Phase transition dynamics and gravitational waves Ariel M´ egevand Universidad Nacional de Mar del Plata Argentina XIII MEXICAN SCHOOL OF PARTICLES AND FIELDS October 2008 A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 1 / 31
Outline Outline Motivation First-order phase transitions Gravitational waves Phase transition dynamics Thermodynamics Bubble nucleation Bubble growth Gravitational waves from a first-order phase transition Turbulence in a first-order phase transition Gravitational waves from turbulence Phase transition dynamics and gravitational waves GWs from detonations and deflagrations Global treatment of deflagration bubbles Results A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 2 / 31
Motivation Motivation Phase transition dynamics Gravitational waves from a first-order phase transition Phase transition dynamics and gravitational waves A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 3 / 31
Motivation First-order phase transitions First-order phase transitions Phase transition dynamics ◮ supercooling ◮ nucleation and expansion of bubbles ◮ bubble collisions ◮ departure form equilibrium Possible consequences ◮ topological defects, magnetic fields ◮ baryogenesis, inhomogeneities ◮ cosmological constant ◮ gravitational waves (GWs) A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 4 / 31
Motivation Gravitational waves Gravitational waves from first-order phase transitions ◮ Since GWs propagate freely, they may provide a direct source of information about the early Universe. The spectrum ◮ The characteristic wavelength of the gravitational radiation is determined by the characteristic length of the source. ◮ The characteristic length is the size of bubbles, which depends on the phase transition dynamics and the Hubble length H − 1 . ◮ For the electroweak phase transition, the characteristic frequency, redshifted to today , is ∼ milli-Hertz. ◮ This is within the sensitivity range of the planned Laser Interferometer Space Antenna (LISA). A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 5 / 31
Phase transition dynamics Motivation Phase transition dynamics Gravitational waves from a first-order phase transition Phase transition dynamics and gravitational waves A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 6 / 31
Phase transition dynamics Thermodynamics Thermodynamics The free energy Thermodynamic quantities ( ρ , p , s ,...) are derived from the free energy density (finite-temperature effective potential). Example: A theory with a Higgs field and particle masses m i ( φ ) F ( φ, T ) = V 0 ( φ ) + V 1-loop ( φ, T ) , V 0 ( φ ) = tree-level potential V 1-loop ( φ, T ) = zero-temperature corrections + finite-temperature corrections A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 7 / 31
Phase transition dynamics Thermodynamics The effective potential F ( φ, T ) = V 0 ( φ ) + V 1-loop ( φ, T ) , where 2 λ v 2 φ 2 + 1 V 0 ( φ ) = − 1 4 λφ 4 V 1-loop ( φ ) = � ± g i � m 2 � � i ( φ ) � � � m 4 − 3 + 2 m 2 i ( φ ) m 2 i ( φ ) log i ( v ) 64 π 2 m 2 2 i ( v ) + � g i T 4 � � m i ( φ ) 2 π 2 I ∓ T 1 ∓ e − √ I ∓ ( x ) = ± � ∞ � y 2 + x 2 � dy y 2 log with 0 A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 8 / 31
Phase transition dynamics Thermodynamics First-order phase transition T > T c T = T c F ( φ, T ) T < T c High T : φ = 0 ( false vacuum ) Low T : φ = φ m ( T ) ( true vacuum ) T c = critical temperature φ Figure: The free energy F ( φ, T ) around the critical temperature A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 9 / 31
Phase transition dynamics Thermodynamics First-order phase transition Thermodynamic quantities are different in each phase T > T c : ⇒ F ( φ = 0 , T ) ≡ F + ( T ) ⇒ ρ + , s + , p + , . . . T < T c : ⇒ F ( φ m ( T ) , T ) ≡ F − ( T ) ⇒ ρ − , s − , p − , . . . High-temperature phase φ = 0 ◮ Energy density: ρ + ( T ) = ρ Λ + g ∗ π 2 T 4 / 30 = false vacuum + radiation Low-temperature phase φ = φ m ( T ) ◮ ρ − ( T ) depends on the effective potential A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 10 / 31
