Gravitational waves from thermal phase transitions, from the bottom - PowerPoint PPT Presentation

Further reading The only nonperturbative calculation: Moore and Rummukainen Basic idea: Langer Nucleation rates and the phase transition duration: (see also Kapusta ) Enqvist, Ignatius, Kajantie and Rummukainen 13

Nucleation basics When the universe drops below the critical temperature, broken phase is the new global minimum. Quantum (or thermal) fluctuations will excite the field over the potential barrier to the new minimum. Consider a single scalar field with Lagrangian ϕ 1 2 ∂ μ ∂ μ  = ϕ ϕ − U ( ϕ ) and equation of motion ∂ 2 ϕ ∇ 2 U ′ + ϕ = ( ϕ ). ∂ t 2 14 . 1

More nucleation basics Want to calculate probability for field to tunnel from false vacuum ϕ to true vacuum . ϕ + ϕ − Like calculating a tunnelling amplitude in quantum mechanics. Solve for trajectory that 'bounces' from to and back again in a ϕ + ϕ − localised region This will give the exponential factor in the nucleation probability. 14 . 2

Computing the bounce At , system has invariance, so change variables T = 0 O(4) to : t 2 x 2 ‾ ‾‾‾‾‾ ‾ ρ = + √ d 2 ϕ 3 d ϕ U ′ + = ( ϕ ). d ρ 2 ρ d ρ with boundary conditions lim ϕ ( ρ ) = ϕ + ρ → ∞ ∣ ∂ ϕ = 0 ∣ ∂ ρ ∣ ρ =0 Solve this by shooting, and then compute the action for S 4 this path. 14 . 3

Fluctuations and finite temperature Add a prefactor given by the contribution of fluctuations about the minimum and also the bounce path. ϕ − However, we are interested in the finite- version of this T calculation, in which case the symmetry is . O(3) We can then use dimensional analysis to guess the prefactor: Γ S 3 ( T ) T 4 ≈ exp ( − ) V T with 2 1 d ϕ r 2 S 3 = 4 π ∫ d r + V e ff ( ϕ , T ) ] . 2 ( ) [ dr 14 . 4

Nucleation rates The full (finite-T) expression is − 1/2 t ′ ∇ 2 V ″ ϕ − 3/2 de [ − + ( , T ) ] Γ ω − S 3 = π ( ) ∇ 2 V ″ ϕ + V 2 π T det [ − + ( , T ) ] [ ] S 3 ( T ) × exp ( − ) . T The above expression is very similar to that for the sphaleron rate - the two processes have much in common 15 . 1

Limiting cases for S 3 As discussed above, solve for bounce profile by shooting. Identify two limiting cases: For small supercooling ( ), bubbles are T c − T N ≪ T c thin-wall type (with walls). tanh For large supercooling, bubbles are close to a Gaussian. 15 . 2

Beyond S 3 Using as the exponential parameter in the S 3 ( T )/ T nucleation rate is a high-temperature approximation. One can also compute the nucleation rate nonperturbatively, both the prefactor and the exponential part. Results suggest that (for SM): True supercooling lies between 1- and 2-loop results 2-loop perturbative surface tension close to true result Unfortunately, nucleation rate only studied at one point in the dimensionally reduced SM theory - so generally still follow the usual anaysis 15 . 3

Making use of Γ The nucleation rate gives the probability of nucleating a Γ bubble per unit volume per unit time. More useful for cosmology is to consider the inverse duration of the phase transition, defined as ˙ dS ( t ) ∣ Γ β ≡ − ∣ ≈ ∣ dt Γ t = t ∗ The phase transition completes when the probability of nucleating one bubble per horizon volume is of order 1 T ∗ S 3 T ∗ T ∗ ( )/ ∼ − 4 log ≈ 100 m Pl 15 . 4

Making further use of Γ Using the adiabaticity of the expansion of the universe the time-temperature relation is dT = − TH dt This gives, for the ratio of the inverse phase transition duration relative to the Hubble rate, β dS ∣ d S 3 ( T ) ∣ = T ∗ ∣ = T ∗ ∣ ∣ ∣ H ∗ dT dT T T = T ∗ T = T ∗ If then the phase transition won't complete... β ≲ 1 H ∗ 15 . 5

