Gravitational Waves from the Electroweak Phase Transition A Semi-Analytic Calculation arxiv:0911.0687 John Kehayias University of California, Santa Cruz And Santa Cruz Institute of Particle Physics November 13, 2009 2009 California APS Meeting
This work was done in collaboration with Stefano Profumo, and has been submitted to JCAP. A preprint is available at arxiv:0911.0687.
Introduction It is possible for a strongly first order phase transition in the early universe to produce an observable gravitational wave (GW) spectrum. A first order transition proceeds by tunneling through a potential energy barrier to a lower energy (symmetry breaking) vacuum state. Once a bubble is nucleated, the bubble wall is accelerated and contains energy, rapidly expanding (near the speed of light). Collisions will break the spherical symmetry. Energy goes into accelerating the wall, reheating the plasma (universe), and even turbulent motions (in many cases more important than the collisions themselves) [1, 2]. 2009 California APS Meeting John Kehayias arxiv:0911.0687 3 / 10
Introduction It is possible for a strongly first order phase transition in the early universe to produce an observable gravitational wave (GW) spectrum. A first order transition proceeds by tunneling through a potential energy barrier to a lower energy (symmetry breaking) vacuum state. Once a bubble is nucleated, the bubble wall is accelerated and contains energy, rapidly expanding (near the speed of light). Collisions will break the spherical symmetry. Energy goes into accelerating the wall, reheating the plasma (universe), and even turbulent motions (in many cases more important than the collisions themselves) [1, 2]. 2009 California APS Meeting John Kehayias arxiv:0911.0687 3 / 10
Introduction It is possible for a strongly first order phase transition in the early universe to produce an observable gravitational wave (GW) spectrum. A first order transition proceeds by tunneling through a potential energy barrier to a lower energy (symmetry breaking) vacuum state. Once a bubble is nucleated, the bubble wall is accelerated and contains energy, rapidly expanding (near the speed of light). Collisions will break the spherical symmetry. Energy goes into accelerating the wall, reheating the plasma (universe), and even turbulent motions (in many cases more important than the collisions themselves) [1, 2]. 2009 California APS Meeting John Kehayias arxiv:0911.0687 3 / 10
Introduction It is possible for a strongly first order phase transition in the early universe to produce an observable gravitational wave (GW) spectrum. A first order transition proceeds by tunneling through a potential energy barrier to a lower energy (symmetry breaking) vacuum state. Once a bubble is nucleated, the bubble wall is accelerated and contains energy, rapidly expanding (near the speed of light). Collisions will break the spherical symmetry. Energy goes into accelerating the wall, reheating the plasma (universe), and even turbulent motions (in many cases more important than the collisions themselves) [1, 2]. 2009 California APS Meeting John Kehayias arxiv:0911.0687 3 / 10
Tunneling Temperature and GW Parameters The symmetry for finite temperature tunneling is O(3), and so the three-dimensional Euclidean action determines the tunneling rate [3, 4]. The tunneling temperature, T t, is defined as the temperature where the probability of nucleating a bubble in a horizon volume is O(1). Typically, T t is calculated numerically from the tunneling rate T 4 e S E3/T. The GW spectrum is parameterized by α, giving the energy change (strength) of the phase transition, and β, characterizing the bubble nucleation rate (the inverse is the duration ) and peak frequency. 2009 California APS Meeting John Kehayias arxiv:0911.0687 4 / 10
Tunneling Temperature and GW Parameters The symmetry for finite temperature tunneling is O(3), and so the three-dimensional Euclidean action determines the tunneling rate [3, 4]. The tunneling temperature, T t, is defined as the temperature where the probability of nucleating a bubble in a horizon volume is O(1). Typically, T t is calculated numerically from the tunneling rate T 4 e S E3/T. The GW spectrum is parameterized by α, giving the energy change (strength) of the phase transition, and β, characterizing the bubble nucleation rate (the inverse is the duration ) and peak frequency. 2009 California APS Meeting John Kehayias arxiv:0911.0687 4 / 10
Tunneling Temperature and GW Parameters The symmetry for finite temperature tunneling is O(3), and so the three-dimensional Euclidean action determines the tunneling rate [3, 4]. The tunneling temperature, T t, is defined as the temperature where the probability of nucleating a bubble in a horizon volume is O(1). Typically, T t is calculated numerically from the tunneling rate T 4 e S E3/T. The GW spectrum is parameterized by α, giving the energy change (strength) of the phase transition, and β, characterizing the bubble nucleation rate (the inverse is the duration ) and peak frequency. 2009 California APS Meeting John Kehayias arxiv:0911.0687 4 / 10
A Generic Effective Potential V eff (φ, T) = λ(t) 4 φ4 (ET e)φ 3 + D(T 2 T 2 0)φ 2 This potential is modeled after the form of the high temperature expansion of the SM Higgs potential. There is one additional term, motivated by gauge singlet models, e, which can enhance the phase transition strength by increasing the barrier. We used an approximation for S E3 for tunneling with general quartic potentials, found through numerical fitting [5]. We expand the action near T c to derive an expression for the tunneling temperature, and use that to evaluate the GW parameters. 2009 California APS Meeting John Kehayias arxiv:0911.0687 5 / 10
