Imprints of Cosmic Phase Transition on Gravitational Waves (GWs) - PowerPoint PPT Presentation

Imprints of Cosmic Phase Transition on Gravitational Waves (GWs) Takeo Moroi (Tokyo) Refs: Jinno, TM and Nakayama, arXiv:1112.0084 [hep-ph]

1. Introduction

Early universe: environment with high-energy particles High temperature ∼ high energy How “deep” can we probe with various objects? • Last scattering of photon: ∼ 1 eV • Last scattering of neutrino: ∼ 1 MeV • “Last scattering” of GWs: inflation The history of our universe is imprinted in GWs ⇒ We may be able to extract information about high energy physics which cannot be probed by colliders ⇒ GW spectrum may be precisely measured in (far) future by, for e.g., DECIGO

Today’s subject: phase transition (or SSB) • QCD phase transition • EW symmetry breaking • Peccei-Quinn symmetry • GUT • · · · In models with SSB, cosmic phase transition may occur ⇒ Is there any effect on observables? ⇒ If yes, what can we learn?

The spectrum of GWs is affected by cosmic phase transition 1. Primordial GWs are produced during inflation (via quan- tum fluctuation) 2. Spectrum of GWs is deformed during the cosmic phase transition Outline 1. Introduction 2. Gravitational Waves: Production and Evolution 3. Phase Transition 4. Effects of Cosmic Phase Transition on GWs 5. Summary

2. GWs: Production and Evolution

Story: 1. Primordial GWs are produced during inflation (via quan- tum fluctuation) 2. Evolution of the amplitudes of GWs depends how the universe expands 3. Spectrum of GWs is deformed during the cosmic phase transition Gravitational wave: • Fluctuation of the metric (propagating mode) • Its evolution is governed by the Einstein equation

Metric: ds 2 = − dt 2 + a 2 ( t )( δ ij + 2 h ij ) dx i dx j ,j = 0 ) Physical mode: transverse and traceless ( h i i = h ij Fourier amplitude (using comoving wave-number � k ) d 3 � 1 k ij e i� ∫ (2 π ) 3 ˜ h ( λ ) k ( t ) ǫ ( λ ) k� x h ij ( t,� x ) = ∑ � M Pl λ =+ , × M Pl ≃ 2 . 4 × 10 18 GeV : Reduced Planck scale ǫ ( λ ) ij : polarization tensor (transverse & traceless) h ( λ ) ˜ k ( t ) : Canonically normalized � d 3 �   L ( flat ) = 1 k  1 ( λ ) ( λ ) ˙ k ˙ ∫ ∫ ˜ ˜ 2 M 2 d 3 xR + · · · ≃ ∑ h h k + · · · � − � Pl  (2 π ) 3 2 λ

Gravitational wave in de Sitter background: a ∝ e H inf t d 3 � ) 2   k  1 k − 1 ( k ( λ ) ( λ ) ˙ k ˙ ∫ ˜ ˜ h ( λ ) ˜ k ˜ h ( λ ) (2 π ) 3 a 3 ∑ L = h h � − � � − �  2 2 a k λ h ( λ ) ⇒ ˜ behaves as massless scalar field � k Quantum fluctuation generated during inflation ) 2  k 3   h ≡ 1 1 1 ( H inf � � | ˜ h ( λ )  × ∆ 2 k | 2 ∑ ∑ inflation ≃ � M 2 M 2 2 π 2 V 2 π λ λ Pl Pl The primordial GW amplitude is proportional to H inf ⇒ The effects of GWs become observable when the energy scale of the inflation is high

The tensor-to-scalar ratio r ≡ ∆ 2 h (= 16 ǫ ) ∆ 2 R R : curvature perturbation ( ∆ 2 R ≃ 2 . 42 × 10 − 9 ) ǫ : Slow-roll parameter • WMAP 7 years ⇒ r < 0 . 24 • PLANCK / Future CMB interferometric observations ⇒ r as small as 0 . 1 − 0 . 01 will be detected • Future experiments to detect GWs (DECIGO, · · · ) ⇒ GW spectrum will be observed if r > ∼ ∼ 10 − 3

