Can we get gravity waves from the nucleation of non-spherical bubbles? Peter Sloan (work done with Neil Barnaby and Lev Kofman) University of Toronto Cosmo '08 Madison, Wisconsin 28/08/08

GW from nucleation of bubbles2 No.

GW from nucleation of bubbles3 Introduction Early in its history, the universe may have undergone a number of phase transitions. First order phase transitions are particularly interesting: bubbles of new phase nucleate and grow until they collide with each other, eventually bringing the universe into the new phase. Single bubbles usually nucleate as spheres (O(3,1) symmetry): do not radiate gravitational waves. Collisions destroy the symmetry: good source of gravitational waves.

GW from nucleation of bubbles4 Bubble collisions Gravity waves from collisions were examined by Kosowsky, Turner and others in a series of papers in the 1990s: Phys. Rev. D 45, (1992) Phys. Rev. D 47, (1993) Phys. Rev. D 49, (1994) The GW peak frequency falls within the LISA band for the EWPT. We looked at another possible source for gravity waves in phase transitions. For this, we go back to the original paper on bubble nucleation and growth.

GW from nucleation of bubbles5 Non-Spherical Bubbles Coleman first studied bubble nucleation in vacuum in a famous '77 paper (PRD, Vol. 15, 2929) He showed that the nucleation rate of bubbles goes like where B is the 4-d Euclidian action of the scalar field The bubble is actually the analytic continuation of the Euclidian 'bounce' solution used to compute the nucleation rate. Coleman notes that non- spherical bounces are possible, but have a higher action (exponential suppression)

GW from nucleation of bubbles6 Non-Spherical Bubbles So we have the possibility of getting non-spherical bubbles, but the asymmetry must be very small or the bubbles will not be nucleated. Does this translate to tiny production of gravity waves? Nucleation is a quantum tunnelling event and happens nearly instantaneously. Energy released as gravity waves is strongly dependent on the timing. Quadrupole approx So we may be able to produce lots of gravity waves even with a small asymmetry.

GW from nucleation of bubbles7 Bounce Solution With a Ф 4 potential + small linear term (thin wall approx), we can solve for approximately a spherically symmetric bounce. At rho =0, infinity, Ф is in the metastable minimum and stable minimum, respectfully. In between, the solution is approximately Here ρ-bar is the radius of the bubble at t=0 (large in the thin wall approx).

GW from nucleation of bubbles8 Non-Spherical Bounce Look for a slightly non-spherical solution that satisfies the linearized Euclidian EOM and bounce boundary conditions and is localized at the bubble wall: Expand δΦ in 4-d hyperspherical harmonics

GW from nucleation of bubbles9 Within the thin wall approximation, we can solve for the non spherical bounce: The dimensionless constant 1/(μρ-bar) is a small parameter proportional to ε. The boundary conditions demand that the bubble be at rest at nucleation. In Euclidian space this translates as: This translates to the demand that b_nlm =0 when n-l is even.

GW from nucleation of bubbles10 Change in action: So the hyperspherical harmonics can have amplitude as large as (λ 1/2 μρ-bar) without significantly lowering the probability of nucleation.

GW from nucleation of bubbles11 So now we can try and calculate the energy amount of gravity waves produced from nucleation. This can be done analytically if we work to lowest order in 1/(μρ- bar), and work within the quadrupole approximation. A step function must be added in to account for the fact that there is no bubble before t=0. This gives us a delta function term in the quadrupole moment of T_00. This term is the effect from nucleation. A problem arises since we have a term d 3 Q_ij/dt 3 d 3 Q ij /dt 3 α δ(t)^2. So the delta function must be regularized. Quadrupole Approximation

GW from nucleation of bubbles12 We make a weighted average over coefficients b_nlm, weighted but their probability of nucleation (series of gaussian integrals) Final answer after some work The question arises: what exactly is Δt? Initially thought to be given by uncertainty relation --> large E_gw But actually somewhat ambiguous, tunnelling should be instantaneous. Maybe there's a better way to think about the problem.

GW from nucleation of bubbles13 An alternative approach The idea of a bubble forming over a time Δt is wrong. The bubble tunnels instantaneously. Any gravity waves produced then tunnel along with the bubble. The bounce profile rotated to Minkowski space serves as the initial bubble profile. Similarly, the Euclidian gravity wave profile from tunnelling serves as the initial conditions to the source-free gravity wave equations. So start with the solution to the Euclidian version of the wave equation (just the Poisson equation). We can avoid computing the transverse traceless components of T_uv at this point by working with the trace reversed metric.

GW from nucleation of bubbles14 We expand the T_uv in hyperspherical harmonics After some math... and rotating back to minkowski space, we can now take the TT part and compute the energy released in gravity waves. Unfortunately, when we try an Order of Magnitude estimate, we see that a significant amount of gravity waves are not produced.

GW from nucleation of bubbles15 So where can we go from here? Slightly non-spherical bubbles can be produced in the thin wall approximation. No appreciable gravity waves from nucleation, BUT Bubble shape may be unstable in certain scenarios Interaction of bubble wall with fluid outside may produce interesting changes in the shape of the bubble Other aspects of the phase transition can produce appreciable gravity waves (turbulence, etc)

GW from nucleation of bubbles16 Message: We should consider all sources of gravitational waves when looking at phase transitions