Nucleation - conclusion Nucleation rate per unit volume per unit time computed Γ from bounce actions S ( T ) = min{ S 3 ( T )/ T , S 4 ( T )} Inverse duration relative to Hubble rate computed from β H ∗ , and controls GW signal Γ To get : β 1. Find effective potential V e ff ( ϕ , T ) 2. Compute (or ) for extremal bubble by S 3 ( T )/ T S 4 ( T ) solving 'equation of motion' 3. Determine transition temperature T ∗ 4. Evaluate at β / H T ∗ Use as input to the GW power spectrum. β / H ∗ 16

3: Wall velocities 17

Motivation Wall velocity connects the electroweak phase transition to the two big unknowns: Baryogenesis (rate of baryon asymmetry production) Gravitational waves ( dependence) v 3 [Almost] at the bottom of a hierarchy of abstraction: wall Can derive friction term for higher-level simulations Check how valid using a single scalar field and ideal fluid really is. 18

Further reading Prokopec and Moore: hep-ph/9503296 and . hep-ph/9506475 Konstandin, Nardini and Rues: . arXiv:1407.3132 Kozaczuk: . arXiv:1506.04741 19

What happens at the bubble wall? Forces in equilibrium: Inside, , latent heat released. ⟨ ϕ ⟩ ≠ 0  = Δ V ( T ) Outside, , friction from everything coupling to . ⟨ ϕ ⟩ = 0 ϕ When there is no net force, wall stops accelerating. Is there a finite below for which this happens? v wall c (Vacuum case: no force on wall - nothing to stop it accelerating to ) c 20 . 1

Free body diagram What does the friction term look like? 20 . 2

Expand Higgs field about classical profile Φ ( x , t ) → Φ cl ( x , t ) + δ Φ ( x , t ) and follow behaviour of . Φ cl In the Standard Model, equation of motion is Φ † ∂ μ ∂ μ Φ cl − μ Φ cl + 2 λ ( cl Φ cl Φ cl ) g 2 ⎯ ⎯ ⎯⎯ Φ 2 Φ † Φ † 4 A 2 + 2 λ ( 2 ⟨ δ δ Φ ⟩ Φ cl + ⟨ δ ⟩ ) − ⟨ ⟩ + ∑ y ⟨ ψ R ϕ L ⟩ = 0 cl Top line - classical bits; bottom line - fluctuations How to treat the fluctuations? Consider one component from ... ϕ Φ = (0, ϕ / 2 ‾ ) √ 20 . 3

Field is slowly varying compared to reciprocal momenta ϕ of particles in plasma ( ) ∝ T ⇒ treat in WKB Write phase space density as f ( k , x ) Separate into equilibrium and nonequilibrium parts, f ( k , x ) → f ( k , x ) + δ f ( k , x ) due to equilibrium thermal fluctuations; absorbed f ( k , x ) into 'finite-temperature effective potential' for Φ cl is the departure from that equilibrium δ f ( k , x ) 20 . 4

Equation of motion is (schematically) d 3 dm 2 k ∂ μ ∂ μ i V ′ ϕ + ( ϕ , T ) + δ ( k , x ) = 0 f i e ff ) 3 E i ∑ ∫ d ϕ (2 π 2 i : gradient of finite- effective potential V ′ ( ϕ ) T e ff : deviation from equilibrium phase space density f i ( k , x ) of th species i : effective mass of th species: m i i Leptons: m 2 y 2 ϕ 2 = /2 Gauge bosons: m 2 g 2 w ϕ 2 = /4 Also Higgs and pseudo-Goldstone modes 20 . 5

After some algebra: Force on particles Force on ϕ      d 3 ⏞ k ∂ μ T μν F ν − f ( k ) = 0 (2 π ) 3 ∫ This equation is the realisation of this idea: 20 . 6

Another interpretation: Fluid part Field part   d 3    ⏞ k ∂ μ T μν F ν − f ( k ) = 0 (2 π ) 3 ∫ i.e.: ∂ μ T μν ∂ μ T μν + = 0 ϕ fl uid We will return to this later! 20 . 7

Layers of abstraction 21 . 1

Layers of abstraction We have so far been using field theory equations of motion. Less tricky, but more abstract, are: Boltzmann equations Hydrodynamic equations In particular, the hydrodynamic equations we get are a valuable motivation for the rest of today's lectures We will now look at how to arrive at these higher-level approximations 21 . 2

Boltzmann equations: a reminder What is a Boltzmann equation? Phase space is positions and momenta . x k Tells us how our distribution functions evolve. f i ( x , k ) Consists of four parts: Time evolution ∂ t f i ( x , k ). Streaming terms in momentum and position space ˙ x ˙ ⋅ ∇ x f + p ⋅ ∇ p f Collision C [ f ] 22 . 1