GW evolution after inflation k 2 ( λ ) ( λ ) H = ˙ a ¨ + 3 H ˙ h ( λ ) ˜ ˜ ˜ h h + = 0 with � � k k � a 2 ( t ) k a Before the horizon-in: k ≪ aH ˜ h � k ∼ const. After the horizon-in: k ≫ aH k 2 k 2  + H k 2   2 k = d  1 k + 1 2 − 3 H ˙ k = ˙ ¨ ˙ ˙ ˜ ˜ ˜ ˜ k ˜ ˜ a 2 ˜ a 2 ˜ h 2 h 2 h h � h � k + h � h � h � � � � k k k a 2 ( t ) dt 2 2 � k 2 � k 2 � k 2 d ≃ − 4 ˙ a 2 � � � ˙ � � a 2 ˜ a 2 ˜ ˜ a 2 ˜ h 2 h 2 h 2 ⇒ ⇔ h ≃ � � � � k k k k dt a osc osc osc osc � ˜ h 2 osc ∼ a − 2 � ⇒ � k

Physical Wavelength Amplitude of GWs Scale Inflation RD ⇒ k > ∼ aH : ˜ h � k ∼ const. Horizon ⇒ k < ∼ aH : � ˜ h 2 k � osc ∼ a − 2 � Scale factor a − 2    a ( t ) � � � � h ( λ ) h ( λ ) | ˜ | ˜ k | 2 k | 2 osc ≃ inflation × � �  a | k = aH

∫ Energy density: ρ GW ( t ) ≡ d ln k ρ GW ( t ; k ) ) 2  k 3     1  1 ( λ ) ( λ ) k + 1 ( k ˙ k ˙ h ( λ ) h ( λ ) ˜ ˜ ˜ k ˜  × ∑ ρ GW ( t ; k ) = h h � − � � − �  2 π 2 k V 2 2 a λ ) 2 ∑  k 3   1 ( k � � h ( λ ) | ˜  × k | 2 ≃ � 2 π 2 V a osc λ ( H inf − 4 ) 2    a ( t ) M 2 Pl H 2 ≃  k = aH 2 πM Pl a | k = aH For modes which enter the horizon at the RD epoch: ( H inf ) 2 ρ GW ( t ; k ) ≃ ρ rad ( t ) 2 πM Pl

Present GW spectrum: GW = ρ ( tot ) GW ( t NOW ) ∫ Ω ( tot ) ≡ d ln k Ω GW ( k ) ρ crit In the case without phase transition (i.e., standard case): Ω ( SM ) GW ( k ) ≃ 1 . 7 × 10 − 15 r 0 . 1 γ : k EW ≪ k ≪ k RH . r 0 . 1 : the tensor-to-scalar ratio in units of 0 . 1 4 / 3 ( k ) n t      g ∗ ( T in ( k )) g ∗ s 0 γ =    g ∗ 0 g ∗ s ( T in ( k )) k 0 Ω ( SM ) GW ( k ) is insensitive to k

In future, GW spectrum may be measured ⇒ BBO / DECIGO Expected sensitivity

3. Phase Transition

The spectrum of GWs is affected by phase transitions ⇔ There may exist significant entropy production at the time of phase transition Model: two real scalar fields φ and χ V ( φ ) = g φ ) 2 + h 24( φ 2 − v 2 2 χ 2 φ 2 φ : scalar field responsible for symmetry breaking χ : degrees of freedom in thermal bath “Thermal mass” is generated for φ in the thermal bath ⇒ Cosmic phase transition occurs

