Phase-Transition Gravitational Waves from Bubble Collisions

What gravitational-wave spectrum would a cosmic phase transition leave behind? A first-order transition would fill the universe with colliding bubbles whose sound waves stir space-time. Today that would be a broad background, peaking near a millihertz for transitions at a few hundred GeV. Its height depends on how long the sound lasts before shocks or expansion stop it.

Bubbles of a new phase grow, collide and set the plasma ringing. Set the transition speed and wall speed, then read the spectrum today against LISA noise.

Mean bubble separation

H_*R_*

$0.0527$

Sound lifetime

H_*\tau_{\rm sw}

$0.368$

Peak today

h^2\Omega_{\rm p}

$1.6\times10^{-11}$

LISA SNR, 4 years $634$

Set the transition

Released vacuum energy

\alpha

0.2

How fast bubbles appear

\beta/H_*

50

Bubble wall speed

\xi_w

0.9

Drag the gold handle on a bubble wall to scrub time.

Temperature and more numbers

Temperature after the transition

T_*

200\,\mathrm{GeV}

Temperature only shifts frequencies. The number of relativistic species is held at $g_*=100$ .

$\kappa=0.274,\quad K=0.0274$

$f_1=0.125\,\mathrm{mHz},\quad f_2=0.873\,\mathrm{mHz}$

$H_*\tau_{\rm sh}=0.368,\quad\Upsilon=0.241$

The expansion factor of Guo et al. (2021) is shown for comparison only; the template uses the minimum rule.

Model assumptions

The bubble slice comes from one frozen synthetic Monte Carlo record of exponential nucleation, with lengths in units of the wall speed times the nucleation timescale. Sound shells are drawn from our own bag-model profile solutions. The spectrum is the parametric sound-wave template of Caprini et al. (2024), not a solution of the Einstein or fluid equations. It is shown only for subsonic deflagrations below the sound speed minus 0.05 and for detonations faster than the Jouguet speed, when a deflagration profile exists and the mean bubble separation is below the Hubble radius. Turbulence, bubble-collision and foreground contributions are omitted. The SNR uses LISA A and E channels for four years without foregrounds.

Spectrum today against LISA noiseParametric template · instrument model

How much false vacuum is leftSynthetic offline Monte Carlo · ideal reference

How long the sound lastsTwo lifetime rules · model comparison

How much energy reaches the fluidPublished fit · our offline profile solutions

Gold: the sound-wave template, shown only inside its stated range. Grey: LISA A-channel noise without foregrounds. Teal dots: measured on the slice of the synthetic record; white curve: $h(t)=e^{-e^{\beta(t-t_*)}}$ .

Bubbles, sound and a background of gravitational waves

BackgroundIf the hot early universe cooled through a first-order transition, the new phase appeared as bubbles. Their walls pushed on the plasma, and after the bubbles merged the plasma kept ringing with sound waves. Moving sound shells change the stress of matter, and changing stress radiates gravitational waves.

Why it mattersThree numbers set the signal: how much energy is released, how quickly bubbles appear, and how fast the walls move. Expansion then stretches the waves to today.

Focus question: Which parameter moves the peak, and which one raises it?
One-sentence intuition: Bubble size sets the frequency; the fluid kinetic energy and the time the sound lasts set the height.

Core mathematical model

Mean bubble separation

H_*R_*=(8\pi)^{1/3}\,\frac{\max(\xi_w,c_s)}{\beta/H_*}

With an exponential nucleation rate the bubble number density is $\beta^3/(8\pi\xi_w^3)$ . Faster nucleation, larger $\beta$ , means more and smaller bubbles.

Kinetic energy and sound lifetime

K\simeq0.6\,\frac{\kappa\alpha}{1+\alpha},\qquad H_*\tau_{\rm sw}=\min\!\left[\frac{H_*R_*}{\sqrt{K/\Gamma}},\,1\right]

The single-bubble efficiency $\kappa$ depends on wall speed and strength. Sound lasts until shocks form, about one bubble size divided by the fluid speed, but at most one Hubble time.

The template today

h^2\Omega_{\rm int}=h^2F_{\rm GW,0}\,A_{\rm sw}\,K^2\,(H_*\tau_{\rm sw})(H_*R_*),\qquad f_1\simeq\frac{0.2\,H_{*,0}}{H_*R_*}

Here $A_{\rm sw}\simeq0.11$ comes from simulations, and $H_{*,0}$ and $F_{\rm GW,0}$ redshift frequency and energy density to today. The spectrum rises as $f^3$ , runs as $f$ between the two breaks and falls as $f^{-3}$ .

Common difficulties

Faster walls do not always mean a louder signal

Typical misconceptionThe fastest walls always give the largest spectrum.

Better mental modelEfficiency peaks near the Jouguet speed and drops for very fast detonations, where most energy heats the plasma. Sweep the wall speed and watch the efficiency curve.

A long-lived source is not free

Typical misconceptionSound always lasts a full Hubble time.

Better mental modelFor strong transitions shocks form within a fraction of a Hubble time. The template then scales with the shorter lifetime, so the amplitude grows more slowly than the square of the kinetic energy.

Run the experiment

01
Watch one transition
Start with Ringing in the LISA band. Watch bubbles appear, merge and leave sound rings.
What to observe: The measured false-vacuum fraction in the slice follows the analytic curve within the scatter of one box.
02
Move the peak
Drag the nucleation-speed slider from 20 to 300.
What to observe: The peak moves to higher frequency and drops, because bubbles get smaller and the sound breaks up sooner.
03
Cross the sound speed
Slowly drag the wall speed from 0.4 to 0.9, or drag the gold wall handle to scrub time.
What to observe: A shock in front of slow walls turns into a rarefaction behind fast walls. Between them the template stays silent and says why.

Sources and model context

Read the course: from bubble nucleation to the LISA spectrum

Caprini et al. (2024) — sound-wave double broken power law, redshift factors, LISA noise and response model.
Espinosa, Konstandin, No and Servant (2010) — single-bubble fluid profiles and efficiency fits; our own profile solutions check the fits.
Ellis, Lewicki and No (2020) — finite lifetime of the sound-wave source.
Guo, Sinha, Vagie and White (2021) — expansion suppression factor, shown for comparison.
Robson, Cornish and Liu (2019) — LISA sensitivity curve used as an independent noise check.

Phase-Transition Gravitational Waves from Bubble Collisions

Bubbles, sound and a background of gravitational waves

Core mathematical model

Mean bubble separation

Kinetic energy and sound lifetime

The template today

Common difficulties

Faster walls do not always mean a louder signal

A long-lived source is not free

Run the experiment

Watch one transition

Move the peak

Cross the sound speed

Sources and model context

Related experiments and articles

Articles

From bubble nucleation to the spectrum LISA could see