Boltzmann equation for distribution f The Boltzmann equation is d f ∂ f ˙ = + x ˙ ⋅ ∇ x f + p ⋅ ∇ p f = − C [ f ]. d t ∂ t This is a semiclassical approximation to the quantum Liouville equations for all the fields Only valid when the momenta of the fields is much higher than the inverse wall thickness: 1 p ≳ gT ≫ . L w Very difficult to work with directly, so model the distribution of each particle with a 'fluid' ansatz. f i 22 . 2

Fluid approximation As mentioned, fluid approximation sets the scene for the rest of these lectures on the electroweak phase transition In short, we have d 3 k T fl uid = k μ k ν f i ( k ) = w u μ u ν − g μν p μν (2 π ) 3 E i ∑ ∫ i but we will try to justify this. 23 . 1

Deriving the fluid approximation The flow ansatz is 1 1 f i ( k , x ) = = u μ e X e β ( x )( ( x ) k μ + μ ( x )) ± 1 ± 1 with four-velocity , chemical potential and u μ ( x ) μ ( x ) inverse temperature . β ( x ) Substituting this ansatz into the Boltzmann equations for the system yields (after much algebra!) a (relativistic) Euler momentum equation u μ ∂ μ u ν + ∂ ν p = C . 23 . 2

The field-fluid model Energy conservation requires that ∂ μ T μν T μν ∂ μ T μν = ( + ) = 0. ϕ fl uid We are now ready to present the full model: ∂ V e ff ( ϕ , T ) ∂ μ ∂ μ ∂ ν ∂ ν v w u μ ∂ μ ∂ ν ( ϕ ) ϕ − ϕ = − η ( ϕ , ) ϕ ϕ ∂ ϕ ∂ V e ff ( ϕ , T ) u μ u ν ∂ ν ∂ ν v w u μ ∂ μ ∂ ν ∂ μ ( w ) − p + ϕ = + η ( ϕ , ) ϕ ϕ ∂ ϕ Besides the (dimensionful) definition here, one choice for η η ̃ ϕ 2 that is well motivated is . T This model is the basis of spherical and 3D simulations. One can also obtain steady-state equations. 24 . 1

The field-fluid model: observations Consider the fluid equation: ∂ V e ff ( ϕ , T ) u μ u ν ∂ ν ∂ ν v w u μ ∂ μ ∂ ν ∂ μ ( w ) − p + ϕ = η ( ϕ , ) ϕ ϕ ∂ ϕ Away from the bubble wall, the right hand side goes to zero. The left hand side has no length scale. Therefore any fluid solution must be parametrised by a dimensionless ratio, e.g. radius of the bubble to time since nucleation - define . ξ = r / t Fluid profiles will scale with the bubble radius: they are large, extended objects! 24 . 2

Runaway walls? We have assumed that the wall reaches a terminal velocity (less than ). c But what if it doesn't? Termed a 'runaway wall'. Consequences would include: Less interaction with plasma Lower amplitude of GWs Runaway walls are currently a hot topic - with a recent paper suggesting that they may not exist (due to subleading corrections arising from the treatment of gauge bosons) 25

Wall velocities: conclusion Detailed studies have been carried out of the wall velocity, using thermal field theory techniques. Higher level calculations and simulations use an effective field-fluid model, with the wall velocity as an input parameter. The damping term for field-fluid models (and hence the wall velocity) is generally obtained by a qualitative matching to the Boltzmann equations. 26

4: Thermodynamics 27

Motivation In the previous section we described the various layers of approximation up to the field-fluid model. Now we will use that field-fluid model (and steady-state results) to explore the macroscopic behaviour of the wall. This is important both for baryogenesis and also for the GW power spectrum. 28

Further reading Energy budget: Espinosa, Konstandin, No and Servant arXiv:1004.4187 29

Combustion physics Source: (public domain) Wikimedia Commons 30 . 1

Reaction front At a reaction front, there is a chemical transformation. The fluid is chemically and physically distinct on both sides. Different from a shock front, where the energy density and entropy change. We have a reaction front as before and after ⟨ ϕ ⟩ = 0 ⟨ ϕ ⟩ ≠ 0 30 . 2

Detonations vs deflagrations If the scalar field wall moves supersonically and the fluid enters the wall at rest, we have a detonation If the scalar field wall moves subsonically and the fluid enters the wall at its maximum velocity, we have a deflagration Can also get a hybrid where the wall moves supersonically but some fluid bunches up in front of it, like a deflagration 30 . 3

