aa r X i v : . [ phy s i c s . g e n - ph ] J un Gravitational Waves Created During the EWPT
Leonard S. KisslingerDepartment of Physics, Carnegie Mellon University, Pittsburgh, PA 15213
Abstract
We study gravitational waves generated by bubble expansion created during the Cosmological Elec-troweak Phase Transition (EWPT). The energy radiated via gravitational waves is produced by thestress-energy tensor created by the magnetic field produced by bubble collisions during the EWPT,which occured about 10 −
11 seconds after the Big Bang.
PACS Indices:98.70. Vc, 98.80.-k, 98-62. En, 98.80. CqKeywords:Gravitational Waves,energy radiated, Electroweak Phase Transition
The present work is an extension of the estimate of gravitational waves created during the CosmologicalQuantum Chromodynamic Phase transition (QCDPT)[1] to an estimate of gravitational waves created duringthe Cosmological Electroweak Phase Transition (EWPT). It is also closely related to a recent publication ofgravitational radiation produced by pulsar creation[2].The QCDPT and EWPT are first order cosmological phase transitions so they have latent heat: thequark condensate for the QCDPT and the Higgs mass for the EWPT.For the QCDPT q ( x ) , ¯ q ( x ) are the quark, antiquark fields. | > is the vacuum state. < | ¯ q ( x ) q ( x ) | > = quark condensate < | ¯ q ( x ) q ( x ) | > = 0 in quark gluon plasma phase ≃ − ( . GeV ) in hadron phase . Therefore for the QCDPT the latent heat ≃ − ( . GeV ) .For the EWPT, with φ H the Higgs field, with T c ≃
125 GeV the critical temperature, < | φ H | > = 0 for T c ≥
125 GeV < | φ H | > = 125 GeV for T c ≤
125 GeV . Therefore for the EWPT the latent heat ≃
125 GeV= M H , the Higgs mass.There have been several articles published on gravitational radiation produced by Cosmological PhaseTransitions: the Electroweak Phase Transition (EWPT) at about 10 −
11 seconds after the Big Bang and theQuantum Chromodynamics Phase Transition (QCDPT) at about 10 − seconds.One of the first studies, which is a basis for the present work, was “Gravitational radiation from first-order phase transitions” by Kamionkowski, Kosowsky and Turner[3]. More recent studies were gravitationalradiation from primoidal turbulence[4], gravitationl radiation from cosmological phase ransition magneticfields[5], and polarization of such gravitational radiation[6].The present work also makes use of the stress-energy tensor produced by the magnetic wall during theEWPT[7]. In order to estimate the energy radiated by gravitational waves during the EWPT one needsthe nucleon mass M n in units of f m − , B W , the magnitude of the magnetic field at the bubble wall duringthe EWPT, and t EW P T = 10 − seconds after the Big Bang, the time of the EWPT. The values of theseparameters used in the present work are taken from estimates in Ref[7].The only experimental detection of gravitational waves was the observation of gravitational waves frombinary black hole mergers Ref[8]. Gravity waves from black hole mergers had been predicted[9, 10] and werein agreement within experimental and theoretical errors with Ref[8].1 Electroweak Phase Transition (EWPT)
The Standard EW Model has fields with quanta:Fermions (quantum spin 1/2 particles) are e − , ν e , the µ and τ leptons, the quarks q u , q d and two otherquark generations. The EW gauge bosons (quantum spin 1) are W + , W − , Z o and photon γ Since the EWPT is a first-order phase transition there is latent heat. The latent heat for the EWPT isthe Higgs boson (quantum spin 0) mass, M H =. 125 GeV. At the LHC[11] is was found that M H ≃ e− ν e W − ν ee νν ee+ W+ ZoW W − Zo+ud ud uu Lepton weak interaction conserves CP−No BaryogenesisQuark weak interaction violates CP−Baryogenesis Possible
Baryogenesis requires a first order EWPT
During the EWPT bubbles form and magnetic fields were created via bubble collisions, as shown in thefigure below.
B field zr
From Ref[5], B W ≃ . × Gauss. Therefore B EP P TW ≃ . × × B QCDP TW .2 Gravitational Radiation From Magnetic Fields Generated bythe EWPT
The energy radiated by gravitational waves with frequency interval dω and solid angle d Ω is[3] dEdωd
Ω = 2 Gω Λ ij,lm (ˆ k ) T ∗ ij ( k, ω ) T lm ( k, ω ) , (1)Λ ij,lm (ˆ k ) = δ il δ jm − δ ij δ lm / δ ij ˆ k l ˆ k m / δ lm ˆ k i ˆ k j / − δ il ˆ k j ˆ k m + ˆ k i ˆ k j ˆ k l ˆ k m / . (2)From Eq(20) in Ref[7], which makes use of Eq(1), with k ≃ k/ √ T ij ( k, ω ) = δ i δ j π √ πM n B W e k / M − n δ ( t − t EW P T ) , (3)where t EW P T is the time of the EWPT, with t EW P T ≃ − seconds (4) B W ≃ . × Gauss M − N ≃ . . From Eqs(1,2,3) the energy radiated by gravitational waves with frequency interval dω , eliminating thesolid angle as in Ref[3], using k ≃ k/ √ dEdω = 8 Gω π B W M N e − k / M − n [ 12 − k k + k k ] . (5)since k ≃ k/ √ k /k ≃ / √
3, from Eq(5) one finds dEdω ≃ Gω π B W M N e − k M N . (6)From Refs([5],[3],[12]) k ≃ π/λ o ≃ π/ (10 f m ) . (7)Therefore, using M − N ≃ . e − k / M N ≃ e = 1 . / ; , (8)from Eq(6) dEdω = 8 Gω π B W M N . (9)Using (with s=second) G ( Gauss) = 8 . − s cm (10)cm = 10 fm , (11)and Eq(4) one obtains our final equation for the energy radiated by gravitational waves during the QCDPT dEdω = 3 . ω s . (12)3 review of Gravitational Wave Physics[13] discusses the generation of gravitational waves, the firstdetection of Gravitational waves by the LIGO detector[14], and many other aspects of Gravitational WavePhysics. An aspect of Ref[13] important for the present work is that the units for Gravitational radiationare explained.These are used for dEdω shown in Figure 1, with s = second. ω d E d ω ( e r g ss ) Figure 1: Gravitational radiation energy produced during the QCDPT as a function of frequency ω .4 Conclusions
We have estimated the gravitational radiation energy produced by gravitational waves during the EWPT,which occured at a time t EW P T ≃ − seconds after the Big Bang. This estimate is based on methodsfound in Ref[3], but the estimate is of gravitational radiation energy produced during a specific CosmologicalPhase Transition, the Electroweak Phase Transition, rather than a more general study of Gravitationalradiation from first-order phase transitions[3].From Figure 1 the Gravitational radiation energy as a function of ω for the most likely values of ω/s are approximately 2 × to 8 × ergs/s. This is approximately 4.3 × larger than the Gravitationalradiation energy produced during the QCDPT[1].Note that gravitational waves from binary black hole mergers were detected in 2016[8]. From the resultsshown in Figure 1, with Gravitational radiation energy ≃ × ergs/s the current gravitational wavedetectors, such as LIGO[14], can detect gravitational waves produced during the EWPT. We hope that suchLIGO experiments are carried out soon. Acknowledgements
The author Leonard S. Kisslinger was a visitor at Los Alamos National Laboratory, Group P25.
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