Potential of φ surrounded by the thermal bath (at φ ∼ 0 ) V T ( φ ) = g φ ) 2 + h 24 T 2 φ 2 + · · · ≡ V 0 + h 24( φ 2 − v 2 24( T 2 − T 2 c ) φ 2 + · · · Critical temperature: temperature for V ′′ T ( φ = 0) = 0 � � 2 g � � T c = h v φ

Approximately, the phase transition occurs when V ′′ T ( φ = 0) = 0 ⇔ Tunneling rate is suppressed when g ≪ 1 Expectation value of φ :  0 : T > T c  � φ � = v φ : T < T c  Entropy is produced due to the phase transition • Temperature just before the phase transition: T c • Temperature just after the phase transition: T PT > T c ⇒ ρ rad ( T PT ) = ρ rad ( T c ) + V 0

Expansion rate at the phase transition: � � ρ rad ( T c ) + V 0 � � H PT ≡ � 3 M 2 Pl The mode which enters the horizon at the phase transition: k PT ≡ a ( t PT ) H PT ⇒ k < k PT : out-of-horizon at t = t PT Present frequency: f PT = k PT / 2 πa NOW T PT ( ) f PT ≃ 2 . 7 Hz × 10 8 GeV  g 1 / 4 v φ   ⇒ [ f PT ] g/h 2 ≪ 1 ≃ 0 . 50 Hz × 10 8 GeV 

Relevant equations to be solved (background): ( ˙ ) 2 a = ρ rad + V 0 θ ( t PT − t ) • H 2 = 3 M 2 a Pl • ˙ ρ rad + 4 Hρ rad = V 0 δ ( t − t PT ) t PT : time of phase transition (i.e., T = T c ) Effects of φ • Deviation from the radiation-domination at t ∼ t PT • Entropy production due to the phase transition

Evolution of the universe (with h = 1 ): H = 1 2 t in RD ρ rad ( T c ) ∼ O (1) × h 2 V 0 g ∗ g g ∗ : Effective number of massless degrees of freedom

4. Imprints of Phase Transition in GWs

Time k < k PT k > k PT k = k PT Behavior of GW amplitudes: • k ≤ k PT : No effect of phase transition • k ≥ k PT : Density is diluted due to the entropy production -1 H Scale Inflation After inflation (RD, ...) k / a = h t g n e l e v a W l a c i s y h P Phase Transition ⇒ Ω GW ( k > ∼ k PT ) is suppressed

“Short wavelength (i.e., high frequency)” mode: k ≥ k PT 1. The amplitude is constant until the horizon-reentry [ ρ GW ( k )] k = aH ≃ ρ GW ( t = 0) 2. ρ GW ( k ) ∝ a − 4 once the mode enters the horizon 4 ) 4   ( a horizon-in  a PT [ ρ GW ( k )] ( t ) ≃ [ ρ GW ( k )] k = aH  a PT a ( t ) − 4 ) − 4   ( T horizon-in  T PT ≃ [ ρ GW ( k )] k = aH  T c T ( t ) ( T c ) 4 ≃ [ ρ GW ( k )] no phase transition T PT

Ω GW ( k > ∼ k PT ) becomes suppressed ( T c ) 4 � R ≡ Ω GW ( k ) ρ rad ( T c ) � = = � � Ω ( SM ) T PT ρ rad ( T c ) + V 0 GW ( k ) � � k ≫ k PT Spectrum of GWs: result of numerical calculation k PT f PT = 2 πa NOW T PT ( ) ≃ 2 . 7 Hz 10 8 GeV

What can we learn from the GW spectrum? • Position of the drop-off ( ∼ f PT ) ⇒ “Reheating temperature” after the phase transition • Magnitude of the drop-off ( R ) ⇒ Entropy production • Slope of the drop-off ( ∼ d Ω GW /d ln k ) ⇒ Time scale of the reheating (instantaneous or ?) GWs from white dwarf binaries are significant for small- f ⇒ It will be difficult to extract the signal of cosmic phase transition in the GW spectrum if f PT < ∼ 0 . 1 Hz [Farmer & Phinney]