transition) However, SU(6) R SU(6) L chiral symmetry breaking is 1st-order for massless quarks [D. Pisarski, F. Wilczek, PRD 29 (1984) 338] Implicit assumption: chiral transition completes quickly More refined analysis currently under investigation: effective field theory for the Higgs, dilaton and pions U(6) U(6) linear sigma model for the pions ( ) [ L = Tr µ ϕ µ ϕ m 2 ϕ ϕ λ 1 Tr ( ϕ ϕ)] 2 λ2 Tr ( ϕ ϕ) 2 + L(ϕ, φ, χ) requires a proper treatment of infrared divergences 43 / 36

122 Gravitational Waves 1st order chiral transition stochastic background of GWs 44 / 36

123 Gravitational Waves 1st order chiral transition stochastic background of GWs Peak frequency roughly given by duration of transition (size of bubbles at collision): f p vr 1 c 44 / 36

124 Gravitational Waves 1st order chiral transition stochastic background of GWs Peak frequency roughly given by duration of transition (size of bubbles at collision): f p vr 1 c observed frequency today: f 0 = f p a(t c ) a(t 0 ) v R c H c T c 100 MeV Hz 10 7 Hz 44 / 36

125 Gravitational Waves 1st order chiral transition stochastic background of GWs Peak frequency roughly given by duration of transition (size of bubbles at collision): f p vr 1 c observed frequency today: f 0 = f p a(t c ) a(t 0 ) v R c H c T c 100 MeV Hz 10 7 Hz possible detection with Pulsar Timing Arrays (EPTA, NANOGrav, SKA) [C. Caprini et al., PRD 82 (2010) ] [A. Kobakhidze et al., EPJ C77 (2017) 570] 44 / 36

126 Gravitational Waves 1st order chiral transition stochastic background of GWs Peak frequency roughly given by duration of transition (size of bubbles at collision): f p vr 1 c observed frequency today: f 0 = f p a(t c ) a(t 0 ) v R c H c T c 100 MeV Hz 10 7 Hz possible detection with Pulsar Timing Arrays (EPTA, NANOGrav, SKA) [C. Caprini et al., PRD 82 (2010) ] [A. Kobakhidze et al., EPJ C77 (2017) 570] precise spectrum and amplitude of the background currently under computation (within linear sigma model) 44 / 36

127 Gravitational Waves [From rhcole.com/apps/gwplotter/] 44 / 36

128 Summary of Backup Scale invariant extensions of the SM motivated by the hierarchy problem 45 / 36

129 Summary of Backup Scale invariant extensions of the SM motivated by the hierarchy problem Low energy effective formulation with a dilaton field 45 / 36

130 Summary of Backup Scale invariant extensions of the SM motivated by the hierarchy problem Low energy effective formulation with a dilaton field Interesting predictions: small dilaton mass: m χ 10 8 ev low temperature QCD-induced electroweak transition potential GW signal in the range of Pulsar Timing Arrays 45 / 36

131 Summary of Backup Scale invariant extensions of the SM motivated by the hierarchy problem Low energy effective formulation with a dilaton field Interesting predictions: small dilaton mass: m χ 10 8 ev low temperature QCD-induced electroweak transition potential GW signal in the range of Pulsar Timing Arrays To investigate further: precise dynamics of the transitions Black Holes production 45 / 36