Phase transition dynamics Thermodynamics First-order phase transition Discontinuities at T = T c ◮ At the critical temperature, F + ( T c ) = F − ( T c ), but ρ + ( T c ) > ρ − ( T c ). ◮ L ≡ ρ + ( T c ) − ρ − ( T c ) = latent heat . The latent heat ◮ L is released during bubble expansion. ◮ Should not be confused with ρ Λ or ∆ F . A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 11 / 31
Phase transition dynamics Bubble nucleation Bubble nucleation ◮ During the adiabatic cooling of the Universe, the temperature T c is reached. ◮ The system is in the φ = 0 phase [i.e., φ ( x ) ≡ 0]. ◮ At T < T c bubbles of the stable phase T = T c [i.e., with φ = φ m inside] F ( φ, T ) begin to nucleate in the T < T c supercooled φ = 0 phase. T = T 0 ◮ At T = T 0 the barrier disappears. ( T 0 ∼ T c .) φ A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 12 / 31
Phase transition dynamics Bubble nucleation Bubble nucleation Nucleation rate Thermal tunneling probability per unit volume per unit time: Γ ≃ T 4 e − S 3 ( T ) / T S 3 ( T ) = three-dimensional instanton action = free energy of the critical bubble Γ is extremely sensitive to temperature: ◮ At T = T c , Γ = 0 ( S 3 = ∞ ) ◮ At T = T 0 , Γ ∼ T 4 ( S 3 = 0) ◮ Nucleation becomes important as soon as Γ ∼ H 4 , and ◮ H 4 ∼ ( T 2 / M Planck ) 4 ≪ T 4 . A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 13 / 31
Phase transition dynamics Bubble growth Bubble growth ◮ Once nucleated, bubbles expand until they fill all space. ◮ The velocity of bubble walls depends on several parameters. ◮ Pressure difference ∆ p = p − − p + Depends on supercooling. (At T = T c , p − = p + ). ◮ Friction of bubble wall with plasma Depends on microphysics (particles-Higgs interactions). ◮ Latent heat L = ρ + − ρ − injected into the plasma. Causes reheating and fluid motions. ◮ Hydrodynamics allows two propagation modes: detonations and deflagrations. A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 14 / 31
Phase transition dynamics Bubble growth Hydrodynamics Detonations ◮ The phase transition front (bubble wall) moves faster than the speed of sound: v w > c s . ◮ No signal precedes the wall. It is followed by a rarefaction wave. ◮ A bubble wall does not influence other bubbles, except in the collision regions Deflagrations ◮ The deflagration front is subsonic ( v w < c s ). ◮ The wall is preceded by a shock wave which moves at a velocity v sh ≈ c s . ◮ Thus, it will influence other bubbles. A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 15 / 31
GWs from a phase transition Motivation Phase transition dynamics Gravitational waves from a first-order phase transition Phase transition dynamics and gravitational waves A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 16 / 31
GWs from a phase transition Possible mechanisms Possible mechanisms Bubble collisions ◮ The walls of expanding bubbles provide thin energy concentrations that move rapidly. Turbulence ◮ In the early Universe, the Reynolds number is large enough to produce turbulence when energy is injected. Magnetohydrodynamics (turbulence in a magnetized plasma) ◮ It develops in an electrically conducting fluid, in the presence of magnetic fields. A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 17 / 31
GWs from a phase transition Turbulence in a first-order phase transition Cosmological turbulence Kolmogoroff-type turbulence ◮ Energy is injected by a stirring source at a length scale L S . ◮ Eddies of each size L break into smaller ones. ◮ When turbulence is fully developed, a cascade of energy is established from larger to smaller length scales. ◮ The cascade begins at the stirring scale L S and stops at the dissipation scale L D ≪ L S . ◮ Energy in the cascade is transmitted with a constant rate ε . ◮ For stationary turbulence, the dissipation rate ε equals the power that is injected by the source . A. M´ egevand (Mar del Plata, Argentina) Phase transition dynamics and GWs San Carlos 2008 18 / 31
Recommend
More recommend
