Gravitational waves from bubble collisions: An analytic derivation

We consider gravitational wave production by bubble collisions during a cosmological first-order phase transition. In the literature, such spectra have been estimated by simulating the bubble dynamics, under so-called thin-wall and envelope approximations in a flat background metric. However, we show that, within these assumptions, the gravitational wave spectrum can be estimated in an analytic way. Our estimation is based on the observation that the two-point correlator of the energy-momentum tensor $⟨T(x)T(y)⟩$ can be expressed analytically under these assumptions. Though the final expressions for the spectrum contain a few integrations that cannot be calculated explicitly, we can easily estimate it numerically. As a result, it is found that the most of the contributions to the spectrum come from single-bubble contribution to the correlator, and in addition the fall-off of the spectrum at high frequencies is found to be proportional to $f^{-1}$. We also provide fitting formulas for the spectrum.

Figure 1

Rough sketch of how the phase transition looks with the envelope and thin-wall approximations. Collided walls are neglected as a source of GWs, and every spacial point is passed by bubble walls only once.

Figure 2

Schematic picture of single- and double-bubble contributions to the correlator $⟨T(x)T(y)⟩$. The red line (the one w/o dashed lines) corresponds to the wall of one single bubble, while the blue line (the one w/ dashed lines) corresponds to intersecting two bubble walls. The dashed lines are neglected in the envelope approximation. Note that, though this figure shows $t_x=t_y$ case for simplicity, contributions from $t_x\ne t_y$ exist in the calculation of $⟨T(x)T(y)⟩$.

Figure 3

Notations for quantities on the past light cones of $x$ and $y$.

Figure 4

How the light cones in Fig. 3 look like in $2+1$ dimensions. The yellow circles represent the nucleation points for single-bubble (on the red central arrows) and double-bubble (on the blue separate arrows) contributions. The red line along the intersection of the two light cones shows $\delta V_{xy}$ in Fig. 3.

Figure 5

Notations for quantities on the constant-time hypersurface ${\mathrm{\Sigma }}_t$. The red diamond-shaped region denotes the one where a bubble nucleate in single-bubble spectrum (see Sec. 3c), while the outer blue regions between the dotted lines denote the ones where bubbles nucleate in double-bubble spectrum (see Sec. 3d).

Figure 6

Plot of the GW spectrum $\mathrm{\Delta }$ (blue). Single- and double-bubble spectra ${\mathrm{\Delta }}^{(s)}$ (red) and ${\mathrm{\Delta }}^{(d)}$ (yellow) are also plotted. Black lines are auxiliary ones proportional to $k^{-1}$ and $k^{-2}$, respectively.

Figure 7

Plot of the GW spectrum $\mathrm{\Delta }$ (blue) for $v=1$, 0.1 and 0.01 from top to bottom. Red and yellow lines correspond to single and double bubble spectrum ${\mathrm{\Delta }}^{(s)}$ and ${\mathrm{\Delta }}^{(d)}$, respectively.

Figure 8

Plot of the expansion coefficient $({\mathrm{\Delta }}^{(d)}/v^3{)}^{(1)}$ in Eq. (73) (blue) and an auxiliary line proportional to $k^{-1}$ (black). This figure shows that $({\mathrm{\Delta }}^{(d)}/v^3{)}^{(1)}$ scales as $k^{-1}$ for high frequencies.

Figure 9

Plot of the peak frequency $f_{\mathrm{peak}}/\beta$ as a function of the bubble wall velocity $v$. The blue line is numerically calculated from the analytic expression (71) and (72), while the red line corresponds to the fitting formula (75).

Figure 10

Plot of the GW amplitude at the peak ${\mathrm{\Delta }}_{\mathrm{peak}}$ scaled by $v^3$. The blue line is numerically calculated from the analytic expression (71) and (72), while the red line corresponds to the fitting formula (76).

Figure 11

Illustration of why the spherical symmetry of a single bubble do not undermine our argument. We set the evaluation time to be $t_x=t_y$ in this figure. We require the bubble walls fragments propagating to $\overrightarrow{x}$ and $\overrightarrow{y}$ to remain uncollided until the evaluation time, while other parts of this bubble can be already collided with others. This automatically takes into account the breaking of the spherical symmetry of a single bubble.