--- title: "Phase-transition gravitational waves" description: "What gravitational-wave spectrum would a first-order phase transition leave? Follow bubbles, sound waves and redshift, then recompute a benchmark." canonical_url: "https://sandboxphysics.com/stories/phase-transition-gravitational-waves/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/phase-transition-gravitational-waves/" last_modified: "2026-11-14" --- # Phase-transition gravitational waves Sandbox Physics · 2026-11-12 If the early universe boiled, bubbles of a new phase grew, collided and set the hot plasma ringing. That sound can leave a faint background of gravitational waves. These six pages follow it from the bubbles to the spectrum LISA could measure, and say where the calculation stops being reliable. 1. ## [What a bubbling early universe would leave in gravitational waves](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/what-a-bubbling-early-universe-leaves-in-gravitational-waves/) If the hot early universe changed state through bubbles, their sound would leave a faint hum of gravitational waves near a millihertz today. 2. ## [How bubbles appear and fill space](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/how-bubbles-appear-and-fill-space/) Derive how much old phase survives under an exponentially growing nucleation rate, and why the mean bubble spacing is set by the wall speed over beta. 3. ## [How bubble walls push the plasma](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/how-bubble-walls-push-the-plasma/) Classify walls as deflagrations, hybrids or detonations, find the Jouguet speed and compute how much released energy becomes fluid motion. 4. ## [From the sound lifetime to the spectrum today](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/from-sound-lifetime-to-the-spectrum-today/) Combine bubble size, fluid energy and sound lifetime into the LISA sound-wave template, then redshift its frequencies and energy density to today. 5. ## [Recompute a benchmark sound-wave spectrum](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/recompute-a-benchmark-sound-wave-spectrum/) Download one stdlib Python script and three fixed benchmarks, recompute the sound lifetime, spectrum today and LISA SNR, and see four refused cases. 6. ## [Read the LISA sound-wave template and its limits](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/read-the-lisa-sound-wave-template-and-its-limits/) Read section 2.2.1 of arXiv:2403.03723v2 with two lifetime papers: what the template assumes about sound lifetime and walls, and what LISA could identify. ## Benchmark tutorial materials [Download the offline tutorial ZIP](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/phase-transition-benchmark-v1.zip) · [Read units, versions and limits](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/README.md) Standard-library Python recomputes the source lifetime, the two break frequencies, the template spectrum today and the LISA signal-to-noise ratio for three fixed benchmarks, and shows how four failure cases are rejected. It evaluates a published parametric template; it does not solve the Einstein or fluid equations. Files, sizes and SHA-256 - [README.md](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/README.md) (6252 B) `296087f7e0c72f7c9cb613563064b85abfbaf3318d4b66feef069f35bb7685e6` - [recompute\_benchmark.py](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/recompute_benchmark.py) (12375 B) `f2c7c8359bff16583431c8457e4c1f1cf1c995a1c5fc65b597ca2729a4ddf098` - [benchmarks.json](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/benchmarks.json) (2329 B) `daf8124eedb3f23491f9964205f7d32bcd632fd12a762f63b89f832fb4258dde` - [expected.json](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/expected.json) (3733 B) `1cf760ff6941739c0a13a77e9d2946d05d85c8e229c367a955130d2a934115d8` - [phase-transition-benchmark-v1.zip](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/phase-transition-benchmark-v1.zip) (9851 B) `541d13d3b1b14734d368dd74d8bfe360c935c893c74dd5d2c10206f64fce4bbf` ## Sources and calculation scope 1. [Caprini et al., Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, arXiv:2403.03723v2 / JCAP 10 (2024) 020](https://arxiv.org/abs/2403.03723v2) Read in the arXiv v2 LaTeX source: abstract and introduction, section 2.2.1 (sound-wave double broken power law, amplitude, mean bubble separation and redshift factors), section 3 and appendix A (two-parameter LISA noise and response) and the opening of section 5 on reconstruction. The noise equation prints the optical-metrology corner as 2e-3 mHz; our materials use 2 mHz and report both results. Figure 2 was read from its caption only. 2. [Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: an update, arXiv:1910.13125v2 / JCAP (2020), doi:10.1088/1475-7516/2020/03/024](https://arxiv.org/abs/1910.13125v2) Read the LaTeX source in two places: the sound-wave signal section (signal.tex, about lines 180 to 260) and the definition of the root-mean-square fluid velocity and kinetic-energy fraction (prelim.tex, lines 171 to 182). Used for the earlier shock-time definition only. 3. [Espinosa, Konstandin, No & Servant, Energy budget of cosmological first-order phase transitions, arXiv:1004.4187v1 / JCAP 1006:028 (2010)](https://arxiv.org/abs/1004.4187v1) Read the fluid equations for a single self-similar bubble, the deflagration, hybrid and detonation profiles, and the appendix-B efficiency fits with their stated precision of better than 15 percent. Our own profile solutions in the bag model check the fits independently. 4. [Ellis, Lewicki & No, Gravitational waves from first-order cosmological phase transitions: lifetime of the sound wave source, arXiv:2003.07360v4 / JCAP 07 (2020) 050](https://arxiv.org/abs/2003.07360v4) Read the LaTeX source for the shock-formation time, the mean fluid velocity written through the efficiency, and the argument that the sound-wave source switches off well before a Hubble time for strong transitions. 5. [Guo, Sinha, Vagie & White, Phase transitions in an expanding universe: stochastic gravitational waves in standard and non-standard histories, arXiv:2007.08537v3 / JCAP 01 (2021) 001](https://arxiv.org/abs/2007.08537v3) Read the main LaTeX file for the expansion suppression factor for a radiation-dominated universe. It is shown beside the template lifetime factor for comparison; the template itself does not use it. 6. [Robson, Cornish & Liu, The construction and use of LISA sensitivity curves, arXiv:1803.01944v2 / Class. Quantum Grav. (2019), doi:10.1088/1361-6382/ab1101](https://arxiv.org/abs/1803.01944v2) Located the optical-metrology and test-mass noise terms and the sensitivity formula in the LaTeX source. Used only as an independent check of the noise curve shape and of the 2 mHz corner; the paper was not read section by section. 7. [Original benchmark tutorial materials (standard-library Python)](https://sandboxphysics.com/stories/phase-transition-gravitational-waves/materials/README.md) Our script, three fixed benchmarks, four failure cases and the expected output. It evaluates the published parametric template, the efficiency fits and the noise model; it does not solve the Einstein or fluid equations and contains no browser Engine code, paper text or LISA data.