A Generic Effective Potential V eff (φ, T) = λ(t) 4 φ4 (ET e)φ 3 + D(T 2 T 2 0)φ 2 This potential is modeled after the form of the high temperature expansion of the SM Higgs potential. There is one additional term, motivated by gauge singlet models, e, which can enhance the phase transition strength by increasing the barrier. We used an approximation for S E3 for tunneling with general quartic potentials, found through numerical fitting [5]. We expand the action near T c to derive an expression for the tunneling temperature, and use that to evaluate the GW parameters. 2009 California APS Meeting John Kehayias arxiv:0911.0687 5 / 10
A Generic Effective Potential V eff (φ, T) = λ(t) 4 φ4 (ET e)φ 3 + D(T 2 T 2 0)φ 2 This potential is modeled after the form of the high temperature expansion of the SM Higgs potential. There is one additional term, motivated by gauge singlet models, e, which can enhance the phase transition strength by increasing the barrier. We used an approximation for S E3 for tunneling with general quartic potentials, found through numerical fitting [5]. We expand the action near T c to derive an expression for the tunneling temperature, and use that to evaluate the GW parameters. 2009 California APS Meeting John Kehayias arxiv:0911.0687 5 / 10
General Results Expanding at T = T c(1 ɛ), we find that S E3 /T 1/ɛ 2. ɛ is expressed in terms of all the parameters of the theory. The constraints of having the potential accurately describe electroweak symmetry breaking (proper Higgs vev, stable minimum, etc.) yield an expression for T 0 = T dest and expressions for λ in terms of the Higgs mass. While each parameter has an affect on the GW parameters, e has the largest effect, possibly greatly enhancing the phase transition strength [6]. This analysis is applicable to top-flavor and extra SU(2) triplet and singlet models, as well as higher dimensional operator effects, and other possible models. 2009 California APS Meeting John Kehayias arxiv:0911.0687 6 / 10
General Results Expanding at T = T c(1 ɛ), we find that S E3 /T 1/ɛ 2. ɛ is expressed in terms of all the parameters of the theory. The constraints of having the potential accurately describe electroweak symmetry breaking (proper Higgs vev, stable minimum, etc.) yield an expression for T 0 = T dest and expressions for λ in terms of the Higgs mass. While each parameter has an affect on the GW parameters, e has the largest effect, possibly greatly enhancing the phase transition strength [6]. This analysis is applicable to top-flavor and extra SU(2) triplet and singlet models, as well as higher dimensional operator effects, and other possible models. 2009 California APS Meeting John Kehayias arxiv:0911.0687 6 / 10
General Results Expanding at T = T c(1 ɛ), we find that S E3 /T 1/ɛ 2. ɛ is expressed in terms of all the parameters of the theory. The constraints of having the potential accurately describe electroweak symmetry breaking (proper Higgs vev, stable minimum, etc.) yield an expression for T 0 = T dest and expressions for λ in terms of the Higgs mass. While each parameter has an affect on the GW parameters, e has the largest effect, possibly greatly enhancing the phase transition strength [6]. This analysis is applicable to top-flavor and extra SU(2) triplet and singlet models, as well as higher dimensional operator effects, and other possible models. 2009 California APS Meeting John Kehayias arxiv:0911.0687 6 / 10
Some Plots 2009 California APS Meeting John Kehayias arxiv:0911.0687 7 / 10
And Some More 2009 California APS Meeting John Kehayias arxiv:0911.0687 8 / 10
Last One 2009 California APS Meeting John Kehayias arxiv:0911.0687 9 / 10
Summary Studying gravitational waves can provide a unique window to early universe physics, and highlights the interesting connection between particle physics and cosmology. We derived an approximate expression for the tunneling temperature, which is usually done numerically. Additionally, we have general expressions for the GW parameters. Finally, plots of the parameter space show where the GW spectrum may be observable, for any class of models that fit this generic potential. 2009 California APS Meeting John Kehayias arxiv:0911.0687 10 / 10
Selected References [1] R. Apreda, M. Maggiore, A. Nicolis, and A. Riotto, Gravitational waves from electroweak phase transitions, Nucl. Phys. B631 (2002) 342 368, arxiv:gr-qc/0107033. [2] A. Nicolis, Relic gravitational waves from colliding bubbles and cosmic turbulence, Class. Quant. Grav. 21 (2004) L27, arxiv:gr-qc/0303084. [3] A. D. Linde, Decay of the False Vacuum at Finite Temperature, Nucl. Phys. B216 (1983) 421. [4] A. D. Linde, On the Vacuum Instability and the Higgs Meson Mass, Phys. Lett. B70 (1977) 306. [5] F. C. Adams, General solutions for tunneling of scalar fields with quartic potentials, Phys. Rev. D48 (1993) 2800 2805, arxiv:hep-ph/9302321. [6] J. Kehayias and S. Profumo, Semi-Analytic Calculation of the Gravitational Wave Signal From the Electroweak Phase Transition for General Quartic Scalar Effective Potentials, arxiv:0911.0687 [hep-ph]. Please see the preprint for the full list.
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