Fluid profile equation As mentioned before, away from the bubble wall, there is no length scale in the fluid equations. Therefore expect that fluid profile around a spherical bubble will scale as : radius/time ξ = r / t Rearrange Euler equation to remove diffusion, using ‾ ‾‾‾‾‾‾‾‾‾‾‾‾ ‾ c s = (d p / dT )(d ϵ / dT ) √ Then, if we know the fluid velocity we can solve μ 2 v ∂ v γ 2 2 = (1 − v ξ ) [ − 1 ] c 2 ξ ∂ ξ s with the Lorentz-boosted fluid velocity μ ( ξ , v ) = ( ξ − v )/(1 − ξ v ). 30 . 4

Fluid profiles Source: Espinosa, Konstandin, No and Servant 30 . 5

Phase transition strength The story so far: 1. Bubbles nucleate (parameter ) β 2. Bubbles expand with finite velocity ( ) v w 3. Extensive fluid shell around bubble 4. Latent heat turned into fluid KE???  Object of this section is to quantify how much of the latent heat ends up as kinetic energy. Define phase transition strength  (  ) latent heat at T α T = = π 2 T 4 g ( T ) /30 radiation energy at T which tell us how much of the energy of the universe was stored as latent heat in the phase transition. 31 . 1

Computing the efficiency Larger ⇒ stronger phase transition α T But it does not tell us how much of ends up as fluid  kinetic energy For that we define the efficiency wu i u i fl uid KE κ f = =  latent heat Then is the fraction of the energy density in the κ f α T universe that ends up as fluid kinetic energy at the transition. ⎯⎯⎯ 2 Very roughly, ⎯ ⎯ , the Lorentz-boosted mean square κ f α T ≈ U f fluid velocity as the transition completes. Can be computed more accurately either from spherical simulations or directly solving. 31 . 2

Efficiency curves Source: Espinosa, Konstandin, No and Servant 31 . 3

An aside: scalar field efficiency One can also define σ S scalar fi eld gradients κ ϕ = =  V latent heat Note that because this scales as , the surface area over S / V the volume, this is suppressed by the inverse bubble radius. Hence for realistic thermal phase transitions, is small. κ ϕ 31 . 4

Thermodynamics: conclusion Thermal first-order transitions have a reaction front Reaction fronts can be deflagrations (generally subsonic), detonations (supersonic) or hybrids (a mixture). The fluid reaches a scaling profile in based on the ξ = r / t available latent heat and wall velocity. From this, one can compute the efficiency and hence κ f how much of the energy in the universe ends up in the fluid . κ f α T 32 . 1

Recap What parameters have we introduced? EWPT introduction: latent heat Nucleation: inverse duration β Wall velocities: v w Thermodynamics: and α T κ That more or less summarises what we need to know about the physics of the phase transition, so we can now talk about the production of GWs. 32 . 2

5: Two approximations 33

Motivation In this section we will briefly look at two widely-used but simple approximations. First, the quadrupole approximation makes a reappearance. We will see why (a version of) the quadrupole formula is a bad approximation for bubbles The next approximation is the envelope approximation This was widely used until recently for studying bubble collisions. It is still important for vacuum transitions where the scalar field walls are all that matters (and can κ ϕ dominate) 34

Further reading "Weinberg formula" Weinberg Early quadrupole and envelope calculations [ ] [ ] Kamionkowski and Kosowsky and Turner and Watkins Later envelope approximation results Huber and Konstandin Recent developments Jinno and Takimoto 35

Preliminaries Starting point is the Weinberg formula dE GW ω 2 Λ ij , lm k ̂ T ∗ ij k ̂ T lm k ̂ = 2 G ( ) ( , ω ) ( , ω ) d ω d Ω with 1 k ̂ T ij e i ω t d 3 e − i ω ⋅ x T ij k ̂ ( , ω ) = ∫ d t x ( x, t) ∫ 2 π and 1 P ij k ̂ P lm k ̂ 2 P ij k ̂ P lm k ̂ Λ ij , lm ≡ ( ) ( ) − ( ) ( ) where P ij k ̂ k ̂ i k ̂ ( ) = δ ij − j 36 . 1

Quadrupole approximation Consider a pair of vacuum scalar bubbles along the -axis z In integral for take , such that k ̂ T ij ⋅ x → 0 1 T Q e i ω t d 3 T ij T ij k ̂ ( , ω ) → ( ω ) ≡ d t x ( x , t ) ij ∫ 2 π Using cylindrical symmetry... T Q T Q T Q T Q ( ω ) = ( ω ) + ( ω ) + ( ω ) = D ( ω ) δ ij + Δ ( ω ) δ iz δ jz zz xx yy ij where only sources gravitational waves. Δ ( ω ) 37 . 1

Quadrupole approximation: result Now note that 2 1 1 2 2 sin 4 k ̂ Λ ij , lm δ iz δ jz δ lz δ mz = Λ zz , zz = 2 ( 1 − = θ z ) So, in the quadrupole approximation d E ω 2 | Δ ( ω ) | 2 sin 4 = G θ d ω d Ω Here can encode details of the Δ ( ω ) bubble walls interacting, and can be found numerically. 37 . 2

O(2,1) simulation Kosowsky, Turner and Watkins 1992 37 . 3

Limitations of the quadrupole approx. Kosowsky, Turner and Watkins 1992 Quadrupole approximation is an overestimate! Unfortunately at higher wavenumbers , ω the higher multipoles dominate Only considered a pair of bubbles! In reality, many bubbles, less symmetry, bubble walls probably microscopic Motivates envelope approximation... 37 . 4

Quadrupole vs full linearised GR Kosowsky, Turner and Watkins, 1992 37 . 5

More about these early simulations: There is a nasty cutoff accounting for the symmetry O(2, 1) Spacetime with symmetry is isomorphic to an O(2, 1) O(3) pseudo-Schwarzschild-de Sitter spacetime Petrov type D - no GWs 37 . 6

Envelope approximation 38 . 1

Envelope approximation Kosowsky, Turner and Watkins Kamionkowski, Kosowsky and Turner ; Thin, hollow bubbles, no fluid Stress-energy tensor on wall ∝ R 3 Solid angle: overlapping bubbles → GWs How is the envelope approximation implemented? 38 . 2

Envelope approximation: derivation The stress energy tensor of the system can be turned T ij ( x , t ) into a sum of uncollided areas of each of the bubbles: S n n 1 k ̂ x n x ̂ T ij , n e i ω t r 2 e − i ω ⋅ ( + r ) T ij ( k , ω ) = ∫ d t d Ω ∫ d r ( r , t ) ∑ n ∫ S n 2 π and then if we assume the walls are thin k ̂ x n x ̂ T ij , n r 2 e − i ω ⋅ ( + r ) 4 π ∫ d r ( r , t ) 4 π k ̂ x n R n x ̂ x ̂ 3 e − i ω ⋅ ( + ( t ) ) j R n ) 3 κρ vac i x ̂ ≈ ( t . ⏟ i.e. σ 38 . 3

Envelope approximation: implementation With the approximation listed above, we get a double oscillatory integral: ρ vac v 3 T ij k ̂ w C ij k ̂ ( , ω ) = κ ( , ω ) 1 k ̂ x n e i ω ( t − ⋅ ) t n ) 3 A n , ij k ̂ C ij k ̂ ( , ω ) = ∫ d t ( t − ( , ω ) 6 π ∑ n t n k ̂ x ̂ x ̂ e − i ω v w ( t − ) ⋅ A n , ij k ̂ i x ̂ ( , ω ) = d Ω j ∫ S n Then evaluate these time-domain Fourier transforms numerically Integrate over uncollided areas at each timestep. S n Note that all , i.e. full result k ̂ ⋅ x ≠ 0 38 . 4

Envelope approximation: implementation Huber and Konstandin 2008 38 . 5

Envelope approximation: results Plot from Huber and Konstandin 2008 . Wall velocities top to bottom . v w = {1, 0.1, 0.01} Total power scales as . v 3 w Peak at . ω / β ≈ 1 Power laws on both sides of peak. 38 . 6

Envelope approximation: results Simple power spectrum: One length scale (average radius ) R ∗ Two power laws ( , ) ω 3 ∼ ω − 1 Amplitude ⇒ 4 numbers define spectral form NB: Used to be applied to shock waves (fluid KE), now only use for bubble wall (field gradient energy) 38 . 7

Envelope approximation 4-5 numbers parametrise the transition: , vacuum energy fraction α T ∗ , bubble wall speed v w , conversion 'efficiency' into gradient energy ( ∇ ϕ ) 2 κ ϕ Transition rate: , Hubble rate at transition H ∗ , bubble nucleation rate β [only matters for vacuum/runaway transitions] 38 . 8

Envelope approximation: comparison with full scalar field simulations 38 . 9

Envelope approximation: comparison with fluid source